The effect of anisotropic wall compliance on boundary-layer stability and transition

Abstract

A fairly simple theoretical model of an anisotropic compliant wall has been developed. It has been used to undertake a comprehensive numerical study of boundary-layer stability over such walls. The study is based on linearized theory, makes the usual quasi-parallel-flow approximation, uses the Blasius profile as the basic undisturbed flow and assumes two-dimensional disturbances. An investigation is carried out of the effects of anisotropic wall compliance on the Tollmien–Schlichting waves and the two previously identified wall modes, namely travelling-wave flutter and divergence. In addition global convergence techniques are used to search for other possible instabilities.An asymptotic theory, valid for high Reynolds numbers, is also presented. This can provide accurate estimates of the eigenvalues. It is applicable to a much wider class of compliant walls than the relatively simple model used for the numerical study. An important use of the asymptotic theory is to help identify and elucidate the various energy-exchange mechanisms responsible for stabilization or destabilization of the instabilities. A reduction in the production of disturbance energy by the Reynolds shear stress is the main reason for the favourable effect of anisotropic wall compliance on instability growth. Other energy-exchange mechanisms, which have been found to make a significant contribution, include energy transfer from the disturbance to the mean flow due to the interaction of the fluctuating shear stress and the displaced mean flow, and the work done by the perturbations in wall pressure and shear stress.It is found that anisotropic wall compliance confers very considerable advantage with respect to reduction in instability growth rate and transition delay. Using a fairly conservative criterion an almost ten-fold rise in transitional Reynolds number is predicted for anisotropic walls having the appropriate properties. Anisotropic wall compliance makes travelling-wave flutter much more sensitive to viscous effects and has a considerable stabilizing influence. The application of global convergence methods has led to the discovery of an anomalous spatially growing eigenmode which, according to conventional interpretation, would represent an instability. Further study of an appropriate initial-value problem has revealed that the new eigenmode is probably not an instability and that, for compliant walls, complex wavenumbers with positive real and negative imaginary parts do not necessarily correspond to an instability.