LOCAL SCOUR DUE TO SUBMERGED HORIZONTAL JET

By S. S. Chatterjee, 1 S. N. Ghosh, 2 and M. Chatterjee 3

ABs'rRACT: The phenomena of local scour and sediment transport due to a hor- izontal jet issuing from a sluice and flowing over a rigid apron, then on to an erodible bed, have been investigated. Based on experimental data, the scour char- acteristics have been correlated through the development of empirical expressions for the time to reach equilibrium stage, the locations of maximum scour depth and peak of dune, and the variation of maximum scour depth with time. Similarity criteria for the predication of development of the scour hole and sediment transport have also been developed and, apart from the flow parameters and sediment prop- erties, the jet diffusion characteristics play an important role in these processes. The empirical relationships developed in this study highlight the necessity of an individual assessment when developing the similarity criteria for a particular flow situation.

INTRODUCTION

Water discharged through a sluice forms a jet that flows over an erodible bed after leaving a rigid apron. The high velocity of the jet causes high local shear stresses that generally exceed the critical shear stress for incipient motion of the bed material, resulting in local scour at the downstream end of the rigid apron. This causes an increase in the local flow depth; conse- quently, the shear stress acting over the bed will be reduced, which in turn will encourage a reduction in the scouring rate. The limiting extent of scour is reached when the shear stress acting over the bed is reduced to the critical shear stress of the bed material. The extent of the scour hole is strongly dependent on time. Initially, the scour development with time is rapid, but it reduces as the equil ibrium stage is reached. A high discharge is usually passed through a hydraulic structure for a limited time, during which the local erosion rate is relatively high. Depending on the shape of the scour hole and the properties of the bed material, the structure in the vicinity of the scour hole may collapse. It is therefore necessary to study the whole process of scour phenomena, not just the identification of the probable maximum scour depth. It is also of interest to 'know the sediment transport rate at the initial stage to allow the formulation of suitable scale modeling laws.

This study is concerned with an experimental investigation of the scour phenomena and sediment transport due to a two-dimensional submerged horizontal jet of water issuing from a sluice and flowing over a rigid apron to an erodible bed, as shown in Fig. 1.

The pioneering investigation on scour due to a jet was done by Rouse (1939). Scour due to a horizontal wall jet was studied by Laursen (1952)

1Prof. and Head of Civ. Engrg. Dept., Jalpaiguri Government Engrg. Coll., Jal- paiguri--735 102, West Bengal, India.

ZProf. of Civ. Engrg., Indian Inst. of Tech., Kharagpur--721 302, West Bengal, India.

3Lecturer of Civ. Engrg., North Eastern Regional Inst. of Sci. and Tech., Itana- gar--791 109, Arunachal Pradesh, India.

Note. Discussion open until January 1, 1995. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this paper was submitted for review and possible publication on January 16, 1992. This paper is part of the Journal of Hydraulic Engineering, Vol. 120, No. 8, August, 1994. 9 ISSN 0733-9429/94/0008-0973/$2.00 + $.25 per page. Paper No. 3296.

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-XD '1

FIG. 1. Definition

and Tarapore (1956). Scour by circular impinging jets was studied by Dod- diah et al. (1953), Poreh and Hefez (1967), Sarma and Sivasankar (1967), Westrich and Kobus (1973), and Rajaratnam and Beltaos (1977). Iwagki et al. (1958) undertook an analytical study of scour promoted by a three- dimensional jet. Carstens (1966) developed an empirical formula for sedi- ment transport rate by analyzing the experimental data of Laursen (1952). Scour caused by impinging plane jets was studied by Altinbilek and Okyay (1973) and by Francis and Ghosh (1974). Scour by circular turbulent wall jets was studied by Rajaratnam and Berry (1977). The scour due to circular wall jets in cross flow was examined by Rajaratnam (1980). The scour due to plane wall jets in the shallow tailwater was examined by Rajaratnam and MacDougall (1983). Hassan and Narayanan (1985) investigated the local scour downstream from rigid aprons. Local scour caused by submerged wall jets was studied by All and Lim (1986). Uyumaz (1988) investigated the scour pattern downstream from vertical gates. Mason (1988) studied plunge pool scour. A study of the scour pattern in shallow tailwater was done by Johnston (1990). Blaisdell and Anderson (1991) analyzed the scour down- stream from a pipe spillway for the design of a plunge-pool energy dissipator.

The studies of the aforementioned investigators have made important contributions to the knowledge of the phenomena of local scour downstream from hydraulic structures in the relevant flow situations. Their studies were aimed mainly at determining the maximum scour depth and the geometry of scour hole. Little attention seems to have been paid to the study of the transport rate of bed materials. Hassan and Narayanan (1985) and Ali and Lim (1986) are more relevant in the context of the current study. Hassan and Narayanan (1985) investigated the flow characteristics and the similarity of scour profiles downstream from a rigid apron due to a water jet issuing from a sluice. They proposed a semiempirical theory based on a character- istic mean velocity in the scour hole to predict the maximum scour depth with respect to time, and the results were compared with experimental data.

