THE UNIVERSITY OF QUEENSLAND

REPORT CH97/15

AUTHORS: Gangfu ZHANG and Hubert CHANSON

HYDRAULICS OF THE DEVELOPING FLOW REGION OF STEPPED CASCADES: AN EXPERIMENTAL INVESTIGATION

SCHOOL OF CIVIL ENGINEERING

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) 3365 4163 Fax: (61 7) 3365 4599 URL: http://www.civil.uq.edu.au/ 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. 978 1 74272 134 7 The University of Queensland, St Lucia QLD, Australia

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, Email: h.chanson@uq.edu.au

HYDRAULIC MODEL REPORT No. CH97/15

ISBN 978 1 74272 134 7

The University of Queensland, School of Civil Engineering,

February 2015

Hinze Dam stepped spillway (Gold Coast QLD, Australia) in operation on 29 January 2013 - Flow

conditions: h = 1.2 m, = 51.3º, q = 16.6 m2/s

ii

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: /di 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 1/4.5th 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.

iii

TABLE OF CONTENTS

Page

Abstract ii

Keywords ii

Table of contents iii

List of symbols iv

1. Introduction 1

2. Experimental facilities and instrumentation 4

3. Flow patterns 8

3.1 Broad crested weir

3.2 Stepped spillway chute

4. Flow properties in the developing flow region 21

4.1 Presentation

4.2 Velocity distributions

4.3 Pressure distributions

4.4 Total head and energy dissipation

5. Conclusion 33

6. 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

iv

LIST OF SYMBOLS

The following symbols are used in this report:

C void fraction;

CD dimensionless discharge coefficient;

d water depth (m);

dc critical flow depth (m);

di flow depth (m) inception point of free-surface aeration;

E specific energy (m): E = Ht - zo;

Emin minimum specific energy (m);

F* dimensionless discharge:

3s

*ksinθg

qF

f (a) Darcy-Weisbach friction factor;

(b) dimensionless boundary shear stress;

fe dimensionless shear stress in fully-developed air-water flows downstream of inception

point of free-surface aeration;

fMI dimensionless shear stress derived from von Karman momentum integral calculations;

fSf dimensionless shear stress derived from friction slope calculations;

fxy spatially-averaged dimensionless Reynolds shear stress along a step cavity;

g gravity acceleration (m/s2): g = 9.80 m/s2 in Brisbane, Australia;

Hdam dam height (m);

Hmax maximum available total head (m);

Hp piezometric head (i.e. static head) (m);

Hr depth-averaged specific energy (m) on the stepped chute;

Ht total head (m);

H1 upstream head above crest (m);

h vertical step height (m);

ks step roughness height (m): ks = hcos;

L longitudinal distance (m) positive downstream measured from step edge 1;

Lcav step cavity length: Lcav = (l2+h2)1/2;

Lcrest broad-crested weir length (m);

Li distance between step edge 1 and inception point of free-surface aeration;

l horizontal step length (m);

P pressure (Pa);

Q water discharge (m3/s);

q water discharge per unit width (m2/s);

R normalised correlation coefficient;

r radius of curvature (m);

So bed slope: So = sin;

v

t time (s);

V flow velocity (m/s) positive downstream;

Vx longitudinal velocity component (m/s);

V0 free-stream velocity (m/s);

vx longitudinal turbulent velocity fluctuation (m/s);

vx normal turbulent velocity fluctuation (m/s);

W channel width (m);

x longitudinal distance (m) positive downstream measured from step edge;

Y90 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);

yub upper bound of the shear layer (m);

y0.9 characteristic distance (m) where Vx = 0.9×V0 ;

zo 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/m3);

o boundary shear stress (Pa);

Ø 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;

vi

IST Instituto Superior Técnico;

LNEC Laboratório Nacional de Engenharia Civil;

PIV particle image velocimetry;

Std standard deviation;

s second;

UDC Universidad de la Corũna;

UQ The University of Queensland;

u/s upstream.

1

1. INTRODUCTION

Historical records have indicated that stepped spillways have been used for thousands of years

(Chanson 1995,2000-2001). 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-

2

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 m2/s, dc/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 m3/s, q = 7.4 m2/s, dc/h = 2.85

Figure 1.1 – Skimming flows on prototype stepped spillways - Red arrows point to inception point

of free-surface aeration

3

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 (m3/s) q (m2/s) dc/h Instrumentation Remarks Present study 45.0 0.10 1.0 Broad-crest with

u/s and d/s rounding

0.085-0.216 0.085-0.216 0.9-1.7 Prandtl-Pitot tube (=3.18 mm), double-tip

conductivity probe (=0.25 mm)

AEB Hydraulics Laboratory,UQ

Amador (2005) 51.3 0.05 0.5 WES ogee with 3 small steps

0.055 0.11 2.15 Particle image velocimetry (PIV)

Setup #2 (UDC)

Meireles et al. (2012)

53.1 0.08 1.0 WES ogee with small steps

0.08 to 0.20 0.08 to 0.20 1.1 to 2.0 Back-flushing Pitot tube, conductivity probe

Experiments at LNEC by Matos (1999)

0.04 1.0 WES ogee with small steps

0.05 to 0.18 0.05 to 0.18 1.6 to 3.7 Back-flushing Pitot tube, conductivity probe

Experiments at LNEC by Meireles (2004)

0.04 1.0 WES ogee with 0.10 & 0.20 0.10 & 0.20 2.5 & 4.0 Back-flushing Pitot tube, Experiments at LNEC 0.02 1.0 small steps 0.10 to 0.20 0.10 to 0.20 5.0 to 8.0 conductivity probe by Renna (2004)

Notes: dc: critical flow depth; h: vertical step height; Q: water discharge; q: water discharge per unit width; W: channel width; : slope angle with

horizontal.

4

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

2.7×5 m2, 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 0.985 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

0.985 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 0.985 m wide.

The discharge was deduced from detailed velocity and pressure measurements performed above the

broad crested weir which gave:

31

crest

1 )H3

2(g)

L

H243.08966.0(

W

Q (2.1)

where W is the crest width (W = 0.985 m), H1 is the upstream total head above crest and Lcrest is the

crest length (Lcrest = 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 AppleTM 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.

5

(A) Stepped spillway in operation for dc/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

6

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 dc/h < 0.4, where dc is the critical flow depth and h is the vertical step height (h

= 0.1 m). A transition flow was observed for 0.4 < dc/h < 0.9, and skimming flows were seen for

dc/h > 0.9. The overflow was characterised by a series of small water nappes cascading down the

stepped chute for dc/h < 0.15. For 0.15 < dc/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 <

dc/h < 0.9. The flow appeared highly chaotic and a large deflecting jet developed between step

edges 2 and 5 for smaller transition discharges (dc/h < 0.6). For dc/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 (m3/s) dc/h Location Flow regime Step edge 0.085 – 0.216 0.9 – 1.7 Step edges 2 to 8 Skimming flow

Step cavity 0.148 – 0.182 1.3 – 1.5 Step cavities 2 to 4 Skimming flow 1 Reference: http://www.dwyer-inst.com/Product/TestEquipment/PitotTubes/Series160. 2 For dc/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.

7

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

8

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 (H1/Lcrest >

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 (H1/Lcrest < 0.11), the water-

surface 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):

dg2

V

d

dyg

Py

g2

V

E2

d

0

2x

(3.1)

where E is the depth-averaged specific energy, Vx 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

0

dydg

P

d

1

2

1 (3.2)

Assuming a uniform velocity profile (β = 1) and a hydrostatic pressure distribution (Λ = 1),

Equation (3.1) reduces to the classical expression:

g2

VdE

2

(3.3)

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).

9

Upstream roundingSmall rounding

0 10 30 60 cm

r = 0.058 mr = 0.012 m

H1

Lcrest = 0.6 m

(A) Definition sketch of broad-crested weir

(B) Flow over broad-crested weir – H1/Lcrest = 0.257

Figure 3.1 – Flow over broad-crested weir - Flow direction from left to right

x/Lcrest

d/L

cres

t

-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45H1/Lcrest = 0.108H1/Lcrest = 0.173H1/Lcrest = 0.257

H1/Lcrest = 0.370H1/Lcrest = 0.465H1/Lcrest = 0.0483

H1/Lcrest = 0.148H1/Lcrest = 0.221H1/Lcrest = 0.273

H1/Lcrest = 0.321pointer gauge databroad crested weir

Figure 3.2 – Dimensionless free-surface profiles above the broad-crested weir - Photographic and

pointer gauge (solid circles) data

10

In an open channel, critical flow conditions occur when the specific energy is minimal: E = Emin

(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 dc, the minimum specific

energy Emin, the dimensionless discharge coefficient CD (4) and the pressure and momentum

coefficients Λ and β was proposed by Chanson (2006b):

0

3

2C

2

11

E

d

E

d32

D

2

min

c

3

min

c

(3.4)

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):

1CC4

3

1 22D

22D

6

(3.5)

These four solutions are:

3

1ΔΛ)ΛCβ2(1

27

1

ΔΛ)ΛCβ2(127

1Λ

E

d

3 622D

3 622D

min

c

for Δ > 0 (3.6)

3

2

E

d

min

c for Δ = 0 (3.7)

3

cos2

1

3

2

E

d

min

c for Δ < 0 (3.8)

2

3cos13

3cos1

3

2

E

d2

min

c

for Δ < 0 (3.9)

where:

22

DC21cos (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 Emin and it equals

the upstream head above crest H1 (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: 3

minD E3

2g/)B/Q(C

(Chanson 2006b).

11

from the present study in terms of the dimensionless critical depth dc×Λ/H1 as a function of the

dimensionless discharge β×CD2×Λ2 for 0.17 < x/Lcrest < 0.83. 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/Lcrest < 0.83. The result was consistent with the

earlier study of Felder and Chanson (2012).

CD22

d

c/H

1

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

Eq, (3.8)Eq. (3.9)H1/Lcrest = 0.108H1/Lcrest = 0.173H1/Lcrest = 0.257

H1/Lcrest = 0.370H1/Lcrest = 0.465H1/Lcrest = 0.148H1/Lcrest = 0.221H1/Lcrest = 0.281

H1/Lcrest = 0.322H1/Lcrest = 0.048Felder & Chanson

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, Lcrest = 1.01 m).

3.2 STEPPED SPILLWAY CHUTE

3.2.1 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

dc/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

dc/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

12

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

3.2.2 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

13

terms of the normalised streamwise distance L/Li, where L is the distance downstream of step edge

1 measured along the pseudo-bottom and Li is the inception point location (Figure 3.4). For the

largest discharge (dc/h = 1.7), the data were checked against the phase-detection probe outputs, by

integrating the void fraction profiles along the water column:

90Y

0dyC)(1d (3.11)

where d is the equivalent clear-water depth, C is the (time-averaged) void fraction, and Y90 is the

depth at which C = 0.90. 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:

0x V0.99Vyδ (3.12)

0 0

x1 dy)

V

V(1δ (3.13)

0 0

x

0

x2 dy)

V

V(1

V

Vδ (3.14)

0

20

2x

0

x3 dy)

V

V(1

V

Vδ (3.15)

where the boundary thickness δ is defined in terms of 99% of the free-stream velocity, Vx is the

longitudinal velocity component, V0 is the free-stream velocity, δ1 is the displacement thickness, δ2

is the momentum thickness and δ3 is the energy thickness. Figure 3.7 compares the longitudinal

variations in boundary layer characteristics, together with the free-surface profiles, for one

discharge dc/h = 1.5. Note the different scales for the left and right vertical axes. As seen in Figure

14

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/ks = 10 for dc/h = 1.5, where ks is the step roughness height: ks

= hcos. At this point the boundary layer thickness was about: /d = /di 0.8 and this reflected

that the outer edge of the boundary layer was irregular and fluctuating (Klebanoff 1955 1966,

Antonia 1972, Phillips and Ratnanather 1990).

L/Li

d/d c

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1dc/h = 0.9dc/h = 1.0

dc/h = 1.1dc/h = 1.3

dc/h = 1.5dc/h = 1.7

dc/h = 1.7 (conductivity probe)

Figure 3.5 – Longitudinal profiles of dimensionless water depth upstream of the inception point of

free-surface aeration

Figure 3.6 – Side view of water surface profile - Flow from left to right, dc/h = 0.9, Aperture: f/5.6,

Shutter speed = 1/50 s, ISO 1600 – Lens distortion correction using PTLens™

Step edge 2

15

L/ks

d/d c

, /d

c

1/d

c,

2/d c

, 3/

d c

0 1 2 3 4 5 6 7 8 9 10 11 120 0

0.1 0.05

0.2 0.1

0.3 0.15

0.4 0.2

0.5 0.25

0.6 0.3

0.7 0.35

0.8 0.4

0.9 0.45

1 0.5d123

Figure 3.7 – Longitudinal water surface profile: comparison with boundary layer thickness,

displacement thickness, momentum thickness and energy thickness data – Flow conditions: dc/h =

1.5, h = 0.10 m – Note different scales of left and right vertical axes

3.2.3 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:

0.37

sk

L0.15

L

δ

(3.16)

where L is the streamwise distance from the first step edge, and ks = hcos is the step cavity

height. Equation 3.16 is valid for 0.9 ≤ dc/h ≤ 1.7 and 0 < L/ks < 15. The complete dataset is plotted

in Figure 3.8A; the data complemented the earlier findings of Amador et al. (2009) and Meireles et

16

al. (2012) (5) respectively on 1V:0.8H and 1V:0.75H ogee-crested stepped chutes, but for a larger

range of L/ks (6). In Figure 3.8A, the empirical correlation of Amador et al. (2009) was extended to

the present data range. For L/ks > 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 ≤ dc/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/ks (L/ks < 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 ReL = VoL/, with the water density, and the water dynamic viscosity.

