Vortex Method for computing high-Reynolds number flows: Increased accuracy with a fully mesh-less formulation.

Abstract

For the applications of high Reynolds number flows, the vortex method presents the advantage of being free from numerically dissipative truncation error. In practice, however, many vortex methods introduce some numerical dissipation in mesh-based spatial adaption stages, or schemes such as vortex particle splitting. The need for spatial adaption in vortex methods arises from the Lagrangian framework, which results in an increase of discretization error over time. Presently, a vortex method is devised that incorporates radial basis function (RBF) interpolation to provide spatial adaption in a fully mesh-less formulation. Numerical experiments show that there is a potential for higher accuracy in comparison with the standard remeshing techniques. The rate of convergence of the new spatial adaption method is exponential, however convection error limits the vortex method to second order convergence. Avenues for future research involve decreasing convection error, for example by means of deformable basis functions. Nevertheless, the RBF-based spatial adaption scheme has various advantages, in addition to a demonstrated higher accuracy and the obvious benefit of not requiring a regular arrangement of particles or mesh. One presently demonstrated advantage is automatic core size control for the core spreading viscous method, without the need for vortex particle splitting. Three applications have been successfully treated with the presently developed vortex method. The relaxation of monopoles under non-linear perturbations has been computed, resulting in noticeable improvements compared to previously published results. The existence of a quasi-steady state consisting of a rotating tripole has been corroborated, for the case of large amplitude perturbations. The second application consists of the early adaptation of two co-rotating vortices at high Reynolds number, characterized by elliptical deformation of the cores, as well as small scale deformation in the weak areas of vorticity. This is considered to pose a severe test on the present method, or indeed any method. Comparison with results using spectral methods demonstrate in practice the potential for high accuracy computations using a mesh-less method, and in addition show that the naturally adaptive vortex method can result in considerably reduced problem sizes. Finally, for the calculation of non-symmetric Burgers vortices, a correction to the core spreading method for out-of-plane strain was developed. The results establish the capability of the vortex method for the computation of vortices under three-dimensional strain.