THE UNIVERSITY OF QUEENSLAND

REPORT CH75/09

AUTHOR: Hubert CHANSON

ADVECTIVE DIFFUSION OF AIR BUBBLES IN HYDRAULIC JUMPS WITH LARGE FROUDE NUMBERS: AN EXPERIMENTAL STUDY

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 3619 Fax: (61 7) 3365 4599 URL: http://www.eng.uq.edu.au/civil/ First published in 2009 by School of Civil Engineering The University of Queensland, Brisbane QLD 4072, Australia © Chanson This book is copyright ISBN No. 9781864999730 The University of Queensland, St Lucia QLD

ADVECTIVE DIFFUSION OF AIR BUBBLES IN HYDRAULIC JUMPS WITH LARGE FROUDE NUMBERS: AN

EXPERIMENTAL STUDY

by

Hubert CHANSON

Professor in Hydraulic Engineering, School of Engineering,

The University of Queensland, Brisbane QLD 4072, Australia

Ph.: (61 7) 3365 3619, Fax: (61 7) 3365 4599, Email: h.chanson@uq.edu.au

Url: http://www.uq.edu.au/~e2hchans/

REPORT No. CH75/09

ISBN 9781864999730

School of Civil Engineering, The University of Queensland

October 2009

Hydraulic jump roller in a prototype culvert inlet on 20 May 2009 - Flow from left to right, Re ~ 3 106

ii

Abstract

A hydraulic jump is a rapid transition from a high-velocity open channel flow to a slower fluvial motion. It is

commonly experienced in streams and rivers, in industrial channels and during manufacturing processes.

Herein new detailed air-water flow characteristics were measured in the developing shear layer of hydraulic

jumps with partially-developed inflow. The measurements were conducted in a relatively large-size facility

with large Froude numbers (5.1 < Fr1 < 11.2). In the developing shear layer, the distributions of void

fractions were modelled by an advective diffusion equation. The experimental data demonstrated a close

agreement with the theoretical developments, and the air bubble diffusivity was observed to be independent

of the Froude and Reynolds numbers although increasing linearly with the distance from the jump toe. The

experimental observations highlighted a strong air entrainment rate as well as some spray and splashing

above the roller. The measurements of jump toe fluctuations were close to earlier studies. The new data

showed that the jump toe oscillation frequency was equal to the production rate of large-scale vortical

structures in the developing shear layer, and the average convection speed of the large coherent structures

was in average Vej/V1 = 0.32. The void fraction distributions presented a local maximum in the air-water

shear layer and its value decreased quasi-exponentially with increasing distance from the jump toe. The shear

zone was also characterised by a maximum in bubble count rate. The depth-averaged void fraction data

demonstrated a large amount of entrained air as well as a rapid de-aeration of the jump roller. The velocity

profiles followed closely some wall jet equations, and the air-water turbulent properties indicated some

increasing turbulence levels with increasing distance from the bed. The bubble chord time distributions

exhibited a broad range of entrained bubble chord times spreading over several orders of magnitudes. A

detailed analysis of the longitudinal structure of the air and water chords suggested a significant proportion

of bubble clustering in the developing shear region, especially immediately downstream of the jump toe. The

data showed further that, in the shear layer, there was no preferential bubble chord time in the cluster

structures. Overall the study highlighted that the convection of air in the mixing zone was an advective

diffusion process, although there was some rapid flow de-aeration for all Froude numbers.

The technical report is supported by a digital appendix (Appendix D) containing three movies available at

the University of Queensland institutional open access repository UQeSpace

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

Keywords: Hydraulic jumps, Air bubble entrainment, Advective diffusion, Physical modelling, Air-water

flow measurements, Large-scale vortical structures, Turbulence.

iii

TABLE OF CONTENTS

Page

Abstract ii

Keywords ii

Table of contents iii

List of Symbols v

1. Introduction 1

2. Advective diffusion of air bubbles in the shear layer

2.1 Presentation

2.2 Basic theory

2.3 Discussion

3. Experimental facility and instrumentation

3.1 Experimental facility

3.2 Air-water flow instrumentation

3.3 Experimental flow conditions

4. Experimental observations. (1) Basic flow patterns

4.1 Presentation

4.2 Flow properties

4.3 Discussion

5. Experimental observations. (2) Basic air-water flow properties

5.1 Presentation

5.2 Void fraction and bubble rate distributions

5.3 Bubble chord distributions

6. Experimental observations. (3) Velocity and turbulent properties

6.1 Presentation

6.2 Turbulence intensity

6.3 Correlation time scales of turbulence

7. Interactions between air bubbles and turbulence: bubble clustering

7.1 Presentation

7.2 Experimental results

iv

8. Conclusion 48

9. Acknowledgments 51

APPENDICES

Appendix A - Air-water flow measurements 52

Appendix B - Experimental summary 63

Appendix C - Bubble clustering in the developing shear region 68

Appendix D - Movies of the experiments 72

REFERENCES 74

Internet references 77

Bibliographic reference of the Report CH75/09 77

v

List of symbols

The following symbols are used in this report:

C void fraction defined as the volume of air per unit volume of air and water; Cmax local maximum in void fraction in the developing shear layer; Cmean depth averaged void fraction:

∫ ×=90Y

0mean dyCC

C* local minimum in void fraction at the boundary between the air-water shear layer and the upper free-surface region;

Dt air bubble diffusivity (m2/s) in the air-water shear layer; D# dimensionless air bubble diffusivity: D# = Dt/(V1×d1); d 1- flow depth (m); 2- equivalent clear-water flow depth (m):

∫ ×−=90Y

0

dy)C1(d

d1 flow depth (m) measured immediately upstream of the hydraulic jump; d1 flow depth (m) measured downstream of the hydraulic jump roller; F bubble count rate (Hz) defined as the number of bubbles impacting the probe sensor per second; Fmax maximum bubble count rate (Hz) in the air-water shear layer; F2 secondary peak in bubble count rate (Hz) typically located in the upper free-surface region; Fr Froude number; Fej production rate (Hz) of large scale vortical structures; Ftoe hydraulic jump toe oscillation frequency (Hz); Fr1 upstream Froude number: 111 dg/VFr ×= ;

Fr2 downstream Froude number: 222 dg/VFr ×= ;

g gravity acceleration (m/s2) : g = 9.80 m/s2 in Brisbane (Australia); h sluice gate opening (m); K dimensionless constant; Lr hydraulic jump roller length (m); Lair hydraulic jump bubbly flow region length (m); Nc number of bubble clusters per second (Hz); Q water discharge (m3/s); Qair air flow rate (m3/s); qej rate of fluid (m2/s) entrained in the large scales vortices per unit width; Re Reynolds number: μ××ρ= /dVRe 11 ;

Rxx normalised auto-correlation function; Rxz normalised cross-correlation function; (Rxz)max maximum cross-correlation coefficient; ur bubble rise velocity (m/s);

vi

T average air-water interfacial travel time (s) between the two probe sensors; Tu turbulence intensity;

Txx auto-correlation integral time scale (s): ( )

∫=τ=τ

=τ

τ=0R

0xxxx

xx

dRT ;

Txz cross-correlation integral time scale (s): ( )

∫=τ=τ

=τ

τ=0R

0xzxz

xz

dRT ;

T0.5 characteristic time lag (s) for which Rxx = 0.5; V air-water velocity (m/s); Vej advection velocity (m/s) of large scales vortices in the developing shear layer; Vmax maximum air-water velocity (m/s) in the shear layer; V1 upstream flow velocity (m/s): V1 = Q/(W×d1); V1 downstream flow velocity (m/s): V2 = Q/(W×d2); W channel width (m); X dimensional variable (m); X' dimensionless variable: X' = X/d1; x longitudinal distance from the upstream sluice gate (m); x1 longitudinal distance from the upstream gate to the jump toe (m); YCmax vertical elevation (m) where the void fraction in the shear layer is maximum (C = Cmax); YFmax distance (m) from the bed where the bubble count rate is maximum (F = Fmax); YF2 distance (m) from the bed where F = F2; YVmax distance (m) from the bed where V = Vmax; Y90 characteristic distance (m) from the bed where C = 0.90; y distance (m) measured normal to the flow direction; y' dimensionless distance: y' = y/d1; y* distance (m) from the bed of the boundary between the air-water shear layer and the upper free-

surface region where C = C*; y0.5 distance (m) from the bed where V = Vmax/2;

Greek symbols Δx longitudinal distance (m) between probe sensors; Δz transverse distance (m) between probe sensors; δ boundary layer thickness (m); δs diameter (m) of large scale vortices; μ dynamic viscosity (Pa.s) of water; νT momentum exchange coefficient (m2/s); ρ density (kg/m3) of water; τ time lag (s);

5.0τ characteristic time lag (s) for which Rxz = (Rxz)max/2 ;

vii

Subscript * boundary between the upper free-surface region and the air-water shear layer; 1 upstream flow conditions; 2 downstream flow conditions 90 location where C = 0.90.

1

1. Introduction

Hydraulic jumps are commonly experienced in streams and rivers, in industrial channels and manufacturing

processes (Fig. 1-1). A hydraulic jump is the rapid transition from a high-velocity open channel flow to a

slower fluvial motion. It is a sharp discontinuity in terms of the water depth as well as the pressure and

velocity fields. For a horizontal rectangular channel and neglecting boundary friction, the equations of

conservation of mass and momentum in their integral form yield (BÉLANGER 1841):

⎟⎠⎞

⎜⎝⎛ −×+×= 1Fr81

21

dd 2

11

2 (1-1)

2/32

1

2/3

1

2

1Fr81

2FrFr

⎟⎠⎞

⎜⎝⎛ −×+

= (1-2)

where d and V are the flow depth and velocity respectively, the subscripts 1 and 2 refer to the upstream and

downstream flow conditions respectively, Fr is the Froude number: dg/VFr ×= and g is the gravity

acceleration.

The hydraulic jump is classified in terms of its inflow Froude number 111 dg/VFr ×= that is always

greater than unity. Some fundamental experimental studies included BIDONE (1819), DARCY and BAZIN

(1865) and BAHKMETEFF (1932) (see reviews by HAGER (1992) and CHANSON (2009)). The physical

observations demonstrated that, at large Froude numbers, a hydraulic jump is characterised by a marked

roller with a large rate of energy dissipation, some spray and splashing, and some air entrainment (Fig. 1-1).

The first successful air-water flow measurements in hydraulic jumps were conducted by RAJARATNAM

(1962). Table 1-1 summarises a number of important contributions, including the milestone study of RESCH

and LEUTHEUSSER (1972) who showed that the bubble entrainment process and energy dissipation are

strongly affected by the inflow conditions. To date, most experimental studies were conducted with partially-

developed inflow conditions, for which CHANSON (1995) highlighted some similarity with the air

entrainment process in plunging jets. For the last 10 years, a number of physical studies investigated

specifically the interactions between the entrained air and turbulent flow field (Table 1-1).

The present study aims to examine in details the air-water flow properties in hydraulic jumps with large

upstream Froude numbers (5.1 < Fr1 < 11.2) in a large size facility (4.0 104 < Re < 8.3 104). A focus of the

work is on the advective diffusion of air bubbles in the developing shear layer (section 2). The experimental

facility and instrumentation are described in section 3. The main results are presented in sections 4, 5 and 6,

and discussed in section 7.

2

Table 1-1 - Experimental studies of air-water flow measurements in hydraulic jumps

Reference d1 Fr1 Re x1 W Instrumentation m m

(1) (2) (3) (4) (5) (6) (7) LABORATORY STUDIES RAJARATNAM (1962) 0.0254 2.7 to 8.7 3.4 104 to

1.1 105 -- 0.31 Conductivity probe

RESCH & LEUTEUSSER (1972)

0.039 & 0.012

3.0 to 8.0 9.7 to 2.4 104

0.39 to 7.8

0.39 Hot-film probe (∅=0.6 mm)

BABB & AUS (1981) 0.035 6.0 1.2 105 -- 0.46 Hot-film probe (∅=0.4 mm) CHANSON (1995) 0.016 to

0.017 5.0 to 8.1 3.1 to 5.0

104 0.7 to 0.96

0.25 Single-tip phase-detection probe (∅=0.35 mm)

MOSSA & TOLVE (1998) 0.0185 to 0.020

6.42 to 7.3 5.2 to 6.2 104

0.90 0.40 Video-imaging

CHANSON & BRATTBERG (2000)

0.014 6.3 & 8.5 3.3 & 4.4 104

0.50 0.25 Dual-tip phase-detection probe (∅=0.025 mm)

MURZYN et al. (2005) 0.021 to 0.059

2.0 to 4.8 8.8 to 4.6 104

0.35 0.30 Dual-tip phase-detection probe (∅=0.010 mm)

CHANSON (2007) 0.013 to 0.029

5.1 to 8.6 2.5 to 9.8 104

0.50 & 1.0

0.25 & 0.50

Single-tip phase-detection probe (∅=0.35 mm)

GUALTIERI & CHANSON (2007)

0.012 5.2 to 14.3 2.4 to 5.8 104

0.50 0.25 Single-tip phase-detection probe (∅=0.35 mm)

KUCUKALI & CHANSON (2008)

0.024 4.7 to 6.9 5.4 to 8.0 104

1.0 0.50 Single-tip & dual-tip phase-detection probes (∅=0.35 & 0.25 mm)

MURZYN & CHANSON (2009)

0.018 5.1 to 8.3 3.8 to 6.2 104

0.75 0.50 Dual-tip phase-detection probe (∅=0.25 mm)

FIELD STUDY VALLE & PASTERNACK (2006)

0.22 (*) 2.8 (*) 9 105 -- 2 Time Domain Reflectometry (TDR)

Present study 0.018 5.1 to 11.2 4.0 to 8.3 104

0.75 0.50 Dual-tip phase-detection probe (∅=0.25 mm)

Notes: d1: upstream flow depth; Fr1: upstream Froude number; Re: Reynolds number (Re=ρ×V1×d1/μ); W:

channel width; x1: distance between the upstream gate and jump toe; (*): corrected data.

(A) Hydraulic jump at Isle de Serre white water stadium (France) in 2006 (Courtesy of Felix BOLLER)

3

(B) Hydraulic jump in a culvert inlet at Ridge Street, Brisbane during some flash flooding on 20 May 2009 -

Flow from left to right, Re ~ 3 106, Shutter speed: 1/80 s

(C) Details of the air entrainment at the jump toe and of the jump roller (same location at Fig. 1-1B) - Flow

from left to right, Re ~ 3 106, Shutter speed: 1/80 s

Fig. 1-1 - Photographs of hydraulic jumps

4

2. Advective diffusion of air bubbles in the shear layer

2.1 Presentation

A hydraulic jump is characterised by a significant amount of air entrainment (RAJARATNAM 1962,1967,

WISNER 1965). The air is entrapped at the jump toe that is a discontinuity between the impinging flow and

the roller. In a hydraulic jump roller, two distinct air-water regions may be distinguished: the air-water shear

region and the upper free-surface layer (Fig. 2-1). The air-water shear layer is characterised by a transfer of

momentum from the high-velocity jet flow to the recirculation region above, as well as by an advective

diffusion of entrained air bubbles. In the upper free-surface region, the air-water flow is characterised by an

uncontrolled exchange of air and water between the recirculation region and the atmosphere.

In hydraulic jumps with partially-developed inflow conditions, the experimental data demonstrated

conclusively that the void fraction distributions exhibited a characteristic shape in the developing shear layer

with a local maximum in void fraction (RESCH and LEUTHEUSSER 1972, THANDAVESWARA 1974,

CHANSON 1995) (Fig. 2-1, Right). In the shear layer, the void fraction distributions followed closely an

analytical solution of the advective diffusion equation (CHANSON 1995,1997). The findings are extended

herein and a more complete theoretical solution is developed.

Fig. 2-1 - Sketch of air bubble entrainment in hydraulic jumps with partially-developed inflow conditions -

(Left) Air water flow regions - (Right) Vertical distribution of void fraction in the hydraulic jump roller

2.2 Basic theory

In a hydraulic jump, the air bubble entrainment is localised at the intersection of the impinging water jet with

the receiving body of water. The air bubbles are entrained locally at the toe of the jump (Fig. 2-1). The

impingement perimeter is a source of air bubbles, as well as a source of vorticity. The air bubble diffusion in

the hydraulic jump is a form of advective diffusion. For a small control volume and neglecting the buoyancy

effects, the continuity equation for air bubbles becomes:

( ) ( )rt uCCgradDdivVCdiv ×−×=× (2-1)

where C is the void fraction, V is the advective velocity, Dt is the air bubble diffusivity, and ru is the

5

bubble rise velocity. Equation (2-1) is valid for steady flow conditions neglecting the compressibility effects.

