DISCUSSIONS AND CLOSURES

Transcription

1 DISCUSSIONS AND CLOSURES Discussion of Hydraulic Design of Stepped Spillways by Robert M. Boes and Willi H. Hager September 2003, Vol. 129, No. 9, pp DOI: / ASCE :9 671 H. Chanson 1 1 Reader, Dept. of Civil Engineering, Univ. of Queensland, Brisbane QLD 4072, Australia In skimming flows down stepped chutes, Chanson et al presented a comprehensive reanalysis of flow resistance based upon more than 38 model studies and 4 prototype investigations totaling more than 700 data points with channel slopes ranging from 5.7 up to 55. Different research facilities yielded different results and researchers continue to disagree on the reasons for these differences Chanson The authors highlighted nicely the difficulties to estimate flow resistance on stepped chutes, although it is a key design parameter. Here the discusser argues that differences in flow resistance data may be linked to different inflow conditions. A careful reanalysis of large-size experimental results suggests that lower flow resistance was observed in experimental facilities with pressurised intake. Skimming flows are characterized by significant form losses. Observations highlighted strong interactions between the main stream turbulence, the step cavity recirculation zones, and the free-surface Chanson and Toombes 2002a; Yasuda and Chanson Flow resistance data for large-size model data s m, R 1E+5 are presented in Fig. 1 in terms of the equivalent Darcy friction factor f, where s is the step height, is the angle between the pseudobottom formed by the step edges and the horizontal, and R is the flow Reynolds number defined in terms of the hydraulic diameter D H. Details of each study are summarized in Table 1. Fig. 1 presents 179 data. For steep chutes 15, the friction factor data presented no obvious correlation with the relative step roughness height s*cos /D H, Reynolds, Froude, nor Weber numbers Chanson et al They compared favorably, however, with a simplified analytical model of the pseudoboundary shear stress that may be expressed, in dimensionless form, as f d = where f d equivalent Darcy friction factor estimate of the form drag; 1/ dimensionless expansion rate of the shear layer. Eq. 1 predicts f d 0.2 for =6, which is close to observed friction factors Fig. 1. However, skimming flow resistance data appeared to be distributed around three dominant values: f 0.105, 0.17, and 0.30 as shown in Fig. 2. Fig. 2 presents the probability distribution function of Darcy friction factor where the histogram columns represent the number of data with friction factors within the interval: e.g., the probability of friction factors from 0.18 to 0.20 is represented by the column labeled The intervals were selected with a constant logarithmic increment. The first and last columns indicate the number of data with friction factors less than 0.08 and greater than 1.0, respectively. The discusser hypothesizes that flow resistance in skimming flows is not an unique function of flow rate and stepped chute geometry, and that there is some analogy with form drag behind bluff bodies. For the flow behind a cylinder, the drag coefficient is known to be a function of the upstream turbulence affecting the boundary layer separation for a given Reynolds number. For infinitely long smooth cylinders, the effect is best observed for Reynolds numbers about 1 E+5 to 1 E+6. For ventilated cavities behind wedges and wings, several regimes were associated with different drag coefficients for the same inflow conditions, depending upon the amount of ventilation Silberman and Song 1961; Fig. 1. Darcy friction factor of skimming flows on stepped chute 179 data JOURNAL OF HYDRAULIC ENGINEERING ASCE / JUNE 2005 / 521

2 Table 1. Reanalyzed Experimental Data of Flow Resistance Legend Reference Flow conditions Remarks Andre Andre et al =30, s=0.06 m, b=0.5 m Air water flow measurements. Pressurized intake inflow. BaCaRa BaCaRa 1991 =53.1, s=0.06 m, b=1.5 m Clear-water nonaerated flow. =53.1, s=0.024 m Uncontrolled ogee inflow with =59, s=0.024 m small steps in ogee development. =63.4, s=0.024 m Boes Boes 2000 =30, s=0.046, m, b=0.5 m Air water flow measurements. =50, s=0.031, m, b=0.5 m Pressurized intake inflow. Chamani Chamani and =51.3, s=0.313, m, b=0.3 m Air water flow measurements. and Rajaratnam Rajaratnam 1999 =59, s=0.313 to m, b=0.3 m Uncontrolled smooth ogee crest inflow. Chanson, Toombes, Chanson and Toombes 2001 =21.8, s=0.10 m, b=1 m Air