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However, they did not study the rate of transport of bed materials. Ali and Lim (1986) presented various expressions to describe the flow characteris- tics, volume of scour, and the depth of scour hole with respect to time due to two- and three-dimensional jets, but they did not investigate the apron condition. Also, in both cases [Hassan and Narayanan (1985) and Ali and Lira (1986)] the experiments were not conducted to reach the equilibrium stage.

Chatterjee and Ghosh (1980) described the results of an investigation on the evaluation of important hydraulic parameters involved in the transport of sediment due to a wall jet, as shown in Fig. 1. In the current paper, the writers have correlated the local scour and sediment transport caused by such a jet with the sediment characteristics and the hydraulic parameters evaluated by Chatterjee and Ghosh (1980). From the experimental data, the empirical relationships for the time required to reach the equilibrium depth, the volume of scour at any time, the location of maximum scour point and peak of dune, time variation of maximum scour depth, maximum scour depth at equilibrium, and the volume rate of sediment transport have been developed. The expressions for time variation of maximum scour depth have been compared with the experimental results presented by Hassan and Narayanan (1985) and the relationships developed by Ali and Lim (1986) and Doddiah et al. (1953). The similarity of scour profiles is presented graphically, and the result is compared with the relationship developed by Hassan and Narayanan (1985). A transport equation correlating weight rate of sediment transport with fluid power of the jet and transport stage is formulated following Bagnold (1973). These empirical relationships are practically useful and the sediment transport equation is particularly ben- eficial when developing the scaling law for sand transport.

FORMULATION OF SCOUR AND SEDIMENT TRANSPORT FUNCTION

The differential equation of transport of bed load (Simons et al. 1965) in two-dimensional flow in a rectangular channel with the x-axis taken in the downstream direction along the bed can be written as

Oz 10i - - + - 0 (1 ) Ot K Ox

where z = riverbed elevation measured from a datum plane; K = bulk unit weight of the bed material; i = weight flow rate of sediment per unit width; and t = duration of time. To predict the scoured-bed profiles from (1) the solution of the relationships governing the flow parameters and sediment characteristics are necessary. Unfortunately, the flow character- istics and the mechanics of sediment transport in the present flow situation are very complicated, which precludes the direct solution of (1). Accord- ingly, from the experimental data, the writers have developed empirical relationships to describe the scour hole geometry and sediment transport.

The most significant parameters that have been used conventionally to describe the scoured-bed profiles are the distance of the maximum scour depth from the end of rigid apron XM, the maximum scour depth at any instant of time hm, and the distance of the peak of the dune from the end of rigid apron Xo, as shown in Fig. 1. Each parameter at any instant of time must be a function of scour volume per unit width Vs and representative grain size dg. The scour volume V,, on the other hand, is a function of time t corresponding to a given flow condition that means the values of effiux

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thickness of jet B0, effiux velocity of jet functional form, Vs can be expressed as

V, = fl(Uo, Bo, d~, t) (2)

Accordingly X~4, hm, and XD can be expressed in alternative forms as

XM = Y2(Vo, Bo, d~, t) (3)

hm = Z3(Uo, Bo, dg, t) (4)

S D --- f4(Uo, Bo, dg, t) (5)

U0, and dg. Therefore, in the

The experimental data shows that the scour profiles are similar in nature. Therefore, Vs can also be expressed in terms of time required to reach equilibrium depth T as

Vs= fs(Uo, Bo, dg, T) (6)

Also, T depends on Uo and dg only, which means

T = f6(Uo, dg) (7)

From the functional relationship of Vs, the volume rate of transport at any instant of time qs can be evaluated from the relationship

dVs (8) q" = dt

and the corresponding weight flow rate i can be evaluated from the expres- sion

i - (1 - n)(S, - 1)pgqs (9)

where p = mass density of water; n = porosity of bed material; Ss = specific gravity of bed material; and g = the acceleration due to gravity.

Following the procedure similar to that of Bagnold (1973), the sediment transport function can now be described by correlating the weight flow rate with the fluid power of the jet w and transport stage (u./u.o) - 1, w being expressed as pqah/Bo, where q is discharge per unit width, Ah is head causing the water to flow, u. is shear velocity, and U.o is critical shear velocity.

EXPERIMENTAL SETUP AND PROCEDURES

Flume The experiments have been conducted in a glass-walled flume, as shown

in Fig. 2. The flume is supplied with water from an overhead tank through an inlet pipe and a stilling pool. The downstream end of the flume is fitted with an outlet pipe that discharges water into a measuring tank that measures the volumetric discharge rate. Two point gauges are mounted on a traveling bridge, one for measuring the water-surface profile and the other for locating a micropitot tube and a preston tube (a pitot tube with a large diameter) that are used to measure velocity and dynamic pressure. A precision ma- nometer is positioned on the traveling bridge to measure pressure differ- ences.

