Thelattice Boltzmann methods (LBM), originated from thelattice gas automata (LGA) method (Hardy-Pomeau-Pazzis andFrisch-Hasslacher-Pomeau models), is a class ofcomputational fluid dynamics (CFD) methods forfluid simulation. Instead of solving theNavier–Stokes equations directly, a fluid density on a lattice is simulated with streaming and collision (relaxation) processes.^[1] The method is versatile^[1] as the model fluid can straightforwardly be made to mimic common fluid behaviour like vapour/liquid coexistence, and so fluid systems such as liquid droplets can be simulated. Also, fluids in complex environments such as porous media can be straightforwardly simulated, whereas with complex boundaries other CFD methods can be hard to work with.

Unlike CFD methods that solve the conservation equations of macroscopic properties (i.e., mass, momentum, and energy) numerically, LBM models the fluid consisting of fictive particles, and such particles perform consecutive propagation and collision processes over a discrete lattice. Due to its particulate nature and local dynamics, LBM has several advantages over other conventional CFD methods, especially in dealing with complex boundaries, incorporating microscopic interactions, and parallelization of the algorithm.^[2] A different interpretation of the lattice Boltzmann equation is that of a discrete-velocityBoltzmann equation. The numerical methods of solution of the system of partial differential equations then give rise to a discrete map, which can be interpreted as the propagation and collision of fictitious particles.

In an algorithm, there are collision and streaming steps. These evolve the density of the fluid${\displaystyle \rho (\overrightarrow{x},t)}$, for${\displaystyle \overrightarrow{x}}$ the position and${\displaystyle t}$ the time. As the fluid is on a lattice, the density has a number of components${\displaystyle f_i,i=0,\dots ,a}$ equal to the number of lattice vectors connected to each lattice point. As an example, the lattice vectors for a simple lattice used in simulations in two dimensions is shown here. This lattice is usually denoted D2Q9, for two dimensions and nine vectors: four vectors along north, east, south and west, plus four vectors to the corners of aunit square, plus a vector with both components zero. Then, for example vector${\displaystyle {\overrightarrow{e}}_4=(0,-1)}$, i.e., it points due south and so has no${\displaystyle x}$ component but a${\displaystyle y}$ component of${\displaystyle -1}$. So one of the nine components of the total density at the central lattice point,${\displaystyle f_4(\overrightarrow{x},t)}$, is that part of the fluid at point${\displaystyle \overrightarrow{x}}$ moving due south, at a speed in lattice units of one.

Then the steps that evolve the fluid in time are:^[1]

The collision step

For the Bhatnagar Gross and Krook (BGK)^[3] model, which leads relaxation to equilibrium via collisions between the molecules of a fluid, we have

${\displaystyle f_i^{\ast }(\overrightarrow{x},t)=f_i(\overrightarrow{x},t)+\frac{f_i^{eq}(\overrightarrow{x},t)-f_i(\overrightarrow{x},t)}{{\tau }_f}}$,

where${\displaystyle f_i^{\ast }(\overrightarrow{x},t)}$ is the new lattice density, and${\displaystyle f_i^{eq}(\overrightarrow{x},t)}$ is the equilibrium density along directioni which can be expressed by using a Taylor expansion (see below, inMathematical equations for simulations):

${\displaystyle f_i^{eq}={\omega }_i\rho \left(1+\frac{3{\overrightarrow{e}}_i\cdot \overrightarrow{u}}{c^2}+\frac{9({\overrightarrow{e}}_i\cdot \overrightarrow{u}{)}^2}{2c^4}-\frac{3(\overrightarrow{u}\cdot \overrightarrow{u})}{2c^2}\right)}$.

The model assumes that the fluid locally relaxes to equilibrium over a characteristic timescale${\displaystyle {\tau }_f}$. This timescale determines thekinematic viscosity, the larger it is, the larger is the kinematic viscosity.

The streaming step

${\displaystyle f_i(\overrightarrow{x}+{\overrightarrow{e}}_i{\delta }_t,t+{\delta }_t)=f_i^{\ast }(\overrightarrow{x},t)}$

As${\displaystyle f_i^{\ast }(\overrightarrow{x},t)}$ is, by definition, the fluid density at point${\displaystyle \overrightarrow{x}}$ at time${\displaystyle t}$, that is moving at a velocity of${\displaystyle {\overrightarrow{e}}_i}$ per time step, then at the next time step${\displaystyle t+{\delta }_t}$ it will have flowed to point${\displaystyle \overrightarrow{x}+{\overrightarrow{e}}_i{\delta }_t}$.

• The LBM was designed from scratch to run efficiently onmassively parallel architectures, ranging from inexpensive embeddedFPGAs andDSPs up toGPUs and heterogeneous clusters and supercomputers (even with a slow interconnection network). It enables complex physics and sophisticated algorithms. Efficiency leads to a qualitatively new level of understanding since it allows solving problems that previously could not be approached (or only with insufficient accuracy).
• The method originates from a molecular description of a fluid and can directly incorporate physical terms stemming from a knowledge of the interaction between molecules. Hence it is an indispensable instrument in fundamental research, as it keeps the cycle between the elaboration of a theory and the formulation of a corresponding numerical model short.
• Automated data pre-processing and lattice generation in a time that accounts for a small fraction of the total simulation.
• Parallel data analysis, post-processing and evaluation.
• Fully resolved multi-phase flow with small droplets and bubbles.
• Fully resolved flow through complex geometries and porous media.
• Complex, coupled flow with heat transfer and chemical reactions.

