l ehrstuhl für modellierung und simulation statistical theory of the isotropic turbulence (k-41...

Lehrstuhl fürModellierung und Simulation

Statistical theory of the isotropic turbulence (K-41 theory)

2. Kolmogorov theory

Lecture 3

UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION

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Kolmogorov Theory K41 Kolmogorov Theory K41 Andrey Nikolaevich Kolmogorovwas a Soviet Russian mathematician, preeminent in the 20th century, who advanced various scientific fields (among them probability theory, topology, intuitionistic logic, turbulence, classical mechanics and computational complexity).

((www.wikipedia.org)www.wikipedia.org)

„„Every mathematician believes he is ahead over Every mathematician believes he is ahead over all others. The reason why they don't say this in all others. The reason why they don't say this in public, is because they are intelligent people“ public, is because they are intelligent people“


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Physical model beyond the K41 Physical model beyond the K41

Most important physical processes areMost important physical processes are• Transfer energy from large scales to small onesTransfer energy from large scales to small ones• Dissipation of the energy in samll vorticesDissipation of the energy in samll vorticesTwo parameters are of importance: kinematic viscosity and dissipation rate Two parameters are of importance: kinematic viscosity and dissipation rate

The size range is referred to as the universal equilibrium rangeThe size range is referred to as the universal equilibrium rangeEIl < l


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Hypothesis of local isotropyHypothesis of local isotropy3/ 2k

L

1/ 2u k

Ll

Macroscale of the flow ,Macroscale of the flow ,

characteristic velocity of macrovorticescharacteristic velocity of macrovortices

Kolmogorov‘s hypothesis of local isotropyKolmogorov‘s hypothesis of local isotropy

At sufficiently high Reynolds numbers , the smallAt sufficiently high Reynolds numbers , the small-scale motion with scales are statistically isotropic.-scale motion with scales are statistically isotropic.

Directional information is lost. The laws describing the Directional information is lost. The laws describing the small-scale motion are universal. small-scale motion are universal.

tRe uL /


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Theory of Kolmogorov (1941) K-41

u u, u ,

In every turbulent flow at sufficiently high Reynolds number, In every turbulent flow at sufficiently high Reynolds number, the statistics of the small-scale motions have a universal the statistics of the small-scale motions have a universal form that is uniquely determined by kinematic viscosity and form that is uniquely determined by kinematic viscosity and turbulent energy dissipation rate turbulent energy dissipation rate

EIl l


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Kolmogorov scale, time and velocityKolmogorov scale, time and velocity

1/ 43

1/ 4

1/ 2

,

u ( ) ,

( /


Some useful estimationsSome useful estimations

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3/ 4

t

1/ 4t

1/ 2t

/ L Re ,

u / u (Re ) ,

/ T (Re )

3/ 2k

L 1/ 2u k


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Distribution of Komogorov scale in jet mixer at Re=10000


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The strongest and simultaneously the most questionable assumption

of the Kolmogorov-41:Dissipation rate is an universal constant forDissipation rate is an universal constant foreach turbulent flow. each turbulent flow.

Comment of Landau (1942): The dissipation rate is a stochastic Comment of Landau (1942): The dissipation rate is a stochastic function, it is not constant.function, it is not constant.


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Inertial subrangeInertial subrange

, i. e.,L l

In every turbulent flow at sufficiently high Reynolds number, there is the In every turbulent flow at sufficiently high Reynolds number, there is the range of scales range of scales l l which are small compared with L, however they are large which are small compared with L, however they are large compared with compared with

Since the vortices of this range are much larger than Kolmogorov‘s vortices, Since the vortices of this range are much larger than Kolmogorov‘s vortices, we can assume that their Reynolds numbers are large we can assume that their Reynolds numbers are large and their motion is little affected by the viscosityand their motion is little affected by the viscosity

/llu


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E(k) k


formform


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Interpretation of different subrangesInterpretation of different subranges

