Numerical solution of three-dimensional velocity-vorticity Navier-Stokes equations by finite difference method

Abstract

This paper describes the finite difference numerical procedure for solving velocity–vorticity form of the Navier–Stokes equations in three dimensions. The velocity Poisson equations are made parabolic using the false-transient technique and are solved along with the vorticity transport equations. The parabolic velocity Poisson equations are advanced in time using the alternating direction implicit (ADI) procedure and are solved along with the continuity equation for velocities, thus ensuring a divergence-free velocity field. The vorticity transport equations in conservative form are solved using the second-order accurate Adams–Bashforth central difference scheme in order to assure divergence-free vorticity field in three dimensions. The velocity and vorticity Cartesian components are discretized using a central difference scheme on a staggered grid for accuracy reasons. The application of the ADI procedure for the parabolic velocity Poisson equations along with the continuity equation results in diagonally dominant tri-diagonal matrix equations. Thus the explicit method for the vorticity equations and the tri-diagonal matrix algorithm for the Poisson equations combine to give a simplified numerical scheme for solving three-dimensional problems, which otherwise requires enormous computational effort. For three-dimensional-driven cavity flow predictions, the present method is found to be efficient and accurate for the Reynolds number range 100⩽Re⩽2000. Copyright © 2004 John Wiley & Sons, Ltd.