Numerical Solution of the Three-Dimensional Equations of Motion for Laminar Natural Convection

Abstract

A method is presented for numerical finite difference solution of the equations of motion in three dimensions. The complete Navier‐Stokes equations are transformed and expressed in terms of vorticity and a vector potential. The transformed equations are solved using an alternating direction method for the parabolic portion of the problem, and successive over‐relaxation for the elliptic portion. The classical problem of convection in fluid layers bounded by solid walls and heated from below is solved in both two and three dimensions. Comparison with other methods and with prior work in two dimensions shows that the new method presented here has important advantages in speed and accuracy. Apparently, there has been no successful prior work in three dimensions except in cases where one component of the equation of motion is highly simplified.