Calculation of supersonic flows with strong viscous-inviscid interaction

Abstract

A procedure for the calculation of strong viscous-inviscid interactions in two-dimension al laminar super- sonic flows with and without separation is described. The equations solved are the so-called parabolized Navier-Stokes equations. The streamwise pressure gradient term is written as a combination of a forward and a backward difference to provide a path for upstream propagation of information. Global iteration is utilized to repeatedly update the pressure field from an initial guess until convergence is achieved. The numerical scheme employed is a new alternating direction explicit (ADE) procedure, which is used as an alternative to the more difficult to program multigrid strategy to accelerate convergence. Results are presented for flows past two flat plate related bodies. HE calculation of supersonic flows in which rapid changes in geometry or boundary condition occur often requires techniques other than classical Prandtl boundary- layer theory or other solution methods suitable for weakly flows. The reason for this is that at high Reynolds number, rapid changes in geometry or boundary condition induce large changes in the properties of the bound- ary layer over a very short length scale. When this occurs, the phenomenon of upstream influence, wherein information is propagated in the upstream direction, becomes important. This type of flow is referred to as a interacting flow. Methods that treat such flows as initial-value problems are not suitable because they are boundary value in nature, in the predominant flow direction. In this paper, a procedure for two-dimensional supersonic strongly flows is developed, which utilizes the parabolized or reduced Navier-Stokes equations. We will employ the former term here. For simplicity it will be as- sumed that x and y are nearly streamwise and stream normal directions, respectively. With special treatment of the pressure gradient term dp/dx, in the streamwise momentum equation, one can develop a procedure which is well suited for calculating weakly flows.14 In order to calculate strongly flows, a means for propagating information in the upstream direction is required. This can be satisfied by utilizing a forward difference for the stream- wise pressure gradient. In subsonic flows, where upstream influence is important even if the viscous-inviscid interaction is weak, a forward difference of px has been shown by Rubin5 to allow the calculation of such flows. Lin and Rubin6 later successfully applied forward differencing of px to a supersonic flow with weak interaction. In that study, forward differencing was utilized throughout the flowfield, including the supersonic flow. In the present technique for- ward differencing of px is applied only in the subsonic por-