E XPERIMENTAL INVESTIGATION OF THE AIR - WATER FLOW PROPERTIES IN THE CAVITY ZONE DOWNSTREAM A CHUTE AERATOR

E XPERIMENTAL 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 THE

AIR-WATER FLOW PROPERTIES IN THE CAVITY ZONE DOWNSTREAM A CHUTE AERATOR

Albin 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 028 Examinator: Petra Jönson

Ämnesgranskare: Per Norrlund

Handledare: James Yang

S AMMANFATTNING

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.

E XECUTIVE 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.

A CKNOWLEDGEMENT

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

N OMENCLATURE

D ENOMINATION SYMBOL UNIT

G REEK

Density 𝜌 𝑘𝑔/𝑚 ³

Downstream chute angle 𝛼 °

Dynamic viscosity 𝜇 𝑁𝑠/𝑚 ²

Kinematic viscosity 𝜈 𝑚 ² /𝑠

Loss coefficient 𝜉 −

Spillway angle 𝜃 ₀ °

Spread angle 𝜓 °

Surface tension 𝜎 𝑁/𝑚

L ATIN

Average air concentration 𝐶 _𝑎 −

Air bubble frequency 𝑓 𝑠 ⁻¹

Air concentration 𝐶 −

Air discharge 𝑄 _𝑎 𝑚 ³ /𝑠

Approach flow Weber number 𝑊 ₀ −

Bottom air concentration 𝐶 _𝑏 −

Bubble diameter 𝑑 𝑐𝑚

Cavity length 𝐿 𝑚

Coefficient of determination 𝑅 ² -

Distance between needle tips 𝑑𝑙 𝑚𝑚

Distance from bottom where C=0 𝑧 ₀ 𝑚

Distance from bottom where C=0.5 𝑧 ₅₀ 𝑚

Distance from bottom where C=0.9 𝑧 ₉₀ 𝑚

Distance from bottom where C=1 𝑧 ₁ 𝑚

Error function 𝑒𝑟𝑓 −

Froude number 𝐹𝑟 −

Flow depth ℎ ₀ 𝑚

Offset height ℎ _𝑠 𝑚

Onset distance 𝐷 𝑐𝑚

Outlet velocity 𝑉 ₀ 𝑚/𝑠

Independent variable 𝑢 −

Maximum height 𝑧 _𝑚 𝑚

Reynolds number 𝑅𝑒 −

Sample frequency 𝑓 _𝑠 𝑘𝐻𝑧

Scanning time 𝑡 _𝑠 𝑠

Time duration for bubble formation 𝑡 𝑚𝑠

Turbulent diffusivity 𝐷 _𝑡 𝑚 ² /𝑠

Turbulent velocity 𝑣′ 𝑚/𝑠

Unit air discharge 𝑞 _𝑎 𝑚 ³ /(𝑚 ∗ 𝑠)

Weber number 𝑊 −

G ^LOSSARY

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.

T ABLE 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 I NTRODUCTION

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 P URPOSE

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 O BJECTIVES

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 L IMITATIONS 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. z

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

BUBBLE DIAMETER.

1.4 M ETHOD

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 W ORK 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 B ^ACKGROUND

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 G ENERAL 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 C AVITATION 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 C AVITATION 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 G LEN C ANYON 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 22 ^nd , 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 m ³ /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 m ³ /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 S ELF - 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, z m , 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 C HUTE 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 T ECHNIQUES 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, Q

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 A IR 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 B OTTOM 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, C b , is studied as a separate parameter instead of studying only the average air concentration in the flow, C a .

In the cavity zone, the bottom air concentration is at a constant C b =1 because the cavity consists only of air. The average air concentration is C a ~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 C b =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/h

IS

DIMENSIONLESS DISTANCE FROM BOTTOM, WHERE h

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 T ^HEORY

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 S CALE 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 M ODEL 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/s ² ] 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 [m ² /s] is the kinematic viscosity, μ [Ns/m ² ] is the dynamic viscosity and ρ

[kg/m ³ ] 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 = 𝜌𝑉 ² ℎ

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 W 0 . The approach flow Weber number is expressed as [17] [3]

𝑊 ₀ = 𝑉

√ 𝜎 𝜌ℎ

(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 W 0 >140 and a Reynolds number of Re>2∙10 ⁵ 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, W 0 and Re should be selected more conservatively.

3.2 A IR 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(𝐶𝑉 ⃗ ) = 𝑑𝑖𝑣(𝐷 _𝑡 ∗ 𝑔𝑟𝑎𝑑 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 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 D t [m ² /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 (

𝑧 _𝑐 2 ∗ √ 𝐷 _𝑡

𝑉 ₀ ∗ 𝑥 ) )

(6)

where x [m] is the distance from the aerator along the flow direction, z c [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, z 50 [m], and V 0 [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(−𝑡 ² ) ∗ 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 z c 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 z c as z c = z-z 50 in equation 6.

