Anisotropic Pressure and Acceleration
Spectra in Shear Flow
Yoshiyuki Tsuji
Nagoya University Japan
Acknowledgement :
Useful discussions and advices were given by Prof. Y. Kaneda
Objective
T. Ishihara, K.Yoshida, and Y.Kaneda,
Anisotropic Velocity Correlation Spectrum at Small Scales in Homogeneous
Turbulent Shear Flow, Phys. Rev., Letter, vol.88,154501,(2002)
Shear effect on inertial-range velocity statistics are directly
investigated .
This idea is applied to the pressure field in the uniform shear
flow, and the shear effect on pressure and pressure gradient
(acceleration) is studied experimentally up to the Reynolds
number based on Taylor micro scale is 800.
2. Pressure Measurements
Pressure measurement
Kolmogorov length scale is   0.19mm for R  700 .
Φ=0.08mm
Φ=0.15mm
2.0
Φ=0.3mm
Φ=0.5mm
0.4mm
d
12mm
l
δ
Microphone
Microphone: 2  10 4  ~p  3.2  10 3 [Pa] 2  101  f  7.0  10 4 [Hz]
1/8 inch
d  3.2 [mm]
pressure measurement inside the boundary layer
Probability density functions
Probability density
100
10-1
10-2
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0
0.0
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0
p/
p/
10-3
10-4
:EXP(R =200)
:DNS(R =164)
:EXP(R =320)
:DNS(R =283)
10-5
-12.0
-8.0
-4.0
0.0
4.0
p/
-12.0
-8.0
-4.0
0.0
4.0
p/
DNS: Kaneda & Ishihara
Nearly homogeneous isotropic field.
Kolmogorov constant
Pressure Spectrum
R  600
R>600
(DNS:Gotoh,2001)
Nearly homogeneous isotropic field.
Pressure measurement in Boundary layer
Epp/[
2
2
u ]
104
slope=-1.0
103
slope=-1.2
102
101
100
10-1
10-2
10-3
+
y =200
R=5875,7420,8925,10515,12070,15205
:Abe et al. (R =2066)
10-4
10-3
10-2
f/u2
10-1
100
-7/3 power-law is not observed in the overlap region of smooth-wall
boundary layer even if the Reynolds number is very high.
Pressure spectrum in the boundary layer
2. Experiment
Experiments: Driving Mixing Layer
y
Mixing layer centerline
Transition region
d=350mm
Potential Core
x/d~5
x/d
Nozzle exit
Mixing layer centerline
L=700mm
In this region, flow reversals are
unlikely and large yaw angles by the
flow are infrequent.
x
Driving Mixing Layer
:x/d=1
:x/d=2
:x/d=3
:x/d=4
:x/d=5
x/d=5
U/UJ
1.0
0.5
0.0
-0.2
x/d=4
-0.1
urms/UJ
y
0.2
0.3
0.2
x/d=2
Nozzle exit
0.1
y/(x-x 0)
x/d=3
x/d=1
0.0
:x/d=1
:x/d=2
:x/d=3
:x/d=4
:x/d=5
0.1
0.0
-0.2
-0.1
0.0
0.1
y/(x-x 0)
Nearly homogeneous shear flow.
0.2
0.3
Reynolds number & Shear parameter
Reynolds number
S*
10-1
R  u ' 

u
x  / u ' 2
2
10-2
Shear parameter
102
S  S  k
*
103
R


1/ 2
1
20

1
S flow
dU dyS k   R  R 1
Simple uniform shear
A 3
*
Driving mixing layer is close to the simple uniform shear flow.
3. Theoretical formula
Shear effect on velocity fluctuation
According to the formula presented by Ishihara, Yoshida and Kaneda
PRL(vol.88,154501,2002), velocity spectrum is defined by

Qij (k , t ) 
1
(2 )3


 

ik r
 dr ui x  r , t u j x, t  e



0
Qij (k , t )  Qij (k , t )  Qij (k , t )
S12  U y :Simple mean shear
  s K1 S2 /:independent
 of wave number k
3 11/ 3
0
Qij (k ) 
 k
Pij (k ) Isotropic part (K41)
 4 u :dependent of wave number k
0
N

Modification due to the
existence
k  1  of mean shear.

