The asymptotic theory of hypersonic boundary-layer stability

Abstract

In this paper the linear stability of the hypersonic boundary layer is considered in the local-parallel approximation. It is assumed that the Prandtl-number ½ < σ < 1 and the viscosity-temperature law is a power function: μ/μ∞ = (T/T∞)ω. The asymptotic theory in the limit M∞ → ∞ is developed.Smith & Brown found for the Blasius base flow and Balsa & Goldstein for the mixing layer that, in this limit, the disturbances of the vorticity mode are located in the thin region between the boundary layer and the external flow. The gas model with σ = 1, ω = 1 was exploited in these studies. Here it is demonstrated that the vorticity mode also exists for gas with ½ < σ 1, ω < 1, but its structure and characteristics are considerably different. The nomenclature is discussed, i.e. what an acoustic mode and a vorticity mode are. The numerical solution of the inviscid instability problem for the vorticity mode is obtained for helium and compared with the solution of the complete Rayleigh equation at finite Mach numbers.The limit M∞ → ∞ in the local-parallel approximation for the Blasius base flow is considered so as to understand the viscous structure of the vorticity mode. The viscous stability problem for the vorticity mode is formulated under these assumptions. The problem contains only a single similarity parameter which is a function of the Mach and Reynolds numbers, the temperature factor and wave inclination angle. This problem is numerically solved for helium. The universal upper branch of the neutral curve is obtained as a result. The asymptotic results are compared with the numerical solutions of the complete problem.