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Bed Materials The experiments have been performed on two erodible beds, one con-

sisting of sand and the other of gravel. The grain-size distribution curves of two bed materials are the same as those used by Chatterjee and Ghosh (1980). The characteristics of the bed materials are summarized in Table 1.

Flow Establishment and Measurements For each type of bed material, experimental runs have been undertaken

for four different gate openings and for various discharges. Before starting a particular run, the initial bed level is marked on the glass wall of the flume corresponding to a suitable height of the downstream control weir and tailwater depth. A two-dimensional submerged horizontal jet is then created by applying a suitable head difference across the sluice opening. In all test runs, the variation of flow depth over the erodible bed is kept relatively shallow by operating the control weir at the downstream end of the flume.

Two identical runs have been made in each test. During the first run, measurements have been taken of the scour profiles at various instants of time, the scour profile at the equilibrium stage, the time required to reach the equilibrium stage, the velocity distribution at the equilibrium stage at several locations along the central plane of the jet, the downstream flow depth in the undisturbed zone, and the discharge. During the second run, the dynamic pressure in a preston tube at the location of maximum scour at various instants of time during the development of scour hole has been measured. The equilibrium stage was assumed to be reached when no grain movement was observed at the location of maximum scour. The experi- mental conditions for the tests are shown in Table 2.

Scour Profiles As soon as the water flowing from the sluice opening reached the erodible

bed, the movement of bed materials from the end of the rigid apron started and the geometry of the scour hole started changing with time. The scour profiles at various times were marked on one of the glass walls of the flume. It was observed that during the initial stage the rate of scouring was very high. It then gradually tapered off as time elapsed. Ultimately the equilib- rium stage was reached when no movement of grains was observed at the location of maximum scour. The time required to reach the equilibrium stage, T, was noted. The scour profiles marked on the glass wall were then traced on to paper on the glass wall. From this, tracing the distance of location of maximum scour and the peak of the dune from the end of rigid apron and the scour depths at any distance for any instant of the time were obtained.

Before starting another run, the disturbed bed was leveled after dewa- tering and drying the bed. The whole procedure was then repeated, and a new set of scour profiles was obtained. The scour profiles have been used to develop various empirical relationships that describe the scour phenom-

TABLE 1. Characteristics of Bed Materials

Bed dg= d~o Wm $ material (mm) % & n (m/s) (degrees)

(1) (2) (3) (4) (5) (6) (7)

Sand 0.76 1.22 2.65 0.430 0.122 29.0 G ravel 4.30 1.43 2.63 0.465 0.488 47.5

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TABLE 2. Significant Test Data (L = 0.66 m)

Bo I Run (m) number (1) (2)

Ah (m) (3)

T (min)

(4______)) (a) Sand a

Uo (m/s)

(5) (mz/S/m)

(6)

D (m) (7)

0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.04 0.04 0.04 0.05 0.05 0.05

1 2 3 4 5 6 7 8 9

10 11 12 13

0.082 0.118 0.172 0.039 0.066 0.137 0.207 0.039 0.051 0.067 0.026 0.036 0.047

(b)

50 30 13

150 80 15 10

120 105 70

180 130 90

1.280 1.560 1.840 0.875 1.138 1.639 2.016 0.875 1.000 1.146 0.715 0.840 0.960

0.0159 0.0204 0.0259 0.0159 0.0210 0.0313 0.0342 0.0160 0.0197 0.0232 0.0216 0.0244 0.0270

0.286 0.291 0.297 0.286 0.287 0.304 0.298 0.291 0.292 0.295 0.292 0.295 0.297

Gravel b

0.02 0.02 0.02 0.02 0.03 0.03 0.03 0.03 0.04 0.04 0.04 0.04 0.05 0.05 0.05

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

0.114 0.142 0.174 0.299 0.089 0.109 0.137 0.162 0.059 0.070 0.086 0.110 0.046 0.055 0.071

aMean value of D = 0.293m. bMean value of D = 0.302m.

135 1.495 95 1.670 65 1.850 20 2.420

155 1.321 135 1.461 100 1.640 80 1.782

240 1.075 220 1.171 200 1.302 160 1.470 280 0.940 230 1.040 195 1.180

0.0220 0.291 0.0246 0.295 0.0275 0.298 0.0425 0.310 0.0264 0.299 0.0306 0.303 0.0330 0.305 0.0378 0.308 0.0283 0.286 0.0343 0.301 0.0333 0.305 0,0419 0.307 0.0312 0.302 0.0350 0.306 0.0410 0.310

ena. The evolution of scour profiles during the development of the scour hole in run 2 is shown in Fig. 3.

ANALYSIS AND DESCRIPTION OF TEST RESULTS

The test data for velocity distribution and dynamic pressure drop were analyzed by Chatterjee and Ghosh (1980). To develop various empirical relationships governing the scour process, the other test data are analyzed in more detail. The development of these relationships is described in the following.

Time Required to Reach Equilibrium Depth The values of T for all test runs are presented in Table 2. To obtain an

empirical expression for T, these values of T have been plotted against U0

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PROFILE t (Hins) 1 I 2 3 3 5 4 8 5 12

RUN NO. 2 s 20 7 30

0,09 8 60

0.06 ORIGINAL BED LEVEL 4 S 6 7 8

0.0~ . . . . . . . . .

o . 0 9 1 1 1 1 p l l I ~ 1 1 I i I I i I ! I I I I I I 0 0.06 0.12 O.l~ 0.24 0.30 0.36 0.~-2 o.~-B 0.54 0.60 0.66 0.72

x(m)

FIG. 3. Evolution of Scour Profiles of Sand Bed in Run 2

300 --

280

240

200

l 60

~ 120

80

40

i I 0o 0.4 o.s 1.2 1.6 2.0 2.4

u0 (Tn/sec)

FIG. 4. Correlation of Equilibrium Time of Scour with Uo

LEGEND RUN Bo Uo ~ RUN Bo UO

No (m) (m/sec) No (m) (m/sec) 0.02 1.280 | r14 0.02 1.495 [3

Z - 1'560 .1~ 115 . 1,670 m 3 " 1,840 ~( |16 ;s 1"850 4 0.03 0.875 ~ }17 , 2.420 5 ,i 1"138 Ill l i b 0.03 1-321 6 , 1.639 0 |19 ,, 1.461 -~

\ 2.0,6~ , ,64o

(Fig. 4). Fig. 4 shows two separate curves confirming the effect of grain diameter on the development of scour hole. The expression of T applicable to both sand and gravel beds is obtained as

T = e4"67d~176176 (10)

where e = the Naperian base.

Volume of Scour at any Time Typical scour profiles during the development of the scour hole for run

2 are shown in Fig. 3. From the recorded scour profiles, the total volume of scour Vs per unit width of mobile bed at any time t has been computed for all test runs, and the value of Vs at any instant of time depends on U0 and B0. On closer scrutiny a relationship can be obtained if the values of V,/U~Bo are plotted against t/T. Accordingly, the values of Vs/U~Bo and t/T have been computed for each individual run, the plots of which are shown in Fig. 5. Thus two relationships are obtained as

0.456

Vs = 0.474Uo2Bo (T) (11)

l/s = 0.374U~Bo - (12)

for the sand and gravel beds, respectively. Using the expression for T [(10)] and incorporating the effect of grain

diameter, (11) and (12) on simplification can be finally expressed as

0.5

~o.s No >

O,I

O

l 0.5

0-3

> 0.t

LEGEND

R- I R -2 ~

SAND . ~ ~ R-3 R-4 A ~

9 u u

~ ~" R-6 0

"" g~ R- R- R-I[ 9

p I i I 1 i I I I i i R -12 ~1{ t iT - R-13 ~(

R-14 R-15 [ ] R-16 B R- J7 nl

R - IB 1~ R- I0 ~-

GRAVEL R-Z0 R-21 E:I

p .-22 cA ~L R-23 (~ ~~~' - LEe..12 R-24 e

R-25 '~ 'lem R-26 A

R-27 ~- i I F I I I I I I I I R-2B &

0.1 0 .3 0.5 0 .7 0.9 Odl t iT b

F IG . 5 . VslU~Bo versus t/T

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V S = 0.456U2Bodg~176176176176176 ) X tO.435dg -~ (13)

Location of Maximum Scour Point and Peak of Dune The measured distance of maximum scour point XM and distance of peak

of dune )to have been plotted against Vs with a logarithmic scale, as shown in Fig. 6 for both sand and gravel beds. Fig. 6 yields

X~ = 0.6V ~ (14)

Xo = 2.684V ~176 (15)

Replacing Vs in (14) and (15) with the parameters suggested in (13) the following expressions for XM and XD are obtained.

X M = 0.447U~176176176176176176176 ) X toa63u~ -~ (16)

X 0 = 1.885U~176176176176176176176176176176176 X t0"196d~ -~ (17)

Similarity of Scour Profiles To study the similarity of scour profiles, the values of h/X D have been

plotted against X/Xo for the sand and gravel beds separately, as shown in Fig. 7. h and Xo are the depth of scour at distance X from the end of rigid apron and the distance of the peak of the dune from the end of rigid apron at any instant of time t, respectively. Fig. 7 indicates that the scour profiles at various times are similar in nature. Fig. 7 also shows that in the case of a sand bed the value of h/XD is different from that of gravel bed for the same values of X/XD, indicating the dependence of the value of h/XD not only on the ratio X/XD but also on the grain-size characteristics.

