hydraulics of plane plunge pool scour - unipi.it · lar plunge pool scour. an air-water jet of...

Hydraulics of Plane Plunge Pool ScourStefano Pagliara1; Willi H. Hager, F.ASCE2; and Hans-Erwin Minor3

Abstract: Plunge pool scour involves a significant risk with trajectory spillways because of structural undermining at a dam foot ordestabilization of adjacent valley slopes. An experimental program towards the understanding of plane plunge pool scour of a completelydisintegrated rock surface was conducted, in which the following items received attention: jet shape, jet velocity, jet air content, tailwaterelevation, granulometry, upstream flow to the scour hole, and the end scour profile in terms of the basic scour features. These effects wereexperimentally investigated based on a systematic variation of the governing scour parameters. The results of this paper allow answeringquestions that have so far not been addressed. Design equations were proposed to sketch the main tendency of the data sets. Thesignificant effect of the densimetric particle Froude number was substantiated. This research may be used to estimate the prominent scourfeatures for nearly two-dimensional jet arrangements involving a pre-aerated high-speed flow.

DOI: 10.1061/�ASCE�0733-9429�2006�132:5�450�

CE Database subject headings: Air water interactions; Hydraulic structures; Scour; Spillways; Pools.

Introduction

Plunge pool scour is known for its destructive action at the foot ofhydraulic structures, originating from a high-speed air-water jetthat produces scour within a short time even in hard rock. Amongother sites, the Kariba Dam in East Africa has experienced con-siderable damage demonstrating the potential of destruction ofsuch energy dissipators. The main hydraulic features of plungepool erosion are currently based on selected model observations.Mason �2002�, Minor et al. �2002�, Rajaratnam and Mazurek�2002�, and Canepa and Hager �2003� have recently added to theunderstanding of these flows. This study furnishes knowledgewith an experimental approach conducted at Versuchsanstalt fürWasserbau, Hydrologie und Glaziologie VAW at ETH Zurich,Switzerland. Whereas Canepa and Hager �2003� included a re-view of present knowledge and a suitable three-phase Froudenumber, this work intends to investigate a number of additionaleffects, namely jet shape, submerged or unsubmerged jet impact,jet velocity, jet impact angle relative to the water surface, and jetair concentration. Further, the effects of tailwater elevation andsediment nonuniformity were investigated. The present experi-ments refer to a quasi-two-dimensional scour formation in whichthe rock surface was considered completely disintegrated and thusretaining no safety resistance against scour.

Experimental Setup

The scour experiments were conducted in a rectangular channelpreviously described by Canepa and Hager �2003�. Its width is

1Professor, Dip. di Ingegneria Civile, Univ. di Pisa, Via Gabba 22,I-56 100 Pisa, Italy.

2Professor, VAW, ETH Zürich, CH-8092 Zurich, Switzerland.3Professor, VAW, ETH Zürich, CH-8092 Zurich, Switzerland.Note. Discussion open until October 1, 2006. Separate discussions

must 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 paper was submitted for review and pos-sible publication on May 25, 2004; approved on July 18, 2005. This paperis part of the Journal of Hydraulic Engineering, Vol. 132, No. 5, May 1,
2006. ©ASCE, ISSN 0733-9429/2006/5-450–461/$25.00.
450 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / MAY 2006

0.500 m; it is 0.70 m high and about 6 m long. The air-watermixture jet was generated with a circular pipe of internal diameterD to which air was added. The air was almost uniformly distrib-uted in the jet as compared to typical prototype conditions withthe air discharge decreasing from the jet contour towards the jetcore. This arrangement had distinctive advantages as comparedwith a standard plunge pool scour setup: The air-water �subscriptAW� mixture velocity VAW was accurately defined within ±5%,the model length was much shorter than a combined spillway andtake-off structure, and all the hydraulic parameters prior to jetimpact onto the tailwater were adequately defined. Fig. 1�a�shows a typical photo of an experimental test. Four jet impactangles relative to the horizontal were investigated, namely,�=30, 45, 60, and 90° �Fig. 2�. The usual jet shape was circular,except for a special series where jets issued from rectangular slitsof 0.029 m were produced with a 0.100 m conduit, with the slitaxis both horizontal and vertical. Plunge pool scour normally in-volves a rock bed, whose resistance against hydraulic impact isdifficult to assess. A completely disintegrated rock bed simulatedwith an incohesive granulate of grain size d50 or d90, and a sedi-ment nonuniformity parameter �= �d84/d16�1/2 offers the least re-sistance against scour. Canepa and Hager �2003� accounted forthe effect of grain size, provided the scour bed was composed ofalmost uniform sediment. This study considers the experimentalSeries I and II: Series I involved almost uniform sediment char-acterized by d16=5.2 mm, d50=6.5 mm, d84=7.8 mm, andd90=8.0 mm, thus �=1.22. The initial sediment thickness variedbetween 0.30 and 0.40 m. Water discharges were up toQW=0.025 m3/s, whereas air discharge was limited toQA=0.045 m3/s for a constant pipe diameter D=0.070 m. SeriesII involves the effects of the tailwater depth, the granulometry, theupstream velocity, and the ridge removal.

