Accurate estimate of disturbance amplitude variation from solution of minimal composite stability theory

Abstract

Minimal composite theory (Proc. R. Soc. Lond., Ser. A, 453 2537–2549, 1979) shows that, at the lowest order in the reciprocal of the local flow Reynolds number R, the stability of a spatially developing similarity flow may be described by an ordinary differential equation in the similarity coordinate. It is, in principle, not possible to determine the dependence of the disturbance amplitude on the streamwise coordinate solely from such an ordinary differential equation. However, noting that, to O(R−2/3), the dependence of the eigenfunction on the normal coordinate is identical in both the full non-parallel and minimal composite theories, and using a method due to Gaster, we show how the streamwise variation of disturbance amplitude can be determined to O(R−1 without solving a partial differential equation, although knowledge of the partial differential operator is required. Comparison with the DNS results of Fasel and Konzelmann shows excellent agreement with the present results. Furthermore, especially in strong adverse pressure gradients, the present amplitude ratio estimates are within 3% of the full non-parallel theory, whereas the Orr–Sommerfeld results show an underestimate by 26%.