APPENDIX A-SUMMARY OF AIR-WATER FLOW PROPERTIES A.1 Presentation

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APPENDIX A-SUMMARY OF AIR-WATER FLOW PROPERTIES

A.1 Presentation

Five discharges were investigated ranging from 0.095 m2_{/s to 0.180 m}2_{/s. Table A-1 summaries the}

various investigations.

Ref. H1

[m] dc/h Probes Type [mm] ∆x [mm] ∆z Step edge No. Location of the inception Step edge No. regime Flow

060323 0.15 1.0 1-tip probe 6 between 5 and 6 TR-SK

060327a 0.15 1.0 1-tip probe 7 between 5 and 6 TR-SK

060327b 0.15 1.0 1-tip probe 8 between 5 and 6 TR-SK

060328a 0.15 1.0 1-tip probe 9 between 5 and 6 TR-SK

060328b 0.15 1.0 1-tip probe 10 between 5 and 6 TR-SK

060329a 0.20 1.33 1-tip probe 7 between 6 and 7 (almost 6) SK

060329b 0.20 1.33 1-tip probe 8 between 6 and 7 (almost 6) SK

060330a 0.20 1.33 1-tip probe 9 between 6 and 7 (almost 6) SK

060330b 0.20 1.33 1-tip probe 10 between 6 and 7 (almost 6) SK

060404a 0.235 1.57 1-tip probe 8 between 7 and 8 SK

060404b 0.235 1.57 1-tip probe 9 between 7 and 8 SK

060405 0.235 1.57 1-tip probe 10 between 7 and 8 SK

060410a 0.173 1.15 2*1-tip probes 8.45 7 between 6and 7 SK

060410b 0.173 1.15 2*1-tip probes 8.45 8 between 6and 7 SK

060410c 0.173 1.15 2*1-tip probes 8.45 9 between 6and 7 SK

060508b 0.173 1.15 2*1-tip probes 3.6 10 between 7 and 8 SK

060509a 0.2175 1.45 2*1-tip probes 3.6 10 between 7 and 8 SK

060530a 0.173 1.15 2-tip probe 7 1.4 10 between 6 and 7 SK

060530b 0.2175 1.45 2-tip probe 7 1.4 10 between 7 and 8 SK

060613a 0.173 1.15 2-tip probe 9.6 1.4 10 between 7 and 8 SK

060613a 0.2175 1.45 2-tip probe 9.6 1.4 10 between 7 and 8 SK

Notes: ∆x streamwise distance between the probe sensors ∆z transverse distance between the probe sensors

2*1-tip probes two single-tip probes fixed on the trolley at a known transverse distance z. TR-SK transition-skimming flow regime

SK skimming flow regime

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Notation

C void fraction defined as the volume of air per unit volume of air and water; Cmean depth-average void fraction defined in terms of Y90: Cmean=1-d/Y90;

d equivalent clear water flow depth [m] defined as:

d

(

1 C

)

dy

90 . 0 y 0 y

∫

= =

−

=

;

dc critical flow depth [m];

Fmax maximum bubble count rate [Hz] at a given cross-section;

fe equivalent Darcy-Weisbach friction factor in air-water skimming flows;

g gravity constant: g=9.80 m/s2_{in Brisbane, Australia;}

h height of steps [m] measured vertically; lstep length of steps [m];

lcav cavity length

l

_cav

=

h

2

+

l

2step

Qw water discharge [m3/s];

qw water discharge per unit width [m2/s];

Tu turbulence intensity defined as: Tu=u’/V Vc critical velocity [m/s]

V

_c

=

3

g

*

q

_w ;

V90 characteristic velocity [m/s] where the air concentration is 90%;

W channel width [m];

x distance along the channel bottom [m];

Y90 characteristic depth [m] where the void fraction is 90%.

θ slope of the channel

∆x

streamwise distance [mm] between the probe sensors

∆z transverse distance [mm] between the probe sensors

A.2 Experimental data

Experimental air-water flow properties distributions for all investigated discharges and locations are presented in the following paragraphs.

A.2.1 Void fraction and bubble count rate distributions

Location: University of Queensland (Australia) Experiments by: G. CAROSI

Experiments characteristics: Channel: W=1 m, θ=22°, 10 steps h=0.10 m lstep=0.25 m

Flow rate dc/h=1.0

Qw=0.0948 m3/s

Location of the inception point: between step edge 5 and 6

Flow regime: transition-skimming flow

Run 060323a, 060327a, 060327b, 060328a, 060328b

Instrumentation: Single-tip probe [Ø=0.35 mm] Step edges investigated No. 6, 7, 8, 9, 10

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0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 C 0.5 0.6 0.7 0.8 0.9 1 y/ Y 90 Step 6 Y90=79 mm Step 7 Y90=64.5 mm Step 8 Y90=63 mm Step 9 Y90=80.5 mm Step 10 Y90=58 mm

Figure A.1-Void fraction distribution as a function of y/Y90.

Flow rate: dc/h=1.0 Instrumentation: single-tip probe [Ø=0.35 mm].

0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 12 14 16 18 20 Fmax*dc/Vc y/ Y 90 Step 6 Y90=79 mm Step 7 Y90=64.5 mm Step 8 Y90=63 mm Step 9 Y90=80.5 mm Step 10 Y90=58 mm

Figure A.2-Bubble count rate distribution as a function of y/Y90.

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Location: University of Queensland (Australia)

Experiments by: G. CAROSI

Flow rate dc/h=1.15

Qw=0.1164 m3/s

Location of the inception point: between step edge 6 and 7 Flow regime: skimming flow

Run 060410a, 060410b, 060410c, 060411a, 060411b, 060411c, 060412a, 060412b, 060412c, 060413a, 060508a, 060508b Instrumentation: Single-tip probe [Ø=0.35 mm]

Step edges investigated No 7, 8, 9, 10

Scan time [s] 45 Scan rate [Hz] 20000 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 _C 0.5 0.6 0.7 0.8 0.9 1 y/ Y 90 Step 7 Y90=74.8 mm Step 8 Y90=64.5 mm Step 9 Y90=63.5 mm Step 10 Y90=61 mm

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0 0.5 1 1.5 2 2.5 0 5 10 _F*dc/Vc 15 20 25 y/ Y 90 Step 7 Y90=74.8 mm Step 8 Y90=64.5 mm Step 9 Y90=63.5 mm Step 10 Y90=61 mm

Figure A.4-Bubble count rate distribution as a function of y/Y90.

Flow rate dc/h=1.33

Qw=0.1431 m3/s

Flow regime: skimming flow

Run 060329a, 060329b, 060330a, 060330b

Instrumentation: Single-tip probe [Ø=0.35 mm] Step edges investigated No 7, 8, 9, 10

Scan time [s] 45

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0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 C 0.5 0.6 0.7 0.8 0.9 1 y/ Y 90 Step 7 Y90=51.37 mm Step 8 Y90=51.9 mm Step 9 Y90=50.48 mm Step 10 Y90=50.66 mm

0 0.5 1 1.5 2 0 5 10 Fmsx*dc/Vc 15 20 y/ Y 90 Step 7 Y90=51.37 mm Step 8 Y90=51.9 mm Step 9 Y90=50.48 mm Step 10 Y90=50.66 mm

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Flow rate dc/h=1.45

Qw=0.1612 m3/s

Run 060509a, 060509b, 060510a, 060510b, 060511a, 060511b, 060511, 060512, 060526a, 060526b Instrumentation: Single-tip probe [Ø=0.35 mm]

Step edges investigated No 8, 9, 10

Scan time [s] 45 Scan rate [Hz] 20000 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 _C 0.5 0.6 0.7 0.8 0.9 1 y/ Y 90 Step 8 Y90=74.5 mm Step 9 Y90=78.3 mm Step 10 Y90=71.5 mm

Figure A.7-Void fraction distribution as a function of y/Y90

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0 0.5 1 1.5 2 0 5 10 F*dc/Vc 15 20 25 y/ Y 90 Step 8 Y90=74.5 mm Step 9 Y90=78.3 mm Step 10 Y90=71.5 mm

Figure A.8-Bubble count rate distribution as a function of y/Y90

Flow rate dc/h=1.57

Qw=0.1798 m3/s

Run 060404a, 060404b, 060405

Instrumentation: Single-tip probe [Ø=0.35 mm] Step edges investigated No 8, 9, 10

Scan time [s] 45

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 _C 0.5 0.6 0.7 0.8 0.9 1 y/ Y 90 Step 8 Y90=79.5 mm Step 9 Y90=86.5 mm Step 10 Y90=84.5 mm

Figure A.9-Void fraction distribution as a function of y/Y90

0 0.5 1 1.5 2 0 2 4 6 8 10 12 14 16 18 20 F*dc/Vc y/ Y 90 Step 8 Y90=79.5 mm Step 9 Y90=86.5 mm Step 10 Y90=84.5 mm

Figure A.10- Bubble count rate distribution as a function of y/Y90

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A.2.2 Interfacial velocity and turbulence intensity distributions Location: University of Queensland (Australia) Experiments by: G. CAROSI

Flow rate dc/h=1.15

Qw=0.1164 m3/s

Run 060526a, 060613a

Instrumentation: Double-tip probe [Ø=0.35 mm, ∆x=7mm and ∆x=9.6mm] Step edge investigated No 10

Scan time [s] 45 Scan rate [Hz] 20000 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 v/V90 y/ Y 90 Step 10 Y90=56.1 mm V90=3.17 m/s, x=7.0mm Step 10 Y90=59.5 mm V90=3.59 m/s, x=9.6mm

Figure A.11- v/V90 as a function of y/Y90

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0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.5 1 Tu 1.5 2 2.5 y/ Y 90 Step 10 Y90=56.1 mm, x=7mm Step 10 Y90=59.5 mm, x=9.6mm

Figure A.12- Tu as a function of y/Y90

Flow rate: dc/h=1.15 Instrumentation: double-tip probe [Ø=0.35 mm ∆x=7mm and ∆x=9.6mm].

