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  • Lattice Boltzmann Method

    Lattice Boltzmann methods (LBMs) are a class of mesoscopic particle based approaches to simulate fluid flows. They are becoming a serious alternative to traditional methods for computational fluid dynamics1,2,9,15. LBMs are especially well suited to simulate flows around complex geometries 3, and they are straightforwardly implemented on parallel machines8. Historically, the lattice Boltzmann approach developed from lattice gases6,7, although it can also be derived directly from the simplified Boltzmann BGK equation5. In lattice gases6,13,14, particles live on the nodes of a discrete lattice. The particles jump from one lattice node to the next, according to their (discrete) velocity. This is called the propagation phase. Then, the particles collide and get a new velocity. This is the collision phase. Hence the simulation proceeds in an alternation between particle propagations and collisions. The two phases can be clearly distinguished in the following animation.
     
     

    It can be shown that lattice gases solve the Navier-Stokes equations of fluid flow4. The major disadvantage of lattice gases for common fluid dynamics applications is the occurence of noise. If the main interest is a smooth flow field one needs to average out over a very large lattice and over a long time. The lattice Boltzmann method solves this problem by pre-averaging the lattice gas. It considers particle distributions that live on the lattice nodes, rather than the individual particles. The general form of the lattice Boltzmann equation is 

    $\displaystyle f_i(x+\Delta t \vec{c}_i, t+{\Delta}t) = f_i(x,t) + \Omega_i$  

    where the $ f_i$ is the concentration of particles that travels with velocity $ \vec{c}_i$. With the discrete velocity $ \vec{c}_i$ the particle distributions travel to the next lattice node in one time step $ \Delta t$

    The collision operator$ \Omega_i$ differs for the many lattice Boltzmann methods. In the lattice Boltzmann Bhatnagher-Gross-Krook method12 (LBGK) that we use, the particle distribution after propagation is relaxed towards the equillibrium distribution $ f_i^{\rm eq}(x,t)$, as 

    $\displaystyle \Omega_i = {\frac{1}{\tau}}(f_i(x,t)-f^{\rm eq}_i(x,t)).$  

    The relaxation $ \tau$ parameter determines the kinematic viscosity $ \nu$ of the simulated fluid, according to, 

    $\displaystyle \nu=(2\tau-1)/6,$  

    The equillibrium distribution $ f_i^{\rm eq}(x,t)$ is a function of the local density $ \rho$ and the local velocity $ \vec{u}$. These are the first and second order moments of the particle distribution as, 

    $\displaystyle \rho(x,t)=\sum_i f_i(x,t),$  

    and 

    $\displaystyle \vec{u}(x,t)={{\sum_i f_i(x,t)\vec{c}_i}\over{\rho(x,t)}}.$  

    The equillibrium density$ f_i^{\rm eq}(\vec{u},\rho)$ is calculated as 

    $\displaystyle f_i^{\rm eq}(\rho,\vec{u})=t_p \rho \left(1 + {{\vec{c_i}.\vec{u}......vec{c_i}.\vec{u})^2}\over{2 c_s^4}} - {{\vec{u}.\vec{u}}\over{2 c_s^2}}\right),$  

    in which $ c_s$ is the speed of sound, the index$ p=\vec{c_i}.\vec{c_i}$ and $ t_p$ is the corresponding equillibrium density for $ \vec{u}=0$. For the three-dimensional, nineteen velocity lattice ($ D_{3}Q_{19}$) that we have used in our simulations,$ t_0={1\over3}$ (rest particle), $ t_1={1\over18}$ (particles streaming to the face-connected neighbours) and $ t_2={1\over36}$ (particles streaming to the edge-connected neighbours). 

    References

    1. Chen, S. & Doolen, G. D. (1998). Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329-364

    2. Chopard, B. & Droz, M. (1998). Cellular Automata Modeling of Physical Systems, Claude Godrèche (ed.): Collection Aléa-Saclay: monographs and texts in statistical physics. Cambridge: Cambridge University Press

    3. For complex geometries see e.g.: Clague, D. S., Kandhai, B. D., Zhang, R., Sloot, P. M. A. (2000). Hydraulic permeability of (un)bounded fibrous media using the lattice Boltzmann method. Phys. Rev. E 61, 616, and Kandhai, D., Vidal, D. J. E., Hoekstra, A. G., Hoefsloot, H., Iedema, P. & Sloot, P. M. A. (1999). Lattice-boltzmann and finite element simulations of fluid flow in a SMRX static mixer reactor. Int. J. Numer. Methods Fluids 31, 1019.

    4. Frisch, U., Hasslacher, B. & Pomeau, Y. (1986). Lattice gas automata for the Navier-Stokes equations. Phys. Rev. Lett. 56, 1505.

    5. He, X. & Luo, L. (1997). Theory of the lattice Boltzmann method: from the Boltzmann equation to the lattice Boltzmann equation. Phys. Rev. E 56, 6811.

    6. Higuera, F. & Jimenez, J. (1989). Boltzmann approach to lattice gas simulations. Europhys. Lett. 9, 663.

    7. Higuera, F., Succi, S. & Benzi, R. (1989). Lattice gas dynamics with enhanced collisions. Europhys. Lett.9, 345.

    8. Kandhai, D., Koponen, A., Hoekstra, A. G., Kataja, M., Timonen, J. & Sloot, P. M. A. (1998). Lattice-Boltzmann hydrodynamics on parallel systems. Comput. Phys. Commun. 111, 14.

    9. Kandhai, D., Vidal, D. J. E., Hoekstra, A. G., Hoefsloot, H., Iedema, P. & Sloot, P. M. A. (1999). Lattice-boltzmann and finite element simulations of fluid flow in a SMRX static mixer reactor. Int. J. Numer. Methods Fluids 31, 1019.

    10. Lowe, C. P. & Frenkel, D. (1995). The super long-time decay of velocity fluctuations in a two-dimensional fluid. Physica A 220, 251.

    11. Merks, R. M. H., Hoekstra, A. G. & Sloot, P. M. A. (2001). The moment propagation method for advection-diffusion in the lattice Boltzmann method: validation and Peclet number limits. Submitted for publication.

    12. Qian Y. H., D'Humieres, D. & Lallemand, P. (1992). Lattice BGK models for Navier-Stokes equation. Europhys. Lett. 17, 479. 

    13. Rivet, J.-P. & Boom, J. P. (2001). Lattice Gas Hydrodynamics. Cambridge Nonlinear Science Series 11, Cambridge: Cambridge University Press

    14. Rothman, Daniel H. & Zaleski, Stéphane (1997). Lattice-Gas Cellular Automata: Simple models of complex hydrodynamics. Claude Godrèche (ed.): Collection Aléa-Saclay: monographs and texts in statistical physics. Cambridge: Cambridge University Press.

    15. Succi S. (2001). The lattice Boltzmann Equation: for Fluid Dynamics and Beyond. Series Numerical Mathematics and Scientific Computation. Oxford New York: Oxford University Press
     


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    August 2014