On the nature of PSE approximation

Abstract

The recently developed method of parabolized stability equations (PSE) offers a fast and efficient way of analyzing the spatial growth of linear and nonlinear (convective) disturbances in shear layers. For incompressible flows, the governing equations may be represented either in primitive variables or by using other formulations obtained by eliminating the pressure gradient (e.g., vorticity-streamfunction formulation). On the other hand, for compressible flows, primitive variables offer a natural and the only choice. We show that primitive-variable formulation is not well-posed due to the ellipticity introduced by the$\partial \hat p/\partial x$ term and the marching solution eventually blows up for a sufficiently small step size. However, it is shown that this difficulty can be overcome if the minimum step size is greater than the inverse of the real part of the streamwise wave number, α_r. An alternative is to drop the$\partial \hat p/\partial x$ term, in which case the residual ellipticity is of no consequence for marching computations with much smaller step sizes. However, the ellipticity cannot be completely removed. Results obtained with streamfunction and vorticity-velocity formulations also show that the numerical difficulties arise for a sufficiently small marching step size. This step-size restriction can be overcome by dropping thedα/dx term from the governing equations. The effect of this term on solution accuracy is negligible for Blasius flow but not so for rotating-disk flow.