EFFECT OF END SILL IN RADIAL BASIN ON CHARACTERISTICS OF FREE HYDRAAULIC JUMPS

The 6th

Int. Conf. on Hydroscience and Engineering (ICHE-2004), May 30-June 3, Brisbane, Australia

EFFECT OF END SILL IN RADIAL BASIN ON CHARACTERISTICS OF FREE

HYDRAAULIC JUMPS

Abdel-Azim M. Negm1, Gamal M. Abdel-Aal

2, Amany A. Habib

3, and Talaat M. Owais

1

ABSTRACT

The end sill may be used in stilling basin to produce double effect on the flow upstream the

sill and downstream of it. Upstream of the end sill, the characteristics of the hydraulic jump

may be affected. Downstream of the end sill, the scour hole may be formed away from the

apron and also may be modified in its dimensions. In this paper, the effect of end sill on the

characteristics of the free hydraulic jump formed upstream of the sill in a radial basin is

investigated experimentally. Sills of different heights were tested under wide range of flow

conditions. It was found that the basic characteristics of the free radial hydraulic jump formed

upstream of end sill are function of the supercritical flow Froude number and the relative

height of the sill. The experimental data was used to estimate the derived functional

relationship using the dimensional analysis. The developed statistical model agreed well with

the experimental data. Moreover, a theoretical model for the energy loss ratio through the

jump was developed and found to be in good agreement with the experimental data.

1. INTRODUCTION

Hydraulic jumps are advantageous for dissipating kinetic energy in stilling basins. It may be

free or submerged depending on both the location and the initial depth of the jump relative to

the gate. The different classifications of jumps were reported in Chow (1959). The hydraulic

jump may be also formed in prismatic or in non-prismatic channels (diverging or sudden

expanded), and may be forced or non-forced. Most of the studies on different types of

hydraulic jump are presented in Hager (1992).

Khalifa and McCorquodale (1979), studied the radial hydraulic jump occurs in stilling

basins with diverging side-walls. They concluded that the sequent depth ratio of the radial jump

is less than that of the rectangular jump, and the length of the radial hydraulic jump is about 70%

of that of rectangular jump with the same flow conditions. Also, the energy loss in a radial

hydraulic jump was 15% higher than that of the rectangular jump. According to the various

methods used in practice, stilling basins were arranged in a variety of geometrical configurations.

1 Professor of Hydraulics, Dept. of Water & Water Structures Eng., Faculty of Engineering, Zagazig University,

Zagazig, Egypt, E-mail: [email protected] 2 Associate Professor, Dept. of Water & Water Structures Eng., Faculty of Engineering, Zagazig University,

Zagazig, Egypt. 3 Assistant Professor, Dept. of Water & Water Structures Eng., Faculty of Engineering, Zagazig University,

Zagazig, Egypt

mailto:[email protected]

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On the other hand, sills or blocks were used in stilling basins to increase the rate of energy

dissipation and to reduce the bed velocity in the region of the hydraulic jump. Many studies had

been conducted to investigate the effect of sills in rectangular basins. The effect of the sill on the

jump characteristics depends on factors such as the sill configuration, sill location and sill

spacing when more than one sill was used. Several investigations dealt with the effect of sill on

the hydraulic jump characteristics when the sill was constructed beneath hydraulic jump such as

Shukry (1958), Rajaratnam (1967), Ohtsu and Yasuda (1991), and Hager and Li (1992). Hager

and Li gave one of these classifications of the forced hydraulic jump due to vertical sill. They

classified the jump over vertical sill into A-jump, B-jump, minimum B-jump and C-jump. The

A-jump was corresponding to the classical hydraulic jump, which was characterized by the

maximum sequent depth ratio for the free jumps. They stated that, A-jump in which the jump

characteristics are not influenced by the presence of sill (or weak effect are present) as the sill

was found at the end of the surface roller and thus it was out side the effective zone for the sill to

affect the jump flow. Other studies on the effect of vertical sill on the jump and different

classification of jumps due to presence of sill could be reviewed in Hager (1992). Wafaie

(2001a,b) investigated experimentally the free rectangular hydraulic jump phenomenon on

roughened channel bed with dentated, solid, zigzagged bed sills, under different flow conditions,

different bed sill heights, and different bed sill locations. Statistical analysis for the experimental

results was made to obtain the best height and location of the bed sill. Recently, few studies were

conducted to discuss the effects of negative step and/or end sill on the characteristics of the

submerged hydraulic jump in radial basins, Negm et al. (2003, 2002b).