As with Navier–Stokes based CFD, LBM methods have been successfully coupled with thermal-specific solutions to enable heat transfer (solids-based conduction, convection and radiation) simulation capability. For multiphase/multicomponent models, the interface thickness is usually large and the density ratio across the interface is small when compared with real fluids. Recently this problem has been resolved by Yuan andSchaefer who improved on models by Shan and Chen, Swift, and He, Chen, and Zhang. They were able to reach density ratios of 1000:1 by simply changing theequation of state. It has been proposed to applyGalilean Transformation to overcome the limitation of modelling high-speed fluid flows.^[4]The fast advancements of this method had also successfully simulatedmicrofluidics,^[5] However, as of now, LBM is still limited in simulating highKnudsen number flows whereMonte Carlo methods are instead used, and high-Mach number flows inaerodynamics are still difficult for LBM, and a consistent thermo-hydrodynamic scheme is absent.^[6]

LBM originated from thelattice gas automata (LGA) method, which can be considered as a simplified fictitious molecular dynamics model in which space, time, and particle velocities are all discrete. For example, in the 2-dimensionalFHP Model each lattice node is connected to its neighbors by 6 lattice velocities on a triangular lattice; there can be either 0 or 1 particles at a lattice node moving with a given lattice velocity. After a time interval, each particle will move to the neighboring node in its direction; this process is called the propagation or streaming step. When more than one particle arrives at the same node from different directions, they collide and change their velocities according to a set of collision rules. Streaming steps and collision steps alternate. Suitable collision rules should conserve theparticle number (mass), momentum, and energy before and after the collision. LGA suffer from several innate defects for use in hydrodynamic simulations: lack ofGalilean invariance for fast flows,statistical noise and poorReynolds number scaling with lattice size. LGA are, however, well suited to simplify and extend the reach ofreaction diffusion andmolecular dynamics models.

The main motivation for the transition from LGA to LBM was the desire to remove the statistical noise by replacing the Boolean particle number in a lattice direction with its ensemble average, the so-called density distribution function. Accompanying this replacement, the discrete collision rule is also replaced by a continuous function known as the collision operator. In the LBM development, an important simplification is to approximate the collision operator with theBhatnagar-Gross-Krook (BGK) relaxation term. This lattice BGK (LBGK) model makes simulations more efficient and allows flexibility of the transport coefficients. On the other hand, it has been shown that the LBM scheme can also be considered as a special discretized form of the continuous Boltzmann equation. FromChapman-Enskog theory, one can recover the governing continuity and Navier–Stokes equations from the LBM algorithm.

Lattice Boltzmann models can be operated on a number of different lattices, both cubic and triangular, and with or without rest particles in the discrete distribution function.

A popular way of classifying the different methods by lattice is the DnQm scheme. Here "Dn" stands for "n dimensions", while "Qm" stands for "m speeds". For example, D3Q15 is a 3-dimensional lattice Boltzmann model on a cubic grid, with rest particles present. Each node has a crystal shape and can deliver particles to 15 nodes: each of the 6 neighboring nodes that share a surface, the 8 neighboring nodes sharing a corner, and itself.^[7] (The D3Q15 model does not contain particles moving to the 12 neighboring nodes that share an edge; adding those would create a "D3Q27" model.)

Real quantities as space and time need to be converted to lattice units prior to simulation. Nondimensional quantities, like theReynolds number, remain the same.

In most Lattice Boltzmann simulations${\displaystyle {\delta }_x}$ is the basic unit for lattice spacing, so if the domain of length${\displaystyle L}$ has${\displaystyle N}$ lattice units along its entire length, the space unit is simply defined as${\displaystyle {\delta }_x=L/N}$. Speeds in lattice Boltzmann simulations are typically given in terms of the speed of sound. The discrete time unit can therefore be given as${\displaystyle {\delta }_t=\frac{{\delta }_x}{C_s}}$, where the denominator${\displaystyle C_s}$ is the physical speed of sound.^[8]

For small-scale flows (such as those seen inporous media mechanics), operating with the true speed of sound can lead to unacceptably short time steps. It is therefore common to raise the latticeMach number to something much larger than the real Mach number, and compensating for this by raising theviscosity as well in order to preserve theReynolds number.^[9]

Simulating multiphase/multicomponent flows has always been a challenge to conventional CFD because of the moving and deformableinterfaces. More fundamentally, the interfaces between differentphases (liquid and vapor) or components (e.g., oil and water) originate from the specific interactions among fluid molecules. Therefore, it is difficult to implement such microscopic interactions into the macroscopic Navier–Stokes equation. However, in LBM, the particulate kinetics provides a relatively easy and consistent way to incorporate the underlying microscopic interactions by modifying the collision operator. Several LBM multiphase/multicomponent models have been developed. Here phase separations are generated automatically from the particle dynamics and no special treatment is needed to manipulate the interfaces as in traditional CFD methods. Successful applications of multiphase/multicomponent LBM models can be found in variouscomplex fluid systems, including interface instability,bubble/droplet dynamics,wetting on solid surfaces, interfacial slip, and droplet electrohydrodynamic deformations.

