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# Application of the Method of Characteristics to the Calculation of Supersonic Nozzles

## 19.1. General principle

Because of its precision and low cost in computation time, the method of characteristics is well suited to determine the flow in supersonic nozzles, planar or axisymmetric, aimed to equip wind tunnels or rocket engines. As we know, a supersonic nozzle has a convergent–divergent shape with a throat (see Chapter 10). It is supplied by a gas (with the stagnation conditions: pressure pstj, temperature Tstj) which accelerates in the convergent part in subsonic, becomes sonic in the throat, and continues its acceleration to supersonic in the divergent part. A supersonic nozzle thus comprises three parts:

– a subsonic region where the flow is governed by an elliptic differential system,

– a transonic region,

– a supersonic region where the system is hyperbolic and which can be treated by the method of characteristics.

We see that the method of characteristics cannot be applied to the entire flow. The calculation procedure proposed here is to adopt a particular treatment for the transonic domain. This will enable us to determine the initial part of the supersonic region from which the method of characteristics can be used.

## 19.2. Calculation of the transonic domain

### 19.2.1. The potential equation in compressible flow

There are various methods, including finite difference-like numerical techniques, to calculate the flow in the throat region. Here, we will present two methods ...

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