Velocity–pressure coupling in finite difference formulations for the Navier–Stokes equations

Abstract

SUMMARYA new numerical procedure for solving the two-dimensional, steady, incompressible, viscous flow equations on a staggered Cartesian grid is presented in this paper. The proposed methodology is finite difference based, but essentially takes advantage of the best features of two well-established numerical formulations, the finite difference and finite volume methods. Some weaknesses of the finite difference approach are removed by exploiting the strengths of the finite volume method. In particular, the issue of velocitypressure coupling is dealt with in the proposed finite difference formulation by developing a pressure correction equation using the SIMPLE approach commonly used in finite volume formulations. However, since this is purely a finite difference formulation, numerical approximation of fluxes is not required. Results presented in this paper are based on first-and second-order upwind schemes for the convective terms. This new formulation is validated against experimental and other numerical data for well-known benchmark problems, namely developing laminar flow in a straight duct, flow over a backward-facing step, and lid-driven cavity flow.

Numerical Computation of a Two-Dimensional Biomagnetic Channel Flow

“…(5)- (7) is essentially based on a new numerical algorithm 21 with additional treatment on the magnetic terms. Also, there are some modifications in the discretisation of the NavierStokes equations which will be explained in a later section.…”

“…To overcome this problem, a new scheme that is implemented with the SIMPLE-type algorithm for the pressure field calculation is proposed. 21 The nonlinear partial differential governing equations will be discretised using the finite difference approximation which is carried out in a staggered grid. The advantages of using the staggered grid over a non-staggered grid are twofold.…”

The effect of magnetisation and Lorentz forces in a two-dimensional biomagnetic channel flow

“…It is constructive to note that the flow previously studied in Rusli et al [18] is driven by velocity gradient. Our original reference which is based on a numerical algorithm proposed by Zogheib [15] can only handle for flow which is driven by velocity gradient.…”

“…However, the formulation by Zogheib [15] is devised for the Navier-Stokes equations only. Therefore, this paper aims to provide a modified scheme that handles the additional magnetic part.…”

“…One of the main challenges of using the FDM is in the handling of the pressure of the flow. To overcome this problem, a new scheme that is implemented with the SIMPLE-type algorithm for the pressure field calculation is proposed by Zogheib [15]. The nonlinear partial differential governing equations will be discretised using the finite difference approximation which is carried out in a staggered grid.…”