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Nian at 11:08, 1 April 2026

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Nian at 19:53, 20 March 2026

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Nian: Created page with "Category:Compressible flow Category:Quasi-one-dimensional flow Category:Inviscid flow TOC \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..."

2026-03-16T14:58:43Z

Created page with "Category:Compressible flow Category:Quasi-one-dimensional flow Category:Inviscid flow __TOC__ \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..."

New page

[[Category:Compressible flow]]
[[Category:Quasi-one-dimensional flow]]
[[Category:Inviscid flow]]

__TOC__

\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}

Area-Mach relation - Revision history

Nian at 13:37, 1 April 2026

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Nian at 21:25, 21 March 2026

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