A lattice Boltzmann model for simulation of gas mixture combustion capable of accommodating significant density variations at low-Mach number regime has been recently proposed.^[10]

To this respect, it is worth to notice that, since LBM deals with a larger set of fields (as compared to conventional CFD), the simulation of reactive gas mixtures presents some additional challenges in terms of memory demand as far as large detailed combustion mechanisms are concerned. Those issues may be addressed, though, by resorting to systematic model reduction techniques.^[11][12][13]

Currently (2009), a thermal lattice-Boltzmann method (TLBM) falls into one of three categories: the multi-speed approach,^[14] the passive scalar approach,^[15] and the thermal energy distribution.^[16]

Starting with the discrete lattice Boltzmann equation (also referred to as LBGK equation due to the collision operator used). We first do a 2nd-orderTaylor series expansion about the left side of the LBE. This is chosen over a simpler 1st-order Taylor expansion as the discrete LBE cannot be recovered. When doing the 2nd-order Taylor series expansion, the zero derivative term and the first term on the right will cancel, leaving only the first and second derivative terms of the Taylor expansion and the collision operator:

${\displaystyle f_i(\overrightarrow{x}+{\overrightarrow{e}}_i{\delta }_t,t+{\delta }_t)=f_i(\overrightarrow{x},t)+\frac{{\delta }_t}{{\tau }_f}(f_i^{eq}-f_i).}$

For simplicity, write${\displaystyle f_i(\overrightarrow{x},t)}$ as${\displaystyle f_i}$. The slightly simplified Taylor series expansion is then as follows, where ":" is the colon product between dyads:

${\displaystyle \frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }t}+{\overrightarrow{e}}_i\cdot \mathrm{\nabla }{f}_i+\left(\frac{1}{2}{\overrightarrow{e}}_i{\overrightarrow{e}}_i:\mathrm{\nabla }\mathrm{\nabla }{f}_i+{\overrightarrow{e}}_i\cdot \mathrm{\nabla }\frac{\mathrm{\partial }{f}_i}{\mathrm{\partial }t}+\frac{1}{2}\frac{{\mathrm{\partial }}^2{f}_i}{\mathrm{\partial }{t}^2}\right)=\frac{1}{\tau }(f_i^{eq}-f_i).}$

By expanding the particle distribution function into equilibrium and non-equilibrium components and using the Chapman-Enskog expansion, where${\displaystyle K}$ is the Knudsen number, the Taylor-expanded LBE can be decomposed into different magnitudes of order for the Knudsen number in order to obtain the proper continuum equations:

${\displaystyle f_i=f_i^{\text{eq}}+K{f}_i^{\text{neq}},}$

${\displaystyle f_i^{\text{neq}}=f_i^{(1)}+K{f}_i^{(2)}+O(K^2).}$

The equilibrium and non-equilibrium distributions satisfy the following relations to their macroscopic variables (these will be used later, once the particle distributions are in the "correct form" in order to scale from the particle to macroscopic level):

${\displaystyle \rho =\sum _i{f}_i^{\text{eq}},}$

${\displaystyle \rho \overrightarrow{u}=\sum _i{f}_i^{\text{eq}}{\overrightarrow{e}}_i,}$

${\displaystyle 0=\sum _i{f}_i^{(k)}\text{for }k=1,2,}$

${\displaystyle 0=\sum _i{f}_i^{(k)}{\overrightarrow{e}}_i.}$

The Chapman-Enskog expansion is then:

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }t}=K\frac{\mathrm{\partial }}{\mathrm{\partial }{t}_1}+K^2\frac{\mathrm{\partial }}{\mathrm{\partial }{t}_2}\text{for }{t}_2(\text{diffusive time-scale})\ll t_1(\text{convective time-scale}),}$

${\displaystyle \frac{\mathrm{\partial }}{\mathrm{\partial }x}=K\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_1}.}$

By substituting the expanded equilibrium and non-equilibrium into the Taylor expansion and separating into different orders of${\displaystyle K}$, the continuum equations are nearly derived.

For order${\displaystyle K^0}$:

${\displaystyle \frac{\mathrm{\partial }{f}_i^{\text{eq}}}{\mathrm{\partial }{t}_1}+{\overrightarrow{e}}_i{\mathrm{\nabla }}_1{f}_i^{\text{eq}}=-\frac{f_i^{(1)}}{\tau }.}$

For order${\displaystyle K^1}$:

