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Turbulence characteristics in skimming flows onstepped spillways

G. Carosi and H. Chanson

Abstract: The stepped spillway design is characterized by an increase in the rate of energy dissipation on the chute associ-ated with a reduction of the size of the downstream energy dissipation system. This study presents a thorough investigationof the air–water flow properties in skimming flows with a focus on the turbulent characteristics. New measurements wereconducted in a large-size facility (q = 228; step height, h = 0.1 m) with several phase-detection intrusive probes. Correla-tion analyses were applied to estimate the integral turbulent length and time scales. The skimming flow properties pre-sented some basic characteristics that were qualitatively and quantitatively in agreement with previous air–water flowmeasurements in skimming flows. Present measurements showed some relatively good correlation between turbulence in-tensities Tu and turbulent length and time scales. These measurements also illustrated large turbulence levels and large tur-bulent time and length scales in the intermediate region between the spray and bubbly flow regions.

Key words: turbulence, stepped spillways, skimming flows, turbulent energy dissipation.

Resume : Les evacuateurs de crues en marches d’escalier sont caracterises par une taux important de dissipation d’energiecinetique sur le coursier, et donc, une reduction de la taille du bassin de dissipation aval. Dans cette etude, on presente desseries de mesures detaillees dans l’ecoulement diphasique eau–air, avec de nouvelles mesures des proprietes turbulentes.Ce travail a ete realise dans une modele physique de grande taille (q = 228, h = 0,1 m) avec plusieurs sondes de mesuresintrusives. L’application d’analyses correlatives fournit des mesures de longueur et temps integrale turbulent. Les resultatsde l’etude sont en accord qualitatifs et quantitatifs avec des etudes precedentes en ecoulements extremement turbulents(« skimming flows »). On montre une correlation relativement bonne entre les intensites turbulentes Tu et les echellesintegrales turbulentes de longueur et de temps. Les resultats suggerent un mecanisme de dissipation turbulente dans laregion intermedaire entre la region d’ecoulement a bulles et la region d’ecoulement a gouttes.

Mots-cles : turbulence, coursier en marches d’escalier, ecoulement extremement turbulent, dissipation d’energie.

Introduction

Stepped spillways have been used for many centuries(Chanson 1995b, 2000, 2001a). The stepped design in-creases the rate of energy dissipation on the chute and re-duces the size of the downstream energy dissipation system(Fig. 1). Figure 1 shows two recent reinforced cement con-crete (RCC) dam stepped spillways with small stilling ba-sins. For the last 20 years, research in the hydraulics ofstepped spillways has been active (Chanson 1995a, 2001a).On a stepped spillway, the waters flow as a succession offree-falling nappes (nappe flow regime) at small discharges(Chamani and Rajaratnam 1994; Chanson 1994a; Toombes2002; El-Kamash et al. 2005). For a range of intermediateflow rates, a transition flow regime is observed (Ohtsu andYasuda 1997; Chanson 2001b; Chanson and Toombes2004). Modern stepped spillways are typically designed for

large discharge capacities corresponding to a skimmingflow regime (Rajaratnam 1990; Chanson 1994b; Chamaniand Rajaratnam 1999). In a skimming flow, the flow isnonaerated at the upstream end of the chute. Free-surfaceaeration occurs when the turbulent shear next to the free sur-face becomes larger than the bubble resistance offered bysurface tension and buoyancy. Downstream of the inceptionpoint of free-surface aeration, some strong air–water mixingtakes place. Large amounts of air are entrained and verystrong interactions between main stream turbulence, step-cavity recirculation zones, and free surface associated withstrong energy dissipation and flow resistance are observed(Chanson and Toombes 2002a; Kokpinar 2005).

The flow resistance is primarily a form drag in skimmingflows (Rajaratnam 1990; Chanson et al. 2002). At each step,the cavity flow is driven by the developing shear layer andthe transfer of momentum across it (Gonzalez and Chanson2004). The energy dissipation mechanisms include cavity re-circulation, momentum exchange with the free stream, andinteractions between free-surface and mainstream turbulence(Fig. 2). The interactions between mixing layer and horizon-tal step face, and the skin friction at the step faces, may con-tribute to further energy dissipation, in particular onmoderate slopes. At each step edge, highly coherent small-scale vortices are formed abruptly at the step corner becauseof the large gradient of vorticity at the corner (Fig. 2). Theinitial region of the mixing layer is dominated by a train of

Received 26 September 2006. Revision accepted 20 February2008. Published on the NRC Research Press Web site atcjce.nrc.ca on 12 August 2008.

G. Carosi and H. Chanson.1 Division of Civil Engineering, TheUniversity of Queensland, Brisbane QLD 4072, Australia.

Written discussion of this article is welcomed and will bereceived by the Editor until 31 January 2009.

