1984) that the flow is characterized by the dimensionless Froude number Fr=υ/√(gy). For Fr<1, flow is said to be subcritical (slow, gentle or tranquil). For Fr=1, flow is critical, with depth equal to yc the critical depth. For Fr>1, flow is supercritical (fast or shooting).
Fig. 6.3 demonstrates these flow conditions. Larger flows have larger values of υc and yc.
Steady flow occurs when the velocity at any point does not change with time. Flow is unsteady in surges and flood waves in open channels (although they may sometimes appear steady to a moving observer). The analytical equations of unsteady flow are complex and difficult to solve (Chapter 16) but the hydrologist is most often concerned with these unsteady flow conditions. With the more simple conditions of steady flow, some open channel flow problems can be solved using the principles of continuity, conservation of energy and conservation of momentum.
6.1.1. Sediment
Natural rivers develop as small streams in their upland headwaters where channel gradients are often steep and erosion of the land and water courses is prevalent. In their middle reaches, rivers tend to be in equilibrium and eroded material from upstream is carried down to the plains where gradients are slack and deposition is the norm. Such general considerations are modified by changes in river discharges when a sudden increase in flow may cause erosion along most of the river channel and subsequent lowering of the discharge results in the deposition of some of the load according to channel gradients and flow velocities.
The sediment load may be subdivided into three components, the bed load, suspended material and a narrow intermediate phase of saltation in which particles separate from the bed load and bounce along in the flow. (For detailed physical definitions, see Raudkivi 1990, Chapter 7.)
Measurement of the sediment load is a complex problem. The simplest component to quantify is the suspended solids and these form part of the physical properties of water-
quality in Chapter 8. The shifting bed load and the variable saltating particles have been the subject of extensive world-wide research to sample the range of conditions in the field and model the bed movements in laboratory flumes and scaled channels. Numerous empirical relationships have been formulated for different types of rivers; these are readily consulted in Raudkivi (1990) and Richards (1982).
For engineers requiring guidance in channel improvement to obtain stable conditions for the transport of a given-amount of water and sediment, an analytical method has been devised (White et al., 1982). The channel variables are related by equations for the continuity of water flow, sediment transport formulae and flow resistance formulae with the condition that either the sediment transport is maximized or the channel slope minimized. The flow is assumed to be steady and uniform and bank material non- cohesive. The method has been developed for both sand and gravel channels.
6.2 River Gauging
As in the measurement of precipitation, measurement of river flow is a sampling procedure. For springs and very small streams, accurate volumetric quantities over timed intervals can be measured. For a large stream, a continuous measure of one variable, river level, is related to the discharge calculated from sampled values of other variables, velocity and depth, so that the final result is strictly an ‘estimated measurement’.
The discharge of a river, Q, is obtained from the summation of the product of mean velocities in the vertical, and related segments, a, of the total cross-sectional area, A. (Fig. 6.4). Thus:
The fixed cross-sectional area is determined with relative ease, but it is much more difficult to ensure consistent measurements of the flow velocities to obtain values of
To obtain a measured estimate of the discharge of a river, it is first necessary to choose a site or short stretch of the channel where variations in discharge will cause the least modification to the cross-section. Ideally, a site where all discharges would be contained within the banks should be used, but almost invariably severe floods exceed the maximum known flow and the river breaks out over an extended flood plain. The second major requisite of a good river gauging site is a well regulated stable bed profile.
A single estimate of river discharge can be made readily on occasions when access to the whole width of the river is feasible and the necessary velocities and depths can be measured. However, such ‘one off values are of limited use to the hydrologist. Continuous monitoring of the river flow is essential for assessing water availability; the continuous recording of velocities across a river is not a practical proposition. It is, however, relatively simple to arrange for the continuous measurement of the river level. A fixed and constant relationship is required between the river level (stage) and the discharge at the gauging site. This occurs along stretches of a regular channel where the flow is slow and uniform and the stage-discharge relationship is under ‘channel control’. In reaches where the flow is usually non-uniform, it is important to arrange a unique relationship between water level and discharge. It is therefore necessary either to find a natural ‘bed control’ as in Fig. 6.3 where critical flow occurs over some rapids with a
tranquil pool upstream, or to build a control structure across the bed of the river making the flow pass through critical conditions (Fig. 6.4(b)). In both cases, the discharge, Q, is a unique function of yc and hence of the water level just upstream of the control. In
establishing a permanent critical section gauging station, care has to be taken to verify that the bed or structural control regulates the upstream flow for all discharges. At very high flows, the section of critical flow may be ‘drowned out’ as higher levels downstream of the control eliminate the critical depth. Then the flow depths will be greater than yc
throughout the control and the relationship between the upstream water level and discharge reverts to ‘channel control’.
At a gauging site, when the flow is contained within the known cross-section and is controlled by a bed structure, then the discharge Q is a function of H (head), the difference in height between the water level upstream and the crest level of the bed control (Fig. 6.4(b)). The functional stage-discharge relationship is established by estimating Q from sampled measurements of velocity across the channel, when it is convenient, for different values of H. Regularly observed or continuously recorded stages or river levels can then be converted to corresponding discharge estimates. For a structural control, e.g. a weir built to standard specifications, the stage-discharge or Q~H relationship is known, and velocity-area measurements are used only as a check on the weir construction and calibration.
After flood flows, cross-sectional dimensions at a gauging station should be checked and if necessary, the river level-discharge relationships amended by a further series of velocity-area measurements.
The type of river gauging station depends very much on the site and character of the river. To a lesser extent, its design is influenced by the data requirements, since most stations established on a permanent basis are made to serve all purposes. Great care must therefore be afforded to initial surveys of the chosen river reach and the behaviour of the flow in both extreme conditions of floods and low flows should be observed if possible. Details of methods used for stage and discharge measurements will be given in the following sections.
6.3 Stage
The water level at a gauging station, the most important measurement in hydrometry, is generally known as the stage. It is measured with respect to a datum, either a local bench mark or the crest level of the control, which in turn should be levelled into the geodetic survey datum of the country (Ordnance Survey datum in the UK). All continuous estimates of the discharge derived from a continuous stage record depend on the accuracy of the stage values. The instruments and installations range from the most primitive to the highly sophisticated, but can be grouped into a few important categories.