I I 1 I I I I I I 2.0

1.0

0.5

0 .3

0 .2

0.1

0.05

0.03

0.02

0.01

~ GRAVE.L/ ~ " A.o L,M

x ~.lg

2 3 5 I0 20 50 I00 200 500 I000

Vs X 10 4 (m 3)

FIG. 6. Variation of Maximum Scour Depth, Location of Maximum Scour, and Location of Peak of Dune as Functions of Scour Volume

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LEGEND

RUN NO. 2 RUN NO.7 RUN NO.8 RUN NO.II t (rains') t (rains) t (rains) t (rains)

i.o o' o.s o- l.o ~ 3.o qi 5.o "D 1.o 6 s.o ,~ 8,o -e 5.o ; ] t . s ~ 15.o ~. 15.o 9 8.o qL z.o 9 3o.o -~ 30.0 e-

iz.o [3 3.0 o" 60.0 ,~ 60.0 e" 20.0 -[El 5.0 ~) 120.0 ~" 120.0 "0 30.0 0 7 .o ~ ~8o.o

I0.0 O.

HASSAN 0.4 I- AND . . . . .

NARAYANAN

I o.2~- APRON LENGTH~ ~ . _

I SAND SED ~ ~ ~c

_02 I I , 1 , I , I , I , I , I , I 0.0 0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5

x /% . . . .

FIG. 7(a). Similarity of Scour Profiles for Sand Bed

L E G E N O

RUN NO.15 RUN NO.21 RUN N0.25 RUN N0.28 1: (mins) t Cmins) t 0~ias) 1: (rajas)

1.0 [ ] 1.0 [ ] 1.0 ~ 1.0 3.0 [ ] 3.0 [ ] 3.0 ~ 5.0

I0.0 [ ] 7.0 [ ] 15.0 ~ 15,0 30.0 ~ 21.0 ~ 45 .0 ~ 45.0 A 50.0 [ ] 45.0 [ ] 90 .0 ~ 120.0 ~k 95.0 [31 80.0 [ ] 160.0 ~ 195.0 ~k

>~0 01--~ _J^ ~ 1 , '~J,_'-_-~ .L , - , , "~, ,~. . _ " ~_~v.~. = ~/ ' - O.B I.O 1.2 I .~

-0 .2 GRAVEL BED

FIG. 7(b). Similarity of Scour Profiles for Gravel Bed

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To compare the scour profiles of this study with those presented by Hassan and Narayanan (1985), the relationships proposed by the latter are super- imposed (dashed lines) on the data of the current study related to the sand bed as shown in Fig. 7(a). Fig. 7(a) shows that the relationship of scour profiles related to sand bed of the current study closely follows the rela- tionships proposed by Hassan and Narayanan (1985). Slight deviation be- yond that location is mainly due to the effect of grain diameter on the scouring process.

Time Variation of Maximum Scour Depth From the scour profiles, the maximum scour depth hm at any time t for

all the test runs has been measured, and for each type of mobile bed, the values of h,, are approximately equal for the same value of Vs. To obtain a relationship between hm and Vs, the values of h,, were initially plotted against corresponding values of Vs for both the sand bed and the gravel bed, Then a few points on the two curves have been selected in close intervals in such a way that if a smooth curve is drawn by joining them, it represents the original curve drawn considering all experimental data for a particular mobile bed. These points have been replotted on a log diagram, as shown in Fig. 6. This figure shows that a single straight line can be fitted covering the whole range of scouring process as

h m = 0.379V ~ (18)

hm = 0.513V ~ (19)

for the sand and gravel beds, respectively. Substituting the value of Vs in terms of (13) in (18) and (19), then incorporating the effect of grain diameter the relationship between h m and t leads to

h., = 0 267d~ U~B d-~ z'O31d~176176 ~176176176 )< tO'219di ~176 9 g I_ 0 0 g \v.. t., 0 7]

(20)

To compare the relationship between hm and Vs proposed in this study with that developed by Ali and Lim (1986), the experimental data of Ali and Lim (1986) for a two-dimensional jet have been analyzed, and the values of h m are plotted against Vs, as shown in Fig. 6. Ali and Lim's (1986) data closely follow the data of the current study at the initial stage of scouring. As the equilibrium depth is approached, their data deviate from the writers' study. Such a data trend is due to the effect of grain diameter on the scouring process.

Ali and Lim (1986) proposed an expression of development of scour hole with respect to time for two-dimensional jets given by

/ \0 .33

o o91 ) R where R = the ratio of jet area to its perimeter. Hassan and Narayanan (1985) also developed a relationship for temporal rate of maximum scour depth9 To develop the expression of time variation of maximum scour depth similar to that proposed by All and Lim (1986), the values of hm/R of a few test runs have been plotted against the corresponding values of Uot/R, as shown in Fig. 8. The experimental data of Hassan and Narayanan (1985) have also been analyzed and plotted in the same figure. The relationship proposed by Ali and Lim (1986) is superimposed on Fig. 8, and the data

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6.0

4.0

2.0

1.0 0.8

I 0.6

"~0.4

0.2

0.1 0 2

I t I I I ~ I ~ I I L I t I ' ; i

[ ]

Q

| % iP ~ 9 f ig~,1~ " I "

9 ' ~ ~ P R

Zx (~A | 4 , 9 9

F_SENT STUOY

I AND , z:~ - ALl AND ElM A NARAYANAN 4

EXPT. 66 9 69 @ e6 9

I i I t I , I l l 2 4 6 '8 10 3

Uot /R

8~ II 9 t4 [ ]

22 @ 26 A

2 4 6 8 10 4

FIG. 8. h,,/R versus Uot/R

of this study and that of Hassan and Narayanan (1985) remain in a wide band, and a relationship as proposed by All and Lim (1986) cannot be formed. Such a data trend of the current study and of Hassan and Narayanan (1985) is due to the introduction of a rigid apron and. the effect of grain diameter in the scouring phenomena.

