**Hydraulic Jump: Formula, Rectangular, Triangular Channel and Table!**

**The Hydraulic Jump or Standing Wave****: **

We know that for a given discharge per unit width of a channel, for a given value of the specific energy head E there can be two possible depths of flow d_{1} and d_{2}.

For instance corresponding to specific energy head E = OG [Fig. 14.82], the depth of flow can be d_{1} = GH or d_{2} = GI. The depth d_{1 }is less than the critical depth and the depth d_{2} is greater than the critical depth.

ADVERTISEMENTS:

When the depth of flow is d_{1} (less than critical depth) the flow is a shooting flow. When the depth of flow is d_{2} (greater than critical depth) the flow is a streaming flow. Shooting flow is an unstable type of flow. If due to certain forced situation, a shooting flow exists in a certain region, it will ultimately convert itself into the stable streaming flow on the downstream side. During such a transformation there will occur a sudden rise in water surface. Such a sudden rise in water surface is called a standing wave or a hydraulic jump.

Note. For a hydraulic jump to occur, the existing flow should be a shooting flow i.e., the depth of flow should be less than the critical depth or the Froude number should be greater than 1.

**Depth after Hydraulic Jump in a Rectangular Channel****: **

Let q be the discharge per unit width of channel. Consider sections 1-1 and 2-2 before and after hydraulic jump. Let d_{1} and d_{2} be the depths at these sections. Let v_{1} and v_{2} be the velocities at these sections.

ADVERTISEMENTS:

Considering unit width of channel,

The loss of energy head can be determined from equation (iv) or (v). The depths d_{1} and d_{2} on either side of the hydraulic jump are called sequent depths.

We know, that as a hydraulic jump is formed the area of flow suddenly increases. Just as in the case of pipes a loss of head occurs in a sudden enlargement, we find in the case of channel flow also a loss of head occurs due to sudden increase in the area of flow brought about by the hydraulic jump.

If no losses had taken place then the specific energy head would be the same for the depths d_{1} and d_{2} before and after the hydraulic jump. Due to loss of energy head the depth d_{2} reached is less than what would have been reached in the theoretical case.

At section 1-1 the specific energy head is E_{1}. If no loss of head occurs then the depth after the jump should have been E_{1}Q. See Fig. 14.85. But due to losses the specific energy head after the jump is E_{2} and the actual depth after the jump is d_{2} = E_{2}Q’.

ADVERTISEMENTS:

In the figure E_{1}E_{2} = loss of head h_{l} due to hydraulic jump.

**The expressions obtained in the theory of hydraulic jump presented above are based on the following assumptions: **

(i) The bed of the channel is horizontal, i.e., bed slope i = 0

ADVERTISEMENTS:

(ii) Friction at bottom and sides of the channel is ignored.

(iii) The velocity is uniform at the channel section.

(iv) Depth wise pressure variation is hydrostatic.

(v) The hydraulic jump occurs abruptly.

**Hydraulic jumps may be classified based on Fraud’s number Fr _{1} upstream of the jump as given in the table below: **

**Depth of Hydraulic jump as a Function of Froude Number:**

We know that the depth of flow after the hydraulic jump is given by –

**It may be noted that: **

(i) Froude number before jump is always greater than 1.

(ii) Froude number after the jump is always less than 1.

(iii) Higher the pre jump F_{r1} lower will be post jump F_{r2}.

(iv) The hydraulic jump is an irreversible and discontinuous process.

**Height of the Standing Wave or Hydraulic Jump: **

This is the difference of water levels between two sections before and after the hydraulic jump.

Height of standing wave = (d_{2} – d_{1})

**Length of Hydraulic Jump: **

This cannot be calculated analytically. The exact point of commencement of the jump and the exact point where it ends are not well defined. For purposes of analysis we may assume the length of the hydraulic jump to be 5 to 7 times the height of the jump.

**A hydraulic jump occurs in site in the following conditions: **

(i) When water moving in shooting flow impacts with water having a larger depth with streaming flow.

(ii) On the downstream sides of sluices.

(iii) At the foot of spillways.

(iv) Where the gradient suddenly changes from a steep slope to a flat slope.

**Relation between Pre Jump and Post Jump Froude Numbers****: **

Let F_{r1} and F_{r2} be the Froude numbers before and after the hydraulic jump.

For a hydraulic jump to occur, the pre jump Froude number F_{r1 }should be greater than 1. The post jump Froude number F_{r2} will be less than 1.

The table below shows the values of the post jump Froude number F_{r2} for various values of pre jump Froude number F_{r1}.

**Hydraulic Jump in a Triangular Channel****: **

Consider a triangular channel whose vertex angle is 2θ. Let the discharge in the channel be Q. Consider sections 1-1 and 2-2 before and after the hydraulic jump.

When the discharge Q and the depth d_{1} before the hydraulic jump are known we can determine the depth d_{2} after the hydraulic jump by solving the above equation in d_{2} by trial and error.