 :characteristic eddy
 size
Qij (k , t )  Cijmn (k )P(k )S mn  Cijmnkl (k ) R(k ) S mn S kl  Anisotropic part
u :characteristic velocity scale u   
1/ 3






 1/ 3 13/ 3
 k k
Cijmn (k )  A Pim (k ) Pjn (k )  Pin (k ) Pjm (k )  k
 BPij (k ) m 2 n
S ij  d U i dx j
k


2 
for
large
wave
numbers
N
S
Pij   ij  ki k j k
Shear effect on velocity fluctuation

Velocity spectrum is obtained
 by the summation with respect to k over a
spherical shell with radius k .



0
Eij (k )   Qij (k , t )   Q ij (k , t )   Qij (k , t )

k k

k k

ˆ
ˆ
E (k )   k a kbQij (k , t )
ab
ij

k k
Isotropic part (K41)
Anisotropic part
E12 (k1 )  
E11 (k1 ) 

k k
In usual experiments, one-dimensional spectrum
is obtained.
18
5 / 3
K 0 2 / 3 k1
55
4 18
5 / 3
 K 0 2 / 3k1
3 55
is proportional to mean shear
36
 33 A  7 B  1/ 3 S12k17 / 3
1729
432
 2 A  B  1/ 3 S12k17 / 3
E (k1 )  
1729
12
11
E22 (k1 ) 

 u1u2   E12 (k1 )dk1


du1 du1
12
  E11
(k1 )dk1

dx1 dx2
Isotropic velocity spectrum
100
E11(k1 ) , E22(k1)
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10
-8
10-9 -1
10
R=710
:E 11(k1)
:E 22(k1)
100
101
102
103
104
k1
Isotropic part (K41)
18
5 / 3
E11 (k1 ) 
K 0 2 / 3 k1
55
4 18
5 / 3
E22 (k1 )   K 0 2 / 3k1
3 55
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
10-10
10-11
600<R<700
10-12
10-13
10-14 -1
10
100
101
E1112(k1)/[1/3S12]
E12(k1)/[1/3S12]
Anisotropic velocity spectrum
A  0.17
102
k1
103
104
100
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
10-10
10-11
10-12
-13
B10
-140.45600<R <700
10

10-15
10-16 -1
10
100
101
102
k1
103
104
Anisotropic part is proportional to mean shear even if S is changed.
12
E12 (k1 )  
36
 33 A  7 B  1/ 3 S12k17 / 3 E1112 (k1 )   432  2 A  B  1/ 3 S12k17 / 3
1729
1729
Shear effect on pressure
According to the formula presented by Ishihara, Yoshioda and Kaneda
PRL(vol.88,154501,2002), pressure spectrum is defined by

Q p (k , t ) 
1
(2 ) 3

  

 ik  r
 dr px  r , t  px, t  e



0
Qp (k , t )  Qp (k , t )  Qp (k , t )
Isotropic part (K41)
Q p (k )  K p 
0
Anisotropic part
4/3
k
13 / 3

kk
Q p (k )   1 22 k 5 S12
k
S12  U y :Simple mean shear
Modification due to the
existence of mean shear.



Qp (k , t )  Cmn (k ) P(k ) Smn  Cijkl (k ) R(k ) Sij Skl 


C ijkl (k ) :4th order isotropic tensor
C mn (k ) :2nd order isotropic tensor
Shear effect on pressure spectrum

Pressure spectrum is obtained by the summation with respect to k over a

spherical shell with radius k .



0
E pp (k )   Q p (k , t )   Q p (k , t )   Q p (k , t )

k k

k k

k k
Isotropic part (K41)
E pp (k1 ) 
x2
7
7 / 3
K P 4 / 3 k1
6
x1
Anisotropic part
E pp (k1 )  0  k1
9 / 3
S12  Ck1
11/ 3
S12 S12
0
Shear effect on pressure spectrum appears in the second order of S12
Shear effect on pressure spectrum

Pressure spectrum is obtained by the summation with respect to k over a

spherical shell with radius k .



