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

Published online by Cambridge University Press:  19 September 2014

Anubhab Roy Open the ORCID record for Anubhab Roy [Opens in a new window]  and
Ganesh Subramanian
Anubhab Roy
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India
Ganesh Subramanian*
Affiliation:
Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Jakkur, Bangalore 560064, India
*
†Email address for correspondence: [email protected]
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Abstract

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.

JFM classification

Boundary Layers: Boundary layer stability Instability: Instability Instability: Transition to turbulence
Type
Papers
Information
Journal of Fluid Mechanics , Volume 757 , 25 October 2014 , pp. 82 - 113
Copyright
© 2014 Cambridge University Press 

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Footnotes

‡

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

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