Flow rate dc/h=1.33

Qw=0.1431 m3/s

Run 060614a, 060614b, 060615a, 060615b

Instrumentation: Double-tip probe [Ø=0.35 mm, ∆x=7mm and ∆x=9.6mm ] Step edge investigated No 7, 8, 9, 10

Scan time [s] 45

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0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 v/V90 y/ Y 90 Step 7 Y90=66.8 mm V90=3.268 m/s, x=7.0mm Step 8 Y90=62 mm V90=3.27 m/s, x=7.0mm Step 9 Y90=70.25 mm V90=3.415 m/s, x=7.0mm Step 10 Y90=63.3 mm V90=3.415 m/s, x=7.0mm

Flow rate: dc/h=1.33 Instrumentation: double-tip probe [Ø=0.35 mm ∆x=7mm].

0 0.5 1 1.5 2 2.5 0 0.5 1 _Tu 1.5 2 2.5 y/ Y 90 Step 7 Y90=66.8mm, x=7mm Step 8 Y90=62 mm, x=7mm Step 9 Y90=70.25mm, x=7mm Step 10 Y90=63.3mm, x=7mm

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Flow rate dc/h=1.45

Qw=0.1612 m3/s

Run 060526b, 060613b

Instrumentation: Double-tip probe [Ø=0.35 mm, ∆x=7mm and ∆x=9.6mm ] Step edge investigated No 10

Scan time [s] 45 Scan rate [Hz] 20000 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 v/V90 y/ Y 90 Step 10 Y90= 71 mm V90=3.59 m/s, x=7.0mm Step 10 Y90=77 mm V90=4.62 m/s, x=9.6mm

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0 0.5 1 1.5 2 2.5 0 0.2 0.4 0.6 _Tu 0.8 1 1.2 1.4 y/ Y 90 Step 10 Y90=71 mm, x=7mm Step 10 Y90=77 mm, x=9.6mm

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APPENDIX B- COMPARATIVE PERFORMANCES OF THE TWO CONDUCTIVITY PROBE SYSTEMS

Basic air-water flow measurements were performed with two single-tip probes (Ø=0.35 mm) for dc/h=1.15 and dc/h=1.45. For the same flow rates, additional measurements were performed with

double-tip probes (Ø=0.25 mm, ∆x=7 mm and ∆x= 9.6 mm).

In this appendix, the performances of the two probe systems are compared systematically. Basic results include void fraction, dimensionless bubble count rate F*dc/Vc and dimensionless air-water

flow velocity V90/Vc as functions of y/Y90. For dc/h=1.15 it was possible further to compare the present

results with the data of GONZALEZ et al. (2005) for dc/h=1.18. Note that in the following Figures, all

data show the same patterns even though different types and size of probes were used. In GONZALEZ et al. (2005) study, the double-tip probe sensor diameter was 0.025 mm and the streamwise distance between probe sensors was ∆x=7.74 mm. Table B-1 summarises the experimental details.

Data set Present study CMH_05

Flow rate dc/h 1.15 1.5 1.18 Instrumentation Single-tip probe Double-tip probe Double-tip probe Ø sensor [mm] 0.35 0.25 0.025 ∆x [mm] - 7.0 9.6 7.74 Scan rate [Hz] 20,000 20,000 20,000 Scan duration [s] 45 45 20

Table B-1.Summary of the experimental investigations

Air-water flow properties for dc/h=1.15

Flow rate dc/h=1.15

Qw=0.1164 m3/s

Run 060613a, 060526a

Instrumentation: Double-tip probe [Ø=0.25 mm, ∆x=7mm and ∆x=9.6mm] Step edges investigated No. 10

Scan time [s] 45

Scan rate [Hz] 20000

Comparison study GONZALEZ, TAKAHASHI and CHANSON (2005) CMH_05

dc/h=1.18

Qw=0.126 m3/s

Instrumentation: Double-tip probe[Ø=0.025mm, ∆x=7.74 mm] Step edge investigated No.10

Scan time: 20 s Scan rate: 20000 Hz

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0 0.5 1 1.5 2 2.5 3 0 0.1 0.2 0.3 0.4 C 0.5 0.6 0.7 0.8 0.9 1 y/ Y 9 0

void fraction 1-tip probe

void fraction 2-tip probe Leading, x=7 mm void fraction 2-tip robe T railing, x=7 mm void fraction 2-tip probe Leading, x=9.6 mm void fraction 2-tip probe T railing, x=9.6 mm

void fraction 2-tip probe CMH_05 Leading, x=7.74 mm void fraction 2-tip probe CMH_05 T railing, x=7.74 mm

Figure B.1- Void fraction as a function of y/Y90 (dc/h=1.15 and dc/h=1.18)

Instrumentation: for the present study double-tip probe [Ø=0.25 mm ∆x=7 mm and ∆x=9.6 mm], for CMH_05 study double-tip probe [Ø=0.025 mm ∆x=7.74 mm].

0 0.2 0.4 0.6 0.8 1 1.2 0 5 10 15 F*dc/Vc 20 25 30 35 y/ Y 9 0

Bubble count rate 1-tip probe

Bubble count rate 2-tip probe Leading, x=7 mm Bubble count rate 2-tip robe Trailing, x=7 mm Bubble count rate 2-ip probe Leading, x=9.6 mm Bubble count rate 2-tip probe Trailing, x=9.6 mm

Bubble count rate 2-tip probe CMH_05 Leading, x=7.74 mm Bubble count rate 2-tip probe CMH_05 Trailing, x=7.74 mm

Figure B.2- F*dc/Vc as a function of y/Y90 (dc/h=1.15 and dc/h=1.18)

Instrumentation: for the present study double-tip probe [Ø=0.25 mm ∆x=7 mm and ∆x=9.6 mm], for CMH_05 double-tip probe [Ø=0.025 mm ∆x=7.74 mm].

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0 0.5 1 1.5 2 2.5 3 0.5 0.7 0.9 1.1 V/V90 y/ Y 9 0

Velocity 2-tip probe, x=7 mm Velocity 2-tip probe, x=9.6 mm

Velocity 2-tip probe CMH_05, x=7.74 mm 1/10th Power law

Figure B.3- V/V90 as a function of y/Y90 (dc/h=1.15 and dc/h=1.18)

Air-water flow properties for dc/h=1.45

Flow rate dc/h=1.45

Qw=0.1612 m3/s

Run 060526b, 060613b

Instrumentation: Double-tip probe [Ø=0.25 mm, ∆x=7mm and ∆x=9.6mm ] Step edges investigated No. 10

Scan time [s] 45

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0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 _C 0.5 0.6 0.7 0.8 0.9 1 y/ Y 9 0

void fraction 1-tip probe

void fraction 2-tip probe Leading, x=7 mm void fraction 2-tip robe T railing, x=7 mm void fraction 2-tip probe Leading, x=9.6 mm void fraction 2-tip probe T railing, x=9.6 mm

Figure B.4- Void fraction as a function of y/Y90 (dc/h=1.45)

0 0.2 0.4 0.6 0.8 1 1.2 0 5 10 F*dc/vc 15 20 25 y/ Y 9 0

Bubble count rate 1-tip probe

Bubble coutn rate 2-tip probe Leading, x=7 mm Bubble count rate 2-tip robe T railing, x=7 mm Bubble count rate 2-tip probe Leading, x= 9.6 mm Bubble count rate 2-tip probe T railing, x=9.6 mm

Figure B.5- F*dc/Vc as a function of y/Y90 (dc/h=1.45)

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0 0.5 1 1.5 2 2.5 3 0.5 0.7 V/V90 0.9 1.1 y/ Y 9 0

Velocity 2-tip probe, x= 7 mm Velocity 2-tip probe, x=9.6 mm 1/10th Power law

Figure B.6- V/V90 as a function of y/Y90 (dc/h=1.45)

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APPENDIX C- CHORD SIZE DATA WITH SINGLE AND DOUBLE-TIP CONDUCTIVITY PROBES

C.1 Presentation

A double-tip conductivity probe can provide the chord length of bubbles and droplets in addition of the void fraction, velocity, the bubble count rate. A single-tip probe can only provide the void fraction, the number of bubbles or droplets per second, which is the bubble count rate, and the bubbles/droplets chord time. A chord size may be derived using a single-tip conductivity probe. The pseudo-chord size

c

h

~

is calculated as:

a / w w a / w

ch

=

U

⋅

t

where:

Uw is equivalent clear-water flow velocity [m/s]

90 Y

w w w

0

U =q d q=

∫

(1 C)dy− qw is water discharge per unit width [m2/s]

d is characteristic depth [m]

90 Y

0

d=

∫

(1 C)dy− C is the void fraction

y is the distance from the pseudo-bottom (formed by the step edges) measured perpendicular to the flow direction

Y90 is the characteristic depth [m] where the void fraction is 90%

A systematic comparison between true chord size and pseudo-chord size was conducted using the data set of GONZALEZ et al. (2005) to assess the validity of pseudo chord size. The data were collected at step edge number 10 for dc/h=1.18 using a double-tip conductivity probe with a diameter of 0.025 mm.

In the present study, new experiments were conducted by using two different probe systems for the same discharge (dc/h=1.15): a single tip probe, whose sensor diameter was 0.35 mm and a double tip

probe, whose sensor diameter was 0.25 mm. All the experimental details are summarised in Table C-1.