water flow measurements. and Gonzalez Uncontrolled broad-crested weir inflow. Gonzalez and Chanson 2004 =15.9, s=0.05 and 0.10 m, b=1 m Air water flow measurements. Uncontrolled broad-crested weir inflow. Matos Matos 2000 =53.1, s=0.08 m, b=1.0 m Air water flow measurements. Uncontrolled smooth ogee crest inflow. Shvainshtein Shvajnshtejn 1999 =38.7, s=0.05 m, b=0.48 m Clear-water nonaerated flow. =51.3, s= m, b=0.48 m Uncontrolled smooth ogee crest inflow. Toombes and Chanson Toombes and Chanson 2000 =3.4, s=0.143 m, b=0.25 and 0.5 m Air water flow measurements. Pressurized intake inflow. Chanson and Toombes 2002b =3.4, s= m, b=0.5 m Air water flow measurements. Pressurized intake inflow. Yasuda and Ohtsu Ohtsu et al =55, s=0.025 m, b=0.4 m Air water flow measurements. Uncontrolled broad-crested weir inflow. Yasuda and Ohtsu 1999 =5.7, s=0.006 to m, b=0.4 m Measurements in downstream stiling basin. =11.3, s=0.006 to 0.10 m, b=0.4 m Uncontrolled broad-crested weir inflow. =19, s=0.002 to 0.08 m, b=0.4 m =30, s=0.004 to 0.07 m, b=0.4 m =55, s=0.003 to m, b=0.4 m Note: bed slope; s step height; and b channel width. Fig. 2. Probability distribution function of chute friction factor 179 data 522 / JOURNAL OF HYDRAULIC ENGINEERING ASCE / JUNE 2005

3 Laali and Michel 1984; Michel 1984; Verron and Michel On stepped chutes, it is proposed that the form drag process may present several modes of excitation that are functions of the inflow conditions. At each step edge, shear instabilities may generate different cavity wake regimes, associated with different drag coefficients. In Fig. 2, the dominant values f 0.105, 0.17, and 0.30 would correspond to three dominant modes or regimes of excitation induced by different inflow conditions Fig. 3. Fig. 3 illustrates basic inflow configurations. With an uncontrolled ogee profile, the pressure distribution is atmospheric in the entire flow at design flow conditions by definition of the ogee development Henderson 1966; Chanson With an uncontrolled broadcrest, the pressure is hydrostatic at the crest. For a pressurised intake, the inflow pressure distribution is greater than hydrostatic. Fig. 2 shows that experiments with pressurized intake yielded consistently lower flow resistance than for uncontrolled inflow conditions. For example, the reanalysis of data from Boes 2000 and Andre et al gives f 0.1, which is about three times smaller than the third dominant value f =0.30, Fig. 2. Similarly, skimming flow experiments by Chanson and Toombes 2002b down a flat slope =3.4, s=0.07 m with pressurized intake yielded friction factors three times smaller than data of Yasuda and Ohtsu 1999 on a 5.7 stepped slope with uncontrolled broad-crest. Overall flow resistance data ranged typically between 0.1 and 0.3 Figs. 1 and 2, although the friction factor is affected by the inflow conditions and by the rate of air entrainment. The drag reduction process was well-documented in smooth chutes Chanson 1994, and it was recently demonstrated on stepped chutes Chanson 1993, ACKNOWLEDGMENTS The discusser thanks Professor C. J. Apelt University of Queensland for helpful discussions. Fig. 3. Sketch of inflow conditions of stepped chutes References Andre, S., Manso, P. A., Schleiss, A., and Boillat, J. L Hydraulic and stability criteria for the rehabilitation of appurtenant spillway structures by alternative macro-roughness concrete linings. Proc., 21st ICOLD Congress, Montreal, Canada, Q. 82, R. 6, BaCaRa Etude de la dissipation d energie sur les evacuateurs à marches Study of the energy dissipation on stepped spillways. Rapport d Essais, Projet National BaCaRa, CEMAGREF-SCP, Aix-en- Provence, France in French. Boes, R. M Zweiphasenstroömung und Energieumsetzung auf Grosskaskaden. PhD thesis, VAW-ETH, Zürich, Switzerland. Chamani, M. R., and Rajaratnam, N Characteristics of skimming flow over stepped spillways. J. Hydraul. Eng , Chanson, H Stepped spillway flows and air entrainment. Can. J. Civ. Eng., 20 3, Chanson, H Drag reduction in open channel flow by aeration and suspended load. J. Hydraul. Res., IAHR, 32 1, Chanson, H The hydraulics of open channel flows: An introduction, Butterworth-Heinemann, London. Chanson, H Hydraulics of stepped spillways: Current status. J. Hydraul. Eng , Chanson, H Drag reduction in skimming flow on stepped spillways by aeration. J. Hydraul. Res., IAHR, 42 3, Chanson, H., and Toombes, L Experimental investigations of air entrainment in transition and skimming flows down a