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

𝐷 _𝑡 = 1

2 ∗ 𝑉 ₀ ∗ 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 ∗ 𝑉 ₀ ^0.630 (10)

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

𝐶 _𝑎 = 1

𝑧 _𝑢 − 𝑧 _𝑙 ∫ 𝐶(𝑧)𝑑𝑧

𝑧

𝑧

(11)

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

Herein, the flow depth z u -z l is not measured in the experiments and is therefore not known. The flow depth is assumed constant as h 0 . Considering the lower jet, its thickness was defined to cover the region between z 90 and z 0 [m], which are the locations in z-direction where the air concentration, C, is equal to 0.9 respectively zero, thus the equation yields

𝐶 _𝑎 = 1

ℎ ₀ ∫ 𝐶(𝑧)𝑑𝑧

𝑧

𝑧

(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 A IR 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, Q a [m ³ /s], along the x- direction by considering the air concentration profile, which yields

𝑄 _𝑎 = 𝐵 ∫ 𝐶(𝑧) ∗

𝑧

𝑧

𝑉 ₀ (𝑧)𝑑𝑧 (13)

where B [m] is the width of the downstream chute and V 0 [m/s] is the outlet velocity. z 1 [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, q a [m ³ /(m∙s)], in terms of the air concentration as [21]

𝑞 _𝑎 = ∫ 𝐶 ∗ 𝑉 ₀

𝑧

𝑧

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, q a

[m ³ /(m∙s)], is described as [1]

𝑞 _𝑎 = 𝑉 ₀ ∫ 𝐶(𝑧)

𝑧

𝑧

𝑑𝑧 (15)

3.4 B UBBLE 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

𝑓 _𝑚𝑎𝑥 = 4𝐶(1 − 𝐶) (17)

where f max [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/h

, IS A DIMENSIONLESS DISTANCE IN Z-DIRECTION WHERE z IS DISTANCE FROM BOTTOM AND h

IS THE INITIAL FLOW

DEPTH, WHICH IS CONSTANT WHILE z IS INCREASING. THE X-AXIS, (f∙h

)/V

, IS DIMENSIONLESS WHERE f IS THE BUBBLE FREQUENCY AND V

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

[1]

3.5 B ERNOULLI ’ 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 𝑉 ²

2 + 𝑔ℎ + 𝑃

𝜌 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (18)

where V [m/s] is the flow velocity, g [m/s ² ] is the gravitational acceleration, h [m] is the height over some reference level, P [Pa] is the pressure and ρ [kg/m ³ ] is the fluid density. The first term, V ² /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 𝛥ℎ _{𝑙𝑜𝑠𝑠} = 𝜉 ^𝑉

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 𝑉 ₁ ²

2 + 𝑔ℎ ₁ + 𝑃 ₁ 𝜌 = 𝑉 ₂ ²

2 + 𝑔ℎ ₂ + 𝑃 ₂

𝜌 + 𝑔∆ℎ _{𝑙𝑜𝑠𝑠} (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 B ERNOULLI ’ S EQUATION IN THE EXPERIMENTS

Equation 19 is used during the experiments to calculate the outlet velocity V 0 . Equation 19 set up between the water surface in the water tank and the outlet, see Figure 12, is

𝑉 ₁ ²

2 + 𝑔ℎ ₁ + 𝑃 ₁ 𝜌 = 𝑉 ₀ ²

2 + 𝑔ℎ ₂ + 𝑃 ₂

𝜌 + 𝑔∆ℎ _{𝑙𝑜𝑠𝑠} (20)

where P 1 =P 2 and V 1 =0 because the area in point 1 is assumed large enough for the velocity to be neglected. Note that V 2 is replaced by V 0 to keep the same notation used elsewhere in the report. With some rewriting, the equation becomes

ℎ ₁ − ℎ ₂ = 𝑉 ₀ ²

2𝑔 + ∆ℎ _{𝑙𝑜𝑠𝑠} = 𝑉 ₀ ²

2𝑔 + 𝜉 𝑉 ₀ ²

2𝑔 = (1 + 𝜉) 𝑉 ₀ ²

2𝑔 (21)

The height difference h 1 -h 2 is substituted into H. The equation then becomes 𝐻 = (1 + 𝜉) 𝑉 ₀ ²

2𝑔 → 𝑉 ₀ = √2𝑔𝐻

√1 + 𝜉 (22)

The term (√1 + 𝜉) ⁻¹ is substituted into μ 0 , which is constant since the loss coefficient is constant. The water tank has been used in previous experiments at Sichuan university, and therefore μ 0 was already known as μ 0 =0.85. The outlet velocity, V 0 , is calculated with the equation

𝑉 ₀ = 𝜇 ₀ √2𝑔𝐻 (23)

25

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

IS THE OUTLET VELOCITY AND H IS THE HEIGHT BETWEEN THE WATER SURFACE AND THE

OUTLET

3.6 C OEFFICIENT OF DETERMINATION

To investigate how well the experimental values coincide with the theoretical values acquired from equations 6 and 17, the coefficient of determination, R ² , can be calculated. R ² is defined as

𝑅 ² = 1 − ∑ ^𝑛 _𝑖=1 (𝑦 _𝑖 − 𝑦 ̂ ) _𝑖 ²

∑ ^𝑛 _𝑖=1 (𝑦 _𝑖 − 𝑦̅) ² (24)

where y is the experimental value, 𝑦̂ is the theoretical value and 𝑦̅ is the mean value of the

experimental values. The closer R ² is to 1, the better the experimental values match the

theoretical values.

26

4 E ^XPERIMENT

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 S ETUP

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 cm ² 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, h s , was created by the model having a larger height than the steel chute. The model could be moved to alter the offset-height, h s . The steel chute was filled with approaching water in each experiment so that the flow depth was kept constant at h 0 =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, h

IS THE FLOW DEPTH, h

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 V 0 was changed by changing the water discharge. All relevant parameters for each experiment are presented in Table 1 and illustrated in Figure 16.

dl air bubble

sensor bar

first tip

second tip

flow direction