掺气坎下游空腔区气泡特性实验研究 - DiVA portal1135390/...Ebba Wargsjö Gunnarsson UPTEC ES 17 028 Examensarbete 30 hp Juni 2017 Teknisk- naturvet enskaplig fakultet

EXPERIMENTAL INVESTIGATION OF THE

AIR-WATER FLOW PROPERTIES IN THE

CAVITY ZONE DOWNSTREAM A CHUTE

AERATOR

掺气坎下游空腔区气泡特性实验研究

Albin Hedehag Damberg Ebba Wargsjö Gunnarsson

UPTEC ES 17 028

Examensarbete 30 hp

Juni 2017

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

EXPERIMENTAL INVESTIGATION OF THEAIR-WATER FLOW PROPERTIES IN THE CAVITYZONE DOWNSTREAM A CHUTE AERATORAlbin Hedehag Damberg; Ebba Wargsjö Gunnarsson

Chute aerators are widely used in spillways to avoid cavitation damage. When the water flow passes the aerator, two jets form – upper and lower jet.

The purpose of this thesis has been to study the effects from the aerator by conducting experiments in a model with a flow depth large enough to ensure that the upper and lower jet remain separated. This means that the effects from the self-aeration at the upper surface has no effect on the process in the lower jet, thus making it possible to quantify the effects from the aerator. This thesis has also provided information of the bubble formation in the lower jet to aid in the ongoing research at Sichuan University. The following questions were set up for this thesis:

• What is cavitation and how is it harmful?• What is the working principle of an aerator?• How is air concentration and bubble frequency distributed in the flow?• How well do the experimental results coincide with theoretical calculations?• How are air bubbles formed and transported within the flow?

The effects from the aerator have been quantified by measuring the air concentration and bubble frequency throughout the cavity zone. The model was modified and the velocity was varied between the experiments to study how different parameters effected the aeration. The results indicate that much air is being entrapped in the lower surface, but only a small amount of the entrapped air is being entrained into the flow and that the bubble frequency increases with both distance from the aerator and with an increased flow velocity. No difference in behaviour was noticed between the different modifications of the model. The bubble formation was studied by recording the flow with a high-speed camera. These recordings were used to obtain data about important parameters for the ongoing research at Sichuan University.

ISSN: 1650-8300, UPTEC ES17 028Examinator: Petra JönsonÄmnesgranskare: Per NorrlundHandledare: James Yang

SAMMANFATTNING

För att undvika kavitationsskador i utskoven, vid vilken vattennivån i en dam kan kontrolleras

vid ett vattenkraftverk, installeras speciella luftningsanordningar för att tillsätta luft i vattnet.

Denna anordning separerar flödet från utskovsbotten, vilket gör att det bildas ett hålrum mellan

botten och flödet där luft kan komma in via luftkanaler kopplade till atmosfären. Luften har en

dämpande effekt på de krafter som uppstår när kavitationsbubblor imploderar, vilket minskar

den negativa effekten på utskovens bottenyta. När vattnet passerar anordningen så fångas luft

in både via självluftning vid den övre fria vattenytan samt den nedre ytan som har kontakt med

hålrummet. Detta bildar två separata strålar som skiljs åt av ett område med vatten som är fritt

från luft.

Detta arbete har gått ut på att enbart undersöka den nedre strålen för att se vilken effekt dessa

luftningsanordningar har. Detta har genomförts med hjälp av experiment i en modell där

luftkoncentration och bubbelfrekvens har mätts. Genom att ha ett tillräckligt stort flödesdjup

har det säkerställts att de båda strålarna skiljs åt av det luftfria området under hela

luftningsprocessen, vilket innebär att självluftningen vid den fria vattenytan inte har påverkat

den luftning som sker vid den nedre vattenytan. Dessa experiment har utförts vid

vattenlaboratoriet på Sichuan University i Chengdu, Kina och är en del av deras pågående

forskning. Flödet har även filmats med en höghastighetskamera för att kunna studera hur

bubblor bildas och färdas nedströms. Dessa filmer har sedan använts för att kunna mäta

bubblornas diameter och avståndet till vart i flödet de bildas. Dessa mått är även de en del av

den pågående forskningen på Sichuan University.

Resultaten från experimenten visar på att det är mycket luft som fastnar vid den nedre ytan då

luftkoncentrationen i, och strax ovanför, ytan är hög. Däremot sjunker luftkoncentrationen

snabbt med avstånd från botten, vilket visar på att mycket av den luft som fastnar vid ytan inte

färdas inåt i flödet. Resultaten visar även att bubbelfrekvensen ökar med både hastighet och

avstånd från luftningsanordning. De mätningar som gjorts från filmerna har varit för få att några

slutsatser ska kunna dras, förhoppningen är däremot att de ska kunna bidra till forskningen

kring hur bubblorna bildas vid den nedre ytan.

EXECUTIVE SUMMARY

The aim of this thesis was to investigate the effects from an aerator by measuring the air

concentration in the lower jet and to study the bubble formation in the lower jet. The results

indicated that much air is being entrapped in the lower surface and just above, where a high air

concentration could be measured. However, the results also showed that the air concentration

is decreasing quickly with the distance from the bottom, which indicates that much of the

entrapped air in the lower surface does not travel further into the flow. The results also showed

that the amount of bubbles is increasing with both velocity and distance from the aerator.

Regarding the bubble formation in the lower jet additional measurements are required before

any straight conclusions can be drawn.

ACKNOWLEDGEMENT

First, we would like to express our great gratitude to Professor Shanjun Liu for inviting us to

Sichuan University and the department of State Key Laboratory of Hydraulics and Mountain

River Engineering in Chengdu, China.

Our supervisor at Sichuan University, Dr. Ruidi Bai, for whom we are very grateful, has been

the key person during our project process in China. We would like to thank him for all the

guidance and help with the experimental investigations conducted for this thesis and for taking

time to answer questions and give advice. Dr. Rudi Bai and his colleague Dr. WangRu Wei

have also been very supportive with the arrangements of social activities and trips. Furthermore,

we would like to state our great gratitude to Ms. Hera Shi for her friendship and humble

treatment, she has been very helpful in the social life during our stay in China.

Thanks to Professor James Yang at Vattenfall R&D and the Royal Technical Institute who has

been responsible for arranging this trip and thesis project together with the Professor at Sichuan

university. Without him, this project would not have been possible. We are also very thankful

to our supervisor at Uppsala University, Dr. Per Norrlund, for his professional guidance,

feedback and advice on our report during the writing process.

At last, we would like to thank the research and knowledge company Energiforsk who founded

this project and contributed to the implementation of this project and Kammarkollegiet for

providing health insurance during our stay in Chengdu.

Chengdu, May 24th, 2017

Albin Hedehag Damberg and Ebba Wargsjö Gunnarsson

NOMENCLATURE

DENOMINATION SYMBOL UNIT

GREEK

Density 𝜌 𝑘𝑔/𝑚3

Downstream chute angle 𝛼 °

Dynamic viscosity 𝜇 𝑁𝑠/𝑚2

Kinematic viscosity 𝜈 𝑚2/𝑠

Loss coefficient 𝜉 −

Spillway angle 𝜃0 °

Spread angle 𝜓 °

Surface tension 𝜎 𝑁/𝑚

LATIN

Average air concentration 𝐶𝑎 −

Air bubble frequency 𝑓 𝑠−1

Air concentration 𝐶 −

Air discharge 𝑄𝑎 𝑚3/𝑠

Approach flow Weber number 𝑊0 −

Bottom air concentration 𝐶𝑏 −

Bubble diameter 𝑑 𝑐𝑚

Cavity length 𝐿 𝑚

Coefficient of determination 𝑅2 -

Distance between needle tips 𝑑𝑙 𝑚𝑚

Distance from bottom where C=0 𝑧0 𝑚

Distance from bottom where C=0.5 𝑧50 𝑚

Distance from bottom where C=0.9 𝑧90 𝑚

Distance from bottom where C=1 𝑧1 𝑚

Error function 𝑒𝑟𝑓 −

Froude number 𝐹𝑟 −

Flow depth ℎ0 𝑚

Offset height ℎ𝑠 𝑚

Onset distance 𝐷 𝑐𝑚

Outlet velocity 𝑉0 𝑚/𝑠

Independent variable 𝑢 −

Maximum height 𝑧𝑚 𝑚

Reynolds number 𝑅𝑒 −

Sample frequency 𝑓𝑠 𝑘𝐻𝑧

Scanning time 𝑡𝑠 𝑠

Time duration for bubble formation 𝑡 𝑚𝑠

Turbulent diffusivity 𝐷𝑡 𝑚2/𝑠

Turbulent velocity 𝑣′ 𝑚/𝑠

Unit air discharge 𝑞𝑎 𝑚3/(𝑚 ∗ 𝑠)

Weber number 𝑊 −

GLOSSARY

Advection

Used to describes the transport through the flow of the fluid. In hydrology, advection is used to

describe water that are transported with sea currents.

Air-detrainment

The process of the entrained air being transported out of the flow and out to the atmosphere

due to the bubble’s rise velocity.

Air discharge

Amount of air entrained into the water from the air inlet over time.

Air-entrainment

The process of the entrapped air in the surface being transported into the flow.

Air-entrapment

The process of air being trapped into the water body, but only in the surface.

Back water

Upon impact with the bottom, part of the water flow will deflect upstream. The amount of

back water depends on the impact angle.

Bottom rollers

Water at the bottom, just upstream the impact point, gets stuck in a local recirculating motion

and does not travel downstream. Air can become trapped in this motion, which causes the air

to not travel downstream either.

Black water

Water with no air entrained.

Bubble rise velocity

Due to air having lower density than water, the air bubbles experience an elevating force

moving the bubbles upward towards the surface. The speed with which it moves towards the

surface is the rise velocity.

Chute

A sloping channel used for transporting a medium, in this case water, to a lower level.

Chute aerator

Devices used to supply air to the water at the chute bottom to prevent cavitation damage.

Coefficient of determination

Used to evaluate how well experimental data coincides with theoretical values. A value as

close to 1 as possible is desired.

Diffusion

A spontaneous process that occurs when molecules that have characteristics separated from the

surroundings being spread, mixed and evens out. This spreading process occurs most commonly

for liquid or gas.

Froude number

A dimensionless parameter used to describe different flow regimes of open channel flow. The

Froude number is the ratio between the inertial and gravitational forces.

Head

Available energy due to vertical change in elevation between two points in the water-body.

Offset

A type of a bottom aerator which resembles a threshold. The threshold separates the flow

from the bottom as it passes

Onset distance

The distance from the offset to the aeration onset.

Phase-detection needle probe

A measurement instrument consisting of two identical tips with an internal concentric

electrode that uses the conductivity between air and water to obtain data about the air

concentration and the amount of bubbles per second in a certain point.

Reynolds number

A dimensionless parameter in fluid mechanics used to predict the transition from laminar to

turbulent flow. The Reynolds number describes the ratio between inertial forces and viscous

forces in a fluid.

Spillway

A structure used to release water from the water dam so that the water does not reach

dangerous heights.

Turbulence

Sudden changes in pressure and flow velocity. Occurs when the excessive kinetic energy

overcomes the damping effect from the fluid’s viscosity.

Turbulence intensity

A quantity that describes the intensity of sudden changes in pressure and flow velocity within

the water flow. Turbulence intensity is the ratio between the turbulence velocity and the mean

velocity of the flow.

Turbulence velocity

The root-mean-square, RMS, of the velocity fluctuations in a turbulent flow.

Unit air discharge

Air discharge per length unit, in this case metres.

Vapour pressure

The pressure in which a subject’s evaporation is in equilibrium between its liquid and solid

state at any given temperature. When the local pressure is equal to the vapour pressure for a

liquid, the liquid and its vapour are in equilibrium. When the local pressure is lower than the

vapour pressure, evaporation commences.

Weber number

A dimensionless parameter used in fluid mechanics. The Weber number is the ratio between

inertial force and surface tension force, which indicates whether the kinetic or the surface

tension energy is dominant.

TABLE OF CONTENTS

1 Introduction ........................................................................................................................ 1

1.1 Purpose ........................................................................................................................ 1

1.2 Objectives .................................................................................................................... 2

1.3 Limitations and assumptions ....................................................................................... 2

1.4 Method ......................................................................................................................... 3

1.5 Work breakdown ......................................................................................................... 5

2 Background ........................................................................................................................ 6

2.1 General description of cavitation ................................................................................. 6

2.2 Cavitation damage ....................................................................................................... 7

2.2.1 Cavitation damage on surfaces ............................................................................. 8

2.2.2 Glen Canyon dam and hoover dam ...................................................................... 8

2.3 Self-aeration and bubble transportation in water ......................................................... 9

2.4 Chute aerators ............................................................................................................ 11

2.4.1 Techniques and working principle ..................................................................... 12

2.4.2 Air distribution ................................................................................................... 14

2.5 Bottom and average air concentration ....................................................................... 15

3 Theory .............................................................................................................................. 17

3.1 Scale effects in hydraulic models .............................................................................. 17

3.1.1 Model and prototype similarities ........................................................................ 18

3.2 Air bubble entrainment and air concentration ........................................................... 19

3.3 Air discharge .............................................................................................................. 21

3.4 Bubble frequency ....................................................................................................... 22

3.5 Bernoulli’s equation .................................................................................................. 23

3.5.1 application of Bernoulli’s equation in the experiments ..................................... 24

3.6 Coefficient of determination ...................................................................................... 25

4 Experiment ....................................................................................................................... 26

4.1 Setup .......................................................................................................................... 26

4.2 Performance ............................................................................................................... 28

4.2.1 Matlab ................................................................................................................. 29

4.2.2 Microsoft excel ................................................................................................... 30

4.2.3 Motion studio and AutoCAD ............................................................................. 30

5 Results .............................................................................................................................. 32

5.1 Experiments with probe ............................................................................................. 32

5.1.1 Model 1 .............................................................................................................. 32

5.1.2 Model 2 .............................................................................................................. 38

5.1.3 Model 3 .............................................................................................................. 45

5.2 Motion Studio ............................................................................................................ 51

6 Discussion ........................................................................................................................ 55

6.1 Future work ................................................................................................................ 57

7 Conclusion ........................................................................................................................ 59

References ................................................................................................................................ 60

Appendix I Short introduction to the research at Sichuan University connected to this thesis I

Appendix II Additional aerator designs ............................................................................... II

Appendix III Matlab-code for air concentration and bubble frequency ............................... IV

Appendix IV Experimental data .......................................................................................... VII

Appendix V More pictures from the high-speed camera .................................................. XV

1

1 INTRODUCTION

In this section, a short introduction to the subject is presented as well as the purpose to why this

thesis is done. To achieve this purpose, certain objectives have been set up and limitations and

assumptions have been made, these is also presented in this section together with the method

chosen for this thesis. In the end, a work breakdown between the two authors is presented.

