Anisotropic Pressure and Acceleration Spectra in Shear Flow

35
Anisotropic Pressure and Acceleration Spectra in Shear Flow Yoshiyuki Tsuji Nagoya University Japan Acknowledgement : Useful discussions and advices were given by Prof. Y. Kaneda

description

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, - PowerPoint PPT Presentation

Transcript of Anisotropic Pressure and Acceleration Spectra in Shear Flow

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

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.

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)

2. Pressure Measurements

Pressure measurement

Φ=0.15mm

d

0.4mm

Φ=0.5mm

l

2.0

12mmδ

Microphone: [Pa]   [Hz]

[mm]

34 102.3~102 p 41 100.7102 f2.3d1/8 inch

Microphone

Kolmogorov length scale is for .700Rmm19.0

Φ=0.3mm

Φ=0.08mm

pressure measurement inside the boundary layer

Probability density functions

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.00.0

0.1

0.2

0.3

0.4

0.5

p/

-12.0 -8.0 -4.0 0.0 4.0

10-5

10-4

10-3

10-2

10-1

100

p/

Prob

abili

ty d

ensi

ty

:EXP(R=200)

:DNS(R=164)

-12.0 -8.0 -4.0 0.0 4.0

p/

:DNS(R=283)

:EXP(R=320)

-3.0 -2.0 -1.0 0.0 1.0 2.0 3.00.0

0.1

0.2

0.3

0.4

0.5

p/

Nearly homogeneous isotropic field.

DNS: Kaneda & Ishihara

Pressure Spectrum

Nearly homogeneous isotropic field.

R>600

(DNS:Gotoh,2001)

Kol

mog

orov

con

stan

t 600R

Pressure measurement in Boundary layer

10-4 10-3 10-2 10-1 10010-3

10-2

10-1

100

101

102

103

104

f/u2

Epp

/[2 u

2 ]

:Abe et al. (R=2066)

slope=-1.0

slope=-1.2

R=5875,7420,8925,10515,12070,15205y+=200

Pressure spectrum in the boundary layer

-7/3 power-law is not observed in the overlap region of smooth-wall boundary layer even if the Reynolds number is very high.

2. Experiment

Experiments: Driving Mixing Layer

x/d

Nozzle exitPotential Core x/d~5

d=350mm

Mixing layer centerline

Mixing layer centerline

Transition region

y

L=700mm

In this region, flow reversals are unlikely and large yaw angles by the flow are infrequent.

-0.2 -0.1 0.0 0.1 0.2 0.30.0

0.5

1.0

y/(x-x0)

U/U

J

:x/d=1:x/d=2:x/d=3:x/d=4:x/d=5

Driving Mixing Layer

-0.2 -0.1 0.0 0.1 0.2 0.30.0

0.1

0.2

y/(x-x0)

u rm

s/UJ

:x/d=1:x/d=2:x/d=3:x/d=4:x/d=5

Nozzle exit

x/d=5

x

x/d=4

x/d=3

x/d=2

x/d=1

Nearly homogeneous shear flow.

y

Reynolds number & Shear parameter

11*

3

201 RRA

SSimple uniform shear flow

102 103

10-2

10-1

R

S*

Driving mixing layer is close to the simple uniform shear flow.

'uR 22 '/ uxu

Reynolds number

kSS *

dydUS

Shear parameter

2/1 k

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

rkijiij etxutrxurdtkQ

,,

)2(

1),(

3

),(),(),( 0 tkQtkQtkQ ijijij

Ss 1 :independent of wave number

k

uN :characteristic eddy size

:characteristic velocity scaleu

3/1 u

:dependent of wave number k

SN for large wave numbers

1kIsotropic part (K41)

klmnijmnklmnijmnij SSkRkCSkPkCtkQ )()()()(),( Anisotropic part

)(4

)( 3/113/200 kPkK

kQ ijij

2

3/133/1 )()()()()()(k

kkkBPkkPkPkPkPAkC nm

ijjminjnimijmn

Modification due to the existence of mean shear.

:Simple mean shearyUS 12

jiij dxUdS 2kkkP jiijij

Velocity spectrum is obtained by the summation with respect to over a spherical shell with radius .

Shear effect on velocity fluctuation

k

k

kk

ijkk

ij

kkijij tkQtkQtkQkE

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

Anisotropic part

Isotropic part (K41) 3/51

3/20111 55

18)( kKkE 3/5

13/2

0122 55

18

3

4)( kKkE

kk

ijbaabij tkQkkkE

),(ˆˆ)(

3/7112

3/1112 733

1729

36)( kSBAkE

3/7112

3/11

1211 2

1729

432)( kSBAkE

111221 )( dkkEuu

11

1211

2

1

1

1 )( dkkEdx

du

dx

du

is proportional to mean shear

In usual experiments, one-dimensional spectrum is obtained.

Isotropic velocity spectrum

Isotropic part (K41)3/5

13/2

0111 55

18)( kKkE 3/5

13/2

0122 55

18

3

4)( kKkE

10-1 100 101 102 103 10410-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

k1

E11

(k1

) , E

22(k

1)

R=710

:E11(k1)

:E22(k1)

Anisotropic velocity spectrum

Anisotropic part

10-1 100 101 102 103 10410-1610-1510-1410-1310-1210-1110-1010-910-810-710-610-510-410-310-210-1100

k1

E11

12(k

1)/[1/

3 S12

]

600<R<700

10-1 100 101 102 103 10410-1410-1310-1210-1110-1010-910-810-710-610-510-410-310-210-1

k1

E12

(k1)

/[1/

3 S12

]

600<R<700

3/7112

3/1112 733

1729

36)( kSBAkE 3/7

1123/1

11211 2

1729

432)( kSBAkE

17.0A 45.0B

is proportional to mean shear even if is changed.

12S

Shear effect on pressure

According to the formula presented by Ishihara, Yoshioda and Kaneda PRL(vol.88,154501,2002), pressure spectrum is defined by

rkip etxptrxprdtkQ

,,

)2(

1),(

3

),(),(),( 0 tkQtkQtkQ ppp

)(kCmn

:2nd order isotropic tensor )(kCijkl

:4th order isotropic tensor

Isotropic part (K41)

3/133/40 )( kKkQ pp

klijijklmnmnp SSkRkCSkPkCtkQ )()()()(),( Anisotropic part

Modification due to the existence of mean shear.

:Simple mean shearyUS 12

125

221)( Sk

k

kkkQp

Pressure spectrum is obtained by the summation with respect to over a spherical shell with radius .

Shear effect on pressure spectrum

k

k

12123/11

1123/9

11 0)( SSkCSkkEpp

kk

pkk

p

kkppp tkQtkQtkQkE

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

Isotropic part (K41)

Anisotropic part

Shear effect on pressure spectrum appears in the second order of 12S

3/71

3/41 6

7)( kKkE Ppp

1x

2x

0

Pressure spectrum is obtained by the summation with respect to over a spherical shell with radius .

Shear effect on pressure spectrum

k

k

kk

pkk

p

kkppp tkQtkQtkQkE

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

Isotropic part (K41)

)(2)3/5(

2)(

),(216/7

6/7

2113/721

3/43/7

2

21 xkxkC

xkKx

xkEP

pp

1x

2x

2x

Anisotropic part

)(2)()3/5(

2)(

),(212/5

2/5

21

21

1321123

2

21 xkxk

xkd

dCxkS

x

xkEpp

Shear effect on pressure spectrum

Isotropic part (K41) Anisotropic part

10-3 10-2 10-1 100 10110-3

10-2

10-1

100

101

102

103

104

105

106

107

108

k1x2

Epp

(k1,x

2)/

x 27/

3

10-3 10-2 10-1 100 10110-410-310-210-1100101102103104105106107108109

k1x2

Epp

(k1,x

2)/

x 23

45.2

)(2)3/5(

2)(

),(216/7

6/7

2113/721

3/43/7

2

21 xkxkC

xkKx

xkEP

pp

)(2)()3/5(

2)(

),(212/5

2/5

21

21

1321123

2

21 xkxk

xkd

dCxkS

x

xkEpp

IYK formula is well satisfied in this experiment.

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

rkiii etxutrxprdtkR

,,

)2(

1),(

3

),(),(),( 0 tkRtkRtkR iii

)(kCimnkl

:5th order isotropic tensor

Isotropic part (K41)

0)(0 kRi

klmnimnklmnimni SSkskCSkrkCtkR )()()()(),(

Anisotropic part

k

k

k

k

k

kb

k

kkkakC m

nii

mnn

imnmi

imn ˆˆˆˆ)(

3

Pressure-velocity spectrum is obtained by the summation with respect to over a spherical shell with radius .

Shear effect on velocity&pressure spectrum

k

k

kk

ikk

i

kkipu tkRtkRtkRkE

i

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

Anisotropic part

0)( 11kEpu

)1( i

kk

ijjpu tkRkkE

i

),(ˆ)(

0

111

11

1 )(1

dkkEkdx

dup pu

0

112

12

1 )(1

dkkEkdx

dup pu

0)( 11

1kEpu

)1,1( ji

3/8112

3/21 11

6

187

18)(

2

kSbakEpu

)2( i3/8

1123/2

12

48

9

140

9)(

1

kSbakEpu

)1,2( ji

Shear effect on velocity&pressure spectrum

3/8112

3/21 11

6

187

18)(

2

kSbakEpu

)2( i3/8

1123/2

12

48

9

140

9)(

1

kSbakEpu

)1,2( ji

101 102 103 10410-16

10-15

10-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

10-4

k1

Epu

2(k 1

)/[

2/3 S

12]

