Generation of spanwise momentum in a simple shear layer

1
Generation of spanwise momentum in a simple
shear layer
47th British Applied Mechanics Colloquium
Liverpool, 4 7 April 2005

• M. Nagata, T. Itano R. Nakamura
• Department of Aeronautics and Astronautics,
  Graduate School of Engineering
• and
• Advanced Research Institute of Fluid Science and
  Engineering
• Kyoto University, JAPAN

2
Background
Basic conductive state

experiments
Vest Arpaci(1969) air, oil
Hart(1971) air, water
Cold
Hot
theory
3
Organisation

• 1. Mathematical formulation
• 2. Linear stability analysis
• 3. 2-dimensional nonlinear analysis
• 4. Stability of 2D nonlinear solutions
• 5. 3-dimensional nonlinear analysis
• 6. Summary

4
Mathematical formulation
Viscous incompressible fluids with Boussinesq
approximation
Equation of continuity
8
Equation of momentum
Equation of energy
Equation of energy
Boundary Condition No-slip, Fixed temperatures
8
- 8
- 8
Equation of state
? coeff. of thermal expansion


5
Mathematical formulation
Non-dimensional parameters
Prandtl number
Grashof number
?thermal diffusivity
?kinematic viscosity
Basic state
6
Linear analysis
perturbations
Poloidal and toroidal decomposition
Boundary Conditions
7
Linear analysis
Normal mode
8
Linear analysis (results)
Pr0.00001 (liquid metals)
Gr
a
9
Linear analysis (results)
Pr0.00001 (liquid metals)
unstable
Gr
stable
a
10
Linear analysis (results)
Pr0.00001 (liquid metals)
unstable
Gr
stable
a
11
Linear analysis
Pr0.00001 (liquid metals)
Critical point
unstable
Gr
495.55
stable
1.34
a
12
Linear analysis (results)
Pr0.00001
Pr0.71 (air)
Pr7.0 (water)
Gr
Pr26 (kerosene)
Pr1000 (engine oil)
a
Critical point
13
Linear analysis (results)
Pr0.00001
Pr0.71 (air)
Pr7.0 (water)
Gr
Pr26 (kerosene)
Pr1000 (engine oil)
a
Pr0.00001, 0.71, 7.0
Pr26, 1000
14
Organisation

• 1. Mathematical formulation
• 2. Linear stability analysis
• 3. 2-dimensional nonlinear analysis
• 4. Stability of 2D nonlinear solutions
• 5. 3-dimensional nonlinear analysis
• 6. Summary

15
2D nonlinear analysis
Separation of disturbances
into
mean parts
and residuals
16
2D nonlinear analysis
Nonlinear algebraic equation
(1) Direct numerical simulation
(2)
Newton-Raphson method
Momentum transport
17
2D steady transverse rolls
Pr0.00001
18
2D steady transverse rolls
Pr0.00001

19
2D steady transverse rolls
Pr0.00001

Grc495.55
DNS
supercritical bifurcation of 2D steady
transverse rolls
Gr510 a1.34
20
2D steady transverse rolls
Pr0.71 (air)
Pr0.00001
Pr7.0 (water)
21
2D steady transverse rolls
Pr0.71 Gr530 a1.41
Cold
Hot
x
x
x
g
z
cold
y
z
z
z-1
z1
22
Organisation

• 1. Mathematical formulation
• 2. Linear stability analysis
• 3. 2-dimensional nonlinear analysis
• 4. Stability of 2D nonlinear solutions
• 5. 3-dimensional nonlinear analysis
• 6. Summary

23
Stability of 2D transverse rolls
perturbations
on 2D transverse rolls
24
Stability of 2D transverse rolls
Eigenvalue problem
25
Stability of 2D transverse rolls (results)
Pr0.00001
Supercritical bif. of 2D transverse rolls
d
b
(positive s in grey)
26
Stability of 2D transverse rolls (results)
Pr0.00001
Supercritical bif. of 2D transverse rolls
Gr510
d
b
2D transverse rolls stable
27
Stability of 2D transverse rolls (results)
Pr0.00001
Supercritical bif. of 2D transverse rolls
Gr520
d
b
Unstable to subharmonic perturbations
28
Stability of 2D transverse rolls (results)
Pr0.00001
Supercritical bif. of 2D transverse rolls
Gr540
d
b
Unstable to harmonic perturbations
29
Stability of 2D transverse rolls (results)
Pr0.00001
Supercritical bif. of 2D transverse rolls
Gr570
d
b
30
Stability of 2D transverse rolls (results)
Pr0.00001
Supercritical bif. of 2D transverse rolls
Gr600
d
b
31
Stability of 2D transverse rolls (results)
Pr0.00001
Supercritical bif. of 2D transverse rolls
Gr600
d
b
32
Organisation

