WRE 301: Open Channel Hydraulics

Chapter 3
Hydraulic Jumps
Hydraulic Jumps
 Hydraulic Jumps

o Hydraulic jump is a phenomena of open channel flow.

o When unstable rapid supercritical flow transforms to

stable uniform subcritical flow then the result is usually an
abrupt rise of water surface. This feature is known as
hydraulic jump.

o It usually occurs when two controls one from u/s and the
other from d/s operate on the same reach.

o The u/s control produces supercritical flow downstream of

the control and the d/s control produces subcritical flow
upstream of the control.
 Hydraulic Jumps

Subcritical
flow

u/s control
spillway Supercritical
flow
 Hydraulic Jumps
o Hydraulic jumps normally occur
o At the foot of the spillway
o Behind a sluice gate
o When a steep slope channel is followed by a mild or horizontal or adverse slope

o Initial Depth: the depth of flow before the jump is known as the initial
depth

o Sequent depth: the depth of flow after the jump is known as sequent
depth or conjugate depth.

o The strength of hydraulic jump is determined by the Froude number of

the flow before the jump i.e supercritical flow.

o The hydraulic jump is accompanied by huge energy dissipation

o Kinetic energy transforms to potential energy

Hydraulic Jumps
Hydraulic Jumps
 Application of Hydraulic Jumps
Practical application of hydraulic jumps in the field of open channel flow
includes
o The dissipation of kinetic energy in high velocity flows over weirs,
spillways, gates and other hydraulic structures to prevent scouring
downstream.

o The increase of depth of water in channels for irrigation and water

distribution purposes

o The increase of discharge of a sluice gate by repelling the tailwater so

it works under free flow condition

o The reduction of uplift pressure under a structure by increasing weight

on it’s apron

o The mixing of chemicals for water purification and wastewater

treatment
 Classification of hydraulic jumps
The United States Bureau of Reclamation (USBR)
classified the hydraulic jumps in horizontal rectangular
channels into the following five categories according to
the Froude number of the incoming flow.

1. Undular jump
2. Weak jump
3. Oscillating jump
4. Steady jump
5. Strong jump
 Classification of hydraulic jumps
1. Undular Jump:

o For 1.0< 𝐹𝑟1 <1.7

o Water surface shows slight undulations
o The two sequent depths are close
o Transition is not abrupt, gradual transition from initial depth to final
depth
o Slightly ruffled water surface
o Energy loss less than 5%
 Classification of hydraulic jumps
2. Weak Jump:

o For 1.7< 𝐹𝑟1 <2.5

o Eddies and rollers are formed on the surface
o The ratio of final to initial depth are is between 2.1 to 3.0
o Low energy dissipation
o Fraction of energy dissipation 5-15%
 Classification of hydraulic jumps
3. Oscillating Jump:

o For 2.5< 𝐹𝑟1 <4.5

o The entering jet oscillates back and forth from the bottom to the
surface and back again
o These oscillations are very common in canals and can travel a
considerable distance damaging earthen banks
o The ratio of final to initial depth are is between 3.1 to 5.0
o Fraction of energy dissipation 15-45%
 Classification of hydraulic jumps
4. Steady Jump:

o For 4.5<𝐹𝑟1 <9.0

o Stable jump
o The position of jump is fixed regardless of the downstream conditions
o Considerable rise in d/s water level
o Intense eddy motion , high level energy dissipation
o The ratio of final to initial depth are is between 5.9 to 12.0
o Fraction of energy dissipation 45-70%
 Classification of hydraulic jumps
5. Strong Jump:

o For 𝐹𝑟1 >9.0

o Jump becomes increasingly rough
o The Froude number should not be allowed to exceed 12 otherwise the
required stilling basin would be massive
o The ratio of final to initial depth > 12.0
o Ability of jump to dissipate energy is massive
o Fraction of energy dissipation 70-85%
Classification of hydraulic jumps
Momentum in open channel
Specific Force
Specific Force
 Specific Force
➢ Specific force F is a function of depth of flow, channel
geometry and discharge and is given by,

