NEI-NO--796

MORTEN KJELDSEN N09705227

CAVITATION IN HYDRAULIC MACHINERY

gGEiVEDJOL 21O ST l

NTNUNorges teknisk- natumtenskapelige universitet Trondheim

DOKTORINGENI0RAVHANDLING 1996:63 INSTITUTE FOR TERMISK ENERGIOG VANNKRAFT TRONDHEIM

ITE-rapport 1996:01

The Norwegian University of Technology and Science Faculty of Mechanical Engineering

November 1996

DiS™BUn0N OF tub DOCUMSfl- B tSUUfTH)

This thesis has been submitted to the Faculty of Mechanical Engineering

at the Norwegian University of Science and Technology in partial fulfillment of the

requirements for the Norwegian academic degree

Doktor Ingenior

November 1996

DISCLAIMER

Portions of tins document may be illegible in electronic image products. Images are produced from the best available original document

Preface

Motivation behind this work

This thesis work is a natural continuation of the ongoing research at the Hydropower laboratory in Trondheim aimed towards the understanding of non-stationary flow phenomena. Simply stated, the question is what kind of cavitation is to be expected regarding the mean quantities of the flow when there are varying pressures oscillating at a mean value close to cavitation inception condition.

This work emerged from a general interest in increasing our knowledge on the erosion of structures in flow systems. Thus part of this work was funded by a national research program on erosion of materials in flow systems. However, in an early phase it was realized that an emphasis on erosion mechanisms would give few applicable or even erroneous results due to the problems of scaling the hydrodynamic erosion processes from laboratory trials to real flow systems. Further indications on coupling between material surface properties and cavitation were found which clearly suggested more effort in the hydrodynamic part. Thus the erosion apparatus used raised questions regarding flow processes governing cavitation behavior and hence the hydrodynamic load on the surface.

During this research there was a visit to the Technical University of Denmark. There the formation of numerous cavitation bubbles, which later will be designated by the term cavitation cloud, was investigated in acoustic systems, and a new per­spective on cavitation modulated or governed pressure pulse oscillations in flow systems was gained.

The major hydropower turbine manufacturer in Norway has traditionally avoided cavitation in their machinery by careful design. Actually the onset of cavitation and a developed mode will be formed at the exit of the turbine, during full load operation. The cavitation zone is extended into the draft tube, thus no erosion due to cavitation is observed on the runner. On the other hand, the use of new materials might make new projects favorable regarding production costs if limited cavitation is allowed elsewhere on the runner blade. The hydrodynamic behavior of the turbine regarding both noise, vibration and cavitation is of great interest. It is therefore crucial to establish scientific knowledge of the influence of dynamic

1

pressure pulsing upon cavitation. This thesis serves as a document to provide the guidelines on which further developments and refinements can be based.

The motivation for this thesis work has three aspects.

1. Coincidence with the on-going scientific work at the laboratory to which the candidate is affiliated.

2. Problems experienced in early stages of the present work.

3. External impulses, inspired both by scientists elsewhere and the expected tech­nological progress. Cavitation research should also be founded on new grounds due to the already acquired knowledge on mean flow values at cavitation in­ception.

n

AcknowledgmentsI am grateful to the following institutions for providing funding for this research work, scholarship and visits abroad:

• Norwegian University of Science and Technology (NTNU), Faculty of Mechan­ical Engineering

• Norwegian Electricity Federation, EnFO and Hafslund Energy.

• The Research Council of Norway.

• NORDTEK-program. Funded by the Nordic Council of Ministers

Professor Hermod Brekke is credited for his open-minded attitude toward this study, his function as a window to the ’’real world of technology” and for shielding us doctoral students from most of the financial and administrative struggles found in research work. The personnel at the Hydropower Laboratory are credited for their interest during this study, just to mention the fellow Dr. Ing. students who made this time worth the effort. Special thanks are given to Bjornar Svingen and Jo Jernsletten as ’’senior companions”.

My visits to the Physics Dept, at the Technical University of Denmark were valu­able and I am grateful for the generous hospitality of the scientific/administration staff and current Ph. D. students. Special thanks are given Dr. K. A. Mprch who, in addition of being an excellent host and a valuable source of many of the emerging thoughts in this work, convinced me that new fundamental insights in the nature of cavitation can be found through Physics.

The international cavitation research community has proved to be an inspiration for the work of a ’’freshman”, generously responding to my requests. Special thanks are given Dr. A. Keller who made the authors visit in Obernach a pleasant journey.

The following institutions/companies have also contributed: Kvaerner Energy a.s and SINTEF Corrosion and Surface Technology.

I am grateful to Mr. Stewart Clark at NTNU for his corrections and comments on the English in this thesis.

Trondheim, October 1996

Morten Kjeldsen

m

IV

Summary

Brief summaries are added at the ends of Chapters 1 through 4.

Purpose and scope of workThe main purpose is to change focus towards the coupling of non-stationary flow phenomena and cavitation. In this thesis it is therefore argued that, in addition to turbulence, superimposed sound pressure fluctuations can have a major impact on cavitation. The practical implementation is that these findings will give the designer of a hydraulic device an additional perspective on how to limit the cavitation problems. This is believed to be technological progress because the design tools for predicting stationary low pressure zones are at a high level, though calculations of off-design operation still needs further exploration. This study may further be at the core of the problem why modulated cavitation, i.e. cavitation governed by an external excitation, seems to be the most erosive form of cavitation. The study may also give an explanation of the causes of unexpected cavitation events in prototypes at submersion levels higher than what recommended from laboratory trials.

The purpose of this thesis is therefore to argue that particularly severe erosion can be the result of pressure fluctuations superimposing the convective flow. This thesis will therefore concentrate on the study of sound pressure formation in flow systems and their impact on cavitation. A cavitation system based solely on a sound field has been chosen as a reference study.

Outline of thesisThe purpose of all chapters is to establish conditions for free stream nucleation of cavitation bubbles or cavitation bubble clouds or clusters.

In Chapter 1 a general introduction to cavitation, regarding this work, is given. Special concern is given to cavitation inception, or the initial cavitation event, and the role of nuclei or water quality. Water quality will in this context be equivalent to the properties of water determining its cavitation susceptibility. A contribution to the field of water quality particularly for Norwegian high-head power plants is

v

given based on the work of others. Chapter 1 is basically a review of such work, and serves as a definition of the phenomena which are subsequently covered.

Chapter 2 is devoted to elucidate the existence of pressure fluctuations for sit­uations that are common in hydraulic machinery. The main implication is that cavitation and especially severe cavitation can be traced back to these pressure fluc­tuations. However strong vortices or flow instabilities due to turbulence which are recognized to be some of the main causes of severe erosion are not particularly in­vestigated in this thesis. But it has to be argued that there must necessarily be a border between pressure pulse governed cavitation and flow-turbulence cavitation.

Chapter 3 deals with the actual implications of Chapter 2 on cavitation dynamics. An outline of an algorithm for calculating the nucleation of a cavity cluster or cloud is also presented.

Chapter 4 provides technical information of the systems that are established and investigated in this study, and some additional experimental results. A reason for having a chapter solely on equipment was to avoid distraction in the preceding sections.

Chapter 5 concludes the thesis and indicates the directions for further research.Throughout this thesis work a number of propositions are made. They serve

partly as a summary and a conclusion of the preceding discussion in a given sec­tion, but also as assertions for which subsequent developments / discussions are based upon. Their role will be emphasized in Chapter 5.

A number of examples are also provided. Their function is to serve as an order of magnitude estimate for the specific effects that are currently discussed. It is emphasized that more detailed analysis are considered to be beyond the current level of numerical calculations either due to insufficient knowledge of boundary and initial conditions, or the lack of numerical tools.1

In the appendices the deductions of important relations used for argumentation elsewhere are given. These deductions are mostly taken from references, but are considered important enough to be included u order to enable discussions on the phenomena they cover.

AchievementsCentral achievements in this thesis can be summarized as follows:

In Proposition 4 in Section 1.1.1 it is established that the saturation of air can­not be uncoupled from the nature and number of solid particles in the cavitating medium. This can have a major impact on the role of water quality considerations at the test facilities for cavitation.

1 The development of a Weak Compressible Flow solver is currently in progress at the Hy­dropower Laboratory.

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Section 2.2.4 presents the erosion disc apparatus, and Section 2.2.3 the vibrating horn and Section 3.3.1 shows pictorial studies of the vibrating horn. The hydro­dynamics of cavitation in the erosion disc apparatus and the vibrating horn have been compared. It is asserted that the qualitative observed differences of erosion are mostly due to the sizes of the cavitation bubbles within the cavitation cloud as concluded in Section 3.3.3. The presented view might have a consequence regarding the interpretation of erosion results obtained for various materials, and the scaling of erosivity from one system to an other.

In Section 2.2.1 the formation of compression- or sound waves within a hydraulic turbine has been deduced and compared with experimental data and numerical calculations. As superimposed sound pressure is asserted to have a major impact on the flow cavitation behavior, the sources of these pressure pulses or waves must be established. At present, at least to my knowledge, no numerical flow solver is able to extract the formation of these pressure pulses.

A theoretical model of the flow instability of an attached cavity in Section 2.2.2 has been made. This model is supported by measurements cited in the literature, and experiments done by the present author. A practical implementation could be that by finding the extension of a cavity on a runner blade one can easily check whether modulation of this interfacial instability occurs due to runner inlet pressure pulse formation or other effects. Modulation of the instability is believed to induce severe cavitation and flow-instabilities. Further in Proposition 12 in Section 2.2.2 it is asserted that the above-mentioned instability induces sound pressures that may have significance for the subsequent behavior of free stream cavitation. The qualitative behavior of a cavitation bubble in a pressure perturbed bubble trajectory has been investigated in Section 3.1.

In Section 3.3.3 it has been argued for parameters supposed to determine the formation and the behavior of a cavitation bubble cloud in an ordinary flow super­imposed acoustics. This will have significance regarding cavitation erosion as it is expected that the collapse of a cavitation cluster or cloud largely increases the am­bient pressure of which the bubbles facing the surface of the structure will collapse. This is discussed in Section 1.3.3.

In order to determine the importance of superimposed sound pressure a non- dimensional number has been deduced in Proposition 17, Section 3.2. This number, termed a dynamic cavitation number, is based on time constants for the flow and the sound pressure perturbation in a cavitation susceptible region.

Two systems which induce forced sound pressure oscillations are investigated in Section 2.2.5 and Section 4.3.2. These are the pipe flow system, which is considered suitable for the low Reynolds number regime, and the water tunnel, which uses an underwater loudspeaker that may be suitable for investigating the combined effect of flow turbulence and superimposed sound pressures.

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viii

Contents

Preface iMotivation behind this work............................................................................. iAcknowledgments............................................................................................... iii

Summary vPurpose and scope of work................................................................................ vOutline of thesis .................................... vAchievements..................................................................................................... vi

Nomenclature xii

Abstract xiv

1 Introduction 11.1 Introduction to and definitions and states of cavitation . . ............. 1

1.1.1 Cavitation inception.................................................................... 2Equilibrium considerations of the nucleation of a cavitation bubble 3Water quality- The cavitation nucleus........................................ 4Temperature considerations........................................................... 8Secondary flows............................................................................. 9

1.1.2 States of cavitation....................................................................... 10Developed cavitation.................................................................... 10Cavitation clouds and cavity clusters............................................... 10Acoustic cavitation....................................................................... 10Modulated cavitation.................................................................... 11Cavitation in hydraulic machinery.................................................. 12

1.2 Cavitation bubble dynamics.................................................................... 121.2.1 Bubble dynamics in flow cavitation............................................... 121.2.2 The cavitation bubble collapse.........................................................13

1.3 Multi-bubble dynamics............................................................................. 141.3.1 The homogeneous solution approach . ..........................................141.3.2 Analytical and numerical calculation of multi bubble behavior 14

ix

1.3.3 The collapse of a cavitation cluster........................................... 15Summary................................................................................................................17

2 On the formation of pressure fluctuations 192.1 Discussion of governing fluid equations..................................................... 212.2 On the formation of pressure pulses; sources............................................23

2.2.1 Turbine entrance flow....................................................................24Stagnation point approximation........................................................ 26Water hammer within the runner channel......................................28Illustrations of the presence of pressure pulses in a rotating runner 30 Conclusion on dynamics within a runner.........................................33

2.2.2 The attached cavity.......................................................................35Assumptions....................................................................................35Results............................................................................................. 36Experimental investigations.............................................................. 38Discussion................................................................................... 38Concluding remarks ...........................................................................41

2.2.3 The vibrating horn.......................................................................43Results.............................................................................................44Discussion and concluding remarks.............................................. 46

2.2.4 Correlation of the pressure field and the cavitation erosivityin a rotating disc device............................................................. 49

Experimentally determined pressure fields..................................... 49Discussion and concluding remarks........................................ ... . 53

2.2.5 Cavitation in pipe systems.............................................................. 57Summary................................................................................................................59

3 Flow cavitation in a fluctuating pressure field 613.1 Bubble dynamics in a time dependent pressure field................................613.2 Determination of dynamic effects in cavitation.........................................653.3 On the nucleation of cavitation clouds ..................................................... 68

3.3.1 Cloud formation in a pure sound field............................................68Results............................................................................................. 68

3.3.2 Cloud formation in a spherical focused sound field...................... 743.3.3 Cavitation cloud formation in hydrodynamic flows superim­

posed sound pressures .....................................................................75Statistical consideration of nuclei distribution in a hydrodynamic

flow.......................................................................................................75Pressure in the liquid................................................. 76Argumentation for the formation of cavitation clouds in a hydro-

dynamic flow....................................................................................... 77

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3.3.4 Discussion and concluding remarks on the nucleation of cavi­tation clouds ....................................................................................79

Summary ................................................................................................................80

4 Experiments 814.1 The vibrating horn ....................................................................................... 814.2 The cavitation erosion apparatus.............................................................. 864.3 The water tunnel at the Hydropower Laboratory......................................89

4.3.1 Mechanical design................................................................... ... . 894.3.2 Arrangement for creating sound pressure pulsations ...... 904.3.3 Experimental equipment................................................................. 914.3.4 Discussion on validity of measurements.........................................93

4.4 Measurements of airborne noise................................................................. 93

5 Conclusions 955.1 Conditions preceding cavitation; concluding statements ........ 95

5.1.1 The cavitation nuclei....................................................................... 965.1.2 The formation of pressure pulses ............................................... 965.1.3 The appearance of cavitation ........................................................975.1.4 Methods of investigation................................................................. 97

5.2 Non-stationary flows and cavitation - Summary....................................... 975.3 Limitations................................................................................................... 985.4 Directions of future research....................................................................... 985.5 Closing remarks.............................................................................................99Closing remarks...................................................................................................... 99

Bibliography 100

Appendix 106

A Bubble dynamics 107A.l Small perturbation oscillation of gaseous bubbles ................................. 110A.2 Boiling versus cavitation.......................................................................... IllA.3 Definition of a cavitating medium........................................................... 112

B Rayleigh instabilities at liquid-vapor interface 115

C Digital analysis 119

D Analysis of wake downstream of a guide vane 121

E The cavitation systems at the Hydropower Laboratory 125

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N omenclat ureSymbol Unit Quantityb m Height of runner inletc TB s_1 Celerity of sound (in water)Cm m, s~l Celerity of sound in a two phase mediumD m? s-1 DiffusivityD m Inlet diameter of turbine runnerD m, Diameter of beaker (vibrating horn)E J EnergyE s"1 Cavitation event rate (page 75)f() m2 s~2 Driving term of bubble dynamicsf Hz, s—1 Frequencyfs Hz Sampling frequency (digital measurements)H m Head drop over a hydraulic turbinei Imaginary termk m_1 Wave number (= f)K — Cavitation numberKs — Dynamic cavitation number.1 m Water level in beaker of the vibrating hornh m Immersion depth of the vibrating hornL if m, Length of interface liquid-vaporL m Cavitation susceptible length along streamlineL J kg'1 Latent heat of evaporationM — Mach numbern rpm Turbine runner rotation speedns — Number of samples digital analysisN(R)dR m~3 Number of nuclei of sizes [R —> R + dR] pr. unit volumeP J s~\W Power, P°= Nominal power of turbine unitP Pa PressuresPoo Pa Non perturbated (by bubble motion) pressure in ambient liquidps Pa Stagnation pressuresPr Pa Vapor pressure (Thermodynamic property)IPcrit Pa Treshold pressure for non stable growth o f bubbles (cavitation)r,R m RadiusR m Bubble wall radiusR m s'1 bubble wall radial velocityR m s~2 bubble wall radial acceleration

XU

s kg s2, Nmr1 Surface tension (thermodynamic property)t s Timeu,U,v ms-1 Velocitiesu,v Reduced velocities X— (2gH)o:5 ’ §= acceleration

due to gravity ~ 9.8 ms~2V m3 VolumeW Js-1 Acoustic powerxif m, Running parameter along interface xzy 6 [0, Lif]

SubindexSection

trans

source

0,1

1,2

21

liq,vap

9

b

0lz,lx 2

n,k

oc

Transducer position Acoustic sourceIndexes for distinguishing acoustic and convective effects Indexes for distinguishing single phase and two phase medium Liquid and vapor phase.Non condensable gas Bubble content Initial conditionIndexes for distinguishing componets (radial,tangential) on absolute flow velocity Counters in sum expressionsNon perturbated state of parameter. Reference value

Greek letters-

6 mVp kg m~3u rad s-14> rad

The International System of Units (SI) has been adopted. Parameters used in appendices are not cited in the nomenclature unless they are explicitly used elsewhere in the report.

Void fraction = v»t)OpT viiq

Perturbation of vapor/ liquid interface Efficieny (turbine runner)Density of medium (thermodynamic property)Angular frequencyAngle (tangential direction in a turbine runner system)

xiu

AbstractThis thesis work suggest that a major factor regarding cavitation appearance in hydraulic machinery is the presence of sound pressures. As a consequence it is stated that a particular severe type of cavitation, the cloud- or cluster-cavitation, can be the direct result of the superimposed sound pressure. This indirectly indicates that the presence of sound pressure within a flow system can increase the erosive power of the flowing liquid.

A general introduction to cavitation is given. Within the introductionary section the role of cavitation nuclei exposed to transient pressures is discussed. Further, to validate this approach to cavitation in flow systems and to reveal the mechanisms of sound pressure formation in flow systems selected systems have been investigated both for cavitation and for sound pressure. The impact on cavitation of superim­posed sound pressures is studied. This study, which in addition proposes the use of a dynamic cavitation number, gives a concluding argumentation that the nucleation of a cloud of cavitation bubbles is more likely than the nucleation of a single large cavitation bubble. A system which induce a sound perturbation in a water tunnel has been developed and is presented in this report.

xiv

Chapter 1

Introduction

1.1 Introduction to and definitions and states of cavitation

Cavitation as a phenomenon is covered by numerous scientific disciplines. Cavita­tion is a limiting factor for the performance of ship propulsion, turbo-machinery and hydro-acoustic devices. Cavitation in liquids is also a topic in physics and even medicine. This introduction is mainly restricted to topics that are relevant to hydraulic machinery.

Special emphasis is made on cavitation inception, and particular concern is given the cavitation nuclei at high-head Norwegian power plants.

Terms that are central for the interpretation of the presented work are defined.Several comprehensive works have been written on the subject of cavitation. For

introductory reading, recent books of Young [52] and Brennen [9] are recommended. The work by Knapp et al. [27] covers cavitation research toward 1970 well. Trevena [47] emphasizes the tensile strengths of liquids and is therefore offering an alternative view on cavitation, i.e. as a rupture of the liquid continuum.

A formal definition of cavitation is given by the ASTM standard:1

Definition 1 The formation and collapse, within a liquid, of cavities or bubbles that contain vapor or gas or both

The definition given by ASTM in Definition 1 is, strictly speaking, the sole definition which should be used. However a definition of cavitation* is given by the present author in order to reveal more of the physics involved, and inspired by the views presented by Trevena:

1 ASTM=the American Society for Testing and Materials. Cavitation erosion standard is cov­ered by the ASTM G32

1

Definition 2 Cavitation is the response of liquids, i.e. phase transition, to a de­crease of pressure in the liquid beyond the pressures necessary to sustain a liquid continuum. Cavitation, as a science, also covers the subsequent events of the initial phase transition.

This definition gives rise to questions like: What is the liquid state preceding cavitation? What happens when the liquid ceases to be a continuum? While the first question is of great importance to the designer of a flow or hydro-acoustic device in order to avoid cavitation, the second is what truly can be said, the science of cavitation.

A further definition based on the governing equations is given in Appendix A.3. Also a definition of cavitation erosion is given in the ASTM standards.

In order to distinguish between cavitation and boiling some typical characteristics are listed in Table 1.1, refer to Plesset and Prosperetti [38] and Appendix A.2 for more details.

Table 1.1: Comparison of some characteristics for cavitation and boiling. Note that cavitation is either due to lack of cohesion i.e. rupture in the liquid or adhesion i.e. rupture at or on a material surface.

Boiling CavitationInception At nucleation site having

large thermal fluctuations (near heat source)

At nucleus in the liquid experiencing sufficient low pressure

Growth of bubbles

Thermal properties and diffusion processes control the growth

Impulse governed (condensation and evaporation need negligible thermal energy)

1.1.1 Cavitation inceptionThe initial phase transition is referred to as incipient cavitation. Cavitation incep­tion is a topic of great interest for a large number of scientists around the world. In case there are single events of bubble formation the cavitation itself will not modify the bulk properties of the flow, hence flow and cavitation can be treated as being uncoupled. Cavitation inception is detected either acoustically or visually. Incipient condition is dependent on; the absolute pressure, the flow velocity including turbu­lence levels and boundary layers, the temperature and thermodynamic parameters

2

determined by pressure and temperature such as viscosity and surface tension and the water quality. In this context water quality is the deviation of the working medium from a pure substance, thus water quality refers to pollutants either sus­pended or dissolved all of which can have an influence on cavitation susceptibility. A review article on cavitation inception is given by Rood [41]. Another recent paper, by Keller [22], offers a rigorous empirical deduction of a cavitation number including the most relevant parameters listed.

