The viscous interaction between sound waves and the trailing edge of a supersonic splitter plate

Abstract

We consider the flow resulting from the interaction between the trailing edge of a supersonic splitter plate and sound waves incident on the trailing edge from upstream, as a model problem of relevance to understanding the unsteady flow in the vicinity of a supersonic jet nozzle. Morgan has previously shown that there is only one plausible solution to the outer potential-flow problem in this supersonic system, in which the vortex-sheet deflection close to the trailing edge varies linearly with the distance from the trailing edge, and that, in contrast to the subsonic version of the problem, it is not possible to construct an outer solution in which the vortex sheet leaves the plate smoothly (i.e. with zero gradient). In this paper our aim is to establish that this supersonic potential-theory solution is consistent with the equations governing the viscous flow close to the plate, and to provide a description of the nature of this inner flow, and we proceed by applying asymptotic analysis in the limit of large Reynolds number. For appropriate choices of the incident-wave amplitude and frequency, the canonical triple-deck structure at the trailing edge is realized, and the governing equations are then simplified by linearizing about the steady base flow in the lower deck; upstream of the trailing edge the unsteady flow is calculated analytically, whilst downstream a two-region parabolic scheme is employed. Our inner viscous flow is seen to match onto the outer potential-theory solution, and in particular we verify that the downstream evolution of the lower-deck flow as it emerges into the outer region corresponds exactly to the behaviour of the vortex sheet at the trailing edge in the outer flow. Once the consistency of the outer solution has been established, the dependence on the various flow parameters can be investigated, and we demonstrate in particular that significant unsteady shear-layer disturbances can be generated at the trailing edge over a wide range of values of the incidence angle, and that the amplitude of these disturbances decreases with increasing supersonic flow speed.