Lattice Boltzmann methods

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Lattice Boltzmann methods (LBM), originated from the lattice gas automata (LGA) method (Hardy-Pomeau-Pazzis and Frisch-Hasslacher-Pomeau models), is a class of computational fluid dynamics (CFD) methods for fluid simulation. Instead of solving the Navier–Stokes equations directly, a fluid density on a lattice is simulated with streaming and collision (relaxation) processes. The method is versatile 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.

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Computer simulation in two dimensions, using Lattice Boltzmann method, of a droplet that starts stretched and relaxes to its equilibrium circular shape

Algorithm

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. A different interpretation of the lattice Boltzmann equation is that of a discrete-velocity Boltzmann 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.

Schematic of D2Q9 lattice vectors for 2D Lattice Boltzmann

In an algorithm, there are collision and streaming steps. These evolve the density of the fluid ${\displaystyle \rho ({\vec {x}},t)}$, for ${\displaystyle {\vec {x}}}$ the position and ${\displaystyle t}$ the time. As the fluid is on a lattice, the density has a number of components ${\displaystyle f_{i},i=0,\ldots ,a}$ 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 a unit square, plus a vector with both components zero. Then, for example vector ${\displaystyle {\vec {e}}_{4}=(0,-1)}$, 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, ${\displaystyle f_{4}({\vec {x}},t)}$, is that part of the fluid at point ${\displaystyle {\vec {x}}}$ moving due south, at a speed in lattice units of one.

Then the steps that evolve the fluid in time are:

The collision step

${\displaystyle f_{i}({\vec {x}},t+\delta _{t})=f_{i}({\vec {x}},t)+{\frac {f_{i}^{eq}({\vec {x}},t)-f_{i}({\vec {x}},t)}{\tau _{f}}}\,\!}$

which is the Bhatnagar Gross and Krook (BGK) model for relaxation to equilibrium via collisions between the molecules of a fluid. ${\displaystyle f_{i}^{eq}({\vec {x}},t)}$ is the equilibrium density along direction i at the current density there. The model assumes that the fluid locally relaxes to equilibrium over a characteristic timescale ${\displaystyle \tau _{f}}$. This timescale determines the kinematic viscosity, the larger it is, the larger is the kinematic viscosity.

 

The streaming step

${\displaystyle f_{i}({\vec {x}}+{\vec {e}}_{i},t+\delta _{t})=f_{i}({\vec {x}},t)\,\!}$

As ${\displaystyle f_{i}({\vec {x}},t)}$ is, by definition, the fluid density at point ${\displaystyle {\vec {x}}}$ at time ${\displaystyle t}$, that is moving at a velocity of ${\displaystyle {\vec {e}}_{i}}$ per time step, then at the next time step ${\displaystyle t+\delta _{t}}$ it will have flowed to point ${\displaystyle {\vec {x}}+{\vec {e}}_{i}}$.

Advantages

• The LBM was designed from scratch to run efficiently on massively parallel architectures, ranging from inexpensive embedded FPGAs and DSPs up to GPUs 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.

Limitations

Despite the increasing popularity of LBM in simulating complex fluid systems, this novel approach has some limitations. At present, high-Mach number flows in aerodynamics are still difficult for LBM, and a consistent thermo-hydrodynamic scheme is absent. However, 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 and Schaefer 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 the equation of state. It has been proposed to apply Galilean Transformation to overcome the limitation of modelling high-speed fluid flows. Nevertheless, the wide applications and fast advancements of this method during the past twenty years have proven its potential in computational physics, including microfluidics: LBM demonstrates promising results in the area of high Knudsen number flows.

Development from the LGA method

LBM originated from the lattice 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-dimensional FHP 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 the particle number (mass), momentum, and energy before and after the collision. LGA suffer from several innate defects for use in hydrodynamic simulations: lack of Galilean invariance for fast flows, statistical noise and poor Reynolds number scaling with lattice size. LGA are, however, well suited to simplify and extend the reach of reaction diffusion and molecular 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 the Bhatnagar-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. From Chapman-Enskog theory, one can recover the governing continuity and Navier–Stokes equations from the LBM algorithm.

