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  • Mathematical Model: We will look at the governing equations, boundary conditions, initial field 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:

Wiki Markup
{latex}
1+1=2

{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:

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{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:

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{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}

Wiki Markup
{latex}$\overrightarrow{F}${latex}
 is an additional acceleration (force per unit particle mass) term. 
Wiki Markup
{latex}$F_D(\overrightarrow{u}-\overrightarrow{u_p})${latex}
 is the drag force per unit particle mass.

 

Wiki Markup
{latex}$F_D${latex}
 can be calculated using the formula below:

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

Here, 

Wiki Markup
{latex}$\overrightarrow{u}${latex}
 is the fluid phase velocity, 
Wiki Markup
{latex}$\overrightarrow{u_p}${latex}
 is the particle velocity, 
Wiki Markup
{latex}$\mu${latex}
 is the molecular viscosity of the fluid, 
Wiki Markup
{latex}$\rho${latex}
 is the fluid density, 
Wiki Markup
{latex}$\rho_p${latex}
 is the density of the particle, and 
Wiki Markup
{latex}$d_p${latex}
 is the particle diameter. Re is the relative Reynolds, number, which is defined as 

Wiki Markup
{latex}$Re \equiv \frac{\rho d_p |\overrightarrow{u_p}-\overrightarrow{u}|}{\mu}${latex}
Note
Under Construction

Expected Results

Note

Under Construction

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