Characteristics of Jumps in Case of Symmetric and Un-Symmetric Operations of Regulators

Abdel-Azim M. Negm1

Abstract Past studies have been covered classical and non-classical hydraulic jumps formed in prismatic and non-prismatic channels. However, hydraulic jumps formed in actual channels downstream multi-vents hydraulic structures are not treated efficiently. Only few studies may be available. In case of multi-vents structures like regulators, one or more vents may be opened or closed during maintenance. If the central vent is operated (or closed) while the others are closed (or opened), it is a case of symmetric operation or may be called symmetric expansion. Similarly, if one or more side vents are opened (or closed) and the vents on the other sides are closed (or opened), it is a case of un-symmetric operation or un-symmetric expansion. In this paper, the hydraulic characteristics of different jumps (free and submerged) created at expansion or below it (downstream piers of multi-vents regulators or between them) are presented based on experimental and theoretical analysis. The performance of these jumps is compared with the classical jump. The jumps created between piers (T-jump) are less efficient than other jumps at expansion. Theoretical and empirical prediction equations are presented for computing the characteristics of different jumps in expanding channels. These equations could be applied in cases of symmetric and un-symmetric operations of multi-vents regulators. Keywords: hydraulics, Hydraulic jumps, expansion, regulators, stilling basins, flow, peirs. 1 Introduction Formation of hydraulic jumps in non-prismatic channels is necessary for dissipating the excess energy downstream hydraulic structure. Particularly, if the given tail-water depth in a prismatic channel of constant width is too low that a classical jump is no longer able to form even with the aid of appurtenance; and if it is not possible to depress the basin floor due to difficulties in excavation, lateral expansion of channel remains the only possibility for guaranteeing the necessary energy dissipation, Herbrand (1973). Such spatial jumps may be expected at downstream of hydraulic structures like flow over a weir or spillway, outlet of dam, head regulator, where the incoming supercritical flow occupies a smaller width and leading to a wider tail-water. 2.1 Classification of Jumps in Expanding Channels Depending upon the toe position of the jump in expanding channel, the following types of jumps may be formed (Figure 1): (a) The Classical jump that is formed in the channel between piers upstream the expansion. It is equivalent to the jump that forms in prismatic channel downstream from the gate such that the initial depth is at the vena contracta of the gate, Figure (1a). ______________________________________________

1Associate Professor, Dept. of Water and Water Structures Engineering, Faculty of Engineering, Zagazig University, Zagazig, Egypt.

(b) The Transitional jump (T-jump) which is formed between piers and partly downstream from the end of the pier in the expanding channel. It begins upstream of the end of piers and ends downstream of the piers. This jump is equivalent to the jump that forms upstream from expansion section and ends downstream from the expansion section, Figure (1b) (c) The Spatial jump (S-jump) may be defined as the jump that is formed just at the downstream from the end of pier i.e, it forms at the expansion section. However, Bremen and Hager (1993) defined it as the jump formed downstream from the expansion section with the jump toe positioned at the point where the two cross-waves hit the walls of the downstream channel. However, in the theoretical analysis, the toe of this jump is always assumed just upstream of the expansion section. In case of regulators, this type of jump may form when the emergency gates are in operation while the main gates are out of order, Figure (1c). (d) Submerged Stable Jump that is formed at the expansion section downstream from the pier when the emergency gates are used instead of the main gates. The depth at the downstream from the gate is more than the gate opening. This type of jump requires the highest tailwater depth among all the defined jumps from (a) to (e). If the main gates are in operation and the tailwater depth is more than that need to form the classical jump, the submerged T-jump is formed provided that the jump ends at the wider channed downstream from the end of the pier nose ( see also Figure 1c). (e) The Repelled jump (R-jump) which formed in the downstream channel with the toe position some distance away downstream from the expansion section, Figure (1d). The R-jump requires the lowest tailwater level. It breaks down if the tailwater increases slightly, and free S- jump formed. More increase in the tailwater depth moves the jump upstream from the expansion section and the T-jump forms. The T-jump breaks down and the classical jump forms if further increase in the tailwater depth occurred and the sequent depth of the jump is located in the upstream channel. Further increase in the tailwater depth produces the submerged T-jump if it ends in the expanding channel or the submerged jump if it ends in the upstream channel.