GONZALEZ et al. (2005) Carosi Chanson (2006)

Data set CMH_05 Present study

dc/h 1.18 1.15 1.15

step edge No. 10 10 10

Instrumentation Double-tip conductivity probe

Ø=0.025 mm ∆x= 7.74 mm Single-tip conductivity probe Ø=0.35 mm Double-tip conductivity probe Ø=0.25 mm ∆x= 7 mm

Scan time [s] 20 45 45

Scan rate [Hz] 20,000 20,000 20,000

Table C-1 Summary of the experiment conditions

Two additional comparisons between the previous study of GONZALEZ et al. (2005) and the present study were conducted to investigate how the size of the probe sensor may affect the measurements. All three comparisons performed are summarised in Table C-2. They include:

1. Comparison between true and pseudo chord size of CMH_05 data collected by a double-tip probe [Ø=0.025 mm]

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2. Comparison between pseudo chord size collected with a double-tip probe [Ø=0.025 mm] and pseudo chord size collected with a single-tip probe [Ø=0.35 mm]

3. Comparison between true chord size collected with a double-tip probe [Ø=0.025 mm] and true chord size collected with a double-tip probe [Ø=0.25 mm].

All comparisons were performed in terms of the probability distribution functions. For each histogram, the range of bubbles/droplets chord size extends from less than 0.5 mm to more than 15 mm and each bin is 0.5 mm wide. If there were less than 10 bubbles/droplets in a bin, a bin was not considered in further analysis because the number of the bubbles/droplets was not statistically meaningful. Normalised chord size probability distribution functions are represented in the following paragraphs by charts where each data point represents the probability of a bubble/droplet chord in 0.5 mm intervals: e.g. the probability of bubble/droplet chord size from 1.0 to 1.5 mm is represented by the data point labelled 1.25. The probability of bubble/droplet greater than 15 mm is not shown for clarity.

Comparison Range of void fraction Purpose of the comparison

True/Pseudo chord size

of CMH_05 data set C < 0.3 and C > 0 6 assess the validity of pseudo chord size Pseudo chord size of CMH_05 data set/

Pseudo chord size of the present study C < 0.3 and C > 0 6 due to the different probe sensor’s size investigate the effects in results True chord size of CMH_05 data set/ True

chord size of the present study

C < 0.3 and C > 0 6 investigate the effects in results due to the different probe sensor’s size

Table C-2 Summary of the comparative analyses

C.2 Comparison between true and pseudo-chord sizes

For the data set of GONZALEZ et al. (2005) (Table C-1), a comparison was conducted between true chord size and pseudo chord size data. Both data sets were measured with the same probe system (Ø=0.025 mm).The analysis was done on bubbles chord sizes for void fractions less than 0.3 and on droplet chords for void fractions greater than 0.6 (Table C-3).

CMH_05 Instrumentation: Double-tip probe [Ø=0.025 mm] Filename No. Void Fraction

05 0.107 10 0.218 12 0.327 17 0.817 19 0.893 20 0.922 21 0.943 22 0.961 24 0.973 25 0.98 27 0.987 28 0.99 29 0.994 30 0.996

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The same patterns were observed for both probability distribution functions. They were skewed with a preponderance of small particles relative the mean size and they tended to follow in average a log-normal distribution for C < 0.93 (Figure C-1). When the void fraction is greater than 0.93, the distributions do not follow a log-normal distribution. This might be caused by an insufficient number of droplets. 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.25 1.25 2.25 3.25 4.25 5.25 6.25 7.25 8.25 9.25 10.3 11.3 12.3 13.3 14.3 ch [mm] PD F

True chord size CMH_05 C=0.893 2059 Droplets Psuedo chord size CMH_05 C=0.893 2059 Droplets

Figure C.1 True and pseudo chord size probability distribution functions.

Experimental data by CMH_05, Flow rate: dc/h=1.18, Instrumentation: double tip probe [0.025 mm]

The relative error was estimated to compare true and pseudo chord sizes of CMH_05 data. For each sample, it was calculated as:

pseudo true % true

(p

p )

e

100 p

−

=

⋅

where:

ppseudo is the probability referred to the pseudo chord size of CMH_05 data;

ptrue is the probability referred to the true chord size of CMH_05 data.

A positive value of the relative error means that the pseudo chord size overestimates the true chord size; conversely a negative value means the pseudo chord size underestimates the true chord size. The relative errors are summarised in Table C-4 where the void fraction and the number of bubbles/droplets are also shown. For void fraction less than 0.97, the relative error was small and positive: the pseudo chord size overestimated the true chord size by 1.98% in average. When the void

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fraction was greater than 0.97 the number of bubbles/droplets was not large enough to be a meaningful statistic population and a conclusion could not be achieved.

Filename No. Void Fraction Number of bubbles/droplets Relative error [%] 05 0.107 3550 1.99 10 0.218 4980 -3.72 12 0.327 5601 -3.55 17 0.817 3397 0.82 19 0.893 2059 1.41 20 0.922 1476 -4.36 21 0.943 1119 8.72 22 0.961 785 4.89 24 0.973 518 11.64 25 0.98 409 14.61 27 0.987 274 9.95 28 0.99 198 -28.57 29 0.994 143 N/A 30 0.996 82 -15.38

Note: N/A means that in all bins of the sample there were less than 10 bubbles/droplets and the relative error was not calculated.

Table C-4 Relative error between true and pseudo-chord sizes,

C.3 Effects of the probe sensor size on the chord size data

Two comparisons were conducted to assess any difference induced by using different probe systems: a comparison between true chord sizes for two sensor sizes (Ø=0.025 mm and Ø=0.25 mm) and a comparison between two pseudo chord size data sets for Ø= 0.025 mm and Ø=0.35 mm (Table C-1). They were performed for similar values of void fraction as shown in Table C-5.

CMH_05 Present study Instrumentation: Double-tip probe [Ø=0.025 mm] Instrumentation Single-tip probe [Ø=0.35 mm] Instrumentation Double-tip probe [Ø=0.25 mm] Filename No. Void Fraction Filename No. Void Fraction Filename No. Void Fraction 05 0.107 08 0.106 06 0.104 10 0.218 09 0.182 -- -- 12 0.327 10 0.327 08 0.373 17 0.817 13 0.798 -- -- 19 0.893 14 0.974 12 0.855 20 0.922 15 0.91 14 0.925 21 0.943 16 0.933 15 0.943 22 0.961 18 0.961 17 0.966 24 0.973 20 0.973 18 0.973 25 0.98 21 0.981 19 0.981 27 0.987 22 0.987 21 0.987 28 0.99 24 0.993 22 0.991 29 0.994 27 0.995 24 0.994 30 0.996 28 0.996 26 0.996

Note: -- no files with similar void fraction could be found to be compared with CMH_05 data Table C-5 File name and void fraction of compared data.

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In both comparisons the bubble count rate was calculated for each sample because the scan times were different: 20 s for CMH_05 study and 45 s for the present study (Table C-1).

t N

B b/d

count =

Where

Nb/d is the number of bubbles or droplets recorded

t is the scan time.

C.3.1 Comparison between true chord size of data sets (Ø=0.025 mm and Ø=0.25 mm)

For CMH_05 data set, pseudo chord sizes were obtained with a double tip probe Ø=0.025 mm; for the present data set, pseudo chord sizes were obtained with a double tip probe Ø=0.25 mm (Table C-1). Thus, the probe sensor used in the CMH_05 experiments was ten times smaller than the probe sensor used in the present study. This did affect the sizes of recorded bubbles: the smaller sensor could detect a large number of small bubbles. In Figure C.2 the bubble count rate is shown in dimensionless terms ( F·dc/Vc) as a function of y/dc where dc is the critical depth and Vc is the critical velocity. In the same

Figure, the void fraction distributions are also presented as functions of y/dc. Note that the void

fraction distributions were not influenced by the probe sensors, but the dimensionless bubble count rate distributions showed some differences in bubble count rate magnitude for y/dc < 0.5. F·dc/Vc of

CMH_05 data set was always greater than F·dc/Vc for the present study.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 5 10 F*dc/Vc 15 20 25 30 y/ d c 0 0.1 0.2 0.3 0.4 0.5 0.6 C 0.7 0.8 0.9 1 1.1 1.2 1.3 F*dc/Vc CMH_05 F*dc/Vc Present study Void fraction CMH_05 Void fraction Present study

Figure C.2 F·dc/Vc and C as function of y/dc

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Experimental data of present study, Flow rate: dc/h=1.15, Instrumentation: double tip probe [0. 25

mm]

In Figure C.3 the same conclusions can be achieved in terms of the normalised probability distribution functions. Although the two distributions tend to follow closely a log-normal shape, they show some differences due to the sensor sizes. CMH_05 data show greater values of the normalised probability distribution function for ch < 3 mm. Conversely, the PDF of present data are always greater for ch > 4 mm. These features were observed for all void fractions investigated and they are consistent with the results obtained by calculating the relative error. It was defined as:

true2006 true2005 % true2005

(p

p

)

e

100 p

−

=

⋅

where

Ptrue2005 is the probability referred to CMH_05 study

Ptrue2006 is the probability referred to the present study.

The relative error is ranging between 4% and 33% with an average of 17.75% for C < 0.97. That is, the chord sizes measured with a 0.25 mm probe sensor were in average 18% larger than those measured with a 0.025 mm probe sensor. The relative errors are summarised in Table C-5 where the void fraction, the number of bubbles/droplets and bubble count rate are also shown. For C > 0.97 the relative errors were not considered because the number of droplets recorded was not large enough to be a meaningful statistic population.

0 0.05 0.1 0.15 0.2 0.25 0.25 1.25 2.25 3.25 4.25 5.25 6.25 7.25 8.25 9.25 10.3 11.3 12.3 13.3 14.3 15.3 ch [mm] PD F

True chord size dataCMH 05 Ø=0.025 mm C=0.107 3550 Bubbles True chord size Present study Ø=0.25 mm C=0.106 5315 Bubbles

Figure C.3 True chord size probability functions for both data set.

