This chapter presents the derivation of the basic equations that we will use in analyses and calculations of unsteady free-surface flows in natural or man-made channels, e.g. tidal or fluvial channels, shipping canals and irrigation canals. We deal with a mass balance and a momentum balance integrated across the entire flow cross section, assuming a hydrostatic pressure distribution.

Approach

The principal subject of this book is the class of unsteady free-surface flows of water with a characteristic length scale that is far greater than the depth, the so-called long waves. Tides, storm surges and flood waves in rivers provide good examples of this category (contrary to ship waves or wind-generated waves, whose lengths are usually not large or even small compared with the depth).

We restrict ourselves to flows in relatively narrow, weakly curved conduits such as tidal channels and rivers, in which the main flow direction is determined by the geometry of the boundary, which is assumed to be given beforehand (excluding morphological changes such as meandering of rivers). In these cases, the bulk flow direction is known so that only the flow intensity (the discharge, say) is to be determined, in addition to the water surface elevation. A typical area of application, a long, slowly winding river reach with lateral side basins, is shown in Figure 1.1.

As expressed by their name, long waves are characterized by length dimensions that far exceed the depths. This implies that the curvature of the streamlines in the vertical plane is negligible, for which reason we will assume a hydrostatic pressure distribution in the vertical. This greatly simplifies the schematization and the calculations.

In bends, the flow is forced to change direction through a lateral variation of the water level, being higher at the outer bank and lower at the inner bank. This is essential in detailed computations of the spiral flow in bends, but it is irrelevant for the large-scale computations of longitudinal variations with which we are concerned. So we will ignore lateral variations in surface elevation. The height of this level above the adopted reference plane z = 0 is designated as h. This quantity is a function of the downstream coordinate s (measured along the axis of the conduit) and the time t, or h = h(s, t).