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

Abstract

AbstractThis paper is an extension of work on separation from a downstream moving wall by Ruban et al. (J. Fluid. Mech., vol. 678, 2011, pp. 124–155) and is in particular concerned with the boundary-layer separation in unsteady two-dimensional laminar supersonic flow. In a frame attached to the wall, the separation is assumed to be provoked by a shock wave impinging upon the boundary layer at a point that moves downstream with a non-dimensional speed which is assumed to be of order ${\mathit{Re}}^{\ensuremath{-} 1/ 8} $ where $\mathit{Re}$ is the Reynolds number. In the coordinate system of the shock however, the wall moves upstream. The strength of the shock and its speed are allowed to vary with time on a characteristic time scale that is large compared to ${\mathit{Re}}^{\ensuremath{-} 1/ 4} $. The ‘triple-deck’ model is used to describe the interaction process. The governing equations of the interaction problem can be derived from the Navier–Stokes equations in the limit $\mathit{Re}\ensuremath{\rightarrow} \infty $. The numerical solutions are obtained using a combination of finite differences along the streamwise direction and Chebyshev collocation along the normal direction in conjunction with Newton linearization. In the present study with the wall moving upstream, the evidence is inconclusive regarding the so-called ‘Moore–Rott–Sears’ criterion being satisfied. Instead it is observed that the pressure rise from its initial value is very slow and that a recirculation region forms, the upstream part of which is wedge-shaped, as also observed in turbulent marginal separation for large values of angle of attack.