Thesis for degree: Licentiate of Engineering

(2.24)
where v is the viscosity. For two flows with equivalent geometry and same Re, these will
obey Reynolds transport theorem and provide equivalent solutions which can be converted
from one flow to the other. Finally, the dimensionless system can now be transformed into the
discrete system by defining a reference length step δ x and time step δ t . If the reference
variables, i.e., δ xD and δ tD , in the dimensionless system are defined, t 0D and x 0D will turn out to
have the value of unity, respectively. Then the reference variables in the discrete system are
defined as:
(2.25)
(2.26)
where N is the number of cells and N iter the number of iterations [25]. Now, the conversion is
easily handled through dimensionless analysis and the variables are defined as.
(2.27)
(2.28)
Both δ x and δ t are attached with some constraints in LBM. The value of u LB may not be larger
than the basic lattice speed of sound c even if the fluid is possibly compressible because LBM
does not support supersonic flows [25]. This leads to the relationship of the reference
variables as follows:
(2.29)
The LBM is a quasi-compressible solver which means it enters a slightly compressible regime
when solving the pressure equation without any significant impact on the numerical accuracy.
To specify how to choose δ t , it is important to note that the LB model is of second-order
accuracy. This means that the lattice error ε(δ x ) ~ δ x 2 . The compressibility error would take
over if the lattice error is reduced and therefore the errors should be kept at the same order,
i.e., ε(Ma) ~ ε(δ x ). This leads to a more specific relationship between the reference variables,
namely δ t ~ δ x 2 [25].
2.3.3 Mass diffusion in LBM
The next step here is to show the connection between variables for the mass diffusion. It
should be mentioned that other physical processes can also be handled by LBM. For example
thermal flow would be handled by setting the temperature as T ≡ ρ in equation (2.7).
Mass diffusion in LBM can be divided into two cases, namely; pure diffusion (u = 0) or
advection-diffusion (u ≠ 0) [26]. For LBM, this means that the equilibrium distribution
function will differ by containing the velocity or not in the equation (see equations (2.11) and
(2.12)). The LBM parameters in an equivalent system of lattice units should be such that
diffusion fluxes are in the same ratio as the actual ones, only larger in magnitude [15-16].
Mass diffusion in a mixture can be described by Fick’s law of diffusion. As Fick’s law is only
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