Anisotropic Pressure and Acceleration Spectra in Shear Flow
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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)
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.
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
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