More recently Negm et al. (2003a,b,c) studied theoretically and experimentally the

effect of negative step in radial stilling basin and the effect of its location on the

characteristics of the free hydraulic jump. Also Abdel-Aal et al. (2003) investigated

theoretically the effect of combining a negative step and an end sill in radial stilling basin.

While Habib et al. (2003) studied theoretically the effect of end sill in radial stilling basin on

the characteristics of the free hydraulic jump.

The above review indicated that the effect of end sill on free hydraulic jump in radial

stilling basins was not investigated experimentally in details. This paper investigate the free

hydraulic jump in radial basin with an end sill experimentally. The purpose of this research is

to present the results of an experimental investigation on the effect of end sill on the free

radial hydraulic jump formed in radial stilling basin and to compare the experimental results

with the previous theoretical ones.

2. THEORETICAL BACKGROUND

2.1 Energy Loss Ratio EL/E1

From Figure 1 the specific energy at the beginning of the jump (E1), and at the end of the

jump E2 can be written as:

E1=g2

Vαd

211

1 ; E2=g2

Vαd

222

2 (1)

Keeping in mind that F1=

1

1

gd

V, and V2=

oo

1

dr

V, substituting in equation (1) and

manipulating to obtain:

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E2= )dr2

Fdr2(d

2o

2o

21

3o

2o

1

(2)

Also, E1 can be expressed as follows

E1= )F2

1(1d 2

11 (3)

From equations (2), and (3), the efficiency of the jump E2/E1 is obtained:

1

2

E

E=

)F(2dr

Fdr2

21

2o

2o

21

3o

2o

(4)

The energy loss ratio is EL/E1= 1- E2/E1 which could be expressed as follows

1

L

E

E=

)F(2dr

F)d2F(2dr

21

2o

2o

21o

21

2o

2o

(5)

2.2 Theoretical Modeling of Depth Ratio (d2/d1)

Based on the use of the continuity and momentum equations, the following theoretical

equation was developed, Habib et al. (2002)

0.01S)(drF6

S)(dr)r(rS)(d1rS)(drd)r(1S)(drd

ss

2

1

os

2

so

2

ss*s

2

ss

ssossossosss

2

o

)r(rS)(dr)SSd(Sdr32dr3

(6)

Alternatively, equation (6) could be rearranged to take the easily solved form, Habib et al.

(2003).

1S)(dr6

S)(dr)r(rS)(d1rS)(drd)r(1S)(drd

ss

os

2

so

2

ss*s

2

ss

ssossossosss

2

o

1

)r(rS)(dr)SSd(Sdr32dr3F

(7)

2.3 Length of Jump

Referring to Fig.1, the following functional form for the length of jump formed in radial basin

with end sill could be expressed as follows:

0L,r,r,r,s,V,d,d,d,,g,f j3211321 (8)

in which is the density of water, g is the gravitational acceleration, is the dynamic

viscosity of water, d1 is the initial depth of jump, d2 is the sequent depth of jump, d3 is the

depth of water at the end sill, V1 is the average velocity of flow at the beginning of jump, z is

the height of the negative step, r1 is the radius where the jump begins, r2 is the radius where

the jump ends, r3 is the radius where the end sill is constructed and Lj is the length of jump.

Using the dimensional analysis principle based on the three repeating variables , d1

and V1, Eq. (8) becomes

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0R,F,d

r,

d

r,

d

r,

d

s,

d

d,

d

d,

d

Lf 11

1

3

1

2

1

1

11

3

1

2

1

j

(9)

In Eqn. (9), both d2/d1 and d3/d1 are functions of F1, r2/d1 and r1/d1 gives r2/r1 and r3/d1 and

r1/d1 gives r3/r1 while the effect of R1 is neglected as the viscosity has a negligible effect on

the hydraulic jump characteristics in the present study because the temperature was fixed

during the course of the experimental work. Also, r2/r1 is a function of the length of the jump.

Moreover, rs/r1 is constant because the position of the sill is fixed and the position of the jump

is fixed. Equation (9) becomes

11

1

j

d

s,Ff

d

L (10)

Similar relationships for the depth ratio and for the energy loss ratio could be obtained.

The nature of the function presented in Eq. (10) will be determined based on the experimental

data using the multiple linear regression analysis.

