Lattice Boltzmann methods

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Lattice Boltzmann methods or LBM are CFD methods for fluid simulation. Instead of solving the Navier-Stokes equations, the discrete Boltzmann equation is solved to simulate the flow of Newtonian fluid with collision models such as Bhatnagar-Gross-Krook (BGK). LBM tracks limited number of particles and the viscous flow behavior emerges automatically from the intrinsic particle stream and collisions process.

LBM is a relatively new simulation technique for complex fluid systems and has attracted great interests of researchers in computational physics. Unlike the traditional Computational fluid dynamics (CFD), which solves 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 algorithm parallelization.

Historically, LBM originated from the Lattice Gas Automata (LGA), which can be considered as a simplified, fictitious molecular dynamic in which space, time, and particle velocities are all discrete. Each lattice node is connected to its neighbors by, for example of the hexagonal FHP model, 6 lattice velocities. There can be either 1 or 0 particle at a lattice node moving in a lattice direction. After a time interval, each particle will move to the neighboring node in its direction, and this process is called the propagation. When there are more than one particles arriving at the same node from different directions, they collide and change their directions according to a set of collision rules. Suitable collision rules should conserve the particle number (mass), momentum, and energy before and after the collision. However, it was also found that LGA is suffering from several native defects: lack of Galilean invariance, statistical noise, exponential complexity for three-dimensional lattices, etc.

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 with this replacement, the discrete collision rules also have to be modified to a continuous function – 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. Through a Chapman-Enskog analysis, 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.

Simulating multiphase/multicomponent flows have 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 us a relative 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 the 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.

Despite the increasing popularities of LBM in simulating complex fluid systems, one should also be aware of some limitations of this novel approach. 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 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. 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 (defined by the ratio of geometric length scale to the mean free path between molecules) flows.

[edit] Mathematical Details

The Boltzmann equation is an evolution equation for a single particle probability distribution function $f (x, v, t)$ :

$\partial_t f + v\partial_x f + F\partial_v f=\Omega$

where $F$ is an external force and $Ω$ is a collision integral. The lattice Boltzmann method discretizes this equation by limiting space to a lattice and the velocity space to a discrete set of velocities $v i$ . The discretized Boltmann equation, which is the Lattice Boltzmann equation then reads:

f i (x + v i, t + 1) - f (x, t) + F i = Ω

The collision operator is often approximated by a BGK collision operator:

$\Omega = \frac{1}{\tau} (f_i^0-f_i)$

where $f_i^0$ is the local equilibrium distribution.

The moments of the $f i$ give the local conserved quantities. The density is given by

ρ =	∑	f_i
	i

and the local momentum is given by

ρu =	∑	f_iv_i.
	i

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

ρθ + ρuu =	∑	f_iv_iv_i.
	i

The collision operator has to respect the conservation laws. Therefore the equilibirum distribution $f_i^0$ must have the same conserved moments as the $f i$ .