Thesis for degree: Licentiate of Engineering

(2.6)
where f a is the PDF, e a the velocity and a the collision term at any spatial location x and time
t along the direction a. The time is increased by the time step Δt. The macroscopic fluid
density is [3]:
(2.7)
The macroscopic velocity u is evaluated by the microscopic velocities e a and the PDF f a and
divided by the macroscopic fluid density ρ as [3]:
(2.8)
This allows the LBM to recover the continuum macroscopic parameters from the discrete
microscopic ones, in this case by the velocities. The distribution function presented in
equation (2.6), called the single relaxation time BGK (Bhatnagar-Gross-Krook) LBM, is one
of the simplest models [3, 14]. The BGK is described by using one relaxation time for the
collision term. The collision term consists of the present PDF and the relaxation toward the
local equilibrium. The collision term Ω a and the D2Q9 equilibrium distribution function f a
eq
are defined as [3]:
(2.9)
(2.10)
where w a is 4/9 for the particle a = 0, 1/9 for a = 1, 2, 3, 4 and 1/36 for a = 5, 6, 7, 8, and τ is
the relaxation number [14-16]. In the simplest implementation the basic speed on the lattice c,
which is also called the lattice speed of sound, is 1 lu/ts [16].
When the mass diffusion is modeled in LBM, two approaches are often used; pure diffusion
or advection-diffusion (also called convection-diffusion). Both pure diffusion and advectiondiffusion
is simulated by another equilibrium distribution f ζ,a eq which is very much alike the
normal fluid distributions function but with a simpler equilibrium equation. For the first case
with pure diffusion only the equilibrium function is defined as [3]:
(2.11)
In the second case, advection-diffusion, which is applied here, the equilibrium function will
include a second term to handle the convective velocity. The equilibrium function is defined
as [3]:
(2.12)
The mixing due to density variations and buoyant effects in porous media can here be handled
as advective and diffusive components rather than an input parameter (such as porosity). For a
porous media, the collision term is considered as a second intermediate step after the
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