A unified coordinate system for solving the three-dimensional Euler equations

Two general coordinate systems have been used extensively in computational fluid dynamics: the Eulerian and the Lagrangian. The Eulerian coordinates c ause excessive numerical diffusion across flow discontinuities, slip lines in particular. The Lagrangian coordinates, on the other hand, can resolve s lip lines sharply but cause severe grid deformation, resulting in large err ors and even breakdown of the computation. Recently, Hui et al. (J. Comput. Phys. 153,596 (1999)) have introduced a unified coordinate system which mo ves with velocity hq, q being velocity of the fluid particle. It includes t he Eulerian system as a special case when h = 0 and the Lagrangian when h = I and was shown to be superior to both Eulerian and Lagrangian systems for the two-dimensional Euler equations of gas dynamics when It is chosen to p reserve the grid angles. The main purpose of this paper is to extend the wo rk of Hui et al. to the three-dimensional Euler equations. In this case, th e free function h is chosen so as to preserve grid skewness. This results i n 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 line s, especially for steady flow. (C) 2001 Academic Press.