${\displaystyle \frac{\mathrm{\partial }{f}_i^{(1)}}{\mathrm{\partial }{t}_1}+\frac{\mathrm{\partial }{f}_i^{\text{eq}}}{\mathrm{\partial }{t}_2}+{\overrightarrow{e}}_i\mathrm{\nabla }{f}_i^{(1)}+\frac{1}{2}{\overrightarrow{e}}_i{\overrightarrow{e}}_i:\mathrm{\nabla }\mathrm{\nabla }{f}_i^{\text{eq}}+{\overrightarrow{e}}_i\cdot \mathrm{\nabla }\frac{\mathrm{\partial }{f}_i^{\text{eq}}}{\mathrm{\partial }{t}_1}+\frac{1}{2}\frac{{\mathrm{\partial }}^2{f}_i^{\text{eq}}}{\mathrm{\partial }{t}_1^2}=-\frac{f_i^{(2)}}{\tau }.}$

Then, the second equation can be simplified with some algebra and the first equation into the following:

${\displaystyle \frac{\mathrm{\partial }{f}_i^{\text{eq}}}{\mathrm{\partial }{t}_2}+\left(1-\frac{1}{2\tau }\right)\left[\frac{\mathrm{\partial }{f}_i^{(1)}}{\mathrm{\partial }{t}_1}+{\overrightarrow{e}}_i{\mathrm{\nabla }}_1{f}_i^{(1)}\right]=-\frac{f_i^{(2)}}{\tau }.}$

Applying the relations between the particle distribution functions and the macroscopic properties from above, the mass and momentum equations are achieved:

${\displaystyle \frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \rho \overrightarrow{u}=0,}$

${\displaystyle \frac{\mathrm{\partial }\rho \overrightarrow{u}}{\mathrm{\partial }t}+\mathrm{\nabla }\cdot \mathrm{\Pi }=0.}$

The momentum flux tensor${\displaystyle \mathrm{\Pi }}$ has the following form then:

${\displaystyle {\mathrm{\Pi }}_{xy}=\sum _i{\overrightarrow{e}}_{ix}{\overrightarrow{e}}_{iy}\left[f_i^{eq}+\left(1-\frac{1}{2\tau }\right)f_i^{(1)}\right],}$

where${\displaystyle {\overrightarrow{e}}_{ix}{\overrightarrow{e}}_{iy}}$ is shorthand for the square of the sum of all the components of${\displaystyle {\overrightarrow{e}}_i}$ (i. e.${\displaystyle {\left(\sum _x{\overrightarrow{e}}_{ix}\right)}^2=\sum _x\sum _y{\overrightarrow{e}}_{ix}{\overrightarrow{e}}_{iy}}$), and the equilibrium particle distribution with second order to be comparable to the Navier–Stokes equation is:

${\displaystyle f_i^{\text{eq}}={\omega }_i\rho \left(1+\frac{{\overrightarrow{e}}_i\overrightarrow{u}}{c_s^2}+\frac{({\overrightarrow{e}}_i\overrightarrow{u}{)}^2}{2c_s^4}-\frac{{\overrightarrow{u}}^2}{2c_s^2}\right).}$

The equilibrium distribution is only valid for small velocities or smallMach numbers. Inserting the equilibrium distribution back into the flux tensor leads to:

${\displaystyle {\mathrm{\Pi }}_{xy}^{(0)}=\sum _i{\overrightarrow{e}}_{ix}{\overrightarrow{e}}_{iy}{f}_i^{eq}=p{\delta }_{xy}+\rho u_x{u}_y,}$

${\displaystyle {\mathrm{\Pi }}_{xy}^{(1)}=\left(1-\frac{1}{2\tau }\right)\sum _i{\overrightarrow{e}}_{ix}{\overrightarrow{e}}_{iy}{f}_i^{(1)}=\nu \left({\mathrm{\nabla }}_x\left(\rho {\overrightarrow{u}}_y\right)+{\mathrm{\nabla }}_y\left(\rho {\overrightarrow{u}}_x\right)\right).}$

Finally, theNavier–Stokes equation is recovered under the assumption that density variation is small:

${\displaystyle \rho \left(\frac{\mathrm{\partial }{\overrightarrow{u}}_x}{\mathrm{\partial }t}+{\mathrm{\nabla }}_y\cdot {\overrightarrow{u}}_x{\overrightarrow{u}}_y\right)=-{\mathrm{\nabla }}_{x}p+\nu {\mathrm{\nabla }}_y\cdot \left({\mathrm{\nabla }}_x\left(\rho {\overrightarrow{u}}_y\right)+{\mathrm{\nabla }}_y\left(\rho {\overrightarrow{u}}_x\right)\right).}$

This derivation follows the work of Chen and Doolen.^[17]

The continuous Boltzmann equation is an evolution equation for a single particle probability distribution function${\displaystyle f(\overrightarrow{x},{\overrightarrow{e}}_i,t)}$ and the internal energy density distribution function${\displaystyle g(\overrightarrow{x},{\overrightarrow{e}}_i,t)}$ (He et al.) are each respectively:

${\displaystyle {\mathrm{\partial }}_{t}f+(\overrightarrow{e}\cdot \mathrm{\nabla })f+F{\mathrm{\partial }}_{v}f=\mathrm{\Omega }(f),}$

${\displaystyle {\mathrm{\partial }}_{t}g+(\overrightarrow{e}\cdot \mathrm{\nabla })g+G{\mathrm{\partial }}_{v}f=\mathrm{\Omega }(g),}$

where${\displaystyle g(\overrightarrow{x},{\overrightarrow{e}}_i,t)}$ is related to${\displaystyle f(\overrightarrow{x},{\overrightarrow{e}}_i,t)}$ by

