Boundary Conditions and Efficient Solution Algorithms For the Potential Function Formulation of the 3-D Viscous Flow Equations

Abstract

SUMMARY For incompressible viscous creeping flows that occur in a number of geophysical situations, the velocity field may be expressed as the curl of a vector potential function, the use of which allows the momentum equation to be written as a biharmonic equation. The 3-D Cartesian formulation for a constant viscosity fluid is summarized here with special reference to two important types of boundary condition: the stress-free boundary (with zero normal velocity and zero tangential stress) and the rigid boundary (with all components of velocity zero). Fast algorithms for inversion of the biharmonic operator with all boundaries stress-free are well established. There also exists a fast method for the solution of the biharmonic equation with a parallel pair of rigid boundaries with the other boundaries stress-free. This method has not previously been applied, but it is a relatively straightforward extension of the Fourier transform based algorithm for the stress-free problem, using an analytical solution to enforce the required boundary conditions for each horizontal harmonic component. The method is easily vectorized and allows solutions to be obtained that compare very favourably in accuracy and solution time with those for the stress-free problem. The errors are of comparable magnitude given the differing harmonic content required by the boundary conditions, and the solution requires between 20 per ce-t (for a 3-D problem) and 30 per cent (for a 2-D problem) more processing time than does the solution of a comparable stress-free problem.