Lattices and the DnQm classification

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. (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 the Reynolds number, remain the same.

Lattice units conversion

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.

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

Simulation of mixtures

Simulating multiphase/multicomponent flows has always been a challenge to conventional CFD because of the moving and deformable interfaces. More fundamentally, the interfaces between different phases (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 various complex 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. 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.

Thermal lattice-Boltzmann method

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

Derivation of Navier–Stokes equation from discrete LBE

See also: Derivation of the Navier–Stokes equations

Starting with the discrete lattice Boltzmann equation (also referred to as LBGK equation due to the collision operator used). We first do a 2nd-order Taylor 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:

${\displaystyle f_{i}({\vec {x}}+{\vec {e}}_{i}\delta _{t},t+\delta _{t})=f_{i}({\vec {x}},t)+{\frac {\delta _{t}}{\tau _{f}}}(f_{i}^{eq}-f_{i}).}$

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

${\displaystyle {\frac {\partial f_{i}}{\partial t}}+{\vec {e}}_{i}\cdot \nabla f_{i}+\left({\frac {1}{2}}{\vec {e}}_{i}{\vec {e}}_{i}:\nabla \nabla f_{i}+{\vec {e}}_{i}\cdot \nabla {\frac {\partial f_{i}}{\partial t}}+{\frac {1}{2}}{\frac {\partial ^{2}f_{i}}{\partial t^{2}}}\right)={\frac {1}{\tau }}(f_{i}^{eq}-f_{i}).}$

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:

${\displaystyle f_{i}=f_{i}^{\text{eq}}+Kf_{i}^{\text{neq}},}$

${\displaystyle f_{i}^{\text{neq}}=f_{i}^{(1)}+Kf_{i}^{(2)}+O(K^{2}).}$

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):

${\displaystyle \rho =\sum _{i}f_{i}^{\text{eq}},}$

${\displaystyle \rho {\vec {u}}=\sum _{i}f_{i}^{\text{eq}}{\vec {e}}_{i},}$

${\displaystyle 0=\sum _{i}f_{i}^{(k)}\qquad {\text{for }}k=1,2,}$

${\displaystyle 0=\sum _{i}f_{i}^{(k)}{\vec {e}}_{i}.}$

The Chapman-Enskog expansion is then:

${\displaystyle {\frac {\partial }{\partial t}}=K{\frac {\partial }{\partial t_{1}}}+K^{2}{\frac {\partial }{\partial t_{2}}}\qquad {\text{for }}t_{2}({\text{diffusive time-scale}})\ll t_{1}({\text{convective time-scale}}),}$

${\displaystyle {\frac {\partial }{\partial x}}=K{\frac {\partial }{\partial x_{1}}}.}$

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

${\displaystyle {\frac {\partial f_{i}^{\text{eq}}}{\partial t_{1}}}+{\vec {e}}_{i}\nabla _{1}f_{i}^{\text{eq}}=-{\frac {f_{i}^{(1)}}{\tau }}.}$

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

${\displaystyle {\frac {\partial f_{i}^{(1)}}{\partial t_{1}}}+{\frac {\partial f_{i}^{\text{eq}}}{\partial t_{2}}}+{\vec {e}}_{i}\nabla f_{i}^{(1)}+{\frac {1}{2}}{\vec {e}}_{i}{\vec {e}}_{i}:\nabla \nabla f_{i}^{\text{eq}}+{\vec {e}}_{i}\cdot \nabla {\frac {\partial f_{i}^{\text{eq}}}{\partial t_{1}}}+{\frac {1}{2}}{\frac {\partial ^{2}f_{i}^{\text{eq}}}{\partial t_{1}^{2}}}=-{\frac {f_{i}^{(2)}}{\tau }}.}$

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

${\displaystyle {\frac {\partial f_{i}^{\text{eq}}}{\partial t_{2}}}+\left(1-{\frac {1}{2\tau }}\right)\left[{\frac {\partial f_{i}^{(1)}}{\partial t_{1}}}+{\vec {e}}_{i}\nabla _{1}f_{i}^{(1)}\right]=-{\frac {f_{i}^{(2)}}{\tau }}.}$

