This article was downloaded by: [University of Southern Queensland]
On: 11 October 2014, At: 19:36
Publisher: Routledge
Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,
37-41 Mortimer Street, London W1T 3JH, UK
Water International
Publication details, including instructions for authors and subscription information:
http://www.tandfonline.com/loi/rwin20
Energy Loss at a Drop Structure with a Step at the Base
a
a
Ismail I. Esen , Jasem M. Alhumoud & Khoanddkar A. Hannan
a
b
Kuwait University , Kuwait
b
Public Authority for Housing Welfare , Kuwait
Published online: 22 Jan 2009.
To cite this article: Ismail I. Esen , Jasem M. Alhumoud & Khoanddkar A. Hannan (2004) Energy Loss at a Drop Structure with
a Step at the Base, Water International, 29:4, 523-529, DOI: 10.1080/02508060408691816
To link to this article: http://dx.doi.org/10.1080/02508060408691816
PLEASE SCROLL DOWN FOR ARTICLE
Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained
in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no
representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the
Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and
are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and
should be independently verified with primary sources of information. Taylor and Francis shall not be liable for
any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever
or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of
the Content.
This article may be used for research, teaching, and private study purposes. Any substantial or systematic
reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any
form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://
www.tandfonline.com/page/terms-and-conditions
International Water Resources Association
Water International, Volume 29, Number 4, Pages 523–529, December 2004
 2004 International Water Resources Association
Technical Note
Energy Loss at a Drop Structure with a Step at the Base
Downloaded by [University of Southern Queensland] at 19:36 11 October 2014
Ismail I. Esen and Jasem M. Alhumoud, Kuwait University, Kuwait, and
Khoanddkar A. Hannan, Public Authority for Housing Welfare, Kuwait
Abstract: The flow over a drop structure placed in a rectangular channel was investigated through
an experimental program. It was noted that the downstream depth of flow was crucial to the formulation
of the problem. Two procedures were developed for the estimation of the downstream depth. The first
procedure was physically based with an empirical component for the estimation of the depth of the pool
formed at the base of the drop. The second was a purely empirical procedure, which resulted in an
equation for the direct estimation of the downstream depth. The parameters of the equation were determined by least-squares techniques. Both procedures resulted in reasonably accurate estimates of the
downstream depth. The energy loss at the drop was then calculated and compared with the results of
previous studies. The investigations were repeated with a single step placed at the base of the drop. It
was observed that the step significantly increased the energy loss at the drop.
Keywords: drop structures, downstream depth, energy dissipation, energy loss, step
Introduction
Vertical drop structures with a free overfall are usually built in irrigation canals where the slope of the canal is
less than the ground slope. Flow over a drop results in an
energy loss due to the mixing of the falling jet with the
recirculating flow in the pool of water formed at the base
of the drop. Previous studies made by Moore (1943), White
(1943), Rand (1955), Gill (1979), and Rajaratnam and Chamani
(1995) contributed towards the evaluation of this loss.
The occurrence of this loss at the base of a drop is
usually beneficial in the hydraulic design of open channels
in which low flow velocities are required. In fact, in certain cases it may be desirable to increase the energy loss
as much as possible. For this reason, we attached a single
step to the base of the drop. This step had a square crosssection and extended through the entire width of the channel. It effectively pushed the overfalling jet of water
downstream, reduced the depth of water in the pool, increased the downstream depth of flow, and increased the
energy loss through the drop.
Previous studies made on the subject mostly emphasized the estimation of the energy loss. In this study, we
used the procedure proposed by Rajaratnam and Chamani
523
(1995) to predict the downstream depth of flow for drops
with and without a step. This procedure required knowledge of the pool depth, and empirical equations were developed for this purpose. Alternately, we developed entirely
empirical equations for the direct determination of the
downstream flow depth. The head loss was then calculated by the energy equation.
Literature Review
Literature survey of the flow over a drop is limited to
cases without a step. As far as the authors know, flows
with a step were not investigated before. A definition sketch
of a vertical drop of height h placed in a rectangular channel is shown in Figure 1. Here we follow the notation
adopted by Rajaratnam and Chamani (1995) and note that
the upstream flow is subcritical, the flow near the drop is
critical with depth yc and velocity Vc, and the flow immediately downstream of the drop is supercritical with depth
y1 and velocity V1 . The nominal depth of the pool formed
at the base of the drop is denoted by yp , and hs is the
height of the step. The length of the step is also hs since
only steps with a square section were investigated. The
free water jet makes an angle of φ when it hits the pool
with velocity V.
524
Technical Notes
The relative energy loss ∆E/Eo is defined as
∆E E o − E1
=
Eo
Eo
Downloaded by [University of Southern Queensland] at 19:36 11 October 2014
Figure 1. Definition sketch of a drop with a step at the base
The first analytical study of the flow over a drop was
reported by Bakhmeteff (1932) who showed that the energy equation could be applied to determine the downstream depth y1 from the equation
V1 = C 2g (Eo − y1 )
(1)
where C is a velocity coefficient, and Eo is the total energy head in the approach channel with respect to the
downstream channel bottom. For rectangular channels E o
is given by the following equation:
3
Eo = h + y c
(2)
2
The value of C was proposed as unity by Bakhmeteff.
Moore (1941) noted, however, that in such a condition there
would be no energy loss associated with the drop. Moore
also cited Bobin (1934) who had proposed a value of 0.95
for C to account for the losses. Nevertheless, Moore (1941)
carried out the first experimental work on the hydraulics
of drops and reported the results in the form of a dimensionless chart. In calculating the downstream velocity head,
Moore relied on the actual velocity measurements rather
than using the continuity equation for the determination of
the average velocity. Moore also used the momentum
equation for the determination of the depth of water in the
pool as
2
y 
y 
=  1  + 2  c  − 3
yc
y
 c
 y1 
yp
(3)
A well known discussion of Moore’s paper was made by
White (1943) in which the flow at the drop was considered to be similar to an inclined jet striking a flat plate. The
water jet is divided into two: one part forming the downstream flow, and the other part flowing towards the pool,
causing circulation in the pool and finally combining with
the initial jet near the pool surface. By applying the energy
and momentum equations, the specific energy E1 at the
downstream section was determined as