${\displaystyle g(\overrightarrow{x},{\overrightarrow{e}}_i,t)=\frac{(\overrightarrow{e}-\overrightarrow{u}{)}^2}{2}f(\overrightarrow{x},{\overrightarrow{e}}_i,t),}$

${\displaystyle F}$ is an external force,${\displaystyle \mathrm{\Omega }}$ is a collision integral, and${\displaystyle \overrightarrow{e}}$ (also labeled by${\displaystyle \overrightarrow{\xi }}$ in literature) is the microscopic velocity. The external force${\displaystyle F}$ is related to temperature external force${\displaystyle G}$ by the relation below. A typical test for one's model is theRayleigh–Bénard convection for${\displaystyle G}$.

${\displaystyle F=\frac{\overrightarrow{G}\cdot (\overrightarrow{e}-\overrightarrow{u})}{RT}{f}^{\text{eq}},}$

${\displaystyle \overrightarrow{G}=\beta g_0(T-T_{avg})\overrightarrow{k}.}$

Macroscopic variables such as density${\displaystyle \rho }$, velocity${\displaystyle \overrightarrow{u}}$, and temperature${\displaystyle T}$ can be calculated as the moments of the density distribution function:

${\displaystyle \rho =\int fd\overrightarrow{e},}$

${\displaystyle \rho \overrightarrow{u}=\int \overrightarrow{e}fd\overrightarrow{e},}$

${\displaystyle \frac{\rho DRT}{2}=\rho ϵ=\int gd\overrightarrow{e}.}$

The lattice Boltzmann method discretizes this equation by limiting space to a lattice and the velocity space to a discrete set of microscopic velocities (i. e.${\displaystyle {\overrightarrow{e}}_i=({\overrightarrow{e}}_{ix},{\overrightarrow{e}}_{iy})}$). The microscopic velocities in D2Q9, D3Q15, and D3Q19 for example are given as:

The single-phase discretized Boltzmann equation for mass density and internal energy density are:

${\displaystyle f_i(\overrightarrow{x}+{\overrightarrow{e}}_i{\delta }_t,t+{\delta }_t)-f_i(\overrightarrow{x},t)+F_i=\mathrm{\Omega }(f),}$

${\displaystyle g_i(\overrightarrow{x}+{\overrightarrow{e}}_i{\delta }_t,t+{\delta }_t)-g_i(\overrightarrow{x},t)+G_i=\mathrm{\Omega }(g).}$

The collision operator is often approximated by a BGK collision operator under the condition it also satisfies the conservation laws:

${\displaystyle \mathrm{\Omega }(f)=\frac{1}{{\tau }_f}(f_i^{\text{eq}}-f_i),}$

${\displaystyle \mathrm{\Omega }(g)=\frac{1}{{\tau }_g}(g_i^{\text{eq}}-g_i).}$

In the collision operator${\displaystyle f_i^{\text{eq}}}$ is the discrete,equilibrium particle probability distribution function. In D2Q9 and D3Q19, it is shown below for an incompressible flow in continuous and discrete form whereD,R, andT are the dimension, universal gas constant, and absolute temperature respectively. The partial derivation for the continuous to discrete form is provided through a simple derivation to second order accuracy.

${\displaystyle f^{\text{eq}}=\frac{\rho }{(2\pi RT{)}^{D/2}}{e}^{-\frac{(\overrightarrow{e}-\overrightarrow{u}{)}^2}{2RT}}}$

${\displaystyle =\frac{\rho }{(2\pi RT{)}^{D/2}}{e}^{-\frac{(\overrightarrow{e}{)}^2}{2RT}}{e}^{\frac{\overrightarrow{e}\overrightarrow{u}}{RT}-\frac{{\overrightarrow{u}}^2}{2RT}}}$

${\displaystyle =\frac{\rho }{(2\pi RT{)}^{D/2}}{e}^{-\frac{(\overrightarrow{e}{)}^2}{2RT}}\left(1+\frac{\overrightarrow{e}\overrightarrow{u}}{RT}+\frac{(\overrightarrow{e}\overrightarrow{u}{)}^2}{2(RT{)}^2}-\frac{{\overrightarrow{u}}^2}{2RT}+...\right)}$

Letting${\displaystyle c=\sqrt{3RT}}$ yields the final result:

${\displaystyle f_i^{eq}={\omega }_i\rho \left(1+\frac{3{\overrightarrow{e}}_i\overrightarrow{u}}{c^2}+\frac{9({\overrightarrow{e}}_i\overrightarrow{u}{)}^2}{2c^4}-\frac{3(\overrightarrow{u}{)}^2}{2c^2}\right)}$

${\displaystyle g^{eq}=\frac{\rho (\overrightarrow{e}-\overrightarrow{u}{)}^2}{2(2\pi RT{)}^{D/2}}{e}^{-\frac{(\overrightarrow{e}-\overrightarrow{u}{)}^2}{2RT}}}$

As much work has already been done on a single-component flow, the following TLBM will be discussed. The multicomponent/multiphase TLBM is also more intriguing and useful than simply one component. To be in line with current research, define the set of all components of the system (i. e. walls of porous media, multiple fluids/gases, etc.)${\displaystyle \mathrm{\Psi }}$ with elements${\displaystyle {\sigma }_j}$.