stepped chute: Application to embankment overflow stepped spillways. Research Report No. CE158, Dept. of Civil Engineering, Univ. of Queensland, Brisbane, Australia. Chanson, H., and Toombes, L. 2002a. Air-water flows down stepped chutes: Turbulence and flow structure observations. Int. J. Multiphase Flow 27 11, Chanson, H., and Toombes, L. 2002b. Energy dissipation and air entrainment in a stepped storm waterway: Experimental study. J. Irrig. Drain. Eng , Chanson, H., Yasuda, Y., and Ohtsu, I Flow resistance in skimming flows and its modelling. Can. J. Civ. Eng. 29 6, Gonzalez, C. A., and Chanson, H Interactions between cavity JOURNAL OF HYDRAULIC ENGINEERING ASCE / JUNE 2005 / 523

4 flow and main stream skimming flows: An experimental study. Can. J. Civ. Eng. 31 1, Henderson, F. M Open channel flow, MacMillan, New York. Laali, A. R., and Michel, J. M Air entrainment in ventilated cavities: Case of the fully developed half cavity. J. Fluids Eng., Trans ASME, Sept., 106, Matos, J Hydraulic design of stepped spillways over RCC dams. Intl Workshop on Hydraulics of Stepped Spillways, Zürich, Switzerland, H. E. Minor and W. H. Hager, eds., Balkema, Rotterdam, The Netherlands, Michel, J. M Some features of water flows with ventilated cavities. J. Fluids Eng., Trans ASME, Sept., 106, Ohtsu, I, Yasuda, Y., and Takahashi, M Characteristics of skimming flow over stepped spillways. J. Hydraul. Eng., , Shvajnshtejn, A. M Stepped spillways and energy dissipation. Gidrotekh. Stroit., 5, in Russian. Silberman, E., and Song, C. S Instability of ventilated cavities. J. Ship Res., 5 1, Toombes, L., and Chanson, H Air-water flow and gas transfer at aeration cascades: A comparative study of smooth and stepped chutes. Int. Workshop on Hydraulics of Stepped Spillways, Zürich, Switzerland, Balkema, Rotterdam, The Netherlands, Verron, J., and Michel, J. M Base-vented hydrofoils of finite span under a free surface: An experimental investigation. J. Ship Res., 28 2, Yasuda, Y., and Chanson, H Micro- and macroscopic study of two-phase flow on a stepped chute. Proc., 30th IAHR Biennial Congress, Thessaloniki, Greece, J. Ganoulis and P. Prinos, eds., vol. D, Yasuda, Y., and Ohtsu, I Flow resistance of skimming flow in stepped channels. Proc., 28th IAHR Congress, Graz, Austria, session B14, CD-ROM. Discussion of Hydraulic Design of Stepped Spillways by Robert M. Boes and Willi H. Hager September 2003, Vol. 129, No. 9, pp DOI: / ASCE :9 671 A. D. Ghare 1 ; P. D. Porey 2 ; and R. N. Ingle 3 1 Sr. Lecturer, Civil Engineering Dept., D. C. V. Raman Institute of Technology, Nagpur, India. 2 Professor, Civil Engineering Dept., Visvesvaraya National Institute of Technology, Nagpur, India. 3 Emeritus Fellow, Civil Engineering Dept., Visvesvaraya National Institute of Technology, Nagpur, India. The authors are to be complimented for presenting extensive experimental data on characteristics of aerated skimming flow over stepped spillways along with hydraulic design aspects of stepped spillways. The authors have focused their attention on various aspects, including onset of skimming flow, aeration characteristics, residual energy, and training wall design. Considering the applicability of the design guidelines, the discussers would like to know the height of stepped spillway in the experimental setup for all 3 cases. Further, the authors may clarify regarding the limiting height of prototype stepped spillways up to which the design guidelines presented in this paper could be applied. 524 / JOURNAL OF HYDRAULIC ENGINEERING ASCE / JUNE 2005 Fig. 1. Variation of Manning s n for different H values The discussers would also like to know the number of steps provided in each case and the location of first step along the spillway profile. Can the authors suggest any readily usable explicit guidelines from hydraulic considerations for deciding on the step height, apart from the given RCC lift thickness? Some other investigators, including Rice and Kadavy 1996, Yildiz and Kas 1998, Chamani and Rajaratnam 1999 have indicated that the step height s affects the energy dissipation over stepped spillway. Eq. 24 includes K, the roughness height perpendicular to the pseudobottom, which can be considered to be a representative term for step height s. In the last paragraph on energy dissipation, it is mentioned that Fig. 12 gives an idea of main parameters involved in the expression of relative residual energy. However, Fig. 12 does not indicate effect of any step