Cavitation inception is an important state regarding hydraulic machinery. Defi­nitions of states of inception are given in the relevant standards. Dealing with the traditional approach to cavitation in systems we use the cavitation number K. and variations like the Thoma number defined as the ratio between NPSH and head drop over a hydraulic turbine:

jy Poo Pv

K is dependent on absolute pressures, stagnation energy = + \pU^, and tem­perature through vapor pressure, p^ and can be measured at neutral positions. For example Ux is a function of the flow rate while p00 usually takes into account the submersion of the machinery, i.e. p^ can be the adjustable artificial absolute pressure on the suction side of the turbine model in a closed laboratory testing loop. Increased flow rate reduces the cavitation number which on a blade-profile will lower the pressure locally even though the submersion is held constant. Inception is found at a certain level of K. This K value is scaled in order to give criteria for the turbine­setting of the prototype. However critical remarks can be made about scaling, due to the parameters that are obviously not included:

• Water quality.

• Secondary flow.

• Geometrical discrepancies.

The two first will be further discussed while the latter will be briefly mentioned when regarding cavitation in hydraulic machinery.

Equilibrium considerations of the nucleation of a cavitation bubble

If bubbles grow in the presence of other growing bubbles the growth is hampered due to the relaxation in the liquid caused by the other bubbles. A simple example is given to illustrate this fact:

3

Example 3 A liquid water sample having an isobaric pressure below vapor pressure will eventually release bubbles if nuclei are present. E.g. consider a liquid sample at T = 20 °C and p = 1000 Pa i.e. 1200 Pa below vapor pressure. To obtain a thermodynamic equilibrium, at p = 2200 Pa, a certain amount of liquid is evapo­rated. The density of the liquid is raised Ap % = 0.5 • 10-3^,. Defining the voidfraction a = ry——g— and using mass conservation the rise of a can be determined.

(P Ap) Vliq ~t~ VvapPvap P%aKa = vap Ap

Vnq "1" Pyap P ”1" Ap Pvap10"

It is concluded that the ”natural” existence of vapor in a slightly depressed liquid is negligible. E.g. one litre should contain a single bubble having a radius of the order of 1 mm, to achieve equilibrium given the values above.

Nucleation of several bubbles is likely to occur because relaxation waves travel at a finite speed. If there is an inter-cavity distance, 1, it may be stated that nucleation occurs simultaneous if ^ < trise where trise is the rising time for a bubble to start growing. trise has to be in the order of ps to get ’’reasonable” inter-cavity distances for cloud cavitation.

For flows a liquid displacement is likely to occur due to straining of the bulk flow which explains the often vast number of cavitation bubbles in a confined region.

Water quality - the cavitation nucleus

Considering cavitation as a rupture it is of prime interest to isolate a ’’weak spot” from where rupture of the liquid continuum occurs. This weak spot is referred to as the cavitation nucleus. Traditionally the micro-bubble, initially filled with non­condensable gas, has been recognized as the nucleus for most cavitating engineering flows according to Liu and Brennen [31].

The governing equation for bubble dynamics content a driving or excitation term, f(R,poa(t)) as is deduced in Appendix A on page 108. A condition for non-stable growth is given based on derivation of the driving terms, f(R), for a bubble initially at rest, f(R) = 0:

dR4 S

|/=o> 0 => Rc = r-3 Pv Poo

Thus bubbles that are larger than Rc will behave as cavitation bubbles if the ambient pressure equals paa. For an ordinary flow situation with a given p00 [f] the existence of nuclei of a specific kind in order to have initial conditions for the solution

4

of the governing equation has to be guaranteed. Restricting us to the free-stream nuclei,2 concentration is given to two kinds; the micro-bubble and the solid particle.

The micro-bubble is initially filled with non-condensable gas, pg, and vapor, pv. such that the stability condition, f(Ro) — 0, gives, pg + pv — px [to] + where p0 = const.Rq. The interior gas pressure, pg +pv, has to exceed the exterior, p^, due to the restoring force of the surface tension, S'. This means that the partial pressure of gas inside the bubble can be greater than the pressure making equilibrium towards the concentration of gas in the bulk of the liquid. Though suggesting that the bubble would dissolve away, Young [52] p. 57. The cited reference concludes that a bubble would not exist unless there is a sound field present. This is due to growth by rectified diffusion which is a process explaining both the stability and growth of a periodically expanding and contracting bubble, for which motion is governed by a sound field.

Water in a traditional Norwegian high-head power plant has a relatively long history of pressurizing. This involves bubbles from the reservoir being able to dis­solve into solution. On the other hand, the existence of intakes of small rivers and streams introduce air which can make the liquid supersaturated regarding air.

To summarize:

The existence of micro-bubbles containing gas in the free stream is dependent on the degree of air saturation of the bulk of the liquid and the pressure-time history of the bulk of water containing the gas bubbles. Further, in a super saturated water quality and a pre-history of the pressure allowing bubbles to dissolve no original gas bubbles, according to present theories, enter the low pressure region.3

Assuming an originally bubble-free liquid, the method left for obtaining gas filled bubbles as the nuclei is diffusion of gas towards a nucleation site when the bulk of liquid is swept over a low pressure region. For a free stream nucleus a nucleation site could be the solid particle. Hence our discussion on the solid particle will be twofold; the particle as a nucleation site for dissolved gas and the particle as a site for direct nucleation of a true cavitation bubble.

Scientific progress in this field has to be forced to explain free stream nucleation on a solid particle.4 Suppose liquid water at surfaces has some kind of structure and that it has bonds to the wetted structure at distinct spots, Mprch and Song

2 A note on nuclei on a wetted surface of the structure is given in the concluding session of this thesis.

3 Models have been proposed that pose surface active molecules which stabilize the bubble. See e.g. Young [52] p. 43. Within the same reference a crevice model is quoted, which on a hydrophobic material guarantees a stable gas site in a pressurized liquid.

4Most of the thoughts expressed on these matters are attributed to Dr. K.A. Mprch at the Technical University of Denmark.

5

:'--/7777777777777rf

Figure 1.1: Water bonds on a hydrophilic surface. Rupture at a molecular step, Ah, of the material surface and the water forms a structured layer making a cavity. Reprinted from Proc. IMechE Conf. on Cavitation 1992, Cambridge, UK. Mprch and Song [37].

[37]. If the structure has a given curvature (nm-scale) the structure of the water in addition to the nature of the bond can make ” cavities” which in turn can serve as nucleation sites for dissolved gas. In the subsequent discussion distinction is made between pure hydrophobic and pure hydrophilic particles. Water has no bindings to the surface of a hydrophobic particle such that the liquid is totally detached i.e. the particle behave like a bubble in practise. The size of the cavity on a pure hydrophilic particle is governed by irregularities on the solid surface.

The mental picture of the latter phenomenon is that a liquid structured layer is only detached when there are sudden changes of molecular geometry on the surface as illustrated in Figure 1.1. Typical equilibrium radius will be at nano-scales that are at a comparable height of the steps of molecular layers (Ah) but increased by the curvature of the line A-B in Figure 1.1. In order to verify this model Mprch uses an apparatus which measures the tunneling current between a sample and a tip.° The apparatus, a scanning tunneling microscope (STM), is able to detect vertical deviations of the sample surface in the order of Angstroms, but less in the plane normal to the vertical axis. What he measures is that the surface of samples covered with water tend to be smoothened as shown in Figure 1.2. The physical reason is as illustrated in Figure 1.1, however the interpretation of tunneling through the liquid structure is beyond the scope of this work.

Concluding on basis of the discussion above, for any degree of hydrophilicity/ hydrophobicity a cavity will form, which can be the seat of gas release from the liquid and in the next step a site for cavitation bubble nucleation.* 6 When proceeding to propose particles as the origin for micro-bubbles in a true cavitating flow the

°A quantum mechanical effect. Discussion omitted.6 This model could have applications to other branches of nucleation studies e.g. boiling and

phase-transition.

6

---------DRY/DRAINED---------- WATER • |

-------- DRY/DRAINED---------WATER

-------DRY/DRAINED--------WATER

Figure 1.2: Surface topography is smoothened when covered with water according to STM measurements. Note the different length scales for abscissa and ordinate. The figure is provided by Dr. K.A. Mprch.

possibility must be established that the diffusion of gas into the cavity is large enough to create a bubble that is able to nucleate cavitation at reasonable pressures. This process, diffusion in water, is modelled by Tick’s law:

In other words the rate of change of concentration of gas at the boundary be­tween the liquid and the cavity is dependent on the rate of deformation of the boundary, v, and diffusion of air in the bulk of the liquid. Hence with additional knowledge of the diffusion of dissolved air through a liquid/gas interface the growth of a micro-bubble nucleated on a particle can be estimated.7 Calculations revealing some of these thoughts have been done by Arndt [3], and some of the time constants involved in filling an initial bubble to form one that is ten times larger are cited. For supersaturation levels of 1.25 — 5 times equilibrium conditions, the growth time from an initial bubble to ten times its radial size is typically in the range 500 s — 30 s which is calculated for bubbles of initial radii 10 /j,m and 100 \im. The bubble sizes are large such that the calculated time-constants are underestimated due to the fact that smaller initial bubbles will be more dense regarding non-condensable gases due to the increased pressure caused by surface tension. For a turbulent field refer to discussion in Section 3.3.3.

Concerning the amount of dissolved air in the water flowing through the turbine systems the following should be considered. In Norwegian power plants water from

7 The presence of particles, with their cavities, can probably contain trapped gas created at conditions preceding the dissolution process.

7

small mountain streams is often introduced to the head race channel. The intakes often introduce a significant amount of fresh air to the head race channel flow. This air is subsequently compressed and driven into solution. Actually the degree of supersaturation may cause the death of fish downstream the outlet of the power plant because of the slow process of releasing the dissolved air content [45].8

For nucleation to occur directly on the particle it is simply stated that the tension, or the negative absolute pressure, necessary is. All partial pressures are vanishing in the driving term of bubble motion.:

26"Per*- ^

which for an ideal hydrophobic particle gives tensions of 15 kPa for a spherical particle where R = 1 fim, to 1.5Pa, for spherical particle where R = 1 cm,. When an idealised hydrophilic particles can resist tensions of the order 1 MPa (R = lnm) due to the nature of the binding.

To summarize:

It is probable that the existence of free stream gas micro-bubble nuclei, in cavitation susceptible regions in the runner, in a traditional Norwe­gian high head power-plant is debatable due to the shown time constants for gas release on solid particles. One of the consequences of this obser­vation can be that free-stream nucleation of cavitation bubbles suggests pressure pulses below zero absolute pressure, for direct nucleation on a solid particle, being the main reason. Attached cavitation can be formed at absolute pressures close to the equilibrium vapor pressure.9

Proposition 4 Based on the model of M0rch [37], and a possible consequence of ’’micro-bubble” nucleation it is hereby asserted that the air content of the bulk of the liquid cannot be treated uncoupled from the characteristics and density of solid particles. This statement am influence the view on which water quality should be handled doing cavitation inception experiments of hydraulic machinery.

Temperature considerations

The obvious temperature influence on cavitation is the increase of vapor pressure at increasing temperatures. Vapor pressure ranges from 0.9 — 2.3 kPa for the tem­perature range 5 — 20 °C which of course favors Norwegian power plants due to low

8 A Pelton runner at atmospheric conditions is a rather effective way of reducing the air content. It should also be mentioned that cavitation itself is suitable for driving gas out of solution. The latter phenomenon is the reason why many cavitation tunnels have a high rate of pressurization downstream of the cavitation zone.

9 The nucleation of a micro-bubble on the wetted surface of the machinery can take place ac­cording to the mentioned effects; however, the time available for creating a ’’large” gas bubble is ’’unlimited”.

8

temperatures of intake water. Before proceeding it is emphasized that the vapor pressure is an equilibrium pressure at a liquid surface. The consequence is that it is not necessary for cavitation to occur at that pressure level, as has been stated in the preceding section. Theoretically water can have a tensile strength of order 1 GPa, shown by Brennen [9] p. 5. The main issue for not having lower pressures than vapor pressure in a flow system is that if a cavity is formed for any reason it will persist due to the low pressure. The cavity will introduce a water surface where the concept of vapor pressure will have a meaning.

Considering the ’’theoretical” water it has been shown that most tension can persist at 10 °C according to Briggs [10] and that the tensile strength abruptly drops as temperatures fall towards the freezing point. The explanation given here is due to Morch and Song [37] and personal communications with Mprch. The main idea is that water behaves structured when it wets solid surfaces, Figure 1.1, i.e. the same model for explaining cavities at surfaces used above. Mprch states further that a possible explanation for the 10 °C extremum is that the structure is behaving elastic, that increased temperatures decrease the tensile strength due to increased thermal activity and that at lower temperatures the structure behave more ice-like or stiff. He suggests that at low temperatures the water actually ruptures, on solid surfaces, like a solid due to tension of the ”ice-structure”.

The third consideration of temperature versus cavitation nucleation is the time constants involved in filling the bubble with vapor. In the governing equation the vapor pressure is set constant throughout all compressions/expansions, where this constant pressure is one of the driving forces for expansion. This is true when the time-constants allow the condensation and evaporation to occur without thermal delay. In practise this mean that the energy required is negligible viewed as temper­ature fluctuations at the bubble wall. If the pressure pulse is short, growth can be seriously hampered and eventually the bubble will not react to a pressure well be­low a pressure able to nucleate non-stable growth and a subsequent violent collapse. This matter is further explored in Appendix A.2.

Secondary flows

Secondary flows have a significant influence on cavitation inception. Results pro­vided by Keller [21] show that turbulence intensity has a major influence on inception cavitation numbers. The reason could be that small vortices, which are believed to be the result of the grid placed upstream to induce turbulence creates ” low-pressures” convected over the flow obstacle. Several other experiments of the same nature have been performed where different approaches to create flow fluctuations have been used such as an oscillating hydrofoil, Reisman et al. [40].

9

1.1.2 States of cavitationDeveloped cavitation .

Developed cavitation may be defined as a state influencing the performance of the flow object, or as a state giving continuous destruction of the material facing the flow (wetted surface) or being a lasting source of noise. The former state suggests that the vapor phase is influencing the bulk properties of the flow, the latter does not necessarily break down performance but is often highly unwanted in power generating and propulsion machinery.

Cavitation clouds and cavity clusters

The term cavitation cloud is adopted from Brennen [9] and is defined as a collection of cavitation bubbles within a confined region in time and space. The term cavity cluster is adopted from Mprch [35] and has been used by March primarily defined as a collection of homogeneous bubbles the behavior of which is treated as a single system. In this thesis cavitation clouds and/or cavity clusters are central because they are suggested to be one of the consequences of the pressure fluctuations investigated and their subsequent ability to focus energy during their collapse.

No distinction is made between the two terms in this thesis.

Acoustic cavitation

Acoustic cavitation is traditionally viewed as cavitation in a stagnant liquid, where the formation of cavitation is due to pressure fluctuations or sound waves. Acoustic cavitation as a discipline is covered well in the work by Young [52] which also contains references to major parts of significant literature in that field.

The characteristics of cavitation created in a sound field is that bubbles in equi­librium are excited to grow without the influence of convective flow effects even at atmospheric conditions. What also distinguishes acoustic cavitation from flow cavitation is the threshold frequencies and amplitudes of non-linear growth.

A pioneering work on cavitation cluster nucleation was done by means of a sound field. Ellis [17] had a cylindrical beaker with a piezo electric ring pressure transmitter. The various fields were obtained using different beakers and forcing frequencies. In the bottom of the beaker a cavitation cloud was formed in a confined region due to cyclic straining of the liquid.

The same effect, as cited above, was found during the experimental investigations done in this thesis work at the Technical University of Denmark, where the vibrating horn obviously set up a field that was able to nucleate a cavitation cloud in the bottom of the beaker. Actually this particular cavitation eroded the membrane of a pressure transducer mounted in that position.

10

This thesis work considers the formation of compression waves in flow systems as a potential cause for enhanced cavitation. Thus several terms have been used throughout the text which need definitions.

Definition 5 Sound pressure: this means a pressure perturbation propagating at a characteristic velocity, the sonic speed in the liquid = c, different from the charac­teristic hydrodynamic flow velocity = U. The ratio / is termed the Mach number, the use of which is demonstrated in Section 2.1.

Definition 6 Acoustic source: this is a confined area where the sound pressure is being formed, due to a temporal variation of forces acting on the liquid medium, and from where it emits.

Definition 7 Sound field: if the temporal emission extends in time, the sound pres­sure will be a function of the time and the spatial position relative the acoustic source. The sound pressures functional dependency on time and position is therefore defined as the sound field. Further if the source is steady oscillatory and reflecting/ absorb­ing boundary conditions exist, a steady oscillatory sound field can be formed with the possibilities of nodes, i.e. at given positions no sound pressures will be detected.

Definition 8 (Sound) pressure pulse: the formation of the sound pressure may not be sinusoidal, thus pulses can be emitted from the source. On the other hand, if the pulses are steady regarding frequency they can be decomposed into trigonometric functional components.

Modulated cavitation

The term modulated cavitation is to be understood as a varying degree of cavitation due to pressure fluctuations excited elsewhere in the flow systems. When monitoring cavitation in a high frequency domain the typical noise and vibration signature of the cavitation event will be typically > 10 kHz. Knapp et al. [28] showed in Francis turbine model tests that the most distinguishable signature for cavitation versus no cavitation operation was found in the range 250 — 700 kHz. If the amplitudes varies according to a lower frequency that occurs naturally, the events of cavitation are said to be modulated by this excitation. This phenomenon has been investigated recently by Bajic and Keller [6] at a hydro-power plant. Abbot et al. [1] and Farhat [18] have investigated this phenomenon in laboratory trials. It is commonly believed that modulated cavitation induces more severe erosion than other manifestations of cavitation.

Acoustic cavitation as described in Section 1.1.2 is obviously a special type of modulated cavitation. That is, with almost no exceptions a well defined frequency is causing the cavitation events.

11

For hydrodynamic flows the excitation can be due to the convection of periodi­cally shed swirling structures into a cavitation susceptible region or due to temporal variation of mass flow rate into a confined region as was briefly mentioned in Sec­tion 1.1.1. In this thesis modulated cavitation is central due to the investigation of the formation of pressure pulses transmitted at the speed of sound into cavitation susceptible regions.

Cavitation in hydraulic machinery

Two resent papers treating cavitation in hydraulic turbines are briefly reviewed. Brekke [7] discusses the main dimensions of the layout in order to prevent cavitation. The conclusion of that work is that most effort to make a unit that is less susceptible to cavitation must necessarily sacrifice loss of efficiency. McNabb and Desbiens [33] state that both development of design and manufacturing have beneficial impacts on the cavitation behavior. As a consequence the concern for cavitation damage should no longer exist. However, what can be argued is that a more geometrical correct unit allows the establishment of a sound field with pressures of significant amplitudes.

1.2 Cavitation bubble dynamicsThe equation for the dynamics of a bubble is deduced in Appendix A and assumes spherical symmetry, incompressible liquid and negligible thermal effects.

/(#,?«,(<))

hi der to take into account the high wall velocity (M —> 1) at the final stages of the collapse, terms to include the compressibility have been added. Reference should be made to textbooks on cavitation for a further introduction on this subject.

1.2.1 Bubble dynamics in flow cavitationIn his thesis work Gindroz [19] used the governing equation for the dynamics of single bubbles in their flow paths, given the latter from numerical analysis, through a Francis turbine runner and past a NACA profile. The interesting part is that he experimentally had a device that was able to inject pre-deflned bubbles into the flow. He could therefore observe the behavior and specifically the onset of cavitation. As a result of his investigation, and based on assumptions of the nuclei content, he was able to predict the cavitation of the prototype, or the scaling of cavitation behavior. The completeness of his research work is emphasized by his subsequent development of a nuclei counting technique suitable for plant measurements.

12

Liu and Brennen [31] stated that it has been recognized that cavitation is caused by micro-bubbles being convected into low pressure regions. However critical re­marks on this model are given by these authors, these ideas are further discussed in Section 3.3.3 in this thesis.

1.2.2 The cavitation bubble collapseThe very field of cavitation is often attributed to the efforts of Lord Rayleigh in his article in the Philosophical Magazine VI, vol. 34, 1917. Where as a numerical exercise he calculated the pressure during the collapse and found a pressure of 1260 times the far field pressure, i.e. the atmospheric pressure, close to the moving boundary (at r = 1.5877? where R = l/207?o ; Ro being the initial bubble size).

The collapse of a cavitation bubble has been investigated by several scientists; numerically by Plesset and Prosperetti [38], Blake [8], Zhang et al. [53] and experi­mentally (photographic studies) by Lauterborn et al. [30].10

Special attention is given to the article by Blake because he includes the effects of different boundaries close to the imploding bubble. He also found that materials, that is composites and flexible materials, actually can have the same influence on the imploding bubble as a free surface does. In both cases the direction of the jet, that forms during the bubble collapse and assumed to be the main cause of erosion, can be significantly distorted and eventually directed away from the surface in question, i.e. contrary to the rigid surface where the jet is directed towards. This should explain why, during the erosion experiments made on the erosion apparatus used in this work, there were no traceable erosion on a polyester specimen while at the same intensity typical steels were eroded.

10 All articles are recommended as an introduction to the field of bubble dynamics.

13

1.3 Multi-bubble dynamicsFrom considering the cavitation bubble as a distinct event, multi bubble cavitation is commonly treated as a homogeneous mixture due to the difficulties treating it as an N-Body system.

1.3.1 The homogeneous solution approachThe subject of the homogeneous solution of bubbles has been reviewed by Brennen [9]. The specifics of the dynamics for a solution containing bubbles has been worked out by Kumar [29], in a small perturbation approach i.e. no actual cavitation.

One of the results from a homogeneous solution approach is the frequency dis­persion of linear waves (acoustics) as has been reported by Chrighton [16]. Which deduced the sonic speed in a liquid as a function of frequency and of the natural frequency for the bubbles in the solution. This is an effect that is further explored in Section 3.3.3.

1.3.2 Analytical and numerical calculation of multi bubble behavior

Chahine [13] formulated the bubble-bubble interaction in a cavitation cloud by regu­lar perturbation theory expanding the solution for a single bubble adding correction terms due to the neighboring ones. Here the first order term is a spherical modifi­cation to the collapse driving pressure. The second and third order terms including non-spherical perturbations on both pressure and velocity fields. The set of equa­tions deduced for every order has been solved for specific conditions. A step-wise pressure drop of finite duration and a finite number of bubbles, showed a clearly en­hanced pressure peak during the collapse, incompressible approach, of a cavitation cloud than for single bubbles. Further Chahine refers to Ellis11 where Chahine con­cluded, based on pictorial works, that the pressure information is present throughout the whole cloud, contrary to the shock wave theory used by e.g. Mprch [35] (to be cited below) and in this presented work where no information of changes in am­bient pressures are found in front of the shock wave. Thus Chahine supports a view where the dynamic cavitation cloud behavior is due to bubble-bubble interac­tions and cannot be described by the homogeneous approach which is to use void fractions to describe the collective effect of cavitation bubbles. Chahine has fur­ther explored the analysis to numerically calculate the bubble-bubble interactions

11 Ellis, A.T ’’Techniques for Pressure Pulse Measurements and High-Speed Photography in Ul­trasonic Cavitation” Proc. Symposium on Cavitation in Hydrodynamics, Teddington, England, 1955. Not reviewed in the present work, it could not be delivered within the deadline for the submission of this thesis.