To compare the expression for hm developed in this study with those developed by Rouse (1939) and Doddiah et al. (1953), the fall velocity of the particle is now considered as the pertinent sediment characteristic. The similarity criteria for scour due to vertical jets developed by Rouse (1939) is given by

T \~o/ J (22)

where S = the distance of the mobile bed from the jet efflux section; and W = the geometric mean fall velocity of the sediment. The similarity criteria for scour due to vertical jet developed by Doddiah et al. (1953) is given by

b 0.023 log - 0.022 b + 0.4 (23)

where a = the area of the jet; and b = the depth of the pool above the original bed. The fall velocity W~, corresponding to the dso size for the two types of bed materials of this study is shown in Table 1. To obtain an expression for hm similar to those of Rouse (1939) and Doddiah et al. (1953) the experimental data presented here have been studied, and a re- lationship has emerged if hm/L is plotted as a function of log[(Wmt/L)(WmL/ UoBo)5](Uo/Wm) 1 - - L/Bo. Such a plot is shown in Fig. 9, which shows

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0.14

0.12

O, I0

l o. 08

"~E 0.06 t -

0.0z

0.02

m

J EQ. 27

9 ~ ~. .~ ,

m @t /~ ~,--EQ.24

0 1 . I I , I , [ I -25 -20 - I0 0 I0 20

Elocg {V/Wm,L '~5l (W-~'9"~ I) _ L 1 ,,--o-g ~o, j ~-o_J "

.26

I 30

LEGEND ~1 R- I .,~ R-2

R-3 A R -4 9 R-5 0 R -6 | R~-7 [] R -8 )~(R-9

R-10 9 R-rl

R-I2 R-13 R-14

[] R-15 [] R-16 m R-17 [] R-18

a-19 I~ R-20 [] R-21 | R-22 @ R-23 O R-24 (1) R-25 A R-26 ~' R-27 ,~ R-28

FIG. 9. Correlation of Development of Maximum Scour Depth with Jet Charac- teristics

that the proposed relationship is applicable to all experimental condi- tions of present study except for the test runs having high values of U0. Also, the development of scour depth is faster in the initial stage com- pared with the later stages. The expressions for hm in the initial stage are given by

hm 0.00385 log (Wm/~] ]~u0,wo,) 1 L -L- = \U~/ J Bo + 0.0184 (24)

for sand and

h,,L = 0.04 {log L\[(Wmt~(WmL~5]L/\UoBo/J ( UO/Wm) 1

for gravel. Those at the later stage are given by

k} + 0.3380 (25)

h,, _ 0.00205 log + 0.046 (26) L \UoBo/J Bo

for sand and

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5.0

4.O

3 .0

l o 2,0

E .-t-

1,0

/ ,.~"

o

LEGEND

R- I

~ R-2 R-3 A R-4 0 R-5 0 R-6 E) R-7 r~ R-8 J~R-9

R-to 9 R-If ~ R-I?-

R-13 ,,,r [] R-I4 a/ E] R-IS

~o~ o [] R-16 / [] R-17

-0- R-~9 EQ.28 ~ R-20

~. , ,~ 17 R-21 / (~ R-22

/ ~ R-?-3 I I J I I 9 e R-24

1.0 2.0 3.0 4..0 5.0 6.0 (I) R-25

Uo/gV,~n___ = A R-26 F0= A R-27 Ax R-2~9

FIG. 10. Correlation of Maximum Scour Depth at Equilibrium with Froude Number

hm 0.00685. log (WmL ~ (Uo/W) 1 L --L = \UoBo} J - B00 + 0.128 (27)

for gravel. A striking similarity can be observed between (24)-(27), and the relationships developed by Rouse (1939) and Doddiah et al. (1953). Also, in this situation, apart from flow parameters and sediment properties, the diffusion characteristics of the jet play an important role in the scouring process. Furthermore, the scouring process differs from one situation to the other, and a single relationship to predict the scour pattern in all situations is not possible. This highlights the necessity of performing individual re- search for developing similarity criteria corresponding to a particular flow situation.

Maximum Scour Depth at Equilibrium Following the procedure proposed by Shen et al. (1969), the maximum

scour depth at equilibrium H,, in the current study is expressed as a function of the Froude number Fo = Uo/(gBo)I/L Accordingly, the values of H,,,/Bo have been plotted against F0 as shown in Fig. 10, from which a relationship for both the sand and gravel beds is obtained:

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Eq. (28) indicates that Hm is independent of grain-size effect. This contrasts the dependence of maximum scour depth h,, at any time during the scouring process on the grain size of bed material.

