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A Review of Hydraulic Jump Properties in Different Channel Bed Conditions
H.M. Imran
*
,
Shatirah Akib
Department of Civil Engineering, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia
ihosen83@gmail.com
Abstract: The main objective of this study is to investigate the potential use of corrugated and roughened beds for
reducing the hydraulic jump length and sequent depth. The paper presents a comprehensive review of the available
literature on the hydraulic jump properties including different types of corrugated and roughened beds. Hydraulic
jumps are frequently used for excessive kinetic energy dissipation under hydraulic structures and the jumps are often
generated with the assistant of baffle blocks and kept inside the stilling basin. Corrugated and roughened beds
showed considerable energy dissipation at the downstream. The jump length and sequent depth also significantly
reduced with respect to the smooth bed. Consequently, the use of corrugated and roughened beds reduced the
scouring length and scouring depth as well as the stilling basin installation cost. This paper discusses the
implications of corrugated and roughened beds, and highlights their findings in different installation systems by
many researchers. Finally, it is found that the applications of corrugated and roughened beds are always showed
better performance than that of the smooth bed. In addition, this study identified some research needs for the future.
[H.M. Imran, Shatirah Akib. A Review of Hydraulic Jump Properties in Different Channel Bed Conditions. Life
Sci J 2013;10(2):126-130] .(ISSN:1097-8135). http://www.lifesciencesite.com. 20
Keywords:
Hydraulic jump, Corrugated and roughened bed, Jump length, Bed shear stress, Sequent and scour depth
1. Introduction
The hydraulic jump is a common
phenomenon in the branch of hydraulics, which is
generally observed in open channel flow, such as
rivers and spillways. When a high velocity
supercritical flow drops to that of a subcritical flow,
the rapid following flow is abruptly slowed and
increases its height, converting some of the flow's
initial kinetic energy into an increase in potential
energy. This phenomenon is called the hydraulic jump.
The study of hydraulic jump has been going on for
around two centuries. The first investigation was
carried out by Bidone (1819). Thereafter, the subject
continued to receive more attention and a tremendous
amount of experimental as well as theoretical work
was done by many eminent hydraulicians with regard
to free hydraulic jump on horizontal beds. Hydraulic
jumps are frequently used for energy dissipation in the
case of hydraulic structures. A jump formation in the
wide rectangular and horizontal channel with smooth
bed conditions is called classical jump, and has been
widely investigated (Peterka, 1958; Rajaratnam, 1967;
McCorquodale, 1986; Hager, 1992). A wide range of
investigation was conducted to evaluate the
effectiveness of roughened beds (Rajaratnam, 1968;
Hughes and Flack, 1984; Hager, 1992; Alhamid, 1994;
Ead et al., 2000) and corrugated beds (Ead and
Rajaratnam, 2002; Izadjoo and Shafai-Bajestan, 2005)
considering different conditions for reducing the
sequent depth and hydraulic jump length. Mohamad
Ali (1991) conducted a series of experiments to study
the effect of roughened beds using cube blocks and
found that the length of hydraulic jump reduced by
around 27 to 67% for a Froude number range of 4 to
10. Another study was carried out by Pagliara (2008)
for homogeneous and non-homogeneous roughened
bed channels, and a jump equation that accounted for
bed roughness and non-homogeneity. The generalized
solution was proposed by Carollo et al. (2009) for the
sequent depth ratio of hydraulic jump over smooth and
roughened beds introducing a coefficient of shear
force in the momentum equation. This study represents
the results of various studies in which the hydraulic
jump characteristics were measured in different
channel bed conditions, and suggests future research
directions.
