Cavitation due to Rarefaction Waves and the Reflection of Shock Waves from the Free Surface of a Liquid

Transcription

1 Cavitation due to Rarefaction Waves and the Reflection of Shock Waves from the Free Surface of a Liquid Justin Sam A Dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Master of Science in Engineering.

2 ii Declaration I declare that this dissertation is my own, unaided work, excet where otherwise acknowledged. It is being submitted for the degree of Master of Science at the University of the Witwatersrand and has not been submitted before for any degree or examination in any other university... Justin Sam. Date

3 iii Abstract Cavitation was generated in ta water samles by the transmission of tension waves into the liquid, using a hydrodynamic shock tube. The liquid cavitated at absolute negative ressures of about - bar. Simulations of bubble resonses showed qualitative agreement with exerimental observations of oscillatory growth and collase cycles. Pressure records showed secondary ressure ulsations, confirming the oscillatory nature of the collase at each rise in ressure. More quantitative comarison of theory and hotograhic records would require a camera with a higher cature rate. Exeriments using another exerimental facility involved liquid comression waves with eak static ressures of u to about MPa, which were transmitted from a conventional gas shock tube into a liquid section and were intended to be reflected at the free surface as exansion waves. These exeriments were unsuccessful in roducing absolute negative ressures or cavitation that was visible through an otical observation section. This was attributed to transition layer effects and ulse attenuation, which contributed to lowering of the eak negative ressure behind the exansion wave, as well as the deth of the transducer and observation section below the free surface, which may have been too low for the eak tension to be suerimosed on the lower ressure behind the incident comression wave. Pressure records suggested that, for lower driver ressures, cavitation occurred below the observation section, although this could not be verified otically.

4 iv For long you live and high you fly but only of you ride the tide and balanced on the biggest wave you race towards an early grave - Roger Waters

5 v Acknowledgements The author is grateful for the valuable inut of Professor B. W. Skews, whose guidance and assistance, throughout this roject has been indisensable. The author is also indebted to his mother, whose sacrifice, though not often soken of, is always acknowledged.

6 vi Table of Contents Title Page i Declaration ii Abstract iii Acknowledgements v Table of Contents vi Table of Aendices x List of Figures xi List of Tables xv Nomenclature xvi. Introduction.. Definition of Cavitation.. Tyes of Cavitation.3. Effects of Cavitation 3.4. Uses of Cavitation 5.5. The Study of Cavitation Producing Controlled Cavitation Definition of the Cavitation Threshold and Detectable Sizes Detection and Visualisation 9.6. The Purose of this Work 0. Shock Waves and Rarefaction Waves.. Comressible Flow.. The Conservation Laws for a Shock Wave.3. General Equations 4.4. Shock Waves and Rarefaction Waves in Gases General Equations for Gases Sound Waves in Gases Finite, Isentroic Waves in Gases Shock Waves in Gases 8.5. Shock Waves and Rarefaction Waves in Liquids 9

7 vii.5.. General Equations for Liquids Sound Waves in Liquids.5.3. Shock Waves in Liquids.5.4. Finite Waves in Liquids.6. The Shock Tube 3.7. Methods of Analysis The Wave Diagram The Pressure-Velocity Diagram 7.8. Shock and Rarefaction Wave Interactions Interactions of a wave with walls or contact surfaces Reflection from a Rigid Wall Shock or Rarefaction Wave Passing Through a Discontinuous Change in Cross-section Free Surface Reflection Cavitation Liquids Under Negative Pressure Cavity Dynamics of Bubbles Initiated in a Stretched Medium Nucleation Cavitation Nuclei Miscible Liquids and Dissolved Solids Non-miscible Liquids and Undissolved Solids Dissolved Gas Undissolved Gas and Vaour Bubbles The Statistical Nature of Cavitation Sherical Bubble Dynamics The Rayleigh-Plesset Equation The Resonse of Bubbles to Rarefaction Waves Collase of Cavitation Bubbles Literature review Pulse Reflection Exeriments Exlosions Bullet-Piston (B-P) Methods 50

8 viii Pulsed Electron Beam Generator Illustrative Case Intense Cavitation Tube Arrest Methods Hydrodynamic Shock Tube Methods Ultrasonic and Wave-Focussing Methods Gas Nuclei Studies The Effect of Precomression The Effect of Evacuation Reeated Cavitation in a Single Samle The Initial State of Samles of Liquid Water Solid Nuclei Studies Cavitation Thresholds Liquid Purity Rate of Stressing Transition Layer Measurement of Tensile Strengths Overlaing of Tension Pulses Exerimental Facilities and Methods The Mach 3 Shock Tube Instrumentation High-Pressure and Vacuum sulies Flow Visualisation The Mach Shock Tube Instrumentation The High-Pressure Suly The Liquid-Gas Interface The Diahragm Bursting Mechanism Exerimental Procedure The Mach 3 Shock Tube The Mach Shock Tube 85

9 ix 6. Exerimental Observations and Results The Mach 3 Shock Tube Pressure Transducer Records Photograhic Records The Mach Shock Tube 9 7. Theoretical Analysis The Mach 3 Shock Tube Wave Diagram Pressure-Velocity Diagrams Bubble Dynamics Simulations Comutational Fluid Mechanics 7.. The Mach Shock Tube 7... Wave Diagram 7... The Area Change The Motion of the Gas-Liquid Interface The Effect of Pie Wall Elasticity Streeter s Method Method of Characteristics Elementary Calculation 8. Discussion The Mach 3 Shock Tube The Mach Shock Tube 3 9. Conclusions Recommendations Imrovements to the Mach 3 Shock Tube Imrovements to the Mach Shock Tube Further Studies 4. References 45

