The Nonlinear Evolution of Inviscid Görtler Vortices in 3-D Boundary Layers: The Effects of Non-Dominant Viscosity in the Critical Layer

Abstract

In many practical situations where Görtier vortices occur the boundary layers are 3-D. Linear studies have found that this three-dimensionality has a stabilising effect and that, for a weak crossflow, inviscid Görtier modes possess some of the largest growth rates whilst also being neutral at certain other wavenumbers. Furthermore, these inviscid modes are governed by an equation very similar to the Taylor-Goldstein equation which governs the linear stability of stratified shear flows. In [1] this close connection was considered and a generalised Richardson number for such vortex instabilities given. We initially considered, using non-equilibrium critical-layer theory, the nonlinear evolution of modes on an unstable stratified shear layer in [2]. In this study it was found that there were three different base integro-differential equations (IDEs) that could govern the amplitude of a disturbance. The three different equations were an IDE with a cubic non-linearity due to viscous effects, an IDE with a cubic nonlinearity due to a novel mechanism and an IDE with a quintic nonlinearity. The choice as to which of these base IDEs is relevant for a given mode depends on the wa.venumber and the relative sizes of parameters representing viscosity, disturbance amplitude and the growth rate of the disturbance.