Two-Phase Flow Characteristics of Stepped Spillways

Two-Phase Flow Characteristics of Stepped Spillways Robert M. Boes1 and Willi H. Hager, F.ASCE2 Abstract: An experimental study on a large model flume with fiber-optical instrumentation indicated that minimum Reynolds and Weber numbers of about 105 and 100, respectively, are required for viscosity and surface tension effects to become negligible compared to gravitational and inertial forces expressed by Froude similitude. Both the location of and the flow depth at the inception point of air entrainment can be expressed as functions of a so-called roughness Froude number containing the unit discharge, step height and chute angle. The depth-averaged air concentration is found to depend only on a normalized vertical distance from the spillway crest and the chute angle for chute slopes ranging from embankment to gravity dam spillways. Air concentration profiles can be expressed by an air bubble diffusion model. The pseudobottom air concentration allows the assessment of the cavitation risk of stepped chutes and is approximated by a regression function. Finally, a new velocity distribution function is presented consisting of a power law up to 80% of the characteristic nondimensional mixture depth, and a constant value above. DOI: 10.1061/共ASCE兲0733-9429共2003兲129:9共661兲 CE Database subject headings: Two phase flow; Flow characteristics; Spillways; Scale effect.

Introduction

Experimental Setup

Stepped spillways have regained popularity over the last 2 decades with the evolution of the roller compacted concrete 共RCC兲 dam construction technique. A stepped chute can be economically integrated into the downstream face of a RCC gravity dam. Another common application is the use of stepped overlays on the downstream face of embankment dams as emergency spillways to safely pass the probable maximum flood 共PMF兲 over the dam crest. Advantages of stepped spillways include ease of construction, reduction of cavitation risk potential, and reduction of the stilling basin dimensions at the downstream dam toe due to significant energy dissipation along the chute. Overviews of spillways with particular focus on stepped chutes are given by Chanson 共1994兲; Vischer and Hager 共1998兲; and Minor 共2000兲. Hydraulic model investigations were conducted at the Laboratory of Hydraulics, Hydrology and Glaciology 共VAW兲, ETH Zurich, Switzerland to study typical features and the energy dissipation characteristics of two-phase flow on stepped chutes 共Boes 2000; Schla¨pfer 2000兲. The aim was to investigate scale effects in modelling skimming flow down stepped spillways, the inception of air entrainment, and air concentration and velocity distributions.

All experiments were conducted in a prismatic rectangular channel of width 0.50 m and length 5.7 m 共Fig. 1兲 with bottom angles from the horizontal of ␾⫽30, 40, and 50°, corresponding to slopes (V:H) of 1:1.73, 1:1.19, and 1:0.84, respectively. The socalled pseudobottom formed by the outer step edges 共Rajaratnam 1990兲 was used as the reference level for flow depths 共Fig. 2兲. Three step heights s⫽23.1, 46.2, and 92.4 mm were investigated for the 30° cascade, steps of 31.1 and 93.3 mm for the 50° chute and of 26.1 mm for ␾⫽40°. When referring to a standard prototype value of s⫽0.60 m, the scaling factors would be 26.0, 13.0, and 6.5 for ␾⫽30°, 23.0 for ␾⫽40°, and 19.3 and 6.4 for ␾⫽50°. The scaling factor ␭ L is the reciprocal value of the model scale. Only the spillway face with constant bottom slope and step size was considered 共Fig. 1兲. The water discharge was provided with a jetbox 共Schwalt and Hager 1992兲, which transformed the pressurized approach flow to a free surface open channel flow of precalibrated approach flow depth h 0 and approach velocity u 0 共Fig. 1兲. The advantage over a conventional channel invert was the independent variation of both h 0 and the approach Froude number F0 ⫽u 0 /(gh 0 ) 1/2, where g⫽the acceleration of gravity. The inductive discharge measurement had an accuracy of ⫾2%. A two-tip fiber-optical probe was used to locally measure both air concentrations C and flow velocities u m of the air-water mixture 共subscript m兲 in selected cross sections at outer step edges. All measurements were conducted in the skimming flow regime where the water flows as a coherent stream parallel to the pseudobottom 共Figs. 1 and 2兲. Beneath the pseudobottom, recirculating vortices fill the cavities between the main flow and the steps. Based on preliminary observations, the present findings relate to axial parameters exempt from side wall effects 共Boes 2000兲. The fiber-optical instrumentation furnished by RBI Instrumentation et Mesure, Grenoble, France, described by Boes and Hager 共1998兲 and Boes 共2000兲 is based upon the different optical refraction indices of air and water. The errors of both C and u m values up to h 99⫽h(C⫽0.99), i.e., a flow depth with a local air concentration of 99%, were examined in preliminary experiments and did not exceed 5% 共Boes 2000兲. Both C and u m data were repro-

1 Project Manager, TIWAG Hydro Engineering GmbH, A-6020 Innsbruck, Austria. E-mail: [email protected]; formerly, Research Hydraulic Engineer, Laboratory of Hydraulics, Hydrology and Glaciology 共VAW兲, Swiss Federal Institute of Technology 共ETH兲, ETH-Zentrum, CH-8092 Zurich, Switzerland. 2 Professor, Head Hydraulic Engineering Division, Laboratory of Hydraulics, Hydrology and Glaciology 共VAW兲, Swiss Federal Institute of Technology 共ETH兲, ETH-Zentrum, CH-8092 Zurich, Switzerland. E-mail: [email protected] Note. Discussion open until February 1, 2004. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on March 15, 2001; approved on March 28, 2003 . This paper is part of the Journal of Hydraulic Engineering, Vol. 129, No. 9, September 1, 2003. ©ASCE, ISSN 0733-9429/2003/9-661– 670/$18.00.