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Experimental data of present study, Flow rate: dc/h=1.15, Instrumentation: double tip probe [0. 25

mm]

CMH_05

Instrumentation: Double-tip probe [Ø=0.025 mm] Instrumentation: Double-tip probe [Ø=0.25 mm] Present study Filename No. Void fraction Bubble count rate Number bubbles/droplet s Filename No. Void fractio n Bubble count rate Number bubbles/droplet s Relative error [%] 05 0.107 177.5 3550 06 0.104 139.35 6271 29.43 12 0.327 183.6 5601 06 0.373 177.75 8000 6.83 19 0.893 102 2059 12 0.855 100.15 4508 32.46 20 0.922 73 1476 14 0.925 55.26 2488 27.24 21 0.943 55.9 1119 15 0.943 41.98 1890 12.85 22 0.961 39.2 785 17 0.966 25.07 1129 10.60 24 0.973 25.8 518 18 0.973 22.33 1006 4.83 25 0.98 20.4 409 19 0.981 15.75 710 3.65 27 0.987 13.6 274 21 0.987 10.84 489 11.54 28 0.99 9.8 198 22 0.991 8.00 362 18.51 29 0.994 7.05 143 24 0.994 5.62 255 N/A 30 0.996 4.05 82 26 0.996 3.55 161 -31.80

Note: N/A in all bins of the sample there were less than 10 bubbles/droplets and the relative error was not calculated

Table C-5 Void fraction, relative error and bubbles count rate of compared data

C.3.2 Comparison between pseudo-chord size data sets (Ø=0.025 mm and Ø=0.35 mm)

Pseudo chord sizes of CMH_05 data set were compared with pseudo-chord sizes of the present data. For CMH_05 data set, pseudo chord sizes were obtained with a double tip probe whose diameter was 0.025 mm, whereas for the present data set, pseudo chord sizes were obtained with a single tip probe whose diameter was 0.35 mm (Table C-1). In this comparison the difference between the probe sensors was significant. The bubble count rate of CMH_05 data set was systematically greater than the bubble count rate of present data set of 3-36% with an average of 19%. Pseudo chord sizes are presented in terms of probability distribution functions in Figure C-4. CMH_05 data set show greater values of PDF for small chord size and lower values of PDF for greater values of chord size. The data set seems to be shifted ahead.

The relative error was calculated as:

pseudo2006 pseudo2005 % pseudo2005

(p

p

)

e

100 p

−

=

⋅

where:

ppseudo2005 is the probability referred to CMH_05 study

Ppseudo2006 is the probability referred to the present study

The results are summarised in Table C-6. For void fraction less than 0.97, pseudo chord sizes were overestimate by 28% in average with the 0.35 mm probe sensor compared to those measured with the 0.025 mm probe sensor.

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.25 1.25 2.25 3.25 4.25 5.25 6.25 7.25 8.25 9.25 10.3 11.3 12.3 13.3 14.3 ch [mm] PD F

Psuedo chord size CMH_05 C=0.327 5601 Bubbles Pseudo chord size Present study C=0.327,7999 Bubbles

Figure C.4 True chord size probability functions for both data set.

Experimental data of present study, Flow rate: dc/h=1.15, Instrumentation: single tip probe [0.35 mm]

CMH_05

Instrumentation: Double-tip probe [Ø=0.025 mm]

Present study

Instrumentation: Single-tip probe [Ø=0.35 mm] Filename

No. Fraction Void Bubble count rate

Number bubbles/droplet

s

Filename

No. Fraction Void Bubble count rate Number bubbles/droplet s Relative error [%] 05 0.107 177.5 3550 08 0.106 118 5315 46.8 10 0.218 249.1 4980 09 0.182 159 7191 18.7 12 0.327 183.6 5601 10 0.327 177 7999 20.1 17 0.817 169 3397 13 0.798 120 5415 44.2 19 0.893 102 2059 14 0.974 82 3704 49.7 20 0.922 73 1476 15 0.91 57 2590 38.7 21 0.943 55.9 1119 16 0.933 43.6 1962 13.5 22 0.961 39.2 785 18 0.961 27.8 1252 15.5 24 0.973 25.8 518 20 0.973 19.4 873 5.5 25 0.98 20.4 409 21 0.981 15.5 701 8.2 27 0.987 13.6 274 22 0.987 10.5 473 -4.1 28 0.99 9.8 198 24 0.993 7.1 321 N/A 29 0.994 7.05 143 27 0.995 4.0 182 N/A 30 0.996 4.05 82 28 0.996 4.4 197 -12.9

Note: N/A means that in all bins of the sample there were less than 10 bubbles/droplets and the relative error was not calculated.

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C.4 Conclusion

All probability distribution functions of chord size were skewed with a preponderance of small particles relative the mean size and they tended to follow closely a log-normal distribution for void fraction less than 0.93. For larger void fractions, the data did not follow as well a log-normal shape, but the number of particles might be insufficient for the results to be statistically meaningful.

Comparative analyses were conducted to assess the validity of pseudo chord size and to investigate how the size of the probe sensor may affect the measurements. For CMH_05 data set, true chord sizes were compared with pseudo chord sizes, which were both measured with the same double tip probe (Ø=0.025mm). The pseudo chord sizes overestimated the true chord size by 1.98% in average for C < 0.97. For greater values of void fraction, the number of bubbles/droplets was not large enough to be a meaningful statistic population.

The influences of the probe sensor size were analysed by comparing true chord sizes obtained with Ø=0.025 mm and Ø=0.25 mm and by comparing the pseudo chord sizes obtained with Ø=0.025 mm and Ø=0.35 mm. The measurements were sensitive to the size of probe sensor. The smallest probe sensor could detect a larger number of bubbles, especially a greater number of bubbles/droplets with chords less than 1 mm. This is reflected in the probability distributions functions and it can be highlighted by the relative errors. The true chord sizes measured with a 0.25 mm probe sensor were in average 18% larger than those measured with a 0.025 mm probe sensor; the pseudo chord sizes overestimated by 28% in average with the 0.35 mm probe sensor compared to those measured with the 0.025 mm probe sensor.

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APPENDIX D- PROBABILITY DISTRIBUTION FUUNCTIONS OF TRUE CHORD SIZES

Air bubble and water droplet chords were measured for a wide range of flow conditions (Table 3-2). The data sets were analysed in terms of probability distribution functions of pseudo and true chord sizes (Appendix C). Herein, all data sets collected with a double-tip conductivity probe are presented in terms of true chord sizes. The results in the following Figures were analysed from the raw signal for dc/h=1.15, dc/h=1.33 and dc/h=1.45. The bubble/water chord size distributions are shown for each

measurement point for each discharge. The legend provides the flow rate dc/h, the distance from the

pseudo bottom in dimensionless terms y/Y90, the void fraction (C) , the bubble count rate (F) and the

number of bubbles/droplets recorded. In each histogram, the capitation and the legend provide the local air-water flow properties. The histogram columns present each the probability of chord size in a 0.5 mm chord interval. For example, the column labelled 1 represents the probability of chord sizes between 1 and 1.5 mm. Chords larger than 15 mm are regroup in the last column labelled > 15.

Flow rate dc/h=1.15 Step edge 10 Y90=56.1 mm Fmax=226.5 Hz V90=3.17 m/s Instrumentation: double tip probe [Ø=0.25 mm, ∆x= 7mm and ∆z= 1.4 mm]

0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 Air chord [mm] PD F y/Y90=0.249 C=0.061 F=104.7 Hz 4731 Bubbles y/Y90=0.303 C=0.062 F=103.2 Hz 4645 Bubbles y/Y90=0.214 C=0.065 F=108.8 Hz 4844 Bubbles

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0 0.05 0.1 0.15 0.2 0.25 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 Air chord[mm] PD F y/Y90=0.356 C=0.1 F=138.1 Hz 6213 Bubbles y/Y90=0.481 C=0.152 F=165.1 Hz 7429 Bubbles y/Y90=0.570 C=0.373 F=228.5 Hz 8000 Bubbles 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 Air chord [mm] PD F y/Y90=0.659 C=0.551 F=228.5 Hz 8000 Bubbles y/Y90=0.748 C=0.64 F=208.5 Hz 8000 Bubbles 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 W ate r chord [mm] PD

F y /Y90=0.837 C=0.763 F=146.4 Hz 6586 Droplets _{y /Y90=0.926 C=0.855 F=100.2 Hz 4508 Droplets} y /Y90=0.016 C=0.908 F=68.2 Hz 3070 Droplets

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0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 W ate r chord [mm] PD F

y/Y90=1.105 C=0.925 F=55.3 Hz 2488 Drop lets y/Y90=1.194 C=0.943 F= 42 Hz 1890 Droplets y/Y90=1.283 C=0.954 F=34.5 Hz 1554 Droplets 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 W ate r chord [mm] PD

F y/Y90=1.372 C=0.966 F=25.1 Hz 1129 Droplets _{y/Y90=1.461 C=0.973 F=22.4 Hz 1006 Droplets} y/Y90=1.551 C=0.981 F=15.8 Hz 710 Droplets 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 Wate r chord [mm] PD

F y/Y90=1.639 C=0.985 F=12.4 Hz 558 Droplets _{y/Y90=1.729 C=0.987 F=10.9 Hz 489 Droplets} y/Y90=1.818 C=0.991 F=8 Hz 362 Droplets

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 Wate r chord [mm] PD F y/Y90=1.907 C=0.992 F=6.6 Hz 295 Droplets y/Y90=1.996 C=0.994 F=5.7 Hz 255 Droplets y/Y90=2.085 C=0.995 F=4.7 Hz 212 Droplets 0 0.05 0.1 0.15 0.2 0.25 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 W ate r chord [mm] PD F y/Y90=2.174 C=0.996 F=3.6 Hz 161 Droplets y/Y90=2.264 C=0.997 F=2.9 Hz 131 Droplets y/Y90=2.442 C=0.998 F= 1.7 Hz 78 Droplets y/Y90=2.620 C=0.999 F=1.3 Hz 57 Droplets

