Multi-resolution lattice Green's function method for incompressible flows

TL;DR

This paper introduces a multi-resolution lattice Green's function method for efficiently solving incompressible flow equations on unbounded domains, combining adaptive mesh refinement with advanced numerical schemes.

Contribution

It presents a novel adaptive mesh refinement approach compatible with the LGF technique, enabling efficient simulation of complex incompressible flows on unbounded domains.

Findings

Verified with vortex ring simulations

Demonstrated reduced computational cells with adaptive refinement

Achieved efficient high Reynolds number flow simulations

Abstract

We propose a multi-resolution strategy that is compatible with the lattice Green's function (LGF) technique for solving viscous, incompressible flows on unbounded domains. The LGF method exploits the regularity of a finite-volume scheme on a formally unbounded Cartesian mesh to yield robust and computationally efficient solutions. The original method is spatially adaptive, but challenging to integrate with embedded mesh refinement as the underlying LGF is only defined for a fixed resolution. We present an ansatz for adaptive mesh refinement, where the solutions to the pressure Poisson equation are approximated using the LGF technique on a composite mesh constructed from a series of infinite lattices of differing resolution. To solve the incompressible Navier-Stokes equations, this is further combined with an integrating factor for the viscous terms and an appropriate Runge-Kutta scheme for the resulting differential-algebraic equations. The parallelized algorithm is verified through with numerical simulations of vortex rings, and the collision of vortex rings at high Reynolds number is simulated to demonstrate the reduction in computational cells achievable with both spatial and refinement adaptivity.