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Fig. 1. Skimming flow over steps of heights s⫽31.1 mm, ␾⫽50°, F0 ⫽4.9, h 0 ⫽60 mm: view of jetbox, approach flow and inception point

duced with only small deviations, the scatter in C being smaller than in u m 共Fig. 3兲. A single sampling sequence was usually stopped after 30 s, or after detection of 4,000 individual air bubbles by the downstream probe tip, the number of detected bubbles per probe tip being similar. In the fully aerated zone with a bubble frequency on the order of 1 kHz the signal acquisition was accomplished within only a few seconds. Preliminary tests on the influence of the maximum number of bubbles detected and the sampling period, respectively, substantiated that even bubble detection numbers of 1,000, corresponding to sampling periods of roughly 1–2 s in the fully aerated flow region, were statistically representative in terms of void fraction and air–water velocity 共Boes 2000兲. These findings corroborated the results of Cubi-

Fig. 3. Comparison of two air concentration and velocity profiles 共䊉兲 1 and 共䊊兲 2 for ␾⫽30° and 共a兲 C(y) for F0 ⫽8.0, Q w ⫽49.5 L/s, s⫽46.2 mm at x⫺L i ⫽2.20 m and 共b兲 u m (y) for F0 ⫽5.7, Q w ⫽100 L/s, s⫽92.4 mm at x⫺L i ⫽1.58 m

zolles 共1996兲 who performed measurements in pressurized twophase flow using a fiber-optical RBI probe. Moreover, three measurements were always performed at each measuring location and the mean was taken as the relevant value. The deviation of the three values was always small, indicating a steady two-phase flow. The vertical translation of the probe was controlled by a fine adjustment travelling mechanism connected to a metric scale unit. The accuracy on the vertical probe position was ⫾0.25 mm. The probe and travelling mechanism were mounted on a trolley travelling parallel to the channel bottom. The accuracy in the longitudinal probe position was ⫾0.5 mm.

Scale Effects

Fig. 2. Longitudinal section of stepped spillway with origin of streamwise coordinate x at spillway crest: 共---兲 pseudobottom, flow region with equivalent clear water depth h w 共darkly shaded兲 and mixture depth h 90 共lightly shaded兲, 共–•–兲 energy head, and 共䊊兲 inception point

Unlike clear water open channel flow, highly turbulent air–water flow cannot be modeled without scale effects when using Froude similitude, because of the important role of viscosity and surface tension. For a true similarity of the aeration process between model and prototype, the Froude, the Reynolds and the Weber similarity laws would have to be fulfilled simultaneously. If Froude scaling is applied, air bubbles are proportionally too large in a scale model, resulting in a lower transport capacity and a higher detrainment rate as compared to prototype conditions 共Kobus 1984兲. Therefore, care must be taken when upscaling model results to the prototype scale. Kobus 共1984兲 proposed Reynolds numbers with flow depth as reference length of at least 105 to minimize viscous effects. Rutschmann 共1988兲 and Speerli 共1999兲, who investigated spillway aerators and bottom outlets, respectively, concluded that the Weber number, with the flow depth as the reference length, should be at least 110 for surface tension effects to be negligible.

662 / JOURNAL OF HYDRAULIC ENGINEERING © ASCE / SEPTEMBER 2003


A common way to estimate scale effects involves model families, i.e., geometrically similar models with different scaling factors ␭ L . Pinto 共1984兲 suggested scaling factors not exceeding 15 for the investigation of spillway aerators, whereas the limiting scaling factor for two-phase flow in bottom outlets is 20 according to Speerli 共1999兲. Eccher and Siegenthaler 共1982兲 used a model family with 18.75⭐␭ L ⭐30 to investigate spillway aerators. Prototype aeration rates of up to 1.5 times larger than those of their model with ␭ L ⫽25 resulted from extrapolation, whereas Vischer et al. 共1982兲 deduced ␭ L ⭐15 for correct aeration modeling. Regarding stepped spillway models, few attempts were directed to the effects of scale. Based upon the M’Bali stepped spillway in the Central African Republic, four scale models with 10⭐␭ L ⭐50 were built and tested by BaCaRa 共1991兲. Scale effects in terms of flow aeration and flow resistance were observed for ␭ L ⭓22. Wahrheit-Lensing 共1996兲 carried out an experimental model study with a bottom angle of ␾⫽51.3° for three different step heights with 15⭐␭ L ⭐33.7 when referring to standard prototype step height of s⫽0.60 m. Viscosity effects were observed for the discharge coefficient C d ⫽q w /(gh 3max)1/2, where q w is the specific water 共subscript w兲 discharge, and h max the maximum approach flow depth above the spillway crest 共Fig. 2兲. She suggested to increase the model dimensions to attain Reynolds numbers R ⫽q w /␯⭓7.5⫻104 , with ␯ as kinematic viscosity of water. Detailed studies on stepped spillway flow were conducted at Nihon University, Japan, for 2 mm⭐s⭐80 mm and 5.7°⭐␾⭐55° 共Ohtsu and Yasuda 1997; Yasuda and Ohtsu 1999兲. Within the Reynolds number range 1⫻104 ⭐R⭐7.0⫻104 , no scale effects in relation to C and u m were observed 共Yasuda, private communication, 1997兲. Chamani and Rajaratnam 共1999兲 conducted experiments with model step heights between 31.25 and 125 mm, corresponding to 4.8⭐␭ L ⭐19.2 with respect to s⫽0.6 m. Their results indicated that the aerated flow parameters depend on ␭ L , but no information on limiting values for the Reynolds and Weber numbers were provided. Pegram et al. 共1999兲 also studied different model scales with various step heights and concluded ‘‘that models with scales of 1:20 and larger can faithfully represent the prototype behavior of stepped spillways,’’ with results ‘‘converging rather quickly as the scale gets bigger than 1:15.’’ Their conclusions were drawn from the sequent depth of the hydraulic jump at the toe of the spillway, instead of air concentration measurements on the spillway. Because air concentrations or mixture velocities along the chute were not measured in most of the model studies performed so far, two-phase flow characteristics could not be compared for different model scales. One of the purposes of the present experimental study was to fill in this gap by systematically investigating the effects of scale for air concentration and mixture velocity results using model families. Conclusions on the minimum Reynolds and Weber numbers and the maximum scaling factor are drawn hereafter.