The present data were best correlated by:

0.18LRe07.1

L

δ with R = 0.405 (3.17)

0.32L

1 Re58.1L

δ with R = 0.343 (3.18)

0.11L

2 Re0.053L

δ with R = 0.179 (3.19)

0.14L

3 Re0.132L

δ with R = 0.242 (3.20)

for 2×105 < ReL < 2×106, 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:

191.0δ1

(3.21)

126.0δ2

(3.22)

53.1δ

2

1

(3.23)

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).

17

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/ks

/L

0 5 10 15 20 25 30 35 40 45 50 55 60 650.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22Present - dc/h = 0.9 - 1.7, 1V:1HMeireles et al. - dc/h = 1.1 - 8.0, 1V:0.75HAmador et al. (2009), 1V:0.8HEq. (3.16)

(A) Boundary layer thickness data - Comparison with Equation (3.16) and previous studies

(Amador et al. 2009, Meireles et al. 2012)

ReL

/L

2E+5 4E+5 6E+5 8E+5 1E+6 1.2E+6 1.4E+6 1.6E+6 1.8E+6 2E+60

0.05

0.1

0.15

0.2

Smooth plate

dc/h = 0.9dc/h = 1.0

dc/h = 1.1dc/h = 1.3

dc/h = 1.5dc/h = 1.7

1.07/ReL0.18

0.37/ReL0.2

(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

18

spillway

3.2.4 Location of the inception point

The characteristics of the inception point of free-surface aeration were recorded for 0.7 ≤ dc/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 Li/ks as a function of the

dimensionless discharge F*:

3s

*ksinθg

qF

(3.24)

where q is the water discharge per unit width, θ = 45° is the chute slope, and ks = 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 di 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:

592.0*04.0

s

i F)(sin

4034.0

k

d

(3.25)

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: /di 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 di 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).

19

F*

Li/k

s

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 80

5

10

15

20

25

30

35

40Present studyChanson (1994)Carosi & Chanson (2008)Meireles et al. (2012)Prototype data

(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*

d i/k

s

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Present studyChanson (1994) =45Prototype 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

20

Table 3.1 - Inception point of free-surface aeration (Present study, = 45º, h = 0.10 m, W = 0.985

m)

q (m2/s) dc/h F* Li/(h×cosθ) di/(h×cosθ) 0.058 0.7 1.17 4 -- 0.085 0.9 1.71 8 0.471 0.099 1.0 2.00 8 0.578 0.114 1.1 2.31 8 -- 0.147 1.3 1.96 12 0.699 0.182 1.5 3.67 12 0.864 0.219 1.7 4.43 16 0.880

Note: (--): data not available.

21

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

dc/h = 0.9 to 1.7 corresponding to the skimming flow regime, where dc 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):

1/N

0

x

δ

y

V

V

at step edges (4.1)

where Vx is the longitudinal velocity component, V0 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 = 0.97. 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 dc/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.

22

Vx/V0

y/

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1dc/h = 0.9dc/h = 1.0dc/h = 1.1dc/h = 1.3dc/h = 1.5dc/h = 1.7Power law N = 4.5

Figure 4.1 – Dimensionless velocity distributions above step edge in the developing flow region of

skimming flows on stepped spillways

Step edge 3 Step edge 4

V/Vc

Figure 4.2 – Dimensionless velocity contours V/Vc between step edges – Flow conditions: dc/h =

1.5 – Flow direction from left to right

23

0.9ub yy (4.2)

where y0.9 is the depth where Vx = 0.9×V0 and V0 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 dc/h = 1.5, where ks = hcos is the step cavity height and Lcav is the step cavity length: Lcav =

(h2+l2)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/Lcav = 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.

x/Lcav

y ub/

k s

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

0.1

0.2

0.3

0.4

0.5

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2

0.1

0.2

0.3

0.4

0.5

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: dc/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 < dc/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

24

dc/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.

P/(gdcos)

y/d

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1dc/h = 0.9dc/h = 1.0

dc/h = 1.1dc/h = 1.3

dc/h = 1.5dc/h = 1.7

hydrostatic

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/Lcav = -0.1, -0.05, +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 (dc/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

25

pressures were recorded for all but the largest discharge (dc/h = 1.7). Maximum and minimum

pressures appeared to be linked to flow stagnation upstream of and flow separation downstream of

each step edge.

P/(gdcos)

y/d

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1dc/h = 0.9dc/h = 1.0dc/h = 1.1dc/h = 1.3dc/h = 1.5dc/h = 1.7hydrostatic

P/(gdcos)

y/d

0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1 dc/h = 0.9dc/h = 1.0dc/h = 1.1dc/h = 1.3dc/h = 1.5dc/h = 1.7hydrostatic

(A) x/Lcav = -0.1 (B) x/Lcav = -0.05

Figure 4.5 – Dimensionless pressure distributions in skimming flows at locations immediately

surrounding step edge 3

P/(gdcos)

y/d

-1.8 -1.2 -0.6 0 0.6 1.2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.8 -1.2 -0.6 0 0.6 1.2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1dc/h = 0.9dc/h = 1.0dc/h = 1.1dc/h = 1.3dc/h = 1.5dc/h = 1.7hydrostatic

P/(gdcos)

y/d

-1.25 -0.75 -0.25 0.25 0.75 1.25

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.25 -0.75 -0.25 0.25 0.75 1.25

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1dc/h = 0.9dc/h = 1.0dc/h = 1.1dc/h = 1.3dc/h = 1.5dc/h = 1.7hydrostatic

(A) x/Lcav = +0.05 (B) x/Lcav = +0.1

Figure 4.6 – Dimensionless pressure distributions in skimming flows at locations immediately

surrounding step edge 3

26

Figure 4.7 presents typical pressure contours above two step cavities for dc/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 nearly-

constant pressure gradient, implying that the streamlines were parallel despite the non-hydrostatic

pressure distributions.

Step edge 3 Step edge 4

P/(ρ.g.d.cosθ)

Figure 4.7 – Dimensionless pressure field contour plot in skimming flow around step edge 3 – Flow

conditions: dc/h = 1.5, h = 0.10 m, Flow direction from left to right

27

Figure 4.8 – Side view of free-surface in the developing flow region – Flow conditions: dc/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 free-

surface curvature, in the present setup, was observed for smaller skimming flow rates.

4.4 TOTAL HEAD AND ENERGY DISSIPATION

4.4.1 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 Ht/Hdam between step cavities 2 – 4 for dc/h = 1.5, where Ht and

Hdam are the total head and dam height (Hdam = 1.2 m) respectively. At the free-surface, Ht was

calculated using the Bernoulli equation. A dashed line was drawn in Figure 4.9 to show the outer

28

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 dc/h = 1.5 - Yellow dashed line shows the outer edge of the boundary layer for that flow

rate

4.4.2 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:

d

0 0tr dyz(Hd

1H ) (4.3)

where d is the flow depth, Ht is the total head and z0 is the step edge elevation above the datum. The

normalised specific energy, Hr/Hmax, along the stepped chute is shown in Figure 4.10, where Hmax is

the maximum specific energy at any step edge along the spillway, L is the streamwise distance from

the first step edge and Li 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

29

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:

20

o

V

8f

(4.4)

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:

20

MI0102

20 V

8

f

x

VV)V(

x

(4.5)

L/Li

Hr/H

max

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dc/h = 0.9dc/h = 1.0

dc/h = 1.1dc/h = 1.3

dc/h = 1.5dc/h = 1.7

Meireles et al., 1V:0.75H

Figure 4.10 – Longitudinal variation in dimensionless specific energy Hr/Hmax in the developing

flow region of skimming flows on a stepped spillway

30

where the subscript MI indicates calculations based upon the momentum integral equation. The

results in terms of fMI are presented in dimensionless form in Figure 4.11. On average, the

dimensionless boundary shear stress deduced from the momentum integral equation equalled: fMI

0.182 (Table 4.1). The result was close to the air-water flow friction factor estimates fe 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 fMI 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 fMI 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

vxvy/(/8V02) between 0 and 0.056 along the pseudo-bottom formed by the step edges in the

developing flow region. His experimental observations yielded a spatially-averaged dimensionless

shear stress fxy 0.27 over the entire step cavity. For comparison, Gonzalez and Chanson (2004)

measured a spatially-averaged dimensionless shear stresses fxy 0.34, in the fully-developed air-

water flow region (Table 4.1).

Lastly a gross estimate of the friction factor might be derived from the friction slope Sf (Sf = -

H/x) assuming a fully-developed flow in a wide channel:

3

cfSf d

dS8f

(4.6)

where d is the flow depth, dc is the critical flow depth: dc = (q2/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 < dc/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.

31

dc/h

f MI,

f Sf,

f e

0.7 0.9 1.1 1.3 1.5 1.7 1.90

0.1

0.2

0.3

0.4

fMI = 0.182 on average

momentum integral friction slope air-water flow region

Figure 4.11 – Dimensionless boundary shear stress in the developing flow region of skimming

flows on stepped spillways

Table 4.1 - Dimensionless boundary stress f = o/(/8V02) in skimming flows on stepped spillways

- Average experimental results

Reference W h Developing flow region Aerated region fMI fxy fSf fxy fe

(º) (m) (m) Momentum integral eq.

(4.5)

Reynolds stress

Friction slope

Eq. (4.6)

Reynolds stress

Friction slope

Eq. (4.6)Present study 45 1.0 0.10 0.182 -- 0.222 -- 0.227 Gonzalez & Chanson (2004) 15.9 1.0 0.10 -- -- -- 0.34 0.15 Amador (2005) 51.3 0.5 0.05 0.124 0.27 -- -- -- Meireles (2011) (1) 53.1 1.0 0.04 0.087 -- -- -- -- 0.08 0.122 -- -- -- -- Frizell et al. (2013) (2) 21.8 0.203 0.027 -- -- 0.089 -- -- 0.055 -- -- 0.127 -- -- 68.2 0.068 -- -- 0.160 -- -- 0.136 -- -- 0.167 -- --

Notes: fe: dimensionless shear stress in fully-developed air-water flows downstream of inception

point of free-surface aeration; fMI: dimensionless shear stress derived from von Karman momentum

integral calculations; fSf: dimensionless shear stress derived from friction slope calculations; fxy:

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;

32

(1): experimental data by Matos (1999), Meireles (2004) and Renna (2004) (Table 1.1); (2): water

tunnel data by Frizell et al. (2013).

33

5. CONCLUSION

In skimming flow on a stepped spillway, 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: 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: /di 0.8.

In the developing boundary layer, the velocity distributions followed a 1/4.5th 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 rapidly-

varied 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 0.23. It is believed that this is the first

study quantifying the boundary shear stress in both developing clear-water and fully-developed air-

water 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

34

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

the developing flow region.

35

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 DP120100481) and through the

University of Queensland are acknowledged.

A-1

APPENDIX A - PHOTOGRAPHS OF EXPERIMENTS

(A)

(B)

Figure A.1 - Skimming flow conditions: dc/h = 0.9, h = 0.1 m, = 45º

A-2

Figure A.2 - Skimming flow conditions: dc/h = 1.1, h = 0.1 m, = 45º

Figure A.3 - Skimming flow conditions: dc/h = 1.5, h = 0.1 m, = 45º

A-3

(A)

(B)

(C)

Figure A.4 - Skimming flow conditions: dc/h = 1.7, h = 0.1 m, = 45º

A-4

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:

0.37

sk

L0.15

L

δ

(B.1)

where L is the streamwise distance from the first step edge, and ks = h×cos is the step cavity

height. Equation B.1 is valid for 0.9 ≤ dc/h ≤ 1.7 and 0 < L/ks < 15.