10 x Table of Aendices Aendix A: Entroic Relations for Arbitrary Gas States 6 Aendix B: Common Relations Across a Liquid Shock 63 Aendix C: Wave Diagram Considerations 66 Aendix D: One-Dimensional Interaction Problems 69 Aendix E: Collision and Reflection of Waves from a Rigid wall 7 Aendix F: Interaction of a Shock or Rarefaction wave with a Discontinuous Change in Cross-Section 73 Aendix G: Physical and Thermodynamic Proerties of Various Substances 76 Aendix H: Summary of Exerimental Cavitation Thresholds Values 80 Aendix I: The Initial States of Samles of Liquid Water 86 Aendix J: Assembly Drawings of the Mach 3 and Mach Shock Tubes 88 Aendix K: Transducer Secifications and Gauge Factors 9 Aendix L: The Mach 3 Shock Tube: Additional Pressure Records 94 Aendix M: The Mach 3 Shock Tube: Photograhic Records of the Lower Section of the Tube 96 Aendix N: The Mach 3 Shock Tube: Photograhic Records of the Lower Section of the Tube 0 Aendix O: The Mach 3 Shock Tube: Photograhic Records of the Free Surface of the Liquid Column 06 Aendix P: The Mach Shock Tube: Additional Pressure Records Aendix Q: Values of State Parameters in the Regions of the Wave Diagrams of the Mach 3 and Mach shock tube 5 Aendix R: Results of Simulations of Bubble Resonse to Alied Pressure Variations 8 Aendix S: Overview of Cavitation Models Emloyed by Available CFD Codes 9 Aendix T: The Taylor Analysis of a Finite, Non-rigid Plate Subjected to a Square Pressure Wave 3

11 xi List of Figures Figure.. The hase diagram of water. 3 Figure.. Moving and fixed shock reference frames. Figure.. Required diahragm ressure ratio for generating shock of strength γ in shock tube with air in the driver section and air, helium and hydrogen in the driver section. 4 Figure.3. Required diahragm ressure ratio for generating shock of Mach number M s in shock tube with nitrogen in the driver section and helium and nitrogen in the driver section. 4 Figure.4. The wave diagram resulting from diahragm ruture in a shock tube. 5 Figure.5. Collision of a wave with a contact surface: initial and final conditions. 8 Figure.6. Resultant absolute ressure vs. time at some deth below the free surface. 3 Figure 3.. Cavitation robability as a function of transducer driving voltage. 4 Figure 3.. A sherical bubble in an infinite liquid. 4 Figure 3.3. Regions of the P-V o lane. 46 Figure 4.. Cavitation zone develoment due to underwater exlosion of a. g charge near the free surface of a liquid. 50 Photograhs of cavity clusters that develoed in distilled water samles tested Figure 4. immediately (uer frame) and one hour after (lower frame) being laced into a cuvette. 53 Figure 4.3. Sequence of x-ray frames of cavitation zone dynamics resulting from shock wave reflection at a free surface. 54 Figure 4.4. Process of intense cavitation in a liquid dro. 55 Figure 4.5. Cavitation develoment in a tube due to downward acceleration of a cavitation tube. 59 Figure 4.6. Side view of test section, showng the result of roagation of an evaoration wave through a suerheated refrigerant samle. 6 Figure 5.. Photograh of the Mach 3 shock tube. 76 Figure 5.. Photograh of the Mach 3 shock tube test (driver) section. 77 Figure 5.3. Photograhs of the Mach shock tube. 8 Figure 5.4. Photograh of the diahragm bursting mechanism of the Mach 3 shock tube. 83 Figure 6.. Pressure trace from test using the Mach 3 shock tube. 87 Figure 6.. Cavitation observed at the bottom of the Mach 3 shock tube. 89 Figure 6.3. Cavitation observed at the middle section of the Mach 3 shock tube. 9

12 xii Figure 6.4. Cavitation observed at the uer section of the Mach 3 shock tube. 9 Figure 6.5. Pressure trace from test using the Mach shock tube. 93 Figure 7.. Wave diagram of the Mach 3 shock tube. 98 Figure 7.. Pressure-velocity diagram for the collision of the exansion waves with the gasliquid interface. 00 Figure 7.3. Pressure-velocity diagram for the collision of the exansion waves with the base of the Mach 3 shock tube. 0 Figure 7.4. Matlab Simulink model of the Rayleigh-Plesset equation 04 Figure 7.5. Variation of bubble radius with time for growth of bubble (R o 0.5 mm) subjected to ste-function ressure dro from 7.93 to - bar at time zero. 05 Figure 7.6. Variation of bubble wall velocity with time for growth of bubble (R o 0.5 mm) subjected to ste-function ressure dro from 7.93 to - bar at time zero. 05 Figure 7.7. Variation of bubble wall acceleration with time for growth of bubble (R o 0.5 mm) subjected to ste-function ressure dro from 7.93 to - bar at time zero. 06 Figure 7.8. Regions of the P-V o lane (initial ressure 7.93 bar, surface tension J.m - ) 07 Figure 7.9. Variation of bubble radius with time for collase of bubble (R o 0.5 mm) subjected to ste-function ressure rise from - to bar at time zero. R(0)=mm, R & (0)=8.3 m.s Figure 7.0. Variation of bubble wall velocity with time for collase of bubble (R o 0.5 mm) subjected to ste-function ressure rise from - to bar at time zero. R(0) = mm, R & (0) = 8.3 m.s - 09 Figure 7.. Variation of bubble wall acceleration with time for collase of bubble (R o 0.5 mm) subjected to ste-function ressure rise from - to bar at time zero. R(0) = mm, R & (0) = 8.3 m.s -. 0 Figure 7.. Wave diagram of the Mach shock tube (gas driver section: helium at bar, gas driven section: air at 0.83 bar). 3 Figure 7.3. Results from Chester-Chisnell-Whitham analysis and the quasi-steady analysis for the de-amliification of a shock with initial Mach number M Si. 6 Figure 7.4. Results of simulations [46] of roagation of shock wave (M S = 3) through a discontinuity with an area enlargement ratio of /0 at the instant 0. ms. 7 Figure 7.5. Results of analysis of the liquid-air interface using Taylor s theory. 9 Figure 7.6. Wave diagram of uer, liquid-filled section of the Mach shock tube, including the effects of ie stretch. Figure 8.. Suggested ressure rofile of the first comression wave transmitted into the liquid. 34 Figure D.. Collision of waves: initial and final conditions. 69

