Energy Dissipation down a Stepped Spillway with Non-Uniform Step Heights.

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1 See discussions, stats, and author profiles for this publication at: Energy Dissipation down a Stepped Spillway with Non-Uniform Step Heights. Article in Journal of Hydraulic Engineering November 2011 Impact Factor: 1.62 DOI: /(ASCE)HY CITATIONS 19 READS authors, including: Stefan Felder University of New South Wales 39 PUBLICATIONS 212 CITATIONS SEE PROFILE All in-text references underlined in blue are linked to publications on ResearchGate, letting you access and read them immediately. Available from: Stefan Felder Retrieved on: 09 May 2016

2 ENERGY DISSIPATION DOWN A STEPPED SPILLWAY WITH NON- UNIFORM STEP HEIGHTS Stefan Felder and Hubert Chanson School of Civil Engineering, The University of Queensland, Brisbane QLD 4072, Australia, Ph.: (61 7) , Fax: (61 7) , h.chanson@uq.edu.au Abstract: While most stepped spillway design guidelines were developed for uniform step heights, a nonuniform stepped design might be a practical alternative in some cases. Herein a physical study was conducted in a moderate slope stepped chute (1V:2H) and five stepped configurations were tested. Detailed air-water flow measurements were performed for each configuration and the results were compared in terms of flow patterns, energy dissipation and flow resistance. The basic findings showed minor differences between all configurations and indicated that the rate of energy dissipation was about the same for uniform and non-uniform stepped configurations. The observations suggested that the non-uniform stepped configurations might induce however some flow instabilities for smaller flow rates. Keywords: Stepped spillways, Non-uniform step heights, Energy dissipation, Air entrainment, Physical measurements. INTRODUCTION The design of stepped spillways is known for at least 3,500 years (Chanson 2001). Stepped spillways were used as flood release facilities of dams, as drop structures in Roman aqueducts and waterways and in water staircases and public fountains of famous gardens. Throughout the centuries, stepped chutes were a common design of hydraulic structures, but at the beginning of the 20th century some breakthroughs in the design of hydraulic jump stilling basins that led to the disuse of stepped spillways. With the development of new, more efficient construction techniques (e.g. roller compacted concrete (RCC)), the design of stepped spillways regained interest in the 1980s (Hansen and Reinhardt 1991, Chanson 1995,2001, Ditchey and Campbell 2000). This was associated with a substantial amount of physical modelling research (Sorensen 1985, Chamani and Rajaratnam 1999, Gonzalez and Chanson 2004, Carosi and Chanson 2008). Most experiments were conducted on stepped spillways with uniform flat steps to quantify the energy dissipation and to provide some design guidelines (Chanson 1995,2001, Boes and Hager 2003, Meireles and Matos 2009). But some prototype spillways are equipped with non-uniform step heights: e.g. Malmsburry (1870) and Upper Coliban (1903) stepped spillways in Australia (Chanson 2001). These spillways were designed with drops of 2 to 4 m followed by smaller steps of m (Fig. 1). The long operation of the spillways indicate that the design is sound although a nappe flow exists for the large drops and a skimming flow regime for the small step heights for the design flow (Chanson 2002). However, some flow instabilities and shock waves might occur for the non-uniform step heights as reported by Toombes and Chanson (2008) in the nappe flow regime and by Thorwarth and Köngeter (2006) for pooled stepped spillways. The only 1

3 experimental test of non-uniform step heights was conducted by Stephenson (1988) on a model with a slope of 45. In a test with occasional large drops, Stephenson observed an increase in energy dissipation of 10%. In the present study, the effects of non-uniform step heights on the air-water flow properties down a stepped chute are tested systematically for a wide range of discharges. It is the aim of this work to assess the effects of occasional large steps, and alternate large and small steps, on the rate of energy dissipation and flow aeration using a large-size facility with moderate slope (1V:2H). EXPERIMENTAL FACILITY AND INSTRUMENTATION The experimental study was conducted in the Gordon McKay Hydraulic Laboratory of the School of Civil Engineering at the University of Queensland. The physical setup was used in several previous studies (Chanson and Toombes 2002, Gonzalez 2005, Carosi and Chanson 2008, Felder and Chanson 2009a). The facility consisted of a large intake structure supplying a constant discharge through a sidewall convergent with a 4.8:1 contraction ratio to the stepped spillway test section. The test section comprised of a 1 m wide and 0.6 m long broad crest weir with upstream rounded corner followed by a 1 m wide and 1 m high stepped spillway section with a slope of 26.6 (1V:2H). At the downstream end of the spillway, a flat concrete channel discharged the water into a dissipation pit, from where the water was pumped to the upstream feeding basin. The discharge was measured with a pointer gauge from the upstream head above the weir crest using a discharge calibration function (Gonzalez and Chanson 2007). The air-water flow properties were recorded with a double-tip conductivity probe (Ø = 0.25 mm, Δx = 7.2 mm). The probe was supported by a trolley system in the longitudinal direction and the positioning of the probe sensor normal to the pseudo-bottom was controlled with a Mitutoyo digital ruler mounted on a fine adjustment screw-drive mechanism. The error in the translation of the probe in the direction normal to the flow was less than 0.5 mm. The accuracy on the longitudinal probe position was less than 0.5 cm and the transverse direction less than 0.1 mm (Carosi and Chanson 2008). For all experiments, the probe sensors were sampled for 45 s at 20 khz to yield meaningful results for all air-water flow properties including void fraction, interfacial velocity and depth-averaged airwater flow properties. EXPERIMENTAL FLOW CONDITIONS The experiments were conducted for a wide range of discharges between 0.02 and m 3 /s which corresponded to Reynolds numbers between and All experiments were conducted on the same spillway model with a slope of 26.6 but different configurations of step heights were investigated (Table 1). In some experiments, the stepped chute was equipped with uniform steps of 5 and 10 cm heights. In addition several non-uniform stepped configurations with combinations of 5 and 10 cm high steps were investigated. Table 1 summarises the conducted experiments and all the channel configurations are sketched in Figure 2. 2

4 BASIC FLOW PATTERNS On uniform stepped configurations, a nappe flow regime was observed for the smallest flow rates. For some intermediate flows, a transition flow was seen with some strong spray, splashing and flow instabilities. For the largest flow rates, the waters skimmed over the pseudo-bottom formed by the stepped edges. For all stepped spillway configurations, the flow patterns were observed for the full set of discharges. The observations included the locations of the inception point of air entrainment and the changes in flow regimes. Furthermore the flow pattern was captured in a series of photos for all flow regimes. Some photos for typical flows over non-uniform steps are shown in Figure 3. The observations of flow regime changes are listed in Table 2. Note that, for configuration C, the observation of flow regime changes was not definite. At the upstream end of the chute, the 5 cm high steps resulted in identical changes of flow regimes as observed for the stepped channel with uniform step heights of 5 cm, i.e. a change from nappe to transition flow for d c /h = 0.53 and from transition to skimming for d c /h = At the same time, the large drop of 10 cm at the lower end of the chute leads to different flow patterns downstream of the drop, i.e. a transition flow regime exists still for d c /h < 1.73 before the flow regime changes to a skimming flow regime while the change from nappe to transition flow is identical to the uniform step heights (Fig. 3D). In the following sections, the present study is focused on the flow properties in transition and skimming flows. The locations of the inception point of free-surface aeration were recorded for all configurations. They are presented in dimensionless terms as L I /k s as a function of a Froude number F* in Figure 4) Herein L I is the longitudinal distance from the first step edge to the inception point location and k s is the step cavity height normal to the flow direction k s = h cosθ. The Froude number F* is defined in terms of the step roughness: q w F* = (1) 3 g sin θ ks In Figure 4, the data are compared with the empirical correlation of Chanson (1995): LI ks (sin θ) F * = (2) and a simple linear correlation of Carosi and Chanson (2008): LI ks = F * 0.45 < d c /h < 1.6 (3) All data sets were in good agreement and indicated that the non-uniform stepped configurations did not have an impact on the location of the inception point of air entrainment. However, the large drop in configuration C was at the downstream end and a positioning further upstream might have yielded some different air entrainment inception locations for smaller flow rates. All data compared well with both correlation functions (Fig. 4). Downstream of the inception point of free-surface aeration, the flow was highly aerated (Fig. 3). The turbulence acting next to the free-surface was large enough to overcome surface tension and buoyancy to entrap the bubbles and carry them downwards. The air-water flow properties were recorded at each step edge 3

