l ehrstuhl für modellierung und simulation statistical theory of the isotropic turbulence (k-41...
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Lehrstuhl fürModellierung und Simulation
Statistical theory of the isotropic turbulence (K-41 theory)
2. Kolmogorov theory
Lecture 3
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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 ( ) ,
( /
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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
Inertial subrangeInertial subrange
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
Inertial subrangeInertial subrange
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
Inertial subrangeInertial subrange
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:
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