A Unified Coordinate System for Solving the Three-Dimensional Euler Equations

Abstract

Two general coordinate systems have been used extensively in computational fluid dynamics: the Eulerian and the Lagrangian. The Eulerian coordinates cause excessive numerical diffusion across flow discontinuities, slip lines in particular. The Lagrangian coordinates, on the other hand, can resolve slip lines sharply but cause severe grid deformation, resulting in large errors and even breakdown of the computation. Recently, Hui et al. (J. Comput. Phys.153, 596 (1999)) have introduced a unified coordinate system which moves with velocity hq, q being velocity of the fluid particle. It includes the Eulerian system as a special case when h=0 and the Lagrangian when h=1 and was shown to be superior to both Eulerian and Lagrangian systems for the two-dimensional Euler equations of gas dynamics when h is chosen to preserve the grid angles. The main purpose of this paper is to extend the work of Hui et al. to the three-dimensional Euler equations. In this case, the free function h is chosen so as to preserve grid skewness. This results in a coordinate system which avoids the excessive numerical diffusion across slip lines in the Eulerian coordinates and avoids severe grid deformation in the Lagrangian coordinates; yet it retains sharp resolution of slip lines, especially for steady flow.