Area-Mach relation: Difference between revisions

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\section{The Area-Mach-Number Relation}
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=== The Area-Mach-Number Relation ===


\noindent Starting point - the continuity equation (Eqn. \ref{eq:governing:cont}):\\
Starting point - the continuity equation (Eqn. \ref{eq:governing:cont}):


\[d(\rho uA)=0 \Rightarrow \rho u A=const\]\\
{{NumEqn|<math>
d(\rho uA)=0 \Rightarrow \rho u A=const
</math>}}


\noindent This applies everywhere in the nozzle and therefore the sonic conditions can be used as a reference\\
This applies everywhere in the nozzle and therefore the sonic conditions can be used as a reference


\[\rho uA=\rho^*u^*A^*=\left\{u^*=a^*\right\}=\rho^*a^*A^*\]\\
{{NumEqn|<math>
\rho uA=\rho^*u^*A^*=\left\{u^*=a^*\right\}=\rho^*a^*A^*
</math>}}


\noindent divide by $\rho uA^*$ gives\\
divide by <math>\rho uA^*</math> gives


\[\frac{\rho^*}{\rho}\frac{a^*}{u}=\frac{A}{A^*}\]
{{NumEqn|<math>
\frac{\rho^*}{\rho}\frac{a^*}{u}=\frac{A}{A^*}
</math>}}


\noindent $a^*/u=1/M^*$ but $\rho^*/\rho$ is unknown\\
<math>a^*/u=1/M^*</math> but <math>\rho^*/\rho</math> is unknown


\[\frac{\rho^*}{\rho}=\frac{\rho^*}{\rho_o}\frac{\rho_o}{\rho}\]\\
{{NumEqn|<math>
\frac{\rho^*}{\rho}=\frac{\rho^*}{\rho_o}\frac{\rho_o}{\rho}
</math>}}


\noindent and thus\\
and thus


\begin{equation}
{{NumEqn|<math>
\frac{\rho^*}{\rho_o}\frac{\rho_o}{\rho}\frac{1}{M^*}=\frac{A}{A^*}
\frac{\rho^*}{\rho_o}\frac{\rho_o}{\rho}\frac{1}{M^*}=\frac{A}{A^*}
\label{eq:areamach:a}
</math>}}
\end{equation}\\


\noindent Using the isentropic relations, we get\\
Using the isentropic relations, we get


\begin{equation}
{{NumEqn|<math>
\frac{\rho^*}{\rho_o}=\frac{1}{\left[\dfrac{1}{2}(\gamma-1)\right]^{1/(\gamma-1)}}
\frac{\rho^*}{\rho_o}=\frac{1}{\left[\dfrac{1}{2}(\gamma-1)\right]^{1/(\gamma-1)}}
\label{eq:rho:a}
</math>}}
\end{equation}\\


\begin{equation}
{{NumEqn|<math>
\frac{\rho_o}{\rho}=\left[1+\frac{1}{2}(\gamma+1)M^2\right]^{1/(\gamma-1)}
\frac{\rho_o}{\rho}=\left[1+\frac{1}{2}(\gamma+1)M^2\right]^{1/(\gamma-1)}
\label{eq:rho:b}
</math>}}
\end{equation}\\


\noindent Eqns. \ref{eq:rho:a} and \ref{eq:rho:b} in Eqn. \ref{eq:areamach:a} gives\\
Eqns. \ref{eq:rho:a} and \ref{eq:rho:b} in Eqn. \ref{eq:areamach:a} gives


\begin{equation}
{{NumEqn|<math>
\frac{A}{A^*}=\frac{1}{M^*}\left[\frac{2+(\gamma-1)M^2}{\gamma+1}\right]^{1/(\gamma-1)}
\frac{A}{A^*}=\frac{1}{M^*}\left[\frac{2+(\gamma-1)M^2}{\gamma+1}\right]^{1/(\gamma-1)}
\label{eq:areamach:b}
</math>}}
\end{equation}\\


\noindent What remains now is to replace $M^*$\\
What remains now is to replace <math>M^*</math>


\begin{equation}
{{NumEqn|<math>
{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}
{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}
</math>}}
\end{equation}\\


\noindent For a calorically perfect gas $a=\sqrt{\gamma R T}$, which gives\\
For a calorically perfect gas <math>a=\sqrt{\gamma R T}</math>, which gives


\begin{equation}
{{NumEqn|<math>
\frac{a^2}{a_o^2}=\frac{T}{T_o}=\left[1+\frac{1}{2}(\gamma-1)M^2\right]^{-1}
\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}
</math>}}
\end{equation}\\


\begin{equation}
{{NumEqn|<math>
\frac{a_o^2}{{a^*}^2}=\frac{T_o}{T^*}=\frac{1}{2}(\gamma+1)
\frac{a_o^2}{{a^*}^2}=\frac{T_o}{T^*}=\frac{1}{2}(\gamma+1)
\label{eq:a:b}
</math>}}
\end{equation}\\


\noindent Eqns. \ref{eq:a:a} and \ref{eq:a:b} in Eqn. \ref{eq:mstar:a} gives\\
Eqns. \ref{eq:a:a} and \ref{eq:a:b} in Eqn. \ref{eq:mstar:a} gives


\begin{equation}
{{NumEqn|<math>
{M^*}^2=\frac{(\gamma+1)M^2}{2+(\gamma-1)M^2}
{M^*}^2=\frac{(\gamma+1)M^2}{2+(\gamma-1)M^2}
\label{eq:mstar:b}
</math>}}
\end{equation}\\


\noindent Now, rewrite Eqn. \ref{eq:areamach:b} as\\
Now, rewrite Eqn. \ref{eq:areamach:b} as


\begin{equation}
{{NumEqn|<math>
\left(\frac{A}{A^*}\right)^2=\frac{1}{{M^*}^2}\left[\frac{2+(\gamma-1)M^2}{\gamma+1}\right]^{2/(\gamma-1)}
\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}
</math>}}
\end{equation}\\


\noindent and insert ${M^*}^2$ from Eqn. \ref{eq:mstar:b}\\
and insert <math>{M^*}^2</math> from Eqn. \ref{eq:mstar:b}


\[\left(\frac{A}{A^*}\right)^2=\frac{2+(\gamma-1)M^2}{(\gamma+1)M^2}\left[\frac{2+(\gamma-1)M^2}{\gamma+1}\right]^{2/(\gamma-1)} \Rightarrow \]\\
{{NumEqn|<math>
\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
</math>}}


\[\left(\frac{A}{A^*}\right)^2=\frac{1}{M^2}\left[\frac{2+(\gamma-1)M^2}{\gamma+1}\right]^{1+2/(\gamma-1)} \Rightarrow\]\\
{{NumEqn|<math>
\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
</math>}}


\begin{equation}
{{NumEqn|<math>
\left(\frac{A}{A^*}\right)^2=\frac{1}{M^2}\left[\frac{2+(\gamma-1)M^2}{\gamma+1}\right]^{(\gamma+1)/(\gamma-1)}
\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}
</math>}}
\end{equation}\\


\noindent which is the area-Mach-number relation.\\
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, $A^*$.
For a nozzle flow, the area-Mach-number relation gives the Mach number, <math>M</math>, at any location inside the nozzle as a function of the ratio between the local cross-section area, <math>A</math>, and the throat area at choked conditions, <math>A^*</math>.


\[M=f\left(\frac{A}{A^*}\right)\]\\
{{NumEqn|<math>
M=f\left(\frac{A}{A^*}\right)
</math>}}


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\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.
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 <math>A^*</math> will change over the shock.
 


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\label{fig:areaMach:trends}
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