0
E pp (k )   Q p (k , t )   Q p (k , t )   Q p (k , t )

k k

k k

k k
x2
x2
Isotropic part (K41)
E pp (k1 , x2 )
x2
7/3
2C1  k1x2 
4/3
7 / 3
 K P (k1x2 )


(5 / 3)  2 
7/6
 7 / 6 (k1x2 )
Anisotropic part
E pp (k1 , x2 )
x2
3
5/ 2


2
C
d
k

x


3
1
1
2
 S12 (k1x2 )
  5 / 2 (k1x2 )

(5 / 3) d (k1x2 )  2 

x1
Shear effect on pressure spectrum
IYK formula is well satisfied in this experiment.
108
3
106
105
Epp(k1,x2)/x2
Epp(k1,x2)/x27/3
107
104
103
102
101
100
10-1
10-2
10-3 -3
10
10-2
10-1
100
k1x2
Isotropic part (K41)
101
109
108
107
106
105
104
103
102
101
100
10-1
10-2
10-3
10-4 -3
10
  2.45
10-2
10-1
100
101
k1x2
Anisotropic part
5/ 2


7/6
E
(
k
,

x
)
2
C
d
k

x


pp
1
2
3
1
1
2
E pp (k1 , x2 )
2C1  k1x2

4/3
7 / 3
S
(
k

x
)

(
k

x
)
5/ 2
1
2 
3
 K P (k1x2 )
 12 71/ 6 (k21x2)(5 / 3) d (k x )  2 
7/3

x


1
2
(5 / 32 )  2 

x2
Shear effect on velocity&pressure
According to the formula presented by Ishihara, Yoshioda and Kaneda
PRL(vol.88,154501,2002), velocity&pressure spectrum is defined by

Ri (k , t ) 
1
(2 )3

  

ik r
 dr px  r , t ui x, t  e



0
Ri (k , t )  Ri (k , t )  Ri (k , t )
Isotropic part (K41)
Ri (k )  0
0
Anisotropic part



Ri (k , t )  Cimn (k )r (k )Smn  Cimnkl (k )s(k )Smn Skl 

kk k
k
k
k 

Cimn (k )  a i m3 n  b  im n   mn i   ni m 
kˆ
kˆ
kˆ
kˆ 


Cimnkl (k ) :5th order isotropic tensor
Shear effect on velocity&pressure spectrum

Pressure-velocity spectrum is obtained
by the summation with respect to k

over a spherical shell with radius k .



0
E pui (k )   Ri (k , t )   R i (k , t )   Ri (k , t )

k k

k k

k k

ˆ
E (k )   k j Ri (k , t )
j
pui

du1
p
  k1E1pu1 (k1 )dk1
dx1
0

k k
Anisotropic part
(i  1)
E pu1 (k1 )  0
(i  2)

du
2
p 1   k1E pu
(k1 )dk1
1
dx2
0
(i  1, j  1)
E1pu1 (k1 )  0
(i  2, j  1)
6  2/3
9  2/3
 18
 9
8 / 3
8 / 3
2
E pu2 (k1 )  
a  b  S12k1
E pu1 (k1 )  
a  b  S12k1
11 
48 
 187
 140
10-4
10-5
10-6
10-7
10-8
10-9
10-10
10-11
10-12
10-13
10-14
10-15
10-16
(i  2)
Epu12(k1)/[2/3S12]
Epu2(k1)/[2/3S12]
Shear effect on velocity&pressure spectrum
101
102
103
k1
104
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
10-10
10-11
10-12
10-13
10-14
10-15
10-16 0
10
101
102
103
k1
(i  2, j  1)
104
a  0.1 b  0.03
6  2/3
9  2/3
 18
 9
8 / 3
8 / 3
2
E pu2 (k1 )  
a  b  S12k1
E pu1 (k1 )  
a  b  S12k1
11 
48 
 187
 140
Isotropic velocity spectrum
100
E11(k1 ) , E22(k1)
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10
-8
10-9 -1
10
R=710
:E 11(k1)
:E 22(k1)
100
101
102
103
104
k1
Isotropic part (K41)
18
5 / 3
E11 (k1 ) 
K 0 2 / 3 k1
55
4 18
5 / 3
E22 (k1 )   K 0 2 / 3k1
3 55
Acceleration
In a usual notation, pressure relates to acceleration vector ;

2
a   p     u   p  
1  p
ai   
  xi
dp dp (t ) dt dp (t ) 1




dx
dt
dx
dt
Uc



1  p
aj  
  x j




i, j  1,2,3
U c Local mean velocity
dp(t ) nmax
   2f n an sin 2f n t   2f n bn cos2f n t 
dt
n 0
2f n max  1 /  k
Shear effect on acceleration
Similar discussion is possible in case of acceleration  i  p xi .
E i j (k ) 

ki k j

k k
k2


ki k j 0 
ki k j
Q p (k , t )   2 Q p (k , t )   2 Q p (k , t )


k k k
k k k
Isotropic part (K41)
E11 (k1 )  A1 4 / 3k1
1/ 3
E 2 2 (k1 )  A2 4 / 3k1
1/ 3
 0  k1
3 / 3
 0  k1
S12  C1k1
3 / 3
5 / 3
S12  C2k1
Anisotropic part
S12 S12
5 / 3
S12 S12
 2 , there is no
As far as looking
for the variance
of 1 5and
3 / 3
/3
4 / 3 1/ 3
E  (k1 )  0effect
  k1by shear.
 C3k1 S12  0  k1 S12 S12
significant
1 2