13 xiii Figure D.. Overtaking of one wave by another: initial and final conditions. 70 Figure F.. Possible steady flow wave atterns for the assage of a shock wave through a small area enlargement. 73 Figure F.. Incident, reflected and transmitted waves in the interaction of a wave with an area enlargement discontinuity. 75 Figure I.. Histograms of nuclei number densities N o in untreated, degassed and filtered ta water. 87 Figure J.. Assembly drawing of the Mach 3 hydrodynamic shock tube. 89 Figure J.. Assembly drawing of the liquid test section of the Mach shock tube. 90 Figure J.3. Assembly drawing of the modified gas driven-section of the Mach shock tube. 9 Figure K.. Drawing of the PCB transducer, model 3A 9 Figure L.. Pressure trace from test using Mach 3 shock tube; Samling rate MHz, 0000 data oints. 95 Figure L.. Pressure trace from test using Mach 3 shock tube; Samling rate MHz, 0000 data oints. 95 Pressure trace from test using Mach shock tube; Driver section filled with Figure P.. helium at bar; Samle rate 00 ks/s, Trigger level +0. V, source CH- (transducer ), osition 4 V, delay 0. 3 Pressure trace from test using Mach shock tube; Driver section filled with Figure P.. helium at 8 bar; Samle rate 500 ks/s, Trigger level +0. V, source CH- (transducer ), osition 0%, delay 0. 3 Pressure trace from test using Mach shock tube; Driver section filled with Figure P.3. helium at 8 bar; Samle rate 00 ks/s, Trigger level +0. V, source CH- (transducer ), osition 0%, delay 0. 4 Pressure trace from test using Mach shock tube; Driver section filled with Figure P.4. helium at 8 bar; Samle rate 00 ks/s, Trigger level +0. V, source CH- (transducer ), osition 0%, delay 0. 4 Figure R.. Variation of bubble radius R with time for growth of bubble (R o 0. mm) subjected to ste-function ressure dro from 7.93 to bar at time zero. 9 Figure R.. Figure R.3. Figure R.4. Figure R.5. Variation of bubble wall velocity R & with time for growth of bubble (R o 0. mm) subjected to ste-function ressure dro from 7.93 to bar at time zero. 9 Variation of bubble wall acceleration R & with time for growth of bubble (R o 0. mm) subjected to ste-function ressure dro from 7.93 to bar at time zero. 0 Variation of bubble radius R with time for growth of bubble (R o µm) subjected to ste-function ressure dro from 7.93 to bar at time zero. 0 Variation of bubble wall velocity R & with time for growth of bubble (R o µm) subjected to ste-function ressure dro from 7.93 to bar at time zero.

14 xiv Figure R.6. Figure R.7. Figure R.8. Figure R.9. Figure R.0. Figure R.. Figure R.. Figure R.3. Figure R.4. Figure R.5. Figure R.6. Figure R.7. Figure R.8. Variation of bubble wall acceleration R & with time for growth of bubble (R o µm) subjected to ste-function ressure dro from 7.93 to bar at time zero. Time taken t v for bubbles of different initial radii R o to exand to visible size R V in adiabatic and isothermal growth cases (R V was taken as mm). 3 Variation of bubble radius R with time for growth of bubble (R o 0 µm) subjected to ste-function ressure dro from 7.93 to 339 Pa at time zero. 3 Variation of bubble wall velocity R & with time for growth of bubble (R o 0 µm) subjected to ste-function ressure dro from 7.93 to 339 Pa at time zero. 4 Variation of bubble wall acceleration R & with time for growth of bubble (R o 0 µm) subjected to ste-function ressure dro from 7.93 to 339 Pa at time zero. 4 Variation of bubble radius R with time for collase of bubble (R o 0.5 mm) subjected to ste-function ressure rise from - to bar at time zero. R(0) =.5 mm, R & (0) = 8.3 m.s -. 5 Variation of bubble wall velocity R & with time for collase of bubble (R o 0.5 mm) subjected to ste-function ressure dro from - to bar at time zero. R(0) =.5 mm, R & (0) = 8.3 m.s -. 5 Variation of bubble wall acceleration R & with time for collase of bubble (R o 0.5 mm) subjected to ste-function ressure dro from - to bar at time zero. R(0) =.5 mm, R & (0) = 8.3 m.s -. 6 Variation of bubble radius R with time for collase of bubble (R o 0.5 mm) subjected to ste-function ressure rise from - to 6 bar at time zero. R(0) =.5 mm, R & (0) = 8.3 m.s -. 6 Variation of bubble wall velocity R & with time for collase of bubble (R o 0.5 mm) subjected to ste-function ressure dro from - to 6 bar at time zero. R(0) =.5 mm, R & (0) = 8.3 m.s -. 7 Variation of bubble wall acceleration R & with time for collase of bubble (R o 0.5 mm) subjected to ste-function ressure dro from - to 6 bar at time zero. R(0) =.5 mm, R & (0) = 8.3 m.s -. 7 Variation of bubble radius R with time for collase of bubble (R o 0. mm) subjected to ste-function ressure rise from - to bar at time zero. R(0) = mm, R & (0) = 8.3 m.s -. 8 Variation of bubble radius R with time for collase of bubble (R o 0.05 mm) subjected to ste-function ressure rise from - to bar at time zero. R(0) =.5 mm, R & (0) = 8.3 m.s -. 8