1.06 + h + 3 


y
2
E1
2
c

=
+
y1
4
h 3
1. 06 +
+
yc 2
2
and its value can be obtained by Equations 3 and 4. Inspection of these equations shows that the relative energy
loss is a function of only h/yc or yc/h.
In reaching Equation 4, White (1943) made several
important assumptions. These assumptions are summarized by Rajaratnam and Chamani (1995) as: i) the circulating flow in the pool at the bottom of the drop, Qc is the
same as the backward flow in an impinging jet of the same
angle Qb ; ii) the velocity of the supercritical stream immediately downstream of the drop is the same as the uniform
velocity at the side of the pool; iii) the angle of the falling
jet is not affected by the presence of the pool; iv) the
energy loss at the drop is due to mixing with the pool; v)
the presence of the pool does not affect the velocity, V, of
the free water jet; and vi) the horizontal component of the
velocity, V, can be determined by applying the momentum
equation between the critical flow section upstream of the
drop and the water jet.
Rand (1955) investigated the flow geometry at straight
drop spillways followed by a hydraulic jump. Using his
own and Moore’s experimental data, Rand developed a
set of empirical equations for the initial and sequent depths
and the location of the jump, and the depth of water in the
pool in terms of a dimensionless drop number D. The drop
number is defined as
D=
q 2  yc 
= 
gh3  h 
3
(6)
where q is the discharge per unit width in the rectangular
channel, and g is the acceleration of gravity.
Gill (1979) modified several assumptions of White
(1943), but neglected the energy loss in the pool. He also
proposed to use Rand’s (1955) empirical equation for the
pool depth
y 
= D0.22 =  c 
h
 h
yp
0.66
(7)
Gill (1979) then obtained the following equations for the
determination of φ and y1 :
cosφ (1 + cosφ ) =
y1
=
yc
3
h y
3
2  − p + 
 y c yc 2 
(8)
1
(1 + cos φ )2 
2
(4)
(5)

