# Chapter 9

# One-Dimensional, Non-Viscous and Adiabatic Steady Flows

## 9.1. Definition and basic hypotheses

We call a flow depending only on one space variable one-dimensional for example, *x.* Therefore, we concern ourselves with the flow in a stream tube aligned along *Ox* and whose section *A(x)* at a point of the abscissa *x* is normal to *Ox.* The axis *Ox* is often straight but may also be slightly curved (see Figure 9.1).

The basic hypothesis is to assume that the flow properties are constant in any section *A*(*x*). Such a situation may correspond to a very slender stream tube, isolated in a flow. In practice, the one-dimensional approximation applies to a flow in a duct whose area *A*(*x*) is slowly varying along an axis *Ox.*

The one-dimensional approximation is often made in internal aerodynamics when dealing with flows in ducts with a slowly varying section, air inlet, nozzle, diffuser, ejector, etc. Its advantage is that it leads to very simple calculations, which is very convenient for applications where we do not look for great accuracy, for example at the pre-project level. Additionally, one-dimensional theory allows us to establish general results of great importance at the fundamental level. However, we have to be wary of using this approximation in situations where the two-dimensional character of the flow is essential. Assuming a one-dimensional flow can then lead to inconsistent or paradoxical results.

**Figure 9.1.** *One-dimensional stream tube*

Given *p, ρ, V, T*, etc. the pressure, the ...