1.1 PURPOSE

To prevent cavitation damage in high-discharge chutes, they are usually equipped with chute

aerators. Chute aerators separate the flow from the chute bottom and supplies air to the lower

surface through an air-supply system [1]. These are an economic counter-measure that have

proven successful through history [2].

Bai et.al [1] at Sichuan University are conducting research on the lower aeration process

downstream of a chute aerator. Earlier research in the field by Pfister and Hager [3] has failed

to eliminate the effect of the self-aeration occurring at the free surface due to too shallow depths.

This leads to the lower and upper aeration processes mixing together earlier than desired, thus

making it hard to quantify the effects of the aeration from the chute aerator. The purpose of this

project is to study the lower jet to obtain knowledge about the air concentration and bubble

behaviour in the cavity zone. The lower jet is defined as the aerated water that has contact with

the cavity and is separated from the upper jet by a region consisting of unaerated water, so

called black water. The lower jets thickness is defined as the region where the air concentration

ranges from 0.9 to zero. The black water ensures that the lower jet only receives air from the

lower surface. The cavity zone is defined as the region of the chute where the cavity occurs,

which means that it contains both the cavity and the flow above.

This investigation was done by conducting experiments in a model. This project was carried

out as a part of the ongoing research at Sichuan University to establish a better understanding

about the air-water properties downstream of the chute aerator.

2

1.2 OBJECTIVES

To obtain knowledge about the air concentration and the bubble behaviour in the cavity zone,

the following objectives have been set up:

• Perform literature studies regarding aeration and chute aerators before conducting

experiments

• Perform model experiments with high-speed camera to observe formation and migration

of air bubbles in the cavity zone

• Perform model experiments to obtain data regarding air concentration, bubble size and

bubble frequency in the cavity zone

• Analyse and evaluate the data to obtain better knowledge about the mechanisms in the

lower aeration

To meet these objectives, the following questions have been set up:

• What is cavitation and how is it harmful?

• What is the working principle of an aerator?

• How is air concentration and bubble frequency distributed in the flow?

• How well do the experimental results coincide with theoretical calculations?

• How are air bubbles formed and transported within the flow?

1.3 LIMITATIONS AND ASSUMPTIONS

In this thesis, it has been assumed that the air concentration at the centreline is constant in the

transverse direction. Due to this assumption, the side wall effects from the downstream chute

have been neglected.

The dimensions of the model, such as width of the downstream chute and offset height has been

considered reasonable with respect to previous studies. The choice of outlet velocities and the

acceptance regarding scale effects are also based on previous studies conducted by, among

others, Pfister and Chanson [3] [4].

Regarding the air bubbles, studies have been conducted in Motion studio and AutoCAD to

measure the bubble size and to study the aeration process. Herein, the bubble shape has been

assumed spherical. It has also been assumed that the turbulence intensity increases with the

distance from the offset in x-direction, which means that the lower surface becomes more

3

irregular with the distance from the offset, see Figure 1. Because of this, the assumption that

the turbulence velocity increases with the distance from the offset can also be made.

FIGURE 1: ILLUSTRATION OF HOW THE SURFACE BECOMES MORE IRREGULAR WITH DISTANCE IN X-

DIRECTION. zm IS THE MAXIMUM HEIGHT FLUCTUATION OF THE SURFACE WHEN A BUBBLE FORMS. d IS THE

BUBBLE DIAMETER.

1.4 METHOD

This master thesis project consists of a literature study and experimental investigations

conducted in an already constructed chute model. The model is illustrated in Figure 2. The

experiments have been conducted at the State Key Laboratory of Hydraulics and Mountain

River Engineering at Sichuan University in Chengdu, China.

The literature studies were conducted to obtain necessary information about aerators and the

aeration process. Causes of cavitation and the consequences of cavitation damage were studied

to understand the importance for these types of research. Literature studies on relevant theory

were also done to obtain knowledge about the physics behind the cavitation bubbles and the

aeration process.

The experiments were conducted in three parts, in which the offset-height and upstream and

downstream angle of the chute aerator were changed for each part. In the first part, the

experiments were conducted with an upstream angle, 0, of 0° and a downstream angle, , of

5.7°. The second part was conducted with an upstream angle of 0=12.5° and α=18.2°. In the

third part, the angles were kept the same while the offset-height was decreased from 5 cm to 3

cm.

4

The experimental equipment consisted of a phase-detection needle probe (CQY-Z8a

Measurement Instrument) for measuring bubble frequency, bubble size and air concentration

and a high-speed camera (MotionXtra HG-LE) for observing the bubble behaviour. The phase-

detection needle probe is of a double-tip design. The working principle behind the phase-

detection probe is the difference in conductivity between air and water. The probes are designed

to puncture an air bubble and can easily enter the bubbles and thus give accurate information

from the fluctuations in conductivity [5]. The output from the probe is air concentration, bubble

size and number of bubbles. As air concentration is only a measure of air present in the water,

it does not describe bubble sizes and their distribution in the water. Bubble analysis is therefore

done to obtain information for the research on the microscale. The collected data have been

mathematically analysed in Matlab and Microsoft Excel.

FIGURE 2: SKETCH OF CHUTE MODEL, WHERE THE Z-AXIS IS DEFINED PERPENDICULAR TO THE CHUTE BOTTOM

AND THE X-AXIS IS DEFINED ALONG THE FLOW DIRECTION. THE CAVITY ZONE IS DEFINED AS THE REGION

BETWEEN THE OFFSET, WHERE THE FLOW IS SEPARATED FROM THE CHUTE BOTTOM, AND THE IMPACT POINT,

WHERE THE FLOW IMPACTS THE CHUTE BOTTOM.

5

1.5 WORK BREAKDOWN

This master thesis has been conducted by two authors, Albin Hedehag Damberg and Ebba

Wargsjö Gunnarsson. The report writing has therefore been divided between the two authors

to simplify the process. The breakdown was made as follows:

Albin Hedehag Damberg has been responsible for the research and writing of subsections 2.3,

2.4 and 2.5 in Background as well as subsections 3.4, 3.5 and 3.6 in Theory.

Ebba Wargsjö Gunnarsson has been responsible for the research and writing of subsections 2.1

and 2.2 in Background as well as subsections 3.1, 3.2 and 3.3 in Theory.

Both authors have contributed to the sections 1, 4, 5, 6 and 7. Both authors have also contributed

to the calculations in Matlab and Microsoft Excel and the analysis of the images in Motion

studio and AutoCAD. Although Albin has had an overall responsibility for the calculations in

Matlab and Ebba has had an overall responsibility for the images in Motion studio and

AutoCAD.

6

2 BACKGROUND

In this section, relevant background information is presented to provide a deeper understanding

about the cavitation process, how air is entrained into the water flow, the working principle of

a chute aerator and earlier research on the subject.

2.1 GENERAL DESCRIPTION OF CAVITATION

Cavitation is defined as the formation of a bubble or a cavity within a liquid. If the cavity is

filled with water vapour, the process is called vaporous cavitation and if the cavity is filled with

some other gas it is classified as gaseous cavitation [6].

The cavitation process can simply be described by studying the process of boiling. However,

there is a technical difference between these two processes. In terms of boiling, an increase in

temperature will result in an increase of the vapour pressure. When the vapour pressure equals

to the local pressure, boiling will occur. At the boiling point, water is changed into water vapour.

This changing process will primarily be observed as bubbles [6].

The boiling temperature is a function of pressure, which means that, when the pressure

decreases, boiling will occur at lower temperatures. The boiling process is described technically

as passing from the liquid state to the vapour state by changing the temperature, as the local

pressure is kept constant. Unlike the cavitation process, which is the process when passing from

the liquid state to the vapour state by changing the local pressure, as the temperature is kept

constant [6].

An open bottle containing a carbonate liquid is an example of bubble formation within a liquid,

which occurs by reductions in pressure. When opening the bottle, bubbles form within the liquid

and rise to the surface. As the bottle is opened, the pressure will decrease and the liquid becomes

supersaturated relative to the carbon dioxide. Therefore, the carbon dioxide starts to diffuse out

of the liquid. This is an example of gaseous cavitation in which vapour pressure of the liquid

never was reached [6].

In flowing systems, cavitation occurs when the pressure at any location decreases below the

vapour pressure of the liquid at the operating temperature. The pressure decrease is often a

result of irregularities in the chute surface [2]. The resulting vapour bubbles that forms within

7

the liquid are transported by the flow and when the pressure reaches a value above the vapour

pressure, the vapour bubbles will collapse. If this procedure occurs close to a solid boundary,

the surface may be exposed to erosion or even component failure in the long run. Due to the

risk of cavitation damage in flowing systems, extra efforts are made to avoid cavitation [7].

2.2 CAVITATION DAMAGE

As mentioned in section 2.1, damage will occur when a cavitation bubble collapses close to a

solid surface due to the forces from the collapse. A collection of cavitation bubbles can produce

pressure waves with a magnitude of several 100 kPa. These united group of bubbles are called

cavitation clouds. Figure 3 shows the process of cavitation cloud implosion, which begins with

a separation of the cavitation cloud from the attached part of cavitation. After the separation,

the cavitation cloud, which is illustrated as a single bubble in Figure 3, travels with the flow

and collapse in the higher-pressure region. Frame 4 illustrates the formation of the re-entrant

jet, which is caused by the collapse of the bubbles. The re-entrant jet will cause a new cavitation

cloud separation and the process will be repeated [8].

FIGURE 3: COLLAPSE OF A GROUP OF BUBBLES. FRAME 1 SHOWS THE SEPARATION OF THE CAVITATION CLOUD,

FRAME 2 AND 3 SHOWS HOW THE CAVITATION CLOUD TRAVELS WITH THE FLOW, FRAME 4-8 SHOWS HOW

CAVITATION CLOUD SEPARATION IS REPEATED [8]

Various mechanisms are normally involved in the damage of hydraulics structures. For

example, when cavitation forms due to irregularity of surfaces, the damage on the surface will

start at the downstream end of the cloud of the collapsing cavitation bubbles. After a while, an

elongated hole will form within the concrete surface. This hole will get larger with high velocity

flow impacting the downstream end of the hole. This causes a pressure difference between the

impact zone and the surrounding area, which may trigger the aggregate or even small chunks

of concrete to be broken from the surface and swept away by the flow. This damage process is

called erosion. Erosion is defined as abrasion, dissolution or transport process [6]. As the

cavitation damage has formed, the damaged area becomes a new source of cavitation, which

8

then forms damage downstream of another area. The erosion may continue into the underlying

foundation material after the structure’s lining has been penetrated [6].

2.2.1 CAVITATION DAMAGE ON SURFACES

It is possible for a surface to be damaged by cavitation as high flow velocities pass over a

surface. There are several factors that decide whether a surface will be damaged or not. These

factors include [6]:

• The cause of the cavitation

• The intensity of the cavitation

• The magnitude of the flow velocity

• The air content of the water

• The surface’s resistance to damage

• For how long the surface is exposed

Cavitation damage always occurs downstream from the source of cavitation. For a cylinder,

with its end turned towards the flow, the damage begins when the length of the cavitation cloud

is equal to the cylinder diameter [6].

It has been showed that the largest damage occurs near the downstream end of the cavitation

cloud. It was also observed that the distance to the maximal damage would increase when both

the flow and the height of the surface irregularities increased [6].

2.2.2 GLEN CANYON DAM AND HOOVER DAM

Two examples of cavitation causing significant damage to the spillways and their linings are

the accidents at the Glen Canyon dam and the Hoover dam.

In July 1941, the first cavitation damage was detected in the spillways at the Hoover Dam,

located at the border between Nevada and Arizona, USA. The spillways were repaired during

the winter of 1941 to 1942. It was assumed that the damages were ascribable to nothing but

roughness and irregularities in the concrete lining and thus the only measure taken was to

remove surface irregularities. This assumption was proven false during the spillage at Glen

Canyon and Hoover Dam in the summer of 1983 as both dams experienced the same style of

cavitation damage that had previously afflicted Hoover dam in 1941 [9].

9

On June 22nd, 1983, the left spillway at Glen Canyon Dam in Arizona, USA, failed during

flooding in the Colorado river. The cause of the failure was several excavated cavitation holes

in the spillway tunnel [10]. The Glen Canyon Dam consists of spillways that are located on

each abutment. Each spillway tunnel is inclined at 55 degrees and at the reservoir surface the

combined discharge capacity of the spillways is about 7800 m3/s. During the flood year of 1983

the reservoir in the Colorado river system was filled completely for the first time and water

release was required. The cavitation damages were initiated by offsets formed on the tunnel

invert at the upstream end of the bend. Both spillways were operated at discharges up to about

850 m3/s. The worst damage occurred in the left tunnel where the cavitation damage resulted in

hole about 11 m deep and 41 m long, which was eroded at the bend into the soft sandstone [11].

After these incidents, the Bureau of Reclamation undertook an extensive program to rebuild the

high dams by installing aeration slots [9]. These extensive repair works and installations of

aeration slots were required to bring the spillways back into service and to prevent potential

future damage [11]. The reparations and modifications of the spillways at the Glen Canyon

Dams achieved a cost of about 20 million dollars [10].

FIGURE 4: CAVITATION DAMAGE AT THE HOOVER DAM IN THE ARIZONA SPILLWAY IN THE YEAR OF 1941 [9]

2.3 SELF-AERATION AND BUBBLE TRANSPORTATION IN WATER

10

When the turbulent boundary layer from the bottom reaches the free water stream, it is possible

for the surrounding air to become entrained into the water body. This process is known as self-

aeration and commences if the turbulence is high enough. This phenomenon can be observed

as the water goes from clear to white, so called “white water” [2].

When the water flow is turbulent enough, the surface becomes irregular and eventually a

separation occurs in the free water surface and droplets of water leave the water body [2]. When

these droplets return to the water body they bring air with them which then gets entrained into

the water [12]. A higher turbulence intensity results in a higher air entrainment [13]. The

entrained air appears as bubbles in the water. Since air has a lower density than water, the

bubbles will experience an elevating force, giving them a rise velocity. For the bubbles to

remain entrained in the water it is required that the downward velocity component from the

turbulence is larger than the rise velocity of the bubbles [12].

Rein [14] researched the process of self-aeration and found that the bubble diameter, surface

tension, water density and turbulence velocity were vital parameters for the formation of air-

entraining bubbles. Rein [14] also concluded that a bubble will only leave the water body when

the maximum height, zm, is larger than its radius, see Figure 1. Current research conducted at

Sichuan University is aiming towards mathematically describing the formation of bubbles in

the lower jet and it is assumed that it is the same vital parameters for bubble formation in the

lower jet as at the free surface. Therefore, it is also assumed that Rein’s statement about drops

can be applied to bubbles within the flow. A short introduction to the research can be found in

Appendix I.