14

including shape-deformable bubble walls [14].

1.3.3 The collapse of a cavitation clusterAs one of the major purposes of this thesis is to establish criteria for cavitation cluster or cloud nucleation it is necessary to look into details of the collapse of a cavity, this specific deduction is found in Mprch [35].

The volume occupied by cavities in a confined region of liquid can be expressed through the void fraction 8\.

0i =V.vap

vmy vap TThe sonic speed of the medium can be expressed as:

Pi4 =PA (i - A)

(l.i)

(1.2)

Where, for further use, index 1 refers to the interior of the bubble and index 2 refers to the exterior. If the region is subjected to a far field pressure exceeding the one found in the interior, the bubbles start to collapse. A pressure front of a shock wave nature will form and propagate through the cluster, a model which is proposed and proved experimentally by Campbell and Pitcher [11]. The Mach number associated with this shock wave is:

% = (g (1.3)

Assuming /% = 0 (all bubbles annihilate), such that the boundary actually fol­lows the shock wave, it is obtained:

Vsw = cmM\ (1.4)

Assuming semi-spherical cluster shape make, using the continuity equation:

R2vr = 0i R (1.5)

The equation of motion gives:

dvT+ vr

dvr(1-6)

1 dpdt dr pdr

Inserting known quantities the pressure distribution in the liquid is found to be:

p{r)=Poo + p[Pi2RR +R2 R

fta# #

2M(1.7)

15

Using the latter equation combined with the fact that p (R) = p%: the final equation yields:

(1.8)

Thus the algorithm to use is to calculate the pressure, using proper initial con­ditions, until the radius of the cluster approaches the size of a single bubble and utilizing single bubble dynamics and the obtained pressures as the forcing term. The final equation is similar to that of a single bubble which is known to create large pressures in the final stages. In this matter the collapse pressure of the last bubbles is greatly enhanced and indirectly a more erosive medium is suspected.12 Using a slightly different approach, where the characteristics of the cloud is directly related to the single bubble dynamics, Wang and Brennen [49] confirmed the formation of a shock wave in their paper.

12 A critical remark can be given to the model. For a semispherical collapse the final implosion occurs at the center, i.e. a distinct erosion mark should be found in the center which is often not there.

16

Summary• Definitions and explanations of cavitation and manifestations of the phenom­

enon of cavitation have been given.

• Cavitation inception, or the onset of cavitation, has been emphasized in this introductory chapter. This emphasis is validated due to its great influence on the turbine setting, for example, and hence the problems of scaling laboratory trials to field conditions.

• It has been argued that both solid particles (debris, suspended material) and dissolved air in water are of major importance in order to quantify the cavi­tation susceptibility and that these two matters are interconnected as far as free stream nucleation is concerned.

• It has been shown that the temperature of the water can have a significant influence on the cavitation characteristics. It is especially emphasized that the cavitation appearance depends on dynamic stressing as a function of temper­ature.

• The use of the governing equation for cavitation bubble dynamics has been demonstrated for several branches of the science of cavitation.

• It has been shown on a theoretical base that a collapsing cavitation cluster or cavitation cloud can enhance the erosive effect through a significant increase in ambient pressure for the bubbles collapsing close to the solid surface.

17

18

Chapter 2

On the formation of pressure fluctuations

The aim of this chapter is to establish an understanding of the processes forming pressure pulses.

A discussion on governing equations in single phase fluid flows will reveal the discrepancies and similarities of acoustic and traditional incompressible flow formu­lations. Qualitative behavior of theses waves in turbulent fields and pressure wave formation by turbulence will be part of the discussions of the given set of equations.

The actual origin and behavior of the pressure waves propagating at the sonic speed of the liquid will not be submitted to a detailed investigation as it is con­sidered beyond the scope of this work. However, when scaling the findings of this work to general fluid machinery and especially hydraulic water turbines, emphasis will be placed on revealing the qualitative mechanisms for the formation and the propagation of these waves for demonstration purposes.

The following systems are subjected to an analysis in this part, due to their direct or indirect importance for cavitation behavior:

Table 2.1: Hydraulic systems characterized by pressure fluctuations investigated in this thesis work and the motivation for doing so.

System MotivationInlet section of turbines Attached cavity The vibrating horn Rotating disc Piping systems

Modelling of source for pressure pulses Modelling of interface instability For comparison with flow system Investigation of pressure pulse formation Exploiting possibilities of cavitation studies

19

The emphasis on flow dynamics is motivated by another reason from those stated in Table 2.1. From a Norwegian point of view, where hydraulic turbines are impor­tant, the following is to be noticed:

The fact that hydroelectric power plants have reservoirs is highly advantageous for the governing of peak-loads in the electric grid. This is due to the short time constants for increasing load from practically no-load to full-load, or from about 0 MW to 300 MW for typical large single turbine units, where the time constant is of the order of one minute and only dependent on valve opening times to avoid water hammer or large pressure fluctuations in the pipe/tunnel system.

The dimensioning criterion for Norwegian power plants has mostly been to secure energy demands for many years in specific regions of the country. Now free power trading inside Norway is allowed hence a region does not depend on deliveries from nearby power plants. Furthermore; prices of electric power are governed by demand such that the plant owner do have a choice of producing all their potential energy in a short period of time or at the traditional production rate.

Technical problems occur when energy production is only for a short time, mainly because the turbines are designed for a lower flow rate than what is now appreciable. The less expensive solution is to exchange the turbine runner. In doing so one can expect severe noise, vibration and cavitation due to decreased distance between guide vane exits and turbine runner inlet creating pressure pulses, and higher flow velocities throughout the whole unit.

When it comes to pressure pulses and their role for cavitation behavior of the machinery it should be kept in mind that there is a dependency on temperature. A relatively high temperature of the water increases the vapor pressure and obviously makes the unit more susceptible to cavitation, while the resistance to a transient pressure wave will increase due to a more thermic controlled growth of the bubbles see Appendix A. 2. Water at Norwegian high head power plants is normally cold < 10 °C hence a greater resistance to the formation of developed cavitation is ex­pected while its resistance to transient pressure waves is less.1 Taking Norwegian power plants as a base for this particular study, emphasis on the coupling transient pressures and cavitation is of major importance.

1 Remark: Taking temperature as the sole measure for the tensile strength of water it is highest at 10 degrees Celsius (Briggs,1950 [10]) where a recorded resistance of 280 bars (-28MPa) is found.

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2.1 Discussion of governing fluid equationsThe continuity equation, control volume, for a fluid flow may be expressed in vector form as:

— +V{p!?) = 0 (2.1)

Where p is the density of the fluid and V the gradient operator and ~v the flow velocity. The conservation law of the impulse of the fluid gives, neglecting viscous and body force terms:

4- VV(pV) = -Vp (2.2)

Where p is the pressure in the fluid. In order to distinguish convective flow and compressible effect due to a sound field we decompose the dependent variables in­troducing subindexes 0 and 1. Where the former is due to convective or mass flow and the latter is due to compressible effects. The dependent variables are then written:

p = Pot'x'.t) +Pi (x\t)if = 'v'o (~x,t) + vi (lf,t) (2.3)

P = PO + PlC&,t)

Compressibility is introduced as an equation of state, given as:

P~Po = c2(p- Po) (2.4)

Which by using the decomposed variables writes:

Pi = c2pi (2.5)

Remark that parameters indexed 1 have an oscillating nature around the in­stantaneous values of the fluid parameters indexed 0. The indexed 0 variables are allowed a temporal component which in an Eularian frame of reference is due to the fact that mass surging may occur and to the fact the turbulence is a momen­tum/mass mixing phenomenon.2 Letting T*o = 0, that is no convective flow, the resulting equations are:

<? atd{povi+pivi) , —»

at ^ yi V (povt + PiVl) =(2.6)

- v

2 Turbulence, strictly speaking, should not be included in this discussion due to the omission of viscous terms in the governing equation for impulse.

21

In a first approach is assumed that p0 and p0 have fixed values, that ^ < 1 (small perturbation) and the equation reduces to

+ PoJ vi =0_^ (2.7)po^jf + poVi V^ = -Vpi

Now adding the two equations, when taking the gradient of the continuity equation and the temporal derivative of the impulse equation.

(2.8)

A dimensional analysis when introducing quantities relevant for acoustic systems t = z,x = A = gives: O(M) - O(M) = O (^)- Hence if ^ < 1 the

pure wave equation is obtained, which also serves as the dynamical equation for one dimensional non-stationary pipe flow. The equation may be deduced in a similar fashion for the pressure. Proceeding by letting li*o ^ 0 A po 7^ kon,st A po ^ konst, now pi — c2 (p0) pi and po = /(po)- Again ^ « 1 since the density of water, p0, is about 1000 ^ at saturation (T = 20°C, p — 2,4 kPa).

^7^ + ^ ^ = 0^^ + ^V(po^ + A,%+Pi^)+ ^

Vo V (poVi + PqVo + Pi^o) = - v (po + Pi)

Subtracting the acoustic equation, Equation 2.7, and the Euler equation or the incompressible convective flow equation from Equation 2.9 the result is:

v*b”o = 0"i‘v + (Pot? + 5^p«o) - 0 (2.10)

Thus a proposition is made:

Proposition 9 When solving the combined effect of the convective flow and acoustics one can solve the acoustic solution and the convective solution independently and use the coupled Equation 2.10 as an estimate of error function for the lack of interaction of the sound field and the convective flow. Remark: the source term of the acoustic equation must be the loss term of the convective solution.

Now q ~ thus in order to get a particle velocity of 1 m/s a pressure fluc­tuation of pi = pc % 1.5 MPa is necessary. Such pressure amplitudes are not considered realistic unless a resonant sound field establishes. The sound field will in

22

the subsequent sections be treated uncoupled from the hydrodynamic or convective flow, due to their assumed negligible interaction.

A preliminary dimensional analysis suggesting both |?q| and |y0| to be C c, i.e. M C 1 where now M is the Mach number. The coupled equation is, continuity correction term vanish, nothing but.

+ iq V Fo = 0 (2.11)

Evidently the set of equations for Weak- Compressible flows, |?q j A |?.’oj -C c, can be written as:

(2.12)

Viscosity terms have been omitted, but in the present analysis they would have been linear and no coupling would have taken place. However much effort is given today to find both the physics involved and the quantification of the viscosity index in pulsed flows for both pulsed convective flows and acoustic flows. Introducing turbulence, in the indexed 0 parameters, it is clearly noticed that fluctuations at time scales comparable to the sound waves can be affected such as the scattering of sound waves and on the other hand, the opposite may be the modulation of turbulent structures.

2.2 On the formation of pressure pulses; sourcesA lot of effort is nowadays put into the investigation of modulated cavitation, where cavitation events are traced back to the natural occurrence of oscillating fluctuation of single phase flow quantities, e.g. shedding of vortex structures from an attached cavity. Little information has been found on the correlation of the nature of these fluctuations and cavitation, thus one of the major challenges is the interpretation of the appearance of modulated cavitation. A study is therefore included for both the formation and propagation of pressure pulses and later on whether they can induce a specific type of cavitation.

An example is provided, Figure 2.1, where part of that work was to investigate draft tube pressure oscillations at a high specific numbered Francis turbine. The frequency of the fluctuations was in the range of 1 — 5 Hz and of amplitudes which certainly would influence the gross cavitation behavior regarding the cavitation num­ber, but not the actual cavitation event. This opinion is also expressed by Bajic and Keller [6]. Concerning the latter any bubble cavitation event can be considered as an event at a constant cavitation number due to the short time constants involved in single bubble and even multi-bubble rise and collapse. However the processes

23

Seconds0,5 1 1,5 2 2,5 3 3,5 4 4,5

Figure 2.1: Large low frequency draft tube pressure fluctuations in a Francis turbine at half load |P % 9.5 MW , n = 250 rpm, if = 51 m.

involved in the formation and the collapse of an attached cavity when the cavita­tion number is varied [40] may cause extraordinary effects which can enhance the vibration and the erosion. This issue is not investigated in this study.

The very nature of an acoustic source is a difficult task by itself. The details of the non-linear effects involved in the creation of sound pressure pulses will not be discussed. The source strength is deduced, with precautions, when sound pressure formation is considered to be of importance. The principle of stagnation pressure variation due to flow-structure interaction as an acoustic source will be used for the turbine runner entrance and the rotating disc. Considering the vibrating horn the moving tip boundary is automatically treated as an acoustic source as no ’’dominant” convective flow patt is emerge. Also an experimental study of the liquid/ vapor- gas interface of an a, bed cavity strongly indicates that this is a source of sound pressure emission. An introductory approach on induction of water-hammers in a runner vane due to spatial variation of mass flow is also given. An accurate study of this effect should include the three dimensionality and the flexure of the vanes suggesting that this is a matter of fluid-structure interaction. Lastly the specifics of a forced oscillating pipe systems are discussed regarding cavitation.

2.2.1 Turbine entrance flowWhen projecting a full-turbine a specific goal is to decrease the inlet diameter of the runner. In doing so a decrease in runner vane length must be admitted, hence a larger load per unit length along the vane is unavoidable. The main consequence

24

is that the design must allow for a low absolute pressure on the suction side even towards the inlet, i.e. conditions are present for cavitation in general to occur.

At the inlet the distortion of the velocity profile caused by the guide vanes, see Chen [15] and Appendix D for citation of experimental results and discussion on validity, will make the stagnation pressure vary at the inlet due to the rotation of the leading edge compared to the guide vanes. This can be shown through simple analysis, which will be reproduced here, and through 3D non-stationary numerical calculations [20]. It is believed that the non-stationary behavior will cause the formation of pressure pulses that correspond to the indexed 1 parameters in Section 2.1. These pulses are of great interest for the characteristics of cavitation because:

1. Inlet suction side cavitation is close to the leading edge and hence close to one of the sources for pressure pulse formation.

2. Outlet cavitation can also be influenced because a moderate pressure pulse, of the order of 1 kPa or 10 cmWC can have a significant influence on the cavitation response, due to the already low absolute pressures.

Two physical reasons for pressure pulse formation are now proposed and inves­tigated:

1. Temporally fluctuating stagnation pressures will make the inlet an acoustic source.

2. Spatially fluctuating flow rate in an absolute frame of reference can be able to induce water hammer in the runner channel at certain conditions. The same effect may be relevant for an uneven flow distribution in the spiral case.

At the acoustic source non-linear effects [34] transfer energy from hydrodynamic flow to acoustic power. It must be noted that the stagnation point itself, indepen­dent of any fluctuation, represents a loss term due to strong retardation of the flow. Further the propagation can be treated as sound waves as it is considered, at least unless a resonant pattern is established [32], that the propagation of these waves represents a negligible perturbation on the bulk flow. The calculation of resonances within the stationary parts of a hydro-turbine unit has been subjected to investi­gations at the Hydropower Laboratory ([32] and [44]). Raabe [39] suggests both how to estimate the strength of the pressure pulse source and how to calculate the transmission of the waves (private correspondence). The guidelines which he offered were quite rigorous but should be confronted when refinements of the calculations are to be made. A critical remark was given by Raabe on the approach given in this section: •

• The rotation of the runner will also influence upstream flow conditions.

25

At the present time, it is not known to what extent as progress within this field of calculation and experimental research is in an early stage. However, even the simple approximation of the problem presented here shows that there could be a significant upstream influence caused by the runner.

Stagnation point approximation

Morse and Ingard [34], p. 761, state that

”a local fluctuation of the velocity, vQ, will produce a local pressure fluctuation, which (from Bernoulli’s law) will be of the order pv\ . This local pressure fluctuation field acts as a (monopole) source of sound and gives rise to a sound pressure”.

A deduction that is based on varying stagnation point pressures at the inlet will be given.

In this context the pressure perturbation experienced at a relative position on a runner blade is of major concern. Traditional analysis would have given no pressure fluctuations due to the assumed stationary flow field within the runner vanes. The approach here only uses the stagnation pressure, which will vary in a relative frame of reference, as the source of sound or compressible waves. The wake is significantly ” smeared out” by diffusion, or viscosity, such that it is most pronounced at the inlet, contrary to swirling structures which can extend through the runner and into the draft tube.

In order to model the in-flow stagnation point on a rotating runner vane a po­tential flow model, Rankine body, has been chosen, see Figure 2.2. A Rankine body is two potential flows which are superimposed; a source and a uniform flow, which can easily be modified in order to account for acceleration of flow in the vane-less interspace. The flow of the vane-less interspace is often considered as an irrotational vortex i.e. sink-(-line vortex. The deduction of these flows is common practice in basic fluid mechanic textbooks. It is emphasized that this model has limitations for the flow considered. Viscous terms in the flow onto the runner vane cannot be ne­glected and especially at the positions close to the head and bottom cover, also the rotating frame of reference chosen for the analysis can affect the actual flow. Viscous analysis of stagnation point flows further shows that a velocity profile develops at the stagnation point giving a correction term to the inviscid solution, however no viscous corrections are given here due to the already-mentioned limitations of the approach.

The effect of a stagnation point and the temporal variation in stagnation point pressures due to the wakes will be investigated in Example 10.

26

0.1 -

0.0 i—i—r T T T

-0.1 0.0 0.1 0.2[m]

Figure 2.2: Rankine half body that serves as the model for the nose of the runner vane. The ’’thickness” and shape deviate from the example provided.

Example 10 A numerical example is given that includes numerical values for a specific Norwegian power plant where dynamic measurements have been made; noise Figure 2.7 and accelerations/ vibrations on the guide vane link Figure 2.8. The given data are H = 430 m, P° = 72 MW, n = 600 rpm, number of guide vanes = 2f and number of runner vanes = 15+15, the inlet diameter D ~ 2100 mm, and the height b % 250 mm. Reduced circumferential speed is set equal to ux = 0.72, while v~ = ^L.3 The Euler turbine equation will provide the necessary data to calculate the velocity triangles at the inlet: v_lx = ~ 0.67 when rj = 0.96. Data obtainedfrom measurements at a stationary cascade without the runner, see Appendix D, are used in order to validate the velocity distortion due to the wakes off the guide vanes. As mil be discussed; in real turbomachinery flow the runner may significantly change this flow field but as a first approximation this is assumed to be accurate. Also wakes can be analyzed for viscous flows showing the diminishing of the wake as a function of the downstream distance, see White [51] and Appendix D. If a wake distortion corresponding to a minimum velocity of 0.66 the free stream velocity a variation of stagnation pressure mil be 300 kPa, if the minimum velocity is set to 0.9 of the free stream velocity a variation of stagnation pressure will be Aps = 25 kPa. The runner nose act as a source as described above with a pressure amplitude of the oscillations between 25 kPa and 300 kPa, at full load operation, which are the limits to be used further. Note that the oscillations are not sinusoidal due to the nature of the wake. The source extension is set to the approximate radius for which a pressure more than 95% of the stagnation pressure is found which is rsource = 5 mm. In order to take into account the impact a number of monopole sources are said to form and be distributed uniformly at the runner vanes nose, a term adopted from Raabe [39].

(2.13)

3 Analytically it can be shown that the most proper value is Uj — Kjeldsen [25]

P T AP‘ Tsource it hr —ut):--- :—e V )

27

Where r = |r(/?,o) — rsource {h)\ i.e. the spatial vector will change slightly due to the different length from a fixed position. The number of sources is set equal to ^—4—. The first approximation gives:

|p| = Aps^ (2.14)

Inserting numerical values, according to the stagnation point range, £ (25 — 300) kPa, high pressures can be obtained, that is (0.6 — 7.5) % of H if b « r. This pressure perturbation is further assumed to propagate as a plane wave downstream. Subsequently a phase shifted reflection at the outlet is assumed to occur which trans­mits a pressure pulse upstream that is able to nucleate cavitation. This behavior is suggested to be a reason for modulated cavitation within a rotating runner channel. A further result of the potential analysis is the influence upstream along the stagna­tion streamline, which was found to be 0.95(7 at a distance 280 mm upstream the stagnation point. The physical meaning of a stagnation streamline from the rotating runner nose and into a non-rotating flow is hard to interpret. This supports thinking at the Faculty of Mechanical Engineering, NTNU and a remark given by Raabe [39] regarding the upstream influence of the runner. It is further suggested that measure­ments of trailing wakes in systems lacking the runner would give erroneous results to some extent and that numerical calculations should use boundary conditions at a station where the downstream runner has negligible influence.

The sound pressure can be of a significant amplitude as shown in Example 10 and can definitively have a major impact on the cavitation behavior. It is remarked that a simultaneous impact ovr*- the total length of the runner inlet is assumed but not necessarily true. Further, the simultaneous impact is only relevant for turbines having an inlet at a constant radius over the height b, which makes this consideration valid for high head turbines. This is supported by the fact that enhanced dynamics traceable to the guide-vane/runner-vane interaction is seldom found for high specific speed turbines where the inlet radius deviates from bottom to head cover. However, my experience is restricted to accelerometer measurements on the guide vane shafts of the turbines investigated in Figures 2.1 and 2.7. It is further remarked that the excitation does not induce a sinusoidal wave at the passing frequency but is rather limited to shocks in the effective wake width at the inlet.

Water hammer within the runner channel

The essence of this idea is to view the runner channel as a rigid pipe and the inhomogeneous inflow as a varying mass flow. The total mass flux onto the runner channel is found by integration on a surface embedding the runner vane channel.

~v* is due to the guide vanes varying in a tangential direction. As with water hammer the theoretical induced acoustic pressure is p = A m, c = pAv ■ c which is based on an emission within the period t = 2^ , a known sonic speed of the liquid (c), here assumed rigid walls, and that the wake vanishes at the outlet thus all mass flow fluctuations are in the inlet area.

It is stated, using the above assumptions, that every time a flow distortion dis­appears a positive pulse is emitted. Further the propagation of the pressures is very interesting to study especially if the outlet can act as a reservoir giving rise to a phase-shifted reflection as mentioned in Example 10. If so, a period of the excitation, acoustic or water hammer, t = 2 • 2/s should be avoided where L is the effective length of the pressure propagation if resonance is not to occur.1 An example is provided in order to demonstrate how to estimate the theoretical pressure strength.