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

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Fig. 1. Photographs of modern stepped spillways: (a) Pedrogao dam stepped spillway (Portugal) on 4 September 2006 — reinforced cementconcrete (RCC) gravity dam structure completed in March 2006, uncontrolled stepped spillway (h = 0.6 m, 1V:0.75H); (b) Riou damstepped spillway (France) on 11 February 2004 — RCC gravity dam structure completed in 1990, uncontrolled stepped spillway (h =0.43 m, 1V:0.6H).

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sequential small-scale vortices that eventually pair to formlarge-scale vortical structures that are advected downstream.The distance from the step edge to the impingement of theshear layer onto the step face becomes an important lengthbecause some feedback may occur almost instantaneouslyfrom the impingement to the singularity region of the shearlayer in the vicinity of the step edge (Lin and Rockwell2001). Experimental studies of turbulent flows past two-dimensional cavities showed that cavity resonance is pri-marily a function of the ratio of boundary layer thicknessto cavity length. That is, Y90 sinq/h for a stepped chute,where Y90 is the characteristic air–water depth for a voidfraction of 0.90, h is the vertical step height, and q is theangle between the pseudo-bottom formed by the step edgesand the horizontal. Ohtsu et al. (2004) showed that theflow resistance appeared to be maximum for a slope q ofaround 188 to 228. Gonzalez and Chanson (2006) hypothe-sized that some maximum values in flow resistance mustbe related to some flow instability. The three-dimensionalnature of recirculating vortices is believed to play a roleto further the rate of energy dissipation, and Gonzalez andChanson (2005) demonstrated quantitatively the means toenhance the flow resistance with turbulence manipulation.

It is the purpose of this study to investigate thoroughlythe air–water flow properties in skimming flows, with a fo-cus on the turbulent characteristics. New measurements wereconducted in a large-size facility (q = 228, h = 0.1 m) withseveral phase-detection intrusive probes. Detailed air–waterflow properties were recorded systematically for severalflow rates. The results included the distributions of turbu-lence intensity and of integral length scales. They showedthat the rate of energy dissipation on stepped spillways is as-sociated with high turbulence levels and large-scale vorticalstructures.

Experimental setup

New experiments were performed in the Gordon McKayHydraulics Laboratory at the University of Queensland(Table 1). The experimental channel was previously usedby Chanson and Toombes (2002a) and Gonzalez (2005).Waters were supplied from a large feeding basin (1.5 mdeep, surface area 6.8 m � 4.8 m) leading to a sidewall con-vergent with a 4.8:1 contraction ratio. The pump, deliveringthe flow rate, was controlled with an adjustable frequencyAC motor drive that enabled an accurate discharge adjust-ment in the closed-circuit system.

The test section consisted of a broad-crested weir (1 mwide, 0.6 m long) followed by 10 identical steps (h =0.1 m, l = 0.25 m) made of marine plywood. The steppedchute was 1 m wide with perspex sidewalls followed by ahorizontal concrete-invert canal ending in a dissipation pit.A comparison between present experiments and past studiesis given in Table 1. Further details on the experimental fa-cility and data were reported in Carosi and Chanson (2006).

Instrumentation

Clear-water flow depths were measured with a pointgauge. The water discharge was measured from the up-stream head above the crest, and the head-discharge rela-tionship was checked with detailed velocity distributionmeasurements on the crest itself.

Air–water flow properties were measured with single-tipand double-tip conductivity probes (Fig. 3). Basic air–waterflow measurements were performed with single-tip conduc-tivity probes (Fig. 3a). The probe sensor consisted of asharpened rod (Ø = 0.35 mm) coated with a nonconductiveepoxy set into a stainless steel surgical needle acting as the

Fig. 2. Stepped cavity recirculation in skimming flows.

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Table 1. Experimental investigations of stepped chute flows on flat slopes (q < 308).

Reference q (8) Step geometry Flow conditions Instrumentation Remarks

Chanson and Toombes (2002a)W = 1 m

Series 1 16 Smooth horizontal steps(h = 0.1 m, l = 0.35 m)

Double-tip conductivity probe(Ø = 0.025 mm)

Experiments TC201

Series 2a 22 Smooth horizontal steps(h = 0.1 m, l = 0.25 m)

qw = 0.046–0.182 m2/s,Re = 1.8 – 7.3�105

Single-tip conductivity probe(Ø = 0.35 mm)

Experiments EV200

Series 2b 22 Smooth horizontal steps(h = 0.1 m, l = 0.25 m)

qw = 0.058 to 0.182 m2/s,Re = 2.3 – 7.3�105

Double-tip conductivity probe(Ø = 0.025 mm)

Experiments TC200

Chanson and Toombes (2002b)Single-tip conductivity probe

(Ø = 0.35 mm)W = 0.5 m

Series 1 3.4 Smooth horizontal steps(h = 0.143 m, l = 2.4 m)

L = 24 m, 10 steps

series 2 3.4 Smooth horizontal steps(h = 0.0715 m, l = 1.2 m).

