Lattice Boltzmann methods
Encyclopedia
Lattice Boltzmann methods (LBM) (Thermal Lattice Boltzmann methods (TLBM)) is a class of computational fluid dynamics
(CFD) methods for fluid simulation. Instead of solving the Navier–Stokes equations, the discrete Boltzmann equation
is solved to simulate the flow of a Newtonian fluid
with collision
models such as Bhatnagar-Gross-Krook
(BGK). By simulating streaming and collision processes across a limited number of particles, the intrinsic particle interactions evince a microcosm of viscous flow behavior applicable across the greater mass.
In the computer algorithm, the collision and streaming step are defined as follows:
Collision step:
Streaming step:
(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. In addition, the pressure field is also directly available from the density distributions and hence there is no extra Poisson equation to be solved as in traditional CFD methods.
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 three-dimensional Lattice Boltzmann model on a cubic grid, with rest particles present. Each node has a crystal shape, and can deliver particles to each of the six neighboring nodes which share a surface, the eight neighboring nodes sharing a corner, and itself. (The D3Q15 model does not contain particles moving to the twelve neighboring nodes which 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 as the Reynolds number remain the same.
is the basic unit for lattice spacing, so if the domain of length 
has 
lattice units along its entire length, the space unit is simply defined as 
. Speeds in lattice Boltzmann simulations are typically given in terms of the speed of sound. The discrete time unit can therefore be given as 
, where the denominator 
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.
. More fundamentally, the interfaces between different phase
s (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.
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 to 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
.
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.
order Taylor series expansion about the left side of the LBE. This is chosen over a simpler 
order Taylor expansion as the discrete LBE cannot be recovered. When doing the 
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.
For simplicity, write
as 
. The slightly simplified Taylor series expansion is then as follows where ":" is the colon product between dyads.
By expanding the particle distribution function into equilibrium and non-equilibrium components and using the Chapman-Enskog Expansion where
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.
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.
The Chapman-Enskog Expansion is then:
.
By substituting the expanded equilibrium and non-equilibrium into the Taylor expansion and separating into different orders of
, the continuum equations are nearly derived.
For order,
:
For order,
:
Then, the second equation can be simplified with some algebra and the first equation into the following.
Applying the relations between the particle distribution functions and the macroscopic properties from above, the mass and momentum equations are achieved.
The momentum flux tensor,
, has the following form then.
Where
is shorthand for the square of the sum of all the components of 
(i.e. 
) and the equilibrium particle distribution with second order in order to be comparable to the Navier Stokes equation is:
.
The equilibirum distribution is only valid for small velocities or small Mach numbers. Inserting the equilibrium distribution back into the flux tensor leads to:
Finally, the Navier-Stokes equation is recovered under the assumption that density variation is small.
This derivation follows the work of Chen and Doolen.
and the internal energy density distribution function 
(He et al.) are each respectively:
where
is related to 
by:
is an external force, 
is a collision integral, and 
(also labeled by 
in literature) is the microscopic velocity. The external force, 
, is related to temperature external force 
by the relation below. A typical test for one's model is the Rayleigh-Bénard convection for 
.
Macroscopic variables such as density
, velocity 
, and temperature 
can be calculated as the moments of the density distribution function:
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.
). 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:
The collision operator is often approximated by a BGK collision operator under the condition it also satisfies the conservation laws.
In the collision operator,
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.
Letting
yields the final result.
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.)
with elements 
.
The relaxation parameter,
, is related to the kinematic viscosity,
, by the following relationship.
The moments
of the
give the local conserved quantities. The density is given by
and the weighted average velocity,
, and the local momentum are given by
In the above equation for the equilibrium velocity
, the 
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, and 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
as given by several authors.
is the effective mass and 
is Green's function representing the interparticle interaction with 
as the neighboring site. Satisfying 
and where 
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 a
, 
. The equation of state
is also given under the condition of a single component and multiphase.
