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NORTH ATLANTIC TREATY ORGANIZATION

ADVISORY GROUP FOR AEROSPACE RESEARCH AND DEVELOPMENT

(ORGANISATION DU TRAITE DE L'ATLANTIOUE NORD)

AGARDograph No.303

REYNOLDS NUMBER EFFECTS IN TRANSONIC FLOW

ELSC-NA^L
by

A.Elsenaar
National Aerospace Laboratory NLR
Anthony Fokkerweg 2,1059 CM Amsterdam
The Netherlands
T.W.Binion, Jr
Calspan Corporation/AEDC Operations
Arnold Air Force Base, TN 37389, USA
and
E.Stanewsky
Deutsche Forschungs- und Versuchsanstalt
fur Luft- und Raumfahrt e.V.
Institut fur Experimentelle Stromungsmechanik
D-3400 Gottingen, F.R. Germany

Edited by

H.G.Homung
Graduate Aeronautical Laboratories
California Institute of Technology
Pasadena, CA 91125, USA

MINISTRY OF DEFENCl

If^S^
CONDON WC2H I t O

This AGARDograph has been produced at the request of the Fluid Dynamics Panel of AGARD.
 THE MISSION OF AGARD

According to its Charter, the mission of AGARD is to bring together the leading personalities of the NATO nations in
the fields of science and technology relating to aerospace for the following purposes:

— Recommending effective ways for the member nations to use their research and development capabilities for the
common benefit of the NATO community;

— Providing scientific and technical advice and assistance to the Military Committee in the field of aerospace research
and development (with particular regard to its military application);

— Continuously stimulating advances in the aerospace sciences relevant to strengthening the common defence posture;

— Improving the co-operation among member nations in aerospace research and development;

— Exchange of scientific and technical information;

— Providing assistance to member nations for the purpose of increasing their scientific and technical potential;

— Rendering scientific and technical assistance, as requested, to other NATO bodies and to member nations in
connection with research and development problems in the aerospace field.

The highest authority within AGARD is the National Delegates Board consisting of officially appointed senior
representatives from each member nation. The mission of AGARD is carried out through the Panels which are composed of
experts appointed by the National Delegates, the Consultant and Exchange Programme and the Aerospace Applications
Studies Programme. The results of AGARD work are reported to the member nations and the NATO Authorities through
the AGARD scries of publications of which this is one.

Participation in AGARD activities is by invitation only and is normally limited to citizens of the NATO nations.

The content of this publication has been reproduced

directly from material supplied by AGARD or the authors.

Published December 1988

Copyright © AGARD 1988
All Rights Reserved

ISBN 92-835-0492-5

Printed by Specialised Printing Services Limited

40 Chigwell Lane, Loughton, Essex IGIO 3TZ
 PREFACE

Reynolds number effects in transonic flow are critically reviewed. In this review, the following geometries are
considered: Airfoils and high aspect ratio wings typical of transport aircraft, fighter-type low aspect ratio delta wings, two-
and three-dimensional bodies characteristic of missiles and combat aircraft fuselages, and afterbodies. J\1SO discussed are
"pseudo"-Reynolds number effects which may arise, for instance, due to the influence of the Reynolds number on the wind
tunnel environment which may, in rum, affect the flow about a model. As an introduction to the AGARDograph, a brief
retrospect of the "history" of Reynolds number effects is presented. There are two aspects closely related to viscous changes
which are not discussed herein: the extrapolation of low Reynolds number wind tunnel results to flight conditions and
fundamental Reynolds number effects, i.e., for instance, the influence of the Reynolds number on the boundary layer
development and on basic viscous/inviscid interactions such as shock boundary layer interaction and trailing edge
separation. Both topics are comprehensively treated in the report of the AGARD Working Group 09 "Boundary Layer
Simulation and Control in Wind Tunnels" whose task it was to provide a methodology for transonic wind tunnel testing and
the extrapolation of low Reynolds number wind tunnel results to llight conditions. The present AGARDograph, mainly
concerned with a discussion of viscous effects actually observed on realistic configurations, can be considered a supplement
to the report of AGARD WG 09.

Systematic study of Reynolds number (scale) effects, which obtained new impetus with the introduction of advanced
transonic airfoils and wings, frequently revealed "anomalies" which could be traced to the wind tunnel environment and
measuring techniques and their response to Reynolds number changes. These "anomalies", sometimes labelled "unit"-
Reynolds number effects, are best described as "pseudo"-Reynolds number effects. Factors which have the potential of
introducing pseudo-Reynolds number effects include: wall interference, tunnel Mach number calibration, noise, turbulence,
humidity, non-uniform flow, flow contamination, side wall effects in two-dimensional tests, model deformation and transition
fixing. True Reynolds number effects on transport-type airfoils and wings were found to be mainly related to two
phenomena: transition point movement and the development of separation. Especially large variations in pressure
distributions and corresponding force and moment coefficients were observed with varying Reynolds number when
separation extended from the foot of the shock to the airfoil trailing edge and when the flow changed from a separated to an
attached state as Reynolds number was increased (or vice versa). For conditions with attached or almost attached flow,
Reynolds number effects appear to be smaller, though certainly not insignificant. Available data for low aspect ratio fighter-
type configurations depicting Reynolds number effects are sparse. The data which are available, however, suggest for most of
the flight regime of interest that Reynolds number effects are small.

The Reynolds number sensitivity of bodies is also strongly related to flow separation. Considering the classical body-
related Reynolds number regimes — sub-critical, critical, supercritical and hypercritical — viscous effects are especially
strong in the critical Reynolds number range, where large changes in the aerodynamic forces occur due to the sensitivity of
transitional separation to viscous changes. The flow is not very sensitive to Reynolds number variations in the subcritical
(laminar) and hypercritical (fully turbulent) Reynolds number domains in which separation is essentially fixed by pressure
gradient. With increasing Mach number, the Reynolds number sensitivity also diminishes rapidly in the critical and
supercritical domains due to the development of local supersonic regions with terminating shock waves strong enough to
separate even the turbulent boundary layer. Significant changes in afterbody drag were consistently observed for subsonic
Mach numbers above the transonic drag rise as result of the influence of viscous changes on the expansion around the
shoulder of the afterbody and/or the pressure recovery downstream. The direction of the Reynolds number influence, i.e.,
increasing or decreasing boattail drag with Reynolds number, seems to be dependent on whether viscous changes
predominantly alter the shoulder expansion or the pressure recovery, the latter being closely coupled with the development
of separation over the rear of the afterbody.

Cette AGARDographie presente un examen des effets du nombre de Reynolds dans les ecoulements transsoniques.
Les geometries suivantes sont examinees: les profiles et les ailes a grande allongemenl caracteristiques des aeronefs de
transport, les voilures en delta a faible allongement typiques des avions de chasse, les corps bi- et tri-dimensionnels
caracteristiques des fuselages de missiles et d'aeronefs de combat, et les arriere-corps.

Les effets des "pseudo" nombres de Reynolds sont egalement examines. Ces effets peuvent se produire, par exemple, en
raison de 1'influence du nombre de Reynolds sur le millieu ambiant de la soufflerie, ce qui risque, a son tour, de modifier les
ecoulements autour d'un modele. Un bref resume de Thistorique" des effets du nombre de Reynolds est presente en
preambule a I'AGARDographie.

Deux aspects etroitement lies aux changements de viscosite ne sont pas traites: il s'agit de 1'extrapolation aux conditions
de vol des resultats obtenus en soufflerie a des nombres de Reynolds peu eleves et des effets fondamentaux du nombre de
Reynolds, c'est a dire, par exemple: 1'influence du nombre de Reynolds sur ce developpemcnt de la couche limite et sur les
interactions visqueuse/non visqueuse de base, telles que 1'interaction choc/couche limite et le decollement au bord de fuite.
Ces deux sujets sont traites de facon complete dans le rapport du groupe de travail 09, intitule "La simulation et le controle
de la couche limite en soufflerie" qui a pour objectif de foumir une methodologie pour les essais en soufflerie transsonique et
pour 1'extrapolation au condition de vol des resultats obtenus en soufflerie, a des nombres de Reynolds peu eleves.

in
 La prcsente AGARDographie, qui concerne principalement des effets visqueux examines in-situ et produits par des
configurations reelles, peut etre considered comme un supplement au rapport du groupe de travail AGARD 09.

L'etude systematique des effets du nombre de Reynolds (effets d'echelle qui a re?u une nouvelle impulsion avec
I'arrivee de profils aerodynamiques et de voilures transsoniques avancees, a souvent revele des "anomalies" parfois appelees
des effets de nombre de Reynolds "unite" seraient mieux definies par le terme "effets de pseudo-nombre de Reynolds".

Les phenomenes susceptibles de creer des effets de pscudo-nombre de Reynolds sont les suivants: 1'effet de paroi,
1'etallonage du nombre de Mach de la soufflerie, le bruit, la turbulence, I'humidite, 1'ecoulement non-uniforme, la
contamination de 1'ecoulement, les effets des parois laterales lors des essais bi-dimensionnels, les deformations de la
maquette et le dedenchement de la transition.

Les effets des nombres de Reynolds reels sur les profils aerodynamiques et les voilures des aeronefs de transport,
dependent principalement de deux phenomenes: le deplacement du point de transition et le declenchement de la separation.
Des ecarts particulierement importants de la repartition des pressions et des coefficients de force et de moment ont ete
observes en fonction de la variation du nombre de Reynolds, lorsque le decollement s'etendait du pied du choc jusqu'au bord
de fuite du profil aerodynamique et lorsqu'une ecoulement detache changeait d'etat pour devenir un ecoulement attache au
fur ct a mesure de 1'augmentation du nombre de Reynolds (ou vice-versa)

Dans le cas d'une ecoulemcnt attache ou quasi-attache, les effets du nombre de Reynolds semblent moins importants,
sans etre insignifiants pour autant.

Tres peu de donnees sont disponibles concernant les configurations du type avion de chasse a faible allongement
relatant les effcts du nombre de Reynolds. Pourtant, les donnees existantc indiquent que les effets du nombre de Reynolds
sont pcu marques dans la majeur partic du regime de vol en question.

La reaction des corps au nombre de Reynolds est egalement etroitement liee au decollement. En considerant les
regimes classiques du nombre de Reynolds associes au corps, soit, sous-critique, critique, surcritique et hypercritique, les
effets visqueux sont particulierement forts dans le domaine des nombres de Reynolds critiques, ou d'importantes
transformations dans les forces aerodynamiques se produisent, en raison des effets du decollement en regime de transition
provoques par les changements de viscosite.

L'ecoulement n'est que peu sensible a des variations du nombre de Reynolds dans les domaines Reynolds sous-
critiques (laminaires) et hypercritiques (entierement turbulents), ou le point de decollement est essentiellement regi par le
gradient de pression. L'effet du nombre de Reynolds dans les domaines critiques et surcritiques diminue rapidement aussi,
au fur et a mesure de 1'augmentation du nombre de Mach, en raison de 1'etablissement de zones locales supcrsoniques,
engendrant des ondes de choc terminales d'une puissance telle a provoquer meme le decollement de la couche limite
tourbillonnaire.

Des modifications non-negligeables de la trainee de 1'arriere corps ont cte observees de facon systematique, pour des
nombres de Mach sub-soniques qui correspondent a 1'accroissement de la trainee en regime transsonique, par suite de
1'influence des changements de viscosite sur la detente de 1'ecoulement autour de 1'epaulement de Tarriere corps et/ou la
recuperation de pression en aval. La direction de 1'influence du nombre de Reynolds, c'est a dire, 1'augmentation ou la
diminution de la trainee de retreint en fonction du nombre de Reynolds, semble dependre des changements de viscosite, qui
modifient d'une maniere prcdominante soit la detente de 1'ecoulement autour de 1'epaulement, soit la recuperation de
pression, cette derniere etant elroitement liee au developpement du dccollement sur la partie arriere de 1'arriere corps.

IV
 CONTENTS
Page
SUMMARY iii
NOMENCLATURE vi

CHAPTER ll INTRODUCTION
by A. Elsenaar 1
1.1 HISTORICAL REVIEW 1
1.2 SCOPE OF REPORT 2
1.2.1 Classification 2
1.2.2 Direct versus indirect Reynolds number effects 3
1.2.3 The sensitivity to Reynolds number: A matter of definition? 3
CHAPTER 2: POTENTIALS FOR PSEUDO-REYNOLDS NUMBER EFFECTS
by T.W. Binion, Jr. 7
2.1 INTRODUCTION 7
2.2 WALL INTERFERENCE 7
2.3 NOISE AND TURBULENCE 8
2.4 HUMIDITY EFFECTS 9
2.5 TUNNEL CONTAMINATION 10
2.6 TUNNEL CALIBRATION 10
2.7 SIDEWALL EFFECTS IN 2-D TESTS 11
2.8 NON-UNIFORM FLOW 11
2.9 THERMAL NONEQUILIBRIUM 11
2.10MODEL DEFORMATION 12
2.11TRANSITION FIXING 12

CHAPTER 3: OBSERVED REYNOLDS NUMBER EFFECTS: AIRFOILS AND HIGH

ASPECT RATIO WINGS
by A. Elsenaar 17
3.1 INTRODUCTION 17
3.2 EFFECTS RELATED TO PARTLY LAMINAR BOUNDARY LAYER FLOW 17
3.2.1 The direct effect on drag 17
3.2.2 Indirect effects on the pressure distribution 18
3.2.3 Effects on lift and pitching moment 18
3.3 FLOWS WITH (ALMOST) ATTACHED TURBULENT BOUNDARY LAYERS 22
3.3.1 The pressure distribution - overall effects 22
3.3.2 The pressure distribution - local effects 22
3.3.3 Lift and pitching moment 23
3.3.4 Drag and drag-divergence 24
3.4 FLOWS WITH TURBULENT BOUNDARY LAYER SEPARATION 32
3.4.1 The classical distinction between type "A" and "B" separation 32
3.4.2 The flow break-down boundary revised 32
3.4.3 Post stall behaviour 34
3.5 THREE-DIMENSIONAL EFFECTS 39
3.5.1 The correspondence between two- and three-dimensional flow 39
3.5.2 Some examples of three-dimensional effects 39
3.6 CONCLUDING REMARKS: FROM WIND TUNNEL TO FLIGHT 45
CHAPTER 4: OBSERVED REYNOLDS NUMBER EFFECTS: LOW ASPECT RATIO WINGS AND BODIES
by T.W. Binion, Jr. and E. Stanewsky 51
4.1 LOW ASPECT RATIO WINGS 51
4.1.1 Zero-lift-drag 51
4.1.2 Delta wings 51
4.1.3 Aerodynamic derivations 52
4.1.4 Buffet 52
4.1.5 Transonic technology wing program 53
4.1.6 Summary 53
4.2 BODIES 56
4.2.1 Introduction 56
4.2.2 Cylinders in normal flow 56
4.2.3 Cylinders in oblique flow 57
4.2.4 Three-dimensional bodies 58
4.2.5 Afterbody and base flow 60
4.2.6 Conclusion 62

CHAPTER 5: CONCLUDING REMARKS 71

REFERENCES 75
 NOMENCLATURE

b Wing span
Bl/2 Viscous flow parameter, see Ref. [79/1]
C,Cav Average or mean aerodynamic chord
CA Axial-force coefficient
CfCfCo Aerodynamic chord

CB Steady bending moment coefficient

CB Unsteady bending moment coefficient

CD Drag coefficient
C
DAP Afterbody pressure drag coefficient
c
Db Base drag coefficient

cDi Induced drag coefficient

Zero lift drag coefficient
Cd Section drag coefficient
CFF Lockheed Compressible Flow Facility

Cdexp Experimental section drag coefficient

C(j F p Section drag coefficient based on form factor
Cf Skin friction coefficient
C L ,CL Lift coefficient
Ci,C{ Wing section lift coefficient
CnrC m Pitching moment coefficient
C n >CN Normal force coefficient
Ci, Normal force coefficient based on fuselage plan area
Cnp Slope of normal force vs yaw angle curve

Cp* Pressure coefficient at local sonic velocity

cP Pressure coefficient

c Base pressure coefficient

Pb
C p t e ,CPTE Trailing edge pressure coefficient
Cy Side force coefficient
d Porous wall hole diameter
D Diameter of cylinder on body
f Frequency

FF Form factor

H, H Shape factor
Ho Total pressure
HST NLR High Speed Tunnel
Ka "Esch" factor
e Chord
e~ Shock induced separation bubble length
L Lower, or length
L.E. Leading edge
M Mach number
M a , M 0 , MJ= Free stream Mach number
MD Design Mach number
M L fMf,Mi 0 C Local Mach number

MsH Mach number just upstream of shock

M» c Cross flow Mach number

P Pressure

Pof P T Total pressure

Ps" Sound pressure
Per Pressure coefficient at sonic velocity
q, q» Free-stream dynamic pressure
R, R e , R H Reynolds number
RD/ Reo Reynolds number based on body diameter

R & c *• R © c Reynolds number based on chord

VI
RCCM Reynolds number based on tunnel contraction and nozzle surface length
Reeff Effective Reynolds number, ReD • K<»
R eL Reynolds number based on body length
RBT Reynolds number at the beginning of boundary-layer transition
Ref aw Transition Reynolds number with adiabatic wall
Re/; Unit Reynolds number, ft-i
Re6« Reynolds number based on displacement thickness
Re6«0 Reynolds number based on displacement thickness at the beginning of
the test section
Ree Reynolds number based on momentum thickness
Reea Reynolds number based on momentum thickness just upstream of shock
R„ Reynolds number based on body width
R WT Roughness height or corner radius of non-circular cylinder
5 Surface distance
S/C Ratio of model reference area to tunnel cross-section area
ST Strouhal number
STN Station
T.E. Trailing edge
Taw Adiabatic wall temperature
TT Total temperature
T„ Wall temperature
t Thickness
ue Velocity at the edge of the boundary layer
U Upper
u' Fluctuating component of axial velocity
u« Boundary layer edge velocity
V Velocity
X,x Axial distance
XT, X/Ctr Transition location
XCSH, X/CSH Shock location in terms of chord
y Spanwise coordinate
a Angle of attack
o<jiv Incidence angle for divergence of trailing edge pressure
Of Fuselage angle of attack
P Compressibility factor
y Ratio of specific heats
A Sweep angle
X Ratio of test section height to width
6* Displacement thickness
60 Lift interference factor
q Semi-span ratio
6, Momentum thickness just upstream of shock
6 Momentum thickness
9, Azimuth angle at separation
o Specific humidity
t Wall porosity or skin friction
.\. Flow angle at porous wall on roll angle
 1. INTRODUCTION
by
A. Elsenaar
National Aerospace Laboratory NLR
Anthony Fokkerweg 2, 1059 CM Amsterdam
The Netherlands

1.1 Historical review

It Is of considerable interest to examine the lines of thought with respect to Reynolds number ef-
fects in the past. This review, however, is not intended to be a complete coverage, the main intention
being to highlight some of the more Important events and related publications that Increased the under-
standing or have changed the attitude of the scientific community towards what is called quite generally
"Reynolds number effects". In this AGARDograph the discussion of Reynolds number effects Is restricted to
transonic flows. Thus, one should look for the earliest evidence at the end of the second world war when
transonic flight became a topic of systematic scientific study.

Indeed, in the early fifties almost simultaneously a NACA [Ref. 52/2] and a RAE [Ref. 51/1] report
were published that dealt with Reynolds number effects on swept wing configurations. Both studies had
been initiated to investigate the cause of previously observed Reynolds number effects on lift and pitch-
ing moment of balance mounted models. These studies Indicated, on the evidence of measured pressure dis-
tributions, that the region of shock-wave boundary layer interaction decreased with increasing Reynolds
number. In the NACA report it was argued that the typical lambda shock pattern for laminar shock-wave
boundary layer Interaction as observed at a Reynolds number of 2 x 10**6 disappeared at a Reynolds number
of 4 - 6 x 10**6 due to turbulent boundary layer flow. This was less clear in the RAE tests in which lami-
nar boundary layer flow was still observed at the highest Reynolds number of 3.5 x 10**6. However, at
that condition the size of the shock interaction region was much smaller as compared with the Re - 0.8 x
10**6 results (see fig. 1-la). The resulting change In the pressure distribution significantly affected
the section lift and pitching moment which was further amplified in the wing pitching moment due to the
swept wing configuration. The prevailing thought at that time was that, since a laminar boundary was very
susceptible to (pressure or shock Induced) separation, the turbulent boundary layer development at higher
Reynolds number would result in higher lift values. Instead, a decrease in lift was observed.
It was argued in the RAE report that fixing transition at low Reynolds number would remove the main pro-
blem caused by a thick laminar boundary layer. The results might then be comparable with the actual pres-
sure distribution at flight Reynolds number. It was already indicated in the report, however, that for-
ward fixation can produce premature trailing edge separation and a more forward shock (see fig. 1—lb). It
is of Interest to note here that the same figure shows a nice example of what is now called the "aft-
flxatlon" technique.
In 1954 a review written by Pcarcey and Holder [Ref. 54/1] was devoted to a discussion of adverse
aerodynamic effects that severely limited flight handling qualities at transonic speeds. I.e. high speed
buffet, aileron floating, wing-drop and pitch-up. It was argued that the effects were related to shock
Induced separation. The analysis is based largely on the interpretation of trailing edge pressures that
appeared to rise quite suddenly for shock Mach numbers between 1.22 and 1.24. The problem of scale effect
was not specifically addressed, although it was mentioned In a footnote that for a proper comparison with
flight "the tunnel tests should, of course, be made with transition fixed". The case for transition fixing
was discussed in much more detail at about the same by Haines, Holder and Pearcey [Ref. 54/2] who stated
that "the major scale effects at hlghsubsonlc and transonic speeds arise from differences between the
conditions under which laminar and turbulent boundary layers separate, and in how thev behave after separa-
tion".
As can be Inferred from the "diagrammatic" sketch In figure 1-2 taken from Ref. (54/2), it was ex-
pected that scale effects were much less severe with turbulent boundary layer development and the result-
ing message was a clear one: fix the boundary layer in wind tunnel tests. Techniques discussed for doing
so are distributed roughness (carborundum and ballotinl), wires, adhesive tape and blowing, and remind us
that there is nothing new today in this respect.
The noted difference between laminar and turbulent shock-wave boundary layer Interactions was not at
all unknown at that time as fundamental studies [see c.g. Ref. 46/1] had already Indicated the phenomena
years before. The new aspect was the notion that the differences could be related to adverse aerodynamic
effects in high speed flight due to separation and scale effects. This notion stimulated basic research
of shock-wave boundary layer Interaction thereafter as reflected In a number of publications [e.g. Refs.
52/1, 55/1 and 55/2]. At the same time, control of the shock-wave formation on airfoils was attempted
leading finally to the design of shock free supercritical airfoils [see e.g. Ref. 62/1]. The classical
airfoil designs like the NACA 6-serles with laminar flow (provided the wing surface was smooth enough)
and without any appreciable rear loading, were replaced by designs of the "peaky" type (moving transition
near the nose) and with a substantial amount of rear loading.
In 1966 a NASA report [Ref. 66/1] written by Loving was published that showed large differences be-
tween wind tunnel and flight data for the C-141 aircraft (fig. 1-3). As he wrote: "the purpose of the
discussion Is to caution experimenters concerning the use of wind-tunnel results In predicting flight
loads and moments when supercritical separated flow is present". This event (or at least his figures,
Judged by the number of times they have been referenced) swept through the aerodynamic community like a
shock wave, bringing a number of researchers and aircraft designers into a state of buffeting. Loving
himself reported that "the results disclosed herein should not come as a surprise; they are merely addi-
tional evidence of the problem associated with separating flows". Scale effects had again become an area
of considerable concern.
In a 1968 paper by Pearcey, Osborn and Haines [Ref. 68/2], a physical model was postulated that ex-
plained the aspect of the problem In more detail. The flows about the mld-flftles type airfoils were clas-
sified as type "A" separation, dominated by strong shock waves with separation rapidly developing from
the shock to the trailing edge. This kind of flow was considered to be weakly Reynolds number dependent.
However, the larger load carried by the aft part of modern airfoils could provoke "classical" (low speed)
trailing edge separation even In the absence of shock waves which is considered to be Reynolds number
dependent. They noted: "It is not surprising, therefore, to find these sensitivities carried over Into
flows in which rear separation and the local effects of the shock interact with one another, nor indeed
to find them amplified by the Interaction". They named this phenomenon type "B" separation. The publica-
tion was followed by a large numbers of papers that discussed scale effects [see e.g. Refs. 71/1, 71/3]
most often related to shock-Induced boundary layer separation. Some publications focussed attention on
the means to calculate Reynolds number effects [e.g. Ref. 71/7], others to simulate Reynolds number ef-
fects In the wind-tunnel [e.g. Refs. 68/1, 71/4]. It was also argued that the final experimental answer
could only be expected from wind tunnels which could achieve the full scale Reynolds number. In 1971, an
ACARD meeting was organized in Goettlngen specifically concerned with these problems ("Facilities and
Techniques for Aerodynamic Testing at Transonic Speeds and High Reynolds number"). Therefrom plans were
developed both in the US and in Europe, to build a high Reynolds number facility [Refs. 72/1, 72/4].
One other Important change in judgement of the significance of Reynolds number effects should not be
left unsaid in this review. Initially the Interest was mainly concerned with the start and consequencles
of flow separation. It was again Haines who noted In 1976 [Ref. 76/2] that "uncertainties (due to Rey-
nolds number effects) not only affect the flow separation characteristics but also the drag In conditions
where the flow is attached". Its significance Is that scale effects are not only of Interest for off-de-
sign conditions but also for performance prediction at the design condition. The argument was even carried
one step further by Haines in Ref. [79/2] where it was argued that the optimisation of the aircraft design
Is greatly hampered by the Reynolds number gap of the present day wlndtunnels (figs, 1-4).
In the absence of a flight Reynolds number facility and stimulated by the aircraft Industry that
simply can not afford a major design change or a short-fall in aerodynamic performance due to uncertain
scale effects, the study of Reynolds number effects is still continuing today. The most systematic infor-
mation in this respect is obtained from existing [e.g. Ref. 78/1] or newly developed [e.g. Ref. 82/1]
wlndtunnels in which the Reynolds number can be varied over a considerable range. This basic research has
not been restricted to airfoils or high aspect ratio wings but has also widened to other configurations
like slender bodies, afterbodies and delta wings.
Partly as a result of anomalies from these systematic Reynolds number studies, but also in its own
right, a considerable Interest exists with respect to the tunnel environmental and measuring technique
effects (like wall interference, tunnel noise and flow quality, transition tripping devices, etc.) in
relation to Reynolds number effects. As already indicated by Holder et al in 1955 [Ref. 55/1], flow qua-
lity might have a substantial Impact on the pressure distributions as measured in the wind tunnel (see
fig. 1-5 and compare with fig. 1-1). The influence of the tunnel environment on the transition location
appeared to be a hot Item in the sixties in connection with what was called a "unit Reynolds number effect"
[Ref. 69/2]. The problem was more or less settled by Dougherty and Fisher [Ref. 80/4] In 1980. They were
able to correlate transition Reynolds number on a 10 degree cone in both wind tunnel and flight with aero-
dynamic noise. Conflicting results of drag measurements with the AGARD Nozzle Afterbody [Ref. 75/1] could
largely be explained when the Reynolds number effect on tunnel calibration was taken into account as poin-
ted out by Aulehla and Beslgk [Ref. 74/2], Many more examples of this kind give as many warnings that the
"observed Reynolds number effects" are, in some cases, attributable to measurement error or the wind tun-
nel (and/or flight) environment rather than Reynolds number per se.
This historical introduction has shown how the way of thinking with respect to Reynolds number ef-
fects has changed, not because of the physics changed but primarily because the frontiers of aerodynamic
design have been widened. To quote Haines once more: "evidence (from the past) has to be Judged in the
context of the wing designs in the future" [Ref. 76/2]. And that is the intention of this AGARDograph.

1.2 Scope of report

1.2.1 Classification
Some kind of classification is needed for the description of Reynolds number effects. However, it
should be emphasized that not all of the phenomena thnt are loosely called Reynolds number effects, are
necessarily related to the "change In flow development with Reynolds number for a particular configura-
tion in free air". Some of the observed Reynolds number effects in wind tunnels or unexplained differ-
ences between wind tunnel and flight happen to be caused by deficiencies in aerodynamic testing In ground
based facilities. These effects, named hereafter "pseudo Reynolds number effects" have sometimes hampered
the understanding of Reynolds number effects and, as is to be expected, will continue to do so even after
high Reynolds number facilities become available. These effects are considered to be of so much impor-
tance that they are dealt with separately in Chapter 7.
If we now restrict ourselves to "true" as opposite to "pseudo" Reynolds number effects, they can be
discussed either In a fundamental way by Isolating one particular aspect or flow phenomenon or as changes
In aerodynamic characteristics as observed on certain classes of aerodynamic shapes (Chapter 3 and 4 ) .
For the latter, a more precise classification is possible, according to the particular flow characteris-
tics:
. airfoils and high aspect ratio wings with flows that are essentially two-dimensional or very weakly
three-dimensional (such that local strip theory still applies);
. low aspect ratio wings with a highly three-dimensional flow development characterized by free vortex
flow, highly skewed shock waves and/or three-dimensional separated regions;
. slender bodies with free-vortex flow development and/or base flow Interacting with a jet or plume.
It is the purpose of this AGARDograph to review some of the evidence on Reynolds number effects and
to provide a kind of more general frame work that might assist In the understanding of the observed ef-
fects. The Reynolds number enters Into the fluid dynamic equations through the viscous terms of the Na-
vler-Stokes equations. In other words, viscous effects constitute a necessary condition for the occurence
of Reynolds number effects. But not all flows with embedded viscous regions are necessarily Reynolds num-
ber dependent. In a still most relevant review by Hall [Ref. 71/5], scale effects were defined as "the
complex of Interactions between the boundary layer development and the external Inviscid flow". This dis-
tinction between a viscous shear layer and the outer Inviscid flow field can conveniently be used to de-
fine two kinds of Reynolds number effects (see fig. 1-6):
. direct Reynolds number effects which occur as a consequence of a change In boundary layer development
with Reynolds number for a fixed ("frozen") pressure distribution and
. indirect Reybolds number effects which appear as variations in the pressure distribution and hence in
the aerodynamic characteristics due to«a change in the boundary layer and wake development with
Reynolds number.
This distinction is very useful In a practical, operational sense as will be discussed next.
1.2.2 Direct versus Indirect Reynolds number effects
According to its definition direct Reynolds number effects can be calculated or estimated with the
help of boundary layer calculation methods and/or semi-emplrlcal correlations. However, the so calculated
direct Reynolds number effects are only relevant for the prediction of scale effects in so far as the
indirect Reynolds number effects are absent or negligible small. This Is to some extent true for attached
flow conditions but as soon as local effects of separation (e.g. near the trailing edge or at the foot of
a shock wave) are present significant changes in pressure distribution might result. For larger separated
flow regions the distinction is even less relevant: direct and Indirect Reynolds number effects arc es-
sentially coupled.
This is all very similar to the distinction between weak and strong interactions as used by Hall in
the above referenced paper. More recently this classification Is also used in computational fluid dyna-
mics to describe the mathematical approach of the coupling of viscous and Inviscid fluid flow [see e.g.
Ref. 83/2]. Basically, the strong interaction approach removes the so called Goldstein singularity for
separated flows and Involves a (quasi-)simultaneous solution of outer (inviscid) and inner (viscous) flow
fields. In other words: for separated flows the direct and indirect Reynolds number effects are formaly
coupled.
The most simple and very regular direct Reynolds number effect Is related to the change in boundary
layer development for a well developed laminar or turbulent boundary layer. The boundary layer properties
generally vary as Re with n - 2 for laminar and n s 5 for turbulent flow. Skin friction drag, being
the best observable example of a direct Reynolds number effect, varies accordingly. Equally Important is
the change In displacement thickness with Reynolds number. This change modifies the external inviscid
flow field and is therefore, by definition, the cause of indirect Reynolds number effects. It is to be
expected that a regular and continuous change In displacement thickness results in comparable smooth chan-
ges in the outer flow field.
A very significant and irregular direct Reynolds number effect stems from the transition from laminar
to turbulent boundary layer flow. For swept wings the attachment line flow can be either laminar or turbu-
lent, depending on Reynolds number, sweep angle and leading edge shape. For flat plate (subsonic or tran-
sonic) flow, a situation somewhat similar to the boundary layer development ahead of the shock on trans-
port-type wings, the transition Reynolds number varies between 2 x 10 and 5 x 10 depending on noise
level and free-stream turbulence. The corresponding chord Reynolds number is often just in between the
Reynolds numbers of the wind tunnel model and the flying aircraft. These variations in transition location
may interact with the outer field in a highly non-linear way. They are very often clearly discernible In
the development of the aerodynamic characteristics. Even more important: laminar boundary layers fundamen-
tally behave differently compared with turbulent boundary layers. As a typical example, a much smaller
pressure gradient is needed for a laminar boundary layer to separate than for a turbulent boundary layer.
Consequently, transition point changes might trigger significant changes in boundary layer separation as
will be discussed next.
The most Important direct Reynolds number effect is the Reynolds number influence on the start of
separation (separation onset or incipient separation) and the subsequent separation development. Vortex
burst or wake-stall are similar, but probably less critical phenomena in this respect. In all these cases
one deals essentially with an Interaction between the vlcous flow development and the Inviscid outer flow
field, with large consequences for the pressure distribution. They are, for that reason, the cause of
significant Indirect Reynolds number effects. The onset of separation is very often described In terms of
a viscous flow parameter (shape factor, Reynolds number based on displacement thickness etc.), although
strictly speaking, locally a strong interaction approach may be more appropriate. When separations are
confined to a small part of the flow field (e.g. a leading edge separation bubble or a local trailing
edge separation) the effects on the outer flow field arc only felt by a change in Kutta condition. This
in turn affects the circulation and hence the overall flow field. In some cases, the resulting flow deve-
lopes continuously, though sometimes highly non-linearly from the attached to partly separated flow condi-
tions. In other cases, however, the separation might set-off a chain of reactions, leading finally to a
complete and sudden break down of rhe flow field.
The direct Reynolds number effects will not be discussed in great depth in this ACARDograph. A very
thorough discussion of these effects can be found in the report of the Research Committee of the AGARD
working group 09 "Wind Tunnel Boundary Layer Simulation and Control" (Ref. 88/1].

