University of the Philippines Diliman College of Engineering

Institute of Civil Engineering National Hydraulics Research Center

2nd

Term A.Y. 2012-2013

Hydraulics Laboratory Experiment #2:

Describing Hydraulic Jumps

Submitted by:

Arvin Sanchez

2009-24184

Submitted to:

Richmark Macuha C.E.

ABSTRACT

The vast applications of hydraulic jumps are of great importance in civil engineering especially in water

resources engineering. Hydraulic Jumps are usually used as energy dissipators and their turbulent

actions are used for mixing chemicals for treatment of water. The characteristics of hydraulic jumps are

described by the conjugate depth equation. Hence, the goal of this experiment is to verify the validity of

the conjugate depth equation and investigate the head loss developed during jumps. At the end of the

analysis, percentage errors less than 10% were obtained from comparing conjugate depths obtained

experimentally to the ones computed using the conjugate depth equation.

INTRODUCTION

When the spillways of dams are opened mainly by lifting its gates, the fluid that will escape would have

appreciably high velocity associated with high kinetic energy. The escaping fluid tends to erode the

surrounding masses that it will encounter (e.g. soils or parts of the masonry). Due to this problem it is

desirable to transform most of the fluid’s kinetic energy to a high potential energy instead. The solution may be achieved by converting a high velocity flow to a flow of greater depth (i.e. low velocity and

relatively low energy). Such sudden variation in open channel flow produces what we refer as hydraulic

jump.

The hydraulic jump is the phenomenon that occurs when an abrupt shift of flow velocity from high

(supercritical) to low (subcritical) exists. Hydraulic jumps are mainly used to dissipate energy in water

flowing over dams, weirs, and other hydraulic structures to prevent scouring downstream from the

structure as illustrated earlier. They are also use to maintain water level in the irrigation channels or for

other water distribution purposes. In masonry structures, hydraulic jumps are used in raising water depth

in an apron to increase its weight to be able to reduce the force of uplift acting underneath. Other

applications include aeration of water, mixing chemicals for water purification and removing air pockets

from water in supply lines.

Because of the energy losses in a hydraulic jump, analysis of flow using the energy equation can’t be used. However, the linear momentum equation can be used. In the analysis, shear forces were neglected

because of the short length of the jump.

Having noted the importance of hydraulic jump in civil engineering applications, it is of great interest to

have a study of the flow behavior in such a phenomenon. In the study of hydraulic jump in open

channels, flow patterns will be observed by measuring required parameters before and after the jump.

From the obtained parameters, we will check for the validity of the conjugate depth equation by

comparing the values of the observed and theoretical downstream to upstream depth ratio. We will also

quantify the energy lost in the jump. Lastly, conclude whether a jump will occur or not base on the

observed characteristics.

THEORY

The transition from supercritical to subcritical in a hydraulic jump is generally a turbulent process with

non-negligible energy loss. If a supercritical flow occurs (by use of hydraulic controls which in our case is

a sluice gate) with the ambient flow is the subcritical, a hydraulic jump will occur. In horizontal rectangular

channels, the relationship between the upstream and downstream depth is given by the conjugate depth

equation. In addition, the energy loss in a hydraulic jump can be shown to be a function of those two

depth referred in Figure 1 as y1 and y2.

Figure 1 Hydraulic jump in an open channel flow

In the analysis of hydraulic jumps, we assume that the length of the jump is so small that shear forces can

be neglected (i.e. friction head is not considered). The flow is assumed incompressible, uniform and the

pressure distribution is hydrostatic before and after the jump. Also, the grade of the channel’s bed is very gentle so that component of the fluid weight ceases in the direction of the jump.

Let us first define the variables that we will be using throughout this discussion. Let Q i be the volumetric

flow rate in the ith section of the control volume. The uniform velocity in section i will be denoted as vi, yi

will stand for the depth of fluid at the section. Ai is the cross sectional area of the ith section. ρ is the

density of the fluid and g is the constant assumed for the acceleration due to gravity.

Since flow is incompressible, continuity equation holds. Applying continuity equation in the upstream

section (section 1) and in the downstream section (section 2) we have: (1) (2)

Let the width of the rectangular channel be w. The area of the ith section will be wyi. Thus, the continuity

equation becomes (3)

Eliminating the constant width w we have: (4)

Newton’s second law states that the net force experienced by a body is equal to the time rate of change

of its linear momentum in the specified direction. Let x be the axis representing the direction of flow.

Applying the Linear Momentum Equation along x we have:

)(22

12

2

2

2

1

21 vvQy

gy

gFFFx (5)

Simplifying equation (5) and substituting = we have (Q now is the flow rate per unit width):

(6)

Factoring the right hand side gives us: (7)

We take the supercritical velocity as a primary parameter. Thus, we write Q in terms of and as . Equation (7) will be reduced to:

(8)

Moreover, we also take note that so that Equation (8) becomes:

(9)

Multiplying Equation (9) by will yield

(10)

Further simplification will give us: (11)

Let the Froude number denoted by Fr be defined as √ . Equation (11) now becomes

(12)

Multiplying Equation (12) by 8 and completing the square on the right hand side we have [ ] (13)

Taking roots gives us [ ] √ (14)

Algebraic manipulation leads us to the desired form of the conjugate depth equation. Take note that it

shows that the conjugate depth (i.e. the ratio between the upstream and downstream depths) is only a

function of the Froude number, Fr. √ (15)

To obtain the energy loss across a hydraulic jump in a rectangular channel, we need to compute

more precisely than what is required for the difference. A river engineer Professor Frank Henderson

(1966) suggested that a rearrangement of the expression for the energy loss obtained from specific

energy equations so that will be written in a form involving products than differences. Equation

(16) shows this point. [ ] (16)

Making a substitution , and taking out the factor leads us to

(17)

Recalling Equation (12) and setting , the bracketed expression reduces to

[ ] (18)

Simplifying further and substituting r back into the conjugate depth again we have the energy loss as a

function of the upstream and downstream depth to be (19)

EXPERIMENTAL SET-UP

Figure 2 illustrates the hydraulic flume used in this experiment. In the start of the experiment, the channel

is ensured to be in a horizontal position. After checking for a very gentle slope or nearly level channel, the

pump is started. Installed in the flume is a sluice gate that will be used to create a supercritical flow

immediately after the gate, followed by a jump, and then a subcritical flow downstream from the hydraulic

jump. Given that the created jump from controlling the two gates has become stable, the required

parameters should now be obtained. The experimenter should then measure the supercritical depth upstream and the subcritical depth downstream. The experimental group takes about 10 trials

observing the hydraulic jumps formed. The constant flow rate for each run is estimated by timing the

discharge of 15L of the fluid downstream. The time in seconds for the passage of 15L of fluid in the

spillway for every run is recorded so that the discharge in will be a derived quantity. In taking this

procedure, careful readings should be sought to minimize the trial error. If reading the discharge Q

directly via flow meter is possible, the experimenter is recommended to perform it to achieve better

accuracy. The supercritical and subcritical depths are obtained directly using the calibration in the flow

demonstrator. Upon noting the discharge, upstream depth and downstream depth, analysis of data will

include calculations of the conjugate depth and head loss for us to compare it to the theoretical values

computed using Equations (15) and (19) respectively.

Figure 2 The Hydraulic Flow Demonstrator used in the experimental set-up

TEST RESULTS

The experiment is conducted with ten trials creating hydraulic jumps at different flow rates with

corresponding pairs of upstream and downstream depths. Table 1 summarizes the obtained data from

conducting the experiment.

Table 1 Summary of Data Collected during the experiment

From this set of data we calculate the experimental conjugate depth ( from Table 1. The velocity for

each section is obtained by using where w is the width of the channel. From the obtained velocity

of the upstream section, the Froude number, Fr is calculated. Consequently, the theoretical conjugate

depth is obtained from the Froude number using Equation (15). The summary of this phase is shown in

Table 2.

