The Transitional Flow Interface uses the lattice Boltzmann method to solve the Boltzmann BGK equation, which is a simplified form of the more general Boltzmann equation. The BGK equation (in the absence of volume forces) is given below:
where ξ is the particle velocity,
x is the position,
t is the time,
f(ξ ,x ,t) is the distribution function (which represents the total number of particles in the phase space
dξdx),
feq is the local equilibrium distribution,
n is the number density,
u is the bulk velocity, and
τ is the relaxation time. This equation is similar to the Boltzmann equation, except the complex collision term has been replaced by a much simpler term that implies a constant relaxation time for all velocities to return to equilibrium. The equilibrium function in
N dimensions is given by
The equation set is still nonlinear as feq(ξ ,u ,ρ) is a nonlinear function of
f(ξ ,x ,t) (see definitions of
n and
u). As in the Boltzmann equation, the macroscopic thermodynamic variables are defined as moment integrals of the distribution function. For example, the number density and fluid velocity are given by
where there are now i distribution functions,
fi, associated with each of the discrete velocity directions,
ξi, and each having an associated equilibrium function,
. The number density and velocity become:
The velocity lattice and equilibrium functions used are critical in determining the accuracy of the final solution. The approach used to generate the equilibrium functions and velocity lattice used by COMSOL Multiphysics is described in detail in
Ref. 1. The distribution function
f(ξ ,x ,t) is expanded in the Hermite polynomials. The velocity space is approximated by a Gauss–Hermite quadrature. This approach is closely related to Grad’s moment scheme for solving the Boltzmann equation (
Ref. 2). The Hermite polynomials are chosen because the lower-order velocity moments of the distribution functions depend only on the lower-order coefficients of the Hermite polynomials, so that truncating the expansion has a minimal effect on the moments themselves.
where neq is the equilibrium number density (usually the local number density),
ueq is the equilibrium velocity (usually the local velocity), and
H(n)(ξ) is the
nth N-dimensional Hermite polynomial, given by
here wi is the weight associated with the velocity quadrature and
cs is the characteristic speed, which in the BGK model is given by
functions are derived by expanding the drifting Maxwellian distribution in the Hermite polynomials to a given order. Using Equation 3-1 and the orthogonality and completeness of the Hermite basis this equation results:
