Handbook of Compressible Aerodynamics

Chapter 21

One-Dimensional, Non-Viscous and Adiabatic Unsteady Flows

21.1. Introduction

This chapter examines the propagation of pressure disturbances, or waves, in space and time. They may be small disturbances of the acoustic type, but also discontinuities such as shock waves. In Chapters 9–20, we have considered steady flows, i.e. independent of time. But the full Euler equations contain derivatives with respect to time; they reflect the temporal change of a spatial field (see Chapters 4 and 7). If we are given a certain initial state at a given instant in the form of velocity, pressure, and density distributions, the Euler equations govern the subsequent change of this field over time. These equations have the property of being hyperbolic as a function of time, which means they can be integrated progressing in time without any influence on the solution at a time point t of the conditions that may prevail at a future time t + Δt. This property is exploited by the numerical integration methods to find a steady solution to the Navier–Stokes or Euler equations. Starting from a more or less arbitrary initial state, we progress in time (time marching integration method) due to the Euler, or Navier–Stokes, equations to reach a steady state, that is, until the derivatives with respect to time vanish. Of course, it is necessary for a steady solution to exist, which is compatible with the boundary conditions to be satisfied.

The study of unsteady behavior is fundamental to understand ...