On hypersonic self‐induced separation, hydraulic jumps and boundary layers with algebraic growth

Abstract

Summary Analytical and numerical properties are described for the free interaction and separation arising when the induced pressure and local displacement are equal, in reduced terms, for large Reynolds number flow. The interaction, known to apply to hypersonic flow, is shown to have possible relevance also to the origins of supercritical (Froude number > 1) hydraulic jumps in liquid layers flowing along horizontal walls. The main theoretical task is to obtain the ultimate behaviour far beyond the separation. An unusual structure is found to emerge there, involving a backward–moving wall layer with algebraically growing velocity at its outer edge, detached shear layer moving forward and, in between, reversed inertial flow uninfluenced directly by the adverse pressure gradient. As a result the pressure then increases like (distance)m, with m = 2(√(7)–2)/3 (= 0.43050 …), and does not approach a plateau. Some more general properties of (Falkner–Skan) boundary layers with algebraic growth are also described.

High‐order upwinding and the hydraulic jump

“…Their analysis was subsequently extended by Bowles and Smith [24] to large Froude number fully developed layers, and again by Higuera [25; 26]. Despite all this research activity, the hydraulic jump is not yet fully understood: it is, to quote Gajjar and Smith [23], clearly a complicated process.…”

“…Finally, an asymptotic high-Reynolds number analysis of the ow, around the upstream end of a laminar jump in a layer of uniform velocity with a thin viscous sublayer, was carried out by Gajjar and Smith [23] using interacting boundary layer theory. Their analysis was subsequently extended by Bowles and Smith [24] to large Froude number fully developed layers, and again by Higuera [25; 26].…”

Instabilities in a high-Reynolds-number boundary layer on a film-coated surface

“…Some aspects of the problem were also examined within the high-Reynolds-number asymptotic framework akin to that in the present paper, in both steady (e.g. Gajjar & Smith 1983;Gajjar 1987;Bowles 1995) andunsteady (e.g. Mahmudov &Terent'ev 1988;Brotherton-Ratcliffe & Smith 1989;Bagbekov & Terent'ev 1991;Hoyle & Smith 1994) formulations.…”

Lighthill and the triple-deck, separation and transition

“…Since then it has found application in many other areas of fluid mechanics, including for example internal channel flow [17,18] and hydraulic jumps [19,20].…”