# Chapter 10

# Application of the One-Dimensional Theoryto the Calculation of Supersonic Nozzles

## 10.1. General definition of a supersonic nozzle

A supersonic propulsion nozzle with a convergent–divergent shape ensures the expansion of propellants from a subsonic state, in the combustion chamber, turbine outlet, or even after the postcombustion chamber, to a supersonic state in the exit plane (*E*). In most cases, the nozzles are axisymmetric. Such a supersonic nozzle is shown in Figure 10.1. It must include a throat (*C*) where the Mach number is equal to 1. The nozzles of this type also equip supersonic wind tunnels.

**Figure 10.1.** *Layout of a supersonic nozzle*

The nozzle is fed under the initial (stagnation) conditions:

– pressure: *P*_{stj},

– temperature: *T*_{stj}.

In these notations, we have introduced the index *j* to denote a propulsion jet. We denote with the index *E* the conditions in the exit plane whose section has an area *A*_{E}:

– pressure: *p*_{E},

– density : *ρ*_{E},

– temperature: *T*_{E},

– gas velocities: *V*_{E},

– Mach number: *M*_{E}.

The area of the throat is denoted by: *A*_{C}.

Two quantities are of particular importance for the nozzles:

– the mass flow: *q*_{m},

– the thrust: *F*.

These quantities can be calculated using the theory of one-dimensional flows. The results thus obtained are approximate, the flow in a real nozzle being not strictly one-dimensional. A precise calculation of the performance of a nozzle must ...