${\displaystyle f_i^{\sigma }(\overrightarrow{x}+{\overrightarrow{e}}_i{\delta }_t,t+{\delta }_t)-f_i^{\sigma }(\overrightarrow{x},t)+F_i=\frac{1}{{\tau }_f^{\sigma }}(f_i^{\sigma ,eq}({\rho }^{\sigma },v^{\sigma })-f_i^{\sigma })}$

The relaxation parameter,${\displaystyle {\tau }_f^{{\sigma }_j}}$, is related to thekinematic viscosity,${\displaystyle {\nu }_f^{{\sigma }_j}}$, by the following relationship:

${\displaystyle {\nu }_f^{{\sigma }_j}=({\tau }_f^{{\sigma }_j}-0.5)c_s^2{\delta }_t.}$

Themoments of the${\displaystyle f_i}$ give the local conserved quantities. The density is given by

${\displaystyle \rho =\sum _{\sigma }\sum _i{f}_i}$

${\displaystyle \rho ϵ=\sum _i{g}_i}$

${\displaystyle {\rho }^{\sigma }=\sum _i{f}_i^{\sigma }}$

and the weighted average velocity,${\displaystyle \overrightarrow{u^{\prime }}}$, and the local momentum are given by

${\displaystyle \overrightarrow{u^{\prime }}=\left(\sum _{\sigma }\frac{{\rho }^{\sigma }\overrightarrow{u^{\sigma }}}{{\tau }_f^{\sigma }}\right)/\left(\sum _{\sigma }\frac{{\rho }^{\sigma }}{{\tau }_f^{\sigma }}\right)}$

${\displaystyle {\rho }^{\sigma }\overrightarrow{u^{\sigma }}=\sum _i{f}_i^{\sigma }{\overrightarrow{e}}_i.}$

${\displaystyle v^{\sigma }=\overrightarrow{u^{\prime }}+\frac{{\tau }_f^{\sigma }}{{\rho }^{\sigma }}{\overrightarrow{F}}^{\sigma }}$

In the above equation for the equilibrium velocity${\displaystyle v^{\sigma }}$, the${\displaystyle {\overrightarrow{F}}^{\sigma }}$ term is the interaction force between a component and the other components. It is still the subject of much discussion as it is typically a tuning parameter that determines how fluid-fluid, fluid-gas, etc. interact. Frank et al. list current models for this force term. The commonly used derivations are Gunstensen chromodynamic model, Swift's free energy-based approach for both liquid/vapor systems and binary fluids, He's intermolecular interaction-based model, the Inamuro approach, and the Lee and Lin approach.^[18]

The following is the general description for${\displaystyle {\overrightarrow{F}}^{\sigma }}$ as given by several authors.^[19][20]

${\displaystyle {\overrightarrow{F}}^{\sigma }=-{\psi }^{\sigma }(\overrightarrow{x})\sum _{{\sigma }_j}{H}^{\sigma {\sigma }_j}(\overrightarrow{x},{\overrightarrow{x}}^{\prime })\sum _i{\psi }^{{\sigma }_j}(\overrightarrow{x}+{\overrightarrow{e}}_i){\overrightarrow{e}}_i}$

${\displaystyle \psi (\overrightarrow{x})}$ is the effective mass and${\displaystyle H(\overrightarrow{x},{\overrightarrow{x}}^{\prime })}$ is Green's function representing the interparticle interaction with${\displaystyle {\overrightarrow{x}}^{\prime }}$ as the neighboring site. Satisfying${\displaystyle H(\overrightarrow{x},{\overrightarrow{x}}^{\prime })=H({\overrightarrow{x}}^{\prime },\overrightarrow{x})}$ and where${\displaystyle H(\overrightarrow{x},{\overrightarrow{x}}^{\prime })>0}$ represents repulsive forces. For D2Q9 and D3Q19, this leads to

The effective mass as proposed by Shan and Chen uses the following effective mass for asingle-component, multiphase system. Theequation of state is also given under the condition of a single component and multiphase.

${\displaystyle \psi (\overrightarrow{x})=\psi ({\rho }^{\sigma })={\rho }_0^{\sigma }\left[1-e^{(-{\rho }^{\sigma }/{\rho }_0^{\sigma })}\right]}$

${\displaystyle p=c_s^2\rho +c_{0}h[\psi (\overrightarrow{x}){]}^2}$

So far, it appears that${\displaystyle {\rho }_0^{\sigma }}$ and${\displaystyle h^{\sigma {\sigma }_j}}$ are free constants to tune but once plugged into the system'sequation of state(EOS), they must satisfy the thermodynamic relationships at the critical point such that${\displaystyle (\mathrm{\partial }P/\mathrm{\partial }\rho {)}_T=({\mathrm{\partial }}^{2}P/\mathrm{\partial }{\rho }^2{)}_T=0}$ and${\displaystyle p=p_c}$. For the EOS,${\displaystyle c_0}$ is 3.0 for D2Q9 and D3Q19 while it equals 10.0 for D3Q15.^[21]

It was later shown by Yuan and Schaefer^[22] that the effective mass density needs to be changed to simulate multiphase flow more accurately. They compared the Shan and Chen (SC), Carnahan-Starling (C–S), van der Waals (vdW), Redlich–Kwong (R–K), Redlich–Kwong Soave (RKS), and Peng–Robinson (P–R) EOS. Their results revealed that the SC EOS was insufficient and that C–S, P–R, R–K, and RKS EOS are all more accurate in modeling multiphase flow of a single component.

