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

Abstract

SUMMARYA robust, artiÿcial compressibility scheme has been developed for modelling laminar steady state and transient, incompressible ows over a wide range of Reynolds and Rayleigh numbers. Artiÿcial 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 artiÿcial compressibility parameters in the equations are automated to allow ease of use while facilitating improved or comparable convergence rates as compared with the standard artiÿcial compressibility scheme. Memory e ciency is achieved through a multistage, pseudo-timeexplicit time-marching solution procedure. A node-centred dual-cell edge-based ÿnite 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 artiÿcial 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.

A matrix free, partitioned solution of fluid–structure interaction problems using finite volume and finite element methods

“…For the purpose of transient calculations, a dual-time-stepping temporal discretisation [40] is employed such that second-order temporal accuracy is achieved while ensuring that all equations are iteratively solved simultaneously in an implicit fashion.…”

“…where ε is typically chosen as 0.1u max where u max is the peak flow velocity in the domain [40]. The third and final incremental solution step written in semi-discrete form now follows:…”

“…The coupling of a higher-order finite element formulation for the structure and a linear finite volume formulation for the fluid leads to non-matching nodes at the fluid/solid interface and the transfer of information at these nodes will also be addressed in this paper. Dual-timestepping [40] is employed for the purpose of temporal discretisation.…”

Choice of Variables and Preconditioning for Time Dependent Problems

“…When evaluating the artificial viscosity we use Q variables as given by (5). However, when updating the variables we revert to w c variables.…”

“…To overcome this difficulty a dual time step method has been used by many investigators. [3][4][5][6]20 In this approach the solution at the next physical time step is determined as a steady state problem to which preconditioning is applicable. We shall analyze time dependent effects on the preconditioning parameters.…”