Figure (1) Types of jumps at sudden expansion and operations of regulators (One) Classical jump, (b) T-jump (jump between piers), (c) Free and Submerged S-jump, (d)

Repelled jump, (e) symmetric operation and (f) un-summetric operation

Figures (1e & 1f) show typical cases of symmetric and un-symmetric operation of multi-vent regulators to show the similarities between flow at sudden expansion at flow in case of multi-vent regulators with some vents out of order. Also, if the two centeral vents of Figure (1e) is closed while the extreme vents are opened, the case is a symmetric operation. In Figure (1f), if all vent are closed except one only on only of the side, the case is un-symmetric operation. 2.2 Review of Past Studies Type (a) jump was studied by many authors, see e.g. Rajaratnam (1967) and Hager et al. (1989, 1990) Type (b) jumps were studied by Bremen and Hager (1993, 1994) and by Matin et al. (1998). Type (c) jump was studied by Sharama (1964), Herbrand (1973), Hager (1985.1992), Matin et al. (1997). Jumps of type (d) and (e) were studied by Unny (1961), Abramove (Sharama 1964), Rajaratnam and Subramanya (1968), Negm et al. (2000) and Negm (2000). Smith (1989) studied the submerged jump formed between piers and extended downstream from the piers. This type of jump may be called submerged T-jump or submerged L-jump depends on the symmetric or un-symmetric operation of the gates of multi-vents regulators or barrages. 2.2.1 Free S-Jump Herbrand (1973) investigated the spatial free jumps in expanding channels and suggested a simple empirical relation for calculating the sequent depth ratio as

β*D=D (1) Where D is the sequent depth ratio of jumps in expanding channels, D* is the depth ratio of classical jumps in prismatic channels as given by equation (2) or (2a), Bb /=β is the expansion ratio, b is width of the approaching channels, B is the width of the expanded channel. The experiments of Herbrand were confined to free jumps with the toe located just upstream from the expansion section.

( )[ ]1815.02/12

1* −+= FD (2)

for F1>2.5, Eq.(2) may be approximated as follows, Breman and Hager (1993):

21

2 1* −= FD (2a)

Herbrand (1973) adopted the following equation for computing the legnth of free S-jump:

64.064.08.0 ** >=<= βββ forLLandforLL jjjj (3)

where *jL is the length of the classical jump and may be given as follows, Bremen and Hager (1993):

−=

221

220 1

1

* FTgh

d

L j (4)

Recently, Matin et al. (1997) used the momentum and continuity equations and developed a simple model for the free spatial jump formed at sudden expansion in the form:

1)-E8+1(21

=D 21 (5)

Where E1 is a modified Froude number to account for the sudden expansion. E1 is related to the approaching flow Froude number by the following expression:

2

112

1 FkE −= (6) Where k is an expansion factor that is a function of β and F1 as follows:

( ) ( )1lnln8.0 F1=k β− (7) It should be noted that for β=1, K=1 and hence E1=F1, Eq.(5) tends to Eq.(2) for classical jump. 2.2.2 T-jump Bremen and Hager (1993) derived a relation for the depth ratio of T-jump as function of sequent depth ratio of classical jump, expansion ratio and dimensionless toe parameter as follows:

(1.9X)]-)[1-1)(1-(D-D=D tanh** β (8) where X is the non-dimensional toe parameter which describes the location of the jumps at upstream of expansion section and is defined as X=

*/ rLx with x is the distance to the toe of jump measured

positively from the expansion section opposite to the flow direction and *rL is the length of the roller

of the classical jump. The length of roller for the classical jump is given as follows, Hager et al. (1990)