The water flow can be described as having four zones in z-direction, which are shown in Figure

5 [12]:

• Upper zone with flying water droplets

• Mixing zone with continuous water surface where water and air are mixed together

• Underlying zone where air bubbles are entrained into the water body

• Air free zone

11

FIGURE 5: CROSS-SECTION OF AN AIR-ENTRAINING WATER FLOW. IT IS ILLUSTRATED THAT THE WATER

SURFACE IS IRREGULAR ENOUGH TO CAUSE DROPS OF WATER TO EJECT FROM THE WATER BODY [12]

Large bubbles have a higher chance of becoming entrained into the water body but a smaller

chance to be transported downward due to a higher rise velocity. Large enough bubbles collapse

due to experienced shear stresses from the turbulence. Small bubbles experience the opposite;

they have a lower chance of becoming entrained but a higher chance to be transported

downward. They also tend to become entrained in each other’s wake and form into an

agglomerate, which leads to formation of larger bubbles. These processes occur simultaneously

and an equilibrium between them arises [12]. It is also notable that for small bubbles, the surface

tension is the dominating effect on its shape and hence they appear as spheres. As the bubbles

grow larger, the shear forces become dominant and they acquire the shape of a spherical

segment [12].

2.4 CHUTE AERATORS

To prevent the risk of cavitation damage on a surface, installations of aerators in hydraulic

structures is a proven solution. A small amount of air added to the water may prevent these

types of damage. This could be done by installing an aerator in a duct or a chute [6].

The addition of air to the bottom of a water flow is an effective way to avoid cavitation damage

on the water way. Due to air having a lower sonic velocity and higher compressibility than

water, the air near a boundary has a dampening effect on the bubble collapses that occur during

cavitation, which reduces the magnitude of the damage [2]. If the volume of air in water is equal

to 0.1 percent, it will increase the mean compressibility approximately 10 times [15].

12

When the self-aeration process does not satisfy the need of air concentration at the bottom of a

flow, bottom aerators are necessary. These add air directly to the bottom, thus increasing the

bottom air concentration without having to consider the bubble transport from the free water

surface downwards to the bottom [2].

2.4.1 TECHNIQUES AND WORKING PRINCIPLE

A bottom aerator creates a cavity between the water flow and the bottom by separating these as

smoothly as possible with the least disturbance in the chute flow. This cavity is connected to

the outer atmosphere via air canals. There is sub-atmospheric pressure in the cavity, which leads

to an air discharge from the outer atmosphere to the cavity. The air is then entrained in the water

flow, which leads to the pressure in the cavity zone always being sub-atmospheric [2]. Because

of this, there is always air flowing into the cavity zone.

There are different techniques to separate the water flow from the bottom. Three designs have

been proposed as suitable for bottom aeration [15]:

• Deflectors – a ramp that deflects the flow from the bottom

• Offsets – a threshold that separates the flow from the bottom as it passes

• Grooves – a groove in the bottom that the flow passes over

These designs can be combined to create a more effective aerator. The combined designs are

shown in Figure 6.

13

FIGURE 6: THREE AERATOR DESIGNS AND HOW THEY CAN BE COMBINED. AT THE TOP OF THE FIGURE THERE IS

A DEFLECTOR, IN THE MIDDLE THERE IS A GROOVE AND AT THE BOTTOM OF THE FIGURE THERE IS AN OFFSET

AND THE CIRCULAR ILLUSTRATIONS ARE COMBINATIONS OF THESE THREE [15]

When the water flow hits the bottom after passing an offset or deflector, the bottom pressure

quickly rises and reaches a maximum value. This process is illustrated in Figure 7. It then

decreases as the air bubbles rise to the surface. If the water travels far enough for the pressure

to once again drop to dangerous levels, a new aerator is needed to avoid cavitation damage [2].

FIGURE 7: BOTTOM AERATOR WITH THE QUICK RISE IN PRESSURE, Δp, ILLUSTRATED AS DOTTED LINE, Qa IS THE

AIR DISCHARGE AND IS THE CHUTE ANGLE, WHICH HEREIN IS DENOTED α [2].

The chute downstream the aerator can be divided into four zones with respect to the aeration

behaviour [1]:

• Cavity zone

• Impact zone

14

• Equilibrium zone

• Far zone

These four zones are illustrated in Figure 2.

2.4.2 AIR DISTRIBUTION

The air is distributed from the atmosphere to the cavity through air supply systems. The air

should be distributed uniformly over the entire chute width with minimum interference to the

water flow across the chute [2].

The design of the air supply system has vital impact on the air discharge that reaches the flowing

water. Even a small change to the air supply system can have a considerable effect on the air

discharge [15]. There are several types of air supply systems, they can either have a canal that

connects the water flow to the atmosphere or they can supply air directly from the atmosphere

if the chute is not enclosed. Volkart and Rutschmann [15] have proposed two different types of

air supply systems with air canals that manage to provide a uniform distribution. One injects

air into the cavity from an air vent supplying air from the wall and one injects air from below

the ramp as an air duct runs beneath the ramp. The second solution requires an aerator that is

combination of the deflector and the offset [15]. The two air supply systems are shown in

Figure 8.

FIGURE 8: TWO TYPES OF AIR SUPPLY SYSTEMS. A) AIR INJECTION FROM THE WALL; B) AIR INJECTION FROM

BENEATH THE RAMP [2]

Aside from the two air supply systems proposed by Volkart and Rutschmann, there are more

methods that have been invented to vent air from the atmosphere into the cavity. These designs

are described in Appendix II.

15

2.5 BOTTOM AND AVERAGE AIR CONCENTRATION

When the flow passes the chute aerator, it is deflected from the bottom. As air is entrained into

the lower jet via the air inlet, the air concentration in the flow rises. The upper jet entrains air

via self-aeration. As the flow reattaches to the bottom at the impact point, air detrainment begins

because of bottom rollers appearing upstream of the impact point and the air concentration at

the bottom quickly decreases [3] [1]. Bottom rollers is a phenomenon where water recirculates

locally at the bottom. This phenomenon can trap the air already present in the flow so that that

particular volume of air enters the local recirculation instead of traveling downstream, which

results in a decrease in air concentration downstream [1]. As the flow passes down the chute,

the black water will disappear and the upper and lower jet will merge [16] [1]. The air bubbles

at the bottom will then travel upwards because of their rise velocity, thus decreasing the bottom

air concentration [1].

Since the air concentration at the bottom is the most significant parameter for cavitation

protection [17], the bottom air concentration, Cb, is studied as a separate parameter instead of

studying only the average air concentration in the flow, Ca.

In the cavity zone, the bottom air concentration is at a constant Cb=1 because the cavity consists

only of air. The average air concentration is Ca~0.1 at take-off and increases rapidly in the

cavity zone to up to several multiples of the take-off value [3].

Bai et al. [1] have researched the air concentration profiles for the upper and lower jets in the

cavity zone as well as downstream of the impact point. Their research indicates that there is

much air from the air inlet that is entrapped at the lower surface, but not much air that is

entrained into the flow as the air concentration decreases with distance in the z-direction from

the bottom. This is illustrated in Figure 9, where the air concentration quickly decreases with

distance from the bottom. The same can be observed for the upper jet where the air

concentration decreases with distance from the surface. This creates two maximums in the air

concentration profile which coincides with the findings of Volkart and Rutschmann [15]. It is

also visible from Figure 9 that the bottom air concentration is Cb=1 in the entire cavity zone up

until x/L=0.91 where back water decreases the air concentration.

16

FIGURE 9: AIR CONCENTRATION PROFILES ALONG THE CHUTE FROM THE OFFSET TO IMPACT POINT. x/L IS

DIMENSIONSLESS DISTANCE ALONG CHUTE, WHERE L IS DISTANCE FROM OFFSET TO IMPACT POINT. z/h0 IS

DIMENSIONLESS DISTANCE FROM BOTTOM, WHERE h0 IS THE INITIAL WATER DEPTH [1]

Downstream of the impact point, the average air concentration will at first decrease due to the

decrease in bottom air concentration. The air transport along the upper surface will remain

unaffected [3]. After some distance, the jet will deflect from the bottom which leads to water

droplets ejecting from the water surface. This will increase the average air concentration due to

air-bubbles being entrapped into the surface. The bottom air concentration will, however, not

be affected by this and will keep decreasing. As the upper and lower jet merges and the bubbles

begin to rise towards the surface, the bottom air concentration decreases but the average air

concentration will maintain a constant value [3]. As the air concentration decreases, the risk of

cavitation damage increases. When the air concentration reaches a lower limit, a new aerator is

needed to avoid cavitation damage [15]. An air concentration profile a considerable length

downstream of the impact point indicates that the air concentration at the bottom and some

distance upwards from the bottom is close to zero. It is not until near the upper surface that the

air concentration increases drastically because of the air entrained via self-aeration [15] [3] [16].

This is illustrated in Figure 10.

17

FIGURE 10: AIR CONCENTRATION PROFILE AT A CONSIDERABLE LENGTH FROM THE AERATOR. THE Y-AXIS IN

THIS PROFILE CORRESPONDS TO THE Z-AXIS USED HEREIN [15].

3 THEORY

In this section, the equations used during this thesis work is presented together with the theory

behind the equations. The equations are used to make sure that scale effects are negligible, to

calculate theoretical values for air concentration and bubble frequency for comparison with the

experimental values and to calculate the outlet velocity during experiments.

3.1 SCALE EFFECTS IN HYDRAULIC MODELS

To find technical and economical solutions of hydraulic engineering problems it is common to

use a physical hydraulic model that is representing a real-world prototype. However, it is

important to consider the differences between the model and the prototype parameters as it

could result in scale effects. Scale effects will occur due to inability to keep the relevant

parameters between the model and the real-world prototype constant [18] [19].

A challenge for physical modellers is to know whether the scale effects can be neglected or not.

Therefore, several investigations have been conducted to provide researchers with necessary

tools about how to decide under which conditions scale effects can be neglected in typical

hydraulic flow phenomena [18] [19] [20].

18

3.1.1 MODEL AND PROTOTYPE SIMILARITIES

To obtain a physical scale model that is completely similar to its real-world prototype so that

scale effects could be prevented, mechanical similarity is required. Mechanical similarity

involves the criteria; Geometric similarity that requires similarity in shape, such as model

lengths, area and volume; Kinematic similarity requires, in addition to geometric similarity, a

constant ratio of time, velocity, acceleration and discharge in the model and its prototype;

Dynamic similarity requires in addition to geometric and kinematic similarities that all force

ratios in both the model and the prototype are identical [18].

In fluid dynamics, the most significant force is the inertial force and is therefore included in all

common force ratio combinations [18]. The ratio between inertia and gravity force results in

the Froude number, the ratio between inertia and the viscosity force results in the Reynolds

number and the ratio between inertia and surface tension gives the Weber number [19].

In open-channel hydraulics, the Froude similarity is often applied, which means that the Froude

number of the model should be equal to the Froude number of the prototype. In models where

friction effects are negligible or for short highly turbulent phenomena, it is common to use this

similarity. The equation of the Froude number, see equation 1 , includes the gravitational

acceleration, g, and even though the model may be accurate, this parameter is not scaled, which

can result in scale effects [18]. The Froude number is expressed as

𝐹𝑟 =𝑉

√𝑔ℎ (1)

where V [m/s] is the characteristic air water flow velocity of the fluid; g [m/s2] is the

gravitational acceleration and h [m] is the characteristic air water flow depth [18] [19] [21].

Reynolds similarity is commonly used at boundaries resulting in extreme losses in a model

compared with its prototype. If the Reynolds number is applied, the scale effects of the Froude

number may not be negligible, the effect of the gravity force on the fluid flow should therefore

be negligible in a model that uses Reynold similarity [18]. The Reynolds number is written as

𝑅𝑒 =𝑉ℎ

𝜈(2)

where 𝜈 = 𝜇/𝜌 [m2/s] is the kinematic viscosity, μ [Ns/m2] is the dynamic viscosity and ρ

[kg/m3] is the density [18] [19].

19

As mentioned above, the Weber number is the ratio between inertia and the surface tension.

The surface tension is often negligible for prototypes in hydraulic engineering but it is, for

example, relevant in scale models for air entrainment and small water depths. If the surface

tension in the model is dominant it is likely that it will cause larger relative bubbles sizes and

faster air detrainment, thus resulting in smaller volume fraction of air [18]. The Weber number,

W, may be written as

𝑊 =𝜌𝑉2ℎ

𝜎(3)

where σ [N/m] is the surface tension [18] [19] [21]. In physical scale models, it is common to

use the approach flow Weber number when suggesting limited values, which is the square root

of the Weber number, denoted W0. The approach flow Weber number is expressed as [17] [3]

𝑊0 =𝑉

√𝜎𝜌ℎ

(4)

For the force ratio combinations mentioned above, limiting values have been suggested to avoid

scale effects in physical hydraulic models. For a typical high-speed air-water flow with a Froude

number between 5 and 15, an approach flow Weber number of W0>140 and a Reynolds number

of Re>2∙105 should be respected in order to avoid scale effects that are related to air

concentration [19] [22]. According to Pfister and Chanson [19], the limits for Weber and

Reynolds are not sensitive for a Froude number within the range of 5<Fr<15 but for a Froude

number less than 5, 5>Fr, W0 and Re should be selected more conservatively.

3.2 AIR BUBBLE ENTRAINMENT AND AIR CONCENTRATION

When high-velocity water jets discharge into the atmosphere, air bubbles are entrained along

the air-water interfaces [4]. These transports in fluids are called advective diffusion, which

means that physical quantities, such as particles and energy, are transported inside a physical

system due to two processes: diffusion and advection [23]. The advective diffusion of air

bubbles is governed by the continuity equation for air, which is written as

𝑑𝑖𝑣(𝐶�⃗� ) = 𝑑𝑖𝑣(𝐷𝑡 ∗ 𝑔𝑟𝑎𝑑⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ 𝐶) (5)

where C is air concentration defined as the volume of air per unit volume of air and water, V

[m/s] is the velocity of the fluid and Dt [m2/s] is the turbulent diffusivity [4].

20

Equation 5 can be solved for circular and two-dimensional jets. Herein, the solution of two-

dimensional jets is the one of interest. In a partially-aerated flow region and with the

assumptions that the velocity distribution is uniform and that the diffusivity is constant, the

analytical solution of the continuity equation for air is described as follow

𝐶 =1

2∗

(

1 − erf

(

𝑧𝑐

2 ∗ √𝐷𝑡𝑉0∗ 𝑥)

)

(6)

where x [m] is the distance from the aerator along the flow direction, zc [m] is equal to the

perpendicular distance from the bottom, z [m], see Figure 2, minus the distance from bottom

where the air concentration C is equal to 50 percent, z50 [m], and V0 [m/s] is the initial flow

velocity [4].

