A Voyage Through Turbulence PDF
A Voyage Through Turbulence PDF
A Voyage Through Turbulence PDF
Edited by
PETER A. DAVIDSON
University of Cambridge
YUKIO KANEDA
Nagoya University
KEITH MOFFATT
University of Cambridge
KATEPALLI R. SREENIVASAN
New York University
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v
vi Contents
ix
x Contributors
“Will no-one rid me of this turbulent priest?” So, according to tradition, cried
Henry II, King of England, in the year 1170, even then conveying a hint of
present frustration and future trouble. The noun form ‘la turbulenza’ appeared
in the Italian writings of that great genius Leonardo da Vinci early in the 16th
century, but did not appear in the English language till somewhat later, one
of its earliest appearances being in the quotation above from Shakespeare.
In his “Memorials of a Tour in Scotland, 1803”, William Wordsworth wrote
metaphorically of the turmoil of battles of long ago: “Yon foaming flood seems
motionless as ice; its dizzy turbulence eludes the eye, frozen by distance . . . ”.
Perhaps we might speak in similar terms of long-past intellectual battles con-
cerning the phenomenon of turbulence in the scientific context.
Turbulence in fluids, or at least its scientific observation, continued to elude
the eye until Osborne Reynolds in 1883 conducted his brilliant ‘flow visualisa-
tion’ experimental study “of the circumstances which determine whether the
motion of water shall be direct or sinuous, and of the law of resistance in par-
allel channels”. Although the existence and potential importance of ‘eddying’
as opposed to steady streamlined flow had been recognized previously, no-
tably by the great 19th-century French pioneers of hydrodynamics, Barré de
Saint-Venant and his follower Joseph Boussinesq, the study of turbulence as a
recognizable branch of fluid mechanics may be said to date from this famous
1883 investigation of Reynolds, who correctly identified the competing roles
of fluid inertia and viscosity in promoting hydrodynamic instability and the
transition from smooth to irregular flow. He did not use the word ‘turbulent’,
opting rather for the phrase ‘sinuous flow’; but just four years later, William
Thomson (Lord Kelvin) introduced1 the phrase ‘turbulent flow’, and (in a later
paper the same year) the abstraction ‘turbulence’, to the literature of fluid
mechanics.
1 ‘On the propagation of laminar motion through a turbulently moving inviscid liquid’, Phil.
Mag. 24, 342–353 (1887).
xi
xii Preface
Some decades elapsed before the word gained acceptance in the scientific lit-
erature. Even in 1897, Boussinesq used the more eloquent phrase “écoulement
tourbillonnant et tumultueux des liquides” within the title of a book2 devoted
essentially to the phenomenon of turbulent flow as then understood. One is
reminded of the song from the 1970s of Guy Béart:
Tourbillonnaire, tourbillonaire,
Deux pas en avant, quatre en arrière!
which we might perhaps facetiously translate with regard to the history of the
subject, and with some degree of poetic license:
Some among these (e.g. Prandtl, von Kármán, Taylor) have the status of
great founder-figures who interacted during the inter-war years through co-
pious correspondence as well as through the International Congresses of the
period. Others (e.g. Kolmogorov, Corrsin, Batchelor, Dhawan) were pivotal
figures in the development of post-war schools of turbulence, radiating out-
wards from their centres of activity (the Russian school, the Johns Hopkins
school, the Cambridge school, and the school of the Indian Institute of Science,
Bangalore, respectively). Yet others (e.g. Richardson, Townsend, Kraichnan,
Saffman) were individualists, whose brilliant contributions made a profound
impact upon the subject.
Many names of other departed colleagues come to mind, for whom sepa-
rate chapters could well have been justified – J.M. Burgers, Kampé de Fériet,
Klebanoff, S.J. Kline, Kovasznay, Laufer, Liepmann, Lighthill, Loitsianski,
Monin, Obukhov, Perry, O.M. Phillips, W.C. Reynolds, Tani, Yaglom, P.Y.
Zhou, . . . , to name but a few. Their contributions are referred to in chapters
of this book. We beg the indulgence of the reader in the choices we have made,
in the interest of providing a reasonably compact yet balanced picture3 .
Why, it may be asked, should the problem of turbulence exert such enduring
fascination within the scientific community? First perhaps because it is recog-
nized as a prototype of problems in the physical sciences exhibiting both strong
nonlinearity and irreversibility, a combination of circumstances that leads to
great irregularity in both space and time of the fields considered. This is also
why its resolution has eluded the best minds of the 20th century. The role of
vortex structures is seen as of central importance, while a statistical approach
is needed to cope with the irregularity of turbulent flow at all scales. No fully
satisfactory treatment combining these aspects has yet been found. The remark
that “Turbulence is the most important unsolved problem of classical physics”
attributed to Nobel Laureate Richard Feynman (and perhaps originating with
Einstein) remains true to this day. Horace Lamb, author of the great classic
treatise Hydrodynamics, is alleged to have said “When I meet my Creator, one
of the first things I shall ask of Him is to reveal to me the solution to the
problem of turbulence” (or words to that effect – see Sidney Goldstein4 ). Cer-
tainly, von Kármán repeated this sentiment at the meeting Mécanique de la
Turbulence in Marseille (1961)! Meanwhile, Robert Kraichnan, Einstein’s last
postdoc, was mounting a massive theoretical attack on the problem, import-
ing techniques from quantum field theory and developing these techniques in
entirely original ways; nevertheless, despite his efforts, turbulence has remained
impervious to purely theoretical onslaught even after the lapse of another half-
century.
Second, the great span of applications of fluid mechanics has generated an
ever-growing need to achieve a better fundamental understanding of the origins
and effects of turbulence in practical circumstances. This need was first fuelled
by the rapid development of aerodynamics in the early part of the 20th century.
We tend to take air-transport for granted nowadays, but it is salutary to recall
that mastery of flight, arguably the greatest engineering accomplishment of
the 20th century, first required an understanding of flow in the viscous bound-
ary layer on an aircraft wing and of the conditions leading to instability and
turbulence in such boundary layers. Soon, the relevance of turbulence in me-
teorology and oceanography came to be recognized, here with the additional
factors (sometimes complicating, sometimes simplifying!) of density stratifi-
cation and Coriolis effects due to the Earth’s rotation. Then at the planetary,
stellar and inter-stellar levels, the relevance of turbulence for the generation
and evolution of magnetic fields as observed in the cosmos came to be simi-
larly recognized in the post-war years. And of course, turbulence remained all
along of key importance in Mechanical and Chemical Engineering, in which
it is the essential requirement for the effective mixing of fluid ingredients to
promote chemical or combustive interactions.
The authors of the 12 chapters of this volume are all experts in various as-
pects of turbulence, and have detailed (and in some cases personal) knowledge
of the personalities of whom they write, and of their impact on the field. Al-
though influenced by editorial comment in some cases, the opinions expressed
remain those of the authors themselves, and we, as editors of the volume, are
deeply grateful to them all for the care and effort that they have devoted to
their task. We hope that this volume, incomplete though it may be, will give
a balanced perspective of the development of ideas and research in turbulence
over what was in many ways an exceedingly turbulent century!
The original idea for this book arose during the programme on The Na-
ture of High Reynolds Number Turbulence held at the Isaac Newton Institute
for Mathematical Sciences, August–December 2008. We wish to express our
warm thanks to the Director and the staff of the Institute for their unfailing
encouragement and support, and for providing an ideal environment for the
initiation of a project of this kind. By happy chance, the book will be pub-
lished just before the European Turbulence Conference (ETC13) to be held
in Warsaw in September 2011. At the suggestion of Konrad Bajer, this confer-
ence will be followed by a symposium Turbulence – the Historical Perspective,
based on the chapters of this volume. We wish to thank Konrad for taking this
Preface xv
1.1 Introduction
1.1.1 Scope
Articles on Osborne Reynolds’ academic life and published works have ap-
peared in a number of publications beginning with a remarkably perceptive
anonymous obituary notice published in Nature within eight days of his death
(on 21 February 1912) and a more extensive account written by Horace Lamb,
FRS, and published by the Royal Society (Lamb, 1913) about a year later.
More recent reviews have been provided by Gibson (1946), a student of
Reynolds and later an academic colleague, by Allen (1970), who provided the
opening article in a volume marking the passage of 100 years from Reynolds
taking up his chair appointment at Manchester in 1868, and by Jackson (1995),
in an issue of Proc. Roy. Soc. celebrating the centenary of the publication of
Reynolds’ 1895 paper on what we now call the Reynolds decomposition of the
Navier–Stokes equations, about which more will be said later in the present
chapter. A significant portion of the present account is therefore devoted to
Reynolds’ family and background and to hitherto unreported aspects of his
character to enable his contributions as a scientist and engineer to be viewed
in the context of his life as a whole. While inevitably some of what is pre-
sented here on his academic work will be known to those who have read the
articles cited above, archive material held by the University of Manchester and
The Royal Society and other material brought to light in the writers’ personal
enquiries provide new perspectives on parts of his career.
1
2 Launder & Jackson
500 acres of land and much of the property around the small village of Debach
located about 5 miles NNW of Woodbridge (White, 1844). Starting in 1779,
three consecutive rectors of the parish of Debach-with-Boulge came from the
Reynolds family. The Rev. Robert Reynolds was instituted on 6 September
1779 on his own petition. When he retired in September 1817 his son, the
Rev. Osborne Shribb Reynolds, became Rector. Then, on his death in Decem-
ber 1848, his eldest son, the Rev. Osborne Reynolds, took over for a while.
This last named, father of the main subject of this chapter, entered Cambridge
University in 1832 as a fee-paying pensioner. After matriculating from Trinity
College in 1833, he transferred to Queens’ College from whence he graduated
in 1837 as 13th Wrangler (Venn, 1954). At that point it seemed that he was
destined to follow a clerical career like his father and grandfather before him
for he was ordained a deacon in Ely Cathedral the following year and became
a priest a year later.
The Rev. Osborne Reynolds married Jane Bryer, née Hickman, the 22-year-
old widow of the late Rev. Thomas Bryer, at Hampstead Church on June 25th,
1839 (The Times, 29 June 1839). Their first child, Jane, was born in 1840 and,
soon afterwards, they moved to Ireland where the Rev. Reynolds had obtained
a position as principal of the First Belfast Collegiate College, in Donegall Place
(Martins Belfast Directory, 1842–43, p. 82). Their second child, Osborne, was
born on 23 August 1842 (Crisp, 1911).
It seems, however, that the Rev. Reynolds still saw a career in the church as
his goal for in 1843 he returned to England with his family to take up an ap-
pointment as curate at the parish church in Chesham, Buckinghamshire. How-
ever, his tenure of this post proved to be short-lived. On February 6th, 1844, his
wife died as a result of complications following the birth, three weeks earlier,
of their second son Edward (The Times, 12 February 1844), leaving the Rev.
Reynolds with the responsibility of bringing up his three small children alone.
That task, allied with the financial limitations of his post as curate at Chesham,
provided the incentive for him to seek an alternative position. In October 1845
he was appointed headmaster of Dedham Grammar School in Essex.
The Rev. Reynolds took up his post at the end of 1845 and held it for al-
most eight years (Jones, 1907). Besides carrying out his duties as headmaster
he provided the personal tuition of his children, who lived with him at Dedham
(1851 Census of Great Britain). It seems that he was also working on inven-
tions, for while in post he took out the first two of the six patents that would be
registered in his name (Ramsey, 1949).
In fact, the small market town of Dedham is located only some 25 miles
south-west of Debach. This relative proximity meant that the Rev. Reynolds
was able to keep very much in touch with his father, the Rev. Osborne Shribb
1: Osborne Reynolds: a turbulent life 3
Reynolds, and with the family’s farming interests in Debach (White, 1844).
On visits there he occupied a farmhouse on the family estate. Moreover, when
his father died in post in 1848, the Rev. Osborne Reynolds was able to take
over as rector, which he did on his own petition (White, 1844), while remain-
ing headmaster of Dedham Grammar School (Clergy List, 1850, p. 14). This
arrangement lasted until May 1850 when a replacement was appointed rec-
tor by the Church of England (Crockfords Clerical Directory, 1850). Family
misfortune still seemed to stalk the Rev. Reynolds, for the following year his
ten-year-old daughter, Jane, died at Dedham.
In 1854, having inherited much of the land and property in Debach, he re-
signed as headmaster at Dedham Grammar School (or ‘was persuaded to re-
sign’, as one contemporary account (Jones, 1907) seems to imply) to take on
what amounted to the life of a gentleman farmer, managing the family estate
in Debach which then employed some 30 staff (1861 Census of Great Britain).
He lived at Debach House with his two sons whom he continued to educate,
concentrating, it seems, on mathematics and mechanics. As will be seen later,
his elder son Osborne warmly acknowledged his father’s role in stimulating
his own interest in mechanics. By the time of the next census (1871) his sons
had both gone up to Cambridge University leaving Osborne Reynolds Senior
free to concentrate on managing his estate and farming interests which, ap-
parently, continued to flourish. Later, however, in the agricultural depression
of 1879, it appears that Osborne Reynolds Senior encountered financial dif-
ficulties. In 1880, at the age of 66, he relinquished the estate to take up the
post of rector of Rockland St Mary in Norfolk (a placement arranged under the
auspices of his Cambridge College, Queens’ (Crockfords Clerical Directory,
1880, p. 839)). He finally retired in 1889 and moved to Clipston, Northamp-
tonshire, where his younger son, Edward, was then rector (Crockfords Clerical
Directory, 1890, p. 1069) and he died there on June 7th the following year (The
Ipswich Journal, 14 June 1890; The Manchester Guardian, 18 June 1890).
In summary, the talented but ill-fated Rev. Reynolds had an exceptionally
strong influence on the formation and development of his elder son, Osborne
Reynolds, who forms the subject of the remainder of this article. Not only was
he directly responsible for Osborne’s primary and secondary education but he
stimulated in him a fascination for mechanics that was to be the bedrock of
his life’s work. He went on to play a major role in shaping the path and cov-
ering the not inconsiderable cost of the further five years which his son spent
receiving practical training in mechanical engineering and a university edu-
cation in mathematics which were so pivotal to his subsequent career. More-
over, the manner in which the Rev. Reynolds coped with the serious domestic
misfortunes he faced must have provided a source of inspiration for his son
4 Launder & Jackson
of private study. On passing the requisite examination early in his second year
he ceremonially demonstrated his firmness of spirit by burning all his Greek
textbooks despite the pleas of his friend and fellow undergraduate at Queens’,
Arthur Wright, who was studying classics, to pass them on to him (Wright,
1912).
In his specialist subject of mathematics Osborne Reynolds also viewed his
experience at Cambridge with some disappointment. Years later (13 October
1876) at a General Meeting of the Manchester Mechanical & Physical Society
he declared:
The mathematical education given at Cambridge, however much it might develop
the power of mind of its possessors, [was] hardly calculated to forward the study
which was its immediate object. Those who did attempt such a course found
that they had spent several years in learning that which they had to lay aside
on commencing their new work. Mathematics and the theory of mechanics, it is
true, were then as now, the educational base most wanted; but these taught with
a view to their application to the simpler problems of astronomy . . . were about
as much use as the Latin grammar . . . for learning French.
Figure 1.1 The opening page of Osborne Reynolds’ letter of application for the
Chair in Engineering. Reproduced with permission of the University of Manch-
ester.
audacity in applying for the Chair and, equally, the wisdom of the appointing
committee in eventually choosing him. However, Reynolds was by no means
the youngest candidate: seven of the 16 were in their twenties, four of whom
were younger than Reynolds. In the 1860s, a sound knowledge of mechanics
and a vision of where it might lead in engineering applications must have been
qualities predominantly possessed by younger candidates, much as knowledge
and competency in certain aspects of software engineering are today.
1: Osborne Reynolds: a turbulent life 7
W.C. Unwin must have felt that he had a good chance, having been assured
of very strong support from his former employer and mentor, William Fair-
bairn, FRS (Allen, 1970), who was chairman of the committee of industrialists
in Manchester which had raised the money to fund the Chair. The College set
up an appointing committee made up of trustees of Owens College and a se-
lection of professors. Fairbairn was not on that committee but, evidently, he
must have been influential behind the scenes in guiding it with a view to ensur-
ing that a suitable appointment was made. The procedure was for candidates
to submit supporting testimonials and, in Reynolds’ case, at least, the hand-
written versions (of which there were 14) were complemented by a printed
version of the same and of his letter of application itself (University of Manch-
ester Archive). The testimonials included one from Mr Hayes from which the
quotation above was taken, another from Archibald Sandeman, then Professor
of Mathematics at Owens College but who had formerly been Reynolds’ tutor
at Queens’ College, plus four others from Cambridge staff including one from
James Clerk Maxwell, FRS, confirming Reynolds’ standing in the graduation
list and ending with the important observation that
I had to examine Mr. Reynolds’ papers for the Mathematical Tripos, including
his solutions of many questions in mechanics and general physics; and found
that he had knowledge of sound principles which will enable him in the study
and teaching of engineering to exemplify the practical use of sound theoretical
principles, and to show that all his practical rules are founded on general laws
established by experiment.
Another referee, the Rev. W.M. Campion, BD, Fellow and Tutor of Queens’
College, Cambridge, wrote:
Mr. Reynolds is an accomplished Mathematician. But he is not a mere theorist.
He possesses a considerable acquaintance with practical mechanics and engi-
neering. For more than a year before he came to the University he studied the
practice of the profession under Mr. Hayes of Stoney Stratford; and since taking
his degree he has been occupied in like manner with Mr. Lawson of London. It
would be difficult to find a Mathematician who combines such practical experi-
ence with theoretical knowledge.
Fulsome communications were also received from J.C. Challis, FRS, Pro-
fessor of Astronomy, and the mathematics tutor, John Dunn, who commented
that while on entry Reynolds had lacked knowledge in mathematics, “by in-
nate talent and undeviating perseverance Mr Reynolds made the most rapid
progress”.
Perhaps most surprisingly to a 21st-century reader, his father, the Reverend
Osborne Reynolds, also provided a testimonial (at the suggestion of another
referee), a task which, in his words, had surprised and embarrassed him. He
8 Launder & Jackson
nevertheless praised his son’s qualities and concluded: “The only point I can
conceive against him is his youth – he is only in his 26th year. But this is com-
pensated for by his early devotion to Science and the practice of his
profession”.
Despite the considerable number of applications, the minutes of a meeting of
the Owens Committee of Trustees on 30 January 1868 reported reservations on
the part of the appointing committee about the response to the advertisement.
Accordingly, Mr Charles F. Beyer, a German who had come to Manchester as
an impecunious young man to make his fortune (and had certainly done so!),
offered to provide sufficient further funds to enable the post to be re-advertised
with “an additional £250 p.a. for the first five years in the hope that the in-
creased remuneration would enable the Trustees to obtain applications from
gentlemen of higher scientific attainments and greater professional experience
than could be expected under the moderate inducements held out in the earlier
advertisement” (University of Manchester Archive). It is more than likely that
this decision was influenced by the following sarcastic article which had ap-
peared in the professional journal, Engineering, earlier that month (10 January
1868):
Technical Education
For all those who are interested in that subject of paramount national impor-
tance, upon which the future greatness of this country and its position in the
civilised world are now recognised to depend – for all those who are speaking,
and writing, and working for the spread of technical education in this country
– we have gratifying news. The trustees of Owens College, in Manchester, are
advertising for an able-bodied man-servant to act as performing professor of en-
gineering for the rising generation in the metropolis of Manchester, at the liberal
rate of wages of thirteen shillings and eight pence per day.
What a stir this grand opening will create in the scientific world! The greatest
men of Great George street will close their offices and compete with each other;
M. Flachat, Professor Conche, Baron Burg, and Professor Ruhlmann will leave
their respective countries and professors’ chairs; men like Rankine, Scott Rus-
sell, and Clausius will gather in long processions in the streets of Manchester,
and vie with each other to answer the call in the newspapers. Thirteen and eight
pence and a proportion of the fees paid by students (and perhaps the free loan of
a sewing machine for the professor’s wife to earn a little extra) are worth apply-
ing for in a country where the income of the head master at Eton is estimated at
£6000, and that of an assistant master at the same school ranges from £1500 to
£3500 a year.
pursue the matter. Thus the Trustees decided to interview “Mr George Fuller,
C.E., Associate of the Institution of Civil Engineers and Mr Osborne Reynolds,
B.A., Fellow of Queens’ College, Cambridge whom they believe to be the most
eligible” (University of Manchester Archive). Both interviewees were drawn
from the original list of applicants. Thus, with Rankine having eventually de-
clined to become a candidate, the increased offer had served nothing other than
to double the salary of the successful applicant. As the world of fluid mechan-
ics gives thanks, the chosen candidate was Osborne Reynolds, a decision which
has been described by Smith (1997) as “an inspired choice and one of the most
successful gambles ever made by an appointing committee”. A photograph of
part of the formal terms of appointment is reproduced in Figure 1.2. As for
W.C. Unwin, as soon as the Trustees’ decision had been reached, his former
employer wrote to him (Walker, 1938):
My dear Unwin,
I am very sorry I cannot forward to you the agreeable intelligence that you
are elected to the position of professor. I so earnestly wished for you to occupy
that position. It would have exactly suited your tastes, and I had every reason
to believe you would have been an active and excellent professor. . . . In wishing
you better luck in your next undertaking, I am,
Yours,
Wm. Fairbairn.
Bearing in mind the researches on materials and on bridge design which he at that
time had recently completed . . . it is certainly remarkable that no better reason
could be adduced by the College authorities for passing over Unwin in favour of
one whose experience of civil engineering was less, and whose fame rests upon
his work as a physicist rather than as an engineer. Owen’s [sic] College at that
date was, to a very great extent, a municipal undertaking and one cannot help
thinking that, in the lively atmosphere that surrounded its early development,
considerations other than academic may have played some part in the delibera-
tions of the Senate.
It is noted for the record that none of the documents seen in the University of
Manchester’s archives lends any support to Walker’s insinuation that “consid-
erations other than academic may have played some part” in the decision. The
10 Launder & Jackson
creation of the Chair was the outcome of leading industrialists from Manch-
ester recognizing the need to underpin the region’s industrial strengths with
a skilled and knowledgeable professional workforce. Moreover, few if any
would agree with Walker’s suggestion that Reynolds’ subsequent “fame”
1: Osborne Reynolds: a turbulent life 11
The results, however, of the labour and invention of this century are not to be
found in a network of railways, in superb bridges, in enormous guns, or in in-
stantaneous communication. We must compare the social state of the inhabitants
of the country with what it was. The change is apparent enough. The population
is double what it was a century back; the people are better fed and better housed,
and comforts and even luxuries that were only within the reach of the wealthy
can now be obtained by all classes alike . . . But with these advantages there are
some drawbacks. These have in many cases assumed national importance, and it
has become the province of the engineer to provide a remedy.
These remarks made at the outset of his career show the youthful Reynolds,
whose contemporary portrait photograph appears in Figure 1.3, clear in his
mind as to what needed to be done. Here was a figure charged with a sense of
mission, full of ideas and ready to face the challenges ahead of him in serving
the needs of society as Professor of Engineering at Owens College, Manch-
ester.
Figure 1.3 Osborne Reynolds, from the time of his appointment to the Owens
College Chair, c University of Manchester. Reproduced by courtesy of the Uni-
versity Librarian and Director, The John Rylands University Library, The Univer-
sity of Manchester.
In a bid to further extend both his own and the College’s contacts with
the technical and industrial community of the area, Reynolds also actively in-
volved himself with two other local societies, the Manchester Association of
Employers, Foremen and Draughtsmen (a group consisting of men with tech-
nical expertise and experience, first formed in 1856) and the Manchester Sci-
entific and Mechanical Society (formed in 1870 by William Fairbairn with
the intention of linking academics with local industrialists). Between 1871 and
1874 Reynolds addressed the first of these bodies on a number of directly prac-
tical topics: Elasticity and fracture; The use of high pressure steam; and Some
properties of steel as a material for construction. In contrast, his lectures to the
Scientific and Mechanical Society, which he twice served as President, were of
a more general nature, as indicated by titles such as Future progress, Engineers
as a profession and Mechanical advances. In these ways, Osborne Reynolds
set out to address the specific needs of the rather diverse sectors of his ‘local
social and technical constituency’.
At the time of Reynolds’ appointment Owens College (which had been
founded in 1851 following a generous bequest by John Owens) occupied a
building on Quay Street, an early photograph of which appears as Figure 1.4,
the former home of Richard Cobden, the distinguished MP for nearby Stock-
port. The building, now restored, is today used as chambers for barristers. In
1868 the newly formed Engineering Department was accommodated in what
had been the stables at the rear of the building. In his recollections, Thomson
(1936) (who had grown up in Cheetham Hill, Manchester, and enrolled as an
engineering student in 1870 at the age of 14) noted that the stable itself was
converted into a lecture room and the hayloft above it into a drawing office.
Little was available in the way of facilities and equipment for experimental
work and Reynolds had to rely on using other science laboratories in the Col-
lege or performing simple experiments at home. This situation explains why
his very early papers were concerned largely with explaining natural phenom-
ena, what Thomson later termed “out-of-door physics”. The work falling under
this heading has been summarized in Jackson (1995) while the papers them-
selves all appear in Volume I of Reynolds’ Collected Works (Reynolds, 1900).
The tails of comets, the solar corona and the aurora, followed by the inductive
role of the Sun on terrestrial magnetism, the electrical properties of clouds and
the phenomenon of thunderstorms form some of the subjects of the early pa-
pers in this group. A further paper concerned the bursting of trees struck by
lightning, the cause of which Reynolds was able to link to the rapid vapor-
ization of moisture within a tree trunk by the sudden discharge of electricity
through it. There were other papers on the destruction of sound by fog and the
refraction of sound by the atmosphere. Thereafter, he tackled topics such as the
14 Launder & Jackson
calming of seas (both by raindrops and by an oil film on the surface) and the
formation of hailstones and snowflakes. For some of these studies he contrived
simple but effective small-scale experiments using relatively unsophisticated
apparatus. Even after the removal of the College in 1873 to a new purpose-
built campus on a site south of the city centre (today the core of Manchester
University) there was initially only limited scope for experimental work. A
fuller summary of this phase of Reynolds’ research appears in Jackson (1995)
together with reproductions of the diagrams of the apparatus used in each case.
Lamb (1913) records in his obituary notice that for some time after Reynolds’
arrival, while he was concentrating on these out-of-door physics problems.
some shade of disappointment was felt by the eminent practical engineers and
other friends of Owens College, who had worked for the creation of the pro-
fessorship. This happened despite the fact, noted above, that Reynolds was at
that time actively integrating with the societies that served local manufacturing
and industrial management needs.
1: Osborne Reynolds: a turbulent life 15
22 July 1879 to George Stokes (at the time editor of Phil. Trans. Roy. Soc.)
from St Leonards-on-Sea. The letter mainly concerned referees’ comments on
the manuscript of his paper ‘On certain dimensional properties of matter in the
gaseous state’ (Jackson, 2010), but he also commented:
I am here nursing my only child who is very ill and I do not like leaving him even
for a day or I would try to see you and save you some of this writing. I expect to
be here all the Summer.
In fact, despite this love and attention, Reynolds’ son died on 27 September
1879 while they were still at St Leonards-on-Sea. (Yorkshire Post, 30 Septem-
ber 1879). This personal tragedy that again, curiously, paralleled events in his
own father’s life, may be said to mark the end of the first phase of Osborne
Reynolds’ professorial career.
the professors until 2 then lectured to students in the afternoon and walked the
2 miles home at 4pm. I sometimes came out of town by tram and joined him.
He carried his stick or umbrella at the slope over a shoulder and he never knew I
was there. He was far away, and I had to make myself known to take a short cut
when we were nearly home. Then he worked again all the evening, after reading
the [news]papers before dinner, until 2 or 3am. Every day alike, except Saturday
afternoon and Sunday, when he went for long walks with his friends.
The 1883 paper Most readers will be familiar with Reynolds’ study of (to
quote from the title of his Phil. Trans. Roy. Soc. paper) ‘the circumstances
which determine whether the motion of water shall be direct or sinuous’. How-
ever, as noted above, some nine years before that paper appeared, he made a
presentation to the Manchester Literary and Philosophical Society, ‘On the ex-
tent and action of the heating surfaces of steam boilers’ (which is reprinted as
Paper 14 in Vol. 1 of his Collected Works; Reynolds, 1900) in which he clearly
signalled his awareness of the importance of turbulent eddies:
The heat carried off by air, or any fluid, from a surface, apart from the effect of
radiation, is proportional to the internal diffusion of the fluid at and near the sur-
face. Now, the rate of this diffusion has been shown to depend on two things:–
The natural internal [molecular] diffusion of the fluid [and] eddies caused by vis-
ible motion which mixes the fluid up and continually brings fresh particles into
contact with the surface. The first of these [mechanisms], molecular diffusion,
is independent of the velocity of the fluid and may be said to depend only on
the nature of the fluid. The second, the effect of eddies, arises entirely from the
motion of the fluid, and is proportional both to the density of the fluid and the
velocity with which it flows past the surface.
18 Launder & Jackson
Figure 1.5 The Osborne Reynolds tank for visualizing distinctions between lam-
inar and turbulent flow.
c The University of Manchester.
In a paper read to the British Association in 1880, ‘On the effect of oil in de-
stroying waves on the surface of water’, Reynolds associated that phenomenon
with the production of turbulent eddies in the water below the surface gener-
ated as a result of the movement of the oil film under the action of the wind
which caused a shear flow to be produced in the water (see Paper 38 of Vol. I of
his Collected Works; Reynolds, 1900). As he related much later in a letter to his
colleague, Horace Lamb (University of Manchester Archive; also reproduced
in Allen, 1970) on conducting an experiment “on a windy day at Mr. Grundy’s
pond in Fallowfield”, he observed that by throwing a small quantity of oil onto
the surface, instead of waves being formed, eddies were produced in the wa-
ter beneath the oil film (“which took on the appearance of plate glass”). This
acute observation evidently provided him with his early insight into turbulence
production in wall shear flows.
Returning now to the 1883 paper, the original print of the so-called Reynolds
tank experiment has been reproduced in numerous articles and text books
so, instead, Figure 1.5 shows a photograph of the apparatus as it is today
at the University of Manchester. The glass tube with a flared entry which
is itself housed within a tank filled with water is still used to provide stu-
dents with a very clear indication of the starkly contrasting states of motion,
whether ‘direct’ or ‘sinuous’ (or, in today’s terminology, laminar or turbulent).
In Reynolds’ own words:
The internal motion of water assumes one or other of two broadly distinguishable
forms – either the elements of the fluid follow one another along lines of motion
which lead in the most direct manner to their destination, or they eddy about in
sinuous paths the most indirect possible.
1: Osborne Reynolds: a turbulent life 19
The existence of these two modes of fluid flow was, of course, already
widely known. The first description of the transition between laminar and tur-
bulent flow in a pipe had been provided by Hagen (1854) who used sawdust as
a means of flow visualization (later, he recommended the use of filings of dark
amber; Hagen, 1869). Reynolds’ dye-streak studies and other data were, how-
ever, the first to show that, for a range of flow velocities, pipe diameters and
viscosities, transition from the former mode to the latter occurred for roughly
the same value of the unifying dimensionless parameter which today bears his
name, the Reynolds number.
The first step in Reynolds’ discovery of this parameter appears to have been
his observation that ‘the tendency of water to eddy becomes much greater as
the temperature rises’. It occurred to him that this might be related to the fact
that the viscosity of water decreased as the temperature rose. By examining
the governing equations of motion he concluded that the forces involved were
of two distinct types, inertial and viscous, and, further, that the ratio of these
terms was related to the product of the mean velocity of the flow and the tube
diameter divided by the kinematic viscosity. In his paper he states:
This is a definite relation of the exact kind for which I was in search. Of course
without integration the equations only gave the relation without showing at all
in what way the motion might depend upon it. It seemed, however, to be certain,
if the eddies were due to one particular cause, that integration would show the
birth of eddies to depend on some definite value of [that group of variables].
Professor Reynolds has traced with much success the passage from one state
of things to the other, and has proved the applicability under these complicated
conditions of the general laws of dynamic similarity as adapted to viscous fluids
by Professor Stokes. In spite of the difficulties which beset both the theoretical
and experimental treatment, we may hope to attain before long to a better under-
standing of a subject which is certainly second to none in scientific as well as
practical interest.
22 Launder & Jackson
Sir George Stokes served as President of the Royal Society from 1885 to
1890 and in this capacity, in November 1888, he presented the Society’s Royal
Medal to Osborne Reynolds “for his investigations in mathematical and ex-
perimental physics, and on the application of scientific theory to engineering”.
More than half of Stokes’ citation was devoted to a summary of the 1883 paper.
Although the physical significance of the dimensionless parameter we know
as the Reynolds number had thus quickly become widely recognized in Britain,
it was only some years after Reynolds’ retirement that his own name became
attached to it through the publications of various German workers. Rott (1990)
cites Sommerfeld (1908) as being the first to link Reynolds’ name with the
parameter, an attribution followed shortly thereafter by Prandtl (1910) in his
early paper on the Reynolds analogy, while later, in an encyclopaedia entry
on fluid motion, Prandtl (1913) unequivocally announces “The forementioned
quantity, a dimensionless number, is named after the discoverer of this similar-
ity, Osborne Reynolds, [and is called] the Reynolds number”.
have been responsible for choosing the equipment for their laboratories). The
first students were admitted in February 1885 from which time Unwin was ap-
pointed Dean of the Central Institution, with all the associated administrative
responsibilities, on top of the task of teaching in his own department without,
initially, any demonstrators or assistants (Walker, 1938). Thus, it seems at least
questionable whether, had Reynolds been chosen for that position, his major
remaining works on fluid mechanics would ever have been written, at least in
the form we know them. The papers that would have been placed in jeopardy
included not only his follow-up to the 1883 paper to which we shall shortly
turn but also his very important paper on film-lubrication (Reynolds, 1886).
Of that Lord Rayleigh, 32 years after its publication, felt able to remark “it in-
cludes most of what is now known on the subject” and in celebration of which
a centennial international conference was held in 1986 (Dowson et al., 1987).
Reynolds’ disappointment at failing to secure the chair in London must have
been assuaged that summer, by the conferment on him of an honorary degree
by the University of Glasgow, where W.J.M. Rankine had formerly been a
professor and where the Thomson brothers (James, Rankine’s successor, and
Sir William [later Lord Kelvin]) then served. Whether this last distinction had
any bearing on Reynolds’ subsequent action is unknown but, later in 1884,
he applied for the vacant Cavendish Professorship of Experimental Physics at
Cambridge. Despite Reynolds’ numerous distinctions, however, the appoint-
ment went to his former student, J.J. Thomson (then a young man of 27 work-
ing at Cambridge, later to become Sir Joseph Thomson, OM, PRS, Nobel lau-
reate and discoverer of the electron). Although it has already been quoted (Gib-
son, 1946; Allen, 1970), it is worthwhile repeating part of Reynolds’ generous
letter of congratulations sent on Boxing Day, 1884:
My dear Thomson,
I do not like to let the occasion pass without offering you my congratulations,
which are none the less sincere that we could not both hold the chair. Your elec-
tion is in itself a matter of great pleasure and pride for me . . . and I have no doubt
but every hope that you will amply justify the wisdom of the election.
Believe me yours sincerely
Osborne Reynolds
still did not have at his disposal what he considered to be adequate laboratory
facilities. Indeed, Thompson (1886; as reported by Allen, 1970) notes that in
that year (1884) Reynolds drew the attention of the University’s Council to
the urgent need for an engineering laboratory. It seems that, finally, this com-
plaint may well have led in 1887 to the overdue provision of state-of-the-art
laboratories (Gibson, 1946).
the Royal Society for more than 15 and, as noted above, had received major
awards. He was then unquestionably the leading engineering fluid mechanicist
in England and quite possibly more widely than that.
Lord Rayleigh had meanwhile become Editor of the Philosophical Transac-
tions of the Royal Society. Perhaps inevitably, on receiving this second manu-
script on turbulent flow from Reynolds, he sent it for review by Sir George
Stokes. This time, however, the referee’s response was rather unsatisfactory.
After a long period of silence, on 31 October 1894 Sir George, now equipped
with a typewriter with both upper- and lower-case letters, sent his reply
(Figure 1.8), effectively acknowledging that he did not understand the work.
The letter is a copy-book example of the ‘on-the-one-hand . . . yet-on-the-other’
style of review: Reynolds hadn’t made his case – yet, he was an able man
and the 1883 paper was sound; moreover the author had paid to have the
present paper printed so obviously he thought it was important. However, the
reviewer couldn’t confirm that view . . . but neither would he assert that it was
wrong!
Stokes’ concluding sentence seems to imply that he had finished with the
matter, but Lord Rayleigh evidently had other ideas. Although the exchanges
are incomplete it seems that Rayleigh pressed Stokes to go further and, when
Stokes pleaded that he had mislaid the copy of the paper, he arranged for him
to be sent another copy. (Since the paper had been printed, Reynolds had ev-
idently submitted several copies.) On 5 December 1894, Sir George sent this
second copy back indicating that he had now found the copy originally sent
to him. He added his regrets that he was “not yet able to go beyond the rough
indication contained in a letter sent to Lord Rayleigh some time ago” (Royal
Society Archive Ref. 209 from Sir G.G. Stokes to Mr Rix).
Meanwhile, Lord Rayleigh had sent the paper to a second referee, Horace
Lamb, Professor of Mathematics at Manchester, who a decade earlier had been
elected a Fellow of the Royal Society. One can only speculate why Rayleigh
approached the only other senior fluid mechanicist in Manchester to review
his own colleague’s work. Nevertheless, on 21 November 1894 Lamb sent his
longhand assessment which began with the brisk summarizing statement:
I think the paper should be published in the Transactions as containing the views
of its author on a subject which he has to a great extent created, although much
of it is obscure and there are some fundamental points which are not clearly
established.
Figure 1.8 Sir George Stokes’ initial review of the 1895 paper,
c The Royal
Society, reproduced with permission.
The Royal Society holds three further communications from the referees of
which only one is dated. There is thus some doubt as to the actual sequencing
though the most probable seems to be the following. At some point Sir George
Stokes does send his review to Lord Rayleigh, a two-page typed assessment
raising some of the problems with the paper he and, indeed, Lamb had aired
earlier. Thereafter (or, possibly, even before that communication), the refer-
ees had made contact with one another, presumably through the intervention
of Lord Rayleigh, which led Lamb to prepare a joint report that Sir George
attached to his letter of 30 January 1895 (Royal Society Archive Ref. 210):
1: Osborne Reynolds: a turbulent life 27
This head-reeling sentence, 100 words in length, is also remarkable for its
naturalness; its innocent admission of the paper’s weaknesses accompanied by
its ready self-forgiveness. The letter then continues:
1: Osborne Reynolds: a turbulent life 29
I now enclose you in M.S.S. a full preliminary description of this part of the
argument which by permission I shall be glad to substitute for the first two
lines of §5 p. 3. It contains, what I hope will be found, a clear definition of the
terms mean-mean motion and relative-mean motion as well as of mean-motion
and heat-motions and of the geometrical distinctions between these motions.
And although no physical-distinction between mean-molar and relative-molar
is draw[n] other than what is implied by the geometrical distinction that the inte-
grals of ρu, etc, taken over the space determined by the scale or period-in-space
of the relative mean motion ρu , etc, are the components of momentum of the
molar motion of the mechanical system within S while the integrals of ρu, etc,
taken over the same space are zero, it is shown that such physical distinction has
no place in the argument any further than it is suppressed by the terms in the
equations of motion.
This passage, like those cited earlier, brings out Reynolds’ infatuation with
long rambling sentences that stand starkly in contrast to Lamb’s crisply stated
criticisms. He finally acknowledges:
With reference to the difficulties in logic of §8 p. 9, equations 7 and 8a, this is
intirely removed by replacing the bar (u) which has dropped from the u in the
left of equation 4, p. 8.
There are, I am sorry to say, certain other misprints in the paper which must
have increased the inherent difficulties of the subject.
Very truly yours,
Osborne Reynolds
“You have not noticed that just above the critical velocity the resistance d p/dz
varies nearly as the cube of the velocity until d p/dz is about double what it is at
the critical velocity. Of course these dimensional facts . . . are the definite clues
to the physics and mechanics of the problem and the gist of the later research.
Nor is it polite or true to speak of the ‘empirical formulation adopted by Engi-
neers’ since it is Engineers who have done the scientific investigations which
alone have given us accurate data.” Readers may form their own opinion as to
whether Reynolds’ acerbic response was provoked by a perception that Lamb
had been one of the referees of his 1895 paper3 .
Despite its rather lukewarm reception by the two eminent referees, the 1895
paper is seen today as a mighty beacon in the literature of fluid mechanics.
First and foremost was the decomposition of the flow into mean and fluctu-
ating parts leading to the averaged momentum equations (now known as the
Reynolds equations) in which the Reynolds stresses appear as unknowns. In
fact, throughout the analysis Reynolds treated the averaging in a form akin
to what is now known as mass-weighted averaging, 60 years earlier than the
source that is usually quoted for introducing that strategy. It was surely just that
his experiments had used water as the fluid medium that has led to this feature
being ignored. The paper’s other major analytical result was the turbulent ki-
netic energy equation in which he observed that the terms comprising products
of Reynolds stress and mean velocity gradient represented a transfer of kinetic
energy from the mean flow to turbulence. As an indicator of just how far this
discovery was ahead of its time, we note that the corresponding, albeit simpler,
equation for the mean square temperature fluctuations was not published until
the 1950s (Corrsin, 1952).
Reynolds’ purpose in examining the turbulent kinetic energy equation was
to provide an explanation of why the changeover from laminar to turbulent
motion should occur at a particular value of the Reynolds number. Indeed, that
was the driving rationale for the whole paper. For this purpose he considered
fully developed laminar flow between parallel planes on which a small analyt-
ical disturbance was superimposed which permitted him to obtain expressions
for the turbulence energy generation and viscous dissipation rates integrated
over the channel. The relative magnitude of these two processes varied with
Reynolds number and the lower critical Reynolds number he identified as be-
ing that where the overall turbulence energy generation rate had grown to bal-
ance the viscous dissipation rate. That his estimates were inaccurate is now
seen as irrelevant since the paper contained more than enough novelty for the
world of fluid mechanics to absorb over the ensuing decades.
3 In response to Reynolds’ criticism, Lamb replaced ‘empirical formula adopted by Engineers’
in the published book by ‘practical formula adopted by writers on hydraulics’.
1: Osborne Reynolds: a turbulent life 31
difficult ones both for him and his family. There he remained until his death
from influenza on 21 February 1912.
His funeral in St Decuman’s church was attended by Horace Lamb (The
West Somerset Free Press, 2 March 1912) and Reynolds is buried in the church-
yard, his gravestone being an elegant art nouveau cross with his name and the
dates of his arrival and departure beautifully engraved thereon (Figure 1.12).
His wife who lived until 1942 is interred with him while two grandsons (the
sons of Henry Osborne Reynolds, one named Osborne Reynolds), both of
whom were killed in action during the Second World War, are memorialized
on the gravestone.
about the subject, but thought it out for himself from the beginning before read-
ing what others had written about it.
In his lectures Reynolds was often carried away by his subject and got into
difficulties. Some humorous incidents are related with regard to the manner in
which he got out of them. He was once explaining the slide rule to his class;
holding one in his hand, he expounded in detail the steps necessary to perform
a multiplication. “We take as a simple example three times four”, he said, and
after appropriate explanations he continued, “Now we arrive at the result; three
times four is 11.8”. The class smiles. “That is near enough for our purpose”, says
Reynolds.
The final photograph of Reynolds with students and his staff (in which Gib-
son is standing in the row behind Reynolds, immediately to his left) appears in
Figure 1.13.
It is appropriate that the final word here should go to Horace Lamb, who in
his obituary notice wrote (Lamb, 1913):
The character of Reynolds was, like his writings, strongly individual. He was
conscious of the value of his work, but was content to leave it to the mature
judgement of the scientific world. For advertisement he had no taste; and undue
pretensions on the part of others only elicited a tolerant smile. To his pupils
he was most generous in the opportunities for valuable work which he put in
1: Osborne Reynolds: a turbulent life 35
Figure 1.13 Osborne Reynolds with staff and students of the Engineering De-
partment, 1903. c The University of Manchester. Reproduced by courtesy of the
University Librarian and Director, The John Rylands University Library, The Uni-
versity of Manchester.
their way, and in the share of credit which he consigned to them in cases of co-
operation. Somewhat reserved in serious or personal matters and occasionally
combative and tenacious in debate, he was in the ordinary relations of life the
most kindly and genial of companions.
1.3.3 Closure
In closing this chapter it is appropriate to ask why it was that, in his lifetime,
Osborne Reynolds was never awarded any national honour. The obituary no-
tice that appeared in Nature just a week after his death, began: “In Professor
Osborne Reynolds . . . Great Britain has lost its most distinguished scientific
engineer”. Towards the end of the piece, after noting his admission to the
Royal Society, the award of the Society’s Royal Medal and his honorary doc-
torate from Glasgow University, it concluded by remarking that “this was the
only public recognition he ever received”. The tone and positioning of this
last observation clearly leave the impression that the writer at least felt there
was a measure of injustice in Reynolds not receiving other honours; why it
was that he did not end his days as Sir Osborne Reynolds (as, in fact, many
of his web entries do, erroneously, refer to him). One may remark that among
the well-known fluid mechanicists of his time, George Stokes, Horace Lamb
and Thomas Stanton4 were all knighted while William Thomson was, as noted
4 After whom the dimensionless heat-transfer coefficient, the Stanton number, is named.
36 Launder & Jackson
earlier, first knighted and later admitted to the peerage as Lord Kelvin of Largs.
If we exclude the last named who made notable contributions in several other
walks of life, many would argue that none contributed as much to the advance-
ment of fluid mechanics and thermodynamics in all its varied aspects as did
Osborne Reynolds (not just in the particular studies of turbulent flow on which
the present article has focused).
Was he, possibly, offered such an honour and declined it? This seems highly
unlikely, first because, while he would have been at pains to dissociate himself
from the formal trappings and snobbery of such a title, he would probably have
been delighted if somewhat bemused by the award. Secondly, if such an offer
had been made and declined, this fact (while kept secret during his lifetime)
would surely have been disclosed following his death in one or more of the
obituaries written by his colleagues.
Thus, there remains the question as to why he was not knighted, in response
to which the authors offer three possible contributory factors. First, his pub-
lic demeanour may have been perceived as lacking sufficient gravitas. As his
daughter’s letter (quoted in §1.2.3) implies, Reynolds was someone with his
head in the clouds while he was also seen by some as rather eccentric. There are
anecdotes of him setting puzzles for his listeners. According to Lamb (1913),
for example: “He had a keen sense of humour, and delighted in startling para-
doxes, which he would maintain half seriously and half playfully, with as-
tonishing ingenuity and resource.” Collectively, these foibles could well have
made him seem unsuited for holding high office in public institutions which is
so often a precursor to (if not quite a prerequisite for) a knighthood. By way
of contrast, Stokes (like Kelvin) served as President of the Royal Society and
his advancement was assured. Stanton, a student of Reynolds, after holding
the Chair of Engineering at the University of Bristol, was appointed Superin-
tendent of the Engineering Department at the National Physical Laboratory,
a post he filled from 1901 to his retirement in 1930. Lamb served twice as
Vice-President of the Royal Society and as President of the London Mathe-
matical Society. His ability to cut through tricky problems, which must have
served him very well throughout his career, is clearly illustrated by his review
of Reynolds’ 1895 paper cited earlier. Moreover, Lamb also possessed a fur-
ther attribute that Reynolds unfortunately lacked: longevity! He was knighted
only at the age of 82 in 1931. Thus, the relatively young age at which Reynolds
dropped from professional visibility was quite possibly a second contributory
factor in his case being overlooked.
Thirdly, it is incontrovertible that the importance of Reynolds’ major works
was simply not widely recognized until after his death. As his obituary in
Nature observed: “Well in advance of his time, in many cases years elapsed
1: Osborne Reynolds: a turbulent life 37
before the practical bearing of his researches was fully appreciated; even now
the sphere of his influence on engineering progress is still widening.” This
was arguably the major contributor to his failure to receive the accolade of a
knighthood. Indeed, we may note, wryly, the correctness of this assertion in
Nature’s obituary for, while summarizing many of his important research con-
tributions, it made no reference at all to the turbulent flow papers which are
central to the present appreciation of his work. We should be indulgent of that
lapse, however, for, when, in 1895, his strategy for the analysis of turbulent
flows was published in Phil. Trans. Roy. Soc., could anyone, even Osborne
Reynolds, have foreseen that it was destined to shape the direction of research
in engineering fluid mechanics for the next century?
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2
Prandtl and the Göttingen school
Eberhard Bodenschatz and Michael Eckert
2.1 Introduction
In the early decades of the 20th century Göttingen was the center for mathemat-
ics. The foundations were laid by Carl Friedrich Gauss (1777–1855) who from
1808 was head of the observatory and professor for astronomy at the Georg
August University (founded in 1737). At the turn of the 20th century, the well-
known mathematician Felix Klein (1849–1925), who joined the University in
1886, established a research center and brought leading scientists to Göttingen.
In 1895 David Hilbert (1862–1943) became Chair of Mathematics and in 1902
Hermann Minkowski (1864–1909) joined the mathematics department. At that
time, pure and applied mathematics pursued diverging paths, and mathemati-
cians at Technical Universities were met with distrust from their engineering
colleagues with regard to their ability to satisfy their practical needs (Hensel,
1989). Klein was particularly eager to demonstrate the power of mathematics
in applied fields (Prandtl, 1926b; Manegold, 1970). In 1905 he established an
Institute for Applied Mathematics and Mechanics in Göttingen by bringing the
young Ludwig Prandtl (1875–1953) and the more senior Carl Runge (1856–
1927), both from the nearby Hanover. A picture of Prandtl at his water tunnel
around 1935 is shown in Figure 2.1.
Prandtl had studied mechanical engineering at the Technische Hochschule
(TH, Technical University) in Munich in the late 1890s. In his studies he was
deeply influenced by August Föppl (1854–1924), whose textbooks on tech-
nical mechanics became legendary. After finishing his studies as mechanical
engineer in 1898, Prandtl became Föppl’s assistant and remained closely re-
lated to him throughout his life, intellectually by his devotion to technical
mechanics and privately as Föppl’s son-in-law (Vogel-Prandtl, 1993). Under
Föppl’s supervision Prandtl wrote his doctoral dissertation on a problem of
40
2: Prandtl and the Göttingen school 41
Figure 2.1 Ludwig Prandtl at his water tunnel in the mid to late 1930s. Reproduc-
tion from the original photograph DLR: FS-258.
the same vein, Klein had arranged Runge’s call to Göttingen. In autumn 1905,
Klein’s institutional plans materialized. Göttingen University opened a new
Institute for Applied Mathematics and Mechanics under the joint directorship
of Runge and Prandtl. Klein also involved Prandtl as Director in the planning
of an extramural Aerodynamic Research Institute, the Motorluftschiffmodell-
Versuchsanstalt, which started its operation with the first Göttingen design
windtunnnel in 1907 (Rotta, 1990; Oswatitsch and Wieghardt, 1987). Klein
regarded Prandtl’s “strong power of intuition and great originality of thought
with the expertise of the engineer and the mastery of the mathematical ap-
paratus” (Manegold, 1970, p. 232) ideal qualities for what he had planned to
establish at Göttingen.
With these institutional measures, the stage was set for Prandtl’s unique
career between science and technology – and for the foundation of an aca-
demic school with a strong focus on basic fluid dynamics and their applica-
tions. Prandtl directed the Institut für Angewandte Mechanik of Göttingen Uni-
versity, the Aerodynamische Versuchsanstalt (AVA), as the rapidly expanding
Motorluftschiffmodell-Versuchsanstalt (airship model test facility) was
renamed after the First World War, and, after 1925, the associated Kaiser-
Wilhelm-Institut (KWI) für Strömungsforschung. His ambitions and the his-
tory leading to the establishment of the KWI are well summarized in his
opening speech at his institute, which has been translated into English (Prandtl,
1925E).
During the half century of Prandtl’s Göttingen period, from 1904 until his
death, his school extended Göttingen’s fame from mathematics to applied me-
chanics, a specialty which acquired in this period the status of a self-contained
discipline. Prandtl had more than eighty doctoral students, among them Hein-
rich Blasius, Theodore von Kármán, Max Munk, Johann Nikuradse, Walter
Tollmien, Hermann Schlichting, Karl Wieghardt, and others who, like Prandtl,
perceived fluid mechanics in general, and turbulence in particular, as a para-
mount challenge to bridge the gulf between theory and practice. Like Prandtl’s
institutional affiliations, his approach towards turbulence reflects a broad spec-
trum of ‘pure’ and ‘applied’ research (if such dichotomies make sense in turbu-
lence research). We have to consider the circumstances and occasions in these
settings in order to better characterize the approach of the Göttingen school on
turbulence.
this concept during his short industrial occupation when he tried to account
for the phenomenon of flow separation in diverging ducts. Prandtl presented
the concept together with photographs of flow around obstacles in a water
trough at the Third International Congress of Mathematicians in Heidelberg
in August 1904 (Prandtl, 1905). In a summary, prepared at the request of the
American Mathematical Society, he declared1 that the “most important result”
of this concept was that it offered an “explanation for the formation of dis-
continuity surfaces (vortex sheets) along continuously curved boundaries”. In
his Heidelberg presentation he expressed the same message in these words:
“A fluid layer set in rotational motion by the friction at the wall moves into
the free fluid and, exerting a complete change of motion, plays there a similar
role as Helmholtz’ discontinuity sheets” (Prandtl, 1905, p. 578). (For more on
the emergence of Helmholtz’s concept of discontinuity surfaces, see Darrigol,
2005, chapter 4.3).
According to the recollection of one participant at the Heidelberg congress,
Klein recognized the momentousness of Prandtl’s method immediately (Som-
merfeld, 1935). However, if this recollection from many years later may be
trusted, Klein’s reaction was exceptional. The boundary layer concept required
elaboration before its potential was more widely recognized (Dryden, 1955;
Goldstein, 1969; Tani, 1977; Grossmann et al., 2004). Its modern understand-
ing in terms of singular perturbation theory (O’Malley Jr., 2010) emerged
only decades later. The first tangible evidence that Prandtl’s concept provided
more than qualitative ideas was offered by Blasius, who derived in his doc-
toral dissertation the coefficient for (laminar) skin friction from the bound-
ary layer equations for the flow along a flat plate (‘Blasius flow’: Blasius,
1908; Hager, 2003). However, this achievement added little understanding to
what Prandtl had considered the most important result of his concept, namely
how vortical motion, to say nothing of turbulence, is created at the
boundary.
Even before Prandtl arrived in Göttingen, the riddle of turbulence was a re-
current theme in Klein’s lectures and seminars. In a seminar on hydraulics
in the winter semester 1903/04 Klein called it a “true need of our time to
bridge the gap between separate developments”. The notorious gulf between
hydraulics and hydrodynamics served to illustrate this need with many ex-
amples. The seminar presentations were expected to focus on the comparison
between theory and experiment in a number of specific problems with the flow
of water, such as the outflow through an orifice, the flow over a weir, pipe
1 Undated draft in response to a request from 13 August 1904, Blatt 43, Cod. Ms. L. Prandtl 14,
Acc. Mss. 1999.2, SUB.
44 Bodenschatz & Eckert
flow, waves, the water jump (‘hydraulic jump’), or the natural water flow in
rivers.2
In the winter semester 1907/08 Klein dedicated another seminar to fluid me-
chanics, this time with the focus on ‘Hydrodynamics, with particular empha-
sis of the hydrodynamics of ships’. With Prandtl and Runge as co-organizers,
the seminar again involved a broad spectrum of problems from fluid mechan-
ics that Klein and his colleagues regarded as suitable for mathematical ap-
proaches.3 Theodore von Kármán, who made then his first steps towards an
outstanding career in Prandtl’s institute, presented a talk on unsteady potential
motion. Blasius, who was finishing his dissertation on the laminar boundary
layer in 1907, reviewed in two sessions contemporary research on turbulent
flows. Other students and collaborators of Prandtl dealt with vortical motion
(Karl Hiemenz) and boundary layers and the detachment of vortices (Georg
Fuhrmann). Although little was published about these themes at the time,
Klein’s seminar served as a testing ground for debates on the notorious prob-
lems of fluid mechanics like the creation of vorticity in ideal fluids (‘Klein’s
Kaffeelöffelexperiment’: see Klein, 1910; Saffman, 1992, chapter 6).
