Vorticity Boundary Condition and Related Issues for Finite Difference Schemes

context of finite difference schemes in vorticity formulation has a long history, going back at least to the 1930s when This paper discusses three basic issues related to the design of finite difference schemes for unsteady viscous incompressible Thom’s formula (see (2.4)) was derived [20]. Thom’s forflows using vorticity formulations: the boundary condition for mula is generally referred to as being local since vorticity vorticity, an efficient time-stepping procedure, and the relation at the boundary is given by a local relation which does not between these schemes and the ones based on velocity–pressure involve coupling to other points at the boundary. There formulation. We show that many of the newly developed global vorticity boundary conditions can actually be written as some was a resurgence of interest in the 1960s and early 1970s local formulas derived earlier. We also show that if we couple a when many variants of Thom’s formula were derived (see standard centered difference scheme with thirdor fourth-order Section 2 and [18]). But the application of these formulas explicit Runge–Kutta methods, the resulting schemes have no in actual computations met with only limited success. It cell Reynolds number constraints. For high Reynolds number flows, these schemes are stable under the CFL condition given was not clear, for example, whether high order formulas by the convective terms. Finally, we show that the classical MAC such as Pearson’s were actually better than lower order scheme is the same as Thom’s formula coupled with secondones. Since most of the computations at the time were order centered differences in the interior, in the sense that one steady state calculations, these formulas were used in an can define discrete vorticity in a natural way for the MAC scheme iterative procedure, and choosing the right relaxation paand get the same values as the ones computed from Thom’s formula. We use this to derive an efficient fourth-order Runge– rameter for the iteration was an issue that caused a great Kutta time discretization for the MAC scheme from the one for deal of confusion. The status as of 1974 was summarized Thom’s formula. We present numerical results for driven cavity in the review article of Orszag and Israeli [12]. flow at high Reynolds number (10). Q 1996 Academic Press, Inc. The point of view that has been heavily favored in the last decade is that the vorticity boundary condition has to be global; i.e., one has to solve a system of equations