1.2.3 The sensitivity to Reynolds number: A matter of definition?

Some configurations or types of flow are reported to be highly sensitive to changes in Reynolds num-
ber. But what precisely is meant with such a qualification?
Reynolds number sensitivity can be defined as the variation In aerodynamic characteristics (like
lift, drag, pitching moment, pressure distribution ...) with Reynolds number for a particular configura-
tion (shape) at constant flow conditions (like Mach, alpha . . . ) . This sensitivity can in principle be
obtained by plotting the particular aerodynamic characteristic versus Reynolds number. Unfortunately, It
is not an easy matter to do an experiment in which only the Reynolds number is varied and all other flow
conditions are kept constant. Even apart from tunnel environmental effects, that might Introduce the
"pseudo" Reynolds number effects to be discussed In chapter 2, the nominal flow conditions in Mach and
Incidence arc rarely reproduced exactly. Also, in wind tunnel flight comparisons small configuration dif-
ferences are almost inevitable. For these reasons the analysis of Reynolds number effects often requires
some kind of Interpolation or, at least, an estimation of the Involved accuracies.
If it Is assumed that the derivative with respect to Reynolds number can be defined for a particular
aerodynamic characteristic. It still remains to be judged if the Reynolds number sensitivity is large or
small. A very pragmatic criteria is: Reynolds number effects are large when they affect significantly the
design (performance) of an aircraft as derived from sub-scale wind tunnel testing. Three drag-counts vari-
ation in drag-creep will be significant for a transport type aircraft, but Irrelevant for a manoeuvring
condition of a fighter configuration. An other, slightly different view is that Reynolds number effects
should be considered large when the indirect Reynolds number effects are large. Most direct Reynolds num-
ber effects (like skin-frlctlon) follow well established rules. One should only worry when the Indirect
Reynolds number effects are significant. Indeed, very often changes in pressure distribution are presen-
ted to emphasize that a particular kind of flow is Reynolds number sensitive. Unfortunately, such a proof
is not sufficient since some pressure distributions are sensitive to changes in other flow conditions as
well. A typical example is the design condition of certain supercritical airfoils that are known to be
very sensitive to small Mach number and angle of incidence changes (see fig. 1-7; note that these results
have been obtained in an adaptive wall wind tunnel where flow conditions are exceptionally well defined
[Ref. 85/1]).
 The quantification of Reynolds number sensitivity from the change In pressure distribution is clear-
ly not applicable when the particular aerodynamic characteristic is not a continuous funtion of the flow
conditions. Such situations typically occur when the flow changes from one type to the other, connected
with events like:
- the onset of vortex formation on a delta-wing
- the maximum lift boundary
- lift divergence
- vortex break-down
- etc.
A typical example is presented in figure 1-8. At alpha - 2° the pressure distribution hardly changes with
Reynolds number whereas near the maximum lift boundary at alpha - 3° large differences can be observed
(note that this figure has some similarity with the classical Reynolds number case presented in fig. 1-3).
An other example [Ref. 80/1] Is given in figure 1-9 where pressure distributions on a delta wing are pre-
sented just before (high Reynolds) and beyond (low Reynolds) vortex formation. These examples do not real-
ly prove that a particular type of flow is highly Reynolds number sensitive. Instead,, when such disconti-
nuous changes are present in the flow development, the Reynolds number effect should be expressed as the
variation in the boundary between two flow regimes, rather than comparing pressure distributions from two
basically different flow regimes. Only when this variation is significant (e.g. in terms of the flight
envelope) one can speak of a large Reynolds number sensistlvity.
In view of these arguments it might be advantageous to consider the Reynolds number effects in the
llft-Mach number plane (fig. 1-10). In this plane (mapping the flight envelope) typical flight conditions
like cruise, maximum lift, drag-rise, etc. are defined. Off-design boundaries (very often related to dis-
continuities in flow development) are also defined in this plane and the aircraft designer actually wants
to know how these boundaries change with Reynolds number. Also, the aircraft is designed for a particular
lift at a chosen Mach number and the aircraft designer wants to know how the pressure distribution, the
drag, the pitching moment, etc. changes with Reynolds number for that particular llft/Mach number combi-
nation. He will be only indirectly interested in the change of angle of attack for constant lift. He is
most likely not Interested in the change in pressure distribution at constant angle of attack although
this is very conveniently measured In the wind tunnel. For these reasons the analysis and Interpretation
of Reynolds number effects is often preferably done In the llft-Mach number plane, at constant lift, con-
stant Mach number or a combination of both.

TRANSITION FIXED R - 3 . 6 » 10*

R - 0 « x 10*

r
(IB

-S
06

0 4

u
* r
I ,.
Ob

t /
s-J

^
0.2 0.4 ,,. 0.6 0 112 04 . . . 06 08V,, l.l
1 1 \ '
0. \
N 1 1 1 1 l\ REYNOLDS NUMBER

lal DIAGRAMMATIC SKETCH Ibl OBSERVED TRAILING-EDGE PRESSURES

RB-S.SXIO" 0.778 0.377 -O.021 0.821 0359 -0,034

IB 0.778 0.367 -O.0I9 0.820 0328 -0022 Fig. 1-2 The effect of Reynolds number and the
8 0.777 0.353 -0.016 0.820 0275 -O.013
state of the boundary layer on the
• I FREE TRANSITION 01 - 2,7° pressure recovery over an airfoil when
boundary-layer separation occurs at
the shock waves (from ref. 54/2)

SUPERCRITICAL PRESSURE DISTRIBUTION

M-0.85; o . - O '

Hi « t o ' " FIXATION

35 FREE 0.778 0.377 -0.021 — 0.821 0.369 -0.034
3,6 8% 0778 0.321 - 0 007 - - 0 820 0.276 -0 009
08 17* 0,781 0 299 -0.013 — 0.818 0321 -0.022

bl INFLUENCE OF TRANSITION FIXING

Fig. 1-1 Early evidence of the effects of Fig. 1-3 The "classical" example of the effect
Reynolds number on the pressure distri- of Reynolds number on the pressure
bution (from ref. 51/1) distribution: the C-141 aircraft (from
ref. 66/1)
 X/C
i 0.95 1.0
© MACH CONSTANT, REYNOLDS VARIABLE
UPPER SURFACE
TRAILING EDGE
CONDITIONS

• R 6K10 6 SECTION AT R- eicIO 6 C L " 0 70. M-0.75

R 6«10 6 SECTION AT R-36K10 6 C L "0.73. M-0.76

076 \ 0.78= MACH
3 R 35X10 6 SECTION AT R-SSulO 6 : C, -0,79. M-0.75

1 AND 3 : SAME SHOCK STRENGTH ANO VISCOUS STANDARD

Fig. 1-4 Example of the achievable gain in

design lift through design for a higher
Reynolds number (from ref. 79/2)

X/C
® REYNOLDS CONSTANT, MACH VARIABLE

Fig. 1-7 Example of the sensitivity of a

supercritical airfoil nt the design
condition for variations in flow
conditions (from ref. 85/1)

— X On* 100 mntt gauja Idiriyl ami poo. m u y . , .

— I On* 100 nanh gauia I M m l ana poo, tl(a.ghia.....
IB Ona 100 math gaun Icaaanl and no ilfamhtanan
^ * « B * n Two 100 main gaunt Iclaan) and ifflerowad ilra*ohianan

Fig. 1-5 Early example of the effect of flow

quality on tbe pressure distribution
(from ref. 55/1)

EFFECTS
ON SHOCK
STRENGTH
SHOCKWAVE \ \ A N 0
BOUNDARY POSITION
LAYER
INTERACTION

25 %C

S, CM

Fig. 1-9 Reynolds number effect on the pressure

EFFECT ON distribution prior to and beyond vortex
OVERALL CIRCULATION
AND PRESSURE DISTRIBUTION formation Illustrating large differen-
ces due to a Reynolds number effect on
—•> DIRECT REYNOLDS NUMBER EFFECT the boundary for vortex formation (from
^ INDIRECT REYNOLDS NUMBER EFFECT ref. 80/1)
DOMINANT RE NUMBER EFFECT
CHARACTERISTIC DIRECT INDIRECT
LIFT AND PITCHING MOMENT X
VISCOUS DRAG X
WAVE DRAG X
DRAG DIVERGENCE X
BOUNDARY LAYER SEPARATION X
BUFFET BOUNDARY X X

Fig. 1-6 Schematic representation of direct and

indirect Reynolds number effects
 2 -

%.

-1

Fig. l-8a Reynolds number effect on the pressure Fig. l-8b Reynolds number effect on lift and
distribution prior to and beyond maxi- trailing edge pressure near the
mum lift illustrating large differen- maximum lift boundary (from ref. 83/3)
ces due to a Reynolds number effect on
tho maximum lift boundary (from ref.
83/3)

LOW RE LOW RE

MAXIMUM LIFT
DRAG DIVERGENCE
Re EFFECT ON
MAX LIFT
VERY LARGE RE EFFECTS
TO BE EXPECTED ON
PRESSURE DISTRIBUTION
R«EFFECT ON
DRAG DIVERGENCE

ON DRAG CREEP TO BE EXPECTED

Mach

Fig. l-10a Example of evaluation of Reynolds Fig. l-10b Example of evaluation of Reynolds
number effects at constant lift number effects at constant incidence
 2.0 POTENTIALS FOR PSEUDO-REYNOLDS NUMBER EFFECTS
by
T. W. Binion, Jr.
Calspan Corporation/AEDC Operations
Arnold Air Force Base, TN 37389

2.1 INTRODUCTION

Reynolds number effects have been traditionally determined by testing a given model
in a variable Reynolds number tunnel, testing a given model in several wind tunnels with
different Reynolds number capabilities, testing similar models of different scale in a
given or different wind tunnels, or a combination of the three. The data are then
extrapolated to flight using various semi-empirical correction techniques. Almost every
parameter which can introduce spurious effects into wind tunnel data has the potential to
vary from tunnel to tunnel at the same nominal test condition or from one Reynolds number
to another in a given tunnel. Factors which have the potential of introducing pseudo-
Reynolds number effects include wall interference, tunnel Mach number calibration, noise,
turbulence, humidity, nonuniform flow (flow angle, tem-perature and pressure gradients),
flow contamination, sidewall effects in two-dimensional tests, thermal non-equilibrium,
model deformation, and transition fixing. In some instances not only are the effects of
the parameters not well understood, their influences may be coupled and not easily
separated by experimental techniques. At present, the theoretical capability does not
exist to evaluate the individual effects of the above factors with confidence much less
their potential coupling. Thus, some of the discussion below is somewhat speculative.
Nevertheless, the evidence presented is strong enough to elicit a degree of caution in
ascribing the cause of variations of data with Reynolds number solely to Reynolds number.

2.2 WALL INTERFERENCE!

Attempts to experimentally assess the effects of wall interference have been

careful to keep as many parameters constant as possible, e.g., model support, instrumen-
tation, transition location, test conditions, etc. Therefore, there is not much direct
evidence of wall interference varying with tunnel Reynolds number. J. Smith (NLR) has
calculated the wall interference for two Reynolds numbers from boundary conditions
measured between the slots of the NLR High Speed Tunnel with a 50-cm chord full span 2D
airfoil. Although the changes in interference are small as a function of Reynolds number.
Fig. 2-1, "they are still substantial when compared with some observed Reynolds number
effects", Ref. [83/4].

For porous walls it has been established in experiments by Jacocks, Ref. [76/5],
and Chan, Ref. [82/6], that the wall crossflow characteristics are a function of the
local wall boundary-layer characteristics. Thus, not only does this imply that wall
interference is a function of tunnel Reynolds number but also the model imposed pressure
distribution on the wall as it affects the displacement thickness. Subsequent to Ref.
[76/5], Jacocks, has correlated the classical wall porosity parameter, dCp/d6, with a
nondimensional wall parameter, (xt/dpReg.. where T. is the wall porosity, t the wall
thickness, d the hole diameter, and Re8, the Reynolds number based on the porous wall
boundary-layer displacement thickness. The correlation shown in Fig. 2-2 clearly
demonstrates the porous wall boundary condition, hence wall interference, varies with
Reynolds number. It should also be noted that, particularly with a ventilated wall, Re6,
varies with the pressure distribution over a given wall which is a function of test
conditions, model and support configuration, and model attitude. Theory does not yet
allow a detailed computation of the local wall boundary condition, nor has it been
established that such a detailed specification is necessary. However, for the AEDC
tunnels, Jacocks also has correlated ReB* at the tunnel nozzle exit with Reynolds number
based on the contraction nozzle length. Fig. 2-3. If one accepts the variation of Re6*0 as
an indication of the variation of the wall boundary condition with tunnel Reynolds
number, then combining the equations in Figs. 2-2 and 2-3 yields

d
p \
— - = 53.4 - Ke,,.,
CN
dO U\ J
at least for the AEDC tunnels.

iThe analysis presented draws upon an unpublished document by Dr. E. M. Kraft,

Calspan Corporation/AEDC Operations.
8

The potential effect of Re on wall interference may be estimated from the classical
theory of Ref. [69/1). Shown in Fig. 2-4 is the variation of the lift interference factor
with the wall porosity parameter, Q, defined for incompressible flow* as

Q-[l+ l|-
where

- = 1/2-iC /do>.
H P

The change in the lift interference factor caused by variations in unit Reynolds number
for Tunnel 4T is indicated in the figure and accompanying table. The tunnel Reynolds
number effects are substantial.

Experiments conducted on an A-10 aircraft model. Fig. 2-5, in Tunnels 16T and 4T by
J. M. Whoric, yielded some information on the effect of Reynolds number on wall
interference. The model blockage was 0.13 and 2.11 percent in 16T and 4T, respectively.
Because of the small blockage ratio, the 16T data are considered wall interference free.
The two tests were conducted with the same instrumentation, support sting, and test
conditions. The difference between the 16T and 4T angle of attack, pitching moment, and
induced drag at constant lift is shown in Fig. 2-6 for two unit Reynolds and Mach
numbers. Even though the tests were conducted transition free, the transition Reynolds
numbers based on Dougherty's cone data, Ref. [82/7], at the conditions of the comparison
are almost identical in the two tunnels. Furthermore, the A-10 data from 16T exhibited no
Reynolds number effects. Thus, the data variations with Reynolds number are the result of
phenomena in 4T, The Aa is not zero at CL = 0 because the tail incidence is 6 deg with
respect to the wing. Therefore, both the wing and tail are lifting, albeit in opposite
directions at C L = 0, which causes lift interference in Tunnel 4T resulting in an effec-
tive angle change for zero lift. The classical lift interference applicable to the A-10
was shown in the table accompanying Fig. 2-4. While classical theory would not be
expected to produce accurate results for such a large and complicated model or even
necessarily the correct trends at transonic Mach numbers, the theoretical lift
interference slopes, da/dd, agree rather well with the data in Fig. 2-6a, particularly
at low lift. Theory aside, if wall interference is the dominant variable between the two
data sets, then the data in Fig. 2-6 strongly suggest that wall interference is a
function of Reynolds number for both Mach numbers since the variation of the coefficient
differences is considerably different for each Reynolds number. Furthermore, the fact
that Aa is not linear with CL suggests that the wall interference boundary condition is
not constant with CL which is not inconsistent with the hypothesis that the boundary con-
dition is a function of the model imposed pressure distribution.

2.3 NOISE AND TURBULENCE

Pate and Schueler, Ref. [69/2], and Dougherty, Ref. [82/7], have shown that the
location of boundary-layer transition is influenced by aerodynamic noise. Dougherty
devised a correlation function between transition Reynolds number and root-mean-square
pressure fluctuation shown in Fig. 2-7. Recently, however, Murthy and Steinle, Ref.
[86/2], have re-examined Dougherty's and other data and suggested that at transonic
speeds, if the tunnel noise is less than about 1% rms C p , noise may not affect transition
at all, the real mechanism being turbulence. For Tollmien-Schlichting type transition,
the spectral content of the disturbances in relation to the laminar instability frequency
is probably the factor in producing premature transition [87-3]. While it is difficult to
decouple the effects of noise and turbulence in wind tunnel data, free-stream turbulence
by itself can exert a substantial influence on transition location. For example, Mignosi,
Ref. [85/7], has calculated the effects of free-stream turbulence on the skin friction
distribution for a supercritical airfoil. Fig. 2-8. The calculations indicate that
transition location on the upper surface is a strong function of rather small variations
in turbulence. The laminar boundary layer on the lower surface separates at about 60%
chord and transition occurs at the end of the separation zone. In a given non-cryogenic
wind tunnel, the amplitude and spectra of the noise and turbulence, hence transition
location, is a function of unit Reynolds number, i.e., tunnel power. For models with free
transition, variation of free-stream noise and/or turbulence with tunnel power could
produce a pseudo-Reynolds number effect by causing transition location to be a function

lEven though the theories in Ref. [69/1] claim applicability to compressible flow,
experience does not substantiate such a claim.
of tunnel Reynolds number rather than model Reynolds number. Such effects can be
partially avoided by fixing transition forward of the free transition location.

There is no known clear evidence that noise affects boundary-layer properties other
than transition location. However, several investigators have obtained experimental
evidence of the effect of free-stream turbulence on boundary-layer properties. Green,
Ref. [73/2], working with low-speed zero-pressure-gradient data and taking Ree as a
characteristic parameter showed that "not only is the effective Reynolds number highly
sensitive to turbulence - the fractional increment in Ree is 60 times that in u'/ue . . . -
but also this sensitivity increases with increasing Reynolds number," Raghunathan and
McAdams also found a strong influence of turbulence on attached zero-pressure-gradient
turbulent boundary layers up to 0.8 Mach number and Ree from 3- to 10-thousand, Ref.
[82/8]. They conducted further experiments using a 9% thick circular-arc tunnel-wall bump
at Mach numbers from 0.65 to 0.78, Refs. [83/5] and [83/6]. Transition was fixed upstream
of the bump. The variation they obtained of trailing-edge separation length, shock
location, and the ratios of boundary-layer momentum thickness, shape factor, and
displacement thickness upstream and downstream of the shock are shown in Fig. 2-9. The
major conclusions of those experiments were: (1) increasing turbulence intensity causes
the shock position to move downstream, (2) while free-stream turbulence affects Reoand
5* upstream of the shock, it also appears to have a direct effect on the shock boundary-
layer interaction and hence shock position, and (3) increasing turbulence intensity
produces a decrease in the length of the trailing-edge separation region and hence an
increase in pressure recovery.

The data which Green analyzed did not contain any information below turbulence
values of 1%. The values of turbulence intensity on the Raghunthan and McAdams
experiments, 0.3 to 6%, were also larger than encountered in many wind tunnels. Mr. Peter
Bradshaw, has argued, as discussed in Ref. [23/2], that the variation of gross boundary-
layer properties with turbulence should be parabolic such that the effects at low values
of turbulence (< 1%) would be diminished and negligible. Further experiments by
Raghunthan and McAdams, Ref. [85/9], while not specifically confirming Bradshaw's
hypothesis, do show that for fixed transition the effects of turbulence below 1% are
small at least for the 18% circular arc airfoil. With free transition there is, of
course, a coupling between the well-known effects of turbulence on transition location
and the effects depicted in Fig. 2-9. The experimental configurations used in each of the
above cited experiments are not particularly sensitive to small changes in any parameter
relative to a supercritical airfoil for example. Fig. 2-8. Therefore, the results should
not be discounted merely because the turbulence values were large. It seems logical to
conclude that if the flow over the model being tested is sensitive to small changes in
boundary-layer properties and if turbulence intensity varies with tunnel Reynolds number
(power), then even with fixed transition location a pseudo-Reynolds number effect is
possible.

2.4 HUMIDITY EFFECTS

The effects of water vapor condensation in the free-stream flow at supersonic Mach
numbers have been recognized for many years. In 1979 preliminary data were obtained in
Tunnel 16T during the tunnel drying process which indicated lift, pitching moment, and
axial force all varied significantly as a function of specific humidity at subsonic Mach
numbers, provided a portion of the flow over the model was supercritical. Typical data
acquired with two scale models of the same configuration are shown in Fig. 2-10. Note
that the effects begin with the dewpoint temperature 10° below the static temperature.
Other, although rather crude, data were obtained which indicated the aerodynamic
coefficient variation, both in magnitude and form, was a function of model configuration.
It is hypothesized that the variations were caused by water vapor condensation in the
supercritical portion of the flow field and hence are potentially a function of Mach
number, temperature, pressure, model attitude, configuration and, because of condensation
rate kinetics, model size.

A computer program has been devised which solves the Euler equations along with the
energy and specie conservation equations for two and three-dimensional transonic flows,
Ref. [85/8]. The code has been verified with almost exact replication of condensation
data from a 10-cm-diam supersonic nozzle, Ref. [63/1], and unpublished calibration data
from Tunnel 16S nozzle (4.9-meter exit diameter). Therefore, the code appears to have
wide ranging applicability. Results from the code applied to the two- dimensional, 200-mm
chord, CAST 10 airfoil tested by Stanewsky, Ref, [83/7], is presented in Fig. 2-11 for
two Reynolds numbers obtained by changing the tunnel total pressure. The value of
specific humidity at which the dewpoint temperature equals the free-stream static
temperature is 0.07 and 0.018 for the 3- and 1-atm total pressure condition,
10

respectively. Thus, the effects occur at humidity values much below the saturation value.
Three conclusions are obvious:

1. There is a significant variation in each coefficient with specific humidity.

2. There is a significant effect of Reynolds number at a constant humidity.

3. The higher the total pressure the lower the value of specific humidity at which
the effect begins and the greater the effect. The effect is a consequence of the
thermodynamic relationship between saturation specific humidity and pressure and
of the higher energy release per unit weight of air as pressure is increased.

As shown in Fig. 2-12, the humidity effect is manifested first as an increase in pressure
on the airfoil upper surface and, as specific humidity increases, by a forward movement
of the shock position. In each instance, the change in the sign of the axial-force slope
corresponds to the beginning of the forward movement of the shock.

The conclusions expressed in this subsection only apply to the specific conditions
of the calculations. There is not yet enough experience nor experimental variation to
generalize. However, the calculation results are sufficiently compelling to warrant the
measurement of humidity as a parameter which defines the test conditions and an
assessment of its influence when trying to define true Reynolds number effects.

2.5 TUNNEL CONTAMINATION

The contamination of tunnel flow by particles small enough to be airborne and

simultaneously hard enough to cause pitting of model surfaces in stagnation point regions
will lead to premature boundary-layer transition. Aluminum models are particularly
susceptible to such damage. From a Reynolds number effect viewpoint, roughness at the
leading edge would mask any true transition movement with Reynolds number effect, negate
the effectiveness of transition trips which would normally be downstream of the
stagnation region and could conceivably produce an unrealistic variation of transition
location with angle of attack, Mach number and Reynolds number. In all but the most
severe instances, the effects of tunnel contamination can be avoided by using hardened
steel models.

2.6 TUNNEL CALIBRATION

All aerodynamic data exhibit some sensitivity to tunnel calibration parameters. It

has been shown that the calibration of perforated tunnels is a function of Reynolds
number, Ref. [78/3]. Aulehla and Eberle, Ref. [82/9], using computational results, argue
that calibration of a solid wall tunnel is a function of Reynolds number which, if not
corrected, will produce an erroneous variation of shock position with Reynolds number.
Perhaps the parameter most sensitive to errors in tunnel calibration is afterbody drag
whose sensitivity near zero can be shown to be

AC - H . *H (2-3)
A
w M U + 0.2Af2)

where A is the cross-sectional area, A w the wing area, and M the tunnel Mach number. The
afterbody drag error is equal to the error in free-stream pressure acting on the model
cross-sectional area. For a typical fighter aircraft an error of 1 count in afterbody
drag is produced by the Mach number error shown in Fig. 2-13. Such extreme sensitivity to
small errors requires not only very careful calibrations, but also that all factors which
affect the tunnel calibration are properly considered.

One example of the effects of calibration error is shown in Fig. 2-14 wherein the
integrated pressure drag on a body of revolution is presented with and without the
Reynolds number correction to the tunnel calibration, Ref. [80/8]. The fact that the
slopes of the uncorrected forebody and afterbody curves are of opposite sign is caused by
the free-stream pressure-area term acting in the opposite direction for the two integra-
tions. Proper application of the tunnel calibration essentially eliminates the variation
of pressure drag with Reynolds number. There is still a small error in free-stream
pressure at the lowest Re in Fig, 2-14 which was not properly removed in the calibration.

Other examples and an extensive discussion of the effects of not calibrating

variable density wind tunnels as a function of Reynolds number has been published by
Auleha in Hef. [87/1].
 11
2.7 SIDEWALL EFFECTS IN 2-D TESTS

There has been a lot of attention paid to the effect of the sidewall boundary layer
on two-dimensional test data, Refs. [82/10] and [83/8] for example. However, no one seems
to have considered the effect tunnel Reynolds number has on the sidewall boundary layer
and hence its influence on the model data. The effect the sidewall boundary layer can
have is illustrated in Fig. 2-15, which presents data wita "adequate" sidewall boundary
layer removal (according to the authors of Ref. [83/8]) and none. The effects are
obviously large, leading to the hypothesis that unless the boundary layer receives the
proper conditioning or the aspect ratio is large (greater than 2.5), tests defining the
variation of two-dimensional data with Reynolds number contain large pseudo-Reynolds
number effects. The definition of what constitutes proper conditioning has not been
totally agreed upon by the testing community and may vary from facility to facility.
However, the procedures summarized in Ref. [87/2], which are being used in the Langley
0.3m TCT Facility, significantly improve data comparisons.

2.8 NON-UNIFORM FLOW

Obviously a tunnel-empty spatial gradient of any free-stream flow property can

affect model data depending upon its magnitude and the model configuration. There is
little information known which indicates if the gradient of parameters normally con-
sidered in this category, pressure, temperature, and flow angle vary with tunnel Reynolds
number. Considering the mechanisms which could cause gradients to vary with Reynolds
number, only the variation of the wall boundary layer influencing buoyancy appears
plausible. Such is possible in a solid wall wind tunnel but not likely in a ventilated
test section. Unpublished data from a porous tunnel indicate buoyancy is not a function
of Reynolds number. Also, there is no reason to expect either tunnel temperature or flow
angle is a function of Reynolds number. It should be noted that buoyancy-like gradients
in x, y, and z due to wall interference can exist and be a function of Reynolds number.

2.9 THERMAL NONEQUILIBRIUM

Although not as severe a problem in a continuous flow tunnel as in a short duration

facility, heat transfer into or out of the model can produce spurious effects on data
which are sensitive to the boundary-layer properties. In short duration facilities
significant differences are likely to arise between the temperature of the model and the
recovery temperature of the stream. Since the temperature of the stream is influenced by
throttling the flow from the reservoir to tunnel total pressure, the stream temperature
is generally a function of tunnel Reynolds number, i.e., of the throttling pressure
ratio. Unless provisions are made to cool the model prior to each test, the stream will
be appreciably cooler than the model. The resulting heat transfer will significantly
alter the boundary-layer properties and appear as a spurious scale effect. A brief study
of this phenomena, Ref. [72/6], concluded that an increase of 1% in model-to-free-stream
temperature ratio would produce an effect roughly equivalent to a 3i% reduction in
Reynolds number. For Reynolds number variations within a given tunnel, the problem may
not be significant. For example, expanding air from a 300-atm pressure reservoir at 300°K
to 50 atra reduces the stagnation temperature to about 270°K; whereas, expanding to 1 atm
reduces the temperature another 10°K. At a constant Mach number the real gas effects
between conditions at 50 atm and 1 atm would be equivalent to a 13% Reynolds number
change within the tunnel. However, if the data from a blowdown facility with the model at
room temperature were compared to a similar condition wherein the model was in thermal
equilibrium, the data at 1 atm would contain an effect equivalent to a 53% change in
Reynolds number because of the effect of heat transfer on the boundary layer of the model
in thermal non-equilibrium. While the numbers given are very approximate, the effects are
significant and should be considered when comparing Reynolds number data from short-
duration facilities.

Dougherty and Fisher, Ref. [80/4], deduced the effect of non-adiabatic wall
temperature on transition Reynolds number measured on a 10-deg cone both in flight and
the AEDC Tunnel 4T. The empirical relationship derived from their data

Re T IRe T = (T IT V (2-4)
\ w aw)

encompasses at least a Mach number range from 0.55 to 2.0 and temperature ratios from
0.95 to 1.08. Such a large sensitivity could introduce pseudo Reynolds number effects
depending upon how a particular test was conducted and the sequential relationship
between Re, T a w , and time.
12

It has long been recognized that temperature changes in continuous tunnels can
manifest as balance shifts. Strain gages have been temperature compensated to minimize
the effect. However, work by Ewald and Krenz, Ref. [86/6], has shown significant sensi-
tivity (80 uv) of axial-force gages to moderate thermal gradients (8°C) within the
balance structure. Since the mechanism producing the effect is heat transfer into one end
of an internal balance, the time constant is measured in hours requiring long times to
reach thermal equilibrium. Such effects will be very pronounced in cryogenic facilities.
If the tunnel temperature is varied systematically as a function of Reynolds number, the
data from an internal balance may well contain a pseudo-Reynolds number effect.

2.10 MODEL DEFORMATION

While it is recognized that wind tunnel model support systems deflect under load,
the models themselves are generally considered to be rigid. However, to comply with the
present day accuracy requirements such assumptions may not be valid. As an example, the
deflections of the two-dimensional model reported in Ref. [83/7] were computed. The model
has a 200-mm chord and a 510-mm span. It was assumed to be rigidly constrained at the
tunnel walls and made of solid steel. The calculated change in the mid-span angle of
attack caused by torsional bending for Re = 30 x 10 6 and 2-deg incidence varied between
0.07 and 0.10 deg in the Mach number range 0.6 to 0.8. The effect of such changes is
shown in Fig. 2-16. The data at 30 x 10* Reynolds number have been corrected to account
for the computed torsional deflection of the airfoil. The correction reduces the differ-
ence between the 10 x 106 and 30 x 106 curves by more than a factor of two except at Mach
number 0.8 at which the wing is almost completely stalled and there is little variation
of lift with incidence. Model deformation does not significantly influence the data at
the 10 x 106 and 4 x 106 Reynolds number since the loads are relatively low.

As another example, the wing torsional bending of the ONERA calibration model M5,
Fig. 2-17, was also calculated. A solid steel wing was assumed although the wing does
contain pressure orifices. The change in incidence as a function of serai-span location is
shown in Fig. 2-18 for Mach number 0.84, lift coefficient 0.4, and various Reynolds
numbers. The deflections at the higher values are large enough to cause variations which
could be interpreted erroneously as Reynolds number effects. In this instance, the wing
dihedral also changes with Reynolds number. However, the dihedral effects should be less
than those caused by the change in wing twist.

2.11 TRANSITION FIXING

A final word is needed on transition fixing as it relates to Reynolds number

effects. A boundary-layer trip is sized for a particular Reynolds number, Mach number,
pressure gradient condition. Substantial deviation from that condition results in either
an over- or under-fixed boundary layer. The necessary variable range for the deviation to
be substantial is not very well defined. A systematic study has not been published.
Nevertheless, the necessary range seems to be several times less than the range of most
test condition variables. Most studies have indicated one must change grit size with Re.
The consequences of overfixing is that the turbulent boundary layer may be thicker than
desired and the drag higher because of grit drag. The consequences of under-fixing are
that the transition zone is either longer than for a just adequate trip height or the
transition is downstream of the desired location. In each situation consequences can
accrue in all aerodynamic parameters. An example of the consequences of over-, under-
fixing is given in Fig. 2-18 taken from Ref. [84/6]. Obviously, if Reynolds number is
varied with fixed transition without changing grit size appropriately with Re or with a
non-hydraulically smooth surface, the resulting variation in aerodynamic coefficients
with Reynolds number is not a true scale effect.
 13

2-D AtRFOIL IN HST

C = 0.5m
Rec = 3.6 x 1 0 6
Rec = 1 2 . 5 )( 1 0 6

/AMo
Mo = 0.73
X = 1.0
X = 0.8
.1
X = 0.5
/
4 AAO

0.2 0.3 0.4 O.S 0.6 0.7 0.8 0.9 1.0

10.02 POROSITY PARAMFrtR, Qv =
AMo
0 0.4 0.8 1.2 Aa 0.4 0.8 1.2Aa(DEG) A-10, M = 0.3, r = 5%
Re/ex 10" 6 dCp/dfl Q «o Aa = 57.3 (S/C)60CL
Fig 2 - 1 . B l o c k a g e ( A M a c h ) a n d i n c i d e n c e
(Aa) c o r r e c t i o n s c a l c u l a t e d f r o m 1.10 1.90 0.51 -0.020 -0.090 CL
measured top and bottom wall 1.44 1.S9 0.56 -0.035 -0.159q
pressure distributions,[84/3]. 2.16 1.35 0.60 -0.050 -0.227 CL
3.60 1.10 0.64 -0.06S -0.295 CL
10.0 dCp
do f . 2.,03 ± ^.,-1« Fig. 2-4. Variation of the upwash interfer-
5.0 ence with changes in unit
Reynolds number.

2.0
AILERON AND SPEED BRAKE
SYM T t d rt/dt
1.0
o 6.0 0.125 0.125 6.0 TRIM TAB
a 7.0 0.125 0.213 4 1
o 10.0 0.125 0.166 7.5 FLAP
HORIZONTAL STABILIZER
O.S - 0 1.0 0.250 0.166 1.5 — f - RUDDER

0.2
^
v
2.5
5.0
-w 10.0
0.250
0.250
0.250
0.166
0.166
0.166
3.8
7.5
15.1
fflV ^ELEVATOR

4
2x1 0 1 x 10 5 1 106

Fig. 2 - 2 , Correlation of porous wall char- LANDING GEAR POD

acteristics with displacement

thickness Reynolds numbers.

5 x 10 8
Fig. 2-5. The A-10 model.
Re/Ixl0" 6
Res. = 0.0267 (Re CN ) 4/5

1 x 108

2 x 107
1 x 104 1 x 10 s 5 x 10 5
Re 6 .

Fig. 2-3. Correlation of nozzle exit d i s -

placement thickness Reynolds
number with tunnel contraction-
nozzle Reynolds number.

-0.008
0.2 0.4 0.6 0.8 1.0 1.2

a. M = 0.3
Fig. 2 - 6 . Comparison of theoretical a n d
e x p e r i m e n t a l r e s u l t s f o r t h e A-
10 m o d e l .
14

1.5 M = 0.73
o TUNNEL DATA a = -0.25 DEG
o TUNNEL DATA R,. = 4 x l 0 6

3 1.0

0.5 L

b . Local Mach number

0.010 r-

UPPER SURFACE

0.005

0.005 "

0.010

-0.008 a. Skin friction

cr1 o 0.2 0.4 0.6 0.8 1.0 1.2 Fig. 2-8. Effect of free-stream turbulence
on skin friction distribution,
b . M = 0.75 [85/7].
F i g . 2 - 6 . Continued.