With regard to turbulence, the records of Blasius’ presentation from this
seminar illustrate what Prandtl and his collaborators must have regarded as
the main problems at that time. After reviewing the empirical laws, such as
Chezy’s law for channel flow and Reynolds’ findings about the transition to
turbulence in pipe flow (for these and other pioneering 19th-century efforts,
see Darrigol, 2005, chapter 6), Blasius concluded that the problems “addressed
to hydrodynamics” should be sorted into two categories: I. Explanation of in-
stability; and II. Description of turbulent motion. These had to address the di-
chotomy of hydraulic description versus rational hydrodynamic explanations.
Concerning the first category, the onset of turbulence, Blasius reviewed Hen-
drik Antoon Lorentz’s recent approach where a criterion for the instability of
laminar flow was derived from a consideration of the energy added to the flow
by a superposed fluctuation (Lorentz, 1897, 1907). With regard to the second
category, fully developed turbulence, Blasius referred mainly to Boussinesq’s
pioneering work (Boussinesq, 1897) where the effect of turbulence was de-
scribed as an additional viscous term in the Navier–Stokes equation. In con-
trast to the normal viscosity, this additional ‘turbulent’ viscosity term was
due to the exchange of momentum by the eddying motion in turbulent flow.
2 Klein, handwritten notes. SUB Cod. Ms. Klein 19 E (Hydraulik, 1903/04), and the seminar
protocol book, no. 20. Göttingen, Lesezimmer des Mathematischen Instituts. Available online
at librarieswithoutwalls.org/klein.html.
3 Klein’s seminar protocol book, no. 27. Göttingen, Lesezimmer des Mathematischen Instituts.
Available online at librarieswithoutwalls.org/klein.html.
2: Prandtl and the Göttingen school 45
Boussinesq’s concept had already been the subject of the preceding seminar
in 1903/04, where the astronomer Karl Schwarzschild and the mathematicians
Hans Hahn and Gustav Herglotz reviewed the state of turbulence (Hahn et al.,
1904). However, the efforts in the seminar to determine the (unknown) eddy
viscosity of Boussinesq’s approach proved futile. “Agreement between this
theory and empirical observations is not achieved,” Blasius concluded in his
presentation.4
In spite of the emphasis on the riddles of turbulence in these seminars, it is
commonly reported that Prandtl ignored turbulence as a research theme until
many years later. For example, the editors of his Collected Papers dated his first
publication in the category Turbulence and Vortex Formation to the year 1921
(see below). The preserved archival sources, however, belie this impression.
Prandtl started to articulate his ideas on turbulence much earlier. “Turbulence I:
Vortices within laminar motion”, he wrote on an envelope with dozens of loose
manuscript pages. The first of these pages is dated by himself as 3 October
1910, with the heading Origin of turbulence. Prandtl considered there “a vortex
line in the boundary layer close to a wall” and argued that such a vortical mo-
tion “fetches (by frictional action) something out of the boundary layer which,
because of the initial rotation, becomes rolled up to another vortex which en-
hances the initial vortex.” Thus he imagined how flows become vortical due to
processes that originate in the initially laminar boundary layer.5 In the same
year he published a paper on A relation between heat exchange and flow resis-
tance in fluids (Prandtl, 1910) which extended the boundary layer concept to
heat conduction. Although it did not explicitly address turbulence – the article
is more renowned because Prandtl introduced here what was later called the
‘Prandtl number’ – it reveals Prandtl’s awareness for the differences of laminar
and turbulent flow with regard to heat exchange and illustrates from a different
perspective how turbulence entered Prandtl’s research agenda (Rotta, 2000).
Another opportunity to think about turbulence from the perspective of the
boundary layer concept came in 1912 when wind tunnel measurements about
the drag of spheres displayed discrepant results. When Otto Föppl (1885–
1963), Prandtl’s brother-in-law and collaborator at the airship model test fa-
cility, compared the data from his own measurements in the Göttingen wind
tunnel with those from the laboratory of Gustave Eiffel (1832–1921) in Paris,
he found a blatant discrepancy and supposed that Eiffel or his collaborator
had omitted a factor of 2 in the final evaluation of their data (Föppl, 1912).
Provoked by this claim, Eiffel performed a new test series and found that
4 Klein’s seminar protocol book, no. 27, p. 80. Göttingen, Lesezimmer des Mathematischen
Instituts. Available online at http://www.librarieswithoutwalls.org/felixKlein.html.
5 Cod. Ms. L. Prandtl 18 (Turbulenz I: Wirbel in Laminarbewegung), Acc. Mss. 1999.2, SUB.
46 Bodenschatz & Eckert
Figure 2.2 Turbulence behind a sphere made visible with smoke. Reproduction
from the original 1914 photograph. GOAR: GK-0116 and GK-0118.
the discrepancy was not the result of an erroneous data evaluation but a new
phenomenon which could be observed only at higher air speeds than those
attained in the Göttingen wind tunnel (Eiffel, 1912). After inserting a nozzle
into their wind tunnel, Prandtl and his collaborators were able to reproduce
Eiffel’s discovery: at a critical air speed the drag coefficient suddenly dropped
to a much lower value. Prandtl also offered an explanation of the new phe-
nomenon. He assumed that the initially laminar boundary layer around the
sphere becomes turbulent beyond a critical air speed. On the assumption that
the transition from laminar to turbulent flow in the boundary layer is analogous
to Reynolds’ case of pipe flow, Prandtl displayed the sphere drag coefficient as
a function of the Reynolds number, UD/ν (flow velocity U, sphere diameter
D, kinematic viscosity ν), rather than, as did Eiffel, of the velocity; thus he
demonstrated that the effect occurred at roughly the same Reynolds number
even if the individual quantities differed widely (the diameters of the spheres
ranged from 7 to 28 cm; the speed in the wind tunnel was varied between 5 and
23 m/s). Prandtl further argued that the turbulent boundary layer flow entrains
fluid from the wake so that the boundary layer stays attached to the surface
of the sphere longer than in the laminar case. In other words, the onset of tur-
bulence in the boundary layer reduces the wake behind the sphere and thus
also its drag. But the argument that turbulence decreases the drag seemed so
paradoxical that Prandtl conceived an experimental test: when the transition to
turbulence in the boundary layer was induced otherwise, e.g. with a thin ‘trip
wire’ around the sphere or a rough surface, the same phenomenon occured.
When smoke was added to the air stream, the reduction of drag became visible
by the reduced extension of the wake behind the sphere (Wieselsberger, 1914;
Prandtl, 1914) (see Figure 2.2).
2: Prandtl and the Göttingen school 47
9 Page 15 (dated 6 March 1916) in Cod. Ms. L. Prandtl, 18, Acc. Mss. 1999.2, SUB.
2: Prandtl and the Göttingen school 49
of the involved vortex interaction he introduced what he called the rough as-
sumption that the vortex remains unchanged for a certain time ∼ r2 /ν and then
suddenly disappears, whereby it communicates its angular momentum to the
mean flow.10
During the war Prandtl had more urgent items on his agenda (Rotta, 1990,
pp. 115–193). But the riddle of turbulence as a paramount challenge did not
disappear from his mind. Nor from that of his former student, the prodigy
Theodore von Kármán, who returned after the War to the Technische Hoch-
schule Aachen as Director of the then fledgling Aerodynamic Institute. Both
the Aachen and the Göttingen fluid dynamicists pursued the quest for a theory
of turbulence in a fierce rivalry. “The competition was gentlemanly, of course.
But it was first-class rivalry nonetheless,” Kármán later recalled, “a kind of
Olympic Games, between Prandtl and me, and beyond that between Göttingen
and Aachen” (von Kármán, 1967, p. 135). Since they had nothing published
on turbulence, both Prandtl and Kármán pondered how to ascertain their pri-
ority in this quest. In summer 1920, Prandtl supposed that von Kármán used a
forthcoming science meeting in Bad Nauheim to present a paper on turbulence
at this occasion. “I do not yet know whether I can come”, he wrote11 to his
rival, “but I wish to be oriented about your plans. As the case may be I will an-
nounce something on turbulence (experimental) as well. I have now visualized
turbulence with lycopodium in a 6 cm wide channel.” The Aachen–Göttingen
rivalry had not yet surfaced publicly at this occasion. By correspondence, how-
ever, it was further developing. The range of topics encompassed Prandtl’s en-
tire working program. Early in 1921 Prandtl learned that von Kármán was busy
elaborating a theory of fully developed turbulence in the boundary layer along
a flat wall – with “fabulous agreement with observations”. Ludwig Hopf and
another collaborator of the Aachen group had by this time started with hot-
wire experiments. Hopf revealed12 that in Aachen they planned to measure in
a water channel the mean square fluctuation and the spectral distribution of the
fluctuations.
Little seems to have resulted from these experiments, neither in Aachen
by means of the hot-wire technique nor in Prandtl’s laboratory by visualiz-
ing turbulence with lycopodium. Von Kármán’s theoretical effort, however,
appeared promising. “Dear Master, colleague, and former boss”, Kármán ad-
dressed Prandtl in a five-page letter with ideas for a turbulent boundary layer
10 Page 16 in Cod. Ms. L. Prandtl, 18, Acc. Mss. 1999.2, SUB. Apparently r and ν are the radius
of the vortex and the kinematic viscosity of the fluid, respectively. Prandtl did not define the
quantities involved here. His remarks are rather sketchy and do not lend themselves for a precise
determination of the beginnings of his future mixing length approach.
11 Prandtl to Kármán, 11 August 1920. GOAR 1364.
12 Hopf to Prandtl, 3 February 1921. MPGA, Abt. III, Rep. 61, Nr. 704.
50 Bodenschatz & Eckert
theory (see below) and about the onset of turbulence.13 The latter was regarded
as the turbulence problem. The difficulty in explaining the transition from lam-
inar to turbulent flow had been rated as a paramount riddle since the late
19th century. In his dissertation performed under Sommerfeld in 1909, Lud-
wig Hopf had titled the introductory section The turbulence problem, because
neither the energy considerations of Reynolds and Lorentz nor the stability
approaches of Lord Kelvin and Lord Rayleigh were successful. Hopf was con-
fronted with the problem in the wake of Sommerfeld’s own stability approach
to viscous flows, but “the consequent analysis of the problem according to the
method of small oscillations by Sommerfeld is not yet accomplished” (Hopf,
1910, pp. 6–7). In the decade that followed the problem was vigorously at-
tacked by this technique (later labeled as the Orr–Sommerfeld method) – with
the discrepant result that plane Couette flow seemed stable for all Reynolds
numbers (Eckert, 2010).
In comparison with these efforts, Prandtl’s approach as sketched in his work-
ing program appeared like a return to the futile attempts of the 19th century:
“At large Reynolds number the difference between viscous and inviscid fluids
is certainly imperceptible,” Hopf commented14 on Prandtl’s idea to start from
the inviscid limit, but at the same time he regarded it “questionable whether
one is able to arrive at a useful approximation that leads down to the critical
number from this end”. In response to such doubts Prandtl began to execute
his working program about the onset of turbulence in plane flows with piece-
wise linear flow profiles. “Calculation according to Rayleigh’s papers III, p.
17ff,” he noted on a piece of paper in January 1921, followed by several pages
of mathematical calculations.15 Despite their initial reservations, the Aachen
rivals were excited about Prandtl’s approach. Von Kármán immediately rushed
his collaborators to undertake a stability analysis for certain piecewise lin-
ear flow profiles, Hopf confided to Prandtl.16 Prandtl had by this time already
asked a doctoral student to perform a similar study. “Because it deals with a
doctoral work, I would be sorry if the Aachener would publish away part of his
dissertation,” he asked Hopf, so as not to interfere in this effort. Von Kármán
responded that the Aachen stability study was aiming at quite different goals,
namely the formation of vortices in the wake of an obstacle (labeled later as
the ‘Kármán vortex street’ after von Kármán’s earlier theory about this phe-
nomenon; Eckert, 2006, chapter 2). The new study was motivated by “the hope
to determine perhaps the constants that have been left indetermined in my old
theory,” wrote Kármán, attempting to calm Prandtl’s worry. Why not arrange
13 Kármán to Prandtl, 12 February 1921. GOAR 3684.
14 Hopf to Prandtl, 27 October 1919. GOAR 3684.
15 Pages 22–26 in Cod. Ms. L. Prandtl, 18, Acc. Mss. 1999.2, SUB.
16 Hopf to Prandtl, 3 February 1921. MPGA, III, Rep. 61, Nr. 704.
2: Prandtl and the Göttingen school 51
effect, boundary layer flow could also be imagined as fully turbulent. From a
practical perspective, the latter appeared much more important than the former.
Data on fluid resistance in pipes, as measured for decades in hydraulic labo-
ratories, offered plenty of problems for testing theories about turbulent fric-
tion. Blasius, who had moved in 1911 to the Berlin Testing Establishment for
Hydraulics and Ship Building (Versuchsanstalt für Wasserbau und Schiffbau),
published in 1913 a survey of pipe flow data: when displayed as a function of
the Reynolds number Re, the coefficient for ‘hydraulic’ (i.e. turbulent) friction
varied in proportion to Re−1/4 (in contrast to laminar friction at low Reynolds
numbers, where it is proportional to Re−1 ) (Blasius, 1913).
No theory could explain this empirical ‘Blasius law’ for turbulent pipe flow.
But it could be used to derive other semi-empirical laws, such as the distri-
bution of velocity in the turbulent boundary layer along a plane smooth wall.
When Kármán challenged Prandtl in 1921 with the outline of such a theory,
he recalled that Prandtl had told him earlier how one could extrapolate from
pipe flow to the flow along a plate, and that Prandtl already knew that the ve-
locity distribution was proportional to y1/7 , where y was the distance from the
wall. Prandtl responded that he had known this “already for a pretty long time,
say since 1913”. He claimed that he had already in earlier times attempted to
calculate boundary layers in which he had assumed a viscosity enhanced by
turbulence, which he chose for simplicity as proportional to the distance from
the wall and proportional to the velocity in the free flow. But he admitted that
Kármán had advanced further with regard to a full-fledged turbulent boundary
layer theory. “I have planned something like this only for the future and have
not yet begun with the elaboration.” Because he was busy with other work he
suggested21 that Kármán should proceed with the publication of this theory:
“I will see afterwards how I can gain recognition with my different derivation,
and I can get over it if the priority of publishing has gone over to friendly
territory.”
Kármán published his derivation without further delay in the first volume of
ZAMM (von Kármán, 1921) with the acknowledgement that it resulted from
a suggestion “by Mr. Prandtl in an oral communication in Autumn 1920”.
Prandtl’s derivation appeared in print only in 1927 – with the remark that “the
preceding treatment dates back to Autumn 1920” (Prandtl, 1927a).
Johann Nikuradse (1894–1979), whom Prandtl assigned by that time an ex-
perimental study about the velocity distribution in turbulent flows as subject
of a doctoral work, dated Prandtl’s derivation more precisely to a discourse
in Göttingen on 5 November, during the winter semester of 1920 (Nikuradse,
21 Prandtl to Kármán, 16 February 1921. MPGA, Abt. III, Rep. 61, Nr. 792.
54 Bodenschatz & Eckert
1926, p. 15). Indeed, Prandtl outlined this derivation in notices dated (by him-
self) to 28 November 1920.22 Further evidence is contained in the first volume
of the Ergebnisse der Aerodynamischen Versuchsanstalt zu Göttingen, accom-
plished at “Christmas 1920” (according to the preface), where Prandtl offered
a formula for the friction coefficient proportional to Re−1/5 , with the Reynolds
number Re related to the length of the plate (Prandtl, 1921b, p. 136). Although
Prandtl did not present the derivation, he could not have arrived at this friction
coefficient without the 1/7th law for the velocity distribution. (The derivation
was based on the assumption that the shear stress at the wall inside the tube
only depends on the flow in the immediate vicinity of the wall; hence it should
not depend on the radius of the tube. Under the additional assumption that the
velocity grows according to a power law with increasing distance from the
wall, the derivation was straightforward.)
Kármán presented his theory on turbulent skin friction again in 1922 at a
conference in Innsbruck (von Kármán, 1924). He perceived it only as a first
step on the way towards a more fundamental understanding of turbulent fric-
tion. The solution, he speculated at the end of his Innsbruck talk, would prob-
ably come from a statistical consideration. But in order to pursue such an in-
vestigation “a fortunate idea” was necessary, “which so far has not yet been
found” (von Kármán, 1924, p. 167). (For more on the quest for a statistical
theory in the 1920s, see Battimelli, 1984.) Prandtl, too, raised little hope for
a more fundamental theory of turbulence from which empirical laws, such as
that of Blasius, could be derived from first principles: “You ask for the theo-
retical derivation of Blasius’ law for pipe friction,” Prandtl responded23 to the
question of a colleague in 1923. “The one who will find it will thereby become
a famous man!”
by the new method of hot wire anemometry (Burgers, 1925). Prandtl had hesi-
tated in 1921 to publish his derivation of the 1/7th law because, as he revealed
in another letter25 to his rival at Aachen, he aimed at a theory in which the
experimental evidence would play a crucial role. Four years later, with the data
from Burgers’ laboratory, from the dissertation of Nikuradse (1926), and from
other investigations about the resistance of water flow in smooth pipes (Jakob
and Erk, 1924), this evidence was available. The experiments confirmed the
1/7th law within the range of Reynolds numbers for which the Blasius 1/4th
law was valid. But they raised doubts whether it was valid for higher Reynolds
numbers. Prandtl, therefore, attempted to generalize his theoretical approach
so that he could derive from any empirical resistance law a formula for the
velocity distribution. He wrote26 to von Kármán in October 1924 thus:
I myself have occupied myself recently with the task to set up a differential equa-
tion for the mean motion in turbulent flow, which is derived from rather simple
assumptions and seems appropriate for very different cases. . . . The empirical is
condensed in a length which is entirely adjusted to the boundary conditions and
which plays the role of a free path length.
The historic papers on turbulent stress and eddy viscosity by Reynolds and
Boussinesq were of course familiar to Prandtl since Klein’s seminars in 1904
and 1907, but without further assumptions these approaches could not be turned
into practical theories. At a first look, Prandtl had just exchanged one unknown
quantity () with another (l). However, in contrast to the eddy viscosity , the
mixing length l was a quantity which, as Prandtl had written27 to Kármán,
“is entirely adjusted to the boundary conditions” of the problem under con-
sideration. The problem of turbulent wall friction, however, required rather
sophisticated assumptions about the mixing length. Without wall interactions,
the mixing length could be adjusted less arbitrarily. Prandtl resorted to other
phenomena for illustrating the mixing length approach, such as the mixing of
a turbulent jet ejected from a nozzle into an ambient fluid at rest. In this case
the assumption that the mixing length is proportional to the width of the jet in
each cross-section gave rise to a differential equation from which the broaden-
ing of the jet behind the nozzle could be calculated. The theoretical distribution
of mean flow velocities obtained by this approach was in excellent agreement
with experimental measurements (Prandtl, 1927b; Tollmien, 1926).
For the turbulent shear flow along a wall, however, the assumption of propor-
tionality between the mixing length l and the distance y from the wall did not
yield the 1/7th law as Prandtl had hoped. Instead, when he attempted to derive
the distribution of velocity for plane channel flow, he arrived at a logarithmic
law – which he dismissed because of unpleasant behavior at the centerline of
the channel (see Figure 2.3).28 From his notes in summer 1924 it is obvious
that he struggled hard to derive an appropriate distribution of velocity from
one or another plausible assumption for the mixing length – and appropriate
meant to him that the mean flow U(y) ∝ y1/7 , not some logarithmic law.
Three years later (Prandtl, 1930, p. 794) in a lecture in Tokyo in 1929, he
dismissed the logarithmic velocity distribution again. He argued that “l pro-
portional y does not lead to the desired result because it leads to U prop. log y,
which would yield U = −∞ for y = 0.”
This provided an opportunity for Kármán to win the next round in their
‘gentlemanly’ competition.
Figure 2.3 Excerpt of Prandtl’s ‘back of the envelope’ calculations from 1924.
(Fritsch, 1928). He found that the velocity profiles line up with each other
in the middle parts if they are shifted parallel. This suggested that the veloc-
ity distribution in the fluid depends only on the shear stress transferred to the
wall and not on the particular wall surface structure. Kármán derived from this
empirical observation a similarity approach. In a letter to Burgers he praised
this approach for its simplicity: “The only important constant thereby is the
proportionality factor in the vicinity of the wall.” As a result, he was led to
58 Bodenschatz & Eckert
logarithmic laws both for the velocity distribution in the turbulent boundary
layer and for the turbulent skin friction coefficient. “The resistance law fits
very well with measurements in all known regions,” he concluded, with a hint
to recent measurements.29
The recent measurements to which Kármán alluded where those of Fritsch in
Aachen and Nikuradse in Göttingen. The latter, in particular, showed a marked
deviation from Blasius’ law, and hence from the 1/7th law for the distribu-
tion of velocity, at higher Reynolds numbers. Nikuradse had presented some
of his results in June 1929 at a conference in Aachen (Nikuradse, 1930); the
comprehensive study appeared only in 1932 (Nikuradse, 1932). By introduc-
ing a dimensionless
wall distance η = v∗ y/ν and velocity ϕ = u/v∗ , where
v∗ = τ0 /ρ is the friction velocity, τ0 the shear stress at the wall and ρ the
density, Nikuradse’s data suggested a logarithmic velocity distribution of the
form ϕ = a + b log η.
Backed by these results from Prandtl’s laboratory, Kármán submitted a pa-
per entitled Mechanical Similarity and Turbulence to the Göttingen Academy
of Science. Unlike Prandtl, he introduced the mixing length as a characteristic
scale of the fluctuating velocities determined by l = kU /U , where k is a di-
mensionless constant (later called the ‘Kármán constant’) and U , U are the
first and second derivatives of the mean velocity of a plane parallel flow in the
x-direction with respect to the perpendicular coordinate y. He derived this for-
mula from the hypothesis that the velocity fluctuations are similar anywhere
and anytime in fully developed turbulent flow at some distance from a wall.
He had plane channel flow in mind, because he chose his coordinate system so
that the x-axis coincided with the centerline between the walls at y = ±h. The
approach would fail both at the center line and at the walls, but was supposed
to yield reasonable results in between. (For more detail on Kármán’s approach,
see Chapter 3 by Leonard and Peters.) Whereas Prandtl’s approach required a
further assumption about the mixing length, Kármán’s l was an explicit func-
tion of y at any point in the cross-section of the flow. Kármán obtained a loga-
rithmic velocity distribution and a logarithmic formula for the turbulent friction
coefficient (von Kármán, 1930a).
A few months later, Kármán presented his theory at the Third International
Congress of Applied Mechanics, held in Stockholm during 24–29 August 1930.
For this occasion he also derived the resistance formula for the turbulent skin
friction of a smooth plate. “The resistance law is no power law,” hinting at
the earlier efforts of Prandtl and himself. “I am convinced that the form of the
resistance law as derived here is irrevocable.” He presented a diagram about
the plate skin friction where he compared the ‘Prandtl v. Kármán 1921’ the-
ory with the ‘new’ one, and with recent measurements from the Hamburgische
Schiffbau–Versuchsanstalt. “It appears to me that for smooth plates the last
mismatch between theory and experiment has disappeared,” he concluded his
Stockholm presentation (von Kármán, 1930b).
Prandtl was by this time preparing a new edition of the Ergebnisse der Aero-
dynamischen Versuchsanstalt zu Göttingen and eager to include the most re-
cent results.30 The practical relevance of Kármán’s theory was obvious. In
May 1932, the Hamburgische Schiffbau–Versuchsanstalt convened a confer-
ence where the recent theories and experiments about turbulent friction were
reviewed. Kármán was invited for a talk on the theory of the fluid resistance,
but he could not attend so that he contributed only in the form of a paper which
was read by another attendee (von Kármán, 1932). Franz Eisner, a scientist
from the Preussische Versuchsanstalt für Wasserbau und Schiffbau in Berlin,
addressed the same theme from a broader perspective, and Günther Kempf
from the Hamburg Schiffbau–Versuchsanstalt presented recent results about
friction on smooth and rough plates (Eisner, 1932b; Kempf, 1932). Prandtl
and others were invited to present commentaries and additions (Prandtl et al.,
1932). By and large, this conference served to acquaint practitioners, partic-
ularly engineers in shipbuilding, with the recent advances achieved in the re-
search laboratories in Göttingen, Aachen and elsewhere.
Two months after this conference, the Schiffbautechnische Gesellschaft pub-
lished short versions of these presentations in its journal Werft, Reederei, Hafen.
From Eisner’s presentation a diagram about plate resistance was shown which
characterized the logarithmic law “after Prandtl (Ergebnisse AVA Göttingen,
IV. Lieferung 1932” as the best fit of the experimental values. According to
this presentation, the “interregnum of power laws” had lasted until 1931, when
Prandtl formulated the correct logarithmic law (Eisner, 1932a). When Kármán
saw this article he was upset. He felt that his breakthrough for the correct plate
formula in 1930 as he had presented it in Stockholm was ignored. He com-
plained in a letter to Prandtl31 that from the article about the Hamburg confer-
ence “it looks as if I have given up working on this problem after 1921, and that
everything has been done in 1931/32 in Göttingen”. He asked Prandtl to cor-
rect this erroneous view in the Göttingen Ergebnisse, which he regarded as the
standard reference work for all future reviews. “I write so frankly how I think
in this matter because I know you as the role model of a just man,” appeal-
ing to Prandtl’s fairness. But he had little sympathy for “your lieutenants who
30 Prandtl to Kármán’s colleagues at Aachen, 30 October 1930. TKC 23.43; Prandtl to Kármán,
29 November 1930; Kármán to Prandtl, 16 December 1930. MPGA, Abt. III, Rep. 61, Nr. 792.
31 Kármán to Prandtl, 26 September 1932. MPGA, Abt. III, Rep. 61, Nr. 793.
60 Bodenschatz & Eckert
understandably do not know other gods beside you. They wish to claim every-
thing for Göttingen.” He was so worried that he also sent Prandtl a telegram32
with the essence of his complaint.
Prandtl responded immediately. He claimed33 that he had no influence on the
publications in Werft, Reederei, Hafen. With regard to the Ergebnisse der Aero-
dynamischen Versuchsanstalt zu Göttingen he calmed Kármán’s worries saying
that in the publication they would of course refer to the latter’s papers. As in
the preceeding volumes of the Ergebnisse, the emphasis was on experimental
results. The news about the logarithmic laws were presented in a rather short
theoretical part (12 out of 148 pages) entitled On turbulent flow in pipes and
along plates. By and large, Prandtl arrived at the same results as Kármán. He
duly acknowledged Kármán’s publications from the year 1930, but he claimed
that he had arrived at the same results at a time when Kármán’s papers had not
yet been known, so that once more, like ten years before with the same prob-
lem, the thoughts in Aachen and Göttingen followed parallel paths (Prandtl,
1932, p. 637). For the Hamburg conference proceedings, Prandtl and Eisner
(1932) formulated a short appendix where they declared “that the priority for
the formal [formelmässige] solution for the resistance of the smooth plate un-
doubtedly is due to Mr. v. Kármán who talked about it in August 1930 at the
Stockholm Mechanics Congress.”
When Kármán was finally aware of these publications, he felt embarrassed:
“I hope that there will not remain an aftertaste from this debate,” he wrote to
Prandtl34 . Prandtl admitted that he had “perhaps not without guilt” contributed
to Kármán’s misgivings. But he insisted35 that his own version of the theory of
plate resistance was better suited for practical use. Although the final results of
Prandtl’s and Kármán’s approaches agreed with each other, there were differ-
ences with regard to the underlying assumptions and the ensuing derivations.
Unlike Kármán, Prandtl did not start from a similarity hypothesis. There was
no ‘Kármán’s constant’ in Prandtl’s version. Instead, when Prandtl accepted
the logarithmic law as empirically given, he used the same dimensional con-
siderations from which he had derived the 1/7th law from Blasius’ empirical
law. In retrospect, with the hindsight of Prandtl’s notes36 , it is obvious that he
came close to Kármán’s reasoning – but the problem of how to account for the
viscous range close to the wall (which Kármán bypassed by using the center-
line of the channel as his vantage point) prevented a solution. In his textbook
32 Kármán to Prandtl, 28 September 1932. MPGA, Abt. III, Rep. 61, Nr. 793.
33 Prandtl to Kármán, 29 September 1932. MPGA, Abt. III, Rep. 61, Nr. 793.
34 Kármán to Prandtl, 9 December 1932. MPGA, Abt. III, Rep. 61, Nr. 793.
35 Prandtl to Karman, 19 December 1932. MPGA, Abt. III, Rep. 61, Nr. 793.
36 Prandtl, notes, MPGA, Abt. III, Rep. 61, Nr. 2276, 2278.
2: Prandtl and the Göttingen school 61
Figure 2.4 Picture of the Rauhigkeitskanal at the Max Planck Institute for
Dynamics and Self-Organization. It was built in 1935 and reconstituted by
Helmut Eckelmann and James Wallace in the 1970s.
Wieghardt and Tillmann, 1944, 1951E) is the only tunnel that survived the dis-
mantling at the end of the war as it was part of the KWI and not the AVA37 .
turbulence measurements. This set the stage for spectral measurements of tur-
bulent velocity fluctuations.
From two letters between Taylor and Prandtl in August and December 1932,
following the correspondence44 discussed above, it is apparent that both had
started to conduct hotwire experiments to investigate the turbulent velocity
fluctuations: Taylor in collaboration with researchers at the National Physi-
cal Laboratory (NPL) – most likely Simmons and Salter (see Simmons et al.,
1938) – and Prandtl with Reichardt (see Reichardt, 1933; Prandtl and Rei-
chardt, 1934). Taylor responded45 with suggestions for pressure correlation
measurements and argued:
The same kind of analysis can be applied to hotwire measurements and I am hop-
ing to begin some work on those lines. In particular the ‘spectrum of turbulence’
has not received much attention.
Prandtl replied46 :
I do not believe that one can achieve a clear result with pressure measurements,
as there is no instrument that can measure these small pressure fluctuations with
sufficient speed. Instead hotwire measurements should lead to good results. We
ourselves have conducted an experiment in which two hotwires are placed at
larger or smaller distances from each other and are, with an amplifier, connected
to a cathode ray tube such that the fluctuations of the one hotwire appear as hor-
izontal paths, and those of the other as perpendicular paths on the fluorescent
screen47 . . . To measure also the magnitude of the correlation my collaborator
Dr. Reichardt built an electrodynamometer with which he can observe the mean
of u1 , u2 and u1 u2 . In any case, I am as convinced as you that from the study of
those correlations as well as between the direction and magnitude fluctuations,
for which we have prepared a hotwire setup, very important insights into turbu-
lent flows can be gained.
In the same letter Prandtl sketched three pictures of the deflections of the
oscilloscope that are also published in Prandtl and Reichardt (1934). In this
article the authors reported that the hotwire measurements leading to the fig-
ures had been finished in August 1932 (date of the letter of Taylor to Prandtl),
and that in October 1933 a micro-pressure manometer had been developed
to measure the very weak turbulent fluctuations.48 It is very remarkable that
it took less than a year for Prandtl and Reichardt to pick up the pressure
measurement proposal by Taylor. It also shows the technical ingenuity and
the excellent mechanics workshop at the Göttingen KWI. The micro-pressure
44 Prandtl to Taylor, 25 July 1932. MPGA, Abt. III, Rep. 61, Nr. 1653.
45 Taylor to Prandtl, 18 August 1932. MPGA, Abt. III, Rep. 61, Nr. 1653.
46 Prandtl to Taylor, 23 December 1932. MPGA, Abt. III, Rep. 61, Nr. 1653.
47 This way of showing correlations had been used at the KWI since 1930 (Reichardt, 1938b).
48 See the paper Reichardt (1934) which was at that time in preparation.
2: Prandtl and the Göttingen school 65
49 Taylor to Prandtl, 14 January 1933. MPGA, Abt. III, Rep. 61, Nr. 1653.
50 Prandtl to Taylor, 25 January 1933. MPGA, Abt. III, Rep. 61, Nr. 1653.
51 Taylor to Prandtl, 1 June 1934. MPGA, Abt. III, Rep. 61, Nr. 1653.
52 Prandtl to Taylor, 5 June 1934. MPGA, Abt. III, Rep. 61, Nr. 1653.
53 Prandtl to Taylor, 28 February 1935. MPGA, Abt. III, Rep. 61, Nr. 1654.
66 Bodenschatz & Eckert
readjusts itself into a condition where the turbulent velocities are much more
nearly equally distributed in space.
I lately have been doing a great deal of work on turbulence . . . In the course of
my work I have brought out two formulae which seem to have practical interest.
The first concerns the rate of decay of energy in a windstream . . . and I have
compared them with some of Dryden’s measurements behind a honeycomb – it
seems to fit. It also fits Simmons’ measurements with turbulence made on a very
different scale . . .
The second formula was concerned with the “theory of the critical Reynolds
number of a sphere behind a turbulence-grid”, as Prandtl replied in his letter55
pointing him to his own experimental work from 1914. Prandtl also mentioned
that measurement from Göttingen see a signature of the grid. Taylor interpreted
this as the “shadow of a screen”, which according to Dryden’s experiments dies
away after a point, where the turbulence is still fully developed.56 It was this
region where Taylor expected his theory to apply. Taylor submitted his results
in four consecutive papers “On the statistics of turbulence” on 4 July 1935
(Taylor, 1935a, 1935b, 1935d, 1935e). Later Prandtl re-derived Taylor’s decay
law of turbulence (Wieghardt, 1941, 1942E; Prandtl and Wieghardt, 1945). A
detailed discussion of the physics of the decay law of grid-generated turbulence
can be found in Chapter 4 by Sreenivasan.
A month later Taylor57 thanked Prandtl for sending him his paper with Rei-
chardt (Prandtl and Reichardt, 1934) on measurements of the correlations of
turbulent velocity fluctuations that Prandtl had already referred to in his letter58
in 1932. Taylor needed these data for “comparison with my theory of energy
dissipation”.
Again the correspondence with Prandtl surely contributed to Taylor’s un-
derstanding and finally led to the third paper in the 1935 sequence (Taylor,
1935d). After returning from the 5th Volta Congress in Rome (on high speeds
in aviation), which both attended, Prandtl mentioned59 to Taylor that “Mr. Re-
ichardt conducts new correlation measurements this time correlations between
u and v . The results we will send in the future”. Again, just as in 193360 ,
54 Taylor to Prandtl, 2 March 1935. MPGA, Abt. III, Rep. 61, Nr. 1654.
55 Prandtl to Taylor, 12 March 1935. MPGA, Abt. III, Rep. 61, Nr. 1654.
56 Taylor to Prandtl, 14 March 1935. MPGA, Abt. III, Rep. 61, Nr. 1654.
57 Taylor to Prandtl, 21 April 1935. MPGA, Abt. III, Rep. 61, Nr. 1654.
58 Prandtl to Taylor, 23 December 1932. MPGA, Abt. III, Rep. 61, Nr. 1653.
59 Prandtl to Taylor, 12 November 1935. MPGA, Abt. III, Rep. 61, Nr. 1654.
60 Prandtl to Taylor, 25 January 1933. MPGA, Abt. III, Rep. 61, Nr. 1653.
2: Prandtl and the Göttingen school 67
where Prandtl used a similar formulation, the correspondence does not return
to the matter of turbulence until more than a year later.
They resumed the discussion again when Taylor sent Prandtl a copy of
his paper on “Correlation measurements in a turbulent flow through a pipe”
(Taylor, 1936). Prandtl responded61 that “currently we are most interested in
the measurement of correlations between locations in the pipe”, suggesting
that Taylor may consider measurements away from the center of the pipe and
mentioned that “the measurements in Fig. 4 agree qualitatively well with our
u measurements. A better agreement is not to be expected as we measured in
a rectangular channel and you in a round pipe.” Taylor replied on 11 January
1937 and also on62 23 January 1937 when he sent Prandtl “our best measure-
ments so that you may compare with your measurements in a flat pipe”.
Thus by 1937 the stage was set at Göttingen and Cambridge for the most
important measurements about the statistics of turbulent fluctuations. At the
same time, Dryden and his co-workers at the National Bureau of Standards in
Washington measured the decay of the longitudinal correlations behind grids
of different mesh sizes M (Dryden et al., 1937) and calculated from it by in-
tegration of the correlation function the integral scale of the flow, what Taylor
(1938b, p. 296) called the “the scale of turbulence”. By using different grids
they were able to show that the grid mesh size M determined the large-scale
L of the flow, just as Taylor had assumed in 1935. They also found that the
relative integral scale L/M increased with the relative distance x/M from the
grid, independently of M. This was later analyzed in more detail by Taylor
(1938b) with data from the National Physical Laboratory in Teddington. The
paper by Dryden and collaborators (Dryden et al., 1937) was very important
for the further development of turbulence research, as it was data from here
that Kolmogorov used in 1941 for comparison with his theory (see Chapter 6
on the Russian school by Falkovich).
of Simmons for the round pipe. Then he answered a question of Prandtl about
the recent paper on the spectrum of velocity fluctuations (Taylor, 1938a) and
explained to him what we now know as ‘Taylor’s frozen flow hypothesis’, i.e.
that the formula depends only on the assumption that u is small compared to U so
that the succession of events at a point fixed in the turbulent stream is assumed
to be related directly to the Fourier analysis of the (u, x) curve obtained from
simultaneous measurements of u and x along a line parallel to the direction of U.
Six months after this exchange, Prandtl and Taylor met in Cambridge, MA, for
the Fifth International Congress for Applied Mechanics, held at Harvard Uni-
versity and the Massachusetts Institute of Technology during 12–16 September
1938. By now the foundations for seminal discoveries in turbulence research
were set. For the next 60 years, experimental turbulence research was domi-
nated by the Eulerian approach, i.e. spatial and equal time measurements of
turbulent fluctuations as introduced in the period 1932–1938.
At this Fifth International Congress turbulence was the most important topic.
In the view of the great interest in the problem of turbulence at the Fourth
Congress and of the important changes in accepted views since 1934 it was de-
cided to hold a Turbulence Symposium at the Fifth Congress. Professor Prandtl
kindly consented to organize this Symposium and . . . The Organizing commit-
tee is grateful to Professor Prandtl and considers his Turbulence Symposium not
only the principal feature of this Congress, but perhaps the Congress activity that
will materially affect the orientation of future research
wrote Hunsaker and von Kármán in the ‘Report of the Secretaries’ in the con-
ference proceedings. Prandtl had gathered the leading turbulence researchers
of his time to this event. The most important talks, other than the one by
Prandtl, were the overview lecture by Taylor on “Some recent developments
in the study of turbulence” (Taylor, 1938b) and Dryden’s presentation with
his measurements of the energy spectrum (Dryden, 1938). It is interesting to
note that, in concluding his paper, Dryden presented a single hotwire technique
that he intended to use to measure the turbulent shearing stress u v . Prandtl’s
laboratory under Reichardt’s leadership had already found a solution earlier64
64 Taylor to Prandtl, 18 March 1938. MPGA, Abt. III, Rep. 61, Nr. 1654.
2: Prandtl and the Göttingen school 69
and Prandtl presented this data in his talk. Reichardt used hotwire anemome-
try with a probe consisting of three parallel wires, where the center wire was
mounted a few millimeters downstream and used as a temperature probe (Re-
ichardt, 1938a,b; 1951E). The transverse component of velocity was sensed by
the wake of one of the front wires. This probe was calibrated in oscillating lam-
inar flow. During the discussion of Dryden’s paper, Prandtl made the following
important remark concerning the turbulent boundary layer: “One can assume
that the boundary zone represents the true ‘eddy factory’ and the spread to-
wards the middle would be more passive.” In his comment, he also showed a
copy of the fluctuation measurements conducted by Reichardt and Motzfeld
(Reichardt, 1938b, Fig. 3) in a wind tunnel of 1 m width and 24 cm height.
Prandtl’s 1938 paper at the turbulence symposium deserves a closer review
because, on the one hand, it became the foundation for his further work on tur-
bulence and, on the other, as an illustration of Prandtl’s style. It reflects beau-
tifully and exemplarily, what von Kármán, for example, admired as Prandtl’s
“ability to establish systems of simplified equations which expressed the essen-
tial physical relations and dropped the nonessentials”; von Kármán regarded
this ability “unique” and even compared Prandtl in this regard with “his great
predecessors in the field of mechanics – men like Leonhard Euler (1707–1783)
and d’Alembert (1717–1783)”.65
In this paper, Prandtl distinguished four types of turbulence: wall turbulence,
free turbulence, turbulence in stratified flows (see also Prandtl and Reichardt,
1934), and the decaying isotropic turbulence. He first considered the decay law
of turbulence behind a grid using his mixing length approach by assuming that
the fluctuating velocity is generated at a time t and then decays:
t 2
dt dU t − t
u2 = l1 ×f ,
−∞ T dy t T
T ) ≈ T +t−t justified by Dryden’s measurements. With the shear stress
with f ( t−t T
dU
τ = ρu l2 dy and l2 = km, where m is the grid spacing and k ≈ 0.103, he derived
const cUm
u =
= ,
t+T x
where x is the downstream distance from the grid, c is related to the thickness
of the rods, and Um is the mean flow. From the equation of motion to lowest
order,
∂U 1 ∂τ
Um = ,
∂x ρ ∂y
65 See Kármán (1957); Anderson (2005); see also Prandtl’s own enlightening contribution to this
topic (Prandtl, 1948b).
70 Bodenschatz & Eckert
the flow. Later (Lumley and Panofsky, 1964), the location of this maximum
was proposed as a surrogate for the integral scale. As shown in Figure 2.6(c),
Motzfeld also compared his data with the 1938 wind tunnel data in Simmons
et al. (1938) by rescaling both datasets with the mean velocity U and the
72 Bodenschatz & Eckert
(a) Fluctuating quantities measured in a flow (b) Velocity spectra for different distances
between two parallel plates for Remax = 17500. from the wall in turbulent channel flow
(Wandturbulenz).
(c) Log/lin plot of the spectrum in (a) (d) Velocity spectrum compared
between shear-generated and grid-
generated decaying turbulence
channel height L or, for the wind tunnel, with the grid spacing L. As we can
see both datasets agree reasonably well. Prandtl remarked about the surprising
collapse of the data in Figure 2.6(a) (Fig. 3 in Prandtl, 1938): “The most re-
markable about these measurements is that de facto the same frequency distri-
bution was found.” From the perspective of experimental techniques, the elec-
tromechanical measurement technique employed by Motzfeld and Reichardt
is also remarkable. As shown in Figure 2.7, they used the amplitude of an
electromechanically driven and viscously damped torsion wire resonant oscil-
lator to measure, by tuning resonance frequencies and damping, the frequency
components of the hotwire signal.
2: Prandtl and the Göttingen school 73
As we will see, these results would lead (Prandtl and Wieghardt, 1945)
not only to the derivation of what is now known as the ‘one equation model’
(Spalding, 1991), but also to the assumption of a universal energy cascade of
turbulence cut off at the dissipation scale (Prandtl, 1945, 1948a). Thus, at age
70, Prandtl had finally found what he had been looking for all his life albeit at
74 Bodenschatz & Eckert
the worst time – when the Second World War ended and he was not allowed to
conduct scientific research.66
Prandtl’s sojourn in the USA in September 1938 was also remarkable in
another respect – because it marked the beginning of his, and for that mat-
ter Germany’s, alienation from the international community. When he tried to
convince the conference committee to have him organize the next Congress in
Germany, he encountered strong opposition based on political and humanitar-
ian reasons. Against many of his foreign colleagues, Prandtl defended Hitler’s
politics and actions. Taylor attempted to cure Prandtl of his political views.67
As discussed also in Chapter 4 by Sreenivasan, Taylor was concerned with the
humanitarian situation of the Jewish population and the political situation in
general. However, Taylor’s candor (he called Hitler “a criminal lunatic”) did
not bode well with Prandtl, who responded again by defending German pol-
itics.68 Only a few days before Prandtl wrote his letter (on 18 October 1938)
12,000 Polish-born Jews were expelled from Germany. On 11 November 1938,
the atrocities of the ‘Kristallnacht’ started the genocide and Holocaust (Gilbert,
2006). Taylor replied on 16 November 1938 with a report about the very bad
experiences in Germany of his own family members.69 Nevertheless he ended
his letter still quite amicably:
You will see that we are not likely to agree on political matters so it would be
best to say no more about them. Fortunately there is no reason why people who
do not agree politically should not be best friends.
Then he continued to make a remark that he does not understand why Prandtl
plotted n f (n) instead of f (n) (shown in Figure 5b) (Fig. 4 in Prandtl, 1938). As
far as we know Prandtl never replied. After this correspondence Prandtl wrote
one more letter to Mrs Taylor.70 Only a month later World War II started and
cut off their communication. Prandtl tried71 to resume contact with Taylor after
the war, but there is no evidence that Taylor ever responded to this effort.
the very important discovery by Motzfeld and Reichardt was not recognized
abroad. We have found no reference to Motzfeld’s 1938 publication other than
in the unpublished 1945 paper by Prandtl (see below). As described above their
discovery showed that the spectrum of the streamwise velocity fluctuation in a
channel flow did not depend on the location of the measurements in the chan-
nel and did agree qualitatively with those by Simmons and Salter for decaying
isotropic turbulence. The 1938 Göttingen results show beautifully the univer-
sal behavior that Kolmogorov postulated in his revolutionary 1941 work (see
Chapter 6 on the Russian school by Falkovich).
Prandtl and his co-workers were not aware of the developments in Russia
and continued their program in turbulence at a slower pace. According to a
British Intelligence report72 after the war, based on an interrogation of Prandtl,
due to more urgent practical problems little fundamental work, either experimen-
tal or theoretical, had been conducted during the war. No work had been done in
Germany similar to that of G.I. Taylor or Kármán and Howarth on the statistical
theory of turbulence. Experiments had been planned on the decay of turbulence
behind grids in a wind tunnel analogous to those undertaken by Simmons at the
National Physical Laboratories, but these were shelved at the outbreak of the
war.
Indeed as far as fully developed turbulence was concerned the progress was
mostly theoretical and mainly relying on measurements made before the war.
In his response to the military interrogators, Prandtl was very modest. From
late autumn of 1944 till the middle of 1945 he worked on the theory of fully
developed turbulence almost daily (see Figure 2.8). This was his most active
period in which he pulled together the threads outlined earlier.
We will now review briefly the development from 1939 to 1944 that led to
this stage. The status of knowledge of turbulence in 1941 is well summarized
in Wieghardt (1941; 1942E), and that between 1941 and 1944 in Prandtl’s
(1948a) FIAT article entitled Turbulenz. Prandtl reviewed in 23 tightly written
pages the work at the KWI in chronological order and by these topics:
(i) Turbulence in the presence of walls
(a) Pipeflow
(b) Flat plates
(c) Flow along walls with pressure increase and decrease
(ii) Free turbulence
(a) General laws
72 British Intelligence Objectives Sub-Committee report 760 that summarizes a visit to the KWI,
26–30 April 1946.
76 Bodenschatz & Eckert
"
Figure 2.8 Prandtl worked continuously on the topic of fully developed turbu-
lence. T1 marks Prandtl’s work on the energy equation of turbulence, T2 his in-
vestigations on the effect of dissipation, V a derivation of the vorticity equation
in a plane shear flow, and TS his attempts to develop a statistical theory of ve-
locity fluctuations. The other letters mark important dates: on 31 October 1944
he formulated for the first time the ‘one equation model’ (E); on 4 January 1945
he presented it at a theory seminar (S) and on 26 January 1945 at a meeting of
the Göttingen Academy of Science (A); 29 January 1945 marks his discovery of
what is known as the Kolmogorov length scale (K41); on 4 February 1945 he had
his 70th birthday (B); on 11 February 1945 he formulated for the first time his
cascade model (C); on 27 March 1945 he is reworking the draft for the paper on
dissipation (R2); on 8 April 1945 Göttingen was occupied by American forces; on
4 July 1945 Prandtl entered remarks on the already typewritten draft revisions of
the dissipation paper. The period in May, where he had no access to the Institute
as it was used by American forces, is light gray – the Institute reopened on 4 June
1945 to close again briefly thereafter.
2: Prandtl and the Göttingen school 77
His drafting of the paper fell right into the end of WWII. By July 1945 the
Institute was under British administration and had74 “many British and Ameri-
can visitors”. Prandtl was still allowed75 “to work on some problems that were
not finished during the war and from which also reports were expected. Start-
ing any new work was forbidden.” By 11 October 1945 the chances for publi-
cation were even worse because76 “all research was shelved completely” and
“any continuation of research was forbidden by the Director of Scientific Re-
search in London”. So it seems that as of August 1945 Prandtl followed orders
and stopped writing the paper and stopped working on turbulence (see Fig-
ure 2.8). In addition, in that period the AVA was being disassembled and the
parts were sent to England. Then in January 1946, Heisenberg and Weizsäcker
returned to Göttingen from being interned in England and brought along their
calculations that superseded Prandtl’s work. So by January 1946 the window
of opportunity for publication had passed. In addition, he was busily writing
the FIAT report on turbulence – and that is where he at least mentioned his
work.
Let us now discuss briefly Prandtl’s last known work on turbulence. His
very carefully written notes77 cover the period from 14 October 1944 until 12
August 1945; they allow us to understand his achievement better. These notes
comprise 65 numbered pages, 5 pages on his talk in a Theory Colloquium
on 4 January 1945 where he presented his energy equation of turbulence, 7
pages on an attempt at understanding the distribution function of velocity from
probability arguments, and 19 pages of sketches and calculations. Figure 2.8
summarizes the days he entered careful handwritten notes in his workbook.
From these entries we see that he devoted a large part of his time to the study
of turbulence. It seems remarkable how much time he was able to dedicate to
this topic, considering that he also directed the research at his Institute and that
he was engaged as an adviser to the Air Ministry concerning the direction of
aeronautical research for the war. How much effort he dedicated to the latter
activity is open to further historical inquiry.
In order to discern the subsequent stages of Prandtl’s approach we proceed
chronologically:
On 31 October 1944 we find the first complete formulation of the ‘one
equation model’ for the evolution of turbulent kinetic energy per unit volume
in terms of the square of the fluctuating velocity (see Figure 2.9). The same
74 Prandtl to Taylor, 18 July 1945. MPGA, Abt. III, Rep. 61, Nr. 1654.
75 Prandtl to Taylor, 18 July 1945. MPGA, Abt. III, Rep. 61, Nr. 1654. See also Prandtl to the
President of the Royal Society, London, 11 October 1945. MPGA, Abt. III, Rep. 61, Nr. 1402.
76 Prandtl to Taylor, 18 October 1945. MPGA, Abt. III, Rep. 61, Nr. 1654.
77 GOAR 3727.
80 Bodenschatz & Eckert
formula now written in terms of kinetic energy per unit volume was presented
by him in his paper at the meeting of the Göttingen Academy of Science on 26
January 1945 (Prandtl and Wieghardt, 1945). There Wieghardt also presented
his determination of the parameters from measurements in grid turbulence
and channel flow and found good agreement with the theory. His differential
equation marked as Eq. 4a in Figure 2.9 determined the change of energy per
2: Prandtl and the Göttingen school 81
unit volume from three terms: the first term on the right-hand side gives the tur-
bulent energy flux for a “bale of turbulence” (in German: Turbulenzballen) of
size l (which he equated with a mixing length), the second term represents the
diffusion of turbulence in the direction of the gradient of turbulent energy and
the third term represents the source of turbulent energy from the mean shear.
Here cu l is the eddy viscosity. Please note that this equation is what is now
called a k-model. This equation was independently derived later by Howard
Emmons in 1954 and by Peter Bradshaw in 1967 (Spalding, 1991). Prandtl
and Wieghardt also pointed out the deficiencies of the model, namely the role
of viscosity at the wall and for the inner structure of a bale of turbulence. For
the latter, Prandtl argued that, as long as the Reynolds number of each bale of
turbulence is large, a three-dimensional version of his equation should also be
applicable to the inner dynamics of the bale of turbulence. He then introduced
the idea of bales of turbulence within bales of turbulence, which we now know
as the turbulent cascade. He called them steps (in German: Stufen) in the se-
quence that goes from large to small. He pointed out that by going from step
to step the Reynolds number will decrease with increasing number of the step
(decreasing size of the turbulent eddy) until viscosity is dominant and all en-
ergy is transformed to heat. Finally he conjectured that a general understanding
of the process can be obtained.
And indeed he discovered it within a short time. On 29 January 1945, only
three days after the presentation at the Academy, he entered in his notes the
derivation of the Kolmogorov length scale that at this time he called in anal-
ogy to Taylor’s smallest length scale λ (Figure 2.10). In Eq. (1) the decay
rate of the kinetic energy per unit mass is equated with the dissipation at the
smallest scales. Eq. (2) connects the final step of the cascade process with the
Kolmogorov velocity. By putting (1) and (2) together Prandtl arrived at the
Kolmogorov length scale given by Eq. (3). This seemed to him so remarkable
that he commented on the side of the page “Checked it multiple times! But
only equilibrium of turbulence.”
At this stage he was almost done, but as we can tell from his typewritten
manuscript (Prandtl, 1945) and from his notes he was not satisfied. He had
to put this on more formal grounds. So only two weeks later, as shown in
Figure 2.11 he used a cascade model for each step of turbulence, with β as
the ratio in length scale from step to step. This allowed him to derive the Kol-
mogorov length more rigorously from a geometrical series.
This became the content of his draft paper from 1945 that we shall dis-
cuss in detail in a separate publication. It is clear that the spectral data from
Motzfeld (1938) were instrumental for his progress (those which Prandtl had
presented at the Cambridge Congress in 1938 and which are reproduced above
82 Bodenschatz & Eckert
Figure 2.10 First known derivation of the Kolmogorov length scale (here
called λ).
in Figure 2.6). Here we close our review with a quote from the introduction to
his unpublished paper “The role of viscosity in the mechanism of developed
turbulence” (Prandtl, 1945) which beautifully reflects his thinking and needs
no further analysis. This is only a short excerpt from the introduction to the pa-
per. A full translation is in preparation and will be published elsewhere. Also,
2: Prandtl and the Göttingen school 83
Figure 2.11 Prandtl’s cascade model for the fluctuating velocities at different
steps in the cascade.
our translation is very close to the original German text and therefore some
sentences are a bit long.
The following analyses, which consider in detail the inner processes of a tur-
bulent flow, will prove that the solution for λ by Taylor that he obtained from
84 Bodenschatz & Eckert
energy considerations does not yet give the smallest element of turbulence. The
mechanism of turbulence generation is not resolved in all details. So much is
however known [here Prandtl referred in a footnote to work by Tollmien pub-
lished in Göttinger Nachr. Heft 1 (1935) p. 79] that flows with an inflection
point in the velocity profile may become unstable at sufficiently large Reynolds
numbers. Therefore one has to expect that at sufficiently high Reynolds number
u l
ν
the motion of an individual bale of turbulence is by itself turbulent, and that
for this secondary turbulence the same is true, and so on. Indeed one observes al-
ready at very modest Reynolds numbers a frequency spectrum that extends over
many decades. That it is mostly the smallest eddies that are responsible for the
conversion of the energy of main motion into heat can easily be understood, as
for them, the deformation velocities ∂u ∂y
, etc. are the largest.