L = 24 m, 18 steps

Toombes (2002)Smooth horizontal steps

(h = 0.143 m, l = 2.4 m)Stepped cascade 3.4 L = 24 m, 10 steps Single-tip conductivity probe

(Ø = 0.35 mm)W = 0.5 m

Single-step chute 3.4 L = 3.2 m, 1 step Double-tip conductivity probe(Ø = 0.025 mm)

W = 0.25 m

Yasuda and Chanson (2003)16 Smooth horizontal steps

(h = 0.05 m)qw = 0.0776 m2/s, Re = 3.1�105 Double-tip conductivity probe

(Ø = 0.025 mm)W = 0.5 m

Ohtsu et al. (2004)5.7, 8.5, 11.3,

19, 23, 30h = 0.006–0.05 m qw = 0.02–0.08 m2/s,

Re = 8�104 – 3.2�105Single-tip conductivity probe

(Ø = 0.1 mm)W = 0.4 m

Gonzalez (2005)Double-tip conductivity probe

(Ø = 0.025 mm)W = 1 m

16 Smooth horizontal steps(h = 0.05 m, l = 0.175 m)



Turbulence manipulationwith triangular vanes

Gonzalez et al. (2005)22 Horizontal steps (h = 0.1 m,

l = 0.25 m)qw = 0.01–0.219 m2/s

Re = 4�104– 9�105Double-tip conductivity probe

(Ø = 0.025 mm)Experiments CMH_05,

W = 1 m

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Table 1 (concluded).

Reference q (8) Step geometry Flow conditions Instrumentation Remarks

Configuration A 22 Rough step faces: 8 mm thickscreens on both vertical andhorizontal step faces

k = 8 mm

Configuration B 22 8 mm thick screens on eachvertical step face

k = 8 mm

Configuration C 22 8 mm thick screens on eachhorizontal step face

k = 8 mm

Configuration S(smooth steps)

22 Smooth horizontal steps

Murillo (2006)Smooth horizontal steps

(h = 0.15 m)Conical hot-film probe

(Dantec 55R42)W = 0.4 m

Model I 16 h = 0.15 m, l = 0.525 m qw = 0.06–0.13 m2/s,Re = 2.4�105– 5.5�105

Model II 11.3 h = 0.15 m, l = 0.75 qw = 0.06–0.14 m2/s,Re = 2.4�105– 5.8�105

Model III 5.7 h = 0.15 m, l = 1.5 m qw = 0.06–0.15 m2/s,Re = 2.4�105– 6.2�105

Thorwarth and Koengeter (2006)Flat horizontal steps and

pooled stepsDouble-tip conductivity probe

(Ø = 0.050 mm)W = 0.5 m, Sorpe dam

spillway model14.6 h = 0.10 m, l = 0.383 m qw = 0.14–0.20 m2/s,

Re = 5.5�105– 7.9�105

14.6 h = 0.05 m, l = 0.192 m qw = 0.025–0.039 m2/s,Re = 1�105– 1.55�105

Present study22 Smooth horizontal steps

(h = 0.1 m, l = 0.25 m)Experiments GH_06,

W = 1 mSeries 1 22 qw = 0.095–0.18 m2/s,

Re = 3.8�105– 7.1�105Single-tip conductivity probes

(Ø = 0.35 mm)Series 2 22 qw = 0.12–0.16 m2/s,

Re = 4.6�105– 6.4�105Double-tip conductivity probe

(Ø = 0.25 mm)

Note: k, screen roughness height; Re, flow Reynolds number defined in terms of hydraulic diameter; Ø, diameter.

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Fig. 3. Photographs of the conductivity probes: (a) Two single-tip conductivity probes side-by-side (�z = 21.7 mm and dc/h = 1.45), flowfrom left to right, shutter speed of 1/80 s; (b) double-tip conductivity probe (�x = 7 mm) in the upper spray region, looking downstream(dc/h = 1.33), shutter speed = 1/500 s.

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second electrode. Additional measurements were performedwith a double-tip conductivity probe (Fig. 3b). The sensorsconsisted of a platinum wire (Ø = 0.25 mm) insulated exceptfor its tip and set into a metal supporting tube. The longitu-dinal spacing between probe sensors was measured with amicroscope and this yielded �x = 7.0 mm. All the probeswere excited by an electronic system (Ref. UQ82.518) de-signed with a response time of less than 10 ms and calibratedwith a square-wave generator. The probe sensors werescanned at 20 kHz per sensor for 45 s.