SYM TUNNEL SYM TUNNEL

o AEDC TUNNEL 16T 0 NASA/AMES 12 PT
•V AEDC TUNNEL 16T (WALLS TAPED) v RAE BEDFORD 8 X 8 SWT
i AEDC TUNNEL 4T - NASA/LANGLEY 16 TT FREON
* AEDC TUNNEL 4T (WALLS WITH TAPE OR SCREEN) * NASA/LANGLEY 16 TDT* TEST
9 ONERA 6 X 6 S-2 MODANE • NASA/LANGLEY 8 TPT | MEDIUM
7 NAS A/AMES 11 TWT • NSR&DC7 X 10 T
/ NASA/AMES 11 TWT (WALLS TAPED) & NASA/LANGLEY 4 SPT
t. NASA/AMES14TWT o RAE BEDFORD 3 X 4 HSST
* NASA/AMES 14 TWT (WALLS TAPED) a NASA /AMES 9 X 7 SWT
« CALSPAN 8 TWT -d NASA/LANGLEY 4 SUPWT (TS NO. 1)
o ARA, LTD. BEDFORD 9 X 8 « FLIGHT DATA, FIG. 18

20 " T

MEAN FAIRED LINE

10

•o 6 -

r
NOTE.
PHANTOM SYMBOLS - SPECTRUM ± 2 0 % Rer BAND -I
OF DISTURBANCES DOMINATED
BY LOW-FREQUENCY COMPONENTS
RAISING OVERALL LEVEL
' i
i . i i i i _l L.
0.02 0.04 0.060.08 0.1 0.2 0.4 0.6 0.8 1.0 2.0 3.0 4.0

Jr\'2l<\oc X 100, PERCENT

F i g . 2 - 7 . C o r r e l a t i o n between Re? and cone s u r f a c e d i s t u r b a n c e measurements,[82/7 ]

 15

a
0 - ZBIO TUBBUltNC!
1 • UPSTREAM Of SMOCK 0.0000 - 0
2 . TRAIL ING [ O G i 0.0015- 1
0.0025 - 2
7 5 S
0.0040 - 3
4 0.0060 - 4
- _- 2 °
SHOCK POSITION
I I I I
1 - 3

15
* . -1.2
n
' L^ss^ '
1

3L 1.00 > - 1.0 0 -0.8

r^-^'yA^:=:::^ \\

PRESSURE COEFFICIENT
• I S ^ A ^

Fig. 2-9. Effects of free-stream turbulence —.- \x-4

W o *
on shock and boundary-layer ^ ^ > ^ >^3
properties on a 9%-thick circu-
lar airfoil, fixed transition
location, [83/5, 83/6].
0.8 •

0.0020 SCALE
A - (WET - DRYIfQR E A C H M0DEL
0 1/4
1 9 —L I I I ,

0.0010 A 1/9
0.2 0.4 0.6
• a£^t&^ 0.8 1.0
PERCENT CHORD
0 ^^C^%r55&£fW>5T3r^oc£trba
•tJgefQg^fl&JV^jrjJSJ^
•0.0010 1 1 1 Fig. 2-12. Effect of humidity variation on
CAST-10 pressure distribution.

12 r
0.01
A
0
/&
-0.01 Iff A A

-0.02
.* *tf 1 i 1 1 1

0.001
5 0
-1 -tt£&&g$B$9&&&&&&^^ Fig# 2 - 1 3 . E rror in Mach number to produce a
-0,001 1-count error in afterbody drag
-10 10 20 30 40 50 for a typical fighter aircraft.
TDP, °F

Fig. 2-10. Effect of moisture content on

MODEL
model aerodynamic coeffi- AFTERBODY
cient, M = 0.9, a = 6 deg.
LENGTH = 330cm
NORMAL FORCE COEFFICIENT M A X I M U M DIAMETER = 41cm
AXIAL FORCE COEFFICIENT
PITCHING MOMENT COEFFICIENT
0.03
3 ATM CALIBRATION INDEPENDENT OF Re
1.3 1 ATM
T j = 310°K
M = 0.765
1.2 a = 2.85, DEG

1.1

1.0

a
£ 0.9
o
D 0.8

0.7

0.6
0.002 0.004 0.006
SPECIFIC HUMIDITY Fig. 2-14. Comparison of pressure-drag coef-
ficient calculated with and
Fig. 2-11, Effect of the humidity on aero- without Reynolds number correc-
dynamic coefficients, two- tions to the tunnel calibration,
dimensional CAST-10 airfoil. [80/8].
16

«oo cN SIDEWALL
SUCTION
0.794 0.826 ADEQUATE"
0.790 0.636 "ZERO"

I I

—I 0.211 6 , 1 -
: 0.4111, [—f- 1-

0.0633 b , - H

MOOR MS

WINGSPAN b. (m) 0 982

REF CHORD, c M 0.137
REF SURF, 5 {mh 0.132
Fig. 2-15. Effect of sidewall boundary-layer FUS. LENGTH, lF H I.OSt
0.124
removal on two-dimensional air- FUS. D I A M ,
MAX
Dp
O K K S SECTION.
M 0.01 M
(i» 2 )
foil pressure distribution,
[83/8].
F i g . 2-17. Onera M-5 model.
6
RexlO"

_^__ 10 FREE TRANSITION

._•_. 301
30 CORRECTED FOR TORSION DEFLECTION
0.8

0.7

UNCORRECTED FOR TORSION

£ 0.6
40 60
8 % SEMI-SPAN

0.5

F i g . 2 - 1 8 . Wing t w i s t , ONERA M-5

0.4 model, Mach number
0.60 0.64 0.68 0.72 0.76 0.80 0.84 0 . 8 4 , CL = 0 . 4 .
MACH NUMBER

F i g . 2-16. V a r i a t i o n of l i f t c o e f f i c i e n t
with Mach number a t 2-deg
i n c i d e n c e and v a r i o u s Reynolds
numbers, 2D CAST-10 a i r f o i l i n
2 0 - i n . span wind t u n n e l .

ADEQUATE TRIP,
ISO GRIT

OVER FIXED, 100 GRIT

OVER FIXED, 120 GRIT UNDER FIXED, 180 GRIT

ADEQUATE TRIP, 150 GRIT
OVER FIXED,
120 GRIT
OVER FIXED
100 GRIT
AIRFOIL 2

LIFT COEFFICIENT ANGLE OF ATTACK

Fig. 2-19. Effect of over, under fixing boundary layer, cruise Mach number,
grit a 7% chord,[84/6].
 17

3. OBSERVED REYNOLDS NUMBER EFFECTS: AIRFOILS AND HIGH ASPECT RATIO WINGS
by
A, Elsenaar
National Aerospace Laboratory NLR
Anthony Fokkerweg 2, 1059 CM Amsterdam
The Netherlands

3.1 Introduction
After the preceding discussion of the pseudo-Reynolds number effect, this and the following chapters
are concerned with the Reynolds number effects as actually observed on various aerodynamic configurations.
In doing this, the classification as discussed in section 1.2.1 will be followed. The experimental infor-
mation has been selected from the open literature and from unpublished results of the National Aerospace
Laboratory NLR In the Netherlands. As far as possible results that were part of a larger systematic study
have been used. In this way the internal consistency of the data could be checked to some extend. More-
over, only results with transition close to the leading edge have been used (with the exception, of cour-
se, of section 3.?). But it can not be excluded completely that part of the data still contain some measu-
rement error or "pseudo-Reynolds number effects", as discussed in chapter 2.
The flow about two-dimensional airfoils has drawn a lot of attention in respect to Reynolds number
effects. The attention is caused by the fact that the first high Reynolds number facilities were, because
of their size, mainly used for two-dimensional testing. But also two-dimensional flow is relevant for
high aspect ratio wings. Thus two-dimensional and high aspect ratio configurations arc discussed together
in one chapter.
To obtain a better understanding of the problems involved, this chapter has been sub-divided into a
number of sections. As a first distinction the state of the boundary layer Is used. Flows with a substan-
tial amount of laminar boundary layer development are discussed separately from flows with a fully turbu-
lent boundary layer. The reason is that transition location variations, laminar shock-wave boundary layer
interactions and, in some cases, laminar separation bubbles introduce typical flow phenomena that are
absent in flight where normally well developed turbulent boundary layers are found. Helicopter blades and
the flow on propeller blades are an exception (due to the low chord Reynolds number), but they can be
tested at their full scale Reynolds number in ground based facilities and do not represent an extrapola-
tion problem. One other class of airfoils is excluded: the (either forced or natural) laminar flow air-
foil. They deserve a very special attention from the point of view of Reynolds number effects which is
beyond the scope of this AGARDograph. In section 3.2 a number of examples related to drag, maximum lift
and pitching moment will be discussed to Illustrate the point that tests with free transition behave quali-
tatively different from flows with mostly turbulent boundary layer development (whether caused by natural
transition close to the leading edge or by artificial means such as fixation). Free transition results
can even by very misleading for flight conditions.
The next two sections are only concerned with Reynolds number effects observed on configurations
with fully turbulent boundary layer flow. This allows a systematic discussion of the problem. Use will
also be made of the distinction (as discussed in section 1.2) between direct Reynolds number effects (re-
sulting from changes in boundary layer development) and Indirect Reynolds number effects (that appear as
changes in pressure distribution). In section 3.5 the question how relevant two-dimensional flows are to
high aspect ratio wings will he addressed. It will be argued that certain conditions must be met before
one can speak of a good correspondence between two- and three-dimensional flows.
The last section of this chapter summarizes the preceding discussions especially in view of the use-
fulness of wind tunnel tests for flight prediction. The AGARD working group WG-09 addresses this topic
specifically and for more information their final report should be consulted (see ref.00/1). However, the
analysis presented in this chapter is hopefully useful in reducing the uncertainty Implied in the full
scale prediction on the basis of sub-scale wind tunnel tests.

3.2 Effects related to partly laminar boundary layer flow

3.2.1 The direct effect on drag

Boundary layer transition appears to be a function of Reynolds number, pressure distribution, sweep
angle, flow quality, etc. On the basis of empirical criteria, It can be determined in principle if the
boundary layer will be laminar or turbulent In the wind tunnel or In flight. For a smooth flat plate boun-
dary layer the transition Reynolds number is 2 to 5x10 depending on noise level and turbulence intensity.
This indicates that an airfoil with a flat pressure,distrlbution might experience substantial regions of
laminar flow for chord Reynolds numbers below 10x10 . This will be even more true for laminar flow air-
foils (like the NACA 6-serles). For "peaky"-type pressure distributions the pressure gradient near the
nose will trigger transition, almost Independent of Reynolds number. Leading edge contamination complica-
tes the situation even more for wings with an appreciable wing sweep. Therefore, at tunnel Reynolds num-
bers of the order of 3x10 laminar or turbulent boundary layer flow will be found, depending on wing geo-
metry and flow conditions. For the same reason appreciable transition location variations might be found
on wind tunnel models as the Mach number or the incidence is changed. On the other hand it is generally
assumed that the boundary layer will be predominantly turbulent at flight Reynolds numbers (say in excess
of 20x10 ) unless the airfoil has been designed specifically for laminar flow. By and large, appreciable
differences in transition location are possible between wind tunnel and flight.
The skin fraction drag of a laminar boundary layer is significantly less than that of a turbulent
boundary layer. Therefore, transition location variations have a very pronounced effect on drag. As a
rule of the thumb one can state that a IZ chord transition point variation corresponds roughly with half
a drag count. Consequently, transition point variations play a dominant role In the variation of drag
with Reynolds number as is well illustrated when comparing test results with fixed and free transition
(Fig. 3.2-1). Along the laminar lower-left branch of the drag,curve In this figure transition moves clo-
ser to the airfoil nose till a Reynolds number of about 10x10 . In this case the transition variation has
been caused by a Reynolds number increase either in a direct way or by an Increase in noise level or tur-
bulence Intensity. At constant Reynolds number the transition point might move as a result of a changing
pressure distribution, e.g. with Mach number. This is typically observed on the upper surface of supercri-
tical airfoils with a roof-type pressure distribution. A rather extreme case (18Z thick supercritical
airfoil) is presented in figure 3.2-2. The transition movement Is coupled with the rearward movement of
the shock (Fig. 3.2-3). The resulting effect on drag Is strongly felt at Mach numbers in excess of 0.7
18

and shows up as a very deep drag bucket around the design Mach number. In this particular case the effect
has been amplified by the fact that the turbulent boundary layer in combination with the shock causes a
trailing edge separation below Mach - 0.7 as the trailing edge pressure development indicates. Less extre-
me examples for two other airfoils are presented In figure 3.2-4. Transition fixing near the nose comple-
tely eliminates the drag bucket near the design condition. The figure is also Included to show how tran-
sition variations on the lower surface can easily be Interpreted as a significant difference in drag
creep between these two airfoils that have identical upper surface pressure distributions and differ only
in the lower surface airfoil geometry. All these examples Illustrate the necessity to measure and/or con-
trol the transition location on airfoils for accurate drag assessment.

3.2.2 Indirect effects on the pressure distribution

It could be argued that the change in airfoil contour through the relatively thin (either laminar or
turbulent) boundary layer will have only a minor effect as far as the pressure distribution Is concerned.
This is, however, not entirely true. The Kutta condition is directly Influenced by the boundary layers
near the trailing edge. And the Kutta condition In turn affects significantly the overall circulation in
addition to the effect of the boundary layer displacement thickness on the effective airfoil thickness
and camber. Moreover, when boundary layer separations are present (e.g. a leading edge bubble, a separa-
tion underneath the shock or at the trailing edge) the interaction with the outer flow is an essential
feature with substantial local or even overall effects on the pressure distribution. The difference be-
tween laminar, transitional or turbulent shock-wave boundary layer Interactions is particularly important.
The dominant effect appears to be a variation of the interaction length as discussed in more detail by
Green (Ref. [71/2]; see also fig. 3.2-5). The extent of the Interaction appears to decrease with increa-
sing Reynolds number (apart from an anomaly at the lower turbulent Reynolds number range). This is all
very well reflected In the pressure distributions as presented in the figures 3.2-6 and 3.2-7. Note that
in the last example hardly any changes are noticeable in the pressure distribution with fixed boundary
layer transition. These cases are very typical for the laminar-type airfoils of the 1950's as discussed
In section 1.1.
Tbe elimination of the high Mach number peak (as found with a turbulent boundary layer) due to the
laminar shock-wave boundary layer interaction as shown in the preceding figures appears to be generally
less pronounced for modern supercritical airfoils with a characteristic "roof-type" pressure distribu-
tion. The interaction region can still be recognized in the form of a pressure plateau ahead of the
shock, but the dominant effect appears to be an upstream movement of the shock Itself as Indicated for
two different airfoils presented in the figures 3.2-8 and 3.2-9. In both examples a change In pressure
distribution can be noticed when the transition point Is moved artificially (in an attempt to simulate
Reynolds number changes) at constant Reynolds number. The very large effect on the shock position (as
much as 30Z chord for airfoil NLR 7301) Is due to the Interaction of the (almost) separated boundary
layer at the trailing edge with the outer flow field, as can be observed from the trailing edge pres-
sures. The upstream movement of transition thickens the boundary layer over the aft part of the airfoil,
thereby hastening separation in addition to the already noted local effect of a laminar versus a turbu-
lent shock-wave/boundary-layer interaction.
Figure 3.2.9 also shows a typical effect on the lower surface in the rear loading region. Just be-
hind 50% chord, a laminar separation bubble can be observed in the case with natural transition. This
bubble disappears with fixed transition but the very thick turbulent boundary layer causes a substantial
reduction in renr-loadlng, especially with 72 and 1531 fixation positions (where the turbulent boundary
layer is possibly separated).
A final example, presented in fig. 3.2-10, is related to the nose region of an airfoil. Such a situ-
ation might occur at intermediate Mach numbers and high angles of attack. In this case transition fixa-
tion near the leading edge suppresses the formation of a laminar separation bubble at the lower Mach num-
bers. The appearance of a shock at higher Mach numbers complicates the situation further with possible
effects on maximum lift development as argued in the report. This type of flow, however, Is still not
well understood (see also [Ref. 76/1]).
In all these cases large differences In pressure distribution between laminar and turbulent boundary
layer development could be observed. Its significance in the context of Reynolds number effects is that
the transition from a laminar to a turbulent boundary layer depends critically on the Reynolds number.
Moreover, the flow quality in the wind tunnel is of considerable Importance as well as discussed In chap-
ter 2. Therefore similar effects as the ones shown here might be observed between wind tunnel and flight
without any artificial boundary layer transition fixation. This can have serious consequencles for an
evaluation of the overall wing characteristics.

3.2.3 Effects on lift and pitching moment

The variation of the pressure distribution with the extent of the laminar flow region as discussed
in the previous section will of course also be reflected in the overall airfoil characteristics. Figure
3.2-11 (unpublished results of NLR) shows a selection of lift curves of a supercritical airfoil at one
Reynolds number. In addition to the free transition case, the transition point has been changed artifi-
cially. It appears that with free transition the approximate linear relation between lift and Incidence
is lost, most notably for the highest Mach number. At Mach - .75 and c. = .25 the natural transition lo-
cation nt the upper surface is close to the leading edge. It moves rapidly downstream, both with decrea-
sing and increasing angle of incidence. In the latter case the transition point movement is coupled with
the downstream movement of the shock, particularly at the higher Mach numbers and similar to that In fi-
gure 3.2-2 and 3.2-3. The transition variations are the cause of significant variations in the local lift
curve slope, as compared with the turbulent case. Very similar effects have also been observed on other
supercritical airfoils like the one presented in figure 3.2-13.
The effect on the pitching moment, figure 3.2-12, is even more dramatic, especially at the higher
Mach numbers. At constant lift the pitching moment varlntlon results from a redistribution of lift over
the airfoil chord. For that reason variations In shock position give rise to significant pitching moment
variations at the higher Mach numbers. But also small variations In the rear loading region at the lower
surface (due to changes in the local boundary layer thickness) result In a significant but now almost
constant shift over the complete Mach number range. Again, the non-llnearltles around a lift of .25 at
Mach - .75 arc eliminated completely with fixed transition.
 19

The variation of maximum lift resulting from different transition point locations is also of some
interest as the figures 3.2-11 and 3.2-13 indicate. With free transition, the more favourable boundary
layer condition ahead of the shock and the improved boundary layer development downstream retards the
separation as compared with the fixed transition case and leads to higher maximum lift values. It is also
worth noting that the c.-max development Is qualitatively different for the higher Mach numbers: with
natural transition the lift loss is much more gradual which is most likely caused by local effects of the
laminar shock-wave/boundary-layer Interaction. It is generally observed that transition fixation reduces
the maximum lift value at the higher Mach numbers, or more specifically, at conditions where the pressure
distribution shows a plateau region terminated by a shock further aft on the airfoil. In the lower Mach
number range the situation is less clear. The local Interaction of the shock with a laminar separation
bubble, as discussed in section 3.2.2 (fig. 3.2-10) seems to result in a slightly higher maximum lift as
compared with a turbulent boundary layer (fig. 3.2-14, related to the same airfoil). In another case how-
ever (fig. 3.2-12) a slightly lower maximum lift is observed In the case of free transition. As was al-
ready noted in Ref. [71/8] the local effects near the nose, and the way In which they interact with the
boundary layer development further downstream, are less clear as compared with a flow which has the shock
wave further downstream. Fortunately, the differences In maximum lift are also appreciable smaller at the
lower Mach numbers.
All these examples Illustrate that the state of the boundary layer (laminar or turbulent) signifi-
cantly affects aerodynamic characteristics like drag, maximum lift or pitching moment. This is even more
so when the transition location varies within a set of experimental results. These variations can be due
to changes in pressure distribution (Mach or angle of attack) but they can also result from Reynolds num-
ber changes or unknown and uncontrolable wind tunnel environmental effects. Generally, such transition
variations will be absent at flight conditions due to the high Reynolds number and surface roughness. For
the evaluation of wind tunnel test results It is advisable to either control transition or measure its
location.
FIXED TRANSITION ( 5 » C I
M 06
O FREE TRANSITION
<• - 0.5° 0.03
ESTIMATE 0 DRAG COEFFICIENT
C
d
UPPER SURFACE LOWER SURFACE
S0NIC
SONIC /
0.02

0.01
--M
ESTIMATED SHOCK
LOSSES

0.3 0.4 0.5 0.6

LJ

Fig. 3.2-1 Variation of subsonic airfoil drag

with Reynolds number (from ref. 68/1)

0.3 0.4 0.6 0.5 0.7 0.8

(X/C),
SHOCK
V
\
0.5 \

© SHOCK WAVE POSITION

(UPPER SURFACE!

o« - o M a , . 0.766 1.0
(EXP. DESIGN CONDITION!

Fig. 3.2-3 Variations in drag and trailing edge

pressure due to transition point
X/C — movement coupled to the shock wave
displacement (see figure 3.2-2)
Fig. 3.2-2 Development of the pressure distribu-
tion in the transonic regime indica-
ting the development of a pressure
plateau with laminar boundary layer
flow (from ref. 78/2)
20

O O AIRFOIL A -06
IDENTICAL UPPER SURFACE,
DIFFERENT LOWER SURFACE
O- 6 AIRFOIL B

T.E.

rj»« B .-TRANSITION POSITION EFFECTED

v
"*v ^ ^ / 7 BY ACENAPHTENE

0.6

L.E. LOWER SURFACE TRANSITION POINT V A R I A T I O N

.7 .8
MACHNUMBER

Fig. 3.2-4 The effects of transition point varla- Fig. 3.2-7 Effect of Reynolds number and arti-
tons on drag creep (from ref. 84/2) ficial boundary layer fixation on
the pressure distribution (from ref.
55/1)

. . ••• Ra I 10* TRANSITION

JJJ FREE
"•" 3.58 FREE
TRANSITIONAL TURBULENT

1-
220K.16%
•9--r
2ZOK30X
TURBULENT *-aa*

v
•-•- 220K, 46V*
LOW HQ
FROflLCAST 1O2/0OA2

a
T-aViE • -C
^wwA^rW^ —•—7TPRESSURE
la|
R|SE V-
«9|
FOR SEPARATION
Ma-0. 66

s V

(b)
S J S C A L E OF 0 1 0 4 0 s r ,8 ^ L 10
^^^SEPARATION «/e

H
1 y , <> • a
10 10* 10 J
10" 10 ue
Fig. 3.2-8 Effect of transition point changes
Fig. 3.2-5 Effect of Reynolds number on some on the pressure distribution of a
properties of shock-induced separa- supercritical airfoil (from ref.
tion at Hach = 1 . 5 (conjectural; 81/1)
ref. 71/2)
-1.2
C
P
H'tPOCtPOO; a - O"
a-9600.000. a - a / * -0.8
C„ CRITICAL

f to 9-AwO 9-tO

^ 1 wwT'\ '
SEPARATION

/ ^
'/-- F-^A-
^ ' t
- N A T . TRANSITION
-^Jjfps-- -30%
0 eo ao to to too o to ao to to too o to ao to t o too •IM
-•7%

Fig. 3.2-6 Effect of a Reynolds number increase Ma-0.75 O - 1°

on the pressure distribution presuma- R « - 2 . 2 . 106

bly caused be a change from laminar 1.2

to turbulent shock-wave/boundary-
Fig. 3.2-9 Effect of transition point changes
layer interaction (from ref. 52/2)
on the pressure distribution of a
supercritical airfoil (from ref.
84/2)
 21

0 3 —•— Tromltiofi traa I 4 14-

- - _ - T r o n a l ( l a n bondl M , w \
04 (0-2%C| |.| 1*
p/p.
05
IO _ Js\ 1 fr
0 61
\^*
\N M 0 -0 30
07
M o -0 40
1

0 «fc —"" 0 */c 0 I

Fig. 3.2-10 Effect of transition fixation near the leading edge at intermediate
Mach numbers (from ref. 71/8)

ol")

Fig. 3.2-11 Effect of transition fixation on Fig. 3.2-13 Effect of transition fixation on
lift development of a supercritical lift development of a supercritical
airfoil for various Mach numbers airfoil for various Hach numbers

Transition free
Transition band I
Transition band IT

SEE FIG. 3 . 2 - 1 0

4 6 « 10 12
I n c i d e n c e , a ldto.1

Fig. 3.2-14 E f f e c t of t r a n s i t i o n f i x a t i o n on
maximum l i f t a t an i n t e r m e d i a t e
Mach number ( s e e a l s o f i g . 3 . 2 - 1 0 )

Fig. 3.2-12 Effect of transition fixation on

pitching moment development of a
supercritical airfoil for various
Mach numbers
22

3.3 Flows with (almost) attached turbulent boundary layers

3.3.1 The pressure distribution - overall effects

In this and the following sections only results with turbulent boundary layer flow (either natural
or artificially tripped) will be considered. They constitute a class of flows with a Reynolds number ef-
fect In the most pure way. A very obvious, but unfortunately not so easy, comparison can be made between
w<nd tunnel and flight. Some examples are shown in the figures 3.3-1, (a) to (e) taken from the referen-
ces [66/1], [68/2] and [72/3]. In the subcritical case (a) and the supercritical cases (c) and (d) the
differences in pressure distribution are small with, for the latter cases, variations in shock position
of 5% chord at most. The cases (b) and (e) seem to indicate a separation in the wind tunnel that was not
observed in flight and vica-versa. In case (b) this separation Is connected with a large shift in shock
wave position: this case is the almost classical example how Reynolds number can change the pressure dis-
tribution. In case (e) the shock position remains virtually unchanged. In reference [72/3] it was repor-
ted thnt Interference effects from the pressure tubing as used in flight were believed to have Influenced
the pressure readings here, although three-dimensional separntlon effects (see section 3.5) were not ex-
cluded. This mny be another example (see chapter 2) that Reynolds number effects and measuring problems
are not always easy to separate. Case (b) seems to suggest that separation of the boundary layer Is of
Importance. Separation effects will be discussed more extensively In the next section. In this section we
will be concerned mainly with attached or almost attached (only local trailing edge separation) flow con-
ditions.
The example (e) Illustrates that wind tunnel/flight comparison is not always an easy matter. Fortu-
nately, pressurized wind tunnels provide the means for systematic studies and some typical results are
presented on the figure 3.3-2 to 3.3-7. They are mainly taken from unpublished work at NLR, studies made
by Stanewsky at DFVLR [Ref. 81/1] and studies made by Blackwell [Ref. 76/4]. Reynolds number effects as
reflected in the pressure distribution are (by definition: see section 1.2.1) "indirect Reynolds number
effects": the change in pressure distribution due to a variation in the (turbulent) boundary layer deve-
lopment. The "regular" part of the pressure distribution change is due to the thinning of the boundary
layer with increasing Reynolds number (the displacement thickness being roughly proportional to Re ).
Consequently the effective airfoil thickness decreases and the effective camber Increases. Of these two
effects, the latter appears to be the most important, especially over the rear part of the nlrfoll, resul-
ting in a local load increase. For a comparison at constant lift, as shown in figure 3.3-2, the Increase
in load over the aft part of the airfoil must be compensated by a decrease in load over the front of the
airfoil, that Is realized by a decrease in angle of Incidence. This feature is common to all pressure
distributions of figure 3.3-2. The shock wave movement appears to be less consistent. With the lessening
of viscous effects over the rear of the airfoil the shock wave moves downstream (In the limiting case of
inviscid flow the shock will be farthest downstream). The decrease In angle of Incidence (to keep the
lift the same) moves the shock, in general, more upstream. The final outcome of these opposite effects
will depend on their relative strength: a downstream movement In the figures 3.3-2 (b) and (d), an up-
stream movement in case (c). The change from an upstream to a downstream shock movement is very regular
as figure 3.3-3 (b) Indicates. In this case the variations in shock position at constant lift appear to
be rather small. Near the design condition, however, larger differences can be obsei-ved as discussed in
section 1.2.3 (figure 1-7). It is due to the fact that the flow is often sensitive to any small change
near the design condition. Figure 3.3-2 (a) provides an other example: a similar change from a double to
a single shock can be obtained at constant Reynolds number with a small Increase in angle of Incidence.
These moderate changes in pressure distribution (except for the design condition) are also reflected In
the small decrease In shock strength at constant lift (fig. 3.3-3 (a)). Is the airfoil of figure 3.3-2
possibly a not so extreme case from the point of view of aerodynamic loading? The pressure distribution
reveals a moderate amount of rear loading and the airfoil does not show appreciable variations in trai-
ling edge pressure with Reynolds number. Such variations do appear to be much larger for another super-
critical airfoil, CAST-10, as reported in [Ref. 81/1] and corresponding with the pressure distribution as
already presented in figure 3.2-8. However, when the variation of shock wave strength and position Is
plotted for the two airfoils as a function of lift and/or Mach number, rather similar variations are ob-
served (see fig. 3.3-3 and 3.3-4). Also the results of other airfoils (this time at constant incidence)
as given in the figures 3.3-5 to 3.3-7 and reported by Blackwell [Ref. 76/4] reflect rather small charges
In pressure distribution. Note also that the Reynolds number effect on the NACA airfoil at high lift is
similar to those observed on the two supercritical airfoils.
These moderate variations In pressure distribution should not come as a surprise. All the examples
are related to attached, or almost attached (limited trailing edge separation at most) flow conditions.
When separations are present the variations in pressure distribution are considerably larger as was al-
ready Illustrated with figure 3.3-lb. Moreover, variations with Reynolds number at constant angle of
incidence may be a little bit larger.

3.3.2 The pressure distribution - local effects

In combination with the rather regular and global Reynolds number effect on the pressure distribu-
tion due to effective thickness nnd camber variations a number of local effects should be noted.

* Local separation underneath the shock

The interaction between the shock wave and the (turbulent) boundary layer will depend on Reynolds
number. A very good review is given by Delery and Marvin [Ref. 85/12] and they Indlcnte that incipient
separation occurs at a shock Mach number close to 1.3 almost Independent of Reynolds number. However,
beyond incipient separation a local separation at the foot of the shock is sometimes clenrly visible in
the pressure distribution (see e.g. figure 3.4-4). F.xtensive correlntlons made by Fulker and Ashlll [Ref.
85/4] indicate that the length of the separation bubble roughly scales with the thickness of the Incoming
boundary layer and will therefore be Reynolds number dependent. The development of this separation bubble
is of prime importance for defining the location of the separation boundaries in the C.-Mach plane (see
3.4).

* Second expansion behind the shock

In some cases, depending on the local curvature of the airfoil, the second expansion behind the
shock can be so strong that locally a second shock is generated. A thick turbulent boundary layer In this
region will reduce the local surface curvature, but at higher Reynolds numbers this second shock may very
 23

well increase In strength, possibly with significant consequencles for the drag development; see fig.
3.3-9 taken from Ref. [76/2] as a typical example. (It was noted later that some of the high Reynolds
number data arc Invalid because of inadequate side-wall suction). The effect might be important when lar-
ge variations in surface curvature occur just downstream of the shock position as measured in the wind
tunnel,

* Lower surface separation

A third local effect In the pressure distribution, already mentioned in section 3.3.2 (fig. 3.2-9),
can be noted on the lower surface of the airfoil in the rear loading region as typically found on modem
supercritical airfoils. The steep local pressure gradient might provoke boundary layer separation at lo-
wer Reynolds numbers. In the concave airfoil contour this separated boundary layer will most likely re-
attach a bit further downstream. For this reason this kind of separation is sometimes hardly visible in
the pressure distribution nnd difficult to detect. In more extreme cases It will cause an appreciable
loss of lift In the lower surface rear loading region. In that case the effect on drag can be consider-
able, in a direct way through lower surface separation drag and in an indirect way through Increased
shock strength at the upper surface to keep the total lift constant. The effect can be assessed experi-
mentally by compnrlng drag of a forward and rear fixation position after a correction for the length of
the laminar boundary layer.

* Local trailing edge separation

The last and most important local effect on the pressure distribution is a limited or local separa-
tion near the trailing edge. When the "load" on the boundary layer increases (higher lift or Mach number,
lower Reynolds number) a small separation will start to develop at the trailing edge and will gradually
move upstream. The tei-m "local" Is used here to distinguish this separation from one that extends from
shock to trailing edge as will be discussed later. The extent of this separation Is difficult to esta-
blish both experimentally and theoretically. In fact one might argue that there Is hardly any difference
between a locally separated boundary layer and a thick turbulent boundary layer in this region. This is
partly due to a stabilizing effect of the interaction with the outer flow: the thick turbulent boundary
layer over the rear of the airfoil decreases the trailing edge pressure and reliefs the severity of the
pressure gradient. This separation will modify the Kutta condition and hence the circulation around the
airfoil. It Is to be expected that the local separation will therefore contribute to some loss of lift at
constant angle of incidence with decreasing Reynolds number. As long as the separation Is restricted to a
small part of the chord (say 10Z of chord), the effects will be gradual. However, at a certain stage in
the separation development a sudden and complete flow brenk-down can occur as indicated by a rapid in-
crease in trailing edge pressure. This situation is complicated further by the interaction between the
trailing edge separation nnd the bubble underneath the shock. This process will be discussed in more de-
tail in section 3.4. Here only conditions with a small and localized separation (cither underneath the
shock or at the trailing edge) will be considered. It is believed, and to some extend justified by the
evidence presented till now and still to come, that such local separations will have. In general, a minor
and gradual effect on the pressure distribution. Some Indication of local trailing edge separation can be
obtained by plotting trailing edge pressure versus lift or Mach number as illustrated in the figures
3.3-10, 11 and 12. The low trailing edge pressures seem to Indicate that the NACA airfoil at 3° angle of
attack (fig. 3.3-10) has developed a trailing edge separation at low Reynolds number that disappears at
higher Reynolds number. This is also the case for the CAST-10 airfoil (fig. 3.3-12) at the lowest Reynolds
number (and prior to flow break-down as Indicated by the rapid decrease on trailing edge pressure). It is
believed that these local separations are of particular significance for the evaluation of drag and pit-
ching moment as will be presented in the next sections.

