Generalized discrete vortex method for cylinders without sharp edges

Abstract

VISCOUS flow around a circular cylinder may be simulated by the fractional step method of Chorin, where the inviscid part of the calculation is solved by convecting discrete vortices in a Lagrangian vortex-in-cell scheme with a radially expanding polar mesh and by superimposing random walks to simulate diffusion. This approach is extended to allow nearly arbitrary cylinder shapes, without sharp edges, by a conformal transformation of the shape to a circle. The attractions of the Lagrangian vortex method concerning numerical diffusion, stability, and efficiency are thus transferred to more general shapes, while maintaining high accuracy where it is required. Forces, pressures, and vortex shedding frequencies for eight different shapes were found to be in reasonable agreement with experiments at subcritical Reynolds numbers. Contents Flows around several cylinders without sharp edges have been computed by the Lagrangian discrete-vortex scheme first proposed by Chorin for the solution of the vorticity equation.1 At each time step, a vortex sheet is created along the cylinder surface in order to satisfy the boundary condition of zero tangential velocity. The sheet is then discretized into new vortices, which are added to those representing the vorticity field of the flow. The process of vorticity diffusion is simulated by adding a random perturbation to the position of each vortex. Finally the vortices are convected in the velocity field. Good flow simulations require a very large number of vortices, and their convection may be handled efficiently by the vortex-in-cell method.2 The computation is performed in a transformed complex plane f. The Theodorsen and Garrick transformation [z =/(£)] is used to map the region outside the cylinder in the physical complex plane z onto the region outside the unit circle If I =1 in the transformed plane. The function / contains a set of mapping coefficients, which are determined for each cylinder by the fast-Fourier-transform method of Ives.3'4 The vorticity field is modeled, in the physical plane, by a distribution of vortices outside the cylinder and, in the transformed plane, by a distribution of vortices of equal circulations at corresponding points outside the unit circle. 5fD, the random perturbation simulating diffusion, is given by