The boundary layer characteristics are summarised in Table B.1, where V0 is the free-stream

velocity, d is the water depth, is the boundary layer thickness defined in terms of 0.99V0, 1 is

the displacement thickness, 2 is the momentum thickness and 3 is the energy thickness. The data

were best correlated by:

0.18LRe07.1

L

δ R = 0.405 (B.2)

0.32L

1 Re58.1L

δ R = 0.343 (B.3)

0.11L

2 Re0.053L

δ R = 0.179 (B.4)

0.14L

3 Re0.132L

δ R = 0.242 (B.5)

for 2×105 < ReL < 2×106, with ReL = VoL/, 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

5/1LRe41.1

L

δ R = 0.402 (B.6)

1/5L

1 Re0.319L

δ R = 0.310 (B.7)

5/1L

2 Re0.169L

δ R = 0.117 (B.8)

For the entire experimental data set, the median data yielded:

191.0δ1

(B.9)

126.0δ2

(B.10)

A-5

53.1δ

2

1

(B.10)

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 = 0.985 m)

dc/h Step edge x (m) d (m) V0 (m/s) (m) 1 (m) 2 (m) 3 (m) 0.9 2 0.14 0.048 1.98 0.0131 0.0020 0.0015 0.0028 0.9 3 0.28 0.041 2.45 0.0183 0.0032 0.0018 0.0032 0.9 4 0.42 0.041 2.82 0.0280 0.0053 0.0037 0.0064 1 2 0.14 0.053 2.04 0.0136 0.0025 0.0018 0.0031 1 3 0.28 0.045 2.50 0.0189 0.0035 0.0017 0.0030 1 4 0.42 0.044 2.87 0.0242 0.0050 0.0035 0.0059

1.1 2 0.14 0.060 2.09 0.0151 0.0029 0.0021 0.0036 1.1 3 0.28 0.051 2.54 0.0204 0.0039 0.0015 0.0029 1.1 4 0.42 0.050 2.90 0.0287 0.0048 0.0035 0.0062 1.3 2 0.14 0.075 2.18 0.0180 0.0047 0.0027 0.0044 1.3 3 0.28 0.063 2.62 0.0230 0.0036 0.0015 0.0028 1.3 4 0.42 0.058 2.98 0.0388 0.0046 0.0034 0.0060 1.3 5 0.57 0.058 3.30 0.0409 0.0100 0.0064 0.0108 1.5 2 0.14 0.089 2.27 0.0229 0.0101 0.0017 0.0035 1.5 3 0.28 0.075 2.70 0.0329 0.0064 0.0043 0.0074 1.5 4 0.42 0.069 3.05 0.0470 0.0106 0.0068 0.0115 1.5 5 0.57 0.068 3.36 0.0515 0.0177 0.0092 0.0144 1.7 2 0.14 0.108 2.34 0.0262 0.0199 -0.0025 0.0007 1.7 3 0.28 0.089 2.77 0.0357 0.0107 0.0060 0.0099 1.7 4 0.42 0.080 3.13 0.0558 0.0085 0.0062 0.0111 1.7 5 0.57 0.078 3.43 0.0479 0.0068 0.0046 0.0083

Note: italic data: erroneous data.

A-6

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 < dc/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

dc critical flow depth

x streamwise distance from step edge 1

xt streamwise distance of the total pressure tapping from step edge 1

y normal distance to the pseudo-bottom

h step height

Hp piezometric head a.k.a. static head

Ht total head

Q flow rate

Figure B1. Sketch of Pitot-tube measurement experiments

A-7

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 0.7 0.057 141.4 0.0 130 7

141.4 3.7 153 11 141.4 6.4 178 13 141.4 9.3 197 15 141.4 12.8 208 17 141.4 15.3 210 18 141.4 18.4 211 18 141.4 21.2 211 18 141.4 24.0 211 20

0.9 0.083 141.4 0.0 124 37 141.4 1.2 131 37 141.4 3.0 149 37 141.4 4.9 173 39 141.4 7.5 201 39 141.4 10.0 223 39 141.4 12.4 235 38 141.4 15.1 240 38 141.4 18.0 241 39 141.4 20.8 241 37 141.4 24.0 240 36 141.4 27.1 241 35 141.4 30.3 241 33 141.4 33.4 241 32 141.4 36.2 241 31

0.9 0.083 264.9 1.9 - 16 264.9 5.1 - 24 264.9 10.8 - 31 264.9 15.5 - 35 264.9 19.7 - 33 264.9 24.4 - 33 264.9 28.1 - 33

0.9 0.083 282.8 0.0 -61 -32 282.8 1.6 96 -32 282.8 3.1 211 -32 282.8 5.0 200 -32 282.8 7.0 209 -29 282.8 10.0 225 -26 282.8 12.8 244 -23 282.8 15.1 266 -19 282.8 19.0 308 -11 282.8 22.2 323 -6 282.8 25.1 336 -1 282.8 30.1 341 9 282.8 35.0 341 17

0.9 0.083 329.3 0.0 281 110 329.3 2.1 305 102 329.3 5.9 322 90 329.3 9.6 335 73 329.3 13.1 343 65 329.3 17.3 345 57 329.3 21.6 345 49 329.3 25.7 345 44 329.3 28.0 345 41

0.9 0.083 336.4 0.0 277 83 336.4 3.5 296 78 336.4 6.6 321 68 336.4 10.2 339 60 336.4 13.3 346 56 336.4 17.6 349 51 336.4 20.6 349 47 336.4 24.0 349 43 336.4 28.1 350 38

0.9 0.083 350.6 2.2 263 -33 350.6 5.9 294 -17 350.6 9.5 328 -9 350.6 13.5 347 7 350.6 17.0 355 12 350.6 21.4 358 20 350.6 25.0 359 25 350.6 28.8 358 27

0.9 0.083 357.7 0.7 243 -33 357.7 4.3 281 -24

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 357.7 8.2 304 -19 357.7 12.1 329 -10 357.7 15.4 349 -3 357.7 20.7 361 6 357.7 24.7 363 15 357.7 29.0 363 22

0.9 0.083 424.3 0.0 70 32 424.3 1.5 184 35 424.3 3.0 187 32 424.3 5.2 208 33 424.3 7.3 236 37 424.3 10.1 263 41 424.3 13.4 305 39 424.3 16.3 352 42 424.3 20.1 393 43 424.3 25.3 420 43 424.3 30.0 441 47

0.9 0.083 470.7 0.6 161 88 470.7 5.6 231 83 470.7 9.9 298 71 470.7 14.3 346 61 470.7 18.2 386 55 470.7 22.5 406 51 470.7 26.9 418 48

0.9 0.083 477.8 1.6 146 81 477.8 6.5 201 63 477.8 11.2 276 53 477.8 15.2 333 48 477.8 19.5 384 48 477.8 24.1 413 44 477.8 28.2 429 41

0.9 0.083 492.0 0.0 156 1 492.0 5.3 248 19 492.0 10.7 326 29 492.0 15.8 402 46 492.0 20.7 435 44 492.0 25.2 431 41

0.9 0.083 499.1 0.6 184 17 499.1 6.6 236 19 499.1 10.7 306 28 499.1 14.9 356 29 499.1 19.0 402 34 499.1 23.1 429 34 499.1 27.5 449 36

1 0.098 141.4 0.0 122 53 141.4 1.5 133 52 141.4 3.3 153 53 141.4 5.6 178 53 141.4 8.1 212 53 141.4 10.6 237 52 141.4 13.6 251 53 141.4 17.3 254 52 141.4 20.6 256 51 141.4 23.7 256 48 141.4 26.7 256 46 141.4 29.6 256 43 141.4 33.4 256 40 141.4 36.3 256 38 141.4 40.1 256 36 141.4 43.4 256 35 264.9 2.6 - 18 264.9 8.5 - 31 264.9 13.7 - 35 264.9 19.1 - 36 264.9 23.5 - 36 264.9 28.5 - 36 264.9 33.3 - 37

1 0.098 282.8 0.0 -70 -40 282.8 1.5 -5 -39 282.8 3.0 227 -39 282.8 5.0 214 -37 282.8 7.2 220 -34 282.8 10.2 239 -30 282.8 13.0 256 -26

A-8

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 282.8 16.2 294 -21 282.8 20.0 324 -14 282.8 24.7 340 -4 282.8 29.9 352 5 282.8 35.0 356 13 282.8 40.0 356 20

1 0.098 329.3 1.8 281 89 329.3 4.6 302 81 329.3 8.9 342 76 329.3 12.5 355 70 329.3 16.9 357 63 329.3 19.5 359 61 329.3 23.2 359 55 329.3 26.7 359 52 329.3 30.7 359 45

1 0.098 336.4 0.4 286 85 336.4 3.0 298 78 336.4 6.7 331 71 336.4 9.8 345 64 336.4 14.0 359 61 336.4 17.3 364 57 336.4 20.8 366 54 336.4 23.9 366 51 336.4 27.5 366 48 336.4 30.1 366 47

1 0.098 350.6 0.0 264 -39 350.6 3.7 294 -29 350.6 7.1 317 -14 350.6 11.0 344 -1 350.6 14.4 353 8 350.6 17.5 361 16 350.6 20.2 363 18 350.6 23.3 364 22 350.6 26.9 364 25 350.6 30.9 364 28

1 0.098 357.7 0.0 246 -34 357.7 3.0 262 -29 357.7 6.7 296 -23 357.7 10.0 316 -16 357.7 14.8 351 -5 357.7 18.6 362 3 357.7 22.7 368 11 357.7 26.5 369 16 357.7 30.3 369 21

1 0.098 424.3 0.0 61 30 424.3 1.6 193 30 424.3 3.1 190 29 424.3 5.0 207 31 424.3 7.6 227 30 424.3 10.2 269 36 424.3 13.3 324 38 424.3 16.3 362 38 424.3 20.1 392 39 424.3 25.1 434 38 424.3 30.1 460 41 424.3 35.2 456 41

1 0.098 470.7 0.9 167 95 470.7 5.7 259 95 470.7 10.3 309 75 470.7 15.4 366 69 470.7 19.6 397 62 470.7 24.2 424 55 470.7 28.6 436 55 470.7 33.3 448 48

1 0.098 477.8 0.6 136 93 477.8 4.5 186 78 477.8 8.9 256 66 477.8 13.1 323 58 477.8 17.7 373 53 477.8 21.8 406 48 477.8 25.8 429 47 477.8 29.7 449 47 477.8 33.6 456 44

1 0.098 492.0 2.0 261 37 492.0 7.0 316 49

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 492.0 11.5 361 56 492.0 16.1 396 54 492.0 20.0 418 57 492.0 24.8 448 51 492.0 28.0 496 48 492.0 33.1 491 44

1 0.098 499.1 0.0 174 22 499.1 5.6 214 31 499.1 11.0 311 37 499.1 16.2 378 42 499.1 20.6 421 45 499.1 25.5 452 44 499.1 31.2 466 41

1.1 0.112 141.4 0.0 131 66 141.4 1.5 143 66 141.4 3.3 151 67 141.4 5.2 181 66 141.4 7.0 205 66 141.4 9.2 229 65 141.4 11.5 247 66 141.4 14.1 262 65 141.4 16.5 268 63 141.4 19.1 269 61 141.4 22.9 270 58 141.4 27.1 270 54 141.4 30.1 270 52 141.4 34.1 270 48 141.4 38.1 270 44 141.4 43.4 270 40 141.4 47.7 270 37

1.1 0.112 264.9 2.0 - 15 264.9 4.8 - 23 264.9 11.2 - 35 264.9 17.6 - 41 264.9 22.7 - 42 264.9 28.8 - 43 264.9 33.5 - 43 264.9 37.7 - 41

1.1 0.112 282.8 0.0 -81 -50 282.8 1.3 -45 -48 282.8 2.8 222 -50 282.8 4.7 225 -46 282.8 6.9 220 -44 282.8 9.2 233 -40 282.8 12.5 250 -33 282.8 15.3 284 -28 282.8 18.6 314 -22 282.8 22.1 335 -15 282.8 26.1 352 -6 282.8 30.1 365 0 282.8 35.2 369 9 282.8 40.2 371 16 282.8 45.4 371 24

1.1 0.112 329.3 1.8 292 94 329.3 4.6 317 90 329.3 8.3 346 84 329.3 11.9 362 80 329.3 14.8 371 79 329.3 18.6 373 71 329.3 23.2 374 66 329.3 28.4 374 60 329.3 33.9 374 51 329.3 37.3 374 49

1.1 0.112 336.4 0.1 289 87 336.4 1.6 311 82 336.4 5.4 337 72 336.4 9.3 363 72 336.4 14.1 377 65 336.4 18.1 379 62 336.4 23.3 381 57 336.4 28.1 381 54 336.4 32.9 381 50 336.4 36.7 381 47

1.1 0.112 350.6 1.1 283 -44 350.6 4.1 294 -32

A-9

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 350.6 6.8 321 -17 350.6 10.5 359 -3 350.6 14.4 378 12 350.6 18.7 383 18 350.6 22.9 389 26 350.6 28.6 389 31 350.6 32.9 390 35 350.6 37.7 390 37 350.6 41.3 389 39

1.1 0.112 357.7 1.3 257 -39 357.7 6.0 287 -29 357.7 10.8 340 -15 357.7 15.6 374 -3 357.7 20.5 383 11 357.7 25.7 392 21 357.7 30.9 393 29 357.7 34.9 393 33 357.7 39.7 393 36

1.1 0.112 424.3 0.0 73 43 424.3 1.5 237 41 424.3 3.0 254 44 424.3 5.0 265 44 424.3 7.1 279 44 424.3 10.1 322 53 424.3 13.2 343 53 424.3 16.2 389 55 424.3 20.2 423 59 424.3 25.2 455 58 424.3 30.4 478 57 424.3 35.4 490 54 424.3 40.2 470 53

1.1 0.112 470.7 0.9 183 118 470.7 6.3 285 108 470.7 11.0 341 91 470.7 16.1 386 75 470.7 20.7 416 67 470.7 25.5 438 60 470.7 30.0 448 55 470.7 34.6 466 51 470.7 38.3 467 51

1.1 0.112 477.8 0.6 159 105 477.8 4.0 191 88 477.8 8.6 286 84 477.8 13.6 346 68 477.8 17.9 391 61 477.8 22.7 431 58 477.8 27.7 448 56 477.8 32.1 464 53 477.8 36.7 473 49 477.8 41.5 476 47

1.1 0.112 492.0 2.0 281 44 492.0 5.8 321 54 492.0 11.1 366 64 492.0 16.3 411 66 492.0 21.3 441 61 492.0 26.7 456 61 492.0 31.8 476 61 492.0 36.5 481 49

1.1 0.112 499.1 4.7 261 40 499.1 9.5 316 49 499.1 14.3 394 64 499.1 18.7 436 68 499.1 23.7 461 67 499.1 28.6 477 54 499.1 33.3 486 48 499.1 38.4 491 48

