## Lattice Boltzmann Method

#### Numerical modeling of fluid flow

Lattice Boltzmann (LB) method is a numerical method for simulating viscous fluid flow.  The LB method approximates the continuous Boltzmann equation by discretizing physical space with lattice nodes and velocity space by a set of microscopic velocity vectors. In the LB method, the physical space is discretized into a set of uniformly spaced nodes (lattice) that represents the voids and the solids (Figure 1), and a discrete set of microscopic velocities is defined for propagation of fluid molecules (Figure 2). The expression D3Q19 in Figure 2 represents the three-dimensional 19 velocity lattice. The time- and space-averaged microscopic movements of particles are modeled using molecular populations called distribution functions, which define the density and velocity at each lattice node. Specific particle interaction rules are set so that the fluid flow Navier-Stokes equations are satisfied.

 Figure 1.  Binary image of       aggregates (black=aggregate,       white=pore) and lattice nodes      at the center of each pore pixel. Figure 2. D3Q19 lattice microscopic velocities

The time dependent movement of fluid particles at each lattice node satisfies the following particle propagation equation:

(1)

where Fi, ei and are the particle distribution function, the microscopic velocity and the collision function at lattice node x, at time t, respectively. The subscript i represent the lattice directions around the node as shown in Figure 1b, and  is the body force and given as   where  is the applied pressure gradient and is the weight factor for ith direction (Martys and Hagedorn 2002). Weight factors (wi) for D3Q19 LB method are: w9 =4/9 for rest particle, wi=1/9 () for particles streaming to the face-connected neighbors and wi=1/36 () for particles streaming to the edge-connected neighbors.  The collision function represents the collision of fluid molecules at each node and has the following form (Bhatnagar et al. 1954):

(2)

where  is the equilibrium distribution function and  is the relaxation time which is related to the viscosity of the fluid (, where is the kinematic viscosity). Equilibrium distribution functions for different models were derived by He and Luo (1997). The function is given in the following form for D3Q19 model:

(3)

where r and u are the density and the macroscopic velocity of the node, wi is the weight factor for ith direction and cs is the lattice speed of sound (=1/ for D3Q19 lattice).

The macroscopic properties, density (r) and velocity (u), of the nodes are calculated using the following relations:

(4)

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