interactions between cavity flow and main stream skimming...

12
Interactions between cavity flow and main stream skimming flows: an experimental study C.A. Gonzalez and H. Chanson Abstract: In the last two decades, research on the hydraulics of skimming flows down stepped chutes was driven by needs for better design guidelines. Skimming flows are characterized by significant momentum transfer from the main stream to the recirculation zones. Investigations are difficult because of the complex nature of the flow, the strong flow aeration, and the interactions between entrained air and turbulence. This study provides a comprehensive database on main stream and cavity flow interactions in skimming flows down a stepped chute. Measurements were conducted on a large facility ( α = 15.9°, h = 0.05 and 0.1 m, W = 1 m) with precise instrumentation based upon a Froude similitude. Air–water velocity and turbulence measurements demonstrated a well-defined mixing layer developing downstream of each step edge in which the velocity profiles had the same shape as classical monophase flow results. A comparative analysis of air–water flow properties for Froude similar flow conditions showed a good agreement between the two step heights in terms of dimensionless distributions of air content, velocity, and turbulence intensity, but dimensionless bubble count rates, turbulence levels, and bubble chord sizes were improperly scaled. Key words: stepped chute, skimming flow, flow recirculation, momentum exchange, physical modelling. Résumé : Durant les dernières deux décennies, les recherches sur l’hydraulique des écoulements extrêmement turbu- lents sur coursiers en marches d’escalier ont été dictées par les besoins de l’industrie pour de meilleures lignes directri- ces de conception. Les écoulements extrêmement turbulents (« skimming flows ») sont caractérisés par un transfert important de quantité de mouvement entre l’écoulement principal et les zones de recirculation dans les marches. L’étude en profondeur de ces processus est compliquée par la complexité de l’écoulement, la présence importante de bulles et les interactions dynamiques entre les bulles d’air et les structures tourbillonnaires. Dans cette étude, on pré- sente une base de données expérimentales complète sur les interactions entre l’écoulement principal et les zones de re- circulation, pour des écoulements extrêmement turbulents sur un coursier en marches d’escalier. Ce travail a été réalisé dans un canal de grande taille ( α = 15,9°, h = 0,05 et 0,1 m, W = 1 m), avec une instrumentation très précise, en se basant sur la loi de Reech-Froude. Les mesures de vitesses du mélange air–eau et d’intensité de turbulence démontrent l’existence d’une couche de mélange bien définie en aval de chaque arête de marche (« step edge »); les profils de vi- tesse présentent une analogie avec les couches de mélange en écoulements monophasiques. En se basant sur la loi de Reech-Froude, une étude avec deux hauteurs de marches démontre des résultats adimensionels comparables, en termes de taux de vide, vitesse et intensité de la turbulence, mais les taux de bulles d’air adimensionels, les niveaux de turbu- lence et les tailles de bulles d’air n’avaient pas le bon ordre de grandeur. Mots clés : coursier en marches d’escalier, écoulement extrêmement turbulent, recirculation, échange de quantité de mouvement, modélisation physique. [Traduit par la Rédaction] Gonzalez and Chanson 44 Introduction During the last two decades, research on the hydraulics of stepped chutes was driven by needs for better design guide- lines (Chanson 1995; Ohtsu and Yasuda 1998; Minor and Hager 2000; Chanson 2001). Most research was conducted for skimming flows corresponding to the largest discharges per unit width. That is, the waters flow down a stepped channel as a coherent stream skimming over the pseudo- bottom formed by step edges (Rajaratnam 1990) (Fig. 1). Beneath the three-dimensional cavity vortices develop and recirculation is maintained through the transmission of shear stress from the main stream (Fig. 2). Small-scale vorticity is also generated at the corner of the steps. Skimming flows are characterized by very significant form losses and mo- mentum transfer from the main stream to the recirculation zones. There is an obvious analogy with skimming flows past large roughness elements and cavities (Townes and Can. J. Civ. Eng. 31: 33–44 (2004) doi: 10.1139/L03-066 © 2004 NRC Canada 33 Received 2 May 2003. Revision accepted 14 July 2003. Published on the NRC Research Press Web site at http://cjce.nrc.ca on 13 January 2004. C.A. Gonzalez and H. Chanson. 1 Department of Civil Engineering, The University of Queensland, Brisbane, QLD 4072, Australia. Written discussion of this article is welcomed and will be received by the Editor until 30 June 2004. 1 Corresponding author (e-mail: [email protected]).

Upload: lamdieu

Post on 12-Jul-2018

215 views

Category:

Documents


0 download

TRANSCRIPT

Interactions between cavity flow and main streamskimming flows: an experimental study

C.A. Gonzalez and H. Chanson

Abstract: In the last two decades, research on the hydraulics of skimming flows down stepped chutes was driven byneeds for better design guidelines. Skimming flows are characterized by significant momentum transfer from the mainstream to the recirculation zones. Investigations are difficult because of the complex nature of the flow, the strong flowaeration, and the interactions between entrained air and turbulence. This study provides a comprehensive database onmain stream and cavity flow interactions in skimming flows down a stepped chute. Measurements were conducted on alarge facility (α = 15.9°, h = 0.05 and 0.1 m, W = 1 m) with precise instrumentation based upon a Froude similitude.Air–water velocity and turbulence measurements demonstrated a well-defined mixing layer developing downstream ofeach step edge in which the velocity profiles had the same shape as classical monophase flow results. A comparativeanalysis of air–water flow properties for Froude similar flow conditions showed a good agreement between the twostep heights in terms of dimensionless distributions of air content, velocity, and turbulence intensity, but dimensionlessbubble count rates, turbulence levels, and bubble chord sizes were improperly scaled.

Key words: stepped chute, skimming flow, flow recirculation, momentum exchange, physical modelling.