Signal correlation analyses

The air–water interfacial velocities were deduced from abasic correlation analysis between the two sensors of thedouble-tip probe (Chanson 1997a, 2002; Crowe et al. 1998).The time-averaged velocity equals

½1� V ¼ �x

T

where T is the air–water interfacial travel time for which thecross-correlation function is maximum and �x is the longi-tudinal distance between probe sensors. Turbulence levelsmay be derived from the relative width of the cross-correla-tion function:

½2� Tu ¼ 0:851

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�0:52 � T0:5

2p

T

where �0.5 is the time scale for which the cross-correlationfunction is half of its maximum value, such asR12RðTþ�0:5Þ¼ 0:5R12ðTÞ, where R12 is the normalizedcross-correlation function; and T0.5 is the characteristic timefor which the normalized auto-correlation function equalsR11ðT0:5Þ¼ 0:5. Equation [2] was derived by Chanson andToombes (2002a). With the single-tip probe, all measure-ments were conducted on the channel centreline (z = 0) anda second identical probe was placed beside the first onewith the probe sensors at the same vertical and streamwisedistances y and x, respectively, but separated by a knowntransverse distance Dz (Fig. 3a). Their signals were analysedin terms of auto-correlation and cross-correlation functionsR11 and R12, respectively. Following Chanson (2006a,2007), the basic correlation analysis results included themaximum cross-correlation coefficient (R12)max and the cor-relation time scales T11 and T12 defined as

½3� T11 ¼� ¼ �ðR11 ¼ 0ÞZ

R11 d�

� ¼ 0

½4� T12 ¼� ¼ �ðR12 ¼ 0ÞZ

R12 d�

� ¼ � ½R12 ¼ ðR12Þmax �

where � is the time lag, T11 is an auto-correlation integraltime scale, and T12 is the cross-correlation time scale.

Identical experiments were repeated with different separa-tion distances. An integral turbulent length scale was calcu-lated as

½5� L12 ¼z ¼ z½ðR12Þmax ¼ 0�Z

ðR12Þmax dz

z ¼ 0

The integral turbulent length scale L12 represents a meas-ure of the transverse scale of the large vortical structures ad-vecting air bubbles and air–water packets. Thecorresponding turbulence integral time scale is

½6� T ¼Rz¼0

ðR12Þmax T12 dz

L12

Additionally, an advection integral length scale is definedas

½7� L11 ¼ V T11

where V is the advective velocity magnitude.

Data accuracyThe water discharge was measured with an accuracy of

about 2%. The translation of the conductivity probes in thedirection normal to the channel invert was controlled withan error of less than 0.5 mm. The accuracy on the longitudi-nal probe position was estimated as ± 0.5 cm. The error onthe transverse position of the probe was less than 0.1 mm.

With the conductivity probes, the error on the void frac-tion measurements was estimated as �C=C ¼ 4% for 0.05 <C < 0.95, �C=C � 0:002=ð1� CÞ for C > 0.95, and �C=C �0:005=C for C < 0.05. The minimum detectable bubblechord time was about 0.05 ms for a data acquisition fre-quency of 20 kHz per channel.

The scan frequency determines the resolution of the intru-sive phase-detection probe, in particular the accuracy ofchord size measurement, minimum detectable air–waterchord length, and the accuracy of the interfacial velocity.Herein, the scan frequency was 20 kHz per sensor and thestreamwise distance between probe sensor was �x =7.0 mm. With the double-tip conductivity probe, the analysisof the velocity field and chord length distributions impliedno slip between the air and water phases. The error on themean air–water velocity measurements was estimated as�V=V ¼ 5% for 0.05 < C < 0.95, �V=V ¼ 10% for 0.01 <C < 0.05, and 0.95 < C < 0.99. The minimum detectablebubble chord length was about 0.15 mm in a 3 m/s flowbased upon a data acquisition frequency of 20 kHz per chan-nel.

The effect of the probe sensor on chord size data weretested for one flow rate (dc/h = 1.15). Present chord data ob-tained with a 0.35 mm probe sensor were compared withsome experimental results obtained by Gonzalez et al.(2005) for dc/h = 1.18 with a 0.025 mm probe sensor. Thepresent data showed consistently larger measured count ratesand a broader range of bubble and (or) droplet sizes with the0.025 mm probe sensor than with the 0.35 mm probe sensor.The chord sizes measured with the 0.35 mm probe sensorwere typically 18% to 50% larger (average: 28%) than thechord lengths measured with the 0.025 mm probe sensor.Chanson and Toombes (2002c) performed a similar compar-ison between two probe sensor sizes (0.025 and 0.35 mm) inskimming flows and obtained comparable results.


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Air–water flow patternsThe basic flow regimes were inspected in a series of vis-

ual observations with discharges ranging from 0.008 to0.180 m3/s. A nappe flow regime was observed for smallflow rates (dc/h < 0.5). For some intermediate discharges(0.5 < dc/h < 0.95), the flow had a chaotic behaviour that ischaracteristic of a transition flow regime. For larger flows(dc/h > 0.95), the waters skimmed above the pseudo-bottomformed by the step edges.

In transition and skimming flows, the location of the in-ception point of free-surface aeration was recorded with dis-charges per unit width above 0.034 m2/s. The data werecompared successfully with earlier studies (Chanson 1995a,2001a). For the present study, the results were best corre-lated eith the following equation with a normalized correla-tion coefficient of 0.95:

½8� LI

h cos �¼ 1:05þ 5:11

qwffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig sin �ðh cos�Þ3

p ;

0:45 <dc

h< 1:6

where qw is the water discharge per unit width, and g is thegravity acceleration. Equation [8] is valid for transition andskimming flows on a 228 stepped chute only.

