Lattice Boltzmann methods
Instead of solving the Navier–Stokes equations directly, a fluid density on a lattice is simulated with streaming and collision (relaxation) processes.
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.
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.
Then the steps that evolve the fluid in time are:[1] 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.
It has been proposed to apply Galilean Transformation to overcome the limitation of modelling high-speed fluid flows.
[4] The fast advancements of this method had also successfully simulated microfluidics,[5] However, as of now, LBM is still limited in simulating high Knudsen number flows where Monte Carlo methods are instead used, and high-Mach number flows in aerodynamics are still difficult for LBM, and a consistent thermo-hydrodynamic scheme is absent.
[6] 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.
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.
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.
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.
[8] 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.
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.
However, in LBM, the particulate kinetics provides a relatively easy and consistent way to incorporate the underlying microscopic interactions by modifying the collision operator.
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.
[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.
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.
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.
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.)
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
[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.
[23] During the last years, the LBM has proven to be a powerful tool for solving problems at different length and time scales.