height parameter on relative residual energy head ratio H res /H max. Fig. 1 shows a plot compiled by discussers based on experimental data obtained by Ghare 2003 and Yildiz and Kas 1998, which show the effect of step height on Manning s equivalent n for a stepped spillway. In this plot H * is considered a ratio of spillway height to step height. Can authors provide any other dimensionless plot that covers all the main parameters including step height s affecting the performance of the stepped spillway under skimming flow regime? Proposed Eq. 24 is based on the results obtained from Eqs. 20 and 21. Hence the use of Eq. 24 appears to be a tedious process. As indicated by the authors in Fig. 12, the variation in relative residual energy head ratio for =40 and 50 is not appreciable; hence a simpler relationship for relative residual energy can be presented eliminating as a variable. The resulting relationship would be applicable for greater than 40. Without a properly designed energy dissipation system on the downstream side, the hydraulic design of a stepped spillway system would be incomplete. The discussers would like to know the opinion of the authors regarding the applicability of the conventional conjugate depth relationship for stilling basin design in case of a stepped spillway where highly aerated flow near the toe of the spillway is encountered. References Chamani, M. R., and Rajaratnam, N Characteristics of skimming flows over stepped spillways. J. Hydr. Engrg ,

5 Ghare, A. D Study of parameters related with the design aspects of stepped spillway. PhD thesis, Visvesvaraya National Institute of Technology, Nagpur, India. Rice, C. E., and Kadavy, K. C Model study of a roller compacted concrete stepped spillway, J. Hydr. Engrg , Yildiz, D., and Kas, I Hydraulic performance of stepped chute spillways. Hydropower Dams, 5 4, Discussion of Hydraulic Design of Stepped Spillways by Robert M. Boes and Willi H. Hager September 2003, Vol. 129, No. 9, pp DOI: DOI: / ASCE :9 671 Jorge Matos 1 1 Asst. Prof., Dept. of Civil Engineering and Architecture, Technical Univ. of Lisbon, IST, Lisbon , Portugal The authors paper, along with Boes and Hager 2003, is considered a significant contribution on the hydraulic design of stepped spillways. Original analysis and findings of the present paper are related to the uniform two-phase flow, the friction factor, and the energy dissipation. In the present discussion, some of the authors results related to the uniform two-phase flow and the friction factor are analzsed in light of some additional experimental data, namely those gathered in a stepped chute assembled at the National Laboratory of Civil Engineering LNEC, Lisbon. A brief reflection is included on the effect of aeration on friction factor. Uniform Two-Phase Flow The authors have carefully analyzed their data in light of the criteria defining the length needed for uniform flow to be attained, namely: 1 similarity of the air concentration profiles near the downstream end of the chute; 2 quasiconstant values of the equivalent clear water and characteristic mixture depths at the downstream spillway end, as given by the drawdown curves developed by Hager and Boes 2000 ; and 3 depth-averaged air concentration values within 20% of those obtained from the formula proposed by Hager 1991, for uniform self-aerated flow on smooth chutes of identical slope. Eq. 13 was then proposed for predicting the relative vertical length needed to attain uniform flow H dam,u /h c, according to which a value of about 20.5 is obtained for 52, typical of gravity dam spillways. Considering that a slight deviation in the adopted uniform flow depth may result in large error in the drawdown length, the results compared fairly well with those proposed by other authors, namely by Yildiz and Kas 1998, Matos and Quintela 1995, or Matos 2000a, and Ohtsu et al In Matos and Quintela 1995 or Matos 2000a, the suggested prediction for the relative vertical length needed to attain uniform flow H dam,u /h c was mostly based upon indirect or nonintrusive estimates of the mean air concentration. Further to these studies, new experimental data on air concentration and velocity were gathered in the 53 sloping stepped chute assembled at LNEC, Lisbon Matos 1999; Matos 2000b. The chute is 2.90 m high from crest to toe, 1.00 m wide, and the step height was 0.08 m. Unit discharges up to 0.2 m 3 /s/m h c =0.16 m were tested. The comparison of the authors formula with the data