5 for all configurations (Table 1). The flow was rapidly varied immediately downstream of the inception point, and it became gradually-varied two to three step cavities downstream. However true uniform equilibrium flow conditions were not achieved since the chute was relatively short. In the gradually-varied flow region, the air concentration profiles were self-similar and followed closely an analytical solution of the air bubble diffusion equation: = yy (yy + 13) C 1 tanh K 2 D o 3 D o 3 (4) where C is the air concentration, y is the distance normal to the pseudo-bottom, Y 90 is the characteristic depth where C = 0.9, K' and D o are functions of the depth-averaged air concentration C mean only (Chanson and Toombes 2002). Equation (4) is compared with some data for a discharge (Fig. 5). The velocity distributions exhibited some self-similarity. The data were best correlated by a power law for y < Y 90 and an uniform profile above: V V y = Y N y <Y 90 (5a) V 1 V = y > Y 90 (5b) 90 where V 90 is the characteristic velocity at y = Y 90. The exponent was about N=10 for skimming flows, although the value may vary from one step edge to the next one for a given flow rate (Felder and Chanson 2009a). Other researchers found slightly different values (Matos 2000, Boes 2000; Bung 2009). ENERGY DISSIPATION AND AERATION The rate of energy dissipation, the flow resistance and some basic depth-averaged flow properties were calculated in both transition and skimming flow regimes for all configurations. For the design of stepped spillways, some basic parameters are the rate of energy dissipation ΔH/H max and the residual energy H res /d c at the downstream end of the stepped spillway. H max is the maximum upstream head above spillway toe H max = H dam + 3/2 d c, with H dam the dam height above spillway toe, ΔH is the total head loss ΔH = H max - H res and H res is the residual head at the measurement section: 2 Uw Hres = d cosθ+ (6) 2 g In Equation (6), the equivalent clear water flow depth d and the flow velocity U w must be calculated from the air-water flow properties. The clear-water flow depth d is defined as Y 90 d = (1 C) dy (7) y= 0 where C is the air concentration. By continuity, the flow velocity is: Y 90 U w = q w (1 C) dy (8) 0 4

