A general optimal formulation for the dynamic Smagorinsky subgrid-scale stress model

Abstract

SUMMARYIn this paper, a general optimal formulation for the dynamic Smagorinsky subgrid-scale (SGS) stress model is reported. The Smagorinsky constitutive relation has been revisited from the perspective of functional variation and optimization. The local error density of the dynamic Smagorinsky SGS model has been minimized directly to determine the model coe cient C S . A su cient and necessary condition for optimizing the SGS model is obtained and an orthogonal condition (OC), which governs the instantaneous spatial distribution of the optimal dynamic model coe cient, is formulated. The OC is a useful general optimization condition, which uniÿes several classical dynamic SGS modelling formulations reported in the literature. In addition, the OC also results in a new dynamic model in the form of a Picard's integral equation. The approximation tensorial space for the projected Leonard stress is identiÿed and the physical meaning for several basic grid and test-grid level tensors is systematically discussed. Numerical simulations of turbulent Couette ow are used to validate the new model formulation as represented by the Picard's integral equation for Reynolds numbers ranging from 1500 to 7050 (based on one half of the velocity di erence of the two plates and the channel height). The relative magnitudes of the Smagorinsky constitutive parameters have been investigated, including the model coe cient, SGS viscosity and ÿltered strain rate tensor. In general, this paper focuses on investigation of fundamental mathematical and physical properties of the popular Smagorinsky constitutive relation and its related dynamic modelling optimization procedure.

Geometrical properties of the vorticity vector derived using large-eddy simulation

“…It can lead to an over-prediction of SGS dissipation if the model coefficient is restricted to be positive, while on the other hand, potential numerical instability due to excessive backscatter of the SGS turbulence kinetic energy (TKE) can occur if the model coefficient is allowed to be negative (Salvetti and Banerjee, 1995;Piomelli, 1993). A systematic analysis of the Smagorinsky constitutive relation and discussion of local optimization procedures for determination of the dynamic coefficient C S can be found in the recent study of Wang and Bergstrom (2005b).…”

Large-eddy simulation of combined forced and natural convection in a vertical plane channel

“…For instance, simulations based on the DM can be numerically unstable due to excessive backscatter of TKE from the subgrid to resolved scales if the model coefficient is not properly bounded [15]. The DM is difficult to be applied locally without plane averaging the model coefficient [13,14] for a plane channel flow; and on the other hand, localization of the DM often results in integral equations which are difficult and costly to solve [16]. Since the above mentioned deficiencies related to the DM stem from its over-simplified constitutive relation, improved dynamic SGS stress models often consider nonSmagorinsky type constitutive relations.…”

Research on complex wake interference of aligned rotors considering the precone variation of the upstream wind turbine

“…A local dynamic Smagorinsky (LDS) sub-grid model (Piomelli and Liu, 1995;Wang and Bergstrom, 2005) is programmed by utilizing the LES method. In contrast to the standard and average dynamic Smagorinsky (ADS) sub-grid model, the LDS sub-grid model can consider the instantaneous and local effects of the flow field, hence greatly improving the near-wall results (Ji et al, 2022).…”