Scales of Turbulent Flow
TURBULENCE: THEORY AND MODELING
What is a ’scale’ ?
Turbulent scales?
• What are the ’turbulent
scales’?
• How to ’detect’?
• How big?
• How small?
• How they evolve?
Lewis Fry Richardson (1881-1953)
• Weather forcasting
• Mathematical analysis of war
• Length of coastlines and
borders
Big whirls have little whirls
Which feed on their velocity
Little whirls have lesser whirls
And so on to viscosity
– in the molecular sense
What is an eddy/whirl?
• How to visualize?
– U, p, w
l2, Q
An eddy eludes precise definition,
but is conceived to be a turbulent
motion localized within a region of
size l, that is at least moderately
coherent over this region. The
region occupied by a large eddy can
also contain smaller eddies.
S.B. Pope
What is an eddy?
Artificial velocity field: uconv + ubig_eddy + usmall_eddy
Artificial vorticityfield.
Autocorrelation functions
• “…at least moderately
coherent…”
• Assume
– Isotropic
– No mean flow
tututR jiij ,,,, rxxxr
U1
U2
U3
x X+r
k, e, Taylor-Green vortices
Longitudinal/Transversal
autocorrelation functions
longitudinal
transversal
Longitudinal autocorrelation function
Transversal autocorrelation function
U1
U2
U3
x X+r
Longitudinal autocorrelation function
• Symmetric
• f’(r)=0
• f’’(r)<0
• How big/small are the
eddies?
– Integral
– Taylor
– Kolmogorov
Integral lengthscales
L11
Taylor lengthscales
• Physical interpretation unclear
• Can be used to estimate
dissipation rate
• Useful in defining a Reynolds
number of universal character
lf
Andrey Nikolaevich Kolmogorov
(1903-1987)
• 5 years old:
• Probability theory
• Statistical theory to artillery
fire
• Turbulence
– 3 hypotheses
1=12
1+3=22
1+3+5=32
1+3+5+7=42
Kolmogorov’s hypothesis of local
isotropy (the 0th hypothesis)
• At sufficiently
high Reynolds
number, the small
scale turbulent
motions are
statistically
isotropic.
The MIT lecture movie can be found at the following link:
https://www.youtube.com/watch?v=1_oyqLOqwnI
Kolmogorov’s first similarity
hypothesis
• In every turbulent flow,
at sufficiently high
Reynolds number, the
statistics of the small
scale motions have a
universal form and are
uniquely determined by
n and e.
Kolmogorov’s second similarity
hypothesis
• In every turbulent flow, at sufficiently high Reynolds
number, there is a range of scales, much smaller than the
largest scales and much larger than the smallest scales,
where the statistics of the motions have a universal form
and are uniquely determined by e idependent of n.
Kolmogorov microscales
• Scales
– Integral
– Taylor
– Kolmogorov
• What happens in-
between?
– Any rule?
Length
Time
Velocity
Reynolds number
Fourier transform
• Determine
frequency/wavenumber
-10
-8
-6
-4
-2
0
2
4
6
8
10
0 50 100 150 200 250 300
Func
0
200
400
600
800
1000
1200
1400
0 0.02 0.04 0.06 0.08 0.1
Ampl
Energy spectrum
• Velocity spectrum tensor
• Two-point correlation
• Wave number vector
• Wave length
κ
κ
l
2
Φ𝑖𝑗 𝜅 =1
(2𝜋)3
−∞
∞
𝑅𝑖𝑗(𝒓)𝑒−𝑖𝜅𝒓𝑑𝒓
𝑅𝑖𝑗 𝒓 =
−∞
∞
Φ𝑖𝑗(𝜅)𝑒−𝑖𝜅𝒓𝑑𝜅
Energy spectrum
• Energy spectrum function
• Simpler than F
• Directional information
removed
• One dimensional spectra:
F κκ dκtκtκE ii ,,
iiii uutRκdtκE
2
1,0
2
1,
0
Turbulent kinetic
energy
Energy-containing
rangeUniversal equilibrium range
Inertial subrange Dissipation
range
Kolmogorov
Hypotheses!
Effect of Reynolds number
Fake vs. real
How do vortices evolve?
Vorticity transport equation
Vorticity: Levi-Civita epsilon:
Taking the curl of the Navier-Stokes equations one gets:
j
ij
jj
i
j
ij
i
x
u
xxxu
t
w
wn
ww 2
j
kijki
x
u
ew
3
2
2
311
x
u
x
u
x
u
j
kjk
ew
equal are indices twoif 0
npermutatio oddan isijk if 1
npermutatioeven an isijk if 1
ijke
What is
what?
Beware missing terms!
Vorticity transport equation
j
ij
jj
i
j
ij
i
x
u
xxxu
t
w
wn
ww 2
It can be shown that
𝜔𝑗Ω1𝑗 = 𝜔1Ω11 +𝜔2 Ω12 + 𝜔3Ω13
=𝜀2𝑗𝑘𝜕𝑢𝑘
𝜕𝑥𝑗Ω12 + 𝜀3𝑗𝑘
𝜕𝑢𝑘
𝜕𝑥𝑗Ω13
= 𝜀213𝜕𝑢3
𝜕𝑥1+ 𝜀231
𝜕𝑢1
𝜕𝑥3
1
2
𝜕𝑢1
𝜕𝑥2−𝜕𝑢2
𝜕𝑥1+
𝜀312𝜕𝑢2
𝜕𝑥1+ 𝜀321
𝜕𝑢1
𝜕𝑥2
1
2
𝜕𝑢1
𝜕𝑥3−𝜕𝑢3
𝜕𝑥1=0
Vorticity transport equation
j
ij
jj
i
j
ij
i
x
u
xxxu
t
w
wn
ww 2
In 2D:
3,0,0 ww j
000
02
1
02
1
2
2
1
2
2
1
1
2
2
1
1
1
x
u
x
u
x
u
x
u
x
u
x
u
Sij
0ijjSw
Vortex stretching
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