3. EXPERIMENTAL WORK

The experimental work of this research was conducted using a re-circulating adjustable flume

of 15.0 m long, 45 cm deep and 30 cm wide, Habib et al. (2003) and Habib (2002). The

discharges were measured using pre-calibrated orifice meter fixed on the feeding pipeline.

The tailgate fixed at the end of the flume was used to control the depth of flow for each run.

The radial basin was made from a clear prespex to enable visual inspection of the

phenomenon being under investigation. The model length was kept constant at 130 cm and a

constant angle of divergence of 5.28o was used. The model was fixed in the middle third of

the flume between its two side-walls as shown in Figure 2. A smooth well painted baffle

block of wood was formed to fit well inside the basin model extending from one side of the

model to the other side at the end of the basin to simulate the end sill. The end sill has an

upstream slope of 1:1 and vertical face from the downstream side. The wood was well painted

by a waterproof material (plastic) to prevent wood from changing its volume by absorbing

water. Three different heights of the end sill (viz 3, 4 and 5 cm) were tested under the same

flow conditions. The range of the experimental data were as follows: Froude numbers (2.0-

7.0), ro (1.2-1.4), and relative height of the end sill, s/d1 (0.0 – 3.4).

Each model was tested using five different gate openings and five discharges for each

gate opening. The measurements were recorded for each discharge. The total number of runs

was 100. A typical test procedure consisted of (a) a gate opening was fixed and a selected

discharge was allowed to pass. (b) the tailgate was adjusted until a free hydraulic jump is

formed. (c) once the stability conditions were reached, the flow rate, length of the jump,

water depths upstream and at the vena contracta downstream of the gate in addition to the tail

water depth and the depth of water above the step were recorded. The length of jump was

taken to be the section at which the flow depth becomes almost fixed. These steps were

repeated for different discharges and different gate openings and so on till the required ranges

of the parameters being under investigation were covered.

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Figure 1 Definition sketch showing the formation of the jump in case of horizontal bed

with end sill

4. ANALYSIS AND DISCUSSIONS OF RESULTS

4.1 Verification of Theoretical Equations Using Experimental Data

The collected experimental data was used to verify the developed theoretical equation for

energy loss ratio (eq. 5). Figure 3 and 4 presents the comparison between the calculated

theoretical values and the experimental values for s/d1=0, 2.3 (2.0-2.6) and 3.0 as a typical

Vertical gate

b=18cm

50cm Lb=130cm

Figure 2 General sketch for radial stilling basin with end sill model in plan

B=30cm

r3

d2

d3

d1

s

r2

Lb

Lj

U.S.

r1

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values. Theoretical values were computed using eq. 5 for energy loss ratio and eq. 7 for depth

ratio using ro=1.3. Clearly, good agreement is obtained.

Figure 3 Relationship between EL/E1 and F1 showing the comparison between the

experimental and the theoretical values of EL/E1 using eq. 5

Figure 4 Relationship between d2/d1 and F1 showing the comparison between the

experimental and the theoretical values of d2/d1 using eq. 7

4.2 Comparison With Previous Studies

Figures 5 presents the comparison between the present results both theoretical and

experimental of depth ratio for smooth stilling basin (free bed) and those of other authors,

theoretical values for rectangular basin, Chow 1)

, theoretical and experimental for smooth

radial basin having divergence angle of 13.5o, Khalifa & McCorquodale

3), theoretical for

smooth rectangular basin of different angles of divergence (5o 27, 7

o 13, 9

o 13.5, 10

o 5, 11

o

49, and 13o 4), Arbhabhirama & Abella (1972) and experimental for smooth radial basin

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0

F1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

EL

/E1 Eqn. (5) s/d1=3.0

Eqn. (5) s/d1=2.0-2.6

Eqn. (5) s/d1=0.0

s/d1=3.0

s/d1=2.0-2.6

Free bed

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0

F1

2

3

4

5

6

7

8

9

10

11

d2

/d1 Eqn. (7) s/d1=0.0

Eqn. (7) s/d1=2.0-2.6

Eqn. (7) s/d1=3.0

Free bed

s/d1=2.0-2.6

s/d1=3.0

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having a divergence angle of 4o 5.6 (expansion ratios (e) of 0.547 and 0.647), Abdel-Aal

