Short-wave instability of an elastic plate in supersonic flow in the presence of the boundary layer

Abstract

Panel flutter is a dangerous aeroelastic instability of the skin panels of supersonic flight vehicles. Though the linear stability of panels in uniform flow has been studied in detail, the influence of the boundary layer is still an open question. Most studies of panel flutter in the presence of the boundary layer are devoted to the ($1/7$)th-power velocity law and yield a stabilising effect of the boundary layer. Recently, Vedeneev (J. Fluid Mech., vol. 736, 2013, pp. 216–249) considered arbitrary velocity and temperature profiles and showed that, for a generalised convex boundary layer profile, a decrease of the growth rates of ‘supersonic’ perturbations (responsible for single-mode panel flutter) is accompanied by destabilisation of ‘subsonic’ perturbations that are neutral in uniform flow. However, this result is not self-consistent, as the long-wave expansion for solutions of the Rayleigh equation was used, whereas subsonic perturbations, generally speaking, cannot be considered as long waves. More surprising results are obtained for the boundary layer profile with a generalised inflection point, where the effect of the layer is destabilising even for ‘supersonic’ perturbations, and such waves can also have short lengths. In order to overcome this inconsistency, in this paper, we solve the Rayleigh equation numerically and investigate the stability of short-wave perturbation of the elastic plate in the presence of the boundary layer. As before, two problem formulations are investigated. First, we study running waves in an infinite plate. Second, we analyse eigenmodes of the plate of large finite length and use Kulikovskii’s global instability criterion. Based on the results of calculations, we confirm that the effect of the boundary layer with a generalised inflection point can be essentially destabilising. On the other hand, for generalised convex boundary layers, calculations show that, unlike the prediction of the long-wave approximation, the finite plate is fully stabilised for sufficiently thick boundary layers.