Control of Chaotic Flows and Fluid Forces (Part 4)

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Control of Chaotic Flows and Fluid Forces (Previous Part 3)

    \[\epsilon\frac{\partial^{(0)}}{\partial t}+\epsilon^{2}\frac{\partial^{(1)}}{\partial t}=\frac{\partial}{\partial t}, \qquad \text{and} \qquad \epsilon\frac{\partial}{\partial \alpha}=\frac{\partial}{\partial \alpha}\]

Using these results in  above equation we get the mass conservation

(1)   \begin{equation*}\frac{\partial \rho}{\partial t}+\frac{\partial (\rho u_{\alpha})}{\partial \alpha}=0\end{equation*}

Now by combining Eq. (??) and (??), we get

    \[\epsilon\left(\frac{\partial^{(0)}(\rho u_{\alpha})}{\partial t}+\frac{\partial (\Pi^{eq}_{\alpha \beta})}{\partial \beta} \right)+\epsilon^{2}\left(\frac{\partial^{(1)}}{\partial t}\left(\rho u_{\alpha} \right)+\frac{\partial}{\partial \beta}\left(1-\frac{\mathit{\Delta}t}{2\tau} \right)\Pi^{1}_{\alpha \beta}\right)=0\]

(2)   \begin{equation*}\left( \epsilon\frac{\partial^{(0)}}{\partial t}+\epsilon^{2}\frac{\partial^{(1)}}{\partial t} \right)\rho u_{\alpha}+\epsilon\frac{\partial (\Pi^{eq}_{\alpha \beta})}{\partial \beta}+\epsilon^{2}\frac{\partial}{\partial \beta}\left(1-\frac{\mathit{\Delta}t}{2\tau} \right)\Pi^{1}_{\alpha \beta}=0 \end{equation*}

As

    \[\epsilon\frac{\partial^{(0)}}{\partial t}+\epsilon^{2}\frac{\partial^{(1)}}{\partial t}=\frac{\partial}{\partial t}, \quad \epsilon\frac{\partial}{\partial \beta}=\frac{\partial}{\partial \beta}, \quad \Pi^{eq}_{\alpha \beta}=\frac{\rho}{3}\delta_{\alpha \beta}+\rho u_{\alpha}u_{\beta}\]

So Eq. (2) becomes

(3)   \begin{equation*}\frac{\partial (\rho u_{\alpha})}{\partial t}+\frac{\partial}{\partial \beta}\left( \frac{\rho}{3}\delta_{\alpha \beta}+\rho u_{\alpha}u_{\beta} \right)= -\epsilon^{2}\frac{\partial}{\partial \beta}\left(1-\frac{\mathit{\Delta}t}{2\tau} \right)\Pi^{1}_{\alpha \beta} \end{equation*}

For D2Q9, we have the gas dynamical relation

    \[p=\rho \vec \zeta^ {2}_{s} \qquad and \qquad \vec \zeta^{2}_{s}=\frac{1}{3}\qquad \Rightarrow p=\frac{\rho}{3}\]

Now by replacing \frac{\rho}{3} by \rho in above equation. We get

(4)   \begin{equation*}\frac{\partial (\rho u_{\alpha})}{\partial t}+\frac{\partial}{\partial \beta}\left(p\delta_{\alpha \beta}+\rho u_{\alpha}u_{\beta} \right)= -\epsilon^{2}\frac{\partial}{\partial \beta}\left(1-\frac{\mathit{\Delta}t}{2\tau} \right)\Pi^{1}_{\alpha \beta}\end{equation*}

Let

(5)   \begin{equation*}-\sigma=\left(1-\frac{\mathit{\Delta}t}{2\tau} \right)\Pi^{1}_{\alpha \beta}\end{equation*}

(6)   \begin{equation*}\frac{\partial (\rho u_{\alpha})}{\partial t}+\frac{\partial}{\partial \beta}\left(p\delta_{\alpha \beta}+\rho u_{\alpha}u_{\beta} \right)= -\epsilon^{2}\frac{\partial}{\partial \beta}(-\sigma) \end{equation*}

Read Control of Chaotic Flows and Fluid Part 1

We use the relation for the tensor

    \[\Pi^{1}_{\alpha \beta}=-\frac{\rho}{3}\tau\left( \frac{\partial}{\partial \beta}u_{\alpha}+\frac{\partial}{\partial \alpha}u_{\beta}\right)+\tau \frac{\partial}{\partial \gamma}(\rho u_{\alpha}u_{\beta}u_{\gamma})\]

So from Eq. (5) we have

    \[-\sigma=-\left(1-\frac{\mathit{\Delta}t}{2\tau} \right)\frac{\rho}{3}\tau\left( \frac{\partial}{\partial \beta}u_{\alpha}+\frac{\partial}{\partial \alpha}u_{\beta}\right)+\left(1-\frac{\mathit{\Delta}t}{2\tau} \right)\tau \frac{\partial}{\partial \gamma}(\rho u_{\alpha}u_{\beta}u_{\gamma})\]

Consider

 

(7)   \begin{equation*} \begin{split}-\epsilon^{2}\frac{\partial}{\partial \beta}(-\sigma)=&\epsilon\frac{\partial}{\partial \beta}\left( \left(1-\frac{\mathit{\Delta}t}{2\tau} \right)\frac{\rho}{3}\tau\left(\epsilon \frac{\partial}{\partial \beta}u_{\alpha}+\epsilon\frac{\partial}{\partial \alpha}u_{\beta}\right)\right)\\ &- \frac{\partial}{\partial \beta}\epsilon^{2} \left(1-\frac{\mathit{\Delta}t}{2\tau} \right)\tau \frac{\partial}{\partial \gamma}(\rho u_{\alpha}u_{\beta}u_{\gamma})\end{split} \end{equation*}

