Pre-Analysis & Start-Up

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In the simulations considered for this tutorial, the fluid flow is a 2D perturbed periodic double shear layer as described in the first section. The geometry is Lx = 59.15m, Ly = 59.15m, and the mesh size is chosen as {latex}$\Delta x = L_x / n_x${latex} 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:
-Momentum Equations
{latex}
\begin{eqnarray*} 
\rho_f (\frac{d \textbf{u}_f}{dt}+\textbf{u}_f \cdot \nabla \textbf{u}_f)=- \nabla p + \mu \nabla ^2 \textbf{u}_f + \textbf{f} 
\end{eqnarray*} 
{latex}
-Continuity Equation
{latex} 
\begin{align*} 
\frac{\partial \rho_f}{\partial t} + \nabla \cdot (\rho \textbf{u}_f)=0
\end{align*} 
{latex}

where {latex}$\textbf{u}${latex} is the fluid velocity, {latex}$p${latex} the pressure, {latex}$\rho_f${latex} the fluid density and {latex}$\textbf{f}${latex} is a momentum exchange term due to the presence of particles. When the particle volume fraction and the particle mass loading {latex}$M=\phi \rho_p/\rho_f${latex} are very small, it is legitimate to neglect the effects of the particles on the fluid: {latex}$\textbf{f}${latex} can be set to zero. This type of coupling is called one-way. In these simulations the fluid phase is air, while the dispersed phase is constituted of about 400 glass beads of diameter a few dozens of micron. This satises both conditions {latex}$\phi \ll 1${latex} and {latex}$M \ll 1${latex}

The suspended particles are considered as rigid spheres of same diameter d, and density {latex}$\rho_p${latex}. Newton's second law written for the particle i stipulates:
{latex}$m_p \frac{d \textbf{u}_p^i}{dt}=\textbf{f}_{ex}^i${latex}
where {latex}$\textbf{u}_p^i${latex} is the velocity of particle i, {latex}$\textbf{f}_{ex}^i${latex} the forces exerted on it, and {latex}$m_p${latex} its mass.
In order to know accurately the hydrodynamic forces exerted on a particle one needs to resolve the flow to a scale significantly smaller than the particle diameter. This is computationally prohibitive. Instead, the hydrodynamic forces can be approximated roughly to be proportional to the drift velocity:
{latex}$\frac{d \textbf{u}_p^i}{dt}=\frac{\textbf{v}-\textbf{u}_p^i}{\tau_p}${latex}
where {latex}$\tau_p${latex} is known as the particle response time. This equation needs to be solved for all particles present in the domain. This is done in Fluent via the module: Discrete Phase Model(DPM).

The particle response time measures the speed at which the particle velocity adapts to the local flow speed. Non-inertial particles, or tracers, have a zero particle response time: they follow the fluid streamlines. Inertial particles with {latex}$\tau_p \neq 0${latex} might adapt quickly or slowly to the fluid speed variations depending on the relative variation of the flow and the particle response time.


This rate of adaptation is measured by a non-dimensional number called Stokes number representing the ratio of the particle response time to the flow characteristic time scale.
{latex}$\St = \frac{\tau_p}{\tau_f}${latex}
 
In these simulations, the characteristic flow time is the inverse of the growth rate of the vortices in the shear layers. This is also predicted by the Orr-Sommerfeld equation. For the particular geometry and configuration we used in this tutorial, the growth rate is {latex}$\gamma = 0.1751 s^{-1} = \frac{1}{\tau_f}. When St = 0 the particles are tracers. They follow the streamlines and, in particular, they will not be able to leave a vortex once caught inside.

When {latex}$St \gg 1${latex}, particles have a ballistic motion and are not affected by the local flow conditions. They are able to shoot through the vorticies without a strong trajectory deviation. Intermediate cases {latex}$St \approx 1${latex} have a maximum coupling between the two phases: particles are attracted to the vorticies, but once they reach the highly swirling vortex cores they are ejected due to their non zero inertia.

In this tutorial, we will consider a nearly tracer case St = 0.2, an intermediate case St = 1 and a nearly ballistic case St = 5. 

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