3.3.3 Lift and pitching moment

The lift curves related to the two supercritical airfoils as discussed in the previous section (see
fig. 3.3-2 to 3.3-4) are presented in the figures 3.3-13 and 3.3-14. The decrease in angle of Incidence
for constant lift is more pronounced when the viscous effects are stronger e.g. at higher lift and Mach
numbers. This appears as a shift and a rotation of the c,-alpha curves. Note also that the largest varia-
tions in o with Reynolds number are found at low Reynolds numbers. The results of airfoil CAST-10 and
model "A" are similar in this respect.
The variation of lift, pitching moment and trailing edge pressure with Mach number (this time at
constant incidence) for both airfoils is presented in the figures 3.3-11 and 3.3-12. Both lift and pitch-
ing moment show an appreciable variation with Reynolds number, this being the most extreme for CAST 10-2.
This is only partly due to the Indirect Reynolds number effect on shock wave position. An equally Impor-
tant contribution arises from a loss of lift with decreasing Reynolds number over the aft part of the
airfoil due to the local decambering effect of the thick boundary layer. As discussed before, the trail-
ing edge pressure development might be viewed as an indicator for the severity of the viscous effects.
A comparison of Its development for the two airfoils Is therefore of interest. In both cases a very regu-
lar variation of trailing edge pressure with Mach number is observed. The CAST-10 airfoil, being closer
to separation at the design condition as boundary layer calculations reveal, shows at the lowest Reynolds
number even a decrease in trailing edge pressure with Increasing Mach number (where the Prandtl-Glauert
rule would suggest an Increase) , a situation that is often Indicative of local trailing edge separation.
It Is of Interest to note here that the lift and pitching moment variations correlate well with the trai-
ling edge pressure development before as well as after flow breakdown as indicated by the rapid decrease
in trailing edge pressure.
The increase in lift at constant incidence with Reynolds number has already been discussed. With
respect to Its variation with Mach number, theory suggests a development that follows approximately the
Prandtl-Glauert rule. The results as presented In the top of figures 3.3-11 and 3.3-12 illustrate very
well that viscous effects modify this development in a Reynolds dependent way and for the CAST-10 airfoil
(low Re) even to such an extent that an almost constant lift is found. Also the pitching moment (up to
lift-divergence) is in this case almost constant. One can conclude that the viscous effects appear to be
stronger on the CAST-10 airfoil. Nevertheless, the two airfoils behave qualitatively very similar.
The study of Blackwell [Ref. 76/4] supports this view as the figures 3.3-6 and 3.3-7 and 3.3-10 in-
dicate. In all cases the lift and pitching moment varies very regular with Reynolds number. The largest
changes with Reynolds number are observed at higher lift or Marh number conditions and at the lowest Rey-
nolds number range where viscous effects are more pronounced.
24

3.3.4 Drag and drag-divergence

In the analysis of drag it is useful to distinguish between flat-plate skin friction drag, subsonic
minimum airfoil drag, subsonic lift dependent drag, compressibility drag and drag-divergence.

* Flat plate skin friction drag

There are many expressions for the Reynolds number dependence of flat piste skin friction drag as
indicated in fig. 3.3-15. Skin friction is approximately proportional to Re with n • 2 for a laminar
and n 5 for a turbulent boundary layer. As a result of this large difference in Reynolds number depen-
dence, the transition location must be accurately known for drag evaluation as already discussed in sec-
tion 3.2. All results presented in this section are obtained with artificial fixation near the nose or
for conditions at which the Reynolds number was so high that natural transition could be expected in the
nose region. When necessary corrections for a (small) region of laminar flow have been made.

* Subsonic minimum airfoil drag

It is common practice in drag evaluation studies to split-up the subsonic drag in a zero lift and a
lift-dependent (viscous and Induced drag) contribution. However, modern airfoils are designed for a par-
ticular lift and for that reason far from symmetric. Consequently, zero-lift drag is not necessarily mini-
mum drag. The airfoil might even show separation at zero lift! Also the induced drag of a wing of finite
span is not necessarily zero at zero lift. For that reason the term subsonic minimum airfoil drag is used
in a rather pragmatic way.
Minimum airfoil drag is often estimated with so-called form factor methods. They express essentially
the airfoil drag as the flat plate skin friction drag multiplied by the form factor that depends on the
airfoil thickness. The form factor accounts, in fact, for some kind of average of the flow velocity over
the airfoil. Typical examples of the variation of experimentally observed drag values are presented in
figure 3.3-16. Although drag is not accurately predicted, the variation with Reynolds number follows close-
ly a skin friction law.

* Subsonic lift dependent drag

A summary of some lift dependent drag values is presented in figure 3.3-17. In this figure the mini-
mum drag as estimated with a from factor method has been subtracted from the experimental values. More-
over, the results have been divided into a low and a (moderately) high Reynolds number range. This parti-
cular representation has been chosen to show the much larger variation in lift dependent drag for the low
Reynolds number cases. It should also be noted that the most "extreme" airfoils (CAST-10 and model "B",
as established from boundary layer calculations) show the largest deviations. For that reason it is sug-
gested that local trailing edge separation (or at least a very thick boundary layer) increases the drag
significantly. This is of some practical importance when low Reynolds number wind tunnel data have to be
extrapolated to flight values on the basis of a well established skin friction trend; the drag might then
be over-estimated by as much as 10%. Also, a particular airfoil, disregarded since it had a relatively
high drag at the tunnel Reynolds number might very well be acceptable or even superior to other airfoils
that apparently performed much better in the tunnel. This is another Illustration of the point raised by
Haines [Ref. 79/2] In his plea that high Reynolds number facilities are Important for aircraft design and
drag optimization. Although trailing edge separation can be detected in the wind tunnel or (approximately)
calculated from boundary layer theory, it seems hardly possible, at present, to either correct or predict
the resulting drag penalty of such a local separation.

* Compressibility drag
The compressibility drag is often defined as the drag increase at constant lift relative to a sub-
sonic condition. Compressibility drag is partly the result of a (slight) decrease in flat plate skin fric-
tion drag with increasing Mach number and an adverse effect on the boundary layer development due to sub-
stantial changes in the pressure distribution as a result of transonic effects. Both effects are represen-
ted rather crudely in some form factor methods (by a compressible skin friction law and by Prandtl- Glau-
ert scaling of the airfoil thickness). This drag creep contribution can be calculated more accurately
from a boundary layer calculation. However, when the boundary layer is separated or close to separation
(as is very often the case) auch a simple calculation will under-estlmate the drag and hence also its
Reynolds number dependence due to the pressure gradient "relief" of the thick boundary layer. Only more
recently, calculation methods that model In a fundamental way the interaction between the boundary layer
(including separation) and the outer flow field (so called strong Interaction) are able to give good quan-
titative results for not so strong shock waves [see Ref. 85/10 and 85/11], The other major contribu-
tion to compressibility drag results from the shock wave development and the interaction with the bound-
ary layer. The Reynolds number effect on the shock wave development is by definition an Indirect Reynolds
number effect. In the figures 3.3-2 to 3.3-7 it was already noted that the Reynolds number effect on shock
wave strength at constant lift is small. Nevertheless, in view of the large sensitivity of the wave drag
to the shock Mach number, quite small changes in pressure distribution may lead to a substantial increase
or decrease in compressibility drag. The existing evidence does not indicate that such large variations
in compressibility drag are generally found as the figures 3.3-5, - 7 , -8, -18, -19 Illustrate. Supercriti-
cal nlrfolls nre designed for a low drag creep by restricting through design the shock wave development
with Mach number. Figure 3.3-18 suggests that the drag creep closely follows the form factor estimate for
lift values slightly below the design lift, with little room for wave drag. The wave drag contribution
Increases with lift, but even then the Reynolds number dependence appears to be almost nil for airfoil
"A". The CAST-10 airfoil on the other hand (fig. 3.3-19) reveals a much larger dependence on Reynolds
number. Since, however, the shock wave development is not to different for these two airfoils (see fig.
3.3-3 and 3.3-4) it is suggested that local trailing edge separation is also of Importance for this dif-
ferent behaviour. Like the lift dependent subsonic drag, the effect is most pronounced for the lowest
Reynolds number.
The drag creep results as presented in the figures 3.3-5 up to 3.3-8 generally seem to confirm this
view. The variation with Reynolds number is small except for the 21Z thick airfoil of figure 3.3-7. Note
however that for this extreme thick airfoil a subsonic (below Mach - .5) drag Increase of 10 counts (re-
lative to a form factor estimate) is obseirved when the Reynolds number is changed. This is somewhat simi-
lar to the observations of the lift dependent drag for the CAST-10 airfoil.
 25

Although these results seem to indicate a rather consistent view with respect to compressibility
drag, one should be careful In generalizing these findings. The airfoil discussed In section 3.3.2 and
for which the pressure distribution is presented in figure 3.3-9 experiences an unfavourable Reynolds
number effect on drag creep due to a very pronounced secondary expansion at higher Reynolds numbers. An
other extreme case is presented in the figures 3.3-20 to -22. This airfoil (the same one as shown in fi-
gure 1-8) shows a dramatic Increase in drag with Reynolds number when evaluated at constant Incidence (ot
• 1.3°), whereas a favourable development is found at constant lift (fig. 3.3-20). This behaviour can be
understood from the development of the pressure distributions (fig. 3.3-21 and - 2 2 ) . Due to trailing edge
separation at the lowest Reynolds number the shock moves aft when the Reynolds number Is increased. This
is accompanied with a pronounced Increase in shock strength at constant incidence lending to an Increase
In wave drag thnt, nt the highest lift values, more than compensates the decrease In viscous drag. At
constant lift, however, the shock still moves aft, but the lower incidence (to keep the lift constant)
reduces the shock strength such that the total drag decreases (note the revised trend in suction peak
level with Increasing Reynolds number). The example nicely Illustrates the Importance of evaluating Rey-
nolds number effects, especially as far as drag is concerned, at constant lift as discussed in section
1.2.3. (fig. 1-10). In all cases a very careful examination of the pressure distribution is required
before a high Reynolds number estimate can be made with some confidence.

* Drag divergence
The sharp rise In drag with Increasing Mach number as experienced In the transonic regime Is often
called drag divergence. Its boundary in the C -Mach number plane limits the region of economic flight.
Drag divergence is distinct from drag creep: the gradual increase of drag prior to drag divergence and
nlso due to compressibility effects. The drag divergence boundary can be derived from plots similar to
figure 3.3-18. Various definitions are possible here like d C /d Mach (at constant lift) - 0.1 as used in
figure 3.3-23. Drag divergence is basicnlly an Inviscid flow phenomenon caused by a rapid, non-linear
increase in shock strength. Viscous effects will modify the drag divergence boundary to the extent that
they modify the pressure distribution. For thnt reason Reynolds number effects on drag divergence will be
small when the Indirect Reynolds number effects (changes In pressure distribution due to Reynolds number)
are small. Figure 3.3-23 shows some typical examples of the Reynolds number effect on drag divergence In
addition to the drag results as shown in the figures 3.3-5, -7 and - 8 . In general, the effects appear to
be very small. Only the results of the CAST-10 airfoil show • much stronger effect. This is not well un-
derstood, but it is possible that local trailing edge separntlon is also to blame.

SUPERCRITICAL PRESSURE DISTRIBUTION

SUBCRITICAL PRESSURE DISTRIBUTION MB0.85; <.f«0*
MBO.75; a f .-0.6*

-W1N0 TUNNEL (TRANSITION FIXED) -WIND TUNNEL (TRANSITION FIXED)

- FULL-SCALE FLIGHT
-FULL-SCALE FLIGHT
-1.0

-.8

Cp .6

-.4

-.2

Fig. 3.3-la Comparisons of pressure distribu- Fig. 3.3-lb Comparisons of pressure distribu-
tions for wind tunnel and flight tions for wind tunnel and flight
- C-141, Ruber itical - - C-141, supercritical -

02

03
^w-^, tar \
-1A
1.3
04 H.u
1.2
a a**" \ V 1.1
1.0

06

0 /
W I N D - T U N N E L . AEROFOIL
0B

Oil

1 I) l l 1
0.1 0.2 0.3 0.4 O.B 0,6 0.7 08 0.9 1.0
X/C

Fig. 3.3-lc Comparisons of pressure distribu-

tions for wind tunnel and flight
- X-1 aircraft -
26

"J a C, Cri R. • to ° Ms a C, Cm Cd R« . Iff*

O 743 95 SO O 742 137 50 -091 .0007 12.4
CL & 740 1 46 51 t . 730 1 83 59 -077 0127 3.0
0 421 yVg. * ti

Tunne J = ^ = 0 466
MODEL"A"
MEASURED (HST) • H MEASURED (HSTI
I * 0 517
@
7J i
1 1
Fl i g h I • 0 482
12
Cb 10 ']
*j^y
y.
•10 ' 9? , a «v
\\ * \
\\• *1 a w, \
i

l
•

0 8 I Bi I ^~jssr».gr>-lfJ"*.~ 11>
7
-
\Tfc \ ^
(TRANSITION AT SKI ^V»#«
06 I I I I I

V\l

Ma a C. Cm Cs Ra. 10* "• <* c l C

m C
9 "••">''
0 4 • 7» 1.37 56 -.087 .0092 176 • 757 179 .61 -.102 .0127 12.5
* 7M 190 .57 -.075 .0117 3.6 * 768 m .61 - 0 9 0 .0173 3.7

0-2 • .to. 11 MOOEL"A-

1
MEASURED IHSTI

i
r 04 06 " >«SN 1 0x/c

f
0-2 ,

0-4

Station 576 M-085B

0 6

Fig. 3.3-2 Example of Indirect Reynolds number

Fig. 3.3-ld Comparisons of pressure distribu- effects on a supercritical airfoil
tions for wind tunnel and flight
- VC-10 -
MODEL"A- M a - 726
MEASURED IHSTI - H a - 12.5 « 1
- • 3.6 x 1

cL
0392
•

-14 0438
0 479
Flight . _ _ • — _ 0-431
-12

-10

-0 8 F d 9^ f*
M0

\v
1
—u—t- -I- O U / I ,
1.0 "^ .4
1_

Fig. 3.3-3 Development of shock strength and

-0 6 position with lift for two Reynolds
1
numbers (unpublished results of
NLR/HST, model "A")
•0.
\ R. Kl( •8
//
CAST 10-2
4 tranulion fi wt\ MEASURED ICFFI
*y&
_
a-K^fe;.
N
\ E| fixed &
nttiur.il
-0 2
fJmf N
5^8. ^ ^ \ >
X N tmnsmon

r-J
tyr0 2 04 06 .%^ 10 x/e

0-2

o
04
. Station 576 M-o 883
©
0 6

Fig. 3.3-le Comparisons of pressure distribu-

tions for wind tunnel and flight Fig. 3.3-4 Development of shock strength and
w.
- VC-10 - position with Mach number (CAST 10-2
airfoil)
 27

© M-O.
N -2.0
Q a 4< ©
J. - 7 x 1 0 s

3 . lo" 6 8
- 22 x 10
32 « I0 6
a. 3"

r\
:\
~/\ \
y T \\
LA^T*^ \ " *. .*<s V
h
.024 -
\.f&*' s^K
~7 ^sN V
A " -^ V*- .020 -

/ 4—^^ ^
' ^ .016 -
0 .2 .4 6 .8 1.0 0 .2 4 6 8 1.0
.012 —

RN-IO6
a-o° 3 « 10 6
OOH
32 x 10 8

©
®
0
R N
7x „10°
6
a 3U 22 x 106

\ .012

**' a
cd
.n n
w>
. /
Fig. 3.3-5 Reynolds number effects for a conven-
tional (NACA 65,- 213) airfoil (from 0
ref. 76/4) .48 .62 .66 .60 .64 .68 .72
M

M-0,60 Fig. 3.3-7 Reynolds number effect for a thick

HN o-2°
© supercritical airfoil (from ref.
4x 106 76/4)
24x 108

.'
DESIGN LIFT

^ 4x ID 6
s V-
/ A -.08 32x10s
\ .012
\
\V
V^ -m
-.04

& C C
- - * w - % . .
0 ac
0 .2 .4 .6 .8 1.0 d
X/C R N -10« .004
f
//
M - 0.76 •Jv
a.jO ®
60 64 .68 .72 .76 .80 .84
N
" W .

4x10s
23x108
^^_,_^ Fig. 3.3-8 Reynolds number effect on compressi-
bility drag for a supercritical
1 airfoil (from ref. 76/4)

Il l ' ,.' y LS
\
\ is

0 .2 .4
"N-'O6
x/c

Fig. 3.3-6 Reynolds number effect for a super-

critical airfoil (from ref. 76/4)
28

OCFF
D NASA
ONAC

"8- -8-8"
10 50

.13 =-o—B *>—i

dcn
o
da
.11
10 50

-.06
C
-o
-.04
D ^ ft o
o — •
8 o

-.02
10 50

NOTE: Results for R = 24.8 x 10*

-** and 28.3 x 10* are invalid because of
inadequate side-wall suction. -.02 O
d
'm
0 BEFORE DRAGRISE d
S
0 10 50
R N -1fj6
U L M OP £ RxICr ' c 0 ®
V > 0 * « 1 632 0 532 206 00166
f a 0761 18TOJC see 36« 00134

.2
tt = 0° °
,• rmtnf w
:
Pt.e.
TVJ
Cp
•Wf. 10
.4
1

»
T 1
f
i
a - 3° " '
Pt.e.
.2

r
4
.6
T
"-0° o
; S x/c
SHOCK
0J 0t 08 08 I/C .4
<<
T
A

.6
a =3°
r » x/c
—?c^s «^— a
SHOCK
> ,'
10 50
i R N -10°
®
Fig. 3.3-10 Reynolds number e f f e c t of some a e r o -
® AFTER DRAG RISE dynamic c h a r a c t e r i s t i c s f o r a conven-
t i o n a l (NACA 6 5 , - 213) a i r f o i l (from
Fig. 3.3-9 Example of Reynolds number effect ref. 75/3)
on second expansion behind the
shock (from ref. 76/2)
 29

O R«c = 1 2 5 • 1 0 s TRANSITION 1
Q R«c • 7.5 • 106 FIXED AT 5% 1 MODEL"A"
£ Rtc • 3.6 • 10«
MEASURED (HST)

.5
Cl

•4r-

3L

-.06
(:
,n Fig. 3.3-13 Lift-curves of a supercritical air-
-.08 foil for various Reynolds numbers
(unpublished results of NLR/HST
Model
-.10

M-S0 M- IK,

QRa. 75. ID* O R a . 3 0 . 1061

DHa- 1? , IC* Natural transition Q R. 17 • 1081 Natural i r a n j i t . o n

.60 .60 .70 M.i A ORa.

ftRa.
10 . I *
4 . 10/ transition liaad at a/c - 07
O
j \
R a . 1 0 - 10*1
H 9 . 4 • I D * l i a n i j t i o n f i . a d at X / C • 07

Or- <—
"-pQS
.2 L

Fig. 3.3-11 Development of lift, pitching moment

and trailing edge pressure with
Hach number for a supercritical
airfoil (unpublished results of
NLR/HST Model "A")

CAST 10-2
MEASURED (CFF|

Fip. 3.3-14 Lift-curves of a supercritical air-

foil for various Reynolds numbers
OR«c -30> 1C)6

R « c . 1 7 B 106
natural t r a n s i t i o n i a = 1° - CONSTANT
(CAST 10-2 Airfoil)

J
D

0"v •
10.
4 .
IC*
MH transition i x a d at x / c •
(j) SCHOENHERR V KARMAN A L L Ra C p 1 • 4 13log[fla.CFl
2 5 8
CJ PRANDTL - SCHLICHTING A L L Ha C F • .455 lloo Hal -

3) SCHULTZ - GRUNOW A L L Ra C F = .427 l l o g R a - . 4 0 7 1 "

(J> I / T ^ P O W E R LAW tVI0B<Ra<107 CF

® SPALDING Si CHI A L L He
IX, NIKURADSE 1.7 • I 0 6 < H a < 1.8 • 1 0 7

0B

F u l l TuaauUNT FLOW

-.10

(ta, - 10-4

Pig. 3.3-12 Development of lift, pitching moment

and trailing edge pressure with
Macb number for a supercritical Pig. 3.3-15 Summary of flat-plate skin friction
airfoil (Airfoil CAST 10-2) variation with Reynolds number
30

• • •» a> MEASURED. FIXED TRANSITION MODEL"A"

MEASURED IHSTI
o v a MEASURED, NATURAL TRANSITION
R«.10"6 A C
d" , C
d
oio FORM FACTOR ESTIMATE • 12.5.
• 75 TRANSITION FIXED
002 9 36 '
FORM FACTOR ESTIMATE Ra • 3 S . 10*
FORM FACTOR ESTIMATE Ra - 12 5 . 10*
MODEL"A"

Mach = .6

«a
004

NACA 65,-213,8 - J ac„ 0

014
a> NASA Langlay
» CFF 002
e

i
J cr 65 60 70 M.

Fig. 3.3-18 Drag creep at constant lift for

MODEL " B "
different Reynolds numbers (unpu-
blished results of NLR/HST; Model
U ^ a. "A")
1 ' 2 4
Ra, • 10"

Fig. 3.3-16 Development of subsonic minimum

airfoil drag with Reynolds number CAST 10-2
(various sources) MEASURED (CFF|
Re • 1 0 - 6
.006-1- • 30
17 NATURAL TRANSITION
A 18 ACd-(CdMi- Cd
004 M . = .6,C{ = C 0 N S T A N T

• 4 FIXED TRANSITION

FORM FACTOR ESTIMATE

o NACA 65, 713. a b X/C,, - .06 Ra - 3 • 10* Re= 1 0 . 10 6

o MODEL "A" X/C,,-.06 Ra-3.8.10*

C{=.3
6
D CAST 10 1 X/C„ - .07 Hi M l )

tv M O O E L " B "

9 MOOEL "bV
X/Ctf-.07

X/C,, - 0
ft. i 9 • to6

Ra-I.8'10*
°Mr 1ST

004 .002
c
"axp °FF

.70 80

0, V NACA 66. 213. a - . 6 X/C,, - .06 Ra«12-10*

o MODEL " A " X/C,,-.06 Ra-US'lO*

.6 •

o CAST 10 2 NATURAL TRANS, R. 10 • I 0 6

A MODEL " B " NATURAL TRANS. Ra - 10 • 10 8

y .70 .80

Fig. 3.3-17 Subsonic lift-dependent drag for Fig. 3.3-19 Drag creep at constant lift for
low and high Reynolds numbers (vari- different Reynolds numbers (CAST
ous sources) 10-2 Airfoil)
 31

©
SUPERCRITICAL AIRFOIL
Ma • .749
_ A A \
«d

.0300 -
. a^"\K

CONSTANT a
CONSTANT
LIFT
M -0.745,a- 0.6°, FIX 7/7

He c -^.S.Itr'.M,.-0.744, C t -0.48 ,a • 0 58
. = »" Ra c - 7.10 6 . M c -0.745, C t -0.42 a •0.81
-+— 6
Rac - 3.6. tO . Mc - 0.748. C t • 0.30 a •0.69

_1_ 1 1 1
1B» 10* 1B« 10*

Fig. 3.3-20 Supercritical drag development with

Reynolds number: evaluation at con-
stant lift versus constant incidence

R e c - 12.5.106,M(. - 0.740. C j -0.832,0, -1.28

Rec- 7.10 6 ,M C -0.742, C t -0.601,0 -1.29
Rec- 3.6.106. M c -0.744. C t -0,466.0 -1.28

Fig. 3.3-22 Variation of a supercritical pres-

R a c - 12.5.106. M c -0.745, C t -0.46 U -0.54
sure distribution with Reynolds
R, c - 7.0.10*. M c - 0 . 7 4 S . C t -0.42 , a - 0.61 number at constant Incidence
Ra - 3.6.106, M c -0.744, C , -0.48 a - 1.28

0.2 0.8 1.0

X/C

F^ ©
Ao
A o
Ao
.6

st C
t
2-D
4 CAST 10-2 (CFF|
2-D - R f 4 . lOgfixadu/J I
M O D E L "A" (HST) oRt-io.io6 „ , •A'
ORe 12.5. 10f| ' R.-30 . 1 Q 6 [ n , t - t " y
aRe • 3 6 . 10°|
WAKE RAKE WAKE RAKE
= .1
C, CONST. Ma \ XtCONST. Ms
^ 70 .80 ^VT
Ra -12.6.10 6 , M - 0.741, Ca -0.81 ,0 -1.15
Rec • 7.0.10°, M -0.742. C. -0.80 , 0 -1.29
„ *X
Re - 3.6.10°, M - 0.741, C , -0.59 ,U -2.35

I I I
3D *, 3D
MODEL " C " (HST)„ MODEL " D " (HST)
A R«c-2.2 . ID* AR« C .2.5. 10«
Fig. 3.3-21 Variation of a supercritical pres- o H e r . 8 . 106 o Re c -7.5a 106
sure distribution with Reynolds
number at constant lift FORCE MEASUREMENTS FORCE MEASUREMENTS
OA

OA
(4SP-)= 1
L
W2—
A 'C. CONST. Ma A ^CONST. Ma
^Antr^ .80 .80

Fig. 3.3-23 Examples of Reynolds number influ-

ence on drag-divergence boundary
32

3.4 Flows with turbulent boundary layer separation

3.4.1 The classical distinction between type "A" and "B" separation
Most of the earlier work on Reynolds number effects was related to (the onset of) separated flow
which directly Influenced (fighter) aircraft performance through the Reynolds number dependence of pheno-
mena like buffet and wing-rock (see section 1.1). The problem of adequately defining separated flow phe-
nomena is of even more significance to-day In view of the demand of high manoeuvrability at conditions
with partly separated flow. The situation Is somewhat different for transport-type aircraft but neverthe-
less of great significance. The normal operating conditions of a transport aircraft are within the sepa-
ration boundaries as defined in the llft/Mach number plane. They should be known for that reason. It then
turns out that the buffet boundary la one of the critical parameters determining the wing area. For that
reason It is important to predict the buffet boundary from wind tunnel tests.The flow regime beyond sepa-
ration is also of Interest for transport-type aircraft from the point of view of aircraft control and
structural Integrity. Drag Is then of minor importance.
Figure 3.4-1 shows a comparison (in pairs) of pressure distributions for various cases. In each case
the Reynolds number effect on the pressure distribution on the left side is small and the variations are
similar to the ones discussed In section 3.3. On the right side of the figure the variations are much
larger and reflect the theme of this and the following sections. In the latter cases the trailing edge
pressures suggest that the flow has separated at the low Reynolds number which causes a significant shock
movement. At the high Reynolds number the trailing edge flow is attached (possibly with the exception of
a small local separation). Since the boundary between attached and separated flow can be reasonably well
defined (e.g. by using the trailing edge pressure divergence as an Indicator; see fig. 3.3-11 and -12),
the systematic analysis will be mainly concerned with the variation of the separation boundary with Rey-
nolds number. Finally some remarks on Reynolds number effects for separated flows will be made. Note that
this approach is distinct from one in which the wing or airfoil is kept at constant Incidence in order to
follow the development of the flow from separated to attached flow conditions with increasing Reynolds
number.
Just after the second world war It was realized that shock wave boundary layer interaction was one
of the prime causes for separation at transonic flight. Many valuable studies of shock wave boundary
layer interaction have been published since (see [Ref. 85/12]) but it was not untlll 1968 that Pearcey,
Osborne and Haines presented a kind of classification of the shock Induced boundary layer separation on
airfoils [Ref. 68/2]. Some illustrative figures, taken from their report, are reproduced In the figures
3.4-2 to 3.4-5. Since that time the type "A" and type "B" separation are often associated with Reynolds
number effects on airfoils with shock waves present.
The figures 3.4-2 and 3.4-3 help to explain the basic differences between the two types of separation. In
both types the final state is a boundary layer separation from shock to trailing edge. However, in the
type "A" separation this final state is achieved by a growth of the separation bubble underneath the
shock. In the type "B" separation a local trailing edge separation appears before rhe final state is
reached. The trailing edge separation is amplified by the adverse effects on the boundary layer develop-
ment due to the upstream shock wave boundary layer Interaction. The final state is reached as soon as the
two separated areas (at the shock and the trailing edge) merge. Therefore the two types differ with res-
pect to the development up to the condition of a complete separation as can be observed from an analysis
of the pressure distribution (see fig. 3.4-4 and 3.4-5; the latter is a so called "Pearccy-plot").
The Important point for Reynolds number effects Is that, as was argued in reference [68/2], the type
"B" separation was considered to be much more Reynolds number sensitive than the type "A" separation.
This clear distinction in Reynolds number sensitivity should be understood from previous work at NPL in
England (e.g. Ref. [54/1] and [55/1]) showing that the local shock Mach number causing separation (indi-
cated by divergence of trailing edge pressure) was a weak funtlon of the free stream Mach number. In other
words, a single shock Mach number defines the beginning of shock induced separation. (It was noted that
shocks close to the leading edge behave differently). The effect of a Reynolds number variation (with
fixed boundary layer transition!) was not studied In detail at that time. In the 1968 paper by Pearcey et
al this shock-Induced separation criterion is not mentioned anymore, but it Is argued that the local flow
at the foot of the shock (notably the development of a supersonic tongue, see ref. [60/2]) is the domi-
nant factor In the subsequent development of the separation bubble. From these observations the circa
1955 conclusions were that the shock strength was the most important factor and the incoming boundary
layer was less Important.
In the type "B" flow the Interaction with the trailing edge separation is essential and since the
trailing edge separation is a pressure Induced boundary layer separation, Reynolds number is bound to
have an effect as well. Fully In line with this description it was also argued that for a sufficiently
high Reynolds number the rear separation might disappear. In other words, the type "B" separation deve-
lopes Into a type "A" separation at higher Reynolds numbers. This opens the possibility that a type "B"
separation as observed in the wind tunnel will not appear on the full scale aircraft. This situation is
not very comfortable for the aircraft designer.
The distinction betwen a type "A" and "B" was based on a very detailed analysis of the available
Information at that time. The question should be posed in this AGARDograph if the (still) limited infor-
mation that has become available since gives rise to a modification of this basic distinction.

3.4.2 The flow break-down boundary revised

The experimental information on flow break-down or separation boundaries Is very limited and often
confidential. This is even more so for the description of physical flow models that calculate or corre-
late the separation development. Nevertheless, it will be attempted to discuss the major elements of the
separation process on airfoils.
Separation occurs when the flow breaks away from the surface leading to regions of reversed flow.
When these regions are confined to the inner part of the turbulent boundary layer, a situation that was
discussed in some detail in section 3.3.2 In relation to trailing edge separation, the effects on the
outer flow field will be gradual and of limited extent. However, when the reversed flow Is no longer con-
fined to the thin shear layer, the outer flow field will be modified drastically, leading to almost dis-
continuous changes in the pressure distribution, most often resulting in a sudden loss of lift and change
in pitching moment. This situation will be called flow break-down. The flow break-down boundaries mark in
the C.-Mach number plane the regions of normal operating conditions of an airplane. They are closely con-
 33

nected with, but not necessarily identical to the buffet and maximum lift boundaries. Various definitions
are in use like a kink in the C -a curve, a break in pitching moment or tangential force development, or
a rapid divergence of trailing edge pressure.
In the transonic regime the flow break-down boundary of particular Interest Involves shock induced sepa-
ration from shock to trailing edge. In this kind of flow the distinction has been made In the past be-
tween type "A" and "B" separation as discussed before. Separation at the foot of the shock and/or trai-
ling edge separation are the main elements In this process of flow break-down.

The length of the separation bubble L underneath the shock can be written as:

L
sep,shock/c " f
'"shock' " - R
V H
sh' (S/C)
sh' P r e S S " dlstr
""> <I«)
whereas the trailing edge separation can be described as:

L
sept.TE/c = 8 { R
V H
TE' (€,/c)
TE' preSS
- dlstr
---- } (lb)
Flow break-down Is defined as the condition at which the separated boundary layer underneath the shock
falls to re-attach to the airfoil surface (type "A"):

L l
sep.shock/c £
" X sh / C <TIa>
or when the shock bubble and trailing edge separation merge (type "B"):

(L + L
sep,shock sep. T E ) / c * ' " X sh / r <IIb>
The expressions I and II constitute a general flow break-down criterion although much more simplified ex-
pression are actually used. Fundamental studies (e.g. see [Ref. 85/12]) indicate that the upstream shock
Mach number for the start of separation (incipient separation) is a very weak function of the upstream
boundary layer shape factor H and for that reason almost Reynolds number Independent. In the I950's a
single correlation, based on shock strength (and hence Reynolds number Independent) was used to define
flow break-down (fig, 3.4-6 taken from Ref. [55/1]). Also more recently simple correlations can be found
in the literature to define flow break-down that make use of a critical shock Mach number without address-
ing the Reynolds number dependence explicitly (fig. 3.4-7 taken from Ref. [82/3] and also [Ref. 81/3]).
In the mean time other fundamental studies (e.g. Ref. [67/1], [78/3] and [81/1]) do Indicate a Reynolds
number dependence as also discussed In the report of the Research Committee of AGARD WG-09 (Ref. 00/1].
Unpublished (and independent) studies from NLR and ARA suggest a correlation for the separation boundary
with M and 0 ,/c as the dominant .parameters. This correlation does reflect a Reynolds number depen-
dence since (9/c) varies roughly as Re . More recently Fulker and Ashlll [85/4] have published very
detailed studies on the separation length in connection with flow break-down. In their correlation the
separation length L , ./S , 1 s expressed as a function of Re 0 , , (though weakly) and the shock
Mach number (fig. 3.5--8;. when the separation length exceeds a critical value, dependent on the pressure
distribution, the boundary layer will not re-attach and the separation boundary is reached. This most
recent correlation for flow break-down by Fulker and Ashill can be represented schematically:

L . , /c + x /c i (x/c) ,_. . (He)

sep,shock shock critical
where (x/c) . . , depends on the pressure distribution and hence the airfoil type,
critical
Many criteria can be found in the literature to described the more classical type of pressure Induced
trailing edge separation [Ref. OO/ll. Just one of these expressions Is reproduced here (from Ref. [72/5])
to Illustrate a fundamental Reynolds number dependency through the Influence on S;

(Q/c)TI, . d c /d x/c i . 007 (III)

IE, p
The situation is, unfortunately, more complex than this simple correlation seems to Indicate, The separa-
tion onset and the separation length can, from a fundamental point of view, only be calculated from a
simultaneous solution of the boundary layer and the outer flow equations since the development of the
(thick) boundary layer close to separation provides a relief In the external flow pressure gradient that
counteracts the separation development. Moreover, the boundary layer conditions at the trailing edge will
depend on the upstream history with shock wave boundary layer Interaction as a complicating factor.

