Numerical Investigations on the Accuracy of Vortex Methods With and Without Remeshing

Abstract

This article presents progress in the direction of a fully meshless formulation of the vortex method. Given that the Lagrangian formulation results in loss of blob overlap over time, a vortex method needs to incorporate some form of spatial adaption to maintain accuracy of discretization. The standard approach, motivated by the requirements of the particle strength exchange viscous scheme for an ordered particle field, has been the remeshing of vortices onto a grid. This in effect brings back the mesh to an otherwise meshless method, which has been subject of much debate. Presently, we investigate the accuracy of the vortex method with and without the standard remeshing. In addition, we formulate a new spatial adaption scheme, akin to the concept of rezoning, but based on ideas of radial basis function interpolation so that a regular mesh is not needed. Numerical experiments demonstrate the capacity of a considerable increase in accuracy. The vortex method (VM) is a Lagrangian, grid-free approach to solving the Navier-Stokes equations in vorticity formulation. It is based on spatially discretizing the vorticity field with elements (vortex blobs) that move with the local fluid velocity. The Lagrangian nature, however, implies that, due to strain, the blobs grow apart in some directions and cluster together in others. As the blobs cease to overlap they lose their ability to recover the smooth vorticity field and accuracy is lost. The widespread solution to this problem is remeshing the particle field, for which high-order interpolation kernels are used in tensor product formulations. (One should mention that some authors dislike the designation “remeshing” and instead refer to the process as “redistribution”. This may be justified as no mesh connectivity is required in the process; it does incur confusion, however, when there is a viscous scheme called vortex redistribution method.) Standard remeshing effectively controls the error due to Lagrangian distortion and long-time calculations are possible. Numerical investigations, however, demonstrate that the remeshing algorithms themselves introduce noticeable errors. In other words, these schemes bring back the mesh to an otherwise mesh-less method, and with it some numerical dissipation. Presently, experiments with a rezoning strategy based on solving the collocation problem for the blob circulation strengths have proved very encouraging. It is demonstrated that higher accuracy is possible, and in addition opportunities have been generated for providing a correction for the core spreading viscous scheme, as well as for variable resolution in the