Fig. 4. Air concentration profiles C(y) for ␾⫽30°, F0 ⫽4.0, h 0 /s ⫽1.04 and ␭ L ⫽(⽧) 6.5 with R⫽3.70⫻105 , W⬇247; ␭ L ⫽(䊊)13.0 with R⫽1.28⫻105 , W⬇134; and ␭ L ⫽(䉲)26.0 with R ⫽4.63⫻104 , W⬇70 at step 共a兲共⽧,䉲兲 12 and 共䊊兲 13 and 共b兲 18 downstream from inception point

Air Concentration For a constant approach Froude number F0 , air concentration profiles C(y), with the transverse coordinate y originating at the pseudobottom 共Fig. 2兲, were taken in several cross sections at about equal step number downstream from the inception point for different step heights or scaling factors. Fig. 4 shows results for F0 ⫽4.0, ␾⫽30° at about 12 and 18 steps downstream from the inception point, whereas Fig. 5 refers to F0 ⫽3.0, ␾⫽50° at step

Fig. 5. Air concentration profiles C(y) for ␾⫽50°, F0 ⫽3.5, h 0 /s ⫽1.06 and ␭ L ⫽(⽧) 6.4 with R⫽3.53⫻105 , W⬇230; and ␭ L ⫽(〫)19.3 with R⫽6.67⫻104 , W⬇78 at step: 共a兲 16 and 共b兲共⽧兲 34 and 共〫兲 33 downstream from inception point



number 16, and approximately 33 relative to the inception point. The general shape of the concentration profiles is in agreement with those of Ruff and Frizell 共1994兲 and Matos and Frizell 共1997兲 obtained with electrical resistivity probes. All results have been scaled to prototype dimensions, with again a standard step height of s⫽0.60 m as the reference length. The Reynolds and Weber numbers are given in the figure captions, where R ¯ m /(␴/(␳L s )) 1/2 with h w ⫽equivalent ⫽uh w /␯⫽q w /␯ and W⫽u clear water depth that would result without the presence of the entrained air; ¯u m ⫽depth-averaged mixture velocity taken from the velocity profiles; ␴⫽surface tension between air and water; ␳ w ⫽density of water; and L s ⫽s/sin ␾ as distance between two step edges 共Fig. 2兲. ¯ ) is an anaThe term ‘‘equivalent’’ implies that h w ⫽h 90(1⫺C lytically computed value in two-phase flow with the mixture depth h 90⫽h(C⫽0.9) and the depth-averaged air concentration ¯ . The present Reynolds number definition is thus based on clear C water quantities. In contrast, the Weber number involves the spatially varied depth-averaged mixture velocity ¯u m . For the present observations, this variation was confined to ⫾10% and therefore neglected. With increasing ␭ L , there is typically a decrease in air concentration close to the pseudobottom for y/h 90⬍1/3– 1/2 共marked 共1兲 in Figs. 4 and 5兲. An increase in C can be observed, however, for y/h 90⬎1/3 to 1/2 共marked 共2兲兲. The C(y) profiles are virtually ‘‘stretched’’ with decreasing ␭ L , which can be explained by the increasing turbulent mass transfer perpendicular to the main flow direction. With decreasing ␭ L and thus increasing Reynolds and Weber numbers, the air bubbles entrained at the free surface are transported more pronouncedly towards the chute bottom. The reason for this behavior lies in increasing downward forces with increasing turbulence intensity, whereas the resisting forces from bubble uplift and surface tension are almost unaffected due to practically invariable absolute bubble size independent of model scale 共Kobus 1984兲. This is expressed by the entrainment limit given by Ervine and Falvey 共1987兲 and Hager 共1992兲 which states that for air to be entrained into a water flow, the turbulent velocity component ␯⬘ perpendicular to the water surface has to overcome forces both from surface tension and bubble uplift

冑

8␴ ␳ wd B

(1)

v ⬘ ⬎ v B cos ␾

(2)

v ⬘⬎

where d B and v B ⫽bubble diameter and bubble rise velocity, respectively. With increasing ␭ L the bubbles become relatively larger, resulting in an increased detrainment rate and a decreased transport rate of the air phase. The near-bottom air concentration is therefore larger for prototypes, which reduces the risk of cavitation on spillways as compared to model prediction. In the upper flow region, marked 共2兲 in Figs. 4 and 5, the water phase (1⫺C) increases with decreasing ␭ L and thus increasing turbulence intensity, because water droplets are ejected further and in larger quantities from the underlying air–water mixture. For the same reason the characteristic mixture depth h 90 , which is important for the design of chute training walls, also increases with decreasing ␭ L . The water is displaced by the air phase from the near-bottom region 共1兲 to higher elevations for a given prototype discharge q w . In general, analogous to the findings of Chanson 共1997兲 for open channel flow over smooth chutes, three distinct flow regions can be distinguished from the concentration profiles: 共1兲 mainly clear water flow comprising small air bubbles for C⬍0.3 to 0.4,