The air-water mixture jet was investigated both for submerged�S� and for unsubmerged �U� jet flow conditions. The latter cor-responds to the common arrangement with a jet traveling throughthe atmosphere and impacting onto an almost stagnant waterbody. For submerged flow, the pipe length was increased such thatthe jet issued below the water surface. Both configurations mayoccur in applications, and the differences between these two flow
types were investigated herein.

Compared to the standard setup involving a spillway and aski-jump, the present experimental configuration was simple. Atypical experiment lasted for about 1 h. This included preparationof a horizontal sediment surface as described by Canepa andHager �2003�, and air supply to the conduit. Time was set to zeroonce water discharge was added, the axial scour profiles at times1, 5, 10, and normally 20 min after test initiation were recorded,along with the sediment surface for the “dry” conditions, i.e.,after the scour hole was drained at the test end �Fig. 1�b��. Thesediment surface during a test was recorded under “dynamic”conditions, thus when an experiment was in progress, except for1 min after test initiation, where only the maximum scour depthwas measured. This contrasts other works in which the “static”sediment surface was recorded, i.e., the jet was either stopped ordeflected. The present experiments lasted until conditions close toan end-scour resulted, for which the scour hole undergoes practi-cally no more temporal changes.

The instrumentation used in this work was described byCanepa and Hager �2003�. It consisted of a special gauge with a40 mm circular plate at its lower end to detect the sediment sur-face under dynamic flow conditions �Fig. 1�b��. This instrumentallowed for scour surface readings without direct optical contactbecause of the presence of dust and air bubbles in the scour hole.

Fig. 1. �a� Generation of plunge pool scour using a pressurizedair-water mixture jet and �b� scour for “dry” conditions aftercompletion of an experimental run

The axial water surface profile across the scour hole was observed

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with a standard point gauge. Usually, the surface was almost hori-zontal for the relatively high tailwater conditions investigated.

All experiments in Series I were conducted in a narrow rangeof relative tailwater depth between 3 and 4 pipe diameters D. Therelative tailwater elevation has a significant effect on the variousscour parameters and was investigated in Series II.

Experimental Results—Test Series I

Definitions

Two test series were conducted with some hundreds of experi-ments in total. The following relates to Test Series I. Fig. 2 is adefinition sketch of the main parameters involved in plane granu-lar plunge pool scour. An air-water jet of diameter D and mean-cross-sectional velocity VAW= �QA+QW� / ��D2 /4� impinges underthe angle � a water body of initial depth ho. Either this heightsubmerges the conduit outlet to result in submerged �S� jet flow,or produces unsubmerged �U� conduit outflow. The discharges areQW for water and QA for air. The original elevation of the sedi-ment bed is zo. An almost plane scour hole of maximum depth zm

at location xm from the scour origin xo develops under the actionof the high-speed jet. The maximum deposition height is zM atlocation xM. The scour profile z�x� intersects the original bed atlocations xa and xu, respectively. The axial lengths involved in thedefinition of the scour geometry are lm=xm−xo, la=xa−xo,lM =xM −xo, and lu=xu−xo.

For two-phase jet flow involving air and water, the densi-metric particle Froude number is Fd=VW / �g�di�1/2 withVW=QW / ��DW

2 /4� as the water velocity, g�= ��s−�� /��g as thereduced gravitational acceleration with the densities �s and � ofsediment and water, respectively, and di= determining grain size.The significant length scale is either the conduit diameter D or theblack-water jet diameter DW= ��4/��QW /VW��1/2. The air contentof the jet is �=QA /QW. A so-called black-water jet occurs for�=0, whereas white-water jets result for ��0.

Effect of Jet Shape

The effect of jet shape on the maximum scour depth is importantbecause flip-buckets may generate a jet geometry that deviateslargely from the circular shape as used in the present tests. There-fore a special series of experiments was conducted using fourdifferent jet shapes: Circular conduits of internal diameters �1�D=0.100 m �Canepa and Hager 2003� and �2� D=0.070 m; andrectangular jets �3� of width b=0.100 m and height hj =0.029 m,and �4� inverted with b=0.029 m and hj =0.100 m. Only black-water observations were considered because the effect of air isdiscussed below. The equivalent �subscript e� diameters of con-figurations �3� and �4� are De= �4bhj /��1/2=0.061 m. The hosedata of Canepa and Hager �2003� were not reconsidered becausethey followed essentially configuration �1�, and their accuracywas considered too low.

Fig. 3 shows the relative scour depth Zm=zm /D or zm /De as afunction of Fd90=VW / �g�d90�1/2 for both the submerged and theunsubmerged jet flow conditions. In the figure insets, the symbolsBW and WW denote black-water and white-water conditions, re-spectively. BW-U thus refers to unsubmerged black-water jetflow. Note from Fig. 3 that there is practically no jet shape effectprovided that the cross-sectional average jet velocity isV=Q / ��De

2 /4�. This result simplifies the application of the
present research in practice. The data define a slightly curved line
OURNAL OF HYDRAULIC ENGINEERING © ASCE / MAY 2006 / 451

with the origin at �0;0�. For the application range of Fd90�20,the data may be approximated with a straight line.