 y p y1 
h hp
 y − y + 1.5 + 2  y − y 
 c
c

 c
c 
(9)
Rajaratnam and Chamani (1995) carried out an experimental investigation and checked the assumptions made
IWRA, Water International, Volume 29, Number 4, December 2004
Downloaded by [University of Southern Queensland] at 19:36 11 October 2014
Technical Notes
by White (1943) and modifications proposed by Gill (1979).
They made velocity measurements at the base of a drop
including pool flow and also at the base of an impinging
jet. With these measurements it was possible to compute
the backflow rates Qc and Qb , and it was observed that
they were distinctly different from each other for a wide
range of flow conditions. Thus, the first assumption of
White was found not to be valid. Similarly, they observed
that the angle φ also varied with flow conditions and the
pool depth, and the pool depth affected the terms in the
energy equation. Thus, Rajaratnam and Chamani (1995)
showed that White’s third and fifth assumptions were also
not valid. Additionally, they showed that the downstream
velocity head must be calculated with the introduction of
an appropriate kinetic energy correction coefficient.
Using Moore’s, Rand’s, and their own data,
Rajaratnam and Chamani (1995) developed the following
empirical equations for the computation of the relative
energy loss and the pool depth:
∆E
y 
= 0.0896  c 
Eo
h 
y 
= 1.107  c 
h
 h
yp
−0.766
(10)
0 .719
(11)
Rajaratnam and Chamani (1995) also developed a physically-based model to predict the drop characteristics. Referring to Figure 1 and applying the momentum equation
to the control volume between the critical flow section
and the pool (Control Volume 1), they obtained the following:
ρq V cosφ +
1
1
γ y 2p = ρq V1 + γ y12
2
2
(12)
where ρ and γ are the density and specific weight of the
fluid. The momentum and continuity equations for the control volume between the pool and the downstream section
(Control Volume 2) were written as
ρq Vc +
1
γ y 2c = ρq V cosφ
2
(13)
and
h+
3
V2
yc = yp +
2
2g
(14)
The continuity equation was written in the form
q = yc gyc
(15)
and Equation 11 was used for the estimation of yp . Equations
11 through 15 were solved for φ, y1 /h, and V1 for different
values of yc/h and the results were observed to compare well
with the measured values. We shall also use these equations in
our study for the prediction of the downstream depth of flow.
Other studies on the hydraulic characteristics of drop
structures include the investigation of the wave type of
525
flow and the oscillating hydraulic jump downstream of the
drop (Turner and Mulvihill, 1987; Kawagoshi and Hager,
1990, Ohtsu and Yasuda, 1994), submerged flow downstream of the drop (Fiuzat, 1987), supercritical flow upstream of the drop (Chamani, 2002), and flow over drops
placed in trapezoidal channels (Noutsopoulos, 1984). These
topics are not within the scope of the present study.
Description of Equipment and Experimental
Procedure
The tilting Universal Modular Flow Channel manufactured by Plint Partners Ltd., England, was the main
laboratory facility. The channel was 20 m long, 0.60 m
wide, and 0.78 m deep; it had glass walls, stainless steel
bed, a stilling basin at the inlet supporting structure, and an
independent water recirculation system. The pump capacity
was 125 l/s. For flow measurements, a mercury manometer was connected to the orifice meter located on the
water supply pipe. The maximum flow rate used in this
study was 100 l/s, and the range of yc/h values was between 0.07 and 0.56. Three different drop structures with
heights of 0.25 m, 0.35 m, and 0.45 m were used in the
study. A slope was provided at the upstream end of the
drop structure to ensure a smooth transition.
To serve as a step restricting circulation at the base of
the drop, concrete blocks of length 0.595 m having square
