Notches can be of different shapes such as a triangular, rectangular, trapezoidal, stepped notch, etc. the bottom of the notch over which the water flows is known as crest or sill and the thin sheet of water flowing through the notch is known as nappe or vein. The edges of the notch are bevelled on the downstream side to have a sharp-edged side and crest resulting in minimum contact with the flowing fluid.

The discharge over notch is measured by measuring the head acting over the notch. As water approaches the notch, its surface becomes curved. Therefore, the head over the notch is to be measured at the upstream of the notch where the effect of curvature is minimum. Also, it should be close to the notch so that the loss of energy between the head measuring section and notch is negligible. In practice, the head over notch is measured at a distance of 3 to 4 times the maximum head from the notch.

Notches can be classified as follows:

1. Rectangular notch:

It takes its name from the shape of its notch. The discharge through a weir or notch is directly related to the water depth or head (H). This head is affected by the condition of the crest, the contraction, the velocity of the approaching stream, and the elevation of the water surface downstream from the weir.

Rectangular notch

The Cipolletti notch is trapezoidal. The slope of the sides, inclined outwardly from the crest and should be one horizontal to four vertical.

Trapezoidal notch

In this case, the notch is "V" in shape. The depth of water above the bottom of the V is called head (H). The V-notch design causes small changes in discharge hence causing a large change in depth and thus allowing more accurate measurement than with a rectangular notch.

V notch

A stepped notch is a combination of rectangular notches as shown in figure 4-6. It is thus obvious that the discharge over such a notch will be the sum of the discharges over the different rectangular notches.

Stepped notch

Weirs can be constructed out of several different materials, depending on their age and purpose. Wood, concrete, or a mixture of rocks, gravel, and boulders can all be used to construct a weir.

In a weir, the surface over which the water flows is known as the crest. The flow of water that moves over the top of this crest is known as the nappe, which is simply the water that makes it overtop the weir. This nappe does not exist with dams, as dams permit no flow of water over the structure. If this nappe falls a significant distance through the air - meaning that the weir increases the elevation of the water before the weir - the weir is said to have free discharge. However, if water flows partially underwater as a result of little elevation increase from the weir it is said to be submerged or drowned.

Weirs can be classified as follows:

1. Proportional Weir:

This is a weir whose shape is so designed that the discharge over the weir is proportional to the head of water over the crest. The crest of the weir is horizontal.

Proportional weir

2. The Cippoletti Weir:

A weir that is so compensated to make up for the loss of discharging capacity due to end contractions is called a Cippoletti weir.

Cippoletti weir

3. Submerged Weir:

Fig. shows a submerged weir. In this case, the water level on the downstream side also is above the crest of the weir.

Submerged weir

4. Anicut or Raised Weir or Barrage:

An anicut is a masonry dam provided across a river to raise the water level on the upstream side to a sufficient extent in the dry season so that the water can be carried by gravitation to places where it otherwise could not reach.

5. Broad Crested Weir:

This is a weir having a very broad sill so that the flow of water over the sill may be compared to the flow of water in a channel. Consider the broad crested weir shown in Fig.

Broad crested weir

6. Ogee Weir:

We know, in the case of a sharp-crested weir the nappe as it leaves the crest springs or rises slightly at the lower surface. In this way, it reaches a maximum rise of 0.115 H’ above the crest and then falls. (H’ is the head over the sharp crest). Suppose the space below the bottom surface of the nappe is filled with masonry or concrete.

The consequent weir formed is shown in Fig. Such a weir is called an ogee weir. Thus in an ogee weir, the solid boundary of the weir exactly coincides with the bottom surface of the nappe of the sharp-crested weir under the designed head.

Ogee weir

7. Separating Weir:

A separating weir is an arrangement provided in the case of town water supply, where it may become necessary to divert the discoloured floodwater from the supply channel. When the discharge is moderate, the water drops over the lip C into a culvert D which communicates with the supply channel.

But during floods, the velocity of flow will be greater due to greater depth and this causes the water to leap across the opening into the waste channel.

Separating weir

Let,

H = Height of water above the sill of a notch

b = Width or length of the notch

Cd = Coefficient of discharge

We know that,

The theoretical velocity of water through the strip                  

Discharge through the strip,

The total discharge over the whole notch may be found out by integrating the above equation within the limits 0 and H.

The corresponding head due to the velocity of the approach is calculated. Now a more accurate value of the discharge is calculated from the above equation. This process may be repeated to get a more accurate value of the discharge.

Consider a weir provided across a river. The water which is approaching the weir possesses a certain velocity known as the velocity of approach. The discharge over the weir is increased due to the velocity of the approach.

Let va be the velocity of approach. This condition is equivalent to a condition in which the free water surface is static at a higher level, at a height ha = va2/2g. The head ha is called the head due to the velocity of approach. If the head of the actual flowing water level over the crest is H, the weir opening may be taken as equivalent to a large orifice in which the heads of still water above the top and bottom edges are respectively, ha and H + ha.