So far, it appears that
and 
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
and 
. For the EOS, 
is 3.0 for D2Q9 and D3Q19 while it equals 10.0 for D3Q15.
It was later shown by Yuan and Schaefer 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:
Computational fluid dynamics
Computational fluid dynamics, usually abbreviated as CFD, is a branch of fluid mechanics that uses numerical methods and algorithms to solve and analyze problems that involve fluid flows. Computers are used to perform the calculations required to simulate the interaction of liquids and gases with...
(CFD) methods for fluid simulation. Instead of solving the Navier–Stokes equations, the discrete Boltzmann equation
Boltzmann equation
The Boltzmann equation, also often known as the Boltzmann transport equation, devised by Ludwig Boltzmann, describes the statistical distribution of one particle in rarefied gas...
is solved to simulate the flow of a Newtonian fluid
Newtonian fluid
A Newtonian fluid is a fluid whose stress versus strain rate curve is linear and passes through the origin. The constant of proportionality is known as the viscosity.-Definition:...
with collision
Collision
A collision is an isolated event which two or more moving bodies exert forces on each other for a relatively short time.Although the most common colloquial use of the word "collision" refers to accidents in which two or more objects collide, the scientific use of the word "collision" implies...
models such as Bhatnagar-Gross-Krook
Bhatnagar-Gross-Krook
The Bhatnagar–Gross–Krook operator term refers to a collision operator used in the Boltzmann Equation and in the Lattice Boltzmann method, a Computational fluid dynamics technique...
(BGK). By simulating streaming and collision processes across a limited number of particles, the intrinsic particle interactions evince a microcosm of viscous flow behavior applicable across the greater mass.
Algorithm
LBM is a relatively new simulation technique for complex fluid systems and has attracted interest from researchers in computational physics. Unlike the traditional CFD methods, which 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 mesh. Due to its particulate nature and local dynamics, LBM has several advantages over other conventional CFD methods, especially in dealing with complex boundaries, incorporating of 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 gives rise to a discrete map, which can be interpreted as the propagation and collision of fictitious particles.In the computer algorithm, the collision and streaming step are defined as follows:
Collision step:
Streaming step:
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 particles arrive 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 numberParticle number
The particle number of a thermodynamic system, conventionally indicated with the letter N, is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which is conjugate to the chemical potential. Unlike most physical quantities, particle...
(mass), momentum, and energy before and after the collision. LGA suffer from several innate defects for use in hydrodynamic simulations: lack of Galilean invariance
Galilean invariance
Galilean invariance or Galilean relativity is a principle of relativity which states that the fundamental laws of physics are the same in all inertial frames...
for fast flows, statistical noise
Statistical noise
Statistical noise is the colloquialism for recognized amounts of unexplained variation in a sample. See errors and residuals in statistics....
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
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
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
Bhatnagar-Gross-Krook
The Bhatnagar–Gross–Krook operator term refers to a collision operator used in the Boltzmann Equation and in the Lattice Boltzmann method, a Computational fluid dynamics technique...
(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
Chapman-Enskog theory
Chapman-Enskog theory presents accurate formulas for a multicomponent gas mixture under thermal and chemical equilibrium. In elastic gases the deviation from the Maxwell–Boltzmann distribution in the equilibrium is small and it can be treated as a perturbation. This method was aimed to obtain...
, one can recover the governing continuity and Navier-Stokes equations from the LBM algorithm. In addition, the pressure field is also directly available from the density distributions and hence there is no extra Poisson equation to be solved as in traditional CFD methods.
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 three-dimensional Lattice Boltzmann model on a cubic grid, with rest particles present. Each node has a crystal shape, and can deliver particles to each of the six neighboring nodes which share a surface, the eight neighboring nodes sharing a corner, and itself. (The D3Q15 model does not contain particles moving to the twelve neighboring nodes which 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 as the Reynolds number remain the same.
