Numerical Solutions of Cauchy-Riemann Equations for Two and Three-Dimensional Flows

Abstract

For two-dimensional flows, the conservation of mass and the definition of vorticity comprise a generalized Cauchy-Riemann system for the velocity components assuming the vorticity is given. If the flow is compressible, the density is a function of the speed and the entropy, and the latter is assumed to be known. Introducing artificial time, a symmetric hyperbolic system can be easily constructed. Artificial viscosity is needed for numerical stability and is obtained from a least-squares formulation. The augmented system is solved explicitly with a standard point relaxation algorithm which is highly parallelizable. For an extension to three-dimensional flows the continuity equation is combined with the definitions of two vorticity components, and are solved for the three velocity components. Second-order accurate results are compared with exact solutions for incompressible, irrotational, and rotational flows around cylinders and spheres. Results for compressible (subsonic) flows are also included.