Some Comments on the Numerical Integration of the Vorticity Equation and Related Equations

Abstract

It is shown that if the streamfunction is specified over an Eulerian region R (x, y) at t=0, and on the boundaries of R (x, y) at t > 0, a solution of the vorticity equation which satisfies certain conditions of continuity will be unique. If vorticity is also specified at the inflow boundary, such a solution is excluded. In the latter case, an internal discontinuity propagates into the region along a material surface, and in principle solutions should be obtained separately for each sub-region and “matched” kinematically and dynamically at the interface. The associated analytical or numerical problems are prohibitive.

The problem of obtaining stable methods of numerical integration, specifying only the stream function on the boundary, is considered. Numerical integrations of a non-linear first order differential equation indicate that such stable methods may exist, in contradiction to linear theory.

The integration of the vorticity and associated equations for the first time-step is also considered. A method is presented which is not only simple to apply, but also maintains the same accuracy as succeeding time-steps for which centered difference techniques are used.