• 1. Mathematical formulation
• 2. Linear stability analysis
• 3. 2-dimensional nonlinear analysis
• 4. Stability of 2D nonlinear solutions
• 5. 3-dimensional nonlinear analysis
• 6. Summary

33
3D nonlinear analysis
Separation of disturbances
into
mean parts
and residuals
Nonlinear algebraic equation
34
3D nonlinear analysis
Governing equations
35
3D nonlinear analysis (results)
Pr0.00001
36
3D nonlinear analysis (results)
Pr0.00001
Bifurcation point of 3D subharmonic solution
37
3D nonlinear analysis (results)
Pr0.00001
DNS
Gr550 a0.67 ß0.79
38
3d nonlinear analysis (results)
Pr0.00001
DNS
Gr550 a0.67 ß0.79
Supercritical bifurcation of 3D subharmonic
solution
39
3D steady subharmonic solution
x-z cross section
y-z cross section
x-y cross section
40
3D nonlinear analysis (results)
Pr0.00001
Bifurcation point of harmonic solution
41
3D nonlinear analysis (results)
Pr0.00001
DNS
Gr580 a1.34 ß0.79
42
3D nonlinear analysis (results)
Pr0.00001
DNS
Gr580 a1.34 ß0.79
Supercritical bifurcation of 3D periodic solution
(3Dharmonic A)
43
3D nonlinear analysis (results)
Pr0.00001
Supercritical bifurcation of 3D travelling-wave
solution (3DTW) at the same bif. point as
3Dharmonic A
44
3D nonlinear analysis (results)
Pr7.0
Pr0.00001
Pr0.71
45

3D Periodic Solution
(3Dharmonic A)
velocity
vorticity
x
x
z
y
46

3D Travelling-wave solution
(3DTW)
velocity
vorticity
x
x
z
y
47
3D Travelling-wave solution
Pr0.71 Gr650 a1.41 ß0.81
Cold
Hot
x
x
g
z
y
cold
hot
Travelling in the spanwise direction
z
z-1
z1
48
Symmetry of 3D travelling-wave solution
f
, including
2D steady transverse rolls with
n0
and stability eigenmodes with n1,
-ßcIms
moreover V(-z)V(z)
49
Summary

• We have analysed nonlinear solutions bifurcating
  from the 2D steady transverse flow when it loses
  its stability with respect to 3D oscillatory
  perturbations, in a simple shear flow between
  parallel plates.
• It is found that a 3D periodic solution and a 3D
  travelling-wave solution bifurcate simultaneously
  at the same Grashof number.
• The momentum with a symmetric velocity profile in
  the spanwise direction is generated for the 3D
  travelling-wave solutions. (Experimental
  investigation to detect the generation of
  momentum in the azimuthal direction for flows in
  a vertical annulus heated from side is under way.)

50
END
51
????????????
?????????
???????????
????????????
????????????
52
????????????
?????
53
????????????
????????????
54
????????????
????(3Dsubharmonic?)????????
55
????????????
???????????
??????
????(3Dsubharmonic?)????????
56
????????????
57
Appendix

(V?0???)
58
??????
Pr0.71(??) 20?
2d
Grashof?
d5.010-3m????
Gr650??????????5.6K???
????u9.810-3m/s
59
??????????
(?t ???????)
60
??????????

• Heywood,J.G.. Masuda,K., Rautmann,R.
  Solonnikov,S.A. 1991 The Navier-Stokes Equations
  II Theory and Numerical Methods. Lecture Notes
  in Mathematics 1530.

61
?????
3DTW????f,?,??????????????????????????
3DTW
3Dsubharmonic?
3Dharmonic?B
62
?
63
?????

• NagataBusse(1983)
• Pr0(???????)??????????monotone
  instability?oscillatory instability???????????????
  ?monotone instability???????????????????
• ChaitKorpela(1989)
• Pr0.71(??),1000(???????)??????????????????????
• CleverBusse(1995)
• Pr0.71,7.0(?)??????????????

64
?????

• Bratsun(2003)
• Pr26(??)????????????????????????????
• NagataItano(2003)
• ?????DNS????Pr0??????????????????????????????????
  ????????????????????????

65
??????
?????????????????
Pr0.00001
Pr0.71
Pr7.0
Subharmonic,harmonic?????????????????
Pr0.00001,0.71
Pr7.0
???????????????subharmonic,harmonic
?????????????????
66
??????
?????????????????
Pr0.00001
Pr0.71
Pr7.0
67
??????
?????????????????
Pr0.00001
Pr0.71
Pr7.0
Subharmonic,harmonic?????????????????
Pr0.00001,0.71
Pr7.0
Subharmonic??????????????