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

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

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

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

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

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

${\displaystyle f_{i}^{\text{eq}}=\omega _{i}\rho \left(1+{\frac {{\vec {e}}_{i}{\vec {u}}}{c_{s}^{2}}}+{\frac {({\vec {e}}_{i}{\vec {u}})^{2}}{2c_{s}^{4}}}-{\frac {{\vec {u}}^{2}}{2c_{s}^{2}}}\right).}$

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

${\displaystyle \Pi _{xy}^{(0)}=\sum _{i}{\vec {e}}_{ix}{\vec {e}}_{iy}f_{i}^{eq}=p\delta _{xy}+\rho u_{x}u_{y},}$

${\displaystyle \Pi _{xy}^{(1)}=\left(1-{\frac {1}{2\tau }}\right)\sum _{i}{\vec {e}}_{ix}{\vec {e}}_{iy}f_{i}^{(1)}=\nu \left(\nabla _{x}\left(\rho {\vec {u}}_{y}\right)+\nabla _{y}\left(\rho {\vec {u}}_{x}\right)\right).}$

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

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

This derivation follows the work of Chen and Doolen.

Mathematical equations for simulations

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

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

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

where ${\displaystyle g({\vec {x}},{\vec {e}}_{i},t)}$ is related to ${\displaystyle f({\vec {x}},{\vec {e}}_{i},t)}$ by

${\displaystyle g({\vec {x}},{\vec {e}}_{i},t)={\frac {({\vec {e}}-{\vec {u}})^{2}}{2}}f({\vec {x}},{\vec {e}}_{i},t),}$

${\displaystyle F}$ is an external force, ${\displaystyle \Omega }$ is a collision integral, and ${\displaystyle {\vec {e}}}$ (also labeled by ${\displaystyle {\vec {\xi }}}$ 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 the Rayleigh–Bénard convection for ${\displaystyle G}$.

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

${\displaystyle {\vec {G}}=\beta g_{0}(T-T_{avg}){\vec {k}}.}$

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

${\displaystyle \rho =\int f\,d{\vec {e}},}$

${\displaystyle \rho {\vec {u}}=\int {\vec {e}}f\,d{\vec {e}},}$

${\displaystyle {\frac {\rho DRT}{2}}=\rho \epsilon =\int g\,d{\vec {e}}.}$

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. ${\displaystyle {\vec {e}}_{i}=({\vec {e}}_{ix},{\vec {e}}_{iy})}$). The microscopic velocities in D2Q9, D3Q15, and D3Q19 for example are given as:

${\displaystyle {\vec {e}}_{i}=c\times {\begin{cases}(0,0)&i=0\\(1,0),(0,1),(-1,0),(0,-1)&i=1,2,3,4\\(1,1),(-1,1),(-1,-1),(1,-1)&i=5,6,7,8\\\end{cases}}}$

${\displaystyle {\vec {e}}_{i}=c\times {\begin{cases}(0,0,0)&i=0\\(\pm 1,0,0),(0,\pm 1,0),(0,0,\pm 1)&i=1,2,...,5,6\\(\pm 1,\pm 1,\pm 1)&i=7,8,...,13,14\\\end{cases}}}$

${\displaystyle {\vec {e}}_{i}=c\times {\begin{cases}(0,0,0)&i=0\\(\pm 1,0,0),(0,\pm 1,0),(0,0,\pm 1)&i=1,2,...,5,6\\(\pm 1,\pm 1,0),(\pm 1,0,\pm 1),(0,\pm 1,\pm 1)&i=7,8,...,17,18\\\end{cases}}}$

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

${\displaystyle f_{i}({\vec {x}}+{\vec {e}}_{i}\delta _{t},t+\delta _{t})-f_{i}({\vec {x}},t)+F_{i}=\Omega (f),}$

${\displaystyle g_{i}({\vec {x}}+{\vec {e}}_{i}\delta _{t},t+\delta _{t})-g_{i}({\vec {x}},t)+G_{i}=\Omega (g).}$

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

${\displaystyle \Omega (f)={\frac {1}{\tau _{f}}}(f_{i}^{\text{eq}}-f_{i}),}$

${\displaystyle \Omega (g)={\frac {1}{\tau _{g}}}(g_{i}^{\text{eq}}-g_{i}).}$

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 where D, R, and T 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.

