Scour below a High Vertical Drop

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Scour below a High Vertical DropSubhasish Dey1 and Rajkumar V. Raikar2

Abstract: Experimental results on scour below a high vertical drop �drop height/critical depth �1� in uniform sands and gravels arepresented. The experimental results are used to describe the effects of important parameters, identified from the dimensional analysis, onequilibrium scour depth. The important observations are that the equilibrium scour depth increases with increase in densimetric Froudenumber, whereas the scour depth decreases with increase in sediment size and tailwater depth. The time scale of scour depth that followsan exponential law is determined. The nondimensional time scale decreases with increase in densimetric Froude number.

DOI: 10.1061/�ASCE�0733-9429�2007�133:5�564�

CE Database subject headings: Open channel flow; Nonuniform flow; Steady flow; Drop structures; Erosion; Scour; Sedimenttransport.

Introduction

Drops are grade-control structures provided in channels for low-ering the bed level when the slope of the channel is smaller thanthe natural ground slope. The stream flow running over suchdrops is called a free overfall. For example, the flow running overa weir built across a channel forms a free overfall. In addition, thescour developed downstream of the bed protection provided tocontrol the slope or elevation of the channel bed creates a drop.Considerable portion of the energy of the flowing stream is dis-sipated through turbulent mixing in the pool formed at the down-stream of these drops. Hence, the sediment bed downstream ofthe drop is eroded significantly due to the instability caused to thebed sediment by diffusion of the falling jet, endangering the foun-dation of the drop structures.

Schoklitsch �1932� was the first to propose an empirical equa-tion for the estimation of scour depth downstream of a verticaldrop. Since then many investigators have proposed equations forthe estimation of scour depth below vertical drops �Mason andArumugam 1985; Bormann and Julien 1991�. However, theseequations are recommended for low vertical drop structures. Adrop is considered a low drop when the relative drop height�H /yc; where H�drop height, i.e., the difference in elevation ofupstream and downstream channel bed; and yc�critical depth up-stream of the drop� is equal to or less than one, otherwise it is ahigh drop �Little and Murphey 1982�. Fig. 1�a� shows the sche-matic of scour below a vertical drop. Robinson et al. �2000� mea-sured the velocity and circulation patterns for an aerated straight

1Associate Professor, Dept. of Civil Engineering, Indian Institute ofTechnology, Kharagpur 721302, West Bengal, India. E-mail:[email protected]

2Assistant Professor, Dept. of Civil Engineering, K. L. E. Society’sCollege of Engineering and Technology, Udyambag, Belgaum 590008,Karnataka, India.

Note. Discussion open until October 1, 2007. Separate discussionsmust be submitted for individual papers. To extend the closing date byone month, a written request must be filed with the ASCE ManagingEditor. The manuscript for this technical note was submitted for reviewand possible publication on March 24, 2006; approved on September 12,2006. This technical note is part of the Journal of Hydraulic Engineer-ing, Vol. 133, No. 5, May 1, 2007. ©ASCE, ISSN 0733-9429/2007/5-
564–568/$25.00.
564 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MAY 2007

J. Hydraul. Eng. 2007

drop using an acoustic Doppler velocimeter. Chamani and Bei-rami �2002� derived an empirical equation to estimate the relativeenergy loss at drops for the supercritical flows. Bormann andJulien �1991� proposed an equation of equilibrium scour depth forsloping grade-control structures based on jet diffusion and particlestability in scour holes downstream of grade-control structures.Stein et al. �1993� developed an analytical method to determinethe time-variation of scour depth downstream of a headcut relat-ing the maximum velocity of the diffusing jet with excess shearstress. They reported that the rate of scour depth increases veryrapidly for scour depths below 95% of the equilibrium scourdepth. Scour process in the plunging pool of sediment beds wasnumerically simulated by Jia et al. �2001�. Robinson et al. �2002�measured scour downstream of an overfall in cohesive bed. Lenziet al. �2003� analyzed 73 grade-control structures in alluvialmountain rivers, which included check dams and bed sills. Theyproposed an energy-based normalization method for scour holedimensions to evaluate the role of jet geometry and aeration onscouring efficiency.

