Development of an improved algorithm for solving nonlinear parabolized stability equations for the purpose of relaxing the initial conditions

Abstract

Nonlinear parabolized stability equations (NPSE) have been proven to be a useful and convenient tool for the investigation of the stability and transition problems of boundary layers. However, in its original formulation, as initial conditions, the complex wave number and shape function of each Fourier mode are required for solving nonlinear problems, which result ambiguity and difficulty to set the initial values. In this paper, I propose a modified algorithm to remove its ambiguity and difficulty. The main idea is that, instead of solving all wave numbers with auxiliary condition, all wave numbers except primary mode are kept real and decided by phase-locked rule, and the shape function of each mode is simply solved by using nonlinear SOR method. The validity of the new formulation is illustrated by comparing the results with those from the corresponding original NPSE simulations as applied to a problem of Goertler instability and H-type transition of a flat plate boundary layer.