Area-velocity relation
\section{The Area-Velocity Relation}
\noindent Starting point - the continuity equation (Eqn. \ref{eq:governing:cont}):\\
\[d(\rho uA)=0 \Rightarrow \rho u dA+\rho Adu +uAd\rho=0\]\\
\noindent divide by $\rho uA$ gives\\
\begin{equation} \frac{d\rho}{\rho}+\frac{du}{u}+\frac{dA}{A}=0 \label{eq:governing:cont:b} \end{equation}\\
\noindent As the name suggests, the area-velocity relation is a relation including the area and the flow velocity. Therefore, the next step is to replace the density terms.\\
\noindent This can be achieved using the momentum equation (Eqn. \ref{eq:governing:mom})\\
\[dp=-\rho udu\Leftrightarrow \frac{dp}{\rho}=-udu\]\\
\[\frac{dp}{\rho}=\frac{dp}{d\rho}\frac{d\rho}{\rho}=-udu\]\\
\noindent If we assume adiabatic and reversible flow processes, i.e., isentropic flow\\
\[\frac{dp}{d\rho}=\left(\frac{dp}{d\rho}\right)_s=a^2\Rightarrow a^2\frac{d\rho}{\rho}=-udu\]\\
\[a^2\frac{d\rho}{\rho}=-udu=-u^2\frac{du}{u}\]\\
\begin{equation} \frac{d\rho}{\rho}=-M^2\frac{du}{u} \label{eq:governing:mom:b} \end{equation}\\
\noindent Eqn. \ref{eq:governing:mom:b} inserted in Eqn. \ref{eq:governing:cont:b} gives\\
\[-M^2\frac{du}{u}+\frac{du}{u}+\frac{dA}{A}=0\]\\
or
\begin{equation} \frac{dA}{A}=(M^2-1)\frac{du}{u} \label{eq:governing:av} \end{equation}\\
\noindent which is the area-velocity relation.\\
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\noindent From the area-velocity relation (Eqn. \ref{eq:governing:av}), we can learn that in a subsonic flow, the flow will accelerate if the cross-section area is decreased and decelerate if the cross-section area is increased. It can also be seen that for supersonic flow, the relation between flow velocity and cross-section area will be the opposite of that for subsonic flows, see Fig. \ref{fig:areavelocity}. For sonic flow, $M=1$, the relation shows that $dA=0$, which means that sonic flow can only occur at a cross-section area maximum or minimum. From the subsonic versus supersonic flow discussion, it can be understood that sonic flow at the minimum cross section area is the only valid option (see Fig. \ref{fig:sonic}).\\
\begin{figure}[ht!] \begin{center} \includegraphics[]{figures/standalone-figures/Chapter06/pdf/area-velocity.pdf} \caption{Area-velocity relation - subsonic flow vs. supersonic flow} \label{fig:areavelocity} \end{center} \end{figure}
\begin{figure}[ht!] \begin{center} \includegraphics[]{figures/standalone-figures/Chapter06/pdf/area-velocity-sonic-flow.pdf} \caption{Area-velocity relation - sonic flow} \label{fig:sonic} \end{center} \end{figure}