(1995). Also, Figure 6 shows the comparison between present theoretical and experimental of

the energy loss ratio and those of other authors, theoretical values for rectangular basin, Chow

1), theoretical and experimental for smooth radial basin having divergence angle of 13.5

o,

Khalifa & McCorquodale (1979), and experimental for smooth radial basin having a

divergence angle of 4o 5.6 (expansion ratios of 0.547 and 0.647), Abdel-Aal (1995). Figure 7

presents a similar comparison of the present (Eq. 15) for the length of jump ratio with those of

Chow (1959), Koloseus & Ahmed (1969), and Aabdel-Aal (1995). These figures indicate a

good general agreement with the present results from both theoretical and experimental point

of view with those of other others for smooth radial basin. Also, general agreement between

measured values and theoretical values. Any discrepancy between present experimental

results and those of others are mainly due to the difference in the angle of divergence and the

models setup. The depth ratio and length of jump ratio of jump formed in radial basin are

smaller than those of rectangular at the same Froude number while the energy loss ratio for

radial jump is more than that for rectangular jump. This is mainly due to the lateral and

longitudinal spread of water in radial basin resulting in lesser depth, higher energy dissipation

and hence lesser weight and consequently shorter length of jump compared to the rectangular

one where the flow spreads only in the longitudinal direction.

5. Experimental Results

The experimental results were classified into ranges of s/d1 from 0 to 3.8. Figures 8a,b,c show

the relationship between energy loss ratio, depth ratio and length of jump ratio respectively

and Froude number for different s/d1. Figure 8a indicated that the energy loss ratio increases

with the increase of s/d1 till it reaches 3.0 then it decreases again. Figures 8b,c show that the

depth and the length ratios decreases with the increase of s/d1 till s/d1 =3.0, then they increase

again by increasing s/d1 but still less than the values of s/d1=0.0 (free bed). These results are

confirmed by inspecting Figure 9 which indicate the relationship between the jump properties

and the s/d1 for different Froude number. Evaluating the effect of end sill on the jump

properties indicated that the depth ratio was decreases by about 2.6% for each unit increase in

s/d1, the length of jump ratio decreases by about 4% for each unit increase in s/d1 while the

energy loss ratio increases by about 2.5% for each unit increase in s/d1.

Figure 5 comparison between the present results both theoretical and experimental of depth

ratio for smooth stilling basin (free bed) and those of other authors

2 3 4 5 6 7 8 9 10 11

F1

0

1

2

3

4

5

6

7

8

9

10

11

d2

/d1

Eqn. (7)

The. Khalifa & McCorquodale 1979

The. (rectangular basin) Chow 1959

The. Arbhabhirama & Abella 1971

Exp. data

Exp. Khalifa & McCorquodale 1979

Exp. Abdel-Aal (e=0.547) 1995

Exp. Abdel-Aal (e=0.647) 1995

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Figure 6 comparison between the present results both theoretical and experimental of energy

loss ratio for smooth stilling basin (free bed) and those of other authors

Figure 7 comparison between the present results both of eq. 15 and experimental of length

ratio for smooth stilling basin (free bed) and those of other authors

Figure 8a Relationship between energy loss ratio and Froude number for different s/d1

2 3 4 5 6 7 8 9

F1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

EL

/E1

The. present study Eqn. (5)

The. Khalifa & McCorquodale 1979

Rect. Chow 1959

Exp. data

Exp. Abdel-Aal 1995 (e=0.547)


Exp. Khalifa & McCorquodale 1979

2 3 4 5 6 7 8 9 10 11 12

F1

5

10

15

20

25

30

35

40

45

50

55

60

Lj/d

1

Eqn. (15)

Rect. basin Chow 1959

Koloseus & Ahmed 1969

Exp. data



Free bed

s/d1=(0.9-1.6)

s/d1=(2.0-2.6)

s/d1=3.0

s/d1=3.8

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0

F1

0.3

0.4

0.5

0.6

0.7

0.8

EL

/E1

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Figure 8b Relationship between depth ratio and Froude number for different s/d1

Figure 8c Relationship between length ratio and Froude number for different s/d1

Figure 9 Relationship between the jump properties and the s/d1 for different Froude number