15 xv List of Tables Table G.. Proerties of liquid water. 76 Table G.. Proerties of air (equivalent). 77 Table G.3. Proerties of helium gas. 78 Table G.4. Proerties of Polycarbonate (Grade: Makrolite 303). 79 Table G.5. Proerties of insertion rubber. 79 Table H.. Tensile strength values recorded from ulse reflection exeriments. 80 Table H.. Tensile strength values recorded from tube-arrest exeriments. 8 Table H.3. Tensile strength values recorded from Berthelot tube exeriments. 8 Table H.4. Tensile strength values recorded from centrifugal stressing exeriments. 83 Table H.5. Tensile strength values recorded from suerheating exeriments. 83 Table H.6. Tensile strength values recorded from inclusion exeriments. 84 Table H.7. Tensile strength values recorded from ultrasonic exeriments. 85 Table H.8. Tensile strength values recorded from other exeriments. 85 Table H.9. Values of the tensile strength of Mercury. 85 Table I.. Summary of exerimentally obtained values of the bubble radius. 86 Table I.. Summary of exerimentally obtained values of bubble concentration. 86 Table K.. PCB 3A: ressure transducer erformance. 9 Table Q.. Values of the arameters in the regions of the wave diagram of the Mach 3 shock tube. 7 Table Q.. Values of the arameters in the regions of the wave diagram of the Mach shock tube. 7 Table R.. Results of simulations using Simulink Adiabatic case (R V taken as mm). Table R.. Results of simulations using Simulink Isothermal case (R V taken as mm).

16 xvi List of symbols a Seed of sound m.s - Â Non-dimensional form of a -- A ÂA Cross-sectional area m A W Characteristic constant of a liquid medium Pa c Secific heat of a liquid kj/kg.k c Secific heat at constant ressure kj/kg.k c v Secific heat at constant volume kj/kg.k e Secific internal energy kj/kg E Total energy kj E Young s Modulus of Elasticity Pa E f Foundation Modulus Pa G Fundamental derivative -- h Secific enthaly kj/kg H Pressure head m k Constant olytroic index -- k o Initial volume concentration of gas/vaour in a liquid -- K Bulk modulus (The inverse of β) Pa K S Isentroic bulk modulus Pa K T Isothermal bulk modulus Pa m Mass er unit area of a late kg/m M Mass of late kg M Mach number -- M Molecular mass or weight of the material kg/mol n Adiabatic exonent of a liquid/constant olytroic index -- N Initial bubble number density in a liquid -- Pressure Pa Far-field ressure Pa P Riemann variable or quantity -- q Heat transferred into a system er unit mass J/kg Q Riemann variable or quantity -- r Reflection factor -- R Gas constant for a articular fluid J/kg.K R Radius of a bubble m R o Universal gas constant J/kg.mol.K

17 xvii R o Radius of a bubble nucleus m R Radius of a bubble m R C Critical radius m R C Minimum radius of visible bubble (within the framework of the detection method used) m s Secific entroy J/kg.K S Non-dimensional form of s -- S Surface tension N/m t Time s t V Time taken for a bubble to exand to visible size s t* Rise/fall time of a ressure ulse s T Temerature K u Particle velocity m.s - ÛU Non-dimensional form of u -- U S Shock velocity m.s - V o (R/R V ) 3 -- V Volume m 3.kg - w Finite wave seed m.s - w Work done by the system on its surroundings J x Dislacement m Z Acoustic imedance Pa.s.m -3 β The comressibility or modulus of comressibility of a liquid Pa - ζ Transverse dislacement of a late m γ Adiabatic exonent (secific heat ratio) for a gas -- γ Shock strength -- µ Dynamic viscosity kg/m.s ν Kinematic viscosity ν = µ /ρ m /s ρ Density kg/m 3 σ Surface tension N/m τ Non-dimensional form of t -- Γ Grüneisen coefficient -- Other Subscrits and Qualifiers For any variable X,

18 xviii X B X G X L X L X R X S X S X T X o X ij X T X W X X X & X & δ X Within the bubble Of the gas Of the liquid Conditions on the right side of an interface Conditions on the left side of an interface Shock conditions Isentroic Within the bubble Initial conditions Ratio of quantity X at i over that at j Isothermal Water conditions Initial conditions Final conditions Rate of change of the variable X Second derivative of X Small change in quantity X

