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.
