-1 0 1 0 1 2 -1 -0.5 0 0.5 1 u Streamwise velocity, Re=1000 β =2 t=25 z β / π y -1 0 1 0 1 2 -1 -0.5 0 0.5 1 u Streamwise velocity, Re=1000 β =4 t=40 z β / π y 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, Israel Research 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) 0 1 2 0 1 2 -1 -0.5 0 0.5 1 x α / π z β / π y ω ω Even (2 vortices) 0 1 2 0 1 2 -1 -0.5 0 0.5 1 x α / π z β / π y ω ω ω ω Odd (4 vortices) Re=1000 β =4 α =2 t=40 z β / π y 0 0.5 1 1.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Re=1000 β =2 α =2 t=25 z β / π y 0 0.5 1 1.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Re=1000 β =4 α =2 t=20 z β / π y 0 0.5 1 1.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Secondary disturbance (streamwise component) Sinuous Instability Spanwise Inflection Varicose Instability Wall-normal Inflection Energy during transition ( ) () 0 0 exp , , d t d t iN tyz i A x M d ω α τ τ = - - ∫ ɶ u u ( ) ( ) ( ) 2 ,, , , , , . ˆ .. , NL x L d ty txy y z t z e yz δ ε ε = + + + + u u u u Couette + 4 modes + nonlinear + secondary Amplitude Eigenfunction Streamwise Eigenvalue Long time wavenumber correction Photo by Brooks Martner 0 50 100 150 200 0 100 200 300 400 500 t G(t) DNS (TG) DNS Analytical (TG) Analytical 0 50 100 150 200 0 50 100 150 200 250 300 350 400 t G(t) DNS (TG) DNS Analytical (TG) Analytical 0 50 100 150 200 0 200 400 600 800 1000 t G(t) DNS (TG) DNS Analytical (TG) Analytical Photo by NASA/JPL Isosurfaces of the Q definition during transition 0 1 0 5 10 15 -1 0 1 x Qdef, t = 40 z y 0 1 0 5 10 15 -1 0 1 x Qdef, t = 60 z y 0 1 0 5 10 15 -1 0 1 x Qdef, t = 80 z y 0 1 0 5 10 15 -1 0 1 x Qdef, t = 20 z y 0 1 0 5 10 15 -1 0 1 x Qdef, t = 40 z y 0 1 0 5 10 15 -1 0 1 x Qdef, t = 60 z y 0 2 0 5 10 15 -1 0 1 x Qdef, t = 40 z y 0 2 0 5 10 15 -1 0 1 x Qdef, t = 30 z y 0 2 0 5 10 15 -1 0 1 x Qdef, t = 25 z y