Receptivity of the boundary layer in transonic flow past an aircraft wing

Abstract

This thesis presents a theoretical analysis of the generation of Tollmien- Schlichting waves in a boundary layer on the wing surface due to external acoustic disturbances and elastic vibrations of the wing itself. An asymptotic approach based on the assumption that the Reynolds number, Re, is large is adopted here. In addition, it is assumed that in the spectrum of these disturbances there are harmonics, which come in resonance with the Tollmien-Schlichting wave on the lower branch of the stability curve. This work is restricted to the cases when the Stokes layer interacts with an isolated roughness, and the flow near the roughness is described by the triple-deck theory. The solution of the triple-deck problem is found in an analytic form and the main concern is with the flow behaviour downstream of the roughness. Further, it is found that there are Tollmien-Schlichting waves forming in the boundary layer behind the roughness, and their amplitudes have been expressed in terms of the receptivity coefficients, which represent the efficiency of the Tollmien-Schlichting wave generation process. The analysis of the boundary layer receptivity to the acoustic disturbances is conducted with an assumption that the flow in the free-stream is in the transonic regime. It is shown that in this situation there are two plane acoustic waves with distinctively different characteristics. One wave always travels downstream in the streamwise direction and has O(1) phase velocity. The second wave phase velocity is an order Re−1/9 quantity and it changes the direction of propagation depending on the Mach number. The analysis has shown that the receptivity coefficient depends on the initial frequency of the perturbations and on the free-stream Mach number. It has been shown that the absolute value of the receptivity coefficient is achieved when the Mach number is one. For the negative values of the parameter, the subsonic behaviour of the receptivity coefficient is recovered. The study of the “slow” moving wave shows that the receptivity coefficient now depends on the shape of the roughness itself as well as acoustic wave parameters and the free-stream Mach number. The third problem considered is a receptivity of the boundary layer on the wing surface due to an elastic vibration of the wing itself. It is found that, when the frequency of the wing surface vibrations is high, the perturbations produced by wing surface vibrations can be described in the framework of “piston” theory. In the flow considered there are two physical mechanisms through which an oscillatory motion of the fluid in the Stokes layer is ex- cited. The first one is the same as for the acoustic problem where the pressure gradient forces the fluid to oscillate in the direction along the wing surface. The second mechanism can only be observed in compressible flows. It was found that there are two Tollmien-Schlichting waves forming in the boundary layer behind the roughness. The first receptivity process is similar to the one studied earlier with the difference that now the flow is considered to be in the subsonic regime. The second receptivity process does not have an analogue in the literature, and it was found that it is large as compared with the first receptivity. This suggests that the receptivity to the wing surface vibrations has to play a major role in the laminar-turbulent transition in the boundary layer.