Interdisciplinary Applied Mathematics

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three dimensions    in    which    the    lattice is a    24-bit    or    25-bit    projection    of    a

four-dimensional FCHC lattice onto three-dimensional space.

A more recent version developed by Chen and collaborators (Chen and Doolen, 1998) employs a square lattice in two dimensions with three speeds and nine velocities. Specifically, they have eight nonzero velocities for moving along the edges of the square and one zero velocity for the rest particle as follows:

(±1, 0), (0, ±1), (±1, ±1), (0,0).

The LB equations can be thought of as discrete analogues of the continuous Boltzmann equation we presented in Section 15.4 but in an incomplete velocity space (phase space); more rigorous work has in fact proved this analogy (Abe, 1997). Let us denote by fj(x,t) the distribution function at x, t with velocity cj, and assume that the collision operator can be described by the BGK approximation we described earlier (Section 15.4). Then, the

LBM-BGK equation of motion is given by

fi(x + CiAt,t + At) — fi(x,t) = -(fi -//q),    (15.49)


where fie4, (i = 0,1,…, 8) is the equilibrium distribution function and t is the relaxation time. An equilibrium distribution that approximates the Maxwellian-Boltzmann equilibrium distribution up to second-order was derived in (Qian et al., 1992), and is given by

f eq fi


1 +

CiaPa (Ciapi{3

where % = 0,1,…, 8, while a and /3 are the two Cartesian directions. All fluid velocities are normalized by a/3RT, and thus the speed of sound is cs = 1/a/3; also, го* are weights. The density and velocity are obtained from formulas similar to the Boltzmann equation, where the integral is replaced by summations, i.e.,

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