Example 11 Using the equations deduced in Appendix D and the numerical val­ues as in the previous example, the mass flow fluctuation due to the wake, having a maximum velocity distortion Aumax, can be calculated. Here Awmax = 0.1 v for demonstration purposes, and s = 0.2 m though suggesting the radial clearance be­tween the exit of the guide vane and the inlet of the runner to be 3.5 cm, , according to Equation D.l. 6 is found to be 9 — 7.5 • 10-4 v2 m. Making yi/2 = 3.7 • 10-3 v thus the wake profile is described by the formula:

AvAvn

-0.693[ri d<PV

</2 _ g—5.1-104[rid$]2t> 2

We can calculate the mass inflow onto a runner vane channel when the wake is flowing into the runner vane as:

rh= pb sin a Av (ridf, u)] rid(p

Where nrv is the number of runner vanes (= 30). a = arctan = 9.2° at the runner inlet according to general rules of calculation and the values given in Example 10. In this context v is the unknown, thus suggesting an iteration procedure to obtain the solution. The theoretical Vi at the inlet is calculated to be v\ = 65.2 m,js, thus m— 553 kg/s. The integration is made by doing the following substitution: 5.1 • 104 (rid<p)2 v~2 = -y such that the numerical value can be found from statistical tables of the normal distribution <&. I.e. m can be written, when C = 5.1 • 104, as

rh= pb sin a vr\-2tt

i $n. V2C

s/2 C 7T V Tlrv

4Raabe [39] suggests that the spiral case and the draft tube can be viewed as reflection bound­aries due to their large volume compared to the channels formed by the stationary vanes and the runner vanes.

29

v is found, to be, using the given values v = 69 mjs . The mass flow drop due to the vanishing of the wake, in the relative frame of reference, is then found as the integration over one wake?

A m= pb sin a

2ttngv

J (rid<p)

orffif = pb sin a (??

2tti'org) n----= 36 kg/s

71>rv

Thus the theoretically induced pressure, due to compression of liquid water is found to be, p = A m c = 50 kPa. Note that the whole wake mass distortion should be introduced or withdrawn in less time than the efficient ^.

Illustrations of the presence of pressure pulses in a rotating runner

Jernsletten [20] investigated in his thesis work non-stationary pressure within a rotating pump impeller model in turbine mode. He reported frequency spectra of the measured signal that clearly indicated the dominant role of the runner-vane guide-vane interaction, though the most prominent deviations of pressure were due to the, assumed, uneven flow rate distribution in the spiral case during a revolution. Further Jernsletten calculated the cross-variance of pressure readings between a station close to the vane inlet and another about l = 16 cm downstream along the vane, transducer positions 1 and 3 in Figure 2.3. Maximum correlation was found for a delay time 0.0026 s or 0.54Tgv, Tgv corresponding to the duration between hits on a wake for a runner vane. This is recalculated to a transport velocity of v = 0°jjQ62gs = 61.5 m/s . The nature of the transport velocity is hard to interpret, i.e. if it is of convective nature and/or propagated with a speed corresponding to the sonic speed in the water. Further the specific geometry of the runner also allows the runner nose on the neighboring runner vane to interact with a wake 0.57 Tgv after the original impact. The actual runner vane is the one with transducers 1,2 and 3 mounted on while the neighboring has transducer 4 mounted on, see Figure 2.3. The measurements also found a significant reduction of the oscillating pressure amplitudes when the radial clearance between guide vane exits and impeller inlets was increased.

Based on these specific experiments Jernsletten [20] concluded that:

• The influence of stationary obstacles upstream of a rotating impeller affect the overall dynamics and this effect will fade away if the wakes off the obstacles are allowed to diminish due to farther transport lengths.

°It is remarked that the number of guide vanes is 24 while the number of runner vanes is 30. Thus in a period equal to (g- — ^j) i.e. 1.7 • 10~3s, for the given turbine, the runner channel will not experience the wake or will experience little influence.

30

Figure 2.3: Position of transducers and sketch of modification of impeller geometry [20]. Reprinted with permission of the author. Dimensions are given in mm,

It is further concluded in this thesis work:

• The measured pressure pulses must be enhanced by a ’’runner-vane/guide- vane” interaction due to the fact that the distance between the guide vane exit and the pressure transducers was kept constant while the original wake should have diminished equally. This means that either an acoustic pressure pulse is formed or the vortex generation is greatly enhanced during the passage of the runner vanes through the wakes.

As far as I know few if any pressure recordings have been made at the exit of a runner vane. Strain gauge measurements performed by Kvaerner Energy a.s, mea­surements that have been cited by Brekke [7], have shown the presence of stress pulse amplitudes comparable to absolute pressure head at the exit. These measure­ments were done using strain-gauges directly glued to the vane surface of a pump prototype turbine impeller. In Figure 2.4 it is noticed the peak at 200 Hz which corresponds exactly to the runner-guide vane passing frequency in a relative frame of reference. Also a typical time series signal is shown, Figure 2.5. Regarding the latter, approximate sinusoidal waves seem to be present, however this does not give information of the true nature of the excitation pressure which could be pulse like.

Other investigations done in this work show qualitative differences between tur­bines that are comparable in design for airborne sound pressure. Where the sound spectrum shown in Figure 2.6 indicates peaks at the ’’passing frequency” but not significantly large compared with other frequencies,6 while a dominant peak is found

6If the peak at 50 Hz are due to disturbances of the measurement chain or due to actual noise refer to discussion in Section 4.4.

31

Figure 2.4: Spectrum analysis of vane stresses in a prototype pump turbine impeller vane. P = 150 MW, n = 500 rpm, H = 400 m, 9 impeller vanes and 24 guide vanes.

/vVW'f. - - - ».

vv\ AAsAfTTz::r ----- v _ ^

Figure 2.5: Typical time series stress recording which the frequency analysis is based upon.

32

20

10 -

-10 -

Expected amplitude peak at 350 [Hz]

0 500 1000 1500 2000 2500Frequency [Hz]

Figure 2.6: Spectrum analysis of airborne sound at the top cover of a Francis turbine: P — 53 MW, H = 260 m and n = 500 rpm Normalized such that the amplitude at 350 Hz corresponds to 0 dB.

in the other shown in Figure 2.7.The set-up of the frequencies measurements in Figures 2.6 and 2.7 is discussed

in Section 4.4.

Conclusion on dynamics within a runner

Models have been established for the formation of pressure pulses to be transmitted with the sonic speed of the liquid. It is emphasized that they are limited regarding quantitative values, but crucial in order to establish their significance. The un­derstanding of processes forming these pulses are also crucial in order to interpret and to implement in non-stationary numerical calculations which are in a continu­ous development phase at the Hydropower Laboratory starting with the efforts of Jernsletten [20]. It is further emphasized that the velocity distortion in the flow field onto the leading edge is causing a variation in the angle of attack at a relative frame of reference. This latter phenomenon can induce the oscillating formation and collapse of an attached cavity which might be responsible for deleterious effects like erosion. The upstream influence of the runner rotation has been briefly mentioned but is not investigated in details.

33

-80 —

Expected airptitude peak at 450 [Hz]

400 600Frequency [Hz]

Figure 2.7: Spectrum analysis of airborne sound at the top cover of a Francis turbine at part load: |P° % 49 MW. H = 430 m and n = 600 rpm.. Normalized such that the amplitude at 450 Hz corresponds to 0dB

u c

1E-11

2000.001000.00 1 500.00Frequency [Hz]

500.00

Figure 2.8: Spectrum analysis of accelerometer measurement on the guide vane link of a Francis turbine at part load: |P° ~ 49 MW, H = 430 m and n = 600 rpm. No calibration of original signal.

34

2.2.2 The attached cavityThe second example treated is a familiar one to the cavitation community. A mani­festation of cavitation is the formation of a large bubble or pocket which covers the structure which is facing the flow, this phenomenon is termed an attached cavity. This kind of cavitation can be appreciated for propellers where skin friction may be largely reduced. One condition is necessarily said to be satisfied; the bubble has to exist downstream the propeller in order to prevent erosion.7 These pockets, possibly containing gas driven out of solution thus forming a type of more persistent bubble, are investigated in this section.

A calculation of the pressure pulse emission off the interface based on certain propositions is also presented. The motivation for the model proposed here is to establish conditions whether any instabilities of the gas-vapor/ liquid interface, as described below, are able to create a local pressure field able to enhance the nucle- ation and subsequent collapse of the free-stream nuclei.

The erosive nature of the cavitation pocket is usually attributed to highly erosive complex vortical structures according to Arndt et al. [4], This can also be seen in photographic studies provided by Farhat [18] and by the work in this theses, see Figure 2.9 where vortices of vapor phase like fine threads are shown downstream the cavity. These vortices are obviously the result of complex interactions of the cavity and the ambient flow.

Assumptions

The theory the work in this section is based upon is found in the book by Carey [12] especially the section ” Kelvin-Helmholtz and Rayleigh-Taylor Instabilities” in that reference. In that analysis an infinite liquid/vapor interface, inviscid liquid and vapor phase, and an incompressible liquid phase are assumed. The assumptions made in order to fulfil the requirements of the cavitation appearance on a hydrofoil are as follows:

1. A perturbation is given the interface according to Equation 2.15, see also in Appendix B:

OO

A = A* + (2.15)771=1

2. The restoring motion of the interface is due to both the pressure gradient set up by the convective flow and surface tension.

7Models are proposed that cavitation bubbles are being led into the stagnation point directly downstream of the ventilated cavity, a point having a relatively large pressure. This model is supported by the fact that cavitation erosion is often found at the closure of such a pocket.

35

3. The volume of the gas-vapor is considered constant during the cycles. A sinusoidal shape of the interface, and constant attachment points (for the current analysis) are therefore assumed.

4. NOTE: The unstable interface is a subject in multiphase flows, but I have not found a directly similar deduction in such literature.8 Effects which are not taken into account for the analysis are: three dimensionality of the pocket and the transition from a smooth surface regime into a turbulent, ’’foamy” looking interface. The ” equilibrium curvature” of the interface is not taken into account. The deviations of the current analysis from Carey are shown in Figure B.l.

Results

The deduction of the governing equations for the instabilities of a liquid vapor interface is based on the book by Carey [12] and is cited in Appendix B, including how the presented approach deviates from Carey. The assumption that the restoring forces, in addition to surface tension forces, consist of the pressure gradient due to the pressure field set up by the convective flow does not violate the system of equations if the pressure gradients normal to the interface at any position along the interface are of equal size.

Consider a 2D air pocket of ” quasi-rectangular” shape, where the equilibrium height is Aq, Figure B.l. The discussion is limited to the parameters given in Equation B.10. The assumption of fixed attachment points and constant volume limit the value of a:

am = m-—;m E N (2.16)

The value obtained for (3 is:

i(3 = a (piUi - pvuv) 27rmuiPi — Pv

m E N'if

(2.17)

Which reassembles a scaling law with a high Strouhal-number (St) (3 = 2tt/:

St = IhfL = m;m, N (2.18)Ul

In their experiments Avellan and Dupont [5] estimate the cavity length in the range of 25 — 30 mm. At a flow velocity of ui = 1.2ttoo = 24 m/s, where Ui is the flow velocity in the vicinity of the cavity, which would give a value of /, according to Equation 2.17 , in the range 800 — 960 Hz at n = 1 which is close to the visually

8 Avellan and Dupont [5] mentioned the Kelvin-Helmotz instabilities (on page 734 in the pro­ceedings) as the source for the creation of turbulent cavitating vortices.

36

observed frequency of 1.1 kHz by the cited authors. What deviates in the theoretical approach used by Avellan and Dupont from the one used in this work is the choice of length scale for this problem. Avellan and Dupont use the thickness of the measured shear layer above the attached cavity, while here it is suggested to directly use the cavity length as the characteristic length scale.

Reisman and Brennen [40] present frequency spectra of an oscillating hydrofoil where a marked peak is found at about 3 kHz. In their discussion, this peak is said to deviate somewhat with the cavitation number. Information on the attached cavity length is lacking such that the scaling law found in Equation 2.18 cannot be used directly. However, as it is expected, Lif varies with the cavitation number which could explain the deviations.

Abbot et al. [1] found a modulation, of the high frequency cavitation signature, frequency corresponding to a Strouhal number of 0.3, L being the effective cavity length. It is believed that this frequency is due to instability of the whole pocket, that is the periodical shedding of the content of the cavity, which is seen in the low frequency domain of the spectrum shown in Figure 2.10. If the latter is true the assumption of constant volume and fixed attachments points, assumed above, fails.

Measurements in this work were also made using a pressure transducer above a partially cavitating hydrofoil. The length of the cavity is estimated to be in the range 20 — 40 mm, Ux approximate 8.5 m/s. The frequency / for n = 1 according to Equation 2.18 should then be 210 — 420 Hz. Measurements have shown a power rise at about 380 Hz and furthermore blurred peaks according to n = 2 and n — 3 regarding Equation 2.18. The dominant part of the instabilities is found at a low frequency < 200 Hz, and could be explained by:

• Non-cavitating domain:

— Vortex shedding off the profile.

• Cavitating domain:

— Vortex shedding off the profile.

— Modified frequency due to enlarged angle of attack and partially cavitat­ing profile.

— Shedding of gas from the attached cavity.

It must also be emphasized that the cavity length L*/ is not a typical length scale but rather a quantity dependent on the pressure distribution over the body, the Reynolds-number and the cavitation number

37

Figure 2,9: Photo (shutter ordinary flash) showing both irregular surface struc­ture and shedding of complex structures on a NACA 4412 profile. Angle of attack =3° and JJoo ~ 8.5 m/s.

Experimental investigations

Frequency spectrum measurements have been done over a cavitating NACA 4412 profile by the author. The setup of the facility is discussed in Section 4.3. The size of the cavity has been estimated through visual inspection. Correlations have been made regarding Equation 2.18. A typical time series recording is shown in Figure 2.11 and the similar band pass filtered in Figure 2.12, the latter in order to observe any modulation of the investigated characteristic frequency.

Regarding the band filtered signal in Figure 2.12 it should be noticed that the modulation is assumed due to low frequency activity of the attached cavity. Ac­cording to the discussion in Section 4.3.2 the range 650 — 750 Hz is modulated by the 350 — 450 Hz range which contain the proposed source, as will be discussed.

Discussion

The restriction on a, as found in Equation 2.16, makes the theoretical pressure, at the position of the transducer (sub-indexed trans) , according to Equation B.3 equal to:

PtransR=1— 6

2^| ztrans~zif- 2 • 10~10

Therefore it is pertinent to ask why pressure oscillations at the transducer posi­tion are measured at all. In place of a thorough analysis Proposition 12 states the

38

3.QE-5

Instability for n= 12.0E-5

1.0E-5

O.OE+O

400Frequency [Hz]

Figure 2.10: Digitally derived frequency spectrum (fs = 40960Hz,ns = 16384) of cavitating profile Ly ~0.02m. Angle of attack =3° and Uoo ~ 8.5ms-1 (Uncali­brated transducer)

0.00 0.10 0.20 0.30 0.40Time [s]

Figure 2.11: Y-axis (voltage readings) IV is roughly lmWc = lOOOOPa. X-axis is time in seconds. The time series presented serve as the basis for the spectrum analysis.

39

-1.48 Band filtered (350,450)[Hz]

-1.48Band filtered (650,750) [Hz]

-1.56------ 1 i i i | i i i i | i i i i | i i i i |

0.00 0.10 0.20 0.30 0.40Time [s]

Figure 2.12: The time series has been digitally (LabView) bandpass filtered in order to investigate any modulation of low frequency components <200Hz.

40

formation of sound pressure waves which will not diminish according to the expo­nential law of Equation B.3. Due to the restrictions of whole wave lengths the sound pressure far field should be dipole-like and weak.

Therefore it is suggested that the restoring motion of the interface can also be due to compression and expansion of the liquid column at the interface. This will generate sound waves.9 Thus a proposition is given:

Proposition 12 The liquid/vapor interface of an attached cavity generates sound pressure waves due to successive expansion and contraction of the liquid column close to the interface.

No studies have been made to reveal the source strength, which would be a difficult task because the same instability, according to Avellan and Dupont [5] is said to be responsible for the emission of a turbulent cavitating field which in turn contributes to the overall sound field. If the proposition above holds, the two next follow:

Proposition 13 The pressure time history at a position off the interface will be due to partly sound pressure emission from the interface.

Proposition 14 In the vicinity of the interface the sound pressure radiation may influence the formation and collapse of cavities.

The pressure at a distance off the attached cavity has been calculated when view­ing the interface as being equally distributed, equally sized, point sources where their strengths are given by the instantaneous compression rate. However, with the geo­metrical constraints of the NACA4412 profile in the water tunnel at the Hydropower Laboratory, Lif = 3 cm and the sonic speed equal to c = 1400 m/s no extraordinary phase speeds were found (the added effect of the modelled distributed sources). The instability is sinusoidal 400 Hz and two dimensional. Some calculated time series are shown in Figure 2.13 where x and y correspond to the distance downstream and above the detachment point respectively see Figure B.l. The pressures are calculated in an x, y plane intersecting at the mid-span of the profile (symmetric conditions yields).

Concluding remarks

Correlation has been found between theory and experiments, measurements made in this work and by Avellan and Dupont [5], regarding predictability of the frequency

9 The deduction by Carey [12] is based on incompressible flows. I have not investigated whether an inclusion of compressible terms in the governing equations will make qualitative changes in the results obtained from the analysis.

41

d>t/i(/>i

I0)01

x=0.02[m],y=0.02|m]

x=0.03[m],y=0.02[m]

x=0.03[m]1x=0.04[m]

0.000 0.001 0.001 0.002 0.002 0.003Time [s]

Figure 2.13: Pressure in the bulk of the liquid due to the proposed acoustics induced by the instability of the interface.

of a liquid vapor/gas instability on a profile embedded in flowing water. It is em­phasized that the erected water tunnel is not equipped in order to do careful studies of the phenomenon due to la of ability to vary the cavitation number at constant flow rates. In addition the pr nee of low frequency instabilities, vortex and cavity content shedding will modulate the high frequency domain because Lif will vary. In order to fully verify this phenomenon simultaneous high speed photographic/ visual studies and pressure recordings should be made.

If the interface is able to emit acoustic pressure waves, as is stated above, the time history of cavitation bubbles travelling at a distance off the interface could be influenced. These matters are further discussed in Section 3.1. Discussion on the validity of the experiments in this work is given in Section 4.3.4.

42

12000 16000 20000Frequency [Hz]

Figure 2.14: Measurement of airborne noise at the top cover of a Francis Turbine P = 62 MW, H = 250 m and n = 500 rpm. Strong high frequency components at about 9 and 18 kHz are seen.

2.2.3 The vibrating horn

During this study I spent several months as a research visitor at the Physics Institute at the Technical University of Denmark where Dr. K. A.. Mprch was the kind host. The author concentrated upon cavitation excited by the sound field of the vibrating horn. The characteristics of cavitation in the vibrating horn system, which is explained in the standard ASTM G32, can have a direct similarity to conditions encountered in flow systems. Obviously any high frequent vibrations in the material structures excited by an hydraulic resonance in flow systems will provide conditions similar to that of the vibrating horn. High frequent vibrations are found in hydraulic turbines and by extending the frequency range of the measurements presented in Figure 2.6 we can find examples of those, see Figure 2.14. A recognized phenomenon is the scroll case resonance as stated by Stepanik [44] pp. 18-22. The characteristic frequency is determined as the product of the frequency of the runner rotation and the number of guide vanes and runner vanes: which for this specific turbine would give: 8.333 s~l - 24 • 28 = 5.6 kHz, thus it deviates by a factor 1.6 regarding the frequency peak in Figure 2.14

43

Measurements have been made during this work in order to provide information of the hydrodynamic characteristics of the vibrating horn device. This specific sys­tem is also of a great general interest due to its status as a standardized method for the investigation of cavitation erosion resistance for materials (ASTM G32). It is further emphasized that the cavitation in this specific system is based solely on the sound field, which is suggested to be significant in flow systems as well. Thus it is considered crucial to reveal information on the processes governing the cavitation in this system in order to correlate or scale erosion tests to flow systems. This has also been the scope of research by Arndt et al. [4j.

For this specific investigation some remarks are given.

1. Due to the horn displacement a stress wave will occur within the specimen therefore several erosion tests performed by investigators use a stationary specimen close to the vibrating tip. Such a set-up is not covered in these investigations.

2. During the experiments no correlations were made between actual erosion and hydrodynamic characteristics.

The results have been reported earlier [23] and [24], and additional information is provided in this section.

Results

The specific goal of the investigation was to provide information of resonance con­ditions within the beaker at the given excitation frequency. The submerge of the horn tip, the depth of the water sample, and the diameter of the beaker were inter­changeable. The pressure transducer was kept at a fixed position regarding the tip surface. Refer to Section 4.1 for details.

One of the recorded pressure time series has been manually digitalized and is shown in Figure 2.15, note: X=j = 66.7 mm, c = 1400 m/s, estimated, and / = 20 kHz.

Regarding the gross of experiments only rms and peak-to-peak values were recorded in addition to spectrum analysis. Where the former numerical values were based on similar scans as shown in Figure 2.15. The values are tabulated in Table 2.2, refer to Section 4.1 for a definition of symbols and calibration procedure for the transducer.

All frequency spectra showed the same qualitative behavior, i.e. a ’’noisy” spec­trum in the range 25—40 kHz as in Figure 2.17 and similar to the ’’crude” frequency

44

Figure 2.15: Digitalized time series pressure recording of a transducer close to the cavitating horn. Submerge lh = 17 mm, depth l = 66.7 mm and diameter beaker d = 138 mm. Ppeak-peak = 2.26 bar and Prrns = 0.54 bar.

1.5E-1 -|

1.0E-1 -

5.0E-2

O.OE+O

40000 5000010000 20000 30000Frequency [Hz]

Figure 2.16: Frequency spectrum of digitalized signal presented in the preceding figure. (Frequency step = 2 kHz).

45

Table 2.2: Numerical values are rms pressure/ Peak to peak pressure. Relative pressure levels versus D = 200 mm, l = 100 mm and 1^ — 12 mm

D [mm] —► lh [mm] —>

l [mm]1

13812 18

22012 18

3308 12 18

55 1.1/1.1 1.2/1.3 1.2/1.370 1.0/1.0 l.l/l.l 1.0/1.190 1.2/1.3 1.3/1.3 1.4/1.5 0.9/1.0 1.5/1.6100 1 1.4/1.5120 1.0/1.0 1.0/1.0

analysis in Figure 2.16 of the digitalized signal in Figure 2.15. Scans toward 100kHz did not reveal any additional information. Arndt et al. [4] presented spectrum analysis of a cavitating horn with a stationary specimen close to the horn. Pressure readings were taken at the specimen position, which did not reveal the frequency noise between 20 and 40 kHz. In the experiments covered in this work the presented measurements were not blurred due to the presence of gaseous resistant bubbles on the transducer membrane. Experience showed that whenever bubbles tend to grow on the pressure transducer membrane the signal rapidly dropped and faded away.

