A finite volume formulation for transient convection and diffusion equations with unstructured distorted grids and its applications in fluid flow simulations with a collocated variable arrangement

Abstract

This paper describes a finite volume method for simulating transport processes governed by convection–diffusion type equations. The formulation is based on a cell-centred, unstructured grid. With an edge-based data structure, discretisation is independent of control volume (or cell) shape. By using a surface vector decomposition at the midpoint of the interface between cells, along with a deferred-correction approach, any cross-diffusion due to grid skewness can be readily accounted for when discretising the diffusive flux. For modelling fluid flow processes, a collocated arrangement of variables is employed so that a single coefficient matrix applies for the momentum equations of each velocity component. To avoid ‘checkerboard oscillation’ (arising from pressure–velocity decoupling) occurring under the collocated variable arrangement when a pressure-based solution algorithm is employed, a novel pseudo-flux interpolation method is proposed for unstructured grids, ensuring that the solution is both under-relaxation factor and time-step (for transient calculation) independent. The methodology can be formulated within a framework whereby either a coupled or a decoupled solution algorithm can be employed. The features and advantages of the method are demonstrated by solution of the Navier–Stokes equations for two benchmark flow problems.