Second-order convergence of the deviatoric stress tensor in the standard Bhatnagar-Gross-Krook lattice Boltzmann method

2010 ◽  
Vol 82 (2) ◽  
Author(s):  
T. Krüger ◽  
F. Varnik ◽  
D. Raabe
Keyword(s):  
Lattice Boltzmann Method ◽  
Stress Tensor ◽  
Lattice Boltzmann ◽  
Second Order ◽  
Deviatoric Stress ◽  
Order Convergence ◽  
Second Order Convergence ◽  
Boltzmann Method ◽  
Deviatoric Stress Tensor

On Initial Conditions for the Lattice Boltzmann Method

2015 ◽  
Vol 18 (2) ◽  
pp. 450-468 ◽  
Author(s):  
Juntao Huang ◽  
Hao Wu ◽  
Wen-An Yong
Keyword(s):  
Asymptotic Analysis ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Initial Conditions ◽  
Second Order ◽  
Collision Term ◽  
Order Convergence ◽  
Numerical Examples ◽  
Second Order Convergence ◽  
Boltzmann Method

AbstractIn this paper, we propose two initialization techniques for the lattice Boltzmann method. The first one is based on the theory of asymptotic analysis developed in [M. Junk and W.-A. Yong, Asymptotic Anal., 35(2003)]. By selecting consistent macroscopic quantities, this initialization leads to the second-order convergence for both velocity and pressure. Another one is an improvement of the consistent initial conditions proposed in [R. W. Mei, L.-S. Luo, P. Lallemand and D. d’Humières, Comput. Fluids, 35(2006)]. The improvement involves a modification of the collision term and a reconstruction step. Numerical examples confirm the accuracy and efficiency of our techniques.

What is the Small Parameter ε in the Chapman-Enskog Expansion of the Lattice Boltzmann Method?

2012 ◽  
Vol 134 (1) ◽  
Author(s):  
Minoru Watari
Keyword(s):  
Small Parameter ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Stokes Equations ◽  
Viscosity Coefficient ◽  
Second Order ◽  
Time Step ◽  
Order Accuracy ◽  
The Difference ◽  
Boltzmann Method

The lattice Boltzmann method (LBM) is shown to be equivalent to the Navier-Stokes equations by applying the Chapman-Enskog (C-E) expansion, which has been established by pioneer researchers. However, it is still difficult for elementary researchers. There is no clear explanation of the small parameter ε used in the C-E expansion. There are several expressions for the viscosity coefficient; some are unclear on the relationship with ε. There are two expressions on the LBM evolution equation. Elementary researchers are perplexed as to which is correct. The LBM achieves second order accuracy by including the numerical viscosity within the physical viscosity. This is not only difficult for elementary researchers to understand but also sometimes leads senior researchers into making errors. The C-E expansion of the LBM was thoroughly reviewed and is presented as a self-contained form in this paper. It is natural to use the time step Δt as ε. The viscosity coefficient is expressed as μ∝Δxc(τ − 1/2). The viscosity relationship and the second order accuracy were confirmed by numerical simulations. The difference in the two expressions on the LBM evolution is simply one of perspective. They are identical. The difference between the relaxation parameter τD for the discrete Boltzmann equation and τ for the LBM was discussed. While τD is a quantity of time, τ is genuinely nondimensional, which is sometimes overlooked.

First- and second-order forcing expansions in a lattice Boltzmann method reproducing isothermal hydrodynamics in artificial compressibility form

2012 ◽  
Vol 698 ◽  
pp. 282-303 ◽  
Author(s):  
Goncalo Silva ◽  
Viriato Semiao
Keyword(s):  
Relaxation Time ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Order Term ◽  
Second Order ◽  
Navier Stokes ◽  
Forcing Term ◽  
Boltzmann Method

AbstractThe isothermal Navier–Stokes equations are determined by the leading three velocity moments of the lattice Boltzmann method (LBM). Necessary conditions establishing the hydrodynamic consistency of these moments are provided by multiscale asymptotic techniques, such as the second-order Chapman–Enskog expansion. However, for simulating incompressible hydrodynamics the structure of the forcing term in the LBM is still a discordant issue as far as its correct velocity expansion order is concerned. This work uses the traditional second-order Chapman–Enskog expansion analysis to demonstrate that the truncation order of the forcing term may depend on the time regime in this case. This is due to the fact that LBM does not reproduce exactly the incompressibility condition. It rather approximates it through a weakly compressible or an artificial compressible system. The present study shows that for the artificial compressible setup, as the incompressibility flow condition is singularly perturbed by the time variable, such a connection will also affect the LBM forcing formulation. As a result, for time-independent incompressible flows the LBM forcing must be truncated to first order whereas for a time-dependent case it is convenient to include the second-order term. The theoretical findings are confirmed by numerical tests carried out in several distinct benchmark flows driven by space- and/or time-varying body forces and possessing known analytical solutions. These results are verified for the single relaxation time, the multiple relaxation time and the regularized collision models.

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