Technical Note

Transitional Expanding Hydraulic JumpEsmaeil Kordi, S.M.ASCE1 and Ismail Abustan, A.M.ASCE2

Abstract: This note presents the general features of velocity and variation in length scales of the transition expanding jump, free jump, andwall jet in the rectangular horizontal open channel. The experiments were performed for a range of Froude numbers from 2 to 6 and at anabrupt expansion ratio of 2. The results indicate that the postdepth y2 required to form an expanding jump is distinctly smaller than that for thecorresponding classical jumps. The expanding jump length was 1.25 times the corresponding free jump length. The velocity profiles for thetransition jumps were similar to those for the wall jet. For the transitional expanding jumps, the boundary layer thickness δ, normalized bythe supercritical flow depth y1, grows more rapidly relative to the free jumps. The normalized boundary layer thickness δ in terms of the lengthscale of b equal to the flow depth y, where the velocity u is equal to half of the maximum velocity um, against the longitudinal direction xnormalized by y1, which was stabilized at 0.31 for expanding jumps. The length scale L, equal to x where the maximum velocity is equal tohalf of the initial velocity u0, of the free jump was approximately 0.75 that of the corresponding transitional expanding jump. The growth rateof the length scale (b=y1) for the expanding jumps was approximately 2.4 times that of the corresponding wall jet and free jumps. DOI: 10.1061/(ASCE)HY.1943-7900.0000479. © 2012 American Society of Civil Engineers.

CE Database subject headings: Hydraulic jump; Velocity; Length; Open channels.

Author keywords: Transitional expanding jump; Free jump; Plane wall jet; Velocity scale; Length scale.

Introduction

A classic hydraulic jump is defined as the transition of a supercriti-cal flow into a subcritical flow formed in a smooth, horizontal, wideprismatic rectangular open channel and approaching a Froudenumber F1 > 2 (Bakhmeteff 1932; Peterka 1958; Hager 1992).For a hydraulic jump created by a gate based on uniform flow,hydrostatic pressure distribution, uniform velocity distribution, andingoing boundary shear stress at the beginning and end of thehydraulic jump, the subcritical postjump depth y�2 is given bythe Belanger equation [Eq. (1)]

y�2y2

¼ 12ð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 8F2

1

q� 1Þ ð1Þ

where y1 = supercritical depth of the jump; F1 = supercriticalFroude number = U1=ðgy1Þ0:5; U1 = mean velocity of the super-critical flow; and g = acceleration caused by gravity. Classicalhydraulic jumps have been investigated extensively (Peterka 1958;Rajaratnam 1965; McCorquodale 1986; Hager 1992). In practicalapplications, such as downstream of hydraulic structures (e.g.,spillways), weirs require an enlargement in design with the inwardsupercritical flow creating an enlarged tail-water that creates anexpanding hydraulic jump. The majority of previous studies onthe behavior of expanding jumps have dealt with the general char-acteristics of the jump, emphasizing the prediction of the conjugate

depths (Kusnetzow 1958; Unny 1961; Rajaratnam and Subramanya1968; Herbrand 1973; Hager 1985; Nettleton and McCorquodale1989; Bremen 1990; Ranga Raju 1993; Bremen and Hager 1993,1994), the general characteristics of the probable sequence ofjumps and the velocity distribution (Noseda 1964; Rajaratnam andSubramanya 1967, 1968a; Herbrand and Knauss 1973; Smith 1989;Bremen 1990; Ferrer and Nasello 2002). Nevertheless, thus far nosystematic analysis has been performed on the transitional expand-ing hydraulic jump (T-jump), plane turbulent wall jet and free jump,particularly in recognizing the overall similarities among theseflows. In the present study, an effort was made to better understandthe behavior of T-jumps and the similarities among the properties ofT-jumps, wall jets and free jumps. To find the overall similarities, thefollowing length scales were employed: (1) the maximum velocityum at each station as a function of the velocity scale, (2) the distance bfrom the channel bed, where the local velocity is equal to half the umand δu=δy < 0, and (3) the longitudinal distance L, where the maxi-mumvalue of the time-averaged longitudinal velocity um at the givenstation is equal to one half of the depth-averaged velocity U1 at thelocationwhere the jump starts (Long et al. 1990;Wu andRajaratnam1995; Ead and Rajaratnam 2002) [Fig. 1(a)]. Laboratory experi-ments were performed for a range of Froude numbers from 2 to6 at expansion ratios of 1 and 2. In all of the runs, the T-jump toewas positioned to be slightly upstream from the starting point of theexpansion section, which can be considered as a transition betweenthe S-jump and the classical hydraulic jump.