Résumé : Durant les dernières deux décennies, les recherches sur l’hydraulique des écoulements extrêmement turbu-lents sur coursiers en marches d’escalier ont été dictées par les besoins de l’industrie pour de meilleures lignes directri-ces de conception. Les écoulements extrêmement turbulents (« skimming flows ») sont caractérisés par un transfertimportant de quantité de mouvement entre l’écoulement principal et les zones de recirculation dans les marches.L’étude en profondeur de ces processus est compliquée par la complexité de l’écoulement, la présence importante debulles et les interactions dynamiques entre les bulles d’air et les structures tourbillonnaires. Dans cette étude, on pré-sente une base de données expérimentales complète sur les interactions entre l’écoulement principal et les zones de re-circulation, pour des écoulements extrêmement turbulents sur un coursier en marches d’escalier. Ce travail a été réalisédans un canal de grande taille (α = 15,9°, h = 0,05 et 0,1 m, W = 1 m), avec une instrumentation très précise, en sebasant sur la loi de Reech-Froude. Les mesures de vitesses du mélange air–eau et d’intensité de turbulence démontrentl’existence d’une couche de mélange bien définie en aval de chaque arête de marche (« step edge »); les profils de vi-tesse présentent une analogie avec les couches de mélange en écoulements monophasiques. En se basant sur la loi deReech-Froude, une étude avec deux hauteurs de marches démontre des résultats adimensionels comparables, en termesde taux de vide, vitesse et intensité de la turbulence, mais les taux de bulles d’air adimensionels, les niveaux de turbu-lence et les tailles de bulles d’air n’avaient pas le bon ordre de grandeur.

Mots clés : coursier en marches d’escalier, écoulement extrêmement turbulent, recirculation, échange de quantité demouvement, modélisation physique.

[Traduit par la Rédaction] Gonzalez and Chanson 44

Introduction

During the last two decades, research on the hydraulics ofstepped chutes was driven by needs for better design guide-lines (Chanson 1995; Ohtsu and Yasuda 1998; Minor andHager 2000; Chanson 2001). Most research was conductedfor skimming flows corresponding to the largest dischargesper unit width. That is, the waters flow down a steppedchannel as a coherent stream skimming over the pseudo-

bottom formed by step edges (Rajaratnam 1990) (Fig. 1).Beneath the three-dimensional cavity vortices develop andrecirculation is maintained through the transmission of shearstress from the main stream (Fig. 2). Small-scale vorticity isalso generated at the corner of the steps. Skimming flowsare characterized by very significant form losses and mo-mentum transfer from the main stream to the recirculationzones. There is an obvious analogy with skimming flowspast large roughness elements and cavities (Townes and

Can. J. Civ. Eng. 31: 33–44 (2004) doi: 10.1139/L03-066 © 2004 NRC Canada

33

Received 2 May 2003. Revision accepted 14 July 2003. Published on the NRC Research Press Web site at http://cjce.nrc.ca on13 January 2004.

C.A. Gonzalez and H. Chanson.1 Department of Civil Engineering, The University of Queensland, Brisbane, QLD 4072, Australia.

Written discussion of this article is welcomed and will be received by the Editor until 30 June 2004.

1Corresponding author (e-mail: [email protected]).

I:\cjce\cjce3101\L03-066.vpJanuary 7, 2004 3:14:06 PM

Color profile: DisabledComposite Default screen

Sabersky 1966; Knight and Macdonald 1979; Djenidi et al.1994; Elavarasan et al. 1995; Tantirige et al. 1994; Mansoand Schleiss 2002). In stepped chutes, however, little re-search was conducted on the interactions between the mainstream and the cavity recirculation, with the exception ofpreliminary experiments by Boes (2000), Chanson andToombes (2002a), and Matos et al. (2001), and some crudemodelling by Chanson et al. (2000, 2002). Investigations aredifficult because of the complex nature of the flow, thestrong flow aeration, and the interactions between entrainedair and turbulence.

It is the purpose of this study to provide a comprehensivedatabase on main stream and cavity flow interactions inskimming flows down a stepped chute. Measurements were

conducted on a large facility (α = 15.9°, h = 0.05 and 0.1 m,W = 1 m, where α is the channel slope, h is the height ofsteps measured vertically, and W is the channel width) withprecise instrumentation. The results provide a better under-standing of the momentum exchange processes.

Similitude and dimensional analysisIn skimming flows down a stepped chute, flow resistance

is primarily step form drag. Free-surface aeration is very in-tense, and its effects cannot be neglected. Analytical and nu-merical studies of skimming flows are difficult because ofthe number and complexity of the relevant equations. Exper-imental investigations are preferred, and this study is no ex-

© 2004 NRC Canada

34 Can. J. Civ. Eng. Vol. 31, 2004

Fig. 1. Skimming flow downs Camp Dyer Diversion Dam spillway — unprotected RCC stepped spillway over an old masonry weir(courtesy of the US Bureau of Reclamation).

Fig. 2. Definition sketch of a skimming flow.

I:\cjce\cjce3101\L03-066.vpJanuary 7, 2004 3:14:07 PM

Color profile: DisabledComposite Default screen

ception. In a channel made of flat horizontal steps, acomplete dimensional analysis yields

[1] CV

gd

uV

vV

dd

, , , , ,e

ab

e

′ ′

= ′

F

xd

yd

q g q

gh

Wh

kh

l

e ew

w

w

w

w

w s; ; ; ; ; ; ;ρµ

µρ σ

α4

3 3

where C is the void fraction, V is the velocity (metres persecond), de is an equivalent clear-water depth (metres), g isthe acceleration due to gravity (metres per second squared),u′ is the root mean square of the axial component of turbu-lent velocity (metres per second), v′ is the root mean squareof the lateral component of turbulent velocity (metres persecond), dab is a characteristic bubble size (metres), xl is thelongitudinal distance measured in the flow direction(metres), y is the distance measured normal to the pseudo-bottom formed by the step edges (metres), qw is the waterdischarge per unit width (square metres per second), µw andρw are the dynamic viscosity (pascal second) and density ofwater (kilograms per cubic metre), respectively, σ is the sur-face tension (newtons per metre), α is the angle between thehorizontal and the pseudo-bottom formed by the step edges,and ks′ the equivalent sand roughness height of the step faces(Fig. 2). For air–water flows, de is usually defined as