6 where q w is the water discharge per unit width. The energy dissipation and residual energy were calculated at the last step edge for all configurations, i.e. at a longitudinal distance measured from the weir crest x = 2.01 m. For all flow configurations, a decreasing rate of energy dissipation with increasing discharge was observed which is consistent with earlier studies on stepped spillways (Chanson 1995, Matos 2000). Figure 6 illustrates the rate of energy dissipation in the present study as a function of a dimensionless height from the weir crest to the measured step edge Δz 0 /d c. First the decreasing energy dissipation with increasing discharge is visible. Second all the data sets were in close agreement. Some small differences between the different experimental configurations suggested the largest energy dissipation for a uniform spillway with 10 cm high steps. The results implied that Stephenson's (1988) observation of a 10% increase in energy dissipation for non-uniform step heights was not fulfilled in the present study. However, the differences for uniform step heights of 5 and 10 cm might suggest some scale effects, as observed in earlier studies of stepped spillway flows (Boes 2000, Chanson and Gonzalez 2005, Felder and Chanson 2009a). Similar findings were seen for the dimensionless residual head H res /d c (Fig. 7). Figure 7 shows some differences in the residual head for the different step configurations: the lowest residual head was achieved with uniform step height h = 0.10 m. For the smaller flow rates, the residual head decreased with increasing discharge for all experiments, while it was about constant for the largest flow rates. However, for the largest flow rates, the discharge was not fully developed at the downstream end of the spillway and the residual energy might be overestimated (Chanson 2001, Gonzalez and Chanson 2004, Felder and Chanson 2009b). In Figure 7, the residual head data are shown as a function of the dimensionless flow rate d c /h and compared with some simple design criterion for moderate slope stepped spillways: the upper dotted line expresses the median residual energy for spillway slopes smaller than 15.9 and the lower dashed line the median values for stepped spillways with slopes of 21.8 and 26.6 (Felder and Chanson 2009). The present findings agree very well with previous studies conducted with slopes between 3.4 and 26.6 which were used for the median values shown in Figure 7. A skimming flow on stepped spillways is characterised by significant form losses caused by the steps. The turbulent flow skims over the pseudo-bottom and initiates recirculating vortices in the cavities. Energy dissipation is caused by momentum exchanges between mainstream flow and cavity flow and by skin friction at the downstream half of each step. In the air-water flow region, the flow resistance is often expressed in the form of an equivalent Darcy-Weisbach friction factor (Rajaratnam 1990). The Darcy friction factor in airwater flow f e is the dimensionless average shear stress between the main stream and cavity recirculation, and it may be expressed as (Chanson et al. 2002): f Y 90 8 g S ( 1 C) dy 8 τ 8 g S d f 0 y= 0 f e = = = ρ w Uw Uw Uw where S f is the friction slope: S f = - H/ x, H is the total head, and x is the distance in flow direction. The experimental results are summarised in Figure 8 where the friction factor is plotted as a function of the dimensionless step roughness height h cosθ/d H with D H the hydraulic diameter. In Figure 8, the data are 5 (9)

7 compared with the solution of a mixing length model: f d = 1/(π 1/2 K) where 1/K is the dimensionless rate of expansion of the shear layer (Chanson et al. 2002). For all configurations, the Darcy-Weisbach friction factor varied between 0.12 and The findings were consistent with the re-analyses of flow resistance data showing variations of Darcy friction factors between 0.1 and 0.35 (Chanson 2006, Felder and Chanson 2009a). For comparison, in the non-aerated flow region, Amador et al. (2006) observed an average flow resistance of for d c /h = 2.1 and Re = DISCUSSION The results showed only small differences between all configurations in terms of flow pattern, energy dissipation and flow resistance. The present findings indicate that the design of stepped spillways with nonuniform step heights does not enhance the energy dissipation at the downstream end, as stated by Stephenson (1988). However the results present some useful information for alternative designs of stepped spillways. The introduction of larger steps, for example, might be triggered by no-hydraulic considerations: e.g., large step heights (h > 2 m) may be introduced to limit access on the stepped spillway by individuals or motorcycles (Chanson 2001). Another example is the Gold Creek dam stepped spillway (1890): designed with nineteen 0.76 m high steps, the spillway was built with twelve 1.5 m high steps because of limited cement availability (Cossins 2000). The current observations of the flow pattern indicated that the non-uniform stepped configurations might lead to some flow instabilities for smaller flow rates, larger flow depth and stronger splashing. Figure 3D illustrate a situation in configuration C. Some further analyses might provide some additional information. It is understood that the air-water flows on stepped spillways are affected by scale effects observed in earlier studies with geometrically similar scale models (Boes 2000, Chanson and Gonzalez 2005, Felder and Chanson 2009b). The present results suggested also some scale effects for the two uniform step geometries (h = 0.05 & 0.10 m) in terms of bubble count rate, turbulence levels, and air bubble sizes (not shown here). For the configurations with non-uniform step heights, some scale effect might also be present. CONCLUSION In the past three decades, a number of studies were performed to develop design guidelines for stepped spillways. The physical studies yielded some design guidelines for stepped chutes with flat horizontal steps for a wide range of channel slopes. It is understood that the stepped spillway design may provide some advantageous features in terms of energy dissipation and flow aeration performances. There is however a lack of understanding for stepped chutes with more complex design, in particular stepped spillways with non-uniform step heights, although some prototype stepped spillways have operated successfully for long periods. The present study investigated five configurations down a moderate slope stepped chute (1V:2H), and the comparative performances were discussed. The basic results showed minor differences between all configurations in terms of flow pattern, energy dissipation and flow resistance. The findings indicated that the rate of energy dissipation was about the same for uniform and non-uniform stepped configurations. The present work suggested that the design of stepped 6