Experimental Installation

The laboratory experiments were conducted at the Universiti SainsMalaysia in a rectangular, horizontal open channel that was 10 mlong, 0.6 m deep, and 0.3 m wide, with a hydraulic circuit for flowrecirculation. The sidewalls and bottom of the open channel werecomposed of glass panels and steel sheets. The selected reach forthe measurements was 1.5 m long and positioned 1.5 m down-stream from the channel inlet. The contraction part in the expansion

1Researcher, School of Civil Engineering, USM Engineering Campus,Univ. Sains Malaysia (USM), Nibong Tebal, 14300, Penang, Malaysia(corresponding author). E-mail: esmaeilkordi@gmail.com

2Associated Professor, School of Civil Engineering, USM EngineeringCampus, Univ. Sains Malaysia (USM), Nibong Tebal, 14300, Penang,Malaysia. E-mail: ceismail@eng.usm.my

Note. This manuscript was submitted on January 22, 2011; approved onJune 23, 2011; published online on June 25, 2011. Discussion period openuntil June 1, 2012; separate discussions must be submitted for individualpapers. This technical note is part of the Journal of Hydraulic Engineer-ing, Vol. 138, No. 1, January 1, 2012. ©ASCE, ISSN 0733-9429/2012/1-105–110/$25.00.

JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JANUARY 2012 / 105

J. Hydraul. Eng. 2012.138:105-110.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KY

OT

O U

KO

GA

KU

CH

IKY

U o

n 12

/24/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

ratio e ¼ ðb2=b1Þ ¼ 2 (Fig. 1), was 1 m long and composed ofPlexiglas panels. The expanding jumps were localized to be slightlyupstream from the beginning of the expansion section using atailgate located at the end of the flume. The supercritical flowwas produced by a sluice gate located 500 mm upstream fromthe end of the contraction section. The discharge rate was measuredusing a magnetic flow meter and calibrated using the volume pertime technique. A point gauge with an accuracy of 0.1 mm waspositioned over the channel centerline to measure the flow depth.A Prandtl tube with an outside diameter of 3.0 mm was used inconjunction with a vertical manometer to measure the time-averaged longitudinal velocity. The roller endpoint was recognized

using the stagnation point and visualized using a float material(Hager et al. 1990). The velocities in the air-water flow (boilingzone) region were not recorded. A total of 17 trials (from two seriesof experiments) were performed, and the general properties of theexperiments are shown in Table 1. The discharge rate Q, supercriti-cal flow depth y1, Froude number F1, and Reynolds number R�

1 ¼ðU1y1=νÞ were in the range of 5.24–1.45(l=s), 1–2 cm, 2.42–6.39and 8,754–49,816, respectively. The scale effect does not appear ifmeasured inflow aspect ratio ω ¼ ðy1=b1Þ, in which y1 is inflowdepth and b1 = inflow channel width, is smaller than the criterionfor ωL proposed by Hager and Bremen (1989). In all of the runs,0:067 ≤ ω ≤ 0:12 was smaller than the expected 0:336 ≤ ωL ≤0:474. The jump asymmetry and poor flow conditions were exam-ined using the asymmetry factor ψ proposed by Bremen (1990). Ifψ > 0:28, the T-jumps were not strongly asymmetric; they weresimilar to symmetric jumps when ψ < 0:125ð1� 2=3X1Þ, whereX1 is the nondimensional jump toe position defined as ðx1=L�r Þand x1 is the distance from the beginning of the expansion sectionto the end of the T-jump, and L�r is the corresponding length of theroller of the classical jump [Fig. 1(b)]. The symmetric T-jump wasselected for the present research.

