Lattice Boltzmann methods

Lattice Boltzmann methods (LBM) 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 other CFD methods the complex boundaries there can be hard to work with. Lattice Boltzmann methods (LBM) 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 other CFD methods the complex boundaries there can be hard to work with. 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 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. In an algorithm, there are collision and streaming steps. These evolve the density of the fluid ρ ( x → , t ) {displaystyle ho ({vec {x}},t)} , for x → {displaystyle {vec {x}}} the position and t {displaystyle t} the time. As the fluid is on a lattice the density has a number of components f i , i = 0 , … , a {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 e → 4 = ( 0 , − 1 ) {displaystyle {vec {e}}_{4}=(0,-1)} , i.e., it points due south and so has no x {displaystyle x} component but a y {displaystyle y} component of − 1 {displaystyle -1} . So one of the nine components of the total density at the central lattice point, f 4 ( x → , t ) {displaystyle f_{4}({vec {x}},t)} , is that part of the fluid at point x → {displaystyle {vec {x}}} moving due south, at a speed in lattice units of one. Then the steps that evolve the fluid in time are: Collision step: f i ( x → , t + δ t ) = f i ( x → , t ) + f i e q ( x → , t ) − f i ( x → , t ) τ f {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)}{ au _{f}}},!} which is the Bhatnagar Gross and Krook (BGK) model for relaxation to equilibrium via collisions between the molecules of a fluid. f i e q ( x → , t ) {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 τ f {displaystyle au _{f}} . This timescale determines the kinematic viscosity, the larger it is, the larger is the kinematic viscosity. Streaming step: f i ( x → + e → i , t + 1 ) = f i ( x → , t + 1 ) {displaystyle f_{i}({vec {x}}+{vec {e}}_{i},t+1)=f_{i}({vec {x}},t+1),!} As f i ( x → , t + 1 ) {displaystyle f_{i}({vec {x}},t+1)} is, by definition, the fluid density at point x → {displaystyle {vec {x}}} at time t {displaystyle t} , that is moving at a velocity of e → i {displaystyle {vec {e}}_{i}} per time step, then at the next time step t + 1 {displaystyle t+1} it will have flowed to point x → + e → i {displaystyle {vec {x}}+{vec {e}}_{i}} . 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.