Control of Chaotic Flows and Fluid Forces (Part 2)

Control of Chaotic Flows and Fluid Forces (Previous Part 1)

LBM is a simple concept. After the collision step, the next step is streaming. When these two steps are completed, one step is completed at a time, and these steps are repeated to achieve the desired accuracy.

Equilibrium Distribution Function

The process of solving transport equations is similar. The primary factor that varies from problem to problem, we can use LBM to solve a variety of real-world problems. The Maxwellian EDF is written as

(1) $\begin{equation*} f^{eq}=\frac{\rho}{(2\pi kT)^{\frac{3}{2}}}e^{-\frac{(\vec\xi-\mu)^2}{2kT}}.\end{equation*}$

By ignoring the 3rd order and higher terms and applying Taylor expansion to the above equation, we get

(2) $\begin{equation*} f^{eq}=\frac{\rho}{(2\pi kT)^\frac{3}{2}} e^{-\frac{\vec\xi.\vec\xi}{2kT}} \left(1-\frac{\vec \mu. \vec \mu - 2\vec\xi.\vec \mu}{2kT} -\frac{(\vec\xi.\vec\mu)^{2}}{2k^{2}T^{2}}\right).\end{equation*}$

Using substitutions $\omega = \frac{1}{(2\pi kT)^\frac{3}{2}} e^{-\frac{\vec\xi.\vec\xi}{2kT}}$ and $kT=\vec\xi_s^{2}$ .
$\omega$ is the weighting factors and $\xi_s$ is the speed of sound.

(3) $\begin{equation*} f^{eq}=\rho\omega \left(1+\frac{2\vec\xi.\vec\mu-\vec \mu. \vec \mu}{2\vec\xi_s^{2}} + \frac{(\vec\xi.\vec\mu)^{2}}{2\vec\xi_s^{4}}\right).\end{equation*}$

The following discretized equilibrium distribution function.

(4) $\begin{equation*} f_i^{eq}=\rho\omega_i \left(1+\frac{2\vec\xi.\vec\mu-\vec \mu. \vec \mu}{2\vec\xi_s^{2}} + \frac{(\vec\xi.\vec\mu)^{2}}{2\vec\xi_s^{4}}\right) + O(\mu^{2}).\end{equation*}$

Discrete velocity sets

Discrete velocity set is represented as $D^{*}_{i}Q^{*}_{m}$ , where $i$ is the dimensions, $m$ is the discrete velocities. Commonly used velocity sets are $D^{*}_{1}Q^{*}_{3}$ , $D^{*}_{2}Q^{*}_{9}$ and $D^{*}_{3}Q^{*}_{15}$ .

1-Dimensional

There are $3$ velocity vectors. The distribution function of central partical has $0$ velocity.

2-Dimensional

There are $9$ velocity vectors. Again the distribution function of central partical has $0$ velocity.

$D_2^*Q_9^*$ velocity discrete set ${\vec\zeta_i}$ Perumal and Dass.

Let $p_i = (i-1)\frac{\pi}{4}$

(5) $\begin{eqnarray*} \vec\zeta_i= \left\{ \begin{array}{ll} 0, & i=0;\\ \vec\zeta(\cos(p_i),\sin(p_i)), & i=1,2,3,4;\\ \sqrt{2}\vec\zeta(\cos(p_i),\sin(p_i)), & i=5,6,7,8 \end{array} \right. \end{eqnarray*}$

3-Dimensional

There are $15$ velocity vectors. Again the distribution function of central partical has $0$ velocity.
$D_2^*Q_{15}^*$ velocity discrete set ${\vec\zeta_i}$ Mei $et$ $al$ .,

(6) $\begin{eqnarray*} \vec\zeta_i=\left\{ \begin{array}{ll} (0,0,0), &i=0;\\ \vec\zeta(\mp1,0,0),\vec\zeta(0,\mp1,0),\vec\zeta(0,0,\mp1),&i=1,2,...,6;\\ \vec\zeta(\mp1,\mp1,\mp1),&i=7,8,...,14 \end{array} \right.\end{eqnarray*}$

The Chapman-Enskog Expansion

Chapman and Enskog, both mathematical physicists, revealed the same methods for obtaining macroscopic equations using the Boltzmann equation in 1917. Chapman eventually discovered a new technique called Chapman-Enskog expansion by combining the two ways in his book.
The equilibrium distribution function for $D^{}2Q^{}_9$ , the discrete velocities and weight functions.

(7) $\begin{equation*}\Pi^{eq}=\sum_{i}f_{i}=\rho\end{equation*}$

(8) $\begin{equation*}\Pi^{eq}_{\alpha}=\sum_{i}f_{i}\vec \zeta_{i\alpha}=\rho u_{\alpha}\end{equation*}$

(9) $\begin{equation*}\Pi^{eq}_{\alpha\beta}=\sum_{i}f_{i}\vec \zeta_{i\alpha} \vec \zeta_{i\beta}=\dfrac{\rho}{3}\delta_{\alpha\beta} +\rho u_{\alpha}u_{\beta}\end{equation*}$

(10) $\begin{equation*}\Pi^{eq}_{\alpha\beta\gamma}=\sum_{i}f_{i}\vec \zeta_{i\alpha} \vec \zeta_{i\beta} \vec \zeta_{i\gamma}=\dfrac{\rho}{3}(u_{\alpha}\delta_{\beta\gamma} + u_{\beta}\delta_{\alpha\gamma}+u_{\gamma}\delta_{\alpha\beta})\end{equation*}$