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126

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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128

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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129

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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