Lehrstuhl fürModellierung und Simulation

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

1. Basic definitions of the statistical theory of turbulence

Lecture 3

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Basic definitions. Reynolds averaging Basic definitions. Reynolds averaging

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3

ij i j

ij i j

R ( x,r ) u ( x )u ( x r ),

R ( r ) u ( x )u ( x r ),

Basic definitions. Correlation function Basic definitions. Correlation function

2

0

i iii

i

ij ii j j

u ( x )u ( x r )( x,r ) ,

u ( x )

L ( x ) ( x,x )dx

2

0

i iii

i

i ii

u ( t ,x )u ( t ,x )( ,x ) ,

u ( t ,x )

T ( x ) ( ,x )d

Correlation functionCorrelation functionCorrelation function in homogeneousCorrelation function in homogeneousturbulence turbulence

Autocorrelation functionAutocorrelation function

Integral lengthIntegral length

Autocorrelation temporal functionAutocorrelation temporal function

Integral time Integral time

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SamplesSamples

4

0 20 40 60 80 100 120 140 160point num ber across the m ixer

-0.40

-0.20

0.00

0.20

0.40

0.60

0.80

1.00

Aut

ocor

ella

tion

func

tion

of f

50

mm

50 m

m

Resolution 300 Resolution 300 µµ

2D2D

AA

BB

CC

AABBCC

Typical form of the autocorrelation coefficient. Scalar turbulenceTypical form of the autocorrelation coefficient. Scalar turbulence

PLIF Measurements of the LTT RostockPLIF Measurements of the LTT Rostock

Physical meaning of sign changePhysical meaning of sign change

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SamplesSamples

5

50

mm

50 m

m

Resolution 300 Resolution 300 µµ

2D2D

Typical distribution of the integral length along the jet mixer. Typical distribution of the integral length along the jet mixer. Scalar turbulenceScalar turbulence

PLIF Measurements of the LTT RostockPLIF Measurements of the LTT Rostock

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Typical autocorrelation coefficient Typical autocorrelation coefficient along the jetalong the jet

6

(from Ginevsky et al. (2004) Acoustic control of turbulent jets. Springer)

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1 along upper border of nozzle at x/D=0.51 along upper border of nozzle at x/D=0.52 along the jet axis at x/d=3.02 along the jet axis at x/d=3.0

autocorrelation coefficient of the autocorrelation coefficient of the longitudinal velocity longitudinal velocity

Isotropic turbulenceIsotropic turbulence

7

2 2

2t t

u (x)u (x r) r F(r) G(r) u f ,

u (x)u (x r) u g G(r).

l l

t t

t t

u (x)u (x r) u (x)u (x r)f (r) ,g(r) ,

u (x)u (x) u (x)u (x)

l l

l l

2t t i iu u (x)u (x) u (x)u (x) .....u (x)u (x), l l

ij i j ijR F(r)r r G(r) 2ij i j ij2

f gR u ( r r g )

r1 f

g f r2 r

2t t i iu u (x)u (x) u (x)u (x) 1 / 3u (x)u (x), l l

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8

f

g

Taylor longitudinal length Taylor longitudinal length

Taylor transverse length Taylor transverse length

/Re u / g

Taylor Reynolds number Taylor Reynolds number

2 22 4

2 2 2

2 2

2 2 2

2

2

1 f f 1f (r) 1 (0)r O(r ) (0)

2 r r 2

r g 1f (r) 1 (0)

2 r 2

rg(r) 1 .

f

f g

g

Isotropic turbulenceIsotropic turbulence

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Correlation function in Fourrier spaceCorrelation function in Fourrier space

9

3

1

8

ikr

ikr

ˆf ( r ,t ) f ( k ,t )e dk ,

f̂ ( k ,t ) f ( r ,t )e dr

3

1

8

ikr ikr

ij ij ij ijR ( r ) ( k )e dk , ( k ) R ( r )e dr

1 11 1 1 1 2 3 2 3

10 0

2

ik rij ij ij( k ) R ( r , , )e dr ( k ,k ,k )dk dk

Usually it is possible to measure only the Usually it is possible to measure only the „„One dimensional spectral Function“One dimensional spectral Function“

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Proof

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1 1

1 1

1 1 1

1 1 1

1( ,0,0) ( ,0,0) ,

2

( ,0,0) ( ,0,0)

ik rij ij

ik rij ij

k R r e dr

R r k e dk

1 2 3 1 2 3 1 2 3( , , ) ( , , ) ikrij ijR r r r k k k e dk dk dk

1 1 2 3 2 3( ,0,0) ( , , ) ,ij ijk k k k dk dk

1 11 1 2 3 2 3 1( ,0,0) ( , , ) ik r

ij ijR r k k k dk dk e dk

Spectral density of the kinetic energySpectral density of the kinetic energy

11

0

0

1 1

2 2ii ii ii

ii

k

TKE R (o ) ( k )dk ( k )dk ,

E( k ) ( k )dk ,

TKE( k ) E( k )dk ,

E(k) dk is the contribution of oscillations with the wave numbers k<k<k+dkE(k) dk is the contribution of oscillations with the wave numbers k<k<k+dk to the kinetic energy of the turbulent motion.to the kinetic energy of the turbulent motion.E(k) is the density of the kinetic energy depending on wave numbers. E(k) is the density of the kinetic energy depending on wave numbers. The dependence E(k) isreferred to as the energy spectrumThe dependence E(k) isreferred to as the energy spectrum

3

1

8

ikr ikr

ij ij ij ijR ( r ) ( k )e dk , ( k ) R ( r )e dr ij i jR ( r ) u ( x )u ( x r )

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