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.

Effects of initial conditions on the coalescence of micro-bubbles

2017 ◽  
Vol 232 (3) ◽  
pp. 457-465 ◽  
Author(s):  
Rou Chen ◽  
Huidan(Whitney) Yu ◽  
Likun Zhu
Keyword(s):  
Lattice Boltzmann Method ◽  
Power Law ◽  
Lattice Boltzmann ◽  
Development Time ◽  
Initial Conditions ◽  
Separation Distance ◽  
Bubble Coalescence ◽  
Half Power ◽  
Boltzmann Method ◽  
Method Model

The effects of initial conditions on the coalescence of two equal-sized air micro-bubbles ( R0) in water are studied using the lattice Boltzmann method. The focus is on effects of two initial set-ups of parent bubbles, separated by a small distance d and connected with a neck bridge radius r0, on the neck bridge growth at the early stage of the bubble coalescence. A sophisticated free energy lattice Boltzmann method model based on the Cahn-Hilliard diffuse interface approach is employed. This lattice Boltzmann method model has been demonstrated suitable for handling a large density ratio of two fluids up to 1000 and capable of minimizing the nonphysical spurious current. In both initial scenarios, the neck bridge evolution exhibits a half power-law scaling, [Formula: see text] after a development time. The half power-law agrees with the recent analytical prediction and experimental results. It has been found that smaller initial separation distance or smaller initial neck bridge radius results in faster growth of neck bridge and bubble coalescence, which is similar to the effects of these two initial scenarios on droplet coalescence. The physical mechanism behind each behavior has been explored. For the initial connected case, faster neck growth and longer development time corresponding to smaller initial neck radius is due to the significant bias between the capillary forces contributed by the meniscus curvature and the neck bridge curvature, whereas in the case of initial separated scenario, faster growth and shorter development time corresponding to shorter separation distance is due to the formation of elongated neck bridge. The prefactor A0 that represents the growth of neck bridge radius at the characteristic time ti captured in each case is in good agreement with the experimental results.

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.

Finite Volume Lattice Boltzmann Method for Nearly Incompressible Flows on Arbitrary Unstructured Meshes

2016 ◽  
Vol 20 (2) ◽  
pp. 301-324 ◽  
Author(s):  
Weidong Li ◽  
Li-Shi Luo
Keyword(s):  
Lattice Boltzmann Method ◽  
Finite Volume ◽  
Lattice Boltzmann ◽  
Incompressible Flows ◽  
Unstructured Meshes ◽  
Two Dimensions ◽  
Collision Term ◽  
Efficient Treatment ◽  
Advection Term ◽  
Boltzmann Method

AbstractA genuine finite volume method based on the lattice Boltzmann equation (LBE) for nearly incompressible flows is developed. The proposed finite volume lattice Boltzmann method (FV-LBM) is grid-transparent, i.e., it requires no knowledge of cell topology, thus it can be implemented on arbitrary unstructured meshes for effective and efficient treatment of complex geometries. Due to the linear advection term in the LBE, it is easy to construct multi-dimensional schemes. In addition, inviscid and viscous fluxes are computed in one step in the LBE, as opposed to in two separate steps for the traditional finite-volume discretization of the Navier-Stokes equations. Because of its conservation constraints, the collision term of the kinetic equation can be treated implicitly without linearization or any other approximation, thus the computational efficiency is enhanced. The collision with multiple-relaxation-time (MRT) model is used in the LBE. The developed FV-LBM is of second-order convergence. The proposed FV-LBM is validated with three test cases in two-dimensions: (a) the Poiseuille flow driven by a constant body force; (b) the Blasius boundary layer; and (c) the steady flow past a cylinder at the Reynolds numbers Re=10, 20, and 40. The results verify the designed accuracy and efficacy of the proposed FV-LBM.

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.

The approach to optimization of finite-difference schemes for the advective stage of finite-difference-based lattice Boltzmann method

2020 ◽  
Vol 11 (01) ◽  
pp. 2050002
Author(s):  
Gerasim V. Krivovichev ◽  
Elena S. Marnopolskaya
Keyword(s):  
Finite Difference ◽  
Lattice Boltzmann Method ◽  
Lattice Boltzmann ◽  
Initial Conditions ◽  
Time Derivative ◽  
Numerical Dispersion ◽  
Finite Difference Schemes ◽  
Advection Equation ◽  
Optimal Values ◽  
Boltzmann Method

The approach to optimization of finite-difference (FD) schemes for the linear advection equation (LAE) is proposed. The FD schemes dependent on the scalar dimensionless parameter are considered. The parameter is included in the expression, which approximates the term with spatial derivatives. The approach is based on the considering of the dispersive and dissipative characteristics of the schemes as the functions of the parameter. For the proper choice of the parameter, these functions are minimized. The approach is applied to the optimization of two-step schemes with an asymmetric approximation of time derivative and with various approximations of the spatial term. The cases of schemes from first to fourth approximation orders are considered. The optimal values of the parameter are obtained. Schemes with the optimal values are applied to the solution of test problems with smooth and discontinuous initial conditions. Also, schemes are used in the FD-based lattice Boltzmann method (LBM) for modeling of the compressible gas flow. The obtained numerical results demonstrate the convergence of the schemes and decaying of the numerical dispersion.

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