Hyperbolicity and optimal coordinates for the three-dimensional supersonic Euler equations

The two-dimensional (2-D) steady Euler equations for supersonic flow in Cartesian coordinates (x,y) is hyperbolic except on the singular line u = alpha, where alpha is the speed of sound and u is the x-component of velocity. However, the Cauchy problem for x > x(o) with initial data 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 orthogonal system consisting of streamlines and their orthogonal lines. This gives 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 marching direction is supersonic, in which case it also simultaneously guarantees the well-posedness of the Cauchy problem. Therefore the best coordinate system that one can hope to have is a coordinate system in which stream surfaces are coordinate surfaces. This leads to the generalized 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 direction 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 discussed.