# A Finite Volume approximation of the Navier-Stokes equations with nonlinear filtering stabilization

## TL;DR

This paper introduces a finite volume approach combined with a nonlinear filtering stabilization for simulating incompressible fluid flows at moderate Reynolds numbers, validated against benchmark and experimental data.

## Contribution

It proposes integrating the Evolve-Filter-Relax algorithm with a finite volume method, providing practical guidance for parameter tuning and mesh feature effects.

## Key findings

- Finite volume method yields different results than finite element method at similar mesh levels.
- Parameter tuning can improve model accuracy and stability.
- Mesh features significantly influence simulation accuracy.

## Abstract

We consider a Leray model with a nonlinear differential low-pass filter for the simulation of incompressible fluid flow at moderately large Reynolds number (in the range of a few thousands) with under-refined meshes. For the implementation of the model, we adopt the three-step algorithm Evolve-Filter-Relax (EFR). The Leray model has been extensively applied within a Finite Element (FE) framework. Here, we propose to combine the EFR algorithm with a computationally efficient Finite Volume (FV) method. Our approach is validated against numerical data available in the literature for the 2D flow past a cylinder and against experimental measurements for the 3D fluid flow in an idealized medical device, as recommended by the U.S. Food and Drug Administration. We will show that for similar levels of mesh refinement FV and FE methods provide significantly different results. Through our numerical experiments, we are able to provide practical directions to tune the parameters involved in the model. Furthermore, we are able to investigate the impact of mesh features (element type, non-orthogonality, local refinement, and element aspect ratio) and the discretization method for the convective term on the agreement between numerical solutions and experimental data.