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

Abstract

It is well known that the use of Eulerian coordinates for shock capturing methods results in badly smeared slip lines, and that Lagrangian coordinates, while capable of producing sharp slip line resolution, may result in severe grid deformation, causing inaccuracy and even breakdown of computation. A unified coordinate system is introduced in which the flow variables are considered to be functions of time and of some permanent identification of pseudo-particles which move with velocity hq, q being the velocity of fluid particles. It includes the Eulerian coordinates as a special case when h=0, and the Lagrangian when h=1. For two-dimensional inviscid flow, the free function h is chosen so as to preserve the grid angles. This results in a coordinate system which avoids 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. Furthermore, the two-dimensional unsteady Euler equations of gasdynamics in the unified coordinates are found to be hyperbolic for all values of h, except when h=1 (i.e., Lagrangian). In the latter case the Euler equations are only weakly hyperbolic, lacking one eigenvector, although all eigenvalues are real. The consequences of this deficiency of the Lagrangian coordinates are pointed out in connection with numerical computation.