Geometric Choking - Shock Waves

For steady-state nozzle flow, the massflow is obtained as

\dot{m}=\rho uA=const

Eqn. (6.32) can be evaluated at any location inside the nozzle and if evaluated at sonic conditions we get

\dot{m}=\rho^* u^* A^*

By definition $u^*=a^*$ and thus

\dot{m}=\rho^* a^* A^*

$\rho^*$ and $a^*$ can be obtained using the ratios $\rho^*/\rho_o$ and $a^*/a_o$

\rho^*=\left(\frac{\rho^*}{\rho_o}\right)\rho_o=\frac{p_o}{RT_o}\left(\frac{2}{\gamma+1}\right)^{1/(\gamma-1)}

a^*=\left(\frac{a^*}{a_o}\right)a_o=a_o\left(\frac{2}{\gamma+1}\right)^{1/2}=\sqrt{\gamma RT_o}\left(\frac{2}{\gamma+1}\right)^{1/2}

Eqns. (6.36) and (6.35) in Eqn. (6.34) gives

\dot{m}=\frac{p_o}{RT_o}\left(\frac{2}{\gamma+1}\right)^{1/(\gamma-1)} \sqrt{\gamma RT_o}\left(\frac{2}{\gamma+1}\right)^{1/2} A^*

which can be rewritten as

\dot{m}=\frac{p_o A^*}{\sqrt{T_o}}\sqrt{\frac{\gamma}{R}\left(\frac{2}{\gamma+1}\right)^{(\gamma+1)/(\gamma-1)}}

Eqn. (6.38) valid for:

It should be noted that the choked massflow can be calculated using Eqn. (6.38) even for cases with shocks downstream of the throat.

6.4 Geometric Choking