Assuming an uniform velocity distribution, for a constant diffusivity independent of the longitudinal and

transverse location, Equation (2-1) becomes:

2

2tr1

yCD

yCu

xCV

∂

∂×=

∂∂

×+∂∂

× (2-2)

where x is the longitudinal coordinate, y is the vertical elevation above the channel bed, V1 is the inflow

velocity and the rise velocity ur is assumed constant. With a change of variable ( yV/uxxX 1r1 ×+−= ),

Equation (2-2) becomes a two-dimensional diffusion equation:

2

2t

t

1

yCD

XC

DV

∂

∂×=

∂∂

× (2-3)

Equation (2-3) is a classical diffusion equation (CRANK 1956). In a hydraulic jump, the air bubbles and

packets are entrained at the jump toe acting as a point source located at (x-x1 = 0, y = d1) where d1 is the

upstream flow depth (Fig. 2-1). The strength of the source equals Qair/W where Qair is the entrained air

volume and W is the channel width. Equation (2-3) can be solved by applying the method of images. The

complete analytical solution is:

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

×

+

−+

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

×

−

−×××π×

= #

2

#

2

#

air

D4'X)1'y(

expD4'X)1'y(

exp'XD4

QQ

C (2-4)

where X' = X/d1, y' = y/d1, and D# is a dimensionless diffusivity: )dV/(DD 11t# ×= . In Equation (2-4),

right handside term, the first term is the contribution of the real source (i.e. jump toe) and the second term is

the contribution of the imaginary source located at (x-x1 = 0, y = -d1). Equation (2-4) yields a good

agreement with experimental data in the developing shear layer of hydraulic jumps with partially-developed

inflow conditions.

2.3 Discussion

CHANSON (1995,1997) proposed a simpler solution in the form of:

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

××

−×=

1

1

2

1

maxC

#max

dxx

dYy

D41expCC (2-5)

where YCmax is the location where the void fraction is maximum in the developing shear layer. Equation (2-5)

is a limiting case of Equation (2-4) assuming that the effects of buoyancy are accounted for by YCmax, the

contribution of the imaginary source term is small, and 'XD4/)Q/Q(C #airmax ××π×= .

In practice, the comparison between experimental data and theoretical results showed consistently that (a) the

effects of the imaginary source term were small, and that (b) the effects of buoyancy were underestimated by

6

Equation (2-4) and better accounted for using the measured location of the maximum void fraction (YCmax).

This is illustrated in Figure 2-2 presenting a comparison between some experimental data and Equations (2-

5) and (2-4). Equation (2-4) is presented (a) assuming ur = 0.3 m/s (ur/V1 = 0.078) and (b) selecting a value

of ur/V1 for the best data fit. The comparative results suggest that Equations (2-4) and Equation (2-5) give

very close results with a suitable, empirical estimate of the ratio ur/V1 (Fig. 2-2). Note that the effects of

buoyancy, hence the ratio ur/V1, tended to increase with the longitudinal distance (x-x1)/d1. The trend was

qualitatively physical corresponding to a larger effect of buoyancy as the jump flow is decelerated.

Quantitatively, however, the best data fit yielded a gross overestimate of the buoyancy contribution and

unrealistic values of the ratio ur/V1.

Some typical values of the dimensionless air bubble diffusion coefficients D# = Dt/(V1×d1) are presented in

Figure 2-3A as a function of the Reynolds number Re = ρ×V1×d1/μ where ρ and μ are respectively the

density and dynamic viscosity of water. Despite some scatter, the data yielded an average dimensionless

diffusivity Dt/(V1×d1) = 0.044 for all three data sets and an average value of 0.042 for the present

experiments. Within the range of experiments, the dimensionless diffusivity was found to be independent of

the inflow Froude number and Reynolds number, although the data suggested some increase in D# with

increasing distance (x-x1)/d1 from the jump toe. This is illustrated in Figure 2-3B. For the present study, the

data were best correlated by:

1

1#d

xx0012.001.0D

−×+= for 70

dxx

51

1 <−

< (2-6)

for 7 < Fr1 < 11.5 with a normalised correlation coefficient of 0.925. Equation (2-6) is compared with the

experimental data in Figure 2-3B.

It is noteworthy to compare with the longitudinal variation of the momentum exchange coefficient (or "eddy

viscosity") in a developing shear layer. GOERTLER's (1942) solution of the Navier-Stokes equations

implies a longitudinal distribution of the "eddy viscosity" function:

1

1

11

Td

xxK4

1dV

−×

×=

×ν (2-7)

where K is a constant equal to 9 to 13.5 in monophase flows (RAJARATNAM 1976, SCHLICHTING

1979), and 4 to 10 in air-water flows (CHANSON 1997).

7

C

y/d 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

10DataTheory (CHANSON 1995)Theory ur/V1=0.078Theory ur/V1=18

Cy/

d 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

10DataTheory (CHANSON 1995)Theory ur/V1=0.078Theory ur/V1=35

Left: x-x1 = 0.225 m; Right: x-x1 = 0.35 m

C

y/d 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

10

11

12DataTheory (CHANSON 1995)Theory ur/V1=0.078Theory ur/V1=60

C

y/d 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1

2

3

4

5

6

7

8

9

10

11

12DataTheory (CHANSON 1995)Theory ur/V1=0.078Theory ur/V1=95

Left: x-x1 = 0.45 m; Right: x-x1 = 0.60 m

Fig. 2-2 - Void fraction distributions in a hydraulic jump with partially-developed inflow conditions: x1 =

0.75 m, d1 = 0.018 m, Fr1 = 9.2, Re = 6.9 104, x-x1 = 0.225, 0.30, 0.45 and 0.60 m- Comparison between

experimental data (Present study), Equation (2-5) (CHANSON 1995), Equation (2-4) assuming ur = 0.3 m/s

and Equation (2-4) with larger value of ur/V1

8

Re

Dt/(

V1.d

1)

2000020000 30000 40000 50000 60000 70000 80000 1000000.0050.0060.0070.008

0.01

0.02

0.03

0.04

0.050.060.070.08

0.1

CHANSON (1997)CHANSON & BRATTBERGCHANSON (2007)Present study

(A) Dt/(V1×d1) as a function of the Reynolds number Re = ρ×V1×d1/μ

(x-x1)/d1

Dt/(

V1.d

1)

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 750

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

Fr1=6.3 CHANSON & BRATTBERGFr1=8.5 CHANSON & BRATTBERGFr1=5.1 Present studyFr1=7.5 Present studyFr1=9.2 Present studyFr1=10.0 Present studyFr1=11.2 Present study0.01+0.0012.(x-x1)/d1

(B) Dt/(V1×d1) as a function of dimensionless distance from the jump toe (x-x1)/d1 - Comparison with

Equation (2-6)

Fig. 2-3 - Dimensionless turbulent diffusivity of air bubbles in the developing shear layer of hydraulic jumps

- Experimental data: CHANSON (1997), CHANSON and BRATTBERG (2000), CHANSON (2007) and

Present study

9

3. Experimental facility and instrumentation

3.1 Experimental facility

The new series of experiments were performed in the Gordon McKAY Hydraulics Laboratory at the

University of Queensland (Table 3-1). The channel was horizontal, 3.2 long and 0.5 m wide. The sidewalls

were made of 3.2 m long, 0.45 m high glass panels and the bed was made of 12 mm thick PVC sheets. The

inflow was controlled by an upstream undershoot gate (Fig. 3-1A). The downstream flow conditions were

controlled by a vertical overshoot gate (Fig. 3-1B). The flume was used previously by CHANSON (2007),

KUCUKALI and CHANSON (2008) and MURZYN and CHANSON (2009).

The channel was fed by a constant head tank. The water discharge was measured with a Venturi meter

located in the supply line that was calibrated on-site with a large V-notch weir. The discharge measurements

were accurate within ±2%. The clear-water flow depths were measured using rail mounted point gages with a

0.2 mm accuracy. The inflow conditions were controlled by a vertical gate with a semi-circular rounded

shape (∅ = 0.3 m) and the downstream coefficient of contraction was about unity. The upstream gate

aperture was fixed during all experiments (h = 0.018 m).

Additional information was obtained with a digital camera Panasonic™ Lumix DMC-FZ20GN (shutter

speed: 8 to 1/2,000 s). Some movies of the experiments are available in the form of a digital appendix

(Appendix D).

(A) Looking upstream at the hydraulic jump with the upstream vertical sluice and head tank in the

background - Fr1 = 10.6, Re = 8.0 104, d1 = 0.018 m, x1 = 0.75 m, shutter speed: 1/50 s

10

(B) Details of the downstream overshoot gate - Fr1 = 11.2, Re = 8.3 104, d1 = 0.0178 m, x1 = 0.75 m, shutter

speed: 1/80 s, Flow from right to left

Fig. 3-1 - Photographs of the experimental facility

3.2 Air-water flow instrumentation

The air-water flow properties were measured with a double-tip conductivity probe (Fig. 3-2). The probe was

equipped with two identical sensors with an inner diameter of 0.25 mm. The longitudinal distance between

probe tips was Δx = 6.96 mm while the transverse separation distance between tips was Δz = 2.08 mm (Fig.

3-2). The probe was manufactured at the University of Queensland. A similar probe was previously used in

several studies, including CHANSON and CAROSI (2007), and KUCUKALI and CHANSON (2008) and

MURZYN and CHANSON (2009). The displacement and the position of the probe in the vertical direction

were controlled by a fine adjustment system connected to a Mitutoyo™ digimatic scale unit with a vertical

accuracy of less than 0.1 mm.

The conductivity probe is a phase-detection intrusive probe designed to pierce the bubbles. The probe design

is based on the difference in electrical resistance between air and water (CROWE at al. 1998, CHANSON

2002). The dual-tip probe was excited by an electronic system (Ref. UQ82.518) designed with a response

time of less than 10 μs. During the experiments, each probe sensor was sampled at 20 kHz for 45 s and the

recorded output signal was a voltage ranging from 0 (air) to 4.5 V (water) (Fig. 3-3). Figures 3-3 shows a

typical probe signal output in the developing shear layer. In the figure, each downward drop in voltage

corresponds to an air bubble being pierced by the sensor tip. The analysis of the probe voltage output was

based upon a single threshold technique, with a threshold set between 45% and 55% of the air–water voltage

range. Below this threshold, the probe was in air whereas it was in water for larger voltage outputs (Fig. 3-3).

11

The single-threshold technique is a robust method that is well-suited to free-surface flows (TOOMBES 2002,

CHANSON and CAROSI 2007).

The processing of the probe signal yielded a number of air-water flow properties. These included the void

fraction C defined as the volume of air per unit volume of air and water, the bubble count rate F defined as

the number of bubbles impacting the probe tip per second, and the air chord time distributions where the

chord time is defined as the time spent by the bubble on the probe tip.

The air-water interfacial velocity V was calculated as V = Δx/T where Δx is the longitudinal distance

between both tips (Δx = 6.96 mm) and T is the average air-water interfacial time between the two probe

sensors (CROWE et al. 1998, CHANSON 1997,2002) (Fig. 3-4). T was deduced from a cross-correlation

analysis (Fig. 3-4 Right). The turbulence level Tu characterised the fluctuations of the air-water interfacial

velocity between the probe sensors (CHANSON and TOOMBES 2002, CHANSON 2002). It was deduced

from the shapes of the cross-correlation Rxz and auto-correlation Rxx functions:

T

T851.0Tu

25.0

25.0 −τ

×= (3-1)

where τ0.5 is the time scale for which the normalised cross-correlation function is half of its maximum value

such as Rxz(T+τ0.5) = (Rxz)max/2, (Rxz)max is the maximum cross-correlation coefficient for τ = T, T0.5 is the

time for which the normalized auto-correlation function equals 0.5 (Fig. 3-4). The notations are illustrated in

Figure 3-4. The analysis of the signal auto-correlation function provided further information. The integral

time scales Txx and Txz represented some time scale relative to the longitudinal bubbly flow structure. They

were some characteristic times of the large eddies advecting the air-water interfaces in the longitudinal

direction (CHANSON 2007, CHANSON and CAROSI 2007).

Herein the data processing of correlation functions were conducted on the raw probe output signals. Indeed,

any analysis based upon thresholded signals would tend to ignore the contributions of the smallest air-water

particles (CHANSON and CAROSI 2007). Thus, the correlation functions were calculated for the original

files of 900,000 samples (sampling frequency of 20 kHz for 45 s).

12

Fig. 3-2 - Photograph of the double-tip conductivity probe - Fr1 = 7.5, Re = 5.6 104, d1 = 0.018 m, x1 = 0.75

m, x-x1 = 0.150 m, y = 0.118 m, shutter speed: 1/80 s, Flow from right to left

Time (s)

Vol

tage

(V)

0.01 0.012 0.014 0.016 0.018 0.02 0.022 0.024 0.026 0.028 0.030

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Air voltage

Water voltage

Leading tipTrailing tipThreshold

Fig. 3-3 - Double-tip conductivity probe outputs: Fr1 = 11.2, Re = 8.3 104, d1 = 0.0178 m, x1 = 0.75 m, x-x1 =

0.255 m, y = 0.0135 m, C = 0.093, F = 132.1 Hz, V = 3.026 m/s

13

Fig. 3-4 - Definition sketch of the auto- and cross-correlation function for a dual-tip probe

Remarks

The phase-detection probes are some sensitive devices susceptible to a number of problems. In the present

study, the quality control procedure developed by TOOMBES (2002, pp. 70-72) was applied thoroughly.

Specifically, the probe signal outputs were checked for (a) some long-term signal decays often induced by

probe tip contamination, (b) any short-term signal fluctuations caused by debris and water impurities, (c) any

electrical noise and (d) non-representative samples. The quality control procedure can be automatised but the

human supervision and intervention are essential to validate each quality control step.

Herein the sampling rate and duration were selected based upon the sensitivity analysis results of

CHANSON (2007b). That study showed that that the sampling frequency had little effect on the void

fraction for any sampling rate above 500 Hz, while the bubble count rate was drastically underestimated for

sampling rates below 5 to 8 kHz. Further the sampling duration had little effect on both void fraction and

bubble count rate for scan periods longer than 30 to 40 s. In the present study, the sampling rate and duration

were 20 kHz and 45 s respectively.

3.3 Experimental flow conditions

Two series of experiments were conducted (Table 3-1). The first series focused on the general hydraulic

jump properties: e.g., upstream and downstream depths, jump toe fluctuation frequency. The experiments

were performed with inflow Froude numbers between 3.4 and 12.4 corresponding to Reynolds numbers

between 2.9 104 and 9.3 104 In the second series of experiments, some detailed air-water flow measurements

at the sub-millimetric scale were conducted using the double-tip conductivity probe. The flow conditions

corresponded to Froude numbers between 5.1 and 11.2 and Reynolds numbers between 4 104 and 8.3 104.

For all experiments, the jump toe was located at x1 = 0.75 m and the same upstream rounded gate opening h

= 0.018 m was used for the whole study. For these conditions, the inflow depth ranged from 0.0178 to 0.019

m depending upon the flow rate. Based on previous experiments made with the same experimental facility

(CHANSON 2005), the inflow was characterised by a partially-developed boundary layer.

Lastly MURZYN and CHANSON (2008) conducted some Froude similar experiments (Fr1 = 5.1 & 8.5) with

14

Reynolds numbers between 2.4 104 and 9.8 104. Their results showed some drastic scale effects in the

smaller hydraulic jumps in terms of void fraction, bubble count rate and bubble chord time distributions for

Re < 4 104. Herein the focus of the study was on the hydraulics jumps with large Froude numbers (Fr1 > 7)

and large Reynolds numbers (Re > 5 104) for which some small to moderate scale effects might be expected.

