An asymptotic theory for distributed receptivity of flow fields with pressure gradients

Abstract

A systematic asymptotic analysis is used to devise a model for distributed boundary-layer receptivity for flow fields with pressure gradients. The model predicts the generation of a Tollmien-Schlichting (TS) wave due to the interaction of a time-harmonic free-stream disturbance with distributed wall waviness and the subsequent evolution of this TS wave under the influence of wall waviness. The model is restricted to two dimensions and the Reynolds number is assumed to be large. Unlike previous models, the present model allows a nonzero pressure gradient in the base flow, and accounts for nonparallel-flow effects. A Green's-function approach was employed. The interaction between the free-stream disturbance and a point source at the wall was first examined in a local region near the point source. Only the component corresponding to the largest-growing instability wave was determined. The downstream evolution of this wave was then investigated over a region that extended for many TS wavelengths. For this reason it was necessary to account for the nonparallel-flow effects. By solving for the dispersion relation in this downstream region, the evolution of the eigenmode was determined. A match between the local-region and the downstream solution led to a uniform approximation for the TS wave emerging from the point source. Summing over a region of point-source solutions and approximating the resulting integral using the method of steepest descents yielded the instability wave for the wavy wall. The principal results of the present work included a solution to the dispersion relation for a general base flow with a nonzero pressure gradient beyond leading order. A region of maximal growth for TS waves that were generated from point sources located in a region of width O (Re-3/16) surrounding the lower branch neutral stability point (LBNSP) was identified. Here Re is the global Reynolds number. For adverse pressure gradients, it was also determined that, when the wall waviness and the TS wave are in exact resonance, the wave produced from distributed receptivity is significantly smaller than the wave generated by a point source at the LBNSP. For the case of a strong favorable pressure gradient the reverse is true. Finally, an investigation of the effect of pressure gradient on detuning was carried out. It showed that complete detuning occurs when the relative difference between the wavenumber of the TS wave at the LBNSP and the wavenumber of the sinusoidal wall is O (Re-3/16). It also revealed that the response to detuning varies with pressure gradient. For situations with a strong favorable pressure gradient, growth rates are highly sensitive to an exact match between the wavenumbers of the wall and the TS wave. On the other hand, as the pressure gradient grows less favorable, sensitivity to detuning decreases.