8 spillways with non-uniform step heights does not enhance the energy dissipation at the downstream end, contrarily to Stephenson (1988). The observations showed that the non-uniform stepped configurations might induce some flow instabilities for smaller flow rates. Altogether, the results provide some practical information for alternative designs of stepped spillways with non-uniform step heights. ACKNOWLEDGEMENTS The authors thank Dr Jorge Matos (IST Lisbon) for his valuable comments. They thank further Graham Illidge, Ahmed Ibrahim, and Clive Booth (The University of Queensland) for their technical assistance. The financial support of the Australian Research Council (Grant DP ) is acknowledged. REFERENCES Amador, A., Sanchez-Juny, M., and Dolz, J. (2006). "Characterization of the Nonaerated Flow Region in a Stepped Spillway by PIV." Jl of Fluids Eng., ASME, Vol. 128, No. 6, pp Boes, R.M. (2000). "Zweiphasenströmung und Energieumsetzung an Grosskaskaden." ('Two-Phase Flow and Energy Dissipation on Cascades.') Ph.D. thesis, VAW-ETH, Zürich, Switzerland (in German). (also Mitteilungen der Versuchsanstalt für Wasserbau, Hydrologie und Glaziologie, ETH-Zürich, Switzerland, No. 166.) Boes, R.M., and Hager, W.H. (2003). "Hydraulic Design of Stepped Spillways." Jl of Hydr Engrg., ASCE, Vol. 129, No. 9, pp Bung, D. (2009). "Zur selbstbelüfteten Gerinneströmung auf Kaskaden mit gemässigter Neigung." Ph.D. thesis, University of Wuppertal, Germany, 292 pages (in German). (also Bericht - Lehr- und Forschungsgebiet Wasserwirtschaft und Wasserbau, University of Wuppertal, Germany, No. 16.) Carosi, G., and Chanson, H. (2008). "Turbulence Characteristics in Skimming Flows on Stepped Spillways." Canadian Journal of Civil Engineering, Vol. 35, No. 9, pp (DOI: /L08-030). Chamani, M.R., and Rajaratnam, N. (1999). "Characteristics of Skimming Flow over Stepped Spillways." Jl of Hyd. Engrg., ASCE, Vol. 125, No. 4, pp Chanson, H. (1995). "Hydraulic Design of Stepped Cascades, Channels, Weirs and Spillways." Pergamon, Oxford, UK, Jan., 292 pages. Chanson, H. (2001). "The Hydraulics of Stepped Chutes and Spillways." Balkema, Lisse, The Netherlands, 418 pages. Chanson, H. (2002). "Enhanced Energy Dissipation in Stepped Chutes. Discussion." Proceedings Institution Civil Engineers, Water and Maritime Engineering, UK, Vol. 154, No. 4, pp & Front cover. Chanson, H. (2006). "Hydraulics of Skimming Flows on Stepped Chutes: the Effects of Inflow Conditions?" Journal of Hydraulic Research, IAHR, Vol. 44, No. 1, pp Chanson, H., and Gonzalez, C.A. (2005). "Physical Modelling and Scale Effects of Air-Water Flows on Stepped Spillways." Journal of Zhejiang University SCIENCE, Vol. 6A, No. 3, March, pp Chanson, H., and Toombes, L. (2002). "Air-Water Flows down Stepped Chutes: Turbulence and Flow Structure Observations." Intl Jl of Multiphase Flow, Vol. 28, No. 11, pp

