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.

1.
Tang, C., and Hafez, M., 2003, “Numerical Simulation of Steady Compressible Flows Using a Zonal Formulation,” Comput. Fluids, to appear.
2.
Hughes
,
T.J.R.
,
Franca
,
L.P.
, and
Hulbert
,
G.M.
,
1989
, “
A New Finite Element Formulation for Computational Fluid Dynamics: VIII. The Galerkin/Least Squares Method for Advective-Diffusive Equations
,”
Comput. Methods Appl. Mech. Eng.
,
73
, pp.
173
189
.
3.
Tezduyar, T.E., and Hughes, T.J.R., 1983, “Finite Element Formulations for Convection Dominated Flows With Particular Emphasis on the Compressible Euler Equations,” AIAA (83-0125), January Paper No. 83-0125.
4.
Lerat, A., and Corre, C., 2003, “Residual-Based Compact Schemes for Multidimensional Hyperbolic Systems of Conservation Laws,” Comput. Fluids, to appear.
5.
McCormack
,
R.W.
, and
Paullay
,
A.J.
,
1974
, “
The Influence of the Computational Mesh on Accuracy for Initial Value Problems With Discontinuous or Non-linear Solutions
,”
Comput. Fluids
,
2
, pp.
339
361
.
6.
Jameson, A., Schmidt, W., and Turkel, E., 1981, “Numerical Solutions for the Euler Equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes.” AIAA Paper No. 81-1259.
7.
Roy, J., Hafez, M., and Chattot, J., 2003, “Explicit Methods for the Solution of the Generalized Cauchy Riemann Equations and Simulation of Invicid Rotational Flows,” Comput. Fluids, to appear.
8.
Bachelor, G.K., 1967, An Introduction to Fluid Dynamics. Cambridge University Press, Cambridge, UK.
9.
Briggs, W.L., Hensen, V.E., and McCormick, S.F., 2000, A Multigrid Tutorial. SIAM.