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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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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-Continuity Equation
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where
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is the fluid velocity,
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the pressure,
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the fluid density and
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is a momentum exchange term due to the presence of particles. When the particle volume fraction
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and the particle mass loading
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are very small, it is legitimate to neglect the effects of the particles on the fluid:
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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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and
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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
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. Newton’s second law written for the particle i stipulates:
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where
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is the velocity of particle i,
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the forces exerted on it, and
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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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where
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is known as the particle response time,
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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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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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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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. 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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, 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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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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