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 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}

is an additional acceleration (force per unit particle mass) term.

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

is the drag force per unit particle mass.

{latex}$F_D${latex}

can be calculated using the formula below:

{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}

is the fluid phase velocity,

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

is the particle velocity,

{latex}$\mu${latex}

is the molecular viscosity of the fluid,

{latex}$\rho${latex}

is the fluid density,

{latex}$\rho_p${latex}

is the density of the particle, and

{latex}$d_p${latex}

is the particle diameter. Re is the relative Reynolds, number, which is defined as

{latex}$Re \equiv \frac{\rho d_p |\overrightarrow{u_p}-\overrightarrow{u}|}{\mu}${latex}

Expected Results

Under Construction

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