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.

Development of a moving artificial compressibility solver on unified coordinates

“…This includes the Eulerian coordinates as a special case when h = 0 and the Lagrangian one when h = 1. As we have known from Hui's works [10,11], it is shown that the smooth solutions of the system of two-dimensional Euler equations of gas dynamics written in the classical Lagrangian coordinates are equivalent to the same system written in the unified coordinates with h = 1. The steps of the proof can easily be repeated for the current artificial compressibility flow equations to show that its weak hyperbolicity existed on the classical Lagrangian coordinates.…”

“…Therefore, numerical diffusion across the slip line can be reduced to a minimum with the crisp capturing of the contact discontinuity. This method was also extended to three-dimensional inviscid flow problems [11] and to shallow water wave problems [12]. In addition, Jin and Xu [13] recently provide another alterative to simulate the low Reynolds number flow based on gas-kinetic BGK model.…”

Compressible flow equations based on moving coordinates determined by the wave speed

“…The wave corresponding to the eigenvalue = 1 , 2 , 3 , or 4 is called an expansion wave in the wave frame if * /* >0, a compression wave if * /* <0, and a linearly degenerate one if * /* = 0.…”

“…Hui and his co-workers [3,4], originally proposed a unified coordinate system which includes the Eulerian approach and Lagrangian approach as two special cases. It keeps the advantages of the Lagrangian approach in capturing sliplines while avoiding severe grid deformation.…”

A Note on the Unified Coordinate System for Computing Shock Waves

“…Recently, Hui and his co-workers proposed a unified coordinate system which unifies both approaches [4,[6][7][8]. This unified approach combines the advantages of the Eulerian approach and the Lagrangian approach.…”