Overall, the present results were very close to the obser-vations of Chanson and Toombes (2002a), Gonzalez (2005),Gonzalez et al. (2005), and Thorwarth and Koengeter (2006)with comparable slopes and step heights (Table 1).

Air–water flow properties at step edges

Void fraction and bubble count rate distributionsExperimental observations demonstrated some substantial

free-surface aeration immediately downstream of the incep-tion point of free-surface aeration and the flow aeration wassustained downstream. At the step edges, the advective dif-fusion of air bubbles was described by an analytical modelof air bubble diffusion, as

½9� C ¼ 1� tanh2 K 0 � y=Y90

2Do

þ ½ðy=Y90Þ � ð1� 3Þ�3Do

3� �

where y is distance measured normal to the pseudo-invertand Y90 is the characteristic distance for C = 90%. The rela-tionship between an integration constant, K’, and a functionof the mean void fraction, Do, (Chanson and Toombes2002a) is expressed as

½10� K 0 ¼ 0:32745015þ 1

2Do

� 8

81Do

½11� Cmean ¼ 0:7622½1:0434� expð�3:614DoÞ�

Equation [9] is compared with experimental data inFig. 4. Figure 4 presents an example of dimensionless distri-butions of void fraction and bubble count rate Fdc/Vc asfunctions of y/dc at several step edges for the same flowrate, where dc is the critical flow depth and Vc is the criticalflow velocity. For that discharge, the flow aeration was nilat step edge 6, immediately upstream of the inception point.

Between step edges 6 and 7, some strong self-aeration tookplace, and the amount of entrained air and the mean air con-tent were about constant between the step edges 7 and 10,with Cmean = 0.36 at the last step edge 10.

Figure 4 shows some typical dimensionless distributionsof bubble count rate. The results consistently presented acharacteristic shape with a maximum value observed forvoid fractions between 0.36 and 0.60. A similar pattern wasobserved in smooth chute and stepped spillway flows(Chanson 1997b; Chanson and Toombes 2002a; Gonzalezand Chanson 2004; Kokpinar 2005).

Air–water velocity and turbulence level distributionsTypical distributions of air–water velocity and turbulent

intensity are presented in Fig. 5 for one flow rate. At eachstep edge, the velocity distributions compared favourablywith a power-law function for y/Y90 £ 1 and with an uniformprofile for y/Y90 > 1:

½12� V

V90

¼ y

Y90

� �1=10

; 0 � y

Y90� 1

½13� V

V90

¼ 1; 1 � y

Y90� 2:5

where V90 is the characteristic air–water velocity at y = Y90.Several studies yielded eq. [12] (Matos 2000; Boes 2000;Chanson and Toombes 2002a; Gonzalez and Chanson2004) but a few documented the velocity distribution in theupper spray region. Equation [12] and [13] are comparedwith experimental data in Fig. 5. In the present study, thevelocity power law exponent was 1/10 on average, althoughit varied between adjacent step edges. Such fluctuationswere believed to be caused by some complicated interfer-ence between adjacent shear layers and cavity flows(Fig. 2).

The distributions of turbulence intensity Tu showed highlevels of turbulence in the skimming flows: 0.3 £ Tu £ 2(Fig. 5). The results were comparable with earlier studies inaerated and nonaerated skimming flows (Chanson andToombes 2002a; Amador et al. 2004; Gonzalez and Chan-son 2004). The turbulence intensity maxima were typicallyobserved for 0.3 £ y/dc £0.4 that corresponded typically to0.35 £ C £ 0.6. In the intermediate region defined as 0.3 £C £ 0.7, the air–water flow structure is extremely compli-cated; it is dominated by interactions between particles andturbulent shear. It is hypothesized that the high turbulencelevels in this intermediate region are caused by the continu-ous deformations and modification of the air–water interfa-cial structure.

Probability distribution functions of air bubble andwater droplet chords

The probability distribution functions of chord sizes wereanalysed in terms of bubble chords in the bubbly flow (C <0.3) and in terms of droplet chord lengths in the spray re-gion (C > 0.7). Typical results are presented in Fig. 6. Foreach graph, the caption and legend provide the local air–water flow properties (C, F) and probe details. The histo-gram columns represent the probability of chord size in a0.5 mm chord interval. For example, the probability of bub-

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Fig. 4. Dimensionless distributions of void fraction C and bubble count rate Fdc/Vc for dc/h = 1.15 (single-tip probe, Ø = 0.35 mm) —comparison with eq. [9] (step edge 10).

Fig. 5. Dimensionless distributions of air–water velocity V/Vc and turbulence intensity Tu for dc/h = 1.33 — comparison with eqs. [12] and[13] (step edge 10).


