Global and local analyses of the Görtler instability in hypersonic flow

Abstract

The stationary Görtler instability in hypersonic flow over a concave wall is systematically investigated across a range of geometric and flow parameters using resolvent analysis, which seeks for the forcing and response pair that maximizes the energy amplification. The optimal forcing takes the form of streamwise vortices, while the optimal response is streamwise streaks. The growth of the optimal disturbance is contributed by both the lift-up and centrifugal mechanisms. The latter becomes dominant as the boundary layer develops, and its growth rate agrees well with that predicted by local stability analysis. In terms of changes in geometric parameters, an increase in curvature destabilizes the Görtler instability, as expected, while the effect of the angle subtended by the concave wall (the turning angle) is shown to be negligible. With respect to changes in flow parameters, the Görtler instability is stabilized at low Reynolds numbers, destabilized under the cold-wall effect, and insensitive to the change in Mach number. The most amplified spanwise wavelength scales with the boundary-layer thickness, which remains mostly unchanged when the freestream Mach number is varied from 3 to 10. A new dimensionless wavelength parameter is proposed to predict the wavelength of the most dangerous Görtler vortices in the compressible flow regime. The resolvent analysis results are confirmed by a three-dimensional numerical simulation, where the hypersonic flow is perturbed by a spatial white noise.