Table 3-1 - Experimental flow conditions for measurements in hydraulic jumps with partially-developed

inflow conditions

Ref. Q W x1 V1 d1 Fr1 Re Remarks m3/s m m m/s m

(1) (2) (3) (4) (5) (6) (7) (8) (9) Series 1 General observations

2 0.0147 0.5 0.75 1.55 0.019 3.58 2.9E+4 3 0.0166 0.5 0.75 1.75 0.019 4.05 3.3E+4 1 0.02225 0.5 0.75 2.34 0.019 5.42 4.4E+4 5 0.0282 0.5 0.75 3.13 0.018 7.46 5.6E+4 4 0.03255 0.5 0.75 3.52 0.0185 8.26 6.5E+4 6 0.0367 0.5 0.75 4.08 0.018 9.70 7.3E+4 7 0.0399 0.5 0.75 4.43 0.018 10.55 7.9E+4 8 0.0470 0.5 0.75 5.22 0.018 12.43 9.3E+4

Series 2 Air-water flow measurements 090331 0.02025 0.5 0.75 2.19 0.0185 5.14 4.0E+4 090317 0.02825 0.5 0.75 3.14 0.018 7.47 5.6E+4 090720 0.03481 0.5 0.75 3.87 0.018 9.21 6.9E+4 090713 0.03780 0.5 0.75 4.20 0.018 10.0 7.5E+4 090414 0.04175 0.5 0.75 4.68 0.01783 11.2 8.3E+4

Notes: d1: upstream flow depth; Fr1: upstream Froude number; Q: water discharge; Re: Reynolds number;

V1: upstream flow velocity; W: channel width; x1: distance between the upstream gate and jump toe.

15

4. Experimental observations. (1) Basic flow patterns

4.1 Presentation

A hydraulic jump is the sudden transformation from a supercritical to subcritical open channel flow. The

flow transition is a singularity in terms of the flow depth, and the velocity and pressure fields. The hydraulic

jump is characterised by the development of large-scale turbulence, a significant rate of energy dissipation,

some significant spray and splashing, and air bubble entrainment. At the jump toe, the air bubbles and air

packets are entrained into a developing shear layer that is characterised by some intensive turbulence

production of large coherent vortices with horizontal axes perpendicular to the flow direction (Fig. 4-1).

Some movies of the experiments illustrate the process in a digital appendix (Appendix D). The air

entrainment takes place in the form of air bubbles, pockets and packets entrapped at the impingement of the

upstream supercritical flow with the jump roller. The air packets are broken up in smaller air bubbles as they

are advected downstream in the shear region that is characterised by some large void fractions and bubble

count rates (paragraph 5.). Once the entrained air is advected into some flow regions with lesser shear stress

levels, the bubble collisions and coalescence lead to the formation of larger air bubbles that are driven

upwards by buoyancy and ultimately towards the free-surface.

In the recirculating region above the developing shear layer, some unsteady flow reversal and recirculation

take place. The location of the jump toe is constantly fluctuating around a mean position (x = x1) and some

"vortex shedding" develops in the mixing layer. The jump toe pulsations are caused by the growth, advection

and pairing of large scale vortices in the developing shear layer (LONG et al. 1991, HABIB et al. 1994). The

high-speed photographs show a significant amount of air-water ejections and splashes above the roller (Fig.

4-2). The ejected packets take different forms ranging from elongated fingers to single droplets and air-water

packets.

The mechanisms of air entrainment in a hydraulic jump are complicated and may be affected by the inflow

conditions (THANDAVESWARA 1974, CHANSON 1997). If the upstream flow is aerated, the aerated

layer at the jet free-surface is entrained past the impingement point. This process is called pre-entrainment or

two-phase flow air flux. At the jump toe, an air layer is set into motion by surface shear friction next to the

free-surface of the impinging flow and the air is entrapped at the entrainment point (i.e. jump toe). Another

mechanism is the aspiration of an induction trumpet formed at the intersection of the water jet with the roller.

The air trumpet acts as a ventilated cavity and the closure ("pinching") of the trumpet releases some air

packets into the shear flow (e.g. CHANSON and BRATTBERG 1998). In the present study, the upstream

flow was little aerated (paragraph 5.1) and the air bubble entrainment was predominantly a combination of

the last two mechanisms.

16

(A) Fr1 = 7.5, Re = 5.6 104, d1 = 0.018 m, x1 = 0.75 m, x-x1 = 0.150 m, shutter speed: 1/80 s (Filename:

P1130787.jpg)

(B) Fr1 = 10.0, Re = 7.5 104, d1 = 0.018 m, x1 = 0.75 m, x-x1 = 0.350 m, shutter speed: 1/80 s (Filename:

P1150364.jpg)

17

(C) Fr1 = 11.2, Re = 8.3 104, d1 = 0.0178 m, x1 = 0.75 m, shutter speed: 1/80 s (Filename: P1140148b.jpg) -

Note the air-water projection above the sidewall immediately downstream (on the left) of the trolley system

Fig. 4-1 - Photographs of air entrainment in hydraulic jumps

4.2 Flow properties

Figure 4-3 presents the ratio of the downstream to upstream depths d2/d1 as a function of the inflow Froude

number Fr1. The experimental data are compared with the application of the equation of conservation of

momentum, called the Bélanger equation:

⎟⎠⎞

⎜⎝⎛ −×+×= 1Fr81

21

dd 2

11

2 (4-1)

where Fr1 is the inflow Froude number. Equation (4-1) is compared with the experimental observations in

Figure 4-3 illustrating a good agreement but at the largest Froude number. In that case (Fr1 = 11.2), the jump

roller interfered with the downstream overshoot gate and the downstream conjugate depth d2 was measured

immediately upstream of the gate.

In Figure 4-3, the dimensionless roller length and bubbly flow region length are also presented. Herein the

roller length Lr was defined as the location where the water surface was quasi-horizontal and the downstream

depth was measured. The length Lair of the bubbly flow region was determined through some sidewall

observations of the entrained air bubbles: i.e., Lair was the average length of the bubbly flow region. The

present data were qualitatively in agreement with the correlations of HAGER et al. (1990) and MURZYN et

al (2007), although both correlations tended to underestimate the jump length by 20-30% (Fig. 4-3).

18

(A) Looking downstream at the jump toe with the probe tip located at x-x1 = 0.075 m (Filename

P1140094.jpg) - Fr1 = 5.1, Re = 4.0 104, d1 = 0.0185 m, x1 = 0.75 m, shutter speed: 1/80 s

(B) Looking upstream at the splashes above roller with the probe tip located at x-x1 = 0.150 m (Filename

P1130802.jpg) - Fr1 = 7.5, Re = 5.6 104, d1 = 0.018 m, x1 = 0.75 m, shutter speed: 1/80 s

19

(D) Fr1 = 7.5, Re = 5.6 104, d1 = 0.018 m, x1 = 0.75 m, shutter speed: 1/80 s - From Top Left, anti-clockwise:

(D1) Sideview with the probe tip located at x-x1 = 0.075 m (Filename P1140129.jpg); (D2) Looking

upstream above the roller with the probe tip located at x-x1 = 0.075 m (Filename P1140135.jpg); (D3)

Looking upstream above the roller with the probe tip located at x-x1 = 0.35 m (Filename P1140158.jpg);

(D4) Looking downstream at the jump toe with the probe tip located at x-x1 = 0.35 m (Filename

P1140167.jpg)

Fig. 4-2 - Air-water projections above the hydraulic jump roller - High-shutter speed photographs

The position of the hydraulic jump toe fluctuated rapidly with time within a 0.05 to 0.25 m amplitude range,

depending upon the inflow conditions. See for example the movies available in the digital appendix

(Appendix D).The toe oscillation frequencies Ftoe were typically about 0.3 to 0.8 Hz for the present study

(Table 4-1, Figure 4-4). The jump toe pulsations were believed to be caused by the formation, production

and pairing of large scale vortices in the developing shear layer of the jump. The frequency Fej of the large-

scale vortical structures as well as their advection velocity Vej were also recorded using some video-records

at 30 fps (Table 4-1). The data are presented in Figures 4-4 and 4-5. Herein Fej represents the rate of

20

production of large-scale coherent structures advected in the developing shear layer.

Figure 4-4 summarises the observations in terms of the Strouhal numbers Ftoe×d1/V1 and Fej×d1/V1 as

functions of the Reynolds number ρ×V1×d1/μ. The data are compared with the jump toe fluctuation data of

LONG et al. (1991) MOSSA and TOLVE (1998), CHANSON (2007) and MURZYN and CHANSON

(2009). Noteworthy the jump toe fluctuation frequencies Ftoe were almost equal to the production rate Fej of

large scale vortical structures. The finding supports the assertion that the jump toe oscillations are caused by

the formation and downstream advection of large scale vortices in the shear layer.

Figure 4-5 presents the dimensionless advection speed Vej/V1 of the large scale coherent structures in the

developing shear layer. The advection speed represented the average convection velocity of the large

coherent structures in the mixing layer. The data were nearly independent of the Reynolds number and they

yielded in average: Vej/V1 = 0.32 for 5.1 < Fr1 < 11.2. For comparison, the observations of BROWN and

ROSHKO (1974) gave a convective speed Vej/ΔV ≈ 0.24 in a free shear layer with a transverse velocity

gradient ΔV.

Table 4-1 - Measured flow properties of hydraulic jumps (Present study)

Run No. Q V1 Fr1 Re d1 d2 Lr Lair Ftoe Fej Vej m3/s m/s m m m m Hz Hz m/s

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) 2 0.0147 1.55 3.6 2.9E+4 0.019 0.089 0.75 0.47 0.492 0.533 0.67 3 0.0166 1.75 4.0 3.3E+4 0.019 0.104 0.78 0.55 0.392 0.2 0.75 1 0.02225 2.34 5.4 4.4E+4 0.019 0.138 1.1 0.81 0.509 0.42 0.61

5/090317 0.0282 3.13 7.5 5.6E+4 0.018 0.178 1.3 1.05 0.833 0.733 0.75 4 0.03255 3.52 8.3 6.5E+4 0.0185 0.206 1.8 1.4 -- 0.533 1.06 6 0.0367 4.08 9.7 7.3E+4 0.018 0.23 1.65 1.55 -- 0.793 1.27

090713 0.0378 4.20 10.0 7.5E+4 0.018 -- -- -- 0.714 -- -- 7 0.0399 4.43 10.6 7.9E+4 0.018 0.246 1.85 1.6 - 1.099 1.20

090414 0.04175 4.68 11.2 8.3E+4 0.0178 -- -- -- 0.765 -- 8 0.047 5.22 12.4 9.3E+4 0.018 0.258 2.05 1.95 -- 1 1.69

Notes: d1: upstream flow depth; d2: downstream flow depth; Fr1: upstream Froude number; Fej: large-scale

vortical structure ejection frequency; Ftoe; jump toe horizontal oscillation frequency; Lair: bubbly flow region

length; Lr; roller length; Q: water discharge; Re: Reynolds number; Vej: advection velocity of large-scale

vortical structures; V1: upstream flow velocity; (--): data not available.

21

Fr1

d 2/d

1

L r/d

1, L a

ir/d 1

1 2 3 4 5 6 7 8 9 10 11 12 131 0

3 30

5 60

7 90

9 120

11 150

13 180

15 210d2/d1 Datad2/d1Theory (Momentum)Lr/d1 DataLr/d1 HAGER et al.Lr/d1 MURZYN et al.Lair/d1 Data

Fig. 4-3 - Ratio of the conjugate depths d2/d1, dimensionless roller length Lr/d1 and bubble flow region length

Lair/d1 as functions of the inflow Froude number Fr1 - Comparison between experimental data (present

study), Equation (4-1) and the correlations of HAGER et al. (1990) and MURZYN et al. (2007)

St_Re_HJ_All.grfRe

F toe

.d1/V

1, F e

j.d1/V

1

10000 20000 30000 40000 50000 70000 100000 2000002000000.001

0.002

0.0030.0040.005

0.007

0.01

0.02

0.030.040.05

0.07

0.1Toe LONG et al.Toe MOSSA & TOLVEToe CHANSONToe MURZYN & CHANSONToe Present studyVortical ejection Present study

Fig. 4-4 - Dimensionless relationship between Strouhal number and Reynolds number in hydraulic jumps:

oscillations of the jump toe (Data: LONG et al. 1991, MOSSA and TOLVE 1998, CHANSON 2007,

MURZYN and CHANSON 2009, Present study) and ejection frequency of large scale vortical structures

(Data: Present study)

22

Re

Vej

/V1

10000 20000 30000 40000 50000 70000 100000 2000002000000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Present studyBROWN & ROSHKO

Fig. 4-5 - Dimensionless advection speed of the large scale vortical structures in the developing shear layer

of hydraulic jumps - Comparison between the present data and the observations of BROWN and ROSHKO

(1974) in a free shear layer

4.3 Discussion

In the developing shear layer, the coherent vortical structures may be assumed to be cylindrical with a

diameter δs. The rate of entrained fluid entangled in the large-scale structures equals:

ej2

sej F4

q ×δ×π

= (4-2)

HOYT ad SELLIN (1989) assumed: δs ≈ 0.34×(x-x1) and the present data yielded: Fej ≈ 0.66 Hz in average.

Replacing into Equation (4-4), the rate of entrained air and water per unit width becomes:

21ej )xx(06.0q −×≈ (4-3)

The expression accounts for both air and water and does not distinguish between the two phases.

23

5. Experimental observations. (2) Basic air-water flow properties

5.1 Presentation

A hydraulic jump is characterised by a significant rate of energy dissipation and the air entrainment is

evidenced by the white colour of the jump roller (Fig. 4-1). Some detailed air-water flow measurements were

conducted for five inflow Froude numbers ranging from 5.1 to 11.2 (Table 5-1). In each case, the upstream

flow was little aerated. This is seen in Figure 4-1 and demonstrated in Figure 5-1 showing the vertical

profiles of void fraction and bubble count rate at a location 0.2 m upstream of the jump toe for Fr1 = 11.2.

For this data set, the depth-averaged void fraction was Cmean = 0.11 where:

∫ ×=90Y

0mean dyCC (5-1)

with y the distance normal to the invert, C the local void fraction and Y90 the distance from the bed where C

= 0.9 (Fig. 5-2).

In the following sections, the results are focused on the air-water flow properties in hydraulic jumps with

large Froude numbers: i.e., Fr1 = 7.5 to 11.2.

C, F.d1/V1

y/d 1

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

1.1

1.2

Fr1 = 11.2, Re = 8.3 E+4,d1 = 0.0178 m, x = 0.6 m

CF.d1/V1

Fig. 5-1 - Dimensionless distribution of void fraction and bubble count rate in the upstream flow: Fr1 = 11.2,

Re = 8.3 104, d1 = 0.01783 m, x1 = 0.75 m, x-x1 = -0.2 m

24

Table 5-1 - Air-water flow measurements in hydraulic jumps with partially-developed inflow conditions

Ref. Q W x1 V1 d1 Fr1 Re Remarks m3/s m m m/s m

(1) (2) (3) (4) (5) (6) (7) (8) (9) 090331 0.02025 0.5 0.75 2.19 0.0185 5.14 4.0E+4 x-x1= 0.075 to 0.35 m 090317 0.02825 3.14 0.018 7.47 5.6E+4 x-x1= 0.075 to 0.40 m 090720 0.03481 3.87 0.018 9.21 6.9E+4 x-x1= 0.075 to 1.00 m 090713 0.03780 4.20 0.018 10.0 7.5E+4 x-x1= 0.075 to 1.05 m 090414 0.04175 4.68 0.01783 11.2 8.3E+4 x-x1= 0.075 to 1.40 m

Notes: d1: upstream flow depth; Fr1: upstream Froude number; Q: water discharge; Re: Reynolds number;

V1: upstream flow velocity; W: channel width; x1: distance between the upstream gate and jump toe.

Fig. 5-2 - Air entrainment in a hydraulic ump with partially-developed inflow: definition sketch

25

5.2 Void fraction and bubble rate distributions

A hydraulic jump with partially-developed inflow is characterised by a turbulent shear layer with an

advective diffusion region in which the air concentration distributions exhibit a peak in the turbulent shear

region (RESCH and LEUTHEUSSER 1972, CHANSON 1995). This is illustrated in Figure 5-2 highlighting

the key definitions. Figure 5-3 and 5-4 present some dimensionless distributions of void fraction C and

bubble count rate F×d1/V1 along the hydraulic jump. The characteristic location Y90/d1 is also shown in

Figures 5-3 and 5-4 (thick dashed line). The full data sets are reported in Appendix A.

In the air-water shear layer, the void fraction profiles followed closely an analytical solution of the advective

diffusion equation for air bubbles (section 2):

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

⎟⎟⎠

⎞⎜⎜⎝

⎛ −

××

−×=

1

1

2

1

maxC

#max

dxx

dYy

D41expCC (5-2)

where D# is a dimensionless diffusivity: )dV/(DD 11t# ×= , Dt is the air bubble diffusivity, d1 and V1 are

respectively the inflow depth and velocity, and YCmax is the distance from the bed where C = Cmax (Fig. 5-2).

Equation (5-2) is compared with some experimental data in Figure 5-5 at four longitudinal locations in a

hydraulic jump (Fr1 = 11.2). Further examples are shown in section 2.