Flow rate dc/h=1.33 Step edge 10 -Y90=63.3 mm Fmax=199.7Hz V90=3.415 m/s Instrumentation: double tip probe [Ø=0.25 mm, ∆x= 7mm and ∆z= 1.4 mm]

0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 Ai r chord [mm] PD F y/Y90=0.182 C=0.048 F=88 Hz 3962 Bubbles y/Y90=0.213 C=0.052 F=90 Hz 4049 Bubbles y/Y90=0.260 C=0.055 F=89.8 Hz 4039 Bubbles

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0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 Ai r chord [mm] PD F y/Y90=0.308 C=0.063 F=102 Hz 4590 Bubbles y/Y90=0.418 C=0.081 F=110.8 Hz 4986 Bubbles y/Y90=497 C=0.126 F=145.3 Hz 6537 Bubbles 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 Air chord [mm] PD F y/Y90=0.655 C=0.41 F=199.7 Hz 8000 Bubbles y/Y90=0.734 C=0.571 F=199.2 Hz 8000 Bubbles y/Y90=0.813 C=0.678 F=161.2 Hz 7256 Bubbles 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 Wate r chord [mm] PD F y/Y90=0.892 C=0.814 F=112.5 Hz 5061 Droplets y/Y90=0.971 C=0.866 F=76.8 Hz 3457 Droplets y/Y90=1.050 C=0.924 F=51 Hz 2295 Droplets

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 W ate r chord [mm] PD F y/Y90=1.129 C=0.944 F=38.8 Hz 1744 Droplets y/Y90=1.208 C=0.955 F=31.8 Hz 1430 Droplets y/Y90=1.287 C=0.967 F=23.7 Hz 1067 Droplets 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 W ate r chord [mm] PD F y/Y90=1.603 C=0.989 F=8.3 Hz 375 Droplet s y/Y90=1.682 C=0.991 F=7.1 Hz 319 Droplet s y/Y90=1.761 C=0.993 F=5.5 Hz 247 Droplet s 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 W ate r chord [mm] PD F y/Y90=1.366 C=0.974 F=20 Hz 898 Droplets y/Y90=1.445 C=0.981 F=15.6 Hz 702 Droplets y/Y90=1.524 C=0.985 F=12.3 Hz 554 Droplets

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 W ate r chord [mm] PD F y/Y90=1.919 C=0.995 F=4 Hz 178 Droplets y/Y90=1.998 C=0.997 F=2.8 Hz 128 Droplets y/Y90=2.077 C=0.996 F=2.7 Hz 121 Droplets 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 >15 W ate r chord [mm] PD F y/Y90=2.156 C=0.998 F=1.6 Hz 72 Droplets y/Y90=2.235 C=0.999 F=1.2 Hz 53 Droplets

Flow rate dc/h=1.45 Step edge 10 -Y90=71 mm Fmax=168.9 Hz V90=3.59 m/s Instrumentation: double tip probe [Ø=0.25 mm, ∆x= 7mm and ∆z= 1.4 mm]

0 0.05 0.1 0.15 0.2 0.25 0.3 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 >15 Air chord [mm] PD F y/Y90=0.141 C=0.011 F=23.5 Hz 1056 Bubbles y/Y90=0.169 C=0.013 F=26.5 Hz 1192 Bubbles y/Y90=0.197 C=0.017 F=29.6 Hz 1333 Bubbles

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0 0.05 0.1 0.15 0.2 0.25 0.3 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 >15 Air chord [mm] PD

F y/Y90=0.281 C=0.027 F=48.2 Hz 2170 Bubbles_{y/Y90=0.310 C=0.03 F=50 Hz 2251 Bubbles} y/Y90=0.380 C=0.041 F=58.9 Hz 2650 Bubbles 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 >15 Air chord [mm] PD F y/Y90=0.451 C=0.068 F=81.6 Hz 3671 Bubbles y/Y90=0.521 C=0.124 F=107.1 Hz 4820 Bubbles y/Y90=0.591 C=0.196 F=133.1 Hz 5990 Bubbles 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 >15 Air chord [mm] PD F y/Y90=0.662 C=0.348 F=165.9 Hz 7467 Bubbles y/Y90=0.732 C=0.521 F=168.9 Hz 7600 Bubbles

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 >15 W ate r chord [mm] PD

F y/Y90=0.803 C=0.715 F=136.8 Hz 6158 Droplets_{y/Y90=0.873 C=0.79 F=113.1 Hz 5091 Droplets} y/Y90=0.944 C=0.858 F=83.6 Hz 3763 Droplets 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 >15 W ate r chord [mm] PD F y/Y90=1.014 C=0.913 F=57 Hz 2566 Droplets y/Y90=1.084 C=0.935 F= 44.2 Hz 1990 Droplets y/Y90=1.155 C=0.951 F=36.5 Hz 1641 Droplets 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 >15 W ate r chord [mm] PD F y/Y90=1.225 C=0.965 F=27.7 Hz 1247 Droplets y/Y90=1.296 C=0.976 F=20.2 Hz 909 Droplets y/Y90=1.366 C=0.98 F=16.5 Hz 742 Droplets

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0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 >15 W ate r chord [mm] PD

F y/Y90=1.431 C=0.983 F= 13.9 Hz 625 Droplets_{y/Y90=1.501 C=0.988 F=9.9 Hz 446 Droplets} y/Y90=1.571 C=0.99 F=8.1 Hz 365 Droplets 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 >15 W ate r chord [mm] PD

F y/Y90=1.647 C=0.992 F=6.9 Hz 309 Droplets_{y/Y90=1.718 C=0.995 F=4.8 Hz 215 Droplets} y/Y90=1.929 C=0.997 F=2.9 Hz 178Droplets 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 >15 W ate r chord [mm] PD F y/Y90=2.074 C=0.998 F=2.6 Hz 192 Droplets y/Y90=2.213 C=0.999 F=1.8 Hz 130 Droplets

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APPENDIX E-BASIC RESULTS OF AUTO AND CROSS CORRELATION

ANALYSES

E.1 Presentation

Three discharges were investigated with two sensors separated by either a known transverse separation distance or a measured separation streamwise distance (Table 3-1). For all three flow rates, auto and cross correlation analyses were conducted. The results of auto and cross correlation analyses are detailed in the following paragraphs and they include: the maximum cross-correlation coefficient (Rxy)max, the characteristic time lag τ for which Rxx=0.5 (τ0.5)xx, the characteristic time lag τ for which

Rxy=0.5*(Rxy)max (τ0.5)xy, the time lag for which Rxy=0 (T0)xx, the time lag for which Rxy=0 (T0)xy. and

the integral time scales Txx Txy.

Table E.1 Summary of all discharge investigated with two probe sensor

Notation

C void fraction defined as the volume of air per unit volume of air and water; Cmean depth-average void fraction defined in terms of Y90: Cmean=1-d/Y90;

d equivalent clear water flow depth [m] defined as:

d

(

1 C

)

dy

90 . 0 y 0 y

∫

= =

−

=

;

d critical flow depth [m];

dc/h Instrumentation ∆x [mm] [mm] ∆z Step edge No 0 8.45 7 0 8.45 8 0 8.45 9 0 8.45 10 0 3.6 10 0 6.30 10 0 10.75 10 0 13.70 10 0 16.70 10 0 21.70 10 0 29.50 10 1.15 two single-tip probes [Ø=0.35 mm]

0 40.30 10 0 8.45 8 0 8.45 9 0 8.45 10 0 3.6 10 0 13.7 10 0 21.7 10 0 35.2 10 1.45 two single-tip probes [Ø=0.35 mm]

0 55.7 10 7.0 1.4 10 1.15 double-tip probe [Ø=0.25 mm] 9.6 1.4 10 7.0 1.4 7 7.0 1.4 8 7.0 1.4 9 1.33 double-tip probe [Ø=0.25 mm] 7.0 1.4 10 7.0 1.4 10 1.45 double-tip probe [Ø=0.25 mm] 9.6 1.4 10

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F bubble count rate [Hz] at a given cross-section;

Fmax maximum bubble count rate [Hz] at a given cross-section;

g gravity constant: g=9.80 m/s2_{in Brisbane, Australia;}

h height of steps [m] measured vertically; lstep length of steps [m];

lcav cavity length

l

_cav

=

h

2

+

l

2step

Qw water discharge [m3/s];

qw water discharge per unit width [m2/s];

Rxx normalised auto-correlation function (reference probe);

Rxy normalised cross-correlation function between two probe output signals;

(Rxy)max maximum cross-correlation between two probe output signals;

t scan time [s];

Tu turbulence intensity defined as: Tu=u’/V; Txx auto-correlation integral time scale defined as:

xx (R 0) xx xx 0 T R ( )d τ=τ = τ= =

∫

τ τ;

Txy cross-correlation integral time scale defined as:

xx xy xy max (R 0) xy xy (R (R ) ) T R ( )d τ=τ = τ=τ = =

∫

τ τ; τRxymax the characteristic time lag τ for which Rxy is maximum;

(τ0.5)xx the characteristic time lag τ for which Rxx=0.5;

(τ0.5)xy the characteristic time lag τ for which Rxy=0.5(Rxy)max ;

(τ0)xx the time lag for which Rxy=0 ;

(τ0) xy the time lag for which Rxy=0;

Tu turbulence intensity defined as: Tu=u’/V Vc critical velocity [m/s]

V

_c

=

3

gq

_w ;

V90 characteristic velocity [m/s] where the air concentration is 90%;

W channel width [m];

x distance along the channel bottom [m];

Y90 characteristic depth [m] where the void fraction is 90%.

