Abstract
Conventional lattice Boltzmann models for the simulation of fluid dynamics are restricted by an error in the stress tensor that is negligible only for small flow velocity and at a singular value of the temperature. To that end, we propose a unified formulation that restores Galilean invariance and the isotropy of the stress tensor by introducing an extended equilibrium. This modification extends lattice Boltzmann models to simulations with higher values of the flow velocity and can be used at temperatures that are higher than the lattice reference temperature, which enhances computational efficiency by decreasing the number of required time steps. Furthermore, the extended model also remains valid for stretched lattices, which are useful when flow gradients are predominant in one direction. The model is validated by simulations of two- and three-dimensional benchmark problems, including the double shear layer flow, the decay of homogeneous isotropic turbulence, the laminar boundary layer over a flat plate and the turbulent channel flow.
Highlights
The lattice Boltzmann method (LBM) solves a Boltzmann-type kinetic equation on a discrete velocity set, forming the links of a space-filling lattice
The proposed revision of the lattice Bhatnagar–Gross– Krook (LBGK) model is based on extending the product-form equilibrium such that the anomaly of the diagonal third-order moment is compensated in the hydrodynamic limit by counter terms, which are added to the diagonal of the equilibrium pressure tensor
This demonstrates that the LBGK model, in the presence of a severe anisotropy triggered by stretched velocities, can be used for the simulation of high Reynolds number wall-bounded flows once the corrections are incorporated with the extended equilibrium
Summary
The lattice Boltzmann method (LBM) solves a Boltzmann-type kinetic equation on a discrete velocity set, forming the links of a space-filling lattice. In the standard LB setting, on the other hand, one approach is to alter the relaxation rates and use a multi-relaxation time collision operator [9] Another approach to extend the flow velocity and temperature range of the standard cubic lattices is to add correction terms to the original LBM [4,5,6,7,8,10,11,12,17,18,19,20,21]. Recent work on the stretched LBM restores the isotropy of the stress tensor by using multi-relaxation time LBM models [30,31,32] These approaches do not address the flow velocity and temperature restrictions. We propose to use an extended equilibrium, which restores the Galilean invariance and isotropy of the stress tensor, enabling simulations at higher flow velocities, higher temperatures using both cubic and stretched lattices, yielding increased accuracy and efficiency.
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