| The neutral stability of a zero-pressure gradient hypersonic boundary layer flow over a flat plate is considered. The derivation of the linear stability equations from the Navier-Stokes equations is reviewed, and a formulation of the governing second order linear differential equation for the pressure disturbance is developed that lends itself to application of the WKB method over the entire boundary layer. This formulation provides a high order approximate analytical eigenfunction and eigenvalue relations for the pressure disturbance, and is applicable to several types of flows at moderate and high Mach numbers as well. The solution in this type of formulation is shown to be dependent only on the relative Mach number profile in the boundary layer, with the relative Mach number referring to the Mach number of the disturbance (with wavespeed c) relative to the mean-flow speed of sound. Pressure perturbation solutions and eigenvalues are determined for the non-inflectional cases of the wave speed c = 0 and c = 1 flow over an adiabatic surface, and show good qualitative agreement with numerical computations as well as results in the asymptotic limit of freestream Mach number {dollar}Msb{lcub}infty{rcub}toinfty{dollar}.; The case for a mean flow profile which possess a generalized inflection point and a wavespeed equal to the mean velocity at the generalized inflection point ({dollar}c = csb{lcub}s{rcub}{dollar}) is examined for the cases with and without heat transfer to the surface. These results also show a good qualitative comparison with the numerical results. The eigenvalue relations for these cases are examined in detail and the "near-linking" of the high-Mach number inviscid vorticity and multiple acoustic modes at moderate Mach numbers are investigated. These regions of the finite Mach number eigenvalue relation is shown to be described as a region of eigenvalue loci veering. This characteristic of the eigenvalue relation which, has only been observed previously in numerical computations, is shown to be approximated locally in the region near the interaction of these inviscid modes by a local hyperbolic approximation which qualitatively models the key aspects of this interaction. |