Effect of Jet Impact Angle

The maximum scour depth Zm may be expressed with indepen-dent functions f1 to f6 accounting for the main parameters of thepresent research as

Zm = f1�Fd90� · f2�� · f3�� · f4�T� · f5�� · f6�Fu� · f7�S or U�

�1�

Fig. 4 shows black-water data as Zm+=Zm / f2�� in whichf2=0.38 sin��+22.5° � was fitted for both submerged �S� and un-submerged �U� flows �f7=1� for the four jet impact angles�=30, 45, 60, and 90°. The difference between the S and the Udata was found negligible from a detailed data analysis. For un-submerged jet flow, the vertical distance from the pipe outlet tothe tailwater surface was of the order of 2D. For black-waterflows, scour depths are practically the same for jets impingingonto a water body and those issued below it. This important find-ing applies for a tailwater depth of at least 2.5 times the diameterD. The maximum scour depth for �=0 thus is from Eq. �1�

Zm = − 0.38 sin�� + 22.5 ° �Fd90, 2 � Fd � 20

30 ° � � � 90 ° , ho � 3.5D �2�

The effect of the jet impact angle was originally assumed to fol-low the sine function. However, the data sets indicated that thescour depth is larger for �=60° than for 90°. This may be ex-plained with two reasons: �1� the deposition height ZM �seebelow� is significantly larger for a jet impact angle of �=60° thanof 90° because less sediment is suspended in the more confinedscour hole of a vertical jet; and �2� the ridge erosion is larger for

Fig. 2. Definition sketch for plunge pool s

60° than for 90° jets for otherwise identical conditions.


Effect of Jet Air Content

The jet air content �=QA /Q plays a significant role in plungepool scour �Canepa and Hager 2003�. It may either be analyzedwith the black-water velocity VW=QW / ��DW

2 /4� with DW as theblack-water jet diameter, or with the air-water mixture velocityVAW=QW�1+�� / ��D2 /4�. The related Froude numbers are Fd90

=V / �g�d90�1/2 and Fd90� =VAW / �g�d90�1/2. The function f3�� in Eq.�1� was determined with the parameter Zm++=Zm+ /Fd90�=Zm / �Fd90� · f2��. Fig. 5 compares Zm++ for ��12 with

f3�� = �1 + ��−m �3�

where m=0.75 for the unsubmerged and m=0.50 for the sub-merged jet configuration, respectively. The effect of jet air contentis �slightly� larger for unsubmerged than for submerged jets. Theprovisional equation for the maximum scour depth Zm due to anair-water mixture jet thus reads

nder submerged and unsubmerged jet flow

Fig. 3. Effect of jet shape on maximum scour depth Zm�Fd90� forunsubmerged �closed symbols� and submerged jet flows �opensymbols�, �—� trend lines

cour u

Zm = zm/De = − 0.38 sin�� + 22.5 ° � · Fd� · �1 + ��−m

2 � Fd� � 40, 30 ° � � � 90° �4�

This relation includes the effects of equivalent jet diameter De fornoncircular jets, the jet impact angle �, the mixture velocityVAW, the relative density times the gravitational accelerationg�= ��s−�� /��g between sediment and fluid phases, the grainsize d90, and the jet air content �. Fig. 6 compares the presentobservations with Eq. �4� for the 200 data sets of Series I for bothsubmerged and unsubmerged jet flow conditions. Note that thewhite-water jets were almost uniformly aerated, in contrast totypical prototype jets with a black-water core and increasing airconcentration from the jet axis towards the jet surface. This effect

Fig. 4. Relative scour depth Zm+ as a function of Fd90 for �U;S�black-water �BW� data and jet angles � between 30 and 90°;R2=0.98, �—� Eq. �2�

Fig. 5. Effect of jet air content � on maximum scour depth Zm++ for�a� submerged �S� and �b� unsubmerged �U� flow conditions, �—�Eq. �3�

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was considered relatively small. Both the externally aerated high-speed jet and jet impact geometries deviating strongly from thecircular cross section must be investigated separately to confirmthe present result, however.

Experimental Results—Test Series II

The original jet diameter of D=0.070 m was reduced by insertinga smaller Plexiglas pipe into the supply pipe. Its internal diameterwas D=0.0465 m and its length 0.40 m; the pipe was attachedeither against or in the flow direction allowing one to locate itsoutlet close to the tailwater. The length of this pipe developed anoutflow without any contraction.

Water discharges were up to QW=0.033 m3/s, and the air flowwas absent in this test series. The tests were limited to �=30° andunsubmerged flows, given that the effects of the jet impact angle,the jet air content, and the submergence were considered in SeriesI. Fig. 7 shows the temporal advance of a scour hole. Fig. 7�a�relates to the test beginning and Fig. 7�b� refers to conditionssome seconds later, indicating that scour progress is extremelyfast in the early stage. After some minutes, the end scour hole ispractically formed �Fig. 7�c��. Fig. 7�d� shows conditions after thewater was stopped and points to the uneven sediment distributionto be discussed below.

Three sediment mixtures M1, M2, and M3 were employed inSeries II in addition to the practically uniform sediment M0 inSeries I. These are characterized with d16, d50, d84, and d90, asshown in Table 1. According to Canepa and Hager �2003�, ratherd90 than d50 is commonly used. The Appendix provides the rela-tion between the two sediment sizes. The sediment nonuniformityparameter �= �d84/d16�1/2 was 1.22, 1.52, 1.73, and 2.66 for thesediments M0–M3, respectively.

