Area-Mach relation
\section{The Area-Mach-Number Relation}
\noindent Starting point - the continuity equation (Eqn. \ref{eq:governing:cont}):\\
\[d(\rho uA)=0 \Rightarrow \rho u A=const\]\\
\noindent This applies everywhere in the nozzle and therefore the sonic conditions can be used as a reference\\
\[\rho uA=\rho^*u^*A^*=\left\{u^*=a^*\right\}=\rho^*a^*A^*\]\\
\noindent divide by $\rho uA^*$ gives\\
\[\frac{\rho^*}{\rho}\frac{a^*}{u}=\frac{A}{A^*}\]
\noindent $a^*/u=1/M^*$ but $\rho^*/\rho$ is unknown\\
\[\frac{\rho^*}{\rho}=\frac{\rho^*}{\rho_o}\frac{\rho_o}{\rho}\]\\
\noindent and thus\\
\begin{equation} \frac{\rho^*}{\rho_o}\frac{\rho_o}{\rho}\frac{1}{M^*}=\frac{A}{A^*} \label{eq:areamach:a} \end{equation}\\
\noindent Using the isentropic relations, we get\\
\begin{equation} \frac{\rho^*}{\rho_o}=\frac{1}{\left[\dfrac{1}{2}(\gamma-1)\right]^{1/(\gamma-1)}} \label{eq:rho:a} \end{equation}\\
\begin{equation} \frac{\rho_o}{\rho}=\left[1+\frac{1}{2}(\gamma+1)M^2\right]^{1/(\gamma-1)} \label{eq:rho:b} \end{equation}\\
\noindent Eqns. \ref{eq:rho:a} and \ref{eq:rho:b} in Eqn. \ref{eq:areamach:a} gives\\
\begin{equation} \frac{A}{A^*}=\frac{1}{M^*}\left[\frac{2+(\gamma-1)M^2}{\gamma+1}\right]^{1/(\gamma-1)} \label{eq:areamach:b} \end{equation}\\
\noindent What remains now is to replace $M^*$\\
\begin{equation} {M^*}^2=\frac{u^2}{{a^*}^2}=\frac{u^2}{a^2}\frac{a^2}{{a^*}^2}=\frac{u^2}{a^2}\frac{a^2}{a_o^2}\frac{a_o^2}{{a^*}^2}=M^2\frac{a^2}{a_o^2}\frac{a_o^2}{{a^*}^2} \label{eq:mstar:a} \end{equation}\\
\noindent For a calorically perfect gas $a=\sqrt{\gamma R T}$, which gives\\
\begin{equation} \frac{a^2}{a_o^2}=\frac{T}{T_o}=\left[1+\frac{1}{2}(\gamma-1)M^2\right]^{-1} \label{eq:a:a} \end{equation}\\
\begin{equation} \frac{a_o^2}{{a^*}^2}=\frac{T_o}{T^*}=\frac{1}{2}(\gamma+1) \label{eq:a:b} \end{equation}\\
\noindent Eqns. \ref{eq:a:a} and \ref{eq:a:b} in Eqn. \ref{eq:mstar:a} gives\\
\begin{equation} {M^*}^2=\frac{(\gamma+1)M^2}{2+(\gamma-1)M^2} \label{eq:mstar:b} \end{equation}\\
\noindent Now, rewrite Eqn. \ref{eq:areamach:b} as\\
\begin{equation} \left(\frac{A}{A^*}\right)^2=\frac{1}{{M^*}^2}\left[\frac{2+(\gamma-1)M^2}{\gamma+1}\right]^{2/(\gamma-1)} \label{eq:areamach:c} \end{equation}\\
\noindent and insert ${M^*}^2$ from Eqn. \ref{eq:mstar:b}\\
\[\left(\frac{A}{A^*}\right)^2=\frac{2+(\gamma-1)M^2}{(\gamma+1)M^2}\left[\frac{2+(\gamma-1)M^2}{\gamma+1}\right]^{2/(\gamma-1)} \Rightarrow \]\\
\[\left(\frac{A}{A^*}\right)^2=\frac{1}{M^2}\left[\frac{2+(\gamma-1)M^2}{\gamma+1}\right]^{1+2/(\gamma-1)} \Rightarrow\]\\
\begin{equation} \left(\frac{A}{A^*}\right)^2=\frac{1}{M^2}\left[\frac{2+(\gamma-1)M^2}{\gamma+1}\right]^{(\gamma+1)/(\gamma-1)} \label{eq:areamach:c} \end{equation}\\
\noindent which is the area-Mach-number relation.\\
\noindent For a nozzle flow, the area-Mach-number relation gives the Mach number, $M$, at any location inside the nozzle as a function of the ratio between the local cross-section area, $A$, and the throat area at choked conditions, $A^*$.
\[M=f\left(\frac{A}{A^*}\right)\]\\
\begin{figure}[ht!] \begin{center} \includegraphics[]{figures/standalone-figures/Chapter06/pdf/area-Mach-subsonic.pdf} \caption{Area-Mach-number relation - subsonic nozzle flow} \label{fig:subsonic} \end{center} \end{figure}
\begin{figure}[ht!] \begin{center} \includegraphics[]{figures/standalone-figures/Chapter06/pdf/area-Mach-supersonic.pdf} \caption{Area-Mach-number relation - supersonic nozzle flow} \label{fig:supersonic} \end{center} \end{figure}
\noindent Due to the assumptions made in the derivation, the area-Mach-number relation is only valid for isentropic flows of calorically perfect gases. This means that it cannot be used throughout the divergent part of a convergent-divergent nozzle in case there is a shock within the nozzle. It can, however, be used both upstream and downstream of the shock. Note that $A^*$ will change over the shock.
\begin{figure}[ht!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[]{figures/standalone-figures/Chapter06/pdf/area-Mach-trends_M.pdf}
\caption{Mach number}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[]{figures/standalone-figures/Chapter06/pdf/area-Mach-trends_U.pdf}
\caption{flow velocity}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[]{figures/standalone-figures/Chapter06/pdf/area-Mach-trends_tau.pdf}
\caption{compressibility}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[]{figures/standalone-figures/Chapter06/pdf/area-Mach-trends_T.pdf}
\caption{temperature}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[]{figures/standalone-figures/Chapter06/pdf/area-Mach-trends_P.pdf}
\caption{pressure}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[]{figures/standalone-figures/Chapter06/pdf/area-Mach-trends_R.pdf}
\caption{density}
\end{subfigure}
\caption{Change in flow variables as a consequence of changes in cross-section area. Blue lines represent subsonic solutions and the orange lines represent supersonic solutions.}
\label{fig:areaMach:trends}
\end{figure}