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0

F1

2

3

4

5

6

7

8

9

10d

2/d

1

Free bed

s/d1=(0.9-1.6)

s/d1=(2.0-2.6)

s/d1=3.0

s/d1=3.8

Free bed

s/d1=(0.9-1.6)

s/d1=(2.0-2.6)

s/d1=3.0

s/d1=3.8

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0

F1

0

5

10

15

20

25

30

35

40

45

50

Lj/

d1

0 1 2 3 4

s/d1

5

10

15

20

25

30

35

40

45

Lj/

d1

0 1 2 3 4

s/d1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

EL

/E1

0 1 2 3 4

s/d1

2

3

4

5

6

7

8

9

10

d2/d

1

F1=7

F1=6

F1=5

F1=4

F1=3

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Development of Statistical Prediction for the jump Characteristics

Based on the experimental data and using the simple and multiple linear regression analysis,

the function of Eq.(10) was determined. The following set of equations was obtained. The

values of coefficients of determination R2, the mean relative absolute error MRE and the

correlation coefficient of the residuals (with the predicted values) are given in Table 1.

Relative Energy Loss (EL/E1)

EL/E1= -1.017-0.155 F1+1.042 F10.5

(11)

EL/E1= -0.973-0.153 F1+1.031 F10.5

+0.006 s/d1 (12)

Relative Depth (d2/d1)

d2/d1=1.266 F1-0.291 (13)

d2/d1= -0.322+1.196 F1-0.055 s/d1 (14)

Relative Length (Lj/d1)

Lj/d1 =6.119 F1-3.504 (15)

Lj/d1 = -4.282+5.938 F1-0.145 s/d1 (16)

Table 1 Values of R2, MRE and residuals correlation coefficient for the developed equations

Ratio Basin type R2

MRE R (residuals)

D2/d1 Smooth radial basin 0.987 0.026 1.67E-03

Radial basin with end sill 0.980 0.028 1.07E-06

Lj/d1 Smooth radial basin 0.983 0.034 2.01E-04


EL/E1 Smooth radial basin 0.980 0.025 2.32E-04


Figures 9a,b,c,d,e,f Present the comparison between measured values and predicted

ones using the statistically developed prediction models (11), (12), (13), (14), (15) and (16)

respectively. From both table 1 and these figures, it is clear that good agreement was obtained

which highlighted the use of these equation in the design of radial basins with or without sill.

Figure 10 shows the comparison between measured and predicted values of Lj/d1 for

typical values of s/d1. Also, the values of Lj/d1 due to s/d1=0.0 are shown in the figure. Good

agreement is obtained.

6. CLOSING REMARK

Other studies of similar nature conducted by Hager and Li (1992) on rectangular basin

indicated that the presence of sill at the end of rectangular basin has little effect of

insignificant effect on the depth ratio of the hydraulic jump. Recently, Negm et al. (2002)

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proved both theoretically and experimentally that the end sill in radial basin has little effect on

the characteristics of the submerged hydraulic jump formed inside the basin.

Figure 10 Measured values versus predicted ones for (a) EL/E1 for smooth basin, (b) EL/E1 for

basin with end sill, (c) d2/d1 for smooth basin, (d) d2/d1 for basin with end sill, (e) Lj/d1 for

smooth basin, (f) Lj/d1 for basin with end sill

Figure 11 Comparison between measured and predicted values of Lj/d1 for typical values of

s/d1.

0.2 0.4 0.6 0.8

EL/E1(Exp.)0.2

0.4

0.6

0.8

EL

/E1(

Pre

dict

ed)

0.2 0.3 0.4 0.5 0.6 0.7 0.8

EL/E1(Predicted)-0.04

-0.02

0.00

0.02

0.04

Res

idua

ls

Fig. (7.35) Results of statistical model of (Eqn 7.29)

a) Predicted EL/E1 versus Exp.

b) Residuals versus Predicted

a

b

0.3 0.4 0.5 0.6 0.7 0.8

EL/E1(Exp.)0.3

0.4

0.5

0.6

0.7

0.8

EL

/E1(

Pre

dict

ed)

b

2 3 4 5 6 7 8 9 10 11

d2/d1(Exp.)2

3

4

5

6

7

8

9

10

11

d2/d

1(P

redi

cted

)

c

2 3 4 5 6 7 8 9 10 11

d2/d1(Exp.)2

3

4

5

6

7

8

9

10

11

d2/d

1(P

redi

cted

)

d

10 15 20 25 30 35 40 45 50

Lj/d1(Exp.)10

15

20

25

30

35

40

45

50

Lj/

d1(P

redi

cted

)

