Laminar boundary-layer reattachment in supersonic flow

Abstract

A rational theory is developed to describe the reattachment of a laminar shear layer in supersonic flow. In the neighbourhood of reattachment the flow develops a threetiered or ‘triple-deck’ structure analogous to that which occurs at a point of separation (Stewartson & Williams 1969) and, as in the separation problem, the local flow pattern may be found independently of the flow in the surrounding regions. The fundamental problem of the reattachment triple deck reduces to the solution of the incompressible boundary-layer equations in the lower deck, which is of streamwise and lateral dimensions O(R^−⅜) and O(R^−⅝), where R [Gt ] 1 is a representative Reynolds number for the flow. Pressure variations in this region are O(R^−¼). Asymptotic solutions in terms of x, the scaled streamwise lower-deck variable, are derived to confirm the transition from a reverse flow profile at x = 0+, through reattachment, to a forward flow as x → ∞, the attainment of the required asymptotic form downstream (as x → ∞) being shown to depend crucially upon the correct choice of the finite part of the pressure in the lower deck at x = 0+. The lower-deck solution is singular at x = 0+ and assumes a complicated multi-structured form which is shown to match upstream with the solution in a largely inviscid region of dimension O(R^−½) where the pressure is O(1) and the major part of the flow reversal takes place. Solutions are presented for reattachment at a wall and for symmetric reattachment behind a wedge or bluff body. In the former case the results also explain the apparent ignorance of upstream conditions in the expansive triple-deck solution formulated by Stewartson (1970) in the context of supersonic flow around a convex corner.