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ble chords between 1 and 1.5 mm is represented by the col-umn labelled 1 mm. Chord sizes larger than 15 mm are re-grouped in the last column (>15).

In the bubbly flow region (C < 0.3), the probability dis-tribution functions showed a broad spectrum of bubblechords at each location. The range of bubble chord ex-tended from less than 0.3 mm to more than 15 mm(Fig. 6a). The bubble chord size distributions were skewedwith a preponderance of small bubbles relative to themean. In Fig. 6a, the mode of the probability distributionfunction was observed for chords between 0.5 and 1 mm.The probability distribution functions of bubble chord

tended to follow a log-normal distribution at all locationsand for all discharges. The result was consistent with someearlier studies (Chanson and Toombes 2002a; Gonzalez2005).

In the spray region, the probability distribution functionsof drop sizes also showed a wide range of droplet chords ateach location. While the droplet chord size distributionswere skewed with a preponderance of small droplets, theprobability density function was flatter than that of the bub-ble chords (Fig. 6b). In the upper spray region (C > 0.95),the probability distribution functions were flat and did notfollow a log-normal distribution.

Fig. 6. Probability distribution functions (PDF) of chord sizes in skimming flows for flow conditions with dc/h = 1.45, step 10, double-tipprobe (Ø = 0.25 mm, �x = 7.0 mm): (a) bubble chord size data (C < 0.3); (b) droplet chord size data (C > 0.7).

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Integral turbulent time and length scales inair–water skimming flows

Some experiments were conducted with two identical sen-sors, separated by a transverse distance, and they were re-peated for several separation distances (Table 2). Typicaldistributions of auto- and cross-correlation time scales areshown in Fig. 7 for three separation distances. Note that thecorrelation time scales are presented in seconds. The voidfraction data are also reported for completeness in Fig. 7.

For �z < 20 mm, the experimental results showed similardistributions of auto- and cross-correlation time scales in thebulk of the flow (C < 0.9). The auto-correlation time scalesT11 exhibited, however, a different trend in the upper sprayregion. This is seen in Fig. 7 for y/dc > 0.8. It is suggestedthat the pattern may indicate a change in the spray structurewith the upper spray region. For C > 0.95, the spray con-sisted primarily of ejected droplets that did not interact withthe rest of the flow.

For all flow conditions and at each step edge, the cross-correlation time-scale distributions presented a maximum inthe intermediate region (0.3 < C < 0.7). Observed maximaof correlation time scales are summarized in the sixth col-umn of Table 2.

For some experiments repeated with several transverseseparation distances, the turbulent length and time scales,L12 and T, respectively, were calculated. Typical results interms of dimensionless turbulent length scale L12/h and inte-gral turbulent time scale T(g/h)0.5 are presented in Fig. 8.

The measured void fraction data are also shown for com-pleteness. In bubbly flows, the turbulent length scale L12must be closely linked with the characteristic sizes of thelarge-size eddies, entrapping air bubbles, as shown by high-speed photographs (Hoyt and Sellin 1989; Chanson 1997a).Hence, the turbulent length scale L12 characterizes the trans-verse size of the vortical structures advecting the air bubblesand air–water packets. In the present study, the integralscales were related to the step height h: i.e., L12/h & 0.02to 0.2 (Fig. 8). The result was irrespective of the dimension-less flow rate dc/h within the range of the experiments. Theassociated turbulence time scale T is a measure of the inte-gral time scale of the large eddies. Present results yieldedbasically 0:004 � T

ffiffiffiffiffiffiffig=h

p� 0:04.

Both the integral turbulent length and time scales weremaximum for about C = 0.5 to 0.7 (Fig. 8). The finding em-phasised the existence of larger turbulent structures in theintermediate zone (0.3 < C < 0.7), and it is hypothesizedthat these large vortices may have a preponderant role interms of turbulent dissipation.

Discussion: relationship between integral scales andturbulent intensity

The high-velocity open-channel flows on the steppedchute were highly turbulent. The present results demon-strated that the high levels of turbulence were associated di-rectly with some large-scale turbulence (Fig. 5, 7, 8). Inparticular, the intermediate region (0.3 < C < 0.7) between

Table 2. Experimental measurements in skimming flows with identical probe sensors (present study).

qw (m2/s) dc=h Instrumentation �z (mm) Step edge T12max (s) Comments0.116 1.15 2 single-tip probes 0 7 0.0041 (+) Auto-correlation

(Ø = 0.35 mm) 8.45 7 0.0027 Cross-correlation0 8 0.0035 (+) Auto-correlation8.45 8 0.0034 Cross-correlation0 9 0.0033 (+) Auto-correlation8.45 9 0.0035 Cross-correlation0 10 0.0053 (+) Auto-correlation3.60 10 0.0050 Cross-correlation6.30 10 0.00378.45 10 0.0039

10.75 10 0.003813.70 10 0.004116.70 10 0.003921.70 10 0.002729.50 10 0.002340.30 10 0.0023

0.161 1.45 2 single-tip probes 0 8 0.0038 (+) Auto-correlation(Ø = 0.35 mm) 8.45 8 0.0042 Cross-correlation

0 9 0.0037 (+) Auto-correlation8.45 9 0.0039 Cross-correlation0 10 0.0055 (+) Auto-correlation3.60 10 0.0048 Cross-correlation8.45 10 0.0041

13.70 10 0.004421.70 10 0.004040.30 10 0.002055.70 10 0.0019

Note: (+) is for C < 0.95.