The effect of tailwater level on the plunge pool scour wasinvestigated by a systematical variation of the tailwater depthrelative to the jet diameter. The tailwater depth was adjusted withthe flap gate at the channel end. Depending on the tailwater andthe approach flow conditions, the elevation of the sediment bedallowed for a variation of the relative tailwater depth. For somespecial tests, discharge was added from the upstream channel endto observe its effect on the scour hole. These conditions may arisein prototypes with recirculating flows deflected by a sedimentridge downstream of the scour hole, or if additional discharge

Fig. 6. Comparison of observed �measured� with predicted�calculated� maximum scour depths, �—� line of perfect agreement,R2=0.91

originates from a bottom outlet.


Effect of Tailwater Depth

The effect of tailwater depth T=ho /D was investigated for vari-ous conduit diameters D ranging between 46 and 100 mm therebyusing exclusively black-water test conditions. Fig. 8 showsthe relative scour depth Zm=zm /D as a function ofFd90=V / �g�d90�1/2 for relative tailwater depths T between 0.7 and10 and sediment M0. Two features are noted: �1� the scour depthZm increases linearly with the densimetric Froude number Fd90;

Table 1. Characteristics of Sediments Tested

Sediment M0 M1 M2 M3

d90 �mm� 8 12.5 17.8 53

d84 �mm� 7.8 11.9 16.1 48

d50 �mm� 6.5 7.6 7.7 21

d16 �mm� 5.2 5.15 5.4 6.8

� 1.22 1.52 1.73 2.66

Fig. 7. Typical evolution of plunge pool scour


and �2� the relative scour depth Zm decreases as relative tailwaterT increases, provided 4�Fd90�20; for Fd90�3 practically noscour occurred.

The maximum scour depth is Zm=Fd90 where depends onthe relative tailwater as �Fig. 8�

= 0.12 ln�1/T� + 0.45 for 0.7 � T � 10 �5�

Fig. 9 shows the significant tailwater effect on the scour holegeometry. For T�5, the maximum scour depth �1� is compara-tively small because of a relatively high ridge. For 3�T�4.5, theridge height increases to its absolute maximum �see below�, as-sociated with a small increase of the scour depth �2�, because ofthe shorter distance between jet impingement and the scour hole.Further decreasing T to roughly 2 erodes the ridge, depending onthe grain size and the water velocity in the ridge region, resultingin a deeper scour hole �3�. If T is reduced to values of the order of1, the ridge is eroded and the scour hole is unprotected �4�. Theerosion of the ridge may not thoroughly be analyzed with theparameters introduced herein because sediment transport dependson additional quantities. Such a discussion was considered out ofthe scope of this research. Note that the tests of Canepa and Hager�2003� and those of Series I were conducted in the narrow rangeof 3�T�4.5.

Effect of Granulometry

Fig. 10 relates to the effect of granulometry. The tailwater effectwas eliminated by the reduced scour depth Zm� =Zm /. The scourdepth increases with an increasing nonuniformity parameter

Fig. 8. Maximum scour depth Zm�Fd90� for various tailwater depths0.7�T�10, �—� prediction, �- - -� data lines, �=30°

Fig. 9. Effect �schematic� of tailwater elevation on maximum scourdepth and ridge height

�= �d84/d16�1/2. Scour holes consisting of nonuniform particlestend to become rougher in the region of jet impact, with the finematerial deposited on the ridge and at the scour hole sides. Thiseffect is smaller for scour holes made up of almost uniform sedi-ment. The data in Fig. 10 were fitted with correlation coefficientsR2=0.95 and 0.98, respectively, as

Zm� = • Fd90 �6�

= − �0.33 + 0.57�� for 1 � � � 3 �7�

An alternative approach involving Fd50 is presented in theAppendix.

Effect of Upstream Velocity

The effect of the upstream �subscript u� velocity Vu on the scourfeatures was established with 40 separate tests by systematicalvariation of the upstream Froude number Fu=Vu / �gho�1/2 �Fig. 2�.As Fu is increased, suspended sediment in the scour hole is trans-ported in the tailwater, resulting in a deeper scour hole and usu-ally a smaller ridge because of ridge erosion.

The effects of tailwater f4�T� and sediment nonuniformityf5�� are accounted for with Eqs. �5� and �7�. Fig. 11 relatesto the effect of Fu on the maximum scour depthZm�=Zm / �f1�Fd� · f4�T� · f5�� as

Zm� = 1 + Fu0.50 for Fu � 0.30 �8�

The data scatter is large, but the effect is evident. Note the in-crease of scour depth of more than 50% for Fu=0.30. The up-

Fig. 10. Reduced maximum scour depth Zm� as a function of Fd90 forvarious �

Fig. 11. Effect of upstream Froude number Fu on relative scourdepth Zm�, �—� Eq. �8�, 3�T�4.5

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stream Froude number must be limited unless an artificial ridgeremoval is a concern.

Ridge Removal

In special tests the sediment ridge �subscript r� was constantlyremoved during an experiment to provide a horizontal sedimentsurface downstream of the scour hole, to explore whether thescour increases. As mentioned, a ridge protects a scour hole fromfurther deepening, but this may be undesirable in certain condi-tions.

Fig. 12 shows the maximum scour depth Zm as a function ofFd90 and the relative tailwater T for ridge removal. It demon-strates some increase of scour hole depth as compared with con-ditions when the ridge is not removed. The slope of the curveZm�Fd90� increases by a factor �= +0.07, independent of allother parameters. Eq. �5� may be generalized to

= 0.12 ln�1/T� + Cr for T−1 � 0.05 �9�

with Cr=0.45 for ridge presence, and Cr=0.52 for artificial ridgeremoval.