cross-sections of seven different sizes were used. Each
side of the square sections were 0.050 m, 0.075 m, 0.100
m, 0.150 m, 0.200 m, 0.225 m, and 0.275 m. When placed
in the channel, the edges of the drop structure and the
step were sealed with a sealing compound. The range of
hs /h values was between 0.11 and 0.61. The flow velocities were measured by two probes of Type 403 (for lower
velocities) and Type 404 (for higher velocities) manufactured by Nixon Instruments, England. Water depths were
measured with point gages. At the downstream end of the
drop, the flow became supercritical and was relatively
unstable. Since the determination of the downstream depth
of flow was extremely important, its accurate measurement was essential. For this reason, the channel bottom
was connected to a glass cylinder with a flexible tube and
the representative water level was measured inside the
cylinder. Overall, 123 experimental runs were made. Each
run was repeated twice: first without any step, then with a
step of a specific size. In each run, flow rate, drop height,
step size, and downstream depth of flow with and without
a step were recorded. Other measurements such as upstream depth of flow, depth of water in the pool yp , and the
downstream velocity V1 will not be reported here. Description of the experimental runs is given in Table 1.
Semi-empirical Procedure for the Estimation of y1
Here we shall use Rajaratnam and Chamani’s (1995)
physically-based model with two modifications. First, the
IWRA, Water International, Volume 29, Number 4, December 2004
526
Technical Notes
0.07
Drop height
(mm)
Step size
(mm)
Number of
experimental runs
0.06
250
75 x 75
100 x 100
50 x 50
75 x 75
100 x 100
150 x 150
50 x 50
75 x 75
100 x 100
150 x 150
200 x 200
225 x 225
275 x 275
12
12
10
9
9
8
8
9
9
10
9
9
7
Calculated values of y1 (m)
Table 1. Description of the experimental runs
0.05
Total
123
Figure 2. Cross plot of calculated and measured values of downstream depth of flow y1 for drops without a step (semi-empirical
procedure)
Downloaded by [University of Southern Queensland] at 19:36 11 October 2014
450
q
y1
0.01
0.01
0.02
y 
= 1.215 c 
h
 h
yp
(16)
(
2q
(V1 − Vc ) − y 2c − y 21
g
)
(17)
with the measured values of y and q, yc was calculated by
Equation 15, V1 was calculated by Equation 16, and yp
values were calculated by Equation 17 for both drops without a step and with a step. With the yp values obtained for
drops without a step, a non-linear least-squares procedure
gave the following result:
y 
= 1.215  c 
h
 h
yp
0.02
0.03
0.04
0.05
0.06
0.07
Measured values of y1 (m)
and the downstream velocity head will be calculated with
this value of V1 without using any Kinetic energy correction coefficient. Next, Equation 7, which gives the pool
depth, will be revised. The reason for this is that it is very
difficult to make a definitive measurement of yp . As
Rajaratnam and Chamani (1995) have shown, water surface in the pool oscillates with time, and there is a backwards water surface slope. The difference between the
minimum and maximum pool depths was almost 20 percent.
The simultaneous solution of Equations 12, 13, and 16
for yp yields
yp =
0.03
0
downstream velocity will be directly obtained from the
continuity equation as
V1 =
0.04
0
0 .8124
(18)
This equation is comparable to Equation 11 and predicts
yp /h values within a difference of ± 5 percent from those
predicted by Equation 11 for yc/h values between 0.2 and 0.6.
Whenever there was a step, the pool depth varied with
q, h, and hs . For that reason, we assumed a relationship
such that the expression for yp /h would reduce to Equation 18 when there was no step, and yp /h would be a function of both yc/h and hs /h when there was a step. The resulting
equation thus obtained by least-squares analysis is
0. 8124
1. 4952
0 .1882


