Symmetry Plane Laminar and Turbulent Viscous Flow on Bodies at Incidence

Abstract

METHOD is developed for solving the thin viscous shocklayer equations for flow along the symmetry plane of sharp or blunt bodies at angle of attack in hypersonic flow. The governing equations are obtained from the Navier-Stokes equations by retaining terms up through second order in the inverse square root of the Reynolds number. The resulting equations are solved using a Crank-Nicolson type finite difference scheme. Contents The procedure generally used for this type of problem at zero incidence involves solving the uncoupled in viscid and boundarylayer equations and coupling the solution in an iterative manner. Because of difficulties involved in matching the boundary layer with the inviscid flow for angle-of-attack problems, this method cannot be readily extended. In the present approach, the complete shock-layer flow is obtained from a single set of equations developed from the general steady-state Navier-Stokes equations using effective transport parameters. The matching problem is therefore eliminated, and displacement thickness effects on inviscid flow are included within the shock-layer approximation. Davis1 developed this concept for laminar flow over bodies at zero angle of attack. The method was extended by Eaton and Kaestner2 to the case of laminar flow on the windward planes of sharp cones at angles of attack. The problem is further extended in this paper3 to include combinations of sharp or blunt bodies at angle of attack in laminar or turbulent flow. The analysis is developed for arbitrarily shaped bodies with a minimum of one plane of symmetry. The flow between the bow shock and body surface is calculated by solving the shock-layer equations. Nonlinear terms are locally linearized. The resulting parabolic equations are tridiagonal in form and are solved using the Thomas algorithm.4 To address the turbulent problem, a mean velocity closure scheme is incorporated into the method by using a Van Driest type two-layer mixing length model with exponential damping near the wall. Properties at the shock boundary are calculated using Rankine-Hugoniot equations. The body surface temperature is specified and the surface velocities are set equal to zero. The resulting computer code (VIS) written for the CDC 6600 accommodates blunt or sharp power-law bodies, hyperboloids, and cones with elliptical cross sections. However, results are only presented3 for bodies with axisymmetric cross sections.