Handbook of Compressible Aerodynamics

Chapter 17

Supersonic, Steady, Two-Dimensional Flows and the Theory of Characteristics

17.1. The theory of characteristics: introduction

In this chapter, we will present the key results of the theory of characteristics applied to a steady, two-dimensional planar or axisymmetric flow of a non-viscous fluid. We shall see that this flow must also be supersonic. The theory of characteristics also applies to the three-dimensional, but then its presentation and interpretation become more difficult. In addition, this theory, which leads to a very efficient calculation method in two dimensions, becomes, in the three dimensions, an implementation complex in the plane of algorithms, which makes it preferred over other methods based on direct numerical solution of Euler equations. The method of characteristics based on the theory of characteristics is actually a numerical solution technique of the Euler equations. The practice has, however, established itself in phraseology to distinguish the method of characteristics from the (numerical) resolution of the Euler equations. This is actually a misunderstanding.

There are several ways to introduce the theory of characteristics, either by considering the framework of the wave theory or mathematically from the study of the properties of pseudo-linear differential systems whose equations do not contain partial derivative products. Here, we will adopt a more direct approach by specializing in the two-dimensional case to avoid too heavy a formalism ...