The earlier discussion is the simplest explanation of the fact that in turbulent
motion always the smaller eddies are present next to the larger ones. G.I. Taylor,
1935, used a different explanation. He pointed out that according to general sta-
tistical relations the probability of two particles separating in time is larger than
for them to come closer, and he applied this relationship to two particles on a vor-
tex line. From the well-known Helmholtz theorem it would follow that – as long
as the viscosity does not act in an opposing sense – the increase of the angular ve-
locity of the vortex line is more probable than its decay. He shows this tendency
with an example, whose series expansion clearly shows the evolution towards
smaller eddies. However, it could not be continued, so the processes could only
be followed for short time intervals. One can counter Taylor’s deductions insofar
that through the increase of the angular velocities, pressure fields develop, which
oppose a further increase of the vortex lines. It thus cannot be expected that the
extension would reach the expected strength. It seems, however, that the action
in the sense of Taylor is surely present, if, though, with weaker magnitude than
expected from a purely kinematic study.
For the development of smaller eddy diameters in the turbulence, one can also
note that wall turbulence starts with thin boundary layers and that free turbulence
has equally thin separating sheets. Therefore, in the beginning, only the smallest
vortices are present and the larger ones appear one after another. Opposing this,
however, is the result that in the fully developed channel flow, the frequency spec-
trum de facto does not depend on the distance from the wall (Motzfeld, 1938).
One would not expect this if all of the fine turbulence originated at the wall.
This strongly supports the validity of the conjecture for stationary turbulence
presented here. Further support is given by investigations conducted later, which
concerned isotropic, temporally decaying, turbulence and which have been quite
satisfactorily justified by experiments. The two descriptions of the re-creation of
the smaller eddies by turbulence of second and higher order, and the one that
relates to the Helmholtz theorem, are, by the way, intricately related: they are
both, so to say, descriptions that elucidate one and the same process only from
different perspectives.
In the following, initially temporally stationary turbulence may be assumed,
as is found, for example, in a stationary channel or pipe flow. Of the dissi-
pated power D in a unit volume per unit time, a very small fraction μ( ∂U ∂y
)2
will be dissipated immediately into heat (U is the velocity of the mean flow);
the rest, which one may call D1 , increases the kinetic energy of the turbulent
2: Prandtl and the Göttingen school 85
Figure 2.12 Calculation of the mixing length from the vorticity equation; 1 of 4.
26 February 1948 the Max Planck Society convened its constitutional meeting
in the cafeteria of Prandtl’s Institute.
Prandtl himself retired from the University and Institute’s Directorships in
the fall of 1946 and continued working on problems in meteorology until his
death on 15 August 1953.
2: Prandtl and the Göttingen school 87
Figure 2.13 Calculation of the mixing length from the vorticity equation; 2 of 4.
2.11 Conclusion
Prandtl’s achievements in fluid mechanics generally, and in turbulence in par-
ticular, are often characterized by the label ‘theory’. However, it is important to
note that he did not perceive himself as a theoretician. When the German Phys-
ical Society of the British Zone awarded him honorary membership two years
88 Bodenschatz & Eckert
Figure 2.14 Calculation of the mixing length from the vorticity equation; 3 of 4.
after the war, he used this occasion to clarify his research style in a lecture en-
titled “My approach towards hydrodynamical theories”. With regard to bound-
ary layer theory, for example, he argued that he was guided by a ‘heuristic
principle’ of this kind: “If the whole problem appears mathematically hope-
less, see what happens if an essential parameter of the problem approaches
zero” (Prandtl, 1948b, p. 1606). His notes amply illustrate how he used one
2: Prandtl and the Göttingen school 89
Figure 2.15 Calculation of the mixing length from the vorticity equation; 4 of 4.
to aeronautical war research?78 Or was the renewed interest in the basic rid-
dles of turbulence sparked by the wartime applications? Or, on the contrary,
did Prandtl at the end of the war find the time to work on what he was really
interested in?
Both Prandtl’s advisory role as well as his local responsibilities for fluid dy-
namics research at Göttingen came to a sudden stop when the American and
British troops occupied his Institute and prevented further research – a prohibi-
tion which Prandtl perceived as unwarranted. Not only did he write79 to Taylor
for help, but also he requested80 help from the President of the Royal Society,
of which he had been a Foreign Member since 1928. “The continuation of the
research activity that had to be shelved during the War should not be hindered
any more!” demanded Prandtl in this letter. His request remained unanswered.
This correspondence provokes further questions regarding Prandtl’s political
attitude. Biographical knowledge of Prandtl has been provided by his family
(Vogel-Prandtl, 1993), by admiring disciples (Flügge-Lotz and Flügge, 1973;
Oswatitsch and Wieghardt, 1987), and by reviews on German wartime aero-
nautical research (Trischler, 1994); a more complete view based on the rich
sources preserved in the archives in Göttingen, Berlin and elsewhere seems
expedient.81 Recent historical studies on the war research at various Kaiser-
Wilhelm-Institutes (see, for example, Maier, 2002; Schmaltz, 2005; Sachse
and Walker, 2005; Maier, 2007; Heim et al., 2009; Gruss and Ruerup, 2011)
call for further inquiries into Prandtl’s motivations for research into turbulence.
An important question of course is: What can we learn from the position of
great men like Prandtl and others in the political web of Nazi Germany? What
consequences arise for the responsibilities of scientists or engineers? Another
lacuna which needs to be addressed in greater detail concerns the relationship
of Prandtl with his colleagues abroad and in Germany, in particular with von
Kármán, Taylor, Sommerfeld and Heisenberg. Last, but not least, one may ask
about the fate of turbulence research at Göttingen under Prandtl’s successors
after the war. We leave these and many other questions for future studies.
our gratitude to the Max Planck Society Archives, especially Lorenz Beck,
Bernd Hofmann, Susanne Übele and Simone Pelzer; and to the Archives of
the German Aerospace Center (DLR), especially Jessika Wichner and An-
drea Missling. We are also very grateful to K. Sreenivasan, N. Peters and G.
Falkovich for sharing their insights on this topic and to the editors of this book
for motivating us to write this review and helpful suggestions for improvements
of the text. EB also thanks Haitao Xu and Zellman Warhaft for valuable com-
ments, and Helmut Eckelmann for sharing his memories about the Institute.
We are most grateful for the understanding and support from our families that
were missing their husband and father for long evenings and weekends. This
work was generously supported by the Max Planck Society and the Research
Institute of the German Museum in Munich. Part of this work was written at
the Kavli Institute for Theoretical Physics and was supported in part by the
National Science Foundation under Grant No. NSF PHY05-51164.
Abbreviations
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100 Bodenschatz & Eckert
3.1 Introduction
Theodore von Kármán, distinguished scientist and engineer with many inter-
ests, was born in Budapest on 11 May 1881. His father, Maurice von Kármán,
a prominent educator and philosopher at the University of Budapest, had a
significant influence over his early intellectual development. After graduating
from the Royal Technical University of Budapest in 1902 with a degree in me-
chanical engineering, von Kármán published in 1906 the first of a long string
of papers concerning solid mechanics problems outside the domain of linear
elasticity theory, in this case on the compression and buckling of columns. In
that same year, apparently at the urging of his father, von Kármán left Hungary
for graduate studies at Göttingen. For his 1908 PhD, supervised by Ludwig
Prandtl, he developed the concepts of reduced-modulus theory and their ap-
plication to column behavior such as buckling. Later, with H.-S. Tsien and
others, he developed a nonlinear theory for the buckling of curved sheets. His
final work in solid mechanics was on the propagation of waves of plastic de-
formation published as a classified report in 1942 and in the open literature in
1950. In von Kármán’s words:
It was another version of the problem I had solved for my doctor’s thesis, in
which I had extended Euler’s classical theory of buckling to a situation beyond
the elastic limit. (von Kármán and Edson, 1967, p. 248)
101
102 Leonard & Peters
course, made his famous stability analysis of a vortex street. Von Kármán also
computed the drag of a cylinder with a vortex-street wake by use of a control-
volume method, apparently the first appearance of such an approach in the
literature (Vincenti, 1993). Again, and on a number of other occasions during
his young career, the influence of Prandtl greatly benefitted von Kármán.
In 1912 von Kármán accepted an invitation to organize an Aerodynamical
Institute at the Technical University of Aachen as director and Professor of
Aerodynamics and Mechanics. His work there was interrupted during World
War I when he served in the Austro–Hungarian Army (see Figure 3.1). Recog-
nizing his talents as an aeronautical engineer, the military authorities assigned
to him and his collaborators the task of developing the first stable, captive
(tethered) hovering helicopter to be used as a battlefield observation post. The
effort was successful. After the war he continued building a highly respected,
internationally known reputation for himself and the institute (Figure 3.2). In
1922 von Kármán invited leading researchers in fluid mechanics to meet in
Innsbruck to discuss progress in that field, the first of many instances where
he showed his leadership in scientific cooperation. It was decided then to have
3: Theodore von Kármán 103
Figure 3.2 Von Kármán lecturing at Aachen, c. 1922. (Reprinted with permission
of the Institute of Aerodynamics at the RWTH Aachen University.)
Figure 3.3 Edward Teller, Enrico Fermi, and von Kármán in Hollywood, 1937.
(Courtesy of the Archives, California Institute of Technology.)
Paul Lukas (von Kármán and Edson, 1967, p. 178). Von Kármán’s interest in
rocket research began during his first decade at Caltech and led to the formation
of the Jet Propulsion Laboratory and later to the establishment of the Aero-
jet General Corporation. Von Kármán was director of the former from 1938
to 1945. It was also during this period that von Kármán made considerable
contributions to the theory of isotropic turbulence with notable interactions
with G.I. Taylor as described in Section 3.3.
layer of a flat plate and finds a −1/5 power Reynolds number dependence.
He remarks that Prandtl had found the same Reynolds number dependence for
the friction coefficient earlier. This “cooperative competition”, as von Kármán
called the interaction between his group in Aachen and Prandtl’s group in
Göttingen later, stimulated his further ideas in developing a “theory of tur-
bulence” (von Kármán, 1924). In order to estimate the energy balance in a
turbulent shear flow he assumes periodic fluctuations of the longitudinal and
vertical velocity components, and formulates a maximum principle to calcu-
late the mean wavelength of the fluctuations. As a side condition he uses the
energy budget. Using scaling arguments in the spirit of Prandtl (1921) and Tiet-
jens (1925), he then derives the friction coefficient. In his autobiography (von
Kármán and Edson, 1967), where he entitles Chapter 17 as ‘Turbulence’, he
qualifies his paper (von Kármán, 1924), presented at the first Congress of Ap-
plied Mechanics in Delft, as the one that set “some foundations for a theory”.
But he “was aware that it would take a really ‘happy thought’ to regiment tur-
bulence into a workable theory”. He mentions the “outstanding contribution”
of Prandtl in introducing the mixing length concept in 1926 at the 2nd Interna-
tional Congress of Applied Mechanics in Zürich (Prandtl, 1927), and considers
the search for a universal law of turbulence his goal, to be pursued in competi-
tion with his former professor. For the next Congress in Stockholm 1930 both
players were invited to give papers on turbulence. In order to compete with
Prandtl, von Kármán knew that he first had to “find a method of developing a
simplified physical concept” to arrive “at the universal formula that would fit
the experimental data then available” (von Kármán and Edson, 1967).
On 12 December 1929 he writes a letter to Burgers in Delft where he sketches
the logarithmic law for the maximum velocity in turbulent tube and channel
flows that will later appear in the Stockholm paper. He remarks:
This is not a theory but it shows that one can do a theory of turbulence without
friction and what remains of the physics when one replaces the distribution by
mean values.
Figure 3.4 Von Kármán studying his lecture notes at Caltech, c. 1943. (Courtesy
of the Archives, California Institute of Technology.)
brought from Göttingen. When the last streetcar from Vaals was to leave to
Aachen at midnight, von Kármán escorted Wattendorf to the station, where he
began writing his newly developed theory into the dirt on the side of the wait-
ing streetcar. At first the conductor waited patiently, then looked repeatedly at
his watch, but finally became insistent that the car must leave. In order to save
the formulas from disappearance, Wattendorf had to jump from the car to the
street at each stop, copy a few lines, and jump back on the car as it moved on.
That night, back in his own room, he boiled the experimental data points down
to one master plot, which he brought to von Kármán the next day.
The outcome of that night was published in two papers (von Kármán, 1930,
1931), the first one presented before the Society of Sciences at Göttingen
on Prandtl’s invitation and the second one at the Stockholm meeting. The
Göttingen paper (von Kármán, 1930) focuses on channel flows and derives
a log law for the centerline velocity as a function of the channel width. He
first defines a new mixing length, later called the ‘von Kármán length’, which
is proportional to the ratio of the first to the second derivative of the mean
axial velocity profile. He then compares it to Prandtl’s mixing length and
3: Theodore von Kármán 107
Figure 3.5 Hugh Dryden, Ludwig Prandtl, von Kármán, and Hsue-Shen Tsien in
Göttingen 1945. Dryden, von Kármán, and Tsien were on a US Air Force mission
to assess the state of German aeronautics technology. (Courtesy of the Archives,
California Institute of Technology.)
argues that his mixing length provides more information because it contains
two different derivatives of the axial mean flow. Introducing it into the expres-
sion for the turbulent shear stress he obtains a second-order differential equa-
tion which he solves to obtain the axial mean velocity profile across the chan-
nel. This expression contains the log of a function of the normalized distance
from the centerline. He also shows that his mixing length scale is proportional
to the distance from the channel wall y for small values of y. He determines the
constant of proportionality κ by comparison with the Göttingen experiments
by Nikuradse (1926) as 0.36 and mentions that it is “probably” universal.
The Stockholm paper (von Kármán, 1931), having the same title as the
Göttingen paper, repeats the previous results but in addition presents an ex-
pression for the mean velocity ū as a function of the distance from y the wall:
1
yuτ
ū/uτ = ln +C . (3.2)
κ ν
Here ν is the kinematic viscosity. He discusses and rejects the possibility of a
power-law dependence of the mean velocity on y and expresses his conviction
that the logarithmic form is the final one. He concedes, however, that at not so
large Reynolds numbers there is an influence of viscosity, which he hopes to
be able to explain by looking closer at the region in the vicinity of the wall.
108 Leonard & Peters
Nikuradse then derives the linear velocity profile in the viscous layer. Then
he presents many experimental results and Kármán’s theory of the Göttingen
paper (von Kármán, 1930) in detail. At the end of the report he enters into
the ‘Similarity hypothesis according to Prandtl’. With uτ as the only scaling
parameter available he shows that
∂ū
uτ = κy (3.3)
∂y
with κ being a universal constant. After integration he reproduces (3.2) and
states that the measurements reproduce κ = 0.4.
In a review article Prandtl (1933) presents this derivation, but then acknowl-
edges that the “theory of Kármán” leads to the same result under the assump-
tion of constant shear stress. He notes that both his mixing length, and that of
von Kármán, lack a convincing foundation. (For more details on the compe-
tition with Prandtl see Section 2.6, ‘Skin friction and turbulence II. The loga-
rithmic law and beyond’ in Chapter 2.)
One may ask why Prandtl published a different derivation of a result that von
Kármán had already published. Was it the simplicity of the derivation? Or was
it that he had sensed that there was more to it, namely a general scaling princi-
ple? We will see below that there is a striking similarity between the procedure
that led to the logarithmic law and to the 4/5-law of isotropic turbulence.
The question whether there is a logarithmic or a power law dependence has
been raised again by Barenblatt (1993), who argues in favour of a power law
yuτ α
ū/uτ = C (3.4)
ν
where both the prefactor C and the power α should depend on the log of the
Reynolds number. The logarithmic law would then represent the envelope of
a family of power-type curves, each corresponding to a fixed Reynolds num-
ber. A different point of view is taken by George et al. (1992); see also Wos-
nik et al. (2000). They used an asymptotic invariance principle for the zero-
pressure-gradient boundary layer to suggest that the profiles in an overlap
3: Theodore von Kármán 109
region between the inner and the outer regions are power laws. In the limit
of infinite Reynolds numbers the log law was recovered in the inner region.
Finally, Oberlack (2001), using Lie-group analysis, showed that for turbulent
shear flows both power laws and logarithmic laws are admissible. From experi-
mental data plotted in a semi-log plot he observes a logarithmic scaling region,
the extent of which depends on the Reynolds number.
Figure 3.6 Geometry of two velocity vectors (von Kármán and Howarth, 1938).
Indeed, Taylor (1937) shows that (3.6–3.7) are well satisfied by data taken by
L.F.G. Simmons of the National Physical Laboratory (NPL).
Triple correlation function Von Kármán and Howarth next consider the triple
correlation function for two velocities at x and one at x , separated by ξ = x −x
as above, and obtain the result
q
3/2 k − h − 2q h
ui u j uk = (u2 ) ξ i ξ j ξk + δ ξ
ij k + δ ξ
ik j + δ ξ
jk i , (3.8)
r3 r r
where k(r, t), q(r, t), and h(r, t) are basic triple correlations defined in Figure
3.7. Imposing continuity on this result led von Kármán and Howarth to the
following:
k = −2h,
r ∂h (3.9)
q = −h − 2 ∂r .
112 Leonard & Peters
Thus, as in the case of the double correlations, the triple correlations for
isotropic, homogeneous turbulence can be expressed in terms of a single func-
tion of the distance between the points in question.
As von Kármán and Howarth note, the solution to (3.13) is given in terms of a
confluent hypergeometric function as follows:
5 5 χ2
f (χ) = exp(−χ2 /8)M[10α − , , ], (3.14)
2 4 8
where M[a, b, z] is the Kummer’s hypergeometric function. When α = 1/4
3: Theodore von Kármán 113
they obtain the solution f (χ) = exp(−χ2 /8) which later was found to give a
good fit to the experimental data of Batchelor and Townsend (1948).
Of course the Kármán–Howarth equation has been used by many other
authors as the basis for new results on isotropic turbulence. We give three ex-
amples: two from the period just after the publication of the 1938 paper and
one from recent times.
Loitsyanski (1939) multiplied (3.10) by r4 and integrated over r from 0 to
∞ to find
d 2 ∞ 4 3/2 ∂f
u r f (r, t)dr = u2 r4 h(r, t) + 2ν u2 r4 (r, t) . (3.15)
dt 0 r→∞ ∂r r→∞
known as Loitsyanski’s invariant. For a number of years this result was gen-
erally accepted and incorporated into, for example, the analysis of the low-
wavenumber limit of the energy spectrum of isotropic turbulence. See Lin
(1949b) and further discussion below. Then, especially after Batchelor and
Proudman’s (1956) analysis of the effect of pressure fluctuations on velocity
correlations and Saffman’s (1967) analysis of the dynamics of the large scales
of turbulence, it was widely thought that Loitsyanski’s integral in (3.16) did not
exist. But recent numerical simulations have shown that if the initial condition
for homogeneous, isotropic decay is impulse-free, then the integral exists and,
after an initial transient, it remains constant. See Davidson (2009) and Pullin
and Meiron (2011) for more discussion and analysis.
Kolmogorov apparently used the Kármán–Howarth paper to derive his 4/5
law. In Kolmogorov (1941a) he considers the dynamics of the second-order
structure function B (r, t) in terms of the third-order structure function B (r, t)
defined respectively by
equation
dB
4
d2 B 4 dB
4 + + B = 6ν + . (3.19)
dr r dr2 r dr
Here = −(3/2) du2 /dt is the energy dissipation rate. He integrates the equa-
tion twice and shows that there is a viscous layer for very small r where
1 r2
B (r) = . (3.20)
15 ν
For larger values of r he neglects the viscous term and obtains
4
B (r) = − r. (3.21)
5
In a recent use of the Kármán–Howarth equation, Lundgren (2002) em-
ployed the technique of matched asymptotic expansions and Lin’s (1949a)
modified version of the von Kármán self-preservation hypothesis (see (3.30)
below) on (3.18) to determine explicit Reynolds-number corrections to the 4/5
law.
There is a striking analogy between the log law including the viscous sub-
layer as it was derived by Prandtl and Kolmogorov’s derivation of the 4/5 law.
Both start from a one-dimensional differential equation, the former in terms
of the physical distance from the wall, the latter in terms of the correlation
coordinate r. Both contain an external scaling parameter appearing as a con-
stant in the equation, namely the shear stress τw leading to the friction velocity
uτ in the log law and the energy dissipation rate in the 4/5 law. Dimensional
analysis using the viscosity ν as additional parameter in both cases then leads
to the viscous length scale ν = ν/uτ for the log law and to the viscous scale
η = (ν3 /)1/4 in Kolmogorov’s derivation. At larger values of the respective
coordinate, when viscous effects vanish, there remains only a single scaling
parameter, uτ and , respectively. This is related to symmetry-breaking in the
chapter on the Russian school in this book (Falkovich, 2011). There is an im-
portant difference, however, between in the derivation of the log law and the
4/5 law when it comes to the ‘universal’ constant. While the log law assumes
a proportionality between the mean velocity gradient and uτ /y leading to an
empirical constant that must be determined experimentally (the von Kármán
constant) no such proportionality appears in Kolmogorov’s derivation. There-
fore its coefficient 4/5 is exact. This is not the case for Kolmogorov’s 2/3 law
where again an empirical constant, the Kolmogorov constant, appears.
It is interesting that von Kármán, having set the basis for both derivations,
did not make the final steps to derive these scaling laws. It looks as if he were
more interested in the solution of the entire or nearly entire problem, the mean
3: Theodore von Kármán 115
velocity profile in the case of channel flow and the two-point correlation func-
tion for the energy-containing range of scales together with the scales in the
inertial subrange for the case of isotropic turbulence.
Kármán–Taylor controversy During the first half of 1937, before the revised
version of the Kármán–Howarth paper was submitted, von Kármán and G.I.
Taylor had a serious dispute concerning their respective theories of the decay of
grid turbulence. In Taylor (1935) it is assumed that the large-eddy lengthscale
3/2
in the expression for dissipation = Cu2 / is equal to the meshlength M of
the grid. Combining this result with Taylor’s other expression for dissipation,
= 15νu2 /λ2 led Taylor to the relation between λ and u2
λ ν
=A (3.22)
M M u2
and, thence, to the decay law
U 5x
= 2 + const. (3.23)
u2 A M
where A is “an absolute constant for all grids of a definite type” (Taylor, 1935).
At the time and for the next few years after that the above expression fit the
experimental data very well. On the other hand, von Kármán (1937a, 1937b)
derives a one-parameter family of decay laws in which, as von Kármán claims,
Taylor’s is a special case. In these derivations, however, von Kármán assumes
that the vortex stretching term in the enstrophy equation is zero in the mean or,
equivalently, that the triple correlation term in the Kármán–Howarth equation
is zero. Von Kármán (1937b) states:
Now it can be shown that because of the isotropic feature of the correlation be-
tween the velocity components and their derivatives, [the vortex stretching term]
vanishes in mean value. The proof will be given elsewhere. As a matter of fact,
any value of the [stretching term] different from zero would mean that the vortex
filaments had a permanent tendency stretched or compressed in the direction of
the vorticity axis.
This assumption led von Kármán to a closed evolution equation for f (r, t) and,
assuming similarity or self-preservation for the correlation function f in terms
√
of the variable χ = r/ νt, von Kármán (1937a) derived the following decay
law with parameter α:
5α
U U x
= 1 + , (3.24)
u2 2 Ut0
u0
116 Leonard & Peters
and where f (χ) satisfies (3.13). Von Kármán notes that his result yields Tay-
lor’s result as an exact result if α = 1/5 and that Taylor obtained (3.23)
by assuming rather arbitrarily a relation between the length λ and the linear
dimension, e.g. the spacing M at the grid or mesh.
A similar derivation of the decay law with corresponding remarks must have
appeared in the original version of the Kármán–Howarth paper. In a letter of 24
January 1937 (CIT von Kármán collection, Box 29, Folder 34) to von Kármán ,
acknowledging receipt of the original manuscript, Taylor states:
with regard to my linear law [(3.23) above] I think your statement is a little
misleading. My assumption is that the scale of turbulence remains constant but
the curvature of R (or f ) continually decreases. Thus the R curve is not self-
preserving. [A corresponding sketch is provided by Taylor.] The fact that you
find a self-preserving law that also gives a linear law is an accident. Thus it is
misleading to say that my non-self preserving assumption is a limiting case of a
self-preserving theory.
I did once think of trying a self-preserving distribution but came √ to the con-
clusion that it would be necessary for Re to be constant. Thus if r/ νt √ is con-
stant then u must decrease such that ru /ν is constant therefore u ∼ 1/ t (not
1/t). . . . If you alter very slightly your remarks about the linear law I would be
grateful.
Taylor also encloses a plot of decay data taken at NPL that clearly supports his
linear law. Von Kármán responds in a letter dated 21 February 1937 (CIT von
Kármán collection, Box 29, Folder 34) to Taylor:
I gladly agree to any changes you suggest concerning the linear law of the decay
of turbulence (3.23). However, I am afraid there is a basic difference between
your viewpoint and mine, as far as the scale of turbulence is concerned. . . . you
assume that a certain scale of turbulence remains practically invariable during the
process of decay. I could not see the justification for such an assumption because
I believe that if the correlation function f is given at t = 0 then the process of
decay is given for all subsequent time.
Although von Kármán, at this point in time, assumes that the triple correlation
term is negligible, he seems open to the possibility that the self-similar solu-
tions to (3.10) with h = 0 will not be realized in experiments; for example, in
von Kármán (1937b) he states “In general the shape of the correlation curve
will vary with time”.
But in a telegram dated 15 April 1937 (CIT von Kármán collection, Box 29,
Folder 34) to Taylor, Howarth states:
Hold up publication of joint Royal paper. Essential changes necessary, especially
points relating to your theory.
3: Theodore von Kármán 117
Thus, by early April, von Kármán with possible input from Howarth and/or
Clark Millikan must have realized his error in neglecting the vortex stretching
term or, equivalently, the triple correlation term. And with Howarth he began
development of a correspondingly revised theory. In fact, in a letter dated 30
June 1937 (CIT von Kármán collection, Box 13, Folder 38) to von Kármán
written aboard the Queen Mary, Howarth, on his way back to England, de-
scribes his efforts to determine the various terms in the dynamical equation for
the triple correlation. An appendix to the letter gives his results obtained so far
with the comment “did not complete . . . too many functions with few relations
between them”.
It won’t be until October that von Kármán submits a new version of the
Kármán–Howarth manuscript. Meanwhile, in Taylor (1937), submitted 1 May
for the June issue, Taylor seems particularly intent on showing that his theory
is not a special case of von Kármán’s general theory and that von Kármán’s
theory is, in fact, flawed. After showing that (3.7) is verified experimentally
as mentioned above, Taylor recalls von Kármán’s (1937a) derivation of the
dynamical equation for u2 f (i.e. (3.10) without the triple correlation term) and
states:
It is difficult to see any a priori reason why these inertia terms [corresponding to
the triple correlations] . . . should be zero. . . . von Kármán promises to give the
proof that the mean values of the inertia terms . . . vanish.
To further support his view, Taylor recounts his results with Green (Taylor
and Green, 1937) on the increase in vorticity with time in three-dimensional
flow with Taylor–Green initial conditions demonstrating the importance of the
vortex stretching term for that problem. Taylor had, in fact, given von Kármán
a preview of those results in a letter dated 12 March 1936 (CIT von Kármán
collection, Box 29, Folder 33):
With a student I have just finished an amusing piece of work. Do you recall that
in my statistical theory of turbulence that I get a formula [(3.22)]?
Taylor (1937) then proceeds to show that von Kármán’s self-similar form
of the solution to the dynamical equation for u2 f , i.e. the solution to (3.13),
does not agree with experiment in two ways. First, Taylor argues that because
1/λ2 ∼ U (see (3.12)) the shape of the curve of f near r = 0 will be affected
118 Leonard & Peters
by a change in U but experiment shows little change in the curve away from
r = 0 with variations in U. A self-similar solution does not have this freedom
and, thus, is not in accordance with experimental observation. Next, Taylor
solves von Kármán’s self-similar form of the evolution equation for f , (3.13),
by power series and shows that its solution is in significant disagreement with
experiment.
With the publication of the Kármán–Howarth paper, however, the dispute
between von Kármán and Taylor is essentially over. In a footnote in that paper
von Kármán acknowledges his previous error in assuming that the triple cor-
relation term would vanish in isotropic turbulence and, when von Kármán and
Howarth revisit the small Re approximation, they note that α = 1/5 (for (3.13)
and (3.24)) gives Taylor’s result but the coincidence “is rather formal”.
Section 11 of the paper, written by von Kármán “especially after reading
Taylor’s (1937) contribution”, explores possible solutions for large Re. Von
Kármán finds a one-parameter family of self-similar solutions (see §3.3.2 be-
low) and, again, Taylor’s result is a special case. “It appears that further experi-
mental results will decide whether Taylor’s assumptions are not too narrow and
whether [von Kármán’s more general solution] correspond more closely to the
experimental facts.” This time von Kármán is eventually proven to be closer
to the truth. For example, Batchelor (1953) presents grid turbulence data that
closely follow the decay law,
U
= (C + Dx)β (3.25)
u2
with β = 1/2 rather than 1. Batchelor (1953) states:
The earliest measurements of u2 at different stages of the decay were a little
confused by inadequate corrections . . . , by turbulence, from sources other than
the grid, and by the inclusion of observations [too close to the grid].
In fact the decay law (3.25) with β = 1/2 had been proposed by Dryden in a
letter dated 16 September 1939 to von Kármán (CIT von Kármán collection,
Box 7, Folder 26) and published in Dryden (1943). Recall, even earlier, Taylor
had derived and then discarded the same law; see Kármán–Taylor controversy
above. More recent data taken at higher Re are satisfied by (3.25) with β ≈
0.65; see, for example, Pope (2000, p. 160).
In spite of the sometimes testy exchanges in 1937 described above, von
Kármán and Taylor, in general, had a very cordial, amiable relationship ex-
tending over several decades. There is correspondence between them starting
in 1928 and lasting for 30 years and a number of mutual visits were undertaken.
For example, in a note of 27 March 1930 (CIT von Kármán collection, Box 29,
3: Theodore von Kármán 119
Folder 33) to Pipö, von Kármán’s sister, Taylor thanks her for her hospitality
while he was in Aachen on a visit and continues:
Please remember me kindly to your mother and tell her that I hope for her sake
and for the sake of European Science that your brother will not go to Pasadena
permanently.
And in the letter dated 9 January 1937 to Taylor (CIT von Kármán collection,
Box 29, Folder 34) cited above, von Kármán states:
I am looking forward with great expectations to my visit in Cambridge next
spring. I hope that Mussolini and Hitler will wait with their big war until after
my lecture.
able to solve this problem. About a year earlier, Lin told von Kármán in a let-
ter dated 28 February 1945 (CIT von Kármán collection, Box 18, Folder 22)
that he was to report for a pre-induction exam for the military draft. This time
von Kármán had a Caltech official write to the local draft board urging reclas-
sification for Lin because Caltech was involved with “high priority weapons
research”.
u2 (1 − f ) = v2 β2 (r/η), (3.30)
(u2 )3/2 h = v β3 (r/η),
3
where v and η are velocity and length scales, respectively. Substituting the
above into the Kármán–Howarth equation and requiring that similarity extend
to all values of r, including the viscous range, Lin finds the Kolmogorov rela-
tions,
η = (ν3 /)1/4 , v = (ν)1/4 , (3.31)
According to Lin, the decay law (3.32) is an improvement over (3.25) with β =
1/2, because of the additional factor γ. Indeed it fits the decay data available
at the time very well. In addition, Lin’s theory also leads to a formula for λ2
that is quadratic in t − t0 which also was in good comparison with experimental
data of Townsend, supplied to Lin by Batchelor.
However, the decay law (3.32) is not compatible with the existence of Loit-
syanski’s invariant (3.16). In fact Kolmogorov (1941b) had shown that Kár-
mán’s self-preservation hypothesis (3.27) on 1 − f rather than f with (3.16)
leads to the decay law
u2 = α(t − t0 )− 7 .
10
(3.33)
As a consequence, von Kármán and Lin (1949, 1951), the latter being von
Kármán’s last published effort on isotropic turbulence, proposed a revised the-
ory in which it is assumed that during the early stage of decay all but the largest
scales participate in an equilibrium and Lin’s decay law (3.32) will hold. For
this early stage they are led to the existence of a medium range of eddy sizes
that yield a corresponding portion of the energy spectrum that is ‘linear’, i.e.
E(κ) ∼ κ. In the intermediate stage, at high enough Re and before the final
period of decay, the ‘linear’ regime is gone and one obtains an equilibrium
between the largest scales and the inertial subrange scales (Kármán similar-
ity) and another equilibrium between inertial subrange scales and the smallest
scales (Kolmogorov similarity).
122 Leonard & Peters
Spectral closures In 1948 von Kármán made brief but notable efforts in the
theory of spectral closures for isotropic turbulence (1948a, 1948b, 1948c). Lin
had given him a running start by informing him in a letter dated 6 June 1947
(CIT von Kármán collection, Box 18, Folder 23) that he had determined that
existence of Loitsyanski’s invariant leads to E(κ) ∼Cκ4 for small κ (Lin, 1949b)
and, in the same letter, by pointing out Heisenberg’s (1948) formula relating
1D spectra to 3D spectra. Lin had also sent to von Kármán a copy of his letter
of 26 June 1945 to Lars Onsager (CIT von Kármán collection, Box 18, Folder
22) containing his result relating the spectral transfer term T (κ) in the evolution
equation for the turbulence energy spectrum to the first and second derivatives
of the sine transform of h(r) (Lin, 1949b). As to the latter von Kármán states:
“Unfortunately this relation does not help, as far as determination of f and h is
concerned” (von Kármán, 1948b).
Von Kármán (1948b) postulates that the transfer function has the following
form:
∞
T (κ) = Θ[E(κ), E(κ ), κ, κ ] dκ (3.34)
0
Cu2 (κ/κ0 )4
E(κ) = , (3.37)
κ0 [1 + (κ/κ0 )2 ]17/6
3: Theodore von Kármán 123
where he assumes that “for the time being, Loitsyanski’s result is correct”. The
proposed spectrum (3.37) is known as the von Kármán spectrum.
3.4 Epilogue
After 1951 von Kármán’s research interests centered on a new field he termed
aerothermochemistry, and on magnetofluidmechanics. All the while his non-
research efforts were considerable and included organizing and chairing in
1944 a Scientific Advisory Group for the US Air Force and forming in 1951
AGARD, the Advisory Group for Aeronautical Research and Development
Group for NATO, the North Atlantic Treaty Organization, the latter being an-
other demonstration of his leadership in international cooperation. Because of
the demands of these and other such responsibilities von Kármán resigned from
his directorship at Caltech and became Professor Emeritus in 1949.
However, at least in 1955, von Kármán still had turbulence on his mind and
was optimistic about our ability in the future to understand the same. He was
asked to gather his thoughts about the future in an article entitled ‘The next
fifty years’ (von Kármán, 1955), or as he puts it, to “scan the coffee grounds
in order to find out what will become of aviation and the aeronautical sciences
during the next half-century”. In the last paragraph he comes to turbulence:
Finally, for the relief of the aerodynamicist, I wish to mention a problem related
to fluid mechanics – incidentally, an essentially ‘fluid’ problem – which proved
to be insoluble by scientific means during the first half-century of modern aero-
dynamics: I mean the problem of turbulence, which borders on fluid mechan-
ics as well as statistics. Many years ago, the lamented Arthur Sommerfeld, the
prominent German specialist of theoretical physics, told me in confidence that in
his lifetime he would like to be able to solve two problems related to theoretical
physics: true significance of quantum mechanics and the actual mechanism of
turbulence.1 I did not have the opportunity to ask him whether he felt that the
present interpretation of the quantum theory (complementarity instead of causal-
ity) was of such a nature that it satisfied the spirit. Without a doubt, turbulence
will have to disclose its secret in the next fifty years.
In a 1959 tribute to von Kármán, Hugh Dryden (CIT von Kármán collection,
Box 7, Folder 26) said:
A notable feature of von Kármán’s scientific and engineering work is the breadth
of field covered. His interests extend from mathematics to rocket propulsion,
from fluid mechanics to solid mechanics, from the crystal to the sand dune, from
1 Nearly the same thoughts are attributed to Horace Lamb by Sydney Goldstein (Davidson, 2004,
p. 24).
124 Leonard & Peters
the helicopter to the space vehicle. This breadth of interest is not limited to sci-
ence and engineering but spills over into politics, music, art, and to every activity
of the human spirit.
Theodore von Kármán was awarded the first US Medal of Science by Pres-
ident John F. Kennedy in February 1963. He died on 7 May 1963, four days
before his eighty-second birthday.
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4
G.I. Taylor: the inspiration behind the
Cambridge school
K.R. Sreenivasan
1 Pippard was the Cavendish Professor when Taylor died. The other Cavendish professors with
whom he overlapped are J.J. Thomson, Lord Rutherford (who arrived at Cambridge slightly
after Taylor’s own return in 1919), William Lawrence Bragg and Nevill Mott – all of whom
were Nobel Laureates.
127
128 Sreenivasan
Figure 4.1 The young and dashing Taylor at aged 39. From Batchelor (1996).
Figure 4.2 Taylor in ‘retirement’ in 1955 (age 69) in the Cavendish Laboratory
with his assistant Walter Thompson. From Batchelor (1996).
French originals with English translations, notes that the ideas of Saint-Venant
and Boussinesq came from observations of water flow in canals which, though
important at the time, fell into oblivion until some of the same ideas were
rediscovered in the newly emerging science of aerodynamics; by that time our
understanding of momentum transport in kinetic theory had also advanced to
a level of maturity and directly spurred people like Prandtl. The French work
did not seem to have reached Taylor.
In England, Reynolds (1883) had observed turbulence and the transition be-
tween laminar and turbulent states and had set Rayleigh (1892), Orr (1907) and
Sommerfeld (1908) on a course for investigating the stability of fluid flows. In
1895, in a paper that was poorly understood then but has turned out to be an
inspiration for enormous activity ever since, Reynolds had also formulated his
‘Reynolds averaging’ and ‘Reynolds stresses’. See Chapter 1 by Launder &
Jackson for a description of Reynolds’ contributions. Turbulence in boundary
layers was mentioned by Reynolds (1874), Lanchester (1907) and by Prandtl7 .
Eiffel (1912) had demonstrated that the drag on a sphere could be reduced
by introducing turbulence, and Prandtl (1914) had offered the right expla-
nation for it. Blasius (1913) had provided his famous 1/4th power law for
forecasting frictional drag. The second edition of Lamb’s Hydrodynamics, in
which he first lamented about turbulence as “the chief outstanding difficulty
of our subject”, had appeared in 1895; the section on turbulence was less
than seven pages long and referred primarily to Reynolds’ (1883) work on
transition, and Rayleigh’s work on inviscid instability (with a few remarks
on Lord Kelvin’s tentative efforts to include viscosity). The next edition of
the book, issued in 1906, added significant material from Reynolds (1895) on
his decomposition method8 and the equation for the energy integral with ex-
plicit forms of the dissipation and production integrals. Taylor was exposed
to Lamb’s Hydrodynamics while still at school, in his ‘Uncle Walter Scott’s
library’ (see The Legacy). The mathematical treatment of eddy motion in the
atmosphere was regarded as a problem of great difficulty and, as Taylor (1915)
remarked, this was because attention was chiefly directed “to the behaviour of
eddies considered as individuals rather than to the average effect of a collec-
tion of eddies”. This dichotomy of treatment is still quite alive in turbulence
research.
By my count, Taylor wrote some thirty papers on topics related to turbu-
lence. They spanned two roughly five-year segments of his life (c. 1915–1921
7 Some crucial ideas attributed to Prandtl by both Karman and Nikuradse were published by
Prandtl in 1927.
8 Horace Lamb was, in fact, one of the two referees of the paper; the second was George Gabriel
Stokes. Its editor was Lord Rayleigh. See Chapter 1 by Launder & Jackson.
4: G.I. Taylor: the inspiration behind the Cambridge school 135
and 1935–1940). The first group of papers was written while he was still a
young researcher, while the second was written some dozen years into his
Royal Society Research Professorship. Most papers in the earlier period were
on specific aspects of meteorology and ocean dynamics, incorporating the Sco-
tia expedition, but included the fundamental paper on diffusion by continuous
movements, which he wrote after he moved back to Cambridge. The second
group is focused on basic aspects of turbulence in which Taylor introduced
isotropic turbulence and its methods. In the intervening years, Taylor worked
and wrote on a wide range of topics such as soap-film methods in elasticity,
physics of the large deformation of solids (almost certainly one of Taylor’s
greatest achievements), aeronautical problems including compressible and su-
personic flows, rotating flows and several other problems of fluid mechanics
not directly linked to turbulence, including the path-breaking work on Taylor–
Couette flow (another of Taylor’s major accomplishments, about which more
will be said in Section 4.4.1). The temporal gap of about fifteen years between
the two ‘turbulence’ periods seems to have made no difference to the continu-
ity of his thoughts. In the first of his famous series on the statistical theory of
turbulence (Taylor 1935a), for example, Taylor took off exactly from where he
had left the subject some fifteen years earlier in his diffusion paper of 1921; one
finds this continuity in his mixing length work as well. Taylor (1971) himself
stressed this continuity. He chose not to write about turbulence beyond about
1938, the exceptions being Taylor & Batchelor (1949) and Taylor (1954) on
the dispersion of matter in turbulent flow through a pipe.
The scientific goal of the Scotia expedition that Taylor seems to have set
for himself was the quantitative description of turbulent processes in the at-
mosphere. In Eddy motion in the atmosphere (Taylor 1915), he used his own
kite measurements as well as data interpolated from other ship measurements
to infer vertical distributions of humidity and temperature in six instances. He
showed by heuristic arguments that the turbulent transport needed to account
for the vertical transport of these quantities was of the form w L where w is a
measure of the fluctuation in the vertical component of the velocity and L is a
length connected with turbulence – later called the ‘mixing length’ – analogous
to the mean free path of molecules (“the average height through which an eddy
moves from the layer at which it was at the same temperature as its surround-
ings, to the layer with which it mixes”). This mechanism to mimic molecular
motion was regarded as “a purely hypothetical process”. Taylor deduced the
rough numerical value for the effective thermal diffusivity in the turbulent wind
over the sea to be on the order of 3 × 103 cm2 s−1 , independent of height for a
range of wind speeds. Later (Taylor 1917b), using the measurements made ear-
lier at different heights on the Eiffel tower, he estimated the eddy diffusivity to
be of the order 105 cm2 s−1 , a much higher value than that over the sea (though
it varied with the season and the height). Needless to say, the eddy diffusivity
is many orders of magnitude larger than the molecular diffusivity.
Taylor then considered the vertical transport of momentum. He supposed
that momentum cannot remain constant over the path of the eddy but postulated
that vorticity might. He conjectured that the eddy viscosity is of the same order
as the eddy diffusivity, although there were no measurements to back it up. This
is in the spirit of the so-called Reynolds analogy (more later).
The vorticity transport theory, according to which the dynamics of turbulent
motion can be represented as the diffusion of vorticity rather than as a diffu-
sion of momentum, had its origins in Taylor’s Adams Prize essay,9 and was
put forward first explicitly in Taylor (1915). Taylor regarded his 1915 descrip-
tion to have been so brief that it appeared to him “to have escaped notice”
(Taylor 1932b). He had become well aware of Prandtl’s momentum transport
theory soon after 1925 and, in the 1932a paper, made a test of the compar-
ative merits of the two theories. He noted that “if the motion is limited to
two dimensions the local differences in pressure do not affect the vorticity of
an element, whereas Prandtl has to neglect them or to assume arbitrarily that
9 Among the winners of the Adams Prize, of interest to fluid dynamicists, are James Clerk
Maxwell, Joseph Proudman, Harold Jeffreys, Sydney Chapman, Sydney Goldstein, S. Chan-
drasekhar, George Batchelor, Leslie Howarth, James Oldroyd, Timothy Pedley, Michael McIn-
tyre and, most recently, Jacques Vanneste.
4: G.I. Taylor: the inspiration behind the Cambridge school 137
they do not affect the mean transfer of momentum even though they certainly
affect the momentum of individual elements of fluid”. In the same paper, Tay-
lor showed that the momentum and vorticity transport theories yielded velocity
distributions that were different from temperature distributions in the wake of
a heated obstacle (as expected from the vorticity transport theory). It was also
in this paper that Taylor extended his theory to three-dimensional motion. In
the appendix to this same paper, A. Fage and V.M. Falkner reported tempera-
ture distributions in the wake of a heated cylinder, and claimed that the results
were in better agreement with the vorticity transport theory. In Taylor (1935f),
it was concluded that, in the annular gap between rotating cylinders (the inner
one heated), Prandtl’s momentum transport theory does well to account for the
distribution of the velocity near the surface but the vorticity transport theory
does better in the bulk.
This comparison was revisited in Taylor (1937a), especially when it ap-
peared that Prandtl (1925), Kármán (1930) and Goldstein (1937) had pressed
ahead with Prandtl’s theory rather than with that of Taylor, who thought that his
theory had something superior to offer. Here, he compared the predictions of
a ‘modified’ vorticity transport theory with Nikuradse’s mean velocity distri-
butions in pipe flow and claimed “very good agreement with observation over
the whole range except perhaps very close to the wall”. (When this conclusion
was later questioned by Prandtl in a private letter, Taylor accepted the criticism
and sent him the revised figures.)
Despite considerable effort (and comparable anguish expressed over it in the
correspondence with Prandtl and Ekman; see later), Taylor did not consider –
at least in hindsight – any of the transport theories satisfactory enough. His
objections were not about the use of mixing length as a practical device but
about the underlying physics. Indeed, in Taylor (1970), he reflected that he
was never satisfied with the mixture-length theory, because the idea that a fluid
mass would go a certain distance unchanged and then deliver up its transferable
property, and become identical with the mean condition at that point, is not a
realistic picture of a physical process.
where κ is the diffusivity for θ, and D/Dt is the material derivative. Through-
out much of the nineteenth century, scientists debated the molecular nature of
matter but the experimental observations of Brownian motion suggested that
molecular agitation was the underlying cause of the diffusion. Einstein had
shown that the same diffusion coefficient appears in the stochastic problem at
the microscopic level as in the diffusion problem on the macroscopic level. The
next qualitative breakthrough in the problem occurred in Taylor (1921) where
we find the following sentences as the motivating salvo:
This is the genesis of turbulent diffusion (Taylor 1921), where, for the first
time, an effort was made to link statistically a continuous and stochastic veloc-
ity function to the displacement of a fluid element – an effort towards describ-
ing the effects of turbulence by means other than invoking the mixing length.
The most important result of the paper is the now-famous formula
t
dX 2
= 2u
2
Q(τ)dτ, (4.2)
dt 0
X 2 = u2 t2 , (4.3)
reflecting the fact that a fluid element simply moves, for some small time, with
its initial velocity. For large time, on the other hand, the integral in (4.2) is
simply T L , so we have the result
X 2 = 2u2 T L t, (4.4)
10 In later recollections, Taylor (1970), we find the following statement: “While thinking of these
things [making mixing length concepts more definitive], I became interested in the form which
a smoke trail takes after leaving a chimney . . . This led me to think of other ways than mixture-
length theory to describe turbulent diffusion. The result was my paper, Diffusion by continuous
movements . . . ”
11 Kampé de Fériet (1939), who put Taylor’s theory on a more formal mathematical footing, also
reported measurements, by his associates, of the diffusion of tiny soap bubbles in a turbulent
air stream and claimed the results to be “in complete accordance” with the theory.
140 Sreenivasan
eddy diffusivity. Vast avenues of work were thus opened up by these two re-
markable people who laid the foundations for the statistical theory of turbulent
diffusion.
In the event, Taylor (1923) found remarkable agreement with his stability the-
ory and observations. Indeed, I know some fluid dynamicists who regard this
excellent agreement as the first foolproof confirmation of the correctness of the
no-slip boundary condition in viscous flows.
As is now well known, the Taylor–Couette flow is steady and purely az-
imuthal for low angular velocities. Taylor showed that when the angular veloc-
ity of the inner cylinder is increased above a certain threshold, the basic flow
142 Sreenivasan
stratification in the atmosphere. Taylor knew well that the fluctuations in the
wind increased or decreased depending on the difference between the speeds at
two heights and on the difference between temperatures at these two heights.
Encouraged by S. Goldstein, who was beginning to work on a similar topic,
he published these results without any experimental support – contrary to his
habit. Prandtl and Richardson were also working on this same topic. Taylor
(1931a) assumed an exponential drop in density and a linear velocity distribu-
tion and showed that a steady stream of uniformly sheared flow in stratified
environment is unstable to small disturbances if
gΔρ
(4.5)
ρH(dU/dz)2
exceeds a critical value of order unity. Here Δρ is the decrease in density over a
layer with a height H in the vertical direction, and dU/dz is the mean velocity
gradient above the ground. The numerical value of the constant itself depends
on the assumptions made. This non-dimensional number is now known as the
(gradient) Richardson number. The next paper (Taylor 1931b) considered a
related problem of determining the critical value of the density gradient above
which the mean flow cannot persist in a uniformly sheared state. Taylor was
searching particularly for internal wave structure here. In this paper, Taylor did
make comparisons with experiments. Details of Taylor’s stability calculations
have been superseded, but there is no doubt they gave substantial impetus for
the researchers who followed.
This was the genesis of isotropic turbulence – a subject that has consumed the
attention of a number of turbulence researchers, occasionally inviting some
skepticism.13 Taylor (1939) paid greater attention to hot-wire developments
and mentioned the work of King (1914), van der Hegge Zijnen (1924), the
NACA work (Dryden & Kuethe 1929, Mock & Dryden 1932 and Mock 1937),
Ziegler (1934), Simmons & Salter (1934), Townend (1934) as well as Schu-
bauer (1935). He wrote to Dryden a longish typewritten letter (rare for Taylor,
since handwritten notes were the norm) describing the NPL hot-wire work
(without, however, mentioning Simmons and Salter by name).
Taylor was quite aware of these limitations: he conveyed them in the letter
to Kármán: “The results are not bad but the trouble at the NPL is that they
have initial turbulence of large scale before getting to the grid.” However, the
agreement of his theory with measurements, whatever their limitations, seemed
to have made a powerful impression on Taylor’s thinking.
Whatever reservations can be expressed about these specific results with the
hindsight of many years, there can be no question that the paper has served as
the source of founding ideas for isotropic turbulence, and, more broadly, for
the entire statistical theory of turbulence.
Taylor (1935b,c) makes experimental contact with the ideas expounded in
Taylor (1935a):
At that time measurements of the rate of decay of the energy of turbulence had re-
cently been made by Hugh Dryden at the Bureau of Standards. I therefore asked
the National Physical Laboratory to measure Eulerian correlations in turbulence
produced by obstacles in a wind stream which had the same geometry as those
used by Dryden when he measured the rate of decay of turbulent energy.
the heated wake has been repeated since then in a number of flows other than
grid turbulence, and simulations have also been performed. There is no doubt
that all the later developments, which are too extensive to cite here, owe much
to Taylor’s path-breaking work.
Perhaps the less known part of Taylor (1936c) is the discussion of the pres-
sure microscale λ p defined through
Later work (see, for example, Eyink et al. 2008) has revealed that this behavior
has to do with the possible occurrence of inviscid singular behavior. Instead of
addressing the problem abstractly, Taylor and Green noted that “It is difficult
to express these ideas in a mathematical form without assuming some definite
form for the disturbance . . . ” and chose a special form of the initial motion
represented by
⎫
u = A cos ax sin by sin cz ⎪ ⎪
⎪
⎪
⎬
v = B sin ax cos by sin cz ⎪ ⎪ (4.8)
⎪
⎪
w = C sin ax sin by cos cz ⎭
148 Sreenivasan
with
Aa + Bb + Cc = 0 (4.9)
This remark shows that Kármán (with L. Howarth and C.C. Lin, both working
with Kármán) had begun to take a major interest in the theory. Indeed, the
particular research on vortex stretching was a direct consequence of Taylor’s
interaction with Kármán. This issue has been discussed also by Leonard &
Peters in Chapter 3 and we shall say more about it in Section 4.6.3.
the contraction is so rapid that the relative motion of two neighbouring [points]
due to the contraction is large compared to that due to turbulence.16
16 This letter is important because it is here that Taylor introduced his theory of isotropic turbu-
lence to Prandtl, described the grid and his linear decay law, and asked: “Have you made any
measurements in Göttingen which bear on this point?” He expressed the view that the Reynolds
number ought to be high enough, “u L/ν > 100”, for his theory to hold. This is also the place
that Taylor discussed his critical Reynolds number theory for the sphere (see Section 4.5.5).
Prandtl did not comment on the grid problem but was not particularly impressed by the critical
Reynolds number work. This should not surprise us because an immense amount of first-rate
work on boundary layer stability had been going on in Prandtl’s own institute; see Section 4.5.5.
150 Sreenivasan
The velocity field was represented by the sinusoidally varying model of the
Taylor–Green problem. Taylor computed the vorticity whose evolution through
the contraction he evaluated, after which he inverted the relation to get the ve-
locity field to deduce the effect of contraction. He also presented a comparison
with the measurements made by Simmons, Townend and Fage (using different
techniques). According to Batchelor (see The Legacy, p. 168), Taylor “was held
up by not knowing how to ‘integrate’ the results over all Fourier components
of a turbulent, and hence, random fluid velocity”;17 the task was later under-
taken successfully by Batchelor & Proudman (1954), whose work brought the
subject to the level of having its own acronym RDT, attracting the attention of
later researchers towards qualitative evaluation of the effects of a ‘mean strain’
that is relatively large.
He cited Lord Rayleigh’s work as the source for the harmonic representation of
an arbitrary signal and noted that Dryden et al. (1937) at NACA and Simmons
& Salter (1938) at NPL had measured the spectrum of turbulence by using
band-pass filters. As we already know, Taylor had made Eulerian correlations
the basis of his theory of isotropic turbulence. In Taylor (1938b), it was made
clear that the spectrum and the correlation functions are Fourier transforms of
each other. This is the Wiener–Khintchine theorem. Taylor himself attributed
the general mathematical form of the theorem to Norbert Wiener’s 1933 book
but we know of no concrete examples pre-dating Taylor which showed that
the measurements follow the Fourier transform relation.18 Taylor was to show
more. He used the correlation functions of the band-pass velocity and demon-
strated that the modified spectrum agreed well with the Fourier transform re-
sult. Research of band-passed velocity signals was taken to the next level of
17 Taylor amusingly related (Batchelor 1975) that he was once described as an ‘x, y, z’ man rather
than a modern ‘i, j, k’ person. Townsend (1990) remarked that, in his lectures to a few graduate
students, Taylor derived “in long hand the mean-value ratios of velocity gradients, using the
methods of his 1935 paper. Obviously he saw no point in deriving them in another way to get
the same result.”
18 Wiener was well aware of Taylor’s work and attributed, on more than one occasion (see, for
example, Wiener 1939), much of the advances in the statistical theory of turbulence to Taylor.
There is, however, no record of any correspondence between them in the Taylor Archives in
Trinity, or, according to my brief investigation, in the Wiener Archives at MIT.
4: G.I. Taylor: the inspiration behind the Cambridge school 151
sophistication many years later by Comte-Bellot & Corrsin (1971). This pa-
per of Taylor’s has had an enormous practical impact on later turbulence re-
search because the two-point velocity correlation could be obtained from the
more convenient spectral analysis. This was also the paper in which Taylor
argued that the velocity fluctuation changes relatively slowly as it is carried
past a point of measurement where the spectral density is measured. This so-
called Taylor’s frozen-flow hypothesis has been a working tool for hundreds of
turbulence researchers (sometimes indiscriminately without realizing its lim-
itations). One can see the rough outlines of the hypothesis in Taylor’s earlier
papers as well.
close interest” in the work of B.M. Jones related to this problem. After some
further manipulations using the concepts already summarized, he expressed
the critical Reynolds number Recr as
where D is the sphere diameter, U is the free-stream velocity with the root-
mean-square fluctuation of u and M is the characteristic size of the device that
is generating the turbulence in the free stream; f is an unknown function of
its argument. He used the data available from Dryden to verify the theory in
its broad terms. A better assessment of the theory was made by Dryden et al.
(1937) who correlated the critical Reynolds number data very well with the
expectations of the theory.