+−=

20tanh16012 1

1

* FdLr (9)

for d1/b<0.1 and F1>2.5. Bremen and Hager (1993) stated that T-jumps are less effective energy dissipator than other types of jumps unless appurtenances are used, such as Repelled jumps (R-Jump) and Stable jumps (S-jumps) in expanding channels. The length of the T-jump may be estimated from the following equation:

(1.9X)]-)[1-(1L

L

j

j tanh1*

β+= (10)

2.2.3 R- jump Rajaratnam & Subramanya (1968), Negm et al. (2000) and Negm (2000) proved that the depth ratio of R-jump is function of the initial Froude number, the expansion ratio as aspect ratio, b/yo, as follows:

=

oo

o

t

yb

Ffyy

,,β (11)

In which yt is the tailwater depth; yo is depth at the outlet of the jet below the gate; Fo is the Froude number at the section of yo;

Kuznetsov (1964) presented a momentum equation for depth ratio and introduced a coefficient Cs to account for the integrated bed shear stress that some authors neglected it. His equation reads:

)-F8(2

C=

yy 2

os

o

t βββ +2 (12)

Where Cs is given by the following expression Cs= 0.665 +0.15 β (13)

Rajaratnam and Subramanya concluded that the use of the momentum equation without Cs is not accurate and with Cs means not reliable as prior knowledge on bed shear stress should be known. They suggested the use of the following empirical equation for the R-jump:

yt/yo=(β+0.3)(Fo-0.85)+0.75 (14) Eq.(10) can be used for β in the range 0.3 to 0.90 and Fo ranges between 2 and 9.0 Recently, Negm et al. (2000) found that equation (5) is applicable for R-jump and k of equation (6) is given by the following expression:

( ) ( )oFe=k lnln8.02169.2 77.0 ββ −− (15)

The length of roller of the R-jump is estimated using the following equation, Negm (2000) based on the experimental data of Rajaratnam and Subramanya (1968):

913.0,052.2141.0159.333.9594.8 2 ==+++−= RSEEwithyb

FyL

oo

o

r β (16)

Equation (16) is valid for 2≤ Fo ≤ 8.5, 0.33≤ β ≤ 0.83 and 2.4≤ b/yo ≤ 13.76. 2.2.4 Submerged S-jump Regarding the stable submerged spatial jump (Submerged S-jump), the following equation is proposed by Abramov (as reported in Sharama (1966)) for computing the depth ratio:

−

+++

++

1)14(

26001

)(1014 2

21

1

1

1o

o

t FKK

2KK

=yy

(17)

In which K1 is an empirical factor that could be taken as 3 to 4 for B/b<10 and 5 to 6 for B/b>10, Rajaratnam and Subramanya (1968). Unny (1961) has obtained a semi-empirical formula for the case of b/B=0.5 as follows:

−+

+

+ 1

)2.21(

812.21

21

221

221

o

oo

o

t

Fby

FF

by

=y

y (18)

With yo being the depth at the outlet or under the gate depth. Rajaratnam and Subramanya (1968) have been compared the equations developed by Unny and Abramov for computing sequent depth ratio of submerged S-jump in expanding channels based on their experimental data. They assumed a value of 3.5 for K1 when B/b<10. Then equation (8) becomes:

)-F(=yy 2

o

o

t 18.101318.0 + (19)

They found unsatisfactory performance of the Abramov and Unny when compared with the experimental data. Based on their experimental work, they suggested the following empirical equation for the submerged S-jump:

4.108.1'

' +ot F=

ry

(20)

Where F'o’=Uo/(gr’)1/2 and r'’ =byo/(b+2yo) is a defined characteristic length of the outlet. They found that the data are much better correlated to r’ than yo and the data are represented by a single straight line. Recently, Negm et al. (2000) found that Eq.(5) is applicable for submerged S-jump and k of equation (6) is given by the following expression:

