Page History

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• 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.

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

Initial Field Velocity Function

The initial field function 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 Added

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

{latex}
\begin{align*}
&\tau = \frac{\rho_d d_d ^2}{18 \mu_g} \\
\end{align*}
{latex}

Expected Results

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