SIAM J. ScI. STAT. COMPUT. Vol. 11, No. 3, pp. 399-424, May 1990 1990 Society for Industrial and Applied Mathematics 001 VORTEX METHODS FOR SLIGHTLY VISCOUS THREE-DIMENSIONAL FLOW* DALIA FISHELOV" Abstract. Vortex methods for slightly viscous three-dimensional flow are presented. Vortex methods have been used extensively for two-dimensional problems, though their most efficient extension to three- dimensional problems is still under investigation. A method that evaluates the vorticity by exactly differentiat- ing an approximate velocity field is applied. Numerical results are presented for a flow past a semi-infinite plate, and they demonstrate three-dimensional features of the flow and transition to turbulence. Key words, vortex methods, boundary layers, turbulent flow AMS(MOS) subject classifications. 76D05, 76D10, 35Q10 1. Introduction. Vortex methods as suggested by Chorin [12] were applied to various problems to simulate incompressible flows (see [34] and [32] for a review). These grid-free methods represent complicated flows by concentrating the computa- tional elements in regions where small-scale phenomena predominate and few elements elsewhere. In addition, vortex methods introduce no artificial viscosity, and therefore they are adequate for solving the slightly viscous Navier-Stokes equation. Vortex methods have been used extensively in the last 15 years, especially for two-dimensional flows. Although three-dimensional vortex methods have been con- sidered inherently difficult, we represent a scheme that involves no elaborate computa- tions and is a natural extension of the two-dimensional schemes. We applied this method to a three-dimensional flow past a semi-infinite plate at high Reynolds number. The velocity far away from the plate is assumed to be uniform. If we assume that the flow is independent of the spanwise variable, the problem is two-dimensional, otherwise the flow is three-dimensional. Chorin [11]-[13] solved the two-dimensional problem numerically; he used computational elements, called blobs, with a smoothed kernel. This kernel is obtained by convolving the singular kernel, which connects vorticity and velocity, with a smoothing function (called a cutoff function). The latter approximates a delta function in the sense that a finite number of its moments are identical to those of a delta function. A numerical solution to a three-dimensional problem was introduced in 1980 by Chorin [11] and by Leonard [32]-[34] using different vortex filament methods. In the filament method we approximate the initial velocity and vorticity along vortex lines, whose tangents are parallel to the vorticity vector. Since circulation is conserved along vortex lines, there is no need to update vorticity. Both authors [11], [34] stepped the Navier-Stokes equations in time by splitting them to the Euler and the heat equations. In [33] Leonard introduces one of the earliest vortex methods to solve the inviscid three-dimensional Euler equations numerically. In his computations he was able to simulate the time development of spotlike disturbances in laminar three-dimensional boundary layer. He suggested to split the velocity field into a sum of the velocity at infinity and a perturbed one, and to track vortex lines and compute their curvatures. * Received by the editors April 25, 1988; accepted for publication (in revised form) May 2, 1989. This work was partially supported (at the Lawrence Berkeley Laboratory) by the Applied Mathematical Sciences Subprogram of the Office of Energy Research, U.S. Department of Energy under contract DE-AC03- 76SF00098, and in part by the Office of Naval Research under contract N00014-76-C-0316. f Department of Mathematics and Lawrence Berkeley Laboratory, University of California, Berkeley, California 94720. Present address, Department of Applied Mathematics, The Weizmann Institute of Science, P.O. Box 26, Rehovot 76100, Israel. 399