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

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

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

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

${\displaystyle f_{i}^{eq}=\omega _{i}\rho \left(1+{\frac {3{\vec {e}}_{i}{\vec {u}}}{c^{2}}}+{\frac {9({\vec {e}}_{i}{\vec {u}})^{2}}{2c^{4}}}-{\frac {3({\vec {u}})^{2}}{2c^{2}}}\right)}$

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

${\displaystyle \omega _{i}={\begin{cases}4/9&i=0\\1/9&i=1,2,3,4\\1/36&i=5,6,7,8\\\end{cases}}}$

${\displaystyle \omega _{i}={\begin{cases}1/3&i=0\\1/18&i=1,2,...,5,6\\1/36&i=7,8,...,17,18\\\end{cases}}}$

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.) ${\displaystyle \Psi }$ with elements ${\displaystyle \sigma _{j}}$.

${\displaystyle f_{i}^{\sigma }({\vec {x}}+{\vec {e}}_{i}\delta _{t},t+\delta _{t})-f_{i}^{\sigma }({\vec {x}},t)+F_{i}={\frac {1}{\tau _{f}^{\sigma }}}(f_{i}^{\sigma ,eq}(\rho ^{\sigma },v^{\sigma })-f_{i}^{\sigma })}$

The relaxation parameter,${\displaystyle \tau _{f}^{\sigma _{j}}\,\!}$, is related to the kinematic viscosity,${\displaystyle \nu _{f}^{\sigma _{j}}\,\!}$, by the following relationship:

${\displaystyle \nu _{f}^{\sigma _{j}}=(\tau _{f}^{\sigma _{j}}-0.5)c_{s}^{2}\delta _{t}.}$

The moments of the ${\displaystyle f_{i}\,\!}$ give the local conserved quantities. The density is given by

${\displaystyle \rho =\sum _{\sigma }\sum _{i}f_{i}\,\!}$

${\displaystyle \rho \epsilon =\sum _{i}g_{i}\,\!}$

${\displaystyle \rho ^{\sigma }=\sum _{i}f_{i}^{\sigma }\,\!}$

and the weighted average velocity, ${\displaystyle {\vec {u'}}\,\!}$, and the local momentum are given by

${\displaystyle {\vec {u'}}=\left(\sum _{\sigma }{\frac {\rho ^{\sigma }{\vec {u^{\sigma }}}}{\tau _{f}^{\sigma }}}\right)/\left(\sum _{\sigma }{\frac {\rho ^{\sigma }}{\tau _{f}^{\sigma }}}\right)}$

${\displaystyle \rho ^{\sigma }{\vec {u^{\sigma }}}=\sum _{i}f_{i}^{\sigma }{\vec {e}}_{i}.}$

${\displaystyle v^{\sigma }={\vec {u'}}+{\frac {\tau _{f}^{\sigma }}{\rho ^{\sigma }}}{\vec {F}}^{\sigma }}$

In the above equation for the equilibrium velocity ${\displaystyle v^{\sigma }\,\!}$, the ${\displaystyle {\vec {F}}^{\sigma }\,\!}$ 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.

The following is the general description for ${\displaystyle {\vec {F}}^{\sigma }\,\!}$ as given by several authors.

${\displaystyle {\vec {F}}^{\sigma }=-\psi ^{\sigma }({\vec {x}})\sum _{\sigma _{j}}H^{\sigma \sigma _{j}}({\vec {x}},{\vec {x}}')\sum _{i}\psi ^{\sigma _{j}}({\vec {x}}+{\vec {e}}_{i}){\vec {e}}_{i}\,\!}$

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

${\displaystyle H^{\sigma \sigma _{j}}({\vec {x}},{\vec {x}}')={\begin{cases}h^{\sigma \sigma _{j}}&\left|{\vec {x}}-{\vec {x}}'\right|\leq c\\0&\left|{\vec {x}}-{\vec {x}}'\right|>c\\\end{cases}}}$