θ slope of the channel

τ time lag;

Ø diameter [mm];

∆x

streamwise distance [mm] between the probe sensors

∆z transverse distance [mm] between the probe sensors

E.2 Auto and cross correlation analysis results

(41)

Good data Suspicious data Possibly incorrect data Data to be deleted

Table E-2. Legend

E.2.1 Flow rate dc/h=1.15

Experiments characteristics:

Channel: W=1 m, θ=22°, 10 steps h=0.10 m lstep=0.25 m

Flow rate dc/h=1.15

Qw=0.1164 m3/s

Location of the inception point: between step edge 6 and 7 Flow regime: transition-skimming flow

Run 060410a , 060410b , 060410c , 060411a , 060411b , 060411c , 060412a , 060412b , 060412c , 060413a , 060508a , 060508b , 060530a , 060613a

Instrumentation: Single-tip probe [Ø=0.35 mm]

double-tip probe [Ø=0.25 mm, ∆x= 7 mm, ∆x=9.6 mm and ∆z= 1.4 mm] Step edges

investigated No.

7, 8, 9, 10 Scan time [s] 45 Scan rate [Hz] 20000 E.2.1.1 Step edge 7

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.0 0.1 0.2 0.3 0.4 0.5 _C 0.6 0.7 0.8 0.9 1.0 (R x y )m a x 0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 0.0070 0.0080 Tx x , T x y [ s] (Rxy)max T xx T xy

Figure E.1 (Rxy)max, Txx, Txy distributions. Step edge 7. Flow rate: dc/h=1.15

(42)

File y/Y90 C F (t0)xx (t0.5)xx Txx Rxymax tRxymax (t0.5)xy (t0)xy Txy name [Hz] [s] [s] [s] [s] [s] [s] [s] 700 0.020 0.023 41.20 0.01243 0.00052 0.00114 0.040 -0.001 0.001 0.0131 0.00025 701 0.087 0.039 54.20 0.02785 0.00069 0.00185 0.029 0.000 0.015 0.0168 0.00024 702 0.154 0.045 55.90 0.01444 0.00674 0.00155 0.069 0.000 0.005 0.0155 0.00034 703 0.167 0.046 57.00 0.01746 0.00074 0.00164 0.057 -0.001 0.003 0.0110 0.00020 704 0.180 0.047 58.70 0.01656 0.00071 0.00141 0.054 0.000 0.004 0.0182 0.00024 705 0.221 0.058 63.20 0.02064 0.00080 0.00167 0.095 -0.001 0.004 0.0173 0.00043 706 0.261 0.078 75.20 0.03000 0.00096 0.00191 0.118 0.000 0.004 0.0022 0.00018 707 0.287 0.086 82.30 0.02496 0.00093 0.00198 0.154 0.000 0.005 0.0247 0.00097 708 0.354 0.173 118.00 0.02342 0.00130 0.00308 0.237 0.000 0.006 0.0224 0.00157 709 0.421 0.286 147.90 0.02395 0.00145 0.00367 0.308 0.000 0.007 0.0242 0.00240 710 0.488 0.427 161.90 0.02707 0.00166 0.00405 0.329 0.000 0.007 0.0254 0.00266 711 0.555 0.573 150.00 0.02586 0.00174 0.00381 0.337 0.000 0.006 0.0182 0.00242 712 0.622 0.687 130.80 0.02483 0.00165 0.00356 0.324 0.000 0.006 0.0245 0.00227 713 0.689 0.759 106.80 0.02305 0.00156 0.00331 0.300 0.000 0.005 0.0200 0.00168 714 0.755 0.791 90.40 0.02800 0.00161 0.00307 0.287 0.000 0.005 0.0225 0.00165 715 0.822 0.833 75.40 0.02480 0.00150 0.00305 0.278 0.000 0.004 0.0256 0.00137 716 0.889 0.866 62.90 0.03000 0.00150 0.00316 0.253 0.000 0.004 0.0300 0.00144 717 0.956 0.886 55.80 0.03000 0.00150 0.00305 0.263 0.000 0.004 0.0281 0.00145 718 1.023 0.907 47.50 0.03080 0.01450 0.00292 0.235 0.000 0.004 0.0333 0.00121 719 1.090 0.926 38.20 0.02700 0.00145 0.00306 0.250 0.000 0.004 0.0283 0.00133 720 1.156 0.932 35.40 0.02775 0.00148 0.00292 0.242 0.000 0.004 0.0281 0.00113 721 1.223 0.940 32.00 0.03000 0.00144 0.00303 0.220 0.000 0.004 0.0333 0.00117 722 1.290 0.960 23.10 0.04000 0.00134 0.00290 0.214 0.000 0.003 0.0211 0.00090 723 1.357 0.966 19.70 0.04000 0.00139 0.00278 0.206 0.000 0.003 0.0324 0.00110 724 1.424 0.971 16.40 0.03190 0.00136 0.00312 0.211 0.000 0.003 0.0245 0.00102 725 1.491 0.980 11.80 0.03550 0.00133 0.00288 0.175 -0.001 0.002 0.0211 0.00059 726 1.557 0.986 8.70 0.04000 0.00126 0.00317 0.181 0.000 0.004 0.0350 0.00124 727 1.624 0.986 8.60 0.04000 0.00119 0.00309 0.195 0.000 0.003 0.0319 0.00104 728 1.691 0.987 7.50 0.04000 0.00138 0.00338 0.198 0.000 0.004 0.0277 0.00110 729 1.758 0.991 5.30 0.04000 0.00119 0.00363 0.167 -0.001 0.004 0.0400 0.00125 730 1.758 0.993 4.20 0.04000 0.00135 0.00492 0.144 0.000 0.003 0.0423 0.00087 731 1.825 0.994 3.90 0.05165 0.00130 0.00527 0.178 0.000 0.003 0.0517 0.00078 732 1.892 0.995 3.30 0.04000 0.00142 0.00705 0.180 0.000 0.003 0.0400 0.00175 733 1.959 0.995 3.10 0.05500 0.00124 0.00735 0.134 0.001 0.002 0.0550 0.00085

Table E.3-Basic results at step edge 7; Flow rate: dc/h=1.15

Instrumentation: two single-tip probes [Ø=0.35 mm ∆z=8.45 mm].

E.2.1.2 Step edge 8

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.0 0.1 0.2 0.3 0.4 0.5 C 0.6 0.7 0.8 0.9 1.0 (R xy) m a x 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 T xx, T x y [ s] (Rxy)max T xx T xy

(43)

File y/Y90 C F (t0)xx (t0.5)xx Txx Rxymax tRxymax (t0.5)xy (t0)xy Txy name [Hz] [s] [s] [s] [s] [s] [s] [s] 800 0.0233 0.01 28 801 0.1008 0.022 43.8 802 0.1473 0.031 53.7 0.02107 0.00374 0.0002 0.03411 -0.00115 0.00356 0.01526 0.00023 803 0.1783 0.03 53.2 0.0108 0.0005 0.0006 0.02361 -0.00165 0.00576 0.01065 0.00016 804 0.2093 0.033 56.8 0.01584 0.00051 0.0008 0.04043 -0.00045 0.00405 0.01547 0.00021 805 0.2558 0.036 56.7 0.0163 0.00055 0.0007 0.04385 0.0002 0.00405 0.01573 0.0002 806 0.3023 0.039 59.3 0.0098 0.00056 0.0006 0.04083 -0.00015 0.00303 0.0127 0.00017 807 0.3333 0.044 63.8 0.0155 0.00063 0.0007 0.07021 -0.0007 0.0037 0.01937 0.00035 808 0.4109 0.069 82.5 0.02267 0.00077 0.0011 0.12028 0.00005 0.00464 0.01684 0.00067 809 0.4884 0.129 116.2 0.02735 0.00105 0.0022 0.20808 -0.0002 0.0056 0.0266 0.0016 810 0.5659 0.226 151.1 0.02916 0.00145 0.0028 0.31049 -0.00025 0.00775 0.03 0.00254 811 0.6434 0.452 177.8 0.02934 0.00176 0.0034 0.36818 0 0.00735 0.0291 0.00335 812 0.7209 0.574 173.2 0.02432 0.00186 0.0031 0.37982 0 0.00754 0.02505 0.00328 813 0.7984 0.699 138.5 0.02785 0.0019 0.0035 0.38638 -0.00025 0.00676 0.02785 0.00333 814 0.876 0.815 99.1 0.0239 0.00165 0.0026 0.3348 -0.00045 0.00566 0.0239 0.00259 815 0.9535 0.879 68.1 0.02814 0.00154 0.0023 0.30126 -0.0004 0.0046 0.02814 0.00201 816 1.031 0.915 49.9 0.0242 0.0042 0.0012 0.26996 -0.0005 0.00435 0.02664 0.00177 817 1.1085 0.93 41 0.02508 0.00134 0.0019 0.24285 -0.0004 0.00434 0.01887 0.00152 818 1.186 0.95 29.7 0.03 0.00135 0.0021 0.23491 0 0.00385 0.03 0.00144 819 1.2636 0.959 24.7 0.025 0.00135 0.0017 0.21858 -0.0002 0.00335 0.2094 0.00113 820 1.3411 0.965 20.7 0.03 0.0013 0.0018 0.18631 -0.0004 0.0038 0.03 0.00108 821 1.4186 0.978 14.7 0.03 0.00116 0.0018 0.19919 -0.0002 0.00292 0.02465 0.00108 822 1.4961 0.98 13 0.03 0.0119 0.0003 0.19262 0.00015 0.00268 0.02354 0.00091 823 1.5736 0.986 8.9 0.03 0.0104 0.0004 0.19043 -0.0006 0.01194 0.01683 0.0007 824 1.6512 0.989 7.3 0.025 0.00114 0.0016 0.12837 -0.00075 0.00234 0.02297 0.00063 825 1.7287 0.99 6.1 0.025 0.00125 0.0018 0.1467 -0.00045 0.00272 0.02473 0.00066 826 1.8062 0.992 5.6 0.025 0.0012 0.0017 0.16565 -0.00025 0.00265 0.01595 0.00064 827 1.8837 0.995 3.9 0.025 0.00103 0.0017 0.1186 0.00075 0.00245 0.01255 0.00028 828 1.9612 0.996 3.1 0.025 0.00104 0.0017 0.11617 -0.0005 0.00153 0.02 0.00043 829 2.0388 0.996 3.1 0.03 0.00107 0.002 0.17976 -0.0006 0.00164 0.02168 0.00057 830 2.0388 0.997 2 0.025 0.00118 0.0022 0.1363 -0.0006 0.0017 0.025 0.00048 831 2.1163 0.998 1.7 0.0162 0.0011 0.0019 0.11722 -0.00195 0.0023 0.00585 0.00048 832 2.1938 0.998 1.8 0.02 0.00105 0.0021 0.14758 -0.0008 0.00276 0.0238 0.00087 833 2.2713 0.998 1.4 0.03 0.001 0.0031 0.07367 0.00035 0.00205 0.01 0.00016 834 2.3488 0.999 0.7

Instrumentation: two single-tip probes side by side [Ø=0.35 mm ∆z=8.45 mm].

