Use of a rotated Riemann solver for the two-dimensional Euler equations.

Abstract

A scheme for the two-dimensional Euler equations that uses flow parameters to determine the direction for upwind-differencing is described. This approach exploits the multi-dimensional nature of the equations and reduces the grid-dependence of conventional schemes. Several angles are tested as the dominant upwinding direction, including the local flow, pressure-gradient, and velocity-magnitude-gradient angles. Roe's approximate Riemann solver is used to calculate fluxes in the upwind direction, as well as for the flux component normal to the upwinding direction. The approach is first tested for two-dimensional scalar convection, where data used to calculate cell face fluxes are interpolated to lie along characteristic lines. The scheme is shown to have accuracy comparable to a high-order MUSCL scheme. A method is also implemented to preserve monotonicity. Solutions of the Euler equations are calculated for a variety of test cases, including channel flows and flows about airfoils. Of the upwinding angles tested, the velocity-magnitude-gradient angle is best because it is normal to both shock and shear waves. Channel flow solutions for a first-order scheme show significant improvement over conventional first-order grid-aligned upwinding, and performance almost comparable to a high-order MUSCL scheme. The method works especially well for pure shear waves. The method is extended to higher-order using a larger interpolation template, but the improvements are marginal. It also becomes more difficult to preserve monotone solutions. Results for the flows about airfoils do not show much improvement over the grid-aligned methods. The quality of the solutions for subsonic and transonic speeds is actually degraded, partly due to large entropy generation at the leading edge. The approach is promising in that it uses flow solution features, rather than grid features, in the solution method. However, the method is computationally expensive, and needs improvement before it can be used on nonuniform grids.