${\displaystyle H^{\sigma \sigma _{j}}({\vec {x}},{\vec {x}}')={\begin{cases}h^{\sigma \sigma _{j}}&\left|{\vec {x}}-{\vec {x}}'\right|=c\\h^{\sigma \sigma _{j}}/2&\left|{\vec {x}}-{\vec {x}}'\right|={\sqrt {2c}}\\0&{\text{otherwise}}\\\end{cases}}}$

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

${\displaystyle \psi ({\vec {x}})=\psi (\rho ^{\sigma })=\rho _{0}^{\sigma }\left[1-e^{(-\rho ^{\sigma }/\rho _{0}^{\sigma })}\right]\,\!}$

${\displaystyle p=c_{s}^{2}\rho +c_{0}h[\psi ({\vec {x}})]^{2}\,\!}$

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's equation of state(EOS), they must satisfy the thermodynamic relationships at the critical point such that ${\displaystyle (\partial P/\partial {\rho })_{T}=(\partial ^{2}P/\partial {\rho ^{2}})_{T}=0\,\!}$ 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.

It was later shown by Yuan and Schaefer^[19] 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:

${\displaystyle \rho \theta +\rho uu=\sum _{i}f_{i}{\vec {e}}_{i}{\vec {e}}_{i}.}$

Applications

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
• Biomedical Flows
• Earth sciences (Soil filtration).
• Energy Sciences (Fuel Cells).

Judaism and politics

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The relationship between Judaism and politics is a historically complex subject, and has evolved over time concurrently with both changes within Jewish society and religious practice, and changes in the general society of places where Jewish people live. In particular, Jewish political thought can be split into four major eras: Biblical (prior to Roman rule), Rabbinic (from roughly the 100 BCE to 600 CE), Medieval (from roughly 600 CE to 1800 CE), and Modern (18th century to the present day).

Several different political models are described across its canon, usually composed of some combination of tribal federation, monarchy, a priestly theocracy, and rule by prophets. Political organization during the Rabbinic and Medieval eras generally involved semi-autonomous rule by Jewish councils and courts (with council membership often composed purely of rabbis) that would govern the community and act as representatives to secular authorities outside the Jewish community. Beginning in the 19th century, and coinciding with the expansion of the political rights accorded to individual Jews in European society, Jews would affiliate with and contribute theory to a wide range of political movements and philosophies.

Biblical models

Stuart Cohen has pointed out that there are three separate power centers depicted in the Hebrew Bible: the priesthood, the royal throne, and the prophets.

One model of biblical politics is the model of the tribal federation, where power is shared among different tribes and institutions. Another is the model of limited constitutional monarchy.

The Hebrew Bible contains a complex chronicle of the Kings of Israel and Judah. Some passages of the Hebrew Bible contain intimate portrayals of the inner workings of the royal households of Saul, David, and Solomon; the accounts of subsequent monarchs are frequently more distanced and less detailed, and frequently begin with the judgement that the monarch "did evil in the sight of the Lord".

Daniel Elazar has argued that the concept of covenant is the fundamental concept in the biblical political tradition and in the later Jewish thought that emerges from the Bible.

Rabbinic period

The Sanhedrin

In Roman Judea, Jewish communities were governed by rabbinical courts known as Sanhedrin. Lesser Sanhedrins composed of 23 judges were appointed to each city, while a Great Sanhedrin with 71 judges was the highest authority, taking cases appealed from the lower courts. The Sanhedrin served as the leadership of the Jewish community under Roman rule, and served as emissaries to the imperial authorities in addition to overseeing religious practice and collecting taxes. The Sanhedrin was the highest Jewish governing body of the Second Temple period.

Talmudic sources on political philosophy

A statement by Judah bar Ilai in the Babylonian Talmud (Sanhedrin 20b) depicts monarchy as the ideal form of Jewish governance, following the Book of Deuteronomy statement that, "When you come into the land that the Lord your God is about to give you, and you take hold of it and dwell in it, and you say, 'Let me put a king over me like all the nations that are around me', you shall surely put over you a king whom the Lord your God chooses..." (Deut. 17:14–15). But the Talmud also brings a different interpretation of this verse from Eleazar ben Arach, who is quoted as explaining that, "This section was spoken only in anticipation of their future murmurings, as it is written, and you say, Let me put a king over me..." (Sanhedrin 20b). In many interpretations, Rabbi Nehorai does not think of appointing a king as a strict obligation, but as a concession to later "murmurings" from Israel.