Accordingly, the discharge is given by,

The quantity H + ha is called still water head

Broad crested weirs are robust structures that are generally constructed from reinforced concrete and which usually span the full width of the channel.  They are used to measure the discharge of rivers and are much more suited for this purpose than the relatively flimsy sharp-crested weirs.  Additionally, by being a critical depth meter, the broad crested weir has the advantage that it operates effectively with higher downstream water levels than a sharp-crested weir.

Only rectangular broad crested weirs will be considered, although there are a variety of possible shapes: triangular, trapezoidal, and round crested all being quite common.  If a standard shape is used when there is a large body of literature available relating to their design, operation, calibration, and coefficient of discharge.  However, if a unique design is adopted, then it will have to be calibrated either in the field by river gauging or using a scaled-down model in the laboratory.

Head-discharge relationship 

A rectangular broad crested weir is shown above.  When the length, L, of the crest is greater than about three times the upstream head, the weir is broad enough for the flow to pass through critical depth somewhere near its downstream edge.  Consequently, this makes the calculation of the discharge relatively straightforward.  Applying the continuity equation to the section on the weir crest where the flow is at critical depth gives:                               

Q = Ac Vc.

Now assuming that the breadth of the weir (b) spans the full width (B) of the channel and that the cross-sectional area of flow is rectangular, then: 

                        Ac = b x Dc and Vc = (g x Dc)1/2    

 Thus from the continuity equation, 

However, equation 1 does not provide a very practical means of calculating Q.  It is much easier to use a stilling well located in a gauging hut just upstream of the weir to measure the head of water, H1, above the crest than to attempt to measure the critical depth on the crest itself.  To eliminate Dc from the equation, we can use the fact that in a rectangular channel .  Using the weir crest as the datum level, and assuming no loss of energy, the specific energy at an upstream section equals that at the critical section: 

            So,                             and      

            Thus,               

If you substitute this expression into Eqn 1, it gives: 

The term in the above equation is the velocity head of the approaching flow.  As with the rectangular sharp-crested weir, the problem arises that the velocity of approach, V1 cannot be calculated until Q is known, and Q cannot be calculated until V1 is known.  A way around this is to involve an iterative procedure, but in practice, it is often found that the velocity head is so small as to be negligible.  Alternatively, a coefficient of discharge, C, can be introduced into the equation to allow for the velocity of approach, non-parallel streamlines over the crest, and energy losses.  C varies between about 1.4 and 2.1 according to the shape of the weir and the discharge, but frequently has a value of about 1.6. Thus: 

The formulation for the discharge coefficient was derived from experimental data, which should be a function of the dimensionless parameters listed in the discharge equation given by:

According to experimental data distribution and Equation, the best fit function of the discharge coefficient was assumed to be of the following form:

 Cd = k1(h/P)k2                                                                                                                                  

where k1 and k2 are parameters that are calculated by r/b. It can be seen that when r/b increases from 0.30 to 0.75, k1 increases, and k2 decreases. b is the half-width of the round weir, and the half-width of the weir near the downstream was not in contact with the flow and did not affect the nappe. The equations for k1 and k2 can be written as follows:

k1 = f1(r/b)                                                                                            

k2 = f2(r/b)                                                                                             

Based on a multivariate optimization approach, the best-fit parameters can be derived by using five data sets of the present experiment in the range of a round-crested weir, and two calculated expressions can be obtained:

k1 = 0.05 r/b + 0.645                                                                            

k2 = −0.075 r/b + 0.0055                                                                      

Incorporating the previous two equations into the flow rate equation, the formula for the discharge coefficient can be written as

Cd = (0.05 r/b + 0.645) × (h/P) (−0.075 r/b +0.0055)                                 

where h = 0.05–0.20 m, and r/b = 0.30–0.75.

Let a reservoir of plan area A be provided with a rectangular weir of length I. Let it be required to find the time taken for the head of water over the weir to fall from a value H1 to a value H2.

Let at any instant the head of water over the weir be h. Let the fall in water level be dh in a small interval of time dt

 Time of Emptying a Reservoir by a Triangular Notch:

 Let a reservoir of plan area A be provided with a triangular notch of angle θ. Let   it be required to find the time taken for the head of water over the notch to fall from a value H1 to a value H2. Let at any instant the head of water over the notch be h. Let the fall in water level be dh in a small interval of time dt.

Solution:

The first weir is contracted, i.e. both ends contracted. 

Given, 

           Length of the weir (L) = 120 cm

           Depth of water over the weir (H) = 30 cm

           So, Discharge of flow   = 

                                              = 344.67 L/sec

In the second weir, length of the weir (L) =150 cm

Discharge through the first weir and second weir is the same.

Let assume, depth of water over the second weir is = H cm

Now,                                                

H= 24.97 cm

Solution:

Given,

Crest width (L) = 60 cm

                 Head of flow over the crest (H) = 30 cm

So, discharge through the weir (Q) = 

  = 183.37 L/sec

The discharge of the Cipoletti weir is 183.37 L/sec