Numerical Solutions for the Pressure Poisson Equation with Neumann Boundary Conditions Using a Non-staggered Grid, I

Numerical solutions are obtained for the pressure Poisson equation with Neumann boundary conditions using a non-staggered grid. The existence of a solution for this equation requires the satisfaction of a compatibility condition which relates the source of the Poisson equation and the Neumann boundary conditions. This compatibility condition is not automatically satisfied on non-staggered grids. Failure to satisfy the compatibility condition leads to non-convergent iterative solutions. Consistent finite-difference approximations for the pressure equation with Neumann boundary conditions are developed to satisfy the compatibility condition on non-staggered grids. The method is applied to calculate the pressure coefficient in a driven cavity when given the velocity field. The velocity is computed from the stream function-vorticity formulation of the Navier-Stokes equations.