In this paper, we review some recent progress made in [4, 5, 6] on finite difference schemes for viscous incompressible flows using vorticity formulation. The main purpose of this series of papers [4, 5, 6] is to resurrect the idea of using local vorticity boundary condition for unsteady calculation. The emphasis is on simplicity of the methods. Three main issues will be discussed: efficient time-stepping procedures and cell Reynolds number constraints, efficient methods in 3D on non-staggered grids and efficient high order methods using compact differencing.

This paper discusses three basic issues related to the design of finite difference schemes for unsteadyviscous incompressible flows using vorticity formulations: the boundary condition for vorticity, an efficient time-stepping procedure, and the relation between these schemes and the ones based on velocity-pressure formulation. We show that many of the newly developed global vorticity boundary conditions can actually be written as some local formulas derived earlier. We also show that if we couple a standard centered difference scheme with third- or fourth-order explicit Runge-Kutta methods, the resulting schemes have nocell Reynolds number constraints. For high Reynolds number flows, these schemes are stable under the CFL condition given by the convective terms. Finally, we show that the classical MAC scheme is the same as Thom's formula coupled with second-order centered differences in the interior, in the sense that one can define discrete vorticity in a natural way for the MAC scheme and get the same values as the ones computed from Thom's formula. We use this to derive an efficient fourth-order Runge-Kutta time discretization for the MAC scheme from the one for Thom's formula. We present numerical results for driven cavity flow at high Reynolds number (10 ).

A new fourth-order accurate finite difference scheme for the computation of unsteady viscous incompressible flows is introduced. The scheme is based on the vorticity-stream function formulation. It is essentially compact and has the nice features of a compact scheme with regard to the treatment of boundary conditions. It is also very efficient, at every time step or Runge–Kutta stage, only two Poisson-like equations have to be solved. The Poisson-like equations are amenable to standard fast Poisson solvers usually designed for second order schemes. Detailed comparison with the second-order scheme shows the clear superiority of this new fourth-order scheme in resolving both the boundary layers and the gross features of the flow. This efficient fourth-order scheme also made it possible to compute the driven cavity flow at Reynolds number 106on a 10242grid at a reasonable cost. Fourth-order convergence is proved under mild regularity requirements. This is the first such result to our knowledge

This is the second of a series of papers on the subject of projection methods for viscous incompressible flow calculations. The purpose of the present paper is to explain why the accuracy of the velocity approximation is not affected by (1) the numerical boundary layers in the approximation of pressure and the intermediate velocity field and (2) the noncommutativity of the projection operator and the laplacian. This is done by using a Godunov–Ryabenki type of analysis in a rigorous fashion. By doing so, we hope to be able to convey the message that normal mode analysis is basically sufficient for understanding the stability and accuracy of a finite-difference method for the Navier–Stokes equation even in the presence of boundaries. As an example, we analyze the second-order projection method based on pressure increment formulations used by van Kan and Bell, Colella, and Glaz. The leading order error term in this case is of $O(\Delta t)$ and behaves as high frequency oscillations over the whole domain, compared with the $O(\Delta t^{{1 / 2}} )$ numerical boundary layers found in the second-order Kim–Momn method.

Simple, efficient, and accurate finite difference methods are introduced for 3D unsteady viscous incompressible flows in the vorticity–vector potential formulation on nonstaggered grids. Two different types of methods are discussed. They differ in the implementation of the normal component of the vorticity boundary condition and consequently the enforcement of the divergence free condition for vorticity. Both second-order and fourth-order accurate schemes are developed. A detailed accuracy test is performed, revealing the structure of the error and the effect of how the convective terms are discretized near the boundary. The influence of the divergence free condition for vorticity to the overall accuracy is studied. Results on the cubic driven cavity flow at Reynolds number 500 and 3200 are shown and compared with that of the MAC scheme.