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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} |
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} |
One way-coupling is legitimate here. See ANSYS documentation (16.2) for further details about the momentum exchange term.
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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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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
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{latex}{\large$$\tau_p \neq 0$$}{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.
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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.
When
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{latex}{\large$$St \gg 1$$}{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
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{latex}{\large$$St \approx 1$$}{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.
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