In the air-water shear layer, the local maximum in void fraction Cmax decreased with increasing distance (x-

x1) from the impingement point while the diffusion layer broadened (Fig. 5-3 to 5-5). This is seen in Figure

5-5 and the data are summarised in Figure 5-6A where the maximum void fraction Cmax is plotted a function

of the dimensionless longitudinal distance (x-x1)/d1 from the jump toe. The present data are compared with

earlier data sets and all the data followed closely an exponential decay:

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−∝

1

1max d

xxexpC (5-3)

Some typical vertical profiles of bubble count rate are presented also in Figures 5-3 to 5.5. Each profile

exhibited a maximum count rate Fmax in the air-water shear layer and a secondary peak F2 in the upper free-

surface region. Both definitions are sketched in Figure 5-2. The maximum bubble count rate was linked with

a region of maximum shear stress. Noteworthy the location YFmax of the maximum bubble count rate was

consistently below the location YCmax of maximum void fraction in the air-water shear layer. The advective

diffusion layer did not coincide with the momentum shear layer (CHANSON and BRATTBERG 2000,

KUCUKALI and CHANSON 2008) highlighting a double diffusion process whereby air bubbles and

vorticity diffuse in the shear region at different rates and in a different manner. The non-coincidence of Cmax

and Fmax demonstrated that the interactions between the developing shear layer and air diffusion layer were

complex.

26

C, 0.1.(x-x1)/d1

y/d 1

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

2

4

6

8

10

12

14Y90/d1Void fraction

(A) Dimensionless distribution of void fraction

F.d1/V1, 0.1.(x-x1)/d1

y/d 1

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

2

4

6

8

10

12

14Y90/d1Bubble count rate

(B) Dimensionless distribution of bubble count rate

Fig. 5-3 - Dimensionless distributions of void fraction and bubble count rate for Fr1 = 9.2, Re = 6.9 104, d1 =

0.018 m, x1 = 0.75 m

27

C, 0.1.(x-x1)/d1

y/d 1

-1 0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

12

14

16Y90/d1Bubble count rate

(A) Dimensionless distribution of void fraction

F.d1/V1, 0.1×(x-x1)/d1

y/d 1

-1 0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

12

14

16Y90/d1Void fraction

(B) Dimensionless distribution of bubble count rate

Fig. 5-4 - Dimensionless distributions of void fraction and bubble count rate for Fr1 = 11.2, Re = 8.3 104, d1

= 0.01783 m, x1 = 0.75 m

28

C, F×d1/V1

y/d 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16C DataF×d1/V1 DataC Theory

C, F×d1/V1y/

d 10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

2

4

6

8

10

12

14

16C DataF×d1/V1 DataC Theory

C, F×d1/V1

y/d 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16C DataF×d1/V1 DataC Theory

C, F×d1/V1

y/d 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

2

4

6

8

10

12

14

16C DataF×d1/V1 DataC Theory

Fig. 5-5 - Dimensionless distributions of void fraction in the air-water shear layer: comparison between

experimental data and Equation (5-2) - Fr1 = 11.2, Re = 8.3 104, d1 = 0.01783 m, x1 = 0.75 m - From Left to

Right, Top to Bottom: (x-x1)/d1 = 12.6, 19.6, 36.4 and 50.5

29

(x-x1)/d1

Cm

ax

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 800

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7Fr1=5.1 Present studyFr1=7.5 Present studyFr1=9.2 Present studyFr1=10.0 Present studyFr1=11.2 Present studyFr1=7.6 MURZYN&CHANSONFr1=8.3 MURZYN&CHANSONFr1=5.1 CHANSONFr1=8.6 CHANSONFr1=6.3 CHANSON&BRATTBERGFr1=8.5 CHANSON&BRATTBERG

(A) Maximum void fraction in the air-water shear layer

(x-x1)/d1

F max×d

1/V1

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2Fr1=5.1 Present studyFr1=7.5 Present studyFr1=9.2 Present studyFr1=10.0 Present studyFr1=11.2 Present studyFr1=7.6 MURZYN&CHANSONFr1=8.3 MURZYN&CHANSONFr1=5.1 CHANSONFr1=8.6 CHANSONFr1=6.3 CHANSON&BRATTBERGFr1=8.5 CHANSON&BRATTBERGEq. (5-5)

(B) Maximum bubble count rate in the air-water shear layer - Comparison with Equation (5-5)

Fig. 5-6 - Dimensionless longitudinal distributions of maximum void fraction and bubble count rate in the

air-water shear layer - Comparison between the present data set and the data of CHANSON and

BRATTBERG (2000), CHANSON (2007) and MURZYN and CHANSON (2009)

Figure 5-6B presents the longitudinal distribution of the maximum bubble count rate in the hydraulic jump.

The present data series are compared with earlier studies and the results showed an exponential decay in

maximum bubble count rate:

30

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−∝

×

1

1

1

1maxd

xxexp

VdF

(5-4)

It is worthwhile to highlight that, for the present data set, the maximum bubble count rate distributions

seemed to reach an asymptotic profile at the largest Froude numbers (Fr1 > 9):

1

1

asymptot1

1maxd

xx018.008.1

VdF −

×−=⎟⎟⎠

⎞⎜⎜⎝

⎛ × for Fr1 > 9 and (x-x1)/d1 < 60 (5-5)

Equation (5-5) is shown in Figure 5-6B. It is unknown whether the asymptotic trend (Eq. (5-5)) is linked

with a physical process or a limitation of the metrology.

Figure 5-7 presents the longitudinal distributions of depth-averaged void fraction Cmean in the hydraulic

jumps. Cmean is defined by Equation (5-1) and it characterises the amount of entrained air since Cmean =

Qair/(Q+Qair) where Qair is the rate of air entrainment. The present data showed consistently a large rate of air

entrainment in the jump as well as a rapid de-aeration of the flow with increasing distance from the jump toe

(Fig. 5-7A). For the present data set, the longitudinal decay in depth-averaged void fraction was proportional

to:

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−∝

1

1mean d

xxexpC (5-6)

For comparison, the experimental data of RAJARATNAM (1962) and MOSSA and TOLVE (1998) are

shown in Figure 5-7B. Note that RAJARATNAM (1962) and MOSSA and TOLVE (1998) calculated their

mean void fraction as an arithmetic mean rather than using Equation (5-1). The arithmetic mean does not

represent a true depth-averaged void fraction and it is surprising that some studies used the data set

(McCORQUODALE and KHALIFA 1983).

Several characteristic air-water flow parameters are regrouped in Figure 5-8, including the dimensionless

location YCmax/d1 where the void fraction is maximum, YFmax/d1 corresponding to the location where the

bubble count rate is maximum, the location y*/d1 corresponding to the boundary between the air-water shear

layer and the upper free-surface region and Y90/d1 corresponding to the location where C = 0.90. The

notation is explained in Figure 5-2 and the full data are reported in Appendix B. The data showed

systematically that:

1

90

1

*

1

maxC

1

maxFd

Ydy

dY

dY

<<< (5-7)

31

(x-x1)/d1

Cm

ean

0 10 20 30 40 50 60 70 80 900

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Fr1 = 5.1Fr1 = 7.5Fr1 = 9.2Fr1 = 10.0Fr1 = 11.2

(x-x1)/d1C

mea

n

0 10 20 30 40 50 60 70 80 900

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Fr1=4.92 RAJARATNAMFr1=6.35 RAJARATNAMFr1=7.12 RAJARATNAMFr1=7.70 RAJARATNAMFr1=8.05 RAJARATNAMFr1=8.12 RAJARATNAMFr1=6.42 MOSSA & TOLVE

(A, Left) Present study

(B, Right) Experimental data of RAJARATNAM (1962) and MOSSA and TOLVE (1998)

Fig. 5-7 - Dimensionless longitudinal distributions of depth-averaged void fraction in hydraulic jumps

(x-x1)/d1

YFm

ax/d

1, Y

Cm

ax/d

1, y *

/d1,

Y90

/d1

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 800

2

4

6

8

10

12

14

16

YFmax/d1 Fr1=5.1YFmax/d1 Fr1=7.5YFmax/d1 Fr1=9.2YFmax/d1 Fr1=10.0YFmax/d1 Fr1=11.2YCmax/d1 Fr1=5.1YCmax/d1 Fr1=7.5YCmax/d1 Fr1=9.2

YCmax/d1 Fr1=10.0YCmax/d1 Fr1=11.2y*/d1 Fr1=5.1y*/d1 Fr1=7.5y*/d1 Fr1=9.2y*/d1 Fr1=10.0y*/d1 Fr1=11.2Y90/d1 Fr1=5.1

Y90/d1 Fr1=7.5Y90/d1 Fr1=9.2Y90/d1 Fr1=10.0Y90/d1 Fr1=11.2YFmax/d1 CHANSON&BRATTBERGYCmax/d1 CHANSON&BRATTBERGYCmax/d1 MURZYN et aly*/d1 MURZYN et al

Fig. 5-8 - Longitudinal distributions of the dimensionless distances YFmax/d1, YCmax/d1, y*/d1 and Y90/d1 -

Comparison between the present experimental data and the data of CHANSON and BRATBERG (2000) and

32

MURZYN et al. (2005)

5.3 Bubble chord distributions

The bubble chord times were recorded for all investigated flow conditions. The bubble chord time is

proportional to the bubble chord length and inversely proportional to the velocity. In a complicated flow

such as a hydraulic jump, some flow reversal and recirculation exist, and the phase-detection intrusive probes

cannot discriminate accurately the direction nor magnitude of the velocity. Therefore only the bubble chord

time data are presented herein.

Figures 5-9 and 5-10 show some typical normalised bubble chord time distributions for two inflow Froude

numbers. Figure 5-9 presents some data in the air-water shear layer at the characteristic location YFmax where

the bubble count rate was maximum (F = Fmax). Figure 5-10 illustrates some data in the upper free-surface

region at the location of the secondary peak in bubble count rate (F= F2 and y = YF2). For each figure, the

legend provides the location (x-x1, y/d1) and the local air-water flow properties (C, F, V). The histogram

columns represent each the probability of droplet chord time in a 0.25 ms chord time interval. For example,

the probability of bubble chord time from 1 to 1.25 ms is represented by the column labelled 1 ms. Bubble

chord times larger than 10 ms are regrouped in the last column (> 10 ms).

The small bubble chord times corresponded to small bubbles passing rapidly in front the probe sensor, while

large chord times implied large air packet flowing slowly past the probe sensor. For intermediate chord

times, there were a wide range of possibilities in terms of bubble sizes depending upon the bubble velocity.

The experimental data showed systematically a number of features. First note the broad spectrum of bubble

chord times at each location. The range of bubble chord time extended over several orders of magnitude,

including at low void fractions, from less than 0.1 ms to more than 20 ms.

Second the distributions were skewed with a preponderance of small bubble chord time relative to the mean.

In Figure 5-9 corresponding to the air-water shear region, the probability of bubble chord time is the largest

for chord times between 0.5 and 1 ms. In Figure 5-10, the mode is about 0.5 to 2 ms and the result was

typical of the upper free-surface region. The probability distribution functions of bubble chord time tended to

follow in average a log-normal distribution, although a gamma distribution provided also a good fit. Note

that a similar finding was observed by CHANSON (2007).

Third, it is noted that the bubble chord time distributions had a similar shape at most vertical elevations y/d1

although the air-water structures may differ substantially. This is seen by comparing Figures 5-9 and 5-10.

Although the quantitative values differed, the overall shape of the bubble chord time was similar.

Figure 5-10 presents some typical distributions of bubble chord times in the free-surface region. The data

showed a large amount of bubble chord times larger than 10 ms. The results were consistent with the visual

observations indicating some large air bubbles and a foamy bubbly flow structure next to the free-surface

(Fig. 4-2). In Figure 5-10B, the bubble chord time data are also compared with the chord time distribution in

the upstream flow region at (x-x1)/d1 = -11.2. The comparison suggested some similarity, although there

were some basic differences: (a) the upstream flow was little aerated (Fig. 5-1), and (b) the bubble chord

times were smaller in the free-surface region of the upstream flow.

33

Bubble chord time (ms)

PDF

0 1.5 3 4.5 6 7.5 9 10.50

0.03

0.06

0.09

0.12

0.15

0.18

0.21

0.24

0.27

0.3

>10 ms

(x-x1)/d1=4.2, F=Fmax(x-x1)/d1=12.5, F=Fmax(x-x1)/d1=19.4, F=Fmax

(x-x1)/d1 y/d1 C F×d1/V1 V/V1

4.17 1.03 0.407 0.826 0.752 12.50 1.36 0.185 0.645 0.599 19.44 2.31 0.107 0.310 0.414

(A) Fr1 = 7.5, Re = 5.6 104, d1 = 0.018 m, x1 = 0.75 m

Bubble chord time (ms)

PDF

0 1.5 3 4.5 6 7.5 9 10.50

0.03

0.06

0.09

0.12

0.15

0.18

0.21

0.24

0.27

0.3

>10 ms

(x-x1)/d1=8.4, F=Fmax(x-x1)/d1=12.5, F=Fmax(x-x1)/d1=19.4, F=Fmax

(x-x1)/d1 y/d1 C F×d1/V1 V/V1

8.41 1.32 0.486 0.835 0.660 22.43 1.32 0.198 0.688 0.594 50.48 3.00 0.063 0.169 0.242

(B) Fr1 = 11.2, Re = 8.3 104, d1 = 0.01783 m, x1 = 0.75 m

Fig. 5-9 - Bubble chord time distributions in the air-water shear layer at the characteristic location where F =

Fmax

34

Bubble chord time (ms)

PDF

0 1.5 3 4.5 6 7.5 9 10.50

0.03

0.06

0.09

0.12

0.15

0.18

0.21

0.24

0.27

0.3

>10 ms(x-x1)/d1=4.2, F=F2(x-x1)/d1=12.5, F=F2(x-x1)/d1=19.4, F=F2

(x-x1)/d1 y/d1 C F×d1/V1 V/V1

4.17 3.11 0.519 0.248 -- 12.50 3.58 0.422 0.247 -- 19.44 6.42 0.380 0.146 --

(A) Fr1 = 7.5, Re = 5.6 104, d1 = 0.018 m, x1 = 0.75 m

Bubble chord time (ms)

PDF

0 1.5 3 4.5 6 7.5 9 10.50

0.03

0.06

0.09

0.12

0.15

0.18

0.21

0.24

0.27

0.3 >10 ms(x-x1)/d1=-11.2, F=F2(x-x1)/d1=+8.4, F=F2(x-x1)/d1=+12.5, F=F2

(x-x1)/d1 y/d1 C F×d1/V1 V/V1 Remark

-11.2 1.04 0.486 0.260 0.93 Upstream flow 8.41 4.68 0.721 0.112 -- Hydraulic jump roller

22.43 6.37 0.275 0.126 -- Hydraulic jump roller (B) Fr1 = 11.2, Re = 8.3 104, d1 = 0.01783 m, x1 = 0.75 m - Comparison with the upstream flow properties

Fig. 5-10 - Bubble chord time distributions in the upper free-surface region at the characteristic location

where F = F2

35

6. Experimental observations. (3) Velocity and turbulent properties

6.1 Presentation

Some air-water velocity measurements were conducted with the dual-tip conductivity probe based upon the

mean travel time between the probe sensors and the distance between probe sensors (Δx = 6.96 mm). All the

data are reported in Appendix A. Typical results are presented in Figure 6-1 for two Froude numbers (Fr1 =

7.5 & 10.0). The graphs present the dimensionless interfacial velocities V/V1 in the hydraulic jump roller,

where V1 is the inflow velocity. At the channel bed, a no-slip condition imposed V(y=0) = 0. All the velocity

profiles exhibited a similar shape despite some scatter. In the developing shear layer, the velocity

distributions followed some wall jet equations (RAJARATNAM 1965, CHANSON and BRATTBERG

2000). The dimensionless distributions of interfacial velocities were best fitted by :

N1

maxVmax Yy

VV

⎟⎟⎠

⎞⎜⎜⎝

⎛= for 1

yYy

maxV< (6-1)

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

2

5.0

maxV

max yYy

765.121exp

VV for 4to3

Yy1maxV

<< (6-2)

where Vmax is the maximum velocity measured at y = YVmax, y0.5 is the vertical elevation where V = Vmax/2

and N is a constant (N ≈ 6). The present results followed closely the wall jet velocity profile, despite some

data scatter caused by the unsteady and fluctuating nature of the flow (Fig. 6-2). This is illustrated in Figure

6-2 where the data are shown in a self-similar presentation and compared with Equation (6-2).