Discussion and concluding remarks

Summarizing this work, it can be concluded that the sound field is both modified by the geometric parameters regarding beakers, submerge of the horn, level of wa­ter and the cavitation itself. Regarding the interpretation of the results presented in Table 2.2 a simple argumentation will be provided. Regarding a sound field; resonance conditions occur if the wave is admitted a pressure node at the surface of the water and a velocity node at the adjoining material. Thus wavelengths A such that l = (2n + 1) • n e N should be susceptible to axial resonance or in the vertical direction. A is fixed by the driving frequency and the sonic speed in water (A = j = 0.066m). Hence water levels l equal to 17,50,83 and 116 mm should fa­vor resonance conditions. Similar a submerge, close to 17 mm should be chosen because a velocity node should establish at that position. While the driving force is velocity fluctuations of the horn tip, a position close to the node would make a sig­nificant enhancement of the pressure amplitude in the sound field. Regarding radial resonance the diameter, D, of the beaker should be chosen such that a velocity node naturally occurs at the center line. This means that the same relation holds for D as for l, which proposes resonant conditions for D equal to 100,166,232,300 and 370

46

Figure 2.17: Typical frequency scan. Continous frequency spectrum of the pressure measured at a position close to the cavitating region: l = 100 mm, lh = 17 mm and D = 300 mm.

47

ram. Concluding that maximum pressures occur for, according to the parameters used, l = 120 rara (90 ram) , lh = 18 mm and D = 220 ram.. According to Table 2.2 the argumentation fails for the configurations l = 120 rara, lh = 18 ram , D = 220 rara and l = 100 rara, A = 12 rara and D = 330 rara. However the latter could be explained by the resonant sound fields, i.e. the pressures are 180° out of phase for radial and axial resonance.

As preliminary concluding remarks we can state that:

• The amplitudes of the driving pressure for the formation and collapse of the cavitation cloud in a vibrating horn system is, as expected, dependent on the geometrical set- up. This dependency can be eliminated by covering the wetted boundaries with acoustic energy absorbing materials.

• The arrangement is considered simple and suitable for studies regarding cav­itation cloud behavior, though it is expected that devices using ring pressure transmitter [17] for the formation of a sound field are more suited, but not in­vestigated in this study. The ring transmitter arrangement is considered more suitable due to the formation of the cavitation cloud far from the sound field source, i.e. contrary to the vibrating horn.

• This study shows the dominant role that geometrical constraints have upon steady oscillated sound pressure excited cavitation systems. This is a matter which will be of importance when investigating more complex 3D spaces like the ones limited by the runner vanes.

48

2.2.4 Correlation of the pressure field and the cavitation erosivity in a rotating disc device

The cavitation erosion apparatus was erected at the Hydropower Laboratory in order to investigate the erosion resistance for selected materials. Hydrodynamic measurements have been made in order to provide information about the hydrody­namic characteristics for the system. Numerical calculations have also been made of the complex flow pattern in the flow system in order to qualitatively conclude the processes involved in the formation of pressure pulses and indirectly the erosive power of the apparatus. Regarding the set-up of the facility, refer to Section 4.2.

Experimentally determined pressure fields

Measurements of the non-stationary pressure field have been made in order to quan­tify any correlation between cavitation erosion and pressure pulsing. As cavitation inducers the following geometries have been tested, tabulated in Table 2.3:

Table 2.3: Cavitation inducer configurations used in the rotating disc apparatus. Geometry is given as the height of the inducer and the diameter. Ref. refers to the inducer configuration while Exp. is to identify the specific experiments made.

Ref. Dia. [mm] Height [mm] Number of inducers Exp.A 5 5 4 1 &2B 5 3 2 3C 3 5 2 4D 3x3 5 2 5

The clearance between the rotating disc and the break-vanes is about 7 mm, this clearance is indicated to be of great importance regarding the erosion of the specimens. The pressure recordings showed the same qualitative behavior while the amplitudes differed. Selected pressure readings will be presented. Experiment 1 with the common erosion-test configuration including four inducers reference A in Figure 2.18 and experiment 3 having two inducers reference A Figure 2.19.

The minimum pressure is directly after an inducer has left the interspace where the transducer 2 or 3 is stationed, this can be stated due to a fixation of the rotating disc relative the trig signal as shown in Figures 2.18 and 2.19. The interpretation of the measured signals is a difficult task due to lack of knowledge of actual flow processes. However the following is proposed to be true:

49

o>cX3*da>O)(9

4.00 -i

2.00 -

5 o.oo -5

-2.000.02 0.03 0.04

Time [s]0.05 0.06

Trig signal

Center transducer

Trans.2

Trans.3

Figure 2.18: Pressure-time recording. Experiment 1 with the common erosion- test configuration including four inducers, ref. A. The plot is based on 185 samples with a sampling rate of 2 kHz, digitally acquired.

8.00 -|

One revolution Trig signal

Center transducer4.00 -

Trans.2

Trans.3

0.00 -

0.04Time [s]

Figure 2.19: Pressure-time recording: Experiment 3 with two inducers reference A. The plot is based on 185 samples with a sampling rate of 2 kHz, digitally acquired.

50

1. The minimum pressure is due the ’’pumping” effect of the cavitation inducer. An effect that draws water out of the enclosure.

2. The distinct peaks of the measured signal is due to the inducer-rib interaction causing an emission of a sound pressure pressure pulse or a water hammer.

The maximum pressure difference which was mostly due to the 200 Hz compo­nent with four inducers, or 100 Hz with two inducers, and the erosion ranking of the various runs have been tabulated in Table 2.4.10 The relative amplitudes of the subsequent oscillations correlate well to the cited maximum values.

Table 2.4: Peak to peak pressures measured in the rotating disk. T2 and T3 refer to the transducers in the preceding figures.

Exp.# Ap [bar], T2 Ap [bar], T3 Erosion ranking1 1,55 1,9 12 1,15 1,35 53 0,75 0,95 34 0,8 1,0 35 1,65 2,05 2

What is to be noticed is that the difference between the Experiments 1 and 2 was that the static pressure in the latter was at a higher level indicating more through- flow, and that there was no visually observed cavitation or erosion on the painted material in this case. The differences can be explained by the cavitation effect which upon collapsing will, in the confined area limited by the brake vanes, make a sudden drop in density which will be measured as a decline of the absolute pressure. What is also to be noticed is that apart from the mentioned difference the pressure peak- to-peak value follows the expected pumping action of the cylindrical inducers when using the cross section area with its normal parallel to the transport direction as a measure. Based on the preceding discussions on sound pressures versus convective flow pressure variation one should be aware that both effects can be significant for the total pressure formation, the flow because of the shearing action of the rotating disc, and the inducers due to the interaction of brake rib-inducer.

A typical frequency spectrum with four inducers 05, hb reference A is as shown in Figure 2.20. Scaling the frequency axis gives as shown in Figure 2.21

10 The erosion in experiment IDS covered a larger area of the specimen than the case in experiment ID 1.

51

.OE+O

.OE-2

.OE-5

.OE-7

.OE-8

100004000 6000Frequency [Hz]

Figure 2.20: Frequency spectrum of measured signal, based on the cited time series, in the rotating disc Experiment 1.

1.0E+0

1.0E-1

1.0E-2

1.0E-3

1.0E-4

1.0E-5

1.0E-6

1 .OE-7

1 OE-8 i i i—r1000 1 500Frequency [Hz]

Figure 2.21: Frequency spectrum for Experiment 1. Same as the preceding figure, with scaled frequency axis.

52

Direction of disc rotation

Motion of liquid within "break vanes'

Figure 2.22: Model for gross liquid flow behavior within the rotating disc facility. Hatched areas correspond to breaking vanes.

Discussion and concluding remarks

It is noticed that 200 Hz is the frequency for each inducer to pass beneath the trans­ducer. The cavitation itself should be modulated by a 1 kHz component radiating pressure in a higher frequency range which was not covered in these measurements. Obtaining measurements which investigate modulation would require the use of transducers mounted on the rotating disc, such measurements were considered but not realized.

The main criticism of these measurements is the high content of free gas or bubbles in the interspace between the brake vanes. This is due to the use of tap water as the coolant, and the effective release process of the dissolved gas due to cavitation. This will certainly induce a dispersion relation of the sound pressure waves and damping at certain frequencies.

In order to validate the periodical formation of sound pressure pulses as proposed in Proposition 2 in Section 2.2.4 the following argumentation is given.

• The formation of an acoustic source is due to a temporal variation of local pressure during the inducers passage beneath the break vanes. This is the same argumentation as in Section 2.2.1.

Thus an investigation whether the presence of a natural varying stagnation pres­sure is necessary. According to Varga et al. [48] the leakage flow from one chamber, limited by the breaking vanes above the rotating disc, to the other is negligible. This as a consequence of the minimum difference in pressure over the break vanes due to the circulation set up within these as illustrated in Figure 2.22.

53

Table 2.5: Comparison of characteristic values for the numerical model and the prototype of the rotating disc, t refers to the thickness of the break vane while h referes to the hight of the break vane.

Numerical model PrototypeM— 0.2 0.05

105 105105 107

Thus a variation in stagnation pressure is expected on the inducer to approxi­mately ps = \pU2 = 0.5 • 1000 • (402 — 202) = 600 kPa, where the value U = 20 m/s is an estimated relative velocity of the cavitation inducers in the mid section. The interpretation of this stagnation pressure is difficult due to the fact that the stagna­tion point is opposite the side where the cavitation event occurs. Visual inspection using a stroboscopic light indicated that the whole cavitation cloud vanished at a position between the brake vanes. This suggests a modulation as observed in this work in a similar system with holes penetrating the rotating disc as cavitation in­ducers that no particular position during the rotation is lacking the cavitation cloud downstream the hole.

However another kind of modulation has been observed for cavitating flows over a hole. In personal communications with K. A. Mprch I was told that during high speed photographic studies cavitation clouds were observed which formed successively from one side to the other of a solid plate perforated by a cylindrical hole. Dr. Morch could not tell whether this was due to flow instabilities, like periodic vortex shedding, or ii the hole acted as a dipole acoustic source or monopole on each side.

It is further believed that the cited model for the liquid motion is inadequate. Therefore a numerical time dependent analysis which includes viscous terms has been performed by the numerical code used in Jernsletten’s thesis work [20]. This analysis, with a deviation in geometry and no cavitation inducers in order to allow a homogeneous grid space, showed the formation of strong vortices and pressure fluctuations/propagations i.e. a non-stationary solution. A strong Mach number dependency on the flow quantities is expected, which in practice will complicate the calculations due to the presence of free-air/bubbles which would induce a dispersion of the waves, or wave propagation velocity dependency on the frequency. The grid is shown in Figure 2.23, while the contours of constant vorticity are shown in Figures 2.24 and 2.25 at two different time steps. The characteristic values for the numerical model and the prototype is shown in Table 2.5.

Based on the preceding discussion it is stated that:

54

Figure 2.23: Numerical grid model of the rotating disc apparatus. Cross section view at maximum radius of the breaking vanes. The deviations from the real system is the choice of thickness for the breaking vane.

55

Figure 2.24: Numerically calculated in­stantaneous (t=0) vorticity index at a cross section limited by the maximum radius of the break vanes.

Figure 2.25: Numerically calculated in­stantaneous (t=dt) vorticity index at a cross section limited by the maximum radius of the break vanes.

• There is a presence of sound pressures induced by shocks because of the varying flow conditions during the passage of the inducer beneath the breaking vane.

This type of flow where flow velocity disturbances and the emission of compres­sion waves are necessarily formed at the same rate will be hard to interpret regarding the mechanisms for cavitation formation and collapse.

Further, it is asserted that this flow is comparable to leakage flow in Kaplan tur­bines where the same two effects could be significant. The formation of compression waves is due to fluctuating gap size between the circumferential part of the runner vane and the turbine housing.

56

2.2.5 Cavitation in pipe systemsA minor experimental study was done on an open pipe rig at the Hydropower Lab­oratory. The specific system designed by Svingen [46] allowed a varying amplitude and frequency —* 200 Hz of through-flow.. At large amplitudes and minimum mean through-flow, cavitation was expected to occur. The measurements at specific fre­quencies showed a cutting of the low pressure cycle and a ’’metallic hammering noise” was heard. Measurements were taken to investigate the high frequency do­main pressures fluctuations of 10 — 20 kHz, and to investigate the modulation by the excitation. There were no means to investigate where along the pipe length cav­itation or column separation occurred, but it is assumed that pressure information of the cavitation events, is transported onto the pressure transducer.

The subject of column separation has been a subject in pipe transient analysis for years according to Martin [42]. In this particular study the motivation was to analyze the pressure recordings in order to find any high frequency domain pressures which could be attributed to modulated cavitation. Time series and high frequency spectrum analyses for the cavitating domain are shown in Figure 2.26. The ampli­tudes of the high frequency components were small and close to the sensitivity of the transducer. The ~ 15 kHz component is believed to be the cause of a radial resonance, i.e. normal to the flow velocity. In addition, the calculated frequency spectrum at such high frequencies is not reliable because a sampling frequency of 40.96 kHz does not include more than two sample points within a 20 kHz cycle.

However the concept of forced pressure oscillation in a pipe flow system is promis­ing regarding the investigation of pressure pulses superimposed in an ordinary flow. This is mainly due to the ability of this specific system to vary the frequency and the flow rate.

57

4.000--

n nnn ,L.J.UUU

-) nnn J !l. .. J Jr£.UUU” . r vi 2„ E1 f

1 .UUU" ■V

s

A. J.v\ 1 "V'\U.UUU" ■

.1 / A.f

n nnn V ^*4J 1,t \-3.000-

0.7'5 C .76 0177 Seeyc178 0.7 9 0.80

ASt-f*3.5E-7-

3.0E-7-

2.5E-7-

2.0E-7-

1.5E-7-

1.0E-7-

5.0E-8-i iii 1, 1J ijjj

10000.0 12000.0 14000.0 [tiz] [160(1 1 1

30.0 18000.0 20478.

Figure 2.26: Investigation of the dynamic behavior in the pipe system. Time se­ries (voltage reading) and high frequency domain spectrum analysis for pressure oscillations in a pipe system [46] with cavitation (/s = 40.96 kHz and ns = 32768).

58

Summary• It has been shown that political relations and decisions within Norway can

make favorable situations where the power-plant owners operate their ma­chinery at conditions which can increase the problems of cavitation through increased overall dynamics.

• The set of equations used for both acoustics and fluid flow have been deduced in order to compare and distinguish these phenomena that are usually treated distinctively and to enable a discussion of relevant parameters.

• It has been shown that flow irregularities at the inlet section of a low-specific speed numbered Francis-turbine can be the source of compression waves. This has been illustrated by cited measurements of both model turbines and pro­totypes.

• Conditions have been proposed for the investigation of acoustic energy trans­mission from a quasi-stable gas-phase which can influence the instantaneous pressures in the flow.

• It has been shown that the influence of geometrical discrepancies on the fluctu­ating pressure in a vibrating horn system corresponding to the ASTM G32-92 standard.

• An investigation on the rotating disc apparatus has been conducted in order to establish any criteria for cavitation intensity versus pressure pulses. •

• An introductory work has been started on cavitation in forced oscillated pipe flows.

59

60

Chapter 3

Flow cavitation in a fluctuating pressure field

It has been established in Chapter 2 of this thesis that the sound pressures of significant amplitudes can be present in flow systems. This chapter will investigate the impact on the cavitation bubble dynamics.

An example of bubble dynamics will be given when a fluctuating sound pressure field is superimposed. The author also introduce a dimensionless number which is proposed to quantify the relative importance of the sound pressure superimposed the hydrodynamic flow. Further a photographic study with simultaneous pressure readings will be presented, this is a reference for future work on cavitation cloud nucleation in fluid flows which, in this thesis is assumed, to have similarities with sound pressure generated hydrodynamic cavitation. Lastly a discussion on actual cloud formation in a fluid flow is given.

3.1 Bubble dynamics in a time dependent pres­sure field

A classical approach to cavitation along streamlines in a flow device is to numeri­cally calculate the non-linear governing equation for bubble dynamics using the La- grangian varying pressure as the time dependant excitation. Valuable results have been published where Gindroz [19] gave a correlation between the above-mentioned approach and experiments utilizing equipment capable of adding pre-selected (in size) micro-bubbles to the bulk of the flow.

This method is further explored when adding a time dependent sound pres­sure field to the streamlines of the bubbles thus simulating non-stationary effects as presented in Chapter 2 of this thesis. What deviates from a pure Lagrangian consideration is that the trajectory of the bubbles is supposed not to be perturbed

61

by the superimposed sound pressure wave. This is due to the nature of a sound wave which can be viewed as a particle oscillating off an equilibrium position. The interaction of a sound field and the convective flow is investigated in Section 2.1.

Based on the topics of this thesis work some consequences of the pressure pulses discussed in Chapter 2 of this thesis will be investigated. The non-linear nature of the governing equations and the difficulties on obtaining exact knowledge of the at all times pressure field along the flow path makes a general classification of cavitation bubble behavior a difficult task. Thus in order to illustrate the ideas an example is provided.

Example 15 It is given a stationary pressure field caused by a hydrodynamic flow over a wetted surface Table 3.1.

Table 3.1: Lagrangian pressure as a function of time for numerical example.t [m.s] Poo [kPa]0 4.21 32 2.53 1.84 1.85 2.710 2.7

The following parameters are used (bubble initially at rest), Table 3.2:

Table 3.2: Numerical values used in example.Parameter Numerical valuePo = 4.2 kPaRo = 0.3 mm.Pv — 2.2 kPa5 = 7.5 mN/mPamp — 3.5 kPaUJ = 2351 rad/s = 374Hz

The fluctuating pressure p = pampsin(ut) is superimposed on the pressure field given in Table 3.1. The pressure field experienced by the nucleus is therefore {poo + p).The bubble size, during the pressure cycles, is found to be as shown in Figure 3.1. Where the vertical axis is the bubble radius in meters (original equilibrium size at

62

Rq (po = 4:.2kPa) = 0.3 mm. Initial partial pressure of non-condensable gas pg = 2 itPa. Critical pressure for such a bubble is 2166.66 Pa, if originally at equilibrium at that pressure. With no pressure perturbation the bubble radius time history became as shoum in Figure 3.2.

0.002 0.004 0.006 0.008

Time (second)

Figure 3.1: Radius-time history for bubble experiencing both a varying pressure field in a Lagrangian frame of reference and a superimposed sound pressure perturbation. The vertical line serve as a limit for calculation.

The two most important observations are

1. The bubble may grow larger when applied to a varying pressure field before reaching the collapsing stage.

2. The position of the collapse deviates. The equations solved are not suitable to calculate the accurate radius-time behavior during the final stages.

It is emphasized that this approach has no meaning unless the following criteria are established:

• The exact nature of the nucleus.

• The exact nature of the flow pressure field and the sound pressure perturbation and any interactions between those two. •

• That this is a single bubble not perturbed by other bubbles within its near­field.

63

0.002 0.004 0.006 0.003Time (second)

Figure 3,2: Radius-time history for bubble with no superposed pressure perturba­tion. The vertical line serves as a limit of calculation.

It is therefore concluded that the approach of following only one bubble through its path will give few quantitative results and perhaps erroneous results. However the approach does give qualitative information regarding the life cycle of a cavitation bubble according to the assumptions given above.

64

3.2 Determination of dynamic effects in cavita­tion

Among specialists in the cavitation community it is often distinguished between what have traditionally been termed as hydrodynamic cavitation and acoustic cav­itation. However as argued in this thesis an intermediate state must exist because sound waves can be formed in hydrodynamic flows. These waves can be of significant pressure amplitudes, as has been shown in Chapter 2. In order to establish a crite­rion for the influence of sinusoidal sound waves, that superimposes a hydrodynamic flow, the following proposition is made:

Proposition 16 If jj and j are of comparable order, where L is such that pamp ~ Pconv, both effects (sound and Lagrangian pressures) are important for the bubble dynamics, f is the frequency of the superimposed sound wave. Further if jj -C j Lagrangian pressure is of prime importance, and opposite, the sound pressure will be decisive for the bubble dynamics.

Proposition 17 The ratio of the characteristic time scales, as described in Propo­sition 16, defined as:

uis proposed to be interpreted as a dynamic cavitation number, and for which the numerical value describes the number of pressure cycles a fluid particle experiences over a cavitation susceptible length L.

The obvious weakness of such a number is the omission of two and three dimen­sional effects. However this can be included by designating a specific number to each streamline. Examples are provided of calculating the involved parameters.

Take a typical high-head turbine, the numerical Example 10 in Section 2.2.1. L is chosen to be in the range of 0.01 — 0.1 m , 1/ ~ 10ms-1 and f = 240 Hz. It is found that jf = 0.001 — 0.01 s and j = 4 ms, which makes the dynamic cavitation number equal to Kd = 0.25 — 2.5. The second turbine for which pressure recordings are shown in Figure 2.1 might also have a value of jj in the range 0.001 — 0.01 s. Disregarding the guide-vane/ runner-vane interaction, j is found to be in the range 0.2 — Is, and KD is in the range 0.0002 — 0.01. Analogies of every system investigated in this thesis it can be tabulated as follows, Table 3.3.

Explanation of the choice of parameters L, U and / is given:

65

Table 3.3: Comparison of cavitation in specific systems based on the proposed signif­icant parameters. L=Cavitation susceptible length, U=flow rate through the region bounded by L and f=the frequency of the superimposed sound pressure.

System L [m] U [:m/s] / [s'1]%II

£Rotating disc 0.05 20 1000 2.5Pipe system C

/ <1 f e.g.>500NACA profile 0.03 10 300 1Vibrating horn 0.01 0 20000 ooNumerical example £7=0.08s 375 ~3

• Rotating disc: L equals the distance from the cavitation inducer to the material specimen. U is taken as a ’’guessed” mean relative velocity of the disc regarding the liquid motion within the break vanes. / is taken as the number of passages between break vanes. The dynamic cavitation number should then be read as the number of pressure cycles = 2.5 superimposed before the final implosion over the specimen.