[2] d C yy

y Y

e d= −=

=

∫ ( )10

90

where Y90 is the characteristic distance for C = 0.9.For geometrically similar models, it is impossible to sat-

isfy simultaneously more than one similitude, and scale ef-fects will exist when one or more π-terms have differentvalues in the model than the prototype. For example, insmall-sized models based upon a Froude similitude, the airentrainment process may be affected by significant scale ef-fects (Wood 1991; Chanson 1997). Similarly, for steppedchute studies based upon a Froude similitude, scale effectsin terms of flow resistance are small when the Reynoldsnumber and step height satisfy, i.e., ρw qw/µw > 2.5 × 104 andh > 0.02 m (Chanson et al. 2002). In the present study, aFroude similitude was used as for most open channel flowstudies (Henderson 1966; Chanson 1999). Detailed air–watermeasurements were conducted in a large-sized facility to en-sure that the experimental results might be up-scaled withnegligible scale effects (Table 1).

Experimental apparatus and instrumentation

Experiments were conducted at The University of Queens-land in a 1-m wide channel previously used by Chanson andToombes (2002a). The new test section was 4.2 m long andconsisted of a broad crest followed by 9 identical steps of0.10 m height or 18 steps of 0.05 m height (Table 1). Thechute slope was α = 15.9° (l = 0.35 and 0.175 m, respec-tively). The flow rate was supplied by a pump controlledwith an adjustable frequency AC motor drive. The dischargewas measured from the upstream head above crest with anaccuracy of about 2%. Flow visualizations were conducted

with high-shutter-speed digital equipment, i.e., a digital video-camera handycam Sony™ DV-CCD DCR-TRV900 (speed:25 frames/s, shutter: 1/4 to 1/10 000 s) and a digital cameraOlympus™ Camedia C-700 (shutter: 1/2 to 1/1 000 s).

Air–water flow properties were measured using a double-tip conductivity probe (Ø = 0.025 mm for each sensor). Theprobe sensors were aligned in the flow direction and excitedby an air bubble detector (AS25240). The probe signal wasscanned at 20 kHz per sensor for 20 s. Most measurementswere conducted with a probe tip separation of ∆x = 8 mm inthe streamwise direction. (The exact distance ∆x was mea-sured with a microscope Beck/London Model 2294 with anerror of less than 0.00217 mm.) The shear flow region im-mediately downstream of the outer step edge was character-ized by intense turbulent shear and recirculation. A fewmeasurements in that flow region (xs/Lcav < 0.5 and y < 0)were performed with a probe sensor spacing of ∆x =3.18 mm, where xs is the streamwise distance from the stepedge and Lcav is the step cavity length (metres). (Fig. 2). Theshorter probe tip spacing authorized better cross-correlationsamong probe tip signals.

The translation of the probes in the direction normal tothe channel invert was controlled by a fine adjustment travel-ling mechanism connected to a Mitutoyo™ digimatic scaleunit. The error on the vertical position of the probe was lessthan 0.1 mm.

Experimental flow conditionsExperimental investigations were conducted for dimen-

sionless flow rates dc /h ranging from 0.6 to 3.2, where dc isthe critical depth. For dc /h < 0.6, a succession of free-fallingnappes was observed. For 0.6 < dc/h < 1.25, the flow exhib-ited a chaotic flow behaviour associated with strong dropletejection processes downstream of the inception point of free-surface aeration, i.e., transition flow regime. For dc /h ≥ 1.3,the flow skimmed over the pseudo-bottom formed by thestep edges, i.e., skimming flow regime. Irregular-cavity fluidejections were observed that were evidences of momentumtransfer between the main stream and the cavity flows. Therecirculating fluid flowed outwards into the main stream andwas replaced by new fluid. The ejection and inflow pro-cesses took place predominantly near the downstream end ofthe cavity.

The present study focused on the highly aerated skimmingflow regime. For five flow rates with h = 0.1 m and sevenflow rates with h = 0.05 m, detailed air–water flow measure-ments were conducted at all outer step edges downstream ofinception and at several positions xs/Lcav in and above therecirculation cavities.

Air–water flow properties at step edges

At the upstream end of the cascade, the flow was smoothand no air entrainment occurred. After a few steps the flowwas characterized by a strong air entrainment. A similar lon-gitudinal pattern is seen in Fig. 1. Downstream, the two-phase flow behaved as a homogeneous mixture. The exactlocation of the interface became undetermined. There werecontinuous exchanges of air–water and momentum betweenthe main stream and the atmosphere. The air–water flowconsisted of a bubbly flow region (C < 30%), a spray region

© 2004 NRC Canada

Gonzalez and Chanson 35

I:\cjce\cjce3101\L03-066.vpJanuary 7, 2004 3:14:07 PM

Color profile: DisabledComposite Default screen

(C > 70%), and an intermediate flow structure for 0.3 < C <0.7 (Fig. 2). Waves and wavelets could propagate along thefree surface (Toombes 2002).

At the step edges, the advective diffusion of air bubbles maybe described by an analytical model of air bubble diffusion

[3]( )

C h Ky Y

D

y Y

D= − ′ − +

12

1 3

32 90 90

3

tan/ / /

o o

where K′ is an integration constant and Do is a dimensionlessdiffusivity term, which is a function of the mean void frac-tion (Chanson and Toombes 2002a). The data were com-pared successfully with eq. [3] (Fig. 3a). Figure 3a alsopresents dimensionless distributions of bubble count ratesFdc/Vc, where F is the bubble count rate (hertz) defined asthe number of bubbles detected by the probe sensor per sec-ond and Vc is the critical flow velocity. For all flow rates, thedata showed maximum bubble count rates for C ≈ 40% to

50%, while the relationship between bubble count rate andvoid fraction was quasi-parabolic (Fig. 4b).

Two phase flow velocity distributions are presented inFig. 3b in terms of the time-averaged air–water velocity Vand turbulence intensity Tu = u′/V. The latter was deducedfrom the width of the cross-correlation function. The pro-cessing technique was detailed in Chanson and Toombes(2002b). In skimming flows, the velocity data at step edgescompared favourably with a power law

[4]V

Vy

Y

N

90 90

1

=

/

where V90 is the characteristic velocity for C = 90%(Fig. 3b). Overall, N was found to be typically between 5and 12, but the data exhibited some longitudinal oscillationswith a wave length of about 2 to 3 step cavity lengths. Suchlongitudinal oscillations were also observed in terms ofmean air content Cmean and mean flow velocity Uw (Table 2)where

© 2004 NRC Canada

36 Can. J. Civ. Eng. Vol. 31, 2004

Reference α (°) qw (m2/s) h(m) Flow regime Instrumentation Remarks

Chanson andToombes(1997, 2002c)

3.4 0.038–0.163 0.143 Nappe flow Single-tip conductivityprobe (Ø = 0.35 mm)

L = 24 m; W = 0.5 m; super-critical inflow (0.03-m noz-zle thickness)

Tozzi et al.(1998)

52.2 0.23 0.053 Skimming flow Conductivity probe Inflow: uncontrolled smoothWES ogee crest followedby smaller first steps

Chamani andRajaratnam(1999)

51.3 and59

0.07–0.2 0.313–0.125 Skimming flow Conductivity probe andflushed Pitot tube (Ø= 3.2 mm)

W = 0.30 m; inflow: uncon-trolled smooth WES ogeecrest

Matos (2000) 53.1 0.08–0.2 0.08 Skimming flow Conductivity probe andflushed Pitot tube (Ø= 3.2 mm)

W = 1 m; inflow: uncontrolledWES ogee crest, with smallfirst steps built in the ogeedevelopment

Toombes andChanson(2000)

3.4 0.08–0.136 0.143 Nappe flow Double-tip conductivityprobe (Ø = 0.025 mm)

L = 3.2 m; W = 0.25 m;supercritical inflow (nozzlethickness 0.028 to 0.040m); ventilated steps

Boes (2000) 30 and50

0.047–0.38 0.023–0.09 Skimming flow Double-tip optical fibreprobe RBI (Ø = 0.1mm, 2.1 mm spacingbetween sensors)

W = 0.5 m; inflow: pressur-ised intake

Ohtsu et al.(2000)

55 0.016–0.03 0.025 Skimming flow Single-tip optical fibreprobe

W = 0.3 m; inflow: uncon-trolled broad crest

Chanson andToombes(2002a,2002b)

21.8 0.06–0.18 0.1 Transition andskimmingflows

Double-tip conductivityprobe (Ø = 0.025 mm)

L = 3.0 m; W = 1 m; inflow:uncontrolled broad crest;experiments TC200

15.9 0.07–0.19 0.1 Transition andskimmingflows

Double-tip conductivityprobe (Ø = 0.025 mm)

L = 4.2 m; W = 1 m; inflow:uncontrolled broad crest;experiments TC201

Present study 15.9 0.020–0.200 0.05 Transition andskimmingflows

Double-tip conductivityprobe (Ø = 0.025 mm)

L = 4.2 m; W = 1 m; inflow:uncontrolled broad crest;experiments CG202

0.075–0.220 0.10 Transition andskimmingflows

Including detailed measure-ments between step edges

Note: L, chute length; W, chute width.

Table 1. Detailed experimental investigations of air entrainment in stepped chutes.

I:\cjce\cjce3101\L03-066.vpJanuary 7, 2004 3:14:07 PM

Color profile: DisabledComposite Default screen

[5] U

C V y

C y

y

y Y

y

y Yw

d

d

=

=

=

=

=

( )

( )

1

1

0

0

90

90

Although Table 2 presents depth-averaged results for oneflow rate only, the data were relatively typical of all results.Some data of Boes (2000) and Matos (2000) showed similarlongitudinal oscillations in terms of depth-averaged air con-tents. It is believed that these longitudinal waves were the

© 2004 NRC Canada

Gonzalez and Chanson 37

Fig. 3. Air–water flow properties in skimming flow at step edges for qw = 0.0643 m2/s, dc/h = 1.5, h = 0.05 m: (a) void fraction anddimensionless bubble count rate distributions — comparison with eq. [3], (b) air–water velocity distributions — comparison witheq. [4], and (c) turbulence intensity distributions.

I:\cjce\cjce3101\L03-066.vpJanuary 7, 2004 3:14:07 PM

Color profile: DisabledComposite Default screen

© 2004 NRC Canada

38 Can. J. Civ. Eng. Vol. 31, 2004

Fig. 4. Air–water flow properties in skimming flow between step edges for qw = 0.219 m2/s, dc/h = 1.7, h = 0.10 m, between stepedges 8 and 9, xs/Lcav = 0.4: (a) void fraction and air–water velocity distributions: comparison with eqs. [3] and [4] at step edge 8,(b) dimensionless bubble count rate distributions: comparison with a parabolic law, and (c) turbulence intensity distributions.

I:\cjce\cjce3101\L03-066.vpJanuary 7, 2004 3:14:07 PM

Color profile: DisabledComposite Default screen

result of strong interactions between vortex shedding down-stream of each step edge and the free surface.

The distributions of turbulence intensity Tu showed highturbulence levels across the entire air–water flow mixture,i.e., 0 ≤ y ≤ Y90 (Fig. 3c). The trend differed significantlyfrom turbulence intensity profiles observed in turbulentboundary layer flows (Schlichting 1979), although they areclose to the earlier results of Chanson and Toombes (2002a)on a 22° slope stepped chute.

Air–water flow properties between stepedges

Between step edges (0 < xs/Lcav < 1), air–water flow prop-erties exhibited significant differences, particularly fory/Y90 < 0.3. Figures 4 and 5 illustrate a data set for one cav-ity flow. Between step edges, void fraction distributionsshowed greater flow aeration than at step edges (Fig. 4).Matos et al. (2001) reported a similar finding. Mean air con-centrations calculated from the pseudo-bottom formed bythe step edges (y = 0) were typically 20% to 30% larger thanthose observed at the upstream and downstream step edges(Table 3). It was proposed that air bubbles were trapped inthe large-scale vortical structures of the recirculation zoneby inertial effect (Matos et al. 2001).

Dimensionless velocity distributions are shown in Fig. 5,while characteristic parameters are summarized in Table 3for the same data set. Figure 5a shows longitudinal varia-tions of the velocity distribution above the recirculationzone. The data suggest a developing shear layer downstreamof the singularity formed by the step edge. In Fig. 5b, theexperimental data are compared with the theoretical solu-tions of Tollmien and Goertler for plane turbulent shear lay-ers (Rajaratnam 1976; Schlichting 1979). For a free shearlayer, Tollmien’s solution of the equations of motion yields

[6]VVo

Ø Ø/ 2ddØ

e e Ø= − +

00176 0133732

. . cos

+

068732

. sine ØØ/ 2

where Ø ∝ y/(axs); Vo is the free-stream velocity; and a is anempirical constant that equals (2lm

2/xs2)1/3, where lm is

Prandtl’s mixing length (Rajaratnam 1976). Goertler’s solu-tion of the equations of motion is

[7]VV

Kxo

50

s

erfy - y= +

12

1

where y50 is the location where V/Vo = 0.5, K is a constantinversely proportional to the rate of expansion of the mixinglayer, and erf is the error function given by

[8] erf(u) =1 2

0πe d−∫ t

u

t

where u and t are dimensionless variables used to define erf.In monophase flows, a was found to be 0.084 and 0.09 for

the data sets of Liepmann and Laufer (1947) and Albertsonet al. (1950), respectively, while K was equal to 11 for thedata of Liepmann and Laufer. For the experimental data pre-sented in Figs. 4 and 5, values of the coefficients a and K aresummarized in Table 3. (These data were obtained from thebest data fit.) Along one-step cavity, the coefficient K in-creased with xs towards monophase flow values (K = 11),while the values of a decreased with xs towards the reportedvalues for monophase flow (a ≈ 0.09).

Figure 5b demonstrates self-similarity of the velocity pro-files. In this figure, the velocity data are presented as V/Voversus K(y – y50)/xs where Vo was selected such as Vo =0.9V90. Experimental observations agreed well with bothTollmien’s and Goertler’s solutions.

The upper edge of the shear layer (located where V/Vo = 1)and the free velocity V = Vo/2 (y = y50) were recorded for allthe locations in the cavity. The results are reported in Fig. 6.The experimental observations highlight the shape of the de-veloping shear layer downstream of each step edge. Theabove finding provides means to develop a relationship forthe growth of the mixing layer and predict the mean velocitydistribution based upon a suitable shear-stress model.

Turbulence levelsTurbulence intensity distributions in the shear layers are

presented in Fig. 5c. The data showed very high levels ofturbulence in the shear flow. Maximum turbulent intensitiesof more than 60% were observed. These values were consis-tent with turbulence intensity measurements in plunging jetflows by Chanson and Brattberg (1998) and in wake flowsbetween rocks by Sumer et al. (2001). However, the presentdata were significantly larger than turbulence levels ob-served in monophase developing shear flows. In monophasemixing layers, experimental data indicated maximum turbu-lence levels (Tu)max = 15% to 20% for xs/dj ≤ 4 where dj isthe jet flow thickness (Davies 1966; Sunyach and Mathieu1969; Wygnanski and Fiedler 1970).

The present data suggested further greater turbulence inten-sities next to the downstream end of the cavity (xs/Lcav ≥ 0.5).For example, the maximum turbulence levels (Tu)max wereabout 0.8, 1, 1.1, and 1.5 for xs = 0.2, 0.3, 0.6, and 0.75, re-spectively (Fig. 5c). The findings might be consistent with vi-sual observations of cavity fluid ejection and replenishmenttaking place primarily next to the cavity downstream end.

© 2004 NRC Canada

Gonzalez and Chanson 39

Step edge Cmean Uw/Vc N Remarks

8 — — — Inception point9 0.387 2.50 9.7

10 0.285 2.59 5.711 0.419 2.45 1612 0.274 2.58 4.613 0.421 2.48 9.814 0.325 2.64 8.415 0.407 2.83 9.316 0.346 2.60 8.017 0.368 2.62 7.9

Note: Uw = qw/de; Vc, critical flow velocity.

Table 2. Longitudinal air–water flow properties for qw =0.0643 m2/s, dc/h = 1.5, h = 0.05 m.

I:\cjce\cjce3101\L03-066.vpJanuary 7, 2004 3:14:08 PM

Color profile: DisabledComposite Default screen

Discussion

Air–water flow similarityFor two dimensionless flow rates (dc/h = 1.5 and 1.7),

identical experiments were repeated with h = 0.05 and

0.10 m. A detailed comparison of the results obtained withthe two step heights for a similar flow rate showed goodagreement between the two configurations in terms ofdimensionless distributions of air content, velocity, and tur-bulence intensity, as well as in terms of mean air content

© 2004 NRC Canada

40 Can. J. Civ. Eng. Vol. 31, 2004

Fig. 5. Air–water velocity distributions between step edges for qw = 0.219 m2/s, dc/h = 1.7, h = 0.10 m, between step edges 8 and 9:(a) velocity distributions, (b) self-similarity: comparison between experimental data and the theoretical solutions of Tollmien andGoertler, and (c) turbulence intensity distributions in the shear layer.

I:\cjce\cjce3101\L03-066.vpJanuary 7, 2004 3:14:09 PM

Color profile: DisabledComposite Default screen

Cmean, dimensionless flow velocity Uw/Vc, and air–waterflow velocity V90/Vc. However, significant differences wereobserved in terms of dimensionless bubble count ratesFdc/Vc, while lesser maximum turbulence levels were notedwith h = 0.05 m. At each step edge, measurements for h =0.05 m showed lesser dimensionless bubble count rates byabout 30% to 50% than those observed for h = 0.10 m at anidentical step edge for the same dimensionless flow rate.The finding suggests some scale effects in terms of bubblecounts and bubble sizes.

A comparative analysis of bubble chord size distributionsfor identical flow rate, location, and local void fractionshowed consistent differences between the two step heightsthat were not scaled at 2:1, implying that eq. [1] could notbe approximated properly by a Froude similitude (e.g.,Fig. 7). Figure 7 compares bubble chord sizes recorded atthe same dimensionless distance from the inception pointand for the same dimensionless flow rate with the two stepheights (h = 0.1 and 0.05 m). Figure 7a shows air chord sizedistributions in 0.5 mm intervals for C ≈ 0.1. The data showsimilar air chord size distributions independently of the stepheight. Figure 7b presents the distributions of mean airchord sizes at the same locations for the two step heights.

Turbulent shear stressAt each step, the cavity flow is driven by the developing

shear layer and the transfer of momentum across it (Figs. 2and 5). The equivalent boundary shear stress of the cavityflow equals the maximum shear stress τmax in the shear layerthat may be modelled by a mixing length model

[9] τmax = ρνt∂∂

=

Vy

y y50

where νt is the momentum exchange coefficient (Chanson etal. 2000). For Goertler’s solution of the equations of motion,the dimensionless pseudo-boundary shear stress equals

[10]8 2

2

τρ π

max

V Ko

=

where 1/K is the rate of expansion of the mixing layer. Notethat eq. [10] is homogeneous to a Darcy–Weisbach frictionfactor. For the data shown in Figs. 5 and 6, the integration ofeq. [9] along the step cavity yields an average friction factor

© 2004 NRC Canada

Gonzalez and Chanson 41

Fig. 6. Sketch of the developing shear layer and experimental data points for qw = 0.219 m2/s, dc/h = 1.7, h = 0.10 m, between stepedges 8 and 9.

Step edge xs/Lcav Cmean Uw/Vc V90/Vc

Goertler’ssolution, K

Tollmien’ssolution, a (Fmaxdc)/Vc

8 0 0.37 2.42 2.72 na na 30.40.2 0.33 2.27 2.75 2.46 1.192 26.20.25 0.35 2.31 2.66 2.32 0.975 29.30.3 0.39 2.29 2.64 2.36 0.809 26.90.4 0.42 2.33 2.69 2.73 0.615 24.50.5 0.46 2.37 2.67 5.16 0.487 26.90.6 0.53 2.34 2.63 4.17 0.417 24.00.75 0.51 2.34 2.69 5.94 0.325 25.10.8 0.47 2.31 2.75 4.02 0.297 26.7

9 1.0 0.39 2.37 2.75 na na 31.6

Note: Cmean, integrated between y = 0 and Y90; Lcav = 0.364 m; na, not applicable.

Table 3. Air–water flow properties between two adjacent step edges for qw = 0.219 m2/s, dc/h = 1.7, h = 0.10 m,between step edges 8 and 9.

I:\cjce\cjce3101\L03-066.vpJanuary 7, 2004 3:14:09 PM

Color profile: DisabledComposite Default screen

[11]1 8

0 342

0L V

xx

L

cav os

s

cav

dτρ

max .=∫ =

This result represents the average dimensionless shearstress between the cavity flow and the main stream. Inmonophase shear flow above a rectangular cavity,Wygnanski and Fiedler (1970) observed that the maximumshear stress was almost independent from the distance fromthe singularity, and their data yielded 8τmax/(ρVo

2) = 0.18.For cavity flows, Haugen and Dhanak (1966) and Kistlerand Tan (1967) observed similar results, which are of thesame order of magnitude as the present findings.

For the same flow rate as in Figs. 5 and 6, the flow resis-tance estimate, derived from the friction slope, was fe = 0.15(Table 4), where fe is the air–water flow friction factor. Theresult is close to eq. [11]. Generally, present experimentalresults demonstrated that the flow resistance was reasonablywell approximated by the integration of eq. [10] along stepcavities. The finding is an improvement of the gross approx-imation of Chanson et al. (2002), who assumed a constantcoefficient K.

For completeness, the “equivalent” friction factors derivedfrom the measured friction slope are presented in Table 4.The results are similar to the findings of Chanson andToombes (2002a) with a 22° slope and the analysis of Chan-son et al. (2002).

© 2004 NRC Canada

42 Can. J. Civ. Eng. Vol. 31, 2004

Fig. 7. Comparison of measured bubble chord sizes for two step heights h = 0.05 m and h = 0.10 m (dc/h = 1.7, (xl – xI)/dc = 7.5)where I is inception f free-surface aeration: (a) bubble chord size probability distribution functions in the bubbly flow (C ~ 0.1), and(b) distributions of mean bubble chord sizes for C < 0.5.

Vertical stepheight, h (m)

Dimensionlessflow rate, dc/h

Darcy friction factorfor air–water flow, fe

0.10 1.4 0.131.5 0.101.6 0.121.7 0.15

0.05 1.5 0.121.7 0.122.0 0.112.2 0.102.4 0.112.7 0.253.2 0.16

Note: fe, air–water flow resistance deduced from the friction slope(Chanson et al. 2002).

Table 4. Flow resistance in skimming flows.

I:\cjce\cjce3101\L03-066.vpJanuary 7, 2004 3:14:10 PM

Color profile: DisabledComposite Default screen

Summary and conclusion

An experimental investigation of skimming flow down astepped chute was conducted in a large-sized facility (α =15.9°, W = 1 m) with two step heights (h = 0.1 and 0.05 m).The study focused on the air–water flow properties betweenstep edges. Air–water velocity and turbulence measurementsdemonstrated a well-defined mixing layer developing down-stream of each step edge. In the developing shear layer, thevelocity profiles had the same shape as classical monophaseflow results (e.g., Tollmien and Goertler profiles), but therate of expansion of the mixing layer was greater, especiallyimmediately downstream of the step edge. Maximum turbu-lent shear estimates in the shear layer yielded equivalentfriction factors that were consistent with Darcy friction fac-tors deduced from the measured friction slope. Overall, thefindings confirmed the analogy between skimming flows andturbulent flows past cavities.

A comparative analysis of air–water flow properties forFroude similar flow conditions showed good agreement be-tween the two step heights in terms of dimensionless distri-butions of air content, velocity, and turbulence intensity, aswell as in terms of mean air content Cmean, dimensionlessflow velocity Uw/Vc, and air–water flow velocity V90/Vc. Sig-nificant differences were observed in terms of dimensionlessbubble count rates, turbulence levels, and bubble chordsizes. The results highlighted some limitations of the Froudesimilitude for studies of skimming flows.

Acknowledgments

The writers acknowledge the assistance of Mr. GrahamIllidge and Dr. L. Toombes. The first writer acknowledgesthe financial support of the National Council for Science andTechnology of Mexico (CONACYT). The second writer ac-knowledges the help of Mr. John LaBoon.

References

Boes, R.M. 2000. Zweiphasenstroömung und Energieumsetzungauf Grosskaskaden. Ph.D. thesis, VAW-ETH, Zürich, Switzer-land.

Chamani, M.R., and Rajaratnam, N. 1999. Characteristics of skim-ming flow over stepped spillways. ASCE Journal of HydraulicEngineering, 125(4): 361–368.

Chanson, H. 1995. Hydraulic design of stepped cascades, channels,weirs and spillways. Pergamon, Oxford, U.K.

Chanson, H. 1997. Air bubble entrainment in free-surface turbulentshear flows. Academic Press, London, U.K.

Chanson, H. 1999. The hydraulics of open channel flows: an intro-duction. Edward Arnold, London, U.K.

Chanson, H. 2000. Forum article. Hydraulics of stepped spillways:current status. ASCE Journal of Hydraulic Engineering, 126(9):636–637.

Chanson, H. 2001. The hydraulics of stepped chutes and spillways.A.A. Balkema, Lisse, The Netherlands.

Chanson, H., and Brattberg, T. 1998. Air entrainment by two-dimensional plunging jets: the impingement region and the very-near flow field. Proceedings of the 1998 American Society of Me-chanical Engineers Fluids Engineering Conference, FEDSM’98,Washington, D.C., 21–25 June 1998. American Society of CivilEngineers, New York, N.Y. Paper FEDSM98-4806. CD-ROM.

Chanson, H., and Toombes, L. 1997. Energy dissipation in steppedwaterway. Proceedings of the 27th IAHR Congress, San Fran-cisco, Calif., 10–15 August 1997. Edited by F.M. Holly, Jr., andA. Alsaffar. Vol. D. pp. 595–600.

Chanson, H., and Toombes, L. 2002a. Experimental investigationsof air entrainment in transition and skimming flows down astepped chute. Canadian Journal of Civil Engineering, 29(1):145–156.

Chanson, H., and Toombes, L. 2002b. Air–water flows downstepped chutes: turbulence and flow structure observations. In-ternational Journal of Multiphase Flow, 27(11): 1737–1761.

Chanson, H., and Toombes, L. 2002c. Energy dissipation and airentrainment in a stepped storm waterway: an experimentalstudy. ASCE Journal of Irrigation and Drainage Engineering,128(5): 305–315.

Chanson, H., Yasuda, Y., and Ohtsu, I. 2000. Flow resistance inskimming flow: a critical review. In Proceedings of the Interna-tional Workshop on Hydraulics of Stepped Spillways, Zürich,Switzerland, 22–24 March 2000. Edited by H.-E. Minor andW.H. Hager. A.A. Balkema Publisher, Rotterdam, The Nether-lands, pp. 95–102.

Chanson, H., Yasuda, Y., and Ohtsu, I. 2002. Flow resistance inskimming flows and its modelling. Canadian Journal of CivilEngineering, 29(6): 809–819.

Davies, P.O.A.L. 1966. Turbulence structure in free shear layers.AIAA Journal, 4(11): 1971–1978.

Djenidi, L, Anselmet, F., and Antonia, R.A. 1994. LDA measure-ments in a turbulent boundary layer over a D-type rough wall.Experiments in Fluids, 16: 323–329.

Elavarasan, R., Pearson, B.R., and Antonia, R.E. 1995. Visualiza-tion of near wall region in a turbulent boundary layer over trans-verse square cavities with different spacing. Proceedings of the12th Australasian Fluid Mechanics Conference, Sydney, Austra-lia, Vol. 1, pp. 485–488.

Haugen, H.L., and Dhanak, A.M. 1966. Momentum transfer in tur-bulent separated flow past a rectangular cavity. Journal of Ap-plied Mechanics, Transactions American Society of MechanicalEngineers, Sept.: 641–464.

Henderson, F.M. 1966. Open channel flow. MacMillan Company,New York, N.Y.

Kistler, A.L., and Tan, F.C. 1967. Some properties of turbulent sep-arated flows. Physics of Fluids, Part II, 10(9): S165-S173.

Knight, D.W., and Macdonald, J.A. 1979. Hydraulic resistance ofartificial strip roughness. ASCE Journal of Hydraulic Division,105(HY6): 675–690.

Liepmann, H.W., and Laufer, J. 1947. Investigation of free turbu-lent mixing. NACA Technical Note, No. 1257, August. NationalAdvisory Committee for Aeronautics, Washington, D.C.

Manso, P.A., and Schleiss, A.J. 2002. Stability of concrete macro-roughness linings for overflow protection of earth embankmentdams. Canadian Journal of Civil Engineering, 29(5): 762–776.

Matos, J. 2000. Hydraulic design of stepped spillways over RCCdams. In Proceedings of the International Workshop on Hy-draulics of Stepped Spillways, Zürich, Switzerland, 22–24March 2000. Edited by H.-E. Minor and W.H. Hager. A.A.Balkema Publisher, Rotterdam, The Netherlands, pp. 187–194.

Matos, J., Yasuda, Y., and Chanson, H. 2001. Interaction betweenfree-surface aeration and cavity recirculation in skimming flowsdown stepped chutes. Proceedings of the 29th IAHR Congress,Beijing, China, Theme D, Vol. 2. Edited by G. Li. TsinghuaUniversity Press, Beijing, pp. 611–617.