Fig. 6. Velocity profiles u m (y) for ␾⫽30° and: 共a兲 F0 ⫽4.0, h 0 /s ⫽1.04 and ␭ L ⫽(⽧)6.5 with R⫽3.70⫻105 , W⫽259; ␭ L ⫽(䊊)13.0 with R⫽1.28⫻105 , W⫽134; and ␭ L ⫽(䉲)26.0 with R ⫽4.63⫻104 , W⫽69 at step 共⽧兲 18, 共䊊兲 19 and 共䉲兲 15 from inception point and 共b兲 F0 ⫽6.0, h 0 /s⫽1.04 and ␭ L ⫽(䊊)13.0 with R ⫽1.92⫻105 , W⫽164; and ␭ L ⫽(䉲) 26.0 with R⫽6.93⫻104 , W⫽86 at step 共䊊兲 30 and 共䉲兲 29 downstream from inception point

共2兲 mainly air flow comprising water droplets for approximately C⬎0.6– 0.7, and 共3兲 an intermediate region with a true two-phase flow of about equal water and air contents in terms of discharge.

Mixture Velocity Similar to the air concentration profiles, Fig. 6 shows velocity profile results in prototype dimensions for different approach Froude numbers and ␾⫽30° at about step numbers 18 and 30 downstream from the inception point, whereas Fig. 7 refers to F0 ⫽3.5, ␾⫽50° at a downstream distance of 4 and approximately 33 steps. Fig. 6 and corresponding figures of Boes 共2000兲 suggest that the velocity increases with increasing ␭ L for a chute angle of 30°. For all experiments the velocity difference between the largest and the medium-sized model with ␭ L ⫽6.5 and 13.0, respectively, is significantly smaller than between ␭ L ⫽13.0 and 26.0. When the smallest model size is compared to the medium-sized model, the mean depth-averaged velocity increases by 3.6%. For 50°, there is also a slight velocity increase with increasing ␭ L , the average difference remaining small for all experimental data of Boes 共2000兲. Because the experimental runs over the small models with ␭ L ⫽26.0 and 19.3 had maximum Reynolds and Weber numbers of R⫽6.93⫻104 and W⫽86, respectively, compared to R⭓105 and W⭓100 for ␭ L ⭐13.0, the limiting values for modeling twophase cascade flow without significant scale effects are R⬇105 and W⬇100 from the velocity profiles. This corresponds to a maximum scaling factor of about 10–15 for stepped spillways of standard step height.



Fig. 8. Nondimensional vertical distance z i /s from the spillway crest to inception point as function of roughness Froude number F for * different spillway chute angles ␾

Fig. 7. Velocity profiles u m (y) for ␾⫽50°, F0 ⫽3.5, h 0 /s⫽1.06 and ␭ L ⫽(⽧) 6.4 with R⫽3.53⫻105 , W⬇219; and ␭ L ⫽(〫)19.3 with R⫽6.67⫻104 , W⬇74 at step 共a兲 4 and 共b兲共⽧兲 34 and 共〫兲 33 downstream from inception point

Inception Point of Air Entrainment At the inception point the degree of turbulence is large enough to entrain air into the black water flow. In the model experiments, the inception point was mathematically defined as the location where the pseudobottom 共subscript b兲 air concentration was C b ⫽0.01, which agreed well with visual observation of the surface flow conditions. The location of the inception point on stepped spillways is significantly closer to the spillway crest than on smooth chutes due to the larger growth of the boundary layer 共Chanson 1994兲. A main advantage of the significant aeration along stepped spillways is the reduction of the cavitation risk potential. For high velocities, the hydrodynamic pressures on the step surfaces or at the step edges may fall below the vapor pressure, resulting in cavitation which might cause severe damage to the spillway concrete. Based on the fundamental work of Peterka 共1953兲, a bottom air concentration of about 5– 8% is considered sufficient to avoid cavitation damage because the compressibility of the air–water mixture can absorb the impact of collapsing vaporized bubbles. Knowing the location of the inception point is thus important to determine the unaerated spillway zone which is potentially prone to cavitation damage 共Boes and Minor 2000兲. Although steps form large offsets away from the flow direction and inhibit cavitation from residing on the boundary 共Frizell and Mefford 1991兲, the placement of an aerator in the black water region of a stepped spillway to artificially entrain air might be of interest. Local air entrainment can also be achieved by bridge-supporting piles downstream of the spillway crest 共Mateos and Elviro 1992兲 or by a flap-gate on top of the crest 共Minor and Boes 2001兲.