Table 1. General Properties of Experiments

y1 (m) y�2 (m) U1 (m=s) F1 (m=s) R�1 D (-) Lj=y�2 (-) L=y1 (-)

Series A: expansion ratio e ¼ 1, (8 runs)

0.02 0.066–0.13 1.08–2.5 2.42–6.31 21316–49816 0.02–0.12 4.91–6.14 14.59–24.84Series B: expansion ratio e ¼ 2, (9 runs)

0.01–0.018 0.038–0.084 0.96–2.19 3.54–6.39 8754–43729 0.154–0.35 5.62–7.54 9.12–18.1

Note: D = nondimension depth deficit factor ½ðy�2 � y2Þ=y�2�, where kinematic viscosity (ν) is equal to 1:1 × 10�6 (m2=s).

(a)

(b)

Fig. 1. Definition sketches of T-jump and abrupt expansion configura-tion for: (a) plan view; (b) side view; xe = distance from beginning ofenlarged section to end of short lateral eddies where velocity distribu-tion is not fully uniform in lateral direction, and δ = boundary layerthickness; x is longitudinal distance from beginning of expansionsection, and y is vertical distance from bed

(a)

(b)

Fig. 2. (a) Variation of sequent depth ratio ½y2=y1�with supercritical F1;(b) variation of depth deficit factor D with F1 for e ¼ 2, 3:5 ≤ F ≤ 6:5

106 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JANUARY 2012

J. Hydraul. Eng. 2012.138:105-110.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KY

OT

O U

KO

GA

KU

CH

IKY

U o

n 12

/24/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Experimental Results

Sequent Depth Ratio

The comparison between the measured values of the sequent flowdepth ðy2=y1Þ and those calculated by Eqs. (2)–(4) are plotted inFig. 2(a) as a function of the supercritical F1 and the expansionratios (e ¼ 1 and e ¼ 2). Fig. 2(a) indicates that the expansion ratioe exerts a considerable decreasing effect on the sequent depth ratio

ðy2=y1Þ. A good agreement can be observed between the measuredðy2=y1Þ for e ¼ 2 and those calculated by Herbrand (1973)[Eq. (2)], Bremen (1990) [Eq. (3)], and Ranga Raju (1993)[Eq. (4)]. For the e ¼ 2 data, the Blanger equation can be rewrittenas [Eq. (5)]

�y2y�2

�¼

�b1b2

�38ð3:1 < F1 < 9Þ ð2Þ

�y�2 � y2y�2 � y1

�¼ ð1� tanhð1:9X1ÞÞ

�1� 1 ffiffi

1e

q�

ð3Þ

��y2y1

�2� 1

�¼ 2F2

1

���b1b2

�2 y1y2

þ b1b2

�ð4Þ

�y�2y1

�¼ 1

2ð

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4:5F2

1

q� 1Þ ð5Þ

The variation of the depth deficit factorD against F1 is plotted inFig. 2(b). In the range studied here for F1ð3:5 < F1 < 6:5Þ ande ¼ 2, the depth deficit factor D (−) did not change with variationsin the F1 number. Fig. 2(b) also indicates a constant mean value of0.25 for D, which suggests that the tail-water depth y2 required toform a T-Jump was noticeably smaller in comparison with thecorresponding classical jumps y�2.

Jump Length

The variation of the normalized jump length Lj in terms of the se-quent depth of the classical jump y�2 against the supercritical Froudenumber F1 is plotted in Fig. 3. For the T-jump (e ¼ 2), the ðLj=y�2Þratio was independent of the supercritical F1 and was nearly equalto the average value of 6.82, compared to the correspondingclassical jump, which was nearly equal to 5.45 in value. The ex-panding jump length was approximately 1.25 times larger than thatof the corresponding classical jump.

Velocity Profiles of Free Jump and Expanding Jump

A consolidated form of all of the experimental data is plotted inFig. 4. Fig. 4 clearly indicates that the velocity profile of the ex-panding jump in the forward flow is similar to that of the free jump.An increase in the longitudinal distance from the toe of jumpcaused a growth in the boundary layer thickness and an increase

Fig. 3. Variation of normalized length of jump in terms of sequentdepth of classical jump ½Lj=y�2� against F1

Fig. 4. Forward flow similarity profiles for all of experiments includingfree jump and T-jump

(a) (b)

Fig. 5. Longitudinal variation of ðum=U1Þ with; (a) ðx=y1Þ and (b) ðx=LjÞ for free jump and T-Jump results, along with corresponding curve for planewall jet results

JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JANUARY 2012 / 107

J. Hydraul. Eng. 2012.138:105-110.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KY

OT

O U

KO

GA

KU

CH

IKY

U o

n 12

/24/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

in the decay of the maximum velocity um. The thickness δ of theboundary layer is approximately 0.25 times that of b, in comparisonto the value for the corresponding classical plane turbulent wall jetequal to 0.16.

