The Kolmogorov-Obukhov Theory of Turbulence
Transcript of The Kolmogorov-Obukhov Theory of Turbulence
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The Kolmogorov-Obukhov Theory ofTurbulence
Björn Birnir
Center for Complex and Nonlinear Scienceand
Department of Mathematics, UC Santa Barbara
The Courant Institute, Nov. 2011
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Outline
1 The Deterministic versus the Stochastic Equation
2 The Form of the Noise
3 The Kolmogorov-Obukov Scaling
4 The generalized hyperbolic distributions
5 Comparison with Simulations and Experiments.
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Outline
1 The Deterministic versus the Stochastic Equation
2 The Form of the Noise
3 The Kolmogorov-Obukov Scaling
4 The generalized hyperbolic distributions
5 Comparison with Simulations and Experiments.
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The Deterministic Navier-Stokes Equations
A general incompressible fluid flow satisfies theNavier-Stokes Equation
ut +u ·∇u = ν∆u−∇pu(x ,0) = u0(x)
with the incompressibility condition
∇ ·u = 0,
Eliminating the pressure using the incompressibilitycondition gives
ut +u ·∇u = ν∆u +∇∆−1trace(∇u)2
u(x ,0) = u0(x)
The turbulence is quantified by the dimensionlessReynolds number R = UL
ν
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Boundary Layers and Turbulence
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The Stochastic Navier-Stokes Equations
In turbulent fluids the laminar solution is unstableSmall noise is magnified by the fluid instability and thesaturated by the nonlinearities in the flow and in theNavier-Stokes equationsIt was pointed out by Kolmogorov [8] that it is moreuseful in turbulent flow to consider the velocity u(x , t) tobe a stochastic processThen it satisfies the stochastic Navier-Stokes equation
du = (−u ·∇u + ν∆u +∇∆−1trace(∇u)2)dt +dft
u(x ,0) = u0(x)
dft is the noise in fully developed turbulence
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Outline
1 The Deterministic versus the Stochastic Equation
2 The Form of the Noise
3 The Kolmogorov-Obukov Scaling
4 The generalized hyperbolic distributions
5 Comparison with Simulations and Experiments.
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The central limit theorem
We construct the noise using the central limit theoremSplit the torus T3 into little boxes and consider thedissipation to be a stochastic process in each boxBy the central limit theorem the average
Sn =1n
n
∑j=1
pj
converges to a Gaussian random variable as n→ ∞
This holds for any Fourier component (ek ) and theresult is the infinite dimensional Brownian motion
df 1t = ∑
k 6=0c
12k dbk
t ek (x)
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Intermittency and the large deviation principle
In addition we get intermittency of the dissipationIf these excursions are completely random then theyare modeled by Poisson process with the rate λ
Applying the large deviation principle, we getexponentially distributed processes, with rate |k |1/3
This also holds in the direction of each Fouriercomponent and gives the noise
df 2t = ∑
k 6=0dkdν
kt ek (x)
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Intermittency and velocity fluctuation
So far our noise is additive. There is also multiplicativenoise due to velocity fluctuationThe multiplicative noise, models the excursion (jumps)in the velocity gradientIf these jumps are completely random they should bemodeled by a Poisson process ηk
tNk
t denotes the integer number of velocity excursion,associated with k th wavenumber, that have occurred attime t .The differential dNk (t) = Nk (t +dt)−Nk (t) denotesthese excursions in the time interval (t , t +dt ].The process
df 3t =
M
∑k 6=0
ZR
hk (t ,z)Nk (dt ,dz),
gives the multiplicative noise term
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Stochastic Navier-Stokes with Turbulent Noise
Adding the two types of additive noise and themultiplicative noise we get the stochastic Navier-Stokesequations describing fully developed turbulence
du = (ν∆u−u ·∇u +∇∆−1tr(∇u)2)dt
+ ∑k 6=0
c12k dbk
t ek (x)+M
∑k 6=0
dkdνkt ek (x)
+ u(M
∑k 6=0
ZR
hk Nk (dt ,dz))
u(x ,0) = u0(x)
Each Fourier component ek comes with its ownBrownian motion bk
t and Poisson process νkt
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Computation of the measure
Now the linearized equation
dz = (ν∆z−u ·∇z−z ·∇u +2∇∆−1tr(∇u∇z)dt
+ ∑k 6=0
(c12k dbk
t +dkdνkt )ek (x) (1)
+ z( ∑k 6=0
ZR
hk (t ,z)Nk (dt ,dz))
z(0) = z0
has (almost) the same invariant measureas the stochastic Navier-Stokes equation for velocitydifferences.
