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6.5 The Area-Mach-Number Relation

Chalmers University of Technology
Department of Mechanical Engineering
Division of Fluid Dynamics

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

d(ρuA)=0⇒ρuA=constd(\rho uA)=0 \Rightarrow \rho u A=const
(6.39)

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

ρuA=ρ∗u∗A∗={u∗=a∗}=ρ∗a∗A∗\rho uA=\rho^*u^*A^*=\left\{u^*=a^*\right\}=\rho^*a^*A^*
(6.40)

divide by ρuA∗\rho uA^* gives

ρ∗ρa∗u=AA∗\frac{\rho^*}{\rho}\frac{a^*}{u}=\frac{A}{A^*}
(6.41)

a∗/u=1/M∗a^*/u=1/M^* but ρ∗/ρ\rho^*/\rho is unknown

ρ∗ρ=ρ∗ρoρoρ\frac{\rho^*}{\rho}=\frac{\rho^*}{\rho_o}\frac{\rho_o}{\rho}
(6.42)

and thus

ρ∗ρoρoρ1M∗=AA∗\frac{\rho^*}{\rho_o}\frac{\rho_o}{\rho}\frac{1}{M^*}=\frac{A}{A^*}
(6.43)

Using the isentropic relations, we get

ρ∗ρo=1[12(γ−1)]1/(γ−1)\frac{\rho^*}{\rho_o}=\frac{1}{\left[\dfrac{1}{2}(\gamma-1)\right]^{1/(\gamma-1)}}
(6.44)
ρoρ=[1+12(γ+1)M2]1/(γ−1)\frac{\rho_o}{\rho}=\left[1+\frac{1}{2}(\gamma+1)M^2\right]^{1/(\gamma-1)}
(6.45)

Eqns. (6.44) and (6.45) in Eqn. (6.43) gives

AA∗=1M∗[2+(γ−1)M2γ+1]1/(γ−1)\frac{A}{A^*}=\frac{1}{M^*}\left[\frac{2+(\gamma-1)M^2}{\gamma+1}\right]^{1/(\gamma-1)}
(6.46)

What remains now is to replace M∗M^*

M∗2=u2a∗2=u2a2a2a∗2=u2a2a2ao2ao2a∗2=M2a2ao2ao2a∗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}
(6.47)

For a calorically perfect gas a=γRTa=\sqrt{\gamma R T}, which gives

a2ao2=TTo=[1+12(γ−1)M2]−1\frac{a^2}{a_o^2}=\frac{T}{T_o}=\left[1+\frac{1}{2}(\gamma-1)M^2\right]^{-1}
(6.48)
ao2a∗2=ToT∗=12(γ+1)\frac{a_o^2}{{a^*}^2}=\frac{T_o}{T^*}=\frac{1}{2}(\gamma+1)
(6.49)

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

M∗2=(γ+1)M22+(γ−1)M2{M^*}^2=\frac{(\gamma+1)M^2}{2+(\gamma-1)M^2}
(6.50)

Now, rewrite Eqn. (6.46) as

(AA∗)2=1M∗2[2+(γ−1)M2γ+1]2/(γ−1)\left(\frac{A}{A^*}\right)^2=\frac{1}{{M^*}^2}\left[\frac{2+(\gamma-1)M^2}{\gamma+1}\right]^{2/(\gamma-1)}
(6.51)

and insert M∗2{M^*}^2 from Eqn. (6.50)

(AA∗)2=2+(γ−1)M2(γ+1)M2[2+(γ−1)M2γ+1]2/(γ−1)⇒\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
(6.52)
(AA∗)2=1M2[2+(γ−1)M2γ+1]1+2/(γ−1)⇒\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
(6.53)
(AA∗)2=1M2[2+(γ−1)M2γ+1](γ+1)/(γ−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)}
(6.54)

which is the area-Mach-number relation.

For a nozzle flow, the area-Mach-number relation gives the Mach number, MM, at any location inside the nozzle as a function of the ratio between the local cross-section area, AA, and the throat area at choked conditions, A∗A^*.

M=f(AA∗)M=f\left(\frac{A}{A^*}\right)
(6.55)
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∗A^* will change over the shock.

Shock Waves
Geometric Choking
Shock Waves
Nozzle Flow