Report 1998-06

A Spectral Galerkin Method for Hydrodynamic Stability Problems

N.P. Hancke, J.M. Melenk and C. Schwab

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 $\ren/p^2$ is small; here $\ren$ 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.

Keywords: Orr-Sommerfeld equation, hydrodynamic stability, eigenvalue problem, spectral method

Paper: Available as PDF (487 KB) or as hardcopy to order reports@sam.math.ethz.ch.