( ) ( )oF=k lnln8.0422.1 485.2 ββ − (21) The length of the submerged S-jump can be estimated from the following equation, Negm (2000):

96.0,48.397.850.4288.766.5 2 ==++−−= RSEEwithyb

Fyb

y

L

oo

oo

j β (22)

Equation (16) is valid for 2≤ Fo ≤ 8.5, 0.33≤ β ≤ 0.83 and 3.4≤ b/yo ≤ 13.75. 3 Theoretical Background The derivation of the general equation for the jumps at expansion may be derived based on the definition sketch of T-jump presented in Figure (3). Assuming that (a) the pressure distribution is assumed as hydrostatic (b) the velocity distribution is considered as uniform (c) effect of turbulence and air entrainment is negligible (d) effect of wall friction is disregarded. Also, the tail-water depth is assumed as the temporal mean value of its fluctuations. The momentum equation for the control volume between section 1 and 2 of Figure (3) can be written as:

)V-VQ(g

=P-P+P 122e1 12 ββγ (23)

Where γ is the unit weight of water, 12 , ββ are the momentum correction factor ( they are assumed to be unity), Q is the discharge, g is the acceleration due to gravity, P1, V1 are the pressure and mean velocity at section 1, P2, V2 are the pressure and velocity at section 2 and Pe is the pressure distribution at expansion section. Defining P1= 0.5γbd1

2, P2=0.5γBd22 and Pe =0.5γ(B-b)de

2, utilizing the continuity equation Q=V1bd1=V2Bd2, introducing the dimensionless parameter D = d2/d1, Bb /=β , e=de/d1, The equation (23) becomes, Herbrand (1973) and Matin et al. (1998):

0=F2+)F2+e+e-D(-D 21

221

223 ββββ (24) Equation (24) could be re-arranged in the following simplified form

0=E-D+D 21

2 (25) where E1 is modified Froude number which incorporates the effect of expansion and position of jump and is defined:

k+

Fk

F=E2

21

12

2 21

121 (26)

in which k1 and k2 are the defined parameters to account for the jump position and effect of expansion ratio. These parameters can be defined as

)D-2(1)-1)(1-e(

=k 1-

2

1β

(27)

)D-(1D-1

=k 1-

-1

2 ββ (28)

The following approximate relation for e can be used if equation (27) is used to estimate k1, Matin et al. (1998):

≅

ββ

-1F2-De

21

2 1/2

(29)

Figure (2) Definitions sketch for T-jump in expanding channel

The sequent depth ratio D is then obtained by solving equation (25) as

D =12

( 1+ 8E -1)12 (30)

From equations (27) and (28), at 1=β , k1=0 and k2=1 and equation (30) reduces to equation (2). For free S-jump, e=1 and k1=0. Both the parameters (k1 and k2) have to be evaluated based on experimental data: In general, equation (30) can be re-expressed as follows:

1)-KF+1(21

=D 218 (31)

+=

22

1

1 12 kFk

K (32)

For R- and Submerged S-jumps, the value of K is given as follows, Negm et al (2000):

3

1

kK = (33)

Where k3 is a factor to account for the effects of expansion and type of jump. However, it should be

B b V1

Pe

Pe

V2

xX

d2 P1 P2 d1 d

1 e 2

noted that for R- and submerged S-jump in expanding channels, F1 should be replaced by Fo (at the outlet or below the gate). 4 Calibration and Verification of Eq.(31) 4.1 Free S-jumps For free S-jump, K is function of k2 only as k1 is zero (X=0) and k2 is found to be function of the approaching flow Froude number and the expansion ratio. It is given by the following expression, Matin et al (2000):