Location The available literature data on the inception point were analyzed and compared with semiexperimental data obtained at VAW

共Boes 2000; Schla¨pfer 2000兲. The inception 共subscript i兲 point location is usually expressed either in terms of the black water length L i , or by the vertical distance z i from the spillway crest 共Fig. 2兲. The latter is chosen here in analogy to Mateos and Elviro 共1997兲, because besides the advantage of determining the flow velocity directly from the elevation difference a general comparison of data from different spillway slopes is possible. Because of the arrangement without a spillway crest in the VAW experiments, the blackwater drawdown curve upstream of the jetbox was computed by accounting for the friction behavior as observed on the downstream blackwater channel portion 共Hager and Blaser 1998; Hager and Boes 2000兲. The fictitious location of the crest was defined as the point where the nonaerated flow depth was h w ⫽h c •cos ␾ 共Fig. 2, Boes 2000兲. With the so-determined length L i the vertical distance was z i ⬇L i sin ␾ 共Fig. 2兲. Hager and Boes 共2000兲 applied the generalized drawdown equation to stepped chutes in analogy to Hager and Blaser 共1998兲 who compared their drawdown equation with experimental data on a smooth chute. The agreement between prediction and the measured flow dephts downstream of the jetbox in both studies suggests that the drawdown equation is applicable for estimating flow depths downstream of a determined location both for conventional and stepped chutes. The assumption of a similar friction behavior along the ogee crest profile as observed on the constantly sloping channel portion downstream of the jetbox has been checked by comparing the authors’ data with experiments of other researchers. In fact, because the semi-experimental data obtained at VAW for ␾⫽50° agree well with those for ␾⬇50° based on visual observation of Chanson 共1994兲; Wahrheit-Lensing 共1996兲; Mateos and Elviro 共1997兲; and Chamani 共2000兲—see below—the validity of that methodology is believed to be proven from a practical standpoint. It was therefore equally applied for the analysis of VAW data from 30 and 40° models. The interest in this new approach lies in developing a general formula based on this study’s semiexperimental data for ␾⫽30, 40, and 50°. So far the great majority of experimental data available was gathered on steep slopes with ␾⬇50° only. In Fig. 8 the data in terms of the nondimensional parameter z i /s of Boes 共2000兲; Schla¨pfer 共2000兲; and Wahrheit-Lensing 共1996兲 are plotted as a function of a roughness Froude number F ⫽q w /(g sin ␾s3)1/2. Also plotted are proposals by Chanson * 共1994兲; Chamani 共2000兲; Mateos and Elviro 共1997兲; and Matos 共1999兲共see also Matos et al. 2000兲. Where necessary, the alternative roughness Froude number FK ⫽q w /(g sin ␾K3)1/2 used by various authors was transformed adequately, with the roughness



height K⫽s•cos ␾ perpendicular to the pseudobottom 共Fig. 2兲, i.e., F ⫽FK (cos ␾)3/2 or F ⫽0.5FK for ␾⫽51°. * * Most data and approximations stem from model experiments with the gravity dam type slope of about 1:0.8 or ␾⬇51°. Only the data of Boes 共2000兲 and Schla¨pfer 共2000兲 include bottom angles of 30 and 40°, respectively, and the function proposed by Chanson 共1994兲 refers to two data points with ␾⫽26.6°, whereas the remaining 47 experiments of various authors apply to ␾⬇52°. In addition, out of the ten scale models underlying the approximation given by Mateos and Elviro 共1997兲, one featured a steep downstream chute angle of ␾⫽71.6°. Whereas Chamani 共2000兲; Mateos and Elviro 共1997兲; and Wahrheit-Lensing 共1996兲 had maximum Froude numbers of F ⬇10 and Matos et al. 共2000兲 of * only F ⬇7, the data of Boes 共2000兲 and those analyzed by Chan* son 共1994兲 extend to roughness Froude numbers F ⬎40 共Fig. 8兲. * For a given F , the approximation proposed by Chamani 共2000兲 * yields the largest vertical distance from the spillway crest, whereas the flow down stepped chutes becomes aerated further upstream according to Matos 共1999兲 whose data were based on velocity and air concentration measurements. The present data agree well with the data from crest profile spillways, particularly with those of Mateos and Elviro 共1997兲 and Wahrheit-Lensing 共1996兲. Even the results from the 30 and 40° models follow the gravity dam type data closely. For F * ⬍10 the VAW data suggest slightly smaller z i /s values than those of Wahrheit-Lensing 共1996兲 and Chamani 共2000兲. For large roughness Froude numbers, there is also close agreement with Chanson 共1994兲. For chute angles 26°⬍␾⬍75° an approximation is thus 共Fig. 8兲 zi ⫽5.90F0.80 s *

L i⫽

共 sin ␾ 兲 7/5s 1/5

authors than the relative clear water depth h w,i /s. Obviously, the inception-flow depths reported in these other studies were mixture depths h m,i , a conclusion also drawn by Matos et al. 共2000兲, according to whom air is transported between surface waves upstream of the inception point. An approximation for all data relating to mixture depth with 26°⬍␾⬍55° is 共Fig. 9兲 h m,i ⫽0.40F0.60 s *

(5)

Replacing F ⫽q w /(g sin ␾s3)1/2 and inserting the critical depth * h c demonstrates that the absolute inception flow depth varies almost linearly with h c . The effects of both step height and spillway slope are nearly insignificant.

(3)

Replacing z i ⬇L i sin ␾ and inserting the critical flow depth h c , the dimensional black water distance L i becomes 5.90h 6/5 c

Fig. 9. Nondimensional flow depth h i /s at inception point as function of roughness Froude number F for different spillway chute * angles ␾

(4)

Eq. 共4兲 indicates the comparably small influence of the step height s on the location of the inception point L i , whereas the critical flow depth h c or the unit discharge q w predominantly govern the unaerated spillway length L i . By doubling the step height the black water length is reduced by only 13%, whereas L i increases by 74% when the unit discharge is doubled. The steeper the spillway slope, the sooner the water flow becomes aerated. An increase in slope from 1:2 to 1:0.8 共␾⫽26.6 –51.3°兲 reduces L i and z i by about 54 and 20%, respectively.

Depth-Averaged Air Concentration ¯ was analyzed as a funcThe depth-averaged air concentration C tion of the relative vertical distance from the inception point Z i ⫽(z⫺z i )/h c 共Fig. 2兲. By normalizing the difference between the ¯ (Z ) and the mean air concentration in a given cross section C i ¯ (Z ⫽0)⫽C ¯ with mean concentration at the inception point C i i ¯ ¯ ¯ C ⫺C , where C is the uniform 共subscript u兲 value proposed by u

i

u

Hager 共1991兲 for smooth chutes as a function of the spillway ¯ (Z ) slope only, the normalized mean air concentration is c i ⫽ 关 C i ¯ ¯ ¯ ⫺C i 兴 /(C u ⫺C i ). This parameter approaches unity with increasing distance from the inception point and can be expressed as 共Fig. 10兲

Flow Depth The normalized inception depth h i is plotted in Fig. 9 for the data of Boes 共2000兲; Schla¨pfer 共2000兲; and Wahrheit-Lensing 共1996兲 as well as the approximations of Chanson 共1994兲 and Matos 共1999兲. For the VAW data, a distinction is made between the mixture depth h 90,i and the equivalent clear water depth h w,i ¯ ), with C ¯ as the depth-averaged air concentration at ⫽h 90,i (1⫺C i i the inception point. The average difference of 35% between h 90,i and h w,i stems from the definition of the inception point location taken where the pseudo-bottom air concentration was C b,i ¯ ⬇1 ⫽0.01. The flow is then already affected by aeration and C i ⫺h w,i /h 90,i ⫽1⫺1/1.35⫽0.26 on the average for all bottom slopes considered. Fig. 9 shows that the h 90,i /s values of Boes 共2000兲 and Schla¨pfer 共2000兲 are closer to the data of the other

¯ (Z ) Fig. 10. Normalized depth-averaged air concentration c i ⫽ 关 C i ¯ ¯ ¯ ⫺C i 兴 /(C u ⫺C i ) as function of relative vertical distance from inception point Z i ⫽(z⫺z i )/h c and 共兲 Eq. 共6兲 for ␾⫽50°



¯ 共 Z 兲 ⫺C ¯ C i i c i⫽ ¯ ⫽ 关 tanh共 5⫻10⫺4 共 100°⫺␾ 兲 Z i 兲兴 1/3 ¯ C u ⫺C i

(6)

where the chute angle ␾ with 26°⭐␾⭐55° is expressed in degrees, and the nondimensional vertical coordinate Z i originates at the inception point 共Fig. 2兲. The uniform mean air concentration for smooth chutes was taken as the normalizing parameter, be¯ cause the spillway roughness is not a relevant parameter for C u 共Chanson 1994; Boes 2000; Matos 2000兲. The equilibrium depthaveraged air concentration for skimming flow on stepped spillways therefore agrees well with the uniform value of smooth chutes of identical slope. This is further demonstrated by the asymptotic trend of the curve towards unity in Fig. 10, which shows a comparison of the VAW data for ␾⫽50° and the data of Wahrheit-Lensing for ␾⫽51.3° with Eq. 共6兲. Scale effects in the aeration process were excluded by accounting only for the data of the larger model steps. Based on the VAW data, the depthaveraged air concentration at the inception point depends upon the bottom slope only and can be approximated for 26°⭐␾⭐55° by ¯ ⫽1.2⫻10⫺3 共 240°⫺␾ 兲 C i

(7)

¯ ⬇0.22 agrees For the typical gravity dam chute angle ␾⬇53°, C i ¯ with C i ⬇0.20 found by Matos et al. 共2000兲.

Air Concentration Distribution The air concentration profiles were compared with an air bubble diffusion model 共Fig. 11兲 proposed by Chanson 共2000兲, further to the theoretical model developed by Wood 共1984兲 for smooth spillway chutes. Assuming that the air bubble rise velocity v B in a two-phase mixture flow is a function of the local air concentration C and the rise velocity in hydrostatic pressure gradient v B,hyd , the air concentration distribution C(y) is expressed by

冉

C 共 y 兲 ⫽1⫺tanh2 K ⬘ ⫺

y 2D ⬘ h 90

冊

(8)

where K ⬘ ⫽integration constant deduced from the boundary condition C(y⫽h 90)⫽0.90 K ⬘ ⫽arctanh

冉冑

0.1⫹

1 2D ⬘

冊

(9)

and D ⬘ ⫽D t cos ␾h90 / v B,hyd contains the turbulent diffusivity D t normal to the flow direction and the mixture depth h 90 . The ¯ from Eq. 共6兲 is thus 共Chanson depth-averaged air concentration C 2000兲

冉冉

冊

¯ ⫽2D ⬘ tanh arctanh共冑0.1兲 ⫹ 1 ⫺ 冑0.1 C 2D ⬘

冊

Fig. 11. Normalized air concentration distribution C(Y 90) and 共兲 Eq. 共8兲 for 共a兲 ␾⫽30°, F0 ⫽4.0, K⫽40 mm and h 0 /s⫽0.54; 共b兲 ␾⫽40°, F0 ⫽5.0, K⫽20 mm and h 0 /s⫽1.2, 共c兲 ␾⫽50°, F0 ⫽3.5, K ⫽20 mm and h 0 /s⫽1.06

(10)

Fig. 11 demonstrates that the analytical solution is essentially in agreement with the model data obtained for different chute angles, although some deviations apply mainly for small nondimensional depths Y 90⫽y/h 90 . The existence of an air concentration boundary layer is apparent from Fig. 11, particularly for large mean air concentrations, i.e., close to the spillway toe, where approxiation 共10兲 departs from the experimental data for Y 90⬍0.3. This agrees with observations of earlier studies 共Chanson 1996; Matos and Frizell 1997, 2000; Matos et al. 2000兲.

Pseudobottom Air Concentration The pseudobottom air concentration C(y⫽0)⫽C b was obtained by measuring local air concentrations C(y) at two locations near

the outer step edges, the closest being at y⫽1.5 mm. The growth of C b can be expressed as a function of the nondimensional distance from the inception point X i ⫽(x⫺L i )/h m,i , with x as streamwise coordinate originating at the spillway crest 共Fig. 2兲 and L i and h m,i computed from Eqs. 共4兲 and 共5兲, respectively. For 26°⬍␾⬍55°, the function C b 共 X i 兲 ⫽0.015X i冑tan ␾/2

(11)

approximates the air concentration at the pseudobottom 共Fig. 12兲. If C b ⫽0.05 and C b ⫽0.08 are considered minimum values to avoid cavitation damage to the spillway concrete, respectively, based on the findings of Peterka 共1953兲, the required nondimensional distances X i,crit from the inception point are



Fig. 12. Pseudobottom air concentration C b (X i ) for different spillway chute angles ␾

X i,crit⫽X i 共 C b,crit⫽0.05兲 ⫽5.0共 sin ␾ 兲 ⫺2.3 X i,crit⫽X i 共 C b,crit⫽0.08兲 ⫽10.0共 sin ␾ 兲

⫺3.0

(12a) (12b)

Eq. 共12兲 suggest that the distance required to attain a sufficiently high bottom air concentration depends significantly on the critical value C b,crit . Because the aeration tends to be more pronounced in the prototype than on spillway models, as discussed above, Eq. 共12a兲 is suggested for design purposes. For typical gravity dams with a downstream slope of 1:0.8, the critical distance from the inception point amounts to about 9h m,i . Similar to the black water region upstream of the inception point, this downstream developing region down to X i,crit might also be prone to cavitation damage for velocities greater than approximately 20 m/s. The critical velocity of 13 m/s mentioned by Mateos and Elviro 共1992兲 was considered too conservative. Because critical velocities of 20 m/s are attained at X i,crit for unit discharges of the order of 25 m3/s/m, depending on the chute angle and the step height, an aerator placed in the unaerated spillway region downstream from the crest is necessary if cavitation damage is expected.

Fig. 13. Velocity distribution Y 90(U 90) for all experimental data with Y 90⬎0.04

depths to attain the surface velocity u 90 . In a given cross section of a stepped chute the maximum velocity is thus reached at a distance of about 80% of the characteristic mixture depth (y ⬇0.8 h 90), above which the velocity remains constant, in agreement with observations of Tozzi 共1994兲, where maximum values were attained at significantly lower relative distances from the pseudobottom, however, due to use of a Pitot tube. Furthermore, due to the influence of the recirculating vortices in the step niches, the normalized velocities close to the outer step edges are slightly smaller than on smooth chutes 共Fig. 13兲. This is expressed by the exponent 1/n with n⫽4.3 in Eq. 共13a兲, indicating a flatter profile or a less pronounced velocity gradient than on conventional spillways with n⫽6.3 and n⫽6.0 according to Cain 共1978兲 and Chanson 共1989兲, respectively. The present value of 4.3 is larger than n⫽3.5 and 4.0 suggested by Chanson 共1994兲 for the data of Frizell 共1992兲 and Tozzi 共1992兲 for chute angles of 27 and 53°, respectively.

Conclusions Velocity Distribution According to Cain 共1978兲 the slip between the air and water phases is negligible for air concentrations 0⭐C⭐0.9. In the present study, some velocity profiles were taken up to h 99 or Y 90⫽y/h 90⬇1.5. Even in this spray region with few water droplets in air the profiles are smooth and continuous, thus suggesting that the no-slip condition can be extended over almost the entire two-phase depth. By normalizing with the local mixture surface velocity u 90⫽u(y⫽h 90), the dimensionless mixture velocity U 90⫽u m (y)/u 90 can be approximated by U 90⫽1.05Y 1/4.3 90 U 90⫽1

for 0.04⭐Y 90⭐0.8 for Y 90⬎0.8

(13a) (13b)

In Fig. 13 the data from both the developing and the fully developed flow regions are plotted, because no significant effect of the flow region on u m was detected. Velocities at very low relative distances from the outer step edge Y 90⬍0.04 have been neglected due to scatter resulting from the high shear stress near the pseudobottom. Although the relative velocities tend to increase slightly with increasing chute angle 共Fig. 13兲, this influence is considered negligible in the range of the experimental results. According to Eq. 共13兲 the velocity profiles on stepped spillways follow a power law up to Y 90⬇0.8 similar to conventional spillways 共Cain 1978; Chanson 1989兲. In contrast to former studies, the velocity distribution departs from the power law approximation at high flow

The results of the present study shed light on the minimum Reynolds and Weber numbers required to minimize scale effects in physical modeling of two-phase air water flows on stepped spillways. The investigations on the aeration characteristics of skimming flows on stepped spillways allow a redefinition of the location and flow depth of the inception point of air entrainment. The depth-averaged air concentration is expressed as a function of the normalized vertical distance from the spillway crest for chute angles ranging from embankment to gravity dam spillways. The present air concentration analysis compares well to a distribution proposed previously by other researchers. The bottom air concentration, important for the assessment of the cavitation risk of a stepped chute, is a function of the non-dimensional distance from the inception point and the chute angle only. Finally, the velocity distribution in terms of the characteristic non-dimensional mixture depth is given. The following findings apply: 1. To exclude scale effects on stepped spillway models, the minimum Reynolds number R⫽q w /␯ must be roughly 105 , ¯ m /(␴/(␳L s )) 1/2 and the minimum Weber number W⫽u should be about 100. For spillways with the standard step height of s⫽0.6 m, a scaling factor ␭ L ⬍15 is thus required. 2. In addition to present day definition of air inception on stepped spillways, the numerical bottom air concentration C b ⫽0.01 is introduced. Based on present and literature experimental data, Eqs. 共3兲–共5兲 are proposed for inception location and inception mixture depth, respectively.