Kolmogorov‘s law:Kolmogorov‘s law:


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Power law spectrum of KolmogorovPower law spectrum of Kolmogorov


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Experimental confirmationExperimental confirmation

Compensated energy Compensated energy spectrum for different spectrum for different flowsflows


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Statistische Auswertung: räumliches Energiespektrum in der J-Mode

1E+3 1E+4 1E+5 1E+6wave num ber (1/m )

1E-4

1E-3

1E-2

1E-1

1E+0

E_f

/E_f

max

J-m ode

x/D=2

x/D=3

x/D=5

x/D=7

x/D=9

inertial convective subrange

viscouse convectivek^-1

k^-5/3

Measurement of the energy spectrum performed by Measurement of the energy spectrum performed by the LTT Rostock (2007)the LTT Rostock (2007)

Concentration of Concentration of injected liquidinjected liquid


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x/D Kolmogorov -1.6667(-0.03)

2 -1.686

3 -1.684

5 -1.745

7 -1.73

9 -1.68

The slope is between -5/3 and -2The slope is between -5/3 and -2

Estimation of the Kolmogorov power in LTT Rostock Estimation of the Kolmogorov power in LTT Rostock measurementsmeasurements


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Classification of methods Classification of methods for turbulence modellingfor turbulence modelling

RANSRANS Semi-empirical modelingSemi-empirical modeling


Large energy Large energy containing structurescontaining structures

Dis

sip

atio

n r

ange

Dis

sip

atio

n r

ange

LESLES Universal modellingUniversal modelling

DNSDNS


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Classification of methods for turbulence modellingClassification of methods for turbulence modelling

50 m

m5

0 m

mResolution 300 Resolution 300 µµ

2D2D

2.72 mm2.72 mm

2.0

8 m

m2

.08

mm

Kornev N., Zhdanov V. and Hassel E.(2008) Study Kornev N., Zhdanov V. and Hassel E.(2008) Study of scalar macro- and microstructures in a of scalar macro- and microstructures in a confined jetconfined jet. . Int. Journal Heat and Fluid Flow, vol. Int. Journal Heat and Fluid Flow, vol. 29/3.29/3.

RANSRANS Semi empiric ModelSemi empiric Model


Large energy Large energy containing containing structuresstructures

Dis

sip

ati

on

Dis

sip

ati

on

LESLESUniverssl Model.Universsl Model.

DNSDNS

Large vortices Large vortices

Middle vorticesMiddle vortices

Small vorticesSmall vortices


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Kolmogorov - Obukhov law:Kolmogorov - Obukhov law:

2 1q

q l lS ( l ) ( u u )

Structure functionsStructure functions

qqqS ( l ) ( l ) ( l )

Kolmogorov - Obukhov lawKolmogorov - Obukhov law


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IntermittencyIntermittency

Discrepancy between measurementDiscrepancy between measurementand the prediction from the and the prediction from the Kolmogorov-Kolmogorov-Obukhov theory for the exponent Obukhov theory for the exponent of the structure function.of the structure function.

The reason of the discrepancy:The reason of the discrepancy:IntermittencyIntermittency(presence of laminar spots(presence of laminar spotsin every turbulent flows even at very in every turbulent flows even at very high Reynolds numbers).high Reynolds numbers).


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Kolmogorov theory K62

Assumption 1Assumption 1

Assumption 2 Lognormal law of Kolmogorov-Assumption 2 Lognormal law of Kolmogorov-ObukhovObukhov

This assumption is proved to be wrongThis assumption is proved to be wrong

Probability density function distribution for the dissipation rate:Probability density function distribution for the dissipation rate:


l ehrstuhl für modellierung und simulation statistical theory of the isotropic turbulence (k-41...

Documents

simulation slide

theory of kolmogorov

kolmogorov theory lecture

kolmogorov scale

small ones dissipation

large scales

samll vortices dissipation

turbulent flow