In this context, it is pertinent to mention that none of theaforementioned studies focus on the effects of various parameterson scour depth downstream of vertical drops. The present studyaims at a detailed parametric investigation on scour depth down-stream of a high vertical drop in uniform sediments �sands andgravels�. The findings of the experimental investigation are usedto describe the effects of various parameters on equilibrium scourdepth including time variation of scour depth.

Experimental Setup and Procedure

The laboratory experiments were carried out in a flume, havingcross section of 0.3 m wide and 0.7 m deep, at the Hydraulic andWater Resources Engineering Laboratory, Indian Institute ofTechnology, Kharagpur, India. A vertical drop was created at theend of a drop structure, which was 0.5 m high from the flumebottom. In this study, the relative drop height �H /yc� was variedfrom 4.3 to 29.8. An air ventilation was provided to avoid suctionbelow the free nappe facilitating a well-shaped nappe �Fig. 1�a��.A sediment recess of 1.1 m long and 0.3 m wide, constructed atthe downstream of the vertical drop, contained uniform sedi-
ments. Sediments of six sand sizes �median size d50=0.26, 0.49,
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0.81, 1.86, 2.54, and 3 mm� and three gravel sizes �d50=4.1, 5.53,and 7.15 mm� were used in the experiments. The depth of sedi-ment recess was adjusted depending on the required drop height.The degree of uniformity of the particle size distribution of asediment sample is defined by the value of geometric standarddeviation �g�=�d84/d16�0.5�, which is less than 1.4 for uniformsediments. For sediments, the geometric standard deviation �g

and the critical value of Shields parameter �c obtained from theShields diagram are furnished in Table 1. Initially, the flume wasfilled with water issued from a pipe at a low rate until the desir-able tailwater depth was reached. The tailwater depth in the flumewas controlled by an adjustable tailgate in the downstream of theflume. A Vernier point gauge with an accuracy of ±0.1 mm wasused to measure the flow depths. Then the discharge in the flumewas gradually increased to the desired value corresponding to thepredetermined jet thickness of the free overfall. The flow dis-charge, controlled by an inlet valve, was determined using themeasuring tank at the downstream end of the flume, where thewater from the flume was discharged. Importantly, it had an ex-cellent comparison with the discharge determined from the mea-sured end depth at the vertical drop. In the initial stage, the scourhole developed rapidly with time. The scour profiles at regularintervals of time were traced on a transparent perspex sheet at-tached to the outside glass wall. To facilitate the reproduction of

Fig. 1. �a� Schematic of scour below a vertical drop; �b� scourprofiles at different times in bed sediment d50=0.81 mm forht=8 cm �low tailwater depth�; and �c� scour profiles at differenttimes in bed sediment d50=0.81 mm for ht=22 cm �high tailwaterdepth�

scour profiles, square grids of size 1 cm�1 cm were made on the

J


perspex sheet. Similarly, the profiles of the nappe up to the plung-ing zone were obtained. Typical scour profiles at different timesduring the development of the scour hole for low and high tail-water depths are shown in Figs. 1�b and c�, respectively. It isnoticeable that for low tailwater depth �Fig. 1�b��, the crest of thedune was flattened, whereas for high tailwater depth �Fig. 1�c��,the dune was well defined. The present experiments revealed thatthe equilibrium scour depth dse reached after 9 h for sedimentsizes d50�0.81 mm. For coarser sediments, the times to reachequilibrium scour were even less than 9 h. Therefore, in thisstudy, it was adequate to consider equilibrium scour time as 9 h.For a few experiments with different sediment sizes, experimentswere run for an extended time period of 12 h. The maximumscour depths observed at 12 h were the same as those obtained at9 h. Two different sets of experimental data were collected�Tables 1 and 2�. In the first set �Table 1�, experiments were runwith uniform sediments under constant tailwater depth and vary-ing jet thickness. On the other hand, in the second set �Table 2�,experiments were run with varying tailwater depth and constantjet thickness.