3.

The streamwise variations of depth-averaged air concentra¯ and pseudobottom air concentration C depend on tion C b relative distance from inception point, chute angle and ¯ ,C ¯ ) or mixture depth h boundary air concentrations (C i u m,i as given in Eqs. 共6兲 and 共11兲, respectively. 4. The velocity distribution is composed of a power function in the lower flow portion and a practically constant surface layer above 80% of the mixture depth as given in Eq. 共13兲. The present research is a step forward in the design of stepped spillways, particularly regarding two-phase flow characteristics. Given that the experimental data were collected from a hydraulic model with both variable bottom slope and step height, the results pertain to typical conditions met in applications. Based on VAW data plus additional results partly presented at the 2000 VAW workshop on the hydraulics of stepped spillways, some definite expressions for the characteristics of air inception, the development of cross-sectional and of bottom air concentration, and the velocity distribution are now available.

Acknowledgment The present project was financed by the Swiss National Science Foundation, Grant No. 21-45424.95.

Notation The following symbols are used in this paper: ¯ ⫽ 1/h 兰 h 90 C(y)dy depth-averaged air concentration; C 90 y⫽0 C b ⫽ bottom air concentration at outer step edge; C b,i ⫽ bottom inception air concentration at outer step edge; C b,crit ⫽ critical bottom air concentration to avoid cavitation damage; C d ⫽ q w /(gh 3max)1/2 discharge coefficient; ¯ ⫽ depth-averaged air concentration at inception C i point; ¯ ⫽ uniform depth-averaged air concentration; C u C(y) ⫽ local air concentration; ¯ (Z )⫺C ¯ )/(C ¯ ⫺C ¯ ) nondimensional c i ⫽ (C i i u i depth-averaged air concentration; D ⬘ ⫽ D t cos ␾h90 / v B,hyd factor; D t ⫽ turbulent diffusivity normal to flow direction; d B ⫽ air bubble diameter; FK ⫽ q w /(g sin␾K3)1/2 roughness Froude number; F ⫽ q w /(g sin ␾s3)1/2 roughness Froude number; * F0 ⫽ q w /(g h 30 ) 1/2 approach Froude number at jetbox; g ⫽ gravitational acceleration; h c ⫽ critical depth; h i ⫽ flow depth at inception point; h m,i ⫽ mixture depth at inception point; h max ⫽ maximum approach flow depth above spillway crest; ¯ )h clear water depth; h w ⫽ (1⫺C 90 h w,i ⫽ clear water depth at inception point; h 0 ⫽ approach flow depth at jetbox; h 90 ⫽ h(C⫽0.90) characteristic mixture depth with local air concentration of C⫽0.90; h 90,i ⫽ characteristic mixture depth at inception point; h 99 ⫽ h(C⫽0.99) flow depth with local air concentration of C⫽0.99;

K ⫽ s•cos ␾ roughness height perpendicular to pseudobottom; K ⬘ ⫽ integration constant; L i ⫽ black water length from spillway crest to inception point; L s ⫽ s/sin ␾ distance between step edges, roughness spacing; n ⫽ exponent of velocity power law; Q w ⫽ water discharge; q w ⫽ water discharge per unit width; R ⫽ uh w /␯⫽q w /␯ Reynolds number; s ⫽ step height; U 90 ⫽ u m /u 90 normalized mixture velocity; u ⫽ flow velocity in x direction; u m (y) ⫽ local mixture velocity; ¯u m ⫽ 1/h 90兰 h 90 u m (y)dy depth-averaged mixture y⫽0 velocity; u 0 ⫽ approach velocity at jetbox; u 90 ⫽ u(y⫽h 90) mixture surface velocity; v ⬘ ⫽ turbulent velocity component in y direction; v B ⫽ air bubble rise velocity in vertical direction; v B,hyd ⫽ air bubble rise velocity in hydrostatic pressure gradient; W ⫽ ¯u m /(␴/(␳ w L s )) 1/2 Weber number; X i ⫽ (x⫺L i )/h m,i nondimensional streamwise coordinate; X i,crit ⫽ X i (C b,crit) critical nondimensional streamwise coordinate; x ⫽ streamwise coordinate originating at spillway crest; Y 90 ⫽ y/h 90 normalized mixture depth; y ⫽ transverse coordinate originating at pseudobottom; Z i ⫽ (z⫺z i )/h c nondimensional vertical distance from inception point; z ⫽ vertical coordinate originating at spillway crest; z i ⫽ vertical black water length from spillway crest to inception point; ␭ L ⫽ scaling factor 共reciprocal of scale兲; ␯ ⫽ kinematic viscosity of water; ␳ w ⫽ density of water; ␴ ⫽ surface tension between air and water; and ␾ ⫽ chute angle from horizontal.

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