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:
\begin
\frac
+ \nabla \cdot (\rho \textbf
\Rightarrow \nabla \cdot \textbf
= 0
\end
Navier-Stokes Equation:
\begin
\rho (\frac{d \textbf{v}}
+\textbf
)=- \nabla p + \mu \nabla ^2 \textbf
+ \textbf
\end
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:
\begin
&\frac
= F_D(\overrightarrow
-\overrightarrow
)+\frac{\overrightarrow
(\rho_p-\rho)}
+\overrightarrow
\end
$\overrightarrow
$
$F_D(\overrightarrow
-\overrightarrow
)$
\begin
F_D = \frac
\frac
\end
Here,
$\overrightarrow
$
$\overrightarrow
$
\begin
Re \equiv \frac{\rho d_p |\overrightarrow
-\overrightarrow
|}
\end
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:
[
U_
=
\left{
\begin
tanh(32y-8) & : 0 \leq x \leq 0.5
-tanh(32y-24) & : 0.5n < n \leq 1
\end
\right.
]
\begin
&V_
= 0
&\text
&tanh=\frac{e^x-e^{-x}}{e^x+e^{-x}}
\end
The function is plotted below (x velocity versus y) using simple MATLAB codes:
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:
\begin
Stk = \frac
(1)
\end
Thus:
In the case of Stokes flow, where the Reynolds number is sufficiently small the relaxation time can be expressed using the equation below:
\begin
\tau = \frac
(3)
\end
Plug (2) and (3) into (1), we get the formula for calculating the Stokes Number in our case below:
\begin
\boxed{Stk = \frac
{18\mu_g}}
\end
Here,
Expected Results
Under Construction