Table 1. Experimental Data with Constant Tailwater Depth

h0

�cm�ht

�cm�d50

�mm� �g �c

ye

�cm�y0

�cm�dse

�cm�

10 15 0.26 1.21 0.042 1.2 0.69 23.1

0.49 1.13 0.031 1.2 0.69 17.2

0.81 1.34 0.031 0.6 0.25 5

0.9 0.45 6.5

1.2 0.69 9.5

1.5 0.92 11.3

1.7 0.01 14.8

1.9 1.29 19.2

2.1 1.3 23.1

2.4 1.59 26.3

2.7 2 28.3

3 2.2 30.8

1.86 1.27 0.04 1.2 0.69 7.5

2.54 1.07 0.044 1.2 0.69 5.6

1.5 0.92 7

1.7 0.01 7.8

1.9 1.29 9

2.1 1.3 10.6

2.4 1.59 15

2.7 2 18.1

3 2.2 19.8

3 1.19 0.046 1.2 0.69 5.4

4.1 1.13 0.05 1.2 0.69 5.5

1.5 0.92 7.3

1.7 0.01 8

1.9 1.29 8.5

2.1 1.3 9.8

2.4 1.59 12

2.7 2 14.2

3 2.2 18

5.53 1.1 0.055 1.2 0.69 4.5

7.15 1.08 0.055 1.2 0.69 4

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Determination of Jet Thickness

In Fig. 1�a�, Section 1 is considered at upstream of the drop wherethe critical depth occurs and Section 0 at the entry of jet into thetailwater. The continuity equation applied between Sections 1 and0 is

Ucyc = U0y0 �1�

where Uc�critical velocity of the flow upstream of the drop;U0�velocity of jet when it enters the tailwater; and y0�thicknessof jet entering the tailwater. In this context, it is pertinent tomention that y0 is used for scaling the scour depth. According toBakhmeteff �1932�, the jet velocity U0 is given by

U0 = C0�2g�h0 + 1.5yc� �2�

where C0�velocity coefficient and h0�height of drop above thetailwater level.

Using the value of end-depth ratio �=ye /yc; where ye�enddepth� for rectangular channels equaling 0.715 as proposed byRouse �1936�, Eq. �2� becomes

U0 = C0�2g�h0 + 2.1ye� �3�

Inserting Eq. �3� into Eq. �1�, the expression for jet thickness y0

can be written as

y0 =1.17ye

1.5

C0�h0 + 2.1ye�0.5 �4�

Using the present experimental data, the value of C0 is estimatedas 0.672. Therefore, Eq. �4� can be given by

y0 = 1.74ye

1.5

�1 + 2.1ye�0.5 �5�

where y0=y0 /h0 and ye=ye /h0. Fig. 2�a� shows the variation of y0

with ye as computed from Eq. �5� along with the experimentaldata.

Influences of Various Parameters on Scour Depth

The parameters influencing the equilibrium scour depth dse belowa vertical drop can be given by

dse = f1�U0,y0,ht,�,�s,g,�,d50� �6�

where ht�tailwater depth; ��mass density of water; �s�massdensity of sediments; g�gravitational acceleration; and��kinematic viscosity of water �=10−6 m2/s�. In sediment–waterinteraction, it is appropriate to represent the independent param-eters g, �, and �s as a combined parameter g �Dey and Raikar

Table 2. Experimental Data with Constant Jet Thickness

ye

�cm�y0

�cm�h0

�cm�ht

�cm� d50=0.81 mm

2.5 1.6 7 8 28.2

10 20.8

12 19.9

14 17.4

16 14.5

18 13.3

20 13.1

22 11.3

dse �cm�

d50=1.86 mm d50=2.54 mm d50=4.1 mm d50=5.53 mm

21 14.7 13 9.1

— 12.7 12.2 8.2

— 10.5 10.2 7.6

13.6 10.1 7.8 6.9

— 9.6 8.1 6

— 7.9 8.2 5.8

— 7.1 6.2 5.3

9.2 6.6 5.3 5

2005�; where is s−1; and s is the relative density of sediments,



Fig. 2. �a� Variation of y0 with ye computed from Eq. �5� and

experimental data; �b� variation of dse with F0 for different h and

y0=1.6 cm; and �c� variation of dse with d for ht=15 cm andy0=0.69 cm

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that is �s /�. Also, the influence of kinematic viscosity � is con-sidered negligible under a fully turbulent flow over a rough bed.Applying the Buckingham theorem with repeating variables U0

and y0, and rearranging the nondimensional parameters logically,yields

dse = f2�F0, d, h� �7�

where dse�dse /y0; F0=U0 / �gd50�0.5, that is densimetric Froude

number; d�d50/y0; and h�ht /y0.