1.3 0.145 141.4 0.0 117 93 141.4 1.3 129 93 141.4 2.9 144 93 141.4 4.6 165 92 141.4 6.2 183 91 141.4 7.7 208 90 141.4 9.8 235 90 141.4 11.8 259 89 141.4 14.1 279 88

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 141.4 16.9 292 86 141.4 19.6 296 83 141.4 22.6 298 81 141.4 26.3 298 77 141.4 30.0 298 73 141.4 34.2 298 69 141.4 38.1 298 64 141.4 42.4 299 61 141.4 47.1 299 56 141.4 51.1 299 52 141.4 57.4 299 47

1.3 0.145 201.5 2.0 45 65 201.5 5.8 92 62 201.5 12.4 242 61 201.5 18.7 303 58 201.5 25.9 314 56 201.5 32.8 315 53 201.5 39.3 315 51 201.5 45.5 315 49 201.5 52.8 315 48 201.5 59.4 315 46

1.3 0.145 215.7 1.9 111 80 215.7 8.8 219 78 215.7 15.5 307 76 215.7 22.8 325 73 215.7 29.6 326 69 215.7 35.7 326 64 215.7 42.5 326 61 215.7 48.6 325 56 215.7 52.7 326 54 215.7 59.4 326 51

1.3 0.145 244.1 -9.2 122 99 244.1 -3.3 165 98 244.1 2.7 236 96 244.1 8.4 311 96 244.1 14.6 342 96 244.1 21.9 345 90 244.1 29.0 345 83 244.1 36.1 345 77 244.1 43.3 346 69 244.1 49.9 345 60 244.1 56.1 346 55

1.3 0.145 258.3 -25.4 121 97 258.3 -17.5 126 100 258.3 -9.1 149 102 258.3 -1.3 221 102 258.3 6.3 314 102 258.3 13.8 352 101 258.3 22.2 356 97 258.3 29.1 356 91 258.3 36.8 356 82 258.3 44.7 356 71 258.3 49.9 356 63

1.3 0.145 264.9 0.0 - 19 264.9 4.9 - 23 264.9 10.1 - 31 264.9 14.2 - 36 264.9 18.4 - 45 264.9 23.4 - 50 264.9 27.0 - 51 264.9 32.4 - 54 264.9 37.4 - 56 264.9 42.4 - 55 264.9 46.7 - 53 264.9 50.5 - 52

1.3 0.145 272.5 -34.7 115 105 272.5 -28.7 118 105 272.5 -21.3 133 108 272.5 -14.1 152 109 272.5 -7.1 189 109 272.5 0.6 271 110 272.5 8.1 343 109 272.5 15.4 364 108 272.5 22.9 366 102 272.5 29.6 366 94

A-10

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 272.5 36.5 366 73 272.5 42.9 366 73 272.5 49.9 366 62

1.3 0.145 282.8 0.0 -109 -65 282.8 1.4 -49 -63 282.8 3.0 239 -63 282.8 5.0 248 -60 282.8 7.5 256 -58 282.8 10.1 273 -49 282.8 13.1 299 -44 282.8 16.1 332 -32 282.8 20.3 354 -21 282.8 25.2 383 -8 282.8 30.3 399 4 282.8 35.1 402 12 282.8 40.3 402 21 282.8 45.1 402 26 282.8 50.1 402 30 282.8 55.0 402 36 282.8 57.5 402 38

1.3 0.145 286.7 -21.1 145 125 286.7 -16.6 161 128 286.7 -9.8 191 127 286.7 -4.2 236 126 286.7 4.5 326 123 286.7 12.2 369 119 286.7 20.9 376 112 286.7 28.0 376 103 286.7 37.7 376 87 286.7 45.9 376 71 286.7 49.9 376 64

1.3 0.145 300.9 -9.8 208 148 300.9 -2.8 266 144 300.9 4.4 336 138 300.9 11.3 374 128 300.9 18.6 387 118 300.9 25.8 387 107 300.9 33.8 387 93 300.9 41.2 387 78 300.9 49.5 387 66

1.3 0.145 329.3 0.1 326 134 329.3 3.8 356 122 329.3 8.8 384 109 329.3 12.6 397 100 329.3 17.6 404 93 329.3 21.7 405 84 329.3 26.5 406 81 329.3 30.9 406 77 329.3 35.2 406 72 329.3 39.7 406 67 329.3 44.3 406 61 329.3 48.7 406 59

1.3 0.145 336.4 0.0 303 94 336.4 3.4 329 82 336.4 6.5 358 79 336.4 10.7 381 72 336.4 15.6 404 72 336.4 19.4 407 72 336.4 24.7 409 68 336.4 28.4 411 68 336.4 33.3 410 68 336.4 38.3 411 65 336.4 42.3 410 62 336.4 47.6 410 57

1.3 0.145 350.6 1.1 306 -74 350.6 4.7 322 -45 350.6 9.4 359 -10 350.6 14.0 395 10 350.6 18.7 408 24 350.6 23.6 416 34 350.6 27.7 418 40 350.6 32.0 418 41 350.6 36.9 418 48 350.6 42.2 418 48 350.6 46.7 418 47

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 350.6 50.3 419 48

1.3 0.145 357.7 1.5 281 -52 357.7 4.9 321 -43 357.7 10.0 361 -22 357.7 15.1 386 -9 357.7 19.0 408 10 357.7 24.2 420 21 357.7 29.3 423 31 357.7 33.1 424 35 357.7 40.0 424 43 357.7 45.1 424 45 357.7 50.0 424 47

1.3 0.145 424.3 0.0 79 41 424.3 1.5 263 42 424.3 3.3 248 42 424.3 5.0 284 45 424.3 7.2 293 48 424.3 10.3 320 52 424.3 13.8 382 58 424.3 16.4 488 60 424.3 20.0 517 62 424.3 25.3 458 62 424.3 30.4 476 61 424.3 36.6 489 57 424.3 42.4 498 53 424.3 48.9 499 49

1.3 0.145 470.7 0.9 226 140 470.7 6.3 306 121 470.7 11.6 391 103 470.7 17.1 441 94 470.7 21.4 466 88 470.7 26.3 487 80 470.7 31.6 497 73 470.7 37.2 499 65 470.7 42.9 502 61 470.7 47.8 503 54

1.3 0.145 477.8 0.0 226 164 477.8 4.1 296 142 477.8 10.2 376 120 477.8 16.6 446 103 477.8 22.0 479 93 477.8 27.6 488 83 477.8 33.4 504 75 477.8 39.0 508 66 477.8 44.6 508 58 477.8 49.1 508 56

1.3 0.145 492.0 0.0 266 46 492.0 2.3 296 56 492.0 7.5 371 70 492.0 12.8 439 78 492.0 18.1 471 85 492.0 24.1 498 81 492.0 29.4 504 76 492.0 34.4 511 70 492.0 40.4 517 63 492.0 46.7 518 57

1.3 0.145 499.1 0.4 306 51 499.1 4.9 351 61 499.1 10.3 402 68 499.1 14.6 448 74 499.1 19.9 470 75 499.1 26.0 506 75 499.1 31.5 517 71 499.1 37.5 521 64 499.1 43.9 525 58 499.1 48.6 524 55

1.3 0.145 565.7 0.0 6 17 565.7 1.5 132 13 565.7 3.1 304 9 565.7 5.2 284 2 565.7 7.2 239 3 565.7 10.2 280 3 565.7 13.2 210 -1 565.7 16.4 251 -1 565.7 20.2 259 3

A-11

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 565.7 25.1 374 9 565.7 30.2 456 20 565.7 36.1 554 32 565.7 43.6 599 48 565.7 50.7 546 64

1.3 0.145 612.1 0.0 151 55 612.1 4.4 181 48 612.1 8.6 261 45 612.1 13.6 361 45 612.1 18.9 446 45 612.1 24.2 511 43 612.1 28.9 551 46 612.1 32.9 575 53 612.1 38.5 592 48 612.1 43.5 600 46

1.3 0.145 619.2 1.5 146 42 619.2 6.0 176 15 619.2 11.1 241 2 619.2 16.7 386 20 619.2 21.8 456 24 619.2 27.2 506 17 619.2 32.5 553 17 619.2 37.9 580 31 619.2 43.4 603 37

1.3 0.145 633.4 2.4 116 -44 633.4 8.9 196 -25 633.4 13.4 286 -14 633.4 19.0 356 -13 633.4 25.5 479 -1 633.4 31.2 536 7 633.4 38.8 596 22 633.4 44.1 603 27

1.3 0.145 640.5 1.7 131 -32 640.5 5.4 141 -35 640.5 10.7 236 -29 640.5 15.1 281 -24 640.5 19.8 396 -16 640.5 24.9 471 -6 640.5 29.8 511 -4 640.5 34.5 561 7 640.5 39.9 596 20 640.5 45.4 613 32

1.3 0.145 707.1 0.0 61 17 707.1 1.1 236 17 707.1 2.6 324 17 707.1 4.9 299 21 707.1 7.3 291 20 707.1 10.3 361 28 707.1 13.2 371 34 707.1 16.4 408 36 707.1 20.2 423 39 707.1 25.3 443 39 707.1 30.3 502 39 707.1 36.7 527 39 707.1 42.9 545 39 707.1 49.2 437 42

1.5 0.179 141.4 0.0 94 108 141.4 1.5 97 108 141.4 3.0 104 107 141.4 4.7 120 107 141.4 6.9 145 106 141.4 9.0 179 105 141.4 10.9 210 104 141.4 13.2 246 103 141.4 16.1 290 102 141.4 19.0 314 100 141.4 22.1 325 98 141.4 25.8 327 95 141.4 30.2 328 91 141.4 34.7 328 87 141.4 39.3 328 83 141.4 43.7 328 79 141.4 49.0 327 75 141.4 54.1 329 70 141.4 60.0 328 66

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 141.4 65.1 328 61 141.4 70.5 328 58 141.4 75.0 328 56

1.5 0.179 229.9 1.1 176 109 229.9 5.5 156 108 229.9 10.4 227 105 229.9 16.9 333 105 229.9 23.4 362 99 229.9 30.7 363 92 229.9 37.5 363 89 229.9 43.3 363 83 229.9 50.0 363 79 229.9 55.6 363 75 229.9 62.5 364 69 229.9 68.9 363 67

1.5 0.179 244.1 -13.6 129 118 244.1 -6.1 134 118 244.1 -0.4 156 117 244.1 7.6 137 115 244.1 13.9 332 114 244.1 20.6 371 111 244.1 26.0 373 107 244.1 32.8 374 102 244.1 38.5 374 97 244.1 45.6 374 91 244.1 52.6 373 85 244.1 60.0 373 77 244.1 68.9 373 69

1.5 0.179 258.3 -25.7 138 118 258.3 -19.7 138 120 258.3 -11.9 142 121 258.3 -4.4 157 121 258.3 2.9 206 121 258.3 10.5 314 118 258.3 17.2 376 118 258.3 23.4 383 115 258.3 29.2 383 109 258.3 35.5 383 103 258.3 41.8 383 99 258.3 48.2 383 91 258.3 54.5 383 83 258.3 60.7 383 75 258.3 67.7 384 69 258.3 72.1 383 65

1.5 0.179 264.9 0.0 - 21 264.9 3.0 - 14 264.9 7.1 - 14 264.9 11.5 - 28 264.9 16.2 - 40 264.9 20.5 - 54 264.9 24.6 - 62 264.9 29.4 - 68 264.9 35.2 - 72 264.9 40.2 - 74 264.9 44.2 - 77 264.9 49.2 - 75 264.9 53.5 - 72 264.9 58.4 - 69 264.9 62.5 - 66 264.9 67.0 - 64

1.5 0.179 272.5 -32.9 140 120 272.5 -28.0 142 119 272.5 -22.0 146 121 272.5 -15.0 149 122 272.5 -7.0 167 124 272.5 -0.4 195 123 272.5 6.0 272 123 272.5 14.3 374 123 272.5 22.1 394 122 272.5 29.0 394 116 272.5 36.3 394 109 272.5 43.3 393 103 272.5 49.5 394 95 272.5 57.0 393 84 272.5 63.5 394 75

A-12

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 272.5 67.7 394 69

1.5 0.179 282.8 0.0 -91 -50 282.8 1.4 13 -64 282.8 3.0 204 -50 282.8 5.1 200 -41 282.8 7.4 207 -37 282.8 10.0 221 -38 282.8 13.2 258 -30 282.8 16.2 270 -22 282.8 20.1 327 -19 282.8 25.2 366 -8 282.8 30.2 407 8 282.8 35.1 421 19 282.8 40.1 427 33 282.8 46.5 428 40 282.8 53.0 428 44 282.8 60.2 428 46 282.8 66.6 428 49

1.5 0.179 286.7 -22.0 152 128 286.7 -16.0 158 129 286.7 -9.3 166 126 286.7 -2.7 195 126 286.7 3.2 144 129 286.7 10.3 331 125 286.7 16.6 392 124 286.7 22.2 404 123 286.7 27.7 403 118 286.7 33.5 403 117 286.7 38.5 404 109 286.7 43.4 404 103 286.7 48.9 404 94 286.7 54.4 404 86 286.7 59.7 404 79 286.7 66.4 404 69

1.5 0.179 300.9 2.0 241 131 300.9 8.7 314 132 300.9 13.9 375 129 300.9 19.6 407 127 300.9 24.7 413 122 300.9 30.3 414 117 300.9 35.3 414 113 300.9 43.3 414 104 300.9 49.0 415 96 300.9 54.1 415 88 300.9 60.6 414 78 300.9 66.4 414 70