RUN Volume (m3) Time (s)

Flow Rate

( )

Water Level

at the Tank

(mm) y1(mm) y2(mm)

1 0.015 23.67 0.000633714 92 10 38

2 0.015 13.98 0.001072961 88 15 44

3 0.015 13.97 0.001073729 141 13 43

4 0.015 23.2 0.000646552 66 10 36

5 0.015 9.79 0.001532176 293 11 80

6 0.015 12.33 0.001216545 148 13 62

7 0.015 14.36 0.001044568 212 10 62

8 0.015 29.56 0.000507442 88 8 35

9 0.015 22.33 0.000671742 190 6 50

10 0.015 18.98 0.000790306 263 8 54

Table 2 Calculation of the Experimental and Theoretical Conjugate Depths

RUN

(mm)

(mm)

Experimental Fr

Theoretical

%error

1 10 38 0.833833634 0.219429904 3.8 2.662680582 3.298649202 15.19867

2 15 44 0.941194187 0.320861655 2.933333333 2.453992714 3.006302965 2.427221

3 13 43 1.086770668 0.328558574 3.307692308 3.043728385 3.833424161 13.71442

4 10 36 0.850725953 0.236312765 3.6 2.716622817 3.374284329 6.68929

5 11 80 1.83274604 0.252002581 7.272727273 5.580146609 7.407342939 1.817327

6 13 62 1.231320863 0.258180181 4.769230769 3.448571414 4.402579892 8.328091

7 10 62 1.374431901 0.221682565 6.2 4.388972797 5.727051022 8.258159

8 8 35 0.834609358 0.190767853 4.375 2.979736898 3.743543797 16.86787

9 6 50 1.473118533 0.176774224 8.333333333 6.072978315 8.103030351 2.842183

10 8 54 1.299844712 0.192569587 6.75 4.640728277 6.081999535 10.98324

All Froude numbers are greater than 1which means that the upstream section is in the supercritical

velocity. It is also proven that for Fr>1, hydraulic jump is possible otherwise it is not. Most of the test runs

yield percentage errors between the theoretical and experimental ratio of depths to be less than 10%. The

great deviation reflected by percentage errors greater than 10% might come from blunders introduced by

inaccurate reading of depths or loss of accuracy in determining the flow rate. Performing a two-tailed

paired t-test on the theoretical and experimental conjugate depth, a p-value of 0.778076859 is obtained.

That is, the null hypothesis taking the difference between the population means of the theoretical and

experimental conjugate depths to be zero is plausible. To illustrate this test result further, a plot relating

the theoretical and experimental conjugate depths with the theoretical is generated. The plot shows the

deviation between the experimental and theoretical results more explicitly.

Figure 3 Plot relating experimental and theoretical

We also compute the specific energies for the upstream and downstream sections. By taking the

difference between these state functions, we obtain the experimental energy loss. Following Equation

(19) gives us the theoretical head loss. Table 3 summarizes the results of this procedure.

0

2

4

6

8

10

0 2 4 6 8 10

Co

nju

ga

te D

ep

th

Theoretical Conjugate Depth

Conjugate Depth Comparison

Theoretical Conjugate Depth Experimental Conjugate Depth

Table 3 Calculations for the Experimental and Theoretical Energy Losses

RUN

(mm)

(mm)

(m)

(m)

Experimental (m)

Theoretical (m) %error

1 10 38 0.045449339 0.04045494 0.004994399 0.014442105 65.41779122

2 15 44 0.060165602 0.049249101 0.0109165 0.009238258 18.16622638

3 13 43 0.073217836 0.048503956 0.02471388 0.012075134 104.6675421

4 10 36 0.046900198 0.038847237 0.00805296 0.012205556 34.02217416

5 11 80 0.182259199 0.083237869 0.09902133 0.09332642 6.102140492

6 13 62 0.090302191 0.065398561 0.02490363 0.036491625 31.75521917

7 10 62 0.106315411 0.064505604 0.041809807 0.056696774 26.25716748

8 8 35 0.043515328 0.036855495 0.006659833 0.017574107 62.10428685

9 6 50 0.116643197 0.051593262 0.065049935 0.070986667 8.363164738

10 8 54 0.094145436 0.055890709 0.038254727 0.056328704 32.08661999

Large percentage errors between the experimental and theoretical head losses are due to the fact that

friction losses are not negligible. Also, the specific energy equation for each section should include the

pressure term. It must also be noted that in trials 5 and 9, deviations in the conjugate depths and head

losses are the least. We can say from these facts that only the random errors greatly affect the deviation

for this runs. We can classify the kind of hydraulic jump observed per trial according to the U.S. Bureau

