Hydraulics of Skimming Flows Over Stepped Channels-Chanson

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Hydraulics of skimming flows over

stepped channelsand spillwaysHubert Chanson

a

aDepartment of Civil Engineering , The University of

Queensland , Lecturer in Hydraulics/Fluid mechanics, Brisbane,QLD 4072, Australia

Published online: 14 Jan 2010.

To cite this article:Hubert Chanson (1994) Hydraulics of skimming flows over stepped channels

and spillways, Journal of Hydraulic Research, 32:3, 445-460, DOI: 10.1080/00221689409498745

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Hydraulics of skimming flows over stepped channelsand spillwaysHydraulique des coulements extrmement turbulentssur des canaux en marches d'escalier

^ ^ ^ HUBERT CHANSONL ^ Lecturer in Hydraulics/Fluid mechanics,L , J Department ofCivilEngineering,^ ^ P T | The University of Queensland,Tfc Brisbane QLD 4072, Australia

SUMMARYStepped chutes have become recentlyapopular m ethod for discharging flood waters. The steps increasesignificantly the rateofenergy d issipation taking place on th e spillway face and red uce t he size of therequired dow nstream en ergy dissipation basin. This paper reviews the hydraulic characteristics of skimmingflows. The onset of skimming flows is discussed. New results are presented to estimate the flow resistancealong stepped chu tes. The study indicates som e major difference between the flow pattern s on steep and flatstepped ch utes. An analogy with flow over large rough ness is developed. T hen the calculations of the start ofair entrain men t are detailed. The results indicate that free surface aeration occurs much m ore upstream thanon smooth spillways. The effects of flow aeration are later discussed.RESUMERcem ment, il est devenu coura nt de concevoir des vacua teurs de crues en marches d'escalier. Les march esd'escalier augmentent considrablement ladissipation d'nergieaulongducoursier,etpermettentderduire la taille des bassins de dissipation avals. Le prsent article dcrit les caractristiques hydrauliquesdes coulements extrmement turbulents sur des coursiers en marches d'escalier. On discute les conditionsd'apparition des coulements extrmem ent turb ulents. Puis on prsente de nouveaux rsultats, perm ettantd'estimer les coefficient de pertes de charge. Cette tude montre une difference de comportement descoulements entre des pentes faibles et fortes. Une analogie avec des coulements sur de grandes rugositsest aussi prsente. Puis on dcrit les caractristiques du point d 'appariton de l'eau blan che. Enfin, les effetsde l'entranement d'air dans l'coulement sont brivement discuts.

1 Introduction1.1 Applications o stepped chutesFor the last dec ade s , s tepp ed spi l lways have bec om e a pop ular me th od for han dl in g f lood re lease s(Fig . 1). Th e s teps increas e s igni f icant ly th e ra te of energ y diss ip a t ion ta king place a lo ng the spi l lway face and reduce the s ize and the cos t of the downst ream s t i l l ing bas in . The development ofnew cons t ruc t ion m a te r i a l s ( e .g . ro l l e r com pac ted conc re t e RCC, gab ion) has i nc reasedthein teres t for s tepped spi l lways . The const ruc t ion of s tepped spi l lway i s compat ib le wi th the s l ip-fo rm ing and RCC p lac ing m e thods ( e .g . M onksv i l l e dam , U SA , M 'Ba l i dam , RCA ) . A l so gab ionstepped spi l lways are the most common type of spi l lways used for gabion dams (e .g . Rie tsprui toutfal l , SA).Stepped channels can a lso be used to increase the d ischarge capaci ty . In USSR, Sovie t engineersRevision received October 25, 1993. Open for discussion till December 31, 1994.

JOURNAL OF HYDRAULIC RESEARCH, VOL. 32, 1994, NO.3 445


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Fig. 1. Skim ming flow abov e a steppe d spillway.

developed the concept of overflow earth dam (Grinchuk et al. 1977, Pravdivets and Bramley1989).The spillway consists ofarevetment of precast concrete blocks laid on a filter and erosionprotection layer. The channel bed is very flexible and allows differential settlements :individualblocks do not need to be connected to adjacent blocks. More high discharge capacity is achieved(i.e.up to 60 m2/s) . Th e high d egree of safety allows the use of such chan nels as primary spillways(Pravdivets and Bramley 1989).Stepped cascades are utilised also in water treatment plants. As an example, five artificialcascades were designed along a waterway system to help the re-oxygenation of the polluted canal(Gasparotto 1992). The waterfalls were landscaped as leisure parks and combined flow aerationand aesthetics. Aesthetical applications of stepped cascades can include stepped fountains incities (e.g. in Brisbane, Hong Kong, Taipei).In this paper, the author reviews the hydraulic characteristics of skimming flows over steppedchutes. A re-analysis of model and prototype data (Table 1) is presented. The results provideinformation on the o nset of skimm ing flow and on the flow resistance in stepped channels. The nthe flow cond itions at the point of inception of air entrainm ent are described. Considerations onenergy dissipation and the effects of air entrainment are discussed briefly.It must be no ted that this paper presents resu lts obtained for chutes with horizontal steps. Esseryand Horner (1978), Peyras etal.(1992) and Frizell (1992) discussed ex perim ental results ob tainedwith inclined and pooled steps.1.2 History of stepped spillwaysThe world's oldest stepped spillways are probably the spillways of the Khosr River dams, Iraq.The Khosr River dams were built arou nd B.C. 694 by the Assyrian King Sennach erib. The damswere designed to supply water to the Assyrian capital city Nineveh (near the actual Mosul).446 JOURNAL DE RECH ERCHE S HYDRA ULIQU ES, VOL. 32, 1994, NO. 3


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Remains of the dams are still in existence (Smith 1971). Both dams feature a stepped downstreamface and were intended to discharge the river over their crests.Much later, the Romans built stepped overflow dams in Syria and Tunisia (Table2).After the fallof the Roman empire, Moslem civil engineers gained experience from the Nabateans, theRomans and Sabeans. Stepped spillways built by the Moslems can be found in Iraq and in Spain.After the Reconquest of Spain, Spanish engineers benefited from the Roman and Moslemprecedents. In 1791, they built the largest dam with a stepped spillway, but the dam was washedout in 1802 after a foundation failure.

