Pre-Analysis & Start-Up

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In the Pre-Analysis step& Start-Up step, we'll review the following:

• Mathematical model: (e.g.: We'll look at the governing equations + boundary conditions and the assumptions contained within the mathematical model.)
• Numerical solution procedure in ANSYS: (e.g.: We'll briefly overview the solution strategy used by ANSYS and contrast it to the hand calculation approach.)
• Hand-calculations of expected results: (e.g.: We'll use an analytical solution of the mathematical model to predict the expected stress field from ANSYS. We'll pay close attention to additional assumptions that have to be made in order to obtain an analytical solution.)

Mathematical Model 

{latex}
1+1=2

{latex}

Numerical Solution Procedure in ANSYS

Hand-Calculations of Expected Results

Start-Up

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• 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$$\Delta x = L_x / n_x$$}

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

{\large$$p$$}

{\large$$\rho_f$$}

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

{\large$$\phi$$}

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

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

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

{\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$$\rho_p$$}

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

where

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

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

{\large $$m_p$$}

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

where

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

{\large $$\rho_p$$}

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$$\tau_p \neq 0$$}

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

{\large$$St \gg 1$$}

{\large$$St \approx 1$$}

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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