2. Hydraulic Jump Properties
2.1. Sequent Depth Ratio
Hydraulic jump length (L
j
) and tail water
depth (y
2
) over corrugated and roughened beds (Figure
1) mainly depend on the upstream flow characteristics,
such as flow velocity (V
1
), flow depth (y
1
), fluid
density (ρ), fluid viscosity (µ), acceleration of gravity
(g), bed corrugation and roughened amplitude (t), and
shape of the corrugated bed (
). Thus, the jump
length or sequent depth of the jump can be written as a
function of:
or ( , , , , , , )
2 1 1
y L f V y g t
j
…............ (1)
If y
1
, g and ρ are considered as three
repeated variables, and by applying the Pi theorem,
Equation (1) can be written in the following form as
Equation (2):
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/ or / ( / ,R / , / , )
2 1 2 1 1 e 1 1 1
1
y y L y f F V gy V y t y
j
… (2)
where F
1
and R
e
are the Froude number and Reynolds
number, respectively, at the upstream of the hydraulic
jump. For a large Reynolds number, if the viscous
force is neglected (Rajaratnam, 1976; Hager and
Bremen, 1989), then the final expressions of sequent
depth or length of the jump can be written as Equation
(3):
2
o r L / y ( , / , )
j 2 1 1
1
y
f F t y
y
………… (3)
The magnitude of L
j
/y
1
and y
2
/y
1
increased
with the initial Froude number, while the value of L
j
/y
1
and y
2
/y
1
decreased for all Froude numbers with the
increased value of I at 12.5% (I is the ratio between
the area of roughness and the area of basin), and then
started to increase for larger values of I (Ezizah et al.,
2012). Their study also found that the sequent depth
reduced by 14 to 20% with respect to the smooth bed.
The variations of sequent depth ratio (y
2
/y
1
) for the
different Froude numbers were studied using a
corrugated bed. The investigated results showed that
the relative roughness and shape of corrugation had
very little significant effect on the sequent depth ratio
(Izadjoo and Shafai-Bajestan, 2005; Ead and Elsebaie,
2009). Carolo et al. (2009) conducted a study over the
natural roughened bed. Different sizes of cobbles,
ranging from 0.46 to 3.2 cm, were used and the Froude
number ranges laid from 4 to 12. Their results showed
that the roughened bed was more effective for
reducing the jump length and sequent depth ratio, in
which the reduction depends on both relative
roughness (t/y
1
) and the Froude number. The
difference between sequent depth y
2
and sequent depth
of classical jump (y
2
*) have been investigated by some
researchers using the following Equation (4):
*
2 2
*
2
y y
D
y
................... (4)
where D is the dimensionless index. Five shapes –
sinusoidal, triangular, two trapezoidal and rectangular
corrugated beds indicated that the D value was
constant at approximately 0.37. The sequent depth
ratio (y
2
/y
1
) was found to be around 88% of the initial
Froude number. The results confirm that the shape of
corrugation and their relative height (t/y
1
) had less
significant effect on the hydraulic jump characteristics
(Elsebaie and Shabaye, 2010). Abdelhaleem et al.
(2012) found that the D values were 0.14, 0.145 and
0.174 for the semi-circular, trapezoidal and triangular
corrugated beds, respectively. The results indicated
that the tail water depths were, respectively, 86%,
85.5% and 82.6%, of the same variable for the jump
over the smooth bed. These results were similar to the
findings of Peterka (1958), who carried out an
experiment for a stilling basin and obtained D values
0.17 and 0.21 for trapezoidal and triangular corrugated
beds, respectively. For the triangular corrugated bed,
Ead and Rajaratnam (2002) found the D value 0.25,
while Izadjoo and Shafai-Bajestan (2005) obtained a
value of 0.20. Thus, it is clear that the triangular
corrugated bed was the best shape for reducing the tail
water depth.
Figure 1: Typical hydraulic jump over corrugated
bed
2.2. Jump Length
The corrugated and roughened beds have a
significant effect on reducing the hydraulic jump
length, as shown in Figure 1. The relationship between
the dimensionless length of jump Lj/y
2
and initial
Froude number has been established considering
different bed conditions. The semi-circular, trapezoidal
and triangular corrugated beds reduced the jump
length by around 10%, 11% and 14%, respectively
(Abdelhaleem et al., 2012). The U-shape corrugated
bed reduced the jump length by around 28 to 47%
compared to the smooth bed for the range of Froude
numbers 3 to 11 (Ezizah et al., 2012). It also showed
that the corrugated beds had little effect on the jump
length when the Froude number was less than three (F
1
≤ 3). Their study result corresponded well with the
findings of Elevator ski (1959). In another
investigation that was carried out over a roughened
bed using T-shape blocks (Aboulatta et al., 2010), the
results indicated that a T-shaped roughened bed can
reduce the jump length and materials compared to that
of the cubic block. A U- shaped roughened bed is more
effective in appreciably reducing the jump length and
sequent depth compared to the T-shaped roughened
bed for Froude number five, even though the
difference is small for Froude numbers greater than
five (Ezizah et al., 2012). The length of hydraulic
jumps over corrugated and roughened beds are always
smaller than for the smooth bed.
