Pre-Analysis & Start-Up

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

• Mathematical Model: We will look at the governing equations, boundary conditions, initial field velocity function of the jet as well as the formula for calculating Stokes number in this case.
• Expected Results: We will discuss the expected results from this simulation and compare the numerical results with the expected results.

Mathematical Model 

Governing Equations:

In almost any fluid dynamics problem, the most important governing equation has to be the Navier-Stokes equation and continuity equation. The two equations govern the fluid flow. Here we will list the Navier-Stokes equations but we will not go into further details.

Continuity Equation:

{latex}
\begin{align*}
\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \textbf{v})=0 \\
\Rightarrow \nabla \cdot \textbf{v} = 0 \\
\end{align*}
{latex}

Navier-Stokes Equation:

• 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 

{latex}
\begin{eqnarray*} 
\rho_f (\frac{d \textbfmathbf{vu}_f}{dt}+\textbfmathbf{vu}_f \cdot \nabla \textbfmathbf{vu}_f)=- \nabla p + \mu \nabla ^2 \textbfmathbf{vu}_f + \textbfmathbf{f} 
\end{eqnarray*} 
{latex}

In this case, however, we have discrete particles in the flow. Since we are using a one-way coupled scheme, the fluid imposes a force on the particles, but not vice versa. By using the particle force balance equation listed below, the particle movement can be calculated by integrating the acceleration term.

Particle Force Balance:

-Continuity Equation

{\large 

{latex}
\begin{align*} 
&\frac{d\partial u\rho_pf}{dt\partial t} =+ F_D(\overrightarrow{u}-\overrightarrow{u_p})+\frac{\overrightarrow{g}\nabla \cdot (\rho_p-\rho)}{\rho_p}+\overrightarrow{F} \\
\end{align*}
{latex}

{latex}$\overrightarrow{F}${latex}

{latex}$F_D(\overrightarrow{u}-\overrightarrow{u_p})${latex}

{latex}$F_D${latex}

{latex}
\begin{align*}
F_D = \frac{18\mu}{\rho_d d_p ^2} \frac{C_D Re}{24}f \mathbf{u}_f)=0
\end{align*}
{latex} 

Here, 

{latex}$\overrightarrow{u}${latex}

{latex}$\overrightarrow{u_p}${latex}

{latex}$\mu${latex}

{latex}$\rho${latex}

{latex}$\rho_p${latex}

{latex}$d_p${latex}

{latex}
\begin{eqnarray*}
Re \equiv \frac{\rho d_p |\overrightarrow{u_p}-\overrightarrow{u}|}{\mu}
\end{eqnarray*}
{latex}

Initial Field Velocity Function

The initial field setup of a jet is essentially two shear layers. The two shear layers are symmetrical to the centerline. A hyperbolic tangent velocity function is used to initialize velocity field of a shear layer. The hyperbolic tangent velocity functions are preferred over discontinuous piecewise functions because hyperbolic tangent functions are more physical. The initial field velocity function for the entire jet is given below:

{latex}
\[
U_{initial}(y) =
\left\{
  \begin{array}{lr}
    tanh(32y-8) & : 0 \leq x \leq 0.5\\
    -tanh(32y-24) & : 0.5n < n \leq 1
  \end{array}
\right.
\]
{latex}

{latex}
\begin{align*}
&V_{initial} = 0\\
&\text{Tip:}\\
&tanh(x)=\frac{e^x-e^{-x}}{e^x+e^{-x}}\\
\end{align*}
{latex}

The function is plotted below (x velocity versus y) using simple MATLAB codes:

Image Removed

Stokes Number

The Stokes number is a dimensionless number that corresponds to the behavior of particles suspended in a fluid flow. The Stokes number is defined as below:

{latex}
\begin{eqnarray*}
Stk = \frac{\tau U_0}{d_c}  (1)
\end{eqnarray*}
{latex}

{latex}$\tau${latex}

{latex}$U_0${latex}

{latex}$d_c${latex}

Thus: 

{latex}$d_c = d_p = d (2)${latex}

In the case of Stokes flow, where the Reynolds number is sufficiently small the relaxation time can be expressed using the equation below:

{latex}
\begin{eqnarray*}
\tau = \frac{\rho_d d_d ^2}{18 \mu_g}(3) \\
\end{eqnarray*}
{latex}

Plug (2) and (3) into (1), we get the formula for calculating the Stokes Number in our case below:

{latex}
\begin{eqnarray*}
\boxed{Stk = \frac{\rho_d \cdot d U_0}{18\mu_g}}\\
\end{eqnarray*}
{latex}

Expected Results

...

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.

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