obtained at the LNEC chute was considered of interest namely, taking into account the different upstream boundary conditions at the entrance of the chute ogee profile for LNEC chute against the jet box system for the authors flume. In Fig. 1, typical characteristic depth profiles are plotted in function of the relative vertical length H dam /h c, for unit discharges of 0.08 and 0.14 m 2 /s, respectively; that is, h c /s=1.1 and 1.6 after Matos 1999, 2000b. The discusser s values of h w,u and h 90,u were computed after estimating the friction factor in uniform aerated flow as briefly described later. The results show that the authors criterion provides reasonably good estimates of equilibrium condition for both the equivalent clear water depth h w,u and the mixture depth h 90,u, particularly for h c /s=1.1, regardless of the dissimilar upstream boundary conditions of the chute. With regard to the mean air concentration, the values obtained at the LNEC chute for H dam,u /h c 20.5 were respectively 0.57 h c /s=1.1 and 0.55 h c /s=1.6, within 13% of the uniform value for smooth chutes of identical slope, as per Hager On the other hand, introducing Eqs. 3 and 5 on Eq. 2 gives: tan C u = F * 1 In Fig. 2, the relative differences between the application of Eq. 1 of this discussion and the formula proposed by Hager Fig. 1. Characteristic depths down the LNEC stepped chute =53 ; b=1.00 m; s=0.08 m : a h c /s=1.1; a h c /s=1.6; IP =inception point. Results from velocity and air concentration data VCD or visual observation OBS ; h m =mixture or bulked flow depths as per visual obervation on scales attached to the chute sidewalls after Matos 1999, 2000b. JOURNAL OF HYDRAULIC ENGINEERING ASCE / JUNE 2005 / 525

6 large values of f w obtained by Yasuda and Othsu 1999, in their important work, might in part be due to their model scale. In fact, the conditions needed for the exemption of scale effects, according to the findings of Boes 2000 and Boes and Hager 2003, were not completely fulfilled in the experimental work of Yasuda and Othsu 1999 i.e., Re 10 and We 100, as indicated in Matos et al Fig. 2. Relative differences between the mean air concentration obtained from Eq. 1 C u Eq. 1 and from the application of Hager 1991 formula for smooth chutes of identical slope C u Hager given as C u % = C ueq. 1 C uhager /C uhager * for smooth chutes are plotted in function of F *, for slopes of 30, 40, and 50. The results suggest a possible underestimation of the uniform mean air concentration for large F*, particularly on steep slopes e.g., 50. It should however be mentioned that the obtained differences are generally below 20%, in conformity with the criterion adopted by the authors to retain the h u,w values for the calculation of friction factors. It is also worth noting that the authors criterion seem to provide reasonably good estimates of equilibrium condition for the friction factor f w obtained from LNEC chute data f w =8 gh w 3 S/q w 2, where S is the estimate of the friction slope, namely for h c /s=1.1 Fig. 3. For h c /s= 1.6, the larger experimental data of H dam,u /h c is limited to 20.5, but the trend suggests a gradually varied flow for H dam,u /h c The values of f w obtained by the discusser Matos 1999 in the quasiuniform or gradually varied flow were 0.06 h c /s=1.1 and 0.10 h c /s=1.6, of the same order of magnitude of the value given by the authors for the 50 sloping chute taken f w f b Interestingly, the friction factors obtained by the authors for =30 taken f w f b 0.11 are identical to that proposed by Frizell et al. 1994, based on data gathered on a 27 sloping large outdoor flume f w 0.11, as well as to that hypothesized in Matos 1997, after reanalyzing the data of Rice and Kadavy 1996 f w 0.11, for =22. They are also similar to the average value for the data of Tozzi 1992, based on closed conduit air flow f w 0.09 for =27, as in Matos It is believed that the Effect of Aeration on Friction Factor The estimation of the friction factor in skimming flows over stepped spillways has been subject to diverging views over the years. Nevertheless, it is today well accepted that former studies that relied upon bulked flow depth measurements overestimated significantly the friction factor on steep chutes. In Matos 1997, this reasoning has been illustrated using the ratio f w / f bulked = 1 C 3, where f bulked is the friction factor computed on the basis of the bulked flow depth, in the quasi-uniform flow. As stated by