e

2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0

F1

0

5

10

15

20

25

30

35

40

45

50

Lj/d

1

Eqn. 15

Eqn. 16

Eqn. 16

Free bed

s/d1=2.0-2.6

s/d1=3.0

5 10 15 20 25 30 35 40 45 50

Lj/d1(Exp.)5

10

15

20

25

30

35

40

45

50

Lj/

d1

(Pre

dic

ted)

f

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7. CONCLUSIONS

An experimental investigation was conducted to study the effect of using end sills in the radial

stilling basin on the characteristics of free hydraulic jump. Sills of different heights were tested

under wide range of flow conditions, each height was tested under similar flow conditions. It was

found that the characteristics of the free hydraulic jump in radial basin with end sill are a function of

the supercritical Froude number and the relative height of the end sill. The optimal height of end sill

in radial basin is triple the initial depth of jump. This height minimizes both the depth and the length

ratio of the jump and maximizes the energy loss ratio. The analysis of results indicates that a unit

increase in the relative height of sill increases the energy dissipation by about 2.5% and decreases

the depth ratio of the jump by about 2.6% and the length ratio by about 4%. The experimental data

were used to calibrate several proposed regression prediction models and the best ones was

presented to be used in the prediction of the jump characteristics for both radial basins with and

without end sills. Also, a theoretical prediction model for the energy loss ratio was developed based

on the use of energy equation. The present developed theoretical model for energy loss ratio and the

previously developed theoretical model for depth ratio were verified using the experimental data.

Good agreement between results of the developed models and the experimental results.

8. REFERENCES

Abdel-Aal G. M. (1995). Control of Hydraulic Jump in Contracted Streams by Gradual

Expansion, Unpublished Ph. D., Faculty of Engineering, Zagazig University, Egypt, 1995

Abdel-Aal, G.M, Negm, A.M., Owais, T.M. and Habib, A.A. (2003). “Theoretical Modeling Of

Hydraulic Jumps At Negative Step In Radial Stilling Basins With End Sill”, Proc. of 7th Int.

Conf. on Water Technology, IWTC2003, April 1-3, Cairo.

Arbhabhirama, A. and Abella, A.U. (1972). “Hydraulic Jump Within Gradually Expanding

Channel”, Journal of the Hydraulics Division, Vol. 97, No HYI, Jan., 1972, pp. 31-41.

Chow, V.T. (1959). Open Channel Hydraulics, McGraw-Hill Book Co., Inc., New York.

Habib, A.A. (2002). “Characteristics of Flow in Diverging Stilling Basins”, Ph. D. Thesis,

Submitted to the Faculty of Engineering, Zagazig University, Zagazig, Egypt.

Habib, A.A., Abdel-Aal, G.M., Negm, A.M. and Owais, T.M. (2003). “Theoretical Modeling Of

Hydraulic Jumps In Radial Stilling Basins Ended With Sills”, Proc. of 7th

Int. Water and

Technology Conference, IWTC-2003, 1-3 April Cairo, Egypt.

Hager, W.H. (1992). Energy Dissipators and Hydraulic Jumps, Kluwer Academic Publications,

Dordrecht, The Netherlands.

Hager, W.H. and Li, D. (1992). “Sill-Controlled Energy Dissipator”, J. Hydraulic Research,

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9. NOMENCLATURE

B = width of the channel;

b = contracted width of the channel ;

d1 = water depth at vena contracta downstream the gate where the channel width is b1 ;

d2 = sequent water depth where the channel width is b2;

d3 = water depth over end sill where the channel width is b3 ;

do = the relative water depth, d2/d1;

ds = the ratio of d3 to d1;

e = b/B (expansion ratio)

F1 = Froude’s number at the initial depth;

Lj = the length of the hydraulic jump;

Lb= the length of the stilling basin;

Q = discharge;

r1 = radius at the beginning of the jump ;

r2 = radius at the end of the jump ;

ro = the ratio of r2 to r1;

r3 = radius at the end sill;

rs = the ratio of r3 to r1;

R2= the coefficient of determination;

s = the sill height;

S= the ratio of s to d1;

V1= average velocity at the initial depth;

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V2= average velocity at the sequent depth;

γ = the specific weight, and

θ = the angle of divergence.

EFFECT OF END SILL IN RADIAL BASIN ON CHARACTERISTICS OF FREE HYDRAAULIC JUMPS

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