E.2.1.3 Step edge 9

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.0 0.1 0.2 0.3 0.4 0.5 C 0.6 0.7 0.8 0.9 1.0 (R xy) m a x 0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 0.0070 Tx x , Tx y [ s] (Rxy)max T xx T xy

Figure E.3 (Rxy)max, Txx, Txy distributions. Step edge 9. Flow rate: dc/h=1.15

(44)

File y/Y90 C F (t0)xx (t0.5)xx Txx Rxymax tRxymax (t0.5)xy (t0)xy Txy name [Hz] [s] [s] [s] [s] [s] [s] [s] 900 0.0236 0.032 57.6 0.02403 0.00048 0.0012 0.03283 -0.0001 0.00473 0.02937 0.00024 901 0.1024 0.037 62.4 0.02847 0.00057 0.0012 0.0359 -0.0019 0.0031 0.01463 0.00026 902 0.1496 0.044 64.8 0.02703 0.00063 0.0012 0.0516 0.00055 0.00453 0.02665 0.00032 903 0.1811 0.049 70.5 0.01803 0.00065 0.001 0.06293 -0.00075 0.00355 0.01576 0.00044 904 0.2598 0.056 74.2 0.029 0.007 0.0003 0.09586 -0.00025 0.00283 0.02273 0.00047 905 0.3071 0.069 83.9 0.02003 0.00078 0.0012 0.1234 0.00025 0.00393 0.01728 0.00061 906 0.3228 0.072 82.5 0.02637 0.00085 0.0012 0.13688 0.00035 0.00394 0.02537 0.00064 907 0.3386 0.081 90.4 0.01658 0.00085 0.0013 0.1546 -0.00025 0.0047 0.02463 0.001 908 0.4173 0.132 115 0.02093 0.0012 0.0021 0.23339 0 0.0048 0.02123 0.00145 909 0.4961 0.226 149.4 0.02553 0.00146 0.0029 0.33052 0 0.00627 0.02343 0.00266 910 0.5748 0.373 177.8 0.029 0.00175 0.0033 0.38159 -0.00025 0.00733 0.02877 0.00334 911 0.6535 0.524 165.7 0.02687 0.00211 0.0032 0.41647 -0.00005 0.00711 0.02553 0.0035 913 0.7323 0.662 142.4 0.02813 0.00185 0.0027 0.36379 -0.0003 0.0049 0.02243 0.00266 914 0.811 0.768 109.2 0.02117 0.0016 0.0022 0.33134 -0.00025 0.00477 0.01987 0.00213 915 0.8898 0.825 90.7 0.01963 0.0016 0.002 0.33018 -0.00005 0.00436 0.01973 0.00186 916 0.8898 0.867 71.2 0.02364 0.0016 0.0021 0.31408 -0.0002 0.00394 0.02653 0.00179 917 0.9685 0.887 61.3 0.02496 0.02264 0.0019 0.29519 -0.0003 0.00363 0.01577 0.00156 918 1.0472 0.917 48.3 0.02694 0.0014 0.0018 0.27457 -0.0004 0.00386 0.01898 0.00148 919 1.126 0.949 33.2 0.029 0.0014 0.002 0.26372 -0.00005 0.00395 0.02675 0.00139 920 1.2047 0.957 28.6 0.029 0.00136 0.0019 0.27358 -0.00025 0.0033 0.029 0.00153 921 1.2835 0.968 21.5 0.029 0.00137 0.0021 0.23939 -0.0003 0.00383 0.0278 0.00146 922 1.3622 0.977 16.6 0.0285 -0.0013 0.0038 0.23774 -0.0003 0.00313 0.0206 0.00106 923 1.4409 0.976 16.6 0.029 0.00135 0.0018 0.22001 -0.0005 0.00304 0.02167 0.00122 924 1.5197 0.985 12 0.0283 0.00115 0.0019 0.21219 -0.00025 0.0027 0.0283 0.00113 925 1.5984 0.986 11 0.029 0.00123 0.0017 0.16576 -0.00005 0.00268 0.01803 0.00067 926 1.7559 0.99 7.1 0.029 0.00127 0.0018 0.17029 -0.00135 0.003 0.0217 0.00101 927 1.8346 0.993 5.6 0.029 0.0011 0.0018 0.15927 -0.0008 0.00277 0.02264 0.00076 928 1.9134 0.995 4.1 0.029 0.00104 0.0023 0.19602 -0.0002 0.02194 0.01 0.00052 929 1.9921 0.996 3.5 0.029 0.00105 0.0023 0.15834 0.00085 0.00275 0.01515 0.00052 930 2.0709 0.996 2.9 0.029 0.0012 0.0024 0.14063 -0.00005 0.00466 0.029 0.00069 931 2.0709 0.997 2.4 0.029 0.0012 0.0021 0.14211 -0.0005 0.00224 0.024 0.00067 932 2.1496 0.998 1.6 0.029 0.00106 0.0046 0.1496 0.00025 0.00206 0.01 0.00038 933 2.2283 0.999 1.4 0.029 0.00106 0.0035 0.13201 -0.00115 0.0042 0.01 0.0005 934 2.3071 0.999 0.8 0.029 0.00146 0.0065 0.15455 0.0001 0.00387 0.029 0.00095

E.2.1.4 Step edge 10 (∆z=8.45 mm)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.0 0.1 0.2 0.3 0.4 0.5 C 0.6 0.7 0.8 0.9 1.0 (R xy) m a x 0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 0.0070 0.0080 Tx x , Tx y [ s] (Rxy)max T xx T xy

(45)

File y/Y90 C F (t0)xx (t0.5)xx Txx Rxymax tRxymax (t0.5)xy (t0)xy Txy name [Hz] [s] [s] [s] [s] [s] [s] [s] 0 0.0328 0.021 47.8 1 0.1148 0.035 68.2 2 0.1639 0.039 71 0.01825 0.00052 0.0012 0.03986 -0.0005 0.00647 0.01228 0.00023 3 0.1967 0.041 71.8 0.01238 0.00053 0.0011 0.03056 0.0006 0.00365 0.01668 0.00019 4 0.2295 0.044 69.3 0.01485 0.00056 0.0012 0.05807 -0.00065 0.00363 0.01463 0.00027 5 0.2787 0.045 76.9 0.01374 0.00053 0.001 0.0598 0.00045 0.0028 0.01085 0.00019 6 0.3279 0.054 82.3 0.01294 0.0006 0.0013 0.08288 0.00025 0.00608 0.01893 0.00048 7 0.3607 0.069 94.9 0.01678 0.0065 0.0015 0.08935 0.0001 0.00386 0.024 0.00049 8 0.4426 0.106 122.8 0.02568 0.00081 0.0019 0.18125 0 0.00411 0.0177 0.00091 10 0.6066 0.327 194.9 0.029 0.0016 0.0042 0.35924 -0.0003 0.00686 0.029 0.00309 11 0.6885 0.515 194 0.03 0.0065 0.0053 0.41234 0.0001 0.00386 0.03 0.00386 12 0.7705 0.636 171.6 0.03 0.00214 0.005 0.42935 -0.00005 0.00738 0.03443 0.00383 13 0.8525 0.798 120.3 0.02903 0.00176 0.0041 0.379 -0.0001 0.00614 0.0333 0.00285 14 0.9344 0.874 79.4 0.0325 0.0017 0.0042 0.38045 -0.00015 0.00517 0.02984 0.00285 15 1.0164 0.91 56.5 0.0285 0.00158 0.0034 0.31911 0 0.00423 0.02835 0.00184 16 1.0984 0.933 44 0.0296 0.0015 0.0033 0.28944 -0.00025 0.00413 0..0227 0.00151 17 1.1803 0.955 32.5 0.02313 0.00138 0.0027 0.26079 -0.0001 0.00363 0.01983 0.00128 18 1.2623 0.961 28.2 0.0279 0.0013 0.0029 0.25575 -0.0002 0.00384 0.0268 0.00139 19 1.3443 0.965 22.4 0.03135 0.00132 0.0029 0.24785 -0.0006 0.00324 0.03205 0.00127 20 1.4262 0.973 20.5 0.035 0.00122 0.0027 0.22723 -0.0001 0.0024 0.035 0.00111 21 1.5082 0.981 16.7 0.025 0.0012 0.0026 0.18672 0.00035 0.00322 0.01755 0.00081 22 1.5902 0.987 11.2 0.0376 0.0012 0.0028 0.17111 0.00025 0.00305 0.035 0.0009 23 1.6721 0.988 10.1 0.035 0.0012 0.0027 0.1699 -0.00075 0.00428 0.0235 0.00084 24 1.7541 0.993 4.8 0.02675 0.00125 0.003 0.14544 -0.0009 0.0035 0.033 0.00088 25 1.8361 0.993 6.4 0.035 0.00118 0.0028 0.12092 0.00035 0.00214 0.0333 0.0006 26 1.918 0.996 3.7 0.03 0.00116 0.0029 0.15908 -0.0009 0.00217 0.03 0.00069 27 2 0.995 3.6 0.026 0.0013 0.0028 0.1313 0.00075 0.00305 0.026 0.00066 28 2.082 0.997 2.9 0.03 0.00104 0.0033 0.14795 0.00035 0.01867 0.03 0.00058 29 2.1639 0.998 2.3 0.03 0.001 0.0038 0.13718 0.0006 0.00217 0.03 0.00078 30 2.1639 0.999 1.4 0.03 0.00105 0.0057 0.11626 -0.0007 0.00207 0.03 0.00063 31 2.2459 0.998 1.2 0.03 0.0013 0.0062 0.12224 -0.00015 0.00134 0.03 0.00051 32 2.3279 0.998 1.9 0.03 0.00129 0.0069 0.15187 0.0012 0.00282 0.03 0.00074 33 2.4098 0.999 1.4 0.035 0.0011 0.0067 0.08116 -0.0009 0.0017 0.035 0.0005 34 2.4918 0.999 1.2 0.035 0.0012 0.0072 0.09943 -0.00025 0.0016 0.02045 0.00018