How do the various types of separation interact with each other? It Is to be expected that prior to
Incipient separation at the foot of the shock (for shock Mach numbers less than roughly 1.3) trailing
edge Reparation is the dominant phenomenon. Such situations might occur at low lift values for Mach
numbers close to and beyond the design Mach number. The Reynolds number dependence enters Into a direct
way. In a way similar to expression III.
When the shock Mach number is higher than 1.3 a local bubble at the foot of the shock will be for-
med. The extent of this separation will be Reynolds number dependent In a direct way as suggested with
expression I.a. The most recent work of Fulker and Ashlll has shown that the over-ruling factor for the
condition of flow break-down appears to be the growth of the separation bubble underneath the shock, Ir-
respective of a possible trailing-edge separation (expression II-c). This, however, does not mean that
trailing edge separation Is not important at all. Trailing edge separation will modify the pressure dis-
tribution in a Reynolds number dependent way. This Indirect Reynolds number effect, as discussed extensi-
vely In section 3.3, will alter the shock strength and hence the conditions for the separation at the
foot of the shock. This in turn will Influence the conditions of the boundary layer at the trailing edge.
This interaction can be very significant in view of the sensitivity of the length of the separation bubble
underneath the shock for the shock Mach number (fig. 3.4-8). The final result of this Interaction process
from the point of view of Reynolds number sensitivity will depend very much on the pressure distribution
and hence the type of airfoil. Fulker and Ashlll noted already the importance of the type of pressure
distribution for their evaluation. Most airfoils from before the 1960's show a rapid increase in shock
34

strength with increasing Mach number or angle of attack. Viscous effects were small, unless the Reynolds
number was very low. For modern supercritical airfoils the variation In shock wave strength has been re-
stricted through design. Rear loading has increased the pressure gradients over the rear of the airfoil.
Viscous effects and hence the Indirect Reynolds number effects, will be much more Important for these
airfoils. It Is therefore to be expected that the flow break-down boundary will also be more sensitive to
Reynolds number, as compared with the airfoils of the 1950's.
Since the trailing edge separation appears to be only of importance for the (gradual) indirect Rey-
nolds number effects, the original distinction between type "A" and "B" flow might be less relevant for
the flow break-down boundary. This is illustrated very schematically in figure 3.4-9. The distinction
will still be very relevant prior to flow break-down for the evaluation of drag and pitching moment as
discussed In section 3.3. For the flow break-down boundary the Reynolds number sensitivity might very
well depend primarily on the airfoil type rather then the particular Reynolds number range.
This view appears to be supported by the experimental Information as far as available to the author.
The figures 3.4-10 till 3.4-14 show some examples of the variation of the flow break-down boundary with
Reynolds number. All results are related to modern supercritical airfoils (some of them discussed above).
In figure 3.4-10 Reynolds number trends for various models of the same airfoil measured in various wind
tunnels are compared [Ref. 82/4, 83/10]. Note that although the absolute values are very much different,
the variations with Reynolds number are very similar. In figure 3.4-11 the Reynolds number trend for one
airfoil Is presented for a range of Mach numbers. The figure depicts the importance of transition fixa-
tion in the analysis of the separation boundary as discussed in section 3.2. Also, some increase in Rey-
nolds number dependence can be noted for higher Mach numbers. Figure 3.4-12 compares various airfoils for
a typical transonic design Mach number. Apart from some variation in Reynolds number dependence, all mo-
dels, Including the three dimensional model "C" (a high aspect ratio transport-type wing) indicate a re-
gular variation of C -max with Reynolds number.
None of the results show a tendency to level off at higher Reynolds numbers (with a possible excep-
tion of the Mach = .6 data in fig. 3.4-11; for that condition the shock is close to the leading edge).
This is also supported for some of these airfoils by figure 3.4-13 in which the shock Mach number at flow
break-down and for a constant shock position has been plotted versus Reynolds number. Again, a very regu-
lar trend is found. In a last example (figure 3.4-14 taken from Ref. [85/2]) the pressure distributions
of a three-dimensional high aspect ratio wing (the one discussed in Ref. [76/1]) were analyzed In order
to classify the separation as a function of Mach and Reynolds number. Also In that case no change from
type "B" to type "A" separation was actually observed although such a change was tentatively Indicated in
the figure.
From the design point of view it might be reassuring to know that no discontinuous changes in the
development of the flow break-down boundary with Reynolds number are to be expected. However, clearly
more research is required to substantiate this view. Also there will be limits Imposed by the "Inviscid
limit" of the outer flow field and surface roughness effects that render the boundary layer development
Reynolds number Independent.

3.4.3 Post stall behaviour

Only very limited information is available with respect to Reynolds number effects beyond separa-
tion. The problem is nevertheless of considerable importance for stability and control and for the deter-
mination of aerodynamic loads. One extreme appears to be very well defined: a flow that Is separated along
a salient edge is Independent of Reynolds number. But for pressure gradient or shock induced separations
on the smooth surface of an airfoil contour, the Reynolds number will have a strong Influence on the sepa-
ration position and hence the dimensions of the separated flow field. For a good understanding of the
problem It is essential to allow for the Interaction between the viscous flow and the non-viscid outer
flow. Moreover, the complete flow field is often highly unsteady and essentially three-dimensional, even
for two-dimensional airfoils. A sound theoretical treatment seems therefore beyond the present capabili-
ties.
Nevertheless, for conditions not too far away from the flow break-down boundary, there appears to be
a more or less systematic development of the separation. Cahlll [Ref. 79/1] has presented a method to
extrapolate pressure distributions with flow separation as measured In the wind tunnel to flight Reynolds
numbers. The procedure is based on the observation that the relations between trailing edge pressure on
the one hand and a viscous flow parameter ("BV) and shock position on the other hand, preserve their
shape, Independent of Reynolds number. Moreover, the trends with Reynolds number appear to be rather gene-
ral. The principle of the method Is shown schematically in figure 3.4-15 and a typical example of a so
corrected pressure distribution is presented in figure 3.4-16. In a later publication [Ref. 83/1] the
purely empirical viscous shock parameter "Bh" has been replaced by a physically more meaningful parameter
defined by:

(M
K =h 1 .
" / .i. " u with e = /C,/2
(v+l).e.M sh f

that follows from asymptotic theory for shock wave boundary layer Interaction. With this parameter the
correlation improved slightly. The method is reported to be primarily used for the prediction of aerody-
namic loads.
For the same class of flows as described by Cahlll's correlation It might very well be possible
(under certain restrictions of tunnel Reynolds number, shock position and the three-dimensionality of the
flow) to simulate in the wind tunnel the high Reynolds number flow with the help of the so called aft-
flxatlon technique. This technique will be discussed very shortly In section 3.6.
 35

R E - E F F E C T : ONE A I R C R A F T

-2
® ®

os ,/ c io oo as
R E - E F F E C T : ONE W/T MODEL

(bl T Y P E B I 1 I A T E A R L Y STAGE W I T H LOCALISED SHOCK

BUBBLE A N D REAR SEPARATION
•7 01 (• •• ( 2 I A T LATER STAGE W I T H ONE SEPARATION
I II II
E X T E N D I N G FROM SHOCK TO T R A I L I N G EDGE
RE-EFFECT : TWO TYPES (?)
Fig. 3.4-1 Observed Reynolds number effects: Fig. 3.4-2 Some details (schematic) of the re-
before and after the separation boun- gion of interaction between turbu-
dary (from various sources) lent boundary layer and shock wave
(from ref. 68/2)

ORIGINAL MODEL. A VARIANTS OF ALTERNATIVE MODEL. 8

(ONLY REARWARD GROWTH OF BUBBLEI (APPLICABLE WHEN REAR SEPARATION IS INCIPIENT OR PRESENTI

REAR SEPARATION REAR SEPARATION REAR SEPARATION

PROVOKED BY PROVOKED BY ALREADY PRESENT
BUBBLE SHOCK ITSELF
(al INTERACTION Ibl INTERACTION lal INTERACTION Ibl INTERACTION
WITH BUBBLE WITH SHOCK WITH BUBBLE WITH SHOCK

SHOCK
BUBBLE AT FOOT OF SHOCK
REAR SEPARATION IN DOWNSTREAM PRESSURE GRADIENT
SEPARATION FROM SHOCK TO TRAILING EDGE
FIRST DIVERGENCE IN TRAILLING-EDGE PRESSURE (FIRST EFFECT ON CIRCULATION!
RAPID DIVERGENCE IN TRAILING-EDGE PRESSURE (MAJOR EFFECTS ON CIRCULATION!

Fig. 3.4-3 Classification of separation development according to type "A" and "B"
separation (from ref. 68/2)
36

F o r each c o n f i g u r a t i o n tested
R - 1.8 x 10° an u n f i l l e d s y m b o l has been
TRANSITION FIXED plotted to indicate the
1.40 local M a c h n u m b e r , M 1 ,
observed (just u p s t r e a m o f
t h e s h o c k ) w i t h n o sepa-
1.38 ration and a filled symbol
t o i n d i c a t e t h e l o w e s t local
M a c h n u m b e r observed
1.36 w i t h separation
LOCAL
* > I- TESTS ON 1 0 % RAE 103
MACH
No, M , A PLAIN
V FLAP DEFLECTED
1.32 > WITH SPOILERS
a} WITH SPOILERS AND
WITH FLAP DEFLECTED
1.30
D 1 0 % RAE 104
O 6 % RAE 104
1.28 V A L U E O F M , - -w--
FOR SEPARATION ~^7Z
1.26
> • 4

1.24
ir* ^
1.22 ••^1*5

1.20

1.18 -«-^i

1.16

1.14 tf>
n
1.12

1.10
06 07 0.8 0.9 1.0
FREE STREAM MACH NO, M„
Fig. 3.4-4 Development of pressure distribution Fig. 3.4-6 Early correlation of the shock-up-
for type "A" and "B" separations stream Uach number to cause separa-
(from ref. 68/2) tion (from ref. 55/1)

-0.5 -0.5 -
FIRST INFLUENCE OF
TYPE"A" TYPE " B "
SEPARATION A T
,Vl-M? FOOT OF SHOCK

-0.4 -0,4 -
FIRST INFLUENCE OF
SHOCK-US
SEPARATION

-0.3

-0.2

TREND OF REAR SEPARATION

-j I I
04 05 0,6 0.7 0.8 0,9

Fig. 3.4-5 So-called "Pearcey-plots" to distinguish between type "A" and "fl" sepa-
ration (from ref. 68/2)
 37

1.5 1.0 Moo" .6

i:

8
0.8 h -A
o
o
< 1.3
m
Moo- 676
12
ctwtAi- ^TRANSITION REGION
12 - C
l
5
NOTE S E P A R A T I O N ONSET
2 1.0 J * * V& TURBULENT_ct_m»x
I D E F I N E D BY T R A I L I N G EDGE
1.1 -
PRESSURE DIVERGENCE
<
0.8 - BUFFET ONSET
< 1.0 _1_ _1_ _L_ _L _l_
8 o.io 0.20 0.30 0.40 0.50 0.60 0.70 0.80
SHOCK POSITION, X/c

Fig. 3.4-7 A more recent correlation ol the 1.0

shock-upstream Mach number to cause
separation (from ref. 82/3)
0.8 -
Moo-.75

0.6 -
700
'BUBBLE'^SHOCK
MAXIMUM LIFT
0,4
—41 LAMINAR INTERACTION
TURBULENT INTERACTION

0.2 B U F F E T ONSET
• - - - • LAMINAR INTERACTION
& £, TURBULENT INTERACTION
_l I I I
G 10 20 30 Re x 10 6

Fig. 3.4-11 Observed variation of maximum lift

for one model at various Hach num-
bers (unpublished results of NLR)
Fig. 3.4-8 Fulker and Ashlll's shock-bubble
correlation

NO TE SEPARATION . TC SEPARATION
8
BUFFET
BOUNDARY
BUFFET BOUNDARY
-max
f
,6 M O D E L "A" (HST)

M
Ra SHOCK

.8 -

SEPARATION . SEPARATION
CRITERIUM CRITERIUM eL CAST 10-2 (CFF)
SIMPLEFIED " CLASSICAL" VIEW PRESENT •' T E N T A T I V E " VIEW

Fig. 3.4-9 Hypothetical models to describe the

Reynolds number sensitivity of the .6 -
separation boundary

| M , .0.760 |
0.85 M O D E L "B" (PT/CFF)
S " 1 - Slope used to correct
o S3 Ma ''Lmax s to Re = 25 x I t *
a TWB"
S s
D ARA
V TKG 0.80 A
OT2"
° NLR X
a *
•» T U - B 075 A.
1|Flag:c = 200 m m
2)Flag: c = 120mm ¥>,y M O D E L "C" (HST) STN 3o STN 4o STN 5a
0.70 •sr cS
Note: For T2 the free-stream
Half filled symbols Ma = .75
Mach number Is Mj>07S4
denote corrected or o <JD<J fixed transition
flexible wall data 0.65, <• natural transition
\ Re„ . 10 -6
I i
106 10 Re7
10 20 40

Fig. 3.4-10 Observed variation of maximum lift Fig. 3.4-12 Reynolds number effect on maximum
for one model in various wind tun- lift for various models near design
nels (from ref. 83/10) Mach number (unpublished results of
RLR and DFVLR)
38

REYNOLDS
. N U M B E R . ICV6

w w \ * CORRECTION FOR
SWEEP EFFECT CLASS A FLOW

M sh 16

a MODEL " B "

1.2 • O MODEL " E "
NOTATION : X EXPERIMENT
O STN 4
* SEPARATION
D STN 5
MODEL"C"
A STN 6
1.1- CORRECTED M 5 h = M $h cosA

(X/C) sh CLASS 82 FLOW

.40
1.0-
MACH NUMBER
Rex 10" 10

Fig. 3.4-13 Shock-upstream Mach number at maxi- Fig. 3.4-14 Tentative and actually observed
mum lift as a function of Reynolds types of flow separation (from ref.
number for a fixed shock location 85/2)
(unpublished results of NLR)

CPTE F R O M ®
C U R V E S S H I F T W I T H Re Mco-CONSTANT
ACCORDING T O © XCSH F R O M ®
CPTE FROM GIVES®
® 0 G I V E S XCSH
Bl/2' 11

CPTEO

O WIND TUNNEL MEASUREMENTS

® "UNIVERSAL "VALID • 0 EXTRAPOLATED

@ ® F R O M W / T TEST W I T H M , , , - C O N S T A N T

LEGEND
Po/Pp \7M2-I
W I T H Po - FREE S T R E A M S T A T I C PRESSURE
VI-XCSH Pp = PRESSURE IN F R O N T OF SHOCK
A • EMPIRICAL CONSTANT
CPTE - T R A I L I N G EDGE PRESSURE
M j g - FREE S T R E A M M A C H NUMBER
XCSH > SHOCK L O C A T I O N ( X / C I

Fig. 3.4-16 Examples ol extrapolated pressure distributions according

to the method of ref. 79/1

C-141 -1.2
" c- 141
n - .193 M - .85 n • .193 M -.85
E X T R A P O L A T E D SHOCK
OCCURANCE LOCUS
-1.0

C
p P A * I \

-.8- 8 0
A 1 1
o°o2
O 0< AA
A
I
6 ©G -.6 - O

A
A
cP° O
i
i
>
>
4
-•>
• ~ -4)- - ML
A bl

-2

° FLIGHT r v S ^\ O
RN-72x10° "y
O WIND TUNNEL RN • 6 x 1 0 6 • EXTRAPOLATED 1
O
• EXTRAPOLATED RN - 72X10 6 0
\ o

2 1 1 1 1

Fig. 3.4-15 Principle (schematic) of the extrapolation procedure

for pressure distributions with flow separation (from ref. 79/1)
 39

3.5 Three-dimensional effects

3.5.1 The correspondence between two- and three-dimensional flow

The discussion sofar was restricted to two-dimensional flows. But how relevant is the two-dimensio-
nal flow for the general case of a high aspect ratio wing? The design of such a wing is generally based
on one or more so called basic airfoil sections. For the larger part of the wing the pressure distribu-
tion can be derived from these two-dimensional basic sections by assuming that:

C
L.3-D " CL.2-D • c o s 2 A

M
-.3-D " M -.2-D / c o s A

where A is the sweep angle. Approximate corrections can be introduced to allow for wing-taper and the
fuselage Influence (see e.g. ref. [62/2]). The wing tip and the wing root are essentially three-dimens-
ional. The tip Is only a small part of the wing and most likely of minor Influence (unless winglets are
used). The wing root flow is of considerably more Importance. There, very often, a double shock pattern
can be observed with an Intersection point close to the kink-section of the wing (fig. 3.5-1). This in-
fluence restricts the two dimensional flow to a region between the kink-section and the tip. But even
then one has to be careful: it is the local direction of the Isobars (more precisely the shock Mach num-
ber perpendicular to the shockfront) that is of importance from a two-dimensional point of view. This
direction can bc different from the wing generators depending on the particular wing design.
The correspondence between two- and three-dimensional flow Is even less clear for the boundary layer
development. In the ideal case of an infinite swept wing some basic differences with respect to the cor-
responding unswept airfoil should be noted. Only for laminar boundary layers the so called "independence
principle" holds, stating that the velocity components In a direction perpendicular to the leading edge
are Identical for two- and (quasi) three-dimensional flows [Ref. 68/4]. Laminar boundary layer separation
will therefore occur at the same relative chord position for the swept and the unswept wing. This no
longer holds, in principle, for turbulent flows, due to a cross-coupling effect of the turbulent motion.
It is to be expected, however, that for flows with a strong pressure gradient (when turbulent shear
stresses are less Important) the independence principle is still of some value. For a discussion of the
Independence principle In relation with shock wave boundary layer interaction see Ref. [85/5], When the
Isobars on the wing are not parallel (due to taper or other three dimensional effects) the flow will en-
dure an additional effect of streamline con- or divergence, relative to the infinite swept wing case.
This causes an additional in- or decrease In boundary layer thickness (see Ref. [85/3] for a theoretical
treatment). The most Important differences between 2-D and 3-D flows, however, are related to transition
and boundary layer separation.
In two-dimensional flows the Tollmlen-SchllchtJng instability is the primary cause for transition.
In three-dimensional flows two more transition agents can be found: leading edge contamination and cross
flow Instability as discussed in more detail by Michel in ref. 00/1. Thus there are two more mechanisms
to produce discrepancies in transition location between wind tunnel and flight. Figure 3.5-2 summarizes
the various transition causing factors for a typical swept wing.
The process of boundary layer separation Is much more complex in three-dimensional flows as pointed
out by Hall in Ref. [71/5], Theoretically, two dimensional flows will exhibit two-dimensional separations
that appear as closed bubbles (on or behind the airfoil) of re-clrculatlng air. Such closed bubbles can
still be observed in the three-dimensional case of an infinite swept wing (near the nose or at the shock
foot), however with an additional spanwise component (see fig. 3.5-3a). In the more general case of a
three-dimensional wing with finite aspect ratio the closed bubble can also appear as a cell-like struc-
ture (fig. 3.5-5), as is also often found In two-dimensional testing. Much more Important and without
parallel In the two-dimensional case is the open separation with the formation of one or more vortices
(fig. 3.5-3b). Very often, these vortices start from a closed bubble and develop In the spanwise direc-
tion. Depending on the magnitude and growth of this spanwise flow, the Inboard wing might have a large
Influence on the flow over the outboard wing as remarked also by Yoshihara (see fig. 3.5-7 taken from
Ref. [75/4]). In the figures 3.5-4 to 3.5-6 some examples of various separation types as actually observ-
ed on high aspect ratio wings, are presented. From this it is quite clear that the open separation af-
fects the outer flow field in an essentially three dimensional way. For this kind of flow all correspon-
dence with two-dimensional flows is lost.

3.5.2 Some examples of three dimensional effects

One should not conclude from the foregoing discussion that the two-dimensional airfoil is without
relevance for high aspect ratio wings. The larger part of the Reynolds number effects can be carried over
to high aspect ratio wings, provided that:

. Only the quasi two-dimensional part of the wing is considered.

(say from kink-section to near the tip);
. The boundary layer is turbulent (either natural or by tripping);
. The flow has not separated.

The consequencles of the first restriction will depend on the particular wing design. When the as-
pect ratio is high enough (say 8 or more) the larger part of the wing is well outside the kink section.
Also, the flow on the wing root Is often less sensitive for Reynolds number changes due to a higher local
chord Reynolds number and a favourable effect of the double shock on the boundary layer development.
Spanwise variations in boundary layer transition location are of considerable influence. Fig. 3.5-8a
is an old example [Ref. 52/2] of very large spanwise variations in load distribution with Reynolds number.
It Is believed that in this case transition point variations, in combination with shock boundary layer
interactions are to blame. In this particular case, a complete reversal in pitching moment variation with
angle of attack was the dramatic result (fig. 3.5-8(b)). A more recent example (fig. 3.5-9 taken from Ref.
[76/1]) is Included to show that when the transition point variation is suppressed by artificial boundary
layer fixation, a qualitatively much more systematic flow development in accordance with the higher Rey-
nolds number situation, can be obtained. Figure 3.5-10 and 3.5-11 Illustrate a similar observation, also
for the flow break-down boundary. Again, the results with fixed transition appear to be much more syste-
40

matlc as compared with free transition, as discussed In section 3.2. It Is even more Important that the
fixed transition results are similar to the two-dimensional dpta In figure 3.4-12, -13 (the model "C" In
these figures is the same one as used for the results shown in figure 3.5-10 and -11). Also the correla-
tion between shock Mach number and Reynolds number at the flow break-down boundary is well In line with
the available two-dimensional data. Fulker and Ashlll [Ref. 85/4] did not find a systematic difference In
their separation correlation between two- and three-dimensional results. The examples indicate that even
at the onset of flow separation the correspondence between two- and three-dimensional flows can be main-
tained. However, in view of the essential three-dimensional nature of the separation Itself, it is very
unlikely that this correspendance will still be found beyond the flow break-down boundary. The uncertain-
ty with respect to Reynolds number effects for two-dimensional separated flows Is more severe for three-
dimensional configurations.
A word of caution should finally be expressed with respect to drag evaluation for a three-dimension-
al configuration. The concern stems from the sensitivity of compressibility drag to small variations in
shock wave strength and, for the three-dimensional case, the sweep angle of the shock. The shock wave
pattern on a 3-D wing is basically determined by the 3-D Inviscid flow development modified by local vis-
cous effects. In reference [76/2] the risk of over- or under-fixatlon of the boundary layer for a high
aspect ratio wing is discussed. It is argued that as a result of non-optimal fixation, the sweep-angle of
the shock might be influenced (see figure 3.5-12) with serious effects on the overall flow field. The
important point to note here is that similar effects may be introduced by a Reynolds number variation
with fixed transition near the leading edge. The argument goes as follows. When the (corresponding 2-D)
Reynolds number sensitivity of the Inner and outer wing is significantly different (e.g. because one part
of the wing Is closer to separation) the sweep angle of the shock may change in this case due to a Rey-
nolds number increase. When the sweep angle of the shock is increased (because the shock at the wing tip
moves faster downstream than the shock near the kink section) the effect on the overall flow development
will be favourable (because the Mach number component perpendicular to the shock decreases) and the Rey-
nolds number sensitivity of the wing will increase, compared with the mean of the 2-D stations. When the
klnk-sectlon is more critical, the reverse Is true and the Reynolds number sensitivity will be reduced.
Figure 3.5-13 (reported in Ref. [84/3]) further illustrates the Importance of three-dimensional effects
on the wave drag in relation to the total wing drag. In this figure an estimate of the wave drag contri-
bution Is depicted (note that the two shaded regions follow from different approximation to derive wave
drag from the wake rake traverses). At the lowest Reynolds number and highest lift the mid-wing region Is
Just beyond the local drag divergence boundary and experiences for that reason some increase in wave drag.
For the other presented conditions (higher Reynolds number and lower lift) the wave drag contribution is
much smaller with a modest variation over the span.

TRANSITION FOLLOWING
TOLLMIEN-SCHLICHTING
INSTABILITY

GRANVILLES CURVE

FORWARD SHOCK

| < ^ Q U A S I - 2-D FLOW=£)j RE-LAMINAR1SATION ?

K
' LEADING-EDGE CONTAMINATION

n n
"1 "2 3 4 "c

AT R. TRANSITION FOLLOWS FROM TOLLMIEN-SCHLICHTING

INSTABILITY

AT R j LEADING-EDGE CONTAMINATION, FOLLOWED

POSSIBLY BY RE-LAM1NARISATION AND THEN
TRANSITION THROUGH TOLLMIEN-SCHLICHTING
INSTABILITY

AT R3 LEADING-EDGE CONTAMINATION, FOLLOWED

POSSIBLY BY RE-LAMINARISATION ANO THEN
TRANSITION THROUGH CROSS-FLOW INSTABILITY

AT R 4 TRANSITION FOLLOWS FROM CROSS-FLOW

INSTABILITY IF LEADING-EDGE CONTAMINATION
WERE ABSENT

Fig. 3.5-1 Schematic representation of three-di- Fig. 3.5-2 Example of different types of transi-
mensional effects in the flow over a tion on a swept wing (froa ref. 72/2)
high aspect ratio wing (after
Slooff)
 41

ATTACHMENT LINE
ATTACHED FLOW
TRAILING-EDGE
\ SEPARATION

SURFACE STREAMLINE
SECONDARY SEPARATION
REATTACHMENT
SWEPT SHORT BUBBLE ATTACHMENT STREAMLINE
EXTERNAL ATTACHMEN"!
SEPARATION
STREAMLINE AND SEPARATION
STREAMLINE SURFACE
STREAMLINE FLOW PATTERN FOR VORTEX SEPARATION

Fig. 3.5-3 Example o f a c l o s e d and open t h r e e - d i m e n s i o n a l separation (from r e f . 71/5)

SWEPT SHORT BUBBLE

TRAILING-EDGE SEPARATION

H 1,02 x 10'
TRIP WIRE ON

POSSIBLE ATTACHMENT LINE

SECONDARY SEPARATION

TRACK OF VORTEX
SEPARATION

R = 0.51 x 10'
TRIP WIRE ON

TRACK OF VORTEX SEPARATION

-ATTACHED FLOW
SEPARATION

R = 1.02x10'
TRIP WIRE ON

Fig. 3.5-4 Flow and isobar patterns on a threedlmenslonal wing with separation (from ref. 71/5)
4:

,40 y/lb

Fig. 3.5-5 Typical local three-dimensional sepa- Fig, 3.5-6 Typical non-local three-dimensional
ration at the wing-root section separation on a wing with shocks
(from ref. 80/6) (from ref. 75/4)
 43

0
6
*k R
c x 10"

— *•— 2.8
---A--- 5.6
HIGH GLOVE ANO « 11.3
FUSELAGE PRESSURES

div
INBOARD
EROSION '

"" "^^1

SPANWISE | ^
CONTAMINATION

a) LEADING-EDGE TRIP

J^
V
'div
SHOCK AND SEPARATION PATTERN AT 45° SWEEP.
M . - 0 95 AND.O-9 0 . 4 "*•* a *

^— N
») • y/m/21 - 0.3 t)-0.;6

SHOCK-INDUCED SEPARATION SHOCK-INDUCED SEPARATION ~~"~T4

b) FREE TRANSITION
C -|45°l

n-0 5
"div
SHOCK-INDUCED SEPARATION c
SHOCK-INDUCED SEPARATION
3-*
C •|45°l

t 0.5 0.6 0.7

c) MIDTRIP
0.8 n 0.9

Fig. 3.5-7 Typical local three-dimensional sepa- Fig. 3.5-9 Example of spanwlse variation of
ration with spirallng vortices at trailing edge pressure divergence as
the wing root (from ref. 80/5) a function of Reynolds number with
and without boundary layer fixation
(from ref. 76/1)

.It

O.
! _ . — IB"
..-a
\ A^5-
.04 JL _C 1 1
t
M

o
m-Mo
i i i
J|

* - l.OOO.OOO
i
w.
1
\
1
i
>
I l /
V 11
i i i
«.j»0
*~4fiOO,000

I 1
I' / f a
I
da'AJ .
ai'^i
. —^,
^-.
5
^
r+l'S
-..-- i-
iI
•
f\ •
J
p
y —0— t.OOO.000
—o-- *,eoo,ooo
a
>
- -.^ 5\
y su 1 fii A 9
I L_ JIS I. c 1
X -j
O
1
tO .40 SO
...
.90 lOO o t o to so so toov? .4 // .09 .04
Fraction of swmltpan, f
A no/* of attack, m. 4og Pitching-momont coofficttnt, Cm

Fig. 3.5-8 Example of Reynolds number effect on the wing loading (from ref. 52/2)
 • NATURAL TRANSITION
A FIXED AT X/C -0.08

Fig. 3.5-10 Local lift development along wing span at two Reynolds numbers with
and without boundary layer fixation (from ref. 84/2)

a INCREASING
3DHIN0,HSfl

O FREE TRANSITION Ma - .76

A FIXED AT i \

5 = ^ = = = -a

WING SECTIONS

LINES INDICATE START OF SHOCK INDUCED SEPARATION

Fig. 3.5-11 Local maximum lift development with Fig. 3.5-12 Possible distortions of atall pat-
Reynolds number for a high aspect terns by spanwise variation of over-
ratio wing (from ref. 84/2) fixing (from ref. 76/2)

WAVE DRAG:
1 )<JWLOWtSTIMATt
.^JWHIGH ESTIMATE
1)16
® '

cdc/c 014
Tv%v
.012 012
tc- • ir—1 •^1
010 ^ ^ .010
•••/^
% "<<. ..
fefe 0< . ^
008

.006
®
C L «.40

R«e > U l l o
6
uq
s ^s.
*<&
008

l>06
©
C L « .47
Re£ = 2 . 2 x l 0 6
" • <

^%
*^
0O4
A R,
c * 8.1 x 10°
004 a\ Re e =:8.lK 108

1
.002 002

0 0
0 .2 .4 .6 .8 t) 1.0 0 .2 .4 .6 17 1.0
Fig. 3.5-13 Spanwise drag variation as a function of Reynolds number for a high
aspect ratio wing (from ref. 84/3)
 45

3.6 Concluding remarks: from wind tunnel to flight

Is the wind tunnel still an adequate tool for the design of aircraft in view of the Reynolds number
deficiency of most existing wind tunnels? And If the answer Is yes, what procedure should be followed in
wind tunnel testing to cope with this deficiency? These questions. Inevitable after the discussions of
the previous sections, will not be discussed here in great depth. AGARD assigned to Working Group 09
"Wind Tunnel Boundary Layer Simulation and Control" the task to propose a methodology to estimate flight
performance from wind tunnel tests. For more information the reader is referred to the final report of
this working group [Ref. 00/1U. In this section only a few remarks will be made to conclude the discus-
sion on Reynolds number effects for two-dimensional airfoils and high aspect ratio wings.

The available information suggests that two phenomena are of prime Importance for the understanding
of differences between wind tunnel and flight:

transition location variations also In connection with laminar versus turbulent shock wave boundary
layer Interaction

and

flow separation either locally (near the leading or trailing edge, at the shock) or of large scale
(between shock and trailing edge).

The first phenomenon appears to be the easiest one to deal with for transport-type configurations at tran-
sonic speeds if one assumes that transition in flight is near the leading edge. There is no doubt that
large transition location variations and laminar shock wave boundary layer Interactions should be avoided
In wind tunnel testing. Numerous examples (see section 3.2) show clearly misleading variations In aerody-
namic characteristics if the transition point can move around freely. This requirement means that the
boundary layer must be tripped artificially. This is a technique in Itself that will not be discussed
here (see e.g. Rcf. [84/2].
When transition has been fixed artificially, the next most Important problem Is related to flow sepa-
ration, either limited In extent or from shock to trailing edge as discussed In the sections 3.3 and 3.A.
Only in the latter case are large variations in pressure distribution observed with Reynolds number. For
conditions with attached or almost attached flow the effect on the pressure distribution (the so called
"indirect Reynolds number effect"; see section 1.2.2) appears to be small, though certainly not Insigni-
ficant. For example a limited region of trailing edge separation tends to Increase this indirect Reynolds
number effect. Some aerodynamic characteristics, notably drag and pitching moment, are very sensitive to
these small variations. Very large changes in pressure distribution are observed when the flow changes
from an attached to a separated flow condition when Reynolds number Is decreased. The separation boundary
(flow break-down, characterized by separation from shock to trailing edge) defines the boundary between
the attached flow and conditions with large scale separations. If the Indirect Reynolds number effects
are small for attached flow conditions and if the Reynolds number enters e separation criteria In an unam-
blglous way (as appears to be the case at least beyond a certain Reynolds number; see fig. 3.4-8), it
then follows that the separation boundary Itself varies in a systematic way with Reynolds number. This,
in principle, provides a basis for an extrapolation procedure of the separation boundary in the lift-Mach
number plane. A careful analysis of the results (in terms of pressure distributions, separation develop-
ment and wake drag analysis) is still needed to prove the validity of such a procedure for a particular
configuration. The preceedlng chapters, in fact, are intended to provide information ("rules" and "excep-
tions to the rules" as a warning) that might be helpful with such an extrapolation procedure. A good under-
standing of the basic flow mechanism Involved is equally important In this respect.

