The Area-Mach-Number Relation - Shock Waves

Starting point - the continuity equation (Eqn. (4.1)):

d(\rho uA)=0 \Rightarrow \rho u A=const

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^*

divide by $\rho uA^*$ 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}{\left[\dfrac{1}{2}(\gamma-1)\right]^{1/(\gamma-1)}}

\frac{\rho_o}{\rho}=\left[1+\frac{1}{2}(\gamma+1)M^2\right]^{1/(\gamma-1)}

Eqns. (6.44) and (6.45) in Eqn. (6.43) 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 $a=\sqrt{\gamma R T}$ , which gives

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

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

Eqns. (6.48) and (6.49) in Eqn. (6.47) gives

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

Now, rewrite Eqn. (6.46) 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. (6.50)

\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)

Area-Mach-number relation - subsonic nozzle flow — Figure 6.4:Area-Mach-number relation - subsonic nozzle flow

Area-Mach-number relation - supersonic nozzle flow — Figure 6.5:Area-Mach-number relation - supersonic nozzle flow

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.

quasi-one-dimensional flow trends - M — Figure 6.6: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.

quasi-one-dimensional flow trends - U — Figure 6.7: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.

quasi-one-dimensional flow trends - tau — Figure 6.8: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.

quasi-one-dimensional flow trends - T — Figure 6.9: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.

quasi-one-dimensional flow trends - P — Figure 6.10: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.

quasi-one-dimensional flow trends - R — Figure 6.11: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.