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.
Numerical Solutions of Cauchy-Riemann Equations for Two and Three-Dimensional Flows
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, Aug. 24, 2001; final revision, June 11, 2002. Associate Editor: T. E. Tezduyar. Discussion on the paper should be addressed to the Editor, Prof. Robert M. McMeeking, Department of Mechanical and Environmental Engineering University of California–Santa Barbara, Santa Barbara, CA 93106-5070, and will be accepted until four months after final publication of the paper itself in the ASME JOURNAL OF APPLIED MECHANICS.
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Hafez , M., and Housman, J. (January 23, 2003). "Numerical Solutions of Cauchy-Riemann Equations for Two and Three-Dimensional Flows ." ASME. J. Appl. Mech. January 2003; 70(1): 27–31. https://doi.org/10.1115/1.1530632
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