As previously mentioned in Chapter 1, the particle dynamics can be modeled by the Boltzmann equation, which represents the time evolution of the particle distribution function f (BOLTZMANN, 1896 apud KREMER, 2010): for the change in the particle trajectory due to collisions with other groups of particles1. Which assumes a rarefied gas, negligible external force during collision, assumption of molecular chaos and a small variation of the distribution function over time (BOLTZMANN, 1896 apud KREMER, 2010). If enough time is given, a point in the space will reach an equilibrium state. This equilibrium is achieved when the number of particles in a particular state that enters and leaves the space is equal (Ω = 0). Applying this condition in the Eq. 4.1 is possible to determine the particle distribution function as (PHILIPPI et al., 2006): where d is the number of Euclidean dimensions, k is the Boltzmann constant, T is the temperature, ξ is the particle velocity, u is the particle mean velocity, and m is the particle’s mass. The Eq. 4.2 is denominated the Maxwell-Boltzmann distribution function and represent the probability equilibrium distribution function of particles to be in a given velocity state. The lattice Boltzmann method is based on the discrete form of the Boltzmann equation, Eq. 4.1. Bhatnagar et al. (1954) proposed a simplification of the collision operator. In their model, currently called the BGK collision operator,Ωis defined as:
Ω = feq−f
τ (4.3)
whereτ is a relaxation time parameter, andfeq is the particle equilibrium distribution function. There are other collision operators, for example, the two relaxation time (TRT)(GINZBURG et al., 2008) and the multiple relaxation time (MRT)(HUMIERES, 1994). Both models increase the number of relaxation parameters when compared with BGK in order to increase the stability of the model, but they also might raise the simulations computational cost.
1 f′ andf1′ represent post collision values of two particle which have the respectively pre-collision velocities:ξ andξ1.
Chapter 4. Numerical Method 39
4.1.1 Discretization of Boltzmann equation
The discretization of the Boltzmann equation, Eq. 4.1, requires a transformation of the phase space into a discrete domain. Because the LBM originates from the lattice gas automata (LGA), the phase space is divided into lattices that accommodate the particle populations between each time step. These particles can only travel to other lattices by determined paths defined by velocity sets.
The velocity space can be discretized by using Hermite polynomials (HE and LUO, 1997), (SHAN et al., 2006), (PHILIPPI et al., 2006). The discretization then returns a group of lattice directions on which the particles can move, denominated velocity sets. The velocity sets are commonly classified based on the number of spatial dimensions and discrete velocities. For two dimensional space and isothermal case, the most common velocity set that preserves the mass and momentum is the D2Q9 (2 dimensions and 9 directions). While for three-dimensional space and isothermal, it is the D3Q19 (3 dimensions and 19 directions), shown in Fig. 4.1.a , and the D3Q27( 3 dimensions and 27 directions), Fig. 4.1.n.
Figure 4.1 – a) D2Q9 and b) D3Q19 velocity sets.
0 1
Another important part related to the velocity space is the discretization of the equilibrium distribu-tion funcdistribu-tion, 4.2, which requires that the first moments in the velocity space of the discrete funcdistribu-tion being equal to the moments in the continuous velocity space. By using Hermite polynomials expansions (PHILIPPI et al., 2006): where cs is the speed of sound,uis the fluid velocity, ci is the velocity vector and wi is the direction weight. For the velocity sets previously mentioned (D2Q9, D3Q19, D3Q27), the speed of sound is equal to1/√
3, while the other constants (ciandwi), are exposed in Table 4.1.
The Eq. 4.1 in the discrete DdQq velocity space can be described as (HE et al., 1998):
∂tfi+ciα∂αfi = Ωi+Fi, i= 0, . . . , q−1 (4.5)
Table 4.1 – Weight values for D2Q9 and D3Q19 velocity schemes.
using the BGK collision operator and with a discretization over time:
fi(x+ci∆t, t+ ∆t)−fi(x, t) = with a second order discrete integration scheme is:
fi(x+ci∆t, t+ ∆t)−fi(x, t) =−∆t
adopting the variable change proposed by He et al. (1998):
f¯i =fi+∆t
2τ (fi−fieq)−∆t
2 Fi (4.8)
considering∆t= 1the Eq. 4.7 becomes:
f¯i(x+ci∆t, t+ ∆t) = (1−ω∆t) ¯fi(x, t) +ω∆tfieq(x, t) + whereω is the relaxation frequency defined as the inverse of the relaxation time. Equation 4.9 can be divided into two operations denominated "Streaming" and "Collision". The collision part is represented as: where the particle collisions occur, and the populations on the lattices are calculated, represented by Fig. 4.2.a. In the second part of the equation denominated streaming, the populations propagate to neighboring lattices, the right side of Fig. 4.2.b. Both of these operations happen at the same time step.
f¯i(x+ci∆t, t+ ∆t) =fi∗(x, t) (4.11) Knowing the particle populations is not enough to solve engineering problems. It is also necessary to know the macroscopic properties: pressure and velocity. These macroscopics can be obtained by calculating the lower order moments in the velocity space (SILVA and SEMIAO, 2012):
ρ= Π0 =X
Chapter 4. Numerical Method 41 Figure 4.2 – Collision–streaming process. On the collision (a) the particles collide and change direction. While on
the streaming step (b) the particles propagate to neighbor lattices.
a) Collision b) Streaming
Source: Own elaboration.
It is also necessary to define the force term presented in Eq. 4.9 and Eq. 4.12. The force can be obtained the same way as the equilibrium distribution function, through Hermite polynomials expansions (SHAN et al., 2006). For a second-order discretization, the force term can be approximate as (GUO et al., 2002):
A critical aspect is that the lattice Boltzmann method can be used to model fluid flow. If the BGK col-lision operator is utilized, the Boltzmann equation returns the Navier-Stokes equations if the relaxation time is correlated with the fluid viscosity as (GUO et al., 2002):
η =ρc2s
One of the methods to increase the accuracy and stability of the lattice Boltzmann method is the pre-collision regularization proposed by Latt and Chopard (2006). On this approach, instead of propagating the particle distribution function, it is used the regularized populations. The regularized distribution function with a force term is defined as:
fireg =fieq+fi(1) (4.15)
wherefi(1)is the second term of expansionfi =fi(0)+ǫfi(1)+ǫ2fi(2)+. . . from the Chapman-Enskog expansion (BUICK and GREATED, 2000). The term fi(1) can be correlated with the force term of the Boltzmann equation as (LUGARINI et al., 2020):
ǫfi(1) ≈ wi
wherefineq is the non-equilibrium distribution function defined as:
fineq =fi−fieq (4.17)
using the definition of regularized population, Eq. 4.15, into Eq. 4.10, is obtained the regularized lattice Boltzmann equation with force term: Another advantage of this method is the local calculation ofΠneqαβ , which correlates to the viscous stress tensor (SILVA and SEMIAO, 2012), which will be necessary to model a non-Newtonian fluid behavior.