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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}{\large$$\Delta x = L_x / n_x$$}{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}
{\large 
\begin{eqnarray*} 
\rho_f (\frac{d \mathbf{u}_f}{dt}+\mathbf{u}_f \cdot \nabla \mathbf{u}_f)=- \nabla p + \mu \nabla ^2 \mathbf{u}_f + \mathbf{f} 
\end{eqnarray*} 
}
{latex}
-Continuity Equation
{latex}
{\large 
\begin{align*} 
\frac{\partial \rho_f}{\partial t} + \nabla \cdot (\rho_f \mathbf{u}_f)=0
\end{align*}
} 
{latex}

where {latex}{\large$$\mathbf{u}$$}{latex} is the fluid velocity, {latex}{\large$$p$$}{latex} the pressure, {latex}{\large$$\rho_f$$}{latex} the fluid density and {latex}{\large$$\mathbf{f}$$}{latex} is a momentum exchange term due to the presence of particles. When the particle volume fraction {latex}{\large$$\phi$$}{latex} and the particle mass loading {latex}{\large$$M=\phi \rho_p/\rho_f$$}{latex} are very small, it is legitimate to neglect the effects of the particles on the fluid: {latex}{\large$$\mathbf{f}$$}{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 satisfies both conditions {latex}{\large$$\phi \ll 1$$}{latex} and {latex}{\large$$M \ll 1$$}{latex}