Area-Mach relation

The Area-Mach-Number Relation

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

$d(\rho uA)=0\Rightarrow \rho uA=const$

(Eq. 5.32)

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

$\rho uA={\rho }^{*}{u}^{*}{A}^{*}=\{u^{*}=a^{*}\}={\rho }^{*}{a}^{*}{A}^{*}$

(Eq. 5.33)

divide by $\rho u{A}^{*}$ gives

$\frac{{\rho }^{*}}{\rho }\frac{a^{*}}{u}=\frac{A}{A^{*}}$

(Eq. 5.34)

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

$\frac{{\rho }^{*}}{\rho }=\frac{{\rho }^{*}}{{\rho }_o}\frac{{\rho }_o}{\rho }$

(Eq. 5.35)

and thus

$\frac{{\rho }^{*}}{{\rho }_o}\frac{{\rho }_o}{\rho }\frac{1}{M^{*}}=\frac{A}{A^{*}}$

(Eq. 5.36)

Using the isentropic relations, we get

$\frac{{\rho }^{*}}{{\rho }_o}=\frac{1}{{[\frac{1}{2}(\gamma -1)]}^{1/(\gamma -1)}}$

(Eq. 5.37)

$\frac{{\rho }_o}{\rho }={[1+\frac{1}{2}(\gamma +1)M^2]}^{1/(\gamma -1)}$

(Eq. 5.38)

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

$\frac{A}{A^{*}}=\frac{1}{M^{*}}{\left[\frac{2+(\gamma -1)M^2}{\gamma +1}\right]}^{1/(\gamma -1)}$

(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 $a=\sqrt{\gamma RT}$, which gives

$\frac{a^2}{a_o^2}=\frac{T}{T_o}={[1+\frac{1}{2}(\gamma -1)M^2]}^{-1}$

(Eq. 5.41)

$\frac{a_o^2}{{a^{*}}^2}=\frac{T_o}{T^{*}}=\frac{1}{2}(\gamma +1)$

(Eq. 5.42)

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

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

(Eq. 5.43)

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

${\left(\frac{A}{A^{*}}\right)}^2=\frac{1}{{M^{*}}^2}{\left[\frac{2+(\gamma -1)M^2}{\gamma +1}\right]}^{2/(\gamma -1)}$

(Eq. 5.44)

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

${\left(\frac{A}{A^{*}}\right)}^2=\frac{2+(\gamma -1)M^2}{(\gamma +1)M^2}{\left[\frac{2+(\gamma -1)M^2}{\gamma +1}\right]}^{2/(\gamma -1)}\Rightarrow$

(Eq. 5.45)

${\left(\frac{A}{A^{*}}\right)}^2=\frac{1}{M^2}{\left[\frac{2+(\gamma -1)M^2}{\gamma +1}\right]}^{1+2/(\gamma -1)}\Rightarrow$

(Eq. 5.46)

${\left(\frac{A}{A^{*}}\right)}^2=\frac{1}{M^2}{\left[\frac{2+(\gamma -1)M^2}{\gamma +1}\right]}^{(\gamma +1)/(\gamma -1)}$

(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, $A^{*}$.

$M=f\left(\frac{A}{A^{*}}\right)$

(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.