In addition to imagining ideal forms of governance, the rabbis accept a principle to obey the government currently in power. The Talmud makes reference to the principle of dina de-malkhuta dina ("the law of the land is law"), a principle recognizing non-Jewish laws and non-Jewish legal jurisdiction as binding on Jewish citizens, provided that they are not contrary to any laws of Judaism.

Medieval period

The Qahal

During the Middle Ages, some Ashkenazi Jewish communities were governed by qahal. The qahal had regulatory control over Jewish communities in a given region; they administered commerce, hygiene, sanitation, charity, Jewish education, kashrut, and relations between landlords and their tenants. It provided a number of community facilities, such as a rabbi, a ritual bath, and an interest-free loan facility for the Jewish community. The qahal even had sufficient authority that it could arrange for individuals to be expelled from synagogues, excommunicating them.

Medieval Jewish political philosophy

Some medieval political theorists such as Maimonides and Nissim of Gerona saw kingship as the ideal form of government. Maimonides' views the commandment in Deuteronomy to appoint a king as a clear positive ideal, following the Talmudic teaching that "three commandments were given to Israel when they entered the land: to appoint a king, as it says, 'You shall surely put over you a king'..." A large section of Maimonides' legal code, the Mishneh Torah, titled "The Laws of Kings and their Wars", deals with the ideal model of kingship, especially in the messianic era, and also concerning ruling over non-Jewish subjects through the Noachide laws. Other sections of Maimonides' Mishneh Torah (mostly also in the Book of Judges, where the laws of kingship are also found) is dedicated to the laws relating to legislators and judges.

Whereas Maimonides' idealized kingship, other medieval political theorists, such as Abravanel, saw kingship as misguided. Later on, other Jewish philosophers such as Baruch Spinoza would lay the groundwork for the Enlightenment, arguing for ideas such as the separation of church and state. Spinoza's writings caused him to be excommunicated from the Jewish community of Amsterdam, although his work and legacy has been largely rehabilitated, especially among secular Jews in the 20th and 21st centuries.

Modern period

With Jewish emancipation, the institution of the qahal as an autonomous entity was officially abolished. Jews increasingly became participants in the wider political and social sphere of larger nations. As Jews became citizens of states with various political systems, and argued about whether to found their own state, Jewish ideas of the relationship between Judaism and politics developed in many different directions.

In Europe

In the nineteenth century and early twentieth century, when there was a large Jewish population in Europe, some Jews favored various forms of liberalism, and saw them as connected with Jewish principles. Some Jews allied themselves with a range of Jewish political movements. These included Socialist and Bundist movements favored by the Jewish left, Zionist movements, Jewish Autonomist movements, Territorialist movements, and Jewish Anarchism movements. Haredi Jews formed an organization known as World Agudath Israel which espoused Haredi Jewish political principles.

21st century

In the 21st century, shifts are occurring. The Jewish community in Great Britain, one of the largest in the Jewish diaspora, is leaning conservative, as a poll published by the Jewish Chronicle in early 2015 shows. Of British Jews polled, 69% would vote for the Conservative Party, while 22% would vote for the Labour Party. This is in stark contrast to the rest of the voter population, which, according to a BBC poll, had Conservatives and Labor almost tied at about a third each. Jews have typically been a part of the British middle class, traditional home of the Conservative Party, although the number of Jews in working class communities of London is in decline. The main voting bloc of poorer Jews in Britain now, made up primarily of ultra-Orthodox, votes en masse for the Conservatives. Attitudes toward Israel influence the vote of three out of four of British Jews. A shift toward conservatism has also been exhibited in France, where about half of the Jewish population is Sephardic. Jérôme Fourquet, director "Public opinion and corporate strategy" department at the polling organization IFOP, notes that there is a "pronounced preference" for right-wing politics among French Jews. During the 2007 election, Jews (Orthodox or not) represented the strongest pillar of support for Sarkozy after observant Catholics.

