Area-Mach relation: Difference between revisions

Latest revision as of 13:37, 1 April 2026

The Area-Mach-Number Relation

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

d (ρ u A) = 0 \Rightarrow ρ u A = c o n s t

(Eq. 5.32)

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

ρ u A = ρ^{*} u^{*} A^{*} = {u^{*} = a^{*}} = ρ^{*} a^{*} A^{*}

(Eq. 5.33)

divide by $ρ u A^{*}$ gives

\frac{ρ^{*}}{ρ} \frac{a^{*}}{u} = \frac{A}{A^{*}}

(Eq. 5.34)

$a^{*} / u = 1 / M^{*}$ but $ρ^{*} / ρ$ is unknown

\frac{ρ^{*}}{ρ} = \frac{ρ^{*}}{ρ_{o}} \frac{ρ_{o}}{ρ}

(Eq. 5.35)

and thus

\frac{ρ^{*}}{ρ_{o}} \frac{ρ_{o}}{ρ} \frac{1}{M^{*}} = \frac{A}{A^{*}}

(Eq. 5.36)

Using the isentropic relations, we get

\frac{ρ^{*}}{ρ_{o}} = \frac{1}{{[\frac{1}{2} (γ - 1)]}^{1 / (γ - 1)}}

(Eq. 5.37)

\frac{ρ_{o}}{ρ} = {[1 + \frac{1}{2} (γ + 1) M^{2}]}^{1 / (γ - 1)}

(Eq. 5.38)

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

\frac{A}{A^{*}} = \frac{1}{M^{*}} {[\frac{2 + (γ - 1) M^{2}}{γ + 1}]}^{1 / (γ - 1)}

(Eq. 5.39)

What remains now is to replace $M^{*}$

{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}}

(Eq. 5.40)

For a calorically perfect gas $a = \sqrt{γ R T}$ , which gives

\frac{a^{2}}{a_{o}^{2}} = \frac{T}{T_{o}} = {[1 + \frac{1}{2} (γ - 1) M^{2}]}^{- 1}

(Eq. 5.41)

\frac{a_{o}^{2}}{{a^{*}}^{2}} = \frac{T_{o}}{T^{*}} = \frac{1}{2} (γ + 1)

(Eq. 5.42)

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

{M^{*}}^{2} = \frac{(γ + 1) M^{2}}{2 + (γ - 1) M^{2}}

(Eq. 5.43)

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

{(\frac{A}{A^{*}})}^{2} = \frac{1}{{M^{*}}^{2}} {[\frac{2 + (γ - 1) M^{2}}{γ + 1}]}^{2 / (γ - 1)}

(Eq. 5.44)

and insert ${M^{*}}^{2}$ from Eqn. \ref{eq:mstar:b}

{(\frac{A}{A^{*}})}^{2} = \frac{2 + (γ - 1) M^{2}}{(γ + 1) M^{2}} {[\frac{2 + (γ - 1) M^{2}}{γ + 1}]}^{2 / (γ - 1)} \Rightarrow

(Eq. 5.45)

{(\frac{A}{A^{*}})}^{2} = \frac{1}{M^{2}} {[\frac{2 + (γ - 1) M^{2}}{γ + 1}]}^{1 + 2 / (γ - 1)} \Rightarrow

(Eq. 5.46)

{(\frac{A}{A^{*}})}^{2} = \frac{1}{M^{2}} {[\frac{2 + (γ - 1) M^{2}}{γ + 1}]}^{(γ + 1) / (γ - 1)}

(Eq. 5.47)

which is the area-Mach-number relation.

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 (\frac{A}{A^{*}})

(Eq. 5.48)

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.

@@ Line 1: / Line 1: @@
-[[Category:Compressible flow]]
+<!--
-[[Category:Quasi-one-dimensional flow]]
+-->[[Category:Compressible flow]]<!--
-[[Category:Inviscid flow]]
+-->[[Category:Quasi-one-dimensional flow]]<!--
+-->[[Category:Inviscid flow]]<!--
+--><noinclude><!--
+-->[[Category:Compressible flow:Topic]]<!--
+--></noinclude><!--
-__TOC__
+--><nomobile><!--
+-->__TOC__<!--
+--></nomobile><!--
-\section{The Area-Mach-Number Relation}
+--><noinclude><!--
+-->{{#vardefine:secno|5}}<!--
+-->{{#vardefine:eqno|31}}<!--
+--></noinclude><!--
+-->
+=== 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^*}
-\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)}}
-\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)}
-\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)}
-\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}
-\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}
-\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)
-\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}
-\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)}
-\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)}
-\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>}}
+<!--
 \begin{figure}[ht!]
 \begin{center}
@@ Line 114: / Line 130: @@
 \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.
+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.
+<!--
 \begin{figure}[ht!]
 \begin{subfigure}[b]{0.5\textwidth}
@@ Line 152: / Line 169: @@
 \label{fig:areaMach:trends}
 \end{figure}
+-->

Area-Mach relation: Difference between revisions

Latest revision as of 13:37, 1 April 2026

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The Area-Mach-Number Relation

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Area-Mach relation: Difference between revisions

Latest revision as of 13:37, 1 April 2026

The Area-Mach-Number Relation

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