A larger value of 0.12 is observed for the longitudinal growthrate of the boundary layer thickness δ=y1 for the T-jump, comparedto the corresponding free jump value of 0.042 and to the walljet value.

General Features of Velocity and Length ScalesVariation

The variation of um=U1 against x=y1 is plotted in Fig. 5(a). A non-linear decrease observed in um=U1 with variation in longitudinaldistance. For the T-jumps at x=y1 < 30, the value of um=U1decreased relative to that for the corresponding free jumps. TheT-Jump experimental data can be described by the plane turbulent

wall jet equation proposed by Rajaratnam (1976), umU1

¼ 3ffiffiffixy1

q. The

variation of um=U1 against x=Lj is plotted in Fig. 5(b) and indicatessignificant scattering and data mixing. In fact, the length of jumpLj cannot be considered as a scale that yields a good correlationfor the um=U1 variation. A poor agreement was observed betweenthe corresponding plane wall jet curve developed by Rajaratnam(1976), um ∝ 1=

ffiffiffix

pwith α ¼ 0:5, and the experimental results.

The variation of um=U1 against x=L is plotted in Fig. 6. The freejump observations show good correspondence to the free jumpcurve proposed by Wu and Rajaratnam (1995). Fig. 6 indicates thatfor um=u1 > 0:5, the corresponding curve for the plane wall jet(Rajaratnam 1976) was positioned below the T-jump mean curve(Fig. 6). For um=U1 < 0:5, the corresponding curve was locatedabove the free jump curve. Notably, similar to the free jump, thevalue of um=U1 ¼ 0:5 of the T-jump was also in cross point withthe corresponding curve for the plane wall jet (Rajaratnam 1976).Nevertheless, mixing of experimental data in the range of 0 <x=L < 1 can be observed. Moreover, a similar behavior in the varia-tion of um=u1 with respect to x=L for free jumps and T-jumps canalso be observed.

The variation of the normalized length scale L=y1 of the freejumps and T-jumps against the supercritical Froude number F1 isplotted in Fig. 7 and indicates that the length scale L for the T-jumpwas significantly smaller than that of the free jump. The length scaleL normalized by supercritical flow depth L=y1 for the free jumpresults collapsed on the curve proposed by Wu and Rajaratnam(1995) (Fig. 7). The growth rate of L=y1 for T-jump data was smallercompared to the free jump. For all the experiments, the ratio ofLf =LTwas approximately equal to 0.75, where the subscripts f and T referto the free jump and T-jump, respectively.

The variation of the length scale b in terms of y1 against thenormalized longitudinal distance x=y1 is plotted in Fig. 8(a). Theseresults consist of a scattering that indicates that the length scale bnormalized by y1 does not have a good correlation. However, anincreasing trend can be distinguished for the transition T-jumpat different rates of growth relative to the free jump [Fig. 8(a)].For the T-jump, the length scale grew more rapidly (0.166) thanthat observed for the free jump (0.07) and wall jet curve (with a

Fig. 6. Longitudinal variation of ðum=U1Þ against ðx=LÞ for free jumpand T-Jump results along with corresponding curve for plane turbulentwall jet results

Fig. 7. Variation of normalized length scale L=y1 against supercriticalFroude number F1

(a) (b)

Fig. 8. Variation of length scale b against normalized longitudinal distance x=y1 according to: (a) length scale in terms of y1; (b) length scale in termsof ½y1F1�

108 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JANUARY 2012

J. Hydraul. Eng. 2012.138:105-110.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KY

OT

O U

KO

GA

KU

CH

IKY

U o

n 12

/24/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

slope of 0.0728) (Rajaratnam 1976; Wu and Rajaratnam 1995; Eadand Rajaratnam 2002). The growth rate b=y1 of the expandingT-jump was approximately 2.4 times that for the correspondingwall jet and free jumps. The growth of the length scale b forthe free jump observation of Ead and Rajaratnam (2002) wasapproximately the same as that for the wall jet line. A bettercorrelation was observed between the length scale b in terms ofy1:F1, as a new scale, and the nondimensional longitudinal distancex=y1 [Fig. 8(b)].

