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

TL;DR

The paper develops a Finite Volume implementation of the Leray-$\alpha$-NL model with nonlinear filtering stabilization, embedded in the Evolve-Filter-Relax (EFR) algorithm, to simulate incompressible flows on under-resolved meshes. By coupling a nonlinear differential low-pass filter with a velocity–update step and a relaxation, the approach stabilizes marginally resolved scales without overdiffusion and is implemented in OpenFOAM. Numerical experiments on 2D flow past a cylinder and 3D FDA nozzle benchmarks show that the nonlinear EFR (with $\chi$ chosen via a physically motivated formula) yields close agreement with experimental data on coarser meshes, while standard NSE or linear-filter variants may fail or require finer meshes. The results highlight the importance of mesh quality, the choice of the filter radius $\alpha$, and the relaxation parameter $\chi$ in achieving accurate predictions at moderately large $Re$, suggesting practical guidelines for parameter tuning and mesh design. The work paves the way for extensions to reduced-order modeling and fluid–structure interaction within a FV Leray framework.

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.

Figures (17)

• Figure 2: (a) Part of mesh $16k_H$ and (b) velocity magnitude at $t = 8$ computed by the NSE on mesh $16k_H$. The velocity magnitude goes from 0 (blue) to 3.9e-1 (red).
• Figure 3: Evolution of lift and drag coefficients given by DNS with meshes $200k_H$ and $200k_P$ compared against the results in John2004. The legend in (b) is common to both subfigures.
• Figure 4: Evolution of lift and drag coefficients computed from the NSE solutions on the prismatic meshes (top) and all hexahedral meshes (bottom) reported in Table \ref{['tab:1']}. The legend in (b) and (d) holds also for (a) and (c).
• Figure 5: Maximum (a) lift and (b) drag coefficients errors as defined in \ref{['eq:E_c_1']} for the different meshes under consideration. The legend in (b) is common to both subfigures.
• Figure 6: Evolution over time of the lift coefficient given by the EF L (left) and EF NL (right) algorithms for $\alpha = h_{min}, \eta$ on all the hexahedral meshes in Table \ref{['tab:1']}. All the figures report also the NSE results.

Theorems & Definitions (4)

• Remark 2.1
• Remark 2.2
• Remark 4.1
• Remark 4.2