19 xix

20 . Introduction.. Definition of Cavitation Cavitation is defined as the formation and activity of bubbles or cavities in a liquid i.e. the rocess of ruturing a liquid. In this context formation refers, generally, both to the creation of a new cavity and to the exansion of a re-existing one to a visible size []. Cavitation is a dynamic henomenon, as it is concerned with the growth and collase of vaour and gas filled cavities, which occurs only in liquids. The exansion of bubbles, which are filled with gas or vaour or, usually, a mixture of both, in a liquid may be induced by reducing the ambient ressure or increasing the temerature by static or dynamic means. This bubble growth will occur at a nominal rate if it takes lace by means of diffusion of dissolved gases into the cavity (mass transfer) or by exansion of the bubble contents with temerature increase or ressure reduction []. Bubble growth by diffusion is called degassing or gaseous cavitation when induced by dynamic-ressure reduction []. Bubble growth will occur exlosively if it is rimarily the result of vaorisation (mass transfer) into the cavity. This rocess is then called: Boiling if caused by temerature rise. Cavitation if caused by dynamic ressure reduction at constant temerature. This is also known as vaorous cavitation []. The revious descrition has distinguished boiling, vaorous cavitation and gaseous cavitation as related henomena if not identical in all resects. Cavitation is sometimes more loosely described as being cause by static or dynamic means [,3]. The conventional hase diagram [3] for water is shown in figure. (note that the scales are not linear). The curve labeled l-v reresents oints where the liquid and vaour are in equilibrium and is thus a curve of saturated temerature versus ressure. This liquidvaour coexistence line is called the binodal [4]. From the arbitrary liquid state a, below the critical oint, one may reach the vaour state by either increasing the temerature

21 above the saturation state at constant ressure (a to b, where b is a vaour state) or by decreasing the ressure at constant temerature (a to c, where c is a vaour state). These rocesses corresond to the rocesses of boiling and cavitation resectively [5]. Since cavitation results due to ressure variations, it may be controlled by controlling the minimum absolute ressure in a system []. Figure.. The hase diagram of water [3]... Tyes of Cavitation Four tyes of cavitation may be distinguished according to the cause of incetion []:. Hydrodynamic Cavitation is roduced by ressure variations in a flowing liquid due to the geometry of the system or the rotation of a roeller. Each liquid element asses through the cavitation zone once.. Acoustic Cavitation is roduced by sound waves in a liquid due to ressure variations or, as will be discussed later, tensions. 3. Otic Cavitation and Particle Cavitation is roduced by hotons of high intensity (laser) light and any other tye of elementary articles (e.g. a roton) ruturing in a liquid. These tyes of cavitation are achieved by deosition of energy into a limited volume. Energy may also be introduced by very small heating elements.

22 3 Hydrodynamic cavitation includes travelling cavitation, where individual transient cavities form and move in a flowing liquid as they exand, shrink, and then collase. The bubbles grow when assing through low-ressure regions and start to collase shortly after they are swet into regions higher ressure. Alternately, hydrodynamic cavitation may occur on a body moving through a stationary liquid. Another tye of hydrodynamic cavitation is vortex cavitation, where cavities develo in the cores of vortices as often occurs on the tis of shi roeller blades. The oeration of valves and fittings involve changes in liquid flow velocity assing through them and may thus also be affected by cavitation [,6]. In acoustic cavitation, sound waves of the high ultrasonic frequencies generated by a submerged iezoelectric sound transducer drive are used []. The waves roduce a highamlitude, high-frequency, alternating ressure field, thereby subjecting a samle of liquid to cycles of low and high ressure, which, if of a sufficiently large amlitude, may result in cycles of cavitation growth and collase known as vibratory cavitation. This cavitation is often accomanied by low velocity liquid flow, of a time eriods far greater than that of the cavitation cycles, which is in the order of milliseconds. Acoustic cavitation is, in general, a non-linear since changes in bubble radius is not roortional to the sound ressure []. Such acoustic techniques are frequently used to investigate the hysics of the cavitation rocess since accurate variation of the sound field allows amle control of bubble size and distribution [7]..3. Effects of cavitation Cavitation causes various hydrodynamic and other effects, which are often destructive. However, its destructive otential is beneficial in some alications. In hydrodynamic cavitation, [] cavitation alters the flow attern and the dynamics of the interaction between the liquid and its boundaries, thereby restricting and reducing the force that may be alied to the liquid by a solid surface. Thus, the flow roduced by a turning vane is reduced and the thrust roduced by a shi s roellers is limited.

23 4 Noting that cavitation bubbles radiate acoustic ressure waves from their surfaces when they exand or contract, the cavitation, being inherently unsteady, may involve significant fluctuating forces and consequently resonant vibration (if one of the frequency comonents of the fluctuations equals the natural frequency of nearby equiment). Such instabilities resulting from cavitation may even affect liquid-roelled missiles []. Collase of cavitation bubbles occurs imlosively for a vaour-filled bubble with insignificant gas content and less so if the gas content is high []. The asymmetry of cavity collase, which leads to henomena such as liquid jets [8,9], as well as central imlosion and the emission of shock waves are otentially damaging and may result in material destruction [9,0]. Due to the low comressibility of the liquid and the high comressibility of the gas within the bubbles, a large amount of otential energy is stored when the bubbles exand. This energy is concentrated into a very small volume when the bubble collases. Consequently, very high ressures and temeratures are roduced, which may erode solids and initiate chemical reactions. The ressures and temeratures reached may be in the order of MPa and K resectively [,5,9]. The temerature of the material adjacent to a collasing bubble may increase by K [7]. Collase may also, by a rebound effect, emit shock waves with ressures as high as 400 MPa [7] through liquid adjacent to bubble, which may erode adjacent walls. These shock waves result in significant noise, which may aear on ressure sensor readings as white noise covering a wide frequency band. Exeriments have shown that cavitation noise and erosion are roortional [7]. The extreme temeratures in a collasing bubble may also result in a henomenon called sonoluminescence in which secies in the gas hase are excited and relax by emitting hotons [7,9]. The resulting flash of light that is discharged may be detectable for eriods in the order of milliseconds, but are weak and only visible by magnification or in comlete darkness [].