The variation of δ=b against the normalized longitudinal dis-tance x in terms of y1 is plotted in Fig. 9. A constant value of0.31 was observed for the δ=b variation with respect to x=y1 forthe T-jump results.

Conclusions

The overall similarity among the free jump, wall jet, and T-jumpwas examined experimentally on the abrupt enlargement ratio oftwo, for a range of Froude numbers from 2 to 6 on the horizontalsmooth bed. A good agreement was observed between the mea-sured sequence depth ratio y2=y1 for the expanding jump and thosepredicted by Herbrand (1973) [Eq. (2)], Bremen (1990) [Eq. (3)]and Ranga Raju (1993) [Eq. (4)]. The dimensionless depth deficitfactor D was at a constant value of 0.25, indicating that the post-depth y2 required to form an expanding jump was noticeablysmaller than those for the corresponding classical jumps y�2. Theresults indicate that the normalized expanding jump length Lj inthe form of y�2 was independent of the supercritical Froude numberF1 and was stabilized at a value of 6.9. The expanding jump lengthwas equivalent to 1.25 times that of the corresponding free jumps.The velocity profiles in the forward flow were similar to the wall jettype for the expanding jump results. Values of 0.14 and 0.042 wereobserved for the growth rate of the boundary layer thickness δ=y1for T-jumps and for the corresponding free jumps, respectively. Atthe given value of x=y1, the normalized maximum velocity um=U1decreased relative to the corresponding free jump. The variation ofum=U1 with x=y1 for T-jumps can be described by the behavior ofplane turbulent wall jets (Rajaratnam 1976). The normalized lengthscale L in terms of y1 for the T-jump was significantly smaller thanthat for the corresponding free jump. The length scale of the freejump Lf was approximately 0.75 times that for the correspondingT-jump length scale LT . For T-jumps, the growth rate of the lengthscale b=y1 was approximately 2.4 times that for the correspondingwall jet and free jump results. For expanding jumps, an averagevalue of 0.31 for the variation of the normalized boundary layerthickness in terms of the length scale bðδ=bÞ against x=y1 was

observed. A better correlation was found between the normalizedgrowth rate of the scale of b against ðy1:F1Þ and x=y1.

Acknowledgments

The research was conducted with the support of the UniversitiSains Malaysia (USM). The writers appreciate the valuable com-ments that were offered by Professor N. Rajaratnam, Universityof Alberta, Canada. The writers are also grateful to the anonymousreviewers and to the editor(s) for the useful comments and sugges-tions that improved the original manuscript.

Notation

The following symbols are used in this paper:b = length scale equal to y, where u ¼ 0:5 um and δu=δy < 0b1 = inflow channel widthb2 = width of channel in expansion sectionD = nondimension depth deficit factor ðy2 � y�2Þ=y2e = expansion ratio b1=b2

F1 = supercritical Froude number U1=ðgy1Þ0:5g = gravitational accelerationL = length scale equal to value of x, where um ¼ 0:5U1Lr = length of roller of jumpLj = length of hydraulic jumpR�1 = modified Reynolds number U1y1=ν

U1 = velocity immediately downstream of gateum = maximum value of velocity u at any sectionx = longitudinal distance from beginning of expansion

sectiony = vertical distance from bedy0 = gate opening depthy1 = supercritical depth of jumpy2 = conjugate tail water depthy�2 = subcritical sequent depth of classical jump Belanger

equation [Eq. (1)]z = transverse distance from bedδ = boundary layer thicknessν = kinematic viscosity of fluidρ = density of fluidω = inflow aspect ratio y1=b1ωL = limit value for scale effectsX1 = nondimensional jump toeψ = jump asymmetry factor

References

Bakhmeteff, B. A. (1932). Hydraulics of open channels, McGraw-Hill,New York.

Bremen, R. (1990). Expanding stilling basin. Communication, R. Sinniger,ed., Laboratoire de constructions hydrauliques, Ecole polytechniquefédérale de Lausanne, 3.

Bremen, R., and Hager, W. H. (1993). “T-Jump in abruptly expandingchannel.” J. Hydraul. Res., 31(1), 61–73.