the authors and also shown in Fig. 1, the bulked flow depth measurements describe the characteristic flow depth h 90 instead of the clear water depth h w. Fig. 4 shows that the results from the application of the above mentioned f w / f bulked ratio are quite similar to those from Eq. 22, for C 0.3; hence, the range of the C values gathered by the authors in quasiuniform skimming flow over steep slopes of 30, 40, and 50. Fig. 1 also shows that bulked flow depths based on visual observation can even overestimate the characteristic depth h 90, particularly in the vicinity of the rapidly varied flow region. These observations seem to conform with the authors reasoning for the differences on f w / f m obtained via the authors and Wahrheit- Lensing 1996 data Fig. 11. According to Fig. 11, the larger differences were obtained for low values of C, for which uniform flow conditions were likely not attained in the Wahrheit-Lensing chute, corresponding possibly to cross sections in the vicinity of the rapidly varied flow region. A major difficulty when comparing classical formulae for the drag reduction in air water flows in smooth chutes Wood 1985, 1991; Chanson 1994 with that expected for stepped chutes consists in the estimation of the fictitious friction factor of the skimming flow that would occur in the latter chutes, in the absence of air entrainment f. For moderate- to large-scale stepped chutes of practical interest, air entrainment is always present downstream of the inception point. How to estimate then the fictitious friction factor of the unaerated skimming flow on an identical chute, for the same discharge? To bypass the difficulties of defining and measuring water depths over the stepped spillway, Tozzi 1992, Fig. 3. Friction factor f w down the LNEC stepped chute =53 ; b=1.00 m; s=0.08 m, for h c /s=1.1 and h c /s=1.6 after Matos 1999 Fig. 4. Ratio f w / f m or f w / f bulked as function of the depth-averaged air concentration 526 / JOURNAL OF HYDRAULIC ENGINEERING ASCE / JUNE 2005

7 Fig. 5. Drag reduction in skimming flow down the LNEC stepped chute =53 ; b=1.00 m; s=0.08 m; 1.1 h c /s 2.0. Comparison with Chanson 1994 formulae for drag reduction caused by free-surface aeration on smooth chutes and rockfilled channels after Matos investigated the air flow in closed conduits, where the geometry of the roughness in the flow direction was equivalent to that found on a conventional stepped chute. Although such data would not accurately represent the friction factor of the air water flow due to the drag reduction f w inasmuch as considered by Tozzi 1992, 1994, it could be used to estimate the fictitious friction factor of the skimming flow on stepped chutes, f. Adopting this approach after reanalyzing the data of Tozzi 1992 for air flow in closed conduit with roughness geometry corresponding to the 53 sloping stepped chute, and considering the data gathered at the LNEC chute for computing f w f w =8 gh w 3 S/q w 2, where S is the estimate of the friction slope, the so obtained f w / f values are plotted in Fig. 5, along with the respective regression curve Matos Fig. 5 also includes the formulae proposed by Chanson 1994 for drag reduction caused by free-surface aeration on smooth chutes and rockfilled channels. The results suggest that the drag reduction on stepped chutes is larger than that found for smooth chutes, in analogy of the authors conclusion considering the approximation f w / f m, and the shape of the regression curve is similar to that corresponding to Chanson 1994 equation for rockfilled channels, for K/h w 0.2. References Boes, R. M Zweiphasenströmung und Energieumsetzung auf Grosskaskaden. PhD thesis, VAW, ETH Zurich, Switzerland in German. Boes, R. M., and Hager, W. H Two-phase flow characteristics of stepped spillways. J. Hydraul. Eng , Chanson, H Hydraulic design of stepped cascades, channels, weirs and spillways, Pergamon, Oxford, U.K. Frizell, K. H., Smith, D. H., and Ruff, J. F Stepped overlays proven for use in protecting overtopped embankment dams. Proc., ASDO Annual Conference, Boston. Hager, W. H Uniform aerated chute flow. J. Hydraul. Eng., 117 4, Hager, W. H., and Boes, R. M Backwater and drawdown curves in stepped spillway flow. Proc., Int. Workshop on Hydraulics of Stepped Spillways, VAW, ETH Zurich, H.