A naturally upwinded conservative procedure for the incompressible Navier-Stokes equations on non-staggered grids

Upwind methods have been previously adopted in incompressible flow calculations in an attempt to eliminate the need for artificial viscosity while maintaining high accuracy. Standard upwind methods found in the literature are primarily concerned with accurate modeling of the convection terms and generally neglect the viscous terms in the analysis. A new procedure is developed here which seeks to include the viscous terms so that regions of significant gradients will not be over-dissipated. The approach taken here is an extension of the work of Refs [1-5] in which interpolating functions are derived from direct integration of linearized forms of the governing equations. The resulting functions are then used as interpolants for differencing the full equations. During the process, no attempt is made to deliberately upwind bias the differencing. However, because of the manner in which coefficients of the differencing are functions of the cell Reynolds number, the scheme introduces its own upwind bias. This property was first observed by Roscoe [1] in which the observation that this class of schemes preserves the hyperbolic/elliptic nature of the original system of equations was made. The present work yields, for the first time, a fully conservative Roscoe-type method. Although the method approaches the development of the finite difference equations in a different manner than traditional upwind procedures, it achieves higher accuracy in a manner analogous to the MUSCL approach [6] used in standard upwind schemes. The procedure also uses a unique discretization of the continuity equation which involves pressure terms as in Ref. [5]. This equation is fully conservative and satisfies mass conservation to machine zero. The new scheme is first demonstrated on the 1-D, viscous, Burger's equation and results are compared with a first and second-order Roe scheme. It is then applied to the 2-D, incompressible Navier-Stokes equations for non-staggered grids. Results for the driven cavity problem are compared with proven methods and are found to be in excellent agreement. (C) 1997 Elsevier Science Ltd.