Area-Mach relation

The Area-Mach-Number Relation

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

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

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

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

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

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

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

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

and thus

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

Using the isentropic relations, we get

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

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

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

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

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

$${M^{*}}^2=\frac{u^2}{{a^{*}}^2}=\frac{u^2}{a^2}\frac{a^2}{{a^{*}}^2}=\frac{u^2}{a^2}\frac{a^2}{a_o^2}\frac{a_o^2}{{a^{*}}^2}=M^2\frac{a^2}{a_o^2}\frac{a_o^2}{{a^{*}}^2}$$

For a calorically perfect gas $a=\sqrt{\gamma RT}$, which gives

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

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

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

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

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

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

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

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

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

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

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, $A^{*}$.

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

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.