Sediment Transport Relationship The expression for volume rate of transport, qs = dVs/dt, is obtained by

differentiating Vs with respect to t, and the result is expressed in terms of (13). The expression for q~ thus obtained for both sand and gravel beds is given by

qs = O.198U~Bodg~176176176176176176 ~176176 t0435dg-~ 1 (29)

By substituting the values of n, S,, p, and O in (9), the expressions for the weight flow rate of sediment i for the sand and gravel beds are obtained as i = 940 q~ kg/m/min and i = 864 q~ kg/m/min, respectively.

Correlation of Transport Rate with Fluid Power of Jet For sand transport in rivers, Bagnold (1973) developed a transport equa-

tion correlating weight rate of transport with stream power U',tySo and trans- port stage (u./u.o) - 1. 0 is mean flow velocity; y is flow depth; So is riverbed slope; and u./u.o = ('r0/'r0c) 1/2, T0 and x0c being boundary shear stress and critical boundary shear stress for the sediment, respectively. Fran- cis and Ghosh (1974) suggested that in the case of jet scour, the fluid power of jet w can be defined as pqAh/Bo and should be used as the appropriate parameter in place of the stream power considered by Bagnold (1973) in his study of sediment transport in rivers. Following the procedure proposed by Bagnold (1973), the weight rate of transport with transport stage and fluid power of jet as suggested by Francis and Ghosh (1974) have been correlated. Chatterjee and Ghosh (1980) presented four equations corre- lating T,/x0c with t/T, xt being the boundary shear stress at any instant of time, indicating the variation of boundary shear stress with time. For this reason, in this case, the writers consider the transport stage (u.,/u.o) - 1 = (x,/Xoc) 1/2 - 1. The values of T,/Xoc at various times for different test runs have been calculated from the equations developed by Chatterjee and Ghosh (1980). The weight rates of transport at various times have been calculated from (30) and the values of i. The fluid power of the jet w for each test run has been calculated using the test data shown in Table 2. To obtain the desired relationship, the values of i/w were plotted against the values of (u,,/U.o) - 1 using a semilogarithmic scale. The experimental data for the sand and gravel beds fall into two distinct bands, indicating the effect of grain diameter dg. A nondimensional parameter xm/dg, similar to y/dg and x/dg (Bagnold 1973) (x m being the distance of maximum scour point from the sluice) is now considered. In this study, since the variation of flow depth is small, its effect on the sediment transport is not considered. On closer scrutiny, if i/w is plotted against [(u.t/U.o) - 1](xm/d~) m, a meaningful relationship applicable to both sand and gravel beds can be obtained, as shown in Fig. 11, where the data can be fitted by two straight lines, one in the range 20 < [(u.,/U.o) - 1](xm/dg) 1/2

I o0 m

0.5 - -

0.2

0.1--

0,05

0,02

of the peak of the dune from the end of the rigid apron. Each of these variables has successfully been expressed in terms of flow parameters and sediment characteristics.

The process of development of the scour hole is adequately described by the relationship of the total volume of scour at any time with the maximum scour depth in terms of flow and grain-size characteristics. The rate of scour is significantly higher during the first few minutes and slower as scouring continues.

The scour profiles are similar in nature and independent of time but dependent on the grain-size characteristics of the bed.

The maximum scour depth at equilibrium can be expressed in terms of the Froude number based on the effiux thickness of the jet, and it is in- dependent of grain size.

The similarity criterion for the scour pattern in the current case has been investigated following the procedures suggested by Rouse (1939) and Dod- diah et al. (1953). It is possible to express the maximum scour depth as a function of time, thickness and velocity of jet, the distance of modible bed from the sluice opening, and the fall velocity of the particles.

The similarity criterion for sand transport has been investigated following the concept proposed by Bagnold (1973) as used in conventional river sed- iment transport. A generalized transport equation correlating weight rate of sediment transport with the fluid power of the jet, the transport stage (u,,/U.o) - 1, and a nondimensional parameter characterizing the diffusion of the jet can be formulated. Depending on the value of the transport stage, two relationships describe the transport laws.

APPENDIXI. REFERENCES

Ali, K. H. M., and Lim, S. Y. (1986). "Local scour caused by submerged wall jets." Proc., Institution of Civil Engineers, London, England, 81(2), 607-645.

Altinbilek, H. D., and Okyay, S. (1973). "Localized scour in a horizontal sand bed under vertical jets." Proc. IAHR Congress, International Association for Hydraulic Research, 1, 99-106.

Bagnold, R. A. (1973). "The nature of saltation and bed load transport in water." Proc., Royal Society, London, England, 332(A), 473-504.

Blaisdell, F. W., and Anderson, C. L. (1991). "Pipe plunge pool energy dissipator." J. Hydr. Engrg., ASCE, 117(3), 303-323.

Carstens, M. R. (1966). "Similarity laws for local scour." J. Hydr. Div., ASCE, 92(3), 13-36.

