A spectral Garlekin method for hydrodynamic stability problems

Abstract

A spectral Galerkin method for calculating the eigenvalues of the Orr-Sommerfeld equation is presented. The matrices of the resulting generalized eigenvalue problem are sparse. A convergence analysis of the method is presented which indicates that a) no spurious eigenvalues occur and b) reliable results can only be expected under the assumption of {\em scale resolution}, i.e., that $\re/p^2$ is small; here $\re$ is the Reynolds number and p is the spectral order. Numerical experiments support that the assumption of scale resolution is necessary to obtain reliable results. Exponential convergence of the method is shown theoretically and observed numerically.