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

Abstract

This 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 Re −1/8 where 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 Re −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 Re → ∞. 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.

Material spike formation in highly unsteady separated flows

“…While the triple-deck theory (Ruban et al 2011) has partially solved the above limitation, all boundary-layer-singularity techniques are valid in the limit of infinite Reynolds number, as opposed to the finite-Reynolds-number flows arising in practice. Similar limitations apply to the Moore-Rott-Sears (MRS) criterion (Rott 1956;Sears 1956;Moore 1958), as shown in Williams (1977), van Dommelen & Shen (1982 and Yapalparvi & van Dommelen (2012).…”

On quasi-steady boundary-layer separation in supersonic flow. Part 2. Downstream moving separation point

“…In § 4 we presented the results of the numerical solution of the triple-deck equations for the boundary layer on an upstream moving wall. The case of an impinging shock, when the boundary layer is exposed to the adverse pressure gradient, was analysed earlier by Zhuk (1982) and Yapalparvi & van Dommelen (2012). Their calculations (and our theoretical arguments) suggest that the increasing wall speed suppresses the separation.…”

“…Zhuk (1982) was the first to present a numerical solution of the triple-deck equations for the upstream moving wall. More recently the calculations were repeated by Yapalparvi & van Dommelen (2012). The results of these works suggest that increasing wall speed suppresses the separation.…”

A new fluid dynamical model coupling horizontal heat with an application to interior separations

“…Moreover, the boundary layer separation has been extensively and deeply investigated in the past several decades, see Refs. [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. It is worth pointing out that the geometric theory of 2D incompressible flows has been established by Ma and Wang to study the structural stability and transition of 2D incompressible fluid flows in Refs.…”