24 5 Liquid films, usually between 0. and 0µm thick [8], are often raidly deformed between searating sheets, thereby develoing a tensile force within the film due to the relative motion of its bonding surfaces. The resulting cavitation of such films is an imortant asect of rocesses such as lubrication and rinting. Barrow et al. [8] exlored the issue of cavitation damage due to cavity growth using an atomic force microscoe and found evidence that when cavitation occurs in confined saces, the growth of a cavity may be raid enough for it to cause more damage to adjacent structures than its collase. Bubble generation [] during the bubble-jet recording rocess occurs by nucleation of ink vaour at cavitation incetion, followed by instantaneous boiling and ultimate bubble collase. These events are initiated by short, raid ulses of heat. Transient cavitation has been observed near oerating mechanical heart valves in vitro and inferred in vivo via the observation of itting on exlanted clinically used valves [3, 4]. When these valves oen and close abrutly, ressure waves are induced in the fluid. If a rarefaction wave of sufficient strength is roduced, cavitation may occur..4. Uses of Cavitation Though cavitation was first noticed for its disadvantageous effects, in a few alications, some effects may be beneficial and cavitation is emloyed to roduce useful effect. Cavitation is used mostly in chemical rocesses and biological and medical alications. In chemical rocesses, cavitation is usually used in one of two ways [7]:. The cavitation bubbles may act as a mixer, vigorous disturbance and increasing the contact area between two liquids or a liquid and a solid.. High temeratures and ressures during the collase of cavitation bubbles romote and initiate chemical reactions, esecially those occurring in aqueous media. In this henomenon known as sonochemistry, cavitation is used to accelerate and selectively control chemical reactions.

25 6 The first method may romote emulsification (e.g. of immiscible liquids) in manufacturing rocesses and also aids cleaning rocesses. Emulsification is the means of removing oils by breaking them u into tiny drolets such that they may be rinsed away. The mixing effect of ultrasonic cavitation is roven to increase the rate of various rocesses such as: homogenisation, crystallisation, hydrogenation, filtration, high-shear extraction, defoaming, wax disersion, de-agglomeration and article disrution. Sonochemistry has a wide variety of alications, articularly in the chemical industry, as discussed in [9,5,6,7]. Acoustically induced cavitation is the foundation for many aaratus used to clean intricate arts []. In ultrasonic cleaning or ultrasonic chemical degreasing, the collase of cavitation bubbles, formed in a cleaning solution, roduces a scrubbing action. For examle, ultrasonic cleaning is used in dentistry, machining, the chemical industry and in the ower industry e.g. on the walls of heat exchangers [8]. Acoustic cavitation has many other industrial alications [9-]. In metallurgical and machining rocesses, acoustic cavitation is also emloyed in the refining of molten metals, rearation of cast comosite materials and in ultrasonic sharening, cutting, metal welding, surface hardening and stress relief. Other alications of acoustic cavitation in the chemical industry include disersion and coagulation, olymerisation and drug rearation. Cavity collase may lead to oxidation: According to the hot sot theory [9,], the high temeratures and ressures at collase cause dissolution of the liquid according to the chemical reactions shown in [3]. The free hydroxyl radicals formed, OH, are effective oxidising agents and thus readily cause decomosition of organic comounds. Cavitating water jets are well suited to the oxidation, and subsequent degradation of organic comounds [4]. These jets generate cavitation over a much larger volume than ultrasonic methods and are thus, generally, more efficient [3]. Cavitating jets are roduced, using simle aaratuses, by maintaining a high ressure across a nozzle. The high-seed water jet is discharged, through the atmoshere or, when submerged, through a liquid. Vortex cavitation occurs in the low-ressure zone in the vortex core [4]. Cavitating jets are effective in removing chemical contaminants, such as esticides and

26 7 biocides, from water as well as in oxidising arsenic to a form that may be more readily removable by reciitation and filtration [3]. Cavitating jets may also raidly reduce large concentrations of bacteria (e.g. E. Coli) by more than five orders of magnitude. Algae concentrations may be reduced by a factor of 00 in just two hours [3]. Another alication of cavitating jets is described in [5]. Imacts caused by the collase of the cavitation bubbles have been used to introduce beneficial, comressive, residual stresses to materials, resulting in surface hardening and increased fatigue life of comonents. Other alications of cavitating jets [4] include cutting, cleaning, surface finishing and erosion testing. Cavitation has been used in many biological and medical alications [7]. Localised cavitation is generated by the rocess of laser-induced otical breakdown, also known as hotodisrution, allowing for surgery with micron recision beneath the surface of biological tissues (most of the energy in the otical ulses asses through the material) with virtually no collateral damage. Laser ulses used in this rocess are in the order of femtoseconds [6]. In this case, the mechanical effects rather than chemical reactions of cavitation are utilised. Shock Wave Lithotrisy (SWL) involves the destruction of stones by the use of focussed acoustic waves [7]. Acoustic cavitation may also destroy bacteria and yeast cells and has been used in the removal of cell contents such as enzymes and of viruses from infected tissue [7]. In addition to the oxidising reactions described above the effect may be attributed to the effect of cavities inside the bacteria and in art, due to the removal of dissolved gas. Cavitation is emloyed in the delivery of drugs to localised areas [8]. Photomechanical drug delivery involves the use of a laser ulse to generate a cavitation bubble in a blood vessel due to the absortion of laser energy by blood clots or surrounding fluids such as blood. The hydrodynamic ressure arising from the exansion and collase of the cavitation bubble may force drug into the clot or vessel wall.