Table 1. Characteristics of model and prototype studiesCaractristiques d'tudes sur modle et prototype

referenceModel studiesEsseryandHorner (1978)Noori (1984)Sorensen(1985)Diez-Casconetal.(1991)Bayat (1991)

FrizellandMefford (1991)Peyraset al.(1991,1992)BeitzandLawless (1992)Frizell (1992)

Bindoet al.(1993)Christodoulou(1993)

slope(deg.)11to 405.711.352.05

53.151.3

26.6

18.4,26.6,4551.3and48.026.6

51.3455.0

Prototype studiesGrinchuk et al. 8.7(1977)

scale

1/101/251/101/25

1/5

1/60

1/21.31/42.7

N/A

stepheighth(m)0.03to0.050.0040.0130.0610.0240.03-0.060.0240.030.020.051

0.200.02

0.051

0.0380.0190.025

0.41

Nbofsteps4to 1810070115950to100

47

3,4, 510

47

31-434315

12

discharge(m 2/s)

0.007to 0.090.025to 0.20.006to 0.280.006to 0.280.022to 0.280.006to 0.07

0.077

0.04to 0.276E-4to0.093

0.577

0.01to0.1420.007to 0.040.02to 0.09

1.8 to 60

flowregimenappeandskimmingskimming

nappeandskimmingskimming

skimmingflownappeandskimmingnappeandskimming

skimmingskimmingskimming

skimming

remarksCIRIA tests (UK).Include inclined steps.(UK). IF= 0.5m.Monksville damspillway model (USA).1^=0.305m.(Spain). 1^=0.8m.Godar-e-landar spillway model (Iran).W 03mUSBR researchlaboratory (USA).1^=0.457m.Gabion stepped chute(France). IF=0.8m.2-D modelof theBurton Gorgedam(Australia).USBR researchlaboratory (USA).W=0.457m.M'Bali spillway model(France). ^ =0.9m.1^= 0.5m.

Dneipr hydro plant(USSR). Full-scaletest. W = U.2m.

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In all early dams (Table 2), stepped spillways were selected to contribute to the stability of thedam, for their simplicity of shape or for a combination of the two. The spillway of the New Crotondam (1906) is probably the first stepped chute designed specifically to dissipate flow energy.Table2. Historical stepped spillwaysEvacuateursdecruesen marches d'escalier dans l'histoire

nameRiver Khosrdams

yearB.C.694

ref.[1,2]

damheight(m)2.9

>1.4

slope(deg.)

30

constructionMasonryoflimestone, sandstonemortared together.

comm ents, dischargeBuiltby the Assyrian KingSennacheribto supply watertohis capital city Nineveh.Discharge overthe damcrest.Lowerdam.Upper dam.5steps.

Kasserinedam A.D.100?

[1] 10 57 Cutand fittedmasonry blockswith mortaredjoints usedto facea rubbleandearthcore.

Roman dam 220 km SW ofTunis , Tunesia.6stepsfollowed by an overfall.Discharge overthe damcrest.W=\5 0m.Qasr Khubbaz A.D. [3]100/200 6.1 Masonrydamwithlimestone slabs. Roman dam in Syria,on theEuphrate rivver. Reservoircapacity: 9000m3ofwater.Mestelladam A.D. [1] 2.1 27 Rubble masonry960 and mortar corefaced with largemasonry blocksand mortared joints.

Builtby theMoslimsinSpain.Stepped weir: W = Tim.Maximum discharge: around4000m3/s (?).Adheimdam 1300? [1] 15.2 51 Masonry steps. Builtby the Moslims duringthe Sassanian period in Iraq.

In Spain. Discharge overthedam crest. Broad crestfollowed by 14stepsand anoverfall.Almansadam 1384? [1] 15 40 Rubble masonrywithafacingoflarge masonryblocks.Puentesdam 1791 [1] 50 51 Rubble masonrycoreset in mortarand faced with

largecutstones.Dam failurein1802.

Dam acrossthe RioGuadalentin, Spain.Discharge overthecrest.4 steps followed by 59degreeuniform slope. Step height:/2=4.175m.

New Croton dam 1906 [41 53 Masonry.Dam failurein1955.USA. Maximum discharge:1650 nrVs.1F = 305m.

Note: [1]Smith (1971);[2]Forb es (1955);[3]Stein (1940);[4]Weg mann (1907).1.3 Flow regimesThe flow over stepped chutes can be divided into nappe flow regime and skimming flow regime(Fig. 2). In the nappe flow regime, the water proceeds in a series of plunges from one step toanother. At the brink of each step, the flow becomes a free-falling jet before impinging the next

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step. Energy dissipation occursbyjet breakup and jet m ixing on the step, and with the formationof a partial or fully developed hydraulic jump on the step (Fig. 2a). Essery and Horner (1978),Peyras et al. (1991) and Chanson (1993) have detailed the flow properties of nappe flow regime.In a skimming flow regime, the water flows down the stepp ed face as a coherent stream skimm ingover the steps and cushioned by the recirculating fluid trapped between them (Fig. 2b). Theexternal edges of the steps form a pseudo-bottom over which the flow pass. Beneath this,horizontal axis vortices develop and are maintained throu gh the transmissio n of shear stress fromthe fluid flowing past the edge of the steps.For a stepped spillway with skim min g flow reg ime , the free-surface of the flow is sm oo th in th eearly steps and no air entrainment occurs. Next to the boundary however, turbulence is generated. W hen th e outer edge of the turbulen t bound ary layer reaches the free surface, air entrainment at the free surface occurs (Fig. 1). Downstream of the point of inception of air entrainment,the flow becomes rapidly aerated as shown by photographs (Krest'yaninov and Pravdivets 1986,Diez-Cascon et al. 1991, Peyras et al. 1991).

h 1f/I/J///J/JJf 1\^v_\

R e c i r c u l a t i n gv o r t i c e sFig. 2. Flow regime s abov e stepped cha nn el: (a) napp e flow; (b) skimm ing flow.

2 Onset of skimming flowsFor small discharges and flat slopes , the water flows as a succession of waterfalls (i.e. nap pe flowregime). An increase of discharge or of slope might induces the apparition of skimming flowregime. The on set of skimm ing flow isafunction of the discharge, the step height and length. T heJOURNAL OF HYDRAULIC RESEARCH, VOL. 32, 1994, NO. 3 449


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author re-analysed experimental data obtained by Essery and Horner (1978), Degoutte et al.(1992) and Beitz and Lawless (1992) (Fig. 3). For these data, skimming flow regime occurs fordischarges larger than a critical value defined as:( ^ c ) o

h 1.057-0 .465*- / (1)where h is the step height, / is the step length and (dc)onse t is the characteristic critical depth. Itmust be noted that equation[1]was dedu ced for hjl ranging from 0.2 to 1.3. It should not be usedoutside that range without great care.

1.40

1.201.000.800.600.400.20000

X

fullydeveloped

hydraulicjump

SKIMMING FLOW M 0.S (M)//^ /

^~partially developedhydraulic jump

NAPPE FLOW " " - - - - .

x ESSERY and HORNER PEYRAS et al.

BEITZ and LAWLESSEquation [1]Transitionhilly/partiallydeveloped jump

Fig. 3. Onset of skimming flow regime.

In flat channels with strip bottom roughness, Adachi (1964) and Knight and MacDonald (1979)observed also stable circulatory motion in the grooves between adjacent elements for subcriticalflows. But during these ex perim ents, the occurrence of "quasi-smooth flow" was independ ent ofthe flow rate and was a function only of the ratio of the rou ghn ess spacing over roughnes s height(i.e. Spacing/Height < 3.4). O'Loughlin and MacDonald (1964) studied flows past cubical roughness elements and observed a stable recirculatory flow pattern for roughness densities larger than0.25. In pipe flows, Levin (1968) analysed head losses caused by grooves perpendicular to theflow. He found that the effect of the g rooves on the head loss is small if the leng th of the cavityover its depth is less than4.It is interesting to no te the close agreement between th e observationsof Adachi (1964), O'Loug hlin and M acDo nald (1964), Levin (1968) and Knight and MacDo nald(1979).In horizontal channels and pipes, the onset of "quasi-smooth flow" over artificial roughnesscorresponds to the apparition of stable recirculation vortices. On stepped channels, it will beshown (paragraph 3) that skimming flows may occur with stable and unstable recirculation eddiesin the cavities. Hence it is not possible to compare the apparition of stable recirculation vorticeswith the onset of skimming flow and the disappearance of nappe flows.3 Flow resistance of skimming flow s3.1 IntroductionFora skimming flow reg ime, horizontal v ortices develop beneath the pseud o-bottom formed bythe external edges of the steps. Th e vortices are maintained through the transmission of turbulent

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shear stress between the skimming stream and the recirculating fluid underneath (Fig. 2b).Several author studied "quasi-smooth flows" past strip roughness in pipes and open channels.Th eir results indicated tha t the classical flow resistance calcu lations m ust be mod ified to takeinto account the shape of the large roughness elem ents. Th e flow resistance is the sum of the skinresistance and the form resistance of the steps. The geometry of a step is characterised by itsdepth normal to the streamlines (i.e. k s= h* cosa) and th e chann el slope (i.e. tan a = /;//).Neglecting the effects of flow aeration, dimensional analysis yields:

where is the friction factor , k^ is the surface (skin) roughness,Re is the Reynolds number andZ)H is the hydraulic diameter.3.2 Experimental dataIf uniform flow conditions are reached along a constant slope channel, the friction factor can bededuced from the momentum equation as (Chanson 1993):

8*g*sina*d20 D H/ = d *T 3)where g is the gravity constant, d 0 is the uniform flow depth and q w is the discharge per unitwidth. For non-uniform gradually varied flows, the friction factor can be deduced from theenergy equation:

8 g d2 D H AHj = 2 *- T *T ~ ( 4 ) 25 degrees,Fig. 4a), the experimental results show little correlation between the flow resistance, the relativeroughness and the channel slope. With channel slopes ranging from 50 to 55 degrees, Fig. 4aindicates values of the friction factor in the range 0.17 to 5with a mean value of abo ut 1.0. Fig. 4bpresents the data obtained with flat chutes (i.e. a < 12 degrees) and indicates a good ag reeme ntbetween data obtained on model (Noori 1984) and on prototype (Grinchuk et al. 1977). Theresults show also an increase of the friction factor with the relative roughness that appears to beindependent of the channel slope. And equation (5) can be correlated by:- F = 1 .4 2 * ln ( 1 - 1 . 2 5 (for flat s lopes) (6)ff U

JOURNAL OF HYDRAULIC RESEARCH, VOL. 32, 1994, NO. 3 451


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v aV

BAYAT ( 5ldeg . )8INDO(5t deg.)CHRITODOULOU (55deg.)DIEZ-CASCON (53deg.)FRIZELL (27 deg.)PEYRAS (27 deg.)PEYRAS (45 deg.)SORENSEN(52deg.)EQ . [6]

o

GRINCHUK(8.7deg.)NOORI (5.7 deg.)NOORI( ll deg.)EQ . [6]

Fig. 4. Friction factors of stepped ch anne ls: (a) steep slopes (i.e. a > 2 7 d egrees); (b) flat slopes (i.e. a < 1 2degrees); Comparison with flows over rockfilled channels and triangular roughness (reference inTable 1).