Reclamation (USBR). Table 4 shows the classification of jumps according to flow characteristics.

Table 4 Characteristics of Hydraulic Jump [USBR, 1955]

Classification Fr Energy Dissipation

Undular Jump 1.0-1.7 <5%

Weak Jump 1.7-2.5 5-15%

Oscillating Jump 2.5-4.5 15-45%

Steady Jump 4.5-9.0 45-70%

Strong Jump >9.0 70-85%

Now, Table 5 shows the classification of the hydraulic jumps observed in this experiment. Most of the

jumps observed were oscillatory in nature.

Table 5 Classified Hydraulic Jumps Based on Flow Characteristics

RUN Fr

Energy Dissipation

(%) Classification

1 2.662680582 10.98893639 Weak

2 2.453992714 18.14408903 Weak/Oscillating

3 3.043728385 33.75390699 Oscillating

4 2.716622817 17.17041844 Oscillating

5 5.580146609 54.32994894 Steady

6 3.448571414 27.57810125 Oscillating

7 4.388972797 39.32619629 Oscillating

8 2.979736898 15.30456864 Oscillating

9 6.072978315 55.76830588 Steady

10 4.640728277 40.63364973 Oscillating/Steady

Lastly, we want to verify if the energy curve will be asymptotic to a grade 1 line. Figure 4 depicts the non-

asymptotic specific energy curve versus stream depth.

Figure 4 Specific Energy vs. Stream Depth

CONCLUSIONS

From the results obtain from the analyses of the data, relatively low deviations were obtained for the

conjugate depths than that of the energy losses. Frictional losses and shear forces were not negligible as

assumed in the formulation of the theory. Specific energy generally decreases as stream depth increases.

Energy losses in jumps decrease as the difference between the supercritical and subcritical depths

decreases. The obtained plot for E vs. y did not asymptote a 45 degree line which signifies errors in the

obtained data.

If replication of the experimental set-up is needed, careful procurement of parameters should be observed

to attain better accuracy. Error due to parallax should be minimized as possible. Pressure terms should

be included in the calculation of specific energies and effects of shear should not be neglected.

Even though the experimental deviations do not satisfy standards, we can conclude that the objectives

given before the experiment were met. We have successfully verified the conjugate depth equation on the

experimental set-up.

0

0,05

0,1

0,15

0,2

0 0,02 0,04 0,06 0,08 0,1Sp

eci

fic

En

erg

y,

E(m

)

Depth of Stream (m)

Energy vs Depth

Supposed Asymptote Energy Curve

REFERENCES

[1] Chin, D.A. (2006) Water Resources Engineering, 2nd edition, Prentice Hall [2] Sutley, David. (2005) An Instruction Manual of Open Channel Hydraulic Experiments for Water Resources Engineering. Thesis. The University of Alabama [3] Mays, Larry (1999) Hydraulic Design Handbook, McGraw Hill. New York.

[4] Hickin, Edward J. River Hydraulics and Channel Form, Chapter 3: The Momentum Equation for open-Channel

Flow. Available from: http://www.sfu.ca/~hickin/FLUIDS/Chapt3-Momentum.pdf [Accessed: 03/20/13]

[5] Federal Highway Administration. US Department of Transportation; Hydraulic Design of Energy Dissipators for

Culverts and Channels. Hydraulic Engineering Circular No. 14, 3rd

edition Available at:

http://www.fhwa.dot.gov/engineering/hydraulics/pubs/06086/hec14ch06.cfm [Accessed: 03/22/13]