General Order Virtual Element Approximation for the Smagorinsky turbulence model

TL;DR

This work develops a general-order Virtual Element Method discretization for the Smagorinsky turbulence model applied to the 2D incompressible Navier–Stokes equations. It introduces a divergence-free VEM framework with a first-time discretization of the Smagorinsky term, and provides a convergence analysis complemented by numerical tests including convergence tests and lid-driven cavity flows up to $Re=10^4$. The results show that isotropic refinement with hanging nodes yields superior accuracy and robustness of the Newton solver compared with uniform or anisotropic meshes, highlighting VEM's suitability for convection-dominated turbulence problems on polygonal meshes. Overall, the paper extends VEM to turbulence modeling and demonstrates practical mesh strategies that improve efficiency and reliability in 2D turbulent simulations.

Abstract

In this paper, we investigate a Smagorinsky model in a virtual element framework to simulate convection-dominated Navier-Stokes equations. We conduct a two-dimensional numerical investigation to assess the performance of the general order virtual element approximation in this context. First, we examine numerically the convergence of the method with respect to the meshsize to certify the novel virtual element numerical discretization, which includes, for the first time, a discretization of the Smagorinsky term. Moreover, we present a numerical study of a lid-driven cavity for different Reynolds numbers (up to 10000) and meshes (uniform, anisotropic, and isotropic with hanging nodes). The results highlight the main advantage of using the virtual elements method in this context: the isotropic refinement with hanging nodes enhances the accuracy of the solution compared to the anisotropic mesh, uses fewer degrees of freedom with respect to the uniform mesh, and yields the most stable behavior in terms of convergence of the Newton solver.

Figures (7)

• Figure 1: Irrotational force test. Pressure errors $\left\| p-p_h\right\|_{\mathcal{{T}}_h}$ for $Re=1$ and $k \in \{2,3\}$.
• Figure 2: Lid-driven cavity flow. Mesh visualization: $31\times31$ USM (left), $31\times31$ ARM (center), and IMH based on a $22\times22$ USM (right).
• Figure 3: Lid-driven cavity flow. Velocity profiles with various Reynolds numbers: $Re = 1000$ using coarse meshes (left), $Re=2500$ using medium meshes (center), and $Re=3200$ using fine meshes (right). The solid and dotted lines represent the $x$- and $y$-components of the velocity, respectively.
• Figure 4: Lid-driven cavity flow. Streamlines for $Re=3200$ using the three fine meshes, and the USM 100x100 for comparison.
• Figure 5: Lid-driven cavity flow. Streamlines for Reynolds numbers in laminar, transitional, and turbulent regimes, using the fine IMH mesh and the USM 100x100 for comparison.

Theorems & Definitions (2)

• Remark 1: Vector monomial decomposition and scaling
• Remark 2: Anisotropic scaling