11   E  (k1 )dk1
0
1 1

 2 2   E  (k1 )dk1
0
2 2

1 2   E  (k1 )dk1
0
1 2
Kolmogorov scaling for acceleration
Following the Kolmogorov’s idea, acceleration is scaled by
energy dissipation and kinematic viscosity,
and the constant a 0 becomes universal.

2
a   p     u   p  
1  p
ai   
  xi



ai a j  a 0 
1  p
aj  
  x j




i, j  1,2,3
 1 / 2 ij
2/3
a 0 :Universal Constant
Kolmogorov scaling for acceleration
a1 a1  a 0 
a0
101
:(2.5*R0.25 +0.08*R0.11 )/3.0
100
S 0
*
102
:Mixing layer
103
R
:DNS(Vedula&Yeung)
:DNS(Gotoh&Fukayama)
a 0 is not constant but increases as Reynolds number increases.
There is no significant difference between S *  0 and S *  0
 1 / 2
2/3
S*  0
Summary : pressure
• In a simple shear flow, shear effect doe not appear
clearly in a single-point statistics.
x2
Anisotropic part
E pp (k1 )  0  k1
9 / 3
S12  Ck1
11/ 3
S12 S12
x1
Shear effect on pressure spectrum appears in the second order of S12
• Shear effect can be evaluated by two-point statistics.
x2
Anisotropic part
  2.45
E pp (k1 , x2 )
x2
3
5/ 2


2
C
d
k

x


3
1
1
2
 S12 (k1x2 )
  5 / 2 (k1x2 )

(5 / 3) d (k1x2 )  2 

x2
x1
Summary : pressure-velocity correlation
• In a simple shear flow, shear effect on pressure
velocity correlation is evaluated by the relation.
Anisotropic part
(i  2)
6 
 18
8 / 3
E pu2 (k1 )  
a  b  2 / 3 S12k1
11 
 187
(i  2, j  1)
9  2/3
 9
8 / 3
2
E pu
(
k
)

a

b

S
k


1
12 1
1
48 
 140
a  0.1 b  0.03

pu2   E pu2 (k1 )dk1
0

du
2
p 1   k1E pu
(k1 )dk1
1
dx2
0
Summary : Acceleration
• In a simple shear flow, shear effect appears on the
correlation between 1 and  2 .
ai  
Anisotropic part
E11 (k1 )  0  k1
3 / 3
E 2 2 (k1 )  0  k1
E1 2 (k1 )  C3k1
S12  C1k1
3 / 3
3 / 3
5 / 3
S12  C2k1
S12 S12
5 / 3
S12  0  k1
5 / 3
1  p

  xi



S12 S12
S12 S12
• The constant a0 defined by Kolmogorov scaling of
acceleration variance is not affected clearly by shear.
ai a j  a 0 
 1 / 2 ij
2/3
Frozen Flow Hypothesis for Pressure
Wall pressure spectrum
Frozen flow hypothesis
DNS result by H. Abe for Channel Flow
Wall pressure spectra
Probability density function of acceleration
Mixing layer
100
probability
10-1
10-2
10-3
10-4
10-5
:La Porta et al.
10
-6
-20.0
-10.0
0.0
(dp/dx)/
*
small
R
large
S
10.0
20.0
Pressure measurement in cylinder wake
~ /U / d 
ω
3

vorticity
~
P [ Pa]
pressure
Q / U  / d 
2
Second invariance of
velocity gradient tensor
100
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
10-10
10-11
10-12
10-13
10-14
10-15
10-16
100
10-1
slope=-1/3
10-2
slope=-7/3
Spectrum
Spectrum
Spectra of pressure and acceleration
Inertial range
10-3
10-4
Inertial range
10-5
10-6
(a) pressure
10-7
10-8
100
101
102
103
frequency [Hz]
E pp ( f )  f
7 / 3
104
(b) acceleration
100
101
102
103
frequency [Hz]
Epp ( f )  f 1 / 3
104