-E. Minor and W. H. Hager, eds., Balkema, Rotterdam, The Netherlands, Matos, J Discussion of Model study of a roller compacted concrete stepped spillway by C. E. Rice and K. C. Kadavy. J. Hydraul. Eng., , Matos, J Emulsionamento de ar e dissipação de energia do escoamento em descarregadores em degraus Air entrainment and energy dissipation on stepped spillways. Research Report, IST, Lisbon in Portuguese. Matos, J. 2000a. Discussion of Hydraulics of skimming flow on modeled stepped spillways by G. G. S. Pegram, A. K. Officer, and S. R. Mottram. J. Hydraul. Eng., , Matos, J. 2000b. Hydraulic design of stepped spillways over RCC dams. Proc., Int. Workshop on Hydraulics of Stepped Spillways, VAW, ETH Zurich, H. -E. Minor and W. H. Hager, eds., Balkema, Rotterdam, The Netherlands, Matos, J., Yasuda, Y., and Chanson, H Interaction between freesurface aeration and cavity recirculation in skimming flows down stepped chutes. Proc., XXIX IAHR Congress, Beijing, China CD- ROM. Matos, J., and Quintela, A Guidelines for the hydraulic design of stepped spillways for concrete dams. ICOLD Energy Dissipation Bull.. Ohtsu, I., Yasuda, Y., and Takahashi, M Discussion of Hydraulics of skimming flow on modeled stepped spillways by G. G. S. Pegram, A. K. Officer, and S. R. Mottram. J. Hydraul. Eng., , Rice, C. E., and Kadavy, K. C Model study of a roller compacted concrete stepped spillway. J. Hydraul. Eng , Tozzi, M. J Caracterização/comportamento de escoamentos em vertedouros com paramento em degraus. PhD thesis. Univ. of São Paulo, Sao Paulo, Brazil in Portuguese. Tozzi, M. J Residual energy in stepped spillways. Int. Water Power Dam Constr., 5, Wahrheit-Lensing, A Selbstbelūftung und Energieumwandlung beim Abfluss über treppenförmige Entlastungsanlagen. PhD thesis, Univ. of Karlsruhe, Karlsruhe, Germany in German. Wood, I. R Air water flows. Proc., 21st IAHR Congress, Melbourne, Australia, keynote address, Wood, I. R Free-surface air entrainment on spillways. Air entrainment in free surface flows, IAHR hydraulic structures design manual no. 4, Hydraulic design considerations, I. R. Wood, ed., Balkema, Rotterdam, The Netherlands, Yasuda, Y., and Ohtsu, I Flow resistance of skimming flows in stepped channels. Proc., 28th IAHR Congress, H. Bergmann, R. Krainer, and H. Breinhälter, eds. CD-ROM, Graz, Austria, B14. Yildiz, D., and Kas, I Hydraulic performance of stepped chute spillways. Hydropower Dams 5 4, Closure to Hydraulic Design of Stepped Spillways by Robert M. Boes and Willi H. Hager September 2003, Vol. 129, No. 9, pp DOI: / ASCE :9 671 Robert M. Boes 1 and Willi H. Hager 2 1 Project Manager, TIWAG-Tiroler Wasserkraft AG, Hydro Engineering GmbH, A-6020 Innsbruck, Austria. robert.boes@tiwag.at 2 Professor, Head Hydr. Div., Lab. of Hydr., Hydrol. and Glaciol. VAW, Swiss Federal Institute of Technology ETH, ETH-Zentrum, CH-8092 Zurich, Switzerland. hager@vaw.baug.ethz.ch The writers appreciate the valuable comments of all the discussers. They are thankful to Chanson for his analyses of the effect of JOURNAL OF HYDRAULIC ENGINEERING ASCE / JUNE 2005 / 527

8 chute inflow conditions on flow resistance and friction factor; to Ghare, Porey, and Ingle for presenting friction factor data from their experimental model study; and especially to Matos, who carefully reanalyzed experimental data from his large-scale stepped chute model in the light of the authors analytical approach related to uniform two-phase flow, friction factor, and effect of aeration on friction factor. Experimental Configuration In response to Ghare et al. the writers would first like to give the vertical flume height from jetbox outlet to chute toe for =30, 40, and 50 i.e., H chute =2.86, 3.68, and 4.38 m, respectively. From Fig. 12 it may be seen that the writers data points amount up to about H dam /h c =75 on the x-axis. The design guidelines may thus readily be applied for dam heights up to 75 times the critical flow depth, meaning that the chute height is not a limiting factor for the guidelines applicability in practice. In most of the writers experiments, the stepped invert started right at the jetbox. In two-step configurations, a smooth invert connected the jetbox to the stepped chute part as indicated in Table 1. Energy Dissipation The writers experiments showed only a relatively small influence of the step height on energy dissipation, as expressed by the exponent 0.1 of the ratio K/D h,w in Eq. 24a and by the term log K/D h,w in Eq. 21. If in the design example with H dam =60 m H dam,u =70 m the step height is reduced to s=0.6 m, the energy dissipation ratio H/H max =1 H res /H max from Eq. 24a will decrease by only 4.3%, whereas