On the breakdown of Rayleigh’s criterion for curved shear flows: a destabilization mechanism for a class of inviscidly stable flows

Abstract

The stability of a high-Reynolds-number flow over a curved surface is considered. Attention is focused on spanwise-periodic vortices of wavelength comparable with the boundary layer thickness. The wall curvature and the Görtler number for the flow are assumed large, so that stable or unstable vortices of wavelength comparable with the boundary layer thickness are dominated by inviscid effects. The growth or decay rate determined by using the viscous correction to the inviscid prediction is found to give a good approximation to the exact value for a range of wavenumbers. For the stable configuration, it is shown that the flow can be destabilized by arbitrarily small wall waviness, with the critical configuration involving only geometric properties of the flow. This destabilization mechanism has consequences for a number of flows of aerodynamic interest, particularly those where surface waviness under loading occurs. Control strategies based on how the curvature is distributed are discussed, and it is demonstrated how small regions of varying curvature can be used to remove the exponentially growing modes. The development of the parametric instabilities is continued into the strongly nonlinear regime. The nonlinear evolution is found to be governed by a generalized form of the cubic amplitude equation, generally referred to as the Stuart-Landau equation. The novel feature of the new equation is that the nonlinearity is not in the form of a power of the amplitude but is fixed by the solution of an associated nonlinear partial differential equation boundary value problem. Numerical solutions of the nonlinear partial differential system that fix the disturbance amplitude are given. The mechanism of destabilization described is shown to be directly relevant to a number of flows that are inviscidly stable in the presence of body forces. Thus it is shown how Rayleigh-Bénard convection or Taylor vortices can, at high Rayleigh or Taylor numbers, be destabilized by asymptotically small modulation in time or space.

Centrifugal/elliptic instabilities in slowly varying channel flows

“…The leading-order approximation to (2.20) and (2.21) in the limit is which must be solved subject to Note that in the Görtler problem at large Görtler numbers the streamwise length scale shortens to for large and the stability problem in general becomes quasi-parallel in the direction unless the local wall curvature also varies on the same length scale; see Hall (2013). Note further that the presence of the function in the above system means that we cannot look for spatial or temporal normal modes of the form .…”

“…Once again we concentrate on the periodic case . Following Hall (2013) we can adapt the exact solution of DHS to the non-parallel case by seeking a steady solution separable in . Thus we write and it is easy to see that (4.1) and (4.2) are satisfied if the function satisfies The Görtler problem over a wall of constant curvature the corresponding equation has and so for wall jets or boundary layers in a semi-infinite region we obtain a spatial growth rate .…”

“…In another related paper Hall (2013) investigated the inviscid regime of Görtler vortices on wavy walls. It was shown that wall curvature modulation about a zero mean can induce instability through a parametric resonance.…”