2
➢ The term 𝑄 ൗ𝑔𝐴 in the equation represents the momentum of flow
passing through a channel section per unit time per unit weight of
water
➢ The term 𝑧Aҧ represents the pressure force per unit weight of
water
➢ Specific force represents the force per unit weight of water
Sequent depths of Hydraulic Jump
 Sequent depths of Hydraulic Jump
Assumptions:

➢ The flow is uniform, and the pressure distribution is hydrostatic

before and after the jump

➢ Loss of head due to friction is negligible

➢ Channel is horizontal so the weight component in the direction of

flow is zero

➢ The momentum correction factor is unity

 Sequent depths of Hydraulic Jump
Applying the momentum equation in the control volume,

𝑃1 − 𝑃2 + 𝑤𝑠𝑖𝑛𝜃 − 𝐹𝑓 = 𝜌𝑄(𝑉1 − 𝑉2 )

For a short reach prismatic horizontal channel , 𝜃=0, neglecting

friction 𝐹𝑓 =0
Then,
𝑃1 − 𝑃2 = 𝜌𝑄(𝑉1 − 𝑉2 )

Considering, a rectangular channel with unit width discharge per

unit width q,

𝑄
𝑞 = = 𝑉1 𝑦1 = 𝑉2 𝑦2
𝑏
 Sequent depths of Hydraulic Jump
Here,
𝑃1 = ℎ𝑦𝑑𝑟𝑜𝑠𝑡𝑎𝑡𝑖𝑐 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑓𝑜𝑟𝑐𝑒 𝑜𝑛 𝑠𝑒𝑐𝑡𝑖𝑜𝑛 1
= 𝛾𝐴1 𝑧ഥ1
𝑦1
= 𝛾(1. 𝑦1 ) 2
1
= 2
𝛾𝑦12
Similarly ,
1
𝑃2 = 2 𝛾𝑦22
Substituting the values in the equation,
1 1
𝛾𝑦12 − 𝛾𝑦22 = 𝜌𝑞(𝑉1 − 𝑉2 )
2 2
From continuity equation,
𝑞 = 𝑉1 𝑦1 = 𝑉2 𝑦2
𝑞 𝑞
𝑉1 = 𝑦 , 𝑉2 = 𝑦
1 2
 Sequent depths of Hydraulic Jump
Substituting the values of 𝑉1 and 𝑉2 in the equation,
1 1 𝑞 𝑞
2
𝛾𝑦12 − 2 𝛾𝑦22 = 𝜌𝑞(𝑦 − 𝑦 )
1 2
1 1 𝑞 𝑞
2
𝜌𝑔𝑦12 − 2 ρ𝑔𝑦22 = 𝜌𝑞(𝑦 − 𝑦 )
1 2
1 2 2 𝑞 𝑞
2
𝜌𝑔(𝑦1 − 𝑦2 ) = 𝜌𝑞(𝑦 − 𝑦 )
1 2
𝑔 𝑦1 − 𝑦2
(𝑦1 − 𝑦2 )(𝑦1 + 𝑦2 ) = 𝑞2 ( )
2 𝑦1 𝑦2

2𝑞2
= 𝑦1 𝑦2 𝑦1 + 𝑦2
𝑔

Dividing both sides by 𝑦1

2𝑞2
= 𝑦22 + 𝑦1 𝑦2
𝑔𝑦1
2
2𝑞2
𝑦2 + 𝑦1 𝑦2 − =0
𝑔𝑦1
 Sequent depths of Hydraulic Jump
Solving for 𝑦2
2𝑞2
−𝑦1 ± 𝑦12 −4𝑔𝑦
1
𝑦2 =
2

As negative root is not possible

2𝑞2
−𝑦1 + 𝑦12 −4𝑔𝑦
1
𝑦2 =
2

𝑦1 𝑦1 2 2𝑞 2
𝑦2 = − + +
2 2 𝑔𝑦1

This is the relationship between conjugate/sequent depth

 Sequent depths of Hydraulic Jump
Conjugate depth in terms of Froude number
Here,
𝑦1 𝑦1 2 2𝑉12 𝑦12 𝑞 = 𝑉1 𝑦1
𝑦2 = − + +
2 2 𝑔𝑦1 2
2
𝑉 1
𝑦1 𝑦1 2 8𝑉12
𝐹𝑟1 =
𝑦2 = − + 1+ 𝑔𝑦1
Multiplying 2 2 𝑔𝑦1
nominator and 𝑦1 𝑦1 2

