Damped Euler-equation method to compute transonic flow around wing-body combinations

Abstract

Inviscid transonic flows containing either strong shock waves or complex vortex structure call for the Euler equations as a realistic model. Presented here is a computational procedure for solving the Euler equations for transonic flow around a wing and fuselage upon an O-O mesh generated by transfinite interpolation. An explicit time-marching finite-volume procedure solves the flow equations and features a nonreflecting far field boundary condition, an internal mechanism for temporal damping, and use of the local time step, all of which improve the convergence of the computation. Converged after several hundred iterations, results computed on the CYBER 203 vector processor are compared with experimental data and potential-flow computations. The Euler-equation model is found to predict the existence of a tip vortex created by inviscid flow separation in the downstream region of the tip of the M6 wing where the radius of curvature approaches zero. AST year a workshop 1 was held in order to assess the currently used computational procedures, one against the other, in several carefully specified two-dimensional transonic flow problems. While reasonable agreement among the results given by the various full potential and Euler-equation methods was obtained in the subcritical cases, a disparity between the results of these two models was found to grow with cases of increasing shock strength. For the NACA 0012 airfoil and conditions M- 0.80 and a. = 1.25 deg, for example, the lift coefficients CL given by all the Euler methods ranged between 0.30 and 0.38, whereas the range of CL for the potential method was 0.28-1.1. This discrepancy has led to a reconsideration of the validity of the potential model when strong shocks are present in the flow, not only locally in terms of the isentropic shock jumps but also, and perhaps even more importantly, with regard to the correct modeling of vortex phenomena throughout the entire flowfield. In the potential representation the vorticity that is bound to the airfoil is accounted for by a jump in the potential across a line originating at the trailing edge and lying a priori along some chosen coordinate direction downstream, the so-called Kutta condition. The Euler equations, in contrast, admit vorticity in the solution and the equivalent to the Kutta condition evidently does not need to be enforced explicitly. This situation is currently under study by researchers using a variety of numerical methods. In three dimensions the flow past a finite wing is even more complex and less is known. Vortices, for example, are shed continuously from the wing tips and the entire trailing edge. Whether a Kutta condition or its equivalent is necessary in this case has not been in- vestigated, but because vorticity is so crucial to the realism achieved by inviscid flow models, interest has been aroused in questions like this and in numerical methods that solve the Euler equations. A number of methods 2'4 exist to solve the Euler equations for three-dimensional flow, but they have been developed exclusively for and applied only to internal flows. Apparently, the solution of the Euler equations for