General Equation for Maximum Depth of Scour Hole

Based on Canepa and Hager �2003� and the present research, theeffects of all independent parameters influencing the maximumscour depth Zm=zm /D can be assessed with

Densimetric Froude number Fd = VW/�g�d90�1/2, f1�Fd� = Fd

�10�

Jet impact angle �, f2�� = − �0.38 sin�� + 22.5 ° �� 11�

Jet air entrainment �, f3�� = �1 + ��−m �12�

Tailwater effect T, f4�T� = �0.12 ln�1/T� + Cr�/0.30 �13�

Sediment nonuniformity �, f5�� = − �0.33 + 0.57�� 14�

Upstream flow effect Fu, f6�Fu� = 1 + Fu0.50 �15�

The final expression for the maximum depth of a plunge pool

Fig. 12. Effect of ridge removal on Zm�Fd90� �—� prediction, �- -�data lines, , 3�T�4.5

scour thus is


Zm = f1�Fd� · f2�� · f3�� · f4�T� · f5�� · f6�Fu� �16�

All individual effects are independently listed from each other.The exponent in Eq. �12� is m=0.75 for unsubmerged andm=0.50 for submerged jet flow. The coefficient 1/�0.3� in Eq. �13�was introduced to be coherent with Eq. �4� established forT�3.5, i.e., f4�T=3.5�= �0.12 ln�1/3.5�+0.45� /0.3=1. Fig. 13compares all measured with the predicted maximum scour depthsof Series II. The agreement is good for the entire range of experi-ments �R2=0.93� and similar to the experiments of Canepa andHager �2003� and of Series I for a total of 435 experiments.

Further Observations

The effect of sediment nonuniformity � on the dimensional scourhole geometry is shown in Fig. 14. All tests refer to black-waterof discharge Q=21.8 L/s with �=1.22 �almost uniform�, 1.73,and 2.66: A scour hole deepens as granulate becomes uniform. Inall three cases a ridge is formed, whose height increases as thescour hole becomes deeper. Also included are the static scourholes, demonstrating a significant difference with the dynamicscour hole �Pagliara et al. 2004a�.

The effect of the upstream Froude number Fu on the dimen-sional scour hole geometry is shown in Fig. 15. For Fu�0 bothscour depth and ridge height are relatively small. By increasingFu for otherwise identical conditions, the scour hole deepens con-tinuously. This is not true for the ridge, however. As Fu increases,the deposition height first increases to a maximum �second largestdischarge in this example� and then reduces because of the in-

Fig. 13. Comparison of measured with predicted maximum scourdepths Zm, �—� Eq. �16�

Fig. 14. Effect of sediment nonuniformity � on scour hole geometryfor �= �−−� 1.22, �- -� 1.73, �++ � 2.66, and �—� dynamic and �- -�corresponding static scour conditions


creased transport capacity of the combined jet and upstream dis-charges. As mentioned, this aspect needs further attention.

Maximum Ridge Height

The maximum ridge height ZM =zM /D allows predicting tailwaterphenomena for the unsubmerged �U� and the submerged �S� jetflows. The maximum ridge height increases linearly with Fd90.Fig. 16�a� shows the data for �=30° and T�3.5. The maximumridge height for unsubmerged flow is almost twice those for sub-merged flow because the additional air entrained by the unsub-merged plunging jet lifts the sediment particles close to the freesurface from where they are transported to the ridge. A consider-able air entrainment results also for unsubmerged jets with ablack-water approach flow. Fig. 16�b� shows the decrease of ZM

with � for unsubmerged jet flows; even smaller values result forsubmerged jet flow �Fig. 16�c��. These may be expressed withR2=0.94 as

ZM = A + B · �1.2nFd; 2 � Fd � 20

30 ° � � � 90 ° , T � 3.5 �17�

Here A=0, B=16, and n=1 for unsubmerged, and A=0.3,B=438, and n=2 for submerged jet flows, respectively. The ridgeheight may thus be kept small for either a large jet impact angle �or a small densimetric Froude number.

The effect of jet air content � on ZM is analyzed as previouslywas the maximum scour depth. Again, unsubmerged and sub-merged jet flow conditions require a separate treatment. Fig. 17�a�shows ZM+=ZM / f2�� as a function of � for unsubmerged jet flowsimilar to Eq. �3� with F=1 as

f3M�� = F�1 + ��−0.50; � � 12 �18�

For submerged jet flow, the data follow a different trend, with aninitial increase to a maximum of almost 1.4, followed by a de-crease similar to the unsubmerged jet flow �Fig. 17�b��. For��2.5, these data follow Eq. �18� with F=1.5. Note that thescatter in predicting the ridge height is generally larger than forthe maximum scour depth. This is attributed to erosion of theridge for small overflow depths. Eq. �18� represents a logic ex-tension of Eq. �3�.

Fig. 18�a� relates to the maximum deposition heightZM =zM /D as a function of Fd90 for various relative tailwaterdepths T=ho /D. Ridge heights are relatively small for small T,increase for the sediment M0 used to a maximum for T�5 todecrease again due to the reduced jet momentum for large T. Eq.