( )( )12 ln1ln4.01 Fk +−= β (34) Figure (3a) shows the relationship between k2 and F1 for different expansion ratio. Also, Figure (3b) shows the relationship between k2 and expansion ratio for different F1. From these figure it is observed that: - .k2 increases slightly with the increase of F1 at constant expansion ratio. - .k2 increases with the decrease of the expansion ratio at constant F1. - The variations of k2 at smaller expansion ratio is significant while they are insignificant for high

values of expansion ratios. Figure (3c) presents a comparison between Eqs.(31,34) and Eq.(1) due to Herbrand (1973). It is clear that the present equation overestimates the depth ratio compared to Eq.(1). The comparison of Eq.(31,34) with Eq.(8) due to Bremen and Hager (1993) at X=0.0 is also presented in Figure (3c). It is clear that good agreement between the present approach and that due to Bremen and Hager (1993). Figure (3d) presents the relationship between D and F1 for different expansion ratio. Shown also the line due to classical jump (k2=1, k1=0). 4.2 T-JUMP For T-jump K is function of k1 and k2. The factor k1 is function of β , F1 and X as follows:

( ) ( )F)(Xf=k 1.511

1ln +− ββ (35)

where f(X) is given as follows:

21)( XCXCXf o += (35a)

Where the values of Co and C1 depend on the expansion ratio as in table (1): table (1) values of Co and C1 of Eq.(35a) β 0.40 0.50 0.60 0.70 0.80 Co 0.697 0.792 0.693 0.612 0.568 C1 -0.423 -0.501 -0.389 -0.417 -0.400 The variations of k1 with F1, X and β are shown in Figures (4a,4b & 4c). Figure (4a) presents a typical variations of k1 with F1 for different X at constant expansion ratio of 0.6. It indicates that k1 increases with the increase of F1 at constant X at β=0.6. While at constant F1, and fixed expansion ratio, the factor k1 increases with the increase of X. This is also clear from Eqs.(35). Figure (4b) presents the variation of k1 with X for different F1 at β=0.60. It is clear at higher values of F1, the value of k1 tends to decreases at X>0.8. generally, k1 increases with the increase of F1 and with the increase of X. Figure (9c) shows that k1 decreases with the increase of the expansion ratio at constant X=0.6 and at fixed F1. However, at higher values of F1, k1 increases with the increase of F1

upto β=0.6 then it decreases again. Figure (4d) shows the comparison between Eqs.(31,34,35) and Eq.(8). Clearly good agreement is obtained. Figure (4e) presents the variations of D with F1 for different X at expansion ratio =0.6 as a typical example. It is observed that D increases as F1 increase at constant X. At higher value of X (X>0.8), the variations of D with F1 are insignificant. The line of classical jump is also shown (k1=0, k2=1). Physically, the classical jump requires more tailwater depth than other free jumps. Figure (4f) shows the variations of D with X for different F1 at expansion ratio equals 0.6. Slight variations of D with X at constant F1 is observed. Figure (4g) presents the variations of D with the expansion ratio for different F1 at X=0.6. It can be observed that D increases with the increase of the expansion ratio at constant F1 and fixed X. The depth ratio, D, increases till it reaches its maximum for free jumps at β=1 (classical jump). 4.3 R- Jump The factor k3, Eq.(33), is function of the expansion ratio and the under gate Froude number, Fo as follows::

( ) ( ) 177.0

3 lnln8.02169.2 δββ +−−oFe=k (36)

In which δ1 is a correction factor that is function of the expansion ratio β (δ1=1.1 for β=0.33, 0.25 for β>0.33). Figure (5a) presents the variations of k3 and Fo and β . Shown also the values due to Eq.(15), Negm et al. (2000). Figure (5b) shows the comparison between the prediction of the present model and the data due to Rajaratnam and Subramanya (1968). Also, shown, in closed symbols, the values due to the use of Eq.(15). Equation (36) is much better than Eq.(15) when used with Eq.(31). The comparison between Eqs.(31,36) and the empirical Eq.(14) due to Rajaratnam and Subramanya (1968) is shown in Figure (5c). The figure shows that both equations are comparable. This proves the accuracy of the present model to predict the depth ratio of R-jump. 4.4 Submerged S-Jump Regarding the submerged S-jump, k3 is function of Fo and β and it is given by the following expression: ( ) ( ) 25.0lnln8.0422.1 485.2

