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

{\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

{\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*} 
}

-Continuity Equation

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

where

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

is the fluid velocity,

{\large$$p$$}

the pressure,

{\large$$\rho_f$$}

the fluid density and

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

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

{\large$$\phi$$}

and the particle mass loading

{\large$$M=\phi \rho_p/\rho_f$$}

are very small, it is legitimate to neglect the effects of the particles on the fluid:

{\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

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

and

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

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

{\large$$\rho_p$$}

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

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

where

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

is the velocity of particle i,

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

the forces exerted on it, and

{\large $$m_p$$}

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:

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

where

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

is known as the particle response time,

{\large $$\rho_p$$}

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

{\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.

{\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

{\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

{\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

{\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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