Lattice Boltzmann equation

S. Succi, The Lattice Boltzmann Equation for Fluid Dynamics and Beyond, Clarendon Press, New York, 2001. 

Li-Shi, L. 2000 Some recent results on discrete velocity models and ramifications for lattice Boltzmann equation. Computer Physics Communications 129, 63-74. 

As an illustration of the capabilities of 3D numerical approaches, in the next section a deeper insight in the fluid flow and mass transport in packed-bed membrane reactors is given with the help of a 3D reactor model, based on the lattice Boltzmann equation method. 

In the last decade, we have found a rapid development of the method known as the lattice Boltzmann equation (LBE) method. Because of its physical soundness and outstanding amenability to parallel processing, the LBE method has been successfully applied for the simulation of a variety of flow- and mass-transport situations, including flows through porous media, turbulence, advective diffusion, multiphase and reactive flows, to name but a few. In particular, an effective numerical approach to simulate advective-diffusion transport (the moment-... 

In a final section, results from 3D calculations using the lattice Boltzmann equation method were presented. For a low aspect ratio results were obtained using a randomly generated catalyst packing. This still computationally expensive method possesses the potential to evaluate in future in more detail the complicated but most important situation close to the wall of a PBMR. 

Consolini L, Aggarwal SK, Murad S (2003) A molecular dynamics simulation of droplet evaporation. International Journal of Heat and Mass Transfer 46 3179-3188 Sued S (2001) The lattice Boltzmann equation for fluid dynamics and beyond. Clarendon Press, Oxford... 

The lattice Boltzmann equation (LBE) is obtained as a dramatic simplification of the Boltzmann equation, (20.1) along with its associated equations, (20.2-20.4). In particular, it was discovered that the roots of the Gauss-Hermite quadrature used to exactly and numerically represent the moment integrals in (20.4), corresponds to a particular set of few discrete particle velocity directions c in the LBE [11]. Eurther-more, the continuous equihbrium distribution (20.3) is expanded in terms of the fluid velocity m as a polynomial, where the continuous particle velocity is replaced by the discrete particle velocity set ga obtained as discussed above, i.e., becomes f 1. After replacing the continuous distribution function / =f x, t) in (20.1) by the discrete distribution function/ =f x, (20.1) is integrated by considering... 

Lee, T. and C.-L. Lin. A Stable Discretization of the Lattice Boltzmann Equation for Simulation of Incompressible Two-Phase Flows at High Density Ratio. J. Comput. Phys. 206 16 (2005). 

NoiffgaUev, R., T.N. Dinh, T.G. Theofanous, and D. Joseph. The Lattice Boltzmann Equation Method Theoretical Interpretation, Numerics and Implications. Int. J. Multiphase Flow 29 117-169 (2003). 

In LBM, a fluid is modeled as fictitious particles moving in a lattice domain at discrete time steps. The major variable in LBM is the density distribution fi x, t), indicating the quantity of particles moving along the /-th lattice direction at position X and time t. The time evolution of density distributions is governed by the so-called lattice Boltzmann equation with a BGK collision term [1, 2] ... 

An external force F can be included in the above LBM algorithm by adding an extra term to the collision operator (RHS of Eq. 1) and the lattice Boltzmann equation becomes... 

Sued S (2001) The Lattice Boltzmann equation. Oxford University Press, Oxford, UK... 

Guo Z, Zheng C, Shi B (2008) Lattice Boltzmann equation with multiple effective relaxation times for gaseous microscale flow. Phys Rev E 77 036707... 

The continuity and momentum equations can be solved by tracking the movements of molecule ensembles through the evolution of the distribution function [20] using the popular LBM. The lattice Boltzmann equation (LBE) can be derived from the Boltzmann equation. For the flows with external forces, the continuous Boltzmann-BGK equation with an external force term, F, is... 

LBM has been developed to simulate flows in microchannels based on kinetics equations and statistical physics. In LBM, the motion of the fluid is modeled by a lattice-Boltzmann equation for the distribution function of the fluid molecules. The discrete velocity Boltzmann equation corresponding to the Navier-Stokes equations can be written as... 

LBM was originally proposed by McNamara and Zanetti [3] to circumvent the limitations of statistical noise that plagued lattice gas automata (EGA). LBM is a simplified kinetic (mesoscopic) and discretized approximation of the continuous Boltzmarui equation. LBM is mesoscopic in nature because the particles are not directly related to the number of molecules like in DSMC or MD but representative of a collection of molecules. Hence, the computational cost is less demanding compared with DSMC and MD. Typical LBM consists of the lattice Boltzmann equation (LBE), lattice stmcture, transformation of lattice units to physical units, and boundary conditions. 

This chapter will shortly explain the basics of the Boltzmann equation, the derivation of the Lattice Boltzmann equation and the connection to the Navier-Stokes equations. 

From the Boltzmann equation to the lattice Boltzmann equation... 

Lattice Boltzmann equation can be obtain through two ways, first is through of "cellular automaton" and second starting from Boltzmann equation, it was review previously, for carries out derivation of Boltzmann s lattice equation is necessary the space time discretization. Immediately presents brief description of second way, it shows by p>ace series. 

Benzi R., Succi S. and Vergassola M. (1992). The lattice-Boltzmann equation theory and applications. Physics Report, 222(3), pp. 145-197. 

He X. Luo L. S. (1997). Theory of the lattice Boltzmann method From the Boltzmann equation to the lattice Boltzmann equation. Physical Review E. 56(6), pp. 6811-6817. 

Sued S. (2001). The lattice Boltzmann equation for fluid dynamics and beyond. Oxford University Press Inc., New York. 

Shan X, Chen H (1994) Simulation of nonideal gases and liquid-gas phase-transitions by the lattice Boltzmann-equation. Phys RevE49(4) 2941-2948... 

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