Fig. 15. Effect of upstream Froude number Fu on scour hole profilez�x�

�17� thus refers to the absolute maximum ridge height, and may

serve for designing plunge pools �solid top curve in Fig. 18�a��.The lines added connect the experimental data without any com-putational approach because of the complexity of the phenomena.Fig. 18�b� refers to the combined effect of relative tailwater T andupstream Froude number Fu. Except for very small tailwater, allcurves increase with Fu because the material suspended over thescour hole is transported downstream onto the ridge, as men-tioned. The full lines in Fig. 18 are valid for Fu=0 and T�5. Theridge height is considered less important than the scour depth fordesign purposes, such that the major efforts were concentrated tothe latter parameter.

Scour Hole Geometry

Length Parameters

The relative scour hole length Lm= lm /D from the origin to its

Fig. 16. Maximum ridge height �a� ZM�Fd90� for �=30° forunsubmerged �up� and submerged �down� jet flow conditions,ZM�Fd90� for 30° ��90° for �b� unsubmerged and �c� submergedjet flow conditions, �—� Eq. �17�, T�3.5

maximum depth �Fig. 2� is plotted in Fig. 19 as a function of Fd90,

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Fig. 17. Effect of jet air content � on maximum ridge height ZM+ for�a� unsubmerged and �b� submerged jet flows, �—� Eq. �18�, T�3.5

Fig. 18. �a� Effect of relative tailwater T on maximum ridge heightZM for uniform sediment �thick line represents the data envelope� and�b� effect of Fd90 and T on ZM for Fu�0


for �=30° , T�3.5 and unsubmerged black-water flow. The dataof Canepa and Hager �2003� and those of the present study col-lapse on a straight line.

Fig. 20�a� shows that the data of Series I relating to the un-submerged �full lines� are higher than for the submerged �dashedlines� jet flows. The trend lines may be expressed with C=0.90for the submerged, and C=1.0 for the unsubmerged jet flows as

Lm = 1.5 + �C − 0.01��Fd90; 2 � Fd90 � 20, 30 ° � � � 90°

�19�

The length la=xa−xo from the scour origin xo to the point xa

where the scour profile intersects the original bed elevation varieswith both � and Fd90. Fig. 20�b� shows a combined plot relatingLa= la /D to Fd90 for both unsubmerged and submerged jet flows.These data may be expressed with �a=1.05 for unsubmerged and�a=0.85 for submerged jet flows, respectively, as

La = 2.5�1 + �a exp�− 0.030��Fd90�

2 � Fd90 � 20, 30 ° � � � 90° �20�

The nondimensional length LM = lM /D of the ridge maximumfrom the scour origin varies with Fd90 and � with �M =1.5 forunsubmerged and �M =1 for submerged jet flow conditions as�Fig. 20�c��

LM = 2.5�1 + 30�M�−1.25Fd90�

2 � Fd90 � 20, 30 ° � � � 90° �21�

The total length Lu= lu /D from the scour hole origin to the ridgeend may be expressed as �Fig. 20�d��

Lu = 2.5�1 + 38.5�u�−1.25Fd90�

2 � Fd90 � 20, 30 ° � � � 90° �22�

with �u=1.5 for unsubmerged and �u=1 for submerged flow,respectively. These lengths serve as the basis of a generalizedscour profile. The previously established relations for the fourlength scales across a scour hole were further processed by com-paring observations with predictions according to Eqs. �19�–�22�resulting in R2=0.81.

Scour Hole Profile

Once the main scour hole lengths were determined, the profilewas considered. Fig. 21�a� shows the nondimensional plot Z /Zm

versus Xm= �x−xm� / �xa−xo� for �=45°, with all data passingthrough the profile minimum �0;−1�. Note that the falling profile

Fig. 19. Relative length Lm�Fd90� for �=30° and unsubmergedblack-water flow, R2=0.97, �—� trend line

limb is slightly higher for the submerged than for the unsub-


merged jet data. The curves start at Xm=−0.62 and −0.69, respec-tively, and pass through Xm= +0.38 and +0.31 at the other side.Due to the selected data normalization, the ridge profiles arepoorly reproduced. A different treatment is presented below. Notealso the larger ridge heights for the white-water than for theblack-water jet data, as previously stated.

A generalized data analysis for the scour hole profiles of SeriesI for jet angles 30° ��60° indicates the typical scatter of Fig.

Fig. 20. Nondimensional lengths �a� Lm; �b� La; �c� LM; and �d� Lu asfunctions of Fd90 and � for �- -� unsubmerged and �- -� submerged jetflow conditions, �—� Eqs. �19�–�22�, T�3.5

21�a�. All profiles may approximated by

Z/Zm = − 1 + �4/9�Xm + 6.08Xm2 + 4.75Xm

3

for Z � 0, 30 ° � � � 60° �23�

The normalization selected for the generalized scour profile thusallows for a simple data analysis independent from the jet impactangle �, the jet air content �, and the tailwater conditions �U;S�because these effects are contained in the previously establishedscalings. Fig. 22�a� compares all data for �=30, 45, and 60° withEq. �23�, resulting in an acceptable overall fit �R2=0.85�.

Fig. 21�b� relates to the jet impact angle �=90° and shows analmost symmetrical scour hole profile. The latter is accounted forwith the symmetry �subscript s� parameter Xs=2�xo−xm� /�xa−xo� to result in Xs=−1 for perfect symmetry. Fig. 21�c� showsXs�� and demonstrates the previously noted fact according towhich profile symmetry is attained for a vertical jet. The data ofSeries II were also described with Eq. �23� for all tests, resultingin a similar agreement with predictions.

Fig. 22�b� relates to the ridge profiles expressed as Z /ZM�XM�with XM = �x−xa� / �xM −xa� for 30° ��90°. All profiles passthus through points �0;0� and �1;1� and tend to disperse further

Fig. 21. Scour hole profile Z /Zm�Xm� for white- and black-water jets,and unsubmerged and submerged jet flow data of Series I for �a��=45° and �b� �=90°; and �c� scour hole asymmetry Xs��

downstream. The profile equation is the parabola

J

Z/ZM = 2XM − XM2 ; 0 � XM � 2, 30 ° � � � 90° �24�

Eqs. �23� and �24� predict the general scour hole and ridge pro-files provided the tailwater elevation is so high that no appre-ciable erosion occurs on the ridge crest. The erosion pattern of theridge needs additional attention because of the combined deposi-tion and erosion pattern.

Comparison with Existing Knowledge

Pagliara et al. �2004b� compared the present data sets with exist-ing formulas for the prediction of the maximum scour depth zm.The following experimental studies were accounted for in chro-nological order: Schoklitsch �1932�; Veronese �1937�; Kotoulas�1967�; Martins �1975�; Mason and Arumugam �1985�; Mason�1989�; D’Agostino �1994�; and D’Agostino and Ferro �2004�.Basically, two types of scour formula are available, namely thoseof Schoklitsch to Martins involving the sum of the maximumscour depth and the tailwater height �zm+ho� against discharge perunit width q, energy head H, and sediment size d50 or d90; and therecent formulas with a more complex parameter arrangement. No-tably, all formulas except Mason �1989� are dimensionally incor-rect. Several effects were not accounted for such as the jet impactangle, the sediment nonuniformity, and the tailwater submer-gence. Pagliara et al. �2004b� found a reasonable performance ofthe Mason �1989� formula, whereas the remainder except forSchoklitsch �1932�; Veronese �1937�; and partly Kotoulas pro-duced poor results. Today, Mason’s formula is mainly used forpredicting the maximum scour depth thereby accounting for theeffect of jet aeration.

The research of Canepa and Hager �2003� investigatedwhether the Froude similitude may be applied for the three-phasejet flow configuration, and whether an extended version of the

Fig. 22. General scour hole profiles for �a� scour hole Z /Zm�Xm� with�—� Eq. �23� for 30° ��60° and �b� ridge profile Z /ZM�XM� for�=30°, �—� Eq. �24�, Series I

densimetric Froude number accounts correctly for the flow con-


ditions. Their results were slightly modified in the present workusing the additional data. However, all except for the “hose data”of Canepa and Hager �2003� were included in the updated dataanalysis.

Limitations

This research involves 435 experiments including those ofCanepa and Hager �2003�. The main parameters influencingplunge pool scour were varied, namely:1. Densimetric particle Froude number Fd90 as the ratio

between the approach water jet velocity VW relative to thescaling velocity computed as the square root of reducedgravitational acceleration g� times the sediment size d90. Itwas demonstrated that scale effects are absent if the jetvelocity is larger than 1 m/s and the sediment size has aminimum of 1 mm. The densimetric Froude number variedbetween 4 and 20;

2. Jet impact angle � varied between 30 and 90°; its effect onplunge pool scour is not as large as would be expected; scourhole depths increase with the jet angle, whereas the ridgeheight decreases for otherwise identical conditions;

3. Jet air content � as the ratio between air and water dis-charges has a significant effect on the scour hole. The larger�, the smaller is the scour hole. Values of � up to 12 wereconsidered, for which the fluid phase was only a small por-tion of the gas phase;

4. Relative tailwater height T as the ratio between the tailwaterdepth and the jet diameter D was varied between 0.7 and 10.Its effect on the maximum scour depth is significant. A scourhole is deep for small T and reduces for larger T, for other-wise identical conditions. The ridge height has a maximumfor T�3.5 due to the combined effect of jet diffusion andridge erosion;

5. Sediment nonuniformity � was varied up to 2.7. A scour holeis deeper for uniform sediment than for a nonuniform granu-lar bed under otherwise identical conditions; and

6. Upstream Froude number Fu was up to 0.44 and has a lim-ited effect on the scour process. The scour hole increaseswith Fu whereas the ridge may become either smaller orlarger as compared with Fu=0.

A number of other parameters were also systematically accountedfor, such as the continuous ridge removal or jets that are eitherfree or submerged from the tailwater. A major conclusion of thisresearch is that all results apply exclusively for Froude similitude.Further, the scour surface was made up with incohesive granulateonto which an almost plane plunge pool scour developed. Themaximum scour hole depth follows then Eqs. �10�–�16� within anestimated accuracy of ±20%. Other parameters such as the scourhole profile or the ridge height may be subject to a larger varia-tion.

