A Variational Finite Element Model for Large-Eddy Simulations of Turbulent Flows

TL;DR

This paper presents a novel variational finite element Large Eddy Simulation model for turbulent channel flows, incorporating a boundary layer-aware eddy viscosity and proving convergence to steady-state Navier-Stokes solutions.

Contribution

It introduces a new finite element-based LES model with a boundary layer-adapted eddy viscosity and demonstrates convergence to steady Navier-Stokes solutions with Navier boundary conditions.

Findings

Model converges to steady Navier-Stokes solutions

Eddy viscosity depends on boundary layer friction velocity

Uses a finite element projection as filtering operation

Abstract

We introduce a new Large Eddy Simulation model in a channel, based on the projection on finite element spaces as filtering operation in its variational form, for a given triangulation $\{{\cal T}_h \}_{h>0}$. The eddy viscosity is expressed in terms of the friction velocity in the boundary layer due to the wall, and is of a standard sub grid-model form outside the boundary layer. The mixing length scale is locally equal to the grid size. The computational domain is the channel without the linear sub-layer of the boundary layer. The no slip boundary condition (BC) is replaced by a Navier (BC) at the computational wall. Considering the steady state case, we show that the variational finite element model we have introduced, has a solution $(\vv_h, p_h)_{h>0}$ that converges to a solution of the steady state Navier-Stokes Equation with Navier BC.