Moving expansion waves

\subsection{Moving Expansion Waves}

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

\begin{figure}[ht!] \begin{center} \includegraphics[]{figures/standalone-figures/Chapter07/pdf/expansion-region.pdf} \caption{Expansion fan centered at $(x,t)=(0.0,0.0)$} \label{fig:characteristics} \end{center} \end{figure}

\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. \\

\begin{equation*} J^+_a=J^+_b \end{equation*}\\

\noindent $J^+$ invariants constant along $C^+$ characteristics\\

\begin{equation*} J^+_a=J^+_c=J^+_e \end{equation*}\\

\begin{equation*} J^+_b=J^+_d=J^+_f \end{equation*}\\

\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.\\

\noindent $J^-$ invariants constant along $C^-$ characteristics\\

\begin{equation*} J^-_c=J^-_d \end{equation*}\\

\begin{equation*} J^-_e=J^-_f \end{equation*}\\

\begin{equation*} \left. \begin{aligned} &u_e=\frac{1}{2}(J^+_e+J^-_e)\\ &u_f=\frac{1}{2}(J^+_f+J^-_f)\\ &J^-_e=J^-_f\\ &J^+_e=J^+_f \end{aligned} \right\}\Rightarrow u_e=u_f \Rightarrow a_e=a_f \end{equation*}\\

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

\begin{equation*} u+\frac{2a}{\gamma-1}=u_4+\frac{2a_4}{\gamma-1}=0+\frac{2a_4}{\gamma-1} \end{equation*}\\

\noindent and thus\\

\begin{equation} \frac{a}{a_4}=1-\frac{\gamma-1}{2}\left(\frac{u}{a_4}\right) \label{eq:expansion:a} \end{equation}\\

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

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

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

\begin{equation} \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} \end{equation}\\

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