1.5 0.179 315.1 2.0 249 132 315.1 6.6 291 128 315.1 11.7 346 124 315.1 17.4 404 120 315.1 22.3 420 117 315.1 28.3 424 110 315.1 34.2 424 104 315.1 40.2 424 102 315.1 46.2 425 94 315.1 52.5 425 85 315.1 58.4 425 77 315.1 65.5 425 69

1.5 0.179 329.3 0.1 229 92 329.3 1.1 241 111 329.3 4.3 276 90 329.3 8.0 334 91 329.3 8.8 337 101 329.3 12.3 366 92 329.3 16.3 408 99 329.3 17.9 411 95 329.3 23.3 431 94 329.3 24.4 432 97 329.3 30.4 444 93 329.3 32.1 434 94 329.3 37.5 444 92 329.3 40.8 434 93 329.3 46.1 444 88 329.3 50.2 434 82 329.3 54.3 444 77

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 329.3 62.1 445 70 329.3 63.1 434 70 329.3 66.1 445 67

1.5 0.179 336.4 1.9 276 107 336.4 6.4 326 97 336.4 11.0 356 87 336.4 15.9 411 88 336.4 20.5 426 84 336.4 25.3 437 85 336.4 31.7 438 85 336.4 37.5 438 85 336.4 42.3 438 87 336.4 48.9 438 79 336.4 55.8 438 73 336.4 62.4 438 67

1.5 0.179 350.6 0.0 213 -19 350.6 4.6 272 -5 350.6 9.0 331 4 350.6 13.2 377 18 350.6 17.4 411 31 350.6 22.3 439 42 350.6 26.7 447 53 350.6 32.4 449 60 350.6 37.6 449 65 350.6 43.3 449 69 350.6 48.2 449 69 350.6 52.4 449 66 350.6 57.4 449 64 350.6 62.1 449 64

1.5 0.179 357.1 1.5 256 -35 357.1 6.7 271 -25 357.1 12.0 338 -4 357.1 17.9 394 12 357.1 25.6 438 30 357.1 31.4 449 42 357.1 37.4 453 52 357.1 43.1 454 60 357.1 49.2 453 62 357.1 56.5 454 60 357.1 61.5 454 60

1.5 0.179 357.7 1.5 246 -38 357.7 4.5 268 -30 357.7 9.7 294 -15 357.7 15.8 369 9 357.7 20.1 404 15 357.7 25.6 433 31 357.7 30.9 450 42 357.7 39.1 453 54 357.7 46.0 454 64 357.7 52.7 453 62 357.7 59.3 454 62 357.7 65.7 453 62

1.5 0.179 364.2 2.6 231 -30 364.2 9.1 308 -19 364.2 16.3 361 -4 364.2 23.4 427 11 364.2 27.7 449 25 364.2 33.8 456 35 364.2 40.5 459 52 364.2 50.0 459 55 364.2 57.4 459 56 364.2 62.9 459 58

1.5 0.179 385.5 -4.6 -29 -58 385.5 2.7 266 -48 385.5 9.5 286 -29 385.5 14.7 339 -17 385.5 21.5 391 -4 385.5 28.2 452 15 385.5 34.6 467 30 385.5 41.5 474 42 385.5 47.6 474 49 385.5 54.0 474 50 385.5 60.3 474 53

1.5 0.179 392.6 -5.0 -21 -60 392.6 1.0 124 -51

A-13

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 392.6 7.4 308 -34 392.6 14.2 336 -17 392.6 19.8 373 -7 392.6 24.5 419 3 392.6 30.1 450 16 392.6 36.9 474 33 392.6 41.2 479 42 392.6 48.8 479 48 392.6 56.4 480 51 392.6 60.3 480 54

1.5 0.179 399.7 -10.1 -32 -60 399.7 -2.8 11 -53 399.7 4.4 228 -35 399.7 11.9 311 -19 399.7 18.8 371 -3 399.7 24.8 401 -4 399.7 32.2 462 23 399.7 38.7 481 38 399.7 46.7 485 46 399.7 53.9 485 52 399.7 61.3 485 54

1.5 0.179 406.3 0.0 - 91 406.3 3.2 - 104 406.3 6.6 - 106 406.3 9.8 - 105 406.3 13.3 - 103 406.3 17.5 - 104 406.3 22.7 - 97 406.3 27.4 - 93 406.3 31.0 - 94 406.3 36.1 - 90 406.3 40.0 - 86 406.3 44.1 - 81 406.3 49.7 - 75 406.3 55.1 - 67 406.3 59.8 - 64

1.5 0.179 406.8 -24.1 -53 -51 406.8 -15.3 -47 -54 406.8 -7.1 -15 -49 406.8 0.0 83 -36 406.8 6.4 272 -21 406.8 12.9 319 -10 406.8 18.2 361 -1 406.8 23.6 376 8 406.8 28.6 439 17 406.8 36.6 479 35 406.8 41.4 486 41 406.8 48.1 489 48 406.8 56.1 490 53 406.8 61.3 490 55

1.5 0.179 413.9 -29.5 -52 -38 413.9 -22.4 -54 -40 413.9 -15.6 -47 -43 413.9 -8.7 -11 -36 413.9 -1.3 73 -25 413.9 4.1 317 -11 413.9 11.0 306 1 413.9 16.5 323 7 413.9 23.2 386 15 413.9 29.9 429 26 413.9 35.9 467 37 413.9 42.5 493 46 413.9 48.4 496 51 413.9 55.6 496 55 413.9 62.8 495 59

1.5 0.179 421.0 -34.6 -41 -29 421.0 -28.5 -45 -30 421.0 -22.5 -45 -30 421.0 -16.8 -46 -25 421.0 -10.8 -16 -27 421.0 -4.8 30 -17 421.0 1.4 147 -4 421.0 8.1 320 6 421.0 13.5 366 16 421.0 18.6 398 24

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 421.0 25.3 446 34 421.0 32.2 484 44 421.0 38.9 492 49 421.0 46.3 499 54 421.0 52.7 499 56 421.0 60.3 499 58

1.5 0.179 424.3 0.0 40 39 424.3 1.5 265 41 424.3 3.2 263 42 424.3 5.2 268 47 424.3 7.2 277 50 424.3 10.1 299 53 424.3 13.3 329 58 424.3 16.4 338 61 424.3 20.0 359 63 424.3 25.1 384 65 424.3 30.2 426 68 424.3 36.2 480 68 424.3 42.3 504 68 424.3 50.2 522 70 424.3 60.2 528 55

1.5 0.179 428.1 -25.3 -14 -41 428.1 -18.1 -40 -6 428.1 -10.2 -15 -1 428.1 0.4 138 12 428.1 8.7 318 24 428.1 14.8 367 31 428.1 22.2 415 38 428.1 29.5 480 46 428.1 35.9 494 52 428.1 43.7 503 56 428.1 50.4 504 58 428.1 60.3 504 58

1.5 0.179 435.2 -19.8 -31 14 435.2 -16.0 -23 19 435.2 -9.4 16 26 435.2 -4.2 273 33 435.2 1.9 189 36 435.2 7.8 321 44 435.2 13.5 379 50 435.2 20.6 431 54 435.2 25.9 446 55 435.2 31.8 494 61 435.2 37.2 499 62 435.2 43.0 505 64 435.2 48.5 509 63 435.2 55.2 509 62 435.2 61.1 509 61

1.5 0.179 442.3 -14.5 7 54 442.3 -9.1 53 65 442.3 -3.5 131 68 442.3 2.4 227 69 442.3 9.9 351 73 442.3 15.4 394 70 442.3 22.0 223 72 442.3 28.8 455 73 442.3 33.7 493 71 442.3 38.4 507 71 442.3 43.4 509 71 442.3 48.9 512 68 442.3 55.3 514 65 442.3 61.1 514 62

1.5 0.179 449.4 2.4 253 103 449.4 7.2 321 99 449.4 12.4 380 96 449.4 17.8 421 87 449.4 23.9 446 86 449.4 31.1 478 80 449.4 39.1 511 77 449.4 44.8 512 74 449.4 50.8 517 70 449.4 56.8 519 66 449.4 60.6 518 63

1.5 0.179 456.5 2.3 263 142 456.5 7.3 336 129

A-14

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 456.5 11.0 368 123 456.5 16.9 395 112 456.5 21.1 412 102 456.5 26.9 462 98 456.5 32.1 486 90 456.5 38.7 514 85 456.5 44.5 519 80 456.5 50.7 523 74 456.5 56.1 523 68 456.5 60.6 523 65

1.5 0.179 463.6 2.3 297 172 463.6 6.6 342 155 463.6 10.5 370 143 463.6 16.3 402 126 463.6 23.5 426 107 463.6 29.3 451 98 463.6 34.2 490 95 463.6 37.7 509 90 463.6 43.1 514 81 463.6 48.6 527 77 463.6 54.6 529 69 463.6 60.7 529 66

1.5 0.179 470.7 0.9 288 179 470.7 6.3 331 157 470.7 11.8 356 140 470.7 17.7 401 121 470.7 23.4 466 111 470.7 29.8 479 103 470.7 36.4 513 93 470.7 42.4 529 85 470.7 48.3 533 75 470.7 55.5 533 68 470.7 60.1 533 65

1.5 0.179 477.8 0.0 326 190 477.8 4.8 382 167 477.8 9.6 416 144 477.8 15.4 451 129 477.8 20.5 466 116 477.8 26.0 506 107 477.8 32.0 516 99 477.8 36.6 534 94 477.8 41.9 538 86 477.8 50.6 539 73 477.8 56.6 539 66

1.5 0.179 492.0 1.9 344 59 492.0 7.8 371 76 492.0 11.5 384 79 492.0 16.5 418 87 492.0 21.1 412 87 492.0 27.1 464 86 492.0 32.2 483 84 492.0 38.5 519 83 492.0 44.2 541 78 492.0 48.6 544 74 492.0 52.5 548 70 492.0 57.9 548 63

1.5 0.179 499.1 0.4 369 52 499.1 5.4 413 63 499.1 10.8 446 72 499.1 16.2 456 78 499.1 21.9 509 83 499.1 27.7 517 82 499.1 34.0 533 81 499.1 39.9 549 79 499.1 46.7 552 73 499.1 52.3 552 67 499.1 57.8 553 63

1.5 0.179 547.7 0.0 - -11 547.7 2.8 - -23 547.7 5.4 - -34 547.7 8.5 - -38 547.7 12.0 - -38 547.7 17.2 - -28 547.7 22.5 - -12 547.7 27.3 - 7

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 547.7 32.1 - 20 547.7 36.2 - 32 547.7 41.3 - 43 547.7 47.0 - 46 547.7 53.4 - 47

1.5 0.179 565.7 0.0 -31 -33 565.7 1.3 94 -35 565.7 3.2 113 -31 565.7 5.3 74 -27 565.7 7.2 74 -27 565.7 10.1 112 -26 565.7 13.4 69 -26 565.7 16.6 69 -25 565.7 20.4 98 -23 565.7 25.1 205 -15 565.7 30.5 311 -6 565.7 36.2 430 7 565.7 43.6 527 24 565.7 50.4 589 34 565.7 60.3 614 48

1.5 0.179 612.1 3.2 74 0 612.1 8.3 129 -7 612.1 13.9 261 3 612.1 20.8 411 22 612.1 26.7 528 27 612.1 34.3 596 46 612.1 40.7 621 55 612.1 47.1 624 56 612.1 53.7 634 53 612.1 60.0 621 53

1.5 0.179 619.2 1.2 32 6 619.2 1.5 26 10 619.2 5.4 54 16 619.2 6.4 66 -19 619.2 9.5 98 -24 619.2 11.7 126 -27 619.2 16.9 211 -24 619.2 22.5 355 0 619.2 28.5 456 17 619.2 34.6 502 28 619.2 40.8 566 39 619.2 46.5 599 45 619.2 51.5 619 48 619.2 57.9 632 49

1.5 0.179 633.4 1.7 -4 -29 633.4 6.8 26 -38 633.4 13.5 171 -40 633.4 22.1 276 -24 633.4 28.6 468 -2 633.4 35.8 589 21 633.4 42.4 603 32 633.4 48.5 619 41 633.4 53.0 642 45 633.4 58.3 642 48

1.5 0.179 640.5 0.9 -14 -23 640.5 5.1 11 -32 640.5 11.2 93 -36 640.5 17.7 176 -43 640.5 23.5 351 -26 640.5 29.0 449 -9 640.5 33.4 511 1 640.5 38.2 576 17 640.5 42.2 616 26 640.5 46.9 625 37 640.5 52.7 651 43 640.5 57.4 653 47

1.5 0.179 707.1 0.0 47 26 707.1 0.0 233 - 707.1 1.3 245 24 707.1 2.1 255 - 707.1 2.9 320 25 707.1 4.9 260 - 707.1 5.1 294 28 707.1 7.2 296 32 707.1 8.0 296 -

A-15

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 707.1 10.2 303 34 707.1 11.0 304 - 707.1 13.5 345 39 707.1 14.3 309 - 707.1 16.4 347 41 707.1 18.3 345 - 707.1 20.2 345 43 707.1 22.0 364 - 707.1 25.3 390 45 707.1 27.1 388 - 707.1 30.2 409 46 707.1 32.1 431 - 707.1 36.1 462 - 707.1 36.3 470 47 707.1 40.2 526 - 707.1 43.1 548 45 707.1 45.2 565 - 707.1 50.4 617 45 707.1 50.4 614 - 707.1 54.6 581 - 707.1 55.6 581 46

1.7 0.216 141.4 0.0 47 84 141.4 1.1 44 85 141.4 2.5 47 86 141.4 4.0 47 86 141.4 6.1 54 86 141.4 8.2 66 85 141.4 10.1 82 85 141.4 13.1 118 84 141.4 16.1 168 84 141.4 20.2 251 84 141.4 23.6 306 84 141.4 26.6 340 83 141.4 30.1 353 83 141.4 35.1 355 80 141.4 40.0 356 79 141.4 46.6 356 76 141.4 53.1 356 75 141.4 60.0 356 73 141.4 70.1 356 70 141.4 80.2 356 67 141.4 90.3 356 65

141.4

95.8

356

65

1.7 0.216 264.9 0.0 - 100 264.9 3.1 - 114 264.9 7.2 - 119 264.9 10.1 - 119 264.9 14.1 - 117 264.9 18.8 - 117 264.9 22.7 - 115 264.9 26.8 - 87 264.9 31.6 - 93 264.9 36.5 - 92 264.9 41.5 - 109 264.9 47.1 - 107 264.9 52.3 - 96 264.9 57.7 - 92 264.9 61.8 - 90 264.9 67.9 - 84 264.9 75.0 - 77 264.9 79.8 - 75

1.7 0.216 282.8 0.0 -15 16 282.8 1.7 98 8 282.8 3.1 147 8 282.8 5.0 135 8 282.8 7.1 116 12 282.8 10.2 139 14 282.8 13.1 136 15 282.8 16.1 181 17 282.8 20.1 226 19 282.8 25.1 288 25 282.8 30.1 372 35 282.8 37.0 423 48 282.8 43.2 449 57

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 282.8 50.1 452 66 282.8 60.1 453 66 282.8 70.6 454 64 282.8 80.0 454 60

1.7 0.216 329.3 0.8 292 191 329.3 0.8 291 188 329.3 6.7 356 170 329.3 11.6 394 153 329.3 17.2 434 139 329.3 23.4 457 129 329.3 30.3 463 125 329.3 38.1 463 114 329.3 46.1 464 110 329.3 54.1 464 102 329.3 60.8 464 95 329.3 70.7 463 85 329.3 79.1 463 79

1.7 0.216 336.4 0.1 278 171 336.4 3.0 291 171 336.4 6.6 336 162 336.4 10.7 376 144 336.4 14.0 411 145 336.4 19.1 438 144 336.4 25.5 463 126 336.4 30.5 466 118 336.4 37.0 467 118 336.4 42.7 468 115 336.4 47.2 468 111 336.4 53.1 468 105 336.4 59.9 468 99 336.4 67.4 468 89 336.4 74.7 468 82 336.4 80.0 468 77

1.7 0.216 350.6 3.8 201 42 350.6 11.4 311 62 350.6 19.0 408 75 350.6 27.0 466 86 350.6 37.1 478 95 350.6 50.0 478 95 350.6 73.3 478 77

1.7 0.216 357.7 0.0 291 48 357.7 3.2 311 54 357.7 8.6 361 68 357.7 14.2 415 80 357.7 19.2 451 90 357.7 24.5 473 94 357.7 29.8 481 97 357.7 34.6 482 98 357.7 40.6 482 98 357.7 48.7 483 99 357.7 54.7 483 97 357.7 62.2 483 90 357.7 70.1 483 82 357.7 76.0 483 77 357.7 80.4 483 74

1.7 0.216 406.3 0.0 - 62 406.3 3.2 - 75 406.3 6.9 - 75 406.3 10.2 - 74 406.3 14.5 - 74 406.3 18.6 - 76 406.3 23.0 - 76 406.3 27.6 - 79 406.3 31.0 - 78 406.3 35.3 - 92 406.3 39.2 - 92 406.3 44.3 - 91 406.3 49.3 - 89 406.3 55.5 - 84 406.3 60.8 - 77 406.3 65.9 - 72 406.3 71.1 - 67

1.7 0.216 424.3 0.0 -5 -8 424.3 1.3 195 -2 424.3 3.1 179 3

A-16

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 424.3 5.3 302 31 424.3 7.5 299 35 424.3 10.1 306 39 424.3 13.1 351 43 424.3 16.4 342 46 424.3 20.1 365 52 424.3 25.1 397 57 424.3 30.2 492 59 424.3 36.2 477 66 424.3 43.0 505 68 424.3 50.1 534 70 424.3 60.1 554 67 424.3 70.1 556 67

1.7 0.216 470.7 0.0 301 182 470.7 1.5 341 180 470.7 7.8 356 157 470.7 14.4 389 141 470.7 21.0 451 122 470.7 27.2 466 114 470.7 33.2 541 108 470.7 39.4 546 102 470.7 45.7 554 100 470.7 51.7 561 92 470.7 56.3 563 87 470.7 61.6 564 79 470.7 68.8 564 73

1.7 0.216 477.8 0.1 231 128 477.8 4.6 286 103 477.8 5.6 314 101 477.8 9.9 393 94 477.8 13.7 416 118 477.8 14.9 464 88 477.8 21.6 529 91 477.8 22.2 543 125 477.8 28.1 554 86 477.8 35.4 564 89 477.8 36.1 565 105 477.8 42.2 568 88 477.8 45.4 568 96 477.8 48.8 568 86 477.8 56.3 568 79 477.8 62.3 568 77 477.8 69.4 568 71

1.7 0.216 492.0 1.9 356 35 492.0 7.0 366 56 492.0 11.8 426 63 492.0 18.5 461 76 492.0 24.9 551 76 492.0 31.4 539 80 492.0 38.3 566 83 492.0 46.2 574 86 492.0 52.2 578 83 492.0 58.2 579 76 492.0 64.6 578 70 492.0 70.4 578 68

1.7 0.216 499.1 0.0 319 -4 499.1 5.2 356 12 499.1 11.1 431 31 499.1 17.3 503 47 499.1 23.4 546 56 499.1 30.3 575 65 499.1 36.6 582 69 499.1 43.4 583 76 499.1 51.2 584 75 499.1 58.9 584 71 499.1 65.8 584 70 499.1 70.5 584 68

1.7 0.216 547.7 1.3 - 77 547.7 5.9 - 95 547.7 11.3 - 104 547.7 11.3 - 104 547.7 17.5 - 101 547.7 22.7 - 96 547.7 28.9 - 95 547.7 34.2 - 92

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 547.7 39.2 - 92 547.7 39.2 - 94 547.7 44.2 - 87 547.7 49.1 - 82 547.7 54.3 - 75 547.7 58.6 - 71

1.7 0.216 565.7 0.0 11 15 565.7 1.4 122 11 565.7 3.2 331 10 565.7 5.1 358 15 565.7 7.2 370 20 565.7 10.0 396 24 565.7 13.5 426 29 565.7 16.3 466 36 565.7 20.1 496 40 565.7 25.5 540 45 565.7 30.1 579 49 565.7 36.8 616 55 565.7 42.6 642 61 565.7 50.2 651 60 565.7 60.2 655 55 565.7 68.0 631 55

1.7 0.216 612.1 3.2 231 124 612.1 8.6 381 124 612.1 13.0 421 114 612.1 17.6 471 99 612.1 22.5 509 79 612.1 27.9 589 82 612.1 33.1 594 79 612.1 38.2 611 76 612.1 47.8 653 72 612.1 52.7 653 75 612.1 57.3 660 66 612.1 62.9 626 72 612.1 63.1 663 61

1.7 0.216 619.2 1.2 73 42 619.2 7.5 341 110 619.2 13.2 291 33 619.2 17.5 371 50 619.2 25.1 472 45 619.2 32.3 576 68 619.2 40.9 618 74 619.2 48.4 648 74 619.2 54.4 653 65 619.2 61.8 666 63

1.7 0.216 633.4 3.1 166 -9 633.4 9.1 316 23 633.4 16.3 401 38 633.4 23.1 456 35 633.4 29.4 566 60 633.4 35.5 611 55 633.4 41.0 633 67 633.4 46.2 651 65 633.4 52.2 668 67 633.4 58.9 674 65 633.4 65.2 678 65

1.7 0.216 640.5 2.2 171 -9 640.5 6.4 181 -4 640.5 12.3 261 6 640.5 17.8 381 25 640.5 23.5 621 75 640.5 28.4 651 78 640.5 33.5 663 83 640.5 38.7 676 87 640.5 43.3 682 85 640.5 48.0 682 82 640.5 53.0 682 77 640.5 57.6 682 72 640.5 62.5 682 66

1.7 0.216 707.1 0.0 45 -15 707.1 1.5 139 -12 707.1 3.1 154 -12 707.1 5.3 149 -10 707.1 7.6 172 -7 707.1 10.0 194 -5

A-17

dc/h Q (m3/s) xt (mm) y (mm) Ht (mm) Hp (mm) 707.1 13.1 262 -1 707.1 16.2 301 3 707.1 20.0 379 8 707.1 25.1 484 19 707.1 30.0 580 29 707.1 36.5 641 38 707.1 42.6 708 45 707.1 50.1 739 50 707.1 60.4 689 52 707.1 66.0 722 55

1.7 0.216 848.5 0.0 40 16 848.5 1.5 381 16 848.5 3.2 383 16 848.5 5.2 385 22 848.5 7.3 408 28 848.5 10.4 437 31 848.5 13.1 470 36 848.5 16.3 506 40 848.5 20.3 560 45 848.5 25.2 604 49 848.5 30.0 662 53 848.5 36.7 736 55 848.5 42.3 783 55 848.5 50.2 826 55 848.5 60.2 805 53 848.5 62.7 736 52

A-18

APPENDIX D - PRESSURE DISTRIBUTIONS IN DEVELOPING FLOWS ON STEPPED CASCADES

The present study demonstrated that the developing flow region on stepped chutes was rapidly

varied. The pressure distributions were non-hydrostatic and some significant pressure redistribution

about each step edge was shown. This section presents results of pressure measurements conducted

around step edges 3 – 5, and in step cavities 2 – 3 and 3 – 4. Note that the data scatter increased

further downstream because of turbulent fluctuations in the boundary layer.

P/(gdcos)

y/d

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1dc/h = 0.9dc/h = 1.0dc/h = 1.1dc/h = 1.3dc/h = 1.5dc/h = 1.7hydrostatic

P/(gdcos)

y/d

0 0.5 1 1.5 2 2.5 3 3.50

0.2

0.4

0.6

0.8

1 dc/h = 0.9dc/h = 1.0dc/h = 1.1dc/h = 1.3dc/h = 1.5dc/h = 1.7hydrostatic

A) x/Lcav = -0.1 (B) x/Lcav= -0.05

P/(gdcos)

y/d

-1.8 -1.2 -0.6 0 0.6 1.2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.8 -1.2 -0.6 0 0.6 1.2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dc/h = 0.9dc/h = 1.0dc/h = 1.1dc/h = 1.3dc/h = 1.5dc/h = 1.7hydrostatic

P/(gdcos)

y/d

-1.25 -0.75 -0.25 0.25 0.75 1.25

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.25 -0.75 -0.25 0.25 0.75 1.25

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1dc/h = 0.9dc/h = 1.0dc/h = 1.1dc/h = 1.3dc/h = 1.5dc/h = 1.7hydrostatic

(C) x/Lcav = 0.05 (D) x/Lcav= 0.1

Figure D.1 – Dimensionless pressure distribution around step edge 3

A-19

P/(gdcos)

y/d

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1dc/h = 0.9dc/h = 1.0dc/h = 1.1dc/h = 1.3dc/h = 1.5dc/h = 1.7-x+1

P/(gdcos)

y/d

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

dc/h = 0.9dc/h = 1.0dc/h = 1.1dc/h = 1.3dc/h = 1.5dc/h = 1.7-x+1

(A) x/Lcav = -0.1 (B) x/Lcav= -0.05

P/(gdcos)

y/d

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 dc/h = 0.9dc/h = 1.0dc/h = 1.1dc/h = 1.3dc/h = 1.5dc/h = 1.7-x+1

P/(gdcos)

y/d

-0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 dc/h = 0.9dc/h = 1.0dc/h = 1.1dc/h = 1.3dc/h = 1.5dc/h = 1.7-x+1

(C) x/Lcav = 0.05 (D) x/Lcav= 0.1

Figure D.2 – Dimensionless pressure distribution around step edge 4

A-20

P/(gdcos)

y/d

-0.5 0 0.5 1 1.5 2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.5 0 0.5 1 1.5 2

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dc/h = 1.3dc/h = 1.5dc/h = 1.7-x+1

P/(gdcos)

y/d

-0.9 -0.3 0.3 0.9 1.5 2.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.9 -0.3 0.3 0.9 1.5 2.1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

dc/h = 1.3dc/h = 1.5dc/h = 1.7-x+1

(A) x/Lcav = -0.1 (B) x/Lcav= -0.05

P/(gdcos)

y/d

-1.25 -0.75 -0.25 0.25 0.75

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.25 -0.75 -0.25 0.25 0.75

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1dc/h = 1.3dc/h = 1.5dc/h = 1.7-x+1

P/(gdcos)

y/d

-1.25 -0.75 -0.25 0.25 0.75 1.25

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-1.25 -0.75 -0.25 0.25 0.75 1.25

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1dc/h = 1.3dc/h = 1.5dc/h = 1.7-x+1

(C) x/Lcav = 0.05 (D) x/Lcav= 0.1

Figure D.3 – Dimensionless pressure distributions around step edge 5

A-21

P/(dcos)+5x/Lcav

y/d

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1 x/Lcav=0.04 x/Lcav=0.1 x/Lcav=0.3 x/Lcav=0.4 x/Lcav=0.5 x/Lcav=0.6 x/Lcav=0.7 x/Lcav=0.9

(A) dc/h = 1.3

P/(dcos)+5x/Lcav

y/d

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1 x/Lcav=0.2 x/Lcav=0.3 x/Lcav=0.4 x/Lcav=0.5 x/Lcav=0.6 x/Lcav=0.7 x/Lcav=0.8 x/Lcav=0.9

(B) dc/h = 1.5

Figure D.4 – Dimensionless pressure distributions between step cavity 2 – 3

A-22

P/(dcos)+5x/Lcav

y/d

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1x/Lcav=0x/Lcav=00.05

x/Lcav=0.1x/Lcav=0.15

x/Lcav=0.3x/Lcav=0.35

x/Lcav=0.4x/Lcav=0.45

x/Lcav=0.5x/Lcav=0.55

x/Lcav=0.6x/Lcav=0.65

x/Lcav=0.7x/Lcav=0.75

x/Lcav=0.8x/Lcav=0.85

x/Lcav=0.90.95

Figure D.5 – Dimensionless pressure distributions between step cavity 3 – 4: dc/h = 1.5

R-1

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cascades." Ph.D. Thesis, Dept. of Civil Engineering, University of Queensland, Australia.