Minor, H.E., and Hager, W.H. (Editors). 2000. In Proceedings ofthe International Workshop on Hydraulics of Stepped Spillways,

© 2004 NRC Canada

Gonzalez and Chanson 43

I:\cjce\cjce3101\L03-066.vpJanuary 7, 2004 3:14:10 PM

Color profile: DisabledComposite Default screen

Zürich, Switzerland, 22–24 March 2000. A.A. Balkema Pub-lisher, Rotterdam, The Netherlands.

Ohtsu, I., and Yasuda, Y. (Editors). 1998. Hydraulic characteristicsof stepped channel flows. Workshop on Flow Characteristicsaround Hydraulic Structures and River Environment, 13 Novem-ber 1998. University Research Center, Nihon University, Tokyo,Japan.

Ohtsu, I., Yasuda, Y., and Takahashi, M. 2000. Characteristics ofskimming flow over stepped spillways. Discussion. ASCE Jour-nal of Hydraulic Engineering, 126(11): 869–871.

Rajaratnam, N. 1976. Turbulent jets. Development in water sci-ence, 5. Elsevier Scientific, New York, N.Y.

Rajaratnam, N. 1990. Skimming flow in stepped spillways. ASCEJournal of Hydraulic Engineering, 116(4): 587–591. Discussion:118(1): 111–114.

Schlichting, H. 1979. Boundary layer theory. 7th ed. McGraw-Hill,New York, N.Y.

Sumer, B.M., Cokgor, S., and Fredsoe, J. 2001. Suction removal ofsediment from between armor blocks. ASCE Journal of Hydrau-lic Engineering, 127(4): 293–306.

Sunyach, M., and Mathieu, J. 1969. Zone de mélange d’un jet plan.Fluctuations induites dans le cone à potentiel-intermittence. In-ternational Journal of Heat and Mass Transfer, 12: 1679–1697.

Tantirige, S.C., Iribarne, A.P., Ojhas, M., and Trass, O. 1994. Theturbulent boundary layer over single V-shaped cavities. Interna-tional Journal of Heat Mass Transfer, 37: 2261–2271.

Toombes, L. 2002. Experimental study of air–water flow propertieson low-gradient stepped cascades. Ph.D. thesis, Department ofCivil Engineering, The University of Queensland, Brisbane,Australia.

Toombes, L., and Chanson, H. 2000. Air–water flow and gas trans-fer at aeration cascades: a comparative study of smooth andstepped chutes. In Proceedings of the International Workshop onHydraulics of Stepped Spillways, Zürich, Switzerland, 22–24March 2000. Edited by H.-E. Minor and W.H. Hager. A.A.Balkema Publisher, Rotterdam, The Netherlands, pp. 77–84.

Townes, H.W., and Sabersky, R.H. 1966. Experiments on the flowover a rough surface. International Journal of Heat and MassTransfer, 9: 729–738.

Tozzi, M., Taniguchi, E., and Ota, J. 1998. Air concentration inflows over stepped spillways. Proceedings of the 1998 AmericanSociety of Mechanical Engineers Fluids Engineering Confer-ence, FEDSM’98, Washington, D.C., 21–25 June 1998. Ameri-can Society of Civil Engineers, New York, N.Y. PaperFEDSM98-5053, CD-ROM.

Wood, I.R. 1991. Air entrainment in free-surface flows. IAHR hy-draulic structures design manual No. 4. Hydraulic design con-siderations. A.A. Balkema Publisher, Rotterdam, TheNetherlands.

Wygnanski, I., and Fiedler, H.E. 1970. The two-dimensional mix-ing region. Journal of Fluid Mechanics, (Part 2), 41: 327–361.

List of symbols

a empirical constantC air concentration defined as the volume of air per unit

volume, also called void fraction

Cmean depth averaged air concentration defined as(1 – Y90) Cmean = d

Do dimensionless diffusivity termde equivalent depth of clear water (m) defined as

de = ( )10

90

−∫ C yY

d

dj jet flow thickness (m)dab air bubble size (m)dc critical flow depth (m)F bubble count rate, i.e., the number of bubbles detected

by the probe sensor per second (Hz)Fr Froude number defined as Fr = V gd/ e

f Darcy friction factorfe Darcy friction factor for air–water flowg gravity constant (m/s2) or acceleration of gravityH total head (m)h vertical height of steps (m)K inverse of the spreading rate of a turbulent shear layerK′ integration constantks′ surface (skin) roughness height (m)L chute length (m)

Lcav step cavity length (m)l horizontal length of steps (m) (measured perpendicular

to the vertical direction)lm Prandtl mixing length (m)q discharge per unit width (m2/s)

Sf friction slope, Sf = –∂ ∂H x/Tu turbulence intensity, Tu = u′/VUw equivalent clear water flow velocity (m/s)

u′ root mean square of longitudinal component of turbulentvelocity (m/s)

V velocity (m/s)Vc critical flow velocity (m/s)Vo free-stream velocity (m/s)

V90 characteristic velocity (m/s) where C = 0.90v′ root mean square of lateral component of turbulent ve-

locity (m/s)W channel width (m)xl longitudinal distance measured in the flow direction (m)xs streamwise distance measured from the step edge (m)

Y90 characteristic depth where the air concentration is90% (m)

y distance from the pseudo-bottom (formed by the stepedges) measured perpendicular to the flow direction (m)

y50 distance normal to the invert where V = Vo/2 (m)α channel slope∆x probe tip separation in the streamwise direction (m)Ø dimensionless term, Ø = y/(ax)µ dynamic viscosity (N·s/m2)ν t turbulent kinematic viscosity (m2/s)ρ density (kg/m3)σ surface tension between air and water (N/m)τ shear stress (Pa)

τmax maximum shear stress (Pa) in a shear layerτo average bottom shear stress (Pa)

© 2004 NRC Canada

44 Can. J. Civ. Eng. Vol. 31, 2004

I:\cjce\cjce3101\L03-066.vpJanuary 7, 2004 3:14:10 PM

Color profile: DisabledComposite Default screen