Large Eddy Simulation with Energy-Conserving Schemes and the Smagorinsky Model: A Note on Accuracy and Computational Efficiency

Abstract

Despite advances in turbulence modelling, the Smagorinsky model remains a popular choice for large eddy simulation (LES) due to its simplicity and ease of use. The dissipation in turbulence energy that the model introduces, is proportional to the Smagorinsky constant, of which many different values have been proposed. These values have been derived for certain simulated test-cases while using a specific set of numerical schemes, to obtain the correct dissipation in energy simply because an incorrect value of the Smagorinsky constant would lead to an incorrect dissipation. However, it is important to bear in mind that numerical codes may suffer from numerical or artificial dissipation, which occurs spuriously through a combination of spatio-temporal and iterative errors. The latter can be controlled through more iterations, the former however, depends on the grid resolution and the time step. Recent research suggests that a complete energy-conserving (EC) spatio-temporal discretisation guarantees zero numerical dissipation for any grid resolution and time step. Therefore, using an EC scheme would ensure that dissipation occurs primarily through the Smagorinsky model (and errors in its implementation) than through the discretisation of the Navier-Stokes (NS) equations. To evaluate the efficacy of these schemes for engineering applications, the article first discusses the use of an EC temporal discretisation as regards to accuracy and computational effort, to ascertain whether EC time advancement is advantageous or not. It was noticed that a simple non-EC explicit method with a smaller time step not only reduces the numerical dissipation to an acceptable level but is computationally cheaper than an implicit-EC scheme for wide range of time steps. Secondly, in terms of spatial discretisation on uniform grids (popular in LES), a simple central-difference scheme is as accurate as an EC spatial discretisation. Finally, following the removal of numerical dissipation with any of the methods mentioned above, one is able to choose a Smagorinsky constant that is nearly independent of the grid resolution (within realistic bounds, for OpenFOAM and an in-house code). This article provides impetus to the efficient use of the Smagorinsky model for LES in fields such as wind farm aerodynamics and atmospheric simulations, instead of more comprehensive and computationally demanding turbulence models.