Lattice units conversion
In most lattice Boltzmann simulations is the basic unit for lattice spacing, so if the domain of length has lattice units along its entire length, the space unit is simply defined as . Speeds in lattice Boltzmann simulations are typically given in terms of the speed of sound. The discrete time unit can therefore be given as , where the denominator 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
Mach number
Mach number is the speed of an object moving through air, or any other fluid substance, divided by the speed of sound as it is in that substance for its particular physical conditions, including those of temperature and pressure...
to something much larger than the real Mach number, and compensating for this by raising the viscosity
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher 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 interfacesInterface (chemistry)
An interface is a surface forming a common boundary among two different phases, such as an insoluble solid and a liquid, two immiscible liquids or a liquid and an insoluble gas. The importance of the interface depends on which type of system is being treated: the bigger the quotient area/volume,...
. More fundamentally, the interfaces between different phase
Phase (matter)
In the physical sciences, a phase is a region of space , throughout which all physical properties of a material are essentially uniform. Examples of physical properties include density, index of refraction, and chemical composition...
s (liquid
Liquid
Liquid is one of the three classical states of matter . Like a gas, a liquid is able to flow and take the shape of a container. Some liquids resist compression, while others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly...
and vapor
Vapor
A vapor or vapour is a substance in the gas phase at a temperature lower than its critical point....
) or components (e.g., oil
Oil
An oil is any substance that is liquid at ambient temperatures and does not mix with water but may mix with other oils and organic solvents. This general definition includes vegetable oils, volatile essential oils, petrochemical oils, and synthetic oils....
and water
Water
Water is a chemical substance with the chemical formula H2O. A water molecule contains one oxygen and two hydrogen atoms connected by covalent bonds. Water is a liquid at ambient conditions, but it often co-exists on Earth with its solid state, ice, and gaseous state . Water also exists in a...
) 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
Liquid bubble
A bubble is a globule of one substance in another, usually gas in a liquid.Due to the Marangoni effect, bubbles may remain intact when they reach the surface of the immersive substance.-Common examples:...
/droplet dynamics, wetting
Wetting
Wetting is the ability of a liquid to maintain contact with a solid surface, resulting from intermolecular interactions when the two are brought together. The degree of wetting is determined by a force balance between adhesive and cohesive forces.Wetting is important in the bonding or adherence of...
on solid surfaces, interfacial slip, and droplet electrohydrodynamic deformations.
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.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 aerodynamicsAerodynamics
Aerodynamics is a branch of dynamics concerned with studying the motion of air, particularly when it interacts with a moving object. Aerodynamics is a subfield of fluid dynamics and gas dynamics, with much theory shared between them. Aerodynamics is often used synonymously with gas dynamics, with...
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 to 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
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
.
Nevertheless, the wide applications and fast advancements of this method during the past twenty years have proven its potential in computational physics, including microfluidics
Microfluidics
Microfluidics deals with the behavior, precise control and manipulation of fluids that are geometrically constrained to a small, typically sub-millimeter, scale.Typically, micro means one of the following features:* small volumes...
: LBM demonstrates promising results in the area of high Knudsen number
Knudsen number
The Knudsen number is a dimensionless number defined as the ratio of the molecular mean free path length to a representative physical length scale. This length scale could be, for example, the radius of a body in a fluid...
flows.
Derivation of Navier-Stokes Equation from Discrete LBE
Starting with the discrete lattice Boltzmann equation (also referred to as LBGK equation due to the collision operator used). We first do a order Taylor series expansion about the left side of the LBE. This is chosen over a simpler order Taylor expansion as the discrete LBE cannot be recovered. When doing the 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.For simplicity, write
as . The slightly simplified Taylor series expansion is then as follows where ":" is the colon product between dyads.By expanding the particle distribution function into equilibrium and non-equilibrium components and using the Chapman-Enskog Expansion where
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.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.
The Chapman-Enskog Expansion is then:
.By substituting the expanded equilibrium and non-equilibrium into the Taylor expansion and separating into different orders of
, the continuum equations are nearly derived.For order,
:For order,
:Then, the second equation can be simplified with some algebra and the first equation into the following.
