Nozzle Flow - Shock Waves

Exercise 6.1

%pip install --upgrade g3d --quiet

from g3d import CFlow

cflow = CFlow(quiet=True)

Note: you may need to restart the kernel to use updated packages.
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 cflow.py v2.6.0 Mar 2026

 a python-based compressible flow calculator


 Niklas Andersson
 Department of Mechanical Engineering
 Division of Fluid Mechanics
 Chalmers University of Technology

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A convergent-divergent nozzle with a throat area of $A_t=0.2\ m^2$ and an exit area of $A_e=0.6\ m^2$ is connected to a reservoir with the pressure $P_o=100\ kPa$ . Calcuate the nozzle pressure ratios $NPR=P_{o_i}/P_b$ corresponding to:

choked flow
internal shocks
shock at exit
overexpanded flow
supercritical flow
underexpanded flow


Poi = 1.e5

At = 0.2
Ae = 0.6

AR = Ae / At

Me_sub = cflow.nozzle.find_Mach_from_area_ratio(AR, mode='subsonic')
Pe_sub = cflow.stagnation.get_static_pressure(Po=Poi, M=Me_sub)

Me_sup = cflow.nozzle.find_Mach_from_area_ratio(AR, mode='supersonic')
Pe_sup = cflow.stagnation.get_static_pressure(Po=Poi, M=Me_sup)

Pe_shock_at_exit = cflow.normal_shock.get_downstream_pressure(M1=Me_sup, P1=Pe_sup)

print('1. choked flow: NPR >= %f' % (Poi / Pe_sub))
print('2. internal shock: %f < NPR < %f' % (Poi / Pe_sub, Poi / Pe_shock_at_exit))
print('3. normal shock at exit: NPR = %f' % (Poi / Pe_shock_at_exit))
print('4. overexpanded flow: %f < NPR < %f' % (Poi / Pe_shock_at_exit, Poi / Pe_sup))
print('5. supercritical flow: NPR = %f' % (Poi / Pe_sup))
print('6. underexpanded flow: %f < NPR' % (Poi / Pe_sup))

1. choked flow: NPR >= 1.027557
2. internal shock: 1.027557 < NPR < 2.659861
3. normal shock at exit: NPR = 2.659861
4. overexpanded flow: 2.659861 < NPR < 21.142234
5. supercritical flow: NPR = 21.142234
6. underexpanded flow: 21.142234 < NPR

6.6 Nozzle Flow