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Lehrstuhl für
Modellierung
und Simulation
Statistical theory of the isotropic
turbulence (K-41 theory)
2. Kolmogorov theory
Lecture 3
UNIVERSITY of ROSTOCK | CHAIR OF MODELLING AND SIMULATION
Kolmogorov Theory K41
Andrey Nikolaevich Kolmogorov
was 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)
„Every mathematician believes he is ahead over
all others. The reason why they don't say this in
public, is because they are intelligent people“
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Physical model beyond the K41
Most important physical processes are
• Transfer energy from large scales to small ones
• Dissipation of the energy in samll vortices
Two parameters are of importance: kinematic viscosity and dissipation rate
The size range
l < l EI
is referred to as the universal equilibrium range
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Hypothesis of local isotropy
L
k 3/ 2
Macroscale of the flow ,
uk
1/ 2
characteristic velocity of macrovortices
Kolmogorov‘s hypothesis of local isotropy
At sufficiently high Reynolds numbers Ret uL / , the small
-scale motion with scales l
L are statistically isotropic.
Directional information is lost. The laws describing the
small-scale motion are universal.
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Theory of Kolmogorov (1941) K-41
In every turbulent flow at sufficiently high Reynolds number,
l EI have a universal
the statistics of the small-scale motions l
form that is uniquely determined by kinematic viscosity and
turbulent energy dissipation rate
u
u
, u ,
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Kolmogorov scale, time and velocity
1/ 4
,
1/ 4
u ( ) ,
3
( /
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1/ 2
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Some useful estimations
k 3/ 2
L
uk
1/ 2
/ L Re t
3/ 4
u / u (Re t )
/ T (Re t )
,
1/ 4
,
1/ 2
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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 for
each turbulent flow.
Comment of Landau (1942): The dissipation rate is a stochastic
function, it is not constant.
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Inertial subrange
In every turbulent flow at sufficiently high Reynolds number, there is the
range of scales l which are small compared with L, however they are large
compared with ,
i. e., L l
Since the vortices of this range are much larger than Kolmogorov‘s vortices,
we can assume that their Reynolds numbers lul / are large
and their motion is little affected by the viscosity
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Inertial subrange
E(k) k
form
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Interpretation of different subranges
Kolmogorov‘s law:
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Power law spectrum of Kolmogorov
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Experimental confirmation
Compensated energy
spectrum for different
flows
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Measurement
of the Auswertung:
energy spectrum performed by
Statistische
the LTTEnergiespektrum
Rostock (2007)
räumliches
in der J-Mode
inertial convective subrange
1E+0
k^-5/3
E_f /E_fmax
1E-1
k^-1
viscouse convective
1E-2
Concentration of
injected liquid
J-mode
x/D=2
x/D=3
1E-3
x/D=5
x/D=7
x/D=9
1E-4
1E+3
1E+4
1E+5
1E+6
wave number (1/m)
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Estimation of the Kolmogorov power in LTT Rostock
measurements
x/D
2
3
5
7
9
Kolmogorov -1.6667(-0.03)
-1.686
-1.684
-1.745
-1.73
-1.68
The slope is between -5/3 and -2
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Classification of methods
for turbulence modelling
Large energy
containing structures
RANSSemi-empirical modeling
Dissipation range
Inertial subrange
LES Universal modelling
DNS
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Large energy
containing
structures
50 mm
Large vortices
Resolution 300 µ
2D
RANSSemi empiric Model
LES
Universsl Model.
DNS
2.08 mm
Middle vortices
Inertial subrange
Dissipation
Classification of methods for turbulence modelling
2.72 mm
Small vortices
Kornev N., Zhdanov V. and Hassel E.(2008) Study
of scalar macro- and microstructures in a
confined jet. Int. Journal Heat and Fluid Flow, vol.
29/3.
18
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Kolmogorov - Obukhov law:
Structure functions
S q ( l ) ( u2l u1l )q
Kolmogorov - Obukhov law
Sq ( l ) ( l )
q
q
(l )
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Intermittency
Discrepancy between measurement
and the prediction from the
KolmogorovObukhov theory for the exponent
of the structure function.
The reason of the discrepancy:
Intermittency
(presence of laminar spots
in every turbulent flows even at very
high Reynolds numbers).
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Kolmogorov theory K62
Assumption 1
Assumption 2 Lognormal law of KolmogorovObukhov
Probability density function distribution for the dissipation rate:
This assumption is proved to be wrong
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