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1 THE UNIVERSITY OF QUEENSLAND SCHOOL OF CIVIL ENGINEERING REPORT CH97/15 HYDRAULICS OF THE DEVELOPING FLOW REGION OF STEPPED CASCADES: AN EXPERIMENTAL INVESTIGATION AUTHORS: Gangfu ZHANG and Hubert CHANSON

2 HYDRAULIC MODEL REPORTS This report is published by the School of Civil Engineering at the University of Queensland. Lists of recently-published titles of this series and of other publications are provided at the end of this report. Requests for copies of any of these documents should be addressed to the Civil Engineering Secretary. The interpretation and opinions expressed herein are solely those of the author(s). Considerable care has been taken to ensure accuracy of the material presented. Nevertheless, responsibility for the use of this material rests with the user. School of Civil Engineering The University of Queensland Brisbane QLD 4072 AUSTRALIA Telephone: (61 7) Fax: (61 7) URL: First published in 2015 by School of Civil Engineering The University of Queensland, Brisbane QLD 4072, Australia Zhang and Chanson This book is copyright ISBN No The University of Queensland, St Lucia QLD, Australia

3 Hydraulics of the Developing Flow Region of Stepped Cascades: an Experimental Investigation by Gangfu ZHANG Ph.D. Research student, The University of Queensland, School of Civil Engineering, Brisbane QLD 4072, Australia and Hubert CHANSON Professor, The University of Queensland, School of Civil Engineering, Brisbane QLD 4072, Australia, HYDRAULIC MODEL REPORT No. CH97/15 ISBN The University of Queensland, School of Civil Engineering, February 2015 Hinze Dam stepped spillway (Gold Coast QLD, Australia) in operation on 29 January Flow conditions: h = 1.2 m, = 51.3º, q = 16.6 m 2 /s

4 ABSTRACT Stepped spillways have been used for some millenia. The steps act as macro-roughness elements, contributing to enhanced energy dissipation and significant aeration. With typical design flow conditions, the water skims down the stepped chute as one large coherent stream: i.e., a skimming flow regime. The upstream flow motion is non-aerated and the free-surface appears smooth and glossy up to the inception point of free-surface aeration. In the non-aerated flow region, a turbulent boundary layer develops until the outer edge of the boundary layer interacts with the free-surface and air entrainment takes place. In this study, new experiments were performed in the developing flow region on a large size stepped spillway model (1V:1H, h = 0.10 m). The flow properties in the developing flow region were documented in detail. Downstream of the broad-crested weir and upstream of the inception point, the free-surface was smooth, although some significant free-surface curvature was observed for all discharges. The inception point of free-surface aeration was observed when the boundary layer thickness reached 80% of the flow depth: /d i 0.8. The location of the inception point of free-surface aeration and the flow depth at inception were comparable to previous laboratory and prototype results. In the developing boundary layer, the velocity distributions followed a 1/4.5 th power law at step edges. Detailed velocity and pressure measurements showed some rapid flow redistribution between step edges and above step cavities. The application of the momentum integral equation indicated an average friction factor of 0.18, close to the observed air-water flow friction factor of 0.23, suggesting that the spatially-averaged dimensionless shear stress was comparable in the developing flow region and fully-aerated flow region. Keywords: Stepped spillways, Developing turbulent boundary layer, Physical modelling, Velocity distributions, Pressure field, Total head, Boundary shear stress, Inception point of free-surface aeration. ii

5 TABLE OF CONTENTS Page Abstract Keywords Table of contents List of symbols ii ii iii iv 1. Introduction 1 2. Experimental facilities and instrumentation 4 3. Flow patterns Broad crested weir 3.2 Stepped spillway chute 4. Flow properties in the developing flow region Presentation 4.2 Velocity distributions 4.3 Pressure distributions 4.4 Total head and energy dissipation 5. Conclusion Acknowledgments 35 APPENDICES Appendix A - Photographs of experiments A-1 Appendix B - Developing boundary layer characteristics A-4 Appendix C - Velocity, pressure and total head data A-6 Appendix D - Pressure distributions in developing flows on stepped cascades A-18 REFERENCES R-1 Bibliography R-4 Open Access Repositories R-5 Bibliographic reference of the Report CH97/15 R-6 iii

6 LIST OF SYMBOLS The following symbols are used in this report: C void fraction; C D d d c d i dimensionless discharge coefficient; water depth (m); critical flow depth (m); flow depth (m) inception point of free-surface aeration; E specific energy (m): E = H t - z o ; E min minimum specific energy (m); F* dimensionless discharge: q F* 3 g sinθ k f (a) Darcy-Weisbach friction factor; (b) dimensionless boundary shear stress; f e f MI f Sf f xy g H dam H max H p H r H t H 1 h k s s dimensionless shear stress in fully-developed air-water flows downstream of inception point of free-surface aeration; dimensionless shear stress derived from von Karman momentum integral calculations; dimensionless shear stress derived from friction slope calculations; spatially-averaged dimensionless Reynolds shear stress along a step cavity; gravity acceleration (m/s 2 ): g = 9.80 m/s 2 in Brisbane, Australia; dam height (m); maximum available total head (m); piezometric head (i.e. static head) (m); depth-averaged specific energy (m) on the stepped chute; total head (m); upstream head above crest (m); vertical step height (m); step roughness height (m): k s = h cos ; L longitudinal distance (m) positive downstream measured from step edge 1; L cav step cavity length: L cav = (l 2 +h 2 ) 1/2 ; L crest L i l P Q q R r S o broad-crested weir length (m); distance between step edge 1 and inception point of free-surface aeration; horizontal step length (m); pressure (Pa); water discharge (m 3 /s); water discharge per unit width (m 2 /s); normalised correlation coefficient; radius of curvature (m); bed slope: S o = sin ; iv

7 t time (s); V flow velocity (m/s) positive downstream; V x V 0 v x v x W x longitudinal velocity component (m/s); free-stream velocity (m/s); longitudinal turbulent velocity fluctuation (m/s); normal turbulent velocity fluctuation (m/s); channel width (m); longitudinal distance (m) positive downstream measured from step edge; Y 90 characteristic distance (m) where C = 0.90; y distance (m) normal to the invert, measured perpendicular to the pseudo-bottom formed by the step edges (on the stepped section); y ub upper bound of the shear layer (m); y 0.9 characteristic distance (m) where V x = 0.9 V 0 ; z o bed elevation (m); Boussinesq coefficient; boundary layer thickness (m); displacement thickness (m); momentum thickness (m); energy thickness (m); dimensionless variable; pressure correction coefficient; water viscosity (Pa.s); water density (kg/m 3 ); o Ø boundary shear stress (Pa); diameter (m); Subscript c critical flow conditions; i inception point of free-surface aeration; 90 90% void fraction; Abbreviations AEB advanced engineering building; AMCA air movement and control association; ASHRAE American society of heating, refrigerating and air conditioning engineers; CFD computational fluid dynamics; dslr digital single-lens reflex (camera); d/s downstream; fps frames per second; v

8 IST LNEC PIV Std s UDC UQ u/s Instituto Superior Técnico; Laboratório Nacional de Engenharia Civil; particle image velocimetry; standard deviation; second; Universidad de la Corũna; The University of Queensland; upstream. vi

