Flat-plate leading edge receptivity to various free-stream disturbance structures.

Abstract

The receptivity process by which two-dimensional, time-harmonic freestream disturbances generate instability waves in the incompressible Blasius boundary layer is investigated analytically. The importance of the leading edge region and the linear nature of the receptivity process are discussed, and Goldstein's (1983a, 1983b) theoretical framework for the leading edge receptivity problem is reviewed. His approach utilizes asymptotic matching of a region close to the leading edge, which is governed by the linearized unsteady boundary layer equation, with a region further downstream, which is described by an Orr-Sommerfeld type equation. The linearized unsteady boundary layer equation is solved numerically, using the slip velocity and pressure gradient obtained from the inviscid interaction of the freestream disturbance with the semi-infinite plate. A new method is developed to extract the receptivity coefficient from this numerical solution. The receptivity coefficient determines the amplitude of the instability wave--a quantity not available from classical stability theory. The freestream disturbances investigated are oblique plane acoustic waves, vortical gusts of various orientations convected downstream with freestream speed U(∞), and a Karman vortex street passing above the plate surface with speed U(p). In addition, the case of a semi-infinite plate in a channel of finite width subject to an upstream traveling acoustic wave on the upper plate surface is considered. For oblique acoustic waves, the dominant receptivity mechanism is related to scattering of the waves by the leading edge. In contrast, for vortical gusts the receptivity produced by leading edge scattering is very small. The boundary layer receptivity to a Karman vortex street is found to be a strong function of the speed ratio U(p)/U(∞). A pronounced influence of channel walls, which is related to the alternate cut-on of higher modes in the upstream and downstream channel halves, is found. A comparison of the present results with available experiments shows good qualitative and quantitative agreement.