denominator by 4
2
𝑦2 = − + 1 + 8𝐹𝑟1
2 2

𝑦1
𝑦2 = (−1 + 2
1 + 8𝐹𝑟1 )
2
Similarly, for 𝑦1
𝑦2
𝑦1 = (−1 + 2
1 + 8𝐹𝑟2 )
2

These equations are known as Belanger momentum equation

 Energy loss in Hydraulic Jump
Energy loss in a hydraulic jump is the difference between total energies
before and after the jump.
For a jump in horizontal rectangular channel,
 Energy loss in Hydraulic Jump

The ratio of energy loss to upstream specific energy is known as relative

energy loss

∆𝐸 𝐸
= (1 − 2ൗ𝐸 )
𝐸1 1
Efficiency of Hydraulic Jump
Height of Hydraulic Jump
Length of Hydraulic Jump
Length of Hydraulic Jump
 Length of Hydraulic Jump

This curve has a fairly horizontal portion where,

𝐿𝑗 /ℎ2 ≈6.0 to 6.1
In the range of Froude numbers (𝐹𝑟1 ≈ 5 𝑡𝑜 13)

Experimentally for rectangular channel,

Silvester (1964) demonstrated that for free hydraulic jumps

in horizontal rectangular channels
Length of Hydraulic Jump
Submerged Hydraulic Jumps
Submerged Hydraulic Jumps
 Submerged Hydraulic Jumps

The length of a submerged hydraulic jump in a horizontal rectangular

channel is estimated by the empirical equation

The length of submerged jump exceeds the length of free jump

by the amount 4.9S
 Examples

Water flows in a horizontal rectangular channel 6m wide at a depth

of 0.52 and a velocity of 15.2 m/s. If the hydraulic jump forms in
this channel , determine

i. the type of jump

ii. The downstream depth needed to form the jump
iii. The horsepower dissipation in the jump
iv. The efficiency of the jump
v. The relative height of the jump
vi. The length of the jump
Examples
Examples
Jumps in horizontal non-rectangular
channels
Jumps in horizontal non-rectangular
channels
Jumps in horizontal non-rectangular
channels
Jumps in horizontal non-rectangular
channels
Jumps in horizontal non-rectangular
channels
Examples
Examples

𝑦1 𝑦1 2 2𝑞 2
𝑦2 = − + +
2 2 𝑔𝑦1
Examples
Examples
Hydraulic jump behind the sluice gate
Examples
 Jumps on Sloping Channels

➢ The analysis of hydraulic jump in sloping channel can be described

using the general momentum equation

➢ It is essential to consider the weight of the water component in

the direction of flow

➢ There are too many unknowns relative to the number of available

equations

➢ Additional information is needed

➢ Hydraulic jumps on sloping channels occur in various forms

depending on the location shown in figure
 Jumps on Sloping Channels
Let,
ℎ𝑡 =tailwater depth( depth produced by downstream control)
ℎ1 =the supercritical depth of flow on the slope which is assumed to be
constant
ℎ2 =the sequent depth corresponding to ℎ1 when the jump occurs on
horizontal channel
ℎ2∗ =sequent depth when the jump occurs on sloping channel

Type A jump:

When ℎ𝑡 ≤ ℎ2
the jump forms on the horizontal bed .
 Jumps on Sloping Channels
Type B jump:

When ℎ2∗ > ℎ𝑡 > ℎ2

the toe of the jump is on the slope and the end of the jump is on the
horizontal bed .

Type C jump:

When ℎ2∗ = ℎ𝑡 > ℎ2

the end of the jump coincides with the sloping and the horizontal bed .
 Jumps on Sloping Channels
Type D jump:

When ℎ𝑡 > ℎ2∗ > ℎ2

A type D jump occurs completely on the sloping channel.