9 1. INTRODUCTION Historical records have indicated that stepped spillways have been used for thousands of years (Chanson 1995, ). The steps act as macro-roughness elements, contributing to enhanced energy dissipation and significant aeration. Recent developments in construction techniques and materials, in particular roller compacted concrete (RCC) and reinforced gabion mesh wires, led to a regained interest in stepped spillway design (Chanson 2001, Chanson et al. 2015). With typical design flow conditions, the water skims down the stepped chute as one large coherent stream: i.e., in a skimming flow regime (Horner 1969, Rajaratnam 1990, Peyras et al. 1992) (Figure 1.1). The overflow streamlines are about parallel to the pseudo-bottom formed by the step edges. Upstream of the aerated flow region, the flow is smooth and glassy with a developing boundary layer underneath. Once the boundary layer outer-edge becomes close to the free-surface, the turbulent shear stresses overcome the combined buoyancy and surface tension and initiate a process of rapid air entrainment (Figure 1.1). Figure 1.1 presents skimming flows down prototype stepped spillways; the location of the inception point is clearly marked. For high specific discharges, the boundary layer may not reach the surface and aeration may not occur along a stepped chute. This issue is particularly relevant to small to medium dams operating with large unit discharges (Gonzalez and Chanson 2007a, Meireles and Matos 2009). In absence of self-aeration, the spillway might be prone to some cavitation damage, although all the prototype observations indicated an absence of cavitation pitting and damage to the steps (Chanson 2001,2015, Frizell et al. 2013). For the last two decades, the majority of research on stepped chutes focused on the hydraulics of the self-aerated flow region downstream of the inception point. The air-water flow properties were investigated systematically (Chanson 1993a,1994,2006a, Ohtsu and Yasuda 1997, Toombes 2002, Gonzalez 2005, Chanson and Carosi 2007, Felder 2013, Wuthrich and Chanson 2014) and the results yielded some preliminary design guideline. On the other hand, the developing flow region was less studied, but in a few pertinent experimental investigations (Table 1.1). These provided results in terms of the clear water flow depth, boundary layer growth, and velocity profiles (Meireles et al. 2012). Amador et al. (2006, 2009) performed a characterisation of the developing flow using particle image velocimetry (PIV), although for a small range of flow conditions. The time-averaged velocity, boundary layer development and water level data were reproduced successfully using computational fluid dynamics (CFD) modelling (Bombardelli et al. 2011) for a limited number of experimental data. The present study aims to provide new experimental informations on the flow patterns and hydrodynamics of the developing flow region on a 1V:1H stepped spillway model. The boundary layer growth, velocity and pressure distributions, and energy dissipation were studied in the large- 1

10 size facility under controlled flow conditions. The study outcomes provide a better characterisation of the developing flow region on stepped spillways including flow resistance estimates and highlight several challenges faced by the design engineers. (A) Hinze Dam stepped spillway (Australia) in operation on 29 January 2013 (Photograph H. Chanson) - Flow conditions: h = 1.2 m, = 51.3º, q = 16.6 m 2 /s, d c /h = 2.53 (Chanson 2013) (B) Paradise Dam stepped spillway (Australia) in operation on 5 March 2013 (Photograph H. Chanson) - Flow conditions: h = 0.62 m, = 57.4º, Q = 2,300 m 3 /s, q = 7.4 m 2 /s, d c /h = 2.85 Figure 1.1 Skimming flows on prototype stepped spillways - Red arrows point to inception point of free-surface aeration 2

11 Table 1.1 Laboratory studies of the developing flow region on stepped spillways: comparative experimental and flow conditions Reference (º) h (m) W (m) Crest design Q (m 3 /s) q (m 2 /s) d c /h Instrumentation Remarks Present study Broad-crest with Prandtl-Pitot tube AEB Hydraulics u/s and d/s rounding ( =3.18 mm), double-tip Laboratory,UQ conductivity probe ( =0.25 mm) Amador (2005) WES ogee with Particle image Setup #2 (UDC) Meireles et al. (2012) 3 small steps velocimetry (PIV) WES ogee with 0.08 to to to 2.0 Back-flushing Pitot tube, Experiments at LNEC by small steps conductivity probe Matos (1999) WES ogee with 0.05 to to to 3.7 Back-flushing Pitot tube, Experiments at LNEC by small steps conductivity probe Meireles (2004) WES ogee with 0.10 & & & 4.0 Back-flushing Pitot tube, Experiments at LNEC small steps 0.10 to to to 8.0 conductivity probe by Renna (2004) Notes: d c : critical flow depth; h: vertical step height; Q: water discharge; q: water discharge per unit width; W: channel width; : slope angle with horizontal. 3

12 2. EXPERIMENTAL FACILITIES AND INSTRUMENTATION 2.1 EXPERIMENTAL FACILITY New experiments were conducted in a large-size steep spillway physical model (1V:1H) located in advanced engineering building (AEB) at the University of Queensland (UQ). A photograph of the experimental facility and a definition sketch are provided in Figure 2.1. Further photographs are presented in Appendix A. The facility was 12.4 m long. The flow was delivered by three pumps driven by adjustable frequency AC motors. Water was fed into a 1.7 m deep 5 m wide intake basin with a surface area of m 2, leading to a 2.8 m long side-wall convergent with a contraction ratio of 5.08:1, resulting in a smooth and waveless flow in the m wide test section. The inflow upstream of the test section was controlled by a broad crested weir. The weir consisted of a 1.2 m high, 0.6 m long and m wide crest with a vertical upstream wall, an upstream rounded nose (0.058 m radius), and a downstream rounded edge (0.012 m radius, Figure 2.1B inset). The crest was made of smooth, painted marine ply. The test section was equipped with 12 impervious flat steps made of plywood. Each step was 0.1 m long, 0.1 m high and m wide. The discharge was deduced from detailed velocity and pressure measurements performed above the broad crested weir which gave: Q W H 2 ( (2.1) L 1 3 ) g ( H1) crest 3 where W is the crest width (W = m), H 1 is the upstream total head above crest and L crest is the crest length (L crest = 0.60 m). Equation (2.1) takes into account the effects of the developing boundary layer along the finite length of the weir (Isaacs 1981, Gonzalez and Chanson 2007b) (see Section 3.1). Clear-water flow depths were measured with a pointer-gauge on the channel centreline. The free-surface profiles were photographed with an Apple TM iphone 5 for additional checks. Water level data were checked with a dual-tip phase detection probe. The probe was designed to record rapidly varying air-water interfaces based upon changes in resistivity. The probe sensor consisted of a stainless-steel needle encasing a platinum tip ( = 0.25 mm) insulated by a thin layer of epoxy. The probe was excited by an electronic system, and sampled at 20 khz per sensor for 45 s. 4

13 (A) Stepped spillway in operation for d c /h = 0.82 (B) Dimensioned sketch Figure 2.1 Experimental facility at the University of Queensland - h = 0.1 m, θ = 45 Total and static head measurements were performed in the clear water flow region on the channel centreline with a Dwyer 166 Series Prandtl-Pitot tube. The 3.18 mm diameter tube was made of corrosion resistant stainless steel, and featured a hemispherical total pressure tapping (Ø = 1.19 mm) at the tip with four equally spaced static pressure tappings (Ø = 0.51 mm) located 25.4 mm 5