It must be emphasised that the experimental data were analysed neglecting the effects of airentra inm ent. N o information is available on the am oun t of air entrained during the experim ents.Th e auth or believes that the flow depth me asurem ents overestimated the clear water flow depth.As a result the values of the friction factor could be overestimated.3.3 DiscussionA comp arison betw een Figs. 4a and 4b shows a sub stantial increase of the friction factor when thechannel slope increases from around 10 degrees (Fig. 4b) up to about 50 degrees (Fig. 4a). It isbelieved that such a difference is caused by different flow patterns in the recirculating cavitybeneath the skimming stream.For flat channels, the cavity of recirculating fluid between the edges of adjacent steps is oblongand large stable recirculation vortices cannot develop. The recirculating vortices do not fill theentire cavity between the edges, and the wake from one edge interferes with the next step asshown by photographs (Baker 1990) (Fig. 5b and 5c). The vortex generation and the dissipationprocess asso ciated w ith each wake are distu rbed by the next step and might interfere with those ofthe adjacent steps. The flow depth will control in part the vertical extent of the wake and thevortex interference region. Th e main flow param eters becom e the distance between two adjacentedges and the depth of flow. This reasoning is confirmed by the results shown on Fig. 4b whichshow a good correlation between the friction factor and Ars/Z)Hwhich is proportional to the edgespacing to flow depth ratio.

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For very flat slopes , the flow p attern is characterise d by the imp act of the wake on th e next ste p, athree-dimensional unstable recirculationinthe wake and some friction drag on the step downstream of the w ake impact (Fig. 5b). Th is flow p attern is called a "wake-step in terfere nce " reg ime .For larger slopes, the tail of the w ake starts interfering with th e nex t wake (Fig. 5c) and t he frictiondrag component disappears. This pattern iscalleda "wake-wake interference" regime.For steep slopes, a stable recircu lation in the cavities betwe en adjacent steps is obs erved as show nby photographs (Peyrasetal. 1991,Diez-Casconetal. 1991). The recirculating vortices are largetwo-dimensional vortices (Fig. 5a). The flow resistanceis afunction of the energy requiredtomaintainthecirculation of these large-scale vortices.With strip roughness, Adachi (1964) and Knight andM acDo nald (1979) observed a stablerecirculation mechanism when the groove height to length ratio of the cavity was larger than 0.4.Other researchers (Maull and East 1963, Kistler and Tan 1967) studied flows in rectangularcavities with various aspect ratio. They observed stable recirculatory flow patternsforheight-length ratio larger than 0.4 to0.45.For a stepped spillway, the cavity heigh t to length ratio equ als:

stablerecirculationvor tex

Stable recirculat iona)

Unstable reci rculat ion wi th wake-step interfe renceb)

I n te r fe renceof the wakevith the next wake

Unstable reci rculat ion wi th wake-wake interact ion(c)

Fig.5. Recirculation in thecavity between adjacent steps .



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[cos a* sin a] and this v alue of 0.4 would corresp ond to a chann el slope a = 26.6 degrees.The results of Maul1 and East (1963), Adachi (1964), Kistler and Tan (1967) and Knight andMacDonald (1979) would imply that stable recirculation occurs for slopes larger than 26.6degrees on a stepped chute. A comparison between Figs. 4a and 4b indicates clearly a differentflow resistance behaviour for channel slopes larger than 27 degrees, and this is coherent with thefindings of these researchers.For slopes around 27 degrees, the flow pattern is most probably characterised by the interferenceof the wake tail with the next wake (Fig. 5c). Note that there is little data available for slopesbetween 15 to 45 degrees.3.4 Analogywithflows over large rough nessFig. 6 compares experimental data obtained on stepped chutes with flat slopes (i.e. a < 12degrees), on rockfilled channels and over large roughness (Table 3). All the sets of data indicatethe same trend: i.e., the friction factor increases with the relative roughness.Flows over rockfilled channels do not exhibit steady stable recirculating flow motion butunstable vortices behind rocks. These three-dimensional unstable flow patterns have somesimilarity with the flow p atter ns above flat stepped c han nels. Th is may explain the good correlation in values of friction factors (Fig. 6). Fig. 6 shows also a good a gree me nt betwe en th e experiments on stepped channels and experiments over large roughness. It is thought that these flowshave similar patterns and are do min ated by unstable recirculating eddies and wake interferenceprocesses.Considering steep stepped channels, the large-scale recirculating vortices play a major role indissipating the flow energy . As a result, th e friction factor results show an a pparen t lack of correlation with the R eynolds n um ber and the relative rough ness. A similar result was noticed by Perryet al. (1969) who analysed velocity profiles in developing boundary layers in arbitrary pressuregradients.It is worth noting tha t Rajaratnam (1990) compared experim ents in stepped channels with experimental results obtained with steeppass and Denil fishways. His analysis showed that the frictionfactors for fishways are of the same order of magnitude as for stepped channels. For steeppassfishways, Rajaratnam and Katopodis (1991) obtained friction factors in the range 0.4 to 4.3.5 Velocity distributionFrizell (1992) performed velocity measurements for a channel slope of 27 degrees and withhorizontal steps. The measurements were obtained in the gradually varied flow region. A re-analysis of the data indicates that the velocity distributions follow a power law and that theexponent of the velocity distribution is about: N = 3.5.For uniform non-ae rated flows, Chen (1990) derived a theoretical relation between the exponentN and the friction factor as

ifN = K*\- (7)Ifwhere K is the Von Karman constant (K = 0.4). For Frizell's (1992) experiments, equation (7)would imp ly: = 0.10. Such a value is of the sam e order of m agn itud e as the exp erimen tal valuesdedu ced from equ atio n (4) (i.e. betw een 0.09 and 0.18, Fig. 4a).454 JOURNAL DE RECHE RCHES HYDR AUL IQUES, VOL. 32, 1994, NO. 3


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Table 3. Expe riments of flows over large rough nessExperimentations d'ecoulements avec de grandes rugosit

referencerelativeslope discharge roug hnes s(deg.) 200 m.Skoglund (1936)Streeter and Chu(1949) (a) (b)Adachi (1964)Perry et al. (1969)Knight andMacDonald (1979)

0.11

0.055

7.5E-4to0.075

0.008to 0.19

0.008to 0.060.0112 &0.02040.023to 0.48

0.006to 0.021

4E+3 to2.5E+52E+5 to8E+5

2.7E+4 to3.2E+5

Wide rectangular brasss pipes.60-degree V-grooves:fcs= 0.32 mm,L s = 1.52 mm.Circular aluminium pipe(0 =0.114 m) with square threads.Wooden rectangular bars: A:s=5mm,4 = 6.4 mm , W = 0.20 m.Recirculating wind tunnel. k s=3.2,12.7 and 25.4 mm. 4 =25.1, 12.7 and22.3 mm. L s/fc,= 1.83 and 9.Square perspex strip roughness:A.s= 3 mm , 4 = 3 mm, W= 0A6 m.Flow overroughness elementsSayre and Albertso n 0.06(1963) to 0.17 0.02to 0.07 0.05to 0.14 8.3E+4 to2.5E+5 Baffle blocks: /cs=38 mm,L s=76 mm, width= 152 mm.1^= 2.44 m.O'Loughlin andMacDonald (1964) 0 to 2 up to0.125 0.023to0.083 Cubical roughness elements:ks=l2.7 m m . Spacing : 1.4 to 9 ksW = 0.61 m.Flows over triangular roughnessVittal et al. (1977) 0.05 3.2E+4 to Flat open channel with two-to 0.13 2.6E+5 dime nsional triangular >.r ou gh ne ss e le m en ts : ^ ^ wRoughness height: 30deg.A:s=0.03 m. Height-length ratio: 1/5.PF= 0.6 m.Gevorkyan andKalantarova (1992) 0.05to 0.125 Stepped teeth directed against theflow:

Height-length ratio:Zemarin's formula. 1/4.Rockfilled channelsJudd and Peterson(1969)Hartung andScheuerlein (1970)Bathurst (1978)Thompson andCampbell (1979)Bathurst (1985)Thorne andZevenbergen (1985)

0.5to 3.86 to340.5to 10.2to 30.23to 2.20.8to 1.1

0.06to 30.06to 0.370.3to 7.70.02to 4.90.15to 0.9

0.04to 0.720.02to 0.20.11to 0.340.014to 0.150.012to 0.250.065to 0.11

1.9E+5 to8.8E+68.5E+4 to2E+62.1E+5 to1.4E+61.2E+6 to2.8E+79.5E+4 to1.7E+76E+5 to3.5R+6

Field data. Natural torrents in USA.Model study. Artificial rockfilledchannel, Germany.Field data. Natural streams inEngland.Field data. Torrents in USA androckfilled channels in New Zealand.Field data in United Kingdom.Field data. Mountain river inColorado, USA.

Note: Re: Reynolds number: Re=Q *Uw DHjfj.v.(a): circular pipe flows; (b): as reported in Perry et al. (1969).



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4 Point of inception of air entrainmentFo r a steppe d spillway, the a erated flow region follows a region where th e free surface of the flowis smooth and glassy (Fig. 1). However turbulence is generated next to the boundary and theturbulent boundary layer grows until the outer edge reaches the free surface. When the outeredge of the boundary layer reaches the free surface, the turbulence can initiate natural freesurface aeratio n (Fig . 1). Th e loc ation of the start of air entrai nm ent is called the po int of inception, and its characteristics are L x and d\. L t is the distance from the start of the growth of theboundary layer and d xis the depth of flow at the point of inception. For smooth spillways, Woodet al. (1983) showed that the flow properties at the point of inception can be estimated as:

- = 1 3 . 6 * ( s i n a )0 0 7 9 6 * ( F * ) 0 7 1 3 (8)d] 0.223 , snc.t- - (F*) 0M i (9)ks (sin a)0.04

where F* is defined as: F* = q^/Vg sin a *kl.For stepped spillway, most designs of the ogee crest are fitted to a Creager profile (e.g. M'Balidam) or a WES profile (e.g. Monksville dam). Usually few smaller steps are introduced near thecrest to eliminate deflecting jets of water (Fig. 1). With such a geometry, the analysis of thegrowth of the boundary layer becomes extremely complex.The author analysed the flow properties at the point of inception using model data. The resultsare presented on Fig. 7. in summary the flow properties can be estimated as:^ = 9 . 8 * ( s i n r 8 0 * ( J F * ) 0 7 1 (10)

Trwr* {F* 11)where k s= h*cosa. Eq uati on (10) and (11) are shown on Fig. 6. A com parison be tween eq uations (8) and (10) indicates that the smooth spillway calculations (equation (8)) would overestimate the distance of the apparition of "white waters" on stepped chutes.It must be emphasised that equations (10) and (11) were deduced from data obtained withchann el slopes ranging from 27 to 52 degrees. Great care must be taken wh en using these equations with different slopes.Discussion of the effects of air entrainmentDownstream of the point of inception, air is entrained at the free surface. A mixture of air andwater extend gradually through the fluid. Far downstream, the flow will become uniform.The auth or (Chan son 1993) showed that the drag reduction ob served with air entrainm entreduces the energy dissipation above the chute and hence the spillway efficiency. For slopeslarger than 30 degrees, the effects of air entrainment can no longer be neglected for the computations of the residual flow energy at the downstream end of the spillway.6 ConclusionOver the recent decades, stepped chutes have become more popular. But there is a lack ofknowledge on the hydraulics of skimming flows. This paper describes some characteristics of456 JOURNAL DE RECHE RCHES HYDRA ULIQU ES, VOL. 32, 1994, NO. 3


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GRINCHUKNOOR1

Q [6]8ATHUR5THARTUNG( IOdeg)JUDDTHOMPSONTHORNEBAZIN

" GEVORKYANKNIGHT and MACDONALDSAYRESKOGLUNDSTREETER and CHU

Fig. 6. Com parison betw een flow resistance of stepped spillways (fiat slopes), large rou gh nes s and rock-filled channels (references inTable 3).

BEITZ and LAWLESS(50 deg.)BINDO(51dcg.)FRIZELL (27 deg.)

SORENSEN(52deg.)EQ [10] (52 deg.)EQ [8] (52 deg.)

./

,-

BINDO(51 deg.)FRIZELL (27 deg.)SORENSEN (52 deg.)

' EQ. [II ] (52 deg.)EQ . [9] (52 deg.)

Fig. 7. Carac teristicsof the pointofince ption : (a) distan ce from the spillway crest; (b) flow de pth .

JOURNAL OF HYDRAULIC RESEARCH, VOL. 32, 1994, NO.3 457


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skimming flows on stepped chutes. Firstly the conditions of apparition of skimming flow aredetailed (eq. [1]). Then new results are presented to estimate the flow resistance along steppedchutes. The study indicate some distinction between the flow resistance over steep and flatstepped chutes: i.e. slopes smaller or larger than 27 degrees. It is believed that the differencecoincides to different flow patterns in the cavity of recirculating fluid between adjacent steps(Fig. 5). Some analogy with flows over large roughness is also developed. For flat slopes, anempirical correlation is presented (equation (6)). For steep slopes, the data show a wide scatterwith me an friction factor of the order of m agn itud e of unity . L ater the flow con dition s at the startof air entrainment are described. The results indicate that free surface aeration occurs muchupstream than on smooth spillways.This study shows that further experimental work is required. In particular there is no informationon the flow characteristics be tween the inception point of air entrainm ent and the uniform flowregion.More new prototype data are required for steep slopes.Notations

hydrau l i c d i ame t e r ( m )d flow depth ( m )m e a s u r e d n o r m a l t o t h echan nel s lope a t t h eedge o f a stepd\ flow depth a t t he incept ion point( m )dc critical flow de pt h( m )(^c)onset cr iti ca l flow d ep th ( m ) a t t h eonse t o fsk im mi ng flowsd0 unif orm flow dep th ( m )also cal led no rm al de pthF* Frou de nu mb er de f i ned a s :F* = n/ / /w

s curvi l inear coor dina te (m) : i .e .d i s t ance ( m )along t h e channel f rom t h e crestU w flow velo ci ty (m /s ) : /w= q wjdW chann el width (m)a spillway slopeiw dyn ami c v i scosi t y o fwa t e r (N . s / m2 )

W density of water (kg/m3)458 JOURNAL DE RE CHERCH ES HYDRA ULIQUE S, VOL. 32, 1994, NO. 3