2.3. Bed Shear Stress
Corrugated and roughened beds are
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generally installed on the channel bed for increasing
the bed shear stress, which, consequently, reduces the
sequent depth and hydraulic jump length. The
following momentum in Equation (5) is frequently
used to calculate the bed shear stress:
( ) ( )
1 2 1 2
F P P M M
…..….……..... (5)
where P
1
, P
2
, M
1
and M
2
are the integrated pressure
and momentum at the sections prior and after the
hydraulic jump occur. The shear force index (
) is
calculated using Equation (6) as follows (Rajaratnam,
1965):
2
0 . 5
1
F
y
………...…...……………(6)
where is the kinematics viscosity of water. The bed
shear stresses over the semi-circular, trapezoidal and
triangular corrugated beds were around 8, 9 and 11
times that of the smooth bed (Abdelhaleem et al., 2012;
Izadjoo and Shafai-Bajestan, 2005). The hydraulic
jump characteristics were also investigated over
corrugated beds for variable wave steepness and a
Froude number range of 3.8 to 8.6 in which the results
showed that the shear stress for the corrugated bed was
around 10 times that of the smooth bed (Abbaspour et
al., 2009). The corrugated bed also had a significant
effect on reducing the ratio of energy (ΔE/E
1
) by
increasing the bed shear stress. The ranges for relative
loss of energy ratio for semi-circular, trapezoidal and
triangular corrugated beds were found to be from 14%
to 64%, 15% to 65% and 16% to 66%, respectively,
while the smooth bed ranges were found to be from
10% to 62% (Abdelhaleem et al., 2012). Similar
results were found in the study by Chow (1959). The
corrugated beds were effective for energy dissipation
downstream hydraulic structures and can reduce the
cost of stilling basins (Abdelhaleem et al., 2012;
Ezizah et al., 2012; Shafai-Bajestan and Neisi, 2009).
It was found that triangular and U-shape corrugated
beds were most effective in reducing the jump length
and sequent depth.
2.4. Scour Depth and Length
The depth and length of scouring can be
significantly reduced by providing the corrugated bed
at the downstream bed channel. Maximum scour hole
depth (D
s
) and scour length (L
s
) are dependent on the
variables in Equation (7), as follows (Abdelhaleem
et al., 2012):
2
/ / ( , , , )
1 1 1
1 1 1
L
y
t
j
Ds y orLs y f F
y y y
.…(7)
The semi-circular, trapezoidal and triangular
corrugated beds decreased the ranges of scour depth
from 22% to 31%, 25% to 34% and 30% to 36%,
respectively, while the scour length decreased the
ranges from 17% to 24, 23% to 25% and 24% to 30%,
respectively, in comparison with the smooth bed
(Abdelhaleem et al., 2012). Based on their
experimental data and statistical methods, several
models were proposed and their coefficients were
calculated. Considering all the trials, the best
Equations (8 and 9) for predicting the relative scour
depth and length can be written in the following form,
respectively:
2
/ 1.51 0.793 0.115 2.571 0.33
1 1
1 1 1
L
y t
j
Ds y F
y y y
…... (8)
2
/ 17.55 8.05 1.19 51.79
1 1
1 1 1
L
y
t
j
Ls y F
y y y
…............ (9)
2.5. Roller Length
The roller length (L
r
) is the horizontal
distance between the toe section of the flow depth y
1
and the roller end, as shown in Figure 1. This length
can be estimated by a visualization technique, such as
with a float to localize the stagnation point. The
experimental studies (Pietrkowski, 1932; Hager, 1992
and Smetana, 1937) suggested that the relation
between the roller length and differences between the
sequent depths can be written as the following
Equation (10):
2
/ ( 1 )
1
1
y
L r y a
y
…………...…... (10)
where ‘a’ is the coefficient, and the suggested values
are 6 (Smetana, 1937), 5.5 (Citrini, 1939) and 5.2
according to Mavis and Luksch (Hager et al. 1990).