14 behind the tip. The tip design met AMCA and ASHRAE specifications and the tube did not require calibration ( 1 ). Any effect resulting from the longitudinal separation between the total and static tappings were taken into account, by repeating independent total and piezometric head measurements at each location as illustrated in Figure 2.2. The Prandtl-Pitot tube was connected to an inclined manometer, with the tubes opened to the atmosphere to give total head and piezometric head readings. The readings were processed to yield the total head, pressure and velocity data. The vertical movement of the Prandtl-Pitot tube was controlled by a fine adjustment travelling mechanism connected to a Mitutoyo digital scale giving an accuracy of ±0.01 mm in the direction normal to the invert. The accuracies of the longitudinal and transverse positions of the tube were estimated to be ±0.5 cm and ±1 mm respectively. Additional observations were recorded with digital single lens reflex (dslr) cameras. 2.2 EXPERIMENTAL FLOW CONDITIONS Although the present study focused on the skimming flow regime, preliminary tests showed that a nappe occurred for d c /h < 0.4, where d c is the critical flow depth and h is the vertical step height (h = 0.1 m). A transition flow was observed for 0.4 < d c /h < 0.9, and skimming flows were seen for d c /h > 0.9. The overflow was characterised by a series of small water nappes cascading down the stepped chute for d c /h < For 0.15 < d c /h < 0.4, the small nappes were replaced by a clear dense supercritical jet above step edge 2 and a large aerated jet deflecting off step edge 5. The transition flow regime was characterised by large hydrodynamic instabilities and strong splashing for 0.4 < d c /h < 0.9. The flow appeared highly chaotic and a large deflecting jet developed between step edges 2 and 5 for smaller transition discharges (d c /h < 0.6). For d c /h > 0.9, the water skimmed above the pseudo-bottom formed by the step edges. The changes in flow regimes were consistent with the literature data (Chanson 2001). Herein the developing flow measurements were focused on the skimming flow regime ( 2 ). The experimental flow conditions are summarised in Table 2.1. Table 2.1 Experimental flow conditions: stepped chute measurements (Present study) Study location Q (m 3 /s) d c /h Location Flow regime Step edge Step edges 2 to 8 Skimming flow Step cavity Step cavities 2 to 4 Skimming flow 1 Reference: 2 For d c /h < 0.9, the Pitot tube measurements were deemed physically infeasible because of the small nappe thickness in a nappe flow and the intense turbulence and high air content in a transition flow. 6

15 Dynamic tapping Static tappings (4 total, 1 shown) Figure 2.2 Pitot tube positioning during flow measurements at step edges - Flow direction from left to right 7

16 3. FLOW PATTERNS 3.1 BROAD CRESTED WEIR The discharge on the steep stepped chute was controlled by the upstream broad crested weir (Figure 3.1). Visual observations and Prandtl-Pitot tube measurements were conducted to characterise the flow conditions above the weir crest. A definition sketch is provided together with a photograph in Figure 3.1. Quiescent inflow conditions were observed for all investigated discharges above the broad-crested weir. The flow accelerated above the upstream rounded nose. Next to the upstream end of the crest the flow was rapidly varied and characterised by some rapid change in free-surface curvature and pressure and velocity distributions. For moderate to large discharges (H 1 /L crest > 0.17), the free-surface fell continuously in the flow direction (Figure 3.2). The flow was critical over the length of the crest (Figure 3.3). For the smallest discharges (H 1 /L crest < 0.11), the watersurface above the crest showed some characteristic wavy shape and the overflow was subcritical over most of the crest length. This characteristic is most likely in consequence of the effect of a developing boundary layer at low flow rates which caused significant energy loss (Isaacs 1981, Chanson 1996). Within a framework of assumptions ( 3 ) (Liggett 1993), the depth-averaged specific energy may be expressed as (Chanson 2006b): d 2 V P x y dy 2 g g 2 0 V E d (3.1) d 2 g where E is the depth-averaged specific energy, V x is the longitudinal component, P is the pressure, y is the distance normal to the invert, V is the depth-averaged velocity, y is the vertical distance above the crest, ρ is the density of water, d is the flow depth, β is the Boussinesq coefficient (i.e. momentum correction coefficient), and Λ is the pressure coefficient defined as: d d 0 P dy g d Assuming a uniform velocity profile (β = 1) and a hydrostatic pressure distribution (Λ = 1), Equation (3.1) reduces to the classical expression: 2 (3.2) V E d (3.3) 2 g 3 including that the vertical acceleration equal to zero, the channel slope is small, the flow is incompressible and there is no heat transfer (Liggett1993). 8

17 cm H 1 Lcrest = 0.6 m Upstream rounding r = m Small rounding r = m (A) Definition sketch of broad-crested weir (B) Flow over broad-crested weir H 1 /L crest = Figure 3.1 Flow over broad-crested weir - Flow direction from left to right d/lcrest H 1 /L crest = H 1 /L crest = H 1 /L crest = H 1 /L crest = H 1 /L crest = H 1 /L crest = x/l crest 9 H 1 /L crest = H 1 /L crest = H 1 /L crest = H 1 /L crest = pointer gauge data broad crested weir Figure 3.2 Dimensionless free-surface profiles above the broad-crested weir - Photographic and pointer gauge (solid circles) data

18 In an open channel, critical flow conditions occur when the specific energy is minimal: E = E min (Bakhmeteff 1932, Henderson 1966). In real fluid flows, the velocity distributions are not uniform because of bed shear and boundary layer development. Furthermore, the pressure distributions above a broad-crested weir are not always hydrostatic, as shown by Gonzalez and Chanson (2007b) and Felder and Chanson (2012). For a smooth overflow, a unique relationship between the critical depth d c, the minimum specific energy E min, the dimensionless discharge coefficient C D ( 4 ) and the pressure and momentum coefficients Λ and β was proposed by Chanson (2006b): d c d c 1 1 C D 2 0 E min E (3.4) min 2 3 The critical depth must satisfy one of the four physical solutions of Equation (3.4), depending upon the sign of the discriminant Δ (Chanson 2006b): C D C D 3 1 (3.5) These four solutions are: E d d E d E c min c min c min Λ (1 2 β C (1 2 β C 2 D 2 D Λ Λ 2 ) 2 ) Λ Λ 6 6 Δ Δ 1 3 for Δ > 0 (3.6) 2 for Δ = 0 (3.7) cos for Δ < 0 (3.8) where: E d c min cos 2 1 cos 3 1 cos for Δ < 0 (3.9) C D (3.10) To date meaningful solutions exist only for 0 (Chanson 2006b). Assuming no energy loss over the weir crest, the specific energy at the weir crest is the minimum specific energy E min and it equals the upstream head above crest H 1 (Fig. 3.1A). Equations (3.7) to (3.10) provide an expression of the critical flow depth for the general case (Chanson 2006b). Figure 3.3 presents experimental data 4 The discharge coefficient is defined as: C 2 (Q / B) / g (Chanson 2006b). 3 D E min 10 3