Dependency of Scour Depth on Various Parameters

The laboratory experimental data �Table 2� with constant jetthickness y0=1.6 cm, tested for tailwater depth ht=8, 14, and22 cm, and d50=0.81−5.53 mm, are used to plot nondimensional

equilibrium scour depth dse versus F0 in Fig. 2�b�. The nondimen-

sional equilibrium scour depth dse increases with increase in den-

simetric Froude number F0. The rate of increase of dse is more forlower F0, whereas for higher F0 �F0�6�, it drops down consid-

erably. It is also evident that for a given F0, dse decreases with

increase in tailwater depth-jet thickness ratio h.Fig. 2�c� depicts the variation of nondimensional equilibrium

scour depth dse with sediment size—jet thickness ratio d for tail-water depth ht=15 cm and jet thickness y0=0.69 cm �Table 1�. It

is evident that the nondimensional equilibrium scour depth dse

decreases with increase in d. It implies that dse is less for rela-tively coarse sediment. To be more precise, with the developmentof a scour hole, the bed shear stress acting on the scour holereduces. As the coarser sediments require relatively more criticalbed shear stress to initiate motion, the equilibrium scour isreached at a lower scour depth dse. It is interesting to note that for

sands �d�0.44�, dse increases significantly with decrease in d.

However, the rate of decrease in dse with d reduces considerably

for gravels �d�0.44�. dse becomes independent of d for d�0.8.The laboratory experimental data �Table 2� tested for different

h, d50=0.81−5.53 mm, and y0=1.6 cm are used to plot Fig. 3 thatshows the dependency of nondimensional equilibrium scour depth

dse on h for different F0. At lower h, nondimensional equilibrium

scour depth dse decreases significantly with increase in h. How-

ever, as h increases, dse becomes almost independent of

h�h�11�. The reduction in plunging action of jet with increase in

h is the probable cause for the reduction of dse. Also, for smaller

densimetric Froude number �F0=3.48�, scour depth dse is essen-

tially independent of tailwater depth at h.

Time Variation of Scour Depth

The time variation of scour below a vertical drop was tested invarious uniform sediments. Fig. 4�a� shows the variation of in-stantaneous scour depth ds with time t for different ye and d50

=2.54 mm. It is evident from Fig. 4�a� that the equilibrium scourdepth dse was reached well before 9 h as mentioned earlier. Ac-cording to Sumer et al. �1993�, the instantaneous scour depth ds at
time t can be represented in functional form as
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ds = dse�1 − exp�t/T�� 8�

where T�time scale. The quantity T represents the time periodduring which scour depth develops substantially. The time scale Tcan be determined from the ds versus t diagram by estimating theslope of the tangent to the ds�t� curve at t�0, as shown schemati-cally in Fig. 4�b�. The time scale can be expressed in nondimen-sional form as T*�=T�gd50

3 �0.5 /y02�. The nondimensional time

scale T* can be written in the following functional form:

T* = T*�F0,�c� �9�

The variation of nondimensional time scale T* with densimetricFroude number F0 for different sediment sizes d50 or �c are givenin Fig. 4�b�. The resulting trend is a set of nearly parallel curves.The nondimensional time scale T* decreases considerably withincrease in densimetric Froude number F0. It is interesting tomention that T* increases indefinitely for lower values of F0.

Conclusions

Experimental results on scour downstream of a high vertical drop

Fig. 3. Variation of dse with h for different F0

in uniform sediments �sands and gravels� have been analyzed.

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The equilibrium scour depth increases with increase in densimet-ric Froude number. On the other hand, the scour depth decreaseswith increase in sediment size and tailwater depth. To representthe time variation of scour depth, the time scale that follows anexponential law is obtained. The nondimensional time scale de-creases with increase in densimetric Froude number for a fixedvalue of the critical Shields parameter.

