Page 227 - gas transport in porous media

Chapter 13: Lattice Boltzmann Method
and Chen, 1996):
223
eq ρ 3 2
f = 1 − u (13.4)
0
3 2

eq 3 2 2
e
f i = q i ρ 1 + 3 i · u + ((3 i · u) − u ) (13.5)
e
2
where q i = 1/18 for the six vectors along the Cartesian axes, and q i = 1/36 for the
remaining vectors. This distribution, when used in Eq. (13.1), will give back a very
good approximation of incompressible Navier-Stokes behavior for u.
13.2.2 Dispersion
To model dispersion of dilute tracers, the simplest approach is to use one lattice
for a “carrier fluid” that obeys the Navier-Stokes equation, and calculate u for that
carrier fluid only. A separate lattice exists for each tracer, and changes in the tracer
distributions are propagated according to Eqs. (13.1), (13.2), (13.4) and (13.5), using
the u determined from the carrier.
Unlike the Navier-Stokes equations, the advection-dispersion equation (which gov-
erns tracer movement) is linear in u. Flekkøy et al. (1995) and Noble (1996) reasoned
that the tracer equilibrium distributions should be linear in u as well. Thus for a tracer
s, the equilibrium distribution is of the form:

eq
e
f s,i = A + B · ( s, i · u) (13.6)
6 eq
and the A and B are fixed by the requirement that solute is conserved (ρ s = f ),
s,i
and the requirement that the solute flux at equilibrium is due entirely to advection
6 eq
( f e s, i = ρ s · u). Compared to Eq. (13.5), Eq. (13.6) requires fewer floating
s,i
point operations to evaluate the equilibrium distribution, hence is faster to compute.
The real bottleneck in the LB calculation, is the time taken to read all the vectors
f i from memory (for both carrier and tracers), and to write them back after they have
been updated. Thus a great deal of memory traffic can be avoided by using lattices
with fewer vectors for the tracers. In 2D, only four Cartesian vectors are needed,
and in 3D, the six Cartesian vectors will suffice. Wolf-Gladrow (1994) suggested
low-vector lattices for modeling diffusion, and Noble (1996) performed a Chapman-
Enskog expansion for Cartesian lattices, recovering the advection-dispersion equation
and estimating the error. The present author compared the 6-vector and 19-vector
methods for tracer problems in extreme conditions and found the agreement was
extremely close. (It must be emphasized that low-vector lattices, and Eq. (13.6), are
used only for the dilute tracers, not the carrier.)
In many dispersion problems, the flow field u is at steady state, and need not be
recalculated at each timestep. Furthermore, periodic boundary conditions on u are
often appropriate, even when tracer dispersion is not periodic, as in the SC dispersion
problem described in Section 13.3.2. For such problems, one can use a 19-vector
3
carrier fluid in a single repeat unit, N x × N y × N z lu , then save the equilibrium