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the bubbly and spray regions seemed to play a major role inthe development the large eddies and turbulent dissipation.Turbulence level maxima were observed consistently for0.4 < C < 0.5, while maximum integral turbulent scaleswere seen for 0.5 < C < 0.7.

The findings implied that some turbulent energy was dissi-pated in the form of large vortices in the bulk of the flow

while the stepped cavities contributed to intense turbulenceproduction. The dissipated energy contributed to the entrap-ment and advection of air bubbles within the main flow, aswell as to the formation of water droplets and their ejectionabove the free surface. The mechanisms were consistentwith the experimental results of Gonzalez and Chanson(2005), who observed that some increased rate of energy dis-

Fig. 7. Distributions of auto- and cross-correlation time scales in skimming flows (flow conditions: dc/h = 1.45, step edge 10) — compar-ison with the void fraction distribution.

Fig. 8. Dimensionless distribution of integral turbulent length scale L12/h and transverse integral time scale Tffiffiffiffiffiffiffig=h

pin skimming flows (step

edge 10) — comparison with void fraction measurements.

876 Can. J. Civ. Eng. Vol. 35, 2008

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sipation induced by passive turbulent manipulation waslinked with a greater spray production.

The present results showed that, in a cross section, theturbulent intensity Tu and the turbulent length and timescales were closely linked. The relationship betweenturbulence level and turbulent scales were best correlatedby

½14� Tu ¼ 0:372exp 8:73L12

h

� �

½15� Tu ¼ 0:316exp 44:68 T

ffiffiffig

h

r� �

Fig. 9. Dimensionless relationship between turbulence intensity and integral turbulent scales in skimming flows (step edge 10): (a) dimen-sionless relationship between turbulence intensity and integral turbulent length scale L12/h, comparison with eq. [14]; (b) dimensionless re-lationship between turbulence intensity and integral time scale T


p— comparison with eq. [15].


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Equations [14] and [15] are compared with the experi-mental data in Fig. 9.

Flow resistance and turbulent energydissipation

On the stepped chute, the skimming flows were character-ized by significant form losses. Downstream of the inceptionpoint of free-surface aeration, the flow was gradually variedflow in the present study. The average shear stress �o be-tween the skimming flow and the cavity recirculation wasdeduced from the measured friction slope Sf:

½16� fe ¼8�o

pwUw2¼

8g� y ¼ Y90Z

ð1� CÞdyy ¼ 0

�Sf

Uw2

where fe is the equivalent Darcy–Weisbach friction factor ofthe aerated flow, Uw is the water flow velocity: Uw = qw/d,and d is the equivalent clear-water depth (Chanson et al.2002). The friction slope ðSf ¼ �@H=@xÞ is the slope of thetotal head line, where H is the mean total head (Henderson1966; Chanson 1999). Free-surface aeration is always sub-stantial in prototype and laboratory skimming flows, and itseffects must be accounted for using eq. [16].

The flow-resistance results are presented in Table 3. Onaverage, the equivalent Darcy friction factor was fe & 0.14downstream of the inception point of free-surface aeration.Note that in the developing boundary layer region, Amadoret al. (2006) derived the flow resistance by applying an inte-gral momentum method to PIV measurements and their re-sults yielded f = 0.125 for dc/h = 2.1 and Re = 4.4 � 105.

The present results compared favourably with a com-prehensive re-analysis of flow resistance data in skim-ming flows (Chanson 2006b). The friction factor datafollowed closely a simplified analytical model of thepseudo-boundary shear stress, which may be expressed,in dimensionless form, as

½17� fd ¼2ffiffiffi�

p 1

K

� �

where fd is an equivalent Darcy friction factor estimate ofthe form drag and 1/K is the dimensionless expansion rateof the shear layer (Chanson et al. 2002). Equation [17] pre-dicts fd & 0.2 for K = 6, which is close to the observed fric-tion factors (Table 3).

Residual energyFor the present investigation, the rate of energy dissipa-

tion �H/Hmax and the dimensional residual energy Hres/dcwere calculated at the downstream end of the chute (step10), and the results are summarized in Table 3. Thepresent results showed a decreasing rate of energy dissipa-tion on the stepped chute with increasing discharge. Fordesign engineers, however, it is more relevant to considerthe dimensionless residual head Hres/dc. The residual headHres is the total head at the downstream end of the chute,and it equals

½18� Hres ¼ d cos�þ Uw

2g

Present results implied that the dimensionless residualhead was about 2.4 £ Hres/dc £ 3.3, increasing slightly withincreasing flow rate.

