HYPERBOLICITY AND OPTIMAL COORDINATES FOR THE 3-DIMENSIONAL SUPERSONIC EULER EQUATIONS

The two-dimensional (2-D) steady Euler equations for supersonic flow i n Cartesian coordinates (x,y) is hyperbolic except on the singular lin e u = alpha, where alpha is the speed of sound and u is the x-componen t of velocity. However, the Cauchy problem for x > x(o) with initial d ata prescribed at x = x(o) is well-posed only when u > alpha. For the 2-D case the optimal coordinates have been found [W. H. Hui and D. L. Chu, Comput. Fluid Dynamics, 4 (1996), pp. 403-426] to be the orthogon al system consisting of streamlines and their orthogonal lines. This g ives the most robust and accurate computation for supersonic flow. It is commonly thought (e.g., [C. Y. Loh and M. S. Lieu, J. Comput. Phys. , 113 (1994), pp. 224-248]) that the system of three-dimensional (3-D) steady Euler equations for supersonic flow is hyperbolic. The present paper shows that it is hyperbolic only when the velocity in the march ing direction is supersonic, in which case it also simultaneously guar antees the well-posedness of the Cauchy problem. Therefore the best co ordinate system that one can hope to have is a coordinate system in wh ich stream surfaces are coordinate surfaces. This leads to the general ized Lagrangian formulation of Hui and Loh [J. Comput. Phys., 89 (1990 ), pp. 207-240; 103 (1992), pp. 450-464: 103 (1992), pp. 465-471]. The optimal coordinate system must also be one for which the marching dir ection is the flow direction. The paper further shows that a necessary and sufficient condition for the existence of the optimal system is q .del V x q = 0, where q is the flow velocity. The implications of this condition on the design of marching computational schemes are discuss ed.