Three-dimensional upward scheme for solving the Euler equations on unstructured tetrahedral grids

Abstract

A new upwind scheme is developed for solving the three-dimensional Euler equations on unstructured tetrahedral meshes. The method yields solution accuracy and efficiency comparable to that currently available from similar structured-grid codes. The key to achieving this result is a novel cell reconstruction process which is based on an analytical formulation for computing solution gradients within tetrahedral cells. Prior methodology requires the application of cumbersome numerical procedures to evaluate surface integrals around the cell volume. The result is that higher-order differences can now be constructed more efficiently to attain computational times per cell comparable to those of structured codes.

The underlying philosophy employed in constructing the basic flow solver is to draw on proven structured-grid technology whenever possible in order to reduce risk. Thus, spatial discretization is accomplished by a cell-centered finite-volume formulation using flux-difference splitting. Solutions are advanced in time by a 3- stage Runge-Kutta time-stepping scheme with convergence accelerated to steady state by local time stepping and implicit residual smoothing. The flow solver operates at a speed of 34 microseconds per cell per cycle on a CRAY-2S supercomputer and requires 64 words of memory per cell.

Transonic solutions are presented for a broad class of configurations to demonstrate the accuracy, speed, and robustness of the new scheme. Solutions are shown for the ONERA M6 wing, the Boeing 747-200 configuration, a low-wing transport configuration, a high-speed civil transport configuration, and the space shuttle ascent configuration. Computed surface pressure-coefficient distributions on the ONERA M6 wing are compared with structured-grid results as well as experimental data to quantify the accuracy. A further assessment of grid sensitivity and the effect of convergence acceleration parameters is also included for this configuration.

The more complex configurations serve to demonstrate the robustness and efficiency of the new method and its potential for performing routine aerodynamic analysis of full aircraft configurations. For example, the basic transonic flow features are well captured on the space shuttle ascent configuration with only 7 megawords of memory and 142 minutes of CRAY-YMP run time.