A generalized hyperbolic marching technique for three-dimensional supersonic flow with shocks

Abstract

Numerous finite-difference procedures have been developed in recent years to compute supersonic flow fields about relatively simple bodies. These techniques are based on either Eulerian or Lagrangian systems or their combination and take advan- tage of the hyperbolic character of the governing equations to integrate initial Cauchy data in a coordinate direction along which the local velocity component is everywhere supersonic. Common to all these existing techniques (e.g., Babenko et al. [1964], Thomas et al. [1972], Kutler et al. [1973], and Marconi and Salas [1973]) is the requirement that the initial data lie on a plane normal to the marching direction (usually the body axis) and that this plane advance downstream undistorted. This condition simplifies the coordinate geometry and difference equations, but ignores the orientation of the initial data's cones of influence. T For some flows, e.g., those with large incidence angles, the angles between streamlines of the initial data and the marching direction can be so great that the velocity component in that direc- tion is actually subsonic. In such cases, existing finite-difference methods are inappropriate and a more general procedure is needed that admits initial conditions situated on any arbitrary surface and integrates them forward along a general curvi- linear coordinate that nearly conforms to the local streamline, thus distorting and deforming the initial data surface as it advances. This paper introduces such a generalized, nonorthogonal coordinate system and develops finite-difference operators that solve the three-dimensional, steady, inviscid conservation equations of fluid flow referenced to this general frame. The paper also presents a new method of alin- ing the difference mesh to the bow shock wave. This technique eliminates both the necessity of differencing the free-stream flow properties as well as the spurious fluctuations that usually arise in the conservative variables when they are differenced across the discontinuity.