Estimating amplitude ratios in boundary layer stability theory: a comparison between two approaches

We first demonstrate that, if the contributions of higher-order mean flow a re ignored, the parabolized stability equations (Bertolotti et al. 1992) an d the 'full' non-parallel equation of Govindarajan & Narasimha (1995, herea fter GN95) are both equivalent to order R-1 in the local Reynolds number R to Gaster's (1974) equation for the stability of spatially developing bound ary layers. It is therefore of some concern that a detailed comparison betw een Gaster (1974) and GN95 reveals a small difference in the computed ampli tude ratios. Although this difference is not significant in practical terms in Blasius flow, it is traced here to the approximation, in Gaster's metho d, of neglecting the change in eigenfunction shape due to flow non-parallel ism. This approximation is not justified in the critical and the wall layer s, where the neglected term is respectively O(R (-2/3)) and O(R-1) compared to the largest term. The excellent agreement of GN95 with exact numerical simulations, on the other hand, suggests that the effect of change in eigen function is accurately taken into account in that paper.