Numerical Method for Incompressible Vortical Flows with Two Unbounded Directions

Abstract

A new, efficient, and accurate method has been developed for computing unsteady, incompressible, viscous flows in a domain where two dimensions are unbounded, the third dimension is periodic and the vorticity is rapidly decaying in the unbounded directions. We use the term unbounded to mean doubly infinite (no boundaries of any kind). This is an extension of the methods described by others for flows with two periodic and one unbounded direction, where the irrotational velocities outside the vortical domain are treated analytically. The new method is shown to be both accurate and efficient. The method presented here has finite, but arbitrarily high order, formal accuracy, and incurs substantial additional cost for a given mesh. However, this increased cost is more than offset by the reduction in the number of mesh points required for a given accuracy. The result is that for accurate computations, the present method can be orders of magnitude more efficient than others currently in use. This paper presents the method, discusses implementation issues, validates its accuracy, and presents sample calculations.