• Pipe system: L is taken to be of the order j which is to be read as the length for which the negative phase of the oscillations reach a cavitation susceptible pressure level when assuming standing waves. U is the mean flow velocity. Hence a bubble will experience about 500 cycles from when it enters until when it leaves the cavitation susceptible region along the pipe.

• NAG A profile: L is taken as the extension of the attached cavity = Lif. U is taken as the velocity of the liquid at the interface and / is taken as the frequency of the instability. Any bubble will therefore experience about one sound pressure cycle perturbation.

• For an acoustic system pconv should be equal to the equilibrium pressure i.e. no sound field and L should be chosen according to the part of the wavelength, travelling i (cut + k'r')or standing waves, having a pressure comparable to the equilibrium pressure. Thus L must be viewed as the extension of the cavitation zone. Since U is zero, a bubble can experience an infinite number of pressure oscillations.

As an a priori calculation of flow systems, U and pcontl can be found due to numerical calculations while pomp and / could be based on experience, even a simple formula for hydraulic turbines is provided in this thesis in Equation 2.14 or by the developed numerical tools for solving such problems. Thus L can be estimated and the omission of one or the other effect can be deduced.

66

The obvious use of such a number, in addition to the usual cavitation number, is to determine whether effects like rebounds, rectified diffusion, modulated free stream cavitation and the ” Christmas tree effect” are likely to occur for a given flow.1

1The Christmas tree effect, a term the author learned from Dr. A. Keller. Which is a phe­nomenon where the collapse of an initial cloud may enhance the size and (erosive) power of the subsequent one and so on.

67

3.3 On the nucleation of cavitation clouds

3.3.1 Cloud formation in a pure sound fieldA photographic study was done during my study at the Technical University of Denmark under supervision of Dr. K. A. M0rch. Unable to state the overall pressure field, although an argumentation for the sound field in a beaker was given in Section2.2.3, investigations of the system must rely on measurements.

The purpose of this photographic study was to illustrate the cavitation cluster formation and behavior due to the pressure field set up by the vibrating horn. Several photographic series were taken in a set up allowing pictures to be taken from the side of the vessel, i.e. different from the system treated in Section 2.2.3.

Regarding the experimental set-up refer to Section 4.1.

Results

The photographic series was obtained using a high speed camera, the events were covered by 29 pictures within a 100 (is period i.e. each picture contains information for a period equal to 3.57 (is.

The pictures are in turn compared to a simultaneous pressure reading, see Figure3.3, at a transducer positioned close to the cavitating part. Regarding the informa­tion from this work it is observed:

• The cavitation events do not repeat which they should after 50 (is.

• It is also observed that the cavitation cloud size is not according to the ex­pected size regarding the simultaneous raw data pressure readings.

Regarding the latter the phase difference can be found , if the print quality allows, because the large bubbles serve as pressure transducers.

• The rise and collapse times for the cavitation cloud seem to be in the range 3.5—7 /is depending on the definition for ’’complete” nucleation and implosion.

Observing the large bubble we can calculate the phase change or the time lag from the cavitation events to the transducer.

If they react linearly to the driving sound pressure a maximum pressure is found at t 21 /is. Which states that the pressure readings are phase shifted —10 (is i.e. t — 112 (is, contrary to 102 (is in real time, corresponds to t = 0 in the picture series. This shift seems to make the correlation of the pressure readings and the cluster behavior according to what is expected. Further it should be noticed that the size of the distinct bubbles is in the range 0.1 mm , the horn tip is about 16 mm in diameter.

68

Trig signal between 49 and 54 micro secs.

Time [microsecs.]

Figure 3.3: Pressure recording taken close to the cavitating part of a vibrating horn, simultaneously as the preceding pictures. t=0 in picture series equals t = 112 fis in this chart.

69

t=0

t 3^ps

Figure 3.4: High speed photograps of a cavitation cloud at the end of a vibrating (20 kHz) horn. t=0 in the figure corresponds to t — 112 //sin the simultaneous pressure-time recording chart. Qualitatively relating the visually observed event to the chart it is observed (chart time): 0-7 (xs (112-119) fxs p<pmeon f* > 0 ; 7-10.5 Aw(119-122.5) # > 0

70

1=14

Figure 3.5: High speed photographs of cavitation cloud at the end of a cavitating horn. Qualitatively relating the visually observed event to the chart it is observed (chart time): 10.5-13 (122.5-125) ps p>pmean > 0 ; 13-20ps(125-132) p% pmaxg % 0. 20-28(132-140)ps. p>Pm»on and ^ < 0.

71

t=28

_ t=3L5ps

t=3Siis

Figure 3.6: High speed photographic studies of a cavitation cloud at the end of a vibrating horn. Relating the visually observed events to the simultaneous pressure reading it is seen (chart time): 28-40 (140-152)//s p<pmean fjr < 0. p(t^l52/us)=pmin 40-42(l52-154)pS P<Pmean ff > 0.

72

Figure 3.7: High speed photographic studies of a cavitation cloud at the end of a vibrating horn. Relating the visually observed events to the simultaneous pressure reading it is seen (chart time): 42-56(154-168)ps ^ > 0, p<pmean-

73

3.3.2 Cloud formation in a spherical focused sound field

Of special interest for cavitation excited by the use of a sound field, is the spherical focused field. Such a field can be excited by the use of piezo electric ring transducers as has been demonstrated by Ellis [17]. Theoretically Mprch [35] has contributed and some of his conclusions will be quoted because, to my knowledge, the only contribution that considers the prediction of the formation of a cavitation cloud is found in this cited article. Even the sound field considered here is far from what one should expect in fluid machinery some of the findings will be of general nature and applicable to cavitation cluster or cloud formation in hydrodynamic flows as well.

The pressure perturbation in a spherical sound field is found to be:

Ap = —1r

2 PqCqW

7r

12

sin kr cos ut

Where poco are the density and sound speed of the liquid at equilibrium condi­tions, W the acoustic power, k the wave number and w the frequency in rad/s.

If the liquid is strained either by the sound field or by flow through a contraction cavities wall grow from the nuclei. Further straining makes the cavity grow towards a state at which a minimum equilibrium pressure pCTit is reached. This pressure is determined by the surface tension at the liquid/cavity interface and the partial pressures of gas and vapor inside the cavity. Straining beyond the critical condition the growth is continued by straining itself, i.e. bubble interior pressure exceeds the liquid pressure. Subsequently by relaxation of negative pressures in the liquid i.e. liquid molecules add to the interior of the bubble as vapor thus relaxing tension in the remaining liquid because a vapor mass equivalent occupies a larger volume than a corresponding liquid mass. At this latter state |^ < 0 and a cavitating medium is formed according to Appendix A.3.

To explain nucleation of a cluster or cloud of cavities in a spherical focused sound field the following argumentation is provided by Mqrch [35]:

• The focused spherical sound field itself guarantees the existence of a supersonic nucleation front indicating that relaxation waves from nucleated cavities do not hamper further growth for neighboring ones. The extension of the supersonic nucleation front defines the region occupied by cavitation bubbles. •

• The cavitation bubbles within a confined region are shielded from the exterior sound field due to the relaxation of negative pressures making the border of the cluster an effective reflection boundary.

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3.3.3 Cavitation cloud formation in hydrodynamic flows su­perimposed sound pressures

Unlike cavity cluster formation in a stationary liquid submitted to a sound field, the behavior in a hydrodynamic flow will be transient in nature indicating that the cavitation will not necessarily be formed on the same nuclei during subsequent pressure cycles.

A nucleus means a discrete volume in a bulk of liquid that is not able to resist a certain pressure below vapor pressure as stated in Definition 2 in Section 1.1.1. Further it is understood that this volume within the liquid can contain particles, gas bubbles or other pollutants all having the effect of lowering the tensile strength of the liquid. For most calculations in the literature and in engineering situations the nucleus is considered analogous to the gaseous micro bubble.

An argumentation is given for the formation of cavity clouds or clusters in fluid flows. At the time of writing the flow systems utilized in this work have not been visually inspected for cavitation cloud or cavity cluster formation. However an algorithm will be presented and an argumentation on the validity is given. It is emphasized that this outline exclusively deals with free-stream nucleation and does not consider nucleation on the wetted surface.

The way towards an engineering tool for predicting cavitation is through a discus­sion on water quality, as was done in Chapter 1 and a discussion of the nuclei during their passage over the flow body. Further both the knowledge of mean quantities of the flow field as well as non-stationary characteristics should be established.

The distribution of nuclei in a hydrodynamic flow

Liu and Brennen [31] have in a recent paper tried to isolate the conditions that are decisive for what they refer to as the cavitation event rate. The authors provide a formula for the effective nuclei number density distribution, their approach also takes into account the bubble-bubble interaction giving a formula for a ”non-nucleable” sphere surrounding an already nucleated bubble. The number of nuclei consists of the ’’total number” of probable nuclei, i.e. micro-bubbles and particles, of all sizes but the total number is reduced according to their position in the pressure field set up by the hydrodynamic flow, a screening effect due to migration of nuclei onto further offset streamlines at the stagnation point /2 and a boundary layer effect influencing the flow rate of nuclei close to the surface f\. Their expression for the event rate is found to be:

S/M/3 oo

E = J 2irrsU (1 — Cpms)2 fi (y) J0

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Where the interpretation of the various terms should be: = radius of theSchiebe head form used in the study by the cited authors, Cpms =minimum value of Cp. U (1 — CpMs)~2 = a gradient in downstream direction indicating the extension of, and the flow rate into, the cavitation zone. N (R) = actual number of nuclei of sizes between R and R + dR in a confined region. —is a reduction term due to hampering of nucleation of nuclei close to a nucleated one. /s includes a finite bubble size effect.

Turbulence is a major problem in determining cavitation events. Anton [2] gave a theoretical deduction of the turbulence levels influence upon cavitation inception. He concluded that the lowest absolute pressure levels will be reduced and that growth of nuclei due to rectified diffusion will be enhanced. Recently Keller [21] has investigated this experimentally.

Further at a specific point the rate and number of nucleations reach a limit for which the total is considered as a single system as explained in the following. This treatise also opens for simultaneous nucleation of cavitation bubbles in a confined region due to small scale vortices. This latter phenomenon is a recognized cause of cluster/cloud nucleation in flow systems. However as the scope of this thesis is to investigate the role of sound waves emitted from distinct sources a detailed look into that important field is omitted.2

Pressure in the liquid

In order to accurately calculate the pressures affecting the appearance of cavitation it is stated that three origins of pressures have to be taken into account.

1. The absolute pressure, pressure gradients and turbulent fluctuations all caused by the hydrodynamic flow.3

2. Pressure waves caused by temporally varying shocks. The waves may establish a sound field thus indicating the possibility of an amplitude resonance within the system.

3. Pressure emission from cavitation bubbles during their growing and collapsing stage.

Thus the algorithm that is proposed to be used for the calculation of pressures affecting the cavitation appearance is as follows.

Tn the case of cavitation nucleation the strong tangential velocity, in orcier to enable the low pressure at the center, will certainly influence the nucleation and the subsequent behavior of the cavitation bubble. A good example is the thread-like vortex structures originating from turbine runners and existing at considerable a length into the draft tube, and the tip vortex of propellers.

3 This thesis will not discuss large vorticity structures sweeping over the surface. These vorticities may induce deleterious cavitation effects.

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1. Given absolute pressures for the flow paths of the nuclei. This can be found through single phase numerical analysis as it is expected that free stream cloud nucleation does not necessarily disturb the flow field significantly.

2. Given amplitudes and frequency or Fourier components of either a harmonic or a transient pressure perturbation.

The pressure perturbation from each nuclei n upon nucleus k can be given as:OO

Pk = ^2 Pn [t] , Rn [*] n ~ k) (3.2)n= 1

Where capital R is the single bubble characteristics. Thus only the first order perturbation is taken into consideration, i.e. a spherical modification of the ambient pressure, see discussion Section 1.3 or Chahine [13].

The added pressures set up by the convective flow and the incident sound waves, of amplitudes pm, can be approximated by:

yp(x, y) = Psurf(x) + J ^ {y) dy +

surf

In addition to bubble-bubble interactions a nucleated cluster will influence the sound field as shown by Mprch [37]. Regarding the latter reference is made to the discussion in Appendix A.3 where a cavitating medium is totally compliant to incident sound waves. This consideration is a straight-forward task when a simulta­neous nucleation is guaranteed. However in the case of flow nucleation such a kind of nucleation cannot be asserted. Such information is believed to be observed in careful experimental set ups, it is referred to Section 4.3.2 in order to investigate the method suggested here in hydrodynamic flow systems for establishing a sound field that can provide information of the acoustic properties of a cavitating medium.

If free air is present, as in the rotating disc system investigated in this work or through artificial injection in hydroturbines, the sound field can be distorted due to dispersion of the sound waves, Crighton [16].

Argumentation for the formation of cavitation clouds in a hydrodynamic flow

Mprch [36] provides an argumentation of cloud nucleation in flows. He suggests the continuous formation of nuclei at certain locations on the solid body. The nuclei are convected into the wake where they accumulate to form a cluster or cloud. The further development of the clouds is dependent on the far field pressure, e.g. due to an incident sound wave, on the boundary of the cloud.

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As we lack a worked out model on the nucleation of a cavitation cloud, according to the given algorithm, it can be argued that this will occur based on the formula given by Liu and Brennen, Equation 3.1. Rc and Cpms will be a function of time due to the superimposed sound wave. A direct consequence is that the event rate will be time dependent. It is therefore asserted that there can be an intermediate stage at which the event rate forms a cavitation cloud, and which probably will be the maximum number of event rates because the sound field will be significantly distorted by the cavitating medium.

Another view, provided by March [35], is that the amount of straining energy which could not be stored by the liquid continuum, as stated in Definition 2 in Section 1.1, is used for the formation of a cavity cluster. This is a view which can be explored by viewing the energy balance of the acoustic power and the energy required for evaporation of a bubble. In order to illustrate this consideration an example is provided, where it is assumed that all the energy in the sound field can be extracted for the formation of a cavitation bubble.

Example 18 Taking a simple model for the acoustics the power of a sound wave radiated from a monopole source and at sufficient distance, that is the wave can he considered as plane, the power can be expressed as.:

w = ^—pc

I.e. for the turbine, Example 10 Section 2.2.1, p is found from Equation 2.14, such that W = where b is taken to be a typical width of the runner channel.Further the cavitation bubbles are assumed to grow during a fraction of the sound wave, such that the rise time for bubbles is given according to the forced frequency trise = e/-1. Taking e to be r— , which is said to correspond to the part of the wave making cavitation in a susceptible region. The sound wave energy extracted for bubble growth is calculated to be (b = 250 mm and rxcmrce = 5 mm):

E.acoustic —AP2sb\f-1

pcWhich is calculated to be EacoustiC ~ 0.001 J, however the region of interest for cavitation will be covered by a fraction of the wave along the streamlines. The energy required for evaporation is found in Equation A.6.

E = jTTR’Lpl (T)

Thus R can be calculated if the energy required for evaporation is withdrawn from the sound energy.

Eacoustic

Lpev(T)4ir

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Taking for T = 298 K the radius is calculated to be: R ~ 26 m,m, likewise ten bubbles of equal size R = 12 mm, or thousand bubbles of equal size R = 2.6mm. can be formed. Thus at a maximum a bubble of the given order could theoretically be nucleated over the whole cross section and for a given distance along the flow. The calculated value is not realistic due to the obvious fact that the sound field is not focused onto one single bubble. It is stated that a more probable consequence is the nucleation of several bubbles within a confined region.

3.3.4 Discussion and concluding remarks on the nucleation of cavitation clouds

M0rch [35] validated the formation of a cavity cluster by the formation of a super­sonic nucleation front as discussed in Section 3.3.2. The experiments presented in Section 3.3.1 also showed the formation of a cavity cluster. In this specific system it is believed that the formation of such a nucleation front does not occur. The present author [23] investigated numerically the pressure pulses formed beneath the vibrating horn using a similar approach as in Section 2.2.2, i.e. distributed mono­pole sources, and found no supersonic phase speeds. It is therefore asserted that the limitations of cavitation bubble sizes in steady oscillatory systems is due to the thermodynamic constraints as discussed in Appendix A. 2. For flow systems with a sound pressure superimposed it as been argued, using the model of Liu and Brennen cited in Section 3.3.3, that a probable consequence is the formation of a cavitation cloud. Regarding the quantitative information necessary to reveal the cavitation cloud extension a discussion was provided in Section 3.3.3 regarding the nucleation pressures. It is stated in this work that one of the main problems is the interaction of the nucleated region and the sound field.

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Summary• It has been shown by using well-established physical relations for cavitation

bubble dynamics that the superimposed sound pressure perturbation influ­ences the bubble radius history.

• A relation has been proposed in order to quantify the relative importance of sound pressure and flow regarding cavitation.

• An experimental study on cluster cavitation in a vibrating horn has been pre­sented. A discussion on this has been presented in order to extract information significant for flow cavitation.

• Models for the nucleation of a cloud of bubbles in ordinary flows superimposed sound waves have been discussed. This must be viewed in connection with the study of the formation of sound pressure fluctuations in Chapter 2 of this thesis report. It is emphasized that further experimental observations are necessary in order to quantify the proposed model.

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Chapter 4

Experiments

Both the water tunnel and the rotating disc apparatus have been built using my spec­ifications. The experimental investigations of the vibrating horn have been planned in cooperation with Dr. K. A. Mprch. the rack erected for doing parametric studies of the pressure fields at vibrating horn were made according to my specifications.

4.1 The vibrating hornThe design of the horn rack and the beakers is done to fulfill the requirements of the ASTM G32 standard regarding cavitation erosion tests. The rack was made to allow adjustable submerge, A, of the vibrating horn. Further, beakers made of stainless steel and of various sizes could be inserted which all allowed variations of the diameter D and the water level l. The definition of parameters l, lh and D is shown in Figure 4.2.

The horn consist of a piezoelectric material which is brought to resonance at the driving frequency, 20 kHz, the mechanical design is further made such that the largest displacement amplitudes is found at the end of the cavitating surface of the vibrating horn. The power supply was a Branson 184V, no readings of power consumption were taken. The system had recently been calibrated such that the system performance was within the demands of the ASTM standard.

Two piezoelectric transducers (AVL 6QP500a) were used to record the dynamic pressure, with one mounted in a long cylinder making the membrane surface to be maneuvered into any preset position. The resonance frequency of the transducers was above 100 kHz, however no original calibration sheets were available. The transducer at the horn was fixed, but adjustable in the vertical direction, and held at a comparable depth as the horn submerge. The amplifiers utilized were B&K type 2635 and Fribourg TP 200/A power unit, TA 2/C Piezo Amplifier. For calibration procedure refer to Figures 4.3 and 4.4.

The calibration procedure is considered inaccurate and could explain the large

81

Figure 4.1: Picture showing parts of the rack, the horn submerged and the transducer position. The plexiglass window in the bottom of the beaker allowed pictures to be taken from beneath.

82

Vibratin'ducer x exp. horn

D-beaker

Lens

Camera

Figure 4.2: Sketch of the vibrating horn system as utilized for high speed photo­graphic studies. The definitions of the parameters D, 1 and 1/, are also according to the set up of the investigation of the pressure field in the system.

Figure 4.3: Device for dynamic calibration of transducers. A venturi made a pres­sure close to cavitation pressure at its center that was hydraulically connected to the transducer. By a sudden exposure to the atmosphere a voltage reading corre­sponding to a pressure jump of approx. 100 kPa could be made.

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pressure readings cited in the preceding sections. It is shown in Figure 4.4 that there are difficulties in obtaining a measure for the voltage readings versus actual pressure.

The voltage readings were made using a Gould 400 oscilloscope and downloaded on a graphical plotter (Tektronix HC-100). The frequency spectra were recorded by a Rohde & Schwarz type FUA BN48303, and downloaded on a Rhodefe Schwarz Enograph G type ZSG.

The camera used was a Barr & Stroud Ultra-High Speed Framing camera, ca­pable of frame rates of approximate 2-106/romes/s., for which the original devel­opment was done at the Atomic Weapons Research Establishment, Aldermaston U.K. The description of operation is mainly taken from the operation manual. The assembly consist of a main shutter (20 ms opening time) that opens at a rotational speed of the rotating mirror equal to 50 Hz. The acceleration of the mirror is rapid and when the pre-set rotational speed is reached, a trig signal to the light source is given. The light source is decades brighter than incident light such that the events to be captured on the film roll (29 pics.) is essentially the ones during the flash. In order to control the events additional electronics are required: Camera power unit, Frequency Monitor, Automatic trigger unit and a Delay unit. The delay is made to adjust for time delay of trig-signal compared to the rise time of the flash. This in order to ensure that the first picture and the subsequent ones are exposed at maximum illumination during the flash time.

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Figure 4.4: Dynamic calibration of the pressure transducer. In the compensated calibration the value of 0.75 V/bar has been calculated due to an expected expo­nential discharge of the piezo electric element in the transducer. However the value of 0.48 V/bar has been used throughout the text.

85

Figure 4.5: Picture showing (from left) Power supply, flashlight, optics, vibrating horn and the high speed framing camera.

4.2 The cavitation erosion apparatusThe erosion apparatus had a previous counterpart constructed during my work for the siv. ing. degree. A new one was built to make the construction more compact and to allow swapping of cavitation inducer geometries. Several investigations have been performed with a material aspect however these specific experiments will not be reported here.

The set-up is ; 18.3 kW electric motor with a rated speed of 2970 rpm. Original configuration with one tap water inlet and a peripheral outlet at its back. The casing had a diameter of 350 mm and 105 mm depth. The front and back covers had 20 symmetrical, regarding the rotating disc, positioned breaking vanes (130 mm x 40 mm x 5 mm covering a diameter from 80 mm to 340 mm) inserted in order to prevent gross water circulation in the total enclosure. The distance between the rotating disc and the ribs was 10 mm. As cavitation inducers several geometrically shaped ’’obstacles” have been tested, where cylinders of diameter 5 mm and height =5 mm were found to induce most erosive erosion which was observed based on paint erosion trails. Additional illustrations are provided in Appendix E.

The measurements of pressure fluctuations within the enclosures of two ribs were done using the following pressure transducers:

1. 2 x Entran EGX- placed at the same radius and 4 ribs between. These pressure

86

Figure 4.6: Erosion on aluminum specimen mounted in the rotating disc.

transducers were found suitable due to their high eigen-frequency and their low sensibility to vibration, that is no filtering of the signal is necessary.

2. lx Kistler 4043A5 A transducer having a high vibration sensitivity and there­fore used for measuring static system pressure during the runs.