 hs 
 yc 
 
1 − 0.5662 

h
h
 
 


(19)
Equation 19 can be considered as the general equation
that predicts the pool depth. Now, using Equations 12, 13,
14, 15, and 19, the downstream depth of flow y1 can be
determined for given values of q, h, and hs . Using the corresponding values for these parameters in our experimental runs, we calculated y1 for drops without a step and
with a step. Figures 2 and 3 show calculated values of y1
plotted against measured values of y1 for drops without a
step, and for drops with a single step, respectively.
Empirical Procedure for the Estimation of y1
The procedure described in the previous section was
in principle physically based with one exception: the pool
depth yp had to be experimentally determined. At this stage,
one can argue that instead of using a relatively complicated procedure with a component which needs to be es0.09
0.08
0.07
Calculated values of y1 (m)
350
0.06
0.05
0.04
0.03
0.02
0.01
0
0
0.01
0.02
0.03 0.04 0.05 0.06 0.07
Measured values of y1 (m)
0.08
0.09
Figure 3. Cross plot of calculated and measured values of downstream depth of flow y 1 for drops with a step (semi-empirical procedure)
IWRA, Water International, Volume 29, Number 4, December 2004
527
Technical Notes
0.35
0.30
h s/ h=0.6
measured
calculated
h s/ h=0.4
h s/ h=0.2
0.3
0.25
h s/ h=0 ( no step)
0.25
0.20
y1/h
y1/h
0.2
0.15
0.15
0.10
0.1
0.05
0.05
0
0.00
0
0.1
0.2
0.3
0.4
0.5
0
0.6
0.1
0.2
timated by statistical methods, the end result could directly
be obtained by similar techniques. Therefore, it seemed
appropriate to attempt to determine y1 directly by leastsquares procedures. Consequently, for the case of drops
without a step, the following equation was obtained:
1.1854
(20)
The linear correlation coefficient for this equation is R =
0.9986. Figure 4 shows measured and computed values of
y1 /h plotted against yc/h for drops without a step.
For drops with a step, y1 /h would be a function of both yc/
h and hs /h. Additionally, we would require the empirical equation to reduce to Equation 20 when there would be no step.
With these conditions, the resulting regression equation is
1. 1854
y1
y 
= 0.4824  c 
h
 h
1 .4877