9 Chanson, H., Yasuda, Y., and Ohtsu, I. (2002). "Flow Resistance in Skimming Flows and its Modelling." Can Jl of Civ. Eng., Vol. 29, No. 6, pp Cossins, G. (2000). "The Gold Creek Dam Story." Institution of Engineers, Australia, Queensland Div., Brisbane, Australia. Ditchey, E.J., and Campbell, D.B. (2000). "Roller Compacted Concrete and Stepped Spillways." Intl Workshop on Hydraulics of Stepped Spillways, Zürich, Switzerland, H.E. Minor & W.H. Hager Editors, Balkema Publ., pp Felder, S., and Chanson, H. (2009a). "Energy Dissipation, Flow Resistance and Gas-Liquid Interfacial Area in Skimming Flows on Moderate-Slope Stepped Spillways." Environmental Fluid Mechanics, Vol. 9, No. 4, pp (DOI: /s y). Felder, S., and Chanson, H. (2009b). "Turbulence, Dynamic Similarity and Scale Effects in High-Velocity Free-Surface Flows above a Stepped Chute." Experiments in Fluids, Vol. 47, No. 1, pp (DOI: /s ). Gonzalez, C.A. (2005). "An Experimental Study of Free-Surface Aeration on Embankment Stepped Chutes." Ph.D. thesis, Department of Civil Engineering, The University of Queensland, Brisbane, Australia, 240 pages. Gonzalez, C.A., and Chanson, H. (2004). "Interactions between Cavity Flow and Main Stream Skimming Flows: an Experimental Study." Can Jl of Civ. Eng., Vol. 31, No. 1, pp Gonzalez, C.A., and Chanson, H. (2007). "Experimental Measurements of Velocity and Pressure Distribution on a Large Broad-Crested Weir." Flow Measurement and Instrumentation, Vol. 18, No. 3-4, pp (DOI /j.flowmeasinst ). Hansen, K.D., and Reinhardt, W.G. (1991). "Roller-Compacted Concrete Dams." McGraw-Hill, New York, USA, 298 pages. Matos, J. (2000). "Hydraulic Design of Stepped Spillways over RCC Dams." Intl Workshop on Hydraulics of Stepped Spillways, Zürich, Switzerland, H.E. Minor & W.H. Hager Editors, Balkema Publ., pp Meireles, I., and Matos, J. (2009). "Skimming Flow in the Nonaerated Region of Stepped Spillways over Embankment Dams." Jl of Hydr Engrg., ASCE, Vol. 135, No. 8, pp Rajaratnam, N. (1990). "Skimming Flow in Stepped Spillways." Jl of Hyd. Engrg., ASCE, Vol. 116, No. 4, pp Sorensen, R.M. (1985). "Stepped Spillway Hydraulic Model Investigation." Jl of Hyd. Engrg., ASCE, Vol. 111, No. 12, pp Stephenson, D. (1988). "Stepped Energy Dissipators." Proc. Intl Symp. on Hydraulics for High Dams, IAHR, Beijing, China, pp Thorwarth, J., and Köngeter, J. (2006). "Physical Model Tests on a Stepped Chute with Pooled Steps. Investigations of Flow Resistance and Flow Instabilities. Proc. Intl Symp. on Hydraulic Structures, IAHR, Ciudad Guayana, Venezuela, A. Marcano and A. Martinez Ed., pp Toombes, L., and Chanson, H. (2008). "Flow Patterns in Nappe Flow Regime down Low Gradient Stepped Chutes." Jl of Hydraulic Research, IAHR, Vol. 46, No. 1, pp