Bremen, R., and Hager, W. H. (1994). “Expanding stilling basin.” Proc.Inst. Civ. Eng.: Water Marit. Energy, 106(3), 215–228.

Ead, S. A., and Rajaratnam, N. (2002). “Hydraulic jumps on corrugatedbeds.” J. Hydraulic Eng., 128(7), 656–663.

Ferreri, G. B., and Nasello, C. (2002). “Hydraulic jumps at drop and abruptenlargement in rectangular channel.” J. Hydraul. Res., 40(4), 491 –505.

Hager, W. H. (1985). “Hydraulic jump in non-prismatic rectangularchannels.” J. Hydraul. Res., 23(1), 21–35.

Hager, W. H. (1992). Energy Dissipators and hydraulic jump, KluwerAcademic, Dordrecht, Netherlands.

Fig. 9. Variation of δ=b ratio with respect to normalized longitudinaldistance x in terms of y1

JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JANUARY 2012 / 109

J. Hydraul. Eng. 2012.138:105-110.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KY

OT

O U

KO

GA

KU

CH

IKY

U o

n 12

/24/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Hager, W. H., and Bremen, R. (1989). “Classical hydraulic jump: Sequentdepths.” J. Hydraul. Res., 27(5), 565–585.

Hager, W. H., Bremen, R., and Kawagowshi, N. (1990). “Classical hy-draulic jump: Length of roller.” J. Hydraul. Res., 28(5), 591–608.

Herbrand, K. (1973). “The spatial hydraulic jump.” J. Hydraul. Res., 11(3),205–218.

Herbrand, K., and Knauss, J. (1973). “Computation and design of stillingbasins with abruptly or gradually enlarged boundaries.” Int. Commis-sion on Large Dams (ICOLD), 11th Congress, 57–79

Kusnetzow, S. K. (1958). “Die Fliessbewegung bei plotzlicher Verbreire-rung des Strombettes.” Gidrotechniceskoe Stroitelstvo, 27(H6), 34–37(in Russian).

Long, D., Rajaratnam, N., and Steffler, P. M. (1990). “LDA study offlow structure in submerged hydraulic jump.” J. Hydraul. Res., 28(4),437–460.

McCorquodale, J. A. (1986). “Hydraulic jumps and internal flows.” Chap-ter 8, Encyclopedia of fluid mechanics, N. P. Cheremisinoff, ed., Vol. 2,Gulf Publishing, Houston, 120–173.

Nettleton, P. C., and McCorquodale, J. A. (1989). “Radial stilling basinswith baffles locks.” Can. J. Civ. Eng., 16(4), 489–497.

Noseda, G. (1964). “Un fenomeno di instabilita del risalto lungo una

corrente veloce in espansione [An instability phenomenon of hydraulicjump in enlarging supercritical flow],” L’energia elettrica, 41(5),249–254.

Peterka, A. J. (1958). “Hydraulic design of stilling basins and energydissipators.” Eng. Monograph No. 25, U.S. Bureau of Reclamation,Denver.

Rajaratnam, N. (1965). “The hydraulic jump as a wall jet.” J. Hydraul. Div.,91(5), 107–132.

Rajaratnam, N. (1976). Turbulent jets, Elsevier Science, Amsterdam, 304.Rajaratnam, N., and Subramanya, K. (1967). “Diffusion of rectangular wall

jets in wider channels.” J. Hydraul. Res., 5(4), 281–294.Rajaratnam, N., and Subramanya, K. (1968). “Hydraulic jumps below

abrupt symmetrical expansions.” J. Hydraul. Div., 94(2), 481–503.Ranga Raju, K. G. (1993). Flow through open channels, 2nd Ed., McGraw-

Hill, New Delhi, India.Smith, C. D. (1989). “The submerged hydraulic jump in an abrupt lateral

expansion.” J. Hydraul. Res., 27(2), 257–266.Unny, T. E. (1961). “The spatial hydraulic jump.” IX IAHR Congress,

32–42.Wu, S., and Rajaratnam, N. (1995). “Free jumps, submerged jumps and

wall jets.” J. Hydraul. Res., 33(2), 197–212.

110 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / JANUARY 2012

J. Hydraul. Eng. 2012.138:105-110.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KY

OT

O U

KO

GA

KU

CH

IKY

U o

n 12

/24/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.