Arbitrary-Lagrangian-Eulerian finite volume IMEX schemes for the incompressible Navier-Stokes equations on evolving Chimera meshes

TL;DR

This paper introduces a novel semi-implicit IMEX finite volume scheme for incompressible Navier-Stokes equations on evolving Chimera meshes, enabling stable, accurate simulations with efficient handling of moving meshes and divergence-free constraints.

Contribution

The paper develops a second-order accurate semi-implicit IMEX Runge-Kutta scheme combined with a fractional-step method for evolving Chimera meshes, ensuring stability and free-stream preservation.

Findings

Stable and accurate solutions for incompressible flows on moving meshes.

CFL condition depends only on fluid velocity relative to mesh movement.

Successful benchmarks demonstrating the scheme's effectiveness.

Abstract

In this article we design a finite volume semi-implicit IMEX scheme for the incompressible Navier-Stokes equations on evolving Chimera meshes. We employ a time discretization technique that separates explicit and implicit terms which encompass both slow and fast scales. The finite volume approach for both explicit and implicit terms allows to encode into the nonlinear flux the velocity of displacement of the Chimera mesh via integration on moving cells. To attain second-order time accuracy, we employ semi-implicit IMEX Runge-Kutta schemes. These novel schemes are combined with a fractional-step method, thus the governing equations are eventually solved using a projection method to satisfy the divergence-free constraint of the velocity field. The implicit discretization of the viscous terms allows the CFL-type stability condition for the maximum admissible time step to be only defined by the relative fluid velocity referred to the movement of the frame and not depending also on the viscous eigenvalues. Communication between different grid blocks is enabled through compact exchange of information from the fringe cells of one mesh block to the field cells of the other block. Taking advantage of the continuity of the solution and the definition of a minimal compact stencil, the numerical solution of any system of differential equations is characterized by continuous data extrapolation. Free-stream preservation property, i.e. compliance with the Geometric Conservation Law (GCL), is respected. The accuracy and capabilities of the new numerical schemes is proved through an extensive range of test cases, demonstrating ability to solve relevant benchmarks in the field of incompressible fluids.