10 NOTATION C air concentration; C mean D H D o d d c h depth-averaged air concentration; hydraulic diameter (m); dimensionless constant function of the depth-averaged air concentration; clear-water flow depth (m) measured normal to the pseudo-bottom formed by the step edges; critical flow depth (m); vertical step height (m); F* Froude number defined in terms of the step cavity height; f Darcy-Weisbach friction factor; f d f e g H H max H res K K' k s L I l N q w Re S f U w V dimensionless shear stress in a mixing layer; equivalent Darcy-Weisbach friction factor in the air-water flow region; gravity acceleration (m 2 /s); total head (m); upstream total head (m); residual head (m); dimensionless expansion rate of mixing layer; dimensionless constant function of the depth-averaged air concentration; step cavity height (m): k s =h cosθ; longitudinal distance (m) between the first step edge and the inception point of free-surface aeration; horizontal step length (m); exponent; water discharge per unit width (m 2 /s); Reynolds number defined in terms of the flow velocity and hydraulic diameter; friction slope; flow velocity (m/s)' velocity (m/s); V 90 characteristic velocity (m/s) where C = 0.90; Y 90 characteristic distance (m) where C = 0.90; x longitudinal distance (m) from the first step edge; y distance (m) normal to the pseudo-bottom formed by the step edges; Greek symbols ΔH head loss (m); Δx longitudinal spacing (m) between probe sensors; Δz o dam height (m); 9

11 θ bed slope angle with the horizontal, positive downwards; ρ w water density (kg/m 3 ); τ o boundary shear stress (Pa); Subscript c critical flow conditions; 90 flow conditions where C =

12 Table 1 - Summary of experimental configurations with uniform and non-uniform step heights (Present study) C 2 steps with 5 cm height 18 steps with 5 cm height 1 steps with 10 cm height step edge 7 and 8 10 cm step between step edges 13 and d c /h Comment Configuration Steps Characteristic q w [m 2 /s] 10 cm 10 steps with 10 cm height Uniform step heights cm 20 steps with 5 cm height Uniform step heights A 10 steps with 5 cm height Regular alternation of one steps with 10 cm height 10 cm step followed by two cm steps B 9 steps with 10 cm height Two 5 cm steps between Calculation of d c /h with h = 0.1 m 0.7- Calculation of d c /h 3.5 with h = 0.05 m 0.7- Calculation of d c /h 1.85 with h = 0.1 m Calculation of d c /h with h = 0.1 m Calculation of d c /h with h = 0.05 m Table 2 - Summary of flow regime changes for different channel configurations in the present study (θ = 26.6º) Configuration d c /h d c /h Comment NA-TRA TRA-SK 10 cm (uniform step height) Calculation of d c /h with h = 0.1 m 5 cm (uniform step height) Calculation of d c /h with h = 0.05 m A Calculation of d c /h with h = 0.1 m B Calculation of d c /h with h = 0.1 m C (1.07) Calculation of d c /h with h = 0.05 m Notes: NA-TRA: change from nappe to transition flow; TRA-SK: change from transition to skimming flow. 11

13 LIST OF CAPTIONS Fig. 1 - Malmsbury dam stepped spillway (Australia) on 30 January Small steps chute with the large drop in the background Fig. 2 - Sketches of the experimental setup for all step edge configurations and definition of step numbering, θ = 26.6º Fig. 3 - Flow patterns over non-uniform stepped configurations (A) Skimming flows and cavity recirculation for configuration A, d c /h = 1.27, Re = , q w = m 2 /s, h = m, θ = 26.6º - Flow from top left to bottom right (B) Transition flow in large cavities and skimming flows in small cavities for stepped spillway in configuration B, d c /h = 0.85, Re = , q w = m 2 /s, h = steps 0.05 m, θ = 26.6º - Flow from right to left (C) Skimming flows and cavity recirculation in configuration C, d c /h = 1.7, Re = , q w = m 2 /s, h = step 0.1 m, θ = 26.6º - Flow from top right to bottom left (D) Skimming flow and deflected jets in configuration C, d c /h = 1.15, Re = , q w = m 2 /s, h = step 0.1 m, θ = 26.6º - Flow from top right to bottom left Fig. 4 - Inception points of free-surface aeration for uniform and non-uniform step heights - Comparison between experimental data and empirical correlations (Eq. (2) & (3)) Fig. 5 - Dimensionless distributions of air concentration and velocities downstream of the inception of freesurface aeration - Uniform step height configuration: h = 0.10 m, q w = m 2 /s, d c /h = 1/15 Fig. 6 - Energy dissipation for different step configurations in the present study; measurements at last step edge at downstream end (x =2.01 m) Fig. 7 - Residual head for different step configurations in the present study; measurements at last step edge at downstream end: x =2.01 m; median values are shown in dotted/dashed lines Fig. 8 - Flow resistance for different step configurations (Present study) 12