19 from the present study in terms of the dimensionless critical depth d c Λ/H 1 as a function of the dimensionless discharge β C 2 D Λ 2 for 0.17 < x/l crest < Overall a reasonable agreement was obtained between experimental data and Equations (3.8) and (3.9). The finding suggested that critical flow conditions occurred herein for 0.17 < x/l crest < The result was consistent with the earlier study of Felder and Chanson (2012). 1 Eq, (3.8) Eq. (3.9) H 1 /L crest = H 1 /L crest = H 1 /L crest = H 1 /L crest = H 1 /L crest = H 1 /L crest = H 1 /L crest = H 1 /L crest = H 1 /L crest = H 1 /L crest = Felder & Chanson 0.8 dc/h C 2 D 2 Figure 3.3 Dimensionless critical flow depth as a function of pressure, momentum and discharge coefficients above the weir crest - Comparison with analytical solutions (Equations. (3.8) & (3.9)) and rounded broad-crested weir data (Felder and Chanson 2012, L crest = 1.01 m). 3.2 STEPPED SPILLWAY CHUTE Presentation The present investigation focused on the skimming flow regime, typical of most design flow conditions for modern gravity dam stepped spillways. Herein a skimming flow was observed for d c /h > 0.85 to 0.9, in good agreement with the data re-analyses of Chanson and Toombes (2004). The general flow pattern is sketched in Figure 3.4 and photographs are shown in Appendix A. For d c /h > 0.85 to 0.9, the flow skimmed over the pseudo-bottom formed by the step edges and the streamlines were approximately parallel. At the upstream end, the free-surface was smooth and glossy. Some free-surface undulation was observed approximately in phase with the steps, for all 11

20 skimming discharges. Further downstream, the free-surface fluctuated significantly as the boundary layer developed. When the outer edge of the developing boundary layer reached the vicinity of the free-surface, the turbulent shear stresses acting next to the free-surface dominated over the combined effects of surface tension and buoyancy, causing air entrainment (Rao and Rajaratnam 1961, Ervine and Falvey 1987, Chanson 1993b,2008). The instantaneous location of the inception point of free-surface aeration (Section 3.2.3) was influenced by the boundary layer fluctuations. Below the pseudo-bottom, the cavity fluid exhibited a circulatory motion sustained by external momentum transfer from the mainstream flow. A close examination of the cavity vortices revealed patterns showing irregular ejection of fluid from the cavity into the mainstream next to the upper portion of the vertical step face, and replacement of cavity fluid next to the step edge, indicating a high degree of mainstream-cavity interaction, as discussed by Rajaratnam (1990) and Chanson et al. (2002). Downstream of the inception point of free-surface aeration, the flow was self-aerated and air-water flow measurements showed that the velocity profiles were fully-developed (Zhang and Chanson 2015). Figure 3.4 Skimming flow pattern on the stepped spillway Free-surface profiles Water depth measurements were performed along the channel centreline with a pointer gauge for a range of discharges (Table 2.1). The dimensionless free-surface profiles are plotted in Figure 3.5 in 12

21 terms of the normalised streamwise distance L/L i, where L is the distance downstream of step edge 1 measured along the pseudo-bottom and L i is the inception point location (Figure 3.4). For the largest discharge (d c /h = 1.7), the data were checked against the phase-detection probe outputs, by integrating the void fraction profiles along the water column: Y 90 d (1 C) dy (3.11) 0 where d is the equivalent clear-water depth, C is the (time-averaged) void fraction, and Y 90 is the depth at which C = These data are presented in Figure 3.5 with red star symbols. The results showed a close agreement between pointer gauge and phase-detection probe data (Figure 3.5). Some slight difference was observed downstream close to the inception point and might be on account of rapid free-surface flapping induced by turbulence observed in earlier studies (Chamani 2000, Chanson 2001). The data revealed a wavy free-surface for all discharges (Figure 3.5). The curved nature of the free-surface is illustrated in Figure 3.6. The amplitude of the surface waves was the largest above the first few steps and gradually decreased in the flow direction for a given discharge, and the wave length was about two step cavity lengths. The free-surface curvature was most significant for the smaller discharges, with an estimated radius of curvature as small as 0.2 m (i.e. r/h as low as 2). Since the free-surface was the upper streamline, the streamline curvature indicated some vertical acceleration in the upper flow column and non-hydrostatic pressure distributions. The characteristics of the developing boundary layer were derived from the velocity profiles (Section 4.2). The boundary layer, displacement, momentum and energy thicknesses are defined respectively as: δ (3.12) y V 0.99 x V 0 Vx 1 (1 ) dy V 0 0 δ (3.13) Vx Vx 2 (1 ) dy V 0 0 V0 δ (3.14) 2 Vx Vx 3 (1 ) dy V V0 δ (3.15) where the boundary thickness δ is defined in terms of 99% of the free-stream velocity, V x is the longitudinal velocity component, V 0 is the free-stream velocity, δ 1 is the displacement thickness, δ 2 is the momentum thickness and δ 3 is the energy thickness. Figure 3.7 compares the longitudinal variations in boundary layer characteristics, together with the free-surface profiles, for one discharge d c /h = 1.5. Note the different scales for the left and right vertical axes. As seen in Figure 13

22 3.7, the boundary layer thickness increased monotonically towards the free-surface while the flow depth decreased in the downstream direction. The full data set is reported in Appendix B. The onset of air entrainment occurred once the boundary layer outer edge reached the close proximity of the free-surface: e.g., at approximately L/k s = 10 for d c /h = 1.5, where k s is the step roughness height: k s = h cos. At this point the boundary layer thickness was about: /d = /d i 0.8 and this reflected that the outer edge of the boundary layer was irregular and fluctuating (Klebanoff , Antonia 1972, Phillips and Ratnanather 1990). d/dc d c /h = 0.9 d c /h = 1.0 d c /h = 1.1 d c /h = 1.3 d c /h = 1.5 d c /h = 1.7 d c /h = 1.7 (conductivity probe) L/L i Figure 3.5 Longitudinal profiles of dimensionless water depth upstream of the inception point of free-surface aeration Step edge 2 Figure 3.6 Side view of water surface profile - Flow from left to right, d c /h = 0.9, Aperture: f/5.6, Shutter speed = 1/50 s, ISO 1600 Lens distortion correction using PTLens 14

23 d/dc, /dc d L/k s Figure 3.7 Longitudinal water surface profile: comparison with boundary layer thickness, displacement thickness, momentum thickness and energy thickness data Flow conditions: d c /h = 1.5, h = 0.10 m Note different scales of left and right vertical axes /dc, 2/dc, 3/dc Boundary layer growth At the upstream end of the stepped chute, a turbulent boundary layer developed along the stepped invert up to the inception point where the boundary layer outer edge interacted with the free-surface (Wood et al. 1983, Chanson 1994). Upstream of the inception point of aeration, the water column may be divided into a boundary layer of thickness δ and an ideal flow region above. Energy was dissipated in the boundary layer in the form of viscous and turbulent dissipations. The dimensionless boundary layer growth is illustrated in Figure 3.8. Compared to a smooth spillway, the steps acted as macro-roughness elements, causing the onset of aeration to occur earlier than on a smooth chute (Chanson 1994,2001, Amador et al. 2009). The present data set was correlated by: δ L 0.37 L 0.15 k s (3.16) where L is the streamwise distance from the first step edge, and k s = h cos is the step cavity height. Equation 3.16 is valid for 0.9 d c /h 1.7 and 0 < L/k s < 15. The complete dataset is plotted in Figure 3.8A; the data complemented the earlier findings of Amador et al. (2009) and Meireles et 15

24 al. (2012) ( 5 ) respectively on 1V:0.8H and 1V:0.75H ogee-crested stepped chutes, but for a larger range of L/k s ( 6 ). In Figure 3.8A, the empirical correlation of Amador et al. (2009) was extended to the present data range. For L/k s > 5 to 10, the difference between present data, Equation (3.16), the data of Meireles et al. (2012) and the correlation of Amador et al. (2009) tended to be negligible. Altogether Equation (3.16) is seen to provide a reasonable estimate of the boundary layer growth for all data sets (Figure 3.8A). This suggests that the boundary layer development was little influenced by the type of crest and the chute slope for 0.9 d c /h 1.7. Although the present data appeared to contrast with the results of Chanson (2006a) covering a wider range of slope, crest profiles and flow conditions, there was some data scatter for small vales of L/k s (L/k s < 8), possibly caused by the relative influence of the crest depending on the discharge. Figure 3.8B illustrates the boundary layer growth between step edges 2 and 5 as a function of the Reynolds number Re L = V o L/, with the water density, and the water dynamic viscosity. The present data were best correlated by: δ L δ1 L δ2 L δ3 L Re L with R = (3.17) ReL with R = (3.18) ReL with R = (3.19) ReL with R = (3.20) for < Re L < , with R the normalised correlation coefficient. Note the relatively low correlation coefficients, partly caused by the limited data sets, data scatter and some influence of the flow rate. For the entire experimental data set, the median data yielded: δ (3.21) δ (3.22) δ (3.23) 2 where 1 / 2 is the shape factor. The results (Equations (3.21) to (3.23)) compared well with analytical solutions for a velocity power law, with an exponent 1/N = 1/4.5 (Schlichting 1960, 5 Meireles et al. (2012) regrouped experimental data at collected at the LNEC stepped chute (1V:0.75H) by Matos (1999), Meireles (2004) and Renna (2004) (Table 1.1). 6 Depending on the inflow conditions, L either originates from the first step edge (broad-crested weir) or the peak of the crest (ogee-crested weir). 16

25 Chanson 2009). It will be shown that the velocity distributions in the developing boundary layer followed a 1/4.5-th power law at step edges (see Section 4.2). Note that the data showed some effect of the flow rate on the dimensionless boundary layer growth (Figure 3.8B). /L Present - d c /h = , 1V:1H Meireles et al. - d c /h = , 1V:0.75H Amador et al. (2009), 1V:0.8H Eq. (3.16) L/k s (A) Boundary layer thickness data - Comparison with Equation (3.16) and previous studies (Amador et al. 2009, Meireles et al. 2012) 0.2 d c /h = 0.9 d c /h = 1.0 d c /h = 1.1 d c /h = 1.3 d c /h = 1.5 d c /h = /Re L /Re L /L Smooth plate 0 2E+5 4E+5 6E+5 8E+5 1E+6 1.2E+6 1.4E+6 1.6E+6 1.8E+6 2E+6 Re L (B) Details of boundary layer thickness between step edges 2-5 (Present study) - Comparison with Equation (3.17) and smooth plate boundary layer solution (Schlichting 1960, Chanson 2009) Figure 3.8 Experimental observations of boundary layer growth in skimming flows on a stepped 17

26 spillway Location of the inception point The characteristics of the inception point of free-surface aeration were recorded for 0.7 d c /h 1.7 and the results are reported in Figure 3.9. The present data set is presented in Table 3.1 The location of the inception point of free-surface aeration was determined visually ( 7 ). The results are plotted in Figure 3.9A, with the dimensionless length to inception L i /k s as a function of the dimensionless discharge F * : F * q (3.24) g sinθ k 3 s where q is the water discharge per unit width, θ = 45 is the chute slope, and k s = h cosθ is the step roughness. The present data were compared to prototype data and earlier empirical correlations (Chanson 1994, Carosi and Chanson 2008, Meireles et al. 2012). The overall agreement was reasonable, although the correlations tended to overpredict the location of inception point. Herein velocity and pressure measurements suggested that the flow was rapidly varied in the vicinity of the inception point (refer to Section 4). Visually the location of the inception point remained mostly unchanged for 1.7 < F * < 2.3 and 3 < F * < 3.7. Although the visual criterion might reduce the precision of the data, it provides a consistent basis for a comparison with prototype data; Figure 3.9A includes such a comparison between laboratory and prototype data. The water depth d i data at the inception of free-surface aeration are plotted in Figure 3.9B. The present data are compared with prototype observations and Chanson's (1994) correlation: d k i s F 0.04 * (3.25) (sin ) The agreement between experimental data and correlation was overall good. At the inception point of air entrainment, the boundary layer outer edge was close to but smaller than the water depth: i.e,, the boundary layer thickness was about: /d i 0.8 for all data set. The finding was close to Wood's (1985) criterion ( 8 ) for smooth chutes. 7 The inception point location is most commonly determined either by visual observations (Chanson 1994, Carosi and Chanson 2008), or by void fraction measurements (Meireles et al. 2012). The former is the only means to characterise the inception point on prototype stepped spillways. 8 For a smooth chute flow, Wood (1985) considered the inception point of free-surface aeration when d i 1.2. This derived from observations of the boundary layer outer edge being irregular, extending about 1.2 times the mean thickness (Daily and Harleman 1966, Schlichting 1979). 18

27 Present study Chanson (1994) Carosi & Chanson (2008) Meireles et al. (2012) Prototype data 25 Li/ks (A) Location of the inception point of free-surface aeration - Comparison with prototype observations: Brushes Clough dam (UK), Dona Francisca (Brazil), Gold Creek dam (Australia), Hinze dam (2013), Pedrógão (Portugal) (Data re-analysis: Chanson et al. 2015) F * di/ks Present study Chanson (1994) =45 Prototype data (B) Water depth at the inception point of free-surface aeration - Comparison with Equation (3.25) and prototype observations at Dona Francisca (Brazil) (Data re-analysis: Chanson et al. 2015) Figure 3.9 Inception point of free-surface aeration in skimming flows on stepped spillways F * 19

28 Table Inception point of free-surface aeration (Present study, = 45º, h = 0.10 m, W = m) q (m 2 /s) d c /h F * L i /(h cosθ) d i /(h cosθ) Note: (--): data not available. 20

29 4. FLOW PROPERTIES IN THE DEVELOPING FLOW REGION 4.1 PRESENTATION Total and static pressure measurements were conducted in the clear water region upstream of the inception point of free-surface aeration. The data were recorded at selected longitudinal locations along the channel centreline using a Prandtl-Pitot tube for dimensionless discharges ranging from d c /h = 0.9 to 1.7 corresponding to the skimming flow regime, where d c is the critical flow depth and h is the vertical step height. The data were analysed in terms of velocity and pressure distributions, and energy dissipation in the developing flow region. The results are presented below. 4.2 VELOCITY DISTRIBUTIONS The velocity measurements showed that the water column consisted of a developing boundary layer with an ideal flow region above. In the turbulent boundary layer, the velocity profiles at step edges were best described using a power law (Chanson 2001, Amador et al. 2009, Meireles et al. 2012): V V x 0 1/N y at step edges (4.1) δ where V x is the longitudinal velocity component, V 0 is the free-stream velocity, y is the distance normal to the pseudo-bottom formed by the step edges, and is the boundary layer thickness. Typical velocity distributions at step edges 3 and 4 are presented in Figure 4.1. The best data fit yielded N = 4.5, with a normalised correlation coefficient R = The exponent N was close to that suggested by Chanson (2001) for a 1V:2H stepped model typical for embankment dams, but slightly different from N = 3.0 and 3.4 obtained respectively by Amador et al. (2009) and Meireles et al. (2012) in the developing flow region. Figure 4.2 presents typical velocity contours for the entire flow between step edges 2 and 4, for d c /h = 1.5. In Figure 4.2, the locations of velocity measurements ( 9 ) are marked with black dots, except at the free-surface where an ideal flow velocity was assumed. The data showed that the flow was accelerated by gravity in the downstream direction. At each step edge, the boundary layer flow was characterised by a steep velocity gradient. In the ideal flow region, the free-stream velocities matched theoretical predictions based upon the Bernoulli equation (Chanson 1999,2001). Downstream of each step edge, some flow separation occurred and a shear layer developed. Following Pope (2000) and Amador et al. (2006), an upper bound of the shear layer may be: 9 The total and piezometric heads were measured separately to yield the velocity data (see Section 2). When the two readings did not overlap exactly, the velocity at the piezometric head measurement location was calculated by combining the corresponding piezometric head with the total head interpolated within the lexicographically closest bounding triangle formed by three adjacent total head measurements. 21

30 y/ d c /h = 0.9 d c /h = 1.0 d c /h = 1.1 d c /h = 1.3 d c /h = 1.5 d c /h = 1.7 Power law N = V x /V 0 Figure 4.1 Dimensionless velocity distributions above step edge in the developing flow region of skimming flows on stepped spillways V/V c Step edge 3 Step edge 4 Figure 4.2 Dimensionless velocity contours V/V c between step edges Flow conditions: d c /h = 1.5 Flow direction from left to right 22

31 y ub y 0.9 (4.2) where y 0.9 is the depth where V x = 0.9 V 0 and V 0 is the ideal flow velocity. A typical evolution of the upper bound of the shear layer in the flow region surrounding step edge 3 is shown in Figure 4.3 for d c /h = 1.5, where k s = h cos is the step cavity height and L cav is the step cavity length: L cav = (h 2 +l 2 ) 1/2. The data showed that the shear layer upper bound increased in the longitudinal direction downstream of the step edge and reached a maximum at approximately x/l cav = 0.8, close to the finding of Amador et al. (2006). The process was repeated at the next step edge and exhibited a wavy pattern over several steps. Note that the present study was unable to obtain a reliable estimate of the shear layer lower bound in the cavity flow region because of instrumentation limitations yub/ks x/l cav Figure 4.3 Upper bound of mixing layer in skimming flow, with flow direction from left to right and the origin at step edge 3 Flow conditions: d c /h = 1.5, h =0.10 m 4.3 PRESSURE DISTRIBUTIONS The pressure distributions were derived from the piezometric head and water depth measurements, at several locations for 0.9 < d c /h < 1.7. The data showed a rapidly varying pressure field within the developing flow region. Typical dimensionless pressure distributions at step edges are presented in Figure 4.4, where the hydrostatic pressure distribution is shown for reference (red solid line). For all flow rates, the pressure profiles exhibited a linear shape in the ideal flow region immediately below the free-surface. The pressure gradient P/ y appeared to be consistently greater than hydrostatic, and tended to increase with increasing discharge, up to twice the hydrostatic pressure gradient for 23

32 d c /h = 1.7. The deviation from hydrostatic pressure distribution appeared to be a result of vertical flow accelerations linked to the free-surface curvature. Maximum pressures were typically observed in the mid- to lower-flow region, below which a pressure decrease was observed (Figure 4.4). The same trend was seen for all flow rates. y/d d c /h = 0.9 d c /h = 1.0 d c /h = 1.1 d c /h = 1.3 d c /h = 1.5 d c /h = 1.7 hydrostatic P/( g d cos ) Figure 4.4 Dimensionless pressure distributions in skimming flows at step edge 3 Further pressure measurements were conducted at several locations in the developing flow region at intermediate locations between step edges 3 to 5. For each step edge, measurements were conducted at x/l cav = -0.1, -0.05, and +0.1, where x is the streamwise distance from the step edge, positive downstream. Figures 4.5 and 4.6 present typical results upstream and downstream of step edge 3 respectively. Further data are presented in Appendix D. On each side of the step edge, the pressure profiles were characterised by distinctive patterns (Figures 4.5 & 4.6). Upstream of the step edge, the pressure gradient was consistently larger than hydrostatic throughout the entire water column; the findings were consistent with recent numerical data (Bombardelli 2014, Pers. Comm.). The largest normalised bottom pressures were recorded for the smallest discharge (d c /h = 0.9) (Figure 4.5). Downstream of each step edge, the dimensionless pressure increased with increasing distance from the free-surface for a short distance, before a rapid decrease towards the bottom where the pressure was sub-atmospheric (Figure 4.6). Such sub-atmospheric pressures were recorded by Toombes (2002) immediately below the step edges in nappe and transition flows, and by Sanchez-Juny et al. (2008) in skimming flows. Minimum normalised pressures were measured next to the pseudo-bottom and appeared to be inversely proportional to discharge. Negative 24

33 pressures were recorded for all but the largest discharge (d c /h = 1.7). Maximum and minimum pressures appeared to be linked to flow stagnation upstream of and flow separation downstream of each step edge d c /h = 0.9 d c /h = 1.0 d c /h = 1.1 d c /h = 1.3 d c /h = 1.5 d c /h = 1.7 hydrostatic 1 d c /h = 0.9 d c /h = 1.0 d c /h = 1.1 d 0.8 c /h = 1.3 d c /h = 1.5 d c /h = 1.7 hydrostatic 0.6 y/d y/d P/( g d cos ) P/( g d cos ) (A) x/l cav = -0.1 (B) x/l cav = Figure 4.5 Dimensionless pressure distributions in skimming flows at locations immediately surrounding step edge 3 d c /h = 0.9 d c /h = 1.0 d c /h = 1.1 d c /h = 1.3 d c /h = 1.5 d c /h = 1.7 hydrostatic d c /h = 0.9 d c /h = 1.0 d c /h = 1.1 d c /h = 1.3 d c /h = 1.5 d c /h = 1.7 hydrostatic y/d 0.5 y/d P/( g d cos ) P/( g d cos ) (A) x/l cav = (B) x/l cav = +0.1 Figure 4.6 Dimensionless pressure distributions in skimming flows at locations immediately surrounding step edge 3 25

34 Figure 4.7 presents typical pressure contours above two step cavities for d c /h = 1.5. A photograph of the flow is provided in Figure 4.8, illustrating the free-surface curvature. In Figure 4.7, locations of pressure measurements are tagged with black dots, except at the free-surface where the pressure was atmospheric. Figure 4.7 shows a rapid redistribution of the pressure field in the flow direction. The flow was characterised by alternating zones of positive and negative pressures. A large negative pressure zone was observed immediately downstream of step edge 3, with a minimum pressure being below atmospheric. A positive pressure zone was observed next to the horizontal step face, linked to some interaction between the shear layer and the step face, with a change in streamline direction and flow stagnation immediately upstream of step edge 4. The occurrence of higher pressures at the lower edge of the cavity was consistent with numerical simulations (Qian et al. 2009, Bombardelli 2015, Pers. Comm.). Figure 4.7 also shows a few regions with a nearlyconstant pressure gradient, implying that the streamlines were parallel despite the non-hydrostatic pressure distributions. P/(ρ.g.d.cosθ) Step edge 3 Step edge 4 Figure 4.7 Dimensionless pressure field contour plot in skimming flow around step edge 3 Flow conditions: d c /h = 1.5, h = 0.10 m, Flow direction from left to right 26

