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| Particles in a Periodic Double Shear Flow - Panel |
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| Particles in a Periodic Double Shear Flow - Panel |
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Pre-Analysis & Start-Up
In the Pre-Analysis & Start-Up step, we'll review the following:
- Theory for Fluid Phase
- Theory for Particle Phase
- Choosing the Cases
Pre-Analysis:
A particle laden flow is a multiphase flow where one phase is the fluid 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 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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{\large$$\Delta x = L_x / n_x$$} |
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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{\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*}
}
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-Continuity Equation
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{\large
\begin{align*}
\frac{\partial \rho_f}{\partial t} + \nabla \cdot (\rho_f \mathbf{u}_f)=0
\end{align*}
}
|
where
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{\large$$\mathbf{u}$$} |
is the fluid velocity,
the pressure,
the fluid density and
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{\large$$\mathbf{f}$$} |
is a momentum exchange term due to the presence of particles. When the particle volume fraction
and the particle mass loading
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{\large$$M=\phi \rho_p/\rho_f$$} |
are very small, it is legitimate to neglect the effects of the particles on the fluid:
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{\large$$\mathbf{f}$$} |
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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{\large$$\phi \ll 1$$} |
and
One way-coupling is legitimate here. See ANSYS documentation (16.2) for further details about the momentum exchange term.
Particle Phase:
The suspended particles are considered as rigid spheres of same diameter d, and density
. Newton’s second law written for the particle i stipulates:
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{\large $$m_p \frac{d \mathbf{u}_p^i}{dt}=\mathbf{f}_{ex}^i$$} |
where
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{\large$$\mathbf{u}_p^i$$} |
is the velocity of particle i,
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{\large $$\mathbf{f}_{ex}^i$$} |
the forces exerted on it, and
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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{\large $$\frac{d \mathbf{u}_p^i}{dt}=\frac{\mathbf{u}_f-\mathbf{u}_p^i}{\tau_p}$$} |
where
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{\large $$\tau_p=\rho_p D^2/(18\mu)$$} |
is known as the particle response time,
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).
Choosing the Cases:
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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{\large$$\tau_p \neq 0$$} |
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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{\large$$St = \frac{\tau_p}{\tau_f}$$} |
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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{\large$$\gamma = 0.1751 s^{-1} = \frac{1}{\tau_f}$$} |
. 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
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{\large$$St \gg 1$$} |
, 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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{\large$$St \approx 1$$} |
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.
Go to Step 2: Geometry
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