27 8 The effects of cavitation bubble collase may also be used beneficially in gene theray and drug delivery by sonooration or sonohoresis [9,30,3,3]. This is a theraeutic effect of high frequency ultrasound on living cells [9]. The ultrasound may be used to deliberately ruture microbubbles resulting in violent imacts on nearby cell walls and transient, rearable ores through which drug molecules, roteins or foreign genes may enter [9]. Such microbubbles, originally develoed for use as contrast agents for diagnostic imaging [33], are commercially available and may be injected where required. This alication might be articularly useful in (and indeed revolutionise) gene theray and the delivery of toxic and non-toxic drugs [3]. A ossible alication of cavitation is that it may theoretically be used to drive micromechanical systems [,34]. Raid deosition of energy into a liquid causes it to vaorise exlosively at high ressures and exand, thereby erforming work on its surroundings. As sho??wn later, raidly exanding bubbles may accelerate microscoic and nanoscoic articles. Zhao et al. [] generated extremely raid vaour exlosions by heating liquids raidly with micro-heaters. Other alications of cavitation include its use as a flow-control mechanism in a Venturi tube or an orifice [] in water treatment and urification of contaminated soil, in roduction, transortation and rocessing of etroleum and gas. In such cases, cavitation rovides a choking henomenon similar to that encountered in comressible flow. An obvious alication of cavitation is its use for the removal of traed gases in liquids [6]..5. The Study of Cavitation.5.. Producing Controlled Cavitation Exerimental aaratus [,7] such as Venturi tubes (and other conduits with geometrical flow restrictions) and variable-ressure water tunnels, with transarent-test-sections, have allowed researchers to observe the hydrodynamic cavitation and to investigate its effect on ums and turbines.

28 9 When acoustic cavitation is induced by a iezoelectric transducer drive, cavitation eventually develos when the liquid is no longer able to follow the motion of the face of the transducer. Once cavitation occurs, a limit is reached and no more ower may be transmitted to the liquid if the transducer ower is increased []. This is because the cavitation zone then searates the transducer face from the body of liquid..5.. Definition of the Cavitation Threshold and Detectable Sizes Cavitation in liquids is difficult to measure quantitatively. In order to enumerate the rocess of cavitation, the cavitation threshold is defined as the ressure at which a cavitation event, which is characterised by an exlosive growth of bubbles, occurs [35], i.e. it is the breaking tension of a liquid. Various detection methods are used to register this threshold. The concet of a visible or detectable bubble size (one which may be detected within the framework of the technique used) is introduced [36] Detection and Visualisation There are several methods for detecting the resence of cavitation []. They include: Direct observation by visual and hotograhic means. This method necessitates the use of aaratus with transarent viewing, unless used in conjunction with x- ray techniques [36,37,38]. Microscoes may be used for increased satial resolution. Light scattering methods: Indirect observation by allowing cavitation regions to scatter laser-beam light into a hotocell. The occurrence of a cavitation at the interruts the beam and the event is recorded [39,40]. Light absortion methods: Indirect observation of effects of absortion of light exhibited by bubbles [4]. This method is generally less sensitive and simler [36] than light scattering. Caacitance method [36]: Indirect observation of the free-surface dynamics, which may be quantitatively related to the cavity cluster dynamics. This method rovides stable results but is less sensitive than light scattering methods.

29 0 Acoustic method: an acoustic transducer may record the ulses emitted by the imlosion of cavitation bubbles [39]. Acoustic detection may make use of the emitting transducer itself by the echo henomenon [40]. Indirect observation by sensing the noise caused by cavitation []. This method may detect mild cavitation that may be too slight to be observed visually. Indirect observation by measuring the size and number of its in suitably laced, detectors made of, for examle, aluminium foil [7]. For hotograhic methods, reetition rates below 5000 frames er second are, in general most ractical for studying the overall nature of cavitation, whereas higher reetition rates are useful for studying the more obscure details []..6. The Purose of This Work This thesis is concerned with true cavitation brought about by ressure variations alone. The objectives were to demonstrate acoustic cavitation exerimentally and confirm the results using relevant theoretical methods. A related objective was to determine the ressures at which cavitation occurred, i.e. the cavitation threshold, in water. As no acoustic cavitation exeriments had been erformed reviously at the university, test facilities were to be designed and built.

30 . Shock Waves and Rarefaction Waves A violent ressure change caused by the sudden release of chemical, electrical, nuclear, or mechanical energy in a limited sace [0] causes a zone of high ressure called a shock wave to roagate through a medium. [4]. Shock waves, which are irreversible rocesses, may occur in any elastic medium. A shock wave is a very shar, thin front, across which flow roerties, such as ressure, temerature, density, velocity and entroy change. Since, the thickness of a shock wave is tyically in the order of a few angstroms [0], flow roerties may be assumed to change discontinuously across a shock wave. In general, the ratio of the higher ressure behind a shock wave to the lower ressure ahead of the shock is referred to as the shock strength. The ratio of higher density behind a shock to lower density ahead of it is called the shock comression. Similarly, here the density ratio across a rarefaction wave is referred to as the exansion. Shock waves at right angles to the ustream flow that occur in one-dimensional flow [43] are termed normal shock waves. The shock Mach number M S is defined [44] as the Mach number of a suersonic flow in which the shock would be stationary and is thus always ositive... Comressible flow The conservation equations of mass, momentum and energy resectively, for a erfect fluid in one-dimensional flow are (in artial differential equation form) [0]: ρ + ( ρu) = 0 t x (Mass or continuity) (.a) ( + ρ u ) + ( ρu) = 0 (Momentum) (.a) x t E + [ Eρ u + u] = 0 (Energy) (.3a) x x In differential form, the equations are [45]: dρ ρ + dv V + da A = 0 (Mass) (.b)