Applying the relations between the particle distribution functions and the macroscopic properties from above, the mass and momentum equations are achieved.
The momentum flux tensor,
, has the following form then.Where
is shorthand for the square of the sum of all the components of (i.e. ) and the equilibrium particle distribution with second order in order to be comparable to the Navier Stokes equation is:.The equilibirum distribution is only valid for small velocities or small Mach numbers. Inserting the equilibrium distribution back into the flux tensor leads to:
Finally, the Navier-Stokes equation is recovered under the assumption that density variation is small.
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 and the internal energy density distribution function (He et al.) are each respectively:where
is related to by: is an external force, is a collision integral, and (also labeled by in literature) is the microscopic velocity. The external force, , is related to temperature external force by the relation below. A typical test for one's model is the Rayleigh-Bénard convection for .Macroscopic variables such as density
, velocity , and temperature can be calculated as the moments of the density distribution function: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.
). 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:
The collision operator is often approximated by a BGK collision operator under the condition it also satisfies the conservation laws.
In the collision operator,
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.Letting
yields the final result.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.)
with elements .The relaxation parameter,
, is related to the kinematic viscosity,, by the following relationship.The moments
Moment (mathematics)
In mathematics, a moment is, loosely speaking, a quantitative measure of the shape of a set of points. The "second moment", for example, is widely used and measures the "width" of a set of points in one dimension or in higher dimensions measures the shape of a cloud of points as it could be fit by...
of the
give the local conserved quantities. The density is given byand the weighted average velocity,
, and the local momentum are given byIn the above equation for the equilibrium velocity
, the 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, and 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
as given by several authors. is the effective mass and is Green's function representing the interparticle interaction with as the neighboring site. Satisfying and where represents repulsive forces. For D2Q9 and D3Q19, this leads toThe effective mass as proposed by Shan and Chen uses the following effective mass for a
, . The equation of stateEquation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
is also given under the condition of a single component and multiphase.
So far, it appears that
and are free constants to tune but once plugged into the system's equation of stateEquation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
(EOS), they must satisfy the thermodynamic relationships at the critical point such that
and . For the EOS, is 3.0 for D2Q9 and D3Q19 while it equals 10.0 for D3Q15.It was later shown by Yuan and Schaefer 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:
Open Source / free software
- LIMBES: Open source (GPL) code in 2D based on Gauss-Hermite quadrature, parallel (openmp), fortran 90
- OpenLB: Open source (GPLv2) library based on LBM, parallel, C++
- Palabos: Open source (AGPL) parallel C++ code with a large spectrum of lattice Boltzmann models, including thermal, multi-phase, free-surface, and embedded particles.
- Sailfish: Open Source LBM code (LGPL) for Graphics Processing Units (CUDA/OpenCL)
- El'Beem: free CFD code (GPL) which uses LBM
- J-Lattice-Boltzmann: interactive Java applet for experimenting with LBM
- C examples: Some simple example LBM code in C (GPL).
- LBM-C: Open Source (GPLv2) CUDA code
Freeware
- LBSim: software written in C++, proprietary, freely available
Commercial
- PowerFLOW: commercial CFD code which uses LBM, created and distributed by Exa Corp.Exa, Inc.Exa Inc. is a privately-held developer and distributor of Computer-aided engineering software, founded in 1992 by Dr. Kim Molvig. The main product is PowerFLOW, a lattice-boltzmann derived implementation of computational fluid dynamics , which can very accurately simulate internal and external...
- FlowKit: Professional support for the open-source software Palabos.
External links
- palabos.org: A site with various resources related to LBM, including a forum.
- LBM Method
- Lattice Boltzmann summary
- Erlangen Lattice Boltzmann mailing list
- DSFD mailing list: sends information about DSFD and related conferences, opportunities for doctoral, postdoctoral, faculty and industry positions for applicants who have experience with Lattice Boltzmann or other related methods.
- 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