Brief life history of Burgers Johannes Burgers was born in Arnhem, the
Netherlands. His father, a post-office clerk, was a self-educated amateur sci-
entist. In 1914, Burgers entered the University of Leiden, where he came to
know Hendrik Lorentz, Kamerlingh Onnes, Albert Einstein and Niels Bohr,
and was part of a group of students of Ehrenfest. Burgers was, in fact, the first
of his students in Leiden to complete a PhD thesis in 1918. His dissertation
was on the Rutherford–Bohr model of the atom.
At the age of 23, before receiving his PhD, Burgers was appointed as Profes-
sor in the Department of Mechanical Engineering, Shipbuilding and Electrical
Engineering at the Technical University in Delft, where he founded the Lab-
oratory of Aero- and Hydro-dynamics. Somewhat reminiscent of the appoint-
ments of the young and inexperienced Taylor as Reader of Meteorology and of
Reynolds as Professor at Manchester at the age of 26, Burgers was appointed
a professor in a field of study in which he had no experience. Burgers (1975)
noted that the selection committee “considered it desirable to look for a person
with sufficient background in mathematics and its applications who would be
prepared to build up the subject from its fundamentals”. In his characteristi-
cally modest account of his early years in Delft, Burgers wrote that one reason
why he accepted the position was his concern of “having insufficient fantasy
for making fruitful advances in Bohr’s theory”.
Sometime after the Second World War, Burgers played an important role
in establishing the International Union of Theoretical and Applied Mechanics,
which was admitted to the International Council of Scientific Unions in 1947.
He served as general secretary of the Union from 1946 to 1952. This was
another context through which Burgers and Taylor interacted scientifically.
Towards the end of 1955, at age 60, Burgers left Delft to join the faculty of
the University of Maryland. There he developed an interest in the relation of the
Boltzmann equation to the equations of fluid dynamics. His book Flow Equa-
tions for Composite Gases, published in 1969, represents some work of that
154 Sreenivasan
period. He also published, in 1974, the book The Nonlinear Diffusion Equa-
tion; he was then 79 years of age.
approximation and the adhesion model – are used to represent the formation
of large-scale structures of the Universe; see, for example, the review by Shan-
darin 1989).
The one-dimensional and multi-dimensional Burgers equation with stochas-
tic initial conditions (e.g. She et al. 1992, Sinai 1992) and stochastic forcing
(e.g. Chekhlov & Yakhot 1995, Polyakov 1995) has served in recent years as a
testing ground for field-theoretic and advanced probabilistic methods. The lat-
ter has been used as a counterpart of turbulence models where the steady state
is maintained by external forcing, especially for understanding how an inter-
mediate region, where the system exhibits anomalous scaling (not predictable
by dimensional analysis), develops. This anomaly for velocity structure func-
tions can effectively be calculated in the case of the stochastic Burgers equa-
tion. Furthermore, the probability density function of the velocity gradients
can be calculated essentially exactly. These aspects often provide the point of
departure in understanding the structure of the Navier–Stokes turbulence.
In addition to his work on turbulence, Burgers became interested in solid
mechanics in his capacity as Secretary of the Joint Committee on Viscosity
and Plasticity. Taylor had already made pioneering contributions to dislocation
theory in plasticity, and so was a natural correspondent on this topic as well.
Burgers collaborated with his brother, Professor W.G. Burgers, in the work on
dislocations in crystal lattices and introduced, in 1939, the so-called Burgers
vector, which is a measure of the strength of a dislocation in a lattice. He also
studied the fluid dynamics of dilute polymer solutions and wrote some of the
fundamental papers on the intrinsic viscosity of suspensions.
Burgers carried on elaborate correspondence with many leading scientists
of his day (apparently with ease in English, German and Italian as well as his
native Dutch). Much of it is preserved in the Burgers Archives placed in the
Libraries of the University of Maryland.
The first letter found in the archives appears to be a response from Burgers
to Taylor and is dated 24 January 1924. One may infer from Burgers’ letter
that Taylor might have inquired about velocity distributions in turbulent flows.
Burgers responded with his summary of the contributions of Kármán, Prandtl,
Blasius and Taylor himself, mentioned his hot-wire work with van der Hegge
Zijnen and added some figures. The letter is addressed formally as “Dear Sir”
(which turns into “Dear Taylor” by 2 April 1927 – if not sooner). On 3 May
1935, Taylor began his letter to Burgers, apparently without any prompting
from the latter, as follows:
I have lately been working out a theory of the dissipation of energy in turbulent
motion and have got results which agree extremely well with observations. In
the course of the work I have come across some general relationships which I
have not seen before but which it is very possible that some one has noticed. I
am therefore writing to ask you whether you have come across them. They are
concerned with the idea of isotropic turbulence, i.e., turbulence which is isotropic
statistically so that all statistical averages are independent of the direction of the
axis of reference.
He then listed most of the principal results of Taylor (1935a). The measure-
ments with which Taylor claimed excellent agreement were “Dryden’s obser-
vations of decay of turbulence behind honeycombs, NPL observations of the
same type, and recent unpublished observations of Dryden’s on decay behind
similar grids of varying scales”, all of which, he claimed, obeyed the linear law
(4.7), with the value of the constant A “between 2.8 and 3.1 whatever the nature
of the turbulence producing mechanism i.e., square rods, honeycombs, grids of
flat slats arranged in square and grids of round bars arranged in squares”.
In the response of 23 May 1935, Burgers enclosed a 27-page typewritten
work of his own, entitled On the mechanism which determines the intensity
of turbulence, mostly related to shear flows. Burgers made the following three
points: (a) turbulence generated in one place in the flow could be dissipated
elsewhere (he was obviously thinking about inhomogeneous flows such as in
pipes, as one can easily see from his notes just mentioned); (b) that “certain
phase relationships must be present . . . by which the phases of the various terms
are coupled”; and (c) according to his model, the transverse velocity fluctua-
tions would be the largest near the wall – a result from his model that he did
not understand. Shear flows were not Taylor’s concern of the moment, but he
tried to make contact with Burgers’ points of view. In his response of 28 May,
Taylor agreed with (a) and (b) but pointed out that:
isotropic seems to be effective during the transfer of the energy from the place of
origin near the walls to the middle part of the pipe.
correspondence went silent even as Taylor was in the middle of his work
on isotropic turbulence and was carrying on vigorous correspondence with
Prandtl, Kármán and, though less frequently, with Dryden.
Two further letters were sent from Burgers to Taylor immediately towards
the end of the war and were promptly reciprocated. One of them has only mi-
nor relevance to turbulence but are summarized here because of their human
interest. On 16 July 1945 Burgers wrote: “We are anxious to know whether
friends and acquaintances in England are still living . . . My family has not suf-
fered much”, and inquired after some common friends: he also wondered about
the possibility of a visit to England to pursue aeronautical issues, to which
Taylor responded on 8 August 1945: “Very glad to know that you have sur-
vived the war without too much hardship. With regard to the proposed visit,
would be delighted but the security mentality is still with us though it is disap-
pearing in some departments . . . ”. Burgers followed up by sending his papers
and wrote: “The similarity considerations, which seem to have been used by
Kolmogoroff, Onsager and von Weizsäcker, can be applied also to the model
[what we now know as the Burgers equation], so that a similar formula for the
correlation function (depending on r2/3 ) can be deduced”. Taylor knew about
Kolmogorov’s work already through Batchelor (see Section 4.6.6) and was not
particularly impressed by Burgers’ work. In a postscript to his response of
1 March 1947, he wrote: “I have now read your paper on the model of turbu-
lence and it certainly seems to give some interesting results. The main objec-
tion to it from my point of view seems to be that until many of the problems
which are solved in analogue in your paper had been solved with great labour
using the actual equations of a fluid I could not see the usefulness of the ana-
logue. One sees the usefulness of the analogue only when it is no longer so
necessary.” He could very well not have anticipated the resurgence of interest
in Burgers’ equation. These were the last scientific exchanges between the two
men.
In Burgers’ words from a letter to Batchelor on 7 October 1975, “I have
had relatively little contact with Taylor; most of it was in the early period, of
1924 to about 1934. There has been almost nothing since I moved to the US
at the end of 1955.”21 A few brief exchanges did take place sometime after
April 1973: Taylor sent a reprint of his SIAM obituary notice of Kármán to
Burgers with the handwritten notation, “I hope you do not mind my taking
your name in vain!”, to which Burgers responded with great fondness, recalling
Taylor’s stay with Kármán in Burgers’ place, and so forth. Taylor wrote back
21 In fact, he came to know of Taylor’s death through the obituary by Pippard (1975) but later
wrote in Dutch, in response to a request from the Royal Netherlands Academy of Sciences, a
six-page obituary of Taylor.
4: G.I. Taylor: the inspiration behind the Cambridge school 159
with reminiscences of his own, mentioning Stephanie’s death in 1967 and his
stroke in April 1972.22
22 Taylor also wrote that Enrico Volterra had asked him for a contribution celebrating Tullio Levi-
Civita’s centennial. In his response, Burgers mistook Enrico Volterra for Enrico Fermi, and
Taylor corrected it on August 1973 in his exceedingly scrawly hand. The last letter from Burgers
was on 3 October 1973, with apologies for the mix-up and assurance that it didn’t go any further.
160 Sreenivasan
Soc. (Taylor 1937b).23 However, he had a “small point” to bring up; this small
point is what took up most of the letter. It was that Kármán’s attribution that
Taylor’s linear decay was a special case of the self-preserving solutions could
not be correct because self-preservation on all scales was inconsistent with
facts as Taylor knew them.24 He noted that “if you would alter very slightly
your remarks about the linear law for 1/u I would be grateful because I don’t
want to seem, by communicating the paper, to agree with your remarks on all
particulars”. Taylor pointed out that “if I had tried to apply my ideas to the
√
self-preserving distributions, I should have come to 1/u ∝ t – not to t”.
Kármán responded on 21 February 1937:
I gladly agree to any changes you suggest concerning the linear law of the de-
cay . . . in order to accelerate the publication of the paper, please make the correc-
tions to avoid appearance that you agree with all that is said on the matter.
He was clearly anxious to see his paper in print. The suggested change seems to
have been implemented; in its printed version we find: “Hence the coincidence
between these [self-preserving] equations and Taylor’s results is rather formal.”
In the letter, Kármán went on to point out rather firmly that Taylor’s assumption
of the constant length scale cannot be correct. In hindsight, we now know that
Kármán was right on this point.
There indeed was another important aspect on which Taylor was right, and
Kármán not. As far as I can tell, this aspect was not commented on by Tay-
lor in his letter to Kármán but was to occupy a part of one paper (Taylor
1937b) and all of another (Taylor 1938a), for both of which the starting point
was a commentary on Kármán’s (1937b) paper. Taylor (1937b) first verified
the isotropic relation between the longitudinal and transverse correlation func-
tions of isotropic turbulence, earlier discovered by Kármán (“a more general
relationship” than his limiting case, according to Taylor 1973a). He then ques-
tioned Kármán’s assumption of self-similarity of the correlation function and
showed why it is inconsistent with observations. Taylor had recognized very
early that the small scales and the large scales need to follow two different
versions of similarity, and that all the scales cannot be simultaneously self-
preserving. As his final point, Taylor (1937b) dealt with the role of the vortex
stretching terms in the evolution equation for mean-square vorticity. Kármán
(1937b) had assumed that the vortex-stretching term ωi ωk (∂ui /∂xk ) would be
23 In this paper, Taylor cited only the published work of Kármán (1937a,b), and essentially pointed
out their errors, as discussed later.
24 I cannot help admiring Taylor’s conscientiousness in writing to Kármán en route to Switzerland,
one day after he received the paper; and he had recruited his wife to send the paper off by
registered post to the Royal Society the following day.
4: G.I. Taylor: the inspiration behind the Cambridge school 161
zero in the equation for the rate of change of mean-square vorticity. His rea-
soning was that the vortex filaments would otherwise have a permanent ten-
dency to be stretched or compressed along the axis of vorticity. Taylor, who
had thought about this matter since his Adams Essay of 1914, went further now
and showed that such a tendency does indeed exist on the average. He briefly
introduced the Taylor–Green problem, dealt in greater detail in Taylor & Green
(1937), and noted that the vortex-stretching terms are large. He clearly under-
stood that the places where vortex filaments are being stretched are also places
where high vorticity may be present. Using estimates from available measure-
ments, Taylor & Green (1937) deduced that the term neglected by Kármán was
three times as large as the one retained. They also estimated that the vortex-
stretching term increases relative to other terms retained by Kármán as a linear
power of the large-scale Reynolds number.
Taylor, writing to Kármán on 12 March 1936, had already taken the occa-
sion to mention the Taylor–Green work (still unpublished at that point; see
Figure 4.3, which is the reproduction of a page of Taylor’s letter), and con-
cluded that “this puts the whole idea of my dissipation theory of turbulence on
a sounder basis”.
The error in the Kármán–Howarth draft was eventually set right, as de-
scribed in greater detail in Chapter 3. In the printed version of the paper, a new
section was added under the responsibility of the “senior author”, acknowl-
edging the previous error as well as the possibility of small and large scales
possessing two separate laws of self-preservation, and conforming to Taylor’s
earlier request for the slight modification in the interpretation of his linear
law, with Kármán’s addition mentioned earlier: “Hence the coincidence be-
tween these equations and Taylor’s results is rather formal.” All told, Taylor got
his way.
I now wish to add a few remarks on the human element of the exchange of
letters between Taylor and Kármán. It covered a wide range of topics, often
about mutual visits, families and acquaintances, and the topics ranged from
pleasantries to serious scientific exchanges. To get a sense of these friendly
exchanges, we note that the available correspondence begins with a graceful
note from Taylor accepting the award of the honorary degree from the Univer-
sity of Aachen, most likely enabled by Kármán’s intervention, an inquiry from
Kármán asking for a copy of Heisenberg’s 1948 paper on isotropic turbulence,
a note from Taylor informally asking for Kármán’s thinking on “radio waves
coming from beyond our solar system”, Kármán’s cable to Taylor on the oc-
casion of the latter’s 70th birthday: “Dear G.I. First time in life I am ahead of
you. Shall be almost 75 as you reach 70. Cordial congratulations”, and so forth.
162 Sreenivasan
Here, τ is the turbulent stress, which is related to the mean velocity gradient
dU/dy through the eddy diffusivity ε; and is the mixing length. Prandtl noted
that the 1/7th power for the velocity distribution would obtain if β = α/3.
Twenty days later, he was already disenchanted with this proposal because:
the apparent fraction in the inner part of the flow depends only on processes in the
laminar layer of the wall and that the contribution of the first term is compensated
to a large extent at large Reynolds numbers by the second term, which is negative.
This is for me the strangest thing of all and I believe that the formula has no
physical justification.
Taylor took nearly three years to answer Prandtl’s request for his opinion,
and was properly apologetic: He had “thought about the matter but could not
come to any definite conclusions”. Around the same time, Taylor had been car-
rying on a correspondence with V.W. Ekman (see later) about the differences
between the transport of heat and momentum. It is easy to conjecture that this
stimulus must have prompted Taylor to re-examine his transport theory vis-á-
vis Prandtl’s theory. The result is Taylor (1932a,b, 1935f, 1937a), as already
discussed.
Prandtl had great respect for Taylor’s work (“if I had known these papers
[Taylor 1915 and 1921], I would have found my way to turbulence earlier” –
letter of 25 July 1932), but was not hesitant to point out that “The result of your
theory [Taylor 1936a] of critical Reynolds number of a ball behind a turbulence
grid is very strange” (letter of 12 March 1935). Taylor followed up on 14 April
1935 with a summary of his 1935 papers, to which Prandtl responded with
Reichardt’s data in the channel. He was enamored by Taylor’s work but not
entirely sold on isotropic turbulence. On 11 May 1938 (effectively the last
serious scientific letter exchanged between them), Prandtl wrote:
in your new work [Taylor 1938] we were extremely interested in the extremely
cunning trick by which you can obtain the correlation curve from the spectrum
164 Sreenivasan
of time to be written by somebody else who writes more clearly? I hope that you will not take
umbrage against this remark . . . ” Taylor continued to write in his own hand but made a decided
effort to be more neat while writing to Prandtl. However, in this and the subsequent letter, he
made no such effort and his writing returned to being quite messy.
166 Sreenivasan
It seems to me that the only reason for this officially inspired tirade is that the
Nazis want war but Chamberlain’s visit for the first time penetrated the wall of
their censorship and showed them that the people of Germany do not want war.
They are therefore doing their best to incite the populace against us in order that
the desire for war may spread in Germany. You will see that we are not likely to
agree on political matters so it would be best to say no more about them.
He ended the letter and signed it; changing his mind, he crossed it out and
continued on the next page: “Fortunately there is no reason why people who
do not agree politically should not be the best of friends.” This was followed
by a question of clarification on a mundane scientific issue.
It appears that this was the last letter from Taylor to Prandtl. But Prandtl,
who apparently wished to win Taylor over to his point of view, wrote to Steph-
anie Taylor a year later (5 August 1939, just before the Second World War
officially broke out), essentially seeking an ally in her. He lay the blame for
possible war on England, and discussed the political pacts that Germany had
closed for peace. If Prandtl had hoped to convince the Taylors about the situa-
tion in Germany, he must have failed utterly. There is no record of either of the
Taylors having responded to it. There is also no record, either, of any response
to two of Prandtl’s post-war letters using intermediate carriers. The contents of
the letters, as translated by Johanna Vogel-Prandtl, are partially included in the
The Legacy, p. 187, but a few additional comments may be made.
The letter of 28 July 1945 was basically a post-war status report on Prandtl’s
Institute, which was then occupied by Allied Officers. The employees of the In-
stitute were allowed only to make some repairs to the building and write up re-
ports to the Allied Committee on the unfinished work done during
the war. No new work was allowed. Prandtl was hopeful that things would
change for the better but felt that he could not probably last much longer since
he was already 70 years old (though in good health). He mentioned the BBC
programs in German – some of which he thought were useful, especially those
relating to the superiority of the constitutional system of governance – but was
less sanguine about other aspects.
28 This agreement occurred on 29 September 1938, a month before Prandtl’s letter was written,
and is universally regarded as a failed pact.
4: G.I. Taylor: the inspiration behind the Cambridge school 167
In the letter of 11 October 1945, the last one in existence, Prandtl reiterated
that his Institute was prohibited to do any research, listed areas that “await
solution”, and said that he had written to the Royal Society, on the advice of the
British Resident Scientific Officer, to allow further research to be continued.
He attached a copy of that letter, and expressed gratitude if the Royal Society
would allow the request. He was clearly seeking Taylor’s intervention. It is not
known if Taylor did anything. Prandtl also enclosed an improved method for
the calculation of mean profiles in turbulent flows (apparently using his new
closure method – I have not seen that note). Taylor most likely did not respond
to these calculations; his research interests had simply moved on.
Taylor had a good idea of the cascade processes even in his Adams essay of
1914, and had more to say about it in his 1937 paper with A.E. Green, but
the general picture that Kolmogorov (and later Onsager, Heisenberg and von
Weizsc̈aker) brought to bear on the problem had eluded him. In the same way,
one can see the stirrings of small-scale intermittency in Taylor (1917c), but its
statement as a general principle was either not Taylor’s forte – or he was simply
loathe to extending himself beyond the concrete. If he momentarily lapsed into
thinking that Kolmogorov’s results were obvious, we can forgive him for not
seeing their full sweep.
foundation. . . . If the Swedes divide science in the same as we do here then I will
come into as little consideration as mathematicians and could therefore console
myself in the same way mathematicians do.
And the Swedes thought somewhat the same way, as will be seen presently.
The summary is that Taylor and Prandtl were considered together for the
Prize by the Nobel Committee for Physics, which concluded thus:
L. Prandtl and G.I. Taylor have both been active in applied mechanics and occupy
leading positions in this field. The particular conditions prevailing there are due
to the fact that the laws governing the observed phenomena are so complicated
that it is impossible to derive the phenomena from them. For those who cannot
or will not wait for the mathematical methods to be perfected enough to make
today’s impossibilities possible, the task must then be to develop approximate
methods of calculation, which at least apply in special cases. Typical examples
of this kind of approximate methods of calculation are Prandtl’s theories for
boundary layers (1905) and airplanes (1919). Obviously, activities of this kind
cannot in general lead to any ‘discovery or invention in the area of physics’.
No such discovery or invention has been advanced by the two propositioners
who this year recommended Prandtl and Taylor. Therefore, the members of the
Committee cannot recommend that either of them be rewarded.
I should note that Prandtl had been nominated for the 1928 Prize as well
(by Professors Húckel of Zürich, Schiller of Leipzig and Pöschl of Prague).
The conclusion of the Nobel Committee for Physics on that occasion was that
the boundary layer theory “provides no true solution to the hydrodynamical
problem, but Prandtl’s analysis of what actually happens should nevertheless
be considered a valuable advance”. The key person on both occasions was the
committee member C.W. Oseen (1879–1944), who wrote a special opinion in
1928. He stated, in part, that:
The propositioners apparently assume that in order to receive a physics Nobel
Prize, a long-standing, dynamic and successful activity within physics, including
mechanics, suffices. If this assumption were in agreement with the Charter of the
Nobel Foundation, I would have recommended that Professor Prandtl be awarded
a physics Nobel Prize. However, since a prerequisite for receiving such a prize,
according to the Charter, is a discovery or invention in the field of physics, and
since no discovery or invention of the magnitude that would justify a Nobel Prize
has been put forward by the propositioners, I conclude that I cannot second the
nomination.
’18, ’19, ’20) and Gustaf de Laval (’08, also for Chemistry Prize in ’08). The
question marks indicate that there may have been further nominations.
lucky, he felt, the contribution might be useful in a physical view also. I re-
call . . . an earlier conversation between Stewartson and Taylor, Stewartson had
explained to Taylor his personal philosophy for applied mathematics, as de-
scribed above. To Keith’s chagrin, Taylor’s devastating reply had been ‘that
one did not make progress in science in that way’.” However, there is little
doubt that Taylor’s focus was usually a specific physical question rather than
an elegant formulation.
Taylor seems to have placed great value in seeing his theory verified by ex-
periment. Nearly all his papers end with some comparison between his theory
and experiment (or suggestions for the same), even if such comparisons were
sometimes based on scanty evidence. He didn’t seem to have much use for the
paradigm of Karl Popper who argued that a scientific idea cannot be proven
true by observation. It may be wrong even if a large number of observations
agree with it, because there might yet be another that might disprove it: a single
contrary experiment can prove a theory incontrovertibly false. Again, Taylor in
his own words:
I have tried to indicate the kind of interplay which I have used between mathe-
matics and experiment. The late Professor G.H. Hardy regarded all applied math-
ematics as a dull activity, a sort of glorified plumbing, which could not give the
kind of satisfaction he found in pure mathematics. My feeling is that I derive a
rather similar kind of satisfaction from the interplay between applied mathemat-
ics and experiment. It is quite a different kind from the satisfaction one gets in
doing something useful, though one derives an added pleasure when anything
one does turns out accidentally to be of use in engineering.
Corrsin (1968) seems to have thought that the following remark of Taylor re-
flects a characteristic naivete on Taylor’s part:
In California I think less of his [Kármán’s] time must have been absorbed in
administrative duties than it was in Aachen, for he obviously read very widely
and the reviews which he published of the then existing state of knowledge were
particularly valuable.
Taylor, Corrsin thought, must have forgotten about the cadre of intelligent and
eager assistants with whom Kármán had surrounded himself. One can perhaps
speculate that Taylor’s uncomplicated nature is consistent with the fact that he
was not a particularly abstract thinker.
inferred that Taylor had regarded the transport mechanisms of heat and mo-
mentum transport to be the same – a point of view with which he did not agree;
second, Taylor’s neglect of viscosity did not sit well with Ekman. Taylor was
quick with a reply. While acknowledging that “perhaps I was not very clear in
my exposition in 1915”32 , he was adamant that:
It is not entirely accurate to say that Taylor’s interests were confined to his
science. Recall that he learned both to fly and parachute, no doubt in part be-
cause it was easier to make measurements as he flew (rather than have someone
else do his bidding), but also because he loved the open air. His love for sail-
ing is indeed legendary (his next love having been, perhaps, playing golf with
Rutherford, F.W. Aston and R.W. Fowler – mostly “to hear Rutherford talk”,
according to The Legacy). Indeed, one of his letters to his future wife contains
perspicuous comments on the metaphysical poet John Donne, and gives the
impression of having derived much pleasure from reading Shakespeare, Mil-
ton and Shelley. It would seem that he was reading poetry for pleasure, at least
at some point in his life.
It is also not the case that Taylor was indifferent to what others might think of
him even by implication. From The Legacy, p. 211, we learn how incensed he
was when he became aware of a remote resemblance to himself of a character
in the fictionalization of the personalities of some scientists in the atom bomb
era at Los Alamos: he wrote a strongly worded letter to the publishers asking
for an apology. The publisher did tender one.
One further remark: one could not have expected letters of the sort Taylor
wrote Prandtl in 1938 unless he was deeply concerned about politics of his
day and their implications. It is hard to reconcile the heartfelt agony of those
letters with Batchelor’s assessment (The Legacy, p. 110) that Taylor “was not
reflective, and moral or philosophical issues did not often engage his mind”.
These two letters, from which I have already quoted, show that Taylor was
deeply touched by the horrors preceding the Second World War, and that he
could express his views strongly even to the person he once held in the highest
regard. No one alive then could have escaped the imprint of Second World
War, and one may put the fervor of Taylor’s remarks to that aberration. But the
sheer power of those letters suggests that such thoughts could not have been
exceptional to Taylor’s mind.
Two other instances are probably worth quoting. Even later in life (12 De-
cember 1950) when the Royal Society was considering moving to a new site,
he wrote a strong note with: “I most sincerely hope that the matter will be re-
ferred to the Fellows before any irrevocable step is taken”. When a petition was
being prepared in support of Benjamin Levich, who was under persecution by
the Soviet authorities of the time, he wrote a letter thus:
I thought I should say how highly scientists whose work lies in related fields
value his contributions to electrohydrodynamics. Many of us regard him as the
foremost scientist in the field and would feel that a serious blow has been deliv-
ered to science.
176 Sreenivasan
Note that this was after Taylor had suffered his stroke, and that he had never
even met Levich!
I do not not feel competent to open a discussion with such an expert on ex-
plosions . . . feeling that perhaps Cambridge is hardly out of range of the force
of such explosions [at Imperial]. I feel however that any detonation arising in
the Imperial College, London, would have diminished in intensity by the time it
reached Göttingen. I think that a note from you in the subject would be welcomed
if you feel disposed to write one.
If one understands that Prandtl had not convinced Taylor that his assessment
of the work in question was right, one can infer that the latter was being quite
diplomatic here. Recall Taylor’s innocent delight, reported in The Legacy, in
humorously describing how Rutherford could speak no foreign language:
I only remember hearing him speak one sentence in a foreign language. He was
showing his apparatus to a distinguished French mathematician who could speak
no English. I saw him [Rutherford] point to a certain spot and say “ici les α-
particles”.
Acknowledgments Writing this article has been a labor of love but has not
been easy. I am grateful to my co-editors for assigning this task to me. The
material from which I have cited in the article came from archives at the
Libraries of the University of Maryland (Burgers); Caltech (Kármán); Trin-
ity College, Cambridge (Taylor); DLR, Göttingen (Prandtl), and the Nobel
Foundation. Professors J.M. Wallace, A. Leonard, H.K. Moffatt, E. Boden-
schatz, and the officials of the Swedish Academy of Sciences (Nobel Archives)
33 There are some scientists, especially pure mathematicians and theoretical physicists, who do
great work but have short lives. They remain unsullied in our memories because, by the time
fate snuffs out their lives prematurely, they have not had the time to fall behind in their science
and become obsolete, or to accumulate enough academic follies to be a bane to succeeding
generations. No armies of graduate students swarm their labs and no post-docs of theirs are
launched into well-placed positions. They are famous simply because their work is glorious.
We may cite Evariste Galois (who died at 20), Niels Henrik Abel (who died at 27), Srinivasa
Ramanujan (who died at 32), Ettore Majorana (who disappeared and was presumed dead at
32), S. Pancharatnam (who died at 35), etc. They are pure gold and through them shines the
untarnished image of science.
4: G.I. Taylor: the inspiration behind the Cambridge school 179
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5
Lewis Fry Richardson
Roberto Benzi
5.1 Introduction
The nature of turbulent flow has presented a challenge to scientists over many
decades. Although the fundamental equations describing turbulent flows (the
Navier–Stokes equations) are well established, it is fair to say that we do not
yet have a comprehensive theory of turbulence. The difficulties are associ-
ated with the strong nonlinearity of these equations and the non-equilibrium
properties characterizing the statistical behaviour of turbulent flow. Recently,
as predicted by von Neumann 60 years ago, computer simulations of turbu-
lent flows with high accuracy have become possible, leading to a new kind
of experimentation that significantly increases our understanding of the prob-
lem. The largest numerical simulations nowadays use a discretized version
of the Navier–Stokes equations with several billion variables producing many
terabytes of information that may be analyzed by sophisticated statistical tools
and computer visualization. None of these tools were available in the 1920s
when some of the most fundamental concepts in turbulence theory were intro-
duced through the work of Lewis Fry Richardson (1881–1953). Although his
name is not as well-known as other contemporary eminent scientists (e.g. Ein-
stein, Bohr, Fermi) and although his life was spent outside the mainstream of
academia, his discoveries (e.g. the concept of fractal dimension) are now uni-
versally known and essential in understanding the physics of complex systems.
Detailed biographies of Richardson (Ashford (1985); Hunt (1987)) are avail-
able to the interested reader. Our aim here is simply to review his most impor-
tant scientific ideas and to understand how the concepts that he introduced
transformed our view of turbulent flow and, more generally, of complex phe-
nomena. The significance of Richardson’s life has been well summarized by
Hunt (1987) in the following terms:
187
188 Benzi
His work was not at the time much appreciated by the scientific community,
although many distinguished scientists were well aware of his creativity and
his achievements. There are perhaps several reasons why Richardson remained
outside the academic mainstream. As previously remarked, his ideas were sim-
ply too new for most of his contemporaries. In the period 1900–1930, physics
underwent two dramatic and exciting revolutions triggered by the discover-
ies of quantum mechanics and relativity. An equally exciting revolution oc-
curred much later in the 1970s through the discovery of chaotic behavior in
dynamical systems, the statistical mechanics of complex systems, and the uni-
versality properties in second-order phase transitions. Many new discoveries
were triggered by a combination of these new theoretical ideas coupled with
computer simulations based on accurate and efficient numerical schemes for
solving complex nonlinear dynamics. Richardson was an early pioneer in this
field. However, without computers, his numerical schemes could not be imple-
mented, and it was perhaps for this reason that his ideas and pioneering work
were not fully appreciated at the time.
In order to appreciate Richardson’s contribution to the study of turbulence,
one should read his original papers. It is an interesting experience from which
one may recognize that Richardson had a profound respect for any question
that is scientifically meaningful no matter how strange or naive the question
may appear to be. For Richardson, science consisted in investigating any such
question using observation or experimental data and in building a mathematical
framework capable of explaining such data. In all his papers, curiosity was
the driving force, coupled with the motive of providing useful results for real
applications. With these initial conditions satisfied, science (for Richardson)
reaches its highest level and many extraordinary tasks can be accomplished.
After a short period at the Meteorological Office in Benson, Richardson
became a Lecturer at Westminster Training College (then near Westminster
Abbey) and in 1929 he moved north to the Technical College in Paisley, an
industrial town near Glasgow in Scotland. Although Richardson was elected
Fellow of the Royal Society in 1926, there was “at that time . . . no academic
position available for someone wishing to pursue research in meteorology . . . ”
(Hunt (1987)).
Richardson was, like his parents, a Quaker and a committed pacifist, and
this had a profound influence on his whole life: as a Conscientious Objector
during the First World War, he chose as mentioned above to work in France
as an ambulance driver; after the war, he abandoned his studies on dispersion
(even going so far as to burn his notes on the subject) when he discovered that
the military were using his ideas to aid their understanding of the dispersion of
poisonous gas; he resigned from the Meteorological Office in 1926 when it was
190 Benzi
put under military control. Most significantly, after 1929, Richardson’s interest
switched to studying, from a scientific point of view, how to prevent warfare,
a field of research that he virtually created. (This may appear remote from
his research on turbulence; however, in attempting to analyze a great variety of
war-related data, he introduced a new concept with far-reaching consequences,
namely the concept of fractal dimension.) He retired in 1943 and lived in Hill
House at Kilmun, about 25 miles from Glasgow, where he died on 30 Septem-
ber 1953, having spent the last year of his life working on a mathematical
theory of war and conflict.
Richardson’s achievements are now well-established and he is recognized
as one of the leading scientists of the 20th century. In the following discussion,
we shall review his three most important results for turbulence:
• The diffusion law in turbulent flows
• The energy cascade in turbulence and numerical weather forecasting
• The introduction of fractal dimension
In reviewing the ideas underlying these results, we shall outline some basic
physical laws concerning turbulence. It is notable that some of the problems
discussed in Richardson’s original papers are still to this day matters of active
current research.
Einstein’s basic idea was to recognise that diffusivity is the best quantity to
observe in an experiment. Diffusivity is defined as
|X(t) − X(0)|2
K = lim (5.1)
t→∞ 2t
where X(t) denotes the observed displacement of the particle in a fixed direc-
tion at time t. By introducing the concentration density C(x, t) as proportional
to the probability of finding a particle at position x at time t (given the initial
position to be x = 0), the diffusivity enters as a constant in the macroscopic
evolution of C(x, t):
∂C
= K∇2C, (5.2)
∂t
i.e. the well-known diffusion equation. The question raised by Richardson
(1926) starts with the experimental observation that if we assume equation
(5.2) to be valid for turbulent flows, then the so-called ‘constant’ K can vary
by a factor of anything from two to a billion, according to experimental data.
Thus we cannot blindly apply the concept of diffusivity for turbulent flows:
something new and non-trivial is happening which fundamentally modifies the
diffusive process.
Richardson argued that in a turbulent flow, diffusivity is enhanced because
two Lagrangian particles separated at t = 0 by some initial distance R(0) may
behave completely differently as the result of advection by two different ‘wind
gusts’ or ‘eddies’. It is not therefore possible to apply the theory of Brownian
motion as developed by Einstein, without explicitly considering the small-scale
turbulent fluctuations. Today we speak of ‘eddy diffusivity’ whenever the diffu-
sion constant K emerges by some kind of averaging over turbulent fluctuations.
Richardson’s brilliant idea was to discuss diffusion in a turbulent flows in terms
of the separation between different particles. It is an important conceptual step
which is worth discussing in detail. Figure 5.2, from Richardson’s original pa-
per, illustrates the qualitative ideas that he introduced. In his own words:
a small dense cluster of marked molecules, represented by the dots of fig. 1,
which, by molecular diffusion alone, would spread through successive spherical
clusters, shown in fig. 2 and fig. 3, actually seldom passes through the large
spherical stage 3, because it is first sheared in two detached clusters as suggested
in fig. 4. These are carried far away from one another, and are likely to be again
torn into smaller pieces as in fig. 5.
Given two particles whose initial separation is R(0), molecular diffusion and
turbulent advection conspire to increase their separation in time. The sepa-
ration R(t) at time t is a suitable variable for describing both effects. If we
consider a large number, N, of particles, diffusion in turbulent flows can be
192 Benzi
Figure 5.2 The figure shows the qualitative behaviour of particle dispersion in a
turbulent flow. Richardson showed this figure in his original 1926 paper in order
to illustrate the effect of wind gusts on particle dispersion.
Figure 5.3 In his 1926 paper, Richardson computed the diffusivity K(l) as a func-
tion of l using different experimental results available at that time. The ‘scale’ l
in the figure refers to the RMS displacement of the particles. The figure, taken
from the original 1926 Richardson’s paper, is a log–log plot of K versus l and the
straight line is Richardson’s fit to the data leading to the famous 4/3 law.
Reading Richardson’s papers, one may see that he was in general reluctant
to express numbers derived from observations as ‘vulgar’ fractions like 4/3.
Indeed, in his book, Weather Prediction by Numerical Process, we find many
places where vulgar fractions are written by preference in the decimal form,
e.g. 0.2857 rather than 2/7. Actually this is one point that makes Richardson’s
194 Benzi
papers difficult to read. Even in his 1926 paper, Richardson does not use any
vulgar fraction in fitting the data of Figure 5.3. According to his previous work,
one would expect to find the fit given by l1.31 or with some other exponent in
decimal form close to 4/3. The data in the figure do not even indicate that a
power law will provide the best fit! It is therefore all the more remarkable that
Richardson made his first and perhaps only ‘simplification’ by choosing l4/3 ,
a significant choice because it is the only one compatible with the fundamen-
tal ideas leading to the Kolmogorov (1941) theory on homogeneous isotropic
turbulence! Kolmogorov conjectured that in such turbulence the probability
distribution of velocity fluctuations (in the ‘inertial range’ of scales) should
depend only on the average rate of energy dissipation K (the suffix K here cho-
sen to denote Kolmogorov) and the scale l. Kolmogorov further assumed that
K is independent of viscosity (or equivalently of Reynolds number). Since the
physical dimension of K is length2 /time3 and K has dimensions length2 /time,
one immediately obtains for K the estimate K = g 1/3 l4/3 , where g is a nu-
merical constant of order unity. So Richardson’s simplification in fitting the
data of Figure 5.3 is quite special, and we may conjecture that he was aware, at
least intuitively, of possible links between the diffusion constant and the rate of
energy dissipation. It is clear that by assuming the exponent 4/3 as a good fit
for the observational data of Figure 5.3, the numerical prefactor, i.e. the num-
ber 0.2, actually has the physical dimensions length2/3 /time. Knowing how
insistent Richardson was in understanding the physics behind any numerical
results, we may conjecture that he must have thought about the physical mean-
ing of the constant 0.2 and how it is related to turbulence. From an historical
perspective, it is also interesting to remark that the fitting law for Figure 5.2
was written by Richardson as “l4/3 where = 0.2cm2/3 /sec”. The choice of
the Greek letter , anticipating Kolmogorov’s choice of symbol for the rate of
energy dissipation, is also intriguing.
In 1949, together with H. Stommel, Richardson published a short note in the
Journal of Meteorology entitled Note on Eddy Diffusion in the Sea (Richardson
& Stommel (1949)). The idea was to apply the results of the 1926 paper to the
ocean, using a new data set and new information. In their note, Stommel and
Richardson observed that the 1926 results are roughly validated by the new
data, and that “any power between l1.3 and l1.5 would be a tolerable fit to the
observations”. Thus, the requirement for a fit ∼ l4/3 was no longer regarded as
mandatory, nor was it made for simplicity. At the end of the paper, the authors
wrote:
After this manuscript was submitted the writers have read two unpublished manu-
scripts by C.L. von Weizsäcker and W. Heisenberg in which the problem of tur-
bulence for large Reynolds number is treated deductively with the results that
5: Lewis Fry Richardson 195
they arrive at the 4/3 law. The agreement between von Weizsäcker and Heisen-
berg’s deduction and our quite independent induction is a confirmation of both.
These results of von Weizsäcker and Heisenberg come from their version of
the ‘Kolmogorov’ (1941) theory, developed independently but leading to es-
sentially the same results.
Concerning the work with Stommel, Hunt (1987) has related a nice story:
while at Kilmun, Richardson
received the famous visit by H. Stommel, from Woods Hole, which was the op-
portunity to return to the question of how pairs, or a cloud, of particles separate
in a turbulent flow, such as he saw from his house every day on the waters of the
Holy Loch. They threw parsnips from a small pier into Loch Long, and using a
remarkable measuring instrument they confirmed that the rate of spreading in-
creased as the distance between the pairs of parsnips increased, consistent with
the four-thirds law.
in the figure), while at later time R grows much faster (compared to molecu-
lar diffusion alone), namely as t3 . This picture implies that there exists some
characteristic time scale τη such that for t τη the effect of molecular motion
becomes negligible and diffusion is dominated by turbulence. The notion of
τη is not introduced in Richardson’s paper but is an outcome of Kolmogorov’s
theory. In particular, τη can be identified as (ν/K )1/2 where ν is the kinematic
viscosity of the fluid. Notice that in the limit of large Reynolds number, i.e. in
the limit ν → 0, τη → 0, the result R2 (t)
∼ t3 may be expected to hold even
for extremely small R(0).
One way to appreciate the relevance of the above results is to consider the
velocity difference δv(R) ≡ v(x(t) + R(t)) − v(x(t)), where x and x + R are
the positions of the two particles and v(x(t)) and v(x(t) + R(t)) the respective
velocities. Since
d
x = v(x, t), (5.7)
dt
d
(x + R) = v(x + R, t) , (5.8)
dt
by subtracting (5.7) from (5.8) we obtain
dR
= δv(R) . (5.9)
dt
Finally, multiplying (5.9) by R and averaging, we obtain
dR2
= 2δv(R) · R
. (5.10)
dt
Comparing (5.9) with (5.5), we obtain R2
2/3 = 2δv(R)R
. Therefore, for
Richardson’s diffusion to be true, we require that
molecular effects are important) while T L is the longest time-scale of the turbu-
lent flow. Similarly, the range of spatial scales relevant for Richardson scaling
are within the interval [η, L] where η is the Kolmogorov dissipation scale while
L is the largest spatial scale of the turbulent flows. For a turbulent flow at some
Reynolds number Re, the ratio L/η ∼ Re3/4 while T L /τη ∼ Re1/2 , which im-
plies T L /τη ∼ (L/η)2/3 ∼ Reλ , where Reλ is the Reynolds number based on
the Taylor scale. A reasonably good Kolmogorov scaling (5.11) is observed
in numerical simulations and in laboratory experiments for Reλ ∼ 500 which
implies L/η ∼ 104 . However, in order to achieve T/τη ∼ 104 , one needs to
perform experiments or numerical simulations at Reλ ∼ 104 or equivalently
with L/η ∼ 106 , which cannot be achieved at present. The same result is ob-
tained by following the analysis due to Batchelor (1950). Let r0 be the char-
acteristic scale of the initial separation of two Lagrangian particles, with r0
in the inertial range, i.e. η r0 L. Then, by using Kolmogorov’s theory,
Batchelor (1950) showed that Richardson scaling (5.6) should be observed for
t ≥ t0 ≡ r02/3 −1/3 , while for t ≤ t0 one should observe R2 (t)
∼ t2 , usually
referred to as the Batchelor regime. In most laboratory experiments – see for
instance Ouellette et al. (2006) and Salazar & Collins (2009) – t0 lies in the
range 50–100τη which implies that a reasonable Richardson scaling can be
observed for T L ∼ 100t0 ∼ 104 τη .
Using direct numerical simulations of isotropic turbulence, the situation can
be improved as follows: let T n be the time needed for two particles to separate
from distance Rn to distance Rn+1 = ARn , where A > 1 is a fixed number. Then,
upon averaging over many particles, the quantity T n (Rn )
should behave as
R2/3
n . This method has been successfully applied by Boffetta & Sokolov (2002)
and Biferale et al. (2005) whose numerical results are consistent with Richard-
son scaling. In the light of the above discussion, it is all the more remarkable
that Richardson was able to achieve so much, more than 80 years ago.
We may safely conclude by saying that Richardson’s 1926 paper on dif-
fusion is one of the key starting points in our understanding of the physics of
turbulence. It introduces the concept of eddy diffusivity and its scale-dependent
behaviour, and it poses some fundamental questions about turbulence. The
concept of scaling is also introduced in this paper within the context of tur-
bulent flow. Richardson’s findings are still considered to be broadly valid, and
major computational and experimental efforts have been devoted to verifying
his early 1926 results. As we have discussed, he was close to discovering the
role of the rate of energy dissipation in defining the statistical properties of
turbulence, as later developed by Kolmogorov. We infer that Richardson had
a good intuitive understanding of the consequences of his findings and their
wide implications.
5: Lewis Fry Richardson 199
This rhyme appears in a paragraph of the book dealing with the effect of
eddy motion and it conveys the idea that the energy contained at some scale
is transferred to progressively smaller scales until ultimately dissipated by vis-
cosity. In order to obtain the relevant equation of motion for the large-scale
flows of the atmosphere, one needs to parametrize the effect of turbulence in
some way. The concept of the energy cascade indicates that this parametriza-
tion must take account of how energy is dissipated in a turbulent flow.
We note that Richardson’s book was published well before his 1926 paper;
thus, he had already conceived the energy cascade before his famous l4/3 law.
What then is this energy cascade in turbulent flow? The qualitative picture
due to Richardson suggests that the characteristic scale of energy input by
external forces (boundaries or pressure gradients) is much larger than the scale
at which viscous effects, due to molecular motion, become relevant. Therefore,
there must be a flux of energy from large to small scales which is due to the
1 Richardson’s oft-quoted rhyme is a parody of the verse of the Irish satirist Jonathan Swift:
So, naturalists observe, a flea
Hath smaller fleas that on him prey;
And these have smaller fleas to bite ’em
And so proceed ad infinitum.
Thus every poet in his kind,
is bit by him that comes behind.
200 Benzi
Note that Richardson here drew attention to the ‘supply of energy’ which, in
the Bjerknes theory, comes from the discontinuities (i.e. fronts) to the synoptic
scale (cyclone). If such a theory is correct, any attempt to provide weather
forecasts based on finite differences (as is the case for numerical forecasting)
seems doomed to failure since the discontinuities will spoil the precision of
the method. As reported by Hunt (1987), Richardson wrote in 1919 to Napier
Shaw of the Meteorological Office asking whether he could return there in
order to work on upper air ‘sounding’ experiments, with a view to making
“weather predictions by a numerical process . . . a practical system” (Ashford
(1985)). Shaw replied, saying that the ‘graphical scheme’ of V. and J. Bjerknes
had been selected as the operational tool to be developed in the future. So the
point raised by Richardson in his introduction was of crucial relevance.
The scientific controversy, which at that time was not perceived as such, be-
tween Bjerknes’ graphical scheme and Richardson’s numerical scheme is the
first example of a long-standing controversy in dynamical meteorology and,
more generally, in turbulence. To put the controversy in simple words, Bjerk-
nes’ approach was focused on well-defined structures (fronts), while Richard-
son’s approach was based on the equations. This marks the beginning of the
long debate on the importance of coherent structures (such as vortices and
vortex filaments) as opposed to statistical scaling laws underlying the funda-
mental aspects of turbulence. In describing turbulent flows, perhaps the most
important feature to understand is the dynamics of vortices, whose complex
5: Lewis Fry Richardson 201
Due to the scientific authority and impact of von Neumann, the brilliant the-
oretical insight of Charney and Phillips, and the availability (for the time)
of powerful computers, numerical weather prediction was becoming a prac-
tical proposition, as foreseen by Richardson. As a side remark, Phillips’ work
was recognized through the award of the first Napier Shaw Memorial Prize,
202 Benzi
Thus, Richardson knew that his method was opening a new strategy in oper-
ational weather forecasting although many questions and problems remained.
However, “the forecasters within meteorological organizations did not believe
that this provided a practical approach at that time” (Hunt (1987)). In his re-
view paper, Smagorinsky (1981) wrote:
Weather forecasting was quite subjective, . . . based on the experience of having
observed and classified many such evolutions . . . in 1943 I came into contact with
Professor Bernard Haurwitz. When I asked him why physical principles had not
been applied to the practical problem of weather prediction, he quickly pointed
5: Lewis Fry Richardson 203
out the futility of using the tendency equation . . . When queried further, Haurwitz
did recall the work by L.F. Richardson . . .
It is interesting to remark that the entire print run of Richardson’s 1922 book
was a mere 750 copies. When re-issued in 1965 by Dover, it sold 3000 copies.
As mentioned above, an important issue raised in this book was concerned
with the effect of initial conditions and their relevance for the predictability of
atmospheric flows. Due to work by E. Lorenz, we now understand the impor-
tance of Richardson’s findings. Lorenz (1963) was able to provide clear ex-
amples of relatively simple and well-defined dynamical systems whose error
forecast E(t) grew exponentially in time from an initial error E(0), no matter
how small this might be. This feature is what we now refer to as ‘sensitive
dependence on initial conditions’, a feature of any chaotic system. Although
this concept was not available to Richardson, his recognition of the effect of
initial errors, so well summarized in Woolard’s note, was influential in the later
development of operational weather forecasts and in predictability studies in
atmospheric dynamics.
In conjunction with the conceptual steps in his book, Richardson defined a
systematic way to use finite difference methods in dealing with the complex
nonlinear equations describing atmospheric flow; some of his prescriptions are
still used in numerical simulations. Richardson was the first to address the
question of how to improve computational efficiency by suitable organization:
It took me the best part of six weeks to draw up the computing forms and to
work out the new distribution into two vertical columns for the first time . . . With
practice the work of an average computer might go perhaps ten times faster. . . .
Next he analysed in detail how to pass messages and increase the efficiency
of communication between the different areas. What Richardson was describ-
ing in this book was in effect the first architectural design of a massive par-
allel computer as we know it today. Coordination (the task of the man on the
204 Benzi
tall pillar) and message transmission are the core problems in massive parallel
computation: organizing the data flow while parallelizing individual computa-
tions is the key to increased computational efficiency. In most reviews on the
art of parallel computing, Richardson’s ideas are still presented as a way of
introducing the problem.
Richardson made many important contributions to our understanding of at-
mospheric physics. In particular, he obtained a general criterion for the sup-
pression of turbulence by stable density stratification, (Richardson (1920)).
While at Benson, he derived estimates for the relative contributions to the en-
ergy of the turbulence, on the one hand from the buoyancy forces caused by
movement of eddies between levels of different temperatures, and on the other
from acceleration of eddies moving between levels of different wind speed.
The ratio of these two contributions is now called the Richardson number Ri.
Richardson showed that if Ri > 1, then turbulence is suppressed. To quote
Hunt (1987):
this insight into the occurrence and strength of turbulence was soon acknowl-
edged in the scientific world . . . Prandtl (1931) incorporated Richardson’s crite-
rion into his textbook on fluid dynamics only a few years later.
5.5 Conclusions
As observed in the introduction, Richardson’s scientific activity was charac-
terized by his ability to pose deep questions concerning basic concepts. Some
of these questions were answered by introducing new concepts or by develop-
ing new mathematical approaches. Richardson’s results opened a new way of
looking at the complex behaviour of physical systems, in particular turbulence
and turbulent flows.
His character and life were unusual for a scientist of such outstanding cre-
ativity. According to Gold (1971):
Research for Richardson was the inevitable consequence of the tendency of his
mental machine to run almost, but not quite, by itself. So he was a bad listener,
distracted by his thoughts, and a bad driver, seeing his dream instead of the traf-
fic. The same tendency explains why he sometimes appeared abrupt in manner,
otherwise inexplicable in one of his character.
Perhaps the most appropriate comment is that of Taylor (1958) who wrote
“Richardson was a very interesting and original character who seldom thought
on the same lines as his contemporaries and often was not understood by
them”.
5: Lewis Fry Richardson 207
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208 Benzi
The towering figure of Kolmogorov and his very productive school is what was
perceived in the twentieth century as the Russian school of turbulence. How-
ever, important Russian contributions neither start nor end with that school.
What seems to be the first major Russian contribution to the turbulence the-
ory was made by Alexander Alexandrovich Friedman, famous for his work
on non-stationary relativistic cosmology, which has revolutionized our view
of the Universe. Friedman’s biography reads like an adventure novel. Alexan-
der Friedman was born in 1888 to a well-known St. Petersburg artistic family
(Frenkel, 1988). His father, a ballet dancer and a composer, descended from
a baptized Jew who had been given full civil rights after serving 25 years in
the army (a so-called cantonist). His mother, also a conservatory graduate, was
a daughter of the conductor of the Royal Mariinsky Theater. His parents di-
vorced in 1897, their son staying with the father and becoming reconciled with
his mother only after the 1917 revolution. While attending St. Petersburg’s
second gymnasium (the oldest in the city) Friedman befriended a fellow stu-
dent Yakov Tamarkin, who later became a famous American mathematician
and with whom he wrote their first scientific works (on number theory, re-
ceived positively by David Hilbert). In 1906, Friedman and Tamarkin were
admitted to the mathematical section of the Department of Physics and Math-
ematics of Petersburg University where they were strongly influenced by the
209
210 Falkovich
great mathematician V.A. Steklov who taught them partial differential equa-
tions and regularly invited them to his home (with another fellow student V.I.
Smirnov who later wrote the well-known Course of Mathematics, the first
volume with Tamarkin). As his second, informal, teacher Alexander always
mentioned Paul Ehrenfest who was in St. Petersburg in 1907–1912 and later
corresponded with Friedman. Friedman and Tamarkin were among the few
mathematicians invited to attend the regular seminar on theoretical physics
in Ehrenfest’s apartment. Apparently, Ehrenfest triggered Friedman’s interest
in physics and relativity, at first special and then general. During his grad-
uate studies, Alexander Friedman worked on different mathematical subjects
related to a wide set of natural and practical phenomena (among them on poten-
tial flow, corresponding with Joukovsky, who was in Moscow). Yet after get-
ting his MSc degree, Alexander Friedman was firmly set to work on hydrody-
namics and found employment in the Central Geophysical Laboratory. There,
the former pure mathematician turned into a physicist, not only doing theory
but also eagerly participating in atmospheric experiments, setting the measure-
ments and flying on balloons. It is then less surprising to find Friedman flying
a plane during World War I, when he was three times decorated for bravery.
He flew bombing and reconnaissance raids, calculated the first bombardment
tables, organized the first Russian air reconnaissance service and the factory
of navigational devices (in Moscow, with Joukovsky’s support), all the while
publishing scientific papers on hydrodynamics and atmospheric physics. Af-
ter the war ended in 1918, Alexander Alexandrovich was given a professorial
position at Perm University (established in 1916 as a branch of St. Petersburg
University), which boasted at that time Tamarkin, Besikovich and Vinogradov
among the faculty. In 1920 Friedman returned to St. Petersburg. Steklov got
him a junior position at the University (where George Gamov learnt relativity
from him). Soon Friedman was teaching in the Polytechnic as well, where L.G.
Loitsyansky was one of his students. In 1922 Friedman published his famous
work On the curvature of space where the non-stationary Universe was born
(Friedman, 1922). The conceptual novelty of this work is that it posed the task
of describing the evolution of the Universe, not only its structure. The next year
saw the dramatic exchange with Einstein, who at first published the paper that
claimed that Friedman’s work contained an error. Instead of public polemics,
Friedman sent a personal letter to Einstein where he elaborated on the details of
his derivations. After that, Einstein published the second paper admitting that
the error was his. In 1924 Friedman published his work, described below, that
laid down the foundations of the statistical theory of turbulence structure. In
1925 he made a record-breaking balloon flight to the height of 7400 meters to
study atmospheric vortices and make medical self-observations. His personal
6: The Russian school 211
life was quite turbulent at that time too: he was tearing himself between two
women, a devoted wife since 1913 and another one pregnant with his child (“I
do not have enough willpower at the moment to commit suicide” he wrote in a
letter to the mother of his future son). On his way back from summer vacations
by train in the Crimea, Alexander Friedman bought a nice-looking pear at a
Ukrainian train station, did not wash it before eating and died from typhus two
weeks later.
Friedman’s work on turbulence theory was done in conjunction with his stu-
dent Keller and was based on the works of Reynolds and Richardson, both
cited extensively in Friedman and Keller (1925). Recall that Richardson de-
rived the equations for the mean values which contained the averages of non-
linear terms that characterize turbulent fluctuations. Friedman and Keller cite
Richardson’s remark that such averaging would work only in the case of a
so-called time separation when fast irregular motions are imposed on a slow-
changing flow, so that the temporal window of averaging is in between the fast
and slow timescales. For the first time, they then formulated the goal of writing
down a closed set of equations for which an initial value problem for turbulent
flow can be posed and solved. The evolutionary (then revolutionary) approach
of Friedman to the description of the small-scale structure of turbulence paral-
lels his approach to the description of the large-scale structure of the Universe.