The error function, erf, is a function which is common in the solutions of diffusion problems,

such as heat, mass and momentum transfer. The error function is defined as

erf(𝑢) =2

√𝜋∗ ∫ exp(−𝑡2) ∗ 𝑑𝑡

𝑢

0

(7)

where erf is a function of u, and u is equal to the expression in the parenthesis of the erf function

in equation 6. The erf function is defined for all values of u and it is an odd function, since [4]

[24]

erf(𝑢) = −erf(−𝑢) (8)

The fact that the error function is an odd function means that it is symmetrical around the origin.

Using z instead of zc for the theoretical calculations of the air concentration in equation 6 yields

a function that is symmetrical around C=0.5, which means that the curves for each section in

the x-direction of the theoretical calculations will intersect in this point. To make the theoretical

values comparable to the experimental values it is therefore necessary to subtract the point

where the curves intersect, which is done by defining zc as zc = z-z50 in equation 6.

The turbulent diffusivity, Dt, in equations 5 and 6, can be calculated from [4]

𝐷𝑡 =1

2∗𝑉0 ∗ 𝑥

1.2817∗ (tanψ)2 (9)

where Ψ is the initial spread angle of the air bubble diffusion layer in degrees, from which

information on the rate of diffusion of air bubbles can be obtained. The spread angle for two-

dimensional jet experiments may be expressed as [4]

21

𝜓 = 0.698 ∗ 𝑉00.630 (10)

Pfister [3] defined the average air concentration in the cavity zone, Ca, as

𝐶𝑎 =1

𝑧𝑢 − 𝑧𝑙∫ 𝐶(𝑧)𝑑𝑧

𝑧𝑢

𝑧𝑙

(11)

where zu [m] is the upper surface, zl [m] is the lower surface and C(z) is the air concentration.

Herein, the flow depth zu-zl is not measured in the experiments and is therefore not known. The

flow depth is assumed constant as h0. Considering the lower jet, its thickness was defined to

cover the region between z90 and z0 [m], which are the locations in z-direction where the air

concentration, C, is equal to 0.9 respectively zero, thus the equation yields

𝐶𝑎 =1

ℎ0∫ 𝐶(𝑧)𝑑𝑧

𝑧0

𝑧90

(12)

Equation 12 describes the average air concentration for the entire flow, from the lower to the

upper surface. Since the measurements of the air concentration only were conducted on the

lower jet, the water flow was considered unaerated above the lower jet and thus the effect from

the self-aeration at the upper jet is neglected.

3.3 AIR DISCHARGE

To further study the behaviour of the water flow in the cavity zone, the air discharge can be

calculated. Lima et al. [16] presented an equation for the air discharge, Qa [m3/s], along the x-

direction by considering the air concentration profile, which yields

𝑄𝑎 = 𝐵 ∫ 𝐶(𝑧) ∗

𝑧0

𝑧1

𝑉0(𝑧)𝑑𝑧 (13)

where B [m] is the width of the downstream chute and V0 [m/s] is the outlet velocity. z1 [m] is

the location in the z-direction where C is equal to one [16]. Similar to equation 13, Chanson

[21] described the unit air discharge, qa [m3/(m∙s)], in terms of the air concentration as [21]

𝑞𝑎 = ∫ 𝐶 ∗ 𝑉0

𝑧0

𝑧90

𝑑𝑧 (14)

22

In this thesis, the unit air discharge was estimated similar to equation 13 and 14. Using the

same boundaries for the lower jet thickness as in equation 1 , the unit air discharge, qa

[m3/(m∙s)], is described as [1]

𝑞𝑎 = 𝑉0 ∫ 𝐶(𝑧)

𝑧0

𝑧90

𝑑𝑧 (15)

3.4 BUBBLE FREQUENCY

The air bubble frequency is defined as the amount of air-bubbles present at a point in the flow

per second. The equation for the air bubble frequency is

𝑓 =𝑁

𝑡(16)

where f [s-1] is the air bubble frequency, N is the amount of bubbles detected during the scan

period t [s].

Bai et al. [1] conducted research on the bubble frequencies in the cavity zone. It was found that

the bubble frequency was distributed in a similar way as the air concentration as it decreased

when it neared the black water. The distribution of bubble frequencies for different sections in

the cavity zone is illustrated in Figure 11.

Chanson [25] presented an equation to calculate the bubble frequency

𝑓

𝑓𝑚𝑎𝑥= 4𝐶(1 − 𝐶) (17)

where fmax [s-1] is the maximum bubble frequency and C is the local air concentration.

23

FIGURE 11: DISTRIBUTION OF BUBBLE FREQUENCIES IN THE CAVITY ZONE. THE Y-AXIS, z/h0, IS A

DIMENSIONLESS DISTANCE IN Z-DIRECTION WHERE z IS DISTANCE FROM BOTTOM AND h0 IS THE INITIAL FLOW

DEPTH, WHICH IS CONSTANT WHILE z IS INCREASING. THE X-AXIS, (f∙h0)/V0, IS DIMENSIONLESS WHERE f IS THE

BUBBLE FREQUENCY AND V0 IS THE OUTLET FLOW VELOCITY, HERE THE BUBBLE FREQUENCY IS INCREASING

[1]

3.5 BERNOULLI’S EQUATION

The Bernoulli equation is derived from the law about conservation of energy and describes the

steady flow between two points in a flow stream [6]. For the Bernoulli equation to be applicable,

assumptions about the fluid must be made [26]:

• The fluid is incompressible and inviscid

• The flow is stationary

• There is no energy lost or gained

If these assumptions are correct, the energy in the flow can be described as

𝑉2

2+ 𝑔ℎ +

𝑃

𝜌= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (18)

where V [m/s] is the flow velocity, g [m/s2] is the gravitational acceleration, h [m] is the height

over some reference level, P [Pa] is the pressure and ρ [kg/m3] is the fluid density. The first

term, V2/g, is the fluid’s specific kinetic energy, the second term, 𝑔ℎ, is the potential energy

and P/ρ is the energy represented by the pressure [27].

The Bernoulli equation can also be applied with regards to losses. The losses are described as

𝛥ℎ𝑙𝑜𝑠𝑠 = 𝜉𝑉2

2𝑔 [m] where ξ is the loss coefficient. The losses are added to the downstream point,

usually on the right side of the equation, to keep the energy constant [26]. With losses, the

Bernoulli equation between two points is written as

24

𝑉12

2+ 𝑔ℎ1 +

𝑃1𝜌=𝑉22

2+ 𝑔ℎ2 +

𝑃2𝜌+ 𝑔∆ℎ𝑙𝑜𝑠𝑠 (19)

The Bernoulli equation can be used to explain why the pressure drops in a fluid. If the flow

velocity, V, increases, the kinetic energy increases which means that either the potential energy

or pressure energy must decrease for the energy to be constant. Since g and ρ are constants, P

and h are the only variables that can decrease. If the chute does not supply a big enough height

drop to counteract the quadratic increase in kinetic energy, the pressure in the fluid drops and

thus cavitation may occur.

3.5.1 APPLICATION OF BERNOULLI’S EQUATION IN THE EXPERIMENTS

Equation 19 is used during the experiments to calculate the outlet velocity V0. Equation 19 set

up between the water surface in the water tank and the outlet, see Figure 12, is

𝑉12

2+ 𝑔ℎ1 +

𝑃1𝜌=𝑉02

2+ 𝑔ℎ2 +

𝑃2𝜌+ 𝑔∆ℎ𝑙𝑜𝑠𝑠 (20)

where P1=P2 and V1=0 because the area in point 1 is assumed large enough for the velocity to

be neglected. Note that V2 is replaced by V0 to keep the same notation used elsewhere in the

report. With some rewriting, the equation becomes

ℎ1 − ℎ2 =𝑉02

2𝑔+ ∆ℎ𝑙𝑜𝑠𝑠 =

𝑉02

2𝑔+ 𝜉

𝑉02

2𝑔= (1 + 𝜉)

𝑉02

2𝑔(21)

The height difference h1-h2 is substituted into H. The equation then becomes

𝐻 = (1 + 𝜉)𝑉02

2𝑔→ 𝑉0 =

√2𝑔𝐻

√1 + 𝜉(22)

The term (√1 + 𝜉)−1

is substituted into μ0, which is constant since the loss coefficient is

constant. The water tank has been used in previous experiments at Sichuan university, and

therefore μ0 was already known as μ0=0.85. The outlet velocity, V0, is calculated with the

equation

𝑉0 = 𝜇0√2𝑔𝐻 (23)

25

FIGURE 12: SCHEMATIC PICTURE OF WATER TANK. POINT 1 IS LOCATED AT THE WATER SURFACE AND POINT 2

AT THE OUTLET. V0 IS THE OUTLET VELOCITY AND H IS THE HEIGHT BETWEEN THE WATER SURFACE AND THE

OUTLET

3.6 COEFFICIENT OF DETERMINATION

To investigate how well the experimental values coincide with the theoretical values acquired

from equations 6 and 17, the coefficient of determination, R2, can be calculated. R2 is defined

as

𝑅2 = 1 −∑ (𝑦𝑖 − 𝑦�̂�)

2𝑛𝑖=1

∑ (𝑦𝑖 − �̅�)2𝑛𝑖=1

(24)

where y is the experimental value, �̂� is the theoretical value and �̅� is the mean value of the

experimental values. The closer R2 is to 1, the better the experimental values match the

theoretical values.

26

4 EXPERIMENT

In this section, the experiments are explained in detail. Firstly, the setup of the model and the

equipment used during experiments are explained and secondly way the performance of the

experiments is explained.

4.1 SETUP

The experiments were conducted in a model of an offset-aerator in an open conduit-chute

according to Figure 2. The model was 0.25 m wide, 0.3 m high, 3 m long and constructed in

transparent polymethyl methacrylate (PMMA) downstream of the aerator to ensure visibility of

the water flow. The air-supply system was represented with approximately 12 cm2 rectangular

cut-outs at the bottom on each side of the model. The model was connected to a water tank via

a 0.25 m wide and 0.15 m high steel chute at the “offset-end”. The offset-height, hs, was created

by the model having a larger height than the steel chute. The model could be moved to alter the

offset-height, hs. The steel chute was filled with approaching water in each experiment so that

the flow depth was kept constant at h0=15 cm. Figure 13 is a picture of the model. In previous

experiments by Pfister et al. the flow depths have not been large enough to keep the upper and

lower jet separated throughout the cavity zone [22] [28]. By increasing the flow depth, it is

ensured that the black water separates the two jets in every experiment, thus making it possible

to better quantify the effects from the chute aerator since the effects from self-aeration are not

present in the lower jet.

FIGURE 13: PICTURE OF MODEL USED IN EXPERIMENTS. THE WOOD CONSTRUCTION IS A HOLDER FOR THE

MEASUREMENT PROBE, WHICH CAN BE SEEN IN THE BLACK WATER, h0 IS THE FLOW DEPTH, hs IS THE

OFFSET-HEIGHT AND L IS THE LENGTH OF THE CAVITY

For the studies of bubble behaviour, a high-speed camera (MotionXtra HG-LE) with a speed of

1000 frames per second was set up outside of the model. The camera was then able to record

27

the flow in the cavity zone. A ruler with millimetre-precision was fastened on the outside of the

model to create a reference for measurement in the software AutoCAD. For the studies of air

concentration and bubble sizes, the probe (CQY-Z8a Measurement Instrument, Figure 14) was

lowered into the lower jet with the tips of the probes positioned towards the flow direction, as

illustrated in Figure 15, with the length dl between the tips being 12.81 mm. The experimental

data from the probe was obtained with the associated software. The output data was later

analysed using Matlab and Microsoft Excel. The output data is compiled in Appendix IV.

FIGURE 14: PICTURE OF THE NEEDLE PROBE USED IN EXPERIMENTS

FIGURE 15: SKETCH OF THE NEEDLE PROBE RELATIVE TO THE WATER FLOW DIRECTION. dl=12.81 mm.

The model could be modified to change the parameters and 0, this was done to create

different scenarios. The parameter V0 was changed by changing the water discharge. All

relevant parameters for each experiment are presented in Table 1 and illustrated in Figure 16.

dlair bubble

sensor bar

first tip

second tip

flow direction

28

FIGURE 16: ILLUSTRATION OF CHUTE MODEL WITH RELEVANT PARAMETERS. V0 IS THE OUTLET VELOCITY, h0 IS

THE FLOW DEPTH, hs IS THE OFFSET HEIGHT, θ0 IS THE UPSTREAM ANGLE, α IS THE DOWNSTREAM ANGLE AND L

IS THE CAVITY LENGTH.

4.2 PERFORMANCE

The experiments with high-speed camera were only conducted for model 2, experiments 4-6 in

Table 1. The high-speed films were analysed to see how the bubbles form, transport and

collapse in the cavity zone using the software Motion Studio and AutoCAD.

For each version of the model, three experiments were conducted in which both the discharge

and the head were changed for each experiment, thus changing the outlet velocity V0. For each

experiment, the cavity zone was divided into multiple sections in x-direction, each 10 cm long,

for 0<x<L. Measurements were done in the z-direction for each section, where the probe was

inserted at the lower surface and raised 0.5 mm for every measurement until the tips entered the

black water. The probe was then moved in x-direction to the next section and the same

procedure was performed. The experiment was repeated until the impact point was reached.

The software associated to the probe required the input parameters sample frequency, fs [kHz],

and scanning time, ts [s], for every measurement. The sample frequency was set to 40 kHz and

the scanning time was set to 10 s or 40 s. In a sensitivity analysis conducted by Wei et al. [29],

where experiments were performed with a sampling frequency of fs=20-300 kHz and a scanning

time of ts=2-40 s, it was found that the scanning time and the sampling frequency did not have

any significant effect on the experimental result [29]. Given that the scanning time and sampling

frequency are held in a reasonable order of magnitude, the experimental results should be equal

even if different scanning times and sampling frequencies are used.

29

TABLE 1: EXPERIMENT RUNS WITH PARAMETERS

Experiment V0 [m/s] θ0 [°] α [°] hs [m] Fr Re

∗ 10−5

W0

Model 1 1 5.0 0.0 5.7 0.05 4.1 7.2 226.5

2 6.0 0.0 5.7 0.05 5.0 8.6 271.7

3 7.0 0.0 5.7 0.05 5.8 10.1 317.0

Model 2 4 4.3 12.5 18.2 0.05 3.5 6.2 194.7

5 5.0 12.5 18.2 0.05 4.1 7.2 226.5

6 6.0 12.5 18.2 0.05 5.0 8.6 271.7

Model 3 7 6.0 12.5 18.2 0.03 5.0 8.6 271.7

8 7.0 12.5 18.2 0.03 5.8 10.1 317.0

9 7.45 12.5 18.2 0.03 6.0 10.7 337.4

The values of the mechanical and thermal properties that was used to calculate Fr, Re and W0

from equations 1, 2 and 4 were provided from Physics Handbook [30]. These properties were

the dynamic viscosity of water, =1.04∙10-3 [Ns/m2], the density of water, =0.99820∙103

[kg/m3] and the surface tension of water, =73∙10-3 [N/m] [30]. The outlet flow velocity, V0,

was calculated with equation 2 and used as the characteristic air water flow velocity, V, and the

initial flow depth, h0, was used as the characteristic air water flow depth, h.