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R e fe r e n c e s / B i b l i ogr ap h i cADACHI, S . (1 9 6 4 ) , On t h e Ar t i f i c i a l R o u g h n es s , D i s as t e r P rev en t i o n R es ea rch In s t i t u t e B u l l e t i n , No . 6 9 ,Ky o t o Un i v e r s i t y , J ap an , M arch , 2 0 p ag es .BAKER, R . (19 9 0 ), P recas t C o n c re t e B l o ck s fo r H i g h Ve l o c i t y F l o w A p p l i ca t i o n s , J . Iwem , Vo l . 4 , D e c , p p .552-557 , Discuss ion: Vol . 4 , pp . 557-558 .BATHURST, J . C . (1978) , F low Res i s tance of Large-Scale Roughness , J . o f Hyd. Div . , ASCE, Vol . 104 , No.HY12, pp. 1587-1603.BATHURST, J . C . (1 9 85 ) , F l o w R e s i s t an ce E s t i m a t i o n i n M o u n t a i n R i v e r s , J. o f Hy d . En g rg . , AS C E , Vo l . 1 1 1 ,N o . 4 , pp . 625-643.BAYAT, H. O. (1991) , S tepped Spi l lway Feas ib i l i ty Inves t igat ion , Proc. of the 17th ICOLD Congress , Vienna,Aus t r ia , Q.66 , R .98 , pp . 1803-1817.BAZIN, H. (1 8 6 5 ) , R ech erch es Ex p r i m en t a l e s s u r l 'Eco u l em en t d e l 'Eau d an s l e s C an au x Dco u v er t s ,(Ex p er i m en t a l R es ea rch o n Wat e r F l o w i n Op en C h an n e l s ) , M m o i res p rs en t s p a r d i v e r s s av an t s al 'Ac ad mie des Science s , Par i s , F ranc e , Vo l . 19 , pp . 1-494 ( in Fre nc h) .B EITZ , E. and LAWLESS, M . (1 9 9 2 ) , Hy d rau l i c M o d e l S t u d y fo r d am o n GHF L 3 7 9 1 I s aac R i v e r a t B u r t o nG o r g e , W a t e r R e s o u r c e s C o m m i s s i o n R e p o r t , R e f . N o . RE P . 2 4 . 1 , S ep t . B r i s b an e , Au s t r a l i a .BINDO, M ., GAUTIER, J. and LACROIX, F . (1 9 9 3 ) , T h e S t ep p ed S p i l l way o f M 'B a l i Dam , In t l Wat e r Po wer &Da m C o n s t ru c t i o n , Vo l . 4 5 , No . 1, p p . 3 5 -3 6 .CHANSON, H. (1993) , S tepped Spi l lway F lows and Ai r Ent ra inment , Can. J . o f Civ i l Eng. , June.C HEN, C. L . (1990) , Uni f ied Theory on Power Laws for F low Res i s tance , J . o f Hyd. Engrg . , ASCE, Vol . 117 ,N o. 3 , pp . 371-389.CHRISTODOULOU, G. C . (1993) , Energy Diss ipat ion on S tepped Spi l lways , J . o f Hyd. Engrg . , ASCE, Vol . 119 ,N o. 5, pp. 644-650.DEGOUTTE, G ., PEYRAS, L . an d ROYET, P. (1992) , Skimming F low in S tepped Spi l lways - Discuss ion , J . o fHyd . Engrg . , ASC E, Vo l . 118 , N o. 1 . pp . 11 1-114.D I E Z - C A S C O N , J. , B L A N C O , J. L., R E V I L L A , J. and G A R C I A , R. (1991), Studies on the Hydraulic Behaviour ofS t ep p ed S p i ll way s , In t l W at e r P o wer & Da m C o n s t ru c t i o n , S ep t . , p p . 2 26 .THE ENGINEER (1 93 9 ) , T h e L ad ay b o we r R es e rv o i r , T h e En g i n ee r , Vo l . 1 6 8, p p . 4 4 0 -4 4 2 .ESSERY, I. T. S. and HORNER, M . W. (1 9 7 8 ) , T h e Hy d rau l i c Des i g n o f S t ep p ed S p i l l way s , C IR IA R ep o r t No .33 , 3 n d ed i t i o n , J an . , L o n d o n , UK.FORBES, R. J . (1955) , S tudies in Ancient Technology , Leiden , E . J . Br i l l , 9 Vol .FR IZELL , K. H. (1992) , Hydraul ics of S tepped Spi l lways for RCC Dams and Dam Rehabi l i t a t ions , Proc. of the3rd Specia l ty Conf. o n R o l l e r C o m p ac t ed C o n cre t e , AS C E, S an Di eg o C A, US A, p p . 4 2 3 -4 3 9 .FR IZELL , K. H. and M EFFOR D, B . W . (1 9 9 1 ), Des i g n i n g S p i l lway s to P rev en t C av i t a t i o n Da m a g e , C o n cre t eIn te rna t iona l , Vol . 13 , N o. 5 , pp . 58-6 4 .GASPAROTTO, R . (1 9 9 2 ) , Wat e r f a l l Aera t i o n Wo rk s , C i v i l En g i n ee r i n g , AS C E, Oc t . , p p . 5 2 -5 4 .GEVORKYAN, S. G. and KALANTAROVA, Z . K . ( 1 9 9 2 ) , D e s i g n o f T s u n a m i - P r o t e c t i v e E m b a n k m e n t s w i t h aC o m p l ex F ro n t S u r face , G i d ro t ek h n i c h es k o e S t ro i t e l ' s t v o , N o . 5 , M ay , p p . 1 9 -2 1 (Hy d ro t ech n i ca lC o n s t ru c t i o n , 1 9 92 , P l e n u m P u b l . , p p . 2 8 6 -2 9 0 ) .GRINCHUK, A. S., PRAVDIVETS, Y. P . and SHEKHTMAN, N. V . (1 9 7 7 ) , T es t o f Ea r t h S l o p e R ev e t m en t sPe rm i t t i n g F l o w o f W at e r a t L a rg e S p ec i fi c D i s ch a rg es , G i d ro t ek h n i ch es k o e S t ro i t e l ' s t v o , No . 4 , p p . 2 2 -2 6( in R u s s i an ) . (T ran s l a t ed i n Hy d ro t ech n i ca l C o n s t ru c t i o n , 1 9 7 8, P l e n u m Pu b l . , p p . 3 6 7 -3 7 3 ) .HARTUNG, F . and SCHEUERLEIN, H. (1970) , Des ign of Overf low Rockf i l l Dams, Proc. of the 10th ICOLD

C o n g res s , M o n t rea l , C an ad a , Q . 3 6 , R . 3 5 , p p . 5 8 7 -5 9 8 .HENDERSON, F . M . (1 9 6 6 ) , Op en C h an n e l F l o w, M acM i l l an C o m p an y , New Yo rk , US A.JUDD, H. E. and PETERSON, D. F . (1 9 6 9 ) , Hy d rau l i c s o f L a rg e B ed E l em en t C h an n e l s , UWR L R es ea rchRepo r t , PR O 17-6 , Ut ah S ta te Univers i ty , Loga n , USA , 115 pag es .KISTLER, A. L . and T A N ,F . C . (1 9 67 ) , S o m e P ro p er t i e s o f T u rb u l en t S ep a ra t ed F l o w s , Ph y s i c s o f F l u i d s ,Vol . 10 , No. 9 , P t I I , pp . S165-S173.KNIGHT, D . W. an d M AC DONALD, J . A. (1979) , Hydraul ic Res i s tance of Art i f i c ia l S t r ip Roughness , J . o f Hyd.Di v ., AS C E, Vo l . 1 0 5 , No . HY6 , J u n e , p p . 6 7 5 -6 9 0 .KREST'YANINOV, A. N. and PRAVDIVETSS, Y. P . (1 9 8 6) , S t ep p ed S p i ll way s fo r S m al l Da m s , G i d ro t ek h n i ch es koe Mel iorat s iya , No. 8 , pp . 27-30 ( in Russ ian) .LEVIN, L . (1 9 68 ) , F o rm u l a i r e d es C o n d u i t e s F o rc es , O l o d u c t s e t C o n d u i t s d 'Ara t i o n , (Ha n d b o o k o f Pipes ,P i p e l i n es an d Ven t i l a t i o n S h af t s ) , Du n o d , Pa r i s , F ran ce ( i n F ren ch ) .M AULL, F. J . and EAST, L . F . (1 9 6 3 ) , T h ree -Di m en s i o n a l F l o w i n C av i t i e s , J . o f F l u i d M ech . , Vo l . 1 6 , p p .620-632.



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NOOR I ,B. M. A. (1 9 8 4 ) , F o rm Drag R es i s t an ce ofT w o D i m e n s i o n a l S t e p p e d S t e e p O p e n C h a n n e l , P r o c .oft h e 1stIn t lConf. on Hyd.D e s i g n inW a t e r R e s o u r c e s E n g i n e e r i n g , C h a n n e l sandC h a n n e l C o n t ro l S t ru c t u r e s , S o u t h a m p t o n , UK, K. V. H. S m i t h ( ed . ) , S p r i n g e r -Ver l an g Pu b l . , pp. 1.133-1.147.O ' L O U G H L I N , E. M. and M A C D O N A L D , E. G. (1 9 6 4 ) , S o m e R o u g h n es s C o n cen t r a t i o n E f fec t s on B o u n d aryR e s i s t a n c e , J. La H o u i l l e B l a n c h e ,No. 7, pp. 7 7 3 -7 8 3 .PERRY,A. E., SCHOFIELD,W. H. and JOUUERT,P. N. (1 9 69 ) , R o u g h Wal l T u rb u l en t B o u n d ary L ay er s ,J. ofF l u i d M ech . ,Vol. 37, Par t 2, pp.3 8 3 -4 1 3 .PEYRAS,L., ROYET,P. andD E G O U T T E , G. ( 1 9 91 ) , E c o u l e m e n t et Di s s i p a t i o nsur lesD v e r s o i rs en G r a d i n sd e G a b i o n s , ( F l o o w sand Di s s i p a t i o n ofE n e r g yon G a b i o n W e i r s ) ,J. La Ho u i l l e B l an ch e ,No. 1, pp.37-47( i n F ren ch ) .PEYRAS,L., ROYET,P. and DEGOUTTE, G.(1992) , F lowand En erg y Di s s i p a t i o n o v e r S t ep p ed Gab i o n Wei r s ,J. of Hyd. E n g r g . , A S C E , Vol. 118, No. 5, pp. 707-717.PRAVDIVET, Y. P. and BRAMLEY, M. E. (1989) , S tepp ed P ro te ct ion B locks for Dam Spi l lways , In t l WaterP o w e r Dam C o n s t r u c t i o n , J u l y , pp. 4 9 -5 6 .RAJARATNAM, N. (1 9 9 0 ) , S k i m m i n g F l o w in Stepped Spi l lways ,J. of Hy d . En g rg . , AS C E,ol. 116,No. 4, pp.587-591 .RAJARATNAM,N. and KATOPODIS,C.(1 9 9 1 ) ,, Hy d rau l i c sofS t eep p as s F i s h w ay s , C an .J.of Civi l Eng. , Vol .18,

pp . 1024-1032.SAYRE, W. W. and ALBERTSON, M. L. (1 9 6 3 ) , R o u g h n es s S p ac i n g in R i g i d Op en C h an n e l s , T ran s ac t i o n s ,A S C E , Vol. 128, pp. 3 4 3 - 3 7 2 , D i s c u s s i o n : Vol. 128, pp. 372-427.SKOGLUND,V. J. (1936) , Effect of Ro ug hn esson theF r i c t io n C o ef f ic i en t o f a C l o s ed C h an n e l ,J.of Aer o n a u t .S c i e n c e s ,Vol. 4, Nov., pp. 2 8 -2 9 .SMITH, N. (1971) , A Hi s t o ry of D a m s , The C h a u c e r P r e s s , P e t e r D a v i e s , L o n d o n , UK.SORENSEN,R. M. (1 9 8 5 ) , S t ep p e d S p i l lway H y d rau l i c M o d x e l In v es t i g a t i o n ,J. of Hy d . En g rg . , AS C E,Vol.

I l l , No. 12, pp. 1461-1472.STEIN,A. (1990) , Surveyson theR o m a n F r o n t i e r inIraqand T r a n s - J o r d a n , TheG e o g r a p h i c a l J o u r n a l ,Vol.X C V , pp. 4 2 8 -4 3 8 .STREETER, V. L. and C H U , H. (1949) ,F lu id F lowand Hea t T ran s fe r inAri t if i c ia l ly Ro ug hen ed P ipes , Repor tP ro j ec t 4918,A r m o u r R e s e a r c h F o u n d a t i o n , C h i c a g o ,USA.THOMPSON,S. M. andCAMPBELL,P. L.(1 9 7 9 ), Hy d rau l i c s o f a L a rg e C h an n e l Pav ed wi t h B o u l d e r s ,J.of Hyd.

Res., I A H R , Vol. 17, No. 4, pp.3 4 1 -3 5 4 .THORNE, C. R. andZ EVENBERGEN, L. W. (1 9 8 5 ) , Es t im a t i n g V Vel o c i t yin M o u n t a i n R i v e r s,J. of Hyd. Engrg . ,

A S C E , Vol. I l l , No. 4, pp. 6 1 2 -6 2 4 .VITTAL,N., RANGA RAJU,K. G. andGAR DE, R. J. (1977) , Res i s tanceofT w o D i m e n s i o n a l T r i a n g u la r R o u g h -n es s s , J. of H y d . Res.,I A H R , Vol. 15, No. 1, pp. 19-36.W EGM ANN, E. (1 9 0 7 ) , T h e Des s i g n oft h eNewC r o t o n Dam,T r a n s a c t i o n , A S C E ,Vol.L X V I II ,No.1047,pp.3 9 8 -4 5 7 .W O O D ,I. R.,ACKERS,P. and LOVELESS,J. (1 9 8 3 ) , Gen era l M et h o d for Cri t i ca l Poin t onSpi l lways ,J. of Hyd.E n g . , A S C E , Vol. 109, No. 2, pp.3 0 8 -3 1 2 .

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Hydraulics of Skimming Flows Over Stepped Channels-Chanson