A weir is a three-dimensional *channel control*. A control is any channel feature that fixes a relationship between flow rate and depth in its neighborhood. Weirs are also used for flow measurement. The main features of weirs can be explained by considering two-dimensional flows. The cross section of a simple weir is shown in Figure 1. A uniform flow with velocity U approaches a weir of height Z. The level of the free surface above the crest of the weir is z. The length of the crest is 1.

The flow rate per unit width over a weir, defined by , can be written as

Here H = z + U^{2}/2g is the total head, C is the *discharge coefficient*, and g is the acceleration due to gravity. The coefficient C, which depends on the weir geometry, must usually be determined experimentally. For some simple weir geometries, it can be determined numerically. The upstream Froude Number is Fr = U[g(z + Z)]^{–1/2}.

The equation for the flow rate can be explained by considering a weir with a crest long enough to maintain a hydrostatic pressure distribution in the flow over it. Experiment shows that this flow becomes critical, i.e., local Froude number becomes equal to unity. It follows that the height of the free surface above the crest is 2H/3, which is referred to as the critical depth, and that the velocity there is [g(2H/3)]^{1/2}. Thus the flow rate is

for critical flow. This shows that for *critical flow* the discharge coefficient is C = 1.

Weirs providing critical flow over a long enough distance could be used for flow-rate measurement with C = 1. Since practical weirs are usually not long enough, the discharge coefficient C must be determined for them. Clearly C depends upon the two dimensionless parameters z/Z and z/l in agreement with observation.

Typical shapes for weirs are shown in Figure 2. These include thin weirs, rectangular weirs (not shown in Figure 2 since already shown in Figure 1), triangular weirs, round-nosed broad-crested weirs, trapezoidal *flumes*.

**Figure 2. From Ackers, P., White, W. R., Perkins, J. A. and Harrison, A. J. M. (1978), reprinted by permission of John Wiley and Sons, Ltd.**

A detailed description of all these weirs is provided in the book by Ackers, White, Perkins and Harrison (1978). Numerical calculations on weirs have not been much developed yet. It is still a challenge to provide an accurate model for weir flows, which includes the effects of the separation at corners, the three dimensions, the sediment transport, the friction along the crest of the weir.

#### REFERENCES

Ackers, P., White, W. R., Perkins, J. A. and Harrison, A. J. M., (1978) *Weirs and Flumes for Flow Measurement*, Wiley, Chichester.