The Kolmogorov-Obukhov Theory of Turbulence - Bjorn...

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






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






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






Boundary Layers and Turbulence

Turbulence

Birnir






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






Outline






Turbulence

Birnir






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






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






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






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






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






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






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






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






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






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






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






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






Outline






Turbulence

Birnir






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






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






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






Outline






Turbulence

Birnir






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






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






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






Outline






Turbulence

Birnir






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






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






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






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






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






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






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






The Artist by the Water’s EdgeLeonardo da Vinci Observing Turbulence

Turbulence

Birnir






A. Babin, A. Mahalov, and B. Nicolaenko.Long-time averaged euler and navier-stokes equationsfor rotation fluids.In Structure and Dynamics of non-linear waves inFluids, 1994 IUTAM Conference, K. Kirehgassner andA. Mielke (eds), World Scientific, page 145â157, 1995.

O. E. Barndorff-Nilsen.Exponentially decreasing distributions for the logarithmof the particle size.Proc. R. Soc. London, A 353:401–419, 1977.

O. E. Barndorff-Nilsen, P. Blaesild, and JurgenSchmiegel.A parsimonious and universal description of turbulentvelocity increments.Eur. Phys. J. B, 41:345–363, 2004.

Turbulence

Birnir






B. Birnir.The Existence and Uniqueness and Statistical Theoryof Turbulent Solution of the Stochastic Navier-StokesEquation in three dimensions, an overview.Banach J. Math. Anal., 4(1):53–86, 2010.Available at http://repositories.cdlib.org/cnls/.

B. Birnir.The Invariant Measure of Turbulence.CNLS preprint, 2011.Available at http://repositories.cdlib.org/cnls/.

B. Dubrulle.Intermittency in fully developed turbulence: inlog-poisson statistics and generalized scale covariance.Phys. Rev. Letters, 73(7):959–962, 1994.

Turbulence

Birnir






E. Hopf.Statistical hydrodynamics and functional calculus.J. Rat. Mech. Anal., 1(1):87–123, 1953.

A. N. Kolmogorov.The local structure of turbulence in incompressibleviscous fluid for very large reynolds number.Dokl. Akad. Nauk SSSR, 30:9–13, 1941.

A. N. Kolmogorov.A refinement of previous hypotheses concerning thelocal structure of turbulence in a viscous incompressiblefluid at high reynolds number.J. Fluid Mech., 13:82–85, 1962.

R. H. Kraichnan.Turbulent cascade and intermittency growth.

Turbulence

Birnir






In Turbulence and Stochastic Processes, eds. J. C. R.Hunt, O. M. Phillips and D. Williams, Royal Society,pages 65–78, 1991.

A. M. Obukhov.Some specific features of atmospheric turbulence.J. Fluid Mech., 13:77–81, 1962.

B. Oksendal and A. Sulem.Applied Stochastic Control of Jump Diffusions.Springer, New York, 2005.

Z-S She and E. Leveque.Universal scaling laws in fully developed turbulence.Phys. Rev. Letters, 72(3):336–339, 1994.

Z-S She and E. Waymire.

Turbulence

Birnir






Quantized energy cascade and log-poisson statistics infully developed turbulence.Phys. Rev. Letters, 74(2):262–265, 1995.

Michael Wilczek.Statistical and Numerical Investigations of FluidTurbulence.PhD Thesis, Westfälische Wilhelms Universität,Münster, Germany, 2010.

The Kolmogorov-Obukhov Theory of Turbulence - Bjorn...

Documents

Transcript of The Kolmogorov-Obukhov Theory of Turbulence - Bjorn...