Since boundary layer development Is the very origin of Reynolds number effects it might be possible
to manipulate the boundary layer on the model such that the pressure distribution for flight conditions
is simulated, or at least approximated in the wind tunnel. This can be done in various ways: by the so
called aft-flxatlon technique, by energizing the boundary layer with suction, blowing or vortex genera-
tors or by modifying the airfoil contour (relief of pressure gradient through the use of a thick trailing
edge).
Of these methods, the aft-flxation technique Is the best known and most widely used. Most transonic
airfoils show, at tunnel Reynolds numbers, an appreciable region of laminar flow in front of the shock.
The boundary layer can then be tripped such that the boundary layer is turbulent at the shock and res-
sembles the flight condition over the rear part of the airfoil. This is illustrated in fig. 3.6-1 taken
from a study by Blackwell [Ref. 68/1D. This Is one of the earliest publications on this technique, al-
though the method was also suggested by D.L. Loving in his classical paper on shock Induced separated flow
[Ref. 66/1]. The sequence of figures clearly shows that the aft-flxatlon technique helps to suppress the
separation over the rear of the airfoil. In this case transition location was selected such that the high
Reynolds number trailing edge conditions were duplicated with aft-fixation at the lower tunnel Reynolds
number. No large changes are noticeable for attached flow conditions, very much In line with the obser-
vations made before. A more recent Illustration for a three-dimensional configuration is presented in the
figures 3.6-2 and 3.6-3 taken from Ref. [85/4] by Fulker and Ashlll. A more refined simulation criterion
was applied in this case based on the duplication of the relative (to the chord) separation length of the
bubble underneath the shock as discussed In section 3.4.2. Also In this case the aft-flxatlon technique
was used primarily to suppress premature flow break-down In the wind tunnel. The technique can also be
used to simulate a separated flow condition as figure 3.6-4b taken from Ref. [83/3] Illustrates. In this
case details of the flow in the vicinity of the shock were not well represented. Finally, aft-fixatlon
can be used to assess the effects of local separation at the trailing edge or In the lower surface rear
loading region by comparing aft- and forward fixation results. In this way relevant information can be
obtained for an extrapolation procedure.
This powerful technique Is often used to study Reynolds number effects In the wind tunnel. But there
are some limitations as well. First of all, the range of simulated conditions is restricted by the length
of the laminar flow region ahead of the shock wave. In figure 3.6-5 the useable region in the llft-Mach
number plane has been Indicated for a three-dimensional wing with a strip at 3051 (mid wing) chord posi-
tion. In this particular case the useable region appears to be very small and limited by the forward move-
4*5

ment of the shock, either due to decreasing lift and/or Mach number or due to flow break-down at the
higher lift values at the upper surface or the development of the pressure distribution with lift on the
lower surface. The usuable region can bc extended by application of a more forward trip but at the expen-
se of the Reynolds number simulation capability. Thus a particular strip location provides only adequate
simulation In a limited (though very important!) region of the operating conditions as figure 3.6-6 taken
from Ref. [76/1] Illustrates once more. At the wing tip and root, the natural transition is very often
close to the leading edge. In that case a cranked strip must be used (see fig. 3.6-7a) and only the mid
section of the wing is adequately represented. And even this Is questionable when the fixation causes a
significant variation on the local sweep angle of the shock as dlsccused In section 3.5. These restric-
tions may cast some doubts on the ability of the aft-fixation technique to simulate flight conditions
that can only be removed after a careful analysis of the results.
Vortex generators are used extensively to control flow separation in flight (see Ref. [61/6] by Pear-
cey for an excellent review). The same technique can also be used in a wind tunnel to suppress flow break-
down at a lower-than-flight tunnel Reynolds number. Figure 3.6-7b taken from Ref. [74/3] shows that the
effect on pitching moment is similar (though far from Identical) to the effect of aft-flxatlon. This tech-
nique is more difficult to control than the aft-flxation technique and certainly less suitable, for drag
assessment. It was reported [Ref. 85/6] that contour modifications have been used and are still being
studied as a way to simulate the shock wave development (and hence compressibility drag) in drag evalua-
tion studies. The principle is that a thickening of the airfoil contour over the aft part of the airfoil
might provide a pressure relief such that the boundary later thickness and (possibly) the overall circula-
tion is comparable with flight conditions. These and other techniques, like boundary layer suction as
suggested by Green [Ref. 71/2] are difficult to apply and have not yet reached the state of application
for Industrial testing.

Is Computational Aerodynamics (CFD) capable of bridging the Reynolds number gap between wind tunnel
and flight? Such a method must be able to describe direct and indirect Reynolds number effects. The direct
Reynolds number effect (the change In boundary layer development due to a change In Reynolds number for a
"frozen" pressure distribution) can be adequately represented by most boundary layer calculation methods.
In that respect they can be used to extrapolate c.g. viscous drag to higher Reynolds numbers. With local
separations present, the assumption of a "frozen" pressure distribution is no longer valid and the so
called "strong coupling" between the outer flow and the boundary layer flow is essential (c.g, by using
Inverse boundary layer methods; see Ref. [80/7], [82/5] and [84/4] for reviews). For the representation
of the Indirect Reynolds number effect (change in pressure distribution due to the change in boundary
layer development) the trailing edge region (Kutta condition) is extremely Important. This requires for
the calculations the Inclusion of normal pressure gradients in the viscous shear layers and wake curva-
ture effects (see e.g. Ref. [81/4]). Figure 3.6-8 shows, as an illustration, the indirect Reynolds number
effects as calculated by (an old version of) the VGK-method of RAE. The calculations closely reflect the
experimental results for this relatively simple case without trailing edge separation. However, when local
separations are present, the situation becomes increasingly complex. This is even more so beyond flow
break-down when large scale separations are present. In that case the usual thin shear layer assumptions
are no longer valid and the solution of the full Navier Stokes Equations Is required. Recent developments
(see e.g. Ref. [00/1]) indicate that the mathematical tools are "in hand" to solve this problem. Numerical
schemes can be constructed and computers are powerful enough to find a solution in a reasonable time.
This does not mean that the task is a simple one, but the future looks bright. Dark clouds, however, are
still present In the form of turbulence modelling. Due to the complexity of this problem, progress has
been slow over the last decade. Research In this field moves away from "universal turbulence modelling"
Into the direction of specific models for special classes of flow (see e.g. Ref. [81/5] and [84/5]). Empi-
rical information, to determine the "variable constants" Is essential here. For that reason calculation
methods should be validated over a range of flow conditions (from attached to separated flow) rather then
by comparing a few pressure distributions with experiment. In fact, the ability of a calculation method
to describe the Reynolds number effect of an airfoil close to separation would be an ideal test case. The
actual situation is almost paradoxlal In the sense that accurate and reliable (not Influenced by "pseudo
Reynolds number effects" as discussed In chapter 2) experimental Information on Reynolds number effects
Is rather limited as this AGARDograph shows. And this in turn hampers the development of more advanced
calculation methods.
This does not mean that CFD Is of no use for Reynolds number assessment. Computational methods that
can adequately describe attached or almost attached flow conditions are of great help in the interpreta-
tion of the wind tunnel test results. They can be used and are actually used to estimate differences in
transition location between wind tunnel and flight, to determine the simulation parameters for the aft
fixation technique, to estimate at what Reynolds number trailing edge separation will disappear, to extra-
polate drag results to higher Reynolds numbers etc. etc. In this way computational methods are essential
for the enrichment of the information provided by the wind tunnel. The very detailed Information from
CFD-raethods allows the test englngeer to make a much more reliable estimate of flight characteristics
starting from wind tunnel test results.
 47

BOUNDARY-LAYER
PROFILES

TRANSITION TRIP
POSITION UPPER
SURFACE Rcx10
5 1.0 0 .02 .04 .06
x/c z/c o is«yc
EFFECT OF R AND TRANSITION LOCATION. M • 0.70: Of - 0° A 30%

TI=0 89

0.77
.5 1.0
.02 .04 .06 0
x/c z/c
EFFECT OF R AND TRANSITION LOCATION, M = 0.80: 0 = 0°
R xT/c
.05
3.0 .05 065
y\ 3.0 40
10
S 1 1
12
r ^\ cp, sonic
,b I - / v
8 *-
'o f V -.\ q - -4
N
\ = 0.53
r. i i — i ^--.i
.5 1,0
.02 .04 .06 0
x/c z/c
EFFECT OF R AND TRANSITION LOCATION. M • 0.75: a = 3°

Fig. 3.6-1 Illustration of the aft-flxatlon Fig. 3.6-3 Two Independent simulations of full
technique (from ref. 68/1) scale" pressure distributions, outer
wing, Mach = .78, C, = .7 (ref. 85/4)
L

TRANSITION TRIP
'POSITION UPPER
SURFACE
• - - 5
-o- 5
-*- 5
- o ~ 15
"STREAMWISE FROM 0 89
LEADING EDGE%
LOCAL CHORD

Fig, 3.6-2 Scale effects on outer wing pressure

distributions, (a) C = .66; (b) C L
= .70 (from rof. 85/3)
48

| 2-D AIRFOIL]

O PT TRANS. FIXED AT 30% Re • 2.2 x 10',6

-2 -l-CFF R e - 2 0 . 10 6
M -0.75 0-1°

o 0.383 -0.0733
+ 0.392 -0.0771 ©

• TRANSITION FREE
A LEADING-EDGE TRIP R, » 2 . 8 x 10 6
• REAR TRIP )
LEADING-EDGE TRIP R c = 11.7x10 6

-2

w A
C
M a
xc A /

0.5 A 1

| 2-D AIRFOIL | A 7
A
i I —O.Uo
O PT TRANS. FIXED AT 30%, Re = 2.2 • 10 8 A A
+ CFF R e - 2 0 . 10',6 A
A A
M -0.75 a-4°
A
n A
PT 0.830 -0.0805
A
I * A A
k /
CFF 0.600 -0.0795 a
•«
a
a
" \. a
1 f a
• a a
• 1
— —U. 11
1
V" •
•

• i
•
• •
•
l
• •
• •
• • •
X;C
«*** --0.14
0.5

Fig. 3.6-4 Examples of high Reynolds number Fig. 3.6-6 Effect of strip position on pitching
simulation with aft-flxatlon on a moment development with incidence
two-dimensional airfoil (from ref. (Mach = .85) (from ref. 76/1)
83/3)

3-D WING, HST

4 -
UPPER SURFACE TRIP
NOT EFFECTIVE LIFT DIVERGENCE

FIXATION AT30%CHORD FIXATION AT 30% CHORD

L/ _1_
o^V
V^ _i_ • ' '

70 .75 .80 85 70 7 5
M -80 85
Ma Ma

Fig. 3.6-5 Regions of applicability of the aftflxation technique in the lift-Mach-

number plane (from ref. 83/3)
 49

B.L. TRANSITION STRIP a

— B.L. TRANSITION STRIP d

BOUNDARY - LAYER
66
/ TRANSITION STRIP
0.75
0 86

Fig. 3.6-7a The effect of boundary-layer transition position on the

position of the shock wave for a high aspect ratio wing
(from ref. 74/3)
WITHOUT VORTEX GENERATORS
WITH VORTEX GENERATORS

Fig. 3.6-7b The effect of aft-fixatlon and vortex generators on the lift and pit-
ching moment development (from ref. 74/3)

Ma a c, C^ C d Re . 10"6 Ma C, C m Cd R e . 10
O .742 1.32 59-.091 .0097 12.4 .745 .60-1004 0.0095 12.5
A .739 1.83 .59 -077 .0127 3.6 .745 .60-0928 0.0121 3.6
LAA-A-A-
1.3 1
A/
If
lee ll MODEL"A"
1 MEASURED (HST)
l.l l

•1
1
1.0
1
iL
^ B \

.9

1
am
J&tZtt

;i
w

1 i i i
c .2 .4 •6 x/c 1

Fig. 3.6-8 Comparison of measured and calculated indirect Reynolds number effect
on a two-dimensional airfoil (unpublished results of NLR)
50
 51

4.0 OBSERVED REYNOLDS NUMBER EFFECTS: LOW ASPECT RATIO WINGS AND BODIES
by
T. W. Binion, Jr.
Calspan Corporation/AEDC Operations
Arnold Air Force Base, TN 37389
and
E. Stanewsky
Institut f(ir Experimentelle StrOmungsmechanik
Deutsche Forschungs- und Versuchsanstalt fur Luft- und Raumfahrt e.V,
GGttingen, F.R.G.

4.1 Low Aspect Ratio Wings

Reynolds number effects on low aspect ratio configurations are potentially

manifested in four factors: transition location, displacement thickness, shock position,
and separation loci. While in two-dimensional flows, natural transition occurs primarily
because of Tollmien-Schlichting instabilities, transition in highly 3D flows inherent
with low aspect ratio configurations can be the result of leading edge contamination,
cross flow instability, and in some situations Goertler instabilities as well. The
modification of the spatial distribution of displacement thickness due to Reynolds number
is, of course, coupled with the local flow vector distribution. The effects of Reynolds
number on shock position and separation loci are closely coupled with those acting on
displacement thickness. Because of the complex nature of the flows and because low aspect
ratio configurations are used primarily for military applications, definitive information
is not frequently published. Aside from some generic delta wing studies, there have been
no systematic investigations of the basic aerodynamic characteristics of low aspect ratio
configurations reported. The richest source of Reynolds number effects data seems to be
wind tunnel to flight comparisons. Such comparisons are fraught with many difficulties
because of pseudo-Reynolds effects which creep into wind tunnel data and inherent
inaccuracies and measurement difficulties in flight testing. Thus, we feel that the data
which follow should be viewed with some caution; the conclusions depicted may, with
refined techniques and insight, be shown to be incorrect.

4.1.1 Zero-Lift Drag

Saltzman and Bellman 171/9] presented data correlations for the X-15 and the YF-
102 aircraft. Wind tunnel zero-lift drag data were extrapolated using the Karnam-
Schoenherr flat plate skin friction relationship [63/2] which obtains the best
correlations in terms of Reo rather than Rex. Extrapolation of the data. Fig. 4.1.1, is
excellent. Note however, that the X-15 data had to be correlated with the base drag
removed because of the influence of sting interference on the blunt base configuration.
It is further noted that the X-15 airplane and model were both very rigid; there were no
leading edge slats, spoilers, or hinged rudders; no inlet flows, propulsion jets or
bypass airflows to simulate on the model; and, the problem of measuring thrust in flight
was avoided by considering only gliding flight. Thus, all of the pitfalls associated with
the flexible, airbreathing aircraft were avoided. Even at that, it was necessary to
subtract the base drag from both data sets because of sting interference in the wind
tunnel. No explanation was given for the good agreement with the YF-102. However, the
only "Reynolds number effect" for both configurations apparently was that due to skin
friction.

4.1.2 Delta Wings

The formation of stable vortices over swept wings at angle of attack is a useful
lift-generating phenomenon utilized as either the primary lift production mechanism or to
augment the lift produced by a conventional airfoil, see [83/9] for example. From a
phenomenological viewpoint, wings which produce lift can be classified according to their
leading edges. In order to make a statement about Reynolds number effects for delta
wings, it is necessary to specify the type of flow that exists over the wing. It has
become traditional to categorize delta wing flow in terms of the Mach number and the flow
incidence angle in a plane normal to the leading edge. Stanbrook and Squire [64/1] in
evaluating data for sharp leading edge delta wings established demarcation regions for
separated and attached flows at the leading edge in the Mn/an plane. Szodruch [77/1]
extended their idea by defining boundaries of other types of separation. Fig, 4.1.2. Not
only do the specific effects of Reynolds number depend on the region of interest in the
Mn/an plane, but the boundaries themselves seem to be a function of Re, although truly
definitive studies are lacking. Systematic studies for rounded leading edges are also
sparse. Szodruch has summarized the present knowledge with respect to Reynolds number for
52

each class of wings [88/1]. It is felt unnecessary to repeat that summary here. The main
points therefrom are:

(1) With sharp leading edge wings for which the primary flow separation is fixed
at the leading edge, the main vortex position and strength is essentially constant with
Reynolds number. However, the location of the secondary separation lines and hence the
secondary vortex strength appears to be a moderately weak function of Re.

(2) A systematic study of Reynolds number effects for rounded leading edges has
not appeared. Nevertheless, one may deduce from the existing data that the larger the
leading edge radius, the greater the Reynolds number dependency. However, Poll [83/13],
based on the comparison of his work with other results, states "Reynolds number, sweep,
and incidence are insufficient in themselves to determine the type of (vortex) flow which
will occur on a given airfoil section." He suggests the wind tunnel disturbance
environment also may be a significant factor affecting the data.

(3) There is conflicting evidence on the effect of Reynolds number on vortex

breakdown. Again, no systematic investigations have been published. However, Erickson
[80/9] has compiled a group of data from several swept wings tested in both wind and
water tunnels. Fig 4.1.3, which show no correlation between the angle of attack at which
vortex breakdown begins and Reynolds number in the range between 9800 and 40 million.

4.1.3 Aerodynamic Derivatives

Teige, et al. [75/5], have reported a comparison of flight and wind tunnel
obtained aerodynamic derivatives for the SAAB 37 fighter, Fig. 4.1.4. After applicable
corrections, the flight data were reduced through a least-square regression process
involving 75 equations. The results were cross checked using a 6-deg of freedom digital
simulator with the flight control surface pulses as inputs to reproduce the flight
motion. Wind tunnel tests were conducted with 1/30- and 1/50-scale models at Reynolds
numbers of 6- and 12-million. Scatter in the Cnfl wind tunnel data in the transonic regime
was attributed to nonlinear Cn-fl curves. Experience in other tunnels has shown that
nonlinear Cn-B data can be caused by flow non-uniformities in the test section flow if
the pitch-roll technique is used to obtain yaw. However, as the sample data in Fig. 4.1.5
show, while aeroelastic corrections are necessary, no Reynolds number effects were found.
The flight and wind tunnel predictions are in good agreement.

4.1.4 Buffet

Figure 4.1.6 [81/6] depicts a typical flow pattern on a transonic swept wing at
buffet onset and with moderate buffet. At buffet onset, there is a bifurcated shock in
the planform plane, mild shock induced separation, and the beginning of trailing edge
separation in the wing root region. As incidence is increased, the flow pattern becomes
much more complex with thickened shear layers, leading edge as well as shock induced
separated regions, and regions of complete flow breakdown. The flow structure in the
vicinity of the surface has much lower energy and is therefore more unstable. One would
expect that since buffet is associated with such complex phenomena, Reynolds number would
be a strong influencing parameter in buffet characterization. Such, however, does not
appear to be the situation.

Because of the nature of buffet determination in conventional wind tunnels,

aeroelastic effects can mask Reynolds number effects when Re is varied by changing total
pressure. Such a difficulty can be avoided by using a cryogenic facility wherein Re can
be varied by changing temperature keeping the dynamic pressure constant. Boyden [81/7]
has conducted such a test using a 65-deg swept wing. The data. Fig. 4.1.7, are presented
in terms of a steady, C B , and an unsteady, C B , bending moment coefficient. The steady
component is completely independent of Reynolds number as is the dynamic component prior
to vortex breakdown. The reason for the rise in CB beyond flow breakdown at the higher Re
is believed to be associated with the increased frequency parameter rather than Re, see
Mabey [81/6].

Butler and Spavins [77/2] tested a large half model of a 43-deg swept wing at
Reynolds numbers up to the flight value which showed no Re effects and an excellent
agreement with flight data. Fig. 4.1.8.

Buffet tests on the YP-4 showed that, of 8 methods tried to discern buffet, wing-
tip accelerometers were the most sensitive and reliable indicator [70/2]. Wind tunnel
data were obtained with a 5% model at unit Reynolds numbers between 3.4- and 7.8-million
depending upon Mach number. Correlation of flight and wind tunnel data both using wing-
tip accelerometers to detect buffet onset. Fig. 4.1.9, shows that at a given Mach number.
 53

the wind tunnel predicted the onset of flow separation from 0.5- to 1-deg higher angle of
attack than was experienced in flight. Whether that discrepancy is caused by a Reynolds
number mismatch or some other factor was not investigated.

4.1.5 Transonic Technology Wing Program

A transonic technology wing program specifically designed to compare wind tunnel

and flight data was conducted by Dornier in cooperation with several other agencies
[82/11] [82/12]. The experimental aircraft. Fig. 4.1.10, and a 1/10-scale wind tunnel
model were equipped with pressure orifices at identical locations. Comparison of pressure
distributions from flight and Tunnel 16T showed generally very good agreement even though
the Reynolds number differed by a factor of 6 to 10. Typical examples are shown in Fig.
4.1.11. Boundary-layer transition was free in each instance. Drag polars for one TST
configuration are shown in Fig. 4.1.12. The wind tunnel data were obtained in the ONERA
S2Ma at 2.5-million Reynolds number. The agreement is quite good except at Mach number
0.85. The wind tunnel data at Mach number 0.85 probably contains a large increment due to
wall interference since the blockage was 1.2%.

4.1.6 Summary

Available data for low aspect ratio configurations depicting Reynolds number
effects are sparse. No systematic investigations, other than a few for sharp edge delta
wings, have been conducted. The data which are available, however, suggest that for most
of the flight regime of interest, Reynolds number effects are small to nonexistent.

o FLIGHT, FULL SCALE CENTER LINE

a MODEL, SUB SCALE SHOCK REGION LEADING EDGE
— EXTRAPOLATION SEPARATION
WITH SHOCK

2 4 6 10 20 40 60 1 0 0 x 1 0 6
R 1 2
NORMAL MACH NUMBER, M N
F i g . 4 . 1 - 1 . Comparison of e x t r a p o l a t e d
wind tunnel model drag F i g . 4 . 1 - 2 . Leeside flow regimes over
c o e f f i c i e n t s with f u l l - t h i c k Delta wings a t s u p e r -
scale flight results, s o n i c speeds - OR versus Mu
[71/9]. diagram, [ 7 7 / 1 ] .

40
1*I O t *
F\
W
o AA O
REYNOLDS NO. LU
o WATER TUNNEL 4.1 x l O 4
9.8 x 10 3 o
30 !• A °
1 WATER TUNNEL °*o
+ WATER TUNNEL l.Ox 104
D WIND TUNNEL 1.5 x 1 0 6 < X
A WIND TUNNEL 1.3 x l O 6 20
1= A Q
O WIND TUNNEL 9.0 x l O 5
o<
X WIND TUNNEL 1 . 4 x 1 . 7 x 106
2.0 x l O 6
z
< 1
A

o
WIND TUNNEL
FLIGHT 40.0 x 1 0 6 KST
10 : * i1
Z
• WATER TUNNEL l . O x 8.0 x l O 4 o
•* WIND TUNNEL 2.0 x 1 0 6
A WIND TUNNEL l . O x TO6
0 WATER TUNNEL 3.0 x l O 4 60
3.0 x l O 4
55 65 70 75 80 85
t l WATER TUNNEL
WING SWEEP, DEG

Fig. 4.1-3. Effects of wing sweep and Reynolds number on Delta wing
vortex breakdown at the trailing edge, [80/19].
54

MOSS AND MUNDELL

TRAILING-EDGE
SEPARATION ? SHOCK WAVES
TRAILING-EDGE
SEPARATION

COMPLETE FLOW
BREAKDOWN
AT TIP

oc = 8°
MODERATE BUFFETING

Fig. 4.1-6. Transonic flow on swept wing,

M = 0.80, [81/6].

Tf(K) Pt(bar) \\ xlO"6 q(kN/m 2 ) V(m/$)

• 300 1.2 1.8 9.6 120
+ 110 1.2 7.1 9.6 73

-1O.O6
CB
0.10

Fig. 4.1-4. SAAB 37 Viggen, [ 7 5 / 5 ] 0.04

A WT. TESTS RIGID MODEL

0.05
0.02
+ COMPUTED AEROELASTIC
CORRECTION
° FLIGHT TESTS

0J
S
Fig. 4.1-7. Effect of Reynolds number on
buffet, M = 0.35, 65-deg swept
wing, [81/7].

0.5 r BUTLER A N D
SPAVINS

0.4
0 0.5 1.0 1.5 2.0 M
FLIGHT
C„0 0.1 TUNNEL
0.3
H = 6km
i I i cN
0 O.S 1.0 1.5 2.0 M

C, 0.1
H = I km
o—<^n
/K 0.2

0.1
i J i i

0.5 1.0 1.5 2.0 M Rc= 10 TO 18 x 10e

Fig. 4.1-5. Comparison of wind tunnel and

flight derived Cn{3, SAAB-37 LnA- 0.7 -s-

0.8 0.9
fighter, [75/5].
M

Fig. 4.1-8. Flight/tunnel comparison for

buffet onset on a small
fighter aircraft, [77/2].
 55

10 DATA
U SYM SOURCE % BLOCKAGE ALPHA MACH RexlO"6 SECT
UJ
• FLIGHT 2 . 3 3 0.836 23.1 3
I
UJ
8 - FLIGHT TEST
o 16T 0.16 2 . 3 3 0.835 2.5 3

at
a

5
u-

o
ss 0.5 0.6 0.7 0.8 0.9 1.0 CP
MACH NO. - M

Fig. 4.1-9. Wingtip accelerometer, rms indi-

cation of airflow separation,
comparison of wind tunnel and
flight test results, [70/2].

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
X/C

b. M = 0.835
F i g . 4 . 1 - 1 1 . Concluded.

WIND TUNNEL TESTS S2 ONERA

Fig. 4.1-10. Transonic technology wing on

the Alpha jet, [82/12].

DATA Fig 4-1.12. Comparison of forces from wind

SYM SOURCE % BLOCKAGE ALPHA MACH Re x 1 0 " 6 SECT tunnel and flight test, TST
a FLIGHT 2.16 0.603 15.5 3 configuration, [82/11].
o 16T 0.16 2.16 0.600 2.5 3

CP

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
X/C

a. M = 0.6
Fig. 4.1-11. Comparison of wing surface
pressure distribution from
flight and wind tunnel, TST
wing, [82/12].
56

4.2 BODIES

4.2.1 Introduction

Considering the flow about bodies typical of missile configurations and/or fighter
fuselages, there exists an area particularly sensitive to Reynolds number effects,
viz., the high angle of attack domain of freestream conditions. The strong interest
in achieving high maneuverability for both missiles and fighter aircraft has
increased the importance of that domain and hence the need for an accurate pre-
diction of the vehicle performance at such conditions. We will, therefore, discuss
here mainly the Reynolds number dependence of the aerodynamic characteristics of
bodies at high angles of attack but also briefly treat the less sensitive lower
incidence range as well as afterbody and base flow and their contributions to the
overall Reynolds number dependence.

Before proceeding it should be noted that an extensive and thorough review of

viscous effects related to bodies at high angles of attack (with an abundance of
references) was carried out by Polhamus in 1984 [84/7]. We will essentially follow
his discussion - where relevant - but restrict the present discussion to some
characteristic examples of Reynolds number effects supplemented by results on
afterbody and base flow.

4.2.2 Cylinders In normal flow

Many of the Reynolds number dependent viscous flow phenomena encountered by mis-
siles and fuselages at high angles of attack can be related, at least qualitatively,
to the flow about circular cylinders. Here, the various phenomena and regimes
covered as the Reynolds number is increased, say, from the low Reynolds number wind
tunnel tests to full-scale conditions can best be explained. These regimes are
defined in Figure 4.2.1 where the drag coefficient for a circular cylinder is
plotted versus the Reynolds number based on the diameter of the cylinder [84/7].

In the subcritical range, below a Reynolds number of R D =2xl0 5 , the flow is laminar
and a laminar separation occurs at a separation angle of 8 S =80°. Since the flow
does not reattach, a wide wake and a correspondingly high drag coefficient results.
In this range a dominant vortex shedding frequency is present. When increasing
the Reynolds number, the laminar separation is followed by a turbulent reattachment
with transition having occurred in the separated shear layer. In this "critical"
range a renewed but turbulent separation takes place further downstream - for
R D =4xl0 5 , for instance, at 0 S =130° - resulting in a narrowing of the wake with an
associated strong decline in drag coefficient. Other than stated by Polhamus
[84/7], evidence is given, for instance, by Horvath et al. [86/3] and Schewe [83/11]
that a narrowband coherent vortex shedding still exists in the critical Reynolds
number range. Note, that flows with laminar separation and transition taking place
in the separated shear layer are known to give a similarly favorable effect on drag
for airfoils and wings in transonic flow [68/2] [81/1).

In the supercritical range, R Q = 4 X 1 0 ^ to about R Q = 6 X 1 0 5 , transition to turbulent

flow moves upstream to a position ahead of separation and eventually to the vicinity
of the foreward stagnation point. In that range the separation point shifts, due
to the change in boundary layer conditions, also upstream from the extreme down-
stream position 0s=13Oo to about 9 S =115°; there is, due to the widening of the wake,
a corresponding increase in the drag coefficient. Spectra signatures from this
regime are wideband with no predominant frequency except for a small region at the
high Reynolds number end where a narrowband vortex shedding prevails. The flow
development in the supercritical range is very sensitive to freestream turbulence
and surface roughness [86/3].

At the high Reynolds number end of the supercritical range, the drag levels off
since the upstream movement of the transition point eventually comes to a halt.
At still higher Reynolds numbers in a range termed "hypercritical" by Polhamus
[84/7], the boundary layer is fully turbulent and the drag coefficient is likely
to decrease with increasing Reynolds number, predominantly due to a slight downward
shift in the separation location. The vortex wake flow becomes here at least qua-
si-periodic .

Many correlations have been made of the effect of the Reynolds number on the flow
about circular cylinders. These correlations have often shown a wide band of data
uncertainty which was explained as being caused, for instance, by differences in
surface roughness, compressibility and end conditions, i. e., the tunnel width to
cylinder diameter ratio. Polhamus [84/7] has carried out a careful selection of
results from various sources, restricting the data to incompressible flow
 57

(M SO.15)') , which are essentially free from "secondary" effects; they are
depicted in Figure 4.2.2. The data basically exhibit the Reynolds number dependence
described above. The reader is also referred to the investigation of Schewe [83/11]
who covered a Reynolds number range between ReD=2.3xlOc and Re D =7xl0 5 with one
experimental setup by varying the total pressure between one and 51 bar. The
investigation revealed interesting details of the flow behavior in the critical
Reynolds number range where distinct discontinuities in the Reynolds number
dependence were observed. Figure 4.2.3. Here, the discontinuity at A is attributed
to the formation of a laminar separation with turbulent reattachment and subsequent
far aft separation on one side of the cylinder and a laminar separation without
reattachment on the other side. This asymmetric flow development is accompanied
by a steady lift force. Figure 4.2.4, which disappears as soon as the flow re-es-
tablishes its symmetric condition with increasing Reynolds number; the latter
results in the discontinuity at B. Note, that the lift force observed in the cri-
tical range is associated with one mechanism generating side forces (out-off-plane
forces) on, say, a three-dimensional ogive-cylinder body at angle of attack.

It was mentioned earlier that two effects, namely the effects of roughness and
compressibility, frequently obscured a comparison of results on flows about cyl-
inders obtained from different sources. Their individual influence is, therefore,
shown in Figures 4.2.5 and 4.2.6: Roughness of sufficient height essentially
promotes an early transition so that the transition point may always be located
upstream of the separation point as is the situation of the upper two curves of
Figure 4.2.5 (r/Dxlo5=900 and 450, respectively). Here, an increase in Reynolds
number, starting, say, at R D =10 5 , initially moves the transition point further
upstream to the vicinity of the stagnation point resulting in an increase in drag.
At a further increase in Reynolds number the drag gradually decreases with the
behavior of the flow being as in the hypercritical range, although at much lower
Reynolds numbers. As the roughness height is decreased , various stages in the
Reynolds number dependence of the flow development can be observed: At
r/D=110xlc£ , for instance, a very noticeable critical as well as a supercritical
domain still exist; however, both extend over a much smaller Reynolds number range
than is obtained with a smooth surface. Concerning compressibility. Figure 4.2.6,
one observes that the classical Reynolds number dependence persists at least up
to the vicinity of the critical Mach number with the drag increasing as the Mach
number is raised. As soon as shock waves occur, strong enough to separate the
turbulent boundary layer, the characteristic drop in drag in the critical Reynolds
number range is likely to disappear (also see Figure 4.2.12.). Note, that the drag
coefficient decreases in the hypercritical range as was conjectured when discussing
Figure 4.2.1.

Besides the circular cylinder, other cross-sectional shapes are of interest such
as, for instance, a rectangular cross section with rounded corners which might be
selected for a missile body to improve the packing density for submunition [83/12].
Furthermore, such shapes can be found on fuselages of modern combat aircraft which
are required to maneuver at high angles of attack. Figure 4.2.7, again taken from
Ref [84/7], summarizes some typical results concerning the Reynolds number
dependence of the drag coefficient for cylinders with square cross sections having
various values of the normalized corner radius. One observes a strong effect of
the corner radius on both the value of the drag and on the Reynolds number for the
transition from subcritical to supercritical flow. For the smallest radius,
r/w=0.021, separation occurs on the windward corners, essentially fixed by geom-
etry, at least at the low Reynolds numbers considered here, and the strong Reynolds
number effects observed for the circular cylinder cannot be expected. The drag
is similar to that for a flat plate perpendicular to the flow. As the corner radius
is increased, the adverse pressure gradients at the corners are reduced and, as a
consequence, separation is now affected by Reynolds number in a way similar to the
circular cylinder, although the drag of the latter is not reached here.

Ref. 84/7 also considers the influence of the flow angle on drag and side forces
for cylinders with non-circular cross sections and cross sections with other than
rectangular shapes. The results show basically similar Reynolds number dependences
as the ones depicted in Figure 4.2.7. Finally, it should be noted that data for
non-circular cross sections are mainly available for incompressible flow and Rey-
nolds numbers of Re D <2xl0 6 ; there is a considerable need to extend investigations
to compressible flow and to higher Reynolds numbers.

4.2.3 Cylinders in oblique flow

An Interim stage between the cylinder in normal flow and, say, a cone-cylinder body
at angle of attack is the cylinder of infinite length in oblique flow (swept

1
) Applicable to three-dimensional bodies in transonic flow at low angles of attack
5K

cylinder). Results for such a configuration are depicted in Figure 4.2.8 in the
form normal force, C N /sin2a, versus effective Reynolds number, Reeff , for various
sweep angles a [84/7]. Here, the effective Reynolds number corresponds to the one
defined by Esch [75/6], viz..

Reef( = Re D . K^

where

Ka = [0.983 / sino + 0.311 +

0.287 . sina] / 1.581

In this formulation, K a is equal to the ratio of the circumferential length of the

body in the plane parallel to the freestream (ellipse) to that in the cross flow
plane (circle), thus accounting for the actual change in streamline length, say,
to the separation line, as the sweep angle is changed. One observes that this
effective Reynolds number correlates the data of Figure 4.2.8 with regard to the
onset of the critical Reynolds number regime quite well - as a matter of fact,
better than any other correlation. This indicates that the flow component normal
to the cylinder is not Instrumental in determining separation and transition con-
ditions, at least not at sweep angles of a<60°, a fact also supported by the
dependence of the normal force coefficient CN/sin2a on the sweep angle in the
supercritical Reynolds number range. Such a limit in the applicability of the
independence principle, i.e., the independence of the flow components normal and
parallel to the body considered, which holds strictly only for laminar flow, was
also found for shock-induced separation on wings in transonic flow [88/1]. For
further discussions on the Reynolds number dependence of the flow about infinitely
swept cylinders, the reader is referred to Ref. 84/7.