The maximum velocity data Vmax showed a longitudinal decay with increasing distance from the jump toe

(Fig. 6-3). They compared favourably with the observations of CHANSON and BRATTBERG (2000) and

KUCUKALI and CHANSON (2008) and MURZYN and CHANSON (2009) (Fig. 6-3). All the data

followed closely the empirical correlation:

⎟⎟⎠

⎞⎜⎜⎝

⎛ −×−=

1

1

1

maxd

xx028.0exp

VV

(6-3)

Equation (6-3) is compared with the experimental data in Figure 6-3.

In the recirculation region above the mixing zone, the present data indicated some negative time-averaged

velocity (Fig. 6-1 & 6-2). While the probe design was not well suited for some negative velocity

measurements because the signals were adversely affected by the probe support wake, the present findings

demonstrated that some recirculation could be qualitatively observed with the dual-tip probe. Figure 6-4

shows an example of typical auto- and cross-correlation functions in the recirculation region. The cross-

correlation function exhibited a clear peak with a negative time lag (τ = T = 0.00565 s).

36

V/V1, 0.1×(x-x1)/d1

y/d 1

-0.5 0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12Y90/d1(x-x1)/d1=4.2(x-x1)/d1=8.3(x-x1)/d1=12.5(x-x1)/d1=16.7(x-x1)/d1=19.4(x-x1)/d1=22.2

(A) Fr1 = 7.5, Re = 5.6 104, d1 = 0.018 m, x1 = 0.75 m

V/V1, 0.1×(x-x1)/d1

y/d 1

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

2

4

6

8

10

12

14

16Y90/d1(x-x1)/d1=4.2(x-x1)/d1=8.3(x-x1)/d1=12.5(x-x1)/d1=19.4(x-x1)/d1=25.0(x-x1)/d1=33.3(x-x1)/d1=41.7(x-x1)/d1=50.0(x-x1)/d1=55.6

(B) Fr1 = 10.0, Re = 7.5 104, d1 = 0.018 m, x1 = 0.75 m

Fig. 6-1 - Dimensionless velocity distributions in hydraulic jumps with partially-developed inflow conditions

37

V/Vmax

(y-Y

Vm

ax)/y

0.5

-0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 0.9 1.1-0.5

0

0.5

1

1.5

2

2.5

3

3.5Fr1=9.2Wall jet solution(x-x1)/d1=4.2(x-x1)/d1=8.3(x-x1)/d1=12.5(x-x1)/d1=19.4(x-x1)/d1=25.0(x-x1)/d1=33.3

V/Vmax

(y-Y

Vm

ax)/y

0.5

-0.5 -0.3 -0.1 0.1 0.3 0.5 0.7 0.9 1.1-0.5

0

0.5

1

1.5

2

2.5

3

3.5Fr1=10.0Wall jet solution(x-x1)/d1=4.2(x-x1)/d1=8.3(x-x1)/d1=12.5(x-x1)/d1=19.4(x-x1)/d1=25.0(x-x1)/d1=33.3

(A, Left) Fr1 = 9.2, Re = 6.9 104, d1 = 0.018 m, x1 = 0.75 m

(B, Right) Fr1 = 10.0, Re = 7.5 104, d1 = 0.018 m, x1 = 0.75 m

Fig. 6-2 - Dimensionless velocity distributions in hydraulic jumps - Comparison between experimental data

and Equation (6-2)

(x-x1)/d1

Vm

ax/V

1

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2Fr1=5.1 MURZYN&CHANSONFr1=7.6 MURZYN&CHANSONFr1=8.3 MURZYN&CHANSONFr1=6.3 CHANSON&BRATTBERGFr1=8.5 CHANSON&BRATTBERGFr1=6.9 KUCUKALI&CHANSON

Fr1=5.1 Present studyFr1=7.5 Present studyFr1=9.2 Present studyFr1=10.0 Present studyFr1=11.2 Present studyEquation (6-3)

Fig. 6-3 - Longitudinal distribution of dimensionless maximum velocity Vmax/V1 in hydraulic jumps -

Comparison between the present data, the data of CHANSON and BRATTBERG (2000), KUCUKALI and

CHANSON (2008), and MURZYN and CHANSON (2009), and Equation (6-3)

38

τ (s)

Rxx

, Rxy

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

-0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1RxxRxy

Fig. 6-3 - Auto- and cross-correlation functions in the recirculation region: Fr1 =11.2, Re = 8.3 104, d1 =

0.0178 m, x1 = 0.75 m, x-x1 = 0.35 m, y = 0.148 m, C = 0.86, F = 12.3 Hz

6.2 Turbulence intensity

The turbulence intensity Tu was derived from a cross-correlation analysis between the two probe sensor

signals (section 3). This approach was based on the relative width of the auto- and cross-correlation

functions (CHANSON and TOOMBES 2002) and it was restricted to the positive velocity data only. The

turbulence level Tu characterised the fluctuations of the interfacial air-water velocity. Figure 6-4 presents

some typical vertical distributions of turbulence intensity. The results showed some very high levels of

turbulence up to %400 . The turbulence levels increased with increasing distance from the bed y/d1 and with

increasing Froude number. The present results were consistent with those obtained by KUCUKALI and

CHANSON (2008) and MURZYN and CHANSON (2009), while they covered a wider range of flow

conditions, especially for large Froude numbers.

39

Tu

y/d 1

0 0.6 1.2 1.8 2.4 3 3.6 4.2 4.8 5.4 60

0.6

1.2

1.8

2.4

3

3.6

4.2

4.8

5.4

6Fr1=10.0

(x-x1)/d1=8.3(x-x1)/d1=19.4(x-x1)/d1=33.3(x-x1)/d1=50

Tuy/

d 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

1

2

3

4

5

6

7

8

9

10 Fr1=11.2(x-x1)/d1=8.4(x-x1)/d1=16.8(x-x1)/d1=22.4(x-x1)/d1=28(x-x1)/d1=44.9(x-x1)/d1=56.1

(A, Left) Fr1 = 10.0, Re = 7.5 104, d1 = 0.018 m, x1 = 0.75 m

(B, Right) Fr1 =11.2, Re = 8.3 104, d1 = 0.0178 m, x1 = 0.75 m

Fig. 6-4 - Dimensionless distributions of turbulence intensity Tu in hydraulic jumps

6.3 Correlation time scales of turbulence

The data processing of the signal output may provide some information on turbulence structure and its

properties. The analysis of auto- and cross-correlation functions was undertaken on the leading and trailing

tip output signals. Using the technique developed by CHANSON (2007) and CHANSON and CAROSI

(2007), the correlation coefficient functions were calculated to estimate the correlation integral time scales

Txx and Txz (section 3). Herein the focus was on the auto-correlation time scales of turbulence Txx. Note that

the data were restricted to locations where the longitudinal velocity was positive.

The integral time scale Txx characterised the longitudinal coherence of the two-phase flow. It gave an

estimate of the typical longitudinal connection in the air-water flow structure, characterising the large-scale

air-water coherent structures. The auto-correlation function data showed a well-known and well-defined

shape (Fig. 6-3).

Figure 6-5 shows some distribution of dimensionless auto-correlation time scale Txx×V1/d1 for different

distances downstream of the toe. The results showed that the integral time scale increased with increasing

vertical elevation above the bed suggesting that the largest structures developed in the turbulent shear layer.

Towards the downstream end of the roller, the vertical distributions of dimensionless auto-correlation time

scale became more uniform and roughly constant over the whole water column: e.g., Txx×V1/d1 ≈ 0.4 in

Figure 6-4B at (x-x1)/d1 = 22.2. The smallest integral time scales were measured close to the channel bed,

and it is believed that the channel bed prevented the development of large-scale structures. The data were in

agreement with the earlier studies of CHANSON (2007) and MURZYN and CHANSON (2009).

40

Txx×V1/d1

y/d 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.3

0.6

0.9

1.2

1.5

1.8

2.1

2.4

2.7

3

3.3Fr1=7.5

(x-x1)/d1=4.2(x-x1)/d1=8.3(x-x1)/d1=16.7(x-x1)/d1=22.2

Txx×V1/d1y/

d 1

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.250

1

2

3

4

5

6

7

8

9Fr1=9.2

(x-x1)/d1=8.3(x-x1)/d1=19.4(x-x1)/d1=33.3(x-x1)/d1=50

(A, Left) Fr1 = 7.5, Re = 5.6 104, d1 = 0.018 m, x1 = 0.75 m

(B, Right) Fr1 = 9.2, Re = 6.9 104, d1 = 0.018 m, x1 = 0.75 m

Fig. 6-5 - Dimensionless distributions of auto-correlation integral time scales Txx×V1/d1 in hydraulic jumps

41

7. Interactions between air bubbles and turbulence: bubble clustering

7.1 Presentation

In air-water flows, the void fraction and bubble count rate are some gross parameters that cannot describe the

air-water structures nor the interactions between entrained bubbles and turbulent shear. The present

experimental results demonstrated a broad spectrum of bubble chords (section 4). The range of bubble chord

times extended over several orders of magnitude and the distributions of chord times were skewed with a

preponderance of small bubbles relative to the mean. Some signal processing may provide further

information on the longitudinal structure of the air-water flow including bubble clustering. A concentration

of bubbles within some relatively short intervals of time may indicate some clustering while it may be

instead the consequence of a random occurrence. The study of particle clustering events is relevant to infer

whether the formation frequency responds to some particular frequencies of the flow. In turbulent shear

flows, the trapping of bubbles in large-scale vortical structures is a dominant cluster mechanism in the

bubbly region (CHANSON 2007). The clustering index may provide a measure of the vorticity production

rate, of the level of bubble-turbulence interactions and of the associated energy dissipation.

7.2 Experimental results

When two bubbles are closer than a particular time/length scale, they can be considered a group of bubbles:

i.e., a cluster. The characteristic water time/length scale may be related to the water chord statistics or to the

near-wake of the preceding particle (CHANSON and TOOMBES 2002, CHANSON and CAROSI 2007).

Herein the latter approach was applied following CHANSON et al. (2002,2006). Two bubbles were

considered parts of a cluster when the water chord time between the bubbles was less than the bubble chord

time of the lead particle. That is, when a bubble trailed the previous bubble by a short time/length, it was in

the near-wake of and could be influenced by the leading particle.

Figure 7-1 presents some typical characteristics of the bubble clusters in the developing shear layer. All the

data were recorded at the characteristic location y = YFmax where the bubble count rate was maximum (F =

Fmax). Further experimental data are reported in Appendix C. Figure 7-1 includes the longitudinal

distributions of number of clusters per second, the percentage of bubbles in clusters, the average number of

bubbles per cluster, and the probability distribution function of the number of bubbles per cluster for Fr1 =

10.

The experimental results showed systematically a number of trends. The number of clusters per second was

substantial in the air-water shear layer, reaching up to 50 clusters per second for Fr1 = 10 and 11. Further the

number of clusters decreased rapidly with increasing longitudinal distance (Fig. 7-1A). The present data

showed an exponential decay in the number of clusters:

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−∝

×

1

1

1

1cd

xxexpV

dN (7-1)

where Nc is the number of clusters per second.

42

(x-x1)/d1

Nc×

d 1/V

1

0 5 10 15 20 25 30 35 40 45 50 550

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

0.225

0.25Fr1=11.2Fr1=10.0Fr1=9.2Fr1=7.5

(x-x1)/d1%

bub

bles

in c

lust

ers

0 5 10 15 20 25 30 35 40 45 50 550

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Fr1=11.2Fr1=10.0Fr1=9.2Fr1=7.5

(A, Left) Dimensionless number of cluster per second Nc×d1/V1

(B, Right) Percentage of bubbles in clusters

(x-x1)/d1

Nb

bubb

les p

er c

lust

er

0 5 10 15 20 25 30 35 40 45 50 552

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3Fr1=11.2Fr1=10.0Fr1=9.2Fr1=7.5

Number of bubbles per cluster

PDF

2 3 4 5 6 7 8 9 10 11 120

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8(x-x1)/d1=4.2(x-x1)/d1=19.4(x-x1)/d1=33.3

(C, Left) Number of bubbles per cluster

(D, Right) Probability distribution functions of the number of bubbles per cluster for Fr1 = 10.0

Fig. 7-1 - Characteristics of bubble clusters in the air-water shear layer at the locations where F = Fmax (y =

YFmax)

43

The experimental results highlighted that a significant proportion of bubbles were parts of a cluster structure

in the air-water shear zone. That is, more than one third of all bubbles in the beginning of the shear layer ((x-

x1)/d1 < 10) for 7.5 < Fr1 < 11.2. The percentage of bubbles in clusters decreased with increasing longitudinal

distance as seen in Figure 7-1B. The present findings differed from the results of CHANSON (2007) who

found only a small proportion of bubbles in clusters. While a different cluster criterion was used by

CHANSON (2007), it is believed that the key difference was the larger range of inflow Froude numbers

tested in the present study. The present results showed indeed that the proportion of bubbles forming some

clusters was the largest at the largest Froude numbers (9.2 < Fr1 < 11.2) (Fig. 7-1B).

In average, the number of bubbles per cluster ranged from 2.7 down to 2.2 and decreased with increasing

distance from the jump toe (Fig. 7-1C & 7-D). The longitudinal pattern is illustrated in Figure 7-1D showing

the probability distribution function of the number of bubbles per clusters at three longitudinal locations for

one experiment (Fr1 = 10). It is however important to stress that the present study focused on the longitudinal

flow structure and it did not account for bubble travelling side-by side.

7.3 Discussion

A comparative analysis was conducted on the bubble chord times, between all the bubbles and the bubbles in

clusters. A typical comparison is presented in Figure 7-2. The results showed that the distribution of bubble

chord times were comparable and nearly identical for both the whole bubble sample and the bubbles in

cluster. Simply there was no preferential bubble chord in the clusters.

The findings contradict the earlier study of CHANSON (2007) based upon an inter-particle arrival time

analysis. It is believed that a major issue was the assumptions underling the inter-particle arrival time

analysis (EDWARDS and MARX 1995, HEINLEIN and FRITCHING 2006). The method considers an ideal

dispersed flow driven by a superposition of Poisson processes assuming non-interacting particles. The latter

assumption (non-interacting particles) is incorrect in the developing shear layer of a hydraulic jump where

the air bubbles are subjected to a wide range of interactions including bubble trapping in the large-scale

vortices, bubble breakup by turbulent shear, and bubble collisions and coalescence.

44

Bubble chord time (ms)

PDF

0 1.5 3 4.5 6 7.5 9 10.50

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

0.225

0.25

>10 ms

(x-x1)/d1=8.4All bubblesBubbles in clusters

(A) (x-x1)/d1 = 8.4, F = Fmax = 219 Hz, Nc = 46.9 Hz

Bubble chord time (ms)

PDF

0 1.5 3 4.5 6 7.5 9 10.50

0.025

0.05

0.075

0.1

0.125

0.15

0.175

0.2

0.225

0.25

>10 ms

(x-x1)/d1=22.4All bubblesBubbles in clusters

(B) (x-x1)/d1 = 22.4, F = Fmax = 181 Hz, Nc = 30.7 Hz

Fig. 7-2 - Probability distribution functions of bubble chord time in the air-water shear layer - Comparison

between all the bubbles and the bubbles forming parts of a cluster structure - Fr1 =11.2, Re = 8.3 104, d1 =

0.0178 m, x1 = 0.75 m

45

8. Conclusion

An experimental study was performed in some hydraulic jumps with partially-developed inflow and some

detailed air-water flow characteristics were measured in the developing shear layer. The measurements were

conducted in a relatively large-size facility with large Froude numbers (5.1 < Fr1 < 11.2) and Reynolds

numbers (4.0 104 < Re < 8.3 104). The two-phase flow measurements were made with a dual-tip phase

detection probe sampled at 20 kHz for 45 s at each sampling location.

In the developing shear layer, the distributions of void fractions may be modelled by an advective diffusion

equation. The analytical solution of CHANSON (1995) was refined and the experimental data demonstrated

a close agreement with the theoretical developments. The air bubble diffusivity was observed to be

independent of the Froude and Reynolds numbers. However it increased linearly with the distance from the

jump toe in a manner somehow similar to the momentum exchange coefficient in a developing shear layer.

The experimental observations highlighted a lot of air entrainment in the jump roller as well as some spray

and splashing above the roller. The observations of jump toe fluctuations were close to earlier studies, and

the new data showed that the jump toe oscillation frequency was equal to the production rate of large-scale

vortical structures in the developing shear layer. Some video observations highlighted that the average

advection speed of these large coherent structures was in average Vej/V1 ≈ 0.32 in the developing shear layer.

