In the preceding chapters, we have seen that the one-dimensional shallow-water equations (named after De Saint-Venant) form a set of partial differential equations of hyperbolic type. Their mathematical structure is defined by the underlying characteristics, which property is used in the technique of integration. Pressurized flow of a liquid (water, oil, etc.) in closed conduits is mathematically described by a similar set of equations. Therefore, with relatively little extra effort, a solution method for pressurized flow can be obtained. This fact in itself would not be a sufficient justification for the inclusion of this subject in the present book, but since pressurized flow frequently occurs in conjunction with freesurface flow in the context of hydro-engineering projects, it was decided to include the subject, albeit in abbreviated form and in an appendix in order to not interrupt the line of development of the principal subject of this book. The presentation below rests heavily on Chapter 5, to which the reader is referred for more extensive background information. A more detailed account of the subject than is appropriate here can be found in Streeter and Wylie (1967), Jaeger (1977), Fox (1989) and Thorley (1991).

Introduction

In free-surface flows, storage takes place through variations of the free-surface elevation. This is accompanied by pressure variations of a few metres of water column at most, too small to cause appreciable changes in density. The water can therefore be treated as incompressible. Pressurized flows do not have a free surface, so that a corresponding storage cannot occur. In these cases, storage can take place only through elasticity of the pipe wall, allowing profile variations, and compression of the liquid, allowing variations in mass in a given volume.

The abrupt closure or opening of a valve or the abrupt switching on or off of a pump in a pipeline for irrigation, hydropower, drinking water supply, etc., either purposefully or as the result of a failure or an accident, results in rapid variations in flow velocity, accompanied by large pressure variations. This phenomenon is called water hammer because it can sound as if the pipe wall is struck by a hammer. Too large pressures should be avoided, or at least reduced in view of the limited strength of the materials.