An Impulse-formed Navier-Stokes Solver based on Long-range Particle Flow Maps

TL;DR

This paper introduces a novel particle-grid characteristic-mapping framework for Navier-Stokes equations that effectively captures long-range flow dynamics, including viscosity and complex boundaries, by leveraging particle trajectories and impulse mapping.

Contribution

It extends long-range characteristic mapping to viscous Navier-Stokes flows using impulse as the primary transported quantity, integrating geometric and physical effects in a unified solver.

Findings

Accurately models viscous flows with complex boundaries.

Maintains geometric fidelity of flow transport.

Integrates viscosity and body forces via path integrals.

Abstract

We present a particle-grid characteristic-mapping framework that extends long-range characteristic mapping from inviscid flows to general Navier-Stokes dynamics with viscosity, body forces, and complex boundaries. Unlike traditional grid-based and vorticity-centered characteristic methods, our method is built on the observation that particle trajectories naturally provide the long-range flow map, enabling geometric quantities and their gradients to be transported in a direct and effective manner. We identify the impulse, the gauge variable of the velocity field, as the primary quantity mapped along characteristics while remaining compatible with standard velocity-based incompressible solvers. Using the 1-form representation of the impulse equation, we derive an integral formulation that decomposes the impulse evolution into a component transported geometrically along the particle flow map and a complementary component generated by viscosity and body forces evaluated through path integrals accumulated along particle trajectories. These components together yield a unified characteristic-mapping solver capable of handling incompressible Navier-Stokes flows with viscosity and body forces while maintaining the accuracy and geometric fidelity of characteristic transport.