Moving expansion waves: Difference between revisions

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\subsection{Moving Expansion Waves}
==== Moving Expansion Waves ====


\noindent The expansion wave propagation into the driver section in a shock tube can be described using characteristic lines.\\
The expansion wave propagation into the driver section in a shock tube can be described using characteristic lines.


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\noindent The expansion is propagating into stagnant fluid in region four (the driver section), which means that the flow properties ahead of the expansion wave are constant. \\
The expansion is propagating into stagnant fluid in region four (the driver section), which means that the flow properties ahead of the expansion wave are constant.


\begin{equation*}
<math display="block">
J^+_a=J^+_b
J^+_a=J^+_b
\end{equation*}\\
</math>


\noindent $J^+$ invariants constant along $C^+$ characteristics\\
<math>J^+</math> invariants constant along <math>C^+</math> characteristics


\begin{equation*}
<math display="block">
J^+_a=J^+_c=J^+_e
J^+_a=J^+_c=J^+_e
\end{equation*}\\
</math>


\begin{equation*}
<math display="block">
J^+_b=J^+_d=J^+_f
J^+_b=J^+_d=J^+_f
\end{equation*}\\
</math>


\noindent Since $J^+_a=J^+_b$ this also implies $J^+_e=J^+_f$. In fact, since the flow properties ahead of the expansion are constant, all $C^+$ lines will have the same $J^+$ value.\\
Since <math>J^+_a=J^+_b</math> this also implies <math>J^+_e=J^+_f</math>. In fact, since the flow properties ahead of the expansion are constant, all <math>C^+</math> lines will have the same <math>J^+</math> value.


\noindent $J^-$ invariants constant along $C^-$ characteristics\\
<math>J^-</math> invariants constant along <math>C^-</math> characteristics


\begin{equation*}
<math display="block">
J^-_c=J^-_d
J^-_c=J^-_d
\end{equation*}\\
</math>


\begin{equation*}
<math display="block">
J^-_e=J^-_f
J^-_e=J^-_f
\end{equation*}\\
</math>


\begin{equation*}
<math display="block">
\left.
\left.
\begin{aligned}
\begin{aligned}
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\end{aligned}
\end{aligned}
\right\}\Rightarrow u_e=u_f \Rightarrow a_e=a_f
\right\}\Rightarrow u_e=u_f \Rightarrow a_e=a_f
\end{equation*}\\
</math>


\noindent Due to the fact the $J^+$ is constant in the entire expansion region, $u$ and $a$ will be constant along each $C^-$ line.\\
Due to the fact the <math>J^+</math> is constant in the entire expansion region, <math>u</math> and <math>a</math> will be constant along each <math>C^-</math> line.


\noindent The constant $J^+$ value can be used to obtain relations for the variation of flow properties through the expansion region. Evaluation of the $J^+$ invariant at any position within the expansion region should give the same value as in region 4.\\
The constant <math>J^+</math> value can be used to obtain relations for the variation of flow properties through the expansion region. Evaluation of the <math>J^+</math> invariant at any position within the expansion region should give the same value as in region 4.


\begin{equation*}
<math display="block">
u+\frac{2a}{\gamma-1}=u_4+\frac{2a_4}{\gamma-1}=0+\frac{2a_4}{\gamma-1}
u+\frac{2a}{\gamma-1}=u_4+\frac{2a_4}{\gamma-1}=0+\frac{2a_4}{\gamma-1}
\end{equation*}\\
</math>


\noindent and thus\\
and thus


\begin{equation}
<math display="block">
\frac{a}{a_4}=1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right)
\frac{a}{a_4}=1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right)
\label{eq:expansion:a}
</math>
\end{equation}\\


\noindent Eqn. \ref{eq:expansion:a} and $a=\sqrt{\gamma RT}$ gives\\
Eqn. \ref{eq:expansion:a} and <math>a=\sqrt{\gamma RT}</math> gives


\begin{equation}
<math display="block">
\frac{T}{T_4}=\left[1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right)\right]^2
\frac{T}{T_4}=\left[1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right)\right]^2
\label{eq:expansion:b}
</math>
\end{equation}\\


\noindent Using isentropic relations, we can get pressure ratio and density ratio\\
Using isentropic relations, we can get pressure ratio and density ratio


\begin{equation}
<math display="block">
\frac{p}{p_4}=\left[1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right)\right]^{2\gamma/(\gamma-1)}
\frac{p}{p_4}=\left[1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right)\right]^{2\gamma/(\gamma-1)}
\label{eq:expansion:b}
</math>
\end{equation}\\


\begin{equation}
<math display="block">
\frac{\rho}{\rho_4}=\left[1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right)\right]^{2/(\gamma-1)}
\frac{\rho}{\rho_4}=\left[1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right)\right]^{2/(\gamma-1)}
\label{eq:expansion:b}
</math>
\end{equation}\\
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