Achieving closure in the description of turbulence is nontrivial since the hy-
drodynamic equations are nonlinear. Indeed, if v is the velocity of the fluid,
then Newton’s second law gives the acceleration of the fluid particle:
dv ∂v
= + (v∇)v = force per unit mass. (6.1)
dt ∂t
Whatever the forces, the acceleration already contains the second (inertial)
term, which makes the equation nonlinear. Averaging the fluid dynamical equa-
tions, one expresses the time derivative of the mean velocity, ∂v
/∂t, via the
quadratic mean (v∇)v
. Friedman and Keller realized that meaningful closure
can only be achieved by introducing correlation functions between different
points in space and different moments in time. Their approach was intended
for the description of turbulence superimposed on a non-uniform mean flow.
Writing the equation for the two-point function ∂v1 v2
/∂t, they then derived
the closed system of equations by decoupling the third moment via the second
moment and the mean: vi1 v2j vk2
= v1i
v2 j v2k
+ · · · (Friedman and Keller,
1925). It is interesting that Friedman called the correlation functions “moments
of conservation” (Erhaltungsmomenten) as they express “the tendency to pre-
serve deviations from the mean values” in a curious resemblance to the modern
approach based on martingales or zero modes. The work was presented at the
212 Falkovich
6.2 Mathematician
At any moment, there exists a narrow layer between trivial
and impossible where mathematical discoveries are made.
Therefore, an applied problem is either solved trivially or
not solved at all. It is an altogether different story if an
applied problem is found to fit (or made to fit!) the new
formalism interesting for a mathematician.
A.N. Kolmogorov, diary, 1943
Russians managed to continue, well into the twentieth century, the tradition
of great mathematicians doing physics.
Andrei Kolmogorov was born in 1903. His parents weren’t married. The
mother, Maria Kolmogorova, died at birth. The boy was named according
to her wish after Andrei Bolkonski, the protagonist from the novel War and
Peace by Lev Tolstoy. Andrei was adopted by his aunt, Vera Kolmogorova,
and grew up in the estate of his grandfather, district marshal of nobility, near
Yaroslavl. The father, agronomist Nikolai Kataev, took no part in his son’s
upbringing: he perished in 1919, fighting in the Civil War. Vera and Andrei
relocated to Moscow in 1910. In 1920, Andrei graduated from the Madame
6: The Russian school 213
Repman gymnasium (cheap but very good) and was admitted to Moscow Uni-
versity, with which he remained associated for the rest of his life. In a few
months, he passed all the first-year exams and was transferred to the second
year which “gave the right to 16 kg of bread and 1 kg of butter a month –
full material prosperity by the standards of the day” (Kolmogorov, 2001). His
thesis adviser was Nikolai Luzin who ran the famous research group ‘Luzita-
nia’. Apart from him, Kolmogorov was influenced by D. Egorov, V. Stepanov,
M. Suslin, P. Urysohn and P. Aleksandrov, with whom Kolmogorov was close
until the end of his life, sharing a small cottage in Komarovka village where
they regularly invited colleagues and students, who described the unforgettable
atmosphere of science, art, sport and friendship (Shiryaev, 2006). Kolmogorov
completed his doctorate in 1929. In 1931, following a radical restructuring of
the Moscow mathematical community, he was elected a professor. He spent
nine months in 1930–31 in Germany and France, later citing important inter-
actions with R. Courant, H. Weyl, E. Landau, C. Carathéodory, M. Frechet,
P. Levy. Two years later he was appointed director of the Mathematical Re-
search Institute at the university, a position he held until 1939 and again from
1951 to 1953. In 1938–1958 he was a head of the new Department of Proba-
bility and Statistics at the Steklov Mathematical Institute. Between 1946 and
1949 he was also the head of the Turbulence Laboratory in the Institute of
Theoretical Geophysics.
Andrei Nikolaevich Kolmogorov was a Renaissance man: his first scientific
work was on medieval Russian history; he then did research on metallurgy, bal-
listics, biology and statistics of rhythm violations in classical poetry, worked
on educational reform, was the scientific head of a round-the-world oceanolog-
ical expedition and used to make 40 km cross-country ski runs wearing only
shorts. But first and foremost he was one of the greatest and most universal
mathematicians of the twentieth century, if not of all time (Kendall et al.,
1990). Kolmogorov put the notion of probability on a firm axiomatic foun-
dation (Kolmogorov, 1933) and deeply influenced many branches of modern
mathematics, especially the theory of functions, the theory of dynamical sys-
tems, information theory, logics and number theory. Seventy-one people ob-
tained degrees under his supervision, among them several great and quite a
few outstanding scientists. There is a certain grand design in the life work of
Kolmogorov, to which one cannot give justice in this short essay. In his own
words:
I wish to stress the legitimacy and dignity of a mathematician, who understands
the place and the role of his science in the development of natural sciences and
technology, yet quietly continues to develop ‘pure mathematics’ according to its
internal logics.
214 Falkovich
late (viscous) stage of turbulence decay when the size of the turbulence re-
√
gion grows as l(t) νt, one can readily infer the law of the energy decay:
v2 (t) Λl−5 ∝ t−5/2 . Also, neglecting the third moment (as had Friedman and
Keller before), Millionschikov obtained a closed equation and solved it for the
precise r, t dependencies of the second moment (Millionschikov, 1939). To de-
scribe turbulence at large Reynolds number Re, one needs to face eventually the
third moment and account for the nonlinearity of hydrodynamics. Kolmogorov
did that himself, estimating dv2 /dt v3 /l and obtaining l(t) ∝ Λ1/7 t2/7 (Kol-
mogorov, 1941b). While Kolmogorov used his theory of small-scale turbulence
(to be described below) to argue for these estimates, the relation v l/t for in-
tegral quantities seems to be not very sensitive to the details of microscopic
theories. The correction, unexpectedly, came from another direction: conser-
vation of the Loitsyansky integral takes place not universally but depends on
the type of large-scale correlations in the initial turbulent flow. In terms of
6: The Russian school 215
Fourier harmonics
v(p) = v(r) exp(ip · r) dr,
That allowed him to define the viscous (now called Kolmogorov) scale as
η = (ν3 /¯ )1/4 and make the second (correct) assumption that for r12 η
the statistics of velocity differences is independent of the kinematic viscosity
ν. For η r12 L, one uses both assumptions and immediately finds from
dimensional reasoning that v212
= C(¯ r12 )2/3 , where the dimensionless C is
called the Kolmogorov constant (even though it is not, strictly speaking, a con-
stant, as will be clear later).
6: The Russian school 217
Sinai recalls Kolmogorov describing how he inferred the scaling laws after
“half a year analyzing experimental data” on his knees on the apartment floor
covered by papers (Shiryaev, 2006, p. 207). Some thirty years later, we find
Andrei Nikolaevich again in this position on the ship’s cabin floor catching
mistakes in the oceanic data during a round-the-world expedition (Shiryaev,
2006, p. 54). In 1941, the data apparently were from the wind tunnel (Dryden
et al., 1937); they were used in the third 1941 paper (Kolmogorov, 1941c) to
estimate C.
More importantly, in this third paper, Kolmogorov uses the Kármán–Howarth
equation, implicitly assumes that, although proportional to ν, the dissipation
rate ¯ has a finite limit at ν → 0, and derives the elusive third moment.
Schematically, one takes the equation of motion (6.1) at some point 1, mul-
tiplies it by v2 and subtracts the result of the same procedure taken at point
2. All three forces acting on the fluid give no contribution in the interval
η r12 L: viscous friction because r12 η, external force because r12 L
and the pressure term because of local isotropy. This is why that interval is
called inertial, the term so suggestive as to be almost misleading, as we will
see later. In this interval the cubic (inertial) term, which is the energy flux
through the scale r12 , is equal to the time derivative term, which is a constant
rate of energy dissipation: (v12 · ∇)v212
= −2∂v2 /∂t
= −4¯ . Integrating this
one gets
(v12 · r12 /r12 )3
= −4¯ r12 /5 . (6.2)
For many years, the so-called 4/5-law (6.2) was the only exact result in the
theory of incompressible turbulence. It is the first derivation of an ‘anomaly’ in
physics in a sense that the effect of breaking the symmetry (time-reversibility)
remains finite while the symmetry-breaking factor (viscosity) goes to zero; the
next example, the axial anomaly in quantum electrodynamics, was derived by
Schwinger ten years later (Schwinger, 1951).
Obukhov’s approach is based on the equation for the energy spectral density
written as ∂E/∂t+D = T where D is the viscous dissipation and T is the Fourier
image of the nonlinear (inertial) term that describes the energy transfer over
scales (Obukhov, 1941). Obukhov starts his paper by saying that for a given
observation scale l = 1/p, larger-scale velocity fluctuations provide almost
218 Falkovich
Kolmogorov was the first to provide a correct understanding of the local structure
of turbulent flow. As to the equations of turbulent motion, it should be constantly
born in mind . . . that in a turbulent stream the vorticity is confined within a lim-
ited region; qualitatively correct equations should lead to just such a distribution
of eddies.
deep inside other turbulent flows. It is likely that Landau started to have doubts
about Kolmogorov’s description of small-scale structure only later. In 1944, the
sixth volume of the Landau–Lifshitz course, Mechanics of Continuous Media,
appeared (Landau and Lifshitz, 1987). This book firmly set hydrodynamics as
part of physics. The book contained a remark (attributed in later editions to
Landau, 1944), which instantly killed the universality hypothesis:
Alexander Obukhov was soon joined by Andrei Monin and Akiva Yaglom,
the two other key people that established the Kolmogorov school of turbu-
lence. Andrei and Akiva were born the same year, 1921, and died the same
year, 2007. They wrote the book (Monin and Yaglom, 1979) that for several
decades was “the Bible of turbulence”. The triple A of Alexander, Andrei and
Akiva represented very different, and in some respects polar opposite, people.
Alexander and Akiva were never Party members, with the latter even refusing
to work on the nuclear project since he disliked the idea of developing a bomb
for Stalin (Shiryaev, 2006, p. 440), while Andrei was a devoted communist
who joined the Party during the war. That was a stark difference in the Soviet
Union back then. Kolmogorov himself was not a Party member, yet allowed
neither regime critique nor political conversations in his presence (Shiryaev,
2006, p. 442); descent from nobility and homosexuality (criminal under So-
viet penal code) added extra vulnerability for Andrei Nikolaevich in Stalin’s
Russia.
Yaglom grew up in Moscow where his high-school friend was Andrei Sakh-
arov (who later became a friend of Obukhov too). Akiva had a twin brother
Isaak, with whom he shared a first prize at the Moscow Mathematical Olymp-
iad in 1938. The prize was presented by Kolmogorov who never forgot good
students (Yaglom, 1994) and in 1943 invited Yaglom to work on the theory
of Brownian motion. Andrei Monin graduated in 1942 and the same year was
also invited by Kolmogorov to work on probability distributions in functional
spaces (where there is no volume element and thus no density). Both Akiva (in
1941) and Andrei (in 1942) volunteered for military service to fight in the war.
Akiva was rejected because of poor eyesight. Andrei was drafted and spent
the war as an officer-meteorologist serving at military airfields. He returned in
1946 ready to work on turbulence.
222 Falkovich
The first new result after 1941 was, however, obtained by Obukhov whose
Kazan years were important and formative. In addition to Landau, he interacted
there with the physicist M.A. Leontovich, a man of great integrity (who, among
many other things, published with Kolmogorov the paper on Brownian motion
in 1933). Landau and Obukhov were the first to suggest independently the
Lagrangian analog of KO41. If R(t) describes the trajectory of a fluid particle,
then the Lagrangian velocity is defined as V(t) = v(R, t). The relation V(t) −
V(0) (t)1/2 first appeared in the Landau–Lifshitz textbook in 1944. Note
however that the exact Lagrangian relation, which is a direct analog of the
flux law (6.2), is not the (still hypothetical) two-time single-particle relation
|V(t) − V(0)|2 t, but the Lagrangian time derivative of the two-particle
velocity difference: d|δV|2 /dt
= −2¯ (note that > 0 in 3D and < 0 in 2D)
(Falkovich et al., 2001).
From 1946, Kolmogorov arranged a bi-weekly seminar on turbulence which
was a springboard for the explosive development of KO41 and its applications.
Obukhov started to work on the atmospheric boundary layer and dynamic me-
teorology. Already in 1943 he wrote a paper which because of the war was pub-
lished in 1946 and yet was ahead of its time (Obukhov, 1988, p. 96; translated
in Obukhov, 1971). Following Prandtl and Richardson, Obukhov considered
the influence of stable stratification on turbulence. It is clear that turbulence
disturbs stable stratification and increases the potential energy, thus decreasing
the kinetic energy of the fluid. In other words, stratification suppresses tur-
bulence. On the other hand, turbulence influences the vertical profile of the
temperature. Obukhov developed a semi-empirical approach based on a sys-
tematic use of universal dimensionless functions. In addition to the dimension-
less Richardson number that quantifies the relative role of stratification and
wind shear, Obukhov measured the height in units of the sub-layer where the
Richardson number is small and stratification is irrelevant. This defines what
is now called the Obukhov–Monin scale, since the idea of the sub-layer was
systematically exploited by Obukhov and Monin in 1954. In the paper which
is a sequel to that of Obukhov (1988, p. 135), they showed that the profiles
of the wind and the temperature are determined by the vertical fluxes of the
momentum and heat; see Yaglom (1988) for more details.
The year 1949 was exceptionally productive. Kolmogorov applied KO41 to
the problem of deformation and break-up of droplets of one liquid in a tur-
bulent flow of another fluid: flow can break the droplet of the size a if the
pressure difference due to flow ρ(δv)2 ρ(¯ a)2/3 exceeds the surface tension
stress σ/a (Kolmogorov, 1949). Obukhov established the basis of dynamic
meteorology by his famous work on a geostrophic wind, derived what is now
called the Charney–Obukhov equation for the rotating shallow water, known
6: The Russian school 223
the typical turnover time on the scale r12 . This 2/3-law was independently es-
tablished by Corrsin in 1951 and is called the Obukhov–Corrsin law (see also
the Corrsin chapter). The second exact relation in turbulence theory, the flux
expression for a passive scalar analogous to (6.2) for energy, was derived by
Yaglom the same year (Yaglom, 1949a). That same year Obukhov dispelled an
erroneous belief (expressed in Millionschikov, 1941) that pressure fluctuations
are zero in incompressible turbulence (Obukhov, 1949b). By taking the diver-
gence of the Navier–Stokes equation, Obukhov obtained the incompressibility
condition Δp = −∇i ∇ j (vi v j ), which allows one to express the second moment
of pressure via the fourth moment of velocity, which is then decoupled via
the product of the second moments, again assuming Gaussianity: p212
∝ r4/3 .
That 4/3-law together with 5/3, 2/3 and others was the basis for the joke that
Obukhov discovered the fundamental ‘all-thirds law’. There is a truth in every
joke since the number 3 in the denominator of these scaling exponents arises
because of two fundamental reasons: (i) the nonlinearity of the equation of
motion is quadratic and (ii) the fluxes considered are of the quadratic integrals
of motion. Immediately, Yaglom used Obukhov’s approach to derive the mean
pressure gradient and the mean squared fluid acceleration (Yaglom, 1949b).
Remarkably, Yaglom’s estimate for atmosphere showed that typical winds can
make for accelerations exceeding that of gravity. Obukhov, Monin and Yaglom
had a chance to experience that, flying on balloons in turns, thus continuing
Friedman’s tradition; in 1951 the wind data were obtained confirming KO41
224 Falkovich
scaling (Obukhov, 1951) (later, they also observed a layered structure of turbu-
lence, the so-called turbulent ‘pancakes’, predicted by Kolmogorov in 1946 –
Shiryaev, 2006, p. 181). In 1951, Obukhov and Yaglom published together a
detailed paper that presented all the results on pressure and acceleration. Sim-
ilar results were obtained independently by Heisenberg in 1948 and Batchelor
in 1951.
The Kolmogorov turbulence seminar was attended by applied scientists and
engineers as well, and discussions of applied problems went along with the
focus on fundamental issues. In 1951, Kolmogorov accepted the next student,
Gregory Barenblatt, whose name he remembered from the list of the students
whose work won first prizes (following the familiar Obukhov–Yaglom pat-
tern). Barenblatt was given the task of describing the transport of a suspended
sediment by turbulent flows in rivers. Somewhat similarly to stably stratified
flows, turbulence spends energy lifting sediments which, being small, then dis-
sipate energy into heat when descending. Barenblatt built an elegant theory
similar to that of Obukhov–Monin (Barenblatt, 1953).
Important insights into the advection mechanisms were obtained by elimi-
nating global sweeping effects and describing the advected fields in a frame
whose origin moves with the fluid. This picture of the hydrodynamic evolu-
tion, known under the name of quasi-Lagrangian description, was first intro-
duced in Monin (1959). In a kind of a bridge between work on stratification
and passive scalars, Obukhov considered unstable stratification, accounted for
the buoyancy force and defined a new scale above which this force starts to be
important (Obukhov, 1959). Bolgiano discovered this independently the same
year and also suggested KO41-type scaling for turbulent convection at larger
scales (Bolgiano, 1959).
In 1956 the Institute of Geophysics was divided into three parts and Obukhov
was appointed director of the newly created Institute of Atmospheric Physics
which now bears his name. That followed his long conversation with Leon-
tovich which ended with the advice to “avoid administrative zeal” (Obukhov,
1990). In the Soviet Union, the Academy was a huge body that operated hun-
dreds of scientific institutes with tens of thousands of researchers. Academic
institutes worked under the strict Party control and a non-communist director
was a rare bird. Obukhov flouted Party policy in another important respect:
employing numerous Jewish scientists in his Institute. Since the late 1940s
anti-Semitism as a Party policy was steadily gaining ground in Russian so-
ciety and academia. Moscow University was particularly hostile: it was diffi-
cult for a Jew to be accepted as an undergraduate and next to impossible as a
graduate student; this situation further deteriorated at the end of 1960s when
undergraduate studies were closed as well (all the way to the 1970s when
I avoided Moscow and went to Novosibirsk University). The mathematical
6: The Russian school 225
students of Kolmogorov were particularly affected. For example, Sinai was not
accepted for graduate studies after the committee failed him in Marxist philos-
ophy: Kolmogorov was present at the exam but did not interfere (Ya.G. Sinai,
private communications, 2009–2010). Kolmogorov then negotiated for Sinai a
second attempt which succeeded. Turbulence researchers had it easier thanks
to the Institute of Geophysics and later to Obukhov’s Institute. Remarkably,
that quite unusual director did not even fire refuseniks as was required by a di-
rect Party order. Obukhov was universally admired by his co-workers despite
his sometimes harsh style (its acceptance was softened by a common agree-
ment that he was invariably the smartest person in the room, best equipped to
“rule the Earth’s atmosphere”).
Andrei Monin was appointed “to rule the oceans” in 1965 when he was
made director of the Institute of Oceanology. He had not only been a devout
Party member since 1945, but a high-level if somewhat reluctant (Golitsyn,
2009) functionary in the Party hierarchy as an instructor and then the deputy
chairman of the Science Department of the Party Central Committee. While the
Academy kept some marginal degree of independence in electing (or rejecting)
new members, the Department was the body which actually set the policy,
appointed directors, issued permits for visits abroad etc. During the 1950s,
the Department was particularly hostile towards “the group of non-communist
scientists led by Tamm, Leontovich and Landau” (Monin, 1958).
Around 1960–61, Obukhov decided at last to address Landau’s remark on
dissipation rate fluctuations and initiated theoretical and experimental inves-
tigations into the subject (Golitsyn, 2009). Systematic measurements of wind
velocity fluctuations were made by Gurvitz (1960). The calculations of the
fluctuations of the energy dissipation rate , assuming quasi-normality, was
done by Obukhov’s student G.S. Golitsyn, who later extended the approach
of KO41 to the analysis of the dynamics of planetary atmospheres (Golitsyn,
1973) and succeeded Obukhov as director of the Institute. Experimental data
had shown that fluctuations were much stronger than the theoretical estimates.
Strong non-Gaussianity of velocity derivatives was also observed before by
Batchelor and Townsend. Looking for an appropriate model for the statis-
tics of , Obukhov turned to another seminal paper of 1941 by Kolmogorov
(1941d) on a seemingly different subject: ore pulverization. Breaking stones
into smaller and smaller pieces presents a cascade of matter from large to small
scales. A stone that appears after m steps is of size m , which is a product of
the size of an initial large stone and m random factors of fragmentation:
m = e1 . . . em , where ei < 1. If those factors are assumed to be indepen-
dent, then log m is a sum of independent random numbers. As m increases, the
statistics of the sum tends to a normal distribution with the variance propor-
tional to m. In other words, multiplicative randomness leads to log-normality.
226 Falkovich
The USSR delegation included Kolmogorov . . . , his two pre-war students M.D.
Millionschikov and A.M. Obukhov, and me – a war-years student. Such a com-
position had the flavor of Khrushchev’s liberalization (for me it was the first time
I was permitted to attend a meeting in a ‘capitalist country’).
Russians at last had a chance to meet turbulence’s great scholars from all gen-
erations. Most of the heroes of this book were present: von Kármán, Taylor,
Batchelor, Townsend, Corrsin, Saffman and Kraichnan. It is poignant to see
Kolmogorov and Kraichnan (whose names are forever linked by the 2D–3D
5/3-scaling) in the same photograph.
The new theory KO62 gives the same linear scaling for the third moment.
Attempts to estimate μ from experimental data on the variance of dissipation or
velocity structure functions give μ 0.2, so that KO62 only slightly deviates
from KO41 for n < 10 ÷ 12. Its importance must be then mostly conceptual.
The main point is understanding that the relative fluctuations of the dissipa-
tion rate grow unboundedly with the growth of the cascade extent, L/r (in
his paper, Kolmogorov credits that to Landau even though the latter’s 1944 re-
mark did not mention any scale-dependence of the fluctuations – Frisch, 1995).
That understanding opened the way to the description of dissipation concen-
trated on a measure (Novikov and Stewart, 1964), which was later suggested
to be fractal (Mandelbrot, 1974), and shown to be actually multi-fractal (Parisi
and Frisch, 1985; Meneveau and Sreenivasan, 1987). Let us stress another
6: The Russian school 227
conceptual point: the 5/3-law for the energy spectrum is incorrect despite be-
ing (outside of the turbulence community) the most widely known statement
on turbulence. Still, KO62 does not seem to be such a momentous achievement
as KO41. First, it evidently does not make sense for sufficiently high n. Second
and more important, it is still under the spell of two magic concepts of the Kol-
mogorov school: Gaussianity and self-similarity. Compared with KO41, the
new version KO62 somehow pushes these two further down the road: the new
(refined) self-similarity is local and Gaussianity is transfered to logarithms, re-
placing additivity with multiplicativity. Still, KO62 is based on the belief that a
single conservation law (of energy) explains the physics of turbulence and that
the (local) energy transfer rate completely determines local statistics. As we
now believe, direct turbulence cascades (from large to small scales) have, at a
fundamental level, nothing to do with either Gaussianity or self-similarity, even
though these concepts can help to design useful semi-empirical models for ap-
plications. There is more to turbulence than just cascades. Energy conservation
determines only a single moment (the third for incompressible turbulence). To
understand the nature of turbulence statistics, one returns to the old remark
of Friedman that the correlation functions are “moments of conservation”. In
this way, one discovers an infinite number of statistical conservation laws hav-
ing a geometrical nature, each determining its own correlation function; to this
must be added that the exponents are now measured with higher precision and
228 Falkovich
they are neither KO41, nor KO62; see for example Falkovich et al. (2001) and
Falkovich and Sreenivasan (2006).
Note in passing that the interaction between Landau and Kolmogorov was a
two-way street. We described above how Landau’s reaction to KO41 changed
the theory of turbulence. No less fruitful was Kolmogorov’s reaction to Lan-
dau’s suggestion in 1943 that, as the Reynolds number Re grows, the sequence
of instabilities leads to multi-periodic motion; that is, the attractor in the phase
space of the Navier–Stokes equation is a torus whose dimensionality grows
with Re. Superficially, this seems to be very much in the spirit of Kolmogorov’s
own 1941 argument that “at large Re, pulsations of the first order are unstable
in their own turn so that the second-order pulsations appear” (Kolmogorov,
1941a). However, Kolmogorov developed deeper insights into the onset of tur-
bulence and posed the question of whether it is possible that a continuous spec-
trum appears at finite Re. That was answered by work on dynamical systems
theory, which he started in 1953 “because the hope appeared and my spirit up-
lifted” (Stalin died). The resulting KAM theory (after Kolmogorov, Arnold and
Moser) describes which invariant tori survive under a slight change of Hamil-
tonian and forms the basis of understanding Hamiltonian chaos. Later, Kol-
mogorov initiated a great synthesis of the random and deterministic, based on
the notions of entropy and complexity, magnificently carried out by his student
Sinai and others. To overcome the natural prejudice of considering dynamic
systems as deterministic, one needs to be profoundly aware of the finite pre-
cision of any measurement and of the exponential divergence of trajectories
(Kendall et al., 1990). Kolmogorov–Sinai entropy and dynamical chaos are
fundamental to our understanding of numerous phenomena; in particular, re-
lated ideas were used later for describing the statistics of turbulence below the
Kolmogorov–Obukhov scale where the flow is spatially smooth but temporally
random; see for example Falkovich et al. (2001). In addition, Kolmogorov’s
program for the 1958 seminar included the task of developing the theory of
one-dimensional (Burgers) turbulence which was completed by Sinai and oth-
ers some 40 years later.
I find it puzzling though that Kolmogorov himself never applied his powerful
probabilistic thinking and understanding of stochastic processes and complex-
ity to quantum mechanics and statistical physics (it was done by his students
Gelfand and Sinai respectively). It seems that Kolmogorov’s direct contact
with physics was only via classical mechanics and hydrodynamics (Novikov,
2006).
Obukhov started a new chapter in Obukhov (1969) by introducing what he
called systems of hydrodynamic type and what were later known as shell mod-
els. He was inspired by the 1966 work of Arnold on the analogy between the
6: The Russian school 229
Euler equation for incompressible flows and the Euler equation for solid body
motion; see Arnold and Kesin (1998) for the detailed presentation3 . Obukhov
approximated fluid flow by a system of ordinary differential equations with
quadratic nonlinearity and quadratic integrals of motion. Since there was no
consistent way of determining the number of equations for this or that type of
flow, Obukhov initiated laboratory experiments and their detailed comparison
with computations. It is worth noting that Obukhov and his co-workers worked
on few-mode dynamic models (apparently independently of E. Lorentz) as well
as on chains intended to model turbulence cascades (Gledzer et al., 1981).
We conclude this section by referring the reader to the magnificent opus by
Monin and Yaglom where much more can be found on KO41, KO62 and many
other subjects including the field-theoretical approaches of Edwards, Kraich-
nan and others. “If ever a book on turbulence could be called definitive”, de-
clared Science in 1972, “it is this book by two of Russia’s most eminent and
productive scientists in turbulence, oceanography, and atmospheric physics.”
As does the presentation here, it stresses the physics of KO41 and KO62, but
also makes it clear that the theory in its entirety is definitely that of mathe-
maticians. The mathematical foundations were laid before and after 1941 in
the works of Kolmogorov, Obukhov, Gelfand, Yaglom and others. A complete
analysis of stationary processes using the Hilbert space formulation was done
in 1941. Considerable work was done on spectral representations of random
processes; subtle points of legitimacy and convergence were cleared for the
Fourier transform and other orthogonal expansions for translation-invariant
random functions, which physicists take for granted without much thought.
Part 2 of the Monin–Yaglom book was finished in 1966 and published in 1967,
in time to cite the first 1965 paper of Zakharov on wave turbulence. That is the
subject of the next section.
part of the story, Landau himself didn’t work on the theory of developed tur-
bulence, despite his firm belief that the problem belongs in physics. In the
1950s, he was interested in plasma physics and steered in this direction the
young Roald Sagdeev, who went to work in the theoretical division of the Rus-
sian project on controlled thermonuclear fusion. Plasmas are subject to var-
ious instabilities and practically always are turbulent. Inspired by the works
of David Bohm on an anomalous diffusion in plasmas (Bohm et al., 1949)
and the needs of thermonuclear fusion, the theory of plasma instabilities and
turbulence was intensely developed in Russia by B. Kadomtsev, A. Vedenov,
E. Velikhov and R. Sagdeev during the 1950s and 1960s. Sagdeev’s uniform
approach to plasma hydrodynamics (extended then to other continuous media)
was a trademark of the Landau school: at first all dynamical equations of con-
tinuous media were supposed to be written in a canonical Hamiltonian form,
then particular solutions are found and their stability analyzed, then perturba-
tion theory applied to the description of random fields.
To carry on this project, a most unlikely figure appeared: a student expelled
for a fistfight from the Moscow Energy Institute. Vladimir Zakharov was born
in Kazan in 1939 to the Russian family of an engineer. When at elementary
school, he did well and had a slight burr so was considered a Jew by his peers
– an experience conducive to an early formation of personal independence.
Zakharov knew Sagdeev first as a friend of his older brother and he met him
in the Energy Institute where Sagdeev was teaching physics part-time. After
expulsion, Sagdeev brought Zakharov to G. Budker who led parallel experi-
mental projects in two fields (high energy and plasma physics) and two cities
(Moscow and Novosibirsk). In 1957, a new scientific center was created some
3500 kilometers east of Moscow. In 1961 Budker convinced Sagdeev and Za-
kharov to leave Moscow and come to that new center in Novosibirsk. Sagdeev
was to lead the plasma physics department in the newly established Nuclear
Physics Institute (now the Budker Institute) while Zakharov was admitted to
Novosibirsk University, leaving all his troubles behind and starting a new life
in the brave new world of hastily built barrack-style buildings in the middle of
the taiga.
Note in passing that Zakharov’s poetry was published by the main Russian
literary magazines, included in anthologies etc.: there exists a bilingual book
with English translations (Zakharov, 2009a). As a scientist, he grew up inside a
strongly interacting community of physicists and mathematicians, particularly
influenced by M. Vishik, V. Pokrovsky and G. Budker. Zakharov succeeded in
making important advances in the directions usually considered far apart: in-
tegrability and exact solutions on the one hand, and turbulence on the other. In
particular, he was able to find turbulence spectra as exact solutions.
6: The Russian school 231
The right-hand side is a collision term very much like in the Boltzmann kinetic
equation. The idea of phonon collisions was introduced in Peierls (1929); the
collision term was used by Landau and Rumer in 1937 to calculate sound ab-
sorption in solids (Landau and Rumer, 1937). In the article (Zakharov, 1965)
submitted on 28 October 1964, Zakharov took this equation (which he learnt
from Camac et al., 1962) and asked if it has a stationary solution different
from the equilibrium Rayleigh–Jeans distribution nk = T/ωk . Inspired by
the Kolmogorov–Obukhov spectrum he set to look for a power-law solution
nk ∝ k−s . Taking first the case of acoustic waves when the coefficients are rel-
atively simple, ωk ∝ k and W ∝ kpq, Zakharov first checked that the collision
integral tends to −∞ when s → 4 and to +∞ when s → 5, so it has to pass
through zero at some intermediate s. He then bravely substituted s = 4.5 and
obtained for the collision integral 18 gamma-functions that promptly canceled
each other. The first Kolmogorov–Zakharov spectrum was born. Still, it took
some time for Zakharov to appreciate that the spectrum indeed describes a cas-
cade of energy local in k-space and is an exact realization of KO41 ideas: by
checking the convergence of the integrals in the kinetic equation at p → 0
and p → ∞, one can directly establish that the ends of the inertial interval
really do not matter, in contrast with hypotheses about turbulence of incom-
pressible fluids. Interestingly, the position of the Kolmogorov–Zakharov ex-
ponent exactly in the middle of the convergence interval is a general property
now called counterbalanced locality: the contributions of larger and smaller
scales are balanced on the steady spectrum (Zakharov et al., 1992). In 1966,
Zakharov submitted his PhD thesis (under the supervision of Sagdeev) which
232 Falkovich
was devoted to waves on a water surface (Zakharov, 1966; Zakharov and Filo-
nenko, 1966). There, one finds a complete description for the case of capillary
waves: obtaining the spectrum from the flux constancy condition, checking lo-
cality as integral convergence and showing that this is indeed an exact solution
by using conformal transforms that were independently invented by Kraichnan
for his direct interaction approximation at about the same time (see Chapter
10). Then Zakharov √takes on the turbulence of gravity waves whose disper-
sion relation, ωk ∝ k, does not permit three-wave resonances. In this case,
the lowest possible resonance corresponds to four-wave
scattering. Every act
of scattering conserves
not only the energy, E = ωk nk dk, but also the wave
action, Q = nk dk, which can be also called ‘number of waves’. The sit-
uation is thus similar to the two-dimensional Euler equation which conserves
both the energy and squared vorticity. In his thesis, Zakharov derives two exact
steady turbulent solutions of the four-wave kinetic equation, one with the flux
of E and another with the flux of Q. He then argues that the energy cascade
is direct, i.e. towards small scales. While Zakharov derived an exact solution
that describes an inverse cascade, he didn’t explicitly interpret it as such (he
also gave some arguments in the spirit of Onsager about transport of Q to
large scales in a decaying turbulence). After Kraichnan’s 1967 paper was pub-
lished and brought to his attention by B. Kadomtsev, Zakharov realized the
analogy and interpreted the spectra he derived as a double-cascade picture. In
1967, he published the direct-cascade spectrum for Langmuir plasma turbu-
lence (Zakharov, 1967), where the inverse-cascade spectrum was obtained in
1970 by E. Kaner and V. Yakovenko from the Kharkov branch of the Landau
school (Kaner and Yakovenko, 1970). Note that the hypotheses Kolmogorov
formulated in 1941 are true for Zakharov’s direct and inverse cascades of
weak wave turbulence and are probably true for Kraichnan’s inverse cascade
in incompressible two-dimensional turbulence as well. In 2006, Kraichnan
and Zakharov were together awarded the Dirac medal for discovering inverse
cascades.
The early years of the Novosibirsk scientific center were also the years of
Khrushchev’s brief thaw. At that time, there was probably no other place in
the country where academicians, professors and young students lived in such
a close proximity and had so few barriers for scientific and social interaction.
A small town in the forest, “Siberia’s little Athens”, was for a while allowed
some extra degrees of freedom. That was about to end in 1968 when Zakharov
became one of the initiators and signatories of the open letter to the Party
Central Committee protesting the arrests of dissidents. But Brezhnev’s time
was vegetarian compared with that of Stalin: Zakharov’s only punishment was
a ban on foreign travel, then thought to be forever.
6: The Russian school 233
6.5 Epilogue
In twenty years no one will know what
actually happened in our country.
A.N. Kolmogorov, 1943 (Nikol’skii, 2006)
Our story ends (somewhat arbitrarily) in 1970. What followed – study of
shell models by Obukhov’s school, development of the weak turbulence the-
ory by Zakharov’s school, works on Lagrangian formalism and zero modes –
deserves a separate essay which may be too early to write.
In his old age Kolmogorov suffered from Parkinson’s disease and from an
eye illness that made him almost blind. Nevertheless, he tried to work practi-
cally until the end, always surrounded by his former students, who also took
turns in providing necessary help. Landau was seriously injured in an automo-
bile accident in 1962; he was 59 days in a coma and survived with the help of
his students and colleagues in the country and abroad; he lived for six more
years but was unable to work. Obukhov and Yaglom worked until their last
days, monuments of unaging intellect.
Kolmogorov died in 1987 and Obukhov in 1989. That year, the Berlin Wall
fell, the Soviet Union opened the gates and disintegrated within two years. An
exodus of scientists brought substantial parts of the Kolmogorov, Obukhov and
Landau schools to the West. These schools then turned international but also
weakened their links to Russia and started to lose their distinct Russian spirit.
Under one of the most oppressive regimes in the twentieth century, in the
country which lost most of its educated class to emigration, civil war and ter-
ror, and which was often plagued by war, diseases, poverty and hunger, great
mathematical and physical schools flourished. Scholars raised in these schools
had a specific code of behavior. Long corridor chats were the most effective
forums of exchanging the latest ideas. Most seminars had no sharply defined
ends, some even had no clear beginning, as the people came before to discuss
related subjects (Ya.G. Sinai, private communications, 2009–2010). Everyone
worked inside a coherent group of people familiar with the details of each
other’s work (a downside was that some people never had much incentive to
learn how to present their results to the outside world). Much has been said
234 Falkovich
about the aggressive style and interruptions at Russian seminars. One must
however understand the context: in a life which was a sea of official lies, doing
science was perceived as building a small solid island of truth; even uninten-
tional errors risked decreasing the solid ground on which we stand. Landau
used to say: “An error is not a misfortune, it is a shame”. One is reminded of
monastic orders that preserved and advanced knowledge during the dark ages
(though in other respects, most Soviet scientists weren’t monks). A more pro-
saic reason that bonded people within a school was an impaired mobility of
scientists – recall that both Kolmogorov and Landau had a postdoctoral period
abroad, a possibility denied to most of their students4 . Still, the main attraction
of the schools was the personalities of the leaders.
By radically restricting creative activities, a tyrannical society channeled
the creative energy into the narrow sector of natural sciences and mathematics.
Russian society is more open now, and the choice of science as one’s occupa-
tion is rarely placed in the context of morality. Will we ever again be blessed
with universalist geniuses of the caliber of Kolmogorov and Landau?
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7
Stanley Corrsin
Charles Meneveau and James J. Riley
238
7: Stanley Corrsin 239
Figure 7.1 A 22-year-old Stan Corrsin in 1942 at the Graduate Aeronautical Lab-
oratories, Caltech (GALCIT). The verso of the photograph states, in Corrsin’s
handwriting “Man doing research”. The note is addressed to his mother, and adds
for reassurance: “I’m really not so thin, it’s just the light”. Photograph courtesy of
Dr. Stephen D. Corrsin.
program in fundamental and applied fluid dynamics research had been devel-
oping at Caltech’s Guggenheim Aeronautical Laboratory (GALCIT), under the
direction of Theodore von Kármán and Clark B. Millikan. Hans Liepmann,
who in 1939 had just been hired by von Kármán after finishing his PhD in
Zürich, became Corrsin’s main academic adviser. At the time, Liepmann had
begun experimental studies on boundary layers, transition to turbulence, and
various turbulent shear flows. Corrsin began working in Liepmann’s laboratory
and distinguished himself for his dexterity in experimental science.
His first project at Caltech, which became his thesis in partial fulfillment of
the requirements of Aeronautical Engineer, dealt with measurements of the de-
cay of turbulence behind various grids. The subject of isotropic turbulence was
in the air: on a visit to Caltech in 1936 and 1937, Leslie Howarth had collab-
orated with von Kármán and developed the equation for two-point correlation
functions in decaying isotropic turbulence (von Kármán and Howarth, 1938).
Corrsin’s initial experiments provided data on the decay of standard devia-
tions of two of the three turbulent velocity components behind three types of
grids. More will be said later about ingenious measurement techniques of the
time. In a photograph taken in the laboratory (Figure 7.1), he is seen reading
a manometer, pencil tucked behind his ear. The results of the experiments, it
turns out, were rather inconclusive. It was unclear whether turbulence was, or
7: Stanley Corrsin 241
was not, observed to be sufficiently isotropic, or what the decay rate was. At the
end of the thesis, which was never published in journal format, Corrsin writes:
“The [. . . ] conclusions are rather tentative; it is hoped that more certain results,
and in particular the reasons for them, will come out of further investigation”.
Thus were laid the early seeds for Corrsin’s work elucidating the fundamen-
tals of isotropic turbulence. He completed the Masters thesis in 1942 (Corrsin,
1942) but, as further described in §7.6, his definitive experiments on decaying
isotropic turbulence would have to await over two decades to become reality.
Corrsin then began to work in earnest towards his doctoral research and this
work led to important, and no longer tentative or uncertain, results. The 1930s
had seen initial developments in documenting basic properties of what are now
known as the ‘canonical’ turbulent shear flows. By applying Prandtl’s bound-
ary layer concept to turbulent shear layers, thus assuming that they become
asymptotically thin (although many never do), simplified parabolic equations
had been developed describing the mean velocity in plane and round wakes and
jets, in mixing layers, and in turbulent boundary layers along walls. The use
of similarity variables and the eddy-viscosity assumption with local velocity
and length-scales led to further simplifications. A series of experiments, most
notably the measurements of mean velocity profiles in wakes by Townsend,
had already begun to establish the validity and limitations of this approach.
The popular textbooks by Townsend (1956), Hinze (1959) and Tennekes and
Lumley (1972) provide excellent accounts of the accomplishments of that era.
By the early 1940s, after several of the canonical shear flows had been mea-
sured and documented in terms of mean velocity and Reynolds stresses, at-
tention began to turn to the distribution of scalar fields. Examples of scalar
fields include the temperature or the concentration associated with species
being transported by turbulence. They are termed ‘passive scalars’ if they
do not affect the motion, which therefore excludes cases with buoyancy ef-
fects that often occur in geophysical flows, or with strong volumetric expan-
sion that accompany combustion. In the early 1940s, not much was known
about distributions of passive scalars in turbulent shear flows. Of natural im-
portance to propulsion and mixing, the turbulent jet was of great interest to
von Kármán, Millikan, and the National Advisory Committee for Aeronau-
tics (NACA). Thus, Corrsin’s doctoral research project was on detailed mea-
surements of the velocity and temperature fields in round jets. The work was
closely followed by von Kármán, Millikan and supervised by Liepmann.
Financial support was provided by NACA.
Corrsin went to work and designed and, with helpful laboratory techni-
cians, built the experiment out of an existing, open-return 6 1/2 feet diameter
wind tunnel. It was retrofitted with a contracting nozzle unit near its exit, thus
242 Meneveau & Riley
creating a jet. Electrical units upstream would provide heating for the air, and
warm air was also ducted outside the nozzle to improve uniformity of the tem-
perature profile exiting the jet. In characteristic style, he writes:
That this scheme was not completely successful can be seen from the temper-
ature distribution measured at the mouth. It did represent, however, a distinct
improvement over the wooden nozzle first tried.
charged with the instruction of Navy pilots and other military personnel on the
basics of aerodynamics, and thus he remained actively involved in teaching
at Caltech even after the end of the war, all the while writing his doctoral
dissertation. His doctoral degree was awarded in May 1947, for his two-part
dissertation entitled I. Extended Applications of the Hot-Wire Anemometer; II.
Investigations of the Flow in Round Turbulent Jets (Corrsin, 1947).
At Caltech Corrsin had met a young woman, Barbara Daggett, who would
become his wife. She was originally from the Los Angeles area, and worked
as part of the Caltech administrative staff. They were soon married. Then
came the call from Johns Hopkins University to join its faculty as Assistant
Professor.
in 1948, and would remain at the same institution for the rest of his life. The
Corrsins lived several miles north of the Johns Hopkins campus in the subur-
ban, almost rural, Towson area, in a house they bought in 1953. They had two
children, Nancy E. Corrsin and Stephen D. Corrsin. Meanwhile, his parents
Anna and Herman would retire to the state of Florida.
The Johns Hopkins Aeronautics Department continued to grow in the fol-
lowing years. In 1950 Leslie G. Kovasznay was appointed to the faculty, fol-
lowed by Mark Morkovin and Robert Betchov who were appointed as re-
search scientists (Hamburger Archives, JHU, 2009). In one of the most visible
early contributions of the department, the faculty participated in a number of
episodes of the critically acclaimed television series The Hopkins Science Re-
view. Episodes included Flight at Supersonic Speeds, which aired on 2 Febru-
ary 1949, and a series Man Will Conquer Space, that aired in October 1952
and featured Wernher von Braun as the guest.
With him from Caltech, Corrsin brought Mahinder Uberoi, who had re-
ceived his Masters degree there in 1946. Uberoi moved to Baltimore to be-
come Corrsin’s first doctoral student. They also brought along a hot-air jet
unit that they had built at Caltech (Corrsin and Uberoi, 1950). It was more
compact than the original wind tunnel add-on facility Corrsin had used earlier
(Corrsin, 1947). The unit consisted of a centrifugal blower pushing air through
a horizontal chamber with heating coils. A 90 degree elbow then turned the
flow upward and, after passing through further screens and a smooth contrac-
tion, the 1 inch diameter heated jet emanated up into the laboratory. They also
brought hot-wire anemometry from California. The experiment was set up in
a laboratory in the Aeronautics Building (later known as Merryman Hall), an
unassuming, grey concrete block building next to a wooded hillside at the edge
of campus.
The velocity and temperature measurements from this experiment are de-
scribed in some detail in a new NACA report (Corrsin and Uberoi, 1951).
While this report was in press, Corrsin had performed initial analysis of the
velocity data and published a rather remarkable brief communication in the
Journal of the Aeronautical Sciences (Corrsin, 1949). This would be his third
publication in 1949 and since joining Johns Hopkins. [He had written two other
short notes published earlier in 1949, with Kovasznay on a hot-wire length
correction (Corrsin and Kovasznay, 1949) and on transformation formulae be-
tween one and three-dimensional scalar spectra (Kovasznay et al., 1949).] He
begins the short Journal of Aeronautical Sciences note with the statement that
“The most significant idea contributed to the problem of turbulent shear flow in
many years is the hypothesis of local isotropy due to Kolmogorov” (a statement
7: Stanley Corrsin 245
of remarkable longevity still valid, some would say, to this day). He goes on to
present a plot of the correlation coefficient between band-pass filtered signals
of stream and cross-stream components (the normalized cross-spectrum). The
data were taken at the maximum shear region in the jet using X-hot wires and
the voltage readings from both wires were band-pass filtered using analog filter
banks. The difference of their mean-square voltages, evaluated using vacuum
thermocouple units, are proportional to the co-spectrum. The correlation coef-
ficient as function of frequency decays rapidly to zero at the high frequencies
characteristic of small-scale motions. Statistical isotropy demands that the two
fluctuating components be uncorrelated at high frequencies. Corrsin’s obser-
vation, therefore, gave significant and direct support to the notion that small
scales in turbulence are isotropic, in a flow where the large scales clearly are
not isotropic. This would be his first of many direct experimental examinations
of theories pertaining to the small-scale structure of turbulence.
Taking the Fourier transform of the equation for scalar correlations, Corrsin
(1951b) derived the spectral equation for the temperature spectrum, G(k). He
derived the solution in the case of small Péclet number, where the nonlinear
transfer terms are negligible. In this case the problem reduces to the heat equa-
tion with its characteristic exponential decay as ∼ exp(−2γtk2 ), where γ is
the scalar diffusion coefficient and k is the magnitude of the wavenumber. At
low wavenumbers, Corrsin showed that the spectrum grows as k2 , a result in-
timately related to the existence of the invariant N mentioned above. For the
intermediate range of wavenumbers, he went on to generalize the Kolmogorov
(1941) approach using dimensional arguments.
Of crucial relevance are the rates of dissipation of kinetic energy
∞
= 2ν k2 E(k)dk
0
k = L−1
G = LT 2
θ = T 2 T −1
= L2 T −3 ,
where L=Length, T =temperature, and T =time. Hence, the only possible ar-
rangement is
G(k) = Aθ −1/3 k−5/3 , (7.3)
1951) is only one of the very few publications where measured scalar spectra
are reported. Figure 7.2 shows a reproduction of the measured spectra in the
maximum shear region in the heated round jet. The velocity and scalar spectra
display similar decay, not inconsistent with −5/3; but with the scatter of the
data, as well as slightly different results obtained on the jet centerline where
the scalar spectra were a bit flatter, Corrsin never argued that the data really
supported his predictions of −5/3 scaling. The subject of the scaling of power
spectrum continued to elicit many further studies over the subsequent decades,
including several by his students and junior collaborators (Kistler et al., 1954;
Mills et al., 1958; Sreenivasan et al., 1980; Sreenivasan, 1996).
248 Meneveau & Riley
of Taylor’s theory, and an estimate for the turbulent heat transfer coefficient
(turbulent diffusivity) from the Lagrangian analysis. In addition Eulerian and
Lagrangian micro-scales were measured, and a correction and generalization
of a theoretical expression of Heisenberg (1948) relating them was made.
After these ground-breaking results obtained in the late 1940s and early
1950s, institutional challenges called on Corrsin to lead the Mechanical En-
gineering Department. Thus in 1954 he moved from the Aeronautics Depart-
ment in Merryman Hall on the campus periphery, to the more centrally located
Maryland Hall where Mechanical Engineering was housed. As chairman of
ME, he would be expected to devote some part of his time to administrative
duties such as dealing with faculty hirings, teaching assignments, and manag-
ing the departmental infrastructure. There were teaching laboratories and halls
containing large machinery, steam engines and, as remarked by John Lum-
ley, other “examples of man’s ingenuity” (Lumley and Davis, 2003). Lum-
ley had arrived at Johns Hopkins in 1952, and would become Corrsin’s third
PhD student after Uberoi and Kistler. He recalls that one of Corrsin’s major
efforts was to modernize the department by removing the old machines and
replacing them with wind tunnels. This did not occur without some resistance
by more tradition-bound alumni and administrators in the Dean’s office at the
time.
A move away from engineering towards engineering science was to become
one of the hallmarks of mechanics at Johns Hopkins. This move culminated
with the closure of the engineering school altogether and the creation of the
Mechanics Department in 1960. As recalled by Phillips (1986), Corrsin hap-
pily relinquished the chairmanship to George Benton and had a rubber stamp
made that said “let George do it”. He would use it with gusto on the incessant
paperwork that could now proceed to be dealt with somewhere else.
With his family, he would continue to live in their house in Towson, the calm
suburban area north of the city. Every morning he would drive the children to
their school along tree-lined Charles Street, on his way to the Homewood cam-
pus. He never left Baltimore for extended periods of time. He did not absent
himself for sabbatical leaves, preferring to host extended visitors rather than
being a visitor himself. More will be said in §7.9 about the extraordinary en-
vironment at Johns Hopkins at that time, and about Corrsin’s role in shaping it.
He and his students continued to work on the diffusive properties of tur-
bulence through the next three decades. Taylor’s theory, which gives a pre-
diction of the Eulerian heat transfer coefficient, is expressed only in terms of
Lagrangian quantities, namely, the Lagrangian mean-square displacement Y 2
and the Lagrangian velocity time autocorrelation RL (τ) = V(t)V(t + τ), where
V is a Lagrangian velocity and τ the time separation. Almost all data are taken,
250 Meneveau & Riley
however, in an Eulerian frame. This led Corrsin to carefully define and then
address the so-called Euler–Lagrange problem. That is, given the statistical
properties of the Eulerian velocity field, say u(x, t), what are the statistical
properties of interest of the Lagrangian field, in particular the mean-square
displacement and velocity time autocorrelation. Corrsin (1959; and later in
more detail, 1962b) pointed out the exact relationship between the Lagrangian
velocity time correlation, RLi j (τ), the joint Lagrangian displacement/Eulerian
velocity probability density, and the Eulerian space-time velocity autocorre-
lation, RE i j (ζ, τ) = ui (x, t)u j (x + ζ, t + τ). Assuming that, for large time sep-
arations, the Eulerian velocity field becomes independent of the Lagrangian
displacements ζ field, he then obtained
RLi j (τ) = pY (ζ, τ)RE i j (ζ, τ) dζ , (7.4)
where Ψ(σ) is the auto-correlation φ(s)φ(s + σ), and φc is the constant value
of interest of φ. This result can easily be extended to an iso-surface in three
252 Meneveau & Riley
Figure 7.3 Evolution of lines formed with tiny hydrogen bubbles in a turbulent
water channel flow (reprinted from Corrsin and Karweit, 1969).
dimensions. The work was further extended by Corrsin and Phillips (1961) to
include contour lengths and surface areas of multiple-valued random variables.
These results have proven very useful, in particular, in theories of turbulent
combustion, where the area of the flame surface is often directly modeled (see,
for example, Poinsot and Veynante, 2001).
Corrsin and Karweit (1969) were the first to measure the fluid line growth
in turbulence. Michael Karweit was a graduate student pursuing his Masters
degree and would remain at Johns Hopkins as a long-time junior collabora-
tor of Corrsin. Their experiment utilized a water tunnel with a test section of
dimension 8 in. square by 48 in., and approximately homogeneous turbulence
generated by a bi-plane grid of mesh size 12 in. The grid Reynolds number was
1360. They used the ‘hydrogen bubble’ electrolysis method, with a platinum
wire stretched normal to the flow to generate the hydrogen bubble lines. These
lines were photographed at various distances downstream from the wire, and
their lengths were determined using an analysis relating length to the num-
ber of cuts and the angle of the line with respect to a straight reference line
(Corrsin and Phillips, 1961). Photographs and movies from this experiment
are now used in many classes in turbulence throughout the world, and a pho-
tograph is shown in Figure 7.3. Unfortunately, because of the limitation of
the length of the water tunnel, only short-time growth of the lines could be
observed. This growth was consistent with short-time growth estimates, but
7: Stanley Corrsin 253
the measurements could not confirm the conjectured long-time growth of the
lines.
Batchelor (1952) had conjectured that, for long times in stationary, homoge-
neous turbulence, the number of eddies of each size acting to stretch the line is
proportional to the line length. This leads immediately to the conclusion that
the line will grow exponentially. This result was proven more rigorously by
Cocke (1969) and Orszag (1970). Corrsin (1972) offered a simpler geometric
proof of this result. His conclusions were weaker than the previous ones, but
without restrictions to isotropy or constant density being required.
While pointing out the problems of a fundamental nature in the use of a tur-
bulent scalar diffusivity (see below), Corrsin made theoretical and experimen-
tal estimates of this quantity. He extended Taylor (1921)’s theory to include
a homogeneous, isotropic, stationary shear flow (Corrsin, 1953) and found,
for example, for long times, the cubic dependence on time of the streamwise
dispersion, i.e.
2 du1 2 2
X12 ∼ v T 22 t3 , (7.6)
3 dx2 2
where du1 /dx2 is the uniform mean shearing of the u1 component of the veloc-
ity in the x2 direction. This result was extended by Riley and Corrsin (1974)
for the non-isotropic case. In particular, they computed the turbulent diffusivity
tensor Ki j and found it to be non-diagonal, and to depend on the mean shear
and the correlations of v1 and v2 . For example, the K11 component was found
to be
t
dū1 t
K11 (t) = RL11 (τ) dτ + τRL12 (τ) dτ . (7.7)
0 dx2 0
Riley and Corrsin (1971) also performed computer simulations of fluid particle
dispersion of homogeneous shear flows using an artificially constructed Eule-
rian flow field consisting of spatially and temporally varying Fourier modes,
with amplitudes defined so that the statistics of the flow were similar to the
laboratory measurements of Champagne et al. (1970). Their computed results
were consistent with the analysis.
Almost all turbulence models employ, at some point, a linear gradient model,
where the turbulent flux of a quantity (e.g. mass, heat, species concentration,
momentum, kinetic energy) is assumed proportional to the linear gradient of
that quantity. Corrsin would often express skepticism about closure models,
in particular the eddy-diffusivity, gradient-based models and their motivating
analogies to kinetic theory of gases. His arguments about the fundamental lim-
itations of these models have served as motivating force to many researchers
254 Meneveau & Riley
who in subsequent decades have attempted to develop more general and intri-
cate closure models of turbulence.
In an influential paper entitled Limitations of gradient transport models in
random walks and in turbulence, Corrsin (1974) presented a systematic analy-
sis of closure models. Following ideas from continuum mechanics in deriving
a relationship between the molecular flux of the quantity, the properties of the
fluid, and the space and time gradients of the quantity, he assumed a general
functional relationship between the turbulent flux of a quantity in the, say, z
direction, F̄(z), and a functional of the average quantity Γ̄ and the statistical
properties of the velocity field. He then determined the assumptions required
for the turbulent flux to be linearly related to the gradient of the average quan-
tity. With , a length scale of the turbulence, τ, the time scale, and V = /τ the
corresponding turbulent velocity scale, the necessary conditions for the lin-
ear gradient model are found to be the following, where a subscript denotes a
derivative with respect to that quantity:
(i) |F̄zzz /F̄z |2 1, i.e. the turbulent length scale should be much smaller
than the distance over which the curvature of Γ̄ changes appreciably;
(ii) τ|Γ̄tz /Γ̄z | 1, i.e. the turbulence time scale must be much smaller than
the time over which Γ̄ changes appreciably;
(iii) |z / + Vz /V| |Γ̄z /Γ̄|, i.e. the changes in the turbulence properties must
be very small over a distance for which Γ̄ changes appreciably;
(iv) |Vz /V| |z /|, i.e. the turbulent velocity must be appreciably more uni-
form than ; and
(v) the relative change in Γ̄ must be very small over the turbulent time
scale τ.