3 +− oF=k ββ (37) Figure (6a) presents the variation of k3 and Fo and β and shown also the values from Eq.(21) as closed symbols, Negm et al. (2000). Figure (6b) shows the comparison between the prediction of the present model and the data due to Rajaratnam and Subramanya (1968). Also, shown the prediction due to Eq.(31,21) in closed symbols. Clearly, Eq.(37) is much better than Eq.(21). It is found that to compare the empirical equation (20), Eq. (31) could be written in terms of r' and

'oF as follows:

1)-kF

+1(21

=ry ot

3

2'8' (38)

In this case, the value of k3 should be calculated from Eq.(39) which is a modified form of Eq.(37). ( ) ( ) 2

'485.23 lnln8.0422.1 δββ +− oF=k (39)

In which δ2=-0.5 for β=0.33, and δ2=-0.15 for β=0.5 and zero for β more than 0.5.

Figure (6c) shows the variation of k3 with

'oF and β while Figure (6d) presents the comparison

between Eqs.(38,39) and the experimental data due to Rajaratnam and Subramanya (1968). Figure (6e) presents the comparison between Eqs.(38,39) and Eq.(20) for submerged S-jump. These figures prove good performance of the present equations. 5 Applications In Egypt, multi-vent regulators and barrages are widely used to control the flow and discharge in the irrigation canals and across the River Nile. During maintenance programs, some vents may be closed and others are kept opened. The symmetric and un-symmetric operation of the gates yields a case of flow similar to the flow in expanding channels as explained in Figs.(1). Therefore, the application of the developed equations in expanding channels could be used in case of multi-vents regulators for symmetric and un-symmetric operations of the gates. The following applications are possible: 1- The classical jump equations can bee applied if the jump ends upstream of the end of the piers in

case of symmetric or un-symmetric operations. 2- The T-jump equations can be applied for jumps formed between piers provided that the jump

ends downstream from the pier, in both cases of symmetric and un-symmetric operations. 3- The free S-jump equations can be applied when jump formed just upstream the end of the pier.

This is usually occurs when main gates are out of order and the emergency gates are used. Similar to the free S-jump, the submerged S-jump.

4- The R-jump equations can be applied for both cases of symmetric and un-symmetric operations when the jump is formed downstream from the piers.

6 Conclusions The basic characteristics of the different hydraulic jumps formed in expanding channels are presented and discussed. The similarities between the expansion and the symmetric and un-symmetric operations of multi-vents regulators are explained. Equations for predicting the tailwater depth and the length of jump for each jump are introduced in terms of the expansion ratio and the Froude number. The developed equations for sequent depth ratio are compared with data from other sources and with similar equations for other authors. It is possible to use the equations developed for jumps in expanding channels on jumps formed downstream of the gates of the multi-vents regulators in both cases of symmetric and un-symmetric operations. However, it is recommended to compare the performance of the equations in expanding channels with data from prototype or scaled models on multi-vents regulators before applying them in the design. 7 References 1. Bremen, R. and Hager W.H. (1993), T-jump in abruptly expanding channel, J. Hydraulic

Research, Vol.31, No.1, pp.61-73. 2. Bremen, R. and Hager W.H. (1994), Expanding stilling basin, Proc. ICE, Water, maritime

and energy, 106, pp. 215-228 3. Hager, W.H. (1985), Hydraulic jumps in non-prismatic rectangular channels, Journal of

Hydraulic Research. Vol.23, No.1, pp. 21-35 4. Hager, W.H. and Bremen, R. (1989), "Classical Hydraulic Jump: Sequent Depths", Journal

of Hydraulic Research, Vol. 27, No. 5, pp. 565-585. 5. Hager, W.H., Bremen, R and Kawagoshi, N (1990), Classical Hydraulic Jump: Length of