Additionally a home-built infrared diode was mounted in order to track the positions of the cavitation inducers relative the Entran transducers.

Voltage supply arid amplification of the measure signals were delivered from an HBM MA-10 measurement amplifier. The signal was logged using DAQ-cards and Lab View software from National Instruments. For all runs 4096 samples were acquired with a sampling rate of 500 Hz, 2 kHz, 8 kHz and 20 kHz The amount of information did not significantly improve from 2 kHz sampling rate to 20 kHz.

A relatively simplistic, and inaccurate, calibration method was utilized where internal casing pressure was increased when allowing a leakage flow. It was neces­sary to do in-situ calibration of the Entran transducers as tightening of transducer will influence the measure bridge. The calibration results are outlined in Figure 4.7 where the two highest pressures are based on measured signal of the center transducer Kistler calibrated using a calibration device. The first point corresponds to atmospheric condition while the two next are based on measured water levels. The voltage ranges for transducer 1 and transducer 2 are respectively 1.48 — 4.88 V and 2.73 — 4.77 V for the pressure range 0 —> 120 kPa, where 0 corresponds to

87

14000 T

12000

10000 ■■

8000 ■ •

6000 • ■

4000 ■ •

2000 -•

Voltage reading [V]

Figure 4.7: Calibration curves for transducers mounted at a radius corresponding to that of the cavitation inducers.

atmospheric conditions.

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4.3 The water tunnel at the Hydropower Labora­tory

In order to investigate some of the proposed consequences of a superimposed pres­sure perturbation on a hydrodynamic flow a general purpose water tunnel has been erected in order to exploit the permanent installations in the Hydropower laboratory. The water circuit is open to the atmosphere at both the exit and head reservoir. The following criteria served as basis for the choice of setup.

• High Reynolds number test section. Flow velocity to be comparable to absolute outlet velocities of a Francis turbine runner.

• Open loop, water is blown from test section into the sump. Which prevents excessive pump noise, while on the other hand an atmospheric outlet serves as a reflecting condition for other flow instabilities.

• General design in order to utilize it in other branches of research, and as a study for similar installations at the planned new hydraulic laboratory.

4.3.1 Mechanical design

The length of the vertical test section is 1300 mm, and the rectangular cross-section 130 by 190 mm. A NACA 4412 profile of chord length 165 mm is positioned 250 mm, above the outlet to the atmosphere. Upstream of the test section there is a stilling tank of 1000 mm, diameter, containing two flow stabilizing screens of perforated alu­minum plate with 40% aperture, and a 750 mm long pipe of 130 mm, diameter with the loudspeaker described in Section 4.3.2. Two conical flow accelerating sections of 45° and 30° respectively, and a circular-to-rectangular transmission section, each 200 mm long, connect the stilling tank with the test section. Upstream of the tank there is one horizontal bend of 45° and one vertical bend of 90°, fitted in order to utilize the permanent pipe installation in the laboratory, Ahead of the bends there is a butterfly shut-off valve, and further 5 m upstream a flow sluice control valve. See sketch in Appendix E.

The cavitation on the profile is partial, as shown in Figure 2.9, and low frequency pressure/ mass oscillations were detected. This is believed to be due to either the large secondary flows induced by the mentioned bends or by the geometric discrepancies of the plexi-glass profile which was hand crafted. However as the measurements made suggested to be sufficient for the presented work no additional efforts in making a homogeneous cavitation signature were made.

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0.960-1.90 1.91 1.92 1.93 1.94 1.96 1.97 1.98 1.99 2.00

Figure 4.8: Vertical axis represents raw voltage values acquired by the software. Pressure readings of sound emitted from a underwater loadspeaker. Filtered band­pass signal 350—450 Hz, sampling frequency fs = 40.96 kHz and number of samples ns = 8192. Excitation signal ~ 380 Hz. IV(abscissa) corresponds to 10kPa.

4.3.2 Arrangement for creating sound pressure pulsations

In order to do modulation experiments of the instability of the attached cavity as investigated in Section 2.2.2, methods were sought to obtain a fluctuating sound pressure field. In addition these investigations should serve as a conceptional study for cavitation nucleation studies. It must be emphasized that the lack of controlling the water quality at the Hydropower Laboratory would exclude any quantitative investigations of the phenomenon

The one requirement considered important was that the source of the sound field should not distort the convective flow. In order to fulfill that requirement an underwater loudspeaker was mounted in a cylindrica l dpe upstream the rectangular test section of the tunnel. This specific loudspeaker. University sound UW-30 was driven by a sinusoidal pulse generator connected to an ordinary hi-fi amplifier and had an estimated pressure fluctuation amplitude of 50 Pa or 120 dB, at a semi­sphere 1 m off the membrane. While mounting the loudspeaker in a cylindrical pipe the sound energy would be concentrated such that a significant pressure variation could be found over the profile. Measurements were taken to reveal the strength as shown in Figure 4.8:

The forced pressure frequency is calculated to be, according to the accurately calibrated HBM APS transducer at the tunnel wall on the pressure side of the profile, of the order 600 Pa regarding peak-to-peak pressures Figure 4.8. It is also noticed that this is not a standing wave, actually this proved to be a major difficulty and no evidence of a standing wave pattern over a range of frequencies was found. No

90

1.0E-5

8.0E-6-

6.0E-6-

4.0E-6-

2.0E-6-

1.6E-14- i i i i t2.5 200.0 400.0 [Hz] | 600.0 800.0 1000.(

Figure 4.9: Vertical axis represents power density based on raw voltage values aquired by the software. Qualitative nvestigations of modulation of the instabilities for an attached cavity. Excitation frequency ~ 425 Hz, the activity at frequencies others than the excitation frequency seems to diminish suggesting modulation.

investigations have been done close to cavitation inception due to the instabilities of the bulk flow.

In order to investigate whether modulation of the instabilities of the attached cavity reported in Section 2.2.2 could be measured.1 A number of frequency spectra will be shown, for all spectra yields: fs = 40.96 kHz, ns = 16384, high pass filter at 200 Hz, equal flow rate and angle of attack 3°. The measurements done indicates modulation at 425 Hz Figure 4.9, while the expected instability seems to diminish for 690 Hz Figure 4.11.

The frequency peak at about 700 Hz seems to be constant which strongly sug­gests that it is a characteristic frequency for the erected system.

4.3.3 Experimental equipmentA HBM AP/8 transducer is fixed at a given position, connected to an HBM measure­ment amplifier, in order to provide values to a cavitation number. For the present analysis the following physical quantities are used to describe the flow rate.

• Flow discharge. Measured by a calibrated weir in the sump.

• Absolute pressure at fixed station.

xIt is pertinent to cite the ’’perspectives” given by Farhat[18] on page 221 chapter 8.4.2 Controle de la dynamique de la poche de cavitation:..d’un champ de pression acoustique sur le comportement de la poche de cavitation dans le but de controler sa stabilite et la frequence de lacher des cavites transistoirs et par suite reduire le pouvoir erosif de la cavitation.

91

1.0E-5

8.0E-6-

6.QE-6-

4.0E-6-

2.0E-6-

2.7E -12— i— 2.5 200.0 400.0 600.0 [Hz] | 800.0 1000.1

Figure 4.10: Vertical axis represents power density based on raw voltage values aquired by the software. Qualitative investigations of modulation of the instabilities for an attached cavity. Excitation frequency ~ 440 Hz, the activity at frequencies others than the excitation frequency seems to be present though no modulation occurs.

1.0E-5

8.0E-6-

6.0E-6-

4.0E-6-

2.0E-6

5.8E-112.5 200.0 400.0 600.0 [Hz]l 800.0 nooo.t

Figure 4.11: Vertical axis represents power density based on raw voltage values aquired by the software. Qualitative investigations of modulation of the instabilities for an attached cavity. Excitation frequency ~ 690 Hz. The activity seems to diminish over the total frequency range thus suggesting the forcing excitation to silent the instability of the interface.

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1.0E-5

8.0E-6-

B.OE-6-

4.0E-6-

2.0E-6-

3.4E-15400.0 [Hz] | 600.0 1000.C

Figure 4.12: Reference power density frequency spectrum. Neither cavitation (angle of attack = 0°) nor excitation.

• Angle of attack of profile.

The tunnel is design in order to instrument it using several set- ups. The following have been used during the present studies, but measurements are not necessarily reported in the thesis.

• B&K 8103 Hydrophone, Kistler Charge Amplifier.

• Entran EGX Pressure transducer. (For most frequency analysis) Pressure range 0 —> 8 bar, absolute

• Pitot . Fuji differential pressure transducers.

4.3.4 Discussion on validity of measurementsThe measurement chain used in obtaining most frequency spectra is considered accurate for the generic studies made.

The vibration characteristic of the mechanical system has not been subjected to measurements, neither the tunnel nor the profile.

4.4 Measurements of airborne noiseThe measurements presented in Figure 2.7 were done by using a microphone RION UC31N, a Norsonic pre.amp.1201 and acquired on a SONY P CM-2000 DAT recorder sampling frequency 44.1 kHz. The measurements were analyzed using a Toshiba 5200 PC, DAQ-card and software provided by National Instrument.

93

The measurements in Figure 2.6 were using a set- up provided by the acoustics group at the Norwegian University of Science of Technology, the exact data on the equipment is not known. The analysis of the measurement were made as part of this work.

The position of the microphone for both cases was at upper cover of the turbine. The former turbine was operated at a fixed operational point (P ~ 0.65 P°, P° = 72 MW) while the latter was operated over a range of operating points. All operational points showed a qualitative similarity regarding the calculated frequency spectrum.

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Chapter 5

Conclusions

This work supports the view that a multi-bubble (cluster or cloud) collapse is more erosive than single bubble collapse, Section 1.3.3.

This thesis has argued that bubble clouds or clusters will be formed due to compression waves superimposed on a hydrodynamic/ mass flow Section 3.3.3, as can be done by flow shear fields, Avellan and Dupont [5] and Arndt et al [4].

It has been shown that significant sound pressure perturbations probably can be formed in flow systems like high head hydro turbines Section 2.2.1 and in a rotating disc apparatus Section 2.2.4, both systems are susceptible to cavitation erosion.

It is further shown by studies of the vibrating horn, ASTM G32- Standard appa­ratus for the investigation of cavitation resistance, Section 2.2.3 and Section 3.3.1, and argumentation provided in Section 3.3.3 that the qualitative difference between flow cavitation and cavitation caused by a sound field can be the size of the individ­ual bubbles within the clouds due to thermal constraints as discussed in Appendix A.2.

5.1 Conditions preceding cavitation; concluding statements

This thesis proposes that sound waves are of importance also for flow cavitation. It is deduced which effects within specific systems that produces these compression waves in Chapter 2 of this thesis. In order to investigate the impact of compression waves on cavitation a discussion on the cavitation nuclei was given in Section 1.1.1. In Chapter 3 the impact on cavitation was investigated.

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5.1.1 The cavitation nuclei

In order to provide a basis for the development of an engineering tool for the predic­tion of cavitation in sound pressure perturbed flows a discussion on the cavitation nuclei was given in Section 1.1.1. The main conclusions of this discussion were:

• The dissolved gas content cannot be treated uncoupled from the nature and existence of solid particles as is stated in Proposition 4 on page 8.

• It has been argued that temperature might have a significant influence on the appearance of cavitation.

5.1.2 The formation of pressure pulses

An overview of the systems investigated are found in Table 2.1.

• Sound pressure of significant amplitudes regarding the impact on cavitation are found to not necessarily interact with the hydrodynamic flow as shown in Section 2.1. However, the pressure is believed to have a major impact on cavitation. Opposite, cavitation can modify the propagation of pressure fluctuations as is briefly discussed in Section 3.3.3 and Appendix A.3.

• High-head Francis turbine runner, Section 2.2.1: the wake and vortices trailing off the upstream guide vane make a temporal fluctuation of stagnation pressure at the inlet of the runner. A simple deduction of source strength based on potential analysis and fundamental results from acoustics is given, Section 2.2.1. Pressure pulse formation due to a water hammer effect within the runner channel is also proposed within this thesis. Both investigations indicate a significant amplitude of the formed pressure perturbation as found in the Examples 10 and 11.

• Attached cavity, Section 2.2.2. Correspondence is found between a theoretical deduction of instabilities of a liquid/gas-vapor interface and measurements/ observations noted in this work and in the literature. It is further stated, Proposition 12, that these are able to emit pressure waves which in turn can enhance the free stream nucleation of cavitation bubbles. •

• Rotating disc erosion apparatus, Section 2.2.4. Based on the qualitative de­scription of liquid motion within the apparatus and measurements it is stated that there is a presence of sound waves, due to the cavitation inducer-breaking vane interaction.

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5.1.3 The appearance of cavitation• It is found using existing theory that a pressure perturbation due to sound

waves affect the single bubble cavitation behavior, Section 3.1.

• Sound pressure is stated to be able to nucleate clouds of cavitation bubbles that are believed to enhance the erosivity of the flow, Section 3.3.3.

• The difference between cavitation in a sound field and flow cavitation might be less than previously considered, thus it is stated that vital information of cavitation in flow systems can be provided by investigations of the vibrating horn. The main difference between the two is probably the size of the bubbles contained in the cloud, Section 3.3.4. Thus even the erosive power of the two different systems is like the final implosion in high frequency modulated cavitation will make more local impacts on the structure facing the surface.

5.1.4 Methods of investigationRegarding experimental techniques the following is stated.

• The use of an underwater loudspeaker is a promising method to obtain a superimposed sound pressure wave without disturbing the convective flow, Section 4.3.2. It is emphasized that during this study few ways were sought in order to optimize this concept.

• The use of forced oscillating pipe flows is considered to be an interesting method to actually provide the experimental tool for which most of the pro­posals in this thesis can be investigated, Section 2.2.5.

5.2 Non-stationary flows and cavitation - Sum­mary

A listing of dynamics of cavitation presented in this report is made:

• The gaseous micro-bubbles, for which free stream cavitation nucleation is as­sumed to originate, have been shown to have a large time constant in order to grow to a size which gives critical nuclei at ’’reasonable” pressures. Rectified diffusion by a sound field or turbulence may significantly improve the growth rate of a gaseous micro-bubble.. •

• The growth or the response, to transient low pressures, of a cavitation nucleus can be hampered due to thermal constraints. (Refer also to discussion in Trevena [47] p. 119.)

97

• The momentum related resonant frequency for gaseous microbubbles is high until very close to a critical size of the nucleus is reached.

• Attached cavities may induce acoustics as well as the shedding of gaseous/vaporous and turbulent structures.

5.3 LimitationsThroughout the thesis, the issue of turbulence versus cavitation has been avoided. It is emphasized that most situations that are decisive for the formation of sound pressures are also sources for turbulence or swirling structures.

It has also been concentrated upon free stream nucleation which is not necessarily the case for engineering flows. All thoughts expressed on cavitation nuclei in this thesis are also proper for the material of the wetted surface, but its nucleus will have more time available to grow to a sufficient size i.e. critical size. Therefore one must conclude that a lasting negative absolute pressure necessarily must result in the formation of cavitation such as an attached cavity. Thus the critical pressure should not be lower than the vapor pressure.

5.4 Directions of future researchBased on this study it is proposed to continue research along the following directions:

• Water quality

— Establish the quality of the water specifically at high-head Norwegian turbines, and trace any difference when operated with and without intakes of small rivers and streams on the head race channel.

— Establish the effect of various, hydrophobic properties, particles in various water samples saturated by air.

e Dynamic behavior

— The ongoing research on numerical solvers for weakly compressible flows should also be motivated by the possible impact on cavitation appearance.

— The instability of an attached cavity should be submitted to a more detailed investigation, theoretically and experimentally.

— The actual nature of pressure oscillations in any cavitation susceptible flow device should be investigated.

98

— Collecting field experience of the amplitudes and nature of pressure pulses in hydraulic turbines and preferably within the runner vanes.

— The interaction of pressure pulses versus a cavitating medium and/ or free air i.e. bubbles.

• Experimental techniques

— Further investigation of the creation of superimposed sound pressure waves over a cavitating body in order to verify models on cavitation cloud nucleation, and the interaction of flow turbulence and acoustics.

5.5 Closing remarksThis thesis emphasizes the need to merge different fields of science in order to obtain an understanding of the phenomenon of flow cavitation. Two key specializations are acoustics which contributes to the understanding of flow dynamics and physics which contributes to the effect of impurities in the cavitating water.

This study is the first one done, at the Hydropower Laboratory, that considers cavitation as a phenomenon to be understood and not avoided. Disregarding the obvious drawback of researching new ground, cavitation is a grateful field of science which never fails to surprise and for which treasures seems to hide all over.1 Thus the real conclusion is that this thesis is not a closing ’’chapter” within this kind of research but rather the beginning of a longer story.

It is said that history repeats itself. As a student I had the pleasure of spending some weeks at Sulzer Escher Wyss in Zurich. The task was to establish dimensioning criteria for trash-racks. Frustration grew when the maximum structural load was easily taken by steel bars of minimum thickness. Having finished my third year at the Norwegian Institute of Technology I had not acquired much knowledge on fluid mechanics, but more on structural mechanics which did not make this thin structures seem right. Unknown why, it was realized that the dynamic behavior (vortex shedding and eigenmodes of structural vibration) had to be the dimensioning criterion and accordingly the thickness grew towards an intuitive order of size. The same experience has been felt during this cavitation research. Being drawn along the main stream of viewing cavitation as a function of mean quantities like submerge, NPSH and their like it was realized that cavitation was probably more of an effect (than the source) of non-stationary nature of fluid flow; well, this is where I stand at present.

1E.g. Julian Schwinger (Nobel Prize in Physics 1965) ’’Energy transfer in cold fusion and sonoluminescence” (talks at physics colloquia MIT & University of Pennsylvania, 1991) discuss the ’’focusing or amplification of energy of about 11 orders of magnitude” (cavitation bubble collapse), as an proposal to the energy transfer from nuclear energy to an atomic lattice.

99

100

Bibliography

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[2] Anton, I. ’’The effects of turbulence on cavitation inception”, 2eme Journees CAVITATION Societe Hydrotechnique de France, Mars 1992, Paris (Also in ”La Houille Blanche” #5 1993)

[3] Arndt, R. E. A.” Cavitation in fluid machinery and hydraulic structures”, Ann. Rev. Fluid Mech. 1981 (13), pp. 273-328

[4] Arndt, R. E. A.; Ellis, C. R.; Paul, S.” Preliminary Investigation of the Use of Air Injection to Mitigate Cavitation Erosion”, Transactions of the ASME J. Fluid. Eng., September 1995 (117), pp. 498-504

[5] Avellan, F.; Dupont, P. ” Cavitation erosion of hydraulic machines: Generation and dynamics of erosive cavities”, IAHR Symposium- Trondheim 1988, Proc. vol. II pp. 725-738, paper LI

[6] Bajic, B.;Keller, A. ” Spectrum normalization method in vibro-acoustical diag­nostic measurements of hydroturbine cavitation ”, Proc. CAV95 Int. Symp. on Cavitation, Deauville, France, May 1995, pp. 7-13

[7] Brekke, H. ” Challenges for Improving Cavitation Behaviour in Hydraulic Tur­bines” , The 2nd Int. Symp. on Cavitation, April 1994, Tokyo, Japan pp. 31-38

[8] Blake, J. R.; Gibson, D. C. ”Cavitation bubbles near boundaries”, Ann. Rev. Fluid Mech., 1987 (19), pp. 99-123

[9] Brennen, C. E. ”Cavitation and bubble dynamics”, 1995, Oxford University Press

[10] Briggs, L. J. ’’Limiting negative pressure of water”, J. of Applied Physics, July 1950, vol. 21

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[11] Campbell, I. J.; Pitcher, A. S. ’’Shock waves in a liquid containing gas bubbles” Proc. Roy. Soc. A, vol. 243

[12] Carey, V. P. ’’Liquid Vapor Phase Change Phenomena- An introduction to the thermophysics and condensation processes in heat transfer equipement”, Series in Chemical and Mechanical Eng., Hemisphere Publ.Co., 1992

[13] Chahine, G. L. ’’Pressure field generated by the collective collapse of cavitation bubbles”, IAHR Symposium-Amsterdam 1982, proc. vol. 1 paper #2.

[14] Chahine, G. L. ’’Dynamical Interactions in a Multi Bubble Cloud”, Transactions of the ASME, J. Fluids Eng., 1992 (114), pp. 680-686

[15] Chen, X. ’’Theoretical and experimental study of flow through the double cas­cade of a Francis turbine”, Dr. Ing. thesis 1992:81, Department of Hydro and Gas Dynamics, The Norwegian Institute of Technology, Trondheim, Norway

[16] Crighton, D. G. ’’Nonlinear acoustics of bubbly liquids” CISM Courses and Lectures #315 Int.center for Mech. Sci. , in ’’Nonlinear waves in real fluids” ed. Kluwick, A., Springer-Verlag 1991

[17] Ellis, E. T. ” Production of accelerated cavitation damage by an acoustic field in a cylindrical cavity”, The Journal of the Acoustical Society of America, vol. 27 iss. 5, 1955.

[18] Farhat, M. ” Contribution a F etude de l’erosion de cavitatiomMecanismes hy- drodynamiques et prediction”, Thesis no. 1273 (1994), Ecole Poly technique Federate de Lausanne ( TL)

[19] Gindroz, B. ’’Lois de sirmitude dans les essais de cavitation des turbines frau­ds”, (in French) Thesis no. 914 (1991), EPFL, Lausanne

[20] Jernsletten, J. ’’Analysis of Non-Stationary Flow in a Francis Reversible Pump Turbine Runner”. Dr. Ing. thesis 1995:4, Department of Thermal Energy and Hydropower, The Norwegian Institute of Technology, Trondheim, Norway

[21] Keller, A. ’’The effect of flow turbulence on cavitation inception” (in German), Pumpentagung Karlsruhe 30 September-2 October 1996,

[22] Keller, A. ’’New scaling laws for Hydrodynamic Inception”, The 2nd Int. Symp. on Cavitation, April 1994, Tokyo, Japan, pp. 327-334

[23] Kjeldsen, M. ”Cavitation formed by tensile waves in hydrodynamic flows”. Pa­per F-6. IAHR WG On hydraulic machinery under steady oscillating conditions, 7th. int. meeting, 5-7 Sept. 1995, Ljubljana, Slovenia

102

[24] Kjeldsen, M.;Jernsletten, J. ” Cavity clusters in acoustic and flow systems” pp. 269-272. 2nd. Int. Symp. on Cavitation, April 1994, Tokyo, Japan

[25] Kjeldsen, M. ’’The optimal choice of runner inlet diameter for low specific speed Francis runners”, Internal report, Department of Themal Energy and Hydropower, Norwegian University of Science and Technology, Trondheim, Nor­way.

