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An inviscid modal interpretation of the ‘lift-up’ effect

  • Anubhab Roy (a1) and Ganesh Subramanian (a1)


In this paper, we give a modal interpretation of the lift-up effect, one of two well-known mechanisms that lead to an algebraic instability in parallel shearing flows, the other being the Orr mechanism. To this end, we first obtain the two families of continuous spectrum modes that make up the complete spectrum for a non-inflectional velocity profile. One of these families consists of modified versions of the vortex-sheet eigenmodes originally found by Case (Phys. Fluids, vol. 3, 1960, pp. 143–148) for plane Couette flow, while the second family consists of singular jet modes first found by Sazonov (Izv. Acad. Nauk SSSR Atmos. Ocean. Phys., vol. 32, 1996, pp. 21–28), again for Couette flow. The two families are used to construct the modal superposition for an arbitrary three-dimensional distribution of vorticity at the initial instant. The so-called non-modal growth that underlies the lift-up effect is associated with an initial condition consisting of rolls, aligned with the streamwise direction, and with a spanwise modulation (that is, a modulation along the vorticity direction of the base-state shearing flow). This growth is shown to arise from an appropriate superposition of the aforementioned continuous spectrum mode families. The modal superposition is then generalized to an inflectional velocity profile by including additional discrete modes associated with the inflection points. Finally, the non-trivial connection between an inviscid eigenmode and the viscous eigenmodes for large but finite Reynolds number, and the relation between the corresponding modal superpositions, is highlighted.


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Present address: School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA.



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Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover.
Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11, 134150.
Antkowiak, A. & Brancher, P. 2004 Transient growth for the Lamb–Oseen vortex. Phys. Fluids 16, L1L4.
Arnol’d, V. I. 1972 Notes on the three-dimensional flow pattern of a perfect fluid in the presence of a small perturbation of the initial velocity field. Z. Angew. Math. Mech. 36, 236242.
Baines, P. G., Majumdar, S. J. & Mitsudera, H. 1996 The mechanics of the Tollmien–Schlichting wave. J. Fluid Mech. 312, 107124.
Balmforth, N. J. & Morrison, P. J.1995 Singular eigenfunctions for shearing fluids I. Report No. 692, Institute for Fusion Studies, University of Texas, Austin.
Benney, D. J. & Lin, C. C. 1960 On the secondary motion induced by oscillations in a shear flow. Phys. Fluids 3, 656657.
Case, K. M. 1959 Plasma oscillations. Ann. Phys. (NY) 7, 349364.
Case, K. M. 1960 Stability of inviscid plane Couette flow. Phys. Fluids 3, 143148.
Dickinson, R. E. 1970 Development of a Rossby wave critical level. J. Atmos. Sci. 27, 627633.
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.
Ellingsen, T. & Palm, E. 1975 Stability of linear flow. Phys. Fluids 18, 487488.
Fadeev, L. D. 1971 On the theory of the stability of stationary plane-parallel flows of an ideal fluid. Zap. Nauch. Semin. Leningrad. Otdel. Mat. Inst. Akad. Nauk SSSR 21, 164172.
Farrell, B. F. 1987 Developing disturbances in shear. J. Atmos. Sci. 45, 21912199.
Farrell, B. F. & Ioannou, P. J. 1993 Optimal excitation of three-dimensional perturbations in viscous constant shear flow. Phys. Fluids A 5, 13901400.
Friedman, B. 1990 Principles and Techniques of Applied Mathematics. Dover.
Gakhov, F. D. 1990 Boundary Value Problems. Dover.
Grosch, C. E. & Salwen, H. 1978 The continuous spectrum of the Orr–Sommerfeld equation. Part 1. The spectrum and the eigenfunctions. J. Fluid Mech. 87, 3354.
Grosch, C. E. & Salwen, H. 1981 The continuous spectrum of the Orr–Sommerfeld equation. Part 2. Eigenfunction expansions. J. Fluid Mech. 104, 445465.
Jimenez, J. & Pinelli, A. 1999 The autonomous cycle of near wall turbulence. J. Fluid Mech. 389, 335359.
Kelbert, M. & Sazonov, I. 1996 Pulses and Other Wave Processes in Fluids. Kluwer.
Landahl, M. T. 1980 A note on an algebraic instability of inviscid parallel shear flows. J. Fluid Mech. 98, 243251.
Lee, M. J., Kim, J. & Moin, P. 1990 Structure of turbulence at high shear rate. J. Fluid Mech. 216, 561583.
Lighthill, M. J. 1958 An Introduction to Fourier Analysis and Generalized Functions. Cambridge University Press.
Lin, C. C. 1955 The Theory of Hydrodynamic Stability. Cambridge University Press.
Lindzen, R. S. 1988 Instability of plane parallel shear flow (toward a mechanistic picture of how it works). Pure Appl. Geophys. 126, 103121.
Mack, L. M. 1976 A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73, 497520.
Marcus, P. S. & Press, W. H. 1977 On Green’s functions for small disturbances of plane Couette flow. J. Fluid Mech. 79, 525534.
Moffatt, H. K.1965 The interaction of turbulence with rapid uniform shear. Report No. SUDAER-242, Department of Aeronautics and Astronautics, Stanford University, CA.
Murdock, J. W. & Stewartson, K. 1977 Spectra of the Orr–Sommerfeld equation. Phys. Fluids 20, 14041411.
Pradeep, D. S. & Hussain, F. 2006 Transient growth of perturbations in a vortex column. J. Fluid Mech. 550, 251288.
Reddy, S. C., Schmid, P. J. & Henningson, D. S. 1993 Pseudospectra of the Orr–Sommerfeld operator. SIAM J. Appl. Maths 53, 1547.
Reshotko, E. 2001 Transient growth: a factor in bypass transition. Phys. Fluids 13, 10671075.
Rogers, M. M. & Moin, P. 1987 The structure of the vorticity field in homogeneous turbulent flows. J. Fluid Mech. 176, 3366.
Roy, A.2013 Singular eigenfunctions in hydrodynamic stability: the roles of rotation, stratification and elasticity. PhD thesis, Jawaharlal Nehru Centre for Advanced Scientific Research.
Roy, A. & Subramanian, G. 2014 Linearized oscillations of a vortex column: the singular eigenfunctions. J. Fluid Mech. 741, 404460.
Sazonov, I. A. 1989 Interaction of continuous spectrum waves with each other and discrete spectrum waves. Fluid Dyn. Res. 4, 586592.
Sazonov, I. A. 1996 Evolution of three-dimensional wavepackets in the Couette flow. Izv. Acad. Nauk SSSR Atmos. Ocean. Phys. 32, 2128.
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Fluid Flows. Springer.
Stakgold, I. 1968 Boundary Value Problems of Mathematical Physics. Macmillan.
Stewartson, K. 1981 Marginally stable inviscid flows with critical layers. IMA J. Appl. Maths 27, 133175.
Tumin, A. & Reshotko, E. 2001 Spatial theory of optimal disturbances in boundary layers. Phys. Fluids 13, 20972104.
Tung, K. K. 1983 Initial-value problems for Rossby waves in a shear flow with critical level. J. Fluid Mech. 133, 443469.
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An inviscid modal interpretation of the ‘lift-up’ effect

  • Anubhab Roy (a1) and Ganesh Subramanian (a1)


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