Roller", ", Journal of Hydraulic Research, Vol. 28, No. 5, pp. 591-608. 6. Hager, W.H (1992), Energy dissipators & hydraulic jumps, Kluwer Academic publication,

Dordrecht, The Netherlands, pp. 151-173. 7. Herbrand, K (1973), The spatial hydraulic jump, Journal Hydraulic Research, Vol.11, No.3,

pp. 205-218. 8. Kuznelsov, S.K. (1964), Free Horizontal Spread of the Stream Flow in the Tailwater of

Hydro-structure", Soviet Hydro Engineering (English Translation), Israel Program for Scientific Translations, Jerusalem, pp.346-349.

9. Matin, M.A., Negm, A.M., El-Saiad, A.A. and Alhamid, A.A. (1997), Prediction of sequent depth ratio of free hydraulic jump in abruptly enlarged channels”, The Egyptian Journal for Engineering Sciences and Technology, (EJEST Journal), Faculty of Eng., Zagazig University, Zagazig, Egypt, April 1997, Vol. 2, No.1, pp.31-36.

10. Matin, A.M., Alhamid, A.A. and Negm, A.M. (1998), "Prediction of Sequent Depth Ratio of Hydraulic Jump in Abruptly Expanding Channel", Proceedings of International Conference on Hydro-Science and Engineering Hydro-Science and Engineering ICHE98, Advances in Hydro-Science and Engineering, Vol.III, Published on CD-Rom and A Booklet of Abstracts, 31 Aug.-3 Sep. 1998, Cottbus/Berlin, Germany

11. Negm, A.M., Ibrahim, A.A. and Salem, M.N. (2000), "Modeling Of Depth Ratio Of Hydraulic Jumps In Abruptly Enlarged Stilling Basins", Accepted for Publications in Scientific Bulletin, Faculty of Engineering, Ain Shams University, Cairo, Egypt.

12. Negm, A.M. (2000), Empirical Prediction of Properties of R-Jump and Sbumerged S-jump In Abruptly Expanding Stilling Basins", Submitted to The Egyptian Journal for Engineering Sciences and Technology, (EJEST), Faculty of Eng., Zagazig University, Zagazig, Egypt

13. Rajaratnam, N. (1967), "Hydraulic Jumps", Advances in Hydroscience, Ven Te Chow, Ed., Vol. 4, Academic Press, New York, pp. 197-280

14. Rajaratnam, N and Subramanya, K. (1968), Hydraulic jumps below abrupt symmetrical expansions, Proc. ASCE, Journal of Hydraulics Division, Vol.94, HY3, pp. 481-503.

15. Sharama, H.R. (1966), Spatial jump and its problems, Journal of CBIP, New Delhi, India, Oct., pp. 431-440.

16. Smith, G.D. (1989), The submerged hydraulic jump in an abrupt lateral expansion, Journal of Hydraulic Research, Vol. 27, No.2, pp. 257-266.

17. Unny, T.E. (1961), The spatial hydraulic jump, Proc. IX IAHR Congress, Belgrade, pp. 32-42.

8 Notations b width of channel upstream of sudden expansion. B width of channel downstream of sudden expansion. D depth ratio of hydraulic jump. D* depth ratio for classical jump. E1 modified Froude number F1 approaching flow Froude number. Fo approaching flow Froude number at the outlet or under the gate. F’o modified Froude number at the outlet in terms of r’. Lr length or roller of jump with

*rL for the classical jump.

Lj length of jump with *jL for the classical jump.

.r’ a defined parameter for the incoming jet. yo the incoming flow depth below the gate just downstream of the expansion. .yt the tail water depth of R and S-jump. K,k defined parameters X the non-dimensional toe parameter for T-jump. β expansion ratio, β=b/B.