31 d = ρv dv (Momentum) (.b) δ q δw = dh + VdV + gdz (Energy) (.3b) Equations (.b), (.b) and (.3b) are valid for diabatic and adiabatic (without heat addition) rocesses. The momentum equation (.b) is valid only for frictionless flows. In the conservation of energy equation, i.e. first law of thermodynamics, the term δw is the work done by the system on its surroundings ( δ w = VdV )... The Conservation Laws for a Shock Wave Consider a iston, initially at rest, which imulsively acquires a finite velocity u. The fluid adjacent to the iston must move at the same velocity u, while the article seed further downstream remains at its initial value u. These conditions are satisfied by a shock wave, moving at a seed U S, which aears on the iston face and roagates into the fluid. (U S > u ). The velocities behind and ahead of the shock wave are u and u resectively. This discussion [46] is alicable to laboratory frame or moving shock coordinates where the relevant velocities on either side of the shock are u and u. The laws that govern moving shock may be transformed into shock-fixed coordinates, where the relevant velocities on either side of the shock are v and v, using the relations: v = U + S and (.4) v u = u U S (.5) Figure.. Moving and fixed shock reference frames [0].

32 3 If one considers a very small control volume enclosing a shock wave, heat transfer, friction and area changes become negligible. Then, the equations of continuity, momentum and energy for frictionless, adiabatic flow must be satisfied by both states ustream and downstream of a shock wave [45]. The basic conservation equations across a shock wave, derived from the conservation laws of mass, momentum and energy and known as the Rankine-Hugoniot equations, relate the flow roerties across a shock wave discontinuity. The normal shock equations may be derived from the basic conservation equations in differential form [47] or other methods [0]. Being indeendent of state, the conservation equations may be alied to any material. In moving-shock coordinates, the equation of continuity is stated generally, as [46]: U S ( ρ = ρ u ρ u (.6a) ρ) In shock-fixed coordinates, using equations (.) and (.), v ρv ρ = (.6b) In moving-shock coordinates, the equation of momentum conservation is, generally [46], U S ( ρ v ρ v ) = ( + ρ v ) ( + ρ ) (.7a) v In shock-fixed coordinates, using equations (.) and (.), ρ v = + ρv + (.7b) In moving-shock coordinates, the equation of conservation of energy is, in the most general form [46], U S ( ρ + u (.8a) E ρ0e0) = ( + ρe) u ( 0 ρ0e0) 0 and in shock fixed coordinates, by defining the enthaly as h = v + e : h + + (.8b) v = h v Equations (.3b), (.4b) and (.5b) may be combined [45] resulting in exressions for the roerty ratios across the shock. The roerty (ressure, density, sound seed) ratios across a shock wave may be exressed in terms of the ustream Mach number M only [45] for an ideal gas. The strength of a shock may be secified in terms of the ressure,

33 4 density, temerature or article velocity ratio across the shock or the shock Mach number since relationshis between all of those quantities may be derived. The following articularly useful relationshi may be derived [47]: M S γ = γ + γ γ + or, after some maniulation [0], (.9) = S γ ( M γ + ) + (.0) In addition, the ressure ratio may be exressed in terms of the density ratio as [48]: ρ ρ = + γ + γ γ + γ + (.) which is known as the Rankine-Hugoniot relation [0]..3. General Equations Using only the shock mass and momentum relations in shock fixed coordinates, the following equation exressing the seed of a finite wave may be derived [45, 48, 49]: ρ w = (.) ρ ρ ρ For infinitesimal changes in ressure and density ( = δ, ρ ρ = δρ ) equation (.) reduces to the following equation from which the seed of sound may be calculated (assuming that the wave roagation velocity is deendent only on the thermodynamic state of the medium): a = ρ S (.3) The derivative in brackets is isentroic since δρ / ρ <<, imlying an acoustic wave [49]. The adiabatic exonent is defined as [9] a γ = (.4) v

34 5 A olytroic rocess is one, in any medium, which may be described by the relation: ρ = ρ k k k or V = ρ = constant (.5) where the exonent k, the constant olytroic index, may have any value from to -, deending on the rocess. The values of k for isobaric, isothermal and adiabatic (or isentroic) rocesses in an ideal gas are 0, and γ resectively. An isentroic rocess is defined as one, which is adiabatic and reversible [47]. Using Euler s equation of motion for the roagation of a one-dimensional disturbance and the continuity equation for unsteady and uniform one-dimensional flows, the wave equation, a artial differential equation that describes the roagation of waves with seed a, may be derived [0]. The one-dimensional form of the wave equation is [50]: a w = x w t (.6) Since a wave, across which the ressure and density ratios are non-negligible, may still be aroximately isentroic, as shown in [44], 3 tyes of waves may be distinguished: Sound waves (waves of negligible strength and comression or exansion) Weak waves (finite but isentroic) High amlitude waves (including strong shock waves) In the acoustical aroximation, it is assