Weirs are structures consisting of an obstruction across an open channel usually with a specially shaped opening or notch. The weir results in an increase in the water level which is measured upstream of the structure. This increase in water level can be the desired effect, the weir can be used to calculate the flow rate as a function of the head on the weir (this is the most common use of weirs), or the weir can be used to control the release of water. Most weirs are plates (exceptions include the broad-crested weir) or concrete (spillways).

 Understanding Weirs

Spillways, a subset of weirs, are structures consisting of an obstruction across an open channel or body of water that are designed to control the release of water. Spillways are usually concrete and attempt to reduce water separation by taking a form that matches the underside of the nappe. Broad-crested weirs can function as spillways except their form doesn't match the underside of the nappe (like ogee spillways). ### Definitions

• nappe: the water that falls over a weir
• contraction: the decrease in width of the nappe as it falls over the weir; sometimes used to refer to a weir that causes the contraction (i.e. contracted weir)
• suppressed: when the weir extends the full width of the channel, the contractions are suppressed; used to refer to the weir

## Rectangular Weirs

Theoretical flow over a rectangular weir is calculated as:

Q = \frac{2}{3}Cb\sqrt{2g}[(H+h)^{3/2}-h^{3/2}]

where:

• Q = discharge (cfs);
• C = weir coefficient;
• b = width of weir (ft)
• g = acceleration due to gravity (32.2 ft/sec2)
• H = head on weir above weir crest (ft)
• h = velocity head upstream of weir (ft)

Remember, velocity head (ft) is calculated as:

h_v = \frac{v^2}{2g}

where:

• v = velocity (fps)

### Francis Formula

The Francis formula is used to calculate flow over a weir:

Q = 3.3(b-0.1nH)[(H+h)^{3/2}-h^{3/2}]

where:

• n = number of contractions

### Neglecting Approach Velocity

Approach velocity is often negligible (or can be assumed zero and then solved iteratively). In such cases, if the rectangular weir is fully contracted, then the discharge can be calculated as:

Q = 3.33(b-0.2H)H^{3/2}

### When a Weir is Fully Suppressed

and if the rectangular weir is fully suppressed, then the discharge can be calculated as:

Q = 3.33bH^{3/2}

Warning: If the nappe is not ventilated, the water will not jump free on the crest to create a bottom contraction. In this case, the discharge will be greater than calculated.

## Triangular Weirs

The triangular weir has a V-notch opening and is often used for low flow measurements where a rectangular weir wouldn't allow the nappe to spring clear. The theoretical formula is:

Q = \frac{8}{15}C \left(tan{\frac{\theta}{2}}\right) \sqrt{2g} H^{5/2}

where

• θ = vertex angle (degrees)
• C = weir coefficient which depends on head and θ. For θ=90°, C=0.59

## Trapezoidal Weirs

Many configurations of trapezoidal weirs are used, but the most common is the Cipolletti Weir which was a side slope of 0.25 (1/4 horizontal to one vertical) which increases the flow over that of a rectangular weir of the same width. This has the effect of offsetting the decrease in discharge from the end contractions and the formula for flow is:

Q = 3.367bH^{3/2}

## References

1. William George Bligh, "Damns and Weirs", 1915 (Found online here)

Main