Computing cost-effective particle trajectories in numerically calculated incompressible fluids using geometric methods

TL;DR

This paper introduces a novel geometric algorithm for efficiently computing particle trajectories in incompressible fluids, leveraging divergence-free approximations and volume-preserving maps to improve accuracy and computational efficiency.

Contribution

The paper presents a new algorithm that combines matrix-valued radial basis functions with volume-preserving integration to track particles more accurately and efficiently in numerical fluid simulations.

Findings

Allows larger step-sizes in trajectory calculations

Reduces the number of interpolation points needed

Achieves higher accuracy in vortex simulations

Abstract

We present an novel algorithm for tracking massless solid particles in a divergence-free velocity field that is only available at discrete points in space and time such as those arising from a direct numerical simulation of Navier-Stokes. The algorithm creates a divergence-free approximation to the numerical field using matrix valued radial basis functions, which is integrated in time using a volume-preserving map. The resulting method is able to calculate accurate trajectories in a helical vortex using much larger step-sizes and a far lower number of interpolation points which results in a more efficient algorithm compared to a conventional scheme.