2 - One-Dimensional Euler Equations

Summary

Nature confronts the observer with a wealth of nonlinear wave phenomena, not only in the flow of compressible fluids, but also in many other cases of practical interest.

R. Courant and K. O. Friedrichs (1948)

Introduction

The remainder of this book focuses on numerical algorithms for the unsteady Euler equations in one dimension. Although the practical applications of the one-dimensional Euler equations are certainly limited per se, virtually all numerical algorithms for inviscid compressible flow in two and three dimensions owe their origin to techniques developed in the context of the one-dimensional Euler equations. It is therefore essential to understand the development and implementation of these algorithms in their original onedimensional context.

This chapter describes the principal mathematical properties of the onedimensional Euler equations. An understanding of these properties is essential to the development of numerical algorithms. The presentation herein is necessarily brief. For further details, the reader may consult, for example, Courant and Friedrichs (1948) and Landau and Lifshitz (1958).

Differential Forms of One-Dimensional Euler Equations

The one-dimensional Euler equations can be expressed in a variety of differential forms, of which three are particularly useful in the development of numerical algorithms. These forms are applicable where the flow variables are continuously differentiable. However, flow solutions may exhibit discontinuities that require separate treatment, as will be discussed later in Section 2.3.