for s=0.3 m, the energy dissipation will decrease by 8.6%, compared to a reference step height of 1.2 m. For uniform flow at the downstream spillway end e.g., for H dam =75 m, the bottom friction factor f b of Eq. 21 will decrease by 11.8 and 21.7% for step heights of 0.6 and 0.3 m, respectively, while the energy dissipation from Eq. 24b will be reduced by 3.0 and 5.9%, respectively. Because the step height has only a comparatively small effect on H res, it has not been plotted in Fig. 12, but it is instead explicitly expressed in the mentioned equations. The small effect of the roughness height K and thus the step height s on the bottom friction factor f b is shown in Fig. 9 and in Fig. 1 of this closure, where H/H max is plotted versus H dam /h c for different roughness heights K and =30 and 50 taken from Boes Stilling Basin Design The well-known sequent depths equation may be applied to the stilling basin design, provided the uniform equivalent depth h w,u for uniform flow, or the clear water depth at the chute end h w,e for developing flow is considered as the supercritical depth h 1 at the upstream basin end. The design then follows the classical procedure. Table 1. Step Configurations of the writers Experiments s mm Number N of steps Distance from jetbox outlet to first step mm The writers are satisfied with the comments of Matos, corroborating the findings reported in the paper relative to the attainment of uniform flow and to the uniform friction factor. The writers consider this agreement with large-scale experimental data derived from an ogee profile model as a further indication of the validity of their approach to calculate the fictitious location of the spillway crest of their jetbox system model Boes and Hager Using pressurized intake conditions instead of uncontrolled spillway crest conditions as for the majority of existing model studies involves a number of experimental advantages. The authors agree with Matos that the estimation of the friction factor in skimming flows has been subject to diverging views. Whereas Chanson hypothesizes in his discussion that the approach flow conditions e.g., jetbox system versus uncontrolled spillway crest have an effect on the friction factor, Matos s data contradict this statement. In fact, the f w values obtained from Matos s large-scale model with 53 and experimental data of a near-prototype facility with 27 Frizell et al agree well with the writers findings. Similar to Matos s opinion, the authors believe that in some of the experimental model studies reanalyzed by Chanson and listed in his discussion BaCaRa 1991; Chamani and Rajaratnam 1999; Yasuda and Ohtsu 1999, either scale effects could not be excluded due to small model geometries and flow rates, or air concentration and flow depth measurements were inappropriate for highly turbulent two-phase flow. These inaccuracies may lead to incorrect friction factors, to an overestimation of energy dissipation, and finally to a dangerous underdesign of relevant hydraulic structures. The authors note the corroboration of Eq. 22 and Fig. 11 by the analysis of Matos. Uniform Two-Phase Flow and Friction Factor 528 / JOURNAL OF HYDRAULIC ENGINEERING ASCE / JUNE 2005 Fig. 1. Relative energy dissipation H/H max as function of relative spillway height H dam /h c for =30, K= 20, 40, 80 mm; and =50, K= 20 and 60 mm; --- fit for =30 with f b =0.11, fit for =50 with f b =0.08 see Boes 2000

9 References BaCaRa Etude de la dissipation d énergie sur les évacuateurs á marches, Study of the energy dissipation on stepped spillways. Rapport d Essais, Projet National BaCaRa, CEMAGREF-SCP, Aixen-Provence, France in French. Boes, R. M Zweiphasenströmung und Energieumsetzung auf Grosskaskaden, Two-phase flow and energy dissipation on cascades. PhD thesis, Mitteilung Nr. 166, VAW, ETH Zurich, Switzerland in German. Boes, R. M. and Hager, W. H Closure to two-phase flow characteristics of stepped spillways. J. Hydraul. Eng , XX-XX. Chamani, M. R. and Rajaratnam, N Characteristics of skimming flow over stepped spillways. J. Hydraul. Eng , Frizell, K. H., Smith, D. H., and Ruff, J. F Stepped overlays proven for use in protecting overtopped embankment dams. Proc., ASDSO Annual Conference, Boston. Yasuda, Y. and Ohtsu, I Flow resistance of skimming flows in stepped channels. Proc., 28th IAHR Congress, Graz, Austria, H. Bergmann, R. Krainer, H. Breinhälter, eds., CD-ROM, Theme B14. JOURNAL OF HYDRAULIC ENGINEERING ASCE / JUNE 2005 / 529