The basic air-water flow properties presented the same trends as earlier studies performed with lower Froude

numbers. The void fraction distributions presented a local maximum in the air-water shear layer and its value

decreased quasi-exponentially with increasing distance from the jump toe. The air-water mixing layer was

characterised by a maximum in bubble count rate. The depth-averaged void fraction data demonstrated a

large amount of entrained air as well as a rapid de-aeration of the jump roller. The velocity profiles followed

closely some wall jet equations, and the air-water turbulent properties highlighted some increasing

turbulence with increasing distance from the bed.

The bubble chord time distributions showed a broad range of entrained bubble chord times spreading over

two orders of magnitudes. A detailed analysis of the longitudinal structure of the air and water chords

suggested a significant proportion of bubble clustering in the developing shear region, especially close to the

jump toe. In average the number of bubbles per clusters ranged from about 2.7 down to 2.2 with increasing

distance from the jump toe. The data showed further that, in the shear layer, there was no preferential bubble

chord time in the cluster structures.

Overall the study highlighted some seminal features of the air-water shear layer in hydraulic jumps with

large Froude numbers (5.1 < Fr1 < 11.2). The advection of air in the mixing zone was an advective diffusion

process, although there was some rapid flow de-aeration for all Froude numbers.

46

9. Acknowledgements

The author thanks his students Ben HOPKINS and Hugh CASSIDY who conducted carefully and

thoroughly the experiments as part of their CIVL4580 Civil Engineering Research Thesis project. They did

all the experimental measurements and most signal processing: their contribution was indispensable. The

writer acknowledges also the technical assistance of Graham ILLIDGE and Clive BOOTH (The University

of Queensland). The author thanks further Dr Frédéric MURZYN (ESTACA Laval, France) for the detailed

review of the report and his most valuable comments.

47

Appendix A - Air-water flow measurements

A.1 Presentation

New experiments were performed in the Gordon McKAY Hydraulics Laboratory at the University of

Queensland. The channel was horizontal, 3.2 long and 0.5 m wide. The sidewalls were made of 3.2 m long,

0.45 m high glass panels and the bed was made of 12 mm thick PVC sheets. The inflow was controlled by an

upstream undershoot gate (Fig. A-1). The downstream flow conditions were controlled by a vertical

overshoot gate. The flume was used previously by CHANSON (2007) and MURZYN and CHANSON

(2009).

The water discharge was measured with a Venturi meter located in the supply line and it was calibrated on-

site with a large V-notch weir. The discharge measurement was accurate within ±2%. The clear-water flow

depths were measured using rail mounted point gauges with a 0.2 mm accuracy.

The air-water flow properties were measured with a double-tip conductivity probe. The probe sensor size

was 0.25 mm and the longitudinal separation distance between sensors was Δx = 6.96 mm. The probe was

manufactured at the University of Queensland and it was excited by an electronic system (Ref. UQ82.518)

designed with a response time of less than 10 μs. The probe and electronics were previously used by

CHANSON and CAROSI (2007), KUCUKALI and CHANSON (2008) and MURZYN and CHANSON

(2009). During the present experiments, each probe sensor was sampled at 20 kHz for 45 s. The probe

displacement in the vertical direction was controlled by a fine adjustment system connected to a Mitutoyo™

digimatic scale unit with a vertical accuracy Δy of less than 0.1 mm. Table A-1 summarises the experimental

flow conditions.

The flow conditions corresponded to a partially-developed flow at the jump toe. That is, the ratio of bottom

boundary layer thickness to inflow depth δ/d1 was less than unity.

Notation

C void fraction defined as the volume of air per unit volume of air and water; d water depth (m); d1 flow depth (m) measured immediately upstream of the hydraulic jump; Fr1 upstream Froude number: 111 dg/VFr ×= ;

g gravity acceleration (m/s2) : g = 9.80 m/s2 in Brisbane (Australia); Q water discharge (m3/s); Re Reynolds number : μ××ρ= /dVRe 11 ;

V air-water interfacial velocity (m/s); V1 upstream flow velocity (m/s): V1 = Q/(W×d1); W channel width (m); x longitudinal distance from the sluice gate (m); x1 longitudinal distance from the gate to the jump toe (m); Y90 characteristic distance (m) from the bed where C = 0.90;

48

y distance (m) measured normal to the flow direction; μ Dynamic viscosity (Pa.s) of water; ρ density (kg/m3) of water;

Subscript 1 upstream flow conditions; 2 downstream flow conditions.

Fig. A-1 - Definition sketch of the hydraulic jump experiments

Table A-1 - Experimental flow conditions for air-water flow measurements in hydraulic jumps with

partially-developed inflow conditions

Ref. Q W x1 V1 d1 Fr1 Re Remarks m3/s m m m/s m

(1) (2) (3) (4) (5) (6) (7) (8) (9) 090331 0.02025 0.5 0.75 2.19 0.0185 5.14 4.0E+4 Air-water flow measurements. 090317 0.02825 3.14 0.018 7.47 5.6E+4 Upstream gate opening: 090720 0.03481 3.87 0.018 9.21 6.9E+4 h =0.018 m. 090713 0.03780 4.20 0.018 10.0 7.5E+4 090414 0.04175 4.68 0.01783 11.2 8.3E+4

Notes: d1: upstream flow depth; Fr1: upstream Froude number; Q: water discharge; Re: Reynolds number;

V1: upstream flow velocity; W: channel width; x1: distance between the upstream gate and jump toe.

49

A.2 Void fraction and bubble count rate measurements

C, 0.1×(x-x1)/d1

y/d 1

-0.5 0 0.5 1 1.5 2 2.5 30

1

2

3

4

5

6

7

8Y90/d1x-x1/d1=4.1x-x1/d1=5.9x-x1/d1=8.1x-x1/d1=10.3x-x1/d1=12.2x-x1/d1=16.2x-x1/d1=18.9

Fig. A-2 - Void fraction distributions - Fr1 = 5.1, Re = 4.0 104, d1 = 0.0185 m, x1 = 0.75 m

50

F×d1/V1, 0.1×(x-x1)/d1

y/d 1

-0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 30

0.8

1.6

2.4

3.2

4

4.8

5.6

6.4

7.2

8Y90/d1x-x1/d1=4.1x-x1/d1=5.9x-x1/d1=8.1x-x1/d1=10.3x-x1/d1=12.2x-x1/d1=16.2x-x1/d1=18.9

Fig. A-3 - Dimensionless bubble count rate distributions - Fr1 = 5.1, Re = 4.0 104, d1 = 0.0185 m, x1 = 0.75 m

51

C, 0.1×(x-x1)/d1

y/d 1

-0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.50

1

2

3

4

5

6

7

8

9

10

11Y90/d1x-x1/d1=4.2x-x1/d1=8.3x-x1/d1=12.5x-x1/d1=16.7x-x1/d1=19.4x-x1/d1=22.2

Fig. A-4 - Void fraction distributions - Fr1 = 7.5, Re = 5.6 104, d1 = 0.018 m, x1 = 0.75 m

52

F×d1/V1, 0.1×(x-x1)/d1

y/d 1

-0.5 -0.25 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.50

1

2

3

4

5

6

7

8

9

10

11Y90/d1x-x1/d1=4.2x-x1/d1=8.3x-x1/d1=12.5x-x1/d1=16.7x-x1/d1=19.4x-x1/d1=22.2

Fig. A-5 - Dimensionless bubble count rate distributions - Fr1 = 7.5, Re = 5.6 104, d1 = 0.018 m, x1 = 0.75 m

53

C, 0.1.(x-x1)/d1

y/d 1

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

1

2

3

4

5

6

7

8

9

10

11

12

13

14Y90/d1x-x1/d1=4.2x-x1/d1=8.3x-x1/d1=12.5x-x1/d1=19.4x-x1/d1=25x-x1/d1=33x-x1/d1=42x-x1/d1=50x-x1/d1=56

Fig. A-6 - Void fraction distributions - Fr1 = 9.2, Re = 6.9 104, d1 = 0.018 m, x1 = 0.75 m

54

F.d1/V1, 0.1.(x-x1)/d1

y/d 1

-1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

1

2

3

4

5

6

7

8

9

10

11

12

13

14Y90/d1x-x1/d1=4.2x-x1/d1=8.3x-x1/d1=12.5x-x1/d1=19.4x-x1/d1=25x-x1/d1=33x-x1/d1=42x-x1/d1=50x-x1/d1=56

Fig. A-7 - Dimensionless bubble count rate distributions - Fr1 = 9.2, Re = 6.9 104, d1 = 0.018 m, x1 = 0.75 m

55

C, 0.1×(x-x1)/d1

y/d 1

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.50

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15Y90/d1x-x1/d1=4.2x-x1/d1=8.3x-x1/d1=12.5x-x1/d1=19.4x-x1/d1=25x-x1/d1=33.3x-x1/d1=41.7x-x1/d1=50x-x1/d1=55.6

Fig. A-8 - Void fraction distributions - Fr1 = 10.0, Re = 7.5 104, d1 = 0.018 m, x1 = 0.75 m

56

F.d1/V1, 0.1×(x-x1)/d1

y/d 1

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.50

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15Y90/d1x-x1/d1=4.2x-x1/d1=8.3x-x1/d1=12.5x-x1/d1=19.4x-x1/d1=25x-x1/d1=33.3x-x1/d1=41.7x-x1/d1=50x-x1/d1=55.6

Fig. A-9 - Dimensionless bubble count rate distributions - Fr1 = 10.0, Re = 7.5 104, d1 = 0.018 m, x1 = 0.75 m

57

C, 0.1.(x-x1)/d1

y/d 1

-1 0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16Y90/d1x-x1/d1=4.2x-x1/d1=8.4x-x1/d1=12.6x-x1/d1=16.8x-x1/d1=19.6x-x1/d1=22.4x-x1/d1=25.2

x-x1/d1=28.0x-x1/d1=36.5x-x1/d1=44.9x-x1/d1=50.5x-x1/d1=56.1x-x1/d1=61.7x-x1/d1=67.3x-x1/d1=78.5

Fig. A-10 - Void fraction distributions - Fr1 = 11.2, Re = 8.3 104, d1 = 0.01783 m, x1 = 0.75 m

58

F.d1/V1, 0.1×(x-x1)/d1

y/d 1

-1 0 1 2 3 4 5 6 7 8 90

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16Y90/d1x-x1/d1=4.2x-x1/d1=8.4x-x1/d1=12.6x-x1/d1=16.8x-x1/d1=19.6x-x1/d1=22.4x-x1/d1=25.2

x-x1/d1=28.0x-x1/d1=36.5x-x1/d1=44.9x-x1/d1=50.5x-x1/d1=56.1x-x1/d1=61.7x-x1/d1=67.3x-x1/d1=78.5

Fig. A-11 - Dimensionless bubble count rate distributions - Fr1 = 11.2, Re = 8.3 104, d1 = 0.01783 m, x1 = 0.75 m

59

A.3 Velocity measurements

V/V1, 0.1×(x-x1)/d1

y/d 1

-0.5 0 0.5 1 1.5 2 2.5 30

2

4

6

8

10

12Fr1=7.5

Y90/d1(x-x1)/d1=4.2(x-x1)/d1=8.3(x-x1)/d1=12.5(x-x1)/d1=16.7(x-x1)/d1=19.4(x-x1)/d1=22.2

Fig. A-12 - Dimensionless velocity distributions - Fr1 = 7.5, Re = 5.6 104, d1 = 0.018 m, x1 = 0.75 m

60

V/V1

y/d 1

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.50

1

2

3

4

5

6

7

8

9

10

11

12

13Fr1=9.2

Y90/d1(x-x1)/d1=4.2(x-x1)/d1=8.3(x-x1)/d1=12.5(x-x1)/d1=19.4(x-x1)/d1=25.0(x-x1)/d1=33.3(x-x1)/d1=41.7(x-x1)/d1=50.0(x-x1)/d1=55.6

Fig. A-13 - Dimensionless velocity distributions - Fr1 = 9.2, Re = 6.9 104, d1 = 0.018 m, x1 = 0.75 m

61

V/V1, 0.1×(x-x1)/d1

y/d 1

-0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

2

4

6

8

10

12

14

16Fr1=10.0

Y90/d1(x-x1)/d1=4.2(x-x1)/d1=8.3(x-x1)/d1=12.5(x-x1)/d1=19.4(x-x1)/d1=25.0(x-x1)/d1=33.3(x-x1)/d1=41.7(x-x1)/d1=50.0(x-x1)/d1=55.6

Fig. A-14 - Dimensionless velocity distributions - Fr1 = 10.0, Re = 7.5 104, d1 = 0.018 m, x1 = 0.75 m

62

V/V1, 0.1×(x-x1)/d1

y/d 1

-1 0 1 2 3 4 5 6 7 8 90

2

4

6

8

10

12

14

16Fr1=11.2

Y90/d1(x-x1)/d1=4.2(x-x1)/d1=8.4(x-x1)/d1=12.6(x-x1)/d1=16.8(x-x1)/d1=19.6(x-x1)/d1=22.4(x-x1)/d1=25.2

(x-x1)/d1=28.0(x-x1)/d1=36.5(x-x1)/d1=44.9(x-x1)/d1=50.5(x-x1)/d1=56.1(x-x1)/d1=61.7(x-x1)/d1=67.3(x-x1)/d1=78.5

Fig. A-15 - Dimensionless velocity distributions - Fr1 = 11.2, Re = 8.3 104, d1 = 0.01783 m, x1 = 0.75 m

63

Appendix B - Experimental summary

Notation

C void fraction defined as the volume of air per unit volume of air and water; Cmax local maximum in void fraction in the developing shear layer; Cmean depth averaged void fraction:

∫ ×=90Y

0mean dyCC

C* local minimum in void fraction at the boundary between the air-water shear layer and the upper free-surface region;

Dt air bubble diffusivity (m/s2) in the air-water shear layer; D# dimensionless air bubble diffusivity: D# = Dt/(V1×d1); d equivalent clear-water flow depth (m):

∫ ×−=90Y

0

dy)C1(d

d1 flow depth (m) measured immediately upstream of the hydraulic jump; F bubble count rate (Hz) defined as the number of bubbles impacting the probe sensor per second; Fmax maximum bubble count rate (Hz) in the air-water shear layer; F2 secondary peak in bubble count rate (Hz) typically located in the upper free-surface region; Fr1 upstream Froude number: 111 dg/VFr ×= ;

g gravity acceleration (m/s2) : g = 9.80 m/s2 in Brisbane (Australia); Q water discharge (m3/s); Re Reynolds number: μ××ρ= /dVRe 11 ;

V air-water velocity (m/s); Vmax maximum air-water velocity (m/s) in the shear layer; V1 upstream flow velocity (m/s): V1 = Q/(W×d1); W channel width (m); x longitudinal distance from the sluice gate (m); x1 longitudinal distance from the gate to the jump toe (m); YCmax vertical elevation (m) where the void fraction in the shear layer is maximum (C = Cmax); YFmax distance (m) from the bed where the bubble count rate is maximum (F = Fmax); YF2 distance (m) from the bed where F = F2; YVmax distance (m) from the bed where V = Vmax; Y90 characteristic distance (m) from the bed where C = 0.90; y distance (m) measured normal to the flow direction; y* distance (m) from the bed of the boundary between the air-water shear layer and the upper free-

surface region where C = C*; y0.5 distance (m) from the bed where V = Vmax/2;

64

μ dynamic viscosity (Pa.s) of water; ρ density (kg/m3) of water;

Subscript * boundary between the upper free-surface region and the air-water shear layer; 1 upstream flow conditions; 2 downstream flow conditions 90 location where C = 0.90.