35 Figure 4.8 Side view of free-surface in the developing flow region Flow conditions: d c /h = 1.5, h = 0.10 m, Flow direction from left to right Note the significant free-surface curvature Present results demonstrated that the pressure distributions were not hydrostatic in the developing flow region on a stepped spillway and the developing flow was rapidly varied, with rapid longitudinal variations in both pressure and velocity distributions. A close examination of Figure 4.6 showed longitudinal variations in pressure profiles across step cavities. For example, the step cavity 2-3 was generally governed by positive pressures, while a negative pressure core was observed in step cavity 3-4 (Figure 4.7). These observations may be attributed to the stepped geometry. The steps formed a series of expansions and contractions, forcing the flow to redistribute and leading to curved streamlines. The free-surface pattern was influenced by the combination of bottom geometry, slope and inflow conditions. The present findings highlighted the importance of physical modelling during the design process. An improper configuration might lead to the generation of significant free-surface curvatures and adverse negative pressures. A good design should seek to eliminate any rapid flow variations to create a more uniform overflow and pressure field instead. Note that the most significant freesurface curvature, in the present setup, was observed for smaller skimming flow rates. 4.4 TOTAL HEAD AND ENERGY DISSIPATION Total head distributions In the developing flow region above a stepped chute, energy dissipation took place in the boundary layer flow by means of viscous effect and turbulent interactions. Figure 4.9 shows a typical contour plot of dimensionless total head H t /H dam between step cavities 2 4 for d c /h = 1.5, where H t and H dam are the total head and dam height (H dam = 1.2 m) respectively. At the free-surface, H t was calculated using the Bernoulli equation. A dashed line was drawn in Figure 4.9 to show the outer 27

36 edge of the boundary layer for that flow rate. The ideal fluid flow was little affected by the step macro-roughness, and the total head increased rapidly in the downstream direction according to ideal fluid flow calculations. Some energy was dissipated in the boundary layer, as indicated by the steep total head gradient in Figure 4.9. The smallest values were recorded in the step cavity. Figure 4.9 Dimensionless total head distribution contours in skimming flows between step edges 2 4 for d c /h = Yellow dashed line shows the outer edge of the boundary layer for that flow rate Energy dissipation and boundary shear stress The energy dissipation rate in the developing flow region was analysed based on total head measurements. At each step edge, the depth averaged specific energy equals: 1 d H r (Ht z0 ) dy (4.3) d 0 where d is the flow depth, H t is the total head and z 0 is the step edge elevation above the datum. The normalised specific energy, H r /H max, along the stepped chute is shown in Figure 4.10, where H max is the maximum specific energy at any step edge along the spillway, L is the streamwise distance from the first step edge and L i is the distance to the inception point of free-surface aeration. The present results showed a quasi-linear decrease in normalised specific energy as previously reported (Hunt 28

37 and Kadavy 2010, Meireles et al. 2012). In Figure 4.10, the present data are compared to an empirical correlation proposed by Meireles et al. (2012) for an ogee-crested spillway with a 1V:0.75H slope. The agreement between data and correlation was acceptable, although some data scatter was observed, possibly linked to the effects of free-surface curvature and by the distinct facilities. The flow resistance in a skimming flow is commonly estimated using the Darcy-Weisbach friction factor (Rajaratnam 1990, Chanson 2001), although its application to form losses is arguable (Chanson et al. 2002). The friction factor may be however regarded as a dimensionless shear stress between the main streamflow and cavities, namely a spatially-averaged boundary shear stress along the pseudo-bottom since: f 8 (4.4) V o 2 0 where f is the dimensionless boundary shear stress or Darcy-Weisbach friction factor, o is the (spatially-averaged) boundary shear stress and is the fluid density. Herein the dimensionless boundary shear stress was estimated from the von Karman momentum integral equation applied to the developing boundary layer: (V x V x 2 0 MI ) V0 1 V0 f 8 (4.5) Hr/Hmax d c /h = 0.9 d c /h = 1.1 d c /h = 1.5 Meireles et al., 1V:0.75H d c /h = 1.0 d c /h = 1.3 d c /h = L/L i Figure 4.10 Longitudinal variation in dimensionless specific energy H r /H max in the developing flow region of skimming flows on a stepped spillway 29

38 where the subscript MI indicates calculations based upon the momentum integral equation. The results in terms of f MI are presented in dimensionless form in Figure On average, the dimensionless boundary shear stress deduced from the momentum integral equation equalled: f MI (Table 4.1). The result was close to the air-water flow friction factor estimates f e obtained in the fully-developed flow region downstream of the inception point ( 10 ). Amador (2005) and Meireles (2011) presented velocity measurements in the developing flow region of skimming flows. The application of the momentum integral equation gave on average f MI 0.08 to 0.12 (Table 4.1). Both the data of Meireles (2011) and Frizell et al. (2013) ( 11 ) suggested a larger dimensionless shear stress f MI for the largest step height h (Table 4.1), although that these experiments were conducted for different relative step cavity roughness heights. For completeness Amador (2005) reported time-averaged dimensionless tangential stresses v x v y /( /8 V 2 0 ) between 0 and along the pseudo-bottom formed by the step edges in the developing flow region. His experimental observations yielded a spatially-averaged dimensionless shear stress f xy 0.27 over the entire step cavity. For comparison, Gonzalez and Chanson (2004) measured a spatially-averaged dimensionless shear stresses f xy 0.34, in the fully-developed airwater flow region (Table 4.1). Lastly a gross estimate of the friction factor might be derived from the friction slope S f (S f = - H/ x) assuming a fully-developed flow in a wide channel: f Sf 3 d 8 Sf d (4.6) c where d is the flow depth, d c is the critical flow depth: d c = (q 2 /g) 1/3, q is the water discharge per unit width and g is the gravity acceleration. Equation (4.6) was applied in the developing flow region and the results are shown in Figure 4.11 ( 12 ). Altogether the dimensionless shear stress data were close to those derived from Equation (4.5), despite the crude approximation. Figure 4.11 regroups the present experimental data only. In Figure 4.11, the red square data correspond to the application of the von Karman momentum integral equation applied to the developing flow, the blue star data are a crude estimate based upon the friction slope in the developing flow, and the hollow circle data are the data in the fully-developed air-water flow region. Table 4.1 summarises the present data which are compared to previous studies. Altogether the experimental results were close, despite differences in stepped spillway geometry (slope, step height) and instrumentation. 10 The air-water flow measurements were performed for 0.9 < d c /h < 1.7 in the same chute with the dual-tip phase-detection probe system described in Section 2 (Zhang and Chanson 2015). 11 The experiments of Frizell et al. (2013) were conducted in a hydrodynamic water tunnel. 12 Equation (4.6) is not valid in the developing flow region because the flow is not fully-developed. 30

