Numerical methods for hypersonic boundary layer stability

Abstract

Various numerical methods for the solution of linear stability equations for compressible boundary layers are compared. Both the global and local eigenvalue methods for temporal stability analysis are discussed. Global methods are used to compute all the eigenvalues of the discretized system. When a guess for the desired eigenvalue is available, local methods may be used both to purify the eigenvalue and compute the associated eigenfunctions. The extension to spatial stability analysis is also considered. The discretizations studied include: a second-order finite-difference method, a fourth-order accurate two-point compact difference scheme, and a Chebyshev spectral collocation method. Eigenvalue results are presented for Mach numbers up to 10. As the Mach number increases, the performance of the spectral method deteriorates due to the outward movement of the critical layer. To alleviate this problem, a multi-domain spectral collocation method is developed which exhibits better convergence. The overall performance of the fourth-order compact scheme is excellent. Our results also indicate that, in the limit of vanishing Mach number, there exist stable discrete modes in addition to the discrete modes of the Orr-Sommerfeld equation.