4.2.4 Three-dimensional bodies

As a slender body is pitched through the angle of attack range 0°<a<90°, it expe-
riences four distinct flow patterns that reflect the diminishing influence of the
axial flow component eventually leading to conditions discussed in the previous
sections. Figure 4.2.9 [87/4] [79/3] [81/8]. At low angles of attack the axial
component dominates ; the flow is attached and essentially vortex free. As the angle
of attack is increased, cross flow developes and the boundary layer separates on
the leeward side of the body forming a symmetric vortex pair. This condition is
depicted in Figure 4.2.9 (a) for a=30°. Note, that the vortex lifts off the body
at the extreme downstream end. In the next higher o-regime, the cross flow
effects start to dominate and the vortices become asymmetric thereby producing a
side force at zero side slip or yaw. Figure 4.2.9 (a), a=50°. Finally, at very
high angles of attack, the cross flow dominates completely and the vortices are
shed either in a periodic form or in a wideband random fashion dependent on the
Reynolds number as discussed in Section 4.2.2 for the cylinder in normal flow.

It can be expected that the flow development in the various regimes is more or less
strongly influenced by the Reynolds number. To show this the normal force coeffi-
cient, based on freestream conditions and base area, for a typical ogive-cylinder
body is first depicted as function of the angle of attack with the Reynolds number
as parameter, Figure 4.2.10 [78/5]. Also indicated in this figure is the angle
of attack dependence of the normal force according to cross flow theory assuming
a purely laminar and a fully turbulent separation with separation angles corre-
sponding to the maximum drag in the subcritical and the minimum drag in the
supercritical Reynolds number regime, respectively, for the cylinder in normal flow
[51/3]. One observes that the flow development is highly dependent on Reynolds
number, starting already at relatively low angles of incidence, and that a par-
ticularly strong change in normal force occurs as separation shifts from a laminar
to a fully turbulent state.

A still better illustration of the flow development is obtained when cross-plotting

the results of Figure 4.2.10 as function of the Reynolds number with the angle of
attack as parameter. This was carried out by Polhamus [84/7] supplementing these
results by data of Foley [71/101 at an incidence of a=30°. Figure 4.2.11. The
different Reynolds number regimes, discussed in Section 4.2.2 for the cylinder in
normal flow, become clearly distinguishable: the subcritical range where a laminar
separation prevails, the critical Reynolds number domain where a transitional
separation with subsequent turublent reattachment and a renewed separation takes
place - and where the drag coefficient for the cylinder in normal flow has a minimum
- and the super- and hypercritical domains where the separation eventually becomes
fully turbulent. Figure 4.2.11 also illustrates the large reduction in the critical
Reynolds number, based on cylinder diameter and freestream conditions, with
decreasing angle of attack which is associated with the corresponding increase in
streamline length similar to the conditions for the infinitely swept cylinder
discussed earlier. (For the establishment of the characteristic boundaries, marked
 59

by the dashed lines in Figure 4.2.11, from results for the cylinder in normal flow,
the reader is referred to Ref. 84/7.) Note, that the influence of compressibility
on the critical Reynolds number is still minor at the freestream Mach number con-
sidered ( M ^ O . 5 ) , even at the extreme angles of attack.

In order to explore the influence of compressibility on the Reynolds number

dependence, the data of Hartmann [78/5] are cross-plotted in Figure 4.2.12 for
constant Mach numbers at angles of attack of a=30° and o=60° . It can be observed
that the critical Reynolds number increases slightly with Mach number and that,
at the same time, the Reynolds number sensitivity in the critical Reynolds number
range gradually disappears. The latter is due to the fact, already pointed out
in Section 4.2.2, that the cross flow Mach number becomes large enough to establish
supersonic regions on the body with terminating shock waves strong enough to sep-
arate the boundary layer essentially independent of the condition of the boundary
layer upstream of the shock. This is in agreement with evidence presented in Ref.
88/1 which indicates that the onset of shock-induced separation (incipient sepa-
ration) is, for a turbulent boundary layer, only weakly dependent on viscous
effects so that only a marginal decrease in normal force with Reynolds number can
be expected here. Note, that the second set of Reynolds number boundaries given
in Figure 4.2.12, termed "laminar, transitional and fully turbulent", will be
discussed below.

It is well known that side (or out-off-plane) forces on slender bodies at zero
side-slip or yaw angle pose a serious problem to the maneuverability of such con-
figurations so that a closer look at the Reynolds number dependence of side forces
in the various angle of attack domains seems warranted. There are essentially two
mechanisms that may generate side forces: One occurs only in the critical Reynolds
number regime and is related to the steady lift forces on the cylinder in normal
flow caused by different developments of transition and separation on the upper
and lower side of the cylinder (see Figure 4.2.4 and Section 4.2.2). The other
mechanism, referred to at the beginning of this section, also operates in the
subcritical (laminar) and the super- and hypercritical (fully turbulent) separation
regimes; this mechanism is directly related to hydrodynamic instability and is
qualitatively described by the impulsive flow analogy [82/13]. For a more detailed
study of vortex-induced asymmetric loads, the reader is referred to the review of
Ericsson and Reding [81/8] and the papers by Lamont [82/13], Hartmann [87/4] and
Champigny [86/4].

The true effect of Reynolds number and angle of attack on side forces can only be
determined when first considering the variation of the side force with roll angle
which occurs even if the flow is steady and extreme care has been taken to manu-
facture a model without apparent geometric asymmetry [82/13). The phenomenon is
illustrated in Figure 4.2.13 for two angles of attack and Reynolds numbers of
ReD=2.5xl05 and Re D =7.3xl0 5 , corresponding - see Figur 4.2.11 - to the subcritical
and supercritical Reynolds number range, respectively. One observes that in the
test cases considered two side force cycles occur during a complete roll and that,
furthermore, pronounced maxima are present which seem to decrease as the Reynolds
number is raised. Considering henceforth only the absolute maximum during a com-
plete roll as representative of the overall side force, it is indicated in Figure
4.2.14, taken from Ref. 82/13, that there is a strong variation of that force with
angle of attack as well as with Reynolds number. Side forces start to develop,
nearly independent of the Reynolds number, at an angle of attack of about a=30°
and persist almost up to o=90°, especially at Re D =0.4xl0 6 , a Reynolds number which
lies within the critical Reynolds number range where for the cylinder in normal
flow spurious lift forces existed.

The effect of the Reynolds number on the maximum side force is illustrated in Figure
4.2.15 for two test cases [82/13] [85/13]. The maximum side force falls from a
high value at a Reynolds number of approximately R 6 Q = 2 X 1 0 ^ (laminar separation)
to almost zero at the end of the critical Reynolds number range before climbing
again to a higher value at Reynolds numbers of Re D =2xl0 6 to Re D =4xl0 6 at which a
fully turbulent separation exists. The Reynolds number range where the side forces
become almost zero is the same as the one where the cylinder in normal flow exhibits
a minimum in the Reynolds number dependence of the drag coefficient and where the
steady lift force. Figure 4.2.4, falls again to zero. At these conditions sepa-
ration occurs furthest downstream, as indicated by the corresponding circumferen-
tial pressure distribution in Figure 4.2.16, and the symmetric vortex pattern
becomes stable. The results of Figure 4.2.15 are somewhat contrary to a statement
of Ericsson and Reding [81/8] who claim that the largest side forces develop in
the critical Reynolds number regime due to differences in the transition pattern
on opposite sides of the cylinder. Note, that the vortex-induced side loads
decrease with increasing subsonic cross flow Mach number (M^ 2 0.4) and become
insignificant at supersonic cross flow Mach numbers [81/8] [84/7).
60
At various instances we have referred to boundaries identifying characteristic
Reynolds number domains for three-dimensional bodies: One set of boundaries, i.e.,
subcritical, critical and super- and hypercritical, was derived from the conditions
on the cylinder in normal flow utilizing the relation for the effective Reynolds
number K a =Re etf /Re D (see Section 4.2.2 and Figures 4.2.8 and 4.2.11). Another
classification is based simply on the type of separation occurring on the body,
i.e., laminar, transitional - with transition taking place within the separated
region - or fully turbulent separation. Lament derived such boundaries from
pressure distributions similar to the ones depicted in Figure 4.2.16 for an
ogive-cylinder body [82/13]. These boundaries are shown in Figure 4.2.17 in the
Reynolds number / angle of attack plane. One observes that the Reynolds number
boundary between transitional and fully turbulent separation is a strong function
of angle of attack, whereas the laminar-transitional separation boundary is much
less dependent on incidence. Note, that these boundaries may strictly only be
applied to smooth cylinders in low turbulence streams for which they were derived.
It seems likely that freestream turbulence and surface roughness will have an
effect on inclined cylinder flows similar to the one experienced for the cylinder
in normal flow (see Figure 4.2.5).

Also of interest, for reasons outlined in Section 4.2.2, is the Reynolds number
dependence of three-dimensional bodies with non-circular cross sections. Unfortu-
nately, there is only a very limited number of results available and of the few,
just one example is shown here, depicting the effect of the Reynolds number on the
normal force coefficient for the fuselage of an orbiter-type vehicle with various
corner radii. Figure 4.2.18 [84/7] [71/11]. The dependence on Reynolds number and
corner radius is essentially the same as for the corresponding cylinder in normal
flow and, at a given corner radius, say r/w=0.086, the effect of viscosity is very
similar to the one for an ogive-cylinder configuration: a nearly constant normal
force in the subcritical Reynolds number range and a strong decrease in normal force
as the critical Reynolds number is exceeded. Note, that also for the present
configuration, the Reynolds number sensitivity in the critical Reynolds number
range disappears as the normal Mach number component exceeds critical freestream
conditions.

4.2.5 Afterbody and base flow

The investigation of afterbodies, either by detailed studies of the local flow

development or by determination of the overall forces acting on the afterbody, is
an integral part of the flight vehicle design and development process [85/14]. The
importance of such investigations derives, in part, from the fact that afterbodies,
either as a component of the fuselage or as part of the propulsion/exhaust system
of jet-propelled aircraft, may contribute considerably to the overall drag of the
flight vehicle. Accurate prediction of the full-scale flow development requires
the understanding of possible Reynolds number effects on afterbody flow. Note, that
a critical review of computational and experimental techniques concerned with
afterbody/nozzle aerodynamics, jet airframe interference effects, and nozzle
integration is presented in the report of the AGARD Working Group "Aerodynamics
of Aircraft Afterbody" [86/5] (also see 85/14), unfortunately without addressing
the issue of Reynolds number effects to any extent.

A review of the effect of Reynolds number on afterbody drag was given by Pozniak
in Ref. 81/9. Concerning the Reynolds number influence, he comes to the following
conclusions:

• At subsonic Mach numbers and in the absence of major flow separation, signif-
icant but compensating pressure changes are found such that there is little
effect on the afterbody pressure drag of complete afterbodies as Reynolds
number in varied.

• In the presence of flow separation, the effects of Reynolds number tend to be

small when the location of flow separation is fixed as a result of a sudden
change in the boattail contour, but on afterbodies with more continuous con-
tours the point of separation can be affected. Conflicting factors are then
involved and the afterbody drag can increase or decrease by modest amounts or
remain unaffected by Reynolds number changes.

• Significant increases in drag with increasing Reynolds number have been con-
sistently reported for high subsonic Mach numbers above the transonic drag rise
and at supersonic speeds.

• Misleading Reynolds number effects on afterbody drag may occur due to the
influence of the Reynolds number on wall interference and the wind tunnel
environment.
 61

In this section some representative results relating to these conclusions shall

be presented, utilizing data that are considered essentially free of "secondary"
effects caused by changing models, test setups and wind tunnels to cover a wide
Reynolds number range or by the effect of the Reynolds number on the wind tunnel
environment.

Figure 4.2.19 shows the dependence of the pressure drag for a circular-arc-cone
afterbody on the Reynolds number, the latter based on the forebody length, L. The
data were obtained in the NASA-Langley 0.3-m cryogenic tunnel at Mach numbers of
M^O.60 and 0.9 over a wide Reynolds number range [76/6]. Note, that in addition
to the temperature and pressure of the tunnel, the forebody length was varied to
cover the Reynolds number range up to Re L =130xl0 6 , as indicated by the open and
half-filled symbols in Figure 4.2.19.

One observes at both Mach numbers considered and for both body lengths utilized a
moderate increase in afterbody pressure drag, based on maximum body cross section
area, with Reynolds number. There is also a noticeable difference in drag for the
different body lengths. The latter is judged to be due to the differences in the
condition of the boundary layer approaching the afterbody, mainly the displacement
thickness, which is for the same Reynolds number Re, larger in the case of the
longer body. At the lower Mach number, the drag thus seems to increase with dis-
placement thickness, while at M ^ O . 9 0 the pressure drag decreases slightly, at
least in the lower Reynolds number range, which is in accordance with the trend
observed with increasing Reynolds number. This behavior shall be further analyzed
by considering the corresponding pressure distribution.

One of the reasons for the low Reynolds number sensitivity of the afterbody drag
observed here is the compensation .that occurs in the Reynolds number dependence
of the pressure distribution. This is illustrated in Figure 4.2.20 where the minima
and maxima in the pressure distribution, identified in the inset to this figure,
are depicted as function of the Reynolds number for the conditions outlined above.
One notices that the minimum pressure decreases with Reynolds number, which can
essentially be attributed to the reduction in the displacement thickness at the
shoulder of the afterbody; this is confirmed by comparing the pressures for the
two different body lengths. The reduction in displacement thickness results in an
increased curvature of the effective body contour and thus in a stronger expansion.
The pressure recovery improves with Reynolds number and - for the same reason -
with decreasing displacement thickness, similar to the effects observed on tran-
sonic airfoils and wings [81/1], thus compensating, at least in part, the higher
suction peaks. Figure 4.2.20 also provides an explanation why at M ^ O . 6 0 the
pressure drag increases at a given Re, with increasing displacement thickness: the
lower pressures over the rear of the afterbody in case of the thicker boundary layer
are not able to compensate the suction peak, which results in a net increase in
drag. This may be due to a rear separation occurring at these conditions. Generally,
the test cases considered correspond to conditions prior to the transonic drag
rise, and (shock- induced) separation does not play a major role in the flow
development. Overall changes in the pressure drag are correspondingly small.

It was stated in the conclusions cited above (Pozniak 81/9) that the Reynolds number
sensitivity of afterbody drag becomes larger in the vicinity of and especially
above the drag rise boundary. This is illustrated in Figure 4.2.20 for the boattail
"6524", investigated in a comprehensive test program, involving several body
geometries, conducted by Blaha et al. [75/7). The displacement thickness - rather
than the Reynolds number - was here the parameter representing viscous conditions;
it was varied by changing body length and roughness. At Ma)=0.60, for a flow that
is mainly attached, the flow development is similar to the one described above:
increases in the suction peaks and improved pressure recoveries compensate to give
near-zero drag changes with decreasing displacement thickness. Figure 4.2.21 (c).
At the higher Mach numbers, say, M,*, > 0.9, significant areas of separation were
present, which tended to be reduced in extent as the initial displacement thickness
was raised. Figure 4.2.21 (b) . (Note, that the areas denoted "highly turbulent"
in that figure mark regions of intermittent backflow.) As shown by the pressure
distributions at Moo = 0.90, the thinner boundary layer resulted here in a remarkably
strong increase in the suction, indicating the presence of a supersonic pocket with
a terminating shock wave, the latter causing the boundary layer to separate early.
For the larger displacement thickness, the suction pressure barely exceeds the one
for MapO.60. The high suction values and the reduced pressure recoveries due to
separation are, of course, the reason for the large increase in pressure drag
observed for the thinner boundary layers, Figure 4.2.21 (a).

Figure 4.2.22 summarizes the influence of increasing displacement thickness on

boattail pressure drag for the body "6524" just discussed. Also shown is the
Reynolds number dependence for body "2524", characterized by larger pressure gra-
dients, at Ma)=0.90. Due to the larger pressure gradients, the response to viscous
changes is much more severe.
62

The results described in the preceding paragraphs were obtained on simple

forebody/afterbody configurations, and, in general, the afterbody drag increased
with increasing Reynolds number or decreasing displacement thickness. In the fol-
lowing, results for a more complex configuration shall briefly be considered:
Measurements of the installed boattail drag over a wide range of Reynolds numbers
have been made by NASA on boattails mounted behind underwing nacelles on an F-106
type delta aircraft during tunnel and flight tests [75/71. Examples of the measured
pressure drag at M ^ O . 6 0 and 0.90 are depicted in Figure 4.2.23, replotted from
Ref. 81/9, for the flight tests only, being consistent with the initial statement
to restrict the illustration of Reynolds number effects to results obtained with
one configuration and one test setup to avoid "secondary" effects of obscuring the
results.

The flight test data for the various boattail geometries consistently indicate a
reduction in boattail drag with increasing Reynolds number, although the exact rate
is uncertain due to the scatter of data. There are several comments to be made
concerning this Reynolds number dependence: One reason for the drag reduction with
Reynolds number might be due to the dominant effect of the Reynolds number on
separation, hence on pressure recovery, rather than on the suction peak, contrary
to the development observed for the bodies in Figures 4.2.21 and 4.2.22; a favorable
effect of an increase in Reynolds number on separation, determined by wool tufts,
was indicated during the flight tests. Another possible contribution to the
observed variation in drag may arise from the presence of a significant and
favorable interference effect from the closeness of the contoured part of the
afterbody to the wing coupled with a positive influence of increasing Reynolds
number on the wing flow. There is, however, further - indirect - evidence from other
flight tests [74/4] that boattail separation in relation to the suction peak
development may essentially determine this drag behavior: Little effect of Reynolds
number on drag was shown in cases where little movement of the separation point
occurred.

There are many parameters - besides the ones associated with viscous changes - that
may have an influence on afterbody flow, especially for complex geometries. For a
more detailed study of these, the reader is referred to Refs. 85/11 and 81/9 and
the many literature sources given there.

Base pressure is likely to be affected by the boundary layer condition upstream

of the base in a way similar to the afterbody pressure distribution, at least as
far as the expansion around the shoulder is concerned. There seems, therefore,
no need to address here viscous effects on base flow to any great extent. In
addition, a detailed study of the aerodynamics of base flow, including the influ-
ence of combustion, also containing major correlations of base pressure with vis-
cous and geometric parameters, is given in Ref. 76/8, to which the reader is
referred.

A characteristic example of the influence of the state of the boundary layer on

base pressure is depicted in Figure 4.2.24 for a cone-cylinder model with
L/D=5[76/8): In the laminar region, the base pressure coefficient decreases with
increasing Reynolds number, mainly due to the corresponding reduction in dis-
placement thickness at the corner of the base. In the transitional region and the
initial turbulent domain, the base pressure Increases with Reynolds number, prob-
ably due to boundary layer transition progressing upstream towards the apex of the
body with a corresponding increase in the boundary layer thickness. For fully
turbulent flow approaching the corner, the base pressure decreases again as the
Reynolds number is raised; this is illustrated in Figure 4.2.25 by a large body
of experimental data presented in a correlation according to Chapman 158/1).

As a further example of the Reynolds number effect on base flow, some results of
measurements on a space shuttle orbiter-type configuration at transonic speeds -
however, only over a very limited Reynolds number range - are depicted in Figure
4.2.26 [72/7]. The data, obtained for free and forced transition and at various
angles of incidence, all show the same trend: an increase in base-drag coefficient
- due to a drop in base pressure - with increasing Reynolds number similar to the
behavior observed on the simpler geometry discussed above.

4.2.6 Conclusion

The Reynolds number dependence of the flow about bodies, including afterbody and
base flow, has briefly been reviewed. This review showed examples characteristic
of Reynolds number effects for cylinders with circular and non-circular cross
sections in normal flow, sheared cylinders of quasi-infinite span, three-dimen-
sional fuselage- and missile-type bodies, and afterbodies as they exist on fighters
and nacelles as part of the exhaust/propulsion system.
 63

Two-dimensional bodies at subcritical speeds and three-dimensional bodies with

subcritical cross flow velocities exhibit essentially the same behavior with
increasing Reynolds number: At low Reynolds numbers the flow is laminar and laminar
separation occurs, drag and normal force, respectively, are relatively high. As
the Reynolds number is increased, the critical Reynolds number range is entered,
and drag and normal force drop rapidly due to transition occurring in the separated
shear layer with a subsequent turbulent reattachment and renewed separation at a
large circumferential angle. At transitional separation conditions, a steady lift
force may act on the cylinder in cross flow which manifests itself on the three-
dimensional body as a side force. These forces are caused by differences in the
boundary layer, and hence separation development, on opposite sides of the body.
The side force is nearly zero at the transition between the critical and the
supercritical Reynolds number ranges where the drag and the normal force for the
two-dimensional cylinder and the three-dimensional body, respectively, also have
a minimum. Note, that these side forces may occur at any angle of attack and that
they are, at a given incidence and wind tunnel environment, only dependent on
Reynolds number. A further increase in Reynolds number results in an increase in
drag and normal force up to conditions at which the movement of the transition point
on the body ceases; hereafter, a reversal in Reynolds number dependence should
develop; however, very little data is available in the hypercritical Reynolds
number range to confirm this.

A second mechanism generating side forces on a three-dimensional body is related

to hydrodynamic instability. These out-off-plane forces occur at angles of attack
roughly between 30° and 60°, dependent on body length, in all Reynolds number
regimes; they have, however, as indicated above, a minimum in the
critical/supercritical Reynolds number range.

With increasing Mach number, the classical Reynolds number dependence disappears
due to the development of supersonic regions with terminating shock waves strong
enough to separate even the turbulent boundary layer. At such conditions, the flow
development seems not very susceptible to viscous changes The freestream conditions
at which this occurs may, however, dependent on angle of attack, already be
supersonic since the freestream Mach number component normal to the body is the
dominant parameter.

Note, that single or distributed roughness in a form that fixes transition well
upstream of separation also strongly reduces the Reynolds number dependence, as
do bodies with corner radii that essentially fix separation itself.

Concerning afterbody flow, it was found that at subsonic Mach numbers in the absence
of major flow separation significant but compensating pressure changes occur such
that there is little effect of Reynolds number on afterbody pressure drag. However,
strong increases in drag with increasing Reynolds number were consistantly observed
at subsonic Mach numbers above the transonic drag rise and at supersonic speeds.
At the lower Mach numbers, the stronger expansion in the shoulder region of the
afterbody caused by reduced displacement thickness as Reynolds number increases
is compensated by the improved pressure recovery; at the higher Mach numbers,
reducing displacement thickness by increasing Reynolds number causes such a strong
expansion that shock waves develop separating the boundary layer in a way that the
pressure recovery can no longer compensate the expansion and a net increase in drag
results.

There is also evidence obtained on more complex configurations, e.g., boattails

mounted behind underwing nacelles, that the boattail pressure drag may well
decrease with increasing Reynolds number. One reason for this behavior seems to
be the positive effect of the Reynolds number on separation, hence on pressure
recovery, rather than on the suction peak development, as waa the case in the
examples cited above.

The opposing trends in the Reynolds number dependence of afterbody drag suggests
that further research is needed, especially for more complex configurations, to
determine conditions at which either the effect of the Reynolds number on the
suction peak or on separation is dominant, and, furthermore, to determine the
influence of the closeness of the afterbody to adjacent aircraft components. Con-
cerning the Reynolds number effect on the flow about two- and three-dimensional
bodies, more information is needed at Reynolds numbers in the supercritical and
hypercritical domains, especially at Mach numbers above critical freestream con-
ditions and for bodies with non-circular cross sections. Also of interest are
investigations related to the influence of the Reynolds number on the interference
between bodies and control surfaces which was not explicitly discussed here.
64

FLOW
REGIME - SUBCRITICAL-^CRITICAL - SUPERCRITICAL - -HYPERCRITICAL

' = 80* BI30'—~1IS' -•*-? —

-a80'-~l30' NARROW- BAND
RANDOM
QUASI
- W I D E - B A N D RANDOM-
PERIODIC
7? I

*-*3fes
?? S E E F I G U R E w.2.4

DRAG
COEF.

FIGURE 4 . 2 . 1 : Identification of flow

regimes on a c i r c u l a r c y l i n d e r [84/7]
FOR REFERENCES OF THE VARIOUS INVESTIGATORS
SEE [ 8 4 / 7 3
14
r v 'IESELBERGE R
12 1 1 I0 5 10s I0 7
C \ r S T x \CK 0.4
d (a).
1.0 *fe
•-POL HA Mi IS n^a_ ^
0.8 JONES ETAL.
•o
0.6
AIR- X FREON / 02 o
rDELANY &
ISORENSEN _, *>>*£ 9- »••
^ d .
0.4 0.1
L•W& ,M y ' jdxtiatD
0.2 0
0 . . . , i i
I I TT

W-
5 4 6 8 JQ6 2 6 B, 0 7 0.5
I0 J
RD Sr "br.
*
FIGURE 4.2.2: Drag coefficient for smooth
circular cylinders (replotted from 84/7)
a , b , c , d — i d e n t i f y i n d i v i d u a l stages of the s t a t e o f the f l o w
0.3
02
0.1
V-qcrW^MXmAA-^OO I Jor'jr
-rftP-
rf^

ICJ (al
-l l ' ' •

1.0 1 i IT)

Co (c)
0.5 V-ijV
1.0
0 PT(bar)
Sr i b l

0
OA
0.2

FXCN1 Ic)
o 2
• 6
0.5 |— * 15 -
A 30
a 57
i •••-*
^
LLC

o p . =2 bar _j i i i > i
15
to5
• p. -6bar
I0b Re r 10?
I0 l
5
FIGURE 4 . 2 . 3 : C h a r a c t e r i s t i c s of t h e flow
5xl0 5
about a c i r c u l a r c y l i n d e r [83/11]
I0 5 2 4 ffeD

SUBCRITICAL SUPERCRITICAL a. Lift fluctuations (r.m.s.)

b. Strouhal number of t h e l i f t fluc-
FIGURE 4.2.4: Characteristics of the flow tuations
about a circular cylinder: fine structure c. Drag c o e f f i c i e n t
in the critical Reynolds number range
[83/11J
a. Absolute value of the steady lift
force coefficient
b. Strouhal number
c. Drag force
 65

FOR REFERENCES OF THE VARIOUS INVESTIGATORS FOR REFERENCES OF THE VARIOUS INVESTIGATORS
SKW/71 SEE CBi/7]
u

ROSHKO

FIGURE 4.2.5: Effect of distributed FIGURE 4.2.6: Effect of Mach number on

roughness on the drag coefficient of cir- drag coefficients of smooth circular cyl-
cular cylinders (replotted from 84/7) inders (replotted from 84/7)

FOR REFERENCES OF THE VARIOUS INVESTIGATORS FOR REFERENCE "BURSNALL A N D LOFTIN"SEE 184/71
SEE 184/7 J
3.0 POLHAMUS
DELANYS.SORENSEN Al"
r/w CN
0021 \Rw
20
r **
15 0080 .
\

1.0
07671
Q24S
0i2O.
^
,
-i A\
05 T\J
0.370'

v_L,
i_

lO5 w 6 a l0 6 6 8
y

FIGURE 4.2.7: Influence of corner radius Reff

on drag coefficient of modified square
cylinder (replotted from 84/7)
FIGURE 4.2.8: Effect of angle of attack
(yaw) on normal force coefficient for a
circular cylinder (replotted from 84/7)

Symmetric/ Asymmetric/ Asymmetric/

Steady Steady Unsteady

• 1\1
rm*

il

a =80° f
IL
a. Typical conditions visualized by the hydrogen - b u b b l e technique [87/4J

FIGURE 4.2.9: Flow domains of slender

bodies
66

80 WAKE-LIKE FLOW
ocN

60
STEADY ASYMMETRIC
VORTEX FLOW
40

TURBULENT SEPARATION
20 , SYMMETRIC'^ ...
VORTEX FLOW -a, u —•—
VORTEX CALCULATION ACCORDING
FREE FLOW " j " T' ' V ,- TO CROSS FLOW THEORY
0 4 6 12 16 151/3}
BODY LENGTH x/D

(§) B o u n d a r i e s f o r a n ogive - c y l i n d e r b o d y 70 SO 90
C 79/3 J aC-J

FIGURE 4.2.10: Normal force coefficient

FIGURE 4.2.9: Concluded dependent on angle of attack and Reynolds
number [78/5]

12 a • 30'
1
Mm / M m c
10 178/089. ~

T,
1
8 n IO / t i l

o ^Ju^v- 7^ |
6 M 09 / n t

2
4
V ^
* &
wW^''\
0S/02S'
SEE FIGURE 4.2.11

LAMINAR \TRANSITlONAL\ FULLY TURBULENT ISEE FlG. 42 171

SUBCR | CR | SUPERCRITICAL IHYPERCR.
ISEE FIG 4 2 M l
0
10' 2
a. a = 30'
Kfi We,,
wT

FIGURE 4.2.11: Effect of angle of attack

on normal force coefficient for a slender FIGURE 4.2.12: Effect of Mach number on
body. Mro =0.50 normal force coefficient for an ogive-
Data of Hartmann [78/51 and Foley (o=30° cylinder body [78/5]
only [71/10])
a Rt„x 10 s M o . Windtunnel
15 • kt
2.5
CV o - DFVLR
vr
so I <0.2 Goltingen
bcr 7.5 NWG
25

20

15

10

0L
IO 7 R o l l angle dependence of t o t a l side force c o e f f i c i e n t s

FIGURE 4.2.12: Concluded FIGURE 4.2.13: Roll angle dependence of

total side force coefficients [87/4]
 67

1524 mm
(6 in) 12
OGIVE CYLINDER OFREF8S/I0
diam.
'Cyl, I x / D . IS, a . 4 5 ' I
10 wSlDE FORCE -

(C N ) 6

0 1 2 3 4 5 6 7 7.5 diam 6
i — i — i i i i i i i i i _ _i i i

L
0

FIGURE 4.2.15: Variation of overall maxi-

mum side force with Reynolds number,
H„<0.24

10 20 30 40 50 60 70 80 90
ANGLE OF ATTACK, a [ ° ] 5
4 FT FT FTFTFTFIFTFTFTFTFT/f'T
3 FT FT FTFTFTFTFTFTFT7FT7T T T
(FULLY T U R B U L E N T ) , '
FIGURE 4 . 2 . 1 4 : V a r i a t i o n of maximum o v e r - 2 FT FT FTFTFTFTET/TTT T T T
all s i d e force with angle of attack
[82/13] FT FT FTFTFTT T T T T T T
g 1 y ? 1 TRANSITIONAL)
o =30°, x / D =6
s FT FT , F T T T T T T T T T T
* .5 T T T
Re 0 / 10 s
T T T T T T T T T T T T

m m 08 TO 4.0 .2 L L L L L L L LL
I LAMINAR 1
1 1 1 1 1 i _
0 10 20 30 40 50 60 70 80 9 0 '
ANGLE OF ATTACK, ot [ ' ]
LAMINAR
FULLY TURBULENT
FIGURE 4.2.17: Classification of Reynolds
TRANSITIONAL number/angle of attack dependence into
three main flow regimes [82/13]
0 30 60 90 120 150 180
AZIMUTH ANGLE, e [ ° ]

FIGURE 4.2.16: Pressure distribution

around inclined cylinder at different
Reynolds numbers [82/13]

16
C
'N
1.4

1.2

1.0

0.6

0.6

0.4
6 8 W 7

FIGURE 4.2.18: Effect of corner radius on

normal force coefficient for a shuttle
orbiter-type fuselage (84/71 [71/11]
68

FIGURE 4 . 2 . 1 9 : Reynolds number e f f e c t on

afterbody p r e s s u r e drag for two forebody
l e n g t h s (76/6)

-\ AFTERBODY: CIRCULAR
.^ARC/CONE
<33i FOREBODY

040

0.20

OPEN SYMBOLS: L / d m = 76
0.20 Mm=0.60 HALF-FILLED SYMBOLS: L / d m * 8
O-
9 JO —°—-n? *>
-0.40

-0.60

FIGURE 4.2.20: Reynolds number influence

on minimum and maximum afterbody pressures
[76/6]
 69

O.IO 0l2r
\
\ BODY 2524
C
DAp \
\ , /
1
\ —T^ Ju
Tl
0.08
\ u_ N
v BODY 6525
N
v. BODY 2526
N M^-0.90^^^
10.005 ^
0.04

M „ = 060 BODY 6524

0 02 0.4 06
INCREASING Re - sr/d

FIGURE 4.2.22: Effect of displacement

thickness and body geometry on boattail
drag (76/7)

REVERSED FLOW

FIGURE 4.2.21: Influence of Mach number

and displacement thickness on boattail
drag [76/7]

R .RADIUS OF BOATTAIL FLIGHT NOZZLE

SHOULDER TESTS
Rc a RADIUS OF CIRCULAR 100
ARC OF SAME PRO- R/Rw 0 DESCRIPTION
JECTED AREA AND a 25 24 EXTENDED
TERMINAL B0TTAIL a 65 24
ANGLE 0 25 24
wf 25 IS
CIRCULAR ARC-CONIC

FLIGHT FLIGHT
S 0.10 y
y
t 008
kl
O
2 0.06
I
Q 0.04
• * -

9
§ 002
B ov
\ -0.01
10 20 30 40 50 60 0 10 20 30 40 50 60 70
6
REYNOLDS NUMBER,ReLxlQ- REYNOLDS NUMBER,Re L xW 6
(a) MACH NUMBER M „ = 0.6 lb) MACH NUMBER M _ = 09

FIGURE 4.2.23: Reynolds number effect on

pressure drag of circular arc boattails
(75/7) (replotted from 81/9)
70

" Prime " denotes reference condition at corner of base

-Q20
-020
Cpb

-0.15

-CIO

•0.05 -0.08
8x10" 0.16 024 032 0.40 048
REYNOLDS NUMBER, Re, L/[DIRe)"5]

FIGURE 4.2.24: Reynolds number dependence FIGURE 4.2.25: Chapman correlation of base
of base pressure coefficient for cone- pressure coefficient [58/1](replotted
cylinder body [76/8] from 76/8)

K, 51 = caborundum grit at 5% local chord

5e = elevon deflection

REYNOLDS NUMBER. ReL

FIGURE 4.2.26: Dependence of base drag on

Reynolds number for a space shuttle orbi-
ter-type configuration [72/7]
 71

5. CONCLUDING REMARKS

Reynolds number effects in transonic flow were critically reviewed. In this review,
the following geometries were considered: Airfoils and high aspect ratio wings
typical of transport aircraft configurations, fighter-type low aspect ratio delta
wings, two- and three-dimensional bodies characteristic of missiles and combat
aircraft fuselages, and afterbodies. Furthermore included in this review were
pseudo-Reynolds number effects which may arise, for instance, due to the influence
of the Reynolds number on the wind tunnel environment, and, as an introduction to
the present topic, a brief review of the "history" of Reynolds number effects
associated with transonic flow.