Fig. B-1 - Definition sketch of the main notations in the air-water shear layer of hydraulic jumps with

partially-developed inflow conditions

65

B.2 Experimental summary

Ref. Q W x1 V1 d1 Fr1 Re x-x1 Cmax C* m3/s m m m/s m m

(1) (2) (3) (4) (5) (6) (7) (8) (10) (11) (12) 090331 0.02025 0.5 0.75 2.19 0.0185 5.14 4.0E+4 0.075 0.217 0.202

0.11 0.179 0.135 0.15 0.174 0.138 0.19 0.142 0.047 0.225 0.063 0.035 0.30 0.054 0.032 0.35 0.045 0.029

090317 0.02825 0.5 0.75 3.14 0.018 7.47 5.6E+4 0.075 0.684 0.519 0.15 0.348 0.280 0.225 0.271 0.209 0.30 0.186 0.106 0.35 0.116 0.048 0.40 0.120 0.058

090720 0.03481 0.5 0.75 3.87 0.018 9.21 6.9E+4 0.075 0.427 0.338 0.15 0.330 0.214 0.225 0.240 0.144 0.35 0.202 0.109 0.45 0.163 0.092 0.60 0.165 0.062 0.75 0.066 0.039 0.90 0.036 0.021 1.00 0.023 0.027

090713 0.03780 0.5 0.75 4.20 0.018 10.0 7.5E+4 0.075 0.503 0.327 0.15 0.413 0.307 0.225 0.340 0.144 0.35 0.217 0.106 0.45 0.169 0.074 0.60 0.130 0.053 0.75 0.079 0.029 0.90 0.048 0.030 1.05 0.031 0.028

090414 0.04175 0.5 0.75 4.68 0.01783 11.2 8.3E+4 -0.20 -- -- 0.075 0.654 0.587 0.15 0.486 0.340 0.225 0.382 0.212 0.30 0.305 0.167 0.35 0.242 0.141 0.40 0.254 0.137 0.45 0.172 0.084 0.50 0.183 0.088 0.65 0.132 0.058 0.80 0.092 0.029 0.90 0.066 0.021 1.00 0.055 0.027 1.10 0.043 0.023 1.20 0.030 0.024 1.40 0.031 0.027

66

Ref. Fr1 x-x1 YCmax Y90 y* Fmax F2 YFmax YF2 m m m m Hz Hz m m

(1) (7) (10) (13) (14) (15) (16) (17) (18) (19) 090331 5.14 0.075 0.0295 0.0750 0.0355 61.6 28.4 0.0195 0.0405

0.11 0.0355 0.0735 0.0435 51.8 29.8 0.0275 0.0555 0.15 0.0355 0.0910 0.0495 45.9 21.2 0.0275 0.0525 0.19 0.0385 0.1135 0.0625 37.6 19.4 0.0335 0.0935 0.225 0.0485 0.1135 0.0645 23.0 18.0 0.0285 0.0895 0.30 0.0435 0.1200 0.0785 18.1 14.8 0.0435 0.1085 0.35 0.0535 0.1280 0.0785 12.8 8.0 0.0535 0.0935

090317 7.47 0.075 0.0385 0.0785 0.0560 144.1 43.2 0.0185 0.0560 0.15 0.0335 0.0750 0.0385 128.5 48.1 0.0235 0.0485 0.225 0.0355 0.0970 0.0525 112.4 43.1 0.0245 0.0645 0.30 0.0485 0.1230 0.0735 75.1 31.5 0.0355 0.0985 0.35 0.0535 0.1460 0.0935 54.1 25.4 0.0415 0.1155 0.40 0.0615 0.1490 0.0975 59.6 24.6 0.0435 0.1285

090720 9.21 0.075 0.0275 0.0680 0.0355 210.2 38.1 0.0185 0.0435 0.15 0.0275 0.0935 0.0395 203.4 39.4 0.0210 0.0675 0.225 0.0345 0.1180 0.0535 169.9 40.0 0.0225 0.0800 0.35 0.0435 0.1370 0.0815 136.7 41.2 0.0295 0.1035 0.45 0.0535 0.1640 0.0955 103.3 37.5 0.0435 0.1255 0.60 0.0605 0.1820 0.1135 92.0 150.0 0.0435 0.0300 0.75 0.0835 0.2100 0.1585 32.0 15.9 0.0500 0.1835 0.90 0.1000 0.2301 0.1735 44.3 15.5 0.0535 0.2135 1.00 0.1230 0.2250 0.1635 6.8 9.1 0.1135 0.1935

090713 10.0 0.075 0.0210 0.0835 0.0395 245.5 36.8 0.0185 0.0555 0.15 0.0235 0.0915 0.0435 222.9 43.5 0.0210 0.0555 0.225 0.0265 0.1210 0.0535 232.4 37.9 0.0225 0.0825 0.35 0.0395 0.1510 0.0755 183.7 38.7 0.0215 0.0995 0.45 0.0515 0.1700 0.0955 126.3 34.4 0.0295 0.1235 0.60 0.0635 0.2050 0.1135 82.2 26.0 0.0455 0.1600 0.75 0.0635 0.2230 0.1735 47.8 14.2 0.0455 0.1885 0.90 0.0875 0.2485 0.1885 22.6 15.2 0.0795 0.2185 1.05 0.1035 0.2460 0.1835 13.0 12.0 0.0975 0.2135

090414 11.2 -0.20 -- 0.0200 -- 68.4 -- 0.0185 -- 0.075 0.0295 0.0645 0.0405 173.6 22.3 0.0185 0.0735 0.15 0.0235 0.0900 0.0415 219.2 29.3 0.0235 0.0835 0.225 0.0285 0.1135 0.0575 211.9 42.3 0.0235 0.0765 0.30 0.0385 0.1430 0.0685 208.7 42.3 0.0235 0.1035 0.35 0.0435 0.1520 0.0685 178.4 37.2 0.0260 0.1085 0.40 0.0435 0.1635 0.0835 180.8 33.0 0.0235 0.1135 0.45 0.0475 0.1900 0.1135 147.8 29.0 0.0335 0.1485 0.50 0.0555 0.1780 0.1035 131.2 31.4 0.0335 0.1395 0.65 0.0735 0.2150 0.1335 92.9 24.2 0.0435 0.1735 0.80 0.0835 0.2350 0.1535 54.7 20.1 0.0535 0.1935 0.90 0.0885 0.2550 0.1735 44.3 15.5 0.0535 0.2135 1.00 0.1035 0.2700 0.1975 24.7 16.1 0.1035 0.2385 1.10 0.1235 0.2700 0.1835 21.8 14.0 0.1235 0.2435 1.20 0.1585 0.2700 0.1835 10.0 9.4 0.1585 0.1935 1.40 0.2035 0.2700 0.2085 8.4 22.1 0.2035 0.2585

67

Ref. Fr1 x-x1 Vmax YVmax y0.5 d Cmean D# m m/s m m m

(1) (7) (10) (20) (21) (22) (23) (24) (25) 090331 5.14 0.075 1.62 0.0085 0.0386 0.0496 0.339 3.0E-2

0.11 1.47 0.0035 -- 0.0556 0.244 3.5E-2 0.15 1.29 0.0135 -- 0.0664 0.271 2.5E-2 0.19 1.23 0.0115 -- 0.0887 0.219 2.7E-2 0.225 0.97 0.0285 -- 0.0933 0.178 5.0E-2 0.30 0.00 -- -- 0.1032 0.140 3.2E-2 0.35 0.00 -- -- 0.1122 0.123 3.5E-2

090317 7.47 0.075 2.40 0.0165 0.0369 0.0398 0.493 3.0E-2 0.15 2.49 0.0335 0.0390 0.0458 0.390 2.5E-2 0.225 2.10 0.0135 0.0459 0.0620 0.361 2.0E-2 0.30 1.70 0.0235 0.0580 0.0933 0.241 3.0E-2 0.35 1.45 0.0285 0.0589 0.1148 0.214 2.9E-2 0.40 1.30 0.0135 0.1189 0.202 4.0E-2

090720 9.21 0.075 2.96 0.0160 0.0299 0.0389 0.427 2.5E-2 0.15 2.58 0.0160 0.0430 0.0573 0.388 2.2E-2 0.225 2.49 0.0165 0.0548 0.0766 0.351 2.5E-2 0.35 2.11 0.0175 0.0635 0.0984 0.282 4.5E-2 0.45 1.83 0.0335 0.0846 0.1210 0.262 4.5E-2 0.60 1.83 0.0235 0.0921 0.1437 0.211 4.5E-2 0.75 1.55 0.0535 -- 0.1855 0.117 7.5E-2 0.90 1.47 0.0335 -- 0.2069 0.101 7.0E-2 1.00 0.84 0.1335 -- 0.2065 0.082 --

090713 10.0 0.075 3.16 0.0160 0.0369 0.0430 0.485 8.0E-3 0.15 3.03 0.0160 0.0387 0.0549 0.400 8.0E-3 0.225 3.03 0.0165 0.0540 0.0770 0.364 1.0E-2 0.35 2.53 0.0135 0.0449 0.1052 0.303 2.5E-2 0.45 2.25 0.0215 0.0664 0.1205 0.291 4.0E-2 0.60 1.79 0.0255 0.0991 0.1581 0.229 5.0E-2 0.75 1.64 0.0455 -- 0.1920 0.139 4.3E-2 0.90 1.48 0.0335 -- 0.2145 0.137 5.0E-2 1.05 1.30 0.0535 -- 0.2263 0.080 7.0E-2

090414 11.2 -0.20 -- -- -- 0.0178 0.110 -- 0.075 3.98 0.0185 0.0371 0.0343 0.468 2.5E-2 0.15 3.40 0.0160 0.0428 0.0519 0.423 1.0E-2 0.225 3.09 0.0185 0.0506 0.0722 0.364 1.3E-2 0.30 2.58 0.0235 0.0542 0.0922 0.355 2.5E-2 0.35 2.80 0.0145 0.0539 0.1005 0.339 3.5E-2 0.40 2.90 0.0135 0.0678 0.1083 0.338 3.5E-2 0.45 2.40 0.0135 0.0736 0.1428 0.249 3.5E-2 0.50 2.49 0.0235 0.0832 0.1340 0.247 4.0E-2 0.65 2.02 0.0335 0.1177 0.1689 0.214 4.5E-2 0.80 1.70 0.0335 -- 0.2012 0.144 7.2E-2 0.90 1.47 0.0335 0.1235 0.2160 0.153 8.0E-2 1.00 2.40 0.1585 -- 0.2373 0.121 7.5E-2 1.10 1.35 0.0335 -- 0.2420 0.104 9.0E-2 1.20 2.11 0.0435 -- 0.2503 0.073 9.0E-2 1.40 0.74 0.1535 -- 0.2526 0.064 --

68

Appendix C - Bubble clustering in the developing shear region

C.1 Presentation

In air-water flows, an advanced processing of the phase-detection probe signal can provide some information

on the streamwise structure of the air and water including bubble clustering (CHANSON and TOOMBES

2002, CHANSON and CAROSI 2007). A concentration of bubbles within some relatively short intervals of

time may indicate some clustering. In turbulent shear flows, the clustering index may provide a measure of

the vorticity production rate, of the level of bubble-turbulence interactions and of the associated energy

dissipation.

When two bubbles are closer than a particular time/length scale, they can be considered a group of bubbles:

i.e., a cluster. Herein two bubbles were considered parts of a cluster when the water chord time between the

bubbles was less than the bubble chord time of the lead particle. In other words, when a bubble trails the lead

bubble by a short time, it is in the near-wake of and may be influenced by the leading particle (CHANSON

et al. 2006).

The following analysis was conducted in the air-water shear region of hydraulic jumps at the characteristic

location where the bubble count rate was maximum (F = Fmax and y = YFmax).

Notation

d1 flow depth (m) measured immediately upstream of the hydraulic jump; F bubble count rate (Hz) defined as the number of bubbles impacting the probe sensor per second; Fmax maximum bubble count rate (Hz) in the air-water shear layer; Fr1 upstream Froude number: 111 dg/VFr ×= ;

g gravity acceleration (m/s2) : g = 9.80 m/s2 in Brisbane (Australia); Nc number of bubble clusters per second (Hz); Q water discharge (m3/s); Re Reynolds number: μ××ρ= /dVRe 11 ;

V air-water velocity (m/s); V1 upstream flow velocity (m/s): V1 = Q/(W×d1); W channel width (m); x longitudinal distance from the sluice gate (m); x1 longitudinal distance from the gate to the jump toe (m); YFmax distance (m) from the bed where the bubble count rate is maximum (F = Fmax); y distance (m) measured normal to the flow direction; μ dynamic viscosity (Pa.s) of water; ρ density (kg/m3) of water;

Subscript 1 upstream flow conditions.

69

C.2 Experimental summary

Ref. Q W x1 V1 d1 Fr1 Re x-x1 YFmax Fmax Nb clusters per second Nc

Percent. bubbles in

clusters

Average bubble chord time (in

clusters)

Average number of bubbles per

cluster m3/s m m m/s m m m Hz Hz ms

(1) (2) (3) (4) (5) (6) (7) (8) (10) (11) (12) (13) (14) (15) (16) 090317 0.02825 0.5 0.75 3.14 0.018 7.47 5.6E+4 0.075 144.1 0.0185 29.5 0.272 3.80 2.64

0.15 128.5 0.0235 25.8 0.224 2.29 2.54 0.225 112.4 0.0245 18.2 0.147 1.84 2.43 0.30 75.1 0.0355 12.1 0.103 2.67 2.51 0.35 54.1 0.0415 7.7 0.071 2.52 2.37

090720 0.03481 0.5 0.75 3.87 0.018 9.21 6.9E+4 0.075 210.2 0.0185 42.2 0.308 1.62 2.53 0.15 203.4 0.0210 39.4 0.294 1.37 2.52 0.225 169.9 0.0225 29.8 0.250 1.29 2.49 0.35 136.7 0.0295 23.3 0.198 1.61 2.51 0.45 103.3 0.0435 17.2 0.177 2.06 2.54 0.60 92.0 0.0435 14.5 0.121 1.88 2.49 0.75 32.0 0.0500 3.2 0.022 1.96 2.19

090713 0.03780 0.5 0.75 4.20 0.018 10.0 7.5E+4 0.075 245.5 0.0185 45.6 0.383 2.53 2.71 0.15 222.9 0.0210 42.1 0.296 1.31 2.57 0.225 232.4 0.0225 49.5 0.348 1.53 2.63 0.35 183.7 0.0215 32.5 0.262 1.25 2.48 0.45 126.3 0.0295 18.8 0.183 1.44 2.45 0.60 82.2 0.0455 10.3 0.078 1.72 2.35 0.75 47.8 0.0455 5.0 0.036 1.82 2.28

090414 0.04175 0.5 0.75 4.68 0.01783 11.2 8.3E+4 0.075 173.6 0.0185 39.5 0.443 3.97 2.93 0.15 219.2 0.0235 46.9 0.371 2.00 2.74 0.225 211.9 0.0235 40.1 0.302 1.36 2.60 0.30 208.7 0.0235 38.9 0.298 1.30 2.60 0.35 178.4 0.0260 30.7 0.265 1.32 2.54 0.40 180.8 0.0235 30.7 0.269 1.33 2.58 0.45 147.8 0.0335 24.7 0.256 1.53 2.55 0.65 92.9 0.0435 11.6 0.109 1.57 2.40 0.90 44.3 0.0535 4.3 0.115 1.61 2.19

70

C.3 Probability distribution functions of the number of bubbles per cluster

C.3.1 Fr1 = 7.5, y = YFmax and F = Fmax

x-x1 Percent. (PDF) of clusters with 2 3 4 5 6 7 8 9 10 11

m bubbles bubbles bubbles bubbles bubbles bubbles bubbles bubbles bubbles bubbles (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

0.075 0.634 0.233 0.067 0.033 0.014 0.008 0.008 0.002 0.002 0 0.150 0.659 0.215 0.082 0.030 0.005 0.007 0.002 0.001 0 0 0.225 0.713 0.197 0.056 0.022 0.010 0.001 0.001 0 0 0 0.300 0.673 0.211 0.072 0.022 0.018 0.002 0.002 0 0 0 0.350 0.761 0.158 0.049 0.020 0.003 0.006 0.003 0 0 0

C.3.2 Fr1 = 9.2, y = YFmax and F = Fmax

x-x1 Percent. (PDF) of clusters with 2 3 4 5 6 7 8 9 10 11

m bubbles bubbles bubbles bubbles bubbles bubbles bubbles bubbles bubbles bubbles (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

0.075 0.648 0.228 0.086 0.026 0.009 0.002 0.001 0.001 0.001 0 0.150 0.660 0.218 0.085 0.023 0.010 0.003 0.001 0.001 0 0 0.225 0.690 0.200 0.068 0.025 0.012 0.004 0.001 0.000 0.001 0 0.350 0.665 0.221 0.066 0.036 0.009 0.001 0.001 0.001 0 0 0.450 0.653 0.230 0.071 0.027 0.014 0.000 0.003 0.001 0 0 0.600 0.695 0.201 0.063 0.018 0.015 0.005 0.002 0 0.002 0 0.750 0.829 0.151 0.021 0.000 0.000 0.000 0 0 0 0

C.3.3 Fr1 = 10.0, y = YFmax and F = Fmax

x-x1 Percent. (PDF) of clusters with 2 3 4 5 6 7 8 9 10 11

m bubbles bubbles bubbles bubbles bubbles bubbles bubbles bubbles bubbles bubbles (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

0.075 0.592 0.232 0.101 0.049 0.015 0.005 0.004 0.001 0 0.002 0.150 0.644 0.220 0.085 0.034 0.011 0.003 0.001 0.001 0 0 0.225 0.627 0.218 0.086 0.046 0.016 0.007 0.000 0.001 0 0 0.350 0.679 0.215 0.071 0.021 0.011 0.002 0.001 0.001 0 0 0.450 0.701 0.200 0.063 0.026 0.006 0.004 0.001 0 0 0 0.600 0.765 0.170 0.037 0.015 0.009 0.002 0 0 0.002 0 0.750 0.791 0.160 0.040 0.004 0.000 0.004 0 0 0 0

71

C.3.4 Fr1 = 11.2, y = YFmax and F = Fmax

x-x1 Percent. (PDF) of clusters with 2 3 4 5 6 7 8 9 10 11

m bubbles bubbles bubbles bubbles bubbles bubbles bubbles bubbles bubbles bubbles (1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

0.075 0.557 0.235 0.093 0.053 0.024 0.015 0.007 0.006 0.004 0.008 0.150 0.592 0.236 0.094 0.040 0.018 0.011 0.006 0.002 0.002 0 0.225 0.626 0.236 0.084 0.035 0.011 0.004 0.003 0.001 0 0 0.300 0.628 0.233 0.086 0.036 0.011 0.004 0.002 0.001 0 0 0.350 0.680 0.193 0.072 0.034 0.009 0.009 0.001 0.001 0 0 0.400 0.663 0.199 0.081 0.029 0.014 0.006 0.006 0.001 0 0 0.450 0.658 0.206 0.091 0.031 0.009 0.005 0 0 0 0.001 0.650 0.727 0.187 0.056 0.021 0.008 0.002 0 0 0 0 0.900 0.851 0.113 0.036 0.000 0.000 0 0 0 0 0

72

Appendix D - Movies of the experiments

D.1 Introduction

Some new detailed experimental measurements were conducted, and some photographs of the experimental

facility are presented in section 4. A series of short movies were further taken during a number of

experiments. The movie files are deposited with the digital record of the publication at the institutional open

access repository of the University of Queensland: {http://espace.library.uq.edu.au/}. They are listed as part

of the technical report deposit at {http://espace.library.uq.edu.au/list/author_id/193/}. The list of the movies

is detailed in section D.2, including the filenames, file format, and a description of each video.

