The Energy Cascade and The Kolmogorov Scale
The Energy Cascade and The Kolmogorov Scale
The Energy Cascade and The Kolmogorov Scale
The inertial range is the range of scales that are universally (really?) self organized.
Universality comes from the ubiquitous -5/3 energy decay in the power spectrum.
In which case we do not observe (at least a decade in f or k) of -5/3 slope ?
1) at low Reynolds number TBL, there is not enough scale separation between outer
production scale and inner dissipative scales.
2) in complex terrain, or (very) rough wall, wakes and shear layers inject TKE (and
vorticity) at specific scales altering the self organization process.
in df or dk
stop
K41- Kolmogorov (1941) material from “Turbulence,
the legacy of A.N. Kolmogorov”
by Uriel Frisch, 1995
Let us start with two key experimental findings in fully developed turbulence A), B)
A) 2/3 Law: In a turbulent flow at very high Reynolds number, the mean square
velocity increments <( δ v(l) )2 > between two points separated by a distance l (in a
certain range) behaves as the 2/3 power of the distance.
2 2
𝑆2 𝑙 = 𝛿 𝑣 𝑙 = 𝑣 𝑟+𝑙 −𝑣 𝑟 𝑤𝑖𝑡ℎ 𝑙, 𝑓𝑜𝑟 𝑠𝑖𝑚𝑝𝑙𝑖𝑐𝑖𝑡𝑦 ,
𝑜𝑟𝑖𝑒𝑛𝑡𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑖𝑛𝑎𝑙 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛
if the flow is homogeneous, the averaging operator acts on r, for each increment l,
such that S2(r,l) becomes S2(l).
Estimates of the second order (power 2) structure function are obtained using Taylor
hypothesis and time resolved measurements (l=Uτ)
2 2
𝑆2 τ = 𝛿 𝑣 τ = 𝑣 𝑡+τ −𝑣 𝑡 averaged over t
Isotropy the 2/3 law holds also along the transverse velocity : l ⊥ v
even though it is more difficult to measure (no Taylor
hypothesis and multiple sensor simultaneously acquiring)
2 2
𝑆2 𝑙 = 𝛿 𝑣 𝑙 = 𝑣 𝑟+𝑙 −𝑣 𝑟 ~ 𝑙2/3 𝑤𝑖𝑡ℎ 𝑙 𝑜𝑟𝑖𝑒𝑛𝑡𝑒𝑑 𝑖𝑛 𝑎𝑛𝑦 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛
The range of l in which the power law dependency is found is defined as a scaling
range and it is a function of the Reynolds number:
𝑢𝑟𝑚𝑠 𝜆
𝑅𝑒𝜆 = where the Taylor scale 𝜆 is defined as:
𝜐
The larger 𝑅𝑒𝜆 , the larger is the scaling range where 2/3 law applies.
Note that this is in agreement with the extension of the inertial range where k-5/3 applies
http://windtunnel.onera.fr (visit the website!)
B) Law of finite energy dissipation: If in an experiment on turbulent flow all the
control parameters are kept the same , except for viscosity, which is kept as low as
possible, the energy dissipation per unit mass De/dt behaves in a way consistent with
a finite positive limit.
This means that TKE is dissipated at some rate which depends on the energy input,
no matter how small are the scales at which energy is actually dissipated.
At small, but finite scale, dissipation will occur. Even if viscosity is very small.
This statement is consistent with the general dynamics of decay and with the
crude estimate : dE/dt ~ du2/dt ~u2 / (l/u) ~ u3/l
Dimensionally it means that TKE ~u2 is dissipated at a rate dictated by one turnover (or
circulation) time l/u
the energy containing eddies are defined by the integral scale l and the r.m.s velocity u
They can transfer, in one turnover time scale, by non linear interactions, a finite fraction of
TKE to scale sufficiently small for viscosity to be able to remove it into heat.
Now let us introduce the K41 theory
assumptions :
Frish H1 > In the limit of infinite Reynolds number all the possible symmetries of the
Navier stokes equation, usually broken by the mechanisms producing the turbulent
flow, are restored in a statistical sense at the small scales and away from boundaries.
𝑔𝑖𝑣𝑒𝑛 𝛿 𝑣 𝑟, 𝑙 = 𝑣 𝑟 + 𝑙 − 𝑣 𝑟 ,
𝐻1 𝑖𝑚𝑝𝑙𝑖𝑒𝑠: 𝛿 𝑣 𝑟 + 𝜌, 𝑙 = 𝛿 𝑣 𝑟, 𝑙
the statistical property of the velocity increment are invariant with respect to spatial
translation , for 𝜌 < l0 (integral scale)
Note that , additional isotropy assumption at the small scales would imply that:
𝛿 𝑣 𝑟, 𝑙 does not depend on the diretion of l and 𝛿v, i.e.
the statistical property of the velocity increment are independent of simultaneous
rotation of 𝛿 v and l
Frish H2 > Under the same assumptions of H1, the turbulent flow is self similar at
the small scales, i.e. it possesses a unique scaling exponent h
this means that v
𝛿 𝑣 𝑟, 𝜆𝑙 = 𝜆ℎ 𝛿 𝑣 𝑟, 𝑙 for every 𝜆 follows a power law in a
for any r, l, 𝜆𝑙 small compared to the integral scale certain range of increments l
Frish H3 > Under the same assumptions of H1, the turbulent flow has a finite non
vanishing mean rate of dissipation ε per unit mass
2
1) 𝛿𝑣 𝑙 = 𝐶𝜀 2/3 𝑙 2/3 with C universal constant
2) H2 assumption: 𝛿 𝑣 𝑟, 𝜆𝑙 = 𝜆ℎ 𝛿 𝑣 r, l
for every 𝜆𝑙, 𝑟, 𝑙 𝑠𝑚𝑎𝑙𝑙𝑒𝑟 𝑡ℎ𝑎𝑛 𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙 𝑠𝑐𝑎𝑙𝑒
𝛿 𝑣 𝑟, 𝑙 2 α 𝑙 2h
We have only one exponent such that both conditions are matched: h=1/3
Therefore:
1) We can recover the 2/3 law from the K41 universality assumptions, but...
2) we can relax universality assumptions and still recover 2/3 law (K41 paper3)
K41: fourth fifth law
Assuming homogeneity, isotropy and finite dissipation:
3 4
S3( l ) = 𝛿 𝑣∥ 𝑙 = − εl
5
Note that this equation is exact derived from Karman Howarth eq. (derived from
Navire Stokes) and no constant must be introduced. Derivation is shown in Frisch pp.
76-86
Note that :
H2 > : 𝛿 𝑣 𝑟, 𝜆𝑙 = 𝜆ℎ 𝛿 𝑣 r, l
3 4
4/5 law > S3( l ) = 𝛿 𝑣∥ 𝑙 = − εl
5
3 4
𝛿 𝑣∥ 𝜆𝑙 α 𝜆3ℎ − ε𝜆l
5
𝑝
𝑆𝑝 (𝑙) = 𝛿 𝑣∥ 𝑙
Because of self-similarity,
𝑆𝑝 𝑙 𝛼 𝑙 𝑝/3
Sp α (εl)p/3
So we can write:
where Cp are dimensionless, e.g.
𝑆𝑝 (𝑙) = 𝐶𝑝 𝜀 𝑝/3 𝑙 𝑝/3 for p=3, C3= 4/5 which is truly universal
for p=2, C2=2 ± 0.4
and find again the 2/3 law :
STOP
Given a stationary random function, its spectrum E(k) is such that :
The inertial range , where E(k) α k-5/3 applies, extends down to scales comparable with
the Taylor microscale. With the assumption of finite viscosity it can be proven that
the inertial range extends down to the Kolmogorov scale η=(ν3 / ε )-1/4
Phenomenology of turbulence according to K41
let us consider a scale l associated with a velocity scale
2
𝑣𝑙 = 𝛿 𝑣∥ 𝑙 or 𝑣𝑙 = 𝑣 𝑟 + 𝑙 ∗ 𝑣(𝑙)
𝑣𝑙 2 𝑣𝑙 3
Π 𝑙 ~ =
𝑡𝑙 𝑙
In the inertial range, there is no energy input or direct energy dissipation into
heat so the energy flux should be independent of l and equal to finite mean
dissipation rate ε 𝑙
𝑡𝑙 =
3 𝑣𝑙
𝑣𝑙
Π 𝑙 ~ ~𝜖 𝑣𝑙 = 𝑙1/3 𝜖1/3 𝑡𝑙 = 𝑙 2/3 𝜖 −1/3
𝑙
At the lower limit of the
On the upper limit of inertial range, where 𝑙2
the inertial range, viscous effects are 𝑡𝑙 =
dominant, the diffusive 𝜐
for l ~ integral scale
time scale reads:
𝑣0 3
𝜖~
𝑙0
These two term s are equal for
Dimensional analysis 𝑙 = 𝜐 3/4 𝜖 −1/4 = 𝜂
mixing: particle separation in turbulence
note that 𝑡𝑙 = 𝑙 2/3 𝜖 −1/3
implies that 2 particles in a turbulent flow will separate in a way that D l2/Dt ~
𝑙2 2 −1/3 2/3
~𝑙 /𝜖 𝑙 𝜖 ~ 𝜖 1/3 𝑙 4/3
𝑡𝑙
Balance means that we cannot ignore non linear interaction and energy transfer
in the dissipative range. This opens research question on sub-Kolmogorov mixing
K41 : Space filling eddies vs coherent but anisotropic motions
l=l0rn
Violation of K41
any energy flux bypassing the inertial range, as in the case of
complex flow geometry, will result in an unconventional (non K41)
statistical behavior of structure functions
and into a change in the exponents Sp (intermittency). This occurs
for instance in flows over rough or porous terrain, or in a wake,
where the turbulence does not have time-space to adjust itself to
dissipate energy at the required rate, following the standard
cascade.
Landau 1944
It might be thought that the possibility exists in principle of obtaining a universal
formula, applicable to any turbulent flow, which should give S 2(l) for all distance l
smaller than the integral scale l0 . In fact however there can be no such formula:
the instantaneous value of (δv(l))2 might in principle be expressed as a universal
function of the dissipation ε at the instant considered.
When we average that expression however an important part will be played by the
manner of variation of ε over times of the order of the period of the larger eddies, of
order of l0 , and this variation is different in different flows. the result of such averaging
therefore cannot be universal.
Kraichnan 1974
The slope of the structure function power law Cp for p ≠ 3 is not universal as it
depends on the detailed geometry of production of turbulence
< 𝑣4 >
𝐹=
(< 𝑣 2 >)2
self similarity intermittency
Intermittency in turbulence is assessed by checking the slopes of the
structure functions at different order
𝑝
𝑆𝑝 (𝑙) = 𝛿 𝑣∥ 𝑙 Because of self-similarity, 𝑆𝑝 𝑙 𝛼 𝑙 𝑝/3
In the premultiplied form we have:
Benzi 1993 Extended Self Similarity:
Using ESS the linear fit in the log log
phase space is much easier.
(it is almost as it
would compensate
for the different
structure of the
production scale)
Atmospheric
jet flow
Surface Layer
Estimates of the structure functions exponents using Extended Self Similarity
From SLTEST data