Type E jump:

Type E jumps occur on sloping beds which have no break in the slope.
 Jumps on Sloping Channels
Type F jump:

o Rare type of jump

o Forms on adverse slopes channel
o Normally found in stilling basins below drop structures
 Jumps on Sloping Channels

The forces acting on a type E jump on a rectangular channel are

shown below

Considering unit width of the channel and all forces acting parallel to
the channel bottom , the momentum equation may be applied
 Jumps on Sloping Channels
Now,
𝑞 = 𝑈1 𝑑1 = 𝑈2 𝑑2
𝑑1
𝑈2 = 𝑈1
𝑑2
𝐹𝑝1 = 0.5𝜌𝑔𝑑12 𝑐𝑜𝑠𝜃
𝐹𝑝2 = 0.5𝜌𝑔𝑑22 𝑐𝑜𝑠𝜃
𝑑1 = ℎ1 𝑐𝑜𝑠𝜃
𝑑2 = ℎ2∗ 𝑐𝑜𝑠𝜃

Assuming,
The friction force 𝐹𝑓 is negligible
𝛽1 𝑎𝑛𝑑 𝛽2 to be unity
If the jump profile is a straight line, W can be computed easily.
 Jumps on Sloping Channels
The difference between the straight line and the actual profile and the
effect of slope may be corrected by a factor Γ

The equation becomes,

𝑊 = 0.5Γ𝜌𝑔𝐿𝑗 (𝑑1 + 𝑑2 )

Substituting this in the momentum equation and simplifying it can be

shown that

ℎ2∗ 𝑑2 1
= = 1 + 8𝐺 2 − 1
ℎ1 𝑑1 2
Where,

𝐹𝑟1 𝑈1
𝐺= and 𝐹𝑟1 =
Γ𝐿𝑗 𝑠𝑖𝑛𝜃 𝑔𝑑1
𝑐𝑜𝑠𝜃− 𝑑 −𝑑
2 1
 Jumps on Sloping Channels
Here, Γ and 𝐿𝑗 are functions of 𝐹𝑟1 and θ

Hence,
ℎ2∗ 𝑑2
G, and are functions of 𝐹𝑟1 and θ
ℎ1 𝑑1

The term G can be computed using the empirical relationship by

Rajaratnam 1967

𝐺 2 = 𝑘12 𝐹𝑟1
2

Where,
𝑘1 = 100.027𝜃 and θ in degrees
 Jumps on Sloping Channels
Length of the Jump :

The length of the jump on sloping floor is longer than the

corresponding length of the jump on horizontal floor. The variation of
𝐿𝑗
𝑦
with 𝐹𝑟1 for any θ is similar to the variation of θ=0.
2

The variation can be approximated by

In the range of 4.5< 𝐹𝑟1 < 13.0

 Jumps on Sloping Channels
Length of the Jump :
Elevetorski expressed the jump length as

Where, 𝑚𝑠 is a function of θ
The variation of 𝑚𝑠 with tan θ is shown below. for 𝑚𝑠 =6.9 , tan θ is
zero. And the value decreases with increase in channel slope
 Jumps on Sloping Channels
Length of the Jump :

For sloping channels, the length of the jump can be calculated from this
graph
 Jumps on Sloping Channels
Energy loss of the Jump :

Knowing the sequent depths and length of the jump the energy loss
can be calculated
Jumps on Sloping Channels
Jumps on Sloping Channels
Jumps on Sloping Channels
Jumps on Sloping Channels
 Jumps on Sloping Channels

Example:

A rectangular channel is 1 m wide and inclined at an angle of 3.5

degree with the horizontal. Determine the type of the jump when
the discharge is 0.15 𝑚3 /s, the initial depth of flow section is 0.02
m and the tailwater depth is 0.70 m. Also compute the energy loss
in the jump if the length of the jump is 2m
Jumps on Sloping Channels
Jumps on Sloping Channels
Workout Problems
Workout Problems
Workout Problems
Workout Problems
Workout Problems
Stilling Basins
 Stilling Basins
o A stilling basin is a short length of fully paved channel

o Placed at the end of a spillway or any other hydraulic structure

where supercritical flow occurs

o Energy dissipation is deliberately allowed to occur

o The basin has additional appurtenances such as baffle blocks, sills to

aid efficient performance

o Stilling basins are so designed not only a good jump with high energy
dissipation characteristics is formed within the basin but it is also
stable

o For economic considerations the basin should be as small as

practicable
 Stilling Basins: Appurtenances
Chute Blocks:

