On the Görtler instability in hypersonic flows: Sutherland law fluids and real gas effects

Abstract

The Görtler vortex instability mechanism in a hypersonic boundary layer on a curved wall is investigated in this paper. Our aim is to clarify the precise roles of the effects of boundary layer growth, wall cooling and gas dissociation in the determination of stability properties. We first assume that the fluid is an ideal gas with viscosity given by Sutherland’s law. It is shown that when the free-stream Mach number M is large, the boundary layer divides into two sublayers: a wall layer of O ( M 3/2 ) thickness (in terms of the boundary layer variable) over which the basic state temperature is O ( M 2 ), and a temperature adjustment layer of O (1) thickness over which the basic state temperature decreases monotonically to its free-stream value. Görtler vortices which have wavelength comparable with the boundary layer thickness (i.e. have local wavenumber of order M -3/2 ) are referred to as wall modes. We show that their downstream evolution is governed by a set of parabolic partial differential equations and that they have the usual features of Görtler vortices in incompressible boundary layers. As the local wavenumber increases, the neutral Görtler number decreases and the centre of vortex activity moves towards the temperature adjustment layer. Görtler vortices with wavenumber of order one or larger must necessarily be trapped in the temperature adjustment layer, and it is this mode which is the most dangerous. For this mode, we find that the leading-order term in the Görtler number expansion is independent of the wavenumber and is due to the curvature of the basic state. This term is also the asymptotic limit of the neutral Görtler numbers of the wall mode. To determine the higher-order correction terms in the Görtler number expansion, we have to distinguish between two wall curvature cases. When the wall curvature is proportional to (2 x ) -3/2 , where x is the streamwise variable, the Mach number M can be scaled out of the problem and the boundary layer growth takes place over an O (1) lengthscale. The evolution properties of Görtler vortices are then similar to those in incompressible flows. In the more general case when the wall curvature is not proportional to (2 x ) -3/2 , the effect of the curvature of the basic state persists in the downstream development of Görtler vortices; non-parallel effects are important over a larger range of wavenumbers, and they become of second order only when the local wavenumber is of order higher than O ( M 1/4 ). In the latter case the Görtler number expansion has the first two terms independent of non-parallel effects; the first term being due to the curvature of the basic state and the second term due to viscous effects. The second term becomes comparable with the first term when the wavenumber reaches the order M 3/8 , in which case another correction term can also be found independently of non-parallel effects. Next we investigate real gas effects by assuming that the fluid is an ideal dissociating gas. We find that both gas dissociation and wall cooling are destabilizing for the mode trapped in the temperature adjustment layer, but for the wall mode trapped near the wall the effect of gas dissociation can be either destabilizing or stabilizing.