Hydraulic Jump: Formula, Rectangular, Triangular Channel and Table!
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 d1 and d2.
For instance corresponding to specific energy head E = OG [Fig. 14.82], the depth of flow can be d1 = GH or d2 = GI. The depth d1 is less than the critical depth and the depth d2 is greater than the critical depth.
When the depth of flow is d1 (less than critical depth) the flow is a shooting flow. When the depth of flow is d2 (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.
Let q be the discharge per unit width of channel. Consider sections 1-1 and 2-2 before and after hydraulic jump. Let d1 and d2 be the depths at these sections. Let v1 and v2 be the velocities at these sections.
Considering unit width of channel,
The loss of energy head can be determined from equation (iv) or (v). The depths d1 and d2 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 d1 and d2 before and after the hydraulic jump. Due to loss of energy head the depth d2 reached is less than what would have been reached in the theoretical case.
At section 1-1 the specific energy head is E1. If no loss of head occurs then the depth after the jump should have been E1Q. See Fig. 14.85. But due to losses the specific energy head after the jump is E2 and the actual depth after the jump is d2 = E2Q’.
In the figure E1E2 = loss of head hl 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
(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 Fr1 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 Fr1 lower will be post jump Fr2.
(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 = (d2 – d1)
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.
Let Fr1 and Fr2 be the Froude numbers before and after the hydraulic jump.
For a hydraulic jump to occur, the pre jump Froude number Fr1 should be greater than 1. The post jump Froude number Fr2 will be less than 1.
The table below shows the values of the post jump Froude number Fr2 for various values of pre jump Froude number Fr1.
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 d1 before the hydraulic jump are known we can determine the depth d2 after the hydraulic jump by solving the above equation in d2 by trial and error.