Lattice Boltzmann Methods - Mathematical Equations For Simulations

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 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 single-component, multiphase system. 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: