Pre-Analysis & Start-Up

In the Pre-Analysis & Start-Up step, we'll review the following:

• Mathematical Model: We will look at the governing equations, boundary conditions, initial field velocity function of the jet as well as the formula for calculating Stokes number in this case.
• Expected Results: We will discuss the expected results from this simulation and compare the numerical results with the expected results.

Mathematical Model 

Governing Equations:

In almost any fluid dynamics problem, the most important governing equation has to be the Navier-Stokes equation and continuity equation. The two equations govern the fluid flow. Here we will list the Navier-Stokes equations but we will not go into further details.

Continuity Equation:

{latex}
\begin{align*}
\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \textbf{v})=0 \\
\Rightarrow \nabla \cdot \textbf{v} = 0 \\
\end{align*}
{latex}

Navier-Stokes Equation:

{latex}
\begin{eqnarray*}
\rho (\frac{d \textbf{v}}{dt}+\textbf{v} \cdot \nabla \textbf{v})=- \nabla p + \mu \nabla ^2 \textbf{v} + \textbf{f}
\end{eqnarray*}
{latex}

In this case, however, we have discrete particles in the flow. Since we are using a one-way coupled scheme, the fluid imposes a force on the particles, but not vice versa. By using the particle force balance equation listed below, the particle movement can be calculated by integrating the acceleration term.

Particle Force Balance:

{latex}
\begin{align*}
&\frac{d u_p}{dt} = F_D(\overrightarrow{u}-\overrightarrow{u_p})+\frac{\overrightarrow{g}(\rho_p-\rho)}{\rho_p}+\overrightarrow{F} \\
\end{align*}
{latex}

{latex}$\overrightarrow{F}${latex}

{latex}$F_D(\overrightarrow{u}-\overrightarrow{u_p})${latex}

{latex}$F_D${latex}

{latex}
\begin{align*}
F_D = \frac{18\mu}{\rho_d d_p ^2} \frac{C_D Re}{24}
\end{align*}
{latex}

Here, 

{latex}$\overrightarrow{u}${latex}

{latex}$\overrightarrow{u_p}${latex}

{latex}$\mu${latex}

{latex}$\rho${latex}

{latex}$\rho_p${latex}

{latex}$d_p${latex}

{latex}
\begin{eqnarray*}
Re \equiv \frac{\rho d_p |\overrightarrow{u_p}-\overrightarrow{u}|}{\mu}
\end{eqnarray*}
{latex}

Initial Field Velocity Function

The initial field setup of a jet is essentially two shear layers. The two shear layers are symmetrical to the centerline. A hyperbolic tangent velocity function is used to initialize velocity field of a shear layer. The hyperbolic tangent velocity functions are preferred over discontinuous piecewise functions because hyperbolic tangent functions are more physical. The initial field velocity function for the entire jet is given below:

{latex}
\[
U_{initial}(y) =
\left\{
  \begin{array}{lr}
    tanh(32y-8) & : 0 \leq x \leq 0.5\\
    -tanh(32y-24) & : 0.5n < n \leq 1
  \end{array}
\right.
\]
{latex}

{latex}
\begin{align*}
&V_{initial} = 0\\
&\text{Tip:}\\
&tanh(x)=\frac{e^x-e^{-x}}{e^x+e^{-x}}\\
\end{align*}
{latex}

The function is plotted below (x velocity versus y) using simple MATLAB codes:

Stokes Number

The Stokes number is a dimensionless number that corresponds to the behavior of particles suspended in a fluid flow. The Stokes number is defined as below:

{latex}
\begin{eqnarray*}
Stk = \frac{\tau U_0}{d_c}  (1)
\end{eqnarray*}
{latex}

{latex}$\tau${latex}

{latex}$U_0${latex}

{latex}$d_c${latex}

Thus: 

{latex}$d_c = d_p = d (2)${latex}

In the case of Stokes flow, where the Reynolds number is sufficiently small the relaxation time can be expressed using the equation below:

{latex}
\begin{eqnarray*}
\tau = \frac{\rho_d d_d ^2}{18 \mu_g}(3) \\
\end{eqnarray*}
{latex}

Plug (2) and (3) into (1), we get the formula for calculating the Stokes Number in our case below:

{latex}
\begin{eqnarray*}
\boxed{Stk = \frac{\rho_d \cdot d U_0}{18\mu_g}}\\
\end{eqnarray*}
{latex}

Here, 

{latex}$\rho_d${latex}

{latex}$\mu_g${latex}

Expected Results

Go to Step 2: Geometry

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