E.2.1.5 Step edge 10 (∆z=3.6 mm)

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.0 0.1 0.2 0.3 0.4 0.5 C 0.6 0.7 0.8 0.9 1.0 (R xy) m a x 0.0000 0.0010 0.0020 0.0030 0.0040 0.0050 0.0060 T xy [ s] (Rxy)max T xy

Figure E.5 (Rxy)max, Txx, Txy distributions. Step edge 10; Flow rate: dc/h=1.15

(46)

File y/Y90 C F Rxymax tRxymax (t0.5)xy (t0)xy Txy name [Hz] [s] [s] [s] [s] 0 0.0348 0.03 57.5 0.15819 0 0.00245 0.02527 0.00062 1 0.1217 0.031 55.7 0.16927 0 0.0216 0.01777 0.00054 2 0.1739 0.04 65.2 0.1958 0 0.00228 0.01008 0.00058 3 0.2087 0.035 56.8 0.19247 0.00005 0.00194 0.0197 0.00054 4 0.2435 0.043 66.7 0.22197 0.0002 0.00217 0.01513 0.00062 5 0.2957 0.057 77.8 0.28246 0.00005 0.00218 0.01513 0.0009 6 0.3478 0.095 105.5 0.38968 0 0.00293 0.0186 0.00173 7 0.3826 0.114 117.2 0.41402 0.0001 0.00294 0.02455 0.00203 8 0.4696 0.169 135.8 0.50659 0.00005 0.00386 0.02413 0.00303 9 0.5565 0.34 184.1 0.59264 0.00005 0.00493 0.02683 0.00429 10 0.6435 0.514 185.1 0.62735 0.00005 0.00516 0.029 0.00501 11 0.7304 0.68 152.9 0.63393 0.00005 0.00474 0.02423 0.0044 12 0.8174 0.785 108.6 0.62889 0.00005 0.00408 0.02297 0.00396 13 0.9043 0.847 84.3 0.61151 0 0.00336 0.02654 0.00346 14 0.9913 0.896 60.4 0.59416 0.00005 0.00302 0.02785 0.00312 15 1.0783 0.927 47.3 0.57722 0 0.00266 0.02917 0.00262 16 1.1652 0.935 37.8 0.57019 0 0.00266 0.02917 0.00276 17 1.2522 0.953 30.4 0.55456 0 0.00236 0.0251 0.00244 18 1.3391 0.967 23.9 0.52458 0 0.0023 0.0257 0.00208 19 1.4261 0.973 18.8 0.53425 0 0.00213 0.025 0.00209 20 1.513 0.981 14.8 0.51269 0 0.00233 0.025 0.00245 21 1.6 0.984 12.4 0.52067 0.0001 0.00198 0.0289 0.00196 22 1.687 0.988 9.1 0.53526 0 0.00232 0.0289 0.00245 23 1.7739 0.993 4.6 0.46998 0 0.00197 0.0289 0.00206 24 1.8609 0.993 6.6 0.48354 0 0.00195 0.0289 0.00181 25 1.9478 0.995 5 0.4706 0.0001 0.00187 0.029 0.00195 26 2.0348 0.995 4.3 0.46175 0 0.00184 0.029 0.00213 27 2.1217 0.997 2.5 0.47025 -0.0001 0.00153 0.029 0.00217 28 2.2087 0.998 1.8 0.58698 0 0.00157 0.029 0.00244 29 2.2957 0.999 1.3 0.4485 0.0001 0.000173 0.029 0.00217 30 2.2957 0.999 0.7 0.29201 0 0.00197 0.029 0.00212 31 2.3826 0.999 0.9 0.31836 -0.00015 0.00168 0.025 0.00211 32 2.4696 0.999 0.9 0.41464 0.0002 0.00154 0.025 0.00213 33 2.5565 0.999 1 0.28518 0.0002 0.00163 0.025 0.00185 34 2.6435 1 0.5

E.2.1.6 Step edge 10 (∆z=6.30 mm)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.0 0.1 0.2 0.3 0.4 0.5 C 0.6 0.7 0.8 0.9 1.0 (R xy) m a x 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 Tx y [ s] (Rxy)max T xy

(47)

File y/Y90 C F Rxymax tRxymax (t0.5)xy (t0)xy Txy name [Hz] [s] [s] [s] [s] 0 0.0331 0.043 79.8 0.03375 -0.0003 0.00358 0.01675 0.00021 1 0.1157 0.043 78.9 0.02817 -0.0002 0.003358 0.00663 8.2E-05 2 0.1653 0.051 86 0.03918 0.00025 0.0022 0.012256 0.00016 3 0.1983 0.051 87.3 0.05095 -0.00075 0.0024 0.012 0.00018 4 0.2314 0.051 85.5 0.04899 -0.0008 0.00352 0.008158 0.00015 5 0.281 0.062 95.2 0.0705 -0.00005 0.00289 0.01895 0.0003 6 0.3306 0.069 102 0.08587 0.0001 0.0023 0.01095 0.00026 7 0.3636 0.082 110.1 0.10547 0 0.00314 0.02535 0.0005 8 0.4463 0.109 130.7 0.187 -0.0055 0.00397 0.021 0.00093 9 0.5289 0.192 168 0.30861 -0.00045 0.00514 0.02865 0.0021 10 0.6116 0.403 177.8 0.4128 -0.00075 0.00644 0.02796 0.00339 11 0.6942 0.572 177.8 0.44847 -0.00045 0.0067 0.0269 0.00373 12 0.7769 0.709 159.9 0.45044 -0.00045 0.006356 0.02586 0.00329 13 0.8595 0.816 114.1 0.43218 -0.0004 0.00552 0.02738 0.003 14 0.9421 0.867 86.9 0.35142 -0.00035 0.00432 0.0216 0.00212 15 1.0248 0.917 57.1 0.33498 -0.00005 0.00374 0.02135 0.0018 16 1.1074 0.94 43.5 0.29972 -0.00035 0.00346 0.0276 0.00151 17 1.1901 0.949 35.2 0.26185 -0.00065 0.00296 0.0205 0.00103 18 1.2727 0.966 26.3 0.26756 -0.00095 0.00263 0.02373 0.00094 19 1.3554 0.974 21 0.26158 -0.00035 0.00237 0.025 0.00103 20 1.438 0.976 17.7 0.27055 -0.0006 0.00224 0.0154 0.00082 21 1.5207 0.983 14 0.2413 -0.0006 0.00195 0.02585 0.0007 22 1.6033 0.987 11.1 0.22247 -0.00085 0.00186 0.0154 0.00045 23 1.686 0.989 8.6 0.20558 0.00015 0.00232 0.026 0.00081 24 1.7686 0.992 6.6 0.22169 -0.0005 0.00176 0.025 0.00058 25 1.8512 0.994 5.1 0.20861 -0.0007 0.00182 0.0141 0.00061 26 1.9339 0.996 3.4 0.16207 -0.0006 0.0014 0.01305 0.00026 27 2.0165 0.996 3.8 0.17983 -0.00095 0.00128 0.02735 0.00029 28 2.0992 0.997 2.4 0.21685 -0.00015 0.0017 0.025 0.00042 29 2.1818 0.997 2.8 0.19021 0 0.0019 0.0223 0.00053 30 2.2645 0.998 1.6 0.18482 -0.0005 0.00155 0.025 0.00059 31 2.3471 0.998 1.8 0.15723 0.0005 0.00255 0.0175 0.00036 32 2.4298 0.999 1.4 0.11171 0.00045 0.00198 0.025 0.00062 33 2.5124 0.999 1.2 0.13433 0 0.00812 0.02 0.00031 34 2.595 0.999 0.9 0.08693 0.0012 0.00615 0.01795 0.00041 Table E.8-Basic results at step edge 10; Flow rate: dc/h=1.15

E.2.1.7 Step edge 10 (∆z=10.75 mm)

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.0 0.1 0.2 0.3 0.4 0.5 _C 0.6 0.7 0.8 0.9 1.0 (R x y )m a x 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 0.0040 0.0045 Tx y [ s] (Rxy)max T xy

Figure E.7 (Rxy)max, Txx, Txy distributions. Step edge 10; Flow rate: dc/h=1.15