4.2.1 MATLAB

To analyse the output data obtained from the experiments with the needle probe, the data was

exported to Matlab, where it was studied by plotting the air concentration and the bubble

frequency against the distance from bottom, and the air concentration against the bubble

frequency. The relationship between the air concentration and the bubble frequency was

analysed by comparison with the theoretical values calculated from equation 17.

The relationship between the air concentration and distance from bottom was studied by

comparing the theoretical values calculated from equation 6 with the experimental results. As

mentioned in section 3.2, C=0.5 is the point where all the curves from the theoretical function

of the air concentration intersect each other because of the error function. Because of this, the

distance from the bottom, z, was modified to zc by subtracting the point where C=0.5 to make

the theoretical calculations comparable to the experimental results. These calculations can be

seen in the Matlab code in Appendix III. To find the height where the air concentration was

equal to 50 percent, linear interpolation between two points was required, which was done by

hand. Linear interpolation could be performed since the air concentration has an approximately

linear behaviour between 0.2<C<0.8, see for example Figure 18. These heights can be seen in

the tables of the experimental data in Appendix IV, denoted z50.

30

Linear interpolation is performed with the equation

𝑦 = 𝑦0 + (𝑦1 − 𝑦0) ∗𝑥 − 𝑥0𝑥1 − 𝑥0

(25)

where (x, y) is the sought point, with y unknown, located between the known data points

(x0, y0) and (x1, y1).

4.2.2 MICROSOFT EXCEL

Microsoft Excel was used to calculate Ca and qa from equation 12 respectively 15. As there

was no measurement that registered exactly C=0.9, equation 2 was used to calculate the height

where the air concentration was equal to 0.9.

The integral was estimated by calculating the area beneath the curve for C(z). This was done

by implementing the trapezoidal method. The trapezoidal method is defined as

∫ 𝑓(𝑥)𝑑𝑥

𝑥𝑛

𝑥0

≈1

2∑(𝑥𝑖+1 − 𝑥𝑖) ∗ (𝑓(𝑥𝑖) + 𝑓(𝑥𝑖+1))

𝑛−1

𝑖=0

(26)

Note that equations 12 and 15 contain the same integral but is multiplied with a different value,

h0-1 and V0. For each experiment, V0 is constant and h0 is always constant. This means that for

each experiment, the curves for qa and Ca will have the same shape but with different values

because of the different multiplications.

4.2.3 MOTION STUDIO AND AUTOCAD

To analyse the pictures recorded with the high-speed camera, the softwares Motion studio and

AutoCAD were required, in which bubble sizes were measured and the aeration process was

studied. The experiment with the high-speed camera was conducted for model 2, see

experiments 4-6 in Table 1, with an outlet velocity of 4 m/s.

Motion studio was used to handle the files from the high-speed camera, from which several

pictures were selected. The pictures that were selected, were those where a bubble forming and

migrating into the unaerated black water could be seen clearly. A set of pictures was chosen for

every bubble, showing its formation and trajectory in the flow direction.

31

When the pictures had been selected in Motion studio, they were exported to AutoCAD.

AutoCAD provides tools that make it possible to measure distances and sizes in pictures. To

aid in the research conducted at Sichuan University, the air bubble diameters were measured

since it is, as mentioned in section 2.3, assumed a vital parameter in the mathematical

description of bubble formation in the lower jet. Due to the assumption that the turbulence

velocity is increasing with the distance from the offset in the x-direction, the distance between

the offset and the onset of bubble formation was also measured. This distance is herein called

the onset distance.

As mentioned in section 4.1, a 20 cm long ruler was used when the flow was recorded with the

high-speed camera. The ruler was measured in AutoCAD to convert the length of the ruler into

the length unit used in AutoCAD, thus it was known which length unit was corresponding to

20 cm and a relationship between these lengths was provided. Further, this relationship could

be used to get the onset distance and the bubble diameter in centimetres. The bubble was

measured 3-4 times across its cross section to obtain an average diameter, see Figure 17, thus

approximating the bubble to a sphere. Since the same bubble was measured at different

locations in the flow, different diameters were measured so the bubble’s development could be

studied.

FIGURE 17: ILLUSTRATION OF HOW THE ONSET DISTANCE AND BUBBLE DIAMETER WAS MEASURED. THE LEFT

MEASURE POINT IN THE LEFT PICTURE IS THE STEEL CHUTE FROM THE WATER TANK. FOR THIS SET OF

PICTURES, 36.35 LENGTH UNITS CORRESPONDS TO 4 cm.

32

5 RESULTS

5.1 EXPERIMENTS WITH PROBE

In this section, the results from the experiments with the probe are presented. The results are

presented in the same order as the experiments were conducted, that is from model 1 to model

3. Model 1 includes the experiments 1-3, model 2 includes the experiments 4-6 and model 3

includes the experiment 7-9, see Table 1. The coefficients of determination, R2, are presented

for each section in a table below the figure.

5.1.1 MODEL 1

The following graphs show the results from model 1, experiments 1-3 in Table 1, where the

offset height, hs, was set to 0.05 m, the downstream chute angle α=5.7° and the spillway angle

θ0=0°. The Froude numbers for model 1 is within the range of 4<Fr<6, the approach flow

Weber numbers within the range of 220<W0<320 and the Reynolds numbers within the range

of 7∙105<Re<11∙105. These are all within the acceptable limits for avoiding scale effects,

though the Froude number for V0=5 m/s is smaller than 5, see Table 1. This is weighed up by a

Reynolds and Weber number with good margins to their respectable limits.

FIGURE 18: AIR CONCENTRATION IN LOWER JET WHEN V0=5 m/s IN MODEL 1, EXPERIMENT 1. DASHED LINES ARE

THEORETICAL VALUES FROM EQUATION 6 FOR DIFFERENT SECTIONS IN THE CAVITY ZONE AND MARKERS WITH

CORRESPONDING COLOUR ARE EXPERIMENTAL DATA AT DIFFERENT SECTION IN THE CAVITY ZONE.

33

TABLE 2: COEFFICIENTS OF DETERMINATION FOR AIR CONCENTRATION AT EACH SECTION IN EXPERIMENT 1

x=0.1 m x=0.2 m x=0.3 m x=0.4 m

R2 0.9993 0.9963 0.9852 0.9806



CORRESPONDING COLOUR ARE EXPERIMENTAL DATA AT DIFFERENT SECTIONS IN THE CAVITY ZONE.


x=0.1 m x=0.2 m x=0.3 m x=0.4 m

R2 0.9982 0.9976 0.9821 0.9778

34





x=0.1 m x=0.2 m x=0.3 m x=0.4 m

R2 0.9941 0.9927 0.9778 0.9794

Figure 18, 19 and 20 shows the experimental and the theoretical values of the air concentration

in the cavity zone for different outlet flow velocities, V0. From these graphs, it can be seen that

the air concentration decreases with the distance from the bottom, z. Close to the unaerated

black water, the air concentration is nearly zero. It is observed that the bottom air concentration,

Cb, is equal to 1 in the entire cavity zone. A similar trend for the experimental and the theoretical

values is observed from these graphs.

TABLE 5: UNIT AIR DISCHARGE, qa [m3/(m∙s)], FOR EACH SECTION AND OUTLET VELOCITY IN MODEL 1

V0 [m/s] x=0.1 m x=0.2 m x=0.3 m x=0.4 m

5 0.025 0.037 0.043 0.064

6 0.031 0.061 0.061 0.090

7 0.039 0.076 0.076 0.099

Table 5 shows the unit air discharge for model 1 calculated from equation 15. It is observed

that the unit air discharge is increasing for each section along the x-direction and with higher

outlet velocity. Since the unit air discharge is described in terms of the air concentration

distribution and is obtained from the area beneath the curve between C=0.9 to C=0, the increase

in unit air discharge is also notable in Figure 18, 19 and 20. For example, it could be seen that

the area beneath the curve between C=0.9 to C=0 at x=0.4 is larger than the one at x=0.1 in

Figure 18, which also can be seen in Table 5 as qa increases 2.5 times between x=0.1 to x=0.4.

35

As mentioned in section 4.2.2, the average air concentration follows the same trend as the air

discharge because they contain the same integral. Because of this it can also be observed that

Ca will begin at a lower value and rise throughout the entire cavity zone.

FIGURE 21: BUBBLE FREQUENCY IN LOWER JET AT DIFFERENT SECTIONS IN CAVITY ZONE WHEN V0=5 m/s IN

MODEL 1, EXPERIMENT 1



36



Figure 21, 22 and 23 shows the relationship between the bubble frequency and distance from

the bottom in the cavity zone for different outlet flow velocities, V0. From these graphs, it can

be seen that the bubble frequency increases with the distance from the air inlet in the x-direction.

From Figure 18, 19 and 20 it is observed that the air concentration is highest in the lower

surface. An air concentration close to 1 means almost only air and no water, which in turn

means no air bubbles. This is the reason why the bubble frequency is low although the air

concentration is high in the lower surface. As more water occurs, air bubbles form and the

bubble frequency rises. It is observed from the graphs that the bubble frequency continually

rises to a certain point where it then starts decreasing as it nears the unaerated black water. In

comparison to Figure 18, 19 and 20 it is also noted that the lowest and the highest air

concentration occurs at the same bubble frequency.

37

FIGURE 24: RELATIONSHIP BETWEEN BUBBLE FREQUENCY AND AIR CONCENTRATION WHEN V0=5 m/s IN MODEL

1, EXPERIMENT 1. SOLID LINE IS THEORETICAL VALUES FROM EQUATION 17

TABLE 6: COEFFICIENTS OF DETERMINATION FOR RELATIONSHIP BETWEEN BUBBLE FREQUENCY AND AIR

CONCENTRATION AT EACH SECTION IN EXPERIMENT 1

x=0.1 m x=0.2 m x=0.3 m x=0.4 m

R2 0.8754 0.9296 0.7293 0.7460





x=0.1 m x=0.2 m x=0.3 m x=0.4 m

R2 0.7050 0.8492 0.8728 0.5870

38





x=0.1 m x=0.2 m x=0.3 m x=0.4 m

R2 0.7066 0.8850 0.8232 0.8747

Figure 24, 25 and 26 show the relationship between the bubble frequency and the air

concentration of the experimental values and the theoretical values obtained from equation 17.

It can be observed from the theoretical values that the maximum bubble frequency occurs when

the air concentration is equal to 50 percent, which is relatively consistent with the experimental

data at each section. The experimental data follows a similar parabolic trend as the theoretical

values.

5.1.2 MODEL 2

In this section, the graphs for model 2, experiments 4-6 in Table 1, are presented, where the

offset height, hs, was kept the same as in model 1 (hs=0.05 m), the downstream chute angle

α=18.2° and the spillway angle θ0=12.5°. Due to a steeper chute, the water jet will travel further

before impact with the bottom than in model 1, which leads to a longer cavity zone. Because of

the longer cavity zone, more sections in the x-direction could be studied. The Froude numbers

for model 2 is within the range of 3<Fr<6, the approach flow Weber numbers within the range

of 194<W0<272 and the Reynolds numbers within the range of 6∙105<Re<9∙105. Even here,

the values of Fr, W0 and Re are considered to be within the acceptable limits for avoiding scale

effects, though the Froude number for V0=4.3 m/s and V0=5 m/s is smaller than 5, see Table 1.

39

This is weighed up by a Reynolds and Weber number with good margins to their respectable

limits.

FIGURE 27: AIR CONCENTRATION IN LOWER JET WHEN V0=4.3 m/s IN MODEL 2, EXPERIMENT 4. DASHED LINES ARE




x=0.1 m x=0.2 m x=0.3 m x=0.4 m x=0.5 m x=0.6 m

R2 0.9841 0.9922 0.9903 0.9642 0.9459 0.8815




40


x=0.1 m x=0.2 m x=0.3 m x=0.4 m x=0.5 m x=0.6 m x=0.7 m

R2 0.9882 0.9814 0.9596 0.9472 0.9655 0.9474 0.6354





x=0.1 m x=0.2 m x=0.3 m x=0.4 m x=0.5 m x=0.6 m x=0.7 m

R2 0.9914 0.9889 0.9714 0.9481 0.9264 0.9387 0.9130




black water, the air concentration is nearly zero. It is observed that the bottom air concentration,

Cb, is equal to 1 in the cavity zone up until x=0.5 m where back water is present and decreases

the bottom air concentration. The measurements for x=0.7 m in Figure 28 indicates that the

probe was inserted just downstream of the impact point, since the air concentration is so far

from 1 at the bottom.

A similar trend for the experimental and the theoretical values is observed from these graphs.

Despite alterations to the model, the air concentration distribution behaves the same in the z-

direction as in model 1.

41


V0 [m/s] x=0.1 m x=0.2 m x=0.3

m

x=0.4 m x=0.5 m x=0.6 m x=0.7 m

4.3 0.025 0.027 0.037 0.048 0.054 0.083 -

5 0.021 0.032 0.036 0.043 0.076 0.084 0.11

6 0.037 0.042 0.57 0.065 0.073 0.096 0.10


that the unit air discharge is increasing for each section along the x-direction. In comparison to

model 1, additional measurements were conducted in the x-direction for model 2, thus it could

be seen that the unit air discharge continues to increase with the distance from the offset after

x=0.4 m. Even here, the increase in unit air discharge can be observed by studying the area

beneath the curve in Figure 27, 28 and 29. As mentioned in section 4.2.2, the average air

concentration follows the same trend as the air discharge because they contain the same integral.

Because of this it can be observed that Ca will begin at a lower value and rise throughout the

entire cavity zone.

FIGURE 30: BUBBLE FREQUENCY IN LOWER JET AT DIFFERENT SECTIONS IN CAVITY ZONE WHEN V0=4.3 m/s IN


42








From Figure 27, 28 and 29 it is noted that the air concentration is highest in the lower surface.

An air concentration close to 1 means almost only air and no water, which in turn means no air

bubbles. This is the reason why the bubble frequency is low although the air concentration is

43

high in the lower surface. As more water occurs, air bubbles form and the bubble frequency

rises. It is observed from the graphs that the bubble frequency continually rises to a certain point

where it then starts decreasing as it nears the unaerated black water.

In comparison to Figure 27, 28 and 29 it is also noted that the lowest and the highest air

concentration occurs at the same bubble frequency. There are deviations in the last section for

each experiment, as the lowest and highest point does not yield the same bubble frequency. The

last section in Figure 30 and Figure 32 has a higher value at its lowest point, where it then

decreases and follows a similar trend as in the previous sections. This is due to back water

present at the bottom, which gives a higher bubble frequency than if it had been only air present

at the bottom. The last section in Figure 31 does not decrease after the initial high value, instead

it rises to a maximum and then decreases as it nears the unaerated black water. This is due to

the measurement being done just downstream the impact point which coincides with the

observations made from Figure 28.

