secondary instability of transient growth in couette flow · flow theoretical (lst) experimental...
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Sinuous Instability
Spanwise
Inflection
Secondary instability of Transient Growth in Couette Flow
Michael Karp and Jacob Cohen
Faculty of Aerospace Engineering,
Technion - Israel Institute of Technology, Haifa 32000, IsraelResearch supported by the Israeli Science Foundation under Grant No. 1394/11
Motivation• Linear Stability Theory (LST) is unable to predict transition
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 up 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
Flow Theoretical (LST) Experimental
Pipe Poiseuille ∞ (Stable) ~2000
Plane Poiseuille 5772 ~1000
Plane Couette ∞ (Stable) ~360
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
Table 1. Critical Reynolds numbers for transition, Theory vs. Experiment
Research Aim• Study the secondary instability of TG in Couette flow and
utilize it to predict nonlinear transition to turbulence
Results• Optimal disturbances
• Even TG – Sinuous, max. spanwise shear (β≈4≈4≈4≈4)
• Odd TG – Sinuous, max. spanwise shear (β≈4≈4≈4≈4)
• Odd TG – Varicose, max. spanwise shear (β≈≈≈≈2)
• Secondary instability verified by obtaining transition in ‘Channelflow’ DNS (Gibson, 2012)
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Odd (4 vortices)
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Secondary disturbance
(streamwise component)
Sinuous
Instability
Spanwise
Inflection
Varicose
Instability
Wall-normal
Inflection
Energy during transition
( ) ( )0
0 exp, ,d
t
d
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iNt y z iA x
Mdωα τ τ
= − −
∫ɶu u
( ) ( ) ( )2 , ,, , ,, .ˆ . .,NLx L d
t y t x yy zt ze y z δεε= + + + +u u uu
Couette + 4 modes + nonlinear + secondary
AmplitudeEigenfunction Streamwise Eigenvalue Long time
wavenumber correction
Photo by Brooks Martner
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DNS (TG)
DNS
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Analytical
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Photo by NASA/JPL
Isosurfaces of the Q definition during transition
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Qdef, t = 40
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