 hs 
1 + 0.5243 
 h

yc 
 
 h
0. 07571



(21)
0.25
0.2
0.15
0.1
0.05
0
0.1
0.15
0.6
against each other. The solution of Equation 21 is shown
in graphical form in Figure 6 in which y1 /h has been plotted against yc/h for different values of hs /h. Equations 20
and 21 will be referred to as the empirical equations developed for the prediction of y1 .
Comparison of the Two Methods
As a representative selection of the experimental data,
we have used the information available for each twentieth
run and calculated y1 /h values by both procedures described
above. This choice of experimental runs resulted in a wide
range of values for ych and hs /h. The measured and calculated values of y1 /h are given in Table 2 for drops with and
without a step.
The energy loss at the drop is calculated as the difference between the total head in the approach channel Eo
and the total head in the downstream section E 1 . The value
of Eo can readily be determined by Equation 2, and E 1 can
be computed by
E1 = y 1 +
0.05
0.5
Figure 6. Values of y 1 /h estimated by Equation 21
0.3
0
0.4
Energy Loss at the Drop
with a linear correlation coefficient of R = 0.9958. Figure
5 shows measured and computed values of y1 /h plotted
Calculated values of y1 /h
Downloaded by [University of Southern Queensland] at 19:36 11 October 2014
Figure 4. Measured and calculated values of y 1 /h plotted against y c/h
for drops without a step (empirical procedure)
y1
y 
= 0.4824  c 
h
 h
0.3
yc
yc/h
0.2
0.25
0.3
Measured values of y1 /h
Figure 5. Cross plot of calculated and measured values of y1 /h for
drops with a step (empirical procedure)
v12
2g
(22)
where V1 is directly determined by Equation 16. In the
computation of the downstream energy head, no corrections for velocity variations within the cross-section were
made. The relative energy loss can then be computed by
Equation 5.
For drops without a step, we calculated y1 using Equation 2, and the relative energy loss by Equation 5. The
results are shown in Figure 7. On the same figure, relative
energy losses reported by Moore, and Rajaratnam and
Chamani (1995) are also shown. Relative energy losses
for different hs /h values for drops with a step are shown in
Figure 8.
IWRA, Water International, Volume 29, Number 4, December 2004
528
Technical Notes
80
1
yc/h
0.064
21
0.360
41
0.291
61
0.309
81
0.207
101
0.380
121
0.468
hs/h
Measured
0
0.300
0
0.214
0
0.444
0
0.166
0
0.500
0
0.429
0
0.400
0.024
0.025
0.146
0.154
0.111
0.133
0.120
0.131
0.072
0.089
0.157
0.179
0.200
0.216
0.018
0.019
0.144
0.151
0.112
0.127
0.120
0.125
0.075
0.087
0.153
0.173
0.195
0.218
0.019
0.020
0.144
0.151
0.112
0.127
0.119
0.126
0.075
0.087
0.153
0.174
0.196
0.221
Rajaratnam and Chamani
60
50
40
30
20
10
0
0.3
0.4
yc/h
Figure 7. Relative energy loss for drops without a step
0.5
50
40
30
0.1
0.2
0.3
yc
0.4
0.5
0.6
Figure 8. Relative energy loss for drops with a step
Moore
0.2
hs /h=0 (no step)
0
present study
0.1
60
0
80
0
h s/h =0.6
h s /h=0.4
hs /h=0.2
10
Inspection of the values listed in Table 2 shows that in
general the values of y1 calculated by the semi-empirical
and empirical equations are almost identical. However,
comparison of Figures 2 and 3 with Figures 4 and 5 indicates that the semi-empirical procedure predicts higher
than actual y1 values at higher values of yc (or y1 ). This
coupled with the unusually high correlation coefficients
associated with them are points in favor for the use of
Equations 20 and 21 for the prediction of y1 . It is also
relatively easier to apply the empirical procedure.
As can be observed from Figure 6, presence of a step
results in higher y1 values: as the step size is increased, y1
increases at an increasing rate. Figure 7 shows the relative energy loss computed by the empirical equations for
y1 together with the relative energy losses reported by
previous researchers for drops without a step. Although
the fit is reasonably good for a wide range of values of yc/h, at
high values of yc/h, the energy loss reported is higher than
that determined by the proposed procedure. This indicates
70
70
20
Discussion
E/Eo (%)
Downloaded by [University of Southern Queensland] at 19:36 11 October 2014
Run
y1/h
Calculated by Calculated by
semi-empirical
empirical
procedure
procedures
∆E/Eo (%)
Table 2. Measured and calculated values of y1/h for selected
experimental runs
0.6
that corrections for the velocity head should be made at
large flow rates.
The effect of the step on the relative energy loss is
shown in Figure 8. The presence of a step is observed to
significantly increase the head loss. As the flow rate and
step size increase, the effect of the step becomes more
pronounced. For example, at yc/h = 0.2, having a step with
hs /h = 0.6 increases the relative energy loss by about 50
percent. At yc/h = 0.4, a step of the same size increases
the relative energy loss by more than 150 percent.
In this study, the downstream depth of flow was considered to be supercritical. Consequently, there was a hydraulic jump further downstream. Cases in which the jump
moved upstream and affected the flow over the drop structure were not investigated.
The maximum hs /h value used in the study was 0.61.
For larger step sizes, the flow over the drop structure fell