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Solution of the Stochastic LinearizedNavier-Stokes
We solve (1) using the additional help of theFeynmann-Kac formula, and Cameron-Martin (orGirsanov’s Theorem)The solution is
z = eKteR t
0 dqMtz0
+ ∑k 6=0
Z t
0eK (t−s)e
R ts dqMt−s(c
1/2k dβ
ks +dkdν
ks )ek (x)
K is the operator K = ν∆+2∇∆−1tr(∇u∇)K generates a semi-group by the perturbation theory oflinear operators (Kato)
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Cameron-Martin and Feynmann-Kac
Mt is the Martingale
Mt = exp{−Z t
0u(Bs,s) ·dBs−
12
Z t
0|u(Bs,s)|2ds}
Using Mt as an integrating factor eliminates the inertialterms from the equation (1)The Feynmann-Kac formula gives the exponential of asum of terms of the formZ t
sdqk =
Z t
0
ZR
ln(1+hk )Nk (dt ,dz)−Z t
0
ZR
hkmk (dt ,dz),
by a computation similar to the one that produces thegeometric Lévy process, see [12], mk the Lévymeasure.
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The log-Poisson processes
The form of the processese
R t0
RR ln(1+hk )Nk (dt ,dz)−
R t0
RR hk mk (dt ,dz) was found by She
and Leveque [13], for hk = β−1,Nk
t counts the number of jumps, with the meanZR
mk (t ,dz) =−γ ln |k |β−1
,Z t
0
ZR
hkNk (ds,dz) = Nkt ln(β)
It was pointed out by She and Waymire [14] and byDubrulle [6] that they are log-Poisson processes.
eR t
0RR ln(1+hk )Nk (dt ,dz)−
R t0
RR hk mk (dt ,dz) = eNk
t lnβ+γ ln |k |= |k |γβNkt
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The Spectral Theory of the Operator K
Suppose thatE(‖u‖23
2+)≤ C1 (2)
then the operator K generates contration semi-groupsdenoted eKt . We get using the bound [1],[4],
E(‖u‖2116
+(t))≤ C (3)
Lemma (The Inertial Range)
The spectrum of the operators K satisfies the estimate
|λk +ν4π2|k |2| ≤ C|k |2/3 (4)
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Computation of the structure functions
Lemma (The Kolmogorov-Obukov scaling)
The scaling of the structure functions is
Sp ∼ Cp|x −y |ζp ,
whereζp =
p3
+ τp =p9
+2(1− (2/3)p/3)
p3 being the Kolmogorov scaling and τp the intermittencycorrections. The scaling of the structure functions isconsistent with Kolmogorov’s 4/5 law,
S3 =−45
ε|x−y |
to leading order, were ε = dEdt is the energy dissipation
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The first few structure functions
S1(x ,y , t) =2C ∑
k∈Z3\{0}dk
(1−e−λk t)|k |ζ1
sin(πk · (x−y)).
∑k∈Z3\{0}dk < ∞, and for |x−y | small,
S1(x ,y ,∞)∼ 2C ∑
k∈Z3\{0}dk |x−y |ζ1 ,
where ζ1 = 1/3+ τ1 ≈ 0.37. Similarly
S2(x ,y ,∞)∼ 2πζ2
C ∑k∈Z3
[ck +2d2
kC
]|x −y |ζ2 ,
when |x−y | is small, where ζ2 = 2/3+ τ2 ≈ 0.696.