FIGURE 33: RELATIONSHIP BETWEEN BUBBLE FREQUENCY AND AIR CONCENTRATION WHEN V0=4.3 m/s IN MODEL




x=0.1 m x=0.2 m x=0.3 m x=0.4 m x=0.5 m x=0.6 m

R2 0.9470 0.9740 0.9651 0.9601 0.8753 0.5998

44




CONCENTRATION AT EACH SECTION IN EXPERIMENT 5.

x=0.1 m x=0.2 m x=0.3 m x=0.4 m x=0.5 m x=0.6 m x=0.7 m

R2 0.8302 0.9177 0.9450 0.9622 0.7233 0.6062 0.7704





x=0.1 m x=0.2 m x=0.3 m x=0.4 m x=0.5 m x=0.6 m x=0.7 m

R2 0.5313 0.5632 0.4086 0.8071 0.6526 0.1648 0.4240

45

Figure 33, 34 and 35 shows the relationship between the bubble frequency and the air

concentration of the experimental values and the theoretical values. It can be observed from the

theoretical values that the maximum bubble frequency occurs when the air concentration is

equal to 50 percent, which is relatively consistent with the experimental data at each section.

The experimental data follows a similar parabolic trend as the theoretical values, though several

sections in Figure 35 deviates.

5.1.3 MODEL 3

The following graphs show the results from model 3, experiments 7-9 in Table 1, where the

offset height, hs, was set to 0.03 m, the downstream chute angle α=18.2° and the spillway angle

θ0=12.5°. The Froude number for model 3 was found within the range of 5<Fr<6, the approach

flow Weber number within the range of 271<W0<378 and the Reynolds number within the

range of 8∙105<Re<11∙105. These are all within the acceptable limits for avoiding scale effects.





x=0.1 m x=0.2 m x=0.3 m x=0.4 m

R2 0.9981 0.9958 0.9951 0.9873

46





x=0.1 m x=0.2 m x=0.3 m x=0.4 m

R2 0.9576 0.9991 0.9915 0.9697

FIGURE 38: AIR CONCENTRATION IN LOWER JET WHEN V0=7.45 m/s IN MODEL 3, EXPERIMENT 9. DASHED LINES

ARE THEORETICAL VALUES FROM EQUATION 6 FOR DIFFERENT SECTIONS IN THE CAVITY ZONE AND MARKERS

WITH CORRESPONDING COLOUR ARE EXPERIMENTAL DATA AT DIFFERENT SECTIONS IN THE CAVITY ZONE.


x=0.1 m x=0.2 m x=0.3 m x=0.4 m

R2 0.9259 0.9978 0.9853 0.9713

47




black water, the air concentration is nearly zero. It is noted that the bottom air concentration,

Cb, is equal to 1 in the cavity zone up until x=0.2 m where back water is present and decreases

the bottom air concentration. A similar trend for the experimental and the theoretical values is

observed from these graphs.

The air concentration has the same behaviour as the previous models, despite a smaller offset.

The notable difference between model 2 and 3 is that the jet reattaches to the bottom earlier,

resulting in a shorter cavity zone.


V0 [m/s] x=0.1 m x=0.2 m x=0.3 m x=0.4 m

6 0.036 0.060 0.070 0.094

7 0.049 0.072 0.083 0.11

7.45 0.061 0.070 0.098 0.10


that the unit air discharge follows the same trend as in model 1 and 2 where it increases for each

section along the x-direction. This can also be seen by studying the area beneath the curve

between C=0 and C=0.9 in Figure 36, 37 and 38. As mentioned in section 4.2.2, the average

air concentration follows the same trend as the air discharge because they contain the same

integral. Because of this it can be observed that Ca will begin at a lower value and rise

throughout the entire cavity zone.

48





49






From Figure 36, 37 and 38 it is observed that the air concentration is highest in the lower

surface. An air concentration close to 1 means almost only air and no water, which in turn

means no air bubbles. This is the reason why the bubble frequency is low although the air

concentration is high in the lower surface. As more water occurs, air bubbles form and the

bubble frequency rises. It is noted in the graphs that the bubble frequency continually rises to a

certain point where it then starts decreasing as it nears the unaerated black water.

In comparison to Figure 36, 37 and 38 it is also noted that the lowest and the highest air

concentration occurs at the same bubble frequency. A deviation is noted in the last section for

Figure 39, where the lowest and highest points does not yield similar bubble frequencies.

50





x=0.1 m x=0.2 m x=0.3 m x=0.4 m

R2 0.6055 0.6770 0.6290 0.9116





x=0.1 m x=0.2 m x=0.3 m x=0.4 m

R2 0.7945 0.7915 0.7573 0.8617

51

FIGURE 44: RELATIONSHIP BETWEEN BUBBLE FREQUENCY AND AIR CONCENTRATION WHEN V0=7.45 m/s IN MODEL




x=0.1 m x=0.2 m x=0.3 m x=0.4 m

R2 0.7477 0.7518 0.7539 0.8521

Figure 42, 43 and 44 shows the relationship between the bubble frequency and the air

concentration of the experimental values and the theoretical values. It can be observed from the

theoretical values that the maximum bubble frequency occurs when the air concentration is

equal to 50 percent, which is relatively consistent with the experimental data at each section.

The experimental data follows a similar parabolic trend as the theoretical values.

5.2 MOTION STUDIO

In this section, the results from the experiments with the high-speed camera on model 2 are

presented, in which several pictures of air bubbles formation and transportation are shown. The

offset height, hs, was set to 0.05 m, the downstream chute angle α=18.2°, the spillway angle

θ0=12.5° and the outlet velocity was set to V0=4 m/s. The diameter, d [cm], of the air bubbles

and the onset distance, D [cm], are presented in the description for each figure and compiled in

Table 23. The time duration, t [ms], between the first and the last picture for each process is

also presented in each description and compiled in Table 23. The following figures are a series

of cropped pictures to see the process of the bubble formation more clearly. In Appendix V¸

there are pictures of the entire cavity zone. These can be viewed to better see where in the cavity

52

zone the onset aeration is located and how the turbulence intensity develops with the distance

in x-direction.

FIGURE 45: AIR BUBBLE MIGRATING INTO THE BLACK WATER. THE WHITE ARROW IS POINTING AT THE BUBBLE

OF INTEREST. d=0.71-0.89 cm, D=10.73 cm, t=36 ms


OF INTEREST. d=1.54-1.69 cm, D = 7.79 cm, t=29 ms



53




OM INTEREST. d=1.00-1.03 cm, D=16.46 cm, t=28 ms

Figure 45, 46, 47, 48 and 49 shows an air bubble being formed in the cavity zone and then

transported into the black water. The first frame in each figure shows where the formation of

the specific bubble starts, named the onset aeration. Thereafter it is shown how the air bubble

forms, changes in shape, ejects from the lower jet and travels along the flow.

From Figure 45, 46, 47, 48 and 49 it is observed that the air bubbles start to form when the

surface of the lower jet becomes irregular and wavy. The surface irregularity occurs because of

the turbulence intensity and when the turbulence intensity is high enough the air bubbles leave

the lower jet and migrate into the black water.

In Figure 45, 47 and 49 the air bubble diameter is relatively small and it could be observed that

they have a smoother, closer to spherical shape than the air bubbles in Figure 46 and 48, where

the diameter is larger and the shape of the air bubble is more irregular. The reason for this is

that for smaller bubbles, the surface tension is the dominating effect, thus making the spherical

54

shape of the air bubble smoother. For larger bubbles the shear forces are the dominating effect,

which causes the shape of the air bubble to become more irregular.

From Figure 45, 46, 48 and 49 it could be measured that the maximum height, zm, is larger than

the bubble radius right before it leaves the lower jet and migrates into the black water, see Table

23. The bubble radius before migration is assumed as half the bubble diameter after migration.

As discussed in section 2.3 Rein [14] states that “A drop that is projected from the flow will

separate from the bulk liquid only if the maximum height h of its trajectory exceeds its radius”.

Note that the h used in the quotation corresponds to zm used in this report. The working

assumption behind the current research at Sichuan University is that Rein’s statement about

water drops ejecting from the water body can be applied to air bubbles migrating from the lower

jet into the unaerated black water.

TABLE 23: MEASURED BUBBLE DIAMETERS, ONSET DISTANCES, TIME DURATIONS AND MAXIMUM HEIGHTS

Figure d [cm] D [cm] t [ms] zm [cm]

45 0.71-0.89 10.73 36 0.94

46 1.54-1.69 7.79 29 2.20

47 0.50-0.62 11.66 28 -

48 1.55-2.08 16.59 25 1.40

49 1.00-1.03 16.46 28 1.42

55

6 DISCUSSION

The results obtained from the probe experiments regarding air concentration showed that a large

amount of air is being entrapped in the lower surface since the air concentration was high at the

surface and just above it. However, the air concentration quickly decreased in the z-direction,

which indicates that it is not much of the entrapped air that is being entrained into the flow.

These results coincide with the findings of Bai et al. [1]. It was also observed that the inclination

of the curve for air concentration as a function of height rose with increased velocity. A steeper

curve indicates that the rate of the decrease in air concentration is lower, which means that more

air is being entrained into the flow. This result is reasonable since an increased velocity yields

a higher turbulence intensity, which in turn entrains more air [13].

The calculated average air concentrations coincide with the findings of Pfister [3], as Ca began

at a low value and rose to several multiples of the initial value further downstream in the

x-direction, though Pfister [3] observed an increase six times higher than the initial value and

these experiments resulted in increases of approximately three times the initial value. This could

be because Pfister [3] accounted for the self-aeration at the upper surface as well as the aeration

in the lower jet while this thesis neglected the self-aeration and thus the air concentration above

the lower jet was assumed zero. The fact that the flow depth was assumed constant as h0 instead

of being measured can also have a negative impact on the accuracy of the calculated values of

Ca, which can be another source of error.

The measurement of the air concentration matched the theoretical values obtained from

equation 6 unexpectedly good, as can be observed from the coefficients of determination.

Chanson [4] derived the formula from a deflector aerator while this project used an offset

aerator and a deviation was therefore expected. Deviations were noted for the two last sections

in experiments 4, 5 and 6, see Figure 27, 28 and 29. This is due to back water being present,

which means that there was water present at the chute bottom in the cavity zone. Equation 6

does not consider back water as it describes a theoretically perfect, water-free cavity zone which

is hard to achieve in practice. This gives rise to a difficulty in estimating the air concentration

close to the bottom near the impact point. The measurements for x=0.7 m in experiment 5, see

Figure 29, showed a big deviation from the theoretical values. These measurements follow the

same trajectory that Bai et al. [1] observed for the air concentration just downstream the impact

point. Because of this it is assumed that the measurement was accidently performed just

downstream of the impact point. Since equation 6 only describes the air concentration in the

cavity zone, this explains why the deviation is big for this section.

56

Regarding the bubble frequency, it was observed that it increased with the distance in x-

direction from the offset. This means that there were more bubbles present in the flow further

away from the offset. This is reasonable since the air bubbles present earlier in the flow have

been transported downstream, though some will have migrated into the unaerated black water.

At the same time, more bubbles have had time to form and these combined increases the bubble

frequency. These results coincide with the previous results found by Bai et.al. [1]. Deviations

were found in the last sections for experiments 4, 5 and 6, see Figure 30, 31 and 32. It was

observed that for experiments 4 and 6, the bubble frequency started at a higher value, after

which it then decreased and followed a similar trajectory as the previous sections. This is also

due to back water being present at the bottom, as water contains more air bubbles than pure air,

which the cavity is supposed to only consist of. These deviations occurred in the same

experiments as the deviations observed for the air concentration, which further indicates that

back water was present during the experiments. However, the bubble frequency for the last

section in experiment 5 started at a high value and did not decrease, instead it continually

increased until a maximum bubble frequency was reached. This is the same trajectory Bai et.al

[1] observed for the bubble frequency just downstream of the impact point. This is also the

same deviation observed for the air concentration at the last section in experiment 5, which

further indicates that the measurement was accidently performed downstream of the impact

point.

The last section in experiment 7 showed a similar trajectory for the bubble frequency, see Figure

39, as the one observed for the last section in experiment 5. There is, however, no indication

from the air concentration-behaviour for this section that the measurements were performed

downstream of the impact point as the last section in Figure 36 corresponds well to the

theoretical values. This could be because the measurements were not performed at the chute

bottom, the lowest point where a measurement was performed was 0.5 cm above the bottom. If

it is assumed that the bubble frequency had been lower at the bottom, the trajectory would have

been more similar to the trajectory illustrated in Figure 11. Because of this, it is not likely that

the measurements were performed downstream of the impact point.

The only difference between model 2 (experiments 4-6) and model 3 (experiments 7-9) was a

smaller offset for model 3, the chute angle and downstream angle were the same. When

comparing the results from these models, no difference was observed for either the air

concentration or the bubble frequency. The only notable difference was that only four sections

could be measured in model 3, while up to seven sections could be measured in model 2. This

indicates that a smaller offset results in a shorter cavity, which in turn leads to a shorter aeration

process. As Pfister [3] observed, the bottom air concentration decreases after the impact point,

thus a shorter cavity zone leads to a lower air concentration at the bottom some distance

downstream the aerator. Depending on the length of the spillway, a too small offset may result

in cavitation damage or the need of an additional aerator [15].

57

When evaluating the coefficients of determination for equation 17, it is observed that it is not

always applicable to describe how the bubble frequency depends on the air concentration. It

can, however, be observed that f/fmax does follow an approximately parabolic trajectory, just not

always the one proposed by Chanson [25].

When conducting the measurement with the probe, the probe was fastened to a wooden block

with a PMMA-disc that was screwed tight to the block. The wooden block was fastened to the

chute aerator with screw clamps. The water flow produced big forces on the probe, which lead

to it shaking despite its sturdy fastening. This could be a source for measurements errors, but

since the experimental results coincide well with theory and observed deviations can be

explained it is therefore not likely that this had any significant impact on the results.

It should be noted that the measurements done in AutoCAD to obtain the air bubble diameters

and the onset distances provided difficulties in clearly seeing the edges of the bubble surface

and the onset of bubble formation. Therefore, the measurements did not yield any exact

measures, but they gave an approximation which is considered good enough to use in the current

research conducted at Sichuan University.

From Figure 45-49 it could be seen that Rein’s [14] statement that the maximum height should

be larger than the radius for a bubble to eject from the water body can be applied on the observed

bubbles. It was also observed that the bubble shapes were consistent with Falvey’s [12]

statement that smaller bubbles have a spherical shape due to surface tension and larger bubbles

experience increasing shear forces which results in a deformation of the spherical shape.

Regarding the Froude, Weber and Reynolds numbers for each model, these numbers indicate

whether the scale effects can be neglected or not. In these models, the values of Fr, W0 and Re

were found to be within the acceptable limits for avoiding scale effects, thus the experimental

results can be considered as applicable on real-world prototypes [18].

6.1 FUTURE WORK

As this thesis has only studied an offset aerator, further studies should be performed on other

versions of a chute aerator to provide the industry with further knowledge about different

aerators to increase the possibility to avoid cavitation damage. There is a new project about

wide step-aerators where the side walls are too far away to influence the flow as these can suffer

severe cavitation damage. Similar experiments as the ones conducted during this thesis are

58

needed for such a version of an aerator to obtain knowledge about relevant parameters in the

flow to be able to prevent cavitation damage.

Spillways are not always open conduits, they can for example be tunnels underground. If the

air velocity in these tunnels becomes too large, high noises occur which can spread into the

power plant and its surroundings which can have bad effects on both animals and humans. The

air properties in these tunnels have not been researched well enough, which is why the State

Key Laboratory of Hydraulics and Mountain Engineering at Sichuan University would like to

conduct research about it in the future.

59

7 CONCLUSION

The results from the experimental investigations showed that a large amount of air is being

entrapped at the lower surface, but not much of the entrapped air is being entrained into the

flow. It was also observed that the amount of entrained air increases with increased flow

velocity. These results agreed with previous studies with similar scopes. The calculated values

of the Froude number, the approach flow Weber number and the Reynolds number were within

the acceptable limits for avoiding scale effects, thus the conclusion is drawn that the

experimental results are applicable on real-world prototypes.

Although it is difficult to study the bubble mechanism, the results obtained from this study

regarding the air bubbles’ shape and formation coincided with previous research. In addition,

this part of the thesis aimed at contributing data about vital parameters to the ongoing research

at Sichuan University. Hopefully, this study will be supportive for reaching the research

objective. However, in order to draw straight conclusions about the bubble diameter and the

onset distance, additional pictures need to be analysed, in which measurements of these

parameters should be repeated.

60

REFERENCES

[1] R. Bai, F. Zhang, S. Liou och W. Wang, ”Air concentration and bubble characteristics

downstream of a chute aerator,” International journal of multiphase flow, pp. 156-166,

2016.

[2] Subcommittee No. 4 of the Committee on Hydraulics for Dams; W.H. Hager, ”Spillways,

shockwaves and air entrainment - review and recommendations. Bulletin 81,”

Commission Internationale des Grands Barrages, Paris, 1992.

[3] M. Pfister och W. Hager, ”Chute Aerators. I: Air Transport Characteristics,” Journal of

hydraulic engineering, vol. 136, nr 6, 1 June 2010.

[4] H. Chanson, ”Air Bubble entrainment in Free-Surface Turbulent Flows,” Department of

Civil Engineering, Brisbane, 1995.

[5] H. Chanson, ”Air-water flow measurements with intrusive, phase-detection probes. Can

we improve their interpretation?,” Journal of hydrology engineering, pp. 252-255, 2002.

[6] H. Falvey, ”Cavitation in Chutes and Spillways,” Engineering monograph No.42, pp. 1-

135, 1990.

[7] B. Sreedhar, S. Albert och A. Pandit, ”Cavitation damage: Theory and measurements - A

review,” Wear, pp. 177-196, 2016.

[8] M. Dular, B. Bachert, B. Stoffel och B. Sirok, ”Relationship between cavitation structures

and cavitation damge,” Wear, pp. 1176-1184, 2004.

[9] J. D. Rogers, ”Hoover Dam: Operational Milstones, Lessons Learned, and Strategic

Import,” Missouri, 2010.

[10] J. Weisheit, ”Glen Canyon Dam is Broken,” n.d. Web. Mars, 2017.

[11] Bureau of Reclamation, ”Cavitation Damage Induced Failure of Spillways,” 2015.

[12] H. T. Falvey, ”Air-water flow in hydraulic strucutres,” Engineering monograph No.41,

pp. 1-115, 1980.

[13] L. G. Straub och A. G. Anderson, ”Self-aerated flow in open channels,” Journal of the

Hydraulics Division, pp. 456-486, 1958.

[14] M. Rein, ”Turbulent open-channel flows: drop generation and self-aeration,” Journal of

hydraulic engineering, vol. 124, 1998.

61

[15] P. Volkart och P. Rutschman, ”Air Entrainment Devices (Air Slots),” Mitteilungen der

Versuchsanstalt fur Wasserbau, Hydrologie und Glaziologie, nr 72, pp. 1-57, 1984.

[16] A. Lima, H. Schulz och J. Gulliver, ”Air uptake along the lower nappe of a spillway

aerator,” Journal of Hydraulic Research, vol. 46, nr 6, pp. 839-843, 2008.

[17] M. Pfister, ”Chute aerators: Steep deflectors and cavity subpressure,” Journal of

hydraulic engineering, pp. 1208-1215, 2011.

[18] V. Heller, ”Scale effects in physical hydraulic engineering,” Journal of Hydraulic

Research, vol. 49, nr 3, pp. 293-306, 2011.

[19] M. Pfister och H. Chanson, ”Two-phase air-water flows: Scale effects in physical

modelling,” Journal of Hydrodynamics , vol. 26, nr 2, pp. 291-298, 2014.

[20] H. P. Koschitzky och H. Kobus, ”Hydraulics and Design of spillway aerators for

cavitation prevention in high speed flows,” The International Symposium on Hydraulics

for High Dams, pp. 724-733, 1988.

[21] H. Chanson, ”Study of air entrainment and aeration devices,” Journal of Hyrdaulics

Research, vol. 27, nr 3, pp. 301-319, 1989.

[22] M. Pfister och W. Hager, ”Closure of Chute Aerators II: Hydraulic Design,” Journal of

Hydraulic Engineering, pp. 360-367, 2010.

[23] “ceprofs,” 2004. [Online]. Available:

https://ceprofs.civil.tamu.edu/ssocolofsky/cven489/downloads/book/ch2.pdf. [Accessed

14 April 2017].

[24] “mthlab,” [Online]. Available:

http://www.mhtlab.uwaterloo.ca/courses/me755/web_chap2.pdf. [Accessed 14 April

2017].

[25] H. Chanson, ”Measuring Air-Water Interface Area in Supercritical Open Channel Flow,”

Water Research, pp. 1414-1420, 1997.

[26] P. Norrlund, ”Hydropower - Technology and system, 10 c. Lecture 4: Turbines &

Hydraulics,” Uppsala, 2016.

[27] U. Lundin, ”Hydraulics and turbine calculations,” i Hydropower booklet - Technology

and systems, Uppsala, Uppsala University, 2015, pp. 87-110.

[28] M. Pfister, J. Lucas och W. H. Hager, ”Chute aerators: Preaerated approach flow,”

Journal of hydraulic engineering, pp. 1452-1461, 2011.

62

[29] W. Wei, J. Deng och F. Zhang, ”Development of self-aeration for supercritical chute

flows,” International Journal of Multiphase Flow, pp. 172-180, 2016.

[30] C. Nordling och J. Österman, Physics Handbook for Science and Engineering, Lund:

Studentlitteratur AB, 2008.

I

APPENDIX I SHORT INTRODUCTION TO THE RESEARCH AT

SICHUAN UNIVERSITY CONNECTED TO THIS

THESIS

Rein [14] describes the force balance at the maximum height for the fluctuations of the free

surface with the equation

𝜌𝑣′2 =2𝜎

𝑟+ 𝑧𝑚𝜌𝑔 (𝐼)

where v’ [m/s] is the turbulent velocity and r [m] is the radius of the curvature in Figure 1. The

left side of the equation describes the pressure fluctuations. At the maximum height, the

pressure fluctuations are balanced by the surface tension pressure and the hydrostatic pressure,

the first and second term in the right side of the equation.

The research in the State Key Laboratory of Hydraulics and Mountain River Engineering at

Sichuan University is aiming at describing a critical point for the formation of a bubble in the

upper boundary of the lower jet for a two-dimensional flow. The assumption is made that the

same equation can be used if the surface tension pressure of the bubble is used instead of the

surface tension pressure in the water. The bubble is formed due to the fluctuations in the upper

boundary which are produced by the turbulence. As the bubble forms upon the boundary surface

when the maximum height is reached, the bubble diameter, d, must be added to the hydrostatic

pressure. For uniformity purposes, the bubble diameter is used for the surface tension pressure

instead of the radius. These changes yield the equation

𝜌𝑣′2=4𝜎

𝑑+ 𝜌𝑔(𝑧𝑚 + 𝑑) (𝐼𝐼)

which can be rewritten as

𝑣′2 =4𝜎

𝜌𝑑+ 𝑔(𝑧𝑚 + 𝑑) (𝐼𝐼𝐼)

This equation describes the critical point where a bubble leaving the water body can be formed.

Note that this thesis has not been about researching this hypothesis, only providing the research

team with information about bubble diameters and onset distances.

II

APPENDIX II ADDITIONAL AERATOR DESIGNS

Apart from the aerator designs discussed in section 2.4.2 , the following designs are also used

to prevent cavitation damage [6]:

• Ramp or deflectors on sidewalls

• Offset sidewalls

• Pier in the flow

• Slots and ducts in sidewalls

• Duct system underneath the ramp

• Duct system downstream of ramp

These methods are illustrated in Figure I below:

FIGURE I: DIFFERENT METHODS FOR AIR SUPPLY TO AERATORS [6]

Note that duct through sidewall and duct under ramp are the same methods illustrated in Figure

8. Duct through sidewall corresponds to 8a and duct under ramp corresponds to 8b.

Piers in flow, offset sidewalls and deflectors on sidewalls are often used to supply aeration

downstream of the control gates. However, these types of air vents are not suitable for wide

chutes [6].

Slots in sidewalls are suitable where installations in already existing structures is required.

These methods are used in control gate structures. It is important not to have a too small cross-

section area. If that is the case, water will be pulled into a high velocity air stream flowing in

the slot. Together with a deflector, the downstream end of the slot may be offset, making the

cross-section area larger, thus preventing water to enter the slot [6].

III

Ducts through sidewalls are used in wide chutes. The duct is a closed conduit, which can have

a rectangular or a circular cross-section area. In areas where freezing is a problem, the ducts are

routed through an embankment for isolation. This prevents ice plugs from forming in the duct

during times when the water may be standing. Ducts under ramp are also used for wider chutes

or in installation where hydraulic jumps may cover the ramp. The combination of ducts and air

vent system ensures sufficient aeration at the lower nappe of the jet in both cases [6].

When the height of the ramp is too small to give a sufficient air ventilation, a duct downstream

of the ramp is used. However, this system requires a drainage to keep the duct free from water.

Leakage and extremely low flows tends to fill the duct with water if there is no drainage. When

this type of system is used, air enters both via the aerator through the duct and via the drainage

gallery [6].

IV

APPENDIX III MATLAB-CODE FOR AIR CONCENTRATION AND

BUBBLE FREQUENCY

The following code is used to obtain graphs, calculate theoretical values with equations 6 and

17 and to calculate R2 with equation 2. The section presented is used for the experimental data

at V0=4.3 m/s for model 2, see experiment 4 in Table 1. The code is repeated for every

experiment, only with different parameter indexes.

%----------------------------------Model 2-----------------------------

%Create a concentration vector and a frequency vector to compare experiment

data with equation 21

c = 0:0.0001:1;

for i=1:length(c);

F(i) = 4*c(i)*(1-c(i));

end

%Round the values to 4 decimals

C = round(c,4);

%------------------------------------V=4.3 m/s------------------------

filename = 'V=4.3ms.xlsx';

%Read excel-file into Matlab

Values43 = xlsread(filename);

%Remove all NaN-columns

Values43 = Values43(:,~all(isnan(Values43)));

%Get number of rows and columns of Values-matrix

[c, r] = size(Values43);

%Create depth-vector, divide by three since only one third of all columns

are depth-values

DepthSec43 = zeros(c,r/3);

%Create concentration-vector, divide by three since only one third of all

columns are concentration-values

ConcentrationSec43 = zeros(c,r/3);

%Create frequency-vector, divide by three since only one third of all

columns are concentration-values

FreqSec43 = zeros(c,r/3);

%Create f/f_max-vector

FreqByFreqmaxSec43 = zeros(c,r/3);

l = 1;

for i=1:3:r

%Row-number for concentration in values-file

j = i+1;

%Row-number for frequency in values-file

k = i+2;

%Get depth-values from column i and save it in column l

DepthSec43(:,l) = Values43(:,i);

%Get concentration-values from column j and save it in column l

ConcentrationSec43(:,l) = Values43(:,j);

%Get frequency-valuse from column k and save it in column l

FreqSec43(:,l) = Values43(:,k);

V

%Divide every frequency with maximum frequency for Eq.21 and save it in

column l

FreqByFreqmaxSec43(:,l) = FreqSec43(:,l)./max(FreqSec43(:,l));

%Next column in vectors

l=l+1;

end

%----------Obtain theoretical concentration from equation 10----------

%One x-value for each sector in the experiment

x43 = 0.1:0.1:0.6;

%Create z-values for input in Eq. 10

z43 = 0:0.5:9;

%Create theoretical concentration-vector, one column per section and one

row per height-step

C_theory43 = zeros(length(z43),length(x43));

%The calculated z_50-values from data

Z50Sec43 = [5.77, 5.62, 5.2, 4.88,4.09,3.63];

%Create vector for storing R-square for air concentration

ConcErr43 = zeros(1,r/3);

%Create vector for storing R-square for bubble frequency

FreqErr43 = zeros(1,r/3);

%Round the experimental values to 4 decimals to coincide with C, otherwise

find won't work

C43 = round(ConcentrationSec43,4);

%One loop for every section

for e=1:length(x43)

%Obtain theoretical values from function AirConc

[C_theoretical]=AirConc(z43,4.3, Z50Sec43(e), x43(e));

%Store value in vector

C_theory43(:,e) = C_theoretical;

%Obtain values for R-square for concentration from function RSQ

ConcErr43(e) =

RSQ(ConcentrationSec43(:,e),C_theory43(:,e),DepthSec43(:,e),z43);

%Obtain values for R-square for bubble frequency from function RSQ

FreqErr43(e) = RSQ(FreqByFreqmaxSec43(:,e),F,C43(:,e),C);

end

The following code is the function AirConc that is used to calculate theoretical values for the

air concentration.

%----AirConc----

function [C_theoretical]=AirConc(z,V_0,Z50, x)

%Subtract y_50 from height and convert to metres

Z = (z-Z50)./100;

%Calculate spread angle between C=10% and C=90%

psi= 0.698*V_0^0.630;

%Calculate turbulent diffusivity at fre-shear layer air-water interface

D_t=0.5*((V_0*x)/1.2817)*(tand(psi))^2;

%Calculate u for each flow depth

u=Z./(2*sqrt((D_t/V_0)*x));

%Calculate the error function for each element of u

erf_u=erf(u);

%Calculate theoretical value of the air concentration

C_theoretical=0.5*(1-erf_u);

VI

The following code is the function RSQ used to calculate the coefficient of determination, R2.

