Numerical Simulation of Three-Dimensional Incompressible Flow by a New Formulation

SUMMARY A new mathematical formulation, called the pseudovorticity-velocity formulation, of the three-dimensional incompressible NavierStokes equations is presented as an alternative to the vorticity-velocity approach. For the model lid-driven cavity flow problem in two and three dimensions, combined with an explicit mixed spectral/iinite different numerical scheme the proposed formulation is found to be efficient and very accurate as compared with the results available in the literature. In particular, the simulation results demonstrate an attractive feature of the present formulation compared with the vorticity-velocity approach, namely that the divergencefree condition of the velocity field can always be achieved on a non-staggered mesh. The mathematical formulations that are commonly used to simulate three-dimensional incompressible viscous flows include the primitive variables’ (velocity-pressure), vorticity-vector p~tential~,~ and vorticity-velocity4 formulations. As indicated in an overview of these formulations by Gresho: each formulation has its own advantages as well as shortcomings with respect to the others. Both the vorticity-vector potential formulations and the vorticity-velocity approach have a distinct advantage over the velocityTressure formulation in that the pressure need not be calculated explicitly. The vorticity-velocity method, similar to the velocityTressure formulation but unlike the velocity-vector potential approach, suffers from a major difficult in obtaining a solenoidal velocity field so that the continuity equation is satisfied explicitly. For the vorticity-velocity formulation the use of a staggered mesh or alternative boundary conditions for the vorticity has been proposed to provide a divergence-free velocity field.6 In the velocity-vector potential formulation, however, the vector potential is not uniquely defined and a scalar potential is further required for the non-enclosed flow configuration. It follows from the foregoing that it is highly desirable to develop a formulation having the advantages of the velocity-vorticity formulation but avoiding the difficulty of obtaining a divergencefree velocity field. Jia and Nakamura’ recently presented a new formulation in terms of velocity and a new variable q for two-dimensional incompressible flow. It has been demonstrated that their formulation can be applied to both steady and unsteady flow simulations on a non-staggered grid, yielding an essentially divergence-free velocity field. The present study represents a continuing effort in pursuit of this aspect. In this paper we present a new formulation, called the pseudovorticityvelocity formulation, and its application to three-dimensional incompressible viscous flow