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The higher order structure functions
All the structure functions are computed in a similar manner.If p = 2n +1 is odd,
Sp =2p
Cp ∑k∈Z3
dkp (1−e−2λk t)p
|k |ζpsinn(πk · (x−y))
to leading order in the lag variable |x −y |. If p = 2n is even,Sp is
∑k∈Z3
[2n
Cn cnk(1−e−2λk t)n
|k |ζp+
2p
Cp dkp (1−e−λk t)p
|k |ζp]sinp(πk ·(x−y)),
to leading order in |x −y |.
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Outline
1 The Deterministic versus the Stochastic Equation
2 The Form of the Noise
3 The Kolmogorov-Obukov Scaling
4 The generalized hyperbolic distributions
5 Comparison with Simulations and Experiments.
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The Kolmogorov-Obukov scaling hypothesis
The Kolmogorov-Obukov scaling with the intermittencycorrections τp, is
Sn(l) = Cplζp , ζp =p3
+ τp =p9
+2(1− (2/3)p/3) (5)
where l is the lag variable l = |x−y |.The coefficients Cp are not universal but depend on thecks and dks that in turn depend on the large eddies inthe turbulent flowCp = 2pπ
ζp
Cp ∑k∈Z3\{0}dkp or Cp = 2nπ
ζp
Cn ∑k∈Z3 [cnk + 2n
Cn dkp]
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Kolmogorov’s refined scaling hypothesis
In [9, 11] Kolmogorov and Obukhov presented theirrefined similarity hypothesis
Sp = C ′p < εp > lp/3
where l is the lag variable and ε is an averaged energydissipation rateIt can be shown, see [5], that by defining ε
appropriately, this gives
Sp = C ′p < εp > lp/3 = Cplζp
where the coefficients C ′p now are universal
Sp(t ,T , l) = Cplζp +Dp(t)T γp , γp =p6
+3(1− (2/3)p/3)
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The Invariant Measure and the ProbabilityDensity Functions (PDF)
The above computation is a computation of a testinvariant measure, that the real invariant measureshould be absolutely continuous with respect toHopf [7] write down a functional differential equation forthe characteristic function of the invariant measureThe quantity that can be compared directly toexperiments is the PDF
E(δju) = E([u(x +s, ·)−u(x , ·)] · r) =Z
∞
∞
fj(x)dx ,
j = 1, if r = s is the longitudinal direction, and j = 2,r = t , t ⊥ s is a transversal directionUsing Jacobi’s identity and the asymptotics of themoments, we compute the PDF directly
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Outline
1 The Deterministic versus the Stochastic Equation
2 The Form of the Noise
3 The Kolmogorov-Obukov Scaling
4 The generalized hyperbolic distributions
5 Comparison with Simulations and Experiments.
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Computing the PDF from the characteristicfunction
Taking the characteristic functions of the measure ofthe stochastic processes in Equation (1), we get
f (k)∼ k1−ζ1e−δk
Translating this function and taking the inverse Fouriertransform gives
f (x)∼ e−d |x |e−bx
(x− iδ)2−ζ1
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Inserting a Gaussian
The probability density function (PDF) of thecomponents of the velocity increments is a generalizedhyperbolic distribution, see Barndorff-Nilsen [2]Letting α,δ→ ∞, in the formulas for fj(x) below, in sucha way that δ/α→ σ, we get that
fj →e−
(x−µ)22σ
√2πσ
eβ(x−µ).
