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

Abstract

SUMMARYThis paper describes the ÿnite di erence 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 divergencefree velocity ÿeld. The vorticity transport equations in conservative form are solved using the secondorder accurate Adams-Bashforth central di erence scheme in order to assure divergence-free vorticity ÿeld in three dimensions. The velocity and vorticity Cartesian components are discretized using a central di erence 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 simpliÿed numerical scheme for solving three-dimensional problems, which otherwise requires enormous computational e ort. For three-dimensional-driven cavity ow predictions, the present method is found to be e cient and accurate for the Reynolds number range 1006Re62000.

Velocity-Vorticity Formulation for 2D Natural Convection in an Inclined Cavity by the DQ Method

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Effect of thermo‐solutal stratification on recirculation flow patterns in a backward‐facing step channel flow

“…In the present work the authors prefer to use velocity-vorticity formulation to the vorticity-stream function formulation because the dynamic variables, velocities, can be directly obtained. The velocity-vorticity form of Navier-Stokes equations first reported by Fasel [21] for stability analysis of boundary layers in two dimensions, has already been applied for the study of 3D flow problem [22], 3D natural convection problems [23,24] and recently for double diffusive mixed convection in a cavity [16]. In the present work, results are discussed for the effect of thermo-solutal buoyancy forces for (0.1 Ri 10) and (−10 N 10) on the variation of recirculatory flow patterns and heat and mass transfer in a channel with BFS at low Reynolds number.…”

The Method of Lines for Solving the Poisson Boundary Problem for Smectic SmC* an an External Electric Field

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