Equation (10) was verified using the roller length data
for the smooth and rough beds by many investigations
with the coefficient value ‘a’ depending on bed
roughness (Hager et al., 1990; Hughes and Flack, 1984;
Ead and Rajaratnam, 2002 and Carollo and Ferro,
2004) based on the roller length data for the rough and
smooth beds. Moreover, Carollo and Ferro (2004) also
established the applicability of the following
relationships as Equations (11) and (12):
1 .2 7 2
1
/ ( )
1
2
y
L r y a
o
y
………….... (11)
/ ( 1 )
1 1
L r y b F
o
…….... (12)
where a
o
, b
o
are the numerical coefficients depending
on the bed roughness.
3. Future Research Directions
Numerous studies have been conducted to
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investigate the hydraulic jump characteristics
considering the different bed conditions. Different
types of corrugated and roughened bed channel have
been used to identify the efficiency of reduction of the
hydraulic jump length and sequent depths. Moreover,
some equations have been developed to establish the
relationship between different parameters of hydraulic
jump. In addition, further studies can be conducted to
investigate the better performance of the corrugated
bed channel to control the hydraulic jumps. In this
context, some research gaps have been identified for
future research on the hydraulic jump properties in
respect of different bed conditions.
Factors affecting the dynamics of the
boundary shear stress over corrugated beds
can be further investigated.
The effect of larger size boulders used as a
roughened bed material can be investigated
for hydraulic jump properties.
Future studies can be carried out to evaluate
the hydraulic jump characteristics on sloping
bed conditions.
A new circular shape corrugated bed channel
is proposed for further study to investigate the
hydraulic jump characteristics.
Although sensitivity analysis has been carried
out in several studies to investigate the effect
of the change of intensity and roughness
length parameters on the hydraulic jump
length, intensive investigations are needed in
this regard.
More extensive investigations are
recommended to determine the detailed
information concerning the effects of
boundary roughness on hydraulic jump.
4. Conclusions
The following prominent conclusions can be
depicted from the review of the hydraulic jump
properties considering different channel bed conditions:
A corrugated bed always showed better
performance than a smooth bed channel in
reducing hydraulic jump length and sequent
depth by increasing bed shear stress.
Generally, corrugated bed produced more
eddies, and, consequently, increased the bed
shear stress, which reduced the jump length
and sequent depth.
The hydraulic jump length and sequent depth
are significantly reduced by bed shear stress,
which is dependent on the interaction
between the supercritical flow of liquid and
the corrugations of the channel bed.
Among the semi-circular, rectangular,
trapezoidal and triangular corrugated beds,
the most efficient corrugated bed was the
triangular shaped for reducing the sequent
depth and jump length. Conversely, it showed
the best effectiveness for increasing the bed
shear stress.
The reduction in jump length and sequent
depth greatly depended on the Froude number.
For small Froude numbers the amount of
reduction was low while large value Froude
numbers showed a higher reduction.
Corrugated beds confirmed the effectiveness
for energy dissipation at downstream
hydraulic structures and reduce the cost of the
stilling basin.
Boundary resistance greatly depended on the
Reynolds number and Froude number
according to the findings of the smooth bed
channel flow characterized by large Froude
numbers.
Acknowledgments
Financial support by the University of
Malaya (UM), Malaysia under UMRG research grant
number RG 170/12SUS is gratefully acknowledged.
Corresponding Author:
H.M. Imran,
Department of Civil Engineering,Faculty of
Engineering, University of Malaya, 50603 Kuala
Lumpur, Malaysia
E-mail: ihosen83@gmail.com
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