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:

• Theory for Fluid Phase
• Theory for Particle Phase
• Choosing the Cases
  

Pre-Analysis:

A particle laden ow is a multiphase ow where one phase is the uid and the other is dispersed particles. Governing equations for both phases are implemented in Fluent. To run a meaningful simulation, a review of the theory is necessary.

Fluid Phase:

In the simulations considered for this tutorial, the uid ow is a 2D perturbed periodic double shear layer as described in the rst section. The geometry is Lx = XXm, Ly = XXm, and the mesh size is chosen as x = Lx=nx in order to resolve the smallest vorticies. As a rule of thumb. One typically needs about 20 grid points across the shear layers, where the vorticies are going to develop. The boundary conditions are periodic in the x and y  directions. The fluid phase satisfies the Navier Stokes Equations:
- Type here: momentum equations
- Type here: continuity equation
where uf is the uid velocity, p the pressure, rhof the uid density and f is a momentum exchange term due to the presence of particles. When the particle volume fraction and the particle mass loading M = p=f are very small, it is legitimate to neglect the eects of the particles on the uid: f can be set to zero. This type of coupling is called one-way. In these simulations the uid phase is air, while the dispersed phase is constitued of about 400 glass beads of diameter a few dozens of micron. This satises both conditions 1 and M 1. 1
One way-coupling is legitimate here. See ANSYS documentation (link) for further details about the momentum exchange termIn 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}

...

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

{latex}$\tau${latex}

 is the relaxation time of the particle, 

{latex}$U_0${latex}

 is the fluid velocity well away from the particle, and 

{latex}$d_c${latex}

 is the characteristic length scale of the obstacle, in this case it is the diameter of the particle.

...

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

Here, 

{latex}$\rho_d${latex}

 is the density of the particle, 

{latex}$\mu_g${latex}

 is the dynamic viscosity of the fluid, in this case the dynamics viscosity of the gas.

...