[26] Kjeldsen, M. ’’Experience from the construction and use of the blow down water tunnel”, Internal report, Department of Themal Energy and Hydropower, Norwegian University of Science and Technology, Trondheim, Norway.

[27] Knapp, Daily and Hammitt ” Cavitation” McGraw- Hill (1970)

[28] Knapp, W.; Schneider, C.; Schilling, R. ’’Experience with an acoustic cavitation monitor for water turbines”, Paper C453/048 IMechE 1992, Proc. of the IMechE 1992-11 ” Cavitation Int. conf.” pp. 271-275

[29] Kumar, S. ”Some theoretical and experimental studies of cavitation noise”, Thesis. Divison of Engineering and Applied Science, California Institute of Technology (1991). Also presented in J. Fluid Mech. (1993), vol. 253, pp. 565- 591; Kumar, S., Brennen, C. E. ” Some nonlinear interactive effects in bubbly clouds”.

[30] Lauterborn, W.; Philipp, A.;Ohl, C. D. ’’Non linear dynamics of bubbles”, Proc. pp. 121-124 of th 15th International Congress on Acoustics, Trondheim, Norway, 26-30 June 1995

[31] Liu, Z.; Brennen, C. E. ’’Models of cavitation event rates”, Proc. CAV95 Int.Symp. on Cavitation, Deauville, France, May 1995, pp. 321-328

[32] Lund, E. ’’Hydrauliske trykksvingninger i omdreiningshuhommet mellom lppe- hjul og ledeapparat i en Francis turbin” (in Norwegian), Lisentiatarbeid 1966, The Norwegian Institute of Technology

[33] McNabb, J. Y.;Desbiens, E. ”An overview of recent progress in the design analysis and cavitation performance of Francis runners”, Int.symp. on cavita­tion, CAV’95, May 1995, Deauville, France, pp. 37-42

[34] Morse, P. M. & Ingard, K. U. ’’Theoretical acoustics”, 1968, McGraw-Hill

[35] March, K. A. ”On cavity cluster formation in a focused acoustic field”, J. Fluid Mech., vol. 201, pp. 57-76

103

[36] Mprch, K.A. ’’The development of Cavity Clusters in Tensile Stress Fields”, in Meier, G.E.A.; Thompson, P.A. (eds.) ’’Adiabatic Waves in Liquid-Vapor Systems”, IUTAM Symp. Gottingen, 1989. Springer-Verlag 1990.

[37] Morch, K. A.; Song, J. P. ” Cavitation nuclei at solid- liquid interfaces” Paper C453/059 Proc. of the Inst, of Mech. Eng., Cavitation Int. Conf., Dec. 1992, Cambridge (UK)

[38] Plesset, M. S.; Prosperetti, A. ’’Bubble dynamics and cavitation”, Ann. Rev. Fluid Mech., 1977, pp. 145-185

[39] Written correspondance with Dr. Habil. Prof. J. Raabe, March 1996 (Registered at the University Library in Trondheim, NTUB)

[40] Reisman,G.;McKenney, E.;Brennen, C. ” Cloud cavitation on an oscillating Hy­drofoil”, 20th ONR Symp. on Naval Hydrodynamics, UCSB, Aug. 1994

[41] Rood, E. P. ’’Review- Mechanisms of Cavitation Inception”, Transactions of the ASME Journal of Fluids Engineering, vol. 113, June 1991, pp. 163-175

[42] Samuel Martin, C. ’’Keynote speech: Two-phase gas-liquid experiences in fluid transients”, 7th Int.Conf. on Pressure Surges and Fluid Transients in Pipelines and Open Channels (proc.ed.A. Boldy), BHR Group Harrogate 16-18 April 1996., pp. 65-83.

[43] Steller, J. ’’International cavitation erosion test- Test facilities and experimen­tal results”, 2eme Journees CAVITATION Societe Hydrotechnique de France, March 1992, Paris

[44] Stepanik, H. E. ’’Reversible Francis Pump turbinesrlmprovement of Part Load Performance by changes on the design of Impeller Blades”, Dr. Ing. thesis 1991, The Norwegian Institute oi Technology, Trondheim, Norway.

[45] Stokkebq, O. et al. ’’Bekkeinntak pa kraftverkstunneler” (in norwegian), report 5242, Vassdragsregulantenes forening (EnFO)

[46] Svingen, B. ’’Fluid structure interaction in slender pipes”, 7th Int. Conf. on Pressure Surges and Fluid Transients in Pipelines and Open Channels (proc.ed. A. Boldy), BHR Group Harrogate 16-18.April 1996, pp. 385-399.

[47] Trevena, D. H. ’’Cavitation & Tension in Liquids”, 1987, IOP Publishing

[48] Varga, J. J.; Bata, T.; Sebestyen, G. Y. ’’Investigation on cavitation erosion in rotary disc equipment- and some results”, pp. 694-704

104

[49] Wang, Y.C.; Brennen, C.E. ”Shock wave development in the collapse of a cloud of bubbles”, FED-Vol. 194, Cavitation and Multiphase Flow ASME 1994

[50] White, F. M. ’’Fluid Mechanics” 2nded., McGraw-Hill 1986

[51] White, F. M. ’’Viscous fluid flows”, McGraw-Hill 1970; 2”ded. McGraw-Hill 1991

[52] Young, F. R. ’’Cavitation”, 1989, McGraw-Hill Book Company (UK)

[53] Zhang, S.; Duncan, J. H.; Chahine, G. L. ’’The final stage of the collapse of a cavitation bubble near a rigid wall”, J. Fluid Mech. 1993 (257), pp. 147-181

105

106

Appendix A

Bubble dynamics

The equation of single bubble dynamics has been deduced numerous times in text­books. The equation is central in the field of cavitation and is of importance for this work such that the deduction is given.

Consider a single bubble in the bulk of a liquid, at a given time a balance of forces acting on a perfect spherical bubble is taken. Using symmetry it is found that

{Pb — Poo) tt-R2 = S ■ 2xR (A.l)

Pb Poo

Allowing the bubbles surface having a radial velocity component R, neglecting the impulse of the mass within the bubble, the continuity and momentum relations for the liquid mass displaced by the bubble wall spherical movements are calculated. Viscous and compressible terms are neglected:

dp dpu 2 pitM+-s? + ~r 0

du du 1 dp

Boundary conditions at bubble wall, r=R, can be found.

p(r = R) = pb - 2S R

Which is the equilibrium condition found in Equation A.l. Further at infinity

p(r -*• oo) = poo

The first approximation is to take p =const. which indicates an incompressible liquid. From continuity:

dudr

2 it----- => u =

rkonst.

107

Inserting into the linear momentum equation and perform

r—>oc r—>00

Inserting the Emits the result is the equation describing the dynamics of bubble

_ 3 2 1RR+- R = -2 p

Pb — Poo ~ 2 R(A.2)

Now pb is said to consist of the partial sums of vapor pressure and non-condensible gas given as:

M SPb — Pv + ^3k — Poo + 2^

This equation can be studied when applying a perturbation AR to the bubble wall in the driving term. The intension is to see whether R will continue to grow due to the perturbation or if it resists its original shape.

d_OR Pb —3k

MR3k+1

—3kPoo-Pv

RSR2

Solving the inequality:

_3k^-(3.+ 1)|>0

gives the critical size of a cavitation nucleus

■Rc > —-—3/v pv Poo

Which gives the cavitation threshold for an initial microbubble consisting of vapour and gas.

In order to calculate the governing equations in this presentation MATLAB Simulink was used. This program is based on the generation of ” blocks” which clearly show ’’the route of calculation”. Thus Equation A.2 is ’’written” as seen in Figure A.l.

Where the content of the sub-blocks are according to the equation.

108

Calculation of gov.bub.dyn.eq.

juVt.5

Product Integrator Integrator! Radius bubble

Vap pres

Pres Non-Cond gases

Tension

Info. Initial cond.

Ambient pres.

Figure A.l: Copy of display from the symbolic numerical calculation program Simulink. The governing equation for single bubble dynamics is solved according to the ’’expression shown”.

109

A.l Small perturbation oscillation of gaseous bub­bles

Following Plesset and Prosperetti [38] p.147 the natural frequency of an oscillating bubble can be deduced. The basic equation reads, where the viscous term has been added as a boundary condition:

(A.3)

Where the following is assumed considered small perturbations:

Poo (*) = Poo (1 + £ cos art) (A.4)R = Rq [1 + x (£)]

Pi{t) = Pi,eq + — |x=0,i=0 X + -g-r \x=0,i=0X

Inserting Equation A.4 into Equation A.3 neglecting higher order terms of x and x. The equation describing the dynamic of a bubble reads:

x +2(3 x +ujqX = —£Q?e*u'

R = -2 p

Pi Poo2S R

Wdiere

2 _ 16b, 23^ - -par

Assuming the gas follows a polytropic law of compression, gas compression is in-phase with geometric displacement= R0x (t), i.e.:

Pi = Pi,eqRo~R

3k

If further viscosity terms are neglected the term (3 will vanish and a resonance occurs for the value:

Pi,ea 23"3 pag (A.5)

If k = 1 and Ro = Rc , making pi?eq = Pa-u +'^-pv and p^t -pv = it isfound that:

IVn = 0

110

It can be shown that the resonance frequency is high until very close to critical radius for a non-perturbated nucleus. To demonstrate this, the critical radius is tabulated for ambient pressures and resonant frequencies for gaseous nuclei of re­spectively 0.9 and 0.5 of the critical radius. For all T = 293 K. Regarding dynamic pressure perturbation it seems pertinent for most hydrodynamic flows to solely con­centrate on the lowest absolute pressure and neglect the dynamic response.

Table A.l: Resonance frequencies for bubbles of fractional sizes of the critical radius at the given pressure. The resonance frequency for critical radius is identical to zero.

Poo[kPa] Rc[pm\ /o = ^R-l SR‘'l \kHz] /o = [kHz]-20 0.225 1100 5800-2 2.38 88 4700 4.5 33 1801 8.3 13 721.5 14.3 6.0 332 50 0.91 4.92.1 100 0.32 1.7

A.2 Boiling versus cavitationPlesset and Prosperetti [38] also give a deduction of whether thermal energy is of importance for the bubble dynamics.

The energy balance to fill a bubble with vapor, can be expressed as the energy transfer due to evaporation:

15 = (T) (A.6)

R equals the radius of the bubble (final stage), L is the latent heat of evaporation and pi is the density of the vapor phase at a given temperature. The necessary energy is made available through a temperature drop AT in the ambient liquid layer due to diffusion of heat. The thickness of the layer is given by the diffusion length defined as (Dt)1 where D = — =thermal diffusivity pcp the heat capacity ^ —

while k is thermal conductivity making the dimension of D equal toThe energy withdrawn from the layer is therefore:

m3K

m2s

E = AtvR2 (Dt)* pcpAT (A.7)

111

The necessary energy and the available energy must be equal, hence an expression for the temperature drop is obtained:

A T w RLpev (T) 3 (Dt)* pcp

(A.8)

If the temperature drop makes a substantial change of the corresponding vapor pressure the event is referred to as boiling. It is noticed that AT oc t-0"5 such that if the pressure pulse perturbation is sufficiently rapid the maximum radius of single bubbles will be restricted hence for a confined cavitation susceptible region experiencing a pressure pulse the formation of several bubbles will be a more likely outcome.

In the further discussion by the cited authors they deduce asymptotic values for growth rate of an inertia controlled cavitation bubble, assuming pv = const.

Rinerta—

2 Pv Poo \3 p J

And a corresponding maximum bubble size, if the initial pressure perturbation making the bubble reach non-stable growth was an instant pressure drop:

Rmax 1 + -p Rq (Poc — Pv)

For a thermal controlled growth:

R«- ^J Lp; (T) (Dt)j

Tb is the temperature at the bubble wall. An equation that can be integrated once giving R a f05. If Rthermai< Rinerta, the bubble growth is referred to as boiling and vice versa as cavitation.

A.3 Definition of a cavitating mediumThis deduction is given by Mprch [35].

The critical size of a nucleus is found to be:

3 Pv Pcrit

The density of a cavitating medium may be defined as:

P — Pliq (1 0) "t" Pvap0

(A.9)

(A. 10)

112

Further the sonic speed, c, in the medium can be expressed as:

dp 111 . d/3— — — — (I-/?) — + ----- H {Pvap ~ Pliq)dp dp (ATI)

L'0 ^vapIf the nuclei of radius R are homogeneously distributed and assumed to be fixed

in a matrix-like structure say a hexagonal close packed (HCP), the void fraction can be written as:

0 = 9 ( T (A.12)

Where q = 4is the (HCP) matrix constant, and 1 is the intercavity distance. The governing Equation A.2 can be written, if initially at rest, using pg — GT —

GT 22

And similar for the critical condition:GT

Pcrit — j-,3 d* Pv

R

22

(A.13)

(A.14)RL, ' " %

Evaluating GT from Equation A.13, pv from Equation A.9 and substituting it into Equation A.14 the following equation is obtained:

P'r*-p=-iSR'\jp ~2S[r-r:

Thus equals to:

S=2s(b 5RA

(A. 15)

Which further allows ^ in Equation A. 11 to be expressed as :

d(3 d(3 dR dp dR dp

Which makes, using Equations A.15 and A.12:

— — — (I - P) — + (3—----- 1" (Pliq — Pvap)& Ct Cdp eg

For which it is noticed that

vap R 2S (l? ~

—r —> OO C2 R-^Rc

Thus a cavitating medium is defined as a bulk of volume where pressure propa­gation has no meaning.

113

114

Appendix B

Rayleigh instabilities at liquid-vapor interface

Carey [12] decomposes the problem into mean and perturbate components:

u = u + u';w = w + w'; P = P + P'

Where || = ||' = w = 0. The continuity and impulse equations can now be written as (incompressible):

dv! dw’+ 97 = 0

(dv! _8u’\ OPp\W + ua;) “ “a7(dw' _dw'\ dP'

= -dT

(i)

(ii)

(hi)

Derivation of (ii) and (in) with regard to x and z respectively, adding the two equations and substituting with (i) gives:

d2P' d2P' dx2 dw2

Carey further postulates the following functional relationships:

/ = g(z,t) = Ae'°=+# w' (x, z, t) = w(z)eiax+l3tf(z,z,t) = P(z)e™+#

(13.1)

(B.2)

The general solution of Equation B.l using Equation B.2 is

(B.3)

115

For vapor and liquid phase respectively.For the non-perturbed state it is distinguished from Carey by inserting:

Pv = Po (B.4)Pi = Po + f{x)z

Where hydrostatic pressure variation is neglected and that f (x) consists of the pressure gradient (x) set up by the convective flow, 1 is the length of the column which is compressed and e is a weighting parameter for the inclusion of tension and expenasion of the liquid column.1 Both the gradient and the compression will have a restoring effect on the interface.

Further by inserting functions of w' and P' into Equation iii, the following ex­pressions are found:

fyz(z) = -aai[pi(f3 + iaui)} 1 eaz (B.5)wv(z) = —aav \pv ((3 + eaz

By using the continuity equation (i) v! takes the value:

a \dz J

Thus Equations i, ii and iii are satisfied. Proceding by establishing appropriate boundary conditions at the interface, w' must satisfy:

, #w‘i=m+udi

Equation B.5 then reads:

aav [pv (/3 + mitt,)]-1 = /3A + iauvA —aai \pi (f3 + mil;)]-1 = f3A + iauiA

Which is solved for av and ap

av = — (f3 + iauv)2 A (B.6)a

ai = —— ((3 + ioiui)2 AOt

At the interface surface tension is included (Youngs-Laplace equation):

Pz-P^trfl + l) (B.7)rg/

^Note: Carey [12] explicitly states that his deduction is based on an incompressible fluid.

116

Here:dx2 and — = 0

r2(B.8)

\_n _ 3

P = P, + and P, = (B.9)Now substituting Equations B.9 and B.8 into Equation B.7. Equations B.4, B.3

and B.2 substitute the pressure terms and the following equation is obtained:

at — av= [/ (x) + rra2] A

Finally by using Equation B.6 a fundamental relation between 3 and a is obtained:

= {a2 Pipy (Ui - uvf - [cm3 + / (x) a] (p, - pv)} 2 _ ia {pm - pvuv) ^(pi + Pv) (Pi + Pv)

Thus before proceeding it is noticed that 3 has an imaginary term and that the instabilities will grow if the real part is greater than zero. Further the frequency of oscillations is not dependent on the amplitude of the distortion (A).

117

a)This thesis work

u

A(n=1)

rof ileb)Carey

Figure B.l: Discrepancies between the model in this work (a) and the one by Carey (b). Lif measures from detachment point to attachment point along the interface, the curvature is not taken into account. Aq measures from the surface of the profile in (a) toward the non-perturbed interface (solid line).

118

Appendix C

Digital analysis

The discretization of the voltage signal is 10 V/4096, though the lowest recordable fluctuation is 0.00241/.

The spectrum analysis uses the following basic relations:

S„ = ±jF(X)l2ns

Where Sxx is the power density and ns is the number of samples and F (x =

Xk defined as:ns—1

^ At, A: = 0,^ - 1i=0

Where At = j-, fs being the sampling frequency, z,being the sampled voltage reading and j = y/—l.

119

120

Appendix D

Analysis of wake downstream of a guide vane

This section supports the principles which the numerical examples in Sections 2.2.1 and 2.2.1 are based on. The examples, and the discussion preceding those, are crucial in order to make the presence of sound waves of substantial amplitudesprobable in hydraulic machinery. Waves which subsequently are able to nucleate severe erosion. Refer to Figure D.1 for illustration of problem and definition of variables /constants.

In the vaneless interspace the following relations are supposed to hold (impulse and continuity). Note subindexes x and z are tangential and radial directions re­spectively.

vxr — const.

=

In order to track the fluid particles flow path, s, from the guide vane exit to the runner inlet the following relation is proposed to hold:

ds = (dr)2 + (rd4>)2

Where drand d<pare expressed as:drdt d&dt

ds can then be expressed as:

= —vz

— Vx

121

Guide vanesCenter of wake

Flow direction

Figure D.l: Definition of parameters used in the text. Illustration of problem.

= \ cow,st.2 +Q \2dt

2irb ) r

Thus if r can be expressed as a function of t the expression can be integrated using proper limits. It is noticed that only the radial flow vr makes variations in r hence the following substitution is made:

dt = —dr2%rb~Q~

Making the flow path length s equal to (subindexes 0,1 refer to the guide vane exit and the runner inlet respectively):

s

const'2 + (&) 17 tri - r°l

(D.l)

(D.2)

The constant is found for high-head Francis turbines to be (vix = 0.67\j2gH accordin to the Euler equation):

const. = 0.67 \/2gHri

122

E.g. for the numerical example provided in Section 2.2.1 s = 5.6 • [r% — ro]. Basic wake relations according to White [51] pp.478-481, which serve as the base

for calculations in Section 2.2.1, are assumed to hold for a plane turbulent wake.

Av Aumax

Where 1/1/2 and A[/max are defined as:

m/2 =: 0.30(rf)1/2/<n1/2

Avmax % 1.63 ( - j

Where 9 is the momentum thickness of the wake defined as:+00

^ dy = const.

el /

I.e. independent of x.For the numerical Example 11 in Section 2.2.1 Ai;maxcan be postulated while x

is set equal to s. Thus the momentum thickness is found to be:

6 =Anmax

1.63

Subsequently 1/1/2 can be found and hence the profile of the velocity distorton can be found.

Considering the validity of this approach it must be remembered that in the vane­less interspace there is an accelerated flow both in tangential and radial direction. (Refer to the presented discussion in Example 10 on Page 26). The measurements done by Chen [15] clearly shows the diminishing of the wake downstream the sta­tionary guide vanes. The presented measurements are time averaged such that no information of large vortex formation is revealed. White [51] states that a lifting surface will produce eddies. Also the scope of Chen’s study [15] was to investigate the leackage flow from the suction side to the pressure side of the guide vane, an effect which will contribute to the overall non-uniformity of the flow entering the runner.

123

4.# yekwifr ObMWMu# 3 West Aht* Use TraStrig nt tlx- C;#*3f VjmM 5.S«stiaa ?, see f%, -Lt3, Y-dtre-x^ise Kef. F'% M?t

M* 4?i vetoed? sr« Alter titsTmw Wf* oTlSe W* ***** ***#»*< wA* XlT. VxWmtAm W, 3-35)

fee. 43& Veirseil* Stostrifotiticn 25 mm AA*r tbe UtMtlrZ W# *)x lisMe V*im h-r<iteti &■«* 3 %. ,3.3%, Y-dtiXiSsK Be?,

M* 433 PmM#W*W 3% mss *tw t$wTrail®* K<Se* of «ti*f timie Yams SSeesssswi W. we f«. 3A3x*m%em Ret F* llii

Figure D.2: Diminishing of the wake downstream of a stationary guide vane, (reprinted from Chen [15] ,p. 90)

124

Appendix E

The cavitation systems at the Hydropower Laboratory

125

1600

0

Atmosphericexposure

Upper40000

Butterfly valve Shut—off valveSluice—valve/

control—valve6000

Watertunne5000

5000

Rump:<4 ^ H=16m

Q=450l/s10000

Figure E.l: Outline of the Hydropower Laboratory. Measures in ram.

126

S+illi

sc t©e n s

1 2 Profi

Figure E.2: Schematic view of the water tunnel constructed in this thesis work and mounted within the permanent installations at the Hydropower Laboratory. Measures in mm

127

159(100%=163)

Figure E.3: The NACA4412 profile used in the water tunnel at the Hydropower Laboratory. The profile is handcrafted out of a Plexi-glass material. The width of the profile is 130mm. The supporting holes are indicated (each 40mm deep). Measures in mm.

128

Sealed enclosure

Loudspeaker UW30

/— Electrical conn.

about 750

0130

Stilling j tank 01000

Figure E.4: The mounting of a underwater loudspeaker in the stilling tank of the water tunnel at the Hydropower Laboratory. Measures in mm.

129

18i.3kW

V//////A \//////////////>

77A^/^r//////////////>/777A^

Water)inducers

0320,Specimens

Figure E.5: Schematic view of the rotating disc system utilised for the studies presented in this work. Water is fed through four symmetrical positioned inlets at a diameter of about 70 mm, while there is one single outlet at the back of the disc. Measures are in mm.

130

Figure E.6: Picture of the rotating disc apparatus at the Hydropower Laboratory.

131