Conclusions

Plane plunge pool scour for both pre-aerated white-water, andblack-water high-speed jets from a pipe arrangement were con-sidered to determine its prominent features. In contrast to proto-types, the scour hole was simulated with incohesive sedimentwhose resistance against scour action is less than for a rock con-glomerate. It was observed that the jet shape had a small effect on
the scour depth provided the densimetric particle Froude number

Fd90 is employed as the variable involving the average jet veloc-ity, the reduced gravitational acceleration, and the determiningparticle size. The maximum scour depth relative to the jet diam-eter was also related to the jet air content �, the relative tailwaterelevation T, and the sediment characteristics as expressed with d90

and � both for unsubmerged and submerged jet flows. The devia-tion of the experimental data for jet impact angles between 30 and90° from the maximum scour depth prediction �16� was less than±20%. The maximum ridge height was also investigated resultingin differences between the unsubmerged and the submerged jetflow configurations. The scour hole profile was defined using fourtypical lengths. Differences between the submerged and the un-submerged jet configurations were noted.

The densimetric particle Froude number has the main effect onthe plunge pool scour. To reduce scour, the jet velocity must tech-nically be limited, whereas the effect of grain size is relativelysmall. The main features of the nearly plane plunge pool scour arethus available for a range of geometrical and hydraulic configu-rations.

Appendix. Effect of Granulometry Based on MedianGrain Size

All previous results were presented in terms of the densimetricFroude number Fd90 involving the almost largest grain size d90.The purpose of this Appendix is to relate all data to Fd50. Fig.23�a� shows the data of Fig. 8 as Zm�Fd50� for various relativetailwater depths T. As previously, these data follow the expression

Fig. 23. Maximum scour depth Zm versus densimetric Froudenumber Fd50 based on d50 for �=30°: �a� plot analogous to Fig. 8 and�b� reduced plot Zm� �Fd50� with �–� Eq. �28�

Zm=50 ·Fd50, where 50= slope related to that presentation, with

50 = 0.10 ln�1/T� + 0.39 �25�

Then, as previously, the reduced scour depth Zm� was considered.Fig. 23�b� shows Zm� �Fd50� for 1.2��2.7 along with

Zm� = − Fd50 �26�

The sediment nonuniformity has no additional effect on the maxi-mum scour depth if d50 instead of d90 is considered, therefore. Thedata reanalysis based on d50 instead of d90 results in

Zm = f1��Fd� · f2�� · f3�� · f4��T� · f5�� · f6��Fu� �27�

where f3= f3�, f4= f4�, f6= f6�, and

f1� = Fd50 �28�

f2� = − �0.34 sin�� + 22.5 ° �� 29�

f5� = − 1 �30�

The change in f2�� is associated with f1�. The accuracy of thealternative approach based on d50 is comparable to the originaldata analysis based on d90.

Notation

The following symbols are used in this paper:A coefficient;B coefficient;b jet width;C coefficient;

Cr coefficient;D jet diameter;d sediment grain size;F coefficient;

Fd densimetric particle Froude number;f function of;g gravitational acceleration;

g� ��s−�� /��g;h flow depth;

hj jet height;L relative length L= l /D;l length;

m exponent;n exponent;Q discharge;R coefficient of correlation;S submerged flow;T relative tailwater depth;U unsubmerged flow;V cross-sectional velocity;X streamwise nondimensional coordinate;x streamwise coordinate;Z vertical nondimensional coordinate;z vertical coordinate;� jet impact angle;� jet air content;
� coefficient;
J

� water density;�s sediment density;� sediment nonuniformity; nonuniformity parameter; tailwater parameter; and� exponent.

Subscripts

A air;a axis;e equivalent;

M maximum;m minimum;o origin;s sediment;t temporal;u upstream; and

W water.

References

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D’Agostino, V. �1994�. “Indagine sullo scavo a valle di opera trasversalimediante modello fisico a fondo mobile.” Energ. Elettr., 71�2�, 37–51�in Italian�.

D’Agostino, V., and Ferro, V. �2004�. “Scour on alluvial bed downstreamof grade-control structures.” J. Hydraul. Eng., 130�1�, 24–37.

Kotoulas, D. �1967�. “Das Kolkproblem unter besonderer Berücksichti-gung der Faktoren ‘Zeit’ und ‘Geschiebemischung’ im Rahmen derWildbachverbauung.” Schweiz. Anstalt forstliche VersuchswesenMitt., 43�1�, Beer, Zürich, �in German�.

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Minor, H. -E., Hager, W. H., and Canepa, S. �2002�. “Does an aeratedwater jet reduce plunge pool scour?” Rock scour due to falling high-velocity jets, A. J. Schleiss and E. Bollaert, eds., Balkema, Lisse,117–124.

Pagliara, S., Hager, W. H., and Minor, H.-E. �2004a�. “Plunge pool scourin prototype and laboratory.” Int. Conf. Hydraulics of Dams and RiverStructures, Tehran, Balkema, Lisse, 165–172.

Pagliara, S., Hager, W. H., and Minor, H.-E. �2004b�. “Plunge pool scourformulae—An experimental verification.” 29° convegno di idraulica ecostruzioni idrauliche, Trento, Italy, Bios Cosenza, 1, 1131–1138.

Rajaratnam, N., and Mazurek, K. A. �2002�. “Erosion of a polystyrenebed by obliquely impinging circular turbulent air jets.” J. Hydraul.Res., 40�6�, 709–716.

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hydraulics of plane plunge pool scour - unipi.it · lar plunge pool scour. an air-water jet of...

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