The separating flow induced on a ramp under a supersonic main stream is discussed, for high Reynolds numbers, according to the interactive-boundary-layer approach. There are two principal motivations for the study, apart from the recent general upsurge of interest in compressible boundary-layer separation and stall. The first is the need for more progress to be made in the numerical determination of strongly reversed flows than has proved possible in computations hitherto. This is tackled by means of a new computational scheme based in effect on a global Newton iteration procedure but coupled with linearized shooting, second-order accurate windward differencing and linear multi-sweeping. The specific case addressed is the triple-deck version for steady laminar two-dimensional motion, although the present scheme, like the theory described below, also has broader application, for example to subsonic, hypersonic and/or unsteady interactive flows. The second motivation is to compare closely with the recent theoretical prediction (Smith 1988a) of a local breakdown or stall occurring in any interactive boundary-layer solution at a finite value of the controlling parameter, a say, within the reversed-flow region; the breakdown produces a large adverse pressure gradient and minimum negative surface shear locally. The first quantitative comparisons are made between the theory and computational results, derived in this work at values of a, here the scaled ramp angle, greater than those obtainable before, but with a fixed outer boundary. The agreement, while not complete, seems to prove fairly affirmative overall and tends to support the suggestion (in Smith 1988a) that, contrary to most earlier expectations, in general there is a finite upper limit on the extent to which the interacting boundary-layer approach can be taken on its own. A similar conclusion holds for unsteady interactive boundary layers concerning a finite-time breakdown (Smith 19886) and boundary-layer transition, and in the present context the local nonlinear breakdown provides an explanation for the severe computational difficulties encountered previously as well as for a form of airfoil stall.

In the fundamental configuration studied here, a steady hypersonic free stream flows over a thin sharp aligned airfoil or flat plate with a leading-edge shock wave, and the flow field in the shock layer (containing a viscous and an inviscid layer) is steady laminar and two-dimensional, for a perfect gas without real and high-temperature gas effects. The viscous and inviscid layers are analysed and computed simultaneously in the region from the leading edge to the trailing edge, including the upstream-influence effect present, to determine the interactive flow throughout the shock layer and the positions of the shock wave and the boundary-layer edge, where matching is required. Further theoretical analysis of the shock layer helps to explain the computational results, including the nonlinear breakdown possible when forward marching against enhanced upstream influence, for example as the wall enthalpy increases towards its insulated value. Then the viscous layer is computed by sweeping methods, for higher values of wall enthalpies, to prevent this nonlinear breakdown for airfoils including the flat plate. Thin airfoils in hypersonic viscous flow are treated, for higher values of the wall enthalpies and with the upstream-influence effect, as are hypersonic inviscid flows, by modifying the computational methods used for the flat plate. Also, the behaviour of the upstream influence for bodies of relatively large thickness, and under wall velocity slip and enthalpy jump for flat plates, is discussed briefly from a theoretical point of view.Subsequent to the present work, computations based on the Navier–Stokes and on the parabolized Navier–Stokes equations have yielded excellent and good agreement respectively with the present predictions for large Mach and Reynolds numbers.

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