Notation

The following symbols are used in this technical note:C0 � velocity coefficient �M0 L0 T0�;

d � d50/y0 �M0 L0 T0�;ds � scour depth at time t �L�;

dse � dse /y0 �M0 L0 T0�;dse � equilibrium scour depth �L�;d50 � median particle diameter �L�;F0 � densimetric Froude number, U0 / �gd50�0.5

�M0 L0 T0�;g � gravitational acceleration �L T−2�;H � drop height �L�;

Fig. 4. �a� Time variation of scour depth ds with time t ford50=2.54 mm; �b� variation of T* with F0 for different �c



Hc � critical specific energy �L�;

h � ht /y0 �M0 L0 T0�;ht � tailwater depth �L�;h0 � height of drop above tailwater level �L�;s � relative density of sediments �M0 L0 T0�;T � time scale �T�;

T* � nondimensional time scale, T�gd503 �0.5 /y0

2

�M0 L0 T0�;t � time of scouring �T�;

U � average flow velocity upstream of drop �L,T−1�;Uc � critical velocity of flow upstream of drop �L T−1�;U0 � velocity of jet when it enters tailwater �L T−1�;ye � ye /h0 �M0 L0 T0�;y0 � y0 /h0 �M0 L0 T0�;yc � critical depth just upstream of drop �L�;ye � end depth �L�;y0 � thickness of jet entering tailwater �L�; � s−1 �M0 L0 T0�;

�c � critical Shields parameter �M0 L0 T0�;� � kinematic viscosity of water �L2 T−1�;� � mass density of water �M L−3�;

�s � mass density of sediments �M L−3�; and�g � geometric standard deviation �M0 L0 T0�.

References

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

Bormann, N. E., and Julien, P. Y. �1991�. “Scour downstream of grade-control structures.” J. Hydraul. Eng., 117�5�, 579–594.

Chamani, M. R., and Beirami, M. K. �2002�. “Flow characteristics atdrops.” J. Hydraul. Eng., 128�8�, 778–791.

Dey, S., and Raikar, R. V. �2005�. “Scour in long contractions.” J. Hy-draul. Eng., 131�12�, 1036–1049.

Jia, Y., Kitamura, T., and Wang, S. S. Y. �2001�. “Simulation of scourprocess in plunging pool of loose bed-material.” J. Hydraul. Eng.,127�3�, 219–229.

Lenzi, M. A., Marion, A., and Comiti, F. �2003�. “Local scouring atgrade-control structures in alluvial mountain rivers.” Water Resour.Res., 39�7�, 1176–1188.

Little, W. C., and Murphey, J. B. �1982�. “Model study of low drop gradecontrol structures.” J. Hydr. Div., 108�10�, 1132–1146.

Mason, P. J., and Arumugam, K. �1985�. “Free jet scour below dams andflip buckets.” J. Hydraul. Eng., 111�2�, 220–235.

Robinson, K. M., Cook, K. R., and Hanson, G. J. �2000�. “Velocity fieldmeasurements at an overfall.” Trans., Am. Soc. Agri. Biol. Engrs.,43�3�, 665–670.

Robinson, K. M., Hanson, G. J., and Cook, K. R. �2002�. “Scour belowan overfall. Part I: investigation.” Trans., Am. Soc. Agri. Biol. Engrs.,45�4�, 949–956.

Rouse, H. �1936�. “Discharge characteristics of the free overfall.” Civ.Eng. (N.Y.), 6�4�, 125–134.

Schoklitsch, A. �1932�. “Kolkbildung unter uberfallstrahlen.” Wasser-wirtschaft, 24, 341–343.

Stein, O. R., Julien, P. Y., and Alonso, C. V. �1993�. “Mechanics of jetscour downstream of a headcut.” J. Hydraul. Res., 31�6�, 723–738.

Sumer, B. M., Christiansen, N., and Fredsoe, J. �1993�. “Influence ofcross section on wave scour around piles.” J. Waterway, Port,Coastal, Ocean Eng. 119�5�, 477–495.

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