The exponential tails of the PDF are caused byoccasional sharp velocity gradients (rounded of shocks)The cusp at the origin is caused by the random andgentile fluid motion in the center of the ramps leadingup to the sharp velocity gradients, see Kraichnan [10]
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The Probability Density Function (PDF)
Lemma
The PDF is a generalized hyperbolic distribution, λ = 1−ζ1:
f (xj) =(δ/γ)1−ζ1
√2πK1−ζ1
(δγ)
K1−ζ1(α
√δ2 +(xj −µ)2)eβ(x−µ)
(√
δ2 +(xj −µ)2/α)1−ζ1
(6)
where K1−ζ1is modified Bessel’s function of the second
kind, γ =√
α2−β2, ζ1 the scaling exponent of S1,
f (x)∼ (δ/γ)1−ζ1
2πK1−ζ1(δγ)
Γ(1−ζ1)21−ζ1eβµ
(δ2 +(x−µ)2)1−ζ1for x << 1
f (x)∼ (δ/γ)1−ζ1
2πK1−ζ1(δγ)
eβ(x−µ)e−αx
x3/2−ζ1for x >> 1
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Outline
1 The Deterministic versus the Stochastic Equation
2 The Form of the Noise
3 The Kolmogorov-Obukov Scaling
4 The generalized hyperbolic distributions
5 Comparison with Simulations and Experiments.
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
Existence and Uniqueness of the InvariantMeasure
We now compare the above PDFs with the PDFs foundin simulations and experiments.The direct Navier-Stokes (DNS) simulations wereprovided by Michael Wilczek from his Ph.D. thesis, see[15].The experimental results are from EberhardBodenschatz experimental group in Göttingen.We thank both for the permission to use these resultsto compare with the theoretically computed PDFs.A special case of the hyperbolic distribution, the NIGdistribution, was used by Barndorff-Nilsen, Blaesild andSchmiegel [3] to obtain fits to the PDFs for threedifferent experimental data sets.
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The PDF from simulations and fits for thelongitudinal direction
−30 −20 −10 0 10 20 30−8
−7
−6
−5
−4
−3
−2
−1
0
1
2
x
3*P
DF
for
the
long
ditu
dion
al d
irect
ion
Figure: The PDF from simulations and fits for the longitudinaldirection.
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The log of the PDF from simulations and fits forthe longitudinal directionCompare Fig. 4.5 in [15]
−30 −20 −10 0 10 20 30−70
−60
−50
−40
−30
−20
−10
0
x
Log
of P
DF
s fo
r th
e lo
ngdi
tudi
onal
dire
ctio
n
Figure: The log of the PDF from simulations and fits for thelongitudinal direction, compare Fig. 4.5 in [15].
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The PDF from simulations and fits for atransversal direction
−30 −20 −10 0 10 20 30−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
x
PD
Fs
for
diffe
rent
lags
and
thei
r fit
s
Figure: The PDF from simulations and fits for a transversaldirection.
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The log of the PDF from simulations and fits forthe a transversal directionCompare Fig. 4.6 in [15]
−30 −20 −10 0 10 20 30−60
−50
−40
−30
−20
−10
0
x
Log
of P
DF
s fo
r di
ffere
nt la
gs a
nd th
eir
fits
Figure: The log of the PDF from simulations and fits for the atransversal direction, compare Fig. 4.6 in [15].
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The PDF from experiments and fits
−30 −20 −10 0 10 20 30−8
−7
−6
−5
−4
−3
−2
−1
0
1
2
x
3*P
DF
s of
exp
erim
enta
l dat
a
Figure: The PDF from experiments and fits.
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The log of the PDF from experiments and fits
−10 −5 0 5 10
−35
−30
−25
−20
−15
−10
−5
x
Log
of P
DF
for
expe
rimen
tal d
ata
Figure: The log of the PDF from experiments and fits.
Turbulence
Birnir
TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
The Artist by the Water’s EdgeLeonardo da Vinci Observing Turbulence
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TheDeterministicversus theStochasticEquation
The Form ofthe Noise
TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
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TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
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Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
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Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
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TheKolmogorov-ObukovScaling
Thegeneralizedhyperbolicdistributions
ComparisonwithSimulationsandExperiments.
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