Compressible corner flow

Abstract

The viscous compressible flow in the vicinity of a right-angle corner, formed by the intersection of two perpendicular flat plates and aligned with the free stream, is investigated. In the absence of viscous-inviscid interactions and imbedded shock waves, a theory is developed that is valid throughout the subsonic and supersonic Mach number range. Within this limitation and the additional assumptions of unit Prandtl number and a linear viscosity-temperature law, a consistent set of governing equations and boundary conditions is derived. The method of matched asymptotic expansions is applied in order to distinguish the relevant regions in the flow field.In the corner region the Crocco integral is shown to apply, even for a three-dimensional flow field. The equations governing the flow in the corner layer consist of four coupled nonlinear elliptic partial differential equations of the Poisson variety. Since they do not lend themselves to analytic solution, numerical methods are employed. Two such methods used here are the Gauss-Seidel explicit technique and the alternating direction implicit method. The merits of both techniques are discussed with regard to convergence rate, accuracy and stability. The calculations show that in cases where the Gauss-Seidel method fails to give converged solutions, owing to instability, the alternating direction implicit method does provide converged solutions. However, in cases where both methods are convergent, there is no appreciable difference in convergence rates. The numerical calculations were done on a CDC 6600 computer.Results of calculations are presented for representative compressible-flow conditions. The extent of the corner disturbance is controlled by the Mach number and wall temperature ratio in a manner analogous to the two-dimensional boundary layer. A swirling motion is noted in the corner layer which is influenced to a great extent by the asymptotic cross-flow profiles. The skin-friction coefficient is shown to increase monotonically from zero at the corner point to its asymptotic two-dimensional value. For cold wall cases, this value is approached more rapidly. The asymptotic analysis indicates that for even colder wall cases, not considered here, an overshoot is possible.