o These blocks are used prior to the commencement of the stilling

basin
o They are located in the approach channel
o Main purpose is to lift and spilt the oncoming high velocity flow to
enable good interaction in the jump
o These blocks help stabilize the jump, shorten the length and improve
their performance
o Chute blocks are usually provided in one row with uniform gaps
 Stilling Basins: Appurtenances
Baffle Piers:

o These blocks are also known as baffle blocks and floor blocks
o They are specially shaped blocks placed in one row in the middle
region of the channel
o They offer resistive force to high velocity incoming flow and assist
in crating a forced hydraulic jump within the basin

o Out of the three USBR basins only type III basins uses these blocks
 Stilling Basins: Appurtenances
Baffle Piers:

o These blocks are useful in small and medium approach velocities

o In very high approach velocities, they suffer from the possibilities
of cavitation
o Therefore, the baffle piers are used when the incoming velocity is
less than 16m/s
o The height of the baffle piers ℎ3 is a function of approach Froude
number 𝐹1 and depth of flow 𝑦1 and can be calculated
ℎ3
= 0.168𝐹1 + 0.38
𝑦1
 Stilling Basins: Appurtenances
End sill:

o The end sill is provided at the end of the basin

o The end sill assists in controlling the formation of the jump within
the basin
o Also controls the scour in the channel downstream of the basin
o The height of the end sills ℎ4 is a function of approach Froude
number 𝐹1 and depth of flow 𝑦1 and can be calculated
ℎ4
= 0.0536𝐹1 + 1.04
𝑦1
 Stilling Basins: Appurtenances
End sill:
 Stilling Basins of Generalized Design

o Designing a stilling basin for a given hydraulic structure involves

considerations of parameters peculiar to the location of the
structure in addition to the mechanics of flow

o This feature makes the engineering design rely heavily on the

experience of the designer

o Model studies are usually done to achieve efficient design

o To assist preliminary design type designs are available

o Generalized stilling basin designs developed by USBR are the

most extensively used ones.

o USBR basin types II, III, IV will be discussed here

 USBR Basin Type II

o The USBR type II basin is recommended for use in large

structures such as spillways of large dams and large canal
structures
o Incoming Froude number𝐹𝑟1 >4.5
o Incoming velocity 𝑈1 >18m/s
o The length of the basin is reduced by 33% with the provision of
chute blocks at u/s and dentated sill at the d/s end
o The design may be safe for and conservative for spillways with
fall up to 60 m and flows up to 46 𝑚3 /s per m of basin width
o The elevation of the basin should never be designed for less
than sequent depth and a minimum safety factor 5% should be
added
o The length of the basin is obtained from the graph
o The height, width, spacing of chute blocks is equal to ℎ1
o A space equal to 0.5 ℎ1 is provided along the basin wall
o The height of the dentated sill is 0.2ℎ2
o The width and spacing of the dentate is 0.15 ℎ2
USBR Basin Type II
 USBR Basin Type III

o The USBR type III basin is an improved version of type II

o Incoming Froude number𝐹𝑟1 >4.5
o Incoming velocity 𝑈1 <18m/s
o The major change is the introduction of a row of baffle pires
at around the mid length of the basin
o Due to the additional resistance to flow offered by these floor
blocks the length of the type III basin is shorter than that of
typeII
o The basin length can be reduced about 60%
o The basin is meant for use in medium sized spillways
USBR Basin Type III
 USBR Basin Type IV

o The USBR type IV basin is recommended for use with jumps

having 𝐹𝑟1 in the range 2.5 to 4.5

o Used for small size canals and small spillways

o In addition to providing the requisite energy dissipation the

design is specially aimed at reducing the formation of waves
post jump in the channel d/s
USBR Basin Type IV
Example
Example
 Example

1.415m
2.61 m
2.83 m
2.61 m
2.83 m

3.48 m

64.4 m
Dimensions of USBR Stilling Basin type III

Assume a velocity value 15.4 m/s for the previous design

problem and design a Type III basin. You also have to
change other design values to adjust the Froude number.

ℎ3
= 0.168𝐹1 + 0.38
𝑦1

W3=S3=0.75 h3