100 101 102 103 10410-1610-1510-1410-1310-1210-1110-1010-910-810-710-610-510-410-310-210-1

k1

Epu

12 (k1)

/[2/

3 S12

]

1.0a 03.0b

Isotropic velocity spectrum

Isotropic part (K41)3/5

13/2

0111 55

18)( kKkE 3/5

13/2

0122 55

18

3

4)( kKkE

10-1 100 101 102 103 10410-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

k1

E11

(k1

) , E

22(k

1)

R=710

:E11(k1)

:E22(k1)

Acceleration

3,2,1, ji

pupa 2

i

i x

pa

1

In a usual notation, pressure relates to acceleration vector ;

j

j x

pa

1

knf /12 max

Local mean velocity cU

max

0

2cos22sin2)( n

nnnnnnn tfbftfaf

dt

tdp

cUdt

tdp

dx

dt

dt

tdp

dx

dp 1)()(

Similar discussion is possible in case of acceleration .

Shear effect on acceleration

ii xp

kk

pji

kk

pji

kkp

ji tkQk

kktkQ

k

kktkQ

k

kkkE

ji

),(),(),()(

20

22

Isotropic part (K41) Anisotropic part

12123/5

11123/3

13/1

13/4

11 0)(11

SSkCSkkAkE

12123/5

12123/3

13/1

13/4

21 0)(22

SSkCSkkAkE

12123/5

1123/3

133/1

13/4

1 00)(21

SSkSkCkkE

11011 )(11

dkkE

11022 )(22

dkkE

11021 )(21

dkkE

1 2

Kolmogorov scaling for acceleration

0a

3,2,1, ji

ijji aaa 2/13/2

0

:Universal Constant

pupa 2

i

i x

pa

1

Following the Kolmogorov’s idea, acceleration is scaled by energy dissipation and kinematic viscosity, and the constant becomes universal.

j

j x

pa

1

0a

Kolmogorov scaling for acceleration

2/13/2

011 aaa

102 103100

101

R

a 0

:DNS(Vedula&Yeung)

:DNS(Gotoh&Fukayama)

:(2.5*R0.25+0.08*R

0.11)/3.0

is not constant but increases as Reynolds number increases.

There is no significant difference between and

0a

0* S 0* S

0* S

102 103100

101

R

a 0

0* S:Mixing layer

)(2)()3/5(

2)(

),(212/5

2/5

21

21

1321123

2

21 xkxk

xkd

dCxkS

x

xkEpp

Summary : pressure

• In a simple shear flow, shear effect doe not appear clearly in a single-point statistics.

• Shear effect can be evaluated by two-point statistics.

12123/11

1123/9

11 0)( SSkCSkkEpp

Anisotropic part

Shear effect on pressure spectrum appears in the second order of 12S

1x

2x

1x

2x

2xAnisotropic part

45.2

Summary : pressure-velocity correlation

• In a simple shear flow, shear effect on pressure velocity correlation is evaluated by the relation.

Anisotropic part

3/8112

3/21 11

6

187

18)(

2

kSbakEpu

)2( i

3/8112

3/21

2

48

9

140

9)(

1

kSbakEpu

)1,2( ji

0

112 )(2

dkkEpu pu

0

112

12

1 )(1

dkkEkdx

dup pu

1.0a 03.0b

Summary : Acceleration

• In a simple shear flow, shear effect appears on the correlation between and .

• The constant defined by Kolmogorov scaling of acceleration variance is not affected clearly by shear.

Anisotropic part

12123/5

11123/3

11 0)(11

SSkCSkkE

12123/5

12123/3

11 0)(22

SSkCSkkE

12123/5

1123/3

131 0)(21

SSkSkCkE

ijji aaa 2/13/2

0

1 2

i

i x

pa

1

0a

Frozen Flow Hypothesis for Pressure

DNS result by H. Abe for Channel Flow

Frozen flow hypothesis

Wall pressure spectrum

Wall pressure spectra

Probability density function of acceleration

-20.0 -10.0 0.0 10.0 20.010-6

10-5

10-4

10-3

10-2

10-1

100

(dp/dx)/

prob

abili

ty

:La Porta et al.

Mixing layer

S*

R large

small

Pressure measurement in cylinder wake

dU //~3 ω ][

~PaP 2// dUQ

vorticity pressureSecond invariance of velocity gradient tensor

Spectra of pressure and acceleration

100 101 102 103 10410-1610-1510-1410-1310-1210-1110-1010-910-810-710-610-510-410-310-210-1100

Spe

ctru

m

frequency [Hz]

(a) pressure

slope=-7/3

100 101 102 103 10410-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Spe

ctru

m

frequency [Hz]

(b) acceleration

slope=-1/3

3/7)( ffE pp

3/1)( ffE pp

Inertial range

Inertial range