over the step and the flow characteristics changed. This situation resembled cascading flows and was not investigated.
Conclusions
In irrigation systems where the slope of the canal is
less than the ground slope, drop structures are built to reduce the cost of excavation and earth fill. Drops are also
used for prevention of erosion and a grade control structure in drainage channels and as a spillway for small earth
dams. The simplest drop structure is a vertical fall placed
in a rectangular channel. Description of the flow characteristics at drops is important for the design of a stilling
basin downstream of the drop. The stilling basin can be a
flat concrete apron, or a detailed structure which may include chute blocks, baffle piers, and an end sill.
In this study, we proposed placing a concrete step having a square cross-section at the base of the drop running
through the entire width of the channel. To the benefit of
the engineer, this drop increases the energy loss through
the drop, and the downstream depth of flow can be precisely estimated.
The downstream depth of flow y1 over a drop with or
IWRA, Water International, Volume 29, Number 4, December 2004
529
Technical Notes
without a step can be determined by a semi-empirical procedure which requires the solution of a number of equations based on continuity, momentum, and energy principles,
and one additional empirical equation for the depth of pool
at the base of the drop. Alternatively, the downstream depth
of flow can be directly determined by Equation 21, as given
above. Equation 21 can be used for drops with and without a step, and it has a correlation coefficient of R = 0.9958
with the values of y1 predicted by Equation 21. The energy loss at the drop was calculated, and it was observed
that a step placed at the base of the drop significantly
increases the energy loss.
Downloaded by [University of Southern Queensland] at 19:36 11 October 2014
Acknowledgement
The experimental data used in this study were collected during the M.Sc. Thesis work of K.M.A. Hannan
which was supported by the College of Graduate Studies,
Kuwait University. The authors wish to thank the distinguished reviewers of the journal.
Notations
C
D
Eo
E1
g
h
hs
q
Qb
Qc
R
V
V1
Vc
y1
yc
yp
γ
ρ
φ
velocity coefficient
drop number defined by Equation (6)
total energy head in the approach channel
total energy head at the downstream section
acceleration of gravity
height of drop
height of step
discharge per unit width of channel
backward flow of an impinging jet
circulating flow in the pool
linear correlation coefficient
velocity of the free water jet
velocity of flow at the downstream section
critical velocity
downstream depth of flow
critical depth of flow
depth of water in the pool
specific weight of water
density of water
the angle the water jet makes with the pool
neering. He can be reached at E-mail: jasem@kuc01.
kuniv.edu.kw
Khoanddkar A. Hannan. is with the Public Authority for Housing Welfare in Safat, Kuwait.
Discussions open until June 1, 2005.
References
Bakhmeteff, B.A. 1932. Hydraulics of open channels. New York:
McGraw-Hill.
Bobin, P.M. 1934. The design of stilling basins. Leningrad: Transactions, Scientific Research Institute of Hydrotechnics, Vol.
VIII.
Chamani, M. R. 2002. “Flow characteristics at drops.” Journal
of Hydraulic Engineering 128: 788-791.
Fiuzat, A. A. 1987. “Head loss in submerged drop structures.”
Journal of Hydraulic Engineering 113: 1559-1562.
Gill, M.A. 1979. “Hydraulics of rectangular vertical drops structures.” Journal of Hydraulic Research 17: 289-302.
Kawagoshi, N. and W.H. Hager. 1990. “Wave type flow at abrupt
drops.” Journal of Hydraulic Research 28: 235-252.
Moore, W.L. 1943. “Energy loss at the base of free overfall.”
Transactions (ASCE) 108: 1343-1360.
Noutsopoulos, G. C. 1984. “Hydraulic characteristics in a straight
drop structure of trapezoidal cross section.” Channels and
Channels Control Structures, Proceedings of the 1 st International Conference on Hydraulic Design in Water Recourses
Engineering. Southampton, England.
Ohtsu, T and Y. Yasuda. 1994. “Characteristics of flow over drop
structure.” Proceedings of the 1994 International Conference
on Hydraulics in Civil Engineering. Brisbane, Australia.
Rand, W. 1955. “Flow geometry at straight drop spillways.”
Proceedings ASCE 81: 1-13.
Rajaratnam, N. and M.R. Chamani. 1995. “Energy loss at drops.”
Journal of Hydraulic Research 33: 373-384.
Turner, H. O. and M.E. Mulvihill. 1987. “General design for modifications to exiting lower Santa Ana drop structures.” Hydraulic Engineering, Proceedings of the 1987 National
Conference, ASCE. Williamsburg, VA, USA: ASCE.
White, M.P. 1943. Discussion of Moore (1943). Transactions
(ASCE) 108: 1361-1364.
About the Authors
Dr. Ismail I. Esen, is a member of the Department
of Civil Engineering at Kuwait University. Dr. Esen’s research interests include fluid mechanics, hydraulics, environmental engineering, and transport processes.
Dr. Jasem M. Alhumoud, is a member of the Department of Civil Engineering at Kuwait University. Dr.
Alhumoud’s research interests include the principles of
environmental engineering, hydrology and hydrualics, solid
waste management, geoenvironmental and geohydrology
engineering, and the principles of water resources engiIWRA, Water International, Volume 29, Number 4, December 2004