For the popular isothermal Lattice Boltzmann methods these are the only conserved quantities. Thermal models also conserve energy and therefore have an additional conserved quantity:

${\displaystyle \rho \theta +\rho uu=\sum _i{f}_i{\overrightarrow{e}}_i{\overrightarrow{e}}_i.}$

Normally, the lattice Boltzmann methods is implemented on regular grids, However the use of unstructured grid can help with solving complex boundaries, unstructured grids are made of triangles or tetrahedra with variations.

Assuming${\displaystyle {\mathrm{\Omega }}^j}$ is a volume made by allbarycenters of tetrahedra, faces and edges connected to vertex${\displaystyle {\mathit{v}}^j}$, the discrete velocity density function:

${\displaystyle f_i({\mathit{v}}^j,t+\delta t)=f_i({\mathit{v}}^j,t)-\delta t\sum _k{S}_i^{jk}{f}_i({\mathit{v}}^k,t)-\frac{\delta t}{\tau }\sum _k{C}^{jk}(f_i({\mathit{v}}^k,t)-f_i^{eq}({\mathit{v}}^k))}$

where${\displaystyle {\mathit{v}}^k}$ are position of a vertex and its neighbors, and:

${\displaystyle C^{jk}=\frac{1}{V^j}{\int }_{{\mathrm{\Omega }}^j}{w}_k(\mathit{x})d\mathrm{\Omega }}$

${\displaystyle S_i^{jk}=\frac{1}{V^j}{\oint }_{\mathrm{\partial }{\mathrm{\Omega }}^j}(\overrightarrow{e_i}\overrightarrow{n})w_k(\mathit{x})d\mathrm{\Omega }}$

where${\displaystyle w_k(\mathit{x})}$ is wights of a linear interpolation of${\displaystyle \mathit{x}}$ by vertices of triangle or tetrahedra that${\displaystyle \mathit{x}}$ lies within.^[23]

During the last years, the LBM has proven to be a powerful tool for solving problems at different length and time scales. Some of the applications of LBM include:

• Porous Media flows^[24]
• Biomedical Flows
• Earth sciences (Soil filtration).
• Energy Sciences (Fuel Cells^[25]).

This is a barebone implementation of LBM on a 100x100 grid, UsingPython:

