Vorticity-Velocity Formulation for Three-Dimensional Steady Compressible Flows

The vorticity-velocity formulation of the Navier-Stokes equations is extended to the solution of three-dimensional compressible fluid flow and heat transfer problems. The basic governing equations are expressed in terms of three Poisson-like equations for the velocity components together with a vorticity transport equation and an energy equation. The resulting seven coupled partial differential equations are solved by a finite difference method on a single grid and a discrete solution is obtained by combining a steady-state and a time-dependent Newton's method. Once a converged solution is obtained, one of the velocity equations can be removed from the system and replaced by the continuity equation and a "conservative" solution is obtained by using the previous solution as a starting estimate for Newton's method with only a few additional iterations. The numerical procedure is evaluated by applying it to natural and mixed convection problems. The formulation is found to be stable at high Rayleigh numbers and it may be applied to a wide variety of flow and heat transfer problems.