Competition of the multiple Grtler modes in hypersonic boundary layer flows

Competition of multiple Gortler modes in hypersonic boundary layer flows are investigated with the local and marching methods. The wall-layer mode （mode W） and the trapped-layer mode （mode T） both occur in the compressible boundary layer where there exists a temperature adjustment layer near the upper edge. The mode T has the largest growth rate at a lower Gortler number while the mode W dominates at larger G/Srtler numbers. These two modes are both responsible for the flow transition in the hypersonic flows especially when Gortler number is in the high value range in which the crossover of these two modes takes place. Such high Gortler numbers are virtually far beyond the neutral regime. The nonparallel base flows, therefore, cease to influence the stability behavior of the Gortler modes. The effects of the Mach number on the multiple Gortler modes are studied within a chosen Mach number of 0.95, 2, 4 and 6. When the flow Mach number is sufficiently large, e.g., Ma ≥4, the growth rate crossover of the mode T and mode W occurs both in the conventional G-β map as well as on the route downstream for a fixed wavelength disturbance. Four particular regions （Region T, T-W, W-T and W） around the crossover point are highlighted with the marching analysis and the result matches that of the local analysis. The initial disturbance of a normal mode maintains the shape in its corresponding dominating region while a shape-transformation occurs outside this region.

Competition of multiple Grtler modes in hypersonic boundary layer flows are investigated with the local and marching methods.The wall-layer mode(mode W)and the trapped-layer mode(mode T)both occur in the compressible boundary layer where there exists a temperature adjustment layer near the upper edge.The mode T has the largest growth rate at a lower Grtler number while the mode W dominates at larger Grtler numbers.These two modes are both responsible for the flow transition in the hypersonic flows especially when Grtler number is in the high value range in which the crossover of these two modes takes place.Such high Grtler numbers are virtually far beyond the neutral regime.The nonparallel base flows,therefore,cease to influence the stability behavior of the Grtler modes.The effects of the Mach number on the multiple Grtler modes are studied within a chosen Mach number of 0.95,2,4 and 6.When the flow Mach number is sufficiently large,e.g.,Ma 4,the growth rate crossover of the mode T and mode W occurs both in the conventional G-βmap as well as on the route downstream for a fixed wavelength disturbance.Four particular regions(Region T,T-W,W-T and W)around the crossover point are highlighted with the marching analysis and the result matches that of the local analysis.The initial disturbance of a normal mode maintains the shape in its corresponding dominating region while a shape-transformation occurs outside this region.