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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

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{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

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{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

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{latex} {\large  \begin{align*}  \frac{\partial \rho_f}{\partial t} + \nabla \cdot (\rho_f \mathbf{u}_f)=0 \end{align*} }  {latex}

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where

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{latex}{\large$$\mathbf{u}$$}{latex}

is the fluid velocity,

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{latex}{\large$$p$$}{latex}

the pressure,

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{latex}{\large$$\rho_f$$}{latex}

the fluid density and

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{latex}{\large$$\mathbf{f}$$}{latex}

is a momentum exchange term due to the presence of particles. When the particle volume fraction

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{latex}{\large$$\phi$$}{latex}

and the particle mass loading

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{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:

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{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

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{latex}{\large$$\phi \ll 1$$}{latex}

and

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{latex}{\large$$M \ll 1$$}{latex}

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The suspended particles are considered as rigid spheres of same diameter d, and density

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{latex}{\large$$\rho_p$$}{latex}

. Newton’s second law written for the particle i stipulates:

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{latex}{\large $$m_p \frac{d \mathbf{u}_p^i}{dt}=\mathbf{f}_{ex}^i$$}{latex}

where

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{latex}{\large$$\mathbf{u}_p^i$$}{latex}

is the velocity of particle i,

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{latex}{\large $$\mathbf{f}_{ex}^i$$}{latex}

the forces exerted on it, and

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{latex}{\large $$m_p$$}{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 ref3:

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{latex}{\large $$\frac{d \mathbf{u}_p^i}{dt}=\frac{\mathbf{u}_f-\mathbf{u}_p^i}{\tau_p}$$}{latex}

where

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{latex}{\large $$\tau_p=\rho_p D^2/(18\mu)$$}{latex}

is known as the particle response time,

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{latex}{\large $$\rho_p$$}{latex}

the particle density and D the particle diameter. 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).

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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.

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{latex}{\large$$St = \frac{\tau_p}{\tau_f}$$}{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

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{latex}{\large$$\gamma = 0.1751 s^{-1} = \frac{1}{\tau_f}$$}{latex}

. 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.

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