Corrsin then went on to compute these inequalities for several flows, pointing
out that
the archival literature is replete with data showing, either directly or indirectly,
for both scalar and momentum transport, that the mean gradients vary consider-
ably over distances comparable to the length scales characteristic of the ‘eddies’.
He also argued that, for the turbulent flux of a scalar γui (where γ = Γ − Γ̄ is
the scalar fluctuation and ui is the fluctuating part of the velocity vector), the
turbulent diffusivity must be considered as a second-order tensor, i.e.
∂Γ̄
γui = −Ki j . (7.8)
∂x j
He pointed out that there is no reason to assume that the diffusivity matrix is
diagonal, as assumed in many models. In fact, using estimates from various
7: Stanley Corrsin 255
sets of data, he argued that the off-diagonal terms are often comparable to the
diagonal terms.
Several years later, he and co-workers (Sreenivasan et al., 1981) revisited
these conditions and analyzed experimentally obtained turbulent heat flux and
temperature gradients across various types of homogeneous and inhomoge-
neous shear flows. They identified additional conditions and concluded that
there was a need for models based on more than just the mean field properties
of the flow.
Figure 7.4 Professor Stan Corrsin in later years explaining decaying isotropic
turbulence behind a grid in a wind tunnel, including a secondary contraction.
cross-flow heat exchanger installed before the fan. Through the heat exchanger
circulated cooling water siphoned off from a pond which, at the time, graced
the east side of campus. For years, this arrangement would cause friction with
the university’s ground maintenance personnel.
The design also included a secondary contraction which would be located
downstream of the turbulence producing grid. By forcing the initially some-
what anisotropic turbulence to go through the secondary contraction, vorticity
aligned in the streamwise direction would get amplified due to vortex stretch-
ing, and the cross-stream turbulence variance would be increased relative to
the streamwise turbulence component. Corrsin settled for a 1.27:1 secondary
contraction which would greatly reduce the initial anisotropy of the turbulence.
Years later, he would often explain the principle of the secondary contraction
on a blackboard (Figure 7.4).
Geneviève Comte-Bellot arrived to Baltimore in 1963 as a Fulbright and
postdoctoral fellow, having recently obtained her doctorate from the Univer-
sity of Grenoble working with Antoine Craya. She went to work with Corrsin
and implemented various improvements in hot-wire instrumentation and ana-
log data acquisition. In two seminal papers on the decay of isotropic turbulence
that arose from their collaboration, they presented what has become one of the
most celebrated datasets of fluid mechanics. In the first paper (Comte-Bellot
and Corrsin, 1966), they documented the performance of the secondary con-
traction in promoting isotropy of the turbulence behind the grid. They also
7: Stanley Corrsin 257
where u2 and v2 are the variances of streamwise and cross-stream velocity,
respectively, U0 is the mean velocity in the tunnel (that ranged between 10 and
20 m/s), M is the mesh-size of the turbulence-producing grid of bars (M ranged
from 1 to 4 inches), and x − x0 is the downstream distance to a virtual origin. In
the presence of the secondary contraction, the isotropy requirement u2 = v2
was met to a remarkable degree. Moreover, the data yielded decay exponents
that fell mostly in the range between n = −1.2 and n = −1.3, over more than
one decade of scaling. It was the most convincing experimental result showing
that predictions from theories leading to either a t−10/7 or a t−1 decay were not
reproduced.
The second work was published sometime later (Comte-Bellot and Corrsin,
1971) and provided a detailed analysis of measured two-point correlation func-
tions and spectra at various downstream distances from the grid. Hot-wire
probes recorded velocity signals over a wide range of frequencies. Spectra for
high frequencies were obtained using an HP wave analyzer. Since it was sensi-
tive only down to 20 Hz, lower frequencies were captured by recording the sig-
nals to tape and replaying the tapes at higher speeds later on. Measuring corre-
lation functions also involved playing back the tapes with varying time-delays.
Additional analog signal processing included band-pass filters, multipliers and
an electro-chemical integrator whose output finally corresponded to the time-
converged correlation coefficients among narrow band-pass filtered signals.
The results show that correlation functions for band-pass filtered veloci-
ties decay at time-scales commensurate with the eddy-scale highlighted by the
band-pass filtering. Also, all curves could be collapsed by an appropriate time
scale, combining effects at various scales.
Comte-Bellot and Corrsin (1971) also report, in great detail, the precise en-
ergy spectra at various times (distances) during the decay. They used the mea-
sured one-dimensional energy spectrum E11 (k, t) to deduce the radial three-
dimensional energy spectrum using the assumption of isotropy. The resulting
radial spectra E(k, t), carefully tabulated, have been used by many researchers
since to test and validate spectral closures such as eddy-damped quasi-normal
theories and, in recent decades, subgrid-scale models for large eddy simula-
tions (Moin et al., 1991). It has taken three decades for this ground-breaking
experiment to be replicated using direct numerical simulations (de Bruyn Kops
and Riley, 1998) as well as for a similar experiment to be remade at higher
258 Meneveau & Riley
Reynolds number in the same wind tunnel, this time using an active grid (Kang
et al., 2003).
The question of the dynamics of narrow-band effects in turbulence continued
to interest Corrsin for many years. He had been following the theoretical efforts
of R.H. Kraichnan, who at the time lived in relative isolation, north in the New
Hampshire woodlands. Once a year, Corrsin would travel to New Hampshire
to visit with Kraichnan and discuss turbulence. One of the central quantities
of the Kraichnan direct interaction approximation is the response function of
turbulence to a spectrally local disturbance. Partly motivated by the discus-
sions with Kraichnan, Kellogg and Corrsin (1980) performed an experiment in
which the wake of a fine wire stretched across otherwise isotropic grid turbu-
lence introduced a narrow-band disturbance. They recorded its decay and com-
pared it to the linear perturbation response predicted by Kraichnan, noting ‘fair
agreement’. Interest in the dynamics of Fourier modes also led Corrsin to con-
sider early uses of computer simulations. With J. Brasseur, a postdoctoral fel-
low at Hopkins in the early 1980s, they performed numerical experiments and
followed the time-evolution of individual Fourier modes and observed their
interactions within wave-number triads (Brasseur and Corrsin, 1987).
Towards the late 1970s and early 1980s Corrsin directed a concerted ef-
fort to study the most elemental non-isotropic turbulent flow, namely homoge-
neous shear flow in which the mean flow has a linear profile. Champagne et al.
(1970) and Harris et al. (1977) produced such a mean velocity profile by forc-
ing air flow through a set of parallel plates, each channel being associated with
a screen of different solidity. The side with larger solidity corresponds to lower
speeds due to the increased head losses suffered by the flow there. The evo-
lution of turbulence, the growth of length-scales, and the resulting anisotropy
were measured and to this day form a dataset used to calibrate turbulence mod-
els and compare to simulations.
Returning to the question of scalar transport, Tavoularis and Corrsin (1981a)
made direct measurements of the turbulent diffusivity in a homogeneous shear
flow. They used an experimental setup similar to that used in the homogeneous
shear flow experiments of Harris et al. (1977), but with the exit turbulence-
generating rods replaced with heating rods. This produced a uniform temper-
ature gradient in the cross-stream (x2 ) direction, to go along with their uni-
form velocity gradient across the same direction. Detailed measurements were
made of the velocity field and temperature field statistics, including joint tem-
perature/velocity statistics, spectra, autocorrelations, microscales and integral
scales. In particular, with dT̄ /dx2 = constant, and dT̄ /dx1 = dT̄ /dx3 = 0,
from measuring u1 θ and u2 θ they were able to determine K12 = −u1 θ/ dx dT̄
2
existing data for heated turbulent boundary layers and heated pipe flows, they
found approximate values of −2.4 and −2.1, respectively.
This work was extended by Tavoularis and Corrsin (1981b), using the same
flow field, to the case with the mean temperature gradient transverse to the
direction of the mean flow and the mean shear, i.e. dT̄ /dx3 = constant, and
dT̄ /dx1 = dT̄ /dx2 = 0. The only significant heat flux component was u3 θ (the
other two components were approximately zero by symmetry), and gave the
results that K33 = −u3 θ/ dx
dT̄
3
1.6K22 .
(Here r0 is the radial location where the mean axial velocity Ū drops to half of
its peak value.) He went on to note that
the general location of the transition region in the jet is about the same as the
location of the u /Ū maximum. This may mean that a part of the ‘turbulence’
is not due to the usual turbulent velocity fluctuations, but to actual differences
in local mean velocity at a point, as the flow oscillates between the laminar and
turbulent states.
Corrsin had discovered the intermittent layer between a laminar and a turbulent
flow which is now known to be characteristic of any turbulent flow with a free-
stream boundary (i.e. not a solid boundary) such as turbulent boundary layers,
jets, wakes, shear layers, and other related flows.
The first definitive study of the intermittent regions between a laminar and a
turbulent flow was by Corrsin and Kistler (1955), who addressed such regions
for a turbulent boundary layer, a plane wake, and a circular jet. Although inter-
esting experimental data were obtained in this study, one of its principal con-
tributions was conceptual, in defining and clarifying the overall processes in-
volved. The first issue is how to distinguish the turbulent and the non-turbulent
regimes. Corrsin and Kistler realized that it was not the random motion that
distinguished the turbulent region, since the flow in the laminar region was also
quite random. They concluded that the characterizing feature of the turbulent
region was its high vorticity, compared to the essentially irrotational flow of
260 Meneveau & Riley
the non-turbulent region. Thus they concluded to apply “the word ‘turbulent’
to random rotational fields only”.
They surmised that the rotational turbulent region must propagate into the
non-turbulent region, much as “a flame front propagates through a combustible
mixture”. From the vorticity equation they reasoned that
the random vorticity field . . . can propagate only by direct contact, as opposed to
action at a distance, because rotation can be transmitted to irrotational flow only
through direct viscous shearing action. This assures that . . . the turbulent front
will always be a continuous surface; there will be no islands of turbulence out in
the free stream disconnected from the main body of turbulent fluid.
Corrsin and Kistler reasoned that a very thin layer, which they called the
laminar superlayer, separated the turbulent and non-turbulent regions. The tur-
bulent side was characterized by strong vorticity amplification by vortex
stretching, while the superlayer itself was characterized by viscous diffusion
of vorticity across this layer. From simple physical/mathematical arguments,
they concluded that the superlayer was very thin, with a width on the order of
the Kolmogorov scale.
In order to address the intermittency of turbulence in the flow, Corrsin and
Kistler followed Townsend (1948) and defined the intermittency γ as “the frac-
tional time spent by the (fixed) probe in the turbulent fluid”. Experimentally
the intermittency γ was determined by electronically differentiating the hot-
wire signal for the axial component of the velocity, then rectifying, smoothing
and clipping the resulting signal. A signal discriminator was used to determine
whether the resulting signal was strong enough such that the region was tur-
bulent; this signal discriminator was set by comparing the results of the signal
output to a visual oscillogram output. In addition to the usual measurements
of the velocity statistics, they were able to measure the intermittency γ and
the position of the front Y as functions of time and downstream coordinate x.
They were thus able to determine the intermittency γ, which is, in terms of
Y, γ(y) = prob{y ≤ Y(t) ≤ ∞}. In addition, they could determine the average
position of the turbulent front, Ȳ, and its standard deviation σ = {(Y − Ȳ)2 }1/2 ,
which is a measure of the width of the intermittent zone, which they termed
the wrinkle amplitude of the turbulent front.
From their data and using theoretical arguments, they found that the rate of
increase of the wrinkle amplitude of the turbulent front was roughly predicted
by Lagrangian analysis as σ(x) 2(v /Ū)v T L , where v and T L are the local
Lagrangian velocity fluctuation and velocity integral time scale, respectively.
They also found that the downstream growth of the turbulent front, as measured
by Y, was proportional to the growth of the shear-layer thickness.
7: Stanley Corrsin 261
Corrsin and Kistler drew two important additional conclusions from their
study. First, they concluded “that the presence of the turbulent front with its at-
tendant detailed statistical properties will have to be included in basic research
on turbulent shear flows with free-stream boundaries”. Secondly, they specu-
lated that, in considering a scalar (e.g. heat, mass) in the flow for Prandtl and
Schmidt numbers not much smaller than unity, “the front should apply equally
well to heat or chemical composition. Oscillographic observations . . . in a hot
jet show a temperature fluctuation intermittency, presumably coincident with
the vorticity intermittency”.
Corrsin saw indications of outer intermittency in many other fluid dynami-
cal systems. In a noteworthy interview in Sports Illustrated (Terrell, 1959), he
was asked to explain the mechanism underlying the so-called ‘knuckle ball’.
It was the hallmark of Hoyt Wilhelm, a then famous pitcher for the Baltimore
baseball team, the Orioles. Hoyt could throw a ball that would then move in
unpredictable trajectories, thus confusing the opposing team’s batter. A photo-
graph in the article shows Corrsin in front of the blackboard with a sketch of
the flow-field at the rear side of a baseball during flight. A jagged boundary
line encloses the separated turbulent region. It is used to show that the unpre-
dictable trajectories of the knuckle ball can be due to slight changes in lift and
drag forces associated with the complicated geometry of the separated region.
Quoting from the article:
If the separation line was perfectly straight, the ball would go straight, for the
pressure forces would be even. But since the separation line is highly irregular,
so is the course of the ball. And since the separation line is constantly shifting
and changing . . . the course of the knuckle ball can change direction several times
in flight.
Realizing that information about the shape of the fine-scale structures might
eventually help understand the physical processes related to energy transfer
to these scales, Kuo and Corrsin (1972) then attempted to determine the geo-
metric character of the structures. Using the measurement technique of two-
position coincidence functions for the presence of velocity fine-structure, they
tried to distinguish the structures as being ‘blobs’, ‘rods’, or ‘slabs’. Again hot-
wire anemometer measurements were made in nearly isotropic turbulence. In
order to determine the geometry of the structures, Kuo and Corrsin developed
mathematical, geometric models for each structure; these models predicted, for
each assumed structure, the simultaneous detection event rate as well as the si-
multaneous intermittency factor. Comparisons of the experimental results for
these quantities to the predictions of the models then allowed the determination
of the type of fine-scale structures.
Their tentative conclusion was that the fine-scale regions are more rod-like
than blob-like or sheet-like. This implies a tendency for slightly ‘stringy’ struc-
tures, which may overlap with each other. Two other classes of structures were
not eliminated by the measurements, ribbon-like structures, and a mixture of
blobs and rods. Kuo and Corrsin suggested coincidence measurements using
three or more probes to help determine among these alternatives. They also
suggested using similar models for fine-scaled scalar fields to help distinguish
the structures. These detailed results motivated many subsequent publications
by other researchers on the intermittency statistics of turbulence, as well as an
influential paper by Kraichnan (1974). He dealt with an analysis of the energy
cascade along wavenumber bands arranged in octaves in an effort to provide a
possible dynamical explanation for the spatial concentration of energy fluxes
in smaller and smaller subregions of the flow during the cascade. A number of
subsequent developments are recounted in some detail in the book by Frisch
(1995).
heat release on the fluid properties so that, in particular, the fluid density, the
reaction-rate coefficients, and diffusion coefficients remain constant. Further-
more, he assumed that all of the reactant species concentrations but one were
in great excess, so that the concentration of only the latter, say Γ, changes sig-
nificantly in time. Finally, he assumed that the chemical reaction rates were of
simple power-law form, i.e. proportional to Γn , and considered the cases where
n = 1 or n = 2. Therefore, Γ satisfies the following convection–diffusion–
reaction equation:
∂Γ ∂Γ
+ ui = D∇2 Γ − Φ(Γ) , (7.10)
∂t ∂xi
where Φ(Γ) = kn Γn , kn is the reaction-rate constant, D is the molecular diffu-
sivity of Γ, and n is either 1 or 2.
where γm2 (t) is the solution for the nonreacting case (k1 = 0). Therefore, he
found that the effect of the first-order chemical reaction is to cause exponential
decay in both Γ̄ and γ2 .
In a subsequent paper (Corrsin, 1961a), he focused on the spectral behavior
of γ for first-order reactions. Using extensions of the spectral cascade argu-
ments of Onsager (1949), and the mixing theories of Batchelor (1959) and
Batchelor et al. (1959), he derived the following expressions for the energy
spectrum of γ, say G(k), for three different spectral subranges:
! "
(i) inertial–convective subrange, k /νD2 1/4 if ν/D 1, or k
! "
/D3 1/4 if ν/D 1
1 θ∗ 2/3 1
G(k) . (7.15)
3 D k5/3 [Dk2 + k1 ]2
! "
Here B and N are constants determined from the analysis, and kB = /νD2 1/4
is the Batchelor wave number. It is easy to see the effects of the reaction rate
on the spectra. For example, in the inertial–convective subrange, where the
spectrum is proportional to γ −1/3 k−5/3 , the effect of chemical reaction is given
by the factor exp(3k1 −1/3 k−2/3 ). Corrsin points out that, for wave numbers
above kc = k13/2 −1/2 , the effect of the chemical reaction on the spectral shape
is negligible.
Having obtained results for the concentration of the reactant, Γ, Corrsin
(1962a) then addressed the concentration of the product of the reaction, say
P, for first-order reactions. The product concentration for this case satisfies the
following equation:
∂P ∂P
+ ui = DP ∇2 P − k1 Γ . (7.16)
∂t ∂xi
Assuming equal diffusivities for the reactant and the product, i.e. D = DP , the
equation for P̄ is closed and the solution is easily found to be P̄ = P̄0 + Γ̄0 {1 −
exp(−k1 t)}. The equation for the mean-square fluctuations about P̄, say p2 , is
similar to equation (7.11), except that the last term is now +k1 pγ, introducing a
new unknown. Arguing that the p and γ fields are perfectly correlated, Corrsin
was then able to obtain a solution for p2 analogous to equation (7.12):
He also obtained equations for the energy spectra of the product concentra-
tion for the inertial–convective, viscous–convective, and inertial–diffusive sub-
ranges, but these results will not be repeated here. These various predictions
for both mean values and energy spectra are available to the community to
guide experiments and modeling, and have been extensively utilized.
Sirs: I observe with wonder the demands of some college students and some fac-
ulty members (Professor Jerome in “The System Really Isn’t Working,” Life, 1
November) for an education characterized by ‘relevance’. The primary weakness
of the US engineering education from its inception until World War II was its
wholehearted devotion to relevance: spending more time in shop and lab than in
classroom, the students were well trained to cope with the design, operation and
repair of the world’s machinery, possibly to improve it a bit. Then along came
new machines based on scientific principles no one guessed engineers would
ever need to know. The result: many science-trained people had to be hastily re-
cruited into doing engineering work. Their weakness in engineering principles
led to mistakes – but at least they hadn’t been given tunnel vision by a totally
‘relevant’ education. Very truly yours, Stanley Corrsin, Professor.
The department seminar series was particularly interesting and lively. Both
novice and established researchers would visit and present the results of their
work, usually with much lively discussion from the faculty. Douglas Lilly
from the National Center for Atmospheric Research discussed some new
268 Meneveau & Riley
successful example of American urban renewal, the entire inner harbor area
was redeveloped.
Besides political discussions at the coffee period, the latest technical ideas
were argued at length; if the ideas could survive a discussion at coffee some-
what intact, there was some hope for them. And, the coffee period was also a
time of camaraderie and joking, especially if Corrsin were around. Many sto-
ries from this coffee period, perhaps often somewhat embellished, would be
recalled at conference meetings for many years. And if it was an especially
good, or bad, week, some students would go off and buy a supply of wine and
cheese on Friday, and the coffee period would become an even livelier party.
By the late 1970s, however, internal disagreements in the department (by
then called the Department of Mechanics and Materials Science) that had been
developing for some time finally boiled to the surface. Moreover, the recog-
nition that engineering required a separate administrative structure led to the
closing of the department and the reestablishment of a distinct engineering
school at Hopkins. It consisted of several traditional engineering departments
which continued to nurse a distinctly strong science flavor.
Corrsin’s impact on the field has been felt beyond his own scientific contri-
butions, through those he instructed directly and through others he inspired, di-
rectly and indirectly. He had been honored by many professional awards, such
as fellowship in the American Academy of Arts and Sciences, the American
Physical Society and American Society of Mechanical Engineers, membership
of the US National Academy of Engineering, and being named the Theophilus
Halley Smoot Professor of Fluid Mechanics.
During his lifetime, Corrsin saw turbulence research progress from rudimen-
tary single-probe hot-wire measurements in small bench-top shear flow exper-
iments, all the way to large-scale turbulence measurement campaigns in large
wind tunnels, and in the atmosphere and oceans. He saw turbulence theory
develop from simple one-point and two-point closures to path-diagrammatic
methods, and to the first several successful direct numerical and large-eddy
simulations on supercomputers. His own contributions form the backbone of
our present understanding of turbulent scalar transport, of fine-scale structure
of passive scalars in turbulence, and of the phenomenon of outer intermittency.
His contributions to homogeneous turbulence, decaying and sheared, as well
as chemically reacting turbulence, are considered pivotal. Yet, what he called
“the theoretical turbulence problem” (Corrsin, 1961b) remains to this day un-
solved. The lack of systematic methodologies to make analytical predictions
for even the simplest statistical objects continues to pose a serious challenge
to the many fields where turbulence plays a crucial role. In the absence of a
definitive theoretical framework to attack the problem, Corrsin’s approach of
joyful empiricism and fundamental analysis of canonical and carefully cho-
sen example problems remains to this day the best approach to turbulence
research.
Towards the end of his life, on occasion of Liepmann’s 70th birthday in
1984, Corrsin penned the ‘Sonnet to Turbulence’. It was read at the event by
Anatol Roshko and loosely follows the form of William Shakespeare’s Sonnet
#18 and Elizabeth Browning’s poem “How do I love thee? Let me count the
ways . . . ”. In the form received from SC by William K. George (1990), it is
reproduced below as closing words about Corrsin’s life. The sonnet evokes rel-
evant turbulence phenomena and provides insights about his views on several
new approaches that were being proposed at the time. For instance, in juxta-
posing low-dimensional strange attractors versus supercomputing, he correctly
predicted that the latter would be needed due to the very large number of de-
grees of freedom of turbulence (it is useful to recall that at the time the most
powerful supercomputer was the Cray 2).
7: Stanley Corrsin 271
Acknowledgements The authors thank the many friends and colleagues who
have shared their memories. In particular they thank Stephen Davis, Michael
Karweit, Mohamed Gad-El-Hak, K.R. Sreenivasan, and William K. George,
for their comments on an early version of this chapter. They are especially
grateful to Dr. Stephen D. Corrsin, SC’s son, for his valuable recollections of
family and other important events, as well as for comments on this text.
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8
George Batchelor: the post-war renaissance of
research in turbulence
H.K. Moffatt
8.1 Introduction
George Batchelor (1920–2000), whose portrait (1984) by the artist Rupert
Shephard is shown in Figure 8.1, was undoubtedly one of the great figures
of fluid dynamics of the twentieth century. His contributions to two major ar-
eas of the subject, turbulence and low-Reynolds-number microhydrodynamics,
were of seminal quality and have had a lasting impact. At the same time, he
exerted great influence in his multiple roles as founder Editor of the Journal of
Fluid Mechanics, co-Founder and first Chairman of EUROMECH, and Head
of the Department of Applied Mathematics and Theoretical Physics (DAMTP)
in Cambridge from its foundation in 1959 until his retirement in 1983.
I focus in this chapter on his contributions to the theory of turbulence, in
which he was intensively involved over the period 1945 to 1960. His research
monograph The Theory of Homogeneous Turbulence, published in 1953, ap-
peared at a time when he was still optimistic that a complete solution to ‘the
problem of turbulence’ might be found. During this period, he attracted an
outstanding group of research students and post-docs, many from his native
Australia, and Senior Visitors from all over the world, to work with him in
Cambridge on turbulence. By 1960, however, it had become apparent to him
that insurmountable mathematical difficulties in dealing adequately with the
closure problem lay ahead. As he was to say later (Batchelor 1992):
by 1960 . . . I was running short of ideas; the difficulty of making any firm deduc-
tions about turbulence was beginning to be frustrating, and I could not see any
real break-through in the current publications.
Over the next few years, Batchelor focused increasingly on the writing of his
famous textbook An Introduction to Fluid Dynamics (Batchelor 1967), and
in the process was drawn towards low-Reynolds-number fluid mechanics and
276
8: George Batchelor 277
Figure 8.1 Portrait of George Batchelor by Rupert Shephard 1984; this portrait
hangs in DAMTP, Cambridge, the Department founded under Batchelor’s leader-
ship in 1959.
suspension mechanics, the subject that was to give him a new lease of research
life in the decades that followed. After 1960, he wrote few papers on turbu-
lence, but among these few are some gems (Batchelor 1969, 1980; Batchelor,
Canuto & Chasnov 1992) that show the hand of a great master of the subject.
I got to know George Batchelor myself from 1958, when he took me on
as a new research student. George had just completed his work to be pub-
lished in two papers the following year (Batchelor 1959; Batchelor, Howells
& Townsend 1959) on the ‘passive scalar problem’, i.e. the problem of deter-
mining the statistical properties of the distribution of a scalar field which is
convected and diffused within a field of turbulence of known statistical prop-
erties. There was at that time intense interest in the rapidly developing field of
magnetohydrodynamics, partly fuelled by the publication in 1957 of Cowling’s
Interscience Tract Magnetohydrodynamics. Batchelor had himself written a fa-
mously controversial paper “On the spontaneous magnetic field in a conduct-
ing liquid in turbulent motion” (Batchelor 1950a; see also Batchelor 1952b),
and it was natural that I should be drawn to what is now described as the ‘pas-
sive vector problem’, i.e. determination of the statistical evolution of a weak
278 Moffatt
magnetic field, again under the dual influence of convection and diffusion by
a ‘known’ field of turbulence. George gave me enormous encouragement and
support during my early years of research in this area, for which I shall always
be grateful.
My view of Batchelor’s contributions to turbulence is obviously coloured by
my personal interaction with him, and the following selection of what I regard
as his outstanding contributions to the subject has a personal flavour. But I
am influenced also by aspects of his work from the period 1945–1960 that
still generate hot debate in the turbulence community today; among these, for
example, the problem of intermittency which was first identified by Batchelor
& Townsend (1949), and which perhaps contributed to that sense of frustration
that afflicted George (and many others!) from 1960 onwards.
was to endure for the next 15 years during which the foundations of modern
research in turbulence were to be established.
George married Wilma Raetz, also of Melbourne, in 1944, and in January
1945 they set off on an epic sea voyage to Cambridge, via New Zealand,
Panama, Jamaica and New York, and then in a convoy of 90 ships across the
Atlantic to Tilbury docks in London; and thence to Cambridge where George
and Wilma were destined to spend the rest of their lives. George was then just
25 years old.
Alan Townsend came independently to Cambridge, and when he and George
met G.I. Taylor and talked with him about the research that they would under-
take, they were astonished to find that G.I. himself did not intend to resume
work on turbulence, but rather to concentrate on a range of problems – for ex-
ample the rise of large bubbles from underwater explosions, or the blast wave
from a point release of energy – that he had encountered through war-related
research activity. George and Alan were therefore left more-or-less free to de-
termine their own programme of research, with guidance but minimal interfer-
ence from G.I. – and they rose magnificently to this challenge!
I suddenly came across two short articles, each of about four pages in length,
whose quality was immediately clear”. Four pages was the normal limit of
length imposed by the USSR Academy for papers in Doklady, a limit that per-
haps suited Kolmogorov’s minimalist style of presentation, but at the same
time made it exceptionally difficult for others to recognize the significance of
his work. Batchelor did recognize this significance, and proceeded to a full
and thorough discussion of the theory in a style that was to become his hall-
mark: the assumptions of the theory were set out with the utmost care, each
hypothesis being subjected to critical discussion both as to its validity and its
limitations; and the consequences were then derived, and illuminated with a
penetrating physical interpretation at each stage of the argument.
The VIth International Congress of Applied Mechanics, which was held in
Paris in September 1946, provided an early opportunity for Batchelor to an-
nounce his findings to a wide international audience. (This sequence of qua-
drennial Congresses had been established in the 1920s through the forceful
initiative of von Kármán, Prandtl, G.I. Taylor and Jan Burgers, but had been
interrupted by the war. It was a remarkable achievement to reinstate the se-
quence so soon after the war, in the chaotic and straitened conditions that must
still have prevailed in Paris at that time. Paul Germain told me with a degree of
chagrin that the Proceedings of the Congress were duly delivered to Gautiers-
Villars following the Congress, but have never yet appeared, perhaps a record
in publication delay!)
Batchelor wisely published his contribution to the Congress in a brief com-
munication to Nature (Batchelor 1946b). In this, he described the Kolmogorov
theory and he simultaneously drew attention to parallel lines of enquiry of On-
sager, Heisenberg and von Weizsäcker. There is an interesting historical aspect
to this: both Heisenberg and von Weizsäcker were taken, together with other
German physicists, to Britain at the end of the war, and placed under house
arrest in a large country house not far from Cambridge. At their own request,
they were allowed to visit G.I. Taylor, no doubt under surveillance (this was
probably in August 1945 – see Batchelor 1992) to discuss energy transfer in
turbulent flow, and it was in subsequent discussion between Taylor and Batche-
lor that the link with the work of Kolmogorov was recognized. But as Batchelor
said: “The clearest formulation of the ideas was that of Kolmogorov, and it was
also more precise and more general”.
Kolmogorov himself gave due credit to the prior work of Lewis Fry Richard-
son, who had conceived the ‘energy cascade’ mechanism immortalized in a
rhyme reproduced elsewhere in this volume. Kolmogorov’s signal achieve-
ment, on which Batchelor rightly focused, was to identify the mean rate of
dissipation of energy per unit mass and the kinematic viscosity of the fluid ν
8: George Batchelor 283
Figure 8.2 George Batchelor with his mentor Sir Geoffrey (G.I.) Taylor, PhD
graduation day, 1948.
had left Cambridge to take the Chair of Applied Mathematics at the University
of Bristol.
Batchelor’s PhD Examiners were in fact Leslie Howarth and G.I. Taylor
himself (it being still accepted in those days that a Research Supervisor could
also act as Examiner – defending counsel one day, prosecuting the next!).
Batchelor recounted to me many years later that Howarth had asked him during
the oral examination why he had dropped the terms that lack mirror symmetry
in his discussion of the form of the spectrum tensor for isotropic turbulence,
and what these terms might represent if they were retained. These were two
questions to which Batchelor, on his own admission, was unable to give a sat-
isfactory answer! They are in fact the terms that encapsulate the ‘helicity’ of
turbulence, a concept that was to emerge two decades later and find impor-
tant application in two contexts: the Euler equations (for which helicity is a
topological invariant); and turbulent dynamo theory (see below).
The VIIth International Congress of Applied Mechanics was held at Impe-
rial College, London, in September 1948; it was at this Congress that the Inter-
national Union of Theoretical and Applied Mechanics (IUTAM) was formally
established. In the group photograph of some 200 participants, which may be
viewed on the IUTAM website, one may detect G.I. Taylor and Theodore von
Kármán in the front row, and George Batchelor and Alan Townsend perched
together on the plinth of a statue in the back row. I also recognize the un-
mistakable features of James Lighthill, Keith Stewartson and Michael Glauert,
and some others, in this photograph. The Congresses of Paris (1946), London
8: George Batchelor 285
(1948), Istanbul (1952), Brussels (1956), Stresa (1960) and Munich (1964)
provided the opportunity for international contact and exchange of ideas dur-
ing a period when international meetings in theoretical and applied mechanics
were far fewer than they are today; Batchelor attended all of them, and en-
couraged his collaborators and students to do likewise. (I attended the Stresa
Congress myself as a Research Student, and found it wonderfully stimulating
and broadening; I have attended every ICTAM ever since, with just one ex-
ception; these Congresses are like family gatherings, where announcements of
new results are keenly anticipated. G.I. Taylor used to say that he always saved
his best results for these great international Congresses, and recommended that
others should follow his example!)
It is not a very enviable task to follow Dr von Kármán on this subject of turbu-
lence. He explained things so very clearly and he has touched on so many matters
that the list of things which I had to say is now torn to shreds by the crossings out
I have had to make as his talk progressed. But there is one point on what ought
to be called the pure turbulence theory, which I should like to make; this point
concerns the spectrum and will be useful also for Dr von Weizsäcker, in his talk.
After having made that point, I want to plunge straight into the subject that some
of the speakers have lightly touched on and then hastily passed on from, namely
the interaction between the magnetic field and the turbulence. That will perhaps
give us something to talk about. I shall be thinking aloud so that everything may
be questioned.
the velocity field u that convects it, whereas B is free of any such constraint.
Nevertheless, some valid results do follow from the analogy, despite their inse-
cure foundation: in particular, the fact that in the ideal fluid limit, the magnetic
flux through any material circuit is conserved, like the flux of vorticity in an
inviscid non-conducting fluid.
Batchelor applied the analogy to the action of turbulence in a highly con-
ducting cloud of ionized gas on a weak ‘seed’ magnetic field. He argued that,
provided η ν (where η is the magnetic diffusivity and ν the kinematic vis-
cosity), stretching of the field, which is most efficient at the Kolmogorov scale
lv = (ν3 /)1/4 , will intensify it on a time-scale tv = (ν/)1/2 until it reaches
a level of equipartition of energy with the smallest-scale (dissipation-range)
ingredients of the turbulence, i.e. until
B2
/μ0 ρ ∼ (ν)1/2 .
I note that this estimate still attracts credence (see Kulsrud 1999, who writes
“there is equipartition of the small-scale magnetic energy with the kinetic en-
ergy of the smallest eddy” in conformity with Batchelor’s conclusion that “a
steady state is reached when the magnetic field has as much energy as is con-
tained in the small-scale components of the turbulence”). This energy level is
smaller, it should be noted, by a factor Re−1/2 , than the overall mean kinetic
energy u2
/2 of the turbulence.
There were however two reasons, themselves in mutual contradiction, to
doubt Batchelor’s conclusions. On the one hand, intensification of the mag-
netic field through the stretching mechanism may be expected to occur (albeit
relatively slowly) on length-scales much larger than the Kolmogorov scale,
and it is therefore arguable (as urged almost simultaneously by Schlüter &
Biermann 1950) that the range of equipartition should extend ultimately to
the full spectral range of the turbulence. On the other hand, intensification by
stretching is naturally associated with decrease of scale in directions transverse
to the field and hence with accelerated joule dissipation, and it was argued by
Saffman (1964) that this effect would in all circumstances lead to ultimate de-
cay of magnetic energy – as had been previously proved to be the case if all
fields are assumed to be two-dimensional (Zeldovich 1957).
Thus, by 1965, all bets were open, and nothing was certain: in the presence
of homogeneous isotropic turbulence, an initially random magnetic field with
zero mean might grow to equipartition from an infinitesimal level, or might
grow under some subsidiary conditions to some significantly lower level, or
might grow for a while and then decay to zero. Into the gloom of this un-
resolved controversy, there penetrated a shaft of light from beyond the Iron
Curtain, in the work of Steenbeck, Krause & Rädler published in 1966 in the
8: George Batchelor 287
Finally, it may be worthwhile to say a word about the attitude that I have adopted
to the problem of turbulent motion, since workers in the field range over the
whole spectrum from the purest of pure mathematicians to the most cautious of
experimenters. It is my belief that applied mathematics, or theoretical physics, is
a science in its own right, and is neither a watered-down version of pure math-
ematics nor a prim form of physics. The problem of turbulence falls within the
province of this subject, since it is capable of being formulated precisely. The
manner of presentation of the material in this book has been chosen, not with an
eye to the needs of mathematicians or physicists or any other class of people, but
according to what is best suited, in my opinion, to the task of understanding the
phenomenon. Where mathematical analysis contributes to that end, I have used
it as fully as I have been able, and equally I have not hesitated to talk in descrip-
tive physical terms where mathematics seems to hinder the understanding. Such
a plan will not suit everybody’s taste, but it is consistent with my view of the
nature of the subject matter.
290 Moffatt
Figure 8.3 The Fluid Dynamics group at the Cavendish laboratory, 1954; front
row: Tom Ellison, Alan Townsend, G.I. Taylor, George Batchelor, Fritz Ursell,
Milton Van Dyke. Philip Saffman is in the middle of the back row, Stewart Turner
in the top right-hand corner, and Bruce Morton and Owen Phillips third and fifth
from the left in the middle row.
Figure 8.4 George Batchelor, having been recently elected FRS, in his office at
the old Cavendish Laboratory, October 1956.
Figure 8.5 George Batchelor (second from left) at the IXth International Congress
of Applied Mechanics in Brussels, 1956; James Lighthill on the right.
to 1978, but was content throughout that period to leave the running of the
department to Batchelor, who was Head of the Department from its foundation
in 1959 until his retirement in 1983.
Sagaut & Cambon 2008). The paper of Batchelor & Proudman led the way in
this important branch of the theory of turbulence.
A closely related problem which may also be treated by linear techniques
concerns the effect on wind tunnel turbulence of a wire gauze placed across
the stream. This problem was treated by Taylor & Batchelor (1949), interest-
ingly the only paper under their joint authorship. When the turbulence level is
weak, the velocity field on the downstream side of the gauze is linearly related
to that on the upstream side, and so the spectrum tensor of the turbulence im-
mediately downstream of the gauze can be determined in terms of that on the
upstream side. The transverse components of velocity are affected differently
from the longitudinal components, so that turbulence that is isotropic upstream
becomes non-isotropic, but axisymmetric, downstream, a behaviour that was
at least qualitatively confirmed by Townsend (1951a). Batchelor (1946a) had
previously developed a range of techniques appropriate to the description of
axisymmetric turbulence, techniques that here found useful application.
On scales between the Kolmogorov scale and the Batchelor scale, the velocity
gradient is approximately uniform, and on this basis, Batchelor was able to de-
termine the spectrum of the fluctuations of the scalar field in the corresponding
range of wave-numbers k; he found this to be proportional to k−1 . It is worth
remarking that the same technique applied to the corresponding passive vector
problem leads to exponential growth of the energy of the vector field (Moffatt
& Saffman 1964), reflecting in some degree the type of dynamo action that
Batchelor had predicted in 1950.
In the companion paper (Batchelor, Howells & Townsend 1959) the small
Prandtl number situation was considered; in this case, the ‘conduction cut-off’
occurs at wavenumber (/κ3 )1/4 (as previously found by Obukhov 1949), and
the scalar spectrum was determined in the range of wavenumbers between the
conduction cut-off and the viscous cut-off (the k−17/3 -law).
These two papers, coupled with those of Corrsin (1951) and Obukhov (1949),
have provided the starting point for almost all subsequent treatments of the
passive scalar problem, a problem which has attracted renewed attention, with
respect to its intermittency characteristics, in recent years – see Chapter 10
concerning Kraichnan’s contributions to this problem. It is worth noting that,
although Kraichnan’s model involved a ‘delta-correlated’ velocity field, i.e.
one varying infinitely rapidly in time, as opposed to Batchelor’s quasi-steady
model, he still found a k−1 range when ν/κ 1, implying a certain robustness
of this result.
I should not leave the topic of turbulent diffusion without mention of the
interesting paper of Batchelor, Binnie & Phillips (1955) in which it was shown
that the mean velocity of a fluid particle in turbulent pipe flow is equal to
the conventional mean velocity (averaged over the cross-section). This result
was tested experimentally using small neutrally-buoyant spheres injected into
a pipe flow. This was perhaps the first Lagrangian measurement in turbulent
flow. The paper is closely related in spirit to Taylor’s famous (1954) paper
treating the axial diffusion of a scalar in a pipe flow (laminar or turbulent).
comparison with the inner Kolmogorov scale, so that again (shades of his 1959
paper!) he was able to represent the ‘far field’ as a uniform gradient velocity
field. Batchelor assumed that the Péclet number was large and he solved the
advection–diffusion equation in the ‘concentration boundary layer’ around the
particle, in order to calculate the net rate of mass transfer from the particle.
He identified two contributions to this mass transfer, the first related to the
translational motion of the particle relative to the fluid, and the second related
to the local velocity gradient. Batchelor argued that the first of these, for rea-
sons associated with reflectional symmetry, is zero “in common turbulent flow
fields”, and he determined the second contribution in terms of a Nusselt num-
ber Nu, the result being Nu = 0.55(a2 /κν1/2 )1/3 , where a is the radius of the
particle (assumed spherical). The result is in very reasonable agreement with
experiment. Batchelor’s great skill in exploiting what is known of the small-
scale features of turbulent flow is again evident in this paper. His neglect of the
contribution to mass transfer due to particle slip is however debatable, partic-
ularly in turbulence that lacks reflectional symmetry, and still calls for further
investigation.
Batchelor’s final return to turbulence (in collaboration with V.M. Canuto &
J.R. Chasnov) came with his 1992 paper “Homogeneous buoyancy-generated
turbulence”. Still, as ever, Batchelor felt most at home in a statistically homo-
geneous situation. But here a novel variation was conceived, namely a field
of turbulence generated from rest by an initially prescribed, statistically ho-
mogeneous, random density distribution giving a random buoyancy force. The
driven turbulence of course immediately modifies the distribution of buoyancy
forces, and an interesting nonlinear interaction between velocity and buoyancy
fields develops: the buoyancy field itself drives the flow, so we are faced here
with an ‘active’ rather than ‘passive’ scalar field problem. As the authors state
in their abstract, “the analytical and numerical results together give a compre-
hensive description of the birth, life and lingering death of buoyancy-generated
turbulence”. Alas, Batchelor was himself suffering a lingering decline from
Parkinson’s disease which afflicted him with growing intensity from about
1994 until his death at the age of 80 in the millennium year 2000.
Proceedings of the Royal Society. Batchelor saw a need for a journal that would
bring together both experimental and theoretical aspects of the subject, and
planned JFM accordingly, together with a small team of Associate Editors,
George Carrier (Harvard), Wayland Griffith (Princeton), and James Lighthill
(Manchester). A novel feature of the editorial process, maintained to this day,
was that each editor and associate editor should have individual responsibility
to accept (or reject) papers submitted to him: they each acted as individuals
rather than as members of an editorial board, following agreed guidelines on
the scope of the journal and the standards expected. The formula worked ex-
tremely well, as may be seen from the quality of the papers published in the
early years of the journal.
Batchelor was aided by two Assistant Editors, Ian Proudman and Brooke
Benjamin. Their task initially was to prepare accepted papers for the Press,
i.e. to carry out the copy-editing process, a process in which George him-
self led the way, imposing strict rules of style: Fowler’s Dictionary of Mod-
ern English Usage was his constant companion. I served myself as an As-
sistant Editor from 1961; one of my first tasks was the copy-editing of Kol-
mogorov’s (1962) paper, by no means straightforward! One of Batchelor’s
guidelines was that the mathematical equations in a paper should be incor-
porated in such a way as to follow the normal rules of English grammar.
Copy-editing was more than simply correcting grammar or punctuation: it
sometimes involved complete reconstruction of sentences to clarify an author’s
intended meaning; of course, this required the author’s approval at the proof
stage, and extended correspondence could then ensue. But the results of this
painstaking procedure are evident in the quality of the early volumes of the
Journal, more difficult to maintain in later decades as the rate of publication re-
morselessly increased (with corresponding increase in the number of Associate
Editors).
As I have already indicated, Batchelor was heavily committed, from 1959
on, not only to JFM, but also as Head of the new Department of Applied
Mathematics and Theoretical Physics (DAMTP); this was his main adminis-
trative responsibility from 1959 till 1983, when, under the Thatcher regime,
he was induced to take early retirement from his University Professorship. In
the five-year period 1959–1964, there were just two Professors in DAMTP,
the Lucasian Professor (and Nobel Laureate) Paul Dirac and the Plumian Pro-
fessor of Astronomy Fred Hoyle. On his appointment as Head of DAMTP in
1959, Batchelor was promoted from his Lectureship to a Readership in Fluid
Mechanics, and five years later to a newly established Professorship of Ap-
plied Mathematics. These first five years were crucial in the shaping of the
8: George Batchelor 299
Committee and served as its founding Chairman from 1964 till 1987 (when he
was succeeded in this role by the new Cambridge Professor of Applied Math-
ematics, David Crighton.)
This Committee established a series of Colloquia, known as ‘EURO-
MECHs’, the first of which was held in Berlin in April 1965. By 1985 there
were approximately 15 EUROMECHs each year; a total of 230 were held dur-
ing Batchelor’s chairmanship, many in the broad field of turbulence and its
applications. His firm hand and meticulous attention to detail can be seen in
the notes of guidance written for chairmen of Colloquia: three to five days’
duration, participants from all countries of Europe “extending eastwards as
far as Poland and Rumania”, organizing committee of six members for each
Colloquium, not more than 50 participants, topics chosen to be of theoreti-
cal or practical interest but not devoted to details of engineering applications,
introductory lectures to be included, presentation of current and possibly in-
complete work to be encouraged, no obligation to publish Proceedings, and
so on. These Colloquia were run on a financial shoestring, most participants
finding their own individual sources of funding, but this seems to have had lit-
tle adverse effect; on the contrary, the freedom from the constraints often im-
posed by grant-giving bodies gave EUROMECHs a freshness and spontaneity
that was widely valued. The foundation of EUROMECH following on that of
JFM, raised the visibility and impact of research in fluid mechanics, and may
be recognized as an important part of Batchelor’s great legacy to the subject.
The inclusion of Poland was significant. Through IUTAM, Batchelor was a
close friend of Wladek Fiszdon (of the Institute of Fundamental Technological
Research, Warsaw), and had given him encouragement and help in the organi-
zation of the biennial Symposia on Fluid Mechanics in Poland that served so
well to maintain communication between scientists from East and West dur-
ing the long cold-war years. The link with Poland developed with the presence
of Richard Herczynski, a younger colleague of Fiszdon, as a Senior Visitor in
DAMTP in 1960/61.
In 1963 Batchelor attended the Fluid Dynamics Conference held in Zako-
pane, Poland – one of the first at which any Western scientists were present.
He drove across Europe to this meeting with his wife and three young daugh-
ters, Adrienne, Clare and Bryony, and took me along also in the last space in
the family car! Batchelor’s contributed paper on dispersion of pollutant from a
fixed source in a turbulent boundary layer (Batchelor 1964) is a beautiful piece
of work, and deserves to be better known. At this conference, he had prelimi-
nary discussions with Fiszdon and Herczynski about the need for greater Eu-
ropean cooperation in science. He attended several of the subsequent biennial
meetings and was involved much later, through Herczynski, in the traumatic
8: George Batchelor 301
events for Polish scientists during the Solidarity years, always giving moral
support when this was most needed.
8.14 Conclusion
Despite the intensive efforts of mathematicians, physicists and engineers over
the last half century and more, turbulence remains, to paraphrase Einstein, “the
most challenging unsolved problem of classical physics”. The 1950s was a pe-
riod when, under the inspiration of George Batchelor, there was a definite sense
of progress towards a cracking of this problem. But he, like many who followed
him, found that the problem of turbulence was challenging to the point of to-
tal intractibility. Nevertheless, Batchelor established new standards both in the
rigour of the mathematical argument and the depth of physical reasoning that
he brought to bear on the various aspects of the problems that he addressed.
His 1953 monograph is still widely quoted as the definitive introduction to the
theory of homogeneous turbulence, and his later contributions, particularly in
the field of turbulent diffusion, have stood the test of time over the subsequent
fifty years. There can be no doubt that any future treatise on the subject of tur-
bulence will acknowledge Batchelor’s major contributions to the field, coupled
with those of his great mentor G.I. Taylor.
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Saffman, P.G. 1967. The large-scale structure of homogeneous turbulence. J. Fluid
Mech. 27, 581–593.
Saffman, P.G. 1971. On the spectrum and decay of random two-dimensional vorticity
distributions at large Reynolds number. Studies in Applied Mathematics 50, 377–
383.
Sagaut, P. & Cambon, C. 2008. Homogeneous Turbulence Dynamics. Cambridge Uni-
versity Press.
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Feldstärke für ein elektrisch leitendes Medium in turbulenter, durch Coriolis-
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196–212.
Taylor, G.I. 1935. Turbulence in a contracting stream. Z. angew. Math. Mech. 15, 91–96.
Taylor, G.I. 1954. The dispersion of matter in turbulent flow through a pipe. Proc. Roy.
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9
A.A. Townsend
Ivan Marusic and Timothy B. Nickels
In my time, Laby did little research on his own account but he took a very
close interest in the progress of all the work in the laboratory. In 1936, the ac-
tion of Geiger–Müller counters was not well understood and their performance
305
306 Marusic & Nickels
Figure 9.1 Alan Townsend at the age of 21 years, having graduated MSc from the
University of Melbourne.
unpredictable. Burhop and I found that only one-third of those we had made were
reliable. Laby found us making up a large batch in the hope that some of them
would be usable, and, finding out that we had no idea why some would work and
others would not, he ordered us to study their operation in detail. Fortunately,
that batch produced sufficient good counters for our needs and we kept out of his
way until he had forgotten.
scholarship value was only £250 per year which at that time was comfortable
for a young man but not a princely sum (in 1938 the average salary in the UK
was £209).
He went to work at the Cavendish laboratory, then under the directorship of
the Australian Nobel laureate Lawrence Bragg. At this time he worked in the
Cavendish High Voltage and Cyclotron Laboratories run by J.D. Cockcroft.
Cockcroft and his student Walton had recently (April 1932) achieved fame by
splitting the lithium atom with a proton beam (for which they were jointly
awarded the Nobel Prize in physics in 1951). When Townsend arrived Cock-
croft was working on the construction of a new high voltage Phillips set in
the Old Library of the Cambridge Philosophical Society which was intended
to produce two million volts based on the Cockcroft–Walton method of volt-
age amplification for use in cyclotron experiments. While he was waiting for
his own apparatus to be constructed Townsend was occupied in assisting with
the commissioning of the new set. In a letter to his mother in January 1939
Townsend recalls:
There was a big thrill in the lab today as the two million volt set gave two million
volts for the first time, a performance which was considered highly improbable
before Christmas.
Once his own apparatus (a very accurate magnetic spectrometer) was con-
structed he set to work on his thesis research studying the β-ray spectra of light
elements – work which resulted in a paper in the Proceedings of the Royal
Society (Townsend, 1941).
breakthrough they flummoxed the security people by stating “we know the
theory – just show us the wiring diagrams”.
After that he requested permission to return to Australia to work for the
CSIR (Council for Scientific and Industrial Research). He spent six months at
the Mount Stromlo observatory working on metallic surfaces on glass and then
another six months working on lubricants and bearings in the CSIR laboratory
at the University of Melbourne (that was set up by F.P. Bowden). He then went
to the Aeronautics lab at Fisherman’s Bend in Melbourne to run the instru-
ments section. It was here that he first met George Batchelor who was working
in the Aerodynamics section; the two often collaborated on wind tunnel test-
ing. One notable study was their investigation of the ‘singing’ of wind tunnel
corner vanes, where they characterized this aero-elastic resonance effect and
published a short letter in Nature (Batchelor & Townsend, 1945).
If they had hoped to work directly with Taylor on turbulence they were some-
what disappointed. Turner (1997), in his article on G.I. Taylor, notes that:
George Batchelor (G.K.B.) and Alan Townsend (A.A.T.) arrived in 1945 specif-
ically to become research students under G.I.’s supervision and to work respec-
tively on theoretical and experimental problems in turbulence, a field in which
they had been active at the Aeronautical Research Laboratory in Melbourne dur-
ing the war. G.K.B. has admitted since that he was somewhat disconcerted to
discover that G.I. did not immediately return to the subject of turbulent flow to
which he had made such large contributions in the 1930s . . . The research on
turbulence was carried on almost independently by these two students, with G.I.
9: A A. Townsend 309
showing great interest in their results, but suggesting no specific program for
their research. G.K.B. initially followed up Kolmogoroff’s wartime papers on
the statistical equilibrium of the small scale components of turbulent motions,
and A.A.T. began work on turbulence behind grids in the wind tunnel in the
Cavendish Laboratory.
The wind tunnel, designed by Taylor, was located in a large room on the ground
floor of the old Cavendish Laboratory in Free School Lane. The next five years
were an extremely productive time for Townsend. He and Batchelor set about
examining and testing the theories of turbulence, in particular those of Kol-
mogorov and the Russian school that Batchelor was bringing to the attention
of the West. Townsend published a total of eleven papers, five with Batchelor
and six as sole author. Considering he was new to turbulence research when
he arrived in 1945, this must be considered a remarkable output. These papers
established a number of new ideas which he was further to develop throughout
his career.
It was during this extremely productive time, in 1950, that he was appointed
Assistant Director of Research of the Cavendish and married Valerie Dees.
Valerie had come to Cambridge having served in the WAAF during the war
and was working with G.C. Grindley in the Experimental Psychology lab in
Cambridge. They got to know each other through mutual friends on a boat-
ing trip in the Norfolk Broads during which he proposed. They shared a great
love of the outdoors, enjoying all manner of activities, especially camping and
sports. Alan was a very keen sportsman enjoying, in particular, hockey, ten-
nis and skiing. He once commented to his daughter that he became a British
citizen only to save on visa costs for his many European ski trips with his
Cavendish friends. He played hockey for the university and continued play-
ing in the Cambridge third team until well into his sixties (Figure 9.2). He
was also a keen tennis player serving as the university representative for the
Lawn Tennis Association and was president of the Emmanuel College Tennis
club from 1949 to 1968 (after which time it ceased to have a president). He
was still playing regularly with other Fellows of the College in 1998 – at the
age of 81.
Figure 9.2 Hockey. Townsend is on the end at the far right. In the background is
Jesus College.
equations that T. von Kármán and L. Howarth derived in 1938. The concept of
local isotopy of the small-scale eddies had also been advanced by A.N. Kol-
mogorov and A.M. Obukhov in 1941 and, while this was also independently
formulated by C.F. von Weiszacker and L. Onsager, the attention to this was
initially mostly confined to the Russian school led by Kolmogorov. Batchelor,
however, was familiar with Kolmogorov’s papers and saw that they potentially
opened the way to advancing the isotropic theory to inhomogeneous and shear
flows, and he was largely responsible for the dissemination of Kolmogorov’s
ideas to the West after the war. Heisenberg, having been prohibited from work-
ing on nuclear physics problems by the allied forces, returned to his research
on turbulence, and in 1948 proposed a theory for the transfer of energy from
low to high wavenumbers. In the same year L.S.G. Kovasznay proposed an al-
ternative theory for the same problem and the topic saw considerable interest
for many years. However, it was Townsend (together with the theoretical in-
put from Batchelor) who conducted the initial pioneering experiments in this
field.
In homogeneous isotropic turbulence the mean values of all functions of
the components of the velocity fluctuations and their derivatives are indepen-
dent of position and rotations and reflections of the axes defining the frame of
9: A A. Townsend 311
Townsend built the amplifier and compensator stages required for the hot-
wire anemometry as well as the analogue devices required for the statistical
analysis of the hot-wire signals. Hot-wire anemometry involves using fine plat-
inum or tungsten wires (typically 5 microns in diameter) as thermal transduc-
ers with a small sensing length of less than 1 mm. Successfully using them
for turbulence measurement requires circuitry that extends the frequency re-
sponse of the sensor; at that early stage, this was as much an art as science.
Townsend, however, mastered this technique. While others had previously used
hot-wire anemometry (Dryden, Hall, Simmons and colleagues) their measure-
ments were restricted to mean flow, mean-square intensities and double cor-
relations. Townsend pioneered the use of electrical analysis to obtain time
derivatives, triple correlations and other quantities that were required to test
the theoretical prediction of isotopic turbulence.
