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A Vortex-Grid Method for Prandtl's Equations

A fast deterministic vortex-grid method for the solution of the two-dimensional unsteady boundary layer equations is presented. The new method uses a Lagrangian particle formulation for the convective step of the momentum equation, as in Chorin's random vortex sheet method, while the diffusion step is solved with finite differences on an adaptive Eulerian grid. Two algorithms for choosing the grid are proposed and compared. A numerical convergence proof is given on a standard model problem and the rate of convergence is estimated. Finally the scheme is applied to the impulsively started circular cylinder. Our results show that the present scheme is able to describe the formation of the singularity, in agreement with the results of Van Dommelen and Shen [ J. Comput. Phys., 38 (1980), pp. 125--140].