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

Abstract

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 V sh along the body surface. The strength of the shock and its speed V sh are allowed to vary with time t, but not too fast, namely, we assume that the characteristic time scale t Re −1/2 /V 2 w . Here Re denotes the Reynolds number, and V w = −V sh 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 V w = 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 V w = −V sh . 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 V w O(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.

Numerical solution of unsteady boundary-layer separation in supersonic flow: upstream moving wall

“…Let a shock wave be generated by a wedge placed above the plate and let this wedge, hence the shock, move parallel to the plate, as shown in figure 2. For consistency with the notation of Ruban et al (2011), the wedge velocity will be assumed positive if the wedge moves upstream as shown.…”

“…As long as the variation in shock velocity and strength is asymptotically small while it moves over a Re −3/8 triple-deck length scale, the flow can be assumed quasi-steady. The equations governing the flow are the interactive boundary-layer equations, scaled as in Ruban et al (2011):…”

“…The scheme and computational parameters are similar to those used in Ruban et al (2011). As before, we took a value of Y max = 50 to limit the computational domain in the Y-direction.…”

“…Recently Ruban et al (2011) demonstrated the existence of an MRS point and structure of the separated flow region for the downstream moving wall for external supersonic flow. The incompressible computations of Inoue (1981a,b) showed that the MRS conditions were satisfied at a point close to the start of the recirculating region.…”

“…This solution includes viscous-inviscid interaction. See also Ruban et al (2011) and the references therein. The singularity structure is quite different from the Goldstein one envisaged by Moore and Sears & Telionis, however. To understand unsteady separation, the major alternative to studying steady separation from a moving wall has been to study the formation of separation in an initially unseparated boundary layer.…”