(A) relaxation technique of PSE integration and its application on stability analysis of boundary layer over a hump

Abstract

We first derive parabolized stability equations for compressible flows in general curvilinear coordinate system to deal with a broad range of transition prediction problems on complex geometry. A finite differencs PSE code to solve these equations has been developed using an implicit marching procedure. To test the code compressible/imcompressible flat plate flow stability under two-dimensional and three-dimensional disturbances has been investigated. Results of the computation are found to be in good agreement with the multiple scale analysis and DNS data. Stability calculation results by this PSE code for compressible boundary layer at Mach numbers ranging from 0.02 t0 1.5 are also presented and are again seen to be as accurate as the spectral method. Secondly, we propose a new integration scheme as conventional marching procedure for PSE solutions suffers the step size limitation problem, which often leads to divergence when small step size is taken in the marching direction. The proposed scheme is composed of predictor-corretor step. The predictor step employs spatial relaxtion and the corrector step reduces the error due to spatial relaxtion. Numerical stability anlaysis of the proposed method indicates that the step size can be smaller by many times, approximately in proportion to twice the relaxtion factor. The utility of the proposed methods have been demonstrated by stability calculations for the Blasius flow and for the incompressible boundary layer over a hump where the step size limitation is a critical factor. Numerical examples demonstrate that the proposed scheme is robust, accurate and stable even for a very small step size. The applicability of the PSE to the flow over a hump may be in doubt due to separation bubble. This is examined by comparing the result with DNS data. It is found that the PSE can efficiently track the disturbance wave with acceptable accuracy in spite of small separation bubble. A typical evolution scenario of TS w...