On transonic boundary-layer receptivity over a vibrating flat plate coated with a thin liquid film

Abstract

In this paper we study the generation of Tollmien–Schlichting waves initiated by vibrations of a wall where the wall is coated with a thin liquid film in a transonic flow regime. Motion of fluids are described by the two-dimensional Navier–Stokes equations assuming the Reynolds number is large. To find asymptotic solutions of the transonic boundary layer, we conduct an inspection analysis on the affine transformations of the triple-deck model for a subsonic flow and the unsteady full potential equations, with the intention of obtaining the order quantity of the free-stream Mach number in the transonic flow. We construct a modified triple-deck model for the transonic flow by considering the scalings of the perturbations that lead to the viscous–inviscid interaction problem for the flow in a subsonic regime. In particular, we are interested in the region where the subsonic scalings become invalid as the flow approaches transonic regime. We assume the wall oscillates in the vertical direction to the oncoming flow and these vibrations are periodic in time. We outline the process where the flow in the boundary layer converts the wall vibration perturbations into the instability modes which are measured by the receptivity coefficient. The viscous–inviscid interaction problem describes the stability of the boundary layer on the lower branch of the neutral curve. We show that the governing equations for the air viscous sublayer and the film flow are quasi-steady. The equation describing the inviscid layer of the airflow is unsteady and its referred to as the unsteady Kàrmàn–Guderley equation. The influence of the film surface tension is expressed through normal shear stress condition at the interface. We present an analytic formula for the amplitude of the Tollmien–Schlichting waves that are formed in the boundary layer. We analysed our model with different values of surface tension, initial film thickness and Kàrmàn–Guderley parameter. Depending on the value of these parameters, the initial amplitude of the instability waves may grow or decay.