39 0.4 momentum integral friction slope air-water flow region 0.3 fmi, fsf, fe f MI = on average d c /h Figure 4.11 Dimensionless boundary shear stress in the developing flow region of skimming flows on stepped spillways Table Dimensionless boundary stress f = o /( /8 V 0 2 ) in skimming flows on stepped spillways - Average experimental results Reference W h Developing flow region Aerated region f MI f xy f Sf f xy f e (º) (m) (m) Momentum Reynolds Friction Reynolds Friction integral eq. (4.5) stress slope Eq. (4.6) stress slope Eq. (4.6) Present study Gonzalez & Chanson (2004) Amador (2005) Meireles (2011) ( 1 ) Frizell et al. (2013) ( 2 ) Notes: f e : dimensionless shear stress in fully-developed air-water flows downstream of inception point of free-surface aeration; f MI : dimensionless shear stress derived from von Karman momentum integral calculations; f Sf : dimensionless shear stress derived from friction slope calculations; f xy : spatially-averaged dimensionless Reynolds shear stress along a step cavity; h: vertical step height; W: channel width; (--): data not available; Grey shading: approximate estimate, not strictly valid; 31

40 ( 1 ): experimental data by Matos (1999), Meireles (2004) and Renna (2004) (Table 1.1); ( 2 ): water tunnel data by Frizell et al. (2013). 32

41 5. CONCLUSION In skimming flow on a stepped spillway, the upstream flow motion is non-aerated and the freesurface appears smooth and glossy up to the inception point of free-surface aeration. In the nonaerated flow region, a turbulent boundary layer develops until the outer edge of the boundary layer interacts with the free-surface and air entrainment takes place: that is, at and downstream of the inception point. Herein new experiments were performed in the developing flow region on a large size 1V:1H stepped spillway model with step height h = 0.10 m. The flow properties in the developing flow region were documented using a Prandtl-Pitot tube and the results were complemented with phase-detection probe data. Downstream of the broad-crested weir, the water skimmed over the pseudo-bottom formed by the step edges. Upstream of the inception point, the free-surface was smooth over the first few steps, although some significant free-surface curvature was observed for all discharges. The free-surface became increasingly turbulent in the downstream direction. The inception point of free-surface aeration was observed when the boundary layer thickness reached 80% of the flow depth: /d i 0.8. In the developing boundary layer, the velocity distributions followed a 1/4.5 th power law at step edges. The development of the boundary layer was more rapid than that on a smooth chute and the present results were comparable to previous studies (Amador 2005, Meireles et al. 2012). The location of the inception point and the flow depth at inception were compared successfully to previous laboratory and prototype results. In the developing flow region, detailed velocity and pressure measurements showed a rapidlyvaried flow motion. While the free-stream velocity accelerated in the downstream direction as predicted by the Bernoulli principle, the pressure distributions were not hydrostatic. Both velocity and pressure data indicated rapid flow redistributions between step edges and above each step cavity. Three distinctive pressure zones were identified. In the ideal flow zone, the pressure could be predicted by potential flow considerations taking into account the free-surface curvature. Close to the pseudo-bottom, a zone of positive pressures and another of negative pressures were governed by flow stagnation and separation respectively. The spillway engineers should therefore design not only for extreme events, but also consider non design flow conditions for which extreme negative pressures may arise. The application of the von Karman momentum integral equation indicated an average friction factor of 0.18, close to the observed air-water flow friction factor of It is believed that this is the first study quantifying the boundary shear stress in both developing clear-water and fully-developed airwater flow regions, in the same facility for the same range of flow rates. The present finding suggested that the spatially-averaged dimensionless shear stress was comparable in the developing 33

42 flow region and fully-aerated flow region of a stepped spillway, despite the rapidly-varied nature of the developing flow region. 34

43 6. ACKNOWLEDGEMENTS The authors thank Professor Fabian Bombardelli (University of California Davis, USA) and Professor Jorge Matos (IST Lisbon, Portugal) for their detailed review of the report and most valuable comments. They also thank Dr John Macintosh (Water Solutions, Australia) for his comments and involvement as Associate Supervisor. The authors acknowledge the technical assistance of Jason Van Der Gevel and Stewart Matthews (The University of Queensland). The financial supports through the Australian Research Council (Grant DP ) and through the University of Queensland are acknowledged. 35

44 APPENDIX A - PHOTOGRAPHS OF EXPERIMENTS (A) (B) Figure A.1 - Skimming flow conditions: d c /h = 0.9, h = 0.1 m, = 45º A-1

45 Figure A.2 - Skimming flow conditions: d c /h = 1.1, h = 0.1 m, = 45º Figure A.3 - Skimming flow conditions: d c /h = 1.5, h = 0.1 m, = 45º A-2

46 (A) (B) (C) Figure A.4 - Skimming flow conditions: d c /h = 1.7, h = 0.1 m, = 45º A-3

47 APPENDIX B - DEVELOPING BOUNDARY LAYER CHARACTERISTICS At the upstream end of the stepped chute, a boundary layer developed along the spillway invert up to the inception point of free-surface aeration, where the boundary layer outer edge interacts with the free-surface. The boundary layer growth was deduced from the velocity measurements and the results were best correlated by: δ L 0.37 L 0.15 k s (B.1) where L is the streamwise distance from the first step edge, and k s = h cos is the step cavity height. Equation B.1 is valid for 0.9 d c /h 1.7 and 0 < L/k s < 15. The boundary layer characteristics are summarised in Table B.1, where V 0 is the free-stream velocity, d is the water depth, is the boundary layer thickness defined in terms of 0.99 V 0, 1 is the displacement thickness, 2 is the momentum thickness and 3 is the energy thickness. The data were best correlated by: δ L δ1 L δ2 L δ3 L Re L R = (B.2) ReL R = (B.3) ReL R = (B.4) ReL R = (B.5) for < Re L < , with Re L = V o L/, the water density, the water dynamic viscosity and R the normalised correlation coefficient. Using an expression similar to the analytical solution of developing boundary layer on smooth plate with zero pressure gradient, the boundary layer characteristics might be approximated by δ L δ1 L δ2 L 1/ Re L R = (B.6) 1/ ReL R = (B.7) 1/ ReL R = (B.8) For the entire experimental data set, the median data yielded: δ δ (B.9) (B.10) A-4

48 δ (B.10) 2 where 1 / 2 is the shape factor. Equations (B.9) to (B.10) compared well with analytical solutions for a velocity power law, with an exponent 1/N = 1/4.5. The velocity distributions in the developing boundary layer indeed followed a 1/4.5-th power law. Table B.1 - Developing boundary layer characteristics in the developing flow region (Present study, = 45º, h = 0.10 m, W = m) d c /h Step edge x (m) d (m) V 0 (m/s) (m) 1 (m) 2 (m) 3 (m) Note: italic data: erroneous data. A-5

49 APPENDIX C - VELOCITY, PRESSURE AND TOTAL HEAD DATA Total head and piezometric head measurements were conducted at several longitudinal sections in the developing flow region on a stepped chute. Flow rates in the range of 0.7 < d c /h < 1.7 were investigated. The raw data are summarised in this section. A definition sketch is provided in Figure B.1. Relevant notations are summarised below. Notation d c critical flow depth x streamwise distance from step edge 1 x t streamwise distance of the total pressure tapping from step edge 1 y normal distance to the pseudo-bottom h step height H p H t Q piezometric head a.k.a. static head total head flow rate Figure B1. Sketch of Pitot-tube measurement experiments A-6