A multi-moment vortex method for 2D viscous fluids

TL;DR

This paper introduces a new multi-moment vortex method (MMVM) for simulating 2D viscous fluids, leveraging combinatorial formulas and Hermite expansions to efficiently model vortex interactions and deformation.

Contribution

The paper develops a simplified, exact formula-based MMVM that captures vortex deformation without costly non-local computations, advancing vortex simulation techniques.

Findings

Method captures shear diffusion and rapid relaxation to axisymmetry.

Exhibits exponential convergence in spatial accuracy.

Higher Hermite moments improve spatial accuracy.

Abstract

In this paper we introduce simplified, combinatorially exact formulas that arise in the vortex interaction model found in (Nagem, et al., SIAM J. Appl. Dyn. Syst. 2009). These combinatorial formulas allow for the efficient implementation and development of a new multi-moment vortex method (MMVM) using a Hermite expansion to simulate 2D vorticity. The method naturally allows the particles to deform and become highly anisotropic as they evolve without the added cost of computing the non-local Biot-Savart integral. We present three examples using MMVM. We first focus our attention on the implementation of a single particle, large number of Hermite moments case, in the context of quadrupole perturbations of the Lamb-Oseen vortex. At smaller perturbation values, we show the method captures the shear diffusion mechanism and the rapid relaxation (on $Re^{1/3}$ time scale) to an axisymmetric state. We then present two more examples of the full multi-moment vortex method and discuss the results in the context of classic vortex methods. We perform spatial convergence studies of the single-particle method and show that the method exhibits exponential convergence. Lastly, we numerically investigate the spatial accuracy improvement from the inclusion of higher Hermite moments in the full MMVM.