A lattice Boltzmann scheme with a three dimensional cuboid lattice
Date
2016
Authors
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Journal ISSN
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Publisher
University of Delaware
Abstract
The lattice-Boltzmann method (LBM) is a mesoscopic computational scheme which solves the hydrodynamic (continuum) flow by using a set of discrete particle (or model molecular) velocities. The continuum flow evolution such as pressure and hydrodynamic velocity emerges as the average behaviors ( i.e., moments) of the mesoscopic model particles which undergo repeated collisions and propagations on a prescribed lattice. While LBM is now a viable alternative to convectional computational fluid dynamics (CFD) methods, it also has its limitations in terms of computational efficiency, which motivates this thesis work.
In the LBM simulation for fluid flow, the domain is usually discretized with a square lattice in 2D or a cubic lattice in 3D. Some previous studies were made to investigate the possibility of establishing a LBM model with non-standard lattice grid, namely, a rectangular lattice grid in 2D and a non-cubic (cuboid) lattice grid in 3D. The non-standard lattices are computationally more efficient when simulating a non-isotropic and inhomogeneous flows, e.g., the turbulent channel flow.
In some previous non-standard lattice LBM developed by others, the Navier-Stokes equation is not correctly recovered because the resulting viscosity is anisotropic. The anisotropy is caused by the use of different lattice sizes in different directions; and it cannot be fixed without additional degrees of freedom. Recently at the University of Delaware, several new lattice Boltzmann schemes have been developed on a 2D rectangular grid using the multiple-relaxation-time (MRT) collision model, either by redesigning some moments to add a new free parameter, or by extending the equilibrium moments to include higher-order terms. These models can then satisfy all isotropy conditions as required by the Navier-Stokes equations.
In this thesis, based on the similar idea used in the successful design of LBM on a rectangular lattice, we developed a lattice Boltzmann model on a 3D cuboid lattice, namely, a lattice grid with different grid lengths in different spatial directions. Using the multi-scale Chapman-Enskog analysis, we designed the moment equations resulting from our MRT-LBM model, to be fully consistent with the Navier-Stokes equations. A second-order term is added to the equilibrium moments in order to not only satisfy all isotropy conditions but also to better accommodate different values of shear and bulk viscosities. The form of the second-order term and the coefficients of the extended equilibrium moments are determined through an inverse design process. An additional benefit of the model is that the shear viscosity can be adjusted, independent of the stress-moment relaxation parameter, thus improving the numerical stability of the model.
The resulting cuboid MRT-LBM model is then validated through several benchmark simulations, including the transient laminar channel flow, the fully developed single phase turbulent channel flow, and the 3D time-dependent, energy-cascading Taylor-Green vortex flow. In addition, the second-order accuracy of the proposed model is demonstrated in the simulation of 3D Taylor-Green vortex flow and transient laminar channel flow.