Townsend’s first turbulence paper on the decay of isotropic turbulence, co-
authored with Batchelor, appeared in the Proc. Royal Society (Batchelor &
Townsend, 1947), in the same year he submitted his PhD Dissertation on β-
ray spectra of light elements and turbulent flow. It was at this time also that he
was elected a Supernumerary Fellow and College Lecturer in Physics at Em-
manuel College. He remained a Fellow of Emmanuel College for his whole
life having been elected to a life Fellowship. Townsend always gave his affili-
ation in publications as Emmanuel College. This illustrates the important role
312 Marusic & Nickels
the College played in his life and his career. He was Praelector of the College
during 1968–1969 and Vice-Master from 1975 to 1979.
While working on the decay problem Townsend decided to investigate the
validity of the Heisenberg inverse seventh power spectrum for the far viscous
range. In Townsend (1990) he gives a characteristically modest description of
the discovery of an important feature of turbulence as an unexpected result
(Batchelor & Townsend, 1949):
In a moment of inspiration, I decided that the power could be obtained by mea-
suring the flatness factors of a number of high-order velocity derivatives, and
I managed it for the first four. The kurtosis increased both with order of the
derivative and with the Reynolds number of the turbulence. I then realised that
the measurements said nothing about the spectrum, but they did show a consider-
able departure from the predictions of local similarity. The spatial intermittency
of small-scale motion is now well known. Our trouble in writing up the work was
to explain why we had done the measurements in the first place, pride preventing
use of the words, “In a moment of exceptional stupidity, we measured . . . ”.
Spatial intermittency of the small-scale motion refers to the fact that the dis-
sipation of energy (and other characteristics of the flow) is not uniformly dis-
tributed throughout the flow, as had been previously thought, but instead occurs
in very intense events that are sparsely distributed. The spatial mean values
then are averages of very large, very rare events. This is an important funda-
mental result concerning the nature of turbulence, and suggests immediately
that the turbulent motion at these fine scales cannot be totally random and
featureless.
Inspired by this result, Townsend then went on to try to understand what sort
of underlying structure in turbulence could explain it. In Townsend (1951a) he
considered models of the fine-scale motion consisting of
a random distribution of vortex sheets and lines, in which the vorticity distribu-
tion is effectively stationary in time, due to balance between the opposing effects
of vorticity diffusion by molecular viscosity, and vorticity production and con-
vection by the turbulent shear.
He showed that the spectrum produced by a random array of sheets was a bet-
ter fit to the available data than that produced by vortex lines (though they both
gave reasonable results). In order to explain why sheets might be more preva-
lent he derived a remarkable analytical result. He related the principal rates of
strain to the skewness of the velocity fluctuations (which was known, empir-
ically, to be negative) and showed that this implied that the strain field must
locally be of the form which produces sheets (that is, one compressive strain
and two extensive). This particular result, though not unduly emphasized in the
paper, has considerably influenced our understanding of turbulence since.
9: A A. Townsend 313
The sharp turbulent wedge that Townsend noted was later known as the Em-
mons turbulent spot, based on H. Emmons’ 1951 experiments in water table
flows at Harvard, and continues to be studied together in the context of bypass
transition in laminar boundary layers. This initial work on the turbulent wake
sparked Townsend’s interest in turbulent shear flows. While research on grid
turbulence had been very useful for examining the theory, Townsend recog-
nized early on that shear flows are much more common in practice and any
theory of turbulence would need to be able to predict these flows as well.
Townsend made further measurements of the cylinder wake, in particular
considering the distribution of the terms in the equation for the kinetic en-
ergy of the velocity fluctuations (Townsend, 1948). An analysis of the results
showed that there was substantial energy transport up the intensity gradient
314 Marusic & Nickels
across the jet. Townsend realized that this important observation invalidated
local gradient diffusion models for the turbulence and implied the existence
of large-scale motions. He discusses these in terms of jets of turbulent fluid
emitted from the turbulent core but then states:
While the conception of jets of turbulent fluid is more convenient for following
the physical processes in the wake, the alternative but equivalent description that
the turbulent motion consists of a motion of scale small compared with the mean
flow superimposed on a slower turbulent motion whose scale is large compared
with the mean flow may be used.
He further remarks:
It is expected that this type of motion will occur in all systems of turbulent shear
flow with a free boundary, such as wakes, jets and boundary layers.
This paper then marks the birth of his conception of large eddies which he
used to great effect in modelling turbulent shear flows in the first and second
editions of his book. These in effect are ‘coherent structures’ that became the
topic of extensive study in the 1970s and continues to see considerable interest
today. It is important to realize that many models of turbulent flow at that time
were based on ideas such as eddy viscosity in which local stresses were related
to local strains.
In his 1948 paper he also considered the nature of the entrainment process.
Townsend describes:
It may be put that the turbulent diffusion occurs in two stages, the mean jet move-
ment doing the large scale diffusion and the jet turbulence performing small scale
diffusion of the resulting pattern; that is, the jet motion acts to increase the sur-
face area of the turbulent fluid, and so allow a comparatively slow linear diffusion
at the bounding surfaces to achieve a large rate of volume increase.
Thus, using his concept of large eddies (here referred to as jets) he expounded
a view of entrainment that, although controversial, is still very relevant today.
He developed and expounded many these ideas in the first edition of his
monograph The Structure of Turbulent Shear Flow published in 1956. It was
well received and partly on the basis of this and his many seminal papers he
was elected a Fellow of the Royal Society in 1960.
For the authors of this chapter, this quote is extremely familiar, as we both were
introduced to the topic of turbulence by A.E. Perry in his fourth-year fluid
mechanics course at the University of Melbourne. Tony Perry would use an
overhead projector to prominently display Townsend’s quote on a large screen
and have all the students in the class read it out aloud in unison (several times),
and would emphasize the profound nature of the statement. While we students
did not largely understand it at the time, Perry’s passionate delivery made it
clear that the hypothesis was important.
It is interesting that Townsend does not repeat the above statement explicitly
in the second edition of The Structure of Turbulent Shear Flow, but rather the
notion is discussed and extended throughout the book in a subtle and unas-
suming way. For this reason, future references to the hypothesis invariably
include the addition of roughness as well as viscosity affecting the boundary
conditions alone (Perry & Abell, 1977; Raupach et al., 1991; Jimenez, 2004;
etc.), and carry the phrase “mean relative motion” (Perry & Abell, 1977) in
place of “mean motion”, as this is what Townsend meant. Stating “mean rel-
ative motions” (and hence mean velocity derivatives) also leads directly to an
316 Marusic & Nickels
elegant derivation of the logarithmic law in the inertial (or equilibrium) layer
of wall-bounded flows (Rotta, 1962). In the area of turbulent boundary lay-
ers developing on rough walls the hypothesis is particularly useful. It suggests
that the direct effects of roughness are confined to a small region above the
roughness elements whereas the outer part of the flow is influenced only by
the change in the wall shear-stress. This is true in the limit of large Reynolds
numbers and where the roughness elements are small relative to the boundary
layer thickness. Although there is some dispute about its applicability in some
particular situations, the bulk of the evidence indicates that it is true for flows
which satisfy the necessary assumptions (e.g. Schultz & Flack, 2007; Nickels,
2010). Practically it means that in the outer part of the flow the mean velocity
gradient and the stresses are unchanged when scaled with outer flow variables
(the friction velocity the boundary layer thickness). This results in a consider-
able simplification when modelling these flows. The status of this hypothesis
may be judged by the fact that it is now often simply referred to in the liter-
ature as ‘Townsend’s hypothesis’ without requiring a citation of the original
source.
the bulk of the energy-containing eddies are, in a sense, attached to the wall, and
that the dependence of scale of distance from the wall is not a local effect but due
to this attachment of most of the eddies.
He went even further in this paper postulating a possible form of these eddies
that would be consistent with the measurements. As mentioned earlier, many
models of turbulent flow at the time were based on the idea of gradient dif-
fusion, local conditions determining local transport. One would then expect
transport to be down the local gradient by analogy with molecular diffusion.
9: A A. Townsend 317
Townsend immediately saw that the transport up the gradient implied impor-
tant non-local effects.
By 1976 Townsend had developed a unified view of incompressible flow
turbulence that focused on understanding the main energy-containing eddies
or motions. These eddies may be thought of as the velocity fields of some rep-
resentative vortex structures. Townsend had successfully explained his earlier
correlation measurements and those of Grant (1958) in plane wakes by mod-
elling the flow in terms of large eddies, and an obvious extension now was to
wall-bounded flows. Here, however, the key additional feature is the presence
of a wall or surface, which acts as a continuous source (in the presence of a
mean pressure gradient) of vorticity for the flow as it develops along the length
of the surface. By studying the experimental data Townsend concluded that it
was
difficult to imagine how the presence of the wall could impose a dissipation
length-scale proportional to distance from it unless the main eddies of the flow
have diameters proportional to distance of their ‘centres’ from the wall because
their motion is directly influenced by its presence. In other words, the velocity
fields of the main eddies, regarded as persistent, organised flow patterns, extend
to the wall and, in a sense, they are attached to the wall.
Therefore, in essence any eddy with a size that scales with its distance from
the wall may be considered to be attached to the wall. Eddies farther from the
wall are larger in size and hence their velocity fields still extend to the wall (of
course the velocity fields of vortex structures extend to infinity but they decay
rapidly at large distances). These eddies form the basis of the attached eddy
hypothesis. The hypothesis itself is that the main energy-containing motion of
a turbulent wall-bounded flow may be described by a random superposition
of such eddies of different sizes, but with similar velocity distributions. These
eddies are considered as statistically representative structures in that their ge-
ometry and strength are derived from an ensemble average of many different
structures of similar scale.
Townsend then went on to form expressions for the contributions of a ran-
dom superposition of attached eddies of different sizes to the correlation func-
tions and, using the zero-penetration boundary condition at the wall, derived
the distribution of eddy sizes with wall distance necessary to produce invari-
ance of the Reynolds shear stress (−uw/Uτ2 ≈ 1) with distance from the wall, as
observed in the equilibrium layer. (Here, u, v, w refer to the streamwise, span-
wise and wall-normal components of fluctuating velocity respectively, and Uτ
is the friction velocity.) This analysis effectively leads to a population density
of eddies that varies inversely with size and hence with distance from the wall.
That is, the number of eddies of size z per unit wall area is A/z, where A is a
constant. These distributions of eddies also lead to predictions for the variation
318 Marusic & Nickels
centres located at or near z, whereas the inactive motions are due to eddies
much larger than z. Which motions are considered active and inactive de-
pends on the location of the measurement (see Nickels et al., 2007, for further
explanation).
To proceed further with Townsend’s attached eddy hypothesis requires spec-
ifying the detailed structure of the eddies, and this was later done by Perry and
coworkers (Perry & Chong, 1982; Perry et al., 1986; Perry & Marusic, 1995).
Townsend did offer an interpretation of the representative attached eddy in the
form of conical eddies as shown in Figure 9.3. Townsend notes, however, that:
Other possibilities exist, e.g. a distribution of shorter double-roller eddies of var-
ious sizes each with a lateral extent comparable with distance from the wall.
Townsend (1976), where the attached eddy hypothesis is described. Only dis-
cussions with Townsend himself gave us the reassurance that we could stop.
Following Townsend’s thoughts has always been easier in direct discussion
with the author. This is echoed by Stewart & Grant (1999), who note:
He [Townsend] thinks not in words, but in three dimensions. Since he often can-
not put his ideas into words, he frequently has difficulty promulgating them ei-
ther in writing or in speech. One on one, however, he was a joy and inspiration
to work with.
Related to this is an anecdote told by Tony Perry about the attached eddy
hypothesis and k−1 law (where k is streamwise wavenumber). Perry & Abell
(1975) presented the first serious assault, using dynamically calibrated hot-
wires, to uncover the scaling laws of the streamwise fluctuating velocity in
pipe flows, a topic where a compilation of the previous experimental data
highlighted significant disparities and thus considerable uncertainty. The con-
clusions put forward by Perry & Abell (1975) supported Townsend’s simi-
larity hypothesis, and the classical law-of-the-wall description for the turbu-
lence intensities in the logarithmic region where u2 /Uτ2 should be invariant
with Reynolds number. However, this latter result was in contradiction with
Townsend’s theories that were to appear in his book derived using the attached
eddy hypothesis, where u2 /Uτ2 followed a logarithmic function of wall-normal
position (equation 9.1). Further, Perry & Abell were also puzzled as to why
their u-spectra indicated a k−1 behaviour. Fortunately, Townsend was visiting
Melbourne during this time and pointed out that Perry & Abell’s data were
completely consistent with his theories, and that it was all explained in the
latest edition of his book. Perry was very familiar with Townsend’s book but
failed to see the connection and insisted that Alan show him where in the book
this was written. Certainly, nowhere in Townsend (1976) is k−1 mentioned,
while mention of it is made in the first edition (Townsend, 1956) in reference to
the experimental data of Laufer (1955). With this personal ‘translation’ Perry
was able to see that the k−1 behaviour for u-spectra admitted a log law for u2
and he saw the full implication of Townsend’s two hypotheses. Perry & Abell
reconciled their rough and smooth wall data by taking into account pipe de-
velopment length issues, and published another paper (Perry & Abell, 1977)
that revised their previous findings. Here they elegantly showed that the com-
plete spectral behaviour across all wavenumbers in the logarithmic layer could
be postulated using dimensional analysis and Townsend’s hypotheses. Recent
observations of the k−1 behaviour in a high Reynolds number boundary layer
have also been presented in Nickels et al. (2005), and considerable attention to
this issue continues.
9: A A. Townsend 321
Recent advances in the area of wall turbulence have taken advantage of the
extensive new datasets obtained from multiplane particle-image velocimetry
and volumetric data from numerical simulations, as reviewed by Ron Adrian,
a PhD student of Townsend (Adrian, 2007). A key new feature of these studies
has been the documentation of characteristic eddy shapes that take the form of
hairpin-shaped vortices, which appear with a range of scales and a preferred
streamwise organization where the individual vortices align to form hairpin
packets. It is interesting to note that these observations were likely suspected
by Townsend who relied solely on single-point time-series data. In describ-
ing the average eddy shape shown in Figure 9.3, he comments that the cone
eddy could be described as “an organised assembly of simple attached eddies”,
which is in essence a vortex packet.
Townsend’s attached eddy hypothesis has considerably shaped the field of
wall turbulence thus far, and its full impact is probably yet to be realized.
Concepts such as the blocking mechanism (Hunt & Graham, 1978), which
have been used to explain wall turbulence statistics (Hunt & Carlotti, 2001;
Mann, 1996), are inherent in the Townsend attached eddy approach. More-
over, it is worth noting that the Townsend formulations, equations (9.1), are
contradictory to the classical law-of-the-wall description that have been, and
continue to be used, extensively in RANS (Reynolds averaged Navier–Stokes)
calculation schemes. However, new high Reynolds number experiments and
high-fidelity measurements strongly support Townsend’s approach (Perry et
al., 1986; Jimenez & Hoyas, 2008; Nickels et al., 2007; Marusic et al., 2010;
Smits et al., 2011), and as the field enters the phase of dynamical descriptions
required for a fuller understanding Townsend’s ideas will undoubtably con-
tinue to shape the field (Marusic et al., 2010; Smits et al., 2011; Jimenez &
Kawahara, 2011).
it goes against the conventional thinking that underpins the majority of com-
mercial CFD (computational fluid dynamics) schemes. Townsend recognized
that the terms “organized eddy structure” and “turbulent flow” are themselves
descriptive of a “persistent duality in the development of theories of turbu-
lent motion” (Townsend, 1987). This is because much of the modelling that
goes into conventional CFD schemes using the Reynolds averaged momentum
and turbulent kinetic energy, is based on concepts derived from randomness
of motion and on analogies with molecular motion (eddy viscosity concepts).
Townsend inferred that writing equations for mean values at a point in space,
implicitly assumes that the scale of the motion is small compared to the width
of the flow. In practice it was clear that the extent of flow coherence as mea-
sured by velocity correlations is usually as large as the flow is wide, and local-
ized random flow concepts were inherently insufficient to capture the dominant
physics.
9: A A. Townsend 323
Author’s note
Prior to the final editing of this chapter Tim Nickels passed away unexpectedly
at the age of 44 on the 2nd of October 2010. The surviving author wishes to
dedicate this chapter to his co-author, colleague and dear friend.
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328 Marusic & Nickels
10.1 Introduction
Robert Harry Kraichnan (1928–2008) was one of the leaders in the theory of
turbulence for a span of about forty years (mid-1950s to mid-1990s). Among
his many contributions, he is perhaps best known for his work on the inverse
energy cascade (i.e. from small to large scales) for forced two-dimensional
turbulence. This discovery was made in 1967 at a time when two-dimensional
flow was becoming increasingly important for the study of large-scale phe-
nomena in the Earth’s atmosphere and oceans. The impact of the discovery
329
330 Eyink & Frisch
had not yet reached Reynolds numbers high enough to supplement experimen-
tal data on intermittency. Thus a kind of crossing of the desert took place, with
Kraichnan leading a small flock that found help mostly from geophysicists (at
the Wood Hole Oceanographic Institute’s GFD Program and at the National
Center for Atmospheric Research, Boulder).
This crossing of the desert occurred at a time when Kraichnan took him-
self “far from the madding crowd”. After having trod the traditional path of
academia with positions at Columbia University and Courant Institute, he de-
cided in the very early 1960s to become a self-employed turbulence consul-
tant, solely funded by research grants. He moved from New York City to
secluded mountains in New Hampshire, where he lived and worked for al-
most two decades. During those years he warmly welcomed visitors from all
around the world and regularly participated in scientific meetings, workshops
and schools. Eventually, he moved to New Mexico, close to the Los Alamos
National Laboratory.
We shall take the reader on a tour of key contributions to turbulence by
Kraichnan, organized in three main sections, roughly by chronological order:
closure, realizability, the issue of Galilean invariance and MHD turbulence
(Section 10.2); equilibrium statistical mechanics and two-dimensional turbu-
lence (Section 10.3); intermittency and the Kraichnan passive-scalar model
(Section 10.4). In the final section (Section 10.5), we first present two contri-
butions of Kraichnan which do not fit naturally into the three main scientific
sections: his very first published paper, on scattering of sound by turbulence
(Section 10.5.1), and his prediction of the behavior of convection at extremely
high Rayleigh numbers (Section 10.5.2). We then turn to Kraichnan’s impact
on computational turbulence and present concluding remarks.
The emphasis will be on understanding the flow of ideas and how they relate
to the meandering history of the subject. We shall mostly avoid technical ma-
terial. More details may be found in Kraichnan’s numerous papers (over one
hundred), published in journals such as Phys. Rev., Phys. Fluids, J. Fluid Mech.
and J. Atmosph. Sci., and in various reviews and textbooks by other authors,2
but also in proceedings papers by Kraichnan, which are often more elementary
and reader-friendly than those in journals.3
A word of warning: in what follows, standard notions used in turbulence the-
ory, such as the Navier–Stokes and the magnetohydrodynamics (MHD) equa-
tions, statistical homogeneity and isotropy, energy spectra, etc. will be taken
for granted.4
2 Leslie, 1973; Monin & Yaglom, 1975: § 19.6; Orszag, 1977; Rose & Sulem, 1978.
3 See Kraichnan, 1958c, 1972, 1975a.
4 See, e.g., Monin & Yaglom, 1975; Frisch, 1995.
332 Eyink & Frisch
the velocity field at time t, due to a small change in the driving force at times
t < t. In 1958, using a mixture of field-theoretic considerations and heuristic
but plausible simplifying assumptions, Kraichnan proposed the direct inter-
action approximation (DIA).7 When applied to homogeneous, isotropic and
parity-invariant turbulence, the DIA takes the form of two coupled integro-
differential equations for the two-point two-time velocity correlation function
and the response function. These equations are written in Section 10.2.2 in
abstract and concise notation, valid without assuming any particular symme-
try. For homogeneous isotropic turbulence the DIA equations, which resemble
in more complicated form the EDQNM equation (10.3) written below, can be
found in Kraichnan’s papers and in the references mentioned in the introduc-
tion. Such equations are easily extended to anisotropic and/or helical (non-
parity-invariant) turbulence.
As Kraichnan observed himself, the DIA is not consistent with K41. In
particular the inertial-range energy spectrum predicted by DIA is (to leading
order)
E(k) = C (εv0 )1/2 k−3/2 , (10.1)
instead of
E(k) = Cε2/3 k−5/3 , (10.2)
where ε is the kinetic energy dissipation per mass, v0 is the r.m.s. turbulent
velocity, and C (the Kolmogorov constant) and C are dimensionless constants.
The discrepancy has to do with the way the larger eddies from the energy-
containing range of wavenumbers interact with smaller eddies from the inertial
range. Kraichnan expressed his doubts that the former merely
convect local regions of the fluid bodily without significantly distorting their
small-scale (high-k) internal dynamics.
This assumption would not permit the presence of v0 in the inertial-range ex-
pression of the energy spectrum, except through an intermittency effect of a
kind which was not seriously considered until the work by the Russian school
in the early 1960s. Nevertheless, Kraichnan observed the following:
For a while Kraichnan believed that the DIA may be asymptotically exact in
the limit L → ∞ for turbulence endowed with L-periodicity in all the spatial
coordinates. However, after a few months, Kraichnan found that the DIA equa-
tions are actually the exact consequences of a certain random coupling model
obtained from the hydrodynamical equations (Navier–Stokes or MHD) by a
suitable modification of the nonlinear interactions (see Section 10.2.2). The
presence of this model automatically guarantees the realizability of the DIA.
The random coupling model has many properties in common with the origi-
nal hydrodynamical equations, such as symmetries, energy conservation, etc.
But Kraichnan noted that, when various correlation functions are expanded in
terms of Feynman diagrams, the random coupling model retains only a sub-
class of all diagrams; in particular it misses the so-called vertex corrections
which contribute (in field-theoretic language) to the renormalization of the
nonlinear interaction.9
Six years later he discovered a more serious defect: the random coupling
model and the DIA fail invariance under random Galilean transformations.10
Ordinary Galilean invariance – for the Navier–Stokes equations – is the obser-
vation that if (u(x, t), p(x, t)) are the velocity and pressure fields which solve
the Navier–Stokes equations in the absence of boundaries or with periodic
boundary conditions, then (u(x − Vt, t) + V, p(x − Vt, t)) are also solutions for
an arbitrary choice of the velocity V. Random Galilean invariance has the ve-
locity V chosen randomly with an isotropic distribution and independent of the
turbulent velocity-pressure fields. Isotropy ensures that the mean velocity re-
mains zero. In the DIA, when a random Galilean transformation is performed,
the mean square velocity v20 increases and thus the spectrum given by (10.1)
changes. On the one hand this is clearly inconsistent with the Galilean invari-
ance of the Navier–Stokes equations. On the other hand, an influence of the
energy-carrying eddies on the inertial range different from what K41 predicts
(e.g. intermittency effects) cannot be ruled out.
But first, Kraichnan realized that the DIA had to be modified to restore ran-
dom Galilean invariance and, if possible, without losing its other nice features.
To explain how this can be done, we have to become slightly more techni-
cal. Kraichnan derives the following equation from the DIA for the evolution
of the energy spectrum when the turbulence is homogeneous, isotropic and
parity-invariant:
∂t + 2νk E(k, t) =
2
d pdq θkpq ×
k
k # $
b(k, p, q) E(q, t) k2 E(p, t) − p2 E(k, t) + F(k) , (10.3)
pq
(π/2)1/2 p
θkpq ≡ , b(k, p, q) = (xy + z3 ) . (10.4)
v0 (k2 + p2 + q2 )1/2 k
Here E(k, t) is the energy spectrum, F(k) the energy input from random driv-
ing forces, k defines the set of p ≥ 0 and q ≥ 0 such that k, p, q can form a
triangle, x, y, z are the cosines of the angles opposite to sides k, p and q of this
triangle. The factor θkpq can be interpreted as a triad relaxation time. The partic-
ular form given in (10.4) is obtained from the DIA by an asymptotic expansion
valid only when the wavenumber k is in the inertial range. This form implies
that relaxation has a time scale comparable to the convection time 1/(kv0 ) of an
eddy of size 1/k at the r.m.s. velocity v0 . As Kraichnan notes, if v0 is replaced
“by, say, [kE(k)]1/2 , which may be considered the r.m.s. velocity associated
with wavenumbers the order of k only”, then K41 is recovered. Actually this
choice, which also restores random Galilean invariance and is easily shown to
be realizable, became later known as the eddy-damped quasi-normal Marko-
vian (EDQNM) approximation. Note that the EDQNM requires the use of an
adjustable dimensionless constant in front, say, of [kE(k)]1/2 , which can then
be tuned to give a Kolmogorov constant matching experiments and/or simu-
lations. Kraichnan clearly preferred having a systematic theory free of such
constants and then seeing how well it can reproduce experimental data, not
just constants, but also the complete functional forms of spectra. He eventually
developed a version of the EDQNM, called the test field model in which the
triad relaxation time is determined in a systematic way through the introduc-
tion of a passively advected (compressible) test velocity field and a procedure
for eliminating unwanted sweeping effects.11
To overcome the difficulty with random Galilean invariance in a more sys-
tematic way, however, he first developed the Lagrangian history direct inter-
action approximation (LHDIA) and an abridged version thereof (ALHDIA) in
which the correlation functions have fewer time arguments. LHDIA and AL-
HDIA make use of a generalized velocity u(x, t|t ) which has both Eulerian
and Lagrangian characteristics. It is defined as the velocity measured at time
11 Kraichnan, 1958c; for EDQNM, see Orszag, 1966, 1977; and also Rose & Sulem, 1978;
Lesieur, 2008; for the test field model, see Kraichnan, 1971a, 1971b. S.A. Orszag died on
1st May, 2011.
336 Eyink & Frisch
Here u(t) (scalar or vector) collects all the dependent variables, B(·, ·) is a
quadratic form (comprising the inertial and pressure terms for the case of
Navier–Stokes), L is a time-independent linear operator (e.g. the viscous dis-
sipation operator) and f (t) is a prescribed zero-mean random force, taken as
Gaussian for convenience; the prescribed zero-mean initial condition u(0) in
front of the Dirac distribution δ(t) is also taken as random Gaussian. Obvi-
ously, (10.5) can be rewritten as an equivalent integral equation
t t
u(t) + dt eL(t−t ) B(u(t ), u(t )) = dt eL(t−t ) f (t ) + u(0). (10.6)
0 0
u + B(u, u) = F . (10.7)
15 Kraichnan, 1958c, 1961.
338 Eyink & Frisch
Let us now describe the random coupling model in the form used by Herring
and Kraichnan. Imagine that N independent replicas of (10.7) are written with
N independent fields, labeled uα (α = 1, . . . , N) and that these are then coupled
as follows:
1
uα + φαβγ B(uβ , uγ ) = Fα , α = 1, . . . , N. (10.8)
N β,γ
Here, the Fα are N independent identically distributed (iid) replicas of the force
F in (10.7); the ‘random coupling coefficients’ φαβγ are a set of Gaussian
random variables of zero mean and unit variance. These N 3 coefficients are
taken as all independent, except for the requirement that φαβγ be invariant un-
der all permutations of (α, β, γ), a condition crucial for conservation of energy
and other quadratic invariants. When the number of replicas tends to infinity,
closed equations emerge for the two following objects: the correlation func-
tion U ≡ uα ⊗ uα
and the infinitesimal response function G ≡ δuα /δFα
.
Note that off-diagonal terms with α β tend to zero as N → ∞. The an-
gular brackets denote ensemble averages and ⊗ indicates a tensor product:
if we were dealing with a velocity field ui (x, t), then u ⊗ u would stand for
ui (x, t)u j (x , t ). It is also convenient to introduce auxiliary independent zero-
mean Gaussian random fields v̂ and v having the same correlation function U as
any of the uα . With such abstract notation the DIA equations take the compact
form
18 Kraichnan, 1961: § 9.
19 On large-N gauge theory and quantum gravity, G. ’t Hooft, 1974; Makeenko & Migdal, 1979;
Migdal, 1983; Di Francesco, Ginsparg & Zinn-Justin, 1995.
340 Eyink & Frisch
expansion methods might provide a powerful tool for addressing the difficult
problem of inertial-range turbulence scaling.20
In the present hydromagnetic case, it still may be argued plausibly that the action
of the energy range on the inertial range is equivalent to that of spatially uniform
fields. But, in contrast to a uniform velocity field, a uniform magnetic field has a
profound effect on energy transfer.
Indeed, when the equations of MHD are rewritten in terms of the Elsässer
variables z± ≡ v ± b, and a uniform background field of strength b0 is present,
it is found that:
20 For DIA and Wigner matrices, see Frisch & Bourret, 1970. For 1/N expansion in critical phe-
nomena, see Abe, 1973. For large-N shell models, see Pierotti, 1997; Pierotti, L’vov, Pomyalov
& Procaccia, 2000.
21 The
magnetic field can be given the same dimension as the velocity field after division by
4πμρ where μ is the magnetic permeability and ρ the density: these are the units used here.
22 Alfvén, 1942; von Neumann, 1949; Batchelor, 1950; Lee, 1952; Kraichnan, 1958a.
10: Robert H. Kraichnan 341
In the same paper he also suggested that the truncated Euler system could sup-
port an energy cascade just as in the Navier–Stokes system, for “a statistical
ensemble whose initial distribution is multivariate-normal, with all energy con-
centrated in wavenumbers the order of k0 ”. In 1989 he and S. Chen went much
further:
the truncated Euler system can imitate NS fluid: the high-wavenumber degrees
of freedom act like a thermal sink into which the energy of low-wave-number
28 Kraichnan, 1958a, 1959b; Kawasaki, 1970.
10: Robert H. Kraichnan 345
modes excited above equilibrium is dissipated. In the limit where the sink wave-
numbers are very large compared with the anomalously excited wavenumbers,
this dynamical damping acts precisely like a molecular viscosity.
turbulence (randomness, disorder, etc.) For example, J.G. Charney in 1948 for-
mulated the quasi-geostrophic model, in which potential vorticity (like vortic-
ity for 2D Euler) is conserved along every fluid element. A little later, J. von
Neumann made a number of interesting remarks on 2D turbulence, in particu-
lar that it is expected to have far less disorder than in 3D, precisely because of
vorticity conservation; but this material remained unpublished for a long time.
At the very end of his 1953 turbulence monograph, G.K. Batchelor proposed
to investigate the spottiness “of the energy of high wave-number components”
using 2D turbulence. He observed that in a 2D ideal flow the conservation of
the integral of (one half of) the squared vorticity – now called ‘enstrophy’, a
term coined by C.E. Leith, one which we shall use liberally – prevents energy
from solely flowing to high wavenumbers: some energy has to be transferred
also to smaller wavenumbers (larger scales). Batchelor also concurred with
Onsager about the tendency of small vortices of the same sign to merge into
larger vortices. In the same year, R. Fjørtoft showed that, because of the simul-
taneous conservation of energy and enstrophy, it is impossible for 2D dynamics
to change the amplitudes of only two (Fourier) modes. Within a triad of modes,
he showed that the change in energy for the member of the triad with interme-
diate wavenumber is the opposite to that of the other two members and that the
member with lowest wavenumber shows the largest energy change of the two
extreme members.31
It is thus clear that there must be significant inverse transfer of energy in 2D.
However, even in 3D there is some inverse transfer of energy for the case of
freely decaying high Reynolds number flow, where the peak of the energy spec-
trum migrates to smaller wavenumbers. What about the K41 energy cascade
and the presence of an inertial range over which the energy flux is uniform?
Lee showed that a direct energy cascade is not possible in 2D, because it would
violate enstrophy conservation.32 No conjecture can be found in the literature
before 1967 positing any type of 2D power-law cascade range.
Then came Kraichnan. The abstract of his first 1967 paper is worth quoting
in full:
31 See Taylor, 1917: pp. 76–77; Charney, 1947; von Neumann, 1949: §§ 2.3–2.4; Batchelor, 1953:
pp. 186–187; Fjørtoft, 1953.
32 Lee, 1951.
10: Robert H. Kraichnan 347
with zero-vorticity flow. The −3 range gives an upward vorticity flow and zero-
energy flow. The paradox in these results is resolved by the irreducibly triangular
nature of the elementary wavenumber interactions. The formal −3 range gives a
nonlocal cascade and consequently must be modified by logarithmic factors. If
energy is fed in at a constant rate to a band of wavenumbers ∼ ki and the Reynolds
number is large, it is conjectured that a quasi-steady-state results with a − 53 range
for k ki and a −3 range for k ki , up to the viscous cutoff. The total kinetic
energy increases steadily with time as the − 53 range pushes to ever-lower k, until
scales the size of the entire fluid are strongly excited. The rate of energy dissi-
pation by viscosity decreases to zero if kinematic viscosity is decreased to zero
with other parameters unchanged.
Galerkin-truncated to a wavenumber band [kmin , kmax ] with 0 < kmin < kmax ,
the absolute equilibrium e−(αE+βΩ) /Z is Gaussian with an energy spectrum
k
E(k) = , (10.13)
α + βk2
where α and β > 0 are constrained by the knowledge (say, from the initial
kmax
conditions) of the total energy E = k E(k) dk and of the total enstrophy
kmax min
Ω = k k2 E(k) dk ≥ kmin 2
E. If Ω/(kmin
2
E) is very close to unity, it is seen
min
that the ‘inverse temperature’ α must be negative and that the spectrum (10.13)
displays a strong peak near kmin . The situation is in stark contrast to the case
of 3D absolute equilibria: for 3D parity-invariant flow the absolute equilib-
rium energy spectrum is proportional to k2 , as we have seen, and always peaks
at the highest wavenumber. These results suggested to Kraichnan that “a ten-
dency toward equilibrium in an actual physical flow should involve an upward
flow of vorticity and, therefore, by the conservation laws, a downward flow
of energy”. For good measure, Kraichnan gave two independent arguments to
justify the same conclusions. Adapting earlier considerations of Fjørtoft, he
noted that these cascade directions follow if the transfer in each triad is “a
statistically plausible spreading of the excitation in wavenumber: out of the
middle wavenumber into the extremes”. Finally, he showed that such diffusive
spreading in wavenumber indeed develops instantaneously for an initial Gaus-
sian statistical distribution, applying expressions of W.H. Reid and Ogura for
the quasi-normal closure in 2D.34
But are these formal similarity ranges physically realizable? Kraichnan next
turned to this question. Since a k−5/3 range gives a divergent energy at small
wavenumbers and the k−3 range a logarithmically divergent enstrophy, cascade
ranges of arbitrary extent require forcing at some intermediate wavenumber
ki . For finite ranges, Kraichnan noted, there must be some ‘leakage’ of energy
input ε to high wavenumbers and of enstrophy input η to low wavenumbers.
As either of the ranges increases in length it becomes a ‘purer’ cascade, due
to the blocking effect of conservation of the dual invariant. The enstrophy cas-
cade will proceed up to the cutoff wavenumber kd = (η/ν3 )1/6 set by viscosity,
with a vanishingly small energy dissipation εd ∼ ε(ki /kd )2 for kd ki . If there
is no minimum wavenumber k0 , Kraichnan concluded that an inverse cascade
should proceed for ever to smaller and smaller wavenumbers which, on dimen-
sional grounds, scale as a ε−1/2 t−3/2 , where t is the time elapsed. These cas-
cade ranges are only plausible universal states, Kraichnan observed, if the cas-
cade dynamics are scale-local, with the dominant nonlinear interactions among
34 Reid, 1959; Ogura, 1963.
10: Robert H. Kraichnan 349
A further point is that the nonlocalness of the transfer in the −3 range suggests
in itself that [the] cascade there is not accompanied by degradation of the higher
statistics in the fashion usually assumed in a three-dimensional Kolmogorov cas-
cade. This is consistent with a picture of the transfer process as a clumping-
together and coalescence of similarly signed vortices with the high-wavenumber
excitation confined principally to thin and infrequent shear layers attached to the
ever-larger eddies thus formed.
This is the only point in the paper where Kraichnan speculates on physical-
space mechanisms, clearly influenced by the statistical mechanics argument of
Onsager.
Kraichnan considered finally in his 1967 paper the situation that the fluid is
confined to a finite domain with a minimum wavenumber k0 ki . He wrote:
The conjecture is offered here that after the − 53 range reaches down to wavenum-
bers ∼ k0 the downward cascade from ki continues and the energy delivered to
the bottom of the range piles up in the mode k0 . As the energy in k0 rises suf-
ficiently, modification of the − 53 range toward absolute equilibrium is expected,
starting at the bottom and working up to progressively larger wavenumbers.
The present study grew out of an investigation of the approach of a weakly cou-
pled boson gas to equilibrium below the Bose–Einstein condensation tempera-
ture. There is a fairly close dynamical analogy in which the number density and
kinetic-energy density of the bosons play the respective roles of kinetic-energy
density and squared vorticity.
Two very original concepts enter into turbulence theory with Kraichnan’s
landmark work. The first idea is that a single system may support two co-
existing cascades with different spectral ranges: in 2D, the dual cascades of
energy and enstrophy. The second idea is that there may be constant flux spec-
tral ranges that correspond to an inverse cascade, from small to large scales.
Kraichnan’s concept of the 2D inverse energy cascade is very far from the
Richardson–Kolmogorov vision of 3D turbulence, in which energy, introduced
at large scales either through the initial conditions or by suitable forces or by
instabilities, cascades to smaller scales and eventually dissipates by viscosity
into heat. Two other groups were, however, pursuing ideas very closely re-
lated to those of Kraichnan, at this same time. These were G.K. Batchelor and
R.W. Bray at Cambridge, UK, and V. Zakharov in the Soviet Union.
The 2D enstrophy cascade was proposed independently of Kraichnan, and
even somewhat earlier, by Batchelor. This result is reported in the 1966 Cam-
bridge PhD dissertation of Bray. At the beginning of §1.4 he gave his super-
visor, Batchelor, credit for the idea that the enstrophy dissipation rate could
remain finite in the limit of vanishing viscosity. This led Bray to suggest an
enstrophy cascade with a k−1 spectrum and thus a k−3 energy spectrum. Bray
attempted to check this theory by performing a 2D spectral numerical simula-
tion, probably the first of its kind. These results were not made public, however,
until a 1969 paper authored by Batchelor. The analysis of Batchelor and Bray
is remarkably complementary to Kraichnan’s. They considered only decaying
2D turbulence, not forced steady states. Most of their physical discussion was
also in real space, not in spectral space, and focused on the analogy between
stretching of vorticity-gradients in 2D and Taylor’s vortex-stretching mecha-
nism in 3D. Neither in Bray’s thesis nor in Batchelor’s paper was there any
discussion of a separate −5/3 range in 2D with constant flux of energy to large
scales (Batchelor was, by 1969, aware of Kraichnan’s earlier paper and cited it
in his work).36
The notion of two distinct power-law ranges already appears in the 1966
PhD thesis of Zakharov, on the weak turbulence of gravity waves. One of these
ranges was identified as a direct cascade of energy to high-wavenumber, but
the physical interpretation of the other was not clearly identified in the thesis.
Shortly afterward, however, Zakharov hit upon the idea that the second power-
law range corresponds to an inverse cascade of wave action or “quasi-particle
number”. It thus appears that Kraichnan and Zakharov arrived independently
at the idea of an inverse cascade although Kraichnan, it seems, got the idea
slightly earlier into print. Both Kraichnan and Zakharov also clearly realized
the applicability of these notions of dual cascades and inverse cascade to many
other systems, including quantum dynamics of Bose condensation.37
The subject of 2D turbulence continued to interest Kraichnan until the early
1980s, at least, and he wrote several later papers which sharpened the predic-
tions and clarified the physics of his 1967 theory. In a 1971 J. Fluid Mech.
article he applied his TFM closure to both 2D inverse and 3D direct energy
cascades and obtained quantitative results on spectral coefficients. Kraichnan
also studied the 2D enstrophy cascade and, in particular, worked out the log-
arithmic correction mentioned in 1967. A 1976 paper in J. of Atmosph. Sci.
disseminated these results to the community of meteorologists, with special
attention to the phenomenological concept of ‘eddy viscosity’. Kraichnan pro-
posed there a new interpretation of the inverse cascade in terms of a negative
eddy viscosity, an idea that goes back to V. Starr, and he gave a very simple
heuristic explanation for this effect:
If a small-scale motion has the form of a compact blob of vorticity, or an assem-
bly of uncorrelated blobs, a steady straining will eventually draw a typical blob
out into an elongated shape, with corresponding thinning and increase of typical
wavenumber. The typical result will be a decrease of the kinetic energy of the
small-scale motion and a corresponding reinforcement of the straining field.
This idea has been particularly influential in the geophysical literature, where
it has often been invoked to explain inverse energy cascade.38
What is the empirical status of Kraichnan’s dual cascade theory of 2D tur-
bulence? A complete review would be out of place here, but we shall briefly
discuss its verification with an emphasis on the most current work. Only very
recently, in fact, has it become possible to observe both 2D cascades, inverse
energy and direct enstrophy, in a single simulation. This has required herculean
computations at spatial resolutions up to 32,7682 grid points. The simulations
confirm Kraichnan’s predictions for the k−5/3 and k−3 ranges, with less ac-
curacy for the latter due to finite-range effects. Earlier numerical simulations
and laboratory experiments which have focused on a single range have, how-
ever, separately confirmed the predictions of the 1967 paper. A number of
numerical studies of the enstrophy cascade with ‘hyperviscosity’ (powers of
the Laplacian replacing the usual dissipative term) have reported observing
the log-correction to the energy spectrum. The quasi-steady inverse cascade
37 Zakharov, 1966. B. Kadomtsev informed Zakharov of Kraichnan’s 2D paper sometime around
1969 (V. Zakharov, private communication, 2010).
38 For Kraichnan’s application of TFM closure to 2D turbulence cascades, see Kraichnan, 1971b,
1976c; for negative viscosity, see Starr, 1968; Kraichnan, 1976c; Kraichnan & Montgomery,
1980: §4.4; for vortex-thinning mechanism of inverse cascade, see Kraichnan, 1976c; Rhines,
1979; Salmon, 1980, 1998: p. 229. Another notable paper on 2D turbulence is Kraichnan,
1975b.
352 Eyink & Frisch
10.4 Intermittency
Intermittency is a rather general term referring to the spottiness of small-scale
turbulent activity, be it at dissipation-range scales or at inertial-range scales. In
the late 1940s Batchelor and A.A. Townsend observed intermittent behavior
of low-order velocity derivatives; since such derivatives come predominantly
from the transition region between the inertial and dissipation ranges, this inter-
mittency cannot be directly taken as evidence that the self-similarity postulated
for the K41 inertial range is breaking down.41
39 For numerical simulation of simultaneous cascades, see Boffetta, 2007; Boffetta & Musacchio,
2010; for the log-correction in the enstrophy cascade, see Borue, 1993; Gotoh, 1998; Pasquero
& Falkovich, 2002; for quasi-steady cascade, see Smith & Yakhot, 1993; for vortex thinning,
see Chen et al., 2006; Xiao et al. 2008; for condensations, see Chertkov et al., 2007; Bouchet
& Simonnet, 2009.
40 For direct enstrophy cascade in the Earth’s stratosphere, see Cho & Lindborg, 2001; for inverse
energy cascade in the South Pacific, see Scott & Wang, 2005.
41 Batchelor & Townsend, 1949.
10: Robert H. Kraichnan 353
that the full hierarchy of moment or cumulant equations derived for statistical
solutions of the Navier–Stokes equation is compatible with the scale-invariant
K41 theory in the limit of infinite Reynolds numbers. But Kraichnan was also
aware that K41 is equally compatible with the Burgers equation, which def-
initely has no K41 scaling (because of the presence of shocks); he also no-
ticed that the presence of the pressure in the incompressible Navier–Stokes
was likely to reduce the intermittency one would otherwise expect from a sim-
ple vortex-stretching argument. Closure seemed incapable of saying anything
about the breaking of the K41 scale-invariance (one major exception to this
statement is discussed in Section 10.4.3).45
At first Kraichnan examined critically the toy models developed by the Rus-
sian school, observed that εr is not a pure inertial-range quantity and proposed
to study intermittency in terms of more appropriate quantities, such as the local
fluctuations of the energy flux associated with a wavenumber k in the inertial
range. An estimate of this flux is u3r /r, where r ∼ 1/k and ur is, say, the mod-
ulus of the velocity difference between two points separated by a distance r.
With this in mind, he wrote:
If we increase the intermittency by making the fluid into quiescent regions with
√
negligible velocity and active regions, of equal extent, where ur increases by 2,
then the mean kinetic energy in scales√order r is unchanged but the time constant
decreases, and hence ε increases, by 2. This example suggests, first, that if Kol-
mogorov’s theory holds in subregions of the fluid, then the constant f (0) in the
inertial-range law can be universal only if intermittency in the local dissipation
εr , defined as average dissipation over a domain of size r, somewhat tends to a
universal distribution. Second, if intermittency increases as scale size decreases,
and Kolmogorov’s basic ideas hold in local regions, then the cascade becomes
more efficient as r decreases and E(k) must fall off more rapidly than k−5/3 if,
according to conservation of energy, the overall cascade rate is r independent.46
A few years later this remark, together with ideas of Mandelbrot, became a key
ingredient in the development of the β-model, a phenomenological model of
intermittency that uses exclusively inertial-range quantities.47
Kraichnan pursued some of these ideas further himself in an influential 1974
paper in J. Fluid Mech. This paper is pure Kraichnan. A wealth of intrigu-
ing ideas are tossed out, very original model calculations sketched in brief,
and clever counterexamples devised against conventional ideas. At least two
contributions of this paper are now well known. First, Kraichnan proposed a
45 For K41 compatibility of the Navier–Stokes equations, see Orszag & Kruskal, 1966; for the
differences between Burgers and Navier–Stokes turbulence, see Kraichnan, 1974a, 1991.
46 Kraichnan, 1972: p. 213.
47 For using inertial-range quantities, see Kraichnan, 1972: p. 213, 1974a; Frisch, Sulem &
Nelkin, 1978.
10: Robert H. Kraichnan 355
turbulence with large Schmidt number (see below). It was thus quite natural
for Kraichnan to see how well the closure tools he developed for turbulence in
the 1950s and the 1960s were able to cope with passive scalar dynamics. He
applied his LHDIA closure to the passive scalar problem, for example, repro-
ducing Obukhov–Corrsin scaling with precise numerical coefficients.49
In 1968 Kraichnan realized that a closed equation can be obtained for a
scalar field passively advected by a turbulent velocity with a very short correla-
tion time, without any further approximation. The DIA closure is exact for this
special system, reducing to a single equation for the scalar correlation function
at two space points and simultaneous times. The mean Green function reduces
to a Dirac delta because of the zero correlation-time assumption. This 1968
model is now usually called ‘the Kraichnan model’ [of passive scalar dynam-
ics] and has assumed a paradigmatic status for turbulence theory, comparable
to that of the Ising model in the statistical mechanics of critical phenomena. Its
importance stems from a string of major discoveries by Kraichnan and others
on the fundamental mechanism of intermittency, some of which will be de-
scribed only briefly because they took place in the 1990s. Kraichnan showed
that even when the velocity field is not at all intermittent, e.g. a Gaussian ran-
dom field, the passive scalar (henceforth called ‘temperature’ for brevity) can
become intermittent and this in several ways.50
A first mechanism, which applies in the far dissipation range, is basically
the same as described in § 10.4.1 and will not concern us further.
A second mechanism identified by Kraichnan concerns the so-called Batch-
elor regime: when the Schmidt number ν/κ is large, there is a range of scales
for which the velocity field is strongly affected by viscous dissipation, but the
temperature field does not undergo much diffusion; in this regime the velocity
field can be locally replaced by a uniform random shear.51 Tiny, well-separated
temperature blobs are then stretched and squeezed in a way which is amenable
to asymptotic analysis at large times. Actually doing this in a systematic way
would have required all kinds of heavy-duty theoretical tools: path integrals,
large deviation theory, fluctuations of Lyapunov exponents, etc.52 It is then
possible to show that the distributions of spatial derivatives of the tempera-
ture display a log-normal-type intermittency at zero diffusivity53 and a weaker
49 Obukhov, 1949; Corrsin, 1951; Yaglom, 1949; Batchelor, 1959; see also Chapters 8, 7 and 6.
On LHDIA for passive scalars, see Kraichnan, 1965a: § 5–7.
50 Kraichnan, 1968b, 1974b, 1994.
51 Batchelor, 1959, and Chapter 8.
52 See, e.g., the review by Falkovich, Gawȩdzki & Vergassola, 2001.
53 This is a nontrivial variant of the obvious result that when m(t) is a scalar Gaussian ran-
dom function the solution of the differential equation dq(t)/dt = m(t)q(t) with q(0) = 1 is
log-normal.
10: Robert H. Kraichnan 357
form of intermittency in the regime with non-zero diffusivity. Actually, all this
was done – and correctly so – by Kraichnan in a remarkable paper published in
1974, just after the paper on Kolmogorov’s inertial-range theories.54 This paper
is a tour de force, combining very original analytical arguments and deep phys-
ical intuition to reach exact conclusions, without any assistance from the ad-
vanced mathematical methods that were later applied to this problem. Kraich-
nan’s analysis was carried out for general space dimension d – following a
suggestion of M. Nelkin – and one intriguing finding was that intermittency of
the scalar vanished in the limit d → ∞. Kraichnan’s work, which was going
to strongly influence subsequent, more formally rigorous, analyses, showed a
thorough understanding of the mechanism of intermittency in the Batchelor
regime.
The third mechanism identified by Kraichnan was rather close to one of the
Holy Grails of turbulence theory, namely understanding inertial-range anoma-
lous scaling and predicting the scaling exponents. In 1994 Kraichnan conjec-
tured that when the velocity u(x, t) is Gaussian with a power-law spectrum
(K41 would be one instance) and with a very short correlation time (white-in-
time), then for vanishingly small κ the structure functions of the temperature
display anomalous scaling. This is a rather amazing proposal: how can a self-
similar velocity field act on a transported temperature field to endow it with
anomalous scaling and thus with lack of self-similarity? As we shall see, the
qualitative aspects of Kraichnan’s conjecture have been fully corroborated by
later work.55
Now we shall have to become slightly more technical to explain how Kraich-
nan tackled this problem, starting with his 1968 work. Let us rewrite the tem-
perature equation (10.14) in abstract form
1 t
M̃(t) −→ M̃ 2 , → 0, (10.16)
54 Kraichnan, 1974b.
55 Kraichnan, 1994.
358 Eyink & Frisch
Similar closed equations can be derived for p-point moments of the temper-
ature. In 1968 Kraichnan derived the equation for the two-point temperature
correlation functions using this technique and found that the second-order tem-
perature structure functions displayed scaling. The scaling exponent ζ2 can
actually be obtained by simple dimensional analysis. So far no evidence of
anomalous scaling had emerged.
By a method similar to that used in 1968 for the two-point correlations of a
passive scalar, Kraichnan derived in 1994 an equation for the structure function
of order p. This equation is not closed (contrary to the equation for the p-point
correlation function), but Kraichnan proposed a plausible approximate closure
ansatz from which he derived the following scaling exponents ζ p for the pth-
order structure function:
1 1
ζ2p = 4pdζ2 − 2 + (d − ζ2 )2 − (d − ζ2 ). (10.19)
2 2
Since ζ2p is obviously not equal to pζ2 , as would be required by self-similarity,
(10.19) implies anomalous scaling. One year later it was shown that there is in-
deed anomalous scaling, using a zero modes method, borrowed partially from
field theory: the equation for the moments of order 2p has a linear operator L2p
acting on the 2p-point correlation function and an inhomogeneous right-hand
side involving correlation functions of lower order. The zero modes correspond
to certain functions of 2p variables which are killed by L2p . Actually determin-
ing the zero modes turned out to be quite difficult. In most instances it could be
done only perturbatively, using as small parameter either the roughness expo-
nent ξ of the prescribed velocity field or the inverse of the dimension of space
d (as anticipated by Kraichnan’s 1974 paper). The results agreed with numeri-
cal simulations, but did not agree with (10.19) except for a single value ξ 1.
Kraichnan’s prediction (10.19) must cross the numerical curve at one point,
trivially, but it is possibly significant that Kraichnan’s closure ansatz works
best in the regime where the cascade dynamics is scale-local.57
56 Hashminskii, 1966; Frisch & Wirth, 1997.
57 Kraichnan, 1994. For zero-mode methods, see Gawȩdzki & Kupiainen, 1995; Chertkov et al.,
1995; Shraiman & Siggia, 1995; and the review by Falkovich, Gawȩdzki & Vergassola, 2001.
For simulations, see Frisch, Mazzino & Vergassola, 1998; Gat, Procaccia & Zeitak, 1998;
Frisch et al., 1999. The whole story about anomalous scaling for passive scalars is recounted in
www.oca.eu/etc7/work-on-passive-scalar.pdf.
10: Robert H. Kraichnan 359
of the system is not known, suitable averages of certain quantities are known
for a representative ensemble of similar systems”. The article is unusually well
written for a first paper and indicates considerable maturity of the young sci-
entist who had already been active for six years, although he refrained from
publishing.
10.5.4 Conclusions
Our survey has focused on three of Kraichnan’s contributions to turbulence
theory:
(1) spectral closures and realizability;
(2) inverse cascade of energy in 2D turbulence;
(3) intermittency of passive scalars advected by turbulence.
These are, arguably, his most significant achievements which have had the
greatest impact on the field. Spectral closures of the DIA class still have nu-
merous interesting applications when the questions under investigation do not
depend crucially on deviations from K41. Even today an EDQNM calculation,
for example, will often be the first line of assault on a difficult new turbulence
problem. Furthermore, Kraichnan’s criterion of realizability has become part
of the standard toolbox of turbulence closure techniques. Realizability is nec-
essary both for physical meaningfulness and, often, for successful numerical
solution of the closure equations. Kraichnan’s prediction of inverse cascade
has been well verified by experiments and simulations and has relevance in
explaining dynamical processes in the Earth’s atmosphere and oceans. The
concept of an inverse cascade has proved very fruitful in other systems, too,
where similar fluxes of invariants to large scales may occur, such as magnetic
helicity in 3D MHD turbulence, magnetic potential in 2D MHD turbulence,
and particle number in quantum Bose systems. Finally, Kraichnan’s model of
a passive scalar advected by a white-in-time Gaussian random velocity has
become a paradigm for turbulence intermittency and anomalous scaling – an
‘Ising model’ of turbulence. The theory of passive scalar intermittency has
not yet led to a similar successful theory of intermittency in Navier–Stokes
turbulence. However, the Kraichnan model has raised the scientific level of
discourse in the field by providing a nontrivial example of a multifractal field
generated by a turbulence dynamics. It is no longer debatable that anomalous
scaling is possible for Navier–Stokes.64
64 For the inverse magnetic helicity, see Frisch et al., 1975. For the inverse magnetic potential
cascade, see Fyfe & Montgomery, 1976. For Bose condensates, see Semikoz & Tkachev, 1997.
10: Robert H. Kraichnan 363
Regarding intermittency/anomalous scaling, note that there have been many incorrect ‘proofs’
of their absence in Navier–Stokes turbulence, to which the Kraichnan model is a counterexam-
ple; see, e.g., Belinicher & L’vov, 1987.
65 Concerning the various topics that could not be covered in this review, see, for pressure fluc-
tuations: Kraichnan, 1956a, 1956b, 1957a; for shear flow turbulence: Kraichnan, 1964a; for
magnetic dynamo: Kraichnan & Nagarajan, 1967; Kraichnan, 1976a, 1976c, 1979b; for Vlasov
plasma turbulence: Kraichnan & Orszag, 1967c; for predictability and error growth: Kraichnan,
1970b; Kraichnan & Leith, 1972; for Burgers: Kraichnan, 1968a, 1999; Kraichnan & Gotoh,
1993; for quantum turbulence: Kraichnan, 1967a; for path-integrals: Kraichnan, 1958a: § 4.3;
Lewis & Kraichnan, 1962; for self-consistent Langevin models: Kraichnan, 1970c; for varia-
tional approaches: Kraichnan, 1958a: § 4.3; Kraichnan, 1979a; for Wiener chaos expansions:
Kraichnan, 1979a; for Padé approximants: Kraichnan, 1968c, 1970a; for decimation: Kraich-
nan, 1985, 1988; for mapping closure: Kraichnan et al., 1989, Kraichnan, 1991; Kimura &
Kraichnan, 1993; Kraichnan & Gotoh, 1993; for a critique of Tsallis statistics for turbulence:
Gotoh & Kraichnan, 2004.
