Transcript Folie 1 - uni

Lehrstuhl für
Modellierung
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
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Basic definitions. Correlation function
Rij ( x,r )  ui ( x )u j ( x  r ),
Rij ( r )  ui ( x )u j ( x  r ),
ii ( x,r ) 
ui ( x )ui ( x  r )
2
i
u (x)
Correlation function
Correlation function in homogeneous
turbulence
, Autocorrelation function

Lij ( x )   ii ( x,x j )dx j
0
ii (  ,x ) 
Integral length
ui ( t,x )ui ( t   ,x ) Autocorrelation temporal function
,
2
ui ( t,x )

Ti ( x )   ii (  ,x )d
Integral time
0
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Samples
Typical form of the autocorrelation coefficient. Scalar turbulence
1.00
C
B
A
Resolution 300 µ
0.60
A
0.40
B
0.20
0.00
2D
-0.20
50 mm
Autocorellation function of f
0.80
C
-0.40
0
20
40 60 80 100 120 140 160
point number across the mixer
Physical meaning of sign change
PLIF Measurements of the LTT Rostock
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Samples
Typical distribution of the integral length along the jet mixer.
Scalar turbulence
Lf /d
Resolution 300 µ
0,8
50 mm
0,7
0,6
0,5
0,4
2D
0,3
r-mode
j-mode
0,2
0,1
0
2
4
6
8
x/D
10
PLIF Measurements of the LTT Rostock
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Typical autocorrelation coefficient
along the jet
autocorrelation coefficient of the
longitudinal velocity
1 along upper border of nozzle at x/D=0.5
2 along the jet axis at x/d=3.0
(from Ginevsky et al. (2004) Acoustic control of turbulent jets. Springer)
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Isotropic turbulence
u2  ul (x)ul (x)  ut (x)ut (x)  1/
.....u
3ui i(x)u
(x)ui i(x),
(x),
f g
R ij  u ( 2 ri rj  g ij )
Rij  F(r)ri rj  G(r)ij
r
u (x)u l (x  r)
u (x)u t (x  r)
f (r)  l
,g(r)  t
, g  f  1 r f
u l (x)u l (x)
u t (x)u t (x)
2 r
2
u l (x)u l (x  r)  r 2 F(r)  G(r)  u 2f ,
u t (x)u t (x  r)  u 2g  G(r).
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Isotropic turbulence
Taylor longitudinal length  f
Taylor transverse length
g
Taylor Reynolds number
Re  u/ g /
1  2f
 2f
1
2
4
f (r)  1 
(0)r

O(r
)

(0)


2 r 2
r 2
2 f 2
f (r)  1 
g(r)  1 
r2
2 f 2
r2
g 2
 2g
1
 2 (0)  
r
2 g 2
.
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Correlation function in Fourrier space
f ( r ,t ) 
ˆf ( k ,t ) 

1
8 3



ˆf ( k ,t )eikr dk ,

f ( r ,t )e  ikr dr



1
ikr
Rij ( r )   ij ( k )e dk ,ij ( k )  3  Rij ( r )e ikr dr
8 

Usually it is possible to measure only the
„One dimensional spectral Function“
ij ( k1 ) 
1
2



Rij ( r1 ,0, 0 )eik1r1 dr1 
 
   ( k ,k ,k
ij
1
2
3
)dk2dk3
 
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Proof
1
2
ij ( k1 ,0,0) 


Rij ( r1 ,0,0)e  ik1r1 dr1 ,



Rij ( r1 ,0,0) 
ij ( k1 ,0,0)eik1r1 dk1

  
Rij ( r1 , r2 , r3 ) 

ikr

(
k
,
k
,
k
)
e
 ij 1 2 3 dk1dk2dk3
  
 
 ik1r1
Rij ( r1 ,0,0)       ij (k1 , k2 , k3 )dk2dk3 e dk1
   


 
ij (k1 ,0,0) 
 
ij
(k1, k2 , k3 )dk2dk3 ,
 
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Spectral density of the kinetic energy


1
Rij ( r )   ij ( k )e dk ,ij ( k )  3  Rij ( r )e ikr dr
8 

ikr

Rij ( r )  ui ( x )u j ( x  r )

1
1
TKE  Rii ( o )    ii ( k )dk    ii ( k )dk ,
2
2 
0
E( k )    ii ( k )dk ,
k

TKE( k )   E( k )dk ,
0
E(k) dk is the contribution of oscillations with the wave numbers k<k<k+dk
to the kinetic energy of the turbulent motion.
E(k) is the density of the kinetic energy depending on wave numbers.
The dependence E(k) isreferred to as the energy spectrum
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