It was originally envisaged to also present a section on fundamental Reynolds

number effects covering the influence of the Reynolds number on the boundary layer
development, including transition, and on basic viscous / inviscid interactions
such as shock boundary layer interaction and trailing edge separation; however,
since this subject is comprehensively discussed in the report of the AGARD Working
Group 09 "Boundary Layer Simulation and Control in Wind Tunnels", which will be
published concurrent with the present AGARDograph, it was decided to omit that
discussion here.

Systematic study of Reynolds number (scale) effects, which commenced some time
after second world war when transonic flight itself became a matter of thorough
scientific study and which obtained new impetus after Loving of NASA published his
report on the large differences between C-141 wind tunnel and flight results in
1966, frequently reveal "anomalieb" which can be traced to the wind tunnel envi-
ronment and measuring techniques and their response to Reynolds number changes.
Such "anomalies", sometimes labeled as "unit"-Reynolds number effects, are better
described as "pseudo"-Reynolds number effects with the Reynolds number influencing
the wind tunnel environment which, in turn, affects the flow about the model.
Factors which have the potential of introducing pseudo-Reynolds number effects
include wall interference, tunnel Mach number calibration, noise, turbulence,
humidity, non-uniform flow (flow angle, temperature, pressure gradients), flow
contamination, side wall effects in two-dimensional tests, model deformation and
transition fixing. Here are examples of the manifestation of pseudo-Reynolds number
effects: Changing Reynolds number was found to change the characteristics of par-
tially open wind tunnel walls, hence the magnitude of wall interference for a given
model, and therewith, for instance, the effective freestream conditions. Noise and
turbulence may affect the transition location; noise and turbulence tend to
increase due to an increase in wind tunnel power, the latter required to raise
Reynolds number. Transition location may be influenced directly by an increase in
Reynolds number and, superimposed, via the change in turbulence and noise level.
In the same way - although the influence seems to be small - the effect of noise
and turbulence intensity on the turbulent boundary layer development may induce a
pseudo-Reynolds number effect. Humidity alters the pressure distribution including
shock location; humidity effects change with total pressure, the latter utilized
to alter Reynolds number. Wind tunnel contamination by particles small enough to
be airborne and simultaneously hard enough to cause roughening of model surfaces
in the stagnation region may contribute to pseudo-Reynolds number effects in a
different way: Roughness in the stagnation region can cause premature transition
thus obscuring the "true" influence of Reynolds number on transition point move-
ment.

Tunnel calibration must be performed at all total pressure and temperature condi-
tions that are expected during the actual model tests since the relation between
the average test section Mach number and the reference Mach number may be Reynolds
number dependent. Also considered in that regard should be flow non-uniformities.
I.e., spatial gradients of pressure, temperature and angle of attack, for instance,
and their Reynolds number dependence. Especially in two-dimensional flow, the
influence of the Reynolds number on the side wall boundary layer development may
cause spurious effects on the flow about the model which may falsely be interpreted
as real Reynolds number effects; proper treatment of the side wall boundary layer
or a sufficiently large aspect ratio ( £ 2.5) is required to avoid these influences.
Undesireable thermal non-equilibrium, i.e., undesireable deviations of the model
wall temperature from adiabatic wall conditions, may occur in short duration wind
tunnels, for instance, due to the throttling process in the control valve, or in
continuous (cryogenic) wind tunnels when Reynolds number is varied by temperature
changes and time is not allotted for the model to acquire adiabatic wall temper-
ature. Thermal non-equilibrium has a considerable influence on the boundary layer
development and hence on aerodynamic parameters sensitive to viscous changes. Model
deflections may easily introduce pseudo-Reynolds number effects if the deflections
72

are a result of changes in the model load associated with increasing total pressure;
model deflections must be accounted for. Finally, transition fixing may be accom-
panied by pseudo-Reynolds number effects if the height of the roughness element,
utilized to force transition , is not adjusted to the freestream conditions,
including Reynolds number itself, and over- or underfixing occurs resulting in
either too thick a boundary layer or transition taking place downstream of the
tripping device.

Considering "true" Reynolds number effects on transport-type airfoils and wings,

the available information suggests that two phenomena are of primary importance,
especially when regarding the Reynolds number difference between wind tunnel and
flight: transition point movement, itself and in conjunction with laminar versus
turbulent shock boundary layer interaction, and flow separation, either locally
near the trailing edge or of large scale between shock and trailing edge on the
upper airfoil or wing surface. When transition is fixed artificially, Reynolds
number effects are mainly related to flow separation; large variations in the
pressure distribution and correspondingly in the force and moment coefficients were
observed with Reynolds number when separation extended from the foot of the shock
to the trailing edge, and especially, when the flow changed from an attached to a
separated condition as Reynolds number is decreased (or vice versa). For conditions
with attached or almost attached flow, Reynolds number effects appear to be smal-
ler, though certainly not insignificant: A limited region of trailing edge sepa-
ration, which frequently exists on rear loaded wing sections, tends to increase
the Reynolds number effect and some aerodynamic characteristics, notably drag and
pitching moment, are very sensitive to these small variations in the separated
region with Reynolds number. Note, that the boundary between attached and separated
flow as a function of Mach number and angle of attack varies itself with Reynolds
number in a systematic way. If conditions are such that the separation location
does no longer change with Reynolds number, its effect on the overall flow devel-
opment is also small.

The second phenomenon to be considered in conjunction with observed Reynolds number

effects is the transition point movement occurring due to a change in Reynolds
number if transition is left free. Transition point movement has an effect on the
flow development (e.g., separation) similar to a "pure" Reynolds number change.
However, the effect of transition movement is much more pronounced since the change
in the initial boundary layer parameters, which determine the overall flow devel-
opment, is much greater than for the associated Reynolds number change with tran-
sition fixed. Concerning low Reynolds number wind tunnel tests and the scaling of
results to full-scale conditions, uncontrolled transition point movements should
be avoided, e.g., by adequately fixing transition (see the report of the AGARD WG
09), since misleading variations in the aerodynamic characteristics may occur if
the transition point can move around freely.

Available data for low aspect ratio fighter-type configurations depicting Reynolds
number effects are sparse. No systematic investigations, other than few for
sharp-edge delta wings, have been conducted in a sufficiently wide Reynolds number
range to allow firm conclusions to be made. The data which are available, however,
suggest for most of the flight regime of interest that Reynolds number effects are
small to nonexistent, the boundary layer state being controlled by leading edge
contamination or cross flow instability.

The Reynolds number sensitivity of bodies, typical of missiles and fighter aircraft
fuselages, is also strongly related to separation, and one can distinguish essen-
tially four (classical) Reynolds number domains: Subcritical, in which the flow
is laminar and laminar separation occurs without reattachment, critical, in which
separation is transitional with turbulent reattachment and a subsequent far-aft
turbulent separation, supercritical, in which separation is turbulent and the
transition point moves upstream to the vicinity of the stagnation point with
increasing Reynolds number, and hypercritical, in which the flow, including sepa-
ration, is fully turbulent. The flow is not very sensitive to viscous changes in
the eubcritical (laminar) and the hypercritical (fully turbulent) Reynolds number
domains since the separation location is essentially fixed. Especially in the
critical Reynolds number range, large changes in the aerodynamic forces - large
decreases in drag and normal force for the cylinder in cross flow and the three-
dimensional body at angle of attack, respectively - occur due to the sensitivity
of transitional separation to Reynolds number. Note, that for a three-dimensional
body the boundary between the various Reynolds number regimes is angle of attack
dependent.

The Reynolds number sensitivity in the critical and supercritical Reynolds number
domains disappears with increasing Mach number due to the development of supersonic
regions with terminating shock waves strong enough to separate even the turbulent
boundary layer. The freestream velocity at which this happens is, for a three-di-
 73

mensional body, again dependent on incidence. Distributed or single roughness in

a form that fixes transition well upstream of separation also strongly reduces the
Reynolds number sensitivity, as do bodies with corner radii that essentially fix
separation.

At subsonic freestream Mach numbers and in the absence of major flow separation,
significant but compensating pressure changes occur on afterbodies such that there
is little effect on afterbody pressure drag as Reynolds number is varied. Signif-
icant increases in drag with increasing Reynolds number were consistently observed
for subsonic Mach numbers above the transonic drag rise and at supersonic Mach
numbers. However, also observed were flow developments, e.g., on boattails mounted
behind underwing nacelles, where the boattail drag decreased with increasing
Reynolds number. One reason for this behavior may be the positive effect of Rey-
nolds number on separation, hence on pressure recovery, rather than on the value
of the suction peak, as was the case for the configurations for which an increase
in pressure drag with Reynolds number was observed.

The sensitivity of the flow development about transonic flight vehicles to Reynolds
number changes illustrated in the present AGARDograph and the lack of sufficient
high Reynolds number facilities, indicate the necessity to provide generally
accepted means and ways to scale lower-than-flight Reynolds number results to
full-scale conditions. This topic was not addressed to any extent since it was the
task of the AGARD Working Group 09, "Boundary Layer Simulation and Control in Wind
Tunnels", to provide a methodology for transonic wind tunnel testing and the
extrapolation of low Reynolds number wind tunnel results to flight conditions. The
results of the deliberations of that Working Group are published as an AGARD-report
which contains, besides the said methodology, also a review of the present state-
of-the-art in transonic wind tunnel testing, especially with regard to boundary
layer simulation as well as a critical assessment of the physical background of
viscous simulation. The present AGARDograph should be considered as complementary
to the WG 09-report.
74
 75

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76

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70/2 E. G. "Comparison of Wind Tunnel and Flight Test Techniques for

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71/1 E. Stanewsky "Studies of Separation and Reattachment in Transonic Flow."

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71/2 J, E, Green "Some Aspects of Viscous-inviscid Interactions at Tran-

sonic Speeds and Their Dependence on Reynolds Number."
AGARD CP 83, April 1971, Paper 2.

71/3 A. B. Haines Possibilities for Scale Effects on Swept Wings at High

Subsonic Speeds: Recent Evidence from Pressure Plotting
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71/4 J. F. Cahill "Simulation of Full-Scale Flight Aerodynamic Charac-

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AGARD CP 83, April 1971, Paper 20.

71/5 M. G. Hall "Scale Effects in Flows over Swept Wings." AGARD CP 83-
71, 1971.
 77

71/6 J. E. Green "Some Aspects of Viscous-inviscid Interactions at Tran-

sonic Speeds and Their Dependence on Reynolds Number."
AGARD CP 83-71, 1971.
71/7 E. Thomas "A Method for Calculating the Transonic Buffet Boundary
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71/8 J. Osborne "A Type of Stall with Leading-Edge Transonic Flow and
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71/9 E. J. Salzman "A Comparison of Some Aerodynamic Drag Factors as Deter-
D. R. Bellman mined in Full-Scale Flight with Wind Tunnel and Theo-
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71/10 J. E. Foley "Results of a Study of Mach Number and Reynolds Number
on the Cross Flow Drag Characteristics of Ogive-Cylinder
and Ogive-Cylinder-Frustrum-Cylinders at Angles of Attack
to 30 Degrees." NASA CR-61356, October 1971.
71/11 J. J. Brownson "Static Stability Characteristics of Manned Spacecraft
R. E. Graham Center Straight-Wing Space Shuttle Orbiter: Effect of
D. Banducci Reynolds Number and Body Corner Radius at M < 0.5."
NASA TM X-62, 054, 1971.
72/1 LaWs Group "The Need for Large Wind Tunnels in Europe. AGARD AR
No. 60, 1972.
72/2 M, G. Hall "Difficulties in Predicting Boundary-Layer Transition on
D. A. Treadgold Swept Wings." RAE Technical Memorandum Aero 1465, ARC
35160, 1972.
72/3 Ms . G. C. Browne "A Comparison of Wing Pressure Distributions Measured in
T. E. B. Bateman Flight and on a Wind Tunnel Model of the Super VC-10."
H. Pavitt Aeronautical Research Council R & M 3707, 1972,
A. B. Haines
72/4 "The High Reynolds Number Tunnel — A National Need —
Description and Justification." AEDC, February 1972.
72/5 J. P. Hartzuiker "Scale Effects at Transonic Speeds; Basic Considera-
tions." Notes for AGARD/VKI Short Course, "Transonic
Aerodynamic Testing," 1972.

72/6 J. W. Green "Heat Transfer as a Source of Spurious Scale Effects in

D. J. Weeks Subsonic and Transonic Wind Tunnels." AGARD Laws Paper
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72/7 E. Stanewsky "Effect of Reynolds Number and Transition on the Aero-
dynamic Coefficients of the MSC 040 A Orbiter at Tran-
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73/1 J. H. Paterson "A Survey of Drag Prediction Techniques Applicable to
D. G. McWilkenson Subsonic and Transonic Aircraft Design." AGARD CP-124,
W. T. Blackerby April 1973.
73/2 J. E. Green "On the Influence of Free-Stream Turbulence on a Turbu-
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74/1 LaWs Group "The Need for a Large Transonic Tunnel in Europe." AGARD
AR No. 70, 1974.
74/2 F. Aulehla "Fore- and Afterbody Flow-Field Interaction with Consid-
G. Besigk erations of Reynolds Number Effects." AGARD CP 150,
Paper 12, September 1974.

74/3 - S.O,T.H. Han "On the Effect of Variation of Transition Position and
J. P. Hartzuiker of Vortex Generators on the High-Speed Stall Charact-
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TR 74008 0, 1974.

74/4 R. Charaberlin "Flight Reynolds Number Effects on a Contoured Boattail

Nozzle at Subsonic Speeds." NASA TMjX-3053.

75/1 L. L. Galigher "Description of the AGARD Nozzle Afterbody Experiments

P. M. Jackson Conducted by the Arnold Engineering Development Center."
C. E. Robinson AGARD-AG-208, October 1975.

75/2 W. T, Blackerby "High Reynolds Number Tests of a C141A Aircraft Semi-Span

J. F. Cahill Model to Investigate Shock-Induced Separation," NASA CR
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78

75/3 K. P. Burdges "High Reynolds Number Test of a NACA 65i-213, a = 0.5

J. A. Blackwell, Jr Airfoil at Transonic Speeds." NASA CR-2499 (1975).
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75/4 H. Yoshihara "The Many Facets of 3D Transonic Shock-Induced Separa-
Da Zonars tion." AGARD CP 168 Paper 42 (1975).
75/5 S. Teige "Swedish Experience on Correlations of Flight Results
G. Strang with Ground Test Predictions." Paper 21, AGARD-CP-187,
K. E. Stake June 1975.

75/6 H. Esch "The Influence of Reynolds Number on the Normal Force

Characteristics of Slender Cylindrical Bodies." ESA TT-
170, 1975.
75/7 F. A. Wilcox "Reynolds Number Effects on Boattail Drag of Exhaust
R. Chamberlin Nozzles from Wind Tunnel and Flight Tests." AGARD-CP-
150, Paper 21, 1975.
76/1 D. J. Weeks "An Investigation of Scale Effects on the Transonic Flow
over Swept Wings." Part I, Aeronautical Research Council
R 6 M 3842, 1976; Part 2, Aeronautical Research Council
R & M 3842, 1977.
76/2 A. B. Haines "Further Evidence and Thoughts on Scale Effects at High
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43.

76/3 "Workshop on High Reynolds Number Research." NASA CP-

2009 (1976).
76/4 J. A. Blackwell, Jr, "Scale Effects on Advanced Aircraft Wing Designs." Pro-
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76/5 J. L. Jacocks "An Investigation of the Aerodynamic Characteristics of
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Tunnels." Ph.D. Dissertation, University of Tennessee,
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76/6 D. E. Reubush "An Experimental and Analytical Investigation of the

L. E. Putnam Effect on Isolated Boattail Drag of Varying Reynolds
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76/7 B. J. Blaha "Boundary Layer Thickness Effect on Boattail Drag." NASA
R. Chamberlin TM-X-73443, 1976.
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76/8 S. N. B. Murthy "Base Flow Phenomena with and without Injection: Experi-
J. R. Osborn mental Results, Theories and Bibliography." Progress in
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77/1 J. Szodruch "Leeseiten-Stromung bei Schlanken Delta flugeln endlichier
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77/2 G. F. Butler "Preliminary Investigation of a Technique for Prediction
D. C. Conner Loads in Flight from Wind Tunnel Measurements on Models
of Conventional," Construction AGARD CP 204, February
1977.

77/3 J. F. W. Crane "The Effect of Base Cavities on the Drag of an

Axisymmetric Afterbody with a Sonic Nozzle between M<"
= 0.8 and 1.3." RAE TR 77 144, 1977.

78/1 J. A. Blackwell, Jr. "Scale Effects on Supercritical Airfoils." Proceedings of

the XI Congress of ICAS, Lisbon, Portugal, 10-16 September
1978, Vol, 1, Paper No. B3, pp. 370-383.

78/2 J. A. van Egmond "The Design and Aerodynamic Characteristics of an 18%

D. Rozendal Thick Shock-Free Airfoil (NLR 7501)." NLR MP 78016 U,
1978.

78/3 J. W. Kooi "Influence of Free-Stream Mach Number on Transonic

Shock-Wave Boundary-Layer Interaction." NLR MP 78013 U,
1978.

78/4 F. M. Jackson "Calibration of the AEDC-PWT 16-FT Transonic Tunnel with

the Propulsion Test Section at Various Reynolds Num-
bers." AEDC-TR-77-121 (AD-A057877), August 1978.
 79

78/5 K. Hartmann "Uber den Einfluss der Reynolds Zahl auf die
Nonmalkrafte Schlanker Flugkorpernimpfe." Z. Flugwiss,
Weltraum-forscg. 2, 1978, Heft 1, p. 23.
79/1 J. F. Cahill "Correlation of Data Related to Shock Induced Trailing
D. C. Conner Edge Separation and Extrapolation to Flight Reynolds
Number." NASA CR 3178, 1979.
79/2 A. B. Haines "Review of Post-1974 Evidence on Scale Effects at High
Subsonic Speeds." ARA Memo No. 218, 1979.
79/3 G. T. Chapman "The Aerodynamics of Bodies of Revolution of Angles of
E. R. Keener Attack to 90 deg." AIAA Paper 79-23, 1979.
80/1 L. W. McKinney "High Reynolds Number Research - 1980." NASA Conference
D. D. Baals Publication 2183.
80/2 0. M. Pozniak "A Review of the Effect of Reynolds Number on Afterbody
Drag." ARA Report 56, May 1980.
80/3 Y. Brocard "Etude des caracteristiques de 1'ecouleraent tourbillonnaire
F. Manie sur une aile en fleche." L'Aeronautique et L'Astronau-
tique No. 82, 1980-3.
80/4 N. S. Dougherty "Boundary-Layer Transition on a 10-deg Cone: Wind Tunnel/
D. F. Fisher Flight Data Correlation." AIAA Paper No. 80-0154, AIAA
18th Aerospace Sciences Meeting, January 14-16, 1980.
80/5 J.P.F. Lindhout "Comparison of Boundary-Layer Calculations of the Root
B. van den Berg Section of a Wing." The September 1975 Amsterdam Test
A. Elsenaar Case - NLR MP 80028 U.
80/6 D. J. Peake "Three-Dimensional Interactions and Vortical Flows with
M. Tobak Emphasis on High Speeds." NASA TM 81169 (1980).
80/7 "Computation of Viscous-inviscid Interactions. AGARD
CP-291, 1980.
80/8 T. L. Kennedy "An Evaluation of Wind Tunnel Test Techniques for
Aircraft Nozzle Afterbody Testing at Transonic Mach
Numbers." AEDC-TR-80-8 (AD-A091775), November 1980,
80/9 G. E. Erickson "Flow Studies of Slender Wing Vortices." AIAA Paper No.
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July 14-16, 1980.
81/1 E. Stanewsky "Wechselwirkung Zwischen Aussenstromung und Grenzschicht
an Transsonischen Profilen." Dissertation, Techniche
Universitat Berlin, May 1981.

81/2 L. C. Squire "Experimental Work on the Aerodynamics of Integrated

Slender Wings for Supersonic Flight." Progress in
Aerospace Science, Vol. 20, pp. 1-96. Pergamon Press Lt.
1981.

81/3 T. S. Beddoes "A Note on Shock Reversal at High Lift." Westland Heli-
copter Limited, Research Paper 633, 1981.

81/4 R. C. Lock "Survey of Techniques for Estimating Viscous Effects in

M. C. P. Firmin External Aerodynamics." RAE Tech Memo Aero 1900, 1981.

81/5 S. J. Kline "The 1980-1981 AFOSR-HTTM-Stanford Conference on Complex

(Editor) Turbulent Flows." Stanford University, 1981.

81/6 D. G. Mabey "Some Remarks on Buffeting in Unsteady Airloads and

Aeroelastic Problems in Separated and Transonic Flow."
VKI Lecture Series 1981-4.

81/7 R. P. Boyden "Preliminary Results of Buffet Tests in a Cryogenic

Tunnel." NASA TM 81923, July 1981.

81/8 L. E. Ericsson "Review of Vortex-Induced Asymmetric Loads - Part I."

J. P. Reding Z. Fluqwiss, Weltraumforsch, 5, 1981, Heft 3, p. 162.

81/9 0. M, Pozniak "A Review of the Effect of Reynolds Number on Afterbody

Drag." AGARD-CP-301, Aerodynamics of Power Plant Instal-
lation, September 1981, pp. 16-1 to 16-21.

82/1 E. J. Ray "A Review of Reynolds Number Studies Conducted in the

Langley 0.3-ra Transonic Cryogenic Tunnel." AIAA-82-0941.
80

82/2 A. B, Haines "Notes on Transition Fixing and the Interpretation of

Drag Data with Different Transition Fixes." ARA Memo
243, November 1982.
82/3 David Nixon "Transonic Aerodynamic Progress in Astronautics and Aero-
(Editor) nautics." Vol. 81, 1982.
82/4 A. Elsenaar "A Report of a GARTEur Action Group "Two-Dimensional Tran-
E. Stanewsky sonic Testing Methods." AGARD CP No. 335, Paper 5, 1982.
82/5 J. C. le Bal "Viscid-Inviscid Coupling Calculations for Two- and
Three-Dimensional Flows." VKF Lecture Series 1982-4,
1982.

82-6 Y. Y, Chan "Wall Boundary-Layer Effects in Transonic Wind Tunnels."

AGAHD CP-335, September 1982.
82/7 N. S, Dougherty, Jr. "Boundary-Layer Transition on a Slender Cone in Wind
D. P. Fisher Tunnels and Flight for Indications of Plow Quality."
AEDC-TR-81-26 (AD-A111328), February 1982.
82/8 S. Raghunathan "Free-Stream Turbulence and Attached Subsonic Turbulent
R. J. W. McAdams Boundary Layer," AIAA Paper 82-0029, January 1982.
82/9 F. Aulehla "Reynolds Number Effect on Transonic Shock Location, In
A. Eberle Wall Interference in Wind Tunnels." AGARD CP-335,
September 1982.

82/10 R. W. Barnwell "Similarity Rules for Effects of Sidewell Boundary Layer

W. G. Sewall in Two-Dimensional Wind Tunnels." in "Wall Interference
in Wind Tunnels," AGARD CP-335, September 1982.
82/11 D. Jacob "A Comparison between Wind Tunnel and Flight Results for
D. Welte Aerodynamic Performance." AGARD CP-339, October 1982.
H. Wunnenberg
82/12 C. J. Spurlin "Comparison of Flight and Wind Tunnel Data on the Dornier
H. Lueck TST Configuration." AIAA Paper No. 84-0612-CP, March
1984.

82/13 P. J. Lament "Pressures Around an Inclined Ogive Cylinder with

Laminar, Transitional, or Turbulent Separation." AIAA
Journal, Vol. 20, No. 11, November 1982, p. 1492.
83/1 M. M. S. Khan "New Considerations on Scale Extrapolation of Wing Pres-
J. F. Cahill sure Distributions Affected by Transonic Shock Induced
Separation. NASA Contractor Report 166426, March 1983.
83/2 A. E. P. Veldman "A Numerical View on Strong Viscous-inviscid Inter-
action." NLR MP 83049 U, 1983.
83/3 A. Elsenaar "Experience with Transition Fixation in the High-Speed
Regime at NLR." NLR Memorandum AC-83-039 U, 1983.

83/4 S. J. Boersen "Half-Model Testing in the NLR High-Sped Wind Tunnel

A. Elsenaar HST: Its Technique and Application." AGARD CPP 348,
August 1983.

83/5 S. Raghunathan "Free-Stream Turbulence and Transonic Flow overa 'Bump'

R. J. W. McAdams Model" AIAA J o u r n a l , Vol. 21, No. 3, March 1983, pp. 467-
469.
83/6 S. Raghunathan "Boundary-Layer and Turbulence Intensity Measurements in
R. J. W. McAdams a Shock Wave/Boundary-Layer Interaction." AIAA J o u r n a l ,
Vol. 21, No. 9, September 1983, pp. 1349-1350.

83/7 E. Stanewsky "Interaction between the Outer Inviscid Flow and the
Boundary-Layer on Transonic Air Foils." Z. Flugwiss.,
Weltraumforsch 7 (1983), Heft 4, pp. 242-252.

83/8 M. Mokry "Two-Dimensional Wind Tunnel Wall Interference." AGARD

Y. Y. Chan AG-281, November 1983.
D. J. Jones
83/9 A, D. Young "Aerodynamics of Vortical Type Flows in Three Dimen-
(Editor) sions." AGARD CP 342, July 1983.

83/10 GARTEur ACTION GROUP "Two-Dimensional Transonic Testing Methods." GARTEur/TP-

AD(AG-02) 011, also NLR TR 83086U, 1983.

83/11 G. Schewe "On the Force Fluctuations Acting on a Circular Cylinder

in Crossflow from Subcritical up to Transcritical
Reynolds Numbers." Journal of Fluid Mechanics, Vol. 133, p.
265, 1983.
 81

83/12 W. Schneider "Experimental Investigation of Bodies with Non-Circular

Cross Section in Compressible Flow." AGARD-CP-336,
February 1983, pp. 1-19 to 19-19.
83/13 D. I. A. Poll "On the Generation and Subsequent Development of Spiral
Vortex Flow Over a Swept-back Wing." AGARD-CP-342, July
1983, pp. 6-1 to 6-14.
84/1 R. C. Luh "A Numerical Procedure to Predict the Effects of Reynolds
M. D. Mack Number and Trip Strip Variation on Three-Dimensional
Wing Lift and Pitching Moment." AIAA-84-0255, 1984.
84/2 "Grenzschtsteuerung durch transitionfixierung." DFVLR-
Mitt 84-17.
84/3 S. J. Boersen "Half-Model Testing in the NLR High-Speed Wind Tunnel
A. Elsenaar HST: Its Technique and Application." AGARD CP 348, Paper
23, 1984.
84/4 G. Meauze "Viscous-inviscid Flow Interaction Methods/Shock
Boundary-Layer Interaction. ONERA TP 1984-114, 1984.
84/5 D. A. Johnson "A New Turbulence Closure Model for Boundary-Layer Flows
L. S. King with Strong Adverse Pressure Gradients and Separation."
AIAA-84-9175, 1984.
84/6 W. Burgsraueller "Transitionfixierung im Hochgeschwindigkeits-bereich bei
der Fleugelentwicklung fur zivile Flugzeuge, im
Grenzschichtsteuerung durch Transitionsfixierung." DFVLR
Mitteilung 84-17, October 1984.
84/7 E, C. Polhamus "A Review of Some Reynolds Number Effects Related to
Bodies at High Angles of Attack." NASA Contract Report
3809, 1984.
85/1 J. B. Doc "The T2 Cryogenic Induction Tunnel in Toulouse."
AGARD/FDP Special Course "Cryogenic Technology for Wind
Tunnel Testing," V.K.I., 22-26 April 1985.
85/2 A. G. T. Cross Report of the Review Committee of AGARD Working Group
09, Section 2.2.4, "Complex Configurations" (1985) to be
published as part of the final working group report.
85/3 P. R. Ashill "An Integral Methods for Calculating the Effects on Tur-
P. D. Smith bulent Boundary Layer Development of Sweep and Taper."
Aeronautical Journal 882, February 1985.
85/4 J. L. Fulker "A Model of the Flow over Swept Wings with Shock-Induced
P. R. Ashill Separation." Paper presented at the IUTAM Symposium,
"Turbulent Shear-Layer/Shock-Wave Interactions," Septem-
ber 1985, Palaiseau, France.

85/5 G. R. Inger "Supersonic Viscous-inviscid Interaction of a Swept Com-

pression Ramp with a Turbulent Boundary Layer." IUTAM
Symposium.
85/6 A. B. Haines Private Communication.

85/7 A. Mignosi "Fundamental Reflections on Cryogenic Testing." AGARD/

FDP Special Source - Cryogenic Technology for Wind
Tunnel Testing, V.K.I., 22-26 April 1985.

85/8 C. E. Robinson "Estimating Water Vapor Condensation Effects for Tran-

R. C. Bauer sonic and Supersonic Flow Fields." AIAA Paper 85-5020,
R. H. Nichols October 14-16, 1985.

85/9 S. Raghunathan "Relative Effects of Reynolds Number and Free-Stream

R. J. W. McAdams Turbulence in Transonic Flow." AIAA J o u r n a l , Vol. 23, No.
4, April 1985.

85/10 R. Clock "Prediction of Viscous and Wave Drag at High Subsonic

Speeds by Viscous-inviscid Interaction Techniques."
AGARD VKI Special Course, "Aircraft Drag Prediction and
Reduction," May 1985.

85/11 R. E. Melnik "An Improved Inviscid/Inviscid Interaction Procedure for

R. R. Chow Transonic Flows over Airfoils." NASA CR-3805, 1985.
H. R. Mead
A. Jameson

85/12 J. Velery "Turbulent Shock Wave Boundary Layer Interaction, AGARD-

J. G. Marvin ograph No. 280, 1985.
82

85/13 V. Kanagarajan "Force Measurements on a Body-Tail-Combination at Differ-

K. Hartmann ent Reynolds Numbers and Roll Angles up to Incidences of
45 Degrees." DFVLR-Report IB 222-85 A 38, 1985.
85/14 E. E. Covert "Thrust and Drag: Its Prediction and Verification." AIAA
(Editor) Progress in Astronautics a n d Aeronautics, Vol. 98, 1985.

86/1 B. Ewald "The Accuracy Problem of Airplane Development Force Test-

G. Krenz ing in Cryogenic Wind Tunnels." AIAA Paper 86-0776, AIAA
Paper 86-0776, AIAA 14th Aerodynamic Testing Conference,
March 1986.

86/2 S. V. Merthy "Effects of Compressibility and Free-Stream Turbulence

F. W. Steinle, Jr. on Boundary-Layer Transition in High Subsonic and Tran-
sonic Flows." AIAA Paper 86-0764, AIAA 14th Aerodynamic
Testing Conference, March 1986.
86/3 Th. J. Horvath "Coherent Shedding from a Circular Cylinder at Critical,
G. S. Jones Supercritical, and Transcritical Reynolds Numbers." SAE
P. C. Stainback Technical Paper Series, Paper 861768, 1986.
86/4 P. Champigny "Stability of Side Forces on Bodies at High Angles of
Attack." AIAA 4th Applied Aerodynamics Conference, Paper
AIAA 86-1776-CP, 1986.
86/5 "Report of the Working Group on Aerodynamics of Aircraft
Afterbody (Chairman P. Sacher)." AGARD-AR-226, June
1986.

87/1 F. Auleha "Pseudo Reynolds Number Trends, AIAA Paper 87-2612-CP."

Presented at the AIAA 5th Applied Aerodynamics Confer-
ence, Monterey, CA, Aug. 17-19, 1987.
87/2 F. T. Lynch "Wind Tunnel Side Wall Boundary-Layer Effects in Transonic
C. B. Johnson Airfoil Testing — Some Correctable, But Some Not."
AGARD FDP Conference on Aerodynamic Data Accuracy and
Quality: Requirements and Capabilities in Wind Tunnel
Testing, Naples, Italy, September 28-0ctober 1, 1987.
87/3 H. U. Meier "The Influence of Wind Tunnel Turbulence on the Boundary-
U. Michel Layer Transition. Perspectives in Turbulence Studies,"
H. P. Kreplin dedicated to the 75th birthday of Dr. J. C. Rotta.
International Symposium, DFVLR Research Center, Gottingen,
May 11-12, 1987.
87/4 K. Hartmann "Experimental Investigation on an Ogive-Nosed Body at
High Incidence and Different Reynolds Numbers." Inter-
national Conference on Fluid Mechanics, Beijing, China,
July 1-4, 1987.

88/1 AGARD WG09 "Boundary Layer Simulation and Control in Wind Tunnels."
AGARD-AR-224, April 1988.
 REPORT DOCUMENTATION PAGE
1. Recipient's Reference 2. Originator's Reference 3. Further Reference 4. Security Classification
of Document
AGARD-AG-303 ISBN 92-835-0492-5 UNCLASSIFIED

5. Originator Advisory Group for Aerospace Research and Development

North Atlantic Treaty Organization
7 rue Ancelle, 92200 Neuilly sur Seine, France
6. Title REYNOLDS NUMBER EFFECTS IN TRANSONIC FLOW

7. Presented at

8. Author(s) /Editor(s) 9. Date

Various December 1988

10. Author's/Editor's Address 11. Pages

Various 92

12. Distribution Statement This document is distributed in accordance with AGARD

policies and regulations, which are outlined on the
Outside Back Covers of all AGARD publications.
13. Keywords/Descriptors

Reynolds number Fuselages

Transonic flow Missile airframes
Wind tunnels Afterbodies
Airfoils

14. Abstract

Reynolds number effects in transonicfloware critically reviewed. This review, which may be
considered a supplement to AR 224 'Boundary Layer Simulation and Control in Wind Tunnels',
is mainly concerned with a discussion of viscous effects actually observed on realistic
configurations.

The following geometries are considered: Airfoils and high aspect ratio wings typical of transport
aircraft, fighter-type low aspect ratio delta wings, two- and three-dimensional bodies
characteristic of missiles and combat aircraft fuselages, and afterbodies. "Pseudo"-Reynolds
number effects are identified which may arise, for instance due to the influence of the Reynolds
number on the wind tunnel environment and in turn affect theflowabout a model. As an
introduction, a brief retrospect of the "history" of Reynolds number effect is presented.

This AGARDograph has been produced at the request of the Fluid Dynamics Panel of AGARD.
 NATO ^ OTAN
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