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

A robust, arti cial compressibility scheme has been developed for modelling laminar steady state and transient, incompressible ows over a wide range of Reynolds and Rayleigh numbers. Arti cial 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 o er good convergence rates. Free arti cial compressibility parameters in the equations are automated to allow ease of use while facilitating improved or comparable convergence rates as compared with the standard arti cial compressibility scheme. Memory e ciency is achieved through a multistage, pseudo-timeexplicit time-marching solution procedure. A node-centred dual-cell edge-based nite volume discretization technique, suitable for unstructured grids, is used due to its computational e ciency and high-resolution spatial accuracy. In the interest of computational e ciency and ease of implementation, stabilization is achieved via a scalar-valued arti cial 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 II, the scheme will be applied to a number of example problems to solve ows over a wide range of Reynolds and Rayleigh numbers