Extension of the fractional step method to general curvilinear coordinate systems

The extension of the fractional step method to three-dimensional, time-dependent incompressible flaws in non-orthogonal curvilinear coordinate systems is presented. A formulation based on block-LU decomposition is combined with a mixed implicit / explicit treatment of the discretized equations. Using local volume fluxes as dependent variables, the block-LU decomposition enables a unique definition of the sequential operations of the fractional step method for general coordinate systems. In this work a semi-direct scheme is developed for solution of the Poisson equation using series expansion along one coordinate direction that is discretized on a uniform, Cartesian grid. Also investigated in this study is solution of a simplified Poisson equation obtained by neglecting cross derivatives in the full Poisson equation. It is shown that for fractional step methods satisfaction of the zero-divergence constraint is still possible using the simplified Poisson equation, but the associated error is larger than θ(Δt).