Wood, I.R. (1985). "Air water flows." Proc. 21st IAHR Biennial Congress, Melbourne, Australia,

Keynote address, pp. 18-29.

Wood, I. R., Ackers, P., and Loveless, J. (1983). "General method for critical point on spillways."

Journal of Hydraulic Engineering, Vol. 109, No. 2, pp.308-312.

Wuthrich, D, and Chanson, H. (2014). "Hydraulics, air entrainment and energy dissipation on

gabion stepped weir." Journal of Hydraulic Engineering, ASCE, Vol. 140, No. 9, Paper

04014046, 10 pages (DOI: 10.1061/(ASCE)HY.1943-7900.0000919).

Zhang, G., and Chanson, H.(2015). "Broad-crested weir operation upstream of a steep stepped

spillway." Proc. 36th IAHR World Congress, 28 June-3 July, The Hague, the Netherlands, Paper

79662, 11 pages (CD-ROM).

Bibliography

Cain, P., and Wood, I.R. (1981). "Measurements of Self-aerated Flow on a Spillway." Journal of

Hydraulics Division, ASCE, Vol. 107, No. HY11, pp. 1425-1444.

Chanson, H. (1995). "Hydraulic Design of Stepped Cascades, Channels, Weirs and Spillways."

Pergamon, Oxford, UK, Jan., 292 pages.

Chanson, H. (1996). "Prediction of the Transition Nappe/Skimming Flow on a Stepped Channel."

Journal of Hydraulic Research, IAHR, Vol. 34, No. 3, pp. 421-429.

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Chanson, H. (2015). "Energy Dissipation in Hydraulic Structures." IAHR Monograph, CRC Press,

Taylor & Francis Group, Leiden, The Netherlands.

Chanson, H., and Toombes, L. (2002). "Energy Dissipation and Air Entrainment in a Stepped Storm

Waterway: an Experimental Study." Journal of Irrigation and Drainage Engineering, ASCE,

Vol. 128, No. 5, pp. 305-315.

Chanson, H., and Toombes, L. (2002). "Experimental Study of Gas-Liquid Interfacial Properties in

a Stepped Cascade Flow." Environmental Fluid Mechanics, Vol. 2, No. 3, pp. 241-263 (DOI:

10.1023/A:1019884101405).

Chanson, H., and Toombes, L. (2002). "Air-Water Flows down Stepped chutes : Turbulence and

Flow Structure Observations." International Journal of Multiphase Flow, Vol. 27, No. 11, pp.

1737-1761.

Felder, S., and Chanson, H. (2011). "Air-Water Flow Properties in Step Cavity down a Stepped

Chute." International Journal of Multiphase Flow, Vol. 37, No. 7, pp. 732–745 (DOI:

10.1016/j.ijmultiphaseflow.2011.02.009)

Felder, S., and Chanson, H. (2014). "Characterisation of Interfacial Aeration in High-Velocity Free-

Surface Air-Water Flows: What Does 50% Void Fraction Mean?" Proceedings of 19th

Australasian Fluid Mechanics Conference, Melbourne, Australia, Paper 235, 4 pages.

Meireles, I., Cabrita, J., and Matos, J. (2006). "Non-aerated skimming flow properties on stepped

chutes over embankment dams." Proceedings of the International Junior Researcher and

Engineer Workshop on Hydraulic Structures, Jorge Matos and Hubert Chanson Editors, Report

CH61/06, School of Civil Engineering, The University of Queensland, Brisbane, Australia, pp.

91-99.

Pfister, M., and Chanson, H. (2014). "Two-Phase Air-Water Flows: Scale Effects in Physical

Modeling." Journal of Hydrodynamics, Vol. 26, No. 2, pp. 291-298 (DOI: 10.1016/S1001-

6058(14)60032-9).

Wood, I. R. (1991). "Free-surface air entrainment on spillways." in "Air entrainment in free-surface

flows", IAHR Hydraulic Structures Design Manual, Ed. Ian R. Wood, Balkema, Leiden, The

Netherlands.

Zhang, G., and Chanson, H. (2014). "Step Cavity and Gabion Aeration on a Gabion Stepped

Spillway." in "Hydraulic Structures and Society – Engineering Challenges and Extremes", The

University of Queensland, Brisbane, Australia, Proceedings of the 5th IAHR International

Symposium on Hydraulic Structures (ISHS2014), 25-27 June 2014, Brisbane, Australia, H.

Chanson and L. Toombes Editors, 8 pages (DOI: 10.14264/uql.2014.37).

Open Access Repositories

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Bibliographic reference of the Report CH97/15 The Hydraulic Model research report series CH is a refereed publication published by the School of

Civil Engineering at the University of Queensland, Brisbane, Australia.

The bibliographic reference of the present report is:

Zhang, G., and Chanson, H. (2015). "Hydraulics of the Developing Flow Region of Stepped

Cascades: an Experimental Investigation." Hydraulic Model Report No. CH97/15, School of

Civil Engineering, The University of Queensland, Brisbane, Australia, 76 pages (ISBN 978 1

74272 134 7).

The Report CH97/15 is available, in the present form, as a PDF file on the Internet at UQeSpace:

http://espace.library.uq.edu.au/

It is listed at:

http://espace.library.uq.edu.au/list/author_id/193/

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HYDRAULIC MODEL RESEARCH REPORT CH

The Hydraulic Model Report CH series is published by the School of Civil Engineering at the

University of Queensland. Orders of any reprint(s) of the Hydraulic Model Reports should be

addressed to the School Secretary.

School Secretary, School of Civil Engineering, The University of Queensland

Brisbane 4072, Australia - Tel.: (61 7) 3365 3619 - Fax: (61 7) 3365 4599

Url: http://http://www.civil.uq.edu.au// Email: enquiries@civil.uq.edu.au

Report CH Unit price Quantity Total price

SUARA. K., BROWN, R., and CHANSON, H. (2015). "Turbulence and Mixing in the Environment: Multi-Device Study in a Sub-tropical Estuary." Hydraulic Model Report No. CH99/15, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 164pages (ISBN 978 1 74272 138 5).

AUD$60.00

LENG, X., and CHANSON, H. (2015). "Unsteady Turbulence during the Upstream Propagation of Undular and Breaking Tidal Bores: an Experimental Investigation." Hydraulic Model Report No. CH98/15, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 231 pages (ISBN 978 1 74272 135 4).

AUD$60.00

ZHANG, G., and CHANSON, H. (2015). "Hydraulics of the Developing Flow Region of Stepped Cascades: an Experimental Investigation."Hydraulic Model Report No. CH97/15, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 76 pages (ISBN 978 1 74272 134 7).

AUD$60.00

LENG, X., and CHANSON, H. (2014). "Turbulent Advances of Breaking Bores: Experimental Observations." Hydraulic Model Report No. CH96/14, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 40 pages (ISBN 978 1 74272 130 9).

AUD$40.00

WANG, H, MURZYN, F., and D., CHANSON, H. (2014). "Pressure, Turbulence and Two-Phase Flow Measurements in Hydraulic Jumps."Hydraulic Model Report No. CH95/14, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 154 pages (ISBN 97817427206169781742721064).

AUD$60.00

REUNGOAT, D., CHANSON, H., and KEEVIL, C. (2014). "Turbulence, Sedimentary Processes and Tidal Bore Collision in the Arcins Channel, Garonne River (October 2013)." Hydraulic Model Report No. CH94/14School of Civil Engineering, The University of Queensland, Brisbane, Australia, 145 pages (ISBN 9781742721033).

AUD$60.00

LENG, X., and CHANSON, H. (2014). "Propagation of Negative Surges in Rivers and Estuaries: Unsteady Turbulent Mixing including the Effects of Bed Roughness." Hydraulic Model Report No. CH93/13, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 108 pages (ISBN 9781742720944).

AUD$60.00

WUTHRICH, D., and CHANSON, H. (2014). "Aeration and Energy Dissipation over Stepped Gabion Spillways: a Physical Study." Hydraulic Model Report No. CH92/13, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 171 pages and 5 video movies (ISBN 9781742720944).

AUD$60.00

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WANG, H., and CHANSON, H. (2013). "Free-Surface Deformation and Two-Phase Flow Measurements in Hydraulic Jumps". Hydraulic Model Report No. CH91/13, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 108 pages (ISBN 9781742720746).

AUD$60.00

SIMON, B., and CHANSON, H. (2013). "Turbulence Measurements in Tidal Bore-like Positive Surges over a Rough Bed". Hydraulic Model Report No. CH90/12, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 176 pages (ISBN 9781742720685).

AUD$60.00

REUNGOAT, D., CHANSON, H., and CAPLAIN, B. (2012). "Field Measurements in the Tidal Bore of the Garonne River at Arcins (June2012)." Hydraulic Model Report No. CH89/12, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 121 pages (ISBN 9781742720616).

AUD$60.00

CHANSON, H., and WANG, H. (2012). "Unsteady Discharge Calibration of a Large V-Notch Weir." Hydraulic Model Report No. CH88/12, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 50 pages & 4 movies (ISBN 9781742720579).

AUD$60.00

FELDER, S., FROMM, C., and CHANSON, H. (2012). "Air Entrainment and Energy Dissipation on a 8.9° Slope Stepped Spillway with Flat and Pooled Steps." Hydraulic Model Report No. CH86/12, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 82 pages (ISBN 9781742720531).

AUD$60.00

FELDER, S., and CHANSON, H. (2012). "Air-Water Flow Measurements in Instationary Free-Surface Flows: a Triple Decomposition Technique." Hydraulic Model Report No. CH85/12, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 161 pages (ISBN 9781742720494).

AUD$60.00

REICHSTETTER, M., and CHANSON, H. (2011). "Physical and Numerical Modelling of Negative Surges in Open Channels." Hydraulic Model Report No. CH84/11, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 82 pages (ISBN 9781742720388).

AUD$60.00

BROWN, R., CHANSON, H., McINTOSH, D., and MADHANI, J. (2011). "Turbulent Velocity and Suspended Sediment Concentration Measurements in an Urban Environment of the Brisbane River Flood Plain at Gardens Point on 12-13 January 2011." Hydraulic Model Report No. CH83/11, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 120 pages (ISBN 9781742720272).

AUD$60.00

CHANSON, H. "The 2010-2011 Floods in Queensland (Australia): Photographic Observations, Comments and Personal Experience."Hydraulic Model Report No. CH82/11, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 127 pages (ISBN 9781742720234).

AUD$60.00

MOUAZE, D., CHANSON, H., and SIMON, B. (2010). "Field Measurements in the Tidal Bore of the Sélune River in the Bay of Mont Saint Michel (September 2010)." Hydraulic Model Report No. CH81/10, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 72 pages (ISBN 9781742720210).

AUD$60.00

JANSSEN, R., and CHANSON, H. (2010). "Hydraulic Structures: Useful Water Harvesting Systems or Relics." Proceedings of the Third International Junior Researcher and Engineer Workshop on Hydraulic Structures (IJREWHS'10), 2-3 May 2010, Edinburgh, Scotland, R. JANSSEN and H. CHANSON (Eds), Hydraulic Model Report CH80/10, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 211 pages (ISBN 9781742720159).

AUD$60.00

CHANSON, H., LUBIN, P., SIMON, B., and REUNGOAT, D. (2010). "Turbulence and Sediment Processes in the Tidal Bore of the Garonne River: First Observations." Hydraulic Model Report No. CH79/10, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 97 pages (ISBN 9781742720104).

AUD$60.00

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CHACHEREAU, Y., and CHANSON, H., (2010). "Free-Surface Turbulent Fluctuations and Air-Water Flow Measurements in Hydraulics Jumps with Small Inflow Froude Numbers." Hydraulic Model Report No. CH78/10, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 133 pages (ISBN 9781742720036).

AUD$60.00

CHANSON, H., BROWN, R., and TREVETHAN, M. (2010). "Turbulence Measurements in a Small Subtropical Estuary under King Tide Conditions." Hydraulic Model Report No. CH77/10, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 82pages (ISBN 9781864999969).

AUD$60.00

DOCHERTY, N.J., and CHANSON, H. (2010). "Characterisation of Unsteady Turbulence in Breaking Tidal Bores including the Effects ofBed Roughness." Hydraulic Model Report No. CH76/10, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 112pages (ISBN 9781864999884).

AUD$60.00

CHANSON, H. (2009). "Advective Diffusion of Air Bubbles in Hydraulic Jumps with Large Froude Numbers: an Experimental Study."Hydraulic Model Report No. CH75/09, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 89 pages & 3 videos(ISBN 9781864999730).

AUD$60.00

CHANSON, H. (2009). "An Experimental Study of Tidal Bore Propagation: the Impact of Bridge Piers and Channel Constriction."Hydraulic Model Report No. CH74/09, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 110 pages and 5 movies (ISBN 9781864999600).

AUD$60.00

CHANSON, H. (2008). "Jean-Baptiste Charles Joseph BÉLANGER (1790-1874), the Backwater Equation and the Bélanger Equation."Hydraulic Model Report No. CH69/08, Div. of Civil Engineering, The University of Queensland, Brisbane, Australia, 40 pages (ISBN 9781864999211).

AUD$60.00

GOURLAY, M.R., and HACKER, J. (2008). "Reef-Top Currents in Vicinity of Heron Island Boat Harbour, Great Barrier Reef, Australia: 2. Specific Influences of Tides Meteorological Events and Waves."Hydraulic Model Report No. CH73/08, Div. of Civil Engineering, The University of Queensland, Brisbane, Australia, 331 pages (ISBN 9781864999365).

AUD$60.00

GOURLAY, M.R., and HACKER, J. (2008). "Reef Top Currents in Vicinity of Heron Island Boat Harbour Great Barrier Reef, Australia: 1. Overall influence of Tides, Winds, and Waves." Hydraulic Model Report CH72/08, Div. of Civil Engineering, The University of Queensland, Brisbane, Australia, 201 pages (ISBN 9781864999358).

AUD$60.00

LARRARTE, F., and CHANSON, H. (2008). "Experiences and Challenges in Sewers: Measurements and Hydrodynamics." Proceedings of the International Meeting on Measurements and Hydraulics of Sewers,Summer School GEMCEA/LCPC, 19-21 Aug. 2008, Bouguenais, Hydraulic Model Report No. CH70/08, Div. of Civil Engineering, The University of Queensland, Brisbane, Australia (ISBN 9781864999280).

AUD$60.00

CHANSON, H. (2008). "Photographic Observations of Tidal Bores (Mascarets) in France." Hydraulic Model Report No. CH71/08, Div. of Civil Engineering, The University of Queensland, Brisbane, Australia, 104 pages, 1 movie and 2 audio files (ISBN 9781864999303).

AUD$60.00

CHANSON, H. (2008). "Turbulence in Positive Surges and Tidal Bores. Effects of Bed Roughness and Adverse Bed Slopes." Hydraulic Model Report No. CH68/08, Div. of Civil Engineering, The University of Queensland, Brisbane, Australia, 121 pages & 5 movie files (ISBN 9781864999198)

AUD$70.00

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FURUYAMA, S., and CHANSON, H. (2008). "A Numerical Study of Open Channel Flow Hydrodynamics and Turbulence of the Tidal Bore and Dam-Break Flows." Report No. CH66/08, Div. of Civil Engineering, The University of Queensland, Brisbane, Australia, May, 88 pages (ISBN 9781864999068).

AUD$60.00

GUARD, P., MACPHERSON, K., and MOHOUPT, J. (2008). "A Field Investigation into the Groundwater Dynamics of Raine Island." Report No. CH67/08, Div. of Civil Engineering, The University of Queensland, Brisbane, Australia, February, 21 pages (ISBN 9781864999075).

AUD$40.00

FELDER, S., and CHANSON, H. (2008). "Turbulence and Turbulent Length and Time Scales in Skimming Flows on a Stepped Spillway. Dynamic Similarity, Physical Modelling and Scale Effects." Report No. CH64/07, Div. of Civil Engineering, The University of Queensland, Brisbane, Australia, March, 217 pages (ISBN 9781864998870).

AUD$60.00

TREVETHAN, M., CHANSON, H., and BROWN, R.J. (2007). "Turbulence and Turbulent Flux Events in a Small Subtropical Estuary."Report No. CH65/07, Div. of Civil Engineering, The University of Queensland, Brisbane, Australia, November, 67 pages (ISBN 9781864998993)

AUD$60.00

MURZYN, F., and CHANSON, H. (2007). "Free Surface, Bubbly flow and Turbulence Measurements in Hydraulic Jumps." Report CH63/07, Div. of Civil Engineering, The University of Queensland, Brisbane, Australia, August, 116 pages (ISBN 9781864998917).

AUD$60.00

KUCUKALI, S., and CHANSON, H. (2007). "Turbulence in Hydraulic Jumps: Experimental Measurements." Report No. CH62/07, Div. of Civil Engineering, The University of Queensland, Brisbane, Australia, July, 96 pages (ISBN 9781864998825).

AUD$60.00

CHANSON, H., TAKEUCHI, M, and TREVETHAN, M. (2006). "Using Turbidity and Acoustic Backscatter Intensity as Surrogate Measures of Suspended Sediment Concentration. Application to a Sub-Tropical Estuary (Eprapah Creek)." Report No. CH60/06, Div. of Civil Engineering, The University of Queensland, Brisbane, Australia, July, 142 pages (ISBN 1864998628).

AUD$60.00

CAROSI, G., and CHANSON, H. (2006). "Air-Water Time and Length Scales in Skimming Flows on a Stepped Spillway. Application to the Spray Characterisation." Report No. CH59/06, Div. of Civil Engineering, The University of Queensland, Brisbane, Australia, July (ISBN 1864998601).

AUD$60.00

TREVETHAN, M., CHANSON, H., and BROWN, R. (2006). "Two Series of Detailed Turbulence Measurements in a Small Sub-Tropical Estuarine System." Report No. CH58/06, Div. of Civil Engineering, The University of Queensland, Brisbane, Australia, Mar. (ISBN 1864998520).

AUD$60.00

KOCH, C., and CHANSON, H. (2005). "An Experimental Study of Tidal Bores and Positive Surges: Hydrodynamics and Turbulence of the Bore Front." Report No. CH56/05, Dept. of Civil Engineering, The University of Queensland, Brisbane, Australia, July (ISBN 1864998245).

AUD$60.00

CHANSON, H. (2005). "Applications of the Saint-Venant Equations and Method of Characteristics to the Dam Break Wave Problem." Report No. CH55/05, Dept. of Civil Engineering, The University of Queensland, Brisbane, Australia, May (ISBN 1864997966).

AUD$60.00

CHANSON, H., COUSSOT, P., JARNY, S., and TOQUER, L. (2004). "A Study of Dam Break Wave of Thixotropic Fluid: Bentonite Surges down an Inclined plane." Report No. CH54/04, Dept. of Civil Engineering, The University of Queensland, Brisbane, Australia, June, 90 pages (ISBN 1864997710).

AUD$60.00

CHANSON, H. (2003). "A Hydraulic, Environmental and Ecological Assessment of a Sub-tropical Stream in Eastern Australia: Eprapah Creek, Victoria Point QLD on 4 April 2003." Report No. CH52/03, Dept. of Civil Engineering, The University of Queensland, Brisbane, Australia, June, 189 pages (ISBN 1864997044).

AUD$90.00

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CHANSON, H. (2003). "Sudden Flood Release down a Stepped Cascade. Unsteady Air-Water Flow Measurements. Applications to Wave Run-up, Flash Flood and Dam Break Wave." Report CH51/03, Dept of Civil Eng., Univ. of Queensland, Brisbane, Australia, 142 pages (ISBN 1864996552).

AUD$60.00

CHANSON, H,. (2002). "An Experimental Study of Roman Dropshaft Operation : Hydraulics, Two-Phase Flow, Acoustics." Report CH50/02, Dept of Civil Eng., Univ. of Queensland, Brisbane, Australia, 99 pages (ISBN 1864996544).

AUD$60.00

CHANSON, H., and BRATTBERG, T. (1997). "Experimental Investigations of Air Bubble Entrainment in Developing Shear Layers." Report CH48/97, Dept. of Civil Engineering, University of Queensland, Australia, Oct., 309 pages (ISBN 0 86776 748 0).

AUD$90.00

CHANSON, H. (1996). "Some Hydraulic Aspects during Overflow above Inflatable Flexible Membrane Dam." Report CH47/96, Dept. of Civil Engineering, University of Queensland, Australia, May, 60 pages (ISBN 0 86776 644 1).

AUD$60.00

CHANSON, H. (1995). "Flow Characteristics of Undular Hydraulic Jumps. Comparison with Near-Critical Flows." Report CH45/95, Dept. of Civil Engineering, University of Queensland, Australia, June, 202 pages (ISBN 0 86776 612 3).

AUD$60.00

CHANSON, H. (1995). "Air Bubble Entrainment in Free-surface Turbulent Flows. Experimental Investigations." Report CH46/95, Dept. of Civil Engineering, University of Queensland, Australia, June, 368 pages (ISBN 0 86776 611 5).

AUD$80.00

CHANSON, H. (1994). "Hydraulic Design of Stepped Channels and Spillways." Report CH43/94, Dept. of Civil Engineering, University of Queensland, Australia, Feb., 169 pages (ISBN 0 86776 560 7).

AUD$60.00

POSTAGE & HANDLING (per report) AUD$10.00 GRAND TOTAL

OTHER HYDRAULIC RESEARCH REPORTS

Reports/Theses Unit price Quantity Total priceKHEZRI, N. (2014). "Modelling Turbulent Mixing and Sediment Process Beneath Tidal Bores: Physical and Numerical Investigations." Ph.D. thesis, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 267 pages.

AUD$100.00

SIMON, B. (2014). "Effects of Tidal Bores on Turbulent Mixing: a Numerical and Physical Study in Positive Surges." Ph.D. thesis, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 259 pages.

AUD$100.00

FELDER, S. (2013). "Air-Water Flow Properties on Stepped Spillways for Embankment Dams: Aeration, Energy Dissipation and Turbulence on Uniform, Non-Uniform and Pooled Stepped Chutes." Ph.D. thesis, School of Civil Engineering, The University of Queensland, Brisbane, Australia.

AUD$100.00

REICHSTETTER, M. (2011). "Hydraulic Modelling of Unsteady Open Channel Flow: Physical and Analytical Validation of Numerical Models of Positive and Negative Surges." MPhil thesis, School of Civil Engineering, The University of Queensland, Brisbane, Australia, 112 pages.

AUD$80.00

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TREVETHAN, M. (2008). "A Fundamental Study of Turbulence and Turbulent Mixing in a Small Subtropical Estuary." Ph.D. thesis, Div. of Civil Engineering, The University of Queensland, 342 pages.

AUD$100.00

GONZALEZ, C.A. (2005). "An Experimental Study of Free-Surface Aeration on Embankment Stepped Chutes." Ph.D. thesis, Dept of Civil Engineering, The University of Queensland, Brisbane, Australia, 240 pages.

AUD$80.00

TOOMBES, L. (2002). "Experimental Study of Air-Water Flow Properties on Low-Gradient Stepped Cascades." Ph.D. thesis, Dept of Civil Engineering, The University of Queensland, Brisbane, Australia.

AUD$100.00

CHANSON, H. (1988). "A Study of Air Entrainment and Aeration Devices on a Spillway Model." Ph.D. thesis, University of Canterbury, New Zealand.

AUD$60.00

POSTAGE & HANDLING (per report) AUD$10.00 GRAND TOTAL

CIVIL ENGINEERING RESEARCH REPORT CE

The Civil Engineering Research Report CE series is published by the School of Civil Engineering

at the University of Queensland. Orders of any of the Civil Engineering Research Report CE should

be addressed to the School Secretary.

School Secretary, School of Civil Engineering, The University of Queensland

Brisbane 4072, Australia

Tel.: (61 7) 3365 3619 Fax: (61 7) 3365 4599

Url: http://http://www.civil.uq.edu.au// Email: enquiries@civil.uq.edu.au

Recent Research Report CE Unit price Quantity Total priceCALLAGHAN, D.P., NIELSEN, P., and CARTWRIGHT, N. (2006). "Data and Analysis Report: Manihiki and Rakahanga, Northern Cook Islands - For February and October/November 2004 Research Trips." Research Report CE161, Division of Civil Engineering, The University of Queensland (ISBN No. 1864998318).

AUD$10.00

GONZALEZ, C.A., TAKAHASHI, M., and CHANSON, H. (2005). "Effects of Step Roughness in Skimming Flows: an Experimental Study." Research Report No. CE160, Dept. of Civil Engineering, The University of Queensland, Brisbane, Australia, July (ISBN 1864998105).

AUD$10.00

CHANSON, H., and TOOMBES, L. (2001). "Experimental Investigations of Air Entrainment in Transition and Skimming Flows down a Stepped Chute. Application to Embankment Overflow Stepped Spillways." Research Report No. CE158, Dept. of Civil Engineering, The University of Queensland, Brisbane, Australia, July, 74 pages (ISBN 1 864995297).

AUD$10.00

HANDLING (per order) AUD$10.00

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QUEENSLAND and crossed "Not Negotiable".

Orders of any Research Report should be addressed to the School Secretary.

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