14 Fig. 1 - Malmsbury dam stepped spillway (Australia) on 30 January Small steps chute with the large drop in the background 13

15 Fig. 2 - Sketches of the experimental setup for all step edge configurations and definition of step numbering, θ = 26.6º 14

16 Fig. 3 - Flow patterns over non-uniform stepped configurations (A) Skimming flows and cavity recirculation for configuration A, d c /h = 1.27, Re = , q w = m 2 /s, h = m, θ = 26.6º - Flow from top left to bottom right (B) Transition flow in large cavities and skimming flows in small cavities for stepped spillway in configuration B, d c /h = 0.85, Re = , q w = m 2 /s, h = steps 0.05 m, θ = 26.6º - Flow from right to left 15

17 (C) Skimming flows and cavity recirculation in configuration C, d c /h = 1.7, Re = , q w = m 2 /s, h = step 0.1 m, θ = 26.6º - Flow from top right to bottom left (D) Skimming flow and deflected jets in configuration C, d c /h = 1.15, Re = , q w = m 2 /s, h = step 0.1 m, θ = 26.6º - Flow from top right to bottom left 16

18 Fig. 4 - Inception points of free-surface aeration for uniform and non-uniform step heights - Comparison between experimental data and empirical correlations (Eq. (2) & (3)) LI/ks θ = 26.6 ; h = 0.1 m (uniform step height) θ = 26.6 ; h = 0.05 m (uniform step height) θ = 26.6 ; h = m (Config. A) θ = 26.6 ; h = steps 0.05 m (Config. B) θ = 26.6 ; h = step 0.1 m (Config. C) Eq. (2) Eq. (3) F* Fig. 5 - Dimensionless distributions of air concentration and velocities downstream of the inception of freesurface aeration - Uniform step height configuration: h = 0.10 m, q w = m 2 /s, d c /h = 1/ C step edge 6 C step edge 7 C step edge 8 C step edge 9 C step edge 10 C step edge 8 - Theory V/V 90 step edges /10 th power law uniform profile y/y C, V/V 90 17

19 Fig. 6 - Energy dissipation for different step configurations in the present study; measurements at last step edge at downstream end (x =2.01 m) ΔH/Hmax θ = 26.6, h = 0.1 m (uniform step height) θ = 26.6, h = 0.05 m (uniform step height) θ = 26.6, h = m (Config. A) θ = 26.6, h = steps 0.05 m (Config. B) θ = 26.6, h = step 0.1 m (Config. C) Δz 0 /d c Fig. 7 - Residual head for different step configurations in the present study; measurements at last step edge at downstream end: x =2.01 m; median values are shown in dotted/dashed lines Hres/dc θ = 26.6, h = 0.1 m (uniform step height) 1.5 θ = 26.6, h = 0.05 m (uniform step height) 1.0 θ = 26.6, h = m (Config. A) θ = 26.6, h = steps 0.05 m (Config. B) 0.5 θ = 26.6, h = step 0.1 m (Config. C) d c /h 18

20 Fig. 8 - Flow resistance for different step configurations (Present study) fe, fd 0.1 f d, Mixing layer theory θ = 26.6, h = 0.1 m (uniform step height) 0.08 θ = 26.6, h = 0.05 m (uniform step height) 0.07 θ = 26.6, h = m (Config. A) 0.06 θ = 26.6, h = steps 0.05 m (Config. B) θ = 26.6, h = step 0.1 m (Config. C) h cosθ/d H 19