Technion - Israel Institute of Technology, Haifa 32000 ...
Transcript of Technion - Israel Institute of Technology, Haifa 32000 ...
Stability of streaks in shear flows
Michael Karp and Jacob Cohen
Faculty of Aerospace Engineering
Technion - Israel Institute of Technology, Haifa 32000, Israel Research supported by the Israeli Science Foundation under Grant No. 1394/11
Research Aim • Study the secondary instability of TG in Couette flow and utilize it to
predict nonlinear transition to turbulence
References “Tracking stages of transition in Couette flow analytically” Karp, M.,
and Cohen, J., J. Fluid Mech., 748, 2014, pp 896 – 931.
Mathematical Method • Analytical approximation of linear TG using 4 modes
• Calculation of nonlinear interactions between the 4 modes
• Secondary stability analysis of the modified baseflow
• Long time correction of ud using solvability condition
Summary • Maximal growth is not essential for transition
• The role of the TG is to generate inflection points
• Optimal disturbances occur at maximal shear
• Most transition stages are captured analytically
Transient Growth (TG) • A mechanism where infinitesimal disturbances grow in a stable flow.
During this growth, the baseflow can be modified significantly and
instability may occur.
• Most efficient TG occurs for streamwise independent vortices
• Vortical structures (Isosurfaces of the Q def.) during transition
0
1
2 0
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-1
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y
Symmetric (Even, 2 vortices)
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2 0
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Antisymmetric (Odd, 4 vortices)
0
0 expRe
, ,d
t
d
ti
A i xN
Mdt y z
u u
2
0 ,,, , ...ˆ ,, NLx L t y tyey z zz y U u u
Couette + 4 modes + nonlinear
Amplitude
Eigenfunction Streamwise Eigenvalue Long time
wavenumber correction
Photo by Brooks Martner
0 , , , ,, , ..., , , ,d dd t x yx y z zty z u U u u
TG baseflow + Secondary disturbance + correction term
Re=1000 =1.6 =1.2 t=43
z /
y
0 0.5 1 1.5-1
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-0.6
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1Re=1000 =3.5 =1.5 t=36
z /
y
0 0.5 1 1.5-1
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1Re=1000 =3.6 =1.8 t=61
z /
y
0 0.5 1 1.5-1
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DNS
t=75
Theory
t=50 t=85 t=90
DNS
t=30
Theory
t=25 t=35 t=40
0 1 2 0
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0 1 2 0
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-1
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1
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0 1 2 0
2
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-1
0
1
x/
z/
y
0 1 2 0
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-1
0
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0 1 2 0
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-1
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0 1 2 0
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• Secondary instability verified by obtaining transition in
‘Channelflow’ DNS (Gibson, 2012)
(a) Even TG – Sinuous, max. spanwise shear (β=3.6)
(b) Odd TG – Sinuous, max. spanwise shear (β=3.5)
(c) Odd TG – Varicose, max. wall-normal shear (β=1.6)
Even TG (2 vortices) - sinuous
Odd TG (4 vortices) - varicose
0 50 100 150 2000
100
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t
G(t
)
DNS (TG)
DNS
Analytical (TG)
Analytical
0 50 100 150 2000
500
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2000
2500
3000
t
G(t
)
DNS (TG)
DNS
Analytical (TG)
Analytical
• Energy growth during transition
Even TG (2 vortices) - sinuous Odd TG (4 vortices) - varicose
Results • Formation of streamwise velocity streaks containing inflection points
Even TG (2 vortices) Odd TG (4 vortices)
• Secondary disturbance (streamwise component), legend below
(a) (b) (c)