# This is a fluid simulator using the lattice Boltzmann method.# Using D2Q9 and periodic boundary, and used no external library.# It generates two ripples at 50,50 and 50,40.# Reference: Erlend Magnus Viggen's Master thesis, "The Lattice Boltzmann Method with Applications in Acoustics".# For Wikipedia under CC-BY-SA license.importmath# Define some utilitiesdefsum(a):s=0foreina:s=s+ereturns# Weights in D2Q9Weights=[1/36,1/9,1/36,1/9,4/9,1/9,1/36,1/9,1/36]# Discrete velocity vectorsDiscreteVelocityVectors=[[-1,1],[0,1],[1,1],[-1,0],[0,0],[1,0],[-1,-1],[0,-1],[1,-1],]# A Field2D classclassField2D:def__init__(self,res:int):self.field=[]forbinrange(res):fm=[]forainrange(res):fm.append([0,0,0,0,1,0,0,0,0])self.field.append(fm[:])self.res=res# This visualize the simulation, can only be used in a terminal@staticmethoddefVisualizeField(a,sc,res):stringr=""foruinrange(res):row=""forvinrange(res):n=int(u*a.res/res)x=int(v*a.res/res)flowmomentem=a.Momentum(n,x)col="\033[38;2;{0};{1};{2}m██".format(int(127+sc*flowmomentem[0]),int(127+sc*flowmomentem[1]),0)row=row+colprint(row)stringr=stringr+row+"\n"returnstringr# Momentum of the fielddefMomentum(self,x,y):returnvelocityField[y][x][0]*sum(self.field[y][x]),velocityField[y][x][1]*sum(self.field[y][x])# Resolution of the simulationres=100a=Field2D(res)# The velocity fieldvelocityField=[]forDummyVariableinrange(res):DummyList=[]forDummyVariable2inrange(res):DummyList.append([0,0])velocityField.append(DummyList[:])# The density fieldDensityField=[]forDummyVariableinrange(res):DummyList=[]forDummyVariable2inrange(res):DummyList.append(1)DensityField.append(DummyList[:])# Set initial conditionDensityField[50][50]=2DensityField[40][50]=2# Maximum solving stepsMaxSteps=120# The speed of sound, specifically 1/sqrt(3) ~ 0.57SpeedOfSound=1/math.sqrt(3)# time relaxation constantTimeRelaxationConstant=0.5# Solveforsinrange(MaxSteps):# Collision Stepdf=Field2D(res)foryinrange(res):forxinrange(res):forvinrange(9):Velocity=a.field[y][x][v]FirstTerm=Velocity# The Flow VelocityFlowVelocity=velocityField[y][x]Dotted=(FlowVelocity[0]*DiscreteVelocityVectors[v][0]+FlowVelocity[1]*DiscreteVelocityVectors[v][1])# #The taylor expainsion of equilibrium termtaylor=(1+((Dotted)/(SpeedOfSound**2))+((Dotted**2)/(2*SpeedOfSound**4))-((FlowVelocity[0]**2+FlowVelocity[1]**2)/(2*SpeedOfSound**2)))# The current densitydensity=DensityField[y][x]# The equilibriumequilibrium=density*taylor*Weights[v]SecondTerm=(equilibrium-Velocity)/TimeRelaxationConstantdf.field[y][x][v]=FirstTerm+SecondTerm# Streaming Stepforyinrange(0,res):forxinrange(0,res):forvinrange(9):# Target, the lattice point this iteration is solvingTargetY=y+DiscreteVelocityVectors[v][1]TargetX=x+DiscreteVelocityVectors[v][0]# Peiodic BoundaryifTargetY==resandTargetX==res:a.field[TargetY-res][TargetX-res][v]=df.field[y][x][v]elifTargetX==res:a.field[TargetY][TargetX-res][v]=df.field[y][x][v]elifTargetY==res:a.field[TargetY-res][TargetX][v]=df.field[y][x][v]elifTargetY==-1andTargetX==-1:a.field[TargetY+res][TargetX+res][v]=df.field[y][x][v]elifTargetX==-1:a.field[TargetY][TargetX+res][v]=df.field[y][x][v]elifTargetY==-1:a.field[TargetY+res][TargetX][v]=df.field[y][x][v]else:a.field[TargetY][TargetX][v]=df.field[y][x][v]# Calculate macroscopic variablesforyinrange(res):forxinrange(res):# Recompute Density FieldDensityField[y][x]=sum(a.field[y][x])# Recompute Flow VelocityFlowVelocity=[0,0]forDummyVariableinrange(9):FlowVelocity[0]=(FlowVelocity[0]+DiscreteVelocityVectors[DummyVariable][0]*a.field[y][x][DummyVariable])forDummyVariableinrange(9):FlowVelocity[1]=(FlowVelocity[1]+DiscreteVelocityVectors[DummyVariable][1]*a.field[y][x][DummyVariable])FlowVelocity[0]=FlowVelocity[0]/DensityField[y][x]FlowVelocity[1]=FlowVelocity[1]/DensityField[y][x]# Insert to Velocity FieldvelocityField[y][x]=FlowVelocity# VisualizeField2D.VisualizeField(a,128,100)

• Deutsch, Andreas; Sabine Dormann (2004).Cellular Automaton Modeling of Biological Pattern Formation.Birkhäuser Verlag.ISBN 978-0-8176-4281-5.
• Succi, Sauro (2001).The Lattice Boltzmann Equation for Fluid Dynamics and Beyond.Oxford University Press.ISBN 978-0-19-850398-9.
• Wolf-Gladrow, Dieter (2000).Lattice-Gas Cellular Automata and Lattice Boltzmann Models.Springer Verlag.ISBN 978-3-540-66973-9.
• Sukop, Michael C.; Daniel T. Thorne, Jr. (2007).Lattice Boltzmann Modeling: An Introduction for Geoscientists and Engineers.Springer.ISBN 978-3-540-27981-5.
• Jian Guo Zhou (2004).Lattice Boltzmann Methods for Shallow Water Flows.Springer.ISBN 978-3-540-40746-1.
• He, X., Chen, S., Doolen, G. (1998).A Novel Thermal Model for the Lattice Boltzmann Method in Incompressible Limit.Academic Press.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Guo, Z. L.; Shu, C (2013).Lattice Boltzmann Method and Its Applications in Engineering.World Scientific Publishing.
• Huang, H.; M.C. Sukop; X-Y. Lu (2015).Multiphase Lattice Boltzmann Methods: Theory and Application.Wiley-Blackwell.ISBN 978-1-118-97133-8.
• Krüger, T.; Kusumaatmaja, H.; Kuzmin, A.; Shardt, O.; Silva, G.; Viggen, E. M. (2017).The Lattice Boltzmann Method: Principles and Practice.Springer Verlag.ISBN 978-3-319-44647-9.

• LBM Method
• Entropic Lattice Boltzmann Method (ELBM)
• dsfd.org: Website of the annual DSFD conference series (1986 -- now) where advances in theory and application of the lattice Boltzmann method are discussed
• Website of the annual ICMMES conference on lattice Boltzmann methods and their applications

• Computational fluid dynamics
• Lattice models

• Webarchive template wayback links
• Articles with short description
• Short description matches Wikidata
• CS1 maint: multiple names: authors list