The Boltzmann function describes the transport of a particle system described by the velocity distribution function:

equation

Where the term C describes the interaction (collisions) between them.

If the parameters are calculated by averaging over the speed using

equation=9075

the mass density estimation is obtained by:

equation

If the parameters are calculated by averaging over the speed using

equation=9075

the velocity of the flow is calculated by integrating the velocity distribution function on all velocities by weighing the velocities:

equation

If the parameters are calculated by averaging over the speed using

equation=9075

and the equipartition theorem is considered, the temperature can be estimated by integrating the kinetic energy weighted by the velocity distribution divided by the gas constant:

equation

If the parameters are calculated by averaging over the speed using

equation=9075

the flow tensor is calculated by integrating the velocity distribution function on all velocities by weighing the velocity differences:

equation

The problem with macro scale systems that are based on microscopic phenomena is that

- macroscopic models are too simple to correctly reflect the dynamics

- microscopic models to describe the macroscopic reality can not be solved analytically and numerical solutions are cumbersome (= require a lot of computational resources)

The Boltzmann lattice method looks for an intermediate path. It is based on the Boltzmann transport equation, rescues the microscopic part via the collision term and implements a simplified structure to calculate the macroscopic results. We speak of a mesoscopic approach where we can, as required, reduce microscopic effort by losing accuracy but saving resources or improving accuracy by investing more resources.

El modelo D2Q9 es un modelo bidimensional (D2) en que se se conecta el nodo (punto central) en nodos a lo largo de los ejes cartesianos\\n\\nen el origen\\n\\n

$\vec{e}_0=(0,0)$

\\n\\nen las esquinas\\n\\n

$\vec{e}_1=(1,0)$

(E),\\n

$\vec{e}_2=(0,1)$

(N), \\n

$\vec{e}_3=(-1,0)$

(W) y \\n

$\vec{e}_4=(0,-1)$

(S)\\n\\ny en las diagonales\\n\\n

$\vec{e}_5=(1,1)$

(NE), \\n

$\vec{e}_6=(-1,1)$

(SE), \\n

$\vec{e}_7=(-1,-1)$

(SW) y \\n

$\vec{e}_8=(1,-1)$

(NW)

lo que se representa en la siguiente gr fica:

image

The D3Q15 model is a two-dimensional model (D3) in which the node (central point) is connected in nodes along Cartesian axes\\n\\n

$(1,0,0), (-1,0,0), (0,1,0), (0,-1,0), (0,0,1) (0,0,-1)$

\\n\\nand in the corners of the bucket\\n\\n

$(1,0,1), (-1,0,1), (0,1,1) , (0,-1,1), (1,0,-1), (-1,0,-1), (0,1,-1) , (0,-1,-1)$

which is represented in the following graph:

image

It is easy to build models of type D3Q19 (including side edge halves) or D3Q27 (all possible points).

In the case of the discretization in the LBM models we work not with functions of the speed if not with discrete components. In this way the i component is defined by:

equation

where w_i is the relative weight.

In the streaming process the particles are moved according to their velocity directions to neighboring cells

equation

where \vec{x} is the position, t time, \vec{e} _i the direction of the grid and c the speed.

If the collision does not occur at the point of the network, but at a distance \Delta:

image\\n\\nthen the function must consider the offset by weighting the contributions\\n\\n

$f_i(x_f,t+\delta t)=\displaystyle\frac{(1-\Delta)f_{-i}(x_f,t)+\Delta(f_{-i}(x_b,t)+f_{-i}(x_{f2},t)}{1+\Delta}$

If the wall shows an inclination with respect to the network it must be modeled in a more complex form:

image

More general edge

First, an approximate boundary must be defined to allow the necessary edge equations to be established. Then they must be applied in the process of steraming.

In the case of a D2Q9 system we have the 9 values f_i that we have named as O, N, E, S, W, NE, SE, SW, NW. If the number of particles in position (n,m) is denoted as f_i(j,k) we have that the equations are

```

N[x,y] = N[x,y-1]

NW[x,y] = NW[x+1,y-1]

E[x,y] = E[x-1,y]

NE[x,y] = NE[x-1,y-1]

S[x,y] = S[x,y+1]

SE[x,y] = SE[x-1,y+1]

W[x,y] = W[x+1,y]

SW[x,y] = SW[x+1,y+1]

```

In case D2Q9 the term collision is calculated by summing the different factors

equation=9084

so you have for each cell

```

O = O+w(4rho/9)(1-3u2/2) - O)

E = E+w(rho/9)(1 + u_x/3+5u_x^2-3u2/2)-E)

W = W+w(rho/9)(1 - u_x/3+5u_x^2-3u2/2)-W)

N = N+w(rho/9)(1 + u_y/3+5u_y^2-3u2/2)-N)

S = S+w(rho/9)(1 - u_y/3+5u_y^2-3u2/2)-S)

NE = NE+w(rho/36)(1+u_x/3+u_y/3+5(u2+2u_xu_y)/2-3u2/2) - NE)

SE = SE+w(rho/36)(1+u_x/3-u_y/3+5(u2-2u_xu_y)/2-3u2/2) - SE)

NW = NW+w(rho/36)(1-u_x/3+u_y/3+5(u2-2u_xu_y)/2-3u2/2) - NW)

SW = SW+w(rho/36)(1-u_x/3-u_y/3+5(u2+2u_xu_y)/2-3u2/2) - SW)

```

with

```

u2 = u_x^2+u_y^2

```

In the relaxation approximation, it is assumed that the distribution f_i(\vec{x},t) tends to relax at a time \tau to an equilibrium distribution f_i^{eq}(\vec{x},t) according to equation\\n\\n

$\displaystyle\frac{df_i}{dt}=-\displaystyle\frac{f_i-f_i^{eq}}{\tau}$

which has in the discrete approximation the equation

equation

where the term of the differences in the distribution functions represents the collisions.

The equilibrium distribution can be approximated by a distribution of Maxwell Boltzmann

equation

Where m is the mass of the particle, T the system temperature and k the Boltzmann constant.

En la aproximaci n Bhatnagar-Gross-Krook la distribuci n en equilibrio se asume como la de un gas de part culas sin interacci n

equation=9082

con \vec{u} la velocidad del flujo, k la constante de Boltzmann, T la temperatura y m la masa de la particula. Si se desarrolla esta expresi n en el limite de velocidades \vec{u} comparada con la velocidad de las moleculas c\hat{e}_i se tiene que

equation

con \omega_i los pesos dados por

que se determinan asegurando que la distribuci n equilibrio cumpla las leyes de conservaci n.

In the case of particles of a liquid, the method LBM allows to develop simulators as shown in the example:

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