Let

    \[K=\left(1-\frac{\mathit{\Delta}t}{2\tau} \right)\frac{\rho}{3}\tau, \quad E=- \frac{\partial}{\partial \beta}\epsilon^{2} \left(1-\frac{\mathit{\Delta}t}{2\tau} \right)\tau \frac{\partial}{\partial \gamma}(\rho u_{\alpha}u_{\beta}u_{\gamma})\]

    \[\epsilon \frac{\partial}{\partial \alpha}= \frac{\partial}{\partial \alpha}, \qquad \epsilon \frac{\partial}{\partial \beta}=\frac{\partial}{\partial \beta}\]

So Eq. (7) becomes

(8)   \begin{equation*}-\epsilon^{2}\frac{\partial}{\partial \beta}(-\sigma)=\frac{\partial}{\partial \beta}\left(K \left( \frac{\partial}{\partial \beta}u_{\alpha}+\frac{\partial}{\partial \alpha}u_{\beta}\right) \right) + E \end{equation*}

Combining Eq. (6) and (8), we get

    \[\frac{\partial (\rho u_{\alpha})}{\partial t}+\frac{\partial}{\partial \beta}\left(p\delta_{\alpha \beta}+\rho u_{\alpha}u_{\beta} \right)=\frac{\partial}{\partial \beta}\left(K \left( \frac{\partial}{\partial \beta}u_{\alpha}+\frac{\partial}{\partial \alpha}u_{\beta}\right) \right) + E\]

Here \textit{E} represent the error term, we neglect this to get quadratic  Navier Stokes equation.

(9)   \begin{equation*}\frac{\partial (\rho u_{\alpha})}{\partial t}+\frac{\partial}{\partial \beta}\left(p\delta_{\alpha \beta}+\rho u_{\alpha}u_{\beta} \right)=\frac{\partial}{\partial \beta}\left(K \left( \frac{\partial}{\partial \beta}u_{\alpha}+\frac{\partial}{\partial \alpha}u_{\beta}\right) \right)\end{equation*}

Consider the Navier stokes equation

    \[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial (\rho u_{i}u_{j})}{\partial x_{j}}=\frac{\partial p}{\partial x_{j}}+\frac{\partial }{\partial x_{j}}\left(\eta(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}) \right)\]

Comparing above equation with Eq. (9), we get

    \[\eta=K=\left(1-\frac{\mathit{\Delta}t}{2\tau} \right)\frac{\rho}{3}\tau\]

Replacing K with \eta in Eq. (9) we get

    \[\frac{\partial (\rho u_{\alpha})}{\partial t}+\frac{\partial}{\partial \beta}\left(p\delta_{\alpha \beta}+\rho u_{\alpha}u_{\beta} \right)=\frac{\partial}{\partial \beta}\left(\eta \left( \frac{\partial}{\partial \beta}u_{\alpha}+\frac{\partial}{\partial \alpha}u_{\beta}\right) \right)\]

Above equation can also be represented as

    \[\frac{\partial (\rho u_{\alpha})}{\partial t}+\frac{\partial}{\partial \beta}\left(\rho u_{\alpha}u_{\beta} \right)=-\frac{\partial p}{\partial \beta}+\frac{\partial}{\partial \beta}\left(\eta \left( \frac{\partial}{\partial \beta}u_{\alpha}+\frac{\partial}{\partial \alpha}u_{\beta}\right) \right)\]

\bullet We concluded that in macroscopic limit LBE Recovers Navier-Stokes equations

    \[\frac{\partial \rho}{\partial t}+\frac{\partial \rho u_{j}}{\partial x_{j}}=0\]

    \[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial (\rho u_{i}u_{j})}{\partial x_{j}}=\frac{\partial p}{\partial x_{j}}+\frac{\partial }{\partial x_{j}}\left(\eta(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}) \right)\]

Boundary conditions in LBM

PDE solutions are unable to be determined uniquely without using proper initial and BCs. LBM does not allow for the application of BCs directly. It’s necessary to convert the BCs into streaming-in distribution functions. In LBM BCs Kruger et al., there are two types of groups: lattice links and lattice nodes, referred to as link-wise and wet node, respectively.

Convective boundary conditions

The convective BC’s are the linear combination of the function values, and its derivative at the computational domain boundary is a mixed form of BCs from some problems this boundary is also known as the impedance boundary.

The following is a mathematical representation of convective BCs.

(10)   \begin{equation*}u_\infty \partial_x\psi+\partial_t\psi=0.\end{equation*}

Where u_\infty is the uniform and \psi = \rho u, \rho v are in-flow velocity, such type of BC is used in the computational domain outlet, and the velocity is chosen in such a way that one can maintain general conservation.

Lift coefficient

The lift coefficient is the proportion of lift produced by a lifting body to the fluid density around it.

(11)   \begin{equation*}C_{L}= \frac{2F_{L}}{\rho U_{\infty}^{2} D},\end{equation*}

where, F_{L} is the lift force of body.

Drag coefficient

The drag coefficient is an important tool for determining an object’s aerodynamic efficiency, regardless of its size or shape. The flow is streamline when the drag coefficient is low.

(12)   \begin{equation*}C_{D}= \frac{2F_{D}}{\rho U_{\infty}^{2} D},\end{equation*}

where, F_{D} is the drag force of body and \rho is the density of fluid.

Control of Chaotic Flows and Fluid Part 1

Previous part Control of Chaotic Flows and Fluid Forces Part 2

You can also read Control of Chaotic Flows and Part 3

Also, read Control of Chaotic Flows Part 5

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