An Alternative Scheme to Calculate the Strain Rate Tensor for the LES Applications in the LBM

Large eddy simulations (LES) based on the Smagorinsky model can be conveniently used in the lattice Boltzmann method (LBM) because the strain rate tensor, , used to determine the eddy kinematic viscosity can be calculated from the second-order moment of the nonequilibrium distribution function, and the current total nondimensional relaxation time can be determined explicitly. A new method is developed where the distribution function after the relaxation subroutine differs from that after the motion subroutine leading to a similar method to determine , but its application is inconvenient due to the implicit feature. However, the derivation also leads to an alternative explicit scheme for calculating based on physical analysis of the momentum transport process, where the stress tensor, , is calculated first, and then is determined from using the constitutive relationship for Newtonian fluid. The current total nondimensional relaxation time is also given explicitly so that this LES model can be easily used in the LBM. 1. Introduction The lattice Boltzmann method (LBM) [1–3] originated from lattice gas (LG) automata [4–7] can be derived from the Bhatnagar-Gross-Krook (BGK) model equation [8] which is a good approximation to the Boltzmann equation. A general rule for systematically formulating LBM models, which are sufficient for the Navier-Stokes, Burnett fluids, and beyond, was given in [9, 10]. In recent years, many improvements have been made to the LBM with the resulting formulations having computational advantages over traditional continuum methods [11–13]. Many successful models have been proposed to extend the scope of LBM applications, including models for incompressible flows [14–16] and flows involving with thermal energy exchange [17–21]. Some sophisticated solid-fluid boundaries are proposed for regular geometries [22, 23], but there are also general boundary-fitted models [24, 25] available. Large eddy simulations (LES) are useful for numerical predictions of complex turbulent flows and its application in LBM is referred to as the LBM-LES algorithm. Among the successful subgrid stress models used in LES [26, 27], the Smagorinsky model [28] in LBM-LES is known to be quite convenient because the needed strain rate tensor can be calculated directly from the second-order moment of the nonequilibrium distribution function. In this paper, the subgrid stress model used in LBM-LES is the Smagorinsky model unless otherwise stated. There are many successful calculations [29–36] of turbulent flows using LBM-LES, and Yu et al. [31] gave a detailed