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  • Particles in a Periodic Double Shear Flow - Pre-Analysis & Start-Up

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Author(s): Chiyu Jiang, Mohamed Houssem Kasbaoui, Dr. Donald L. Koch, Cornell University

Problem Specification
1. Pre-Analysis & Start-Up
2. Geometry
3. Mesh
4. Physics Setup
5. Numerical Solution
6. Numerical Results
7. Verification & Validation
Exercises
Comments

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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{\large 
\begin

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\rho_f (\frac{d \mathbf

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_f}

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+\mathbf

_f \cdot \nabla \mathbf

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_f + \mathbf

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\end

 
}

-Continuity Equation

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{\large 
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\frac

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+ \nabla \cdot (\rho \mathbf

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_f)=0
\end

} 

where

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{\large$\mathbf

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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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{\large$\mathbf

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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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{\large$\mathbf

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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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{\large $m_p \frac{d \mathbf

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_p^i}

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=\mathbf

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_

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^i$}

where

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{\large$\mathbf

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_p^i$}

is the velocity of particle i,

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{\large $\mathbf

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_

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^i$}

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:

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{\large $\frac{d \mathbf

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_p^i}

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=\frac{\mathbf

_f-\mathbf

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_p^i}

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$}

where

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is known as the particle response time. 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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{\large$St = \frac

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

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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.

Go to Step 2: Geometry

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