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On unsteady boundary-layer separation in supersonic flow. Part 1. Upstream moving separation point

Published online by Cambridge University Press:  15 April 2011

Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2BZ, UK
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada
School of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK
Email address for correspondence:


This study is concerned with the boundary-layer separation from a rigid body surface in unsteady two-dimensional laminar supersonic flow. The separation is assumed to be provoked by a shock wave impinging upon the boundary layer at a point that moves with speed Vsh along the body surface. The strength of the shock and its speed Vsh are allowed to vary with time t, but not too fast, namely, we assume that the characteristic time scale tRe−1/2/Vw2. Here Re denotes the Reynolds number, and Vw = −Vsh is wall velocity referred to the gas velocity V in the free stream. We show that under this assumption the flow in the region of interaction between the shock and boundary layer may be treated as quasi-steady if it is considered in the coordinate frame moving with the shock. We start with the flow regime when Vw = O(Re−1/8). In this case, the interaction between the shock and boundary layer is described by classical triple-deck theory. The main modification to the usual triple-deck formulation is that in the moving frame the body surface is no longer stationary; it moves with the speed Vw = −Vsh. The corresponding solutions of the triple-deck equations have been constructed numerically. For this purpose, we use a numerical technique based on finite differencing along the streamwise direction and Chebyshev collocation in the direction normal to the body surface. In the second part of the paper, we assume that 1 ≫ VwO(Re−1/8), and concentrate our attention on the self-induced separation of the boundary layer. Assuming, as before, that the Reynolds number, Re, is large, the method of matched asymptotic expansions is used to construct the corresponding solutions of the Navier–Stokes equations in a vicinity of the separation point.

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