A simple quadratic relative Mach number model for the generalized inflection point mode of the stability wave in hypersonic boundary layer

The neutral stability of generalized-inflection-point mode in compressible Blasius boundary layer is considered. The derivation of the linear stability equations from the Navier-Stokes equations is reviewed, and a formulation of the governing forth order linear differential equating for pressure disturbance is developed. By considering the inviscid stability theory, the formulation further reduces to a linear second order differential equation, and then lends itself to application of WKB method within the generalized inflection point. To obtain a close form of the acoustic and vorticity modes, the relative Mach number profile is modeled based on the numerical computation. Due to the complexity of the WKB method, a quadric polynomial model is used to examine the phenomenon of the curve veering of these two modes, and then an asymptotic close form of the eigenvalue relation for each mode can be derived explicitly. The results show good qualitative agreement with pioneer works studied by numerical computations as well as analytical studies.; A more accurate quadric polynomial model of relative Mach number is applied to examine the effect of the generalized-inflection-point on the inviscid modes, and a pseudo-dissipation term is added to the second order differential equation which in turns introduces a transition layer that allows the critical layer to join smoothly with the inviscid solutions. The results show that the wave trapped region in fact is not function of free stream Mach number only, but also the wave number, which implies that the wave of generalized-inflection-point mode trapped within the boundary layer possesses higher energy that those with non-inflection-point modes. Furthermore, the weird behavior of, eigenfunction of vorticity mode in Mack's numerical computation is first obtained systematically and analytically.