A Modified Smagorinsky Subgrid Scale Model for the Large Eddy Simulation of Turbulent Flow

TL;DR

This work addresses the wall-damping weakness of the Smagorinsky subgrid scale model in Large Eddy Simulation by introducing a Modified Smagorinsky formulation that adaptively damps eddy viscosity near walls using an instantaneous Reynolds-number criterion, and by comparing its performance with a Dynamic Subgrid Scale approach on channel flow and flow over a backward-facing step. It integrates filtering theory within a finite element framework and analyzes the noncommutation error between filtering and differentiation, showing the interchange error can be second-order under smooth grid grading. The key contributions are (i) a wall-aware damping mechanism for the Smagorinsky model, (ii) a comparative study against DSGS, and (iii) demonstration that the modified approach yields improved mean velocity profiles and wall-turbulence statistics, with practical implications for wall-bounded turbulent flow simulations using finite elements. The findings suggest the proposed method achieves wall-damping with modest overhead, improving accuracy in channel and separation/reattachment scenarios and informing future solver and inlet-extension work for higher-fidelity LES.

Abstract

In the field of Large Eddy Simulation, the Smagorinsky subgrid scale model (in some form) is the most commonly accepted and used subgrid scale model. The purpose of this paper is to address the main weakness of the Smagorinsky model, its poor performance near the wall. The goal is to establish a model that corrects the Smagorinsky model near the walls while at the same time minimizing the computational overhead. A version the Dynamic Subgrid Scale model is also incorporated into the finite element code to facilitate comparisons with the new model near the walls. One of the unique characteristics of Large Eddy Simulations as compared to other methods of dealing with turbulent flows is the idea of filtering. In this paper we define what a filter is and also address an issue related to filters; the error that results when the filtering and differential operations are interchanged. This error is studied under the context of the Finite Element Method which allows us to focus on the function being filtered rather than the filter kernal function, which has been the usual approach in studying this error.

Figures (22)

• Figure 1: Sketch of the channel with relevant dimensions. For our experiment, we use $h=0.5$
• Figure 2: Sketch of the xy--plane face of the graded channel grid.
• Figure 3: Plot of the total kinetic engery for the channel flow using the Smagorinsky, DSGS, and modified Smagorinsky subgrid scale models. At time $t = 150$, a running total of the fluid variables are started to calculate the various mean properties of the flow.
• Figure 4: Time averaged plots of the eddy viscosity normalized by the kinematic viscosity. (a) is a plot along the line $x = 5 h$ and $z = 1.0625 h.$ (b) is a plot along the line $x = 9 h$ and $z = 1.0625 h$.
• Figure 5: Instantaneous snap shot of the eddy viscosity normalized by the kinematic viscosity along the line $x = 5 h$ and $z = 1.0625 h.$ (a) is using the Smagorinsky model, (b) is the DSGS model, and (c) is the modified Smagorinsky model.