In the United States

See also: Jewish views and involvement in U.S. politics

19th century

American Civil War

During the American Civil War, Jews were divided in their views of slavery and abolition. Prior to 1861, there were virtually no rabbinical sermons on slavery. The silence on this issue was probably a result of fear that the controversy would create conflict within the Jewish community. Some Jews owned slaves or traded them. Most southern Jews supported slavery, and few Northern Jews were abolitionists, seeking peace and remaining silent on the subject of slavery. America's largest Jewish community, New York's Jews, were "overwhelmingly pro-southern, pro-slavery, and anti-Lincoln in the early years of the war". However, eventually, they began to lean politically toward Abraham Lincoln's Republican party and emancipation.

20th and 21st centuries

Further information: List of Jewish American politicians

While earlier Jewish immigrants tended to be politically conservative, the wave of Eastern European Jews starting in the early 1880s, were generally more liberal or left-wing, and became the political majority. For most of the 20th century since 1936, the vast majority of Jews in the United States have been aligned with the Democratic Party. Many religious supporters of the Jewish left have argued that left-wing values vis-à-vis social justice can be traced to Jewish religious texts, including the Tanakh and later texts, which include a strong endorsement of hospitality to "the stranger" and the principle of redistribution of wealth in the biblical idea of Jubilee – as well as a tradition of challenging authority, as exemplified by the biblical prophets.

American rabbinic leaders who have advanced a progressive political agenda grounded in Jewish principles have included:

• Michael Lerner (rabbi) of the Jewish Renewal movement
• Arthur Waskow of the Jewish Renewal movement
• Jill Jacobs of Conservative Judaism
• David Saperstein of Reform Judaism

Other prominent Jews who have argued based on Jewish principles for a progressive political agenda have included:

• Ruth Messinger, former President and CEO of American Jewish World Service
• Stosh Cotler, CEO of Bend the Arc

Towards the end of the 20th century, and at the beginning of the 21st century, Republicans began a platform that sought to take the Jewish vote away from the Democrats. While a solid majority of American Jews continues to be aligned with the Democratic Party, many have argued that there is increased Jewish support for political conservatism.

Rabbinic leaders who have advanced a conservative political agenda grounded in Jewish principles have included:

• Seymour Siegel of Conservative Judaism
• David Dalin of Conservative Judaism

Other prominent Jews who have argued based on Jewish principles for a conservative political agenda have included:

• Daniel Lapin of Orthodox Judaism
• David Klinghoffer
• Dennis Prager
• Elliott Abrams
• David Mamet
• Ben Shapiro

Jewish political philosophy in North America

Significant Jewish political philosophers in North America have included:

• David Novak, a rabbi and philosopher at the University of Toronto, associated with the Union for Traditional Judaism
• Alan Mittleman, a rabbi and philosopher at the Jewish Theological Seminary of Conservative Judaism
• Daniel Elazar, a professor of political science at Bar-Ilan University and Temple University
• Michael Walzer, a political theorist at the Institute for Advanced Study in Princeton, New Jersey
• Michael Sandel, a political philosopher at Harvard University
• Robert Nozick, a late political philosopher at Harvard University
• Murray Rothbard, a late anarcho-capitalist philosopher and economist at the University of Nevada, Las Vegas
• Ronald Dworkin, a philosopher of law and political philosopher at New York University

In Israel

Main article: Politics of Israel

The development of a political system in Israel drew largely on European models of governance, rather than on models from the Jewish political tradition. Some political figures in Israel, however, have seen their principles as based in Judaism. This is especially pronounced in political parties that see themselves as religious parties, such as Shas, United Torah Judaism, and The Jewish Home.

Politics in Israel are dominated by Zionist parties. They traditionally fall into three camps, the first two being the largest: Labor Zionism, Revisionist Zionism and Religious Zionism. There are also several non-Zionist Orthodox religious parties, non-Zionist secular left-wing groups as well as non-Zionist and anti-Zionist Israeli Arab parties.

Recent interest in developing political theory grounded in Jewish sources has been spurred on by the activities of the neo-conservative Shalem Center.