An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: Part I. Theory and implementation

A robust, artificial compressibility scheme has been developed for modelling laminar steady state and transient, incompressible flows over a wide range of Reynolds and Rayleigh numbers. Artificial compressibility is applied in a consistant manner resulting in a system of preconditioned governing equations. A locally generalized preconditioner is introduced, designed to be robust and offer good convergence rates. Free artificial compressibility parameters in the equations are automated to allow ease of use while facilitating improved or comparable convergence rates as compared with the standard artificial compressibility scheme. Memory efficiency is achieved through a multistage, pseudo-time-explicit time-marching solution procedure. A node-centred dual-cell edge-based finite volume discretization technique, suitable for unstructured grids, is used due to its computational efficiency and high-resolution spatial accuracy. In the interest of computational efficiency and ease of implementation, stabilization is achieved via a scalar-valued artificial dissipation scheme. Temporal accuracy is facilitated by employing a second-order accurate, dual-time-stepping method. In this part of the paper the theory and implementation details are discussed. In Part 11, the scheme will be applied to a number of example problems to solve flows over a wide range of Reynolds and Rayleigh numbers.

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