All the movies are Copyrights Hubert CHANSON 2009.

D.2 List of movies

Filename Format Description P1140119.MOV Quicktime Air bubble entrainment in hydraulic jump. Side

view. Run 090414, Q= 0.04175 m3/s, d1 = 0.01783 m, Fr1 = 11.2, Re = 8.3 104.

P1140142.MOV Quicktime Air bubble entrainment in hydraulic jump. Spray and splashing above the roller, looking upstream. Run 090414, Q= 0.04175 m3/s, d1 = 0.01783 m, Fr1 = 11.2, Re = 8.3 104.

P1150379.MOV Quicktime Air bubble entrainment in hydraulic jump. Side view. Run 090713, Q= 0.0378 m3/s, d1 = 0.018 m, Fr1 = 18.8, Re = 7.5 104.

D.3 Movie files

The movies files of Appendix D are available in the institutional open access repository of the University of

Queensland (Brisbane, Australia) and they are deposited at UQeSpace {http://espace.library.uq.edu.au/}. The

Digital Files are a series of QuicktimeTM movies. The deposited movie files (Section D.2) were converted to

Flash video for video streaming.

At request, the writer may provide the QuicktimeTM movies as a single compressed file (Filename

Movie_File.7z). The file was prepared with 7-zip version 4.23. The software 7-zip is an open source

software. Most of the source code is under the GNU LGPL license. The unRAR code is under a mixed

license: GNU LGPL + unRAR restrictions. The software 7-zip may be freely downloaded from {www.7-

zip.org}.

The copyrights of the movies remain the property of Hubert CHANSON. Any use of the movies available in

the digital appendix must acknowledge and cite the present report:

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).

73

Further details on the report including the digital appendix may be obtained from Prof. Hubert CHANSON

{h.chanson@uq.edu.au}.

74

References

BABB, A.F., and AUS, H.C. (1981). "Measurements of Air in Flowing Water." Jl of Hyd. Div., ASCE, Vol. 107, No. HY12, pp. 1615-1630.

BAKHMETEFF, B.A. (1932). "Hydraulics of Open Channels." McGraw-Hill, New York, USA, 1st ed., 329 pages.

BÉLANGER, J.B. (1841). "Notes sur l'Hydraulique." ('Notes on Hydraulic Engineering.') Ecole Royale des Ponts et Chaussées, Paris, France, session 1841-1842, 223 pages (in French).

BIDONE, G. (1819). "Le Remou et sur la Propagation des Ondes." ('The Jump and on the Wave Propagation.') Report to Académie Royale des Sciences de Turin, séance 12 Dec., Vol. XXV, pp. 21-112 & 4 plates (in French).

BROWN, G.L., and ROSHKO, A. (1974). "On the Density effects and Large Structure in Two-Dimensional Mixing Layers." Jl Fluid Mech., Vol. 64, pp. 775-816.

CHANSON, H. (1995). "Air Entrainment in Two-dimensional Turbulent Shear Flows with Partially Developed Inflow Conditions." Intl Jl of Multiphase Flow, Vol. 21, No. 6, pp. 1107-1121.

CHANSON, H. (1997). "Air Bubble Entrainment in Free-Surface Turbulent Shear Flows." Academic Press, London, UK, 401 pages (ISBN 0-12-168110-6).

CHANSON, H. (2002). "Air-Water Flow Measurements with Intrusive Phase-Detection Probes. Can we Improve their Interpretation ?" Journal of Hydraulic Engineering, ASCE, Vol. 128, No. 3, pp. 252-255.

CHANSON, H. (2005). "Physical Modelling of the Flow Field in an Undular Tidal Bore." Jl of Hyd. Res., IAHR, Vol. 43, No. 3, pp. 234-244.

CHANSON, H. (2007). "Bubbly Flow Structure in Hydraulic Jump." European Journal of Mechanics B/Fluids, Vol. 26, No. 3, pp.367-384 (DOI: 10.1016/j.euromechflu.2006.08.001).

CHANSON, H. (2007b). "Dynamic Similarity and Scale Effects Affecting Air Bubble Entrainment in Hydraulic Jumps." Proc. 6th International Conference on Multiphase Flow ICMF 2007, Leipzig, Germany, July 9-13, M. SOMMERFIELD Editor, Session 7, Paper S7_Mon_B_S7_Mon_B_3, 11 pages (CD-ROM).

CHANSON, H. (2009). "Current Knowledge In Hydraulic Jumps And Related Phenomena. A Survey of Experimental Results." European Journal of Mechanics B/Fluids, Vol. 28, No. 2, pp. 191-210 (DOI: 10.1016/j.euromechflu.2008.06.004).

CHANSON, H., AOKI, S., and HOQUE, A. (2002). "Similitude of Air Bubble Entrainment and Dispersion in Vertical Circular Plunging Jet Flows. An Experimental Study with Freshwater, Salty Freshwater and Seawater." Coastal/Ocean Engineering Report, No. COE02-1, Dept. of Architecture and Civil Eng., Toyohashi University of Technology, Japan, 94 pages.

CHANSON, H., AOKI, S., and HOQUE, A. (2006). "Bubble Entrainment and Dispersion in Plunging Jet Flows: Freshwater versus Seawater." Journal of Coastal Research, Vol. 22, No. 3, May, pp. 664-677 (DOI: 10.2112/03-0112.1).

CHANSON, H., and BRATTBERG, T. (1998). "Air Entrainment by Two-Dimensional Plunging Jets : the Impingement Region and the Very-Near Flow Field." Proc. 1998 ASME Fluids Eng. Conf., FEDSM'98, Washington DC, USA, June 21-25, Paper FEDSM98-4806, 8 pages (ISBN 0 7918 1950 7) (CD-ROM).

75

CHANSON, H., and BRATTBERG, T. (2000). "Experimental Study of the Air-Water Shear Flow in a Hydraulic Jump." Intl Jl of Multiphase Flow, Vol. 26, No. 4, pp. 583-607.

CHANSON, H., and CAROSI, G. (2007). "Advanced Post-Processing and Correlation Analyses in High-Velocity Air-Water Flows." Environmental Fluid Mechanics, Vol. 7, No. 6, pp. 495-508 (DOI 10.1007/s10652-007-9038-3).

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.

CHOW, V.T. (1959). "Open Channel Hydraulics." McGraw-Hill, New York, USA. CRANK, J. (1956). "The Mathematics of Diffusion." Oxford University Press, London, UK. CROWE, C., SOMMERFIELD, M., and TSUJI, Y. (1998). "Multiphase Flows with Droplets and Particles."

CRC Press, Boca Raton, USA, 471 pages. DARCY, H.P.G., and BAZIN, H. (1865). "Recherches Hydrauliques." ('Hydraulic Research.') Imprimerie

Impériales, Paris, France, Parties 1ère et 2ème (in French). EDWARDS, C.F., and MARX, K.D. (1995). "Multipoint Statistical Structure of the Ideal Spray, Part I:

Fundamental Concepts and the Realization Density." Atomizati & Sprays, Vol. 5, pp. 435-455. GOERTLER, H. (1942). "Berechnung von Aufgaben der freien Turbulenz auf Grund eines neuen

Näherungsansatzes." Z.A.M.M., 22, pp. 244-254 (in German). GUALTIERI, C., and CHANSON, H. (2007). "Experimental Analysis of Froude Number Effect on Air

Entrainment in the Hydraulic Jump." Environmental Fluid Mechanics, Vol. 7, No. 3, pp. 217-238 (DOI: 10.1007/s10654-006-9016-1).

HABIB, E., MOSSA, M., and PETRILLO, A. (1994). "Scour Downstream of Hydraulic Jump." Proc. Conf. Modelling, Testing & Monitoring for Hydro Powerplants, Intl Jl Hydropower & Dams, Budapest, Hungary, pp, 591-602.

HAGER, W.H. (1992). "Energy Dissipators and Hydraulic Jump." Kluwer Academic Publ., Water Science and Technology Library, Vol. 8, Dordrecht, The Netherlands, 288 pages.

HAGER, W.H., BREMEN, R., and KAWAGOSHI, N. (1990). "Classical Hydraulic Jump: Length of Roller." Jl of Hyd. Res., IAHR, Vol. 28, No. 5, pp. 591-608.

HEINLEIN, J., and FRITSCHING, U. (2006). "Droplet Clustering in Sprays." Experiments in Fluids, Vol. 40, No. 3, pp. 464-472.

HOYT, J.W., and SELLIN, R.H.J. (1989). "Hydraulic Jump as 'Mixing Layer'." Jl of Hyd. Engrg., ASCE, Vol. 115, No. 12, pp. 1607-1614.

KUCUKALI, S., and CHANSON, H. (2008). "Turbulence Measurements in Hydraulic Jumps with Partially-Developed Inflow Conditions." Experimental Thermal and Fluid Science, Vol. 33, No. 1, pp. 41-53 (DOI: 10.1016/j.expthermflusci.2008.06.012).

LIGGETT, J.A. (1994). "Fluid Mechanics." McGraw-Hill, New York, USA. LONG, D., RAJARATNAM, N., STEFFLER, P.M., and SMY, P.R. (1991). "Structure of Flow in Hydraulic

Jumps." Jl of Hyd. Research, IAHR, Vol. 29, No. 2, pp. 207-218. McCORQUODALE, J. A., and KHALIFA, A.. (1983). "Internal Flow in Hydraulic Jumps." Jl of Hydraulics

Div., ASCE, Vol. 109, No.5, pp. 684-701.

76

MOSSA, M., and TOLVE, U. (1998). "Flow Visualization in Bubbly Two-Phase Hydraulic Jump." Jl Fluids Eng., ASME, Vol. 120, March, pp. 160-165.

MURZYN, F., and CHANSON, H. (2008). "Experimental Assessment of Scale Effects Affecting Two-Phase Flow Properties in Hydraulic Jumps." Experiments in Fluids, Vol. 45, No. 3, pp. 513-521 (DOI: 10.1007/s00348-008-0494-4).

MURZYN, F., and CHANSON, H. (2009). "Experimental Investigation of Bubbly Flow and Turbulence in Hydraulic Jumps." Environmental Fluid Mechanics, Vol. 9, No. 2, pp. 143-159 (DOI: 10.1007/s10652-008-9077-4).

MURZYN, F., MOUAZE, D., and CHAPLIN, J.R. (2005). "Optical Fibre Probe Measurements of Bubbly Flow in Hydraulic Jumps" Intl Jl of Multiphase Flow, Vol. 31, No. 1, pp. 141-154.

MURZYN, F., MOUAZE, D., and CHAPLIN, J.R. (2007). "Air-Water Interface Dynamic and Free Surface Features in Hydraulic Jumps." Jl of Hydraulic Res., IAHR, Vol. 45, No. 5, pp. 679-685.

RAJARATNAM, N. (1962). "An Experimental Study of Air Entrainment Characteristics of the Hydraulic Jump." Jl of Instn. Eng. India, Vol. 42, No. 7, March, pp. 247-273.

RAJARATNAM, N. (1965). "The Hydraulic Jump as a Wall Jet." Jl of Hyd. Div., ASCE, Vol. 91, No. HY5, pp. 107-132. Discussion : Vol. 92, No. HY3, pp. 110-123 & Vol. 93, No. HY1, pp. 74-76.

RAJARATNAM, N. (1967). "Hydraulic Jumps." Advances in Hydroscience, Ed. V.T. CHOW, Academic Press, New York, USA, Vol. 4, pp. 197-280.

RAJARATNAM, N. (1976). "Turbulent Jets." Elsevier Scientific, Development in Water Science, 5, New York, USA.

RESCH, F.J., and LEUTHEUSSER, H.J. (1972). "Le Ressaut Hydraulique: Mesure de Turbulence dans la Région Diphasique." ('The Hydraulic Jump: Turbulence Measurements in the Two-Phase Flow Region.') Jl La Houille Blanche, No. 4, pp. 279-293 (in French).

SCHLICHTING, H. (1979). "Boundary Layer Theory." McGraw-Hill, New York, USA, 7th edition. THANDAVESWARA, B.S. (1974). "Self Aerated Flow Characteristics in Developing Zones and in

Hydraulic Jumps." Ph.D. thesis, Dept. of Civil Engrg., Indian Institute of Science, Bangalore, India, 399 pages.

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.

VALLE, B.L., and PASTERNACK, G.B. (2006). "Air Concentrations of Submerged and Unsubmerged Hydraulic Jumps in a Bedrock Step-Pool Channel." Jl of Geophysical Res., Vol. 111, No. F3, paper F030l6, 12 pages (DOI: 10.1029/2004JF000140).

WISNER, P. (1965). "Sur le Rôle du Critère de Froude dans l'Etude de l'Entraînement de l'Air par les Courants à Grande Vitesse." ('On the Role of the Froude Criterion for the Study of Air Entrainment in High Velocity Flows.') Proc. 11th IAHR Congress, Leningrad, USSR, paper 1.15 (in French).

77

Internet references

Self-aeration on chute and stepped spillways. Air entrainment and flow aeration in open channel flows

{http://www.uq.edu.au/~e2hchans/self_aer.html}

Open access repository UQeSpace {http://espace.library.uq.edu.au/} Software 7-zip {http://www.7-zip.org/} Software Quicktime {http://www.apple.com/quicktime/download/}

Bibliographic reference of the Report CH75/09 The Hydraulic Model research report series CH is a refereed publication published by the Division of Civil

Engineering at the University of Queensland, Brisbane, Australia.

The bibliographic reference of the present report is :

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).

The Report CH75/09 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/

78

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://www.eng.uq.edu.au/civil/ Email: hodciveng@uq.edu.au

Report CH Unit price Quantity Total price

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 BorePropagation: 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

79

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

80

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 priceTREVETHAN, 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

81

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://www.eng.uq.edu.au/civil/ Email: hodciveng@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 GRAND TOTAL

Note: Prices include postages and processing.

PAYMENT INFORMATION

1- VISA Card

Name on the card :

Visa card number :

82

Expiry date :

Amount :

AUD$ ...........................................

2- Cheque/remittance payable to: THE UNIVERSITY OF QUEENSLAND and crossed "Not Negotiable".

N.B. For overseas buyers, cheque payable in Australian Dollars drawn on an office in Australia of a

bank operating in Australia, payable to: THE UNIVERSITY OF QUEENSLAND and crossed "Not

Negotiable".

Orders of any Research Report 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://www.eng.uq.edu.au/civil/ Email: hodciveng@uq.edu.au