%----Coefficient of determination----

function [RSQ] = RSQ(X_exp,X_theory,Y_exp,Y_theory)

%Remove NaN-values from experimental concentration

X_exp(isnan(X_exp(:,1)),:)=[];

%Remove NaN-values from experimental depth

Y_exp(isnan(Y_exp(:,1)),:)=[];

%Obtain mean value for experimental results

Mean = mean(X_exp);

for i = 1:length(X_exp)

%Find the row where theoretical measure point matches the experimental

measure point, e.g z=4 cm

RadTheory = find(Y_theory==Y_exp(i));

%Obtain the residual

res(i) = (X_exp(i)-X_theory(RadTheory))^2;

%Obtain the total

tot(i) = (X_exp(i)-Mean)^2;

end

%Calculate the sum of all residuals

RES = sum(res);

%Calculate the sum of all totals

TOT = sum(tot);

%Calculate the coefficient of determination

RSQ = 1-RES/TOT;

VII

APPENDIX IV EXPERIMENTAL DATA

TABLE I: AIR CONCENTRATION, C, FOR EXPERIMENT 1. V0=5 m/s

TABLE II: BUBBLE FREQUENCY, f [s-1], FOR EXPERIMENT 1. V0=5 m/s

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m 1.5 - - - 5

2 - - - 14

2.5 - - 4 38

3 - - 12 44

3.5 - 22 77 181

4 23 11 172 252

4.5 12 88 89 122

5 111 165 33 55

5.5 23 66 12 21

6 21 23 6 7

6.5 - 13 4 - 7 - 7 - -

TABLE III: AIR CONCENTRATION, C, FOR EXPERIMENT 2. V0=6 m/s

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m 1.5 - - - 0.982

2 - - - 0.964

2.5 - - 0.995 0.922

3 - - 0.975 0.84

3.5 - 0.995 0.89 0.68

4 0.992 0.99 0.622 0.48

4.5 0.982 0.79 0.32 0.24

5 0.66 0.44 0.14 0.088

5.5 0.08 0.15 0.06 0.022

6 0.014 0.012 0.008 0.0008

6.5 - 0.008 0.0017 - 7 - 0.0018 - -

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m 1.5 - - - 0.984

2 - - - 0.955

2.5 - - 0.982 0.88

3 - 0.995 0.92 0.77

3.5 - 0.92 0.79 0.59

4 0.992 0.79 0.5 0.499

4.5 0.942 0.55 0.26 0.3

5 0.55 0.25 0.11 0.122

5.5 0.12 0.11 0.044 0.041

6 0.011 0.02 0.012 0.01

6.5 - 0.0018 0.006 -

VIII

TABLE IV: BUBBLE FREQUENCY, f [s-1], FOR EXPERIMENT 2. V0=6 m/s

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m 1.5 - - - 5

2 - - - 44

2.5 - - 7 55

3 - - 33 76

3.5 - 22 88 202

4 23 11 200 289

4.5 71 123 122 188

5 133 155 66 66

5.5 67 88 44 44

6 21 61 21 7

6.5 - 32 4 -

TABLE V: AIR CONCENTRATION, C, FOR EXPERIMENT 3. V0=7 m/s

TABLE VI: BUBBLE FREQUENCY, f [s-1], FOR EXPERIMENT 3. V0=7 m/s

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m 1.5 - - - 32

2 - - - 44

2.5 - - 55 66

3 - - 84 222

3.5 - 22 101 309

4 23 44 255 289

4.5 81 111 202 202

5 121 181 133 111

5.5 44 133 81 44

6 33 66 21 47

6.5 - 44 44 33

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m 1.5 - - - 0.995

2 - - - 0.982

2.5 - - 0.982 0.92

3 - 0.995 0.944 0.77

3.5 - 0.92 0.811 0.61

4 0.992 0.84 0.522 0.48

4.5 0.884 0.67 0.392 0.366

5 0.442 0.31 0.166 0.166

5.5 0.088 0.082 0.089 0.061

6 0.021 0.033 0.032 0.032

6.5 - 0.0018 0.011 0.008

IX

TABLE VII: INTERPOLATED HEIGHT, z50 [cm], WHERE C=0.5

V0 [m/s] x=0.1 m x=0.2 m x=0.3 m x=0.4 m

5 5.12 4.90 4.20 3.95

6 5.06 4.60 4.00 4.00

7 4.90 4.70 4.10 3.90

TABLE VIII: AIR CONCENTRATION, C, FOR EXPERIMENT 4. V0=4.3 m/s

TABLE IX: BUBBLE FREQUENCY, f [s-1], FOR EXPERIMENT 4. V0=4.3 m/s

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m x=0.5 m x=0.6 m 0 - - - - 0.9215 0.8087

0.5 - - - - 0.978 0.7597

1 - - - - 0.956 0.8581

1.5 - - - - 0.928 0.908

2 - - - 0.992 0.922 0.863

2.5 - - - 0.985 0.936 0.867

3 - - - 0.955 0.729 0.774

3.5 - - 0.995 0.936 0.621 0.564

4 - - 0.975 0.84 0.559 0.311

4.5 0.9999 0.995 0.89 0.69 0.2122 0.111

5 0.9934 0.9204 0.622 0.44 0.074 0.041

5.5 0.77 0.6 0.311 0.122 0.0314 0.0311

6 0.2793 0.166 0.12 0.0111 0.0021 0.0006

6.5 0.0241 0.0622 0.07 0.0008 - -

7 - 0.0322 0.0314 - - -

7.5 - 0.0018 0.0017 - - -

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m x=0.5 m x=0.6 m 0 - - - - 22 77

0.5 - - - - 12 89

1 - - - - 16 75

1.5 - - - - 18 31

2 - - - 4 21 77

2.5 - - - 14 41 81

3 - - - 21 131 167

3.5 - - 4 33 188 243

4 3 - 8 66 211 171

4.5 4 11 44 162 133 72

5 66 33 155 177 89 44

5.5 88 111 144 66 32 13

6 12 52 66 21 11 4

6.5 3 21 55 4 - -

7 - 14 31 - - -

7.5 - 3 4 - - -

X

TABLE X: AIR CONCENTRATION, C, FOR EXPERIMENT 5. V0=5 m/s

TABLE XI: BUBBLE FREQUENCY, f [s-1], FOR EXPERIMENT 5. V0=5 m/s

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m x=0.5 m x=0.6 m x=0.7 m 0 - - - - - - 0.61

0.5 - - - - - 0.985 0.56

1 - - - - - 0.955 0.62

1.5 - - - - 0.982 0.918 0.69

2 - - - - 0.994 0.88 0.66

2.5 - - - 0.982 0.922 0.79 0.62

3 - - - 0.964 0.89 0.69 0.48

3.5 - - 0.995 0.967 0.79 0.48 0.28

4 - - 0.975 0.92 0.62 0.27 0.11

4.5 - 0.99 0.89 0.722 0.49 0.14 0.07

5 0.992 0.79 0.622 0.44 0.24 0.07 0.031

5.5 0.41 0.32 0.19 0.14 0.11 0.02 0.01

6 0.07 0.08 0.055 0.04 0.04 0.008 0.008

6.5 0.003 0.012 0.0122 0.012 0.011 - -

7 - 0.008 0.008 0.0008 0.008 - -

7.5 - 0.0018 0.0017 - - - -

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m x=0.5 m x=0.6 m x=0.7 m 0 - - - - - - 182

0.5 - - - - - 6 200

1 - - - - - 12 241

1.5 - - - - 4 33 255

2 - - - - 11 44 261

2.5 - - - 5 32 111 311

3 - - - 14 44 211 312

3.5 - - 4 38 87 322 188

4 - - 12 44 199 165 91

4.5 - 11 77 181 288 98 66

5 43 88 211 252 171 66 44

5.5 131 177 89 122 100 33 11

6 32 66 33 55 44 7 7

6.5 22 23 12 21 14 - -

7 - 13 6 7 2 - -

7.5 - 7 4 - - - -

XI

TABLE XII: AIR CONCENTRATION, C, FOR EXPERIMENT 6. V0=6 m/s

TABLE XIII: BUBBLE FREQUENCY, f [s-1], FOR EXPERIMENT 6. V0=6 m/s

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m x=0.5 m x=0.6 m x=0.7 m 0 - - - - - - 0.79

0.5 - - - - - 0.95 0.88

1 - - - - - 0.97 0.932

1.5 - - - - 0.991 0.922 0.91

2 - - - - 0.985 0.908 0.844

2.5 - - - 0.992 0.955 0.863 0.72

3 - - 0.995 0.955 0.87 0.71 0.58

3.5 - - 0.952 0.88 0.71 0.56 0.41

4 - 0.985 0.911 0.69 0.51 0.32 0.26

4.5 0.9999 0.92 0.778 0.42 0.28 0.24 0.17

5 0.77 0.62 0.44 0.25 0.11 0.11 0.094

5.5 0.31 0.25 0.2 0.09 0.055 0.077 0.044

6 0.07 0.08 0.101 0.033 0.022 0.024 0.014

6.5 0.008 0.042 0.021 0.011 0.011 0.012 0.007

7 - 0.01 - - - - -

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m x=0.5 m x=0.6 m x=0.7 m 0 - - - - - - 88

0.5 - - - - - 21 68

1 - - - - - 24 55

1.5 - - - - 14 68 56

2 - - - - 21 44 79

2.5 - - - 12 33 66 211

3 - - 98 33 78 188 382

3.5 - - 61 49 177 400 404

4 - 12 96 233 388 188 222

4.5 44 31 172 342 202 122 89

5 89 117 197 200 133 78 66

5.5 102 122 93 89 66 44 44

6 31 44 76 44 44 33 66

6.5 16 32 27 66 32 24 14

7 - 66 - - - - -

XII

TABLE XIV: AIR CONCENTRATION, C, FOR EXPERIMENT 7. V0=6 m/s

TABLE XV: INTERPOLATED HEIGHT, z50 [cm], WHERE C=0.5

V0 [m/s] x=0.1 m x=0.2 m x=0.3 m x=0.4 m x=0.5 m x=0.6 m x=0.7 m

4.3 5.77 5.62 5.20 4.88 4.09 3.63 -

5 5.42 5.31 5.14 4.89 4.50 3.45 2.93

6 5.29 5.16 4.91 4.35 4.00 3.63 3.24

TABLE XVI:BUBBLE FREQUENCY, f [s-1], FOR EXPERIMENT 7. V0=6 m/s

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m 0.5 - - 44 155

1 - 33 33 215

1.5 - 66 98 283

2 11 99 222 316

2.5 55 144 171 228

3 122 98 98 231

3.5 88 46 66 101

4 33 55 51 28

4.5 - 12 44 22

5 - - - 4

TABLE XVII: AIR CONCENTRATION, C, FOR EXPERIMENT 8. V0=7 m/s

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m 0.5 - - 0.985 0.912

1 - 0.982 0.924 0.82

1.5 - 0.944 0.766 0.664

2 0.985 0.78 0.553 0.501

2.5 0.952 0.55 0.411 0.392

3 0.77 0.33 0.192 0.288

3.5 0.21 0.088 0.066 0.102

4 0.015 0.032 0.01 0.012

4.5 - 0.008 0.006 0.013

5 - - - 0.008

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m 0 - - - 0.955

0.5 - - 0.985 0.942

1 - 0.998 0.902 0.882

1.5 - 0.953 0.744 0.683

2 0.998 0.821 0.55 0.61

2.5 0.722 0.613 0.354 0.5

3 0.479 0.355 0.166 0.341

3.5 0.044 0.144 0.089 0.113

4 0.008 0.014 0.032 0.045

4.5 - 0.008 0.011 0.021

5 - - - 0.011

XIII

TABLE XVIII: BUBBLE FREQUENCY, f [s-1], FOR EXPERIMENT 8. V0=7 m/s

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m 0 - - - 59

0.5 - - 32 88

1 - 19 67 101

1.5 - 34 101 212

2 33 88 232 271

2.5 98 166 211 301

3 132 135 94 221

3.5 51 77 51 104

4 12 68 72 76

4.5 - 13 33 61

5 - - - 68

TABLE XIX: AIR CONCENTRATION, C, FOR EXPERIMENT 9. V0=7.45 m/s

TABLE XX: BUBBLE FREQUENCY, f [s-1], FOR EXPERIMENT 9. V0=7.45 m/s

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m 0 - - - 75

0.5 - - 59 94

1 - - 67 112

1.5 - 35 88 276

2 22 104 202 355

2.5 113 44 294 308

3 198 241 187 187

3.5 144 275 86 133

4 71 198 55 78

4.5 31 122 12 31

5 - 94 - -

z [cm] x=0.1 m x=0.2 m x=0.3 m x=0.4 m 0 - - - 0.977

0.5 - - 0.974 0.921

1 - - 0.932 0.9

1.5 - 0.982 0.844 0.77

2 0.992 0.901 0.65 0.55

2.5 0.702 0.699 0.52 0.41

3 0.511 0.462 0.33 0.26

3.5 0.21 0.198 0.11 0.162

4 0.033 0.042 0.031 0.09

4.5 0.008 0.018 0.011 0.02

5 - 0.007 - -

XIV

TABLE XXI: INTERPOLATED HEIGHT, z50 [cm], WHERE C=0.5

V0 [m/s] x=0.1 m x=0.2 m x=0.3 m x=0.4 m

6 3.24 2.61 2.19 2.00

7 2.96 2.72 2.12 2.50

7.45 3.02 2.92 2.55 2.18

XV

APPENDIX V MORE PICTURES FROM THE HIGH-SPEED CAMERA

The following pictures show the onset distance for each bubble better than the figures in section

5.2. It is also easier to see how the turbulence intensity is increasing with the distance in x-

direction from the offset as the thickness of the lower jet increases within the cavity zone.

FIGURE II: ONSET DISTANCE FOR BUBBLE NUMBER 1. 18.31 LENGTH UNITS CORRESPONDS TO 2 cm

FIGURE III: ONSET DISTANCE FOR BUBBLE NUMBER 2. 36.54 LENGTH UNITS CORRESPONDS TO 4 cm

XVI

FIGURE IV: ONSET DISTANCE FOR BUBBLE NUMBER 3. 18.29 LENGTH UNITS CORRESPONDS TO 2 cm

FIGURE V: ONSET DISTANCE FOR BUBBLE NUMBER 4. 36.72 LENGTH UNITS CORRESPONDS TO 4 cm

XVII

FIGURE VI: ONSET DISTANCE FOR BUBBLE NUMBER 5. 36.24 LENGTH UNITS CORRESPONDS TO 4 cm

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