66 The Germans have aptly called Sitzfleisch the ability to spend endless hours at a desk doing
grueling work. Sitzfleisch is considered by mathematicians to be a better gauge of success than
any of the attractive definitions of talent with which psychologists regale us from time to time
(Gian-Carlo Rota, 1996: p. 64).
67 Los Alamos (May 1998) for Kraichnan’s 70th birthday and Santa Fe (May 2009) and Beijing
(September 2009) after he left us in 2008.
364 Eyink & Frisch
Acknowledgements Many have helped us with their remarks and their own
recollections. We are particularly indebted to B. Castaing, C. Connaughton,
G. Falkovich, H. Frisch, T. Gotoh, J.R. Herring, C.E. Leith, H.K. Moffatt,
S. Nazarenko, S.A. Orszag, I. Procaccia, H. Rose, E.A. Spiegel, K. Sreeni-
vasan, B. Villone and V. Zakharov. GE’s work was partially supported by
NSF Grant Nos. AST–0428325 & CDI–0941530 and UF’s by COST Action
MP0806 and by ANR ‘OTARIE’ BLAN07–2 183172.
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11
Satish Dhawan
Roddam Narasimha
11.1 Introduction
Satish Dhawan was born on 25 September 1920 in Srinagar, Kashmir, the home
town of his mother Lakshmi. His father, Devidayal, was from the North West-
ern Frontier Province; both parents came from professional families, full of
doctors, lawyers and academics – Devidayal retired as a respected judge of the
High Court in Lahore, now in Pakistan. Satish’s education began under pri-
vate tutors at home, as his father kept getting transferred in his early career
from one town to another in the North West (Kipling country to Indo-British
readers). He completed his Indian education at the University of Punjab in La-
hore with an unusual combination of degrees: BA in physics and mathematics
(1938), MA in English literature (1941) and BE (Hons.) in mechanical engi-
neering (1945). In 1946 he sailed to the USA on a government scholarship, and
obtained an MS in aeronautical engineering from the University of Minnesota
the following year. (The summer of 1947 saw much turmoil in the subcontinent
preceding its imminent partition, and Satish’s parents reluctantly left Lahore
for India – never to return – a week before the formal end of colonial rule.) In
the USA Satish moved to the California Institute of Technology where, with
Hans W. Liepmann as his adviser, he obtained the degree of Aeronautical En-
gineer in 1949 and a PhD in aeronautics and mathematics in 1951. Dhawan
made a strong impression, scientifically and otherwise, on everybody he came
in contact with at Caltech. In an obituary he wrote in 2002, Liepmann noted
Dhawan’s “unusual maturity in judging both scientific and human problems”,
his very good sense of humour, and the way he was “immediately accepted and
respected by [the] highly competent and proud group of young scientists” who
worked with Liepmann at the time.
At the end of tenure of the scholarship Dhawan returned to India and joined
the Indian Institute of Science (IISc) as a Senior Scientific Officer. He rose
373
374 Narasimha
Figure 11.1 (Left) Satish Dhawan at Nandidurg, a small hill resort, approximately
45 km from Bangalore; c. 1955. (Right) c. 1985.
The authors defend this admittedly “illogical approach” by saying that the
problem arose naturally from earlier investigations of their own in transonic
flow (Liepmann, 1946) and those of Ackeret, Feldmann & Rott (1946) in su-
personic flow. These two reports, together with LRD, constitute in fact the first
systematic studies of the phenomenon. The effects were seen as dramatic and
clearly of importance even in the general state of inadequate knowledge in su-
personic boundary layers at the time. The reason was that the presence of a
376 Narasimha
viscous layer at the wall can so strikingly alter the classical inviscid flow pic-
ture of reflection because it introduces an additional boundary condition of no
slip at the surface. Even the boundary layer approximation becomes question-
able, because of large pressure gradients across the flow.
Clearly, viscosity can diffuse the effect of the shock wave upstream in sub-
sonic flow near the surface, and the associated adverse pressure gradient can
even cause separation of the flow. Between the laminar–turbulent and attached–
separated states in the boundary layer, and weak–strong, normal–oblique in the
shock wave, many different combinations become possible. The phenomenon
may in particular involve more than one shock system. The pioneering work at
Caltech and Zurich shed much light on the many physical processes operating
in the different flow combinations.
The difference in effect between turbulent and laminar boundary layers had
been observed in transonic flows, but there the incoming shock at some dis-
tance from the boundary layer actually depends on the boundary layer itself as
well as on the flow field, and therefore cannot be controlled independently. On
the other hand in supersonic flow, in particular in the interaction on a flat plate,
the flow phenomena are clearer, and the LRD report provides both pressure
distributions and schlieren pictures of the flow that were so revealing that they
quickly found their way into textbooks (e.g. Schlicting, 1955, pp. 300, 301).
The major result of the investigation is in Figure 11.2, which shows the dra-
matic difference between the pressure distributions on the surface depending
on whether the boundary layer thereon is laminar or turbulent. In the lami-
nar case the effect is felt even 50 boundary-layer thicknesses upstream of the
shock, whereas in the turbulent case it is felt only over 5 boundary-layer thick-
nesses, making it closer to the inviscid limit. As a secondary investigation there
is a brief digression on transition, which is visualized using an oil film tech-
nique. This enabled sketches of turbulent wedges in laminar flow and also of
the patterns resulting from tripping the boundary layer using a wire.
Shock–boundary-layer interactions were extensively studied in the subse-
quent two decades, both experimentally and theoretically. In laminar flows the
physical mechanisms responsible became largely understood, following the
work of Lighthill (1953) and others. The shorter region of upstream influence
in turbulent boundary layers could be interpreted as due to the much thinner
subsonic viscous layer they possess in comparison to that in laminar layers.
Correspondingly, interaction in much of the turbulent layer away from the wall
is inviscid in character. As in any turbulent flow the problem of closure presents
itself here too, and a variety of models have been used with variable results. A
review of the status three decades after the first systematic experimental studies
of the phenomenon was provided by Adamson & Messiter (1980).
11: Satish Dhawan 377
Figure 11.2 Pressure distribution in shock reflection from a flat surface with lam-
inar and turbulent boundary layers on surface. From Liepmann et al. (1951).
Direct measurement of skin friction This was the subject of Dhawan’s doc-
toral thesis. Dhawan once told me that von Kármán would ask how it was that
such a technologically and scientifically important parameter as the skin fric-
tion, for which there was a prediction by the generally successful boundary-
layer theory in laminar flow, had not yet been directly measured. Liepmann
and Dhawan set about tackling that question. Dhawan’s answer is described in
the NACA Report 1121 (completed in 1951, published in 1953). This report –
very well and clearly written, by the way – starts by noting how both wave
drag and induced drag are better understood, theoretically as well as experi-
mentally, than skin friction drag and boundary layer separation. It then reviews
both direct and indirect methods of measuring friction drag, and mentions the
early efforts of Froude and Kempf, and of Schultz-Grunow (1940), to mea-
sure directly the tangential force on a free-moving or floating element of the
surface. The great advantage of the method, as seen in the report, was that
it did not assume the validity of boundary-layer (or any other) theory – and to
that extent would provide an independent test of theory. This might now seem a
378 Narasimha
c−1/2
f = A + B log10 (c f Re)1/2 , (11.1)
where c f is the local skin friction coefficient (wall stress in free-stream dy-
namic pressure units), Re = U x/v is the Reynolds number based on distance x
from the leading edge and free-stream velocity U, and A and B are constants.
However, Dhawan notes that the values that fit his data, A = −0.9, B = 5.06,
were different from von Kármán’s values, A = 1.7 and B = 4.15, and so “this
agreement is believed to be fortuitous to some extent”. [Incidentally the quoted
values of B correspond to von Kármán ‘constants’ (or coefficients) of 0.455 and
0.554 respectively – both higher than the current range of accepted values, but
Dhawan’s is closer.] He further adds that these differences in the constants
are perhaps to be attributed to the fact that A and B are not absolute constants but
depend somewhat on the conditions of the experiment, for example, on Reynolds
number and so forth.
This issue still continues to be debated (see e.g. Marusic et al., 2010). Dhawan
finally concludes that “the logarithmic formula of von Kármán is a fair approx-
imation for incompressible flow”.
Interestingly, the same experimental data of Dhawan are reproduced by
Schlichting (1955, Figure 7.11), except that the theoretical turbulent skin
11: Satish Dhawan 379
Figure 11.3 Local skin friction coefficient in laminar and turbulent boundary lay-
ers. In the former estimates from velocity profile measurements are also included.
From Dhawan (1951).
leading edge, among others. The measurements are again close to von
Kármán’s theory, but Dhawan is (also again) very cautious: despite the close-
ness, “it cannot be concluded that incompressibility lowers the skin friction
coefficient by an amount given by von Kármán’s empirical curve”.
Interestingly, there are preliminary measurements of c f in the region of tran-
sition from laminar to turbulent flow as the Mach number increases from 0.24
to 0.6 (simultaneously with an accompanying increase in the Reynolds number
from about 4.3 × 105 to a little more than 9 × 105 ). These appear to have been
the first ever direct measurements of c f in the transition region of a bound-
ary layer. The data covered the range from laminar nearly-Blasius values to
close to the von Kármán fully turbulent ones. Once again Dhawan is careful
to call the measurements “quite qualitative”, but notes that they cannot be ex-
plained by a steady transition at a critical Reynolds number. Instead, he points
out, Dryden (1936) had already observed intermittent laminar/turbulent flow in
the transition region, and Liepmann (1943) had substantiated this observation.
He then mentions the work of Emmons on turbulent spots, published around
the same time (in fact a month later in July 1951), as providing a reasonable
explanation of the observations.
These two exploratory studies led to the more detailed investigations of Don-
ald Coles (1953) on skin friction in supersonic flow at the Jet Propulsion Lab-
oratory, and the work done in Bangalore on the transition region in low-speed
boundary layers, as I shall shortly describe.
But before doing so I cannot resist the temptation to remark on how
Dhawan’s NACA report brings out the character of his research. Incidentally
the work described there was never published in a journal. There was perhaps
no need, for in those times NACA reports probably had a wider readership than
any aeronautical journal did. One of their advantages was that they were actu-
ally reports – not the highly condensed journal papers of today. So we can see
Dhawan’s method at work: ingenious in design, meticulous in execution and
cautious in interpretation. I have often thought that the same qualities were
responsible for his success as a scientific leader.
11.3 At Bangalore
Dhawan’s first task on his return to India in 1951 was to establish a High Speed
Aerodynamics Laboratory, for which he built two blow-down supersonic tun-
nels, respectively 1 in. by 3 in. and 5 in. by 7 in. test-section size. He also built
a 20 in. by 20 in. low-speed boundary layer tunnel.
I joined the Institute in 1953 for an IISc Diploma (the equivalent of a Mas-
ter’s) in aeronautical engineering, and got to know Dhawan personally as I
11: Satish Dhawan 381
helped him calibrate the 1 in. by 3 in. tunnel and designed its nozzles. At the
end of the two-year course Dhawan asked me if I would like to do research
with him, and I jumped at the chance and registered for the Associateship (a
Master’s by research). I had done a short research project earlier with him
measuring boundary layer velocity profiles in a small low-speed tunnel, and
his suggestion that I should work on boundary layer transition coincided with
my own thoughts. I helped him in setting up the boundary layer lab, built a
hot-wire amplifier and started making turbulence measurements.
Why did Dhawan suggest I work on transition? It is clear from what has
been said before that the subject was very much on his mind when he made
his first preliminary direct skin friction measurements in the transition zone at
Caltech. But, characteristically for Dhawan, there was a strong ‘Indian’ rea-
son as well. He described this in a lecture he gave at the first Asian Congress
of Fluid Mechanics, held in Bangalore in 1980 (Dhawan, 1981). The Depart-
ment had a closed-circuit low-speed wind tunnel with a 5 ft. by 7 ft. elliptic
test-section, with tunnel Reynolds number in an awkward range (1.5 × 106
per ft.) because of transition. Tests were just then being made on a model
of the HF24, a fighter being designed at Hindustan Aircraft (more of this
later), under the leadership of the well-known German designer Kurt Tank
(ex Messerschmitt). Dhawan wanted to explore if we could understand a lit-
tle better how IISc wind tunnel results could be scaled up to flight Reynolds
numbers.
These two motivations were just the kind that appealed to Dhawan, and
drove the transition programme at Bangalore, although it quickly became a
research project entirely in its own right.
Now, intermittent velocity signals, i.e. those showing the presence of turbu-
lent patches (sometimes called bursts), of varying duration and between-patch
time intervals, had frequently been observed in the transition region by various
investigators (Dryden, 1936; Liepmann, 1943). One physical picture inspired
by such observations was that there might be a sharp, fluctuating, jagged front
separating laminar from turbulent flow. Emmons (1951) proposed a radically
different picture in which laminar flow breaks down at isolated points, at each
of which a turbulent ‘spot’ is born. This replaced the laminar-turbulent ‘front’
by a set of ‘islands’ of turbulence in a laminar sea. (The idea was radical be-
cause it permitted a laminar state downstream of the turbulence in the island.
Reynolds (1883) had already observed ‘flashes’ in his pipe where the same
phenomenon occurs; but it was perhaps easier to accept it in a confined duct
than on a semi-open surface.) Emmons further proposed that these spots grow
as they move downstream, eventually leading to fully turbulent flow as they
cover the entire surface. The fraction of time that the flow was turbulent at
any point on the surface (say station x from the leading edge of the surface in
382 Narasimha
turbulent spots and his basic assumptions about their linear propagation char-
acteristics was a powerful trigger for our later work. SK presented the first
intermittency measurements, and was read with great interest by both of us. It
was immediately decided to make similar measurements in the IISc boundary
layer tunnel. We also quickly saw that SK contained an unarticulated puzzle:
why was it that, in spite of confirming Emmons’ physical ideas so convinc-
ingly, the authors did not compare their intermittency measurements with Em-
mons’ theory, but instead fitted them to a ‘Gaussian integral’, i.e. an error func-
tion curve? When I made the comparison, the disagreement between the two
for the intermittency was found to be huge. For example, SK reported that in
natural transition in their famous quiet tunnel at the National Bureau of Stan-
dards (NBS) x was at 6.25 ft; γ was down to 0 already at x = 5.25 ft = 0.84x.
However, at this value of x/x expression (11.3) gives (with m = 0) a value of
γ as high as 0.34 – way beyond any uncertainty in the excellent SK measure-
ments. Basically, therefore, the transition region was much sharper than Em-
mons would predict relative to the distance to the halfway point. The SK data,
and the more extensive datasets we quickly acquired at Bangalore, seemed to
be suggesting that Emmons’ proposal on spot generation across the whole sur-
face of the plate was questionable. Furthermore, the discovery of the ‘calming
effect’, as SK called it, showed that each spot left behind a trail of highly sta-
ble flow, in which further breakdown into spots was unlikely. The function g(x)
may therefore be expected to decrease with increasing x.
After some false starts, it turned out that the simplest explanation of the
data was actually to go to the other extreme and say that all spots were born
at one streamwise location xt , but randomly in time and in spanwise location;
i.e. g is proportional to a Dirac delta function at xt . Upstream of xt laminar
flow is not sufficiently unstable or disturbed, downstream the spots stabilize
the flow.1 This makes xt a strong candidate for being identified with the onset
of transition. The rest of Emmons’ picture stays intact, and (11.3) is replaced
by
⎧
⎪
⎪
⎨ 0& for x ≤ xt ,
γ=⎪ % (11.4)
⎪
⎩ 1 − exp −0.41(x − xt )2 /λ2 for x ≥ xt ,
(Narasimha, 1957), where λ is the distance between points with γ = 0.25 and
0.75. The results agree with observation very well (Figure 11.4) – investing a
precise new meaning to an old length in the problem (namely xt ). In this new
picture transition does not occur everywhere on the plate, and the concept of
an onset location is not only resurrected but indeed found to play a key role.
1 This is not completely true: Wygnanski et al. (1979) found that Tollmien–Schlichting type
waves are excited in the wing-tip regions of isolated spots.
384 Narasimha
Figure 11.5 Boundary-layer thickness in the transition zone, given (except just
downstream of onset) by the fully turbulent value corresponding to a virtual ori-
gin near xt . From Dhawan & Narasimha (1958), reproduced with permission of
Cambridge University Press.
2 Incidentally Coles found that between the three of them Dhawan, Schultz–Grunow and Kempf
had made direct skin friction measurements over more than three decades (roughly one each, by
the way!) in Reynolds number, and displayed this in a figure with a two-page spread in ZAMP
(Coles, 1954b). He “admitted” that his own work was “inspired by the remarkable consistency
of these local friction data . . . with a scatter so small as to be almost unprecedented in bound-
ary layer research”. He introduced one major improvement over Dhawan’s method by using a
nulling technique for force measurement.
386 Narasimha
Figure 11.6 Direct measurements of local skin friction coefficient on a flat plate
during transition from laminar to turbulent flow. Experimental data from Coles
(1954), replotted on a log-linear scale, with the Frankl–Voishel theory for turbu-
lent skin friction in supersonic flow added. Note the overshoot above turbulent
theory (for boundary layer originating at the leading edge), as predicted by spot
theory based on concentrated breakdown at xt (DN). From Dhawan & Narasimha
(1958), reproduced with permission of Cambridge University Press.
c f in supersonic flows, often covering the transition zone. They left no doubt
about the presence of overshoots in c f (Figure 11.6).
Finally, the DN picture predicts that early in the transition zone the displace-
ment thickness δ∗ does not grow monotonically; in fact it has a minimum. This
dip is at first surprising, but the reason from the DN view is clear. For in this
view there is a place where the total boundary-layer thickness δ is the same
for both the laminar layer originating at x ≈ 0 and the turbulent originating
at x = xt (see Figure 11.5). And the γ-weighted mean velocity profile at this
station, reflecting its turbulent part, has to be fuller than the pure laminar flow.
11: Satish Dhawan 387
It follows that δ∗ has to be lower. Estimates of the δ∗ from SK data confirm the
prediction (Figure 11, Narasimha, 1985).
On the whole the broad characteristics of then-recent observations fitted the
DN picture very well. The intermittency data could, I am sure, be made to fit
other curves, including some in the family considered by Emmons and Bryson,
with an appropriate choice of constants (including xc ). Such attempts were in-
deed made later, for example by Abu-Ghannam & Shaw (1980). However, the
conceptual framework of DN not only explains the main features in simple
terms, but has other advantages. For example, it implicitly retains the run-up
to transition onset through instability, which the assumption of constant g ba-
sically renders irrelevant. It also enables predictions of local skin friction or
total drag without ambiguities about what the c f values within the spots are –
as all the spots are born at xt and have (so to speak) about the same age at any
downstream station, their average would to a first approximation be the same
as in the plane turbulent boundary layer with origin at xt . Thus the DN pic-
ture fitted all the observations in the transition zone available at the time into a
coherent scheme which was based on Emmons spots but differed crucially on
their postulated birthplace, i.e. on the assignment of the a priori probability for
g(x).
Assuming concentrated breakdown at one location does however seem ex-
treme, so DN calculated the intermittency when spot formation occurred with
a Gaussian distribution of various widths centred at xt . It became clear that,
while there was a band in x (estimated as being no wider than λ/3 by Narasimha,
1985) in which spots might form, introducing one more length scale describing
the width of the band was generally not worthwhile. This view thus resolved an
old question going back to Prandtl and the rest: is transition abrupt or gradual?
Answer: it is abrupt at onset, occurs at isolated points within a narrow band,
then evolves gradually to an asymptotic state of full-time turbulence.
This work continued for many years at Bangalore with other students, some
working under Dhawan’s supervision, applying the above ideas to pipes, chan-
nels, boundary layers on axisymmetric bodies and under pressure gradients
etc. A review of this work is provided by Narasimha (1985).
Figure 11.8 Two doyens: Satish Dhawan and Zhou Pei-Yuan, at Bangalore in
1980. Both got their PhDs from Caltech, both worked on turbulence (one experi-
mentally, the other theoretically), and both were given high national responsibili-
ties in their respective countries.
Dhawan passed away quietly during the night of 3 January 2002 at home.
His education integrated science, technology and the humanities; so did his
life. A combination of his undoubted technical gifts, unquestioned integrity
and great personal charm enabled him to combine doing and promoting sci-
ence, work for state and society, and manage megatechnology while champi-
oning little science. He left a precious legacy for his country at a special period
in its history.
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effects of turbulence, pressure gradient and flow history. J. Mech. Engg. Sci. 22,
213–228.
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Ackeret, J., Feldmann, F. and Rott, N. 1946. Inst. Aerodyn. ETH, Report no. 10.
Adamson, Jr., T.C. and Messiter, A.F. 1980. Analysis of two-dimensional interactions
between shock waves and boundary layers. Ann. Rev. Fluid Mech. 12, 103–138.
Coles, D.E. 1953. Measurements in the boundary layer on a smooth flat plate in super-
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392 Narasimha
12.1 Introduction
Philip G. Saffman was a leading theoretical fluid dynamicist of the second
half of the twentieth century. He worked in many different sub-fields of fluid
dynamics and, while his impact in other areas perhaps exceeded that in turbu-
lence research, which is the topic of this article, his contributions to the theory
of turbulence were significant and remain relevant today. He was also an in-
cisive and, some might conclude, a somewhat harsh critic of progress or what
he perceived as the lack thereof, in solving ‘the turbulence problem’. This ex-
tended to his own work; he stated in a preface to lectures on homogeneous
turbulence (Saffman, 1968) that
the ideas . . . are new and hopefully important, but are speculative and quite pos-
sibly in serious error.
393
394 Pullin & Meiron
Figure 12.1 Philip G. Saffman (19 March 1931 – 17 August 2008). Photo circa
1975. Source: Caltech Archives.
and the only one of the Saffman children that did not choose to pursue the le-
gal profession. He and his siblings were raised in the Leeds area of England,
but were evacuated to Blackpool during WWII, a move which appeared to
have little effect on his early years. Saffman was something of a prodigy, com-
pleting his high school certificate and taking the Entrance Scholarship Exam
to Cambridge University by the age of fifteen. Cambridge, however, would
not admit him, not because of his age, but because he had yet to complete
his compulsory military service for which of course he was too young. So
he informally attended lectures at Leeds University and learned to play golf.
He then entered the Royal Air Force where he became a teleprinter operator
12: Philip G. Saffman 395
and learned to type, a skill that he found useful in the later age of comput-
ers, email and computational fluid dynamics. Saffman entered Cambridge in
1950, where his aversion to chemistry propelled him to pursue a major in
mathematics.
via collisions of more-or-less equally sized small droplets, these collisions be-
ing driven by the relative motion of random droplets caused by turbulent mo-
tion of the air. Since the droplet sizes are at least an order of magnitude smaller
than the Kolmogorov scale, it is to be expected that the relative motion of two
drops, and hence the binary collision frequency of drops will be governed by
the small-scale turbulent motion. Essentially, it was the spatial variation of
the small-scale velocity field which drives the collision dynamics. Utilizing
ideas from the kinetic theory of gases, Saffman obtained explicit expressions
for the collision rate that accounted for both collisions when drops were mov-
ing with the local air velocity and also when drop motion relative to the air
was included. The key result depended on various parameters including the
mean longitudinal velocity gradient and the mean square acceleration within
the driving turbulent motion. An estimate of the former was provided by Taylor
(1935) while acceleration statistics were derived from Batchelor’s then recent
work on pressure fluctuations in turbulence (Batchelor, 1951).
Saffman’s PhD work did not then contribute to turbulence research, but
it used both well-known and recent advances in understanding of turbulence
kinematics and dynamics. It was clear that Saffman was well versed in these
ideas. While Taylor had long left turbulence research to work in other areas,
Batchelor was highly active in investigations of turbulence dynamics and dif-
fusion, having published his monograph on the theory of homogeneous turbu-
lence in 1953, the same year that Saffman commenced graduate study. Batch-
elor had recently worked on pressure/acceleration statistics (Batchelor, 1951),
Lagrangian motion (Batchelor, 1952), the large-scale structure of turbulence
(Batchelor and Proudman, 1956) along with other topics, and had collaborated
with A.A. Townsend in writing an influential survey on contemporary turbu-
lence research (Batchelor and Townsend, 1956) (see accompanying articles on
A.A. Townsend and G.K. Batchelor, Chapters 9 and 8). As was customary in
supervisor/student relations of the period, Batchelor did not place his name on
his student Saffman’s papers. Indeed, they did not publish jointly, and there is
no evidence that they collaborated on research projects, even though, as will
be seen, they later worked on closely related topics. Interviews of Saffman
by Shirley Cohen for the Caltech Archives in 1989 and 1999 (Cohen, 1999),
which provides much of the anecdotal material above on Saffman’s early life
and Cambridge experience, contains detailed discussion of his collaboration
with Taylor and others. It seems clear that Saffman’s relations with Batchelor
remained cordial but somewhat formal. His work with Taylor is described as a
collaboration where Taylor wanted someone to do the mathematics, for which
Saffman was eminently well-qualified.
12: Philip G. Saffman 397
as
dl j
= Ai j (t)li , (12.2)
dt
where
∂ui
Ai j = (12.3)
∂x j
is the velocity gradient tensor evaluated at the point P. For turbulence, Ai j is
a random function of time whose diagonal components are constrained by the
incompressibility condition. It is of interest to know under what conditions, if
any, the ensemble average (in some sense) of element lengths will be stretched;
that is,
d
|l|
≥ 0, (12.4)
dt
398 Pullin & Meiron
where ·
is some ensemble average. Since 1966 attempted proofs have been
proposed (Cocke, 1969; Orszag, 1970; Etemadi, 1990) for infinitesimal separa-
tion, that the average line element length will increase under certain conditions.
where u(t) is the fluid element velocity in the specified direction, and the in-
tegrand is understood as a Lagrangian autocorrelation of a fluid element at
separated times t and t > t . For statistically stationary turbulence, this is
t−t0
u(t) u(t + τ)
Y (t − t0 ) = 2 u
2 2 (t − t0 − τ)Sp (τ) dτ, Sp (τ) ≡ , (12.6)
0 u2
and where Sp (τ) is the normalized Lagrangian autocorrelation function. For
the dispersion of a conserved substance or quantity (such as temperature in a
mildly heated fluid) that is subject to molecular diffusion but which has no ef-
fect on the turbulence dynamics, when the effects of molecular and turbulent
diffusion are considered to be independent, (12.6) must be modified by the ad-
dition of the strictly diffusive term to give a total dispersion D for the substance
(or heat) (Saffman, 1960):
D2 = Y 2 + 2 κ (t − t0 ). (12.7)
Earlier, Townsend (1954) had considered the growth in the width of a ther-
mal wake behind a line source of heat in turbulence. Arguments based on these
results suggested that, for times such that t −t0 is small compared to the inverse
root-mean-square vorticity of the turbulence,
5
D2 = Y 2 + 2 κ (t − t0 ) + κ (t − t0 )3 ω2 + higher-order terms, (12.8)
9
where ω2 is the mean-square vorticity. This indicates that the rate at which
a volume of heated fluid increases is itself increased by the effect of turbu-
lence. That is, the effect of molecular diffusivity is to accelerate the dispersion
12: Philip G. Saffman 399
over and above the additive independent effects of turbulence and molecular
diffusion.
Saffman (1960) considered this interaction problem, starting from a sub-
stance Lagrangian autocorrelation function that included both the continuum
fluid motion and the underlying random molecular motion,
*
+
R(τ) = U(t) + q(t) U(t + τ) + q(t + τ) , (12.9)
where U(t), which differs from u(t), is the continuum velocity at the point
occupied by a molecule and q(t) is the random velocity. Here angle brackets
denote an average over the random ensemble. The reason that the dispersion
of a substance differs from that of elemental fluid particles is that molecules
of the substance do not move with fluid particles at the local fluid velocity of
the continuum, but rather with this velocity plus a random thermal component
governed by the one-molecule probability distribution of molecular velocities
relative to the mean continuum velocity.
Using a local solution of the passive-scalar advection-diffusion equation in
the presence of the local straining and rotational motion relative to a material
fluid element, and assuming that the random and the continuum motions are
statistically uncorrelated, Saffman obtained
1
D2 = Y 2 + 2 κ (t − t0 ) − κ ω2 (t − t0 )3 + higher-order terms, (12.10)
3
a result that indicates that the effect of diffusivity is to decelerate the disper-
sion, in apparent contradiction to (12.8). The physical interpretation of this
result is that the velocity of a molecule shows a smaller (in magnitude) cor-
relation with its velocity at an earlier time in comparison with the material
element velocity, which decreases the substance dispersion. This has the ef-
fect that the average velocity of a small blob of passive scalar decreases as
the volume of the blob increases under molecular diffusion. Saffman attributed
the discrepancy between (12.10) and (12.8) to the fact that Townsend had as-
sumed that the instantaneous axis of the thermal wake was coincident with
the direction of fluid particles passing through the source, whereas Saffman’s
analysis shows that, owing to the difference between the molecular and the
macroscopic fluid velocity, this is not the case. Although (12.10) is valid only
for small times, Saffman (1960) gave heuristic arguments that at least the sign
of the interaction term would persist for longer. While experimental evidence
(Mickelsen, 1960; Micheli, 1968) is inconclusive, Saffman’s intriguing results
remain interesting and counter-intuitive.
400 Pullin & Meiron
It was also Saffman’s first international meeting on turbulence. The first meet-
ing at Marseilles is often remembered owing to Kolmogorov’s famous pre-
sentation (Kolmogorov, 1962a) of his refined model of turbulent fluctuations
later published as Kolmogorov (1962b) (see Chapter 6 on the Russian school).
Saffman’s contribution was an extension of his earlier work on turbulent dif-
fusion (Saffman, 1962). His presentation was discussed in question time by
Kraichnan, Yaglom and Liepmann.
are considered passive vector fields. In particular, if the Lorentz force, propor-
tional to i jk J j Bk (here Ji is the current density), is neglected in the Navier–
Stokes momentum equations, then Bi behaves as a passive vector field. When
λ = 0, (12.11) governs the evolution of infinitesimal material line elements.
Equation (12.12) gives the evolution of Gi = ∂θ/∂xi , where θ(x, t) is a passive
scalar field whose evolution is described by the advection–(Fickian-)diffusion
equation with scalar diffusivity λ > 0. The evolution of the gradient of φ(x, t)
whose level sets correspond to Lagrangian material surfaces is given by (12.12)
with λ = 0. An issue of interest is the long-time behavior of Fi2 , G2i for small
but finite diffusivity λ ν when fluctuations are introduced at some scale.
The significant example is the turbulent dynamo, where it is hypothesized that
long-time fluctuations in Bi can be sustained by the turbulent motion.
Batchelor (1952) had considered the statistical properties of passive vector
fields satisfying (12.11), concluding that, when λ ν, mean-square values can
grow exponentially in time owing to the predominance of stretching of passive
vectors by the rate-of-strain field of the turbulence. Focusing attention on fine-
scale fluctuations at length scales less than the Kolmogorov length (ν3 /)1/4 ,
where ν is the kinematic viscosity and the mean kinetic energy dissipation,
Saffman (1963) effectively extended to vector fields the earlier study of Batch-
elor (1959) for the sub-Kolmogorov-scale behavior of passive scalars. Starting
from (12.11), (12.12), Saffman worked with equations for the covariance ten-
sors Fi F j and Gi Gj , using their forms for isotropic distributions in terms of
longitudinal and lateral scalar correlation functions. He utlized the idea that the
local rate-of-strain is persistent, and that Fi and Gi tend to align with long-time
(asymptotic) orientations of material lines and the normal to material surfaces
respectively This was used to model the respective third-order correlations,
or transfer terms, for small separation r, in terms containing the eigenvalues
0 < α1 ≥ α2 ≥ α3 < 0, α1 + α2 + α3 = 0 of the local rate-of-strain tensor
S i j , and, in particular, a parameter σ = −α1 /α3 . He argued that the question of
growth or decay appears to depend critically on the role played by the statistics
of the eigenvalues of S i j and on the spatial coherence of the local S i j field.
Saffman concluded that, when perturbations of the vector fields are intro-
duced at some scale, the intensification of the field through stretching and the
production of smaller scales would ultimately be overcome by enhanced ohmic
diffusion with subsequent long-term decay in both Fi2 , G2i . In particular, there
is no sustained turbulent dynamo. Later, however, Moffatt (1970) showed that
Fi2 will grow, with subsequent dynamo action, even when λ ν provided only
that the turbulence lacks reflection symmetry, which occurs when the helicity
ωi ui is nonzero. For the case λ = ν the issue has been further explored using
402 Pullin & Meiron
One of the most important descriptors of the turbulent flow is the energy spec-
trum defined by
E(k) = 1/2 Φii (k)dA(k), (12.15)
|k|=k
and, in particular,
and
∞
ω2 = 2 k2 E(k)dk, (12.19)
0
where
g = f + r f /2, (12.21)
and u1 is the x-component of the velocity. The functions f (r) and g(r) are,
respectively, the longitudinal and lateral velocity correlations. The advantage
of studying homogeneous isotropic turbulence is that one scalar function f (r)
is sufficient to describe the velocity correlation tensor.
The function f (r) has a small distance expansion of the form
r2
f (r) = 1 − + O(r4 ), (12.22)
λ2
404 Pullin & Meiron
= νω2 . (12.32)
it was also possible there to have the third-order correlations vanish only as
r−4 . Using the Kármán–Howarth result (12.29), this implies the second-order
velocity correlations vanish only as r−5 and this makes the Loitsyanski inte-
gral at best conditionally convergent. Indeed, in the discussion of this issue
in their text Fluid Mechanics, Landau and Lifshitz (1987) make the following
uncharacteristic comment:
Doubts have been recently expressed more than once concerning the applicabil-
ity of the conservation law on account of the behavior of the velocity correlation
at very large distances; for example, if this correlation does not decrease suffi-
ciently rapidly, the integral may diverge. This whole subject seems to be as yet
unclear.
Motivated by a discussion with H.W. Liepmann that cast doubt on the k4 law,
Saffman (1967) showed that, depending on the initial conditions, the Loitsyan-
ski invariant might not exist, and that the large-scale energy spectrum could
vary as
E(k) ∼ Ck2 + · · · . (12.42)
Batchelor and Proudman (1956) had assumed that at some initial time t0
the turbulent velocity field was homogeneous and had convergent integral mo-
ments of cumulants of the velocity distribution. Saffman instead considered
the dynamic evolution of a velocity field that had convergent integral moments
of the cumulants of the vorticity. Such an initial condition is less restrictive
because the convergence of velocity moments automatically implies the con-
vergence of vorticity moments, but the converse is not true.
If all the integral moments of the vorticity distribution exist, then it follows
that the spectral tensor of the vorticity, (12.16), has the following form
u = f + ∇φ, (12.48)
∇ φ = −∇ · f ,
2
(12.49)
ω = ∇ × f. (12.50)
He further assumed that the spectral tensor of the force correlation exists and
decreases exponentially with increasing distance. In that case, it is known as a
result of analyticity that, as a function of wavenumber, the spectral correlation
tensor of the force, Mi j , will have the form
Mi j = Mi j + O(k). (12.51)
Using these results, Saffman derived the initial spectral tensors of the vor-
ticity and velocity correlation functions:
are discussed in detail elsewhere in this volume, but to provide Saffman’s per-
spective on the Kolmogorov theory. As the theory states, if an inertial range
exists, a similarity argument indicates that the only relevant dimensional quan-
tities are the turbulent dissipation , and the physical viscosity of the flow ν.
From these a length scale, the Kolmogorov length, and a characteristic velocity
u can be defined respectively as
3 1/4
ν
η= , u = (ν)1/4 . (12.60)
The hypothesis further states that all statistical quantities in this inertial range
at length scales r L p , where L p is the integral scale associated with the
energy spectrum, should be isotropic and should scale only in terms of η and
u. This means for example that
These eddies produce a straining field with rate of strain α ≈ u/L . The strain-
ing field will produce sheets and tubes of vorticity, and Saffman posited that
it is the regions of enhanced dissipation arising from both sets of structures
and their interactions that leads to the bulk of the dissipation. As the sheets are
stretched by the straining flow, they will take on a characteristic length scale
ν νL
δ= = . (12.66)
α u
He further assumed the characteristic velocity of the vortices was also given
by u.
He then considered as an example of subsequent motion the response of a
weak vorticity field packed in a sphere of radius L to a simple velocity field of
the form
u = αxi + βy j + γzk, α + β + γ = 0. (12.67)
Depending on the relative signs of the strains, such a sphere will either stretch
into a sheet or will be pulled into an elliptic cylinder. Because of the stretching,
the vorticity which was originally of size u/L will be amplified to O(u/δ). The
sheet will bend back and forth about itself in the volume and Saffman estimated
this would result in a net area within the box of size (L/η)1/2 L2 = (L5 /δ)1/2 .
Therefore, the volume of fluid taken up by vortical sheets is
This process was termed by Saffman the ‘primary cascade’. He then estimated
the amount of energy dissipation arising from the straining field, the collection
of vortex sheets and the vortex tubes, and showed that the dissipation arising
from sheets scales as
u2 1 u3
sheet ≈ ν σ sheet = 1/4 , (12.70)
δ 2 Re L
and that from tubes as
u2 1 u3
tubes ≈ ν σtubes = 1/2 . (12.71)
δ 2 Re L
Note that there is significant enhancement of dissipation due to sheets, but it
is still dependent on the outer Reynolds number, Re. In addition, there is no
obvious role for the Kolmogorov length.
Saffman hypothesized that secondary instabilities would arise on the vortex
structures. For example, sheets could undergo Taylor–Gortler instabilities and
tubes could undergo vortex breakdown or Couette-type instabilities. In this
case, a secondary structure would arise with a characteristic length scale of δ
such as that of the Gortler cells on the original sheets. These cells would be
separated by internal layers with a characteristic thickness of size
3 1/4
νδ ν L
η= = . (12.72)
u u3
The characteristic vorticity in these internal layers is O(u/η) and the fraction of
volume occupied by these layers is O((η/δ)σ sheet ) since most of the dissipation
is taken up by sheets. This was termed the ‘secondary cascade’ by Saffman.
Note that the total dissipation associated with the secondary cascade is
u2 η u3
secondary ≈ ν σ sheet = , (12.73)
η δ
2 L
the diffusion of a passive scalar in homogeneous turbulence with the result that
the Obukhov–Corrsin ‘constant’ Kθ ∼ Re−1/4 where the Reynolds number Re
is based on the integral scale. The prediction appears to be untested experi-
mentally.
Initial conditions may also be important. Saffman (1977) argues that a realiza-
tion in which the vorticity is piecewise constant at t = 0 will have E ∼ k−4
initially. Since the piecewise constant state must persist when ν = 0, then the
spectrum will remain invariant.
Predictive capability is ranked above rigor but below intelligibility. For accu-
rate DNS of homogeneous turbulence Saffman estimates resolution in each of
three directions as N ∼ Re3/4 11/2
λ and with operation count ∼ Rλ [log Rλ + K]
420 Pullin & Meiron
where Rλ is the Reynolds number based on the Taylor scale. DNS for real-flow
geometries are “awaited with interest”. Jiménez (2003) estimates 2015 for this,
signaling future progress.
Saffman states that
practically everything that is useful in turbulence theory is a scaling law
but that the scaling approach is rarely useful for most complex flows as de-
scribed by Bradshaw (1977). Vortex-based numerical methods, the structural-
vortex approach and Lumley’s POD approach seem to Saffman to have poten-
tial while the main utility of Reynolds-averaged (RANS) methods seems
to lie in the interpolation of experimental data . . .
Saffman is less than sanguine when assessing the dynamical statistical theories
of Kraichnan (but see the accompanying article on R.H. Kraichnan, Chapter
10) remarking that the
absence of a physical basis is unfortunately usually combined with obscurity of
the details.
and Örlü, 2010). Nonetheless there is now overlap between DNS and exper-
iment, and it is predicted that DNS at mid-range laboratory Reynolds num-
bers will be reached within a decade (Jiménez, 2003). The processing of the
ocean of numerical data, perhaps under-anticipated in the 1970s, will remain
a challenge. Long-time existence for the Euler and Navier–Stokes equations
remains open and while there is some experimental support for the indepen-
dence of the rate of energy dissipation on viscosity for grid turbulence at mod-
erate Reynolds numbers (see Sreenivasan, 1984), the issue remains unresolved.
While RANS-type methods are still the principal workhorse of industry, to the
present authors perhaps the most compelling advance made within the topics
covered by Saffman have been in SGS modeling and LES where now quite
realistic unions of complex transport, mixing and reaction physics/chemistry
models have been combined with advances in numerical algorithms and the
treatment of complex bounding geometry to produce an era of increasingly
high-fidelity numerical simulations.
Saffman concludes the written account of his 1966 lectures with
Finally, it would seem that turbulence theory to date can summarized by the
quotation from Macbeth, “full of sound and fury, signifying nothing.” It is to be
hoped that future work will render this quotation inappropriate.
Twelve years later, in his discussion of general principles, Saffman (1978) tem-
pers this as
In searching for a theory of turbulence, perhaps we are looking for a chimera . . . So
perhaps there is no ‘real turbulence problem’, but a large number of turbulent
flows and our problem is the self imposed and possibly impossible task of fitting
many phenomena into the Procrustean bed of a universal turbulence theory.
The reader of this volume, three decades later, may have a response to these
comments.
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12: Philip G. Saffman 425
426
13: Epilogue 427
Table 13.1 Some major events in the history of turbulence
Reference Brief description of the event
Leonardo da Vinci (c. 1500) Used the word ‘turbolenza’ and sketched a
variety of turbulent flows
Katsushika Hokusai (c. 1831) Sketched the “Great wave off Kanagawa”
depicting turbulent broken waves
Hagen (1839) Formally recognized two states of fluid
motion
Saint-Venant (1851), Boussinesq Postulated eddy viscosity
(1870)
Reynolds (1874) Analogy between eddy motion of fluid and
heat transport
Reynolds (1883) Direct and sinuous motion in pipe flows;
Reynolds number
Kelvin (1887) Used ‘turbulence’ in modern scientific
literature
Rayleigh (1892) Inviscid instability
Reynolds (1895) Reynolds decomposition; Reynolds stresses
Prandtl (1904) The concept of the boundary layer; its
separation and control
Orr (1907), Sommerfeld (1908) Equation for viscous stability
Eiffel (1912) Demonstration that turbulence reduces drag
on sphere
Blasius (1913) 1/4-power law for friction in pipe flows
Prandtl (1914) Correct explanation for Eiffel’s observation
King (1914) Hot-wire
Taylor (1915) Eddy motion; introduction of vorticity
transport theory, modified in Taylor
(1937)
Taylor (1921) Diffusion by continuous movements;
Lagrangian autocorrelation function
Richardson (1922) Cascade of scales; possibility of numerical
weather prediction
Taylor (1923) Stability of Couette flow
Keller & Friedman (1924) Eulerian correlation function; moment
equations and an ad hoc closure scheme
Prandtl (1925) Mixing length theory
Richardson (1926) 4/3 dependence of turbulent diffusion on
scale size
Tollmien (1929) Viscous stability solutions
Kármán (1930) Log-law and the outer law for wall flows
Prandtl (1932) Rederivation of the log-law
Nikuradse (1932, 1933) High-Reynolds-number pipe flow
measurements
Leray (1934) Existence of weak solutions of the
Navier–Stokes equations
Taylor (1935) Isotropic turbulence; statistical theory
Taylor (1938) Introduction of spectral analysis
(cont.)
428 13: Epilogue
Table 13.1 Some major events in the history of turbulence (continued)
Reference Brief description of the event
Kármán & Howarth (1938) Self-similarity; Kármán–Howarth equation
Millionshchikov (1939), Proudman Quasi-normal closures, EDQNM models
& Reid (1956), Tatsumi (1957),
Orszag (1970)
von Neumann (1940s) Possibility of electronic computing
Kolmogorov (1941) Local isotropy; universality of small scale;
inertial-range scaling of structure
functions
Obuhkov (1941) Inertial-range scaling of power spectrum
Kolmogorov (1942), Prandtl (1945) Model transport equations for computing
turbulent flows
Landau (1944), Hopf (1948) Successive bifurcations leading to
turbulence
Landau & Lifschitz (1944) Criticism of small-scale universality
Schubauer & Skramstad (1947) Observation of Tollmien–Schlichting waves
Burgers (1948) One-dimensional model-equation
Obukhov (1948), Yaglom (1949), Kolmogorov’s ideas extended to passive
Corrsin (1951) scalars
Onsager (1949) Statistical equilibria of point vortices in two
dimensions
Emmons (1951) Turbulent ‘spots’
Lighthill (1952) Aerodynamically generated noise
Batchelor & Townsend (1951) Dissipation-scale intermittency
Batchelor (1953) The Theory of Homogeneous Turbulence
Dhawan (1953) Direct measurement of skin friction
Kolmogorov (1954), Arnold (1963), KAM theory
Moser (1962)
Feynman (1955) Quantum turbulence
Corrsin & Kistler (1955) Outer intermittency
Batchelor & Proudman (1956), Low wavenumber spectrum
Saffman (1967)
Townsend (1956) Structure of Turbulent Shear Flows
Hinze (1959) Turbulence: An Introduction to Its
Mechanisms and Theory
Batchelor (1959) Passive scalar theory for high Schmidt
number mixing
Kraichnan (1959, 1965) Field theoretic methods (DIA and LHDIA)
Grant et al. (1962) Experimental verification of inertial-range
scaling
Obukhov (1962), Kolmogorov Intermittency; local averaging;
(1962) log-normality; refined similarity
hypotheses
Smagorinsky (1962), Lilly (1967) LES models
Lorenz (1963), Ueda (1960s) Deterministic chaos
Yeh & Cummins (1964) Laser Doppler velocimetry
Favre (1965) Variable density averaging
Steenbeck et al. (1966) Mean field electrodynamics
13: Epilogue 429
Table 13.1 Some major events in the history of turbulence (continued)
Reference brief description of the event
Kraichnan (1967), Batchelor (1969) Two-dimensional turbulence
Kline et al. (1967), Rao et al. (1971) Bursting phenomena
Moreau (1961), Moffatt (1969) Helicity an inviscid invariant
Kovasznay et al. (1971) Conditional sampling
Ruelle & Takens (1971) Strange attractors
Monin & Yaglom (1971, 1975) Statistical Fluid Mechanics, vols 1 and 2
Barenblatt, Zeldovich (1970s) Intermediate asymptotics, incomplete
similarity
Brown & Roshko (1974) Resurgence of coherent structures
Mandelbrot (1974) Application of fractals
Mandelbrot, B.B. 1983. The Fractal Geometry of Nature. W.H. Freeman and Co. New
York.
Millionshchikov, M.D. 1939. Decay of homogeneous isotropic turbulence in viscous
incompressible fluids. Dokl. AN SSSR, 22, 236–240.
Moffatt, H.K. 1969. The degree of knottedness of tangled vortex lines. J. Fluid Mech.
35, 117–129.
Monin, A.S. & Yaglom, A.M. 1971. Statistical Fluid Mech., vol. I. MIT Press (Russian
edition 1965)
Monin, A.S. & Yaglom, A.M. 1975. Statistical Fluid Mech., vol. II. MIT Press (Russian
edition 1965). The two volumes made a valiant effort to bring together much of
the knowledge available at that time.
Moreau, J.-J. 1961. Constants d’un ilôt tourbillonaire en fluide parfait barotrope.
Comptes Rendus, Acad. des Sciences 252, 2810–2813.
Moser, J.K. 1962. On invariant curves of area-preserving mappings of an annulus.
Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II. 1, 1–20.
Nikuradse, J. 1932. Gesetzmässigkeiten der turbulenten Strömung in glatten Röhren.
VDI-Forschungsheft no. 356. The work on rough pipes appeared in 1933 as:
Strömungs gesetze in rauhen Röhren, in VDI-Forschungsheft no. 361.
Obukhov, A.M. 1941. Energy distribution in the spectrum of a turbulent flow. Izv. AN
SSSR Ser. Geogr. Geofiz. 5, 453–466.
Obukhov, A.M. 1949. Structure of temperature fields in a turbulent flow. Izv. AN SSSR
Ser. Geogr. Geofiz. 13, 58–69.
Obukhov, A.M. 1962. Some specific features of atmospheric turbulence. J. Fluid Mech.
13, 77–81.
Onsager, L. 1949. Statistical hydrodynamics. Neuvo Cimento 6, Suppl. no. 2, 279–
287. This article is about both the statistical equilibria of point vortices in two
dimensions and the energy spectrum in three-dimensional turbulence. For a fuller
account of Onsager’s turbulence work, see G.L. Eyink, & K.R. Sreenivasan ‘On-
sager and the theory of hydrodynamic turbulence’. Rev. Mod. Phys. 78, 87–135
(2006).
Orr, W.M. 1907. The stability or instability of the steady motions of a perfect liquid and
of a viscous liquid. Proc. Roy. Irish Acad. A 27, 9–68; 69–138.
Orszag, S.A. 1970. Analytical theories of turbulence. J. Fluid Mech. 41, 363–386.
Prandtl, L. 1904. Über Flüssigkeitsbewegnung bei sehr kleiner Reibung. In Verhand-
lungen des dritten Internationalen Mathematiker-Kongresses in Heidelberg 1904,
edited by A. Krazer, Teubner, Leipzig (1905), 574–584. (English translation in
Early Developments of Modern Aerodynamics, edited by J.A.K. Ackroyd, B.P.
Axcell & A.I. Ruban, Butterworth–Heinemann, Oxford, UK (2001), pp. 77–87.)
For several other lasting contributions to turbulence by Prandtl and his school, see
the accompanying article by E. Bodenschatz & M. Eckert, this volume.
Prandtl, L. 1914. Der Luftwiderstand von Kugelin. Nachrichten der Gesselschaft der
Wissenschaften zu Göttingen, Math.-Phys. Klasse, 177–190.
Prandtl, L., 1925. Bericht uber Untersuchungen zur ausgebildeten Turbulenz. ZAMM 5,
136–139.
Prandtl, L. 1932. Zur turbulenten Strömung in Rohren und längs Platten. Ergebnisse
der Aerodynamischen Versuchsanstalt zu Göttingen. 4, 18–29.
13: Epilogue 433
Prandtl, L. 1945 Über die Rolle der Zähigkeit im Mechanismus der ausgebildete Turbu-
lenz (The role of viscosity in the mechanism of developed turbulence). Göttinger
Archiv des DLR, Göttingen 3712.
Proudman, I. & Reid, W.H. 1954. On the decay of a normally distributed and homoge-
neous turbulent velocity field. Phil. Trans. Roy. Soc. Lond. A 247, 163–189.
Rao, K.N., Narasimha, R. & Badri Narayanan, M.A. 1971. The ‘bursting’ phenomenon
in a turbulent boundary layer. J. Fluid Mech. 48, 339–352.
Rayleigh, Lord. 1892. On the question of stability of the flow of fluids. Phil. Mag. 34,
59–70.
Reynolds, O. 1874. On the extent and action of the heating surface for steam boilers.
Proc. Manchester Lit. Phil. Soc. 14, 7–12. For Reynolds’ contributions to turbu-
lence and his place in history, see the article by B.E. Launder & J.D. Jackson, this
volume.
Reynolds, O. 1883. An experimental investigation of the circumstances which deter-
mine whether the motion of water shall be direct or sinuous, and of the law of
resistance in parallel channels. Phil. Trans. Roy. Soc. Lond. 174, 935–982.
Reynolds, O. 1895. On the dynamical theory of incompressible viscous fluids and the
determination of the criterion. Phil. Tran. Roy. Soc. Lond. 86, 123–164.
Richardson, L.F. 1922. Weather Prediction by Numerical Methods. Cambridge Uni-
versity Press. For Richardson’s other contributions to turbulence and his eclectic
work, see the article by R. Benzi, this volume.
Richardson, L.F. 1926, Atmospheric diffusion shown on a distance–neighbour graph.
Proc. Roy. Soc. Lond. A 110, 709–737.
Ruelle, D. & Takens, F. 1971. On the nature of turbulence. Commun. Math. Phys. 20,
167–192.
Saffman, P.G. 1967. The large-scale structure of homogeneous turbulence. J. Fluid
Mech. 27, 581–593. For Saffman’s other contributions to turbulence, see the arti-
cle by D.I. Pullin & D.I. Meiron, this volume.
Saint-Venant, A.J.C. 1850. Mémoire sur des formulaes nouvelles pour la solution des
problémes relatifs aux eaux courantes. C. R. Acad. Sci. Paris 31, 283–286.
Schubauer, G.B. & Skramstad, H.K. 1947. Laminar boundary-layer oscillations and
stability of laminar flow. J. Aero. Sci. 14, 69–76.
Smagorinsky, J. 1963. General circulation experiments with the primitive equations, I.
The basic experiment. Monthly Weather Rev. 91, 99–164.
Sommerfeld, A. 1908. Ein Beitrag zur hydrodynamischen Erklärung der turbulenten
Flüssigkeitsbewegungen. Proc. 4th Internat. Cong. Math. Rome, 3, 116–124.
Steenbeck, M., Krause, F. & Radler, K.-H. 1966. Berechnung der mittleren Lorentz-
Feldstarke fur ein elektrisch leitendes Medium in turbulenter, durch Coriolis-
Krafte beeinflusster Bewegung. 2. Naturf. 21a, 369–376.
Tatsumi, T. 1957. The theory of decay process of incompressible isotropic turbulence.
Proc. Roy. Soc. Lond. A 239, 16–45.
Taylor, G.I. 1915. Eddy motion in the atmosphere. Phil. Trans. Roy. Soc. Lond. A 215,
1–26.
Taylor, G.I. 1921. Diffusion by continuous movements. Proc. Lond. Math. Soc. 20,
196–212.
Taylor, G.I. 1923. Stability of a viscous liquid contained between two rotating cylinders.
Phil. Trans. Roy. Soc. Lond. A 223, 289–343.
434 13: Epilogue
Taylor, G.I. 1935. Statistical theory of turbulence. I. Proc. Roy. Soc. Lond. A 151, 421–
444. Subsequent parts II–V on this topic appeared in the same journal. For a full
list of references and a more complete description of Taylor’s contributions, see
the article in this volume by K.R. Sreenivasan.
Taylor, G.I. 1937. Flow in pipes and between parallel planes. Proc. Roy. Soc. Lond. A
159, 496–506.
Taylor, G.I. 1938. The spectrum of turbulence. Proc. Roy. Soc. Lond. A 164, 476–481.
Townsend, A.A. 1956. The Structure of Turbulent Shear Flow. Cambridge University
Press. Townsend’s book emphasized the presence of structure within statistical
description. See the article by I. Marusic & T. Nichols, this volume, for an elabo-
ration of this aspect and the other work of Townsend. Similar recognitions of the
importance of flow structures were made by others, e.g., T. Theodorsen, ‘Mecha-
nism of turbulence’, in Proc. Second Midwestern Conf. on Fluid Mech. Ohio State
University, Columbus, Ohio, pp. 1–19 (1952). Townsend initiated the modeling of
small scales through vortex sheets and tubes in ‘On the fine-scale structure of tur-
bulence’, Proc. Roy. Soc. A, 208, 534–642 (1951).
Tollmien, W. 1929. Über die Entstehung der Turbulenz. Nachr. Ges. Wiss. Göttingen
Math-Phys. Kl, II, 21–44
Ueda, Y. 1970. In 1961, Ueda posed a mathematical model on an analog computer that
displayed chaotic dynamics. However, this work was not published until 1970;
see Y. Ueda, C. Hayashi, N. Akamatsu, & H. Itakura, On the behavior of self-
oscillatory systems with external force. Electronics & Communication in Japan
53, 31–39 (1970).
Yaglom, A.M. 1949. Local structure of the temperature field in a turbulent flow. Dokl.
Akad. Nauk. SSSR 69, 743–746.
Yeh, Y. & Cummins, H.Z. 1964. Localized fluid flow measurements with an He–Ne
laser spectrometer. Appl. Phys. Lett. 4, 176–178.