Chatterjee, S. S., and Ghosh, S. N. (1980). "Submerged horizontal jet over erodible bed." J. Hydr. Div., ASCE, 106(11), 1765-1782.

Doddiah, D., Albertson, M. L., and Thomas, R. (1953). "Scour from jets." Proc. IAHR Congress, International Association for Hydraulic Research, 161-169.

Francis, J. R. D., and Ghosh, S. N. (1974). "A new look at local erosions in alluvial rivers." Proc. 5th Australasion Conf. on Hydr. and Fluid Mech., University of Canterbury, Canterbury, New Zealand, 71-77.

Hassan, N. M. K., and Narayanan, R. (1985). "Local scour downstream of an apron." J. Hydr. Engrg., ASCE, 111(11), 1371-1385.

Iwagaki, Y., Smith, G. L., and Albertson, M. L. (1958). "Analytical study of me- chanics of scour for three dimensional jet." Hydr. Conf., ASCE, New York, N.Y.

Johnston, A. J. (1990). "Scourhole developments in shallow tailwater." J. Hydr. Res., IAHR, 28(3), 341-354.

Laursen, E. M. (1952). "Observations on the nature of scour." State Univ. oflowa Bull., 34, 179-197.

Mason, P. J. (1988). "Effects of air entrainment on plunge pool scour." J. Hydr. Engrg., ASCE, 115(3), 385-399.

Porch, M., and Hefez, E. (1967). "Initial scour and sediment motion due to an

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impinging jet." Proc. IAHR Congress, International Association for Hydraulic Research, 3, 8.

Rajaratnam, N. (1980). "Erosion by circular wall jets in cross flow." J. Hydr. Div., ASCE, 106(11), 1867-1883.

Rajaratnam, N., and Beltaos, S. (1977). "Erosion by impinging circular turbulent jets." J. Hydr. Div., ASCE, 103(10), 1191-1205.

Rajaratnam, N., and Berry, B. (1977). "Erosion by circular turbulent wall jets." J. Hydr. Res., 15(3), 277-289.

Rajaratnam, N., and MacDougall, R. K. (1983). "Erosion by plane wall jets with minimum tailwater." J. Hydr. Div., ASCE, 109(7), 1061-1064.

Rouse, H. (1939). "Criteria for similarity in transportation of sediment." State Univ. of Iowa Bull., 20, 33-39.

Sarma, K. V. M., and Sivasankar, R. (1967). "Scour under vertical circular jets." J. Inst. of Engrs., Calcutta, India, 48(3), 568-579.

Shen, H. W., Schneider, V. R., and Karaki, S. (1969). "Local scour around bridge piers." J. Hydr. Div., ASCE, 95(6), 1919-1940.

Simons, D. B., Richardson, E. V., and Nordin, C. F. (1965). "Bed-load equation for ripples and dunes." Prof. Paper 462-H, U.S. Geological Survey.

Tarapore, Z. S. (1956). "Scour below a submerged sluice gate," MS thesis, University of Minnesota, at Minneapolis, Minn.

Uyumaz, A. (1988). "Scour downstream of vertical gate." J. Hydr. Engrg., ASCE, 114(7), 811-816.

Westrich, B., and Kobus, H. (1973). "Erosion of a uniform sand bed by continuous and pulsating jets." Proc. IAHR Congress, International Association for Hydraulic Research, 1, 91-98.

APPENDIX II. NOTAT ION

The following symbols are used in this paper:

B 0

D= O m =

4= dso =

e - -

F 0 = g=

h= h m =

i= K= L = n =

q = qs = R = ss= T= t =

v0= U.~ z

U .O ~-~

width of slot or jet; downstream flow depth; mean value of D; typical grain diameter of surface partic!es, grain diameter corresponding to 50% finer; Naperian base; Froude number = Uo/(gBo)l/2; acceleration due to gravity; maximum scour depth at equilibrium; depth of scour at a distance x from slot; maximum scour depth at any instant of time t; weight flow rate of sediment per meter width per minute; bulk unit weight of bed material; length of rigid apron measured from jet effiux section; porosity of bed material; discharge per meter width; volume rate of transport per meter width; ratio of jet area to its perimeter; specific gravity of bed material; time required to reach equilibrium depth (min); duration of time (min); effiux velocity of jet; shear velocity = ( ' I "o /9 )1 /2 ; critical shear velocity = ('roc/p)l/2;

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U~ t "--

vs= Wm ~-

X =

xM= X m

Z - -

Ah =

p =

%= 9 o =

TOc -~

T t

+=

shear velocity at any time ('r,/p)m; volume of scour per meter width; fall velocity corresponding to dso size; fluid power of jet = (9qAh/Bo) ; distance from end of rigid apron; distance of peak of dune from end of rigid apron; distance of maximum scour point from end of rigid apron; distance of maximum scour point from slot at any time; river bed elevation measured from datum plane; head causing water to flow; mass density of water; standard deviation of grain-size distribution curve; boundary shear stress; critical shear stress for the sediment composing bed; boundary shear stress at time t during scour process; and angle of repose of bed material (degrees).

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