STABILITY OF SPATIALLY DEVELOPING BOUNDARY-LAYERS IN PRESSURE-GRADIENTS

A new formulation of the stability of boundary-layer flows in pressure gradients is presented, taking into account the spatial development o f the flow and utilizing a special coordinate transformation. The form ulation assumes that disturbance wavelength and eigenfunction vary dow nstream no more rapidly than the boundary-layer thickness, and include s all terms nominally of order R(-1) in the boundary-layer Reynolds nu mber R. In Blasius flow, the present approach is consistent with that of Bertolotti et al. (1992) to O(R(-1)) but simpler (i.e. has fewer te rms), and may best be seen as providing a parametric differential equa tion which can be solved without having to march in space. The compute d neutral boundaries depend strongly on distance from the surface, but the one corresponding to the inner maximum of the streamwise velocity perturbation happens to be close to the parallel flow (Orr-Sommerfeld ) boundary. For this quantity, solutions for the Falkner-Skan flows sh ow the effects of spatial growth to be striking only in the presence o f strong adverse pressure gradients. As a rational analysis to O(R(-1) ) demands inclusion of higher-order corrections on the mean flow, an i llustrative calculation of one such correction, due to the displacemen t effect of the boundary layer, is made, and shown to have a significa nt destabilizing influence on the stability boundary in strong adverse pressure gradients. The effect of non-parallelism on the growth of re latively high frequencies can be significant at low Reynolds numbers, but is marginal in other cases. As an extension of the present approac h, a method of dealing with non-similar flows is also presented and il lustrated. However, inherent in the transformation underlying the pres ent approach is a lower-order non-parallel theory, which is obtained b y dropping all terms of nominal order R(-1) except those required for obtaining the lowest-order solution in the critical and wall layers. I t is shown that a reduced Orr-Sommerfeld equation (in transformed coor dinates) already contains the major effects of non-parallelism.