Note that the present results were obtained with a fullydeveloped aerated flow at the chute downstream end. Forlarger discharges, the flow may not be fully developed atthe downstream end of the chute, and that the residual en-ergy could be considerably larger (Chanson 2001b; Meireleset al. 2006).

ConclusionNew measurements were performed in skimming flows on

a large stepped spillway model. Air–water flow measure-ments were performed with some phase-detection intrusiveprobes in the air–water flow region downstream of the in-ception point of free-surface aeration. For some experi-ments, two probe sensors were separated by a knowntransverse, and an advanced signal processing techniquewith signal correlation analyses was applied.

The skimming flow properties presented some basic char-acteristics that were qualitatively and quantitatively inagreement with previous air–water flow measurements inskimming flows. These included the distributions of voidfraction, bubble count rate, and turbulent interfacial velocity.Correlation analyses yielded a further characterization of thelarge eddies advecting the bubbles. Basic results includedthe transverse integral turbulent length and time scales. Theturbulent length scales were closely linked with the charac-teristics of the large-size eddies and their interactions withentrained air bubbles. The transverse integral turbulentlength scales were closely related to the step height: i.e.,L12/h & 0.02 to 0.2, and the integral turbulent time scaleswere within 0:004 � T


p� 0:04:

Table 3. Flow resistance and dimensionless rate of energy dissipation and residual en-ergy at the downstream end of the stepped chute (step 10).

dc=h fe �H/Hmax* Hres=dc* Remarks1.00 0.159 0.58 2.96 Inception point: step edge 5.1.15 1.126 0.54 2.85 Inception point: step edge 6.1.33 0.087 0.52 2.48 Inception point: step edge 6–7.1.45 0.168 0.51 2.35 Inception point: step edge 7–8.1.57 0.160 0.40 3.32 Inception point: step edge 7–8.

*Measured at step edge 10.

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The measurements showed some relatively good correla-tion between turbulence intensities Tu, and turbulent lengthand time scales (Fig. 9). They highlighted further some max-imum turbulence intensities and maximum integral time andlength scales in the intermediate region between the sprayand bubbly flow regions (i.e., 0.3 < C < 0.7). The findingssuggested that turbulent dissipation may be a significantprocess in that intermediate region.

AcknowledgmentsThe writers acknowledge the assistance of Graham Illidge

and Clive Booth.

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List of symbols

C void fraction defined as the volume of air per unitvolume (also called air concentration)

Cmean depth averaged air concentration defined as:ð1� Y90ÞCmean ¼ d

DH hydraulic diameter (m)Do dimensionless diffusivity term

d equivalent clear-water depth (m) defined as:d¼

R Y90

0ð1�CÞdy

dc critical flow depth (m)F bubble count rate (Hz) defined as the number of

bubbles detected by the probe sensor per secondfd form drag equivalent Darcy friction factorfe Darcy friction factor for air–water flowg gravity constant (m/s2) or acceleration of gravityH total head (m)

Hmax upstream head (m) above spillway toeHres residual head (m)

h height of steps (m) (measured vertically)K inverse of the spreading rate of a turbulent shear

layerK’ integration constantk screen roughness height (m)

LI longitudinal distance (m) measured from the weircrest where the inception of free-surface aerationtakes place

L11 air–water advection integral length scale (m)L12 air–water integral turbulent length scale (m)

l horizontal length of steps (m)qw water discharge per unit width (m2/s)

R11 normalized auto-correlation function (referenceprobe)

R12 normalized cross-correlation function between twoprobe output signals

Re flow Reynolds number defined in terms of the hy-draulic diameter

(Rxy)max maximum cross-correlation between two probe out-put signals

Sf friction slope: Sf ¼ �@H=@xT time lag (s) for which R12 = (R12)maxT integral turbulent time scale (s)

T0.5 characteristic time lag � for which Rxx ¼ 0:5T11 an auto-correlation integral time scaleT12 cross-correlation time scaleTu turbulence intensity: Tu ¼ u0=V

Uw equivalent clear water flow velocity (m/s):Uw ¼ qw=d

u’ root mean square of longitudinal component of tur-bulent velocity (m/s)

V velocity (m/s)V90 characteristic velocity (m/s) where C = 0.90Vc critical flow velocity (m/s)W channel width (m)x longitudinal distance (m)Y distance (m) from the pseudo-bottom (formed by

the step edges) measured perpendicular to the flowdirection

Y90 characteristic depth (m) where the air concentrationis 90%

Z transverse distance (m) measured from the chutecentreline

�x probe tip separation (m) in the streamwise direction�z transverse separation distance (m) between sensorØ diameter (m)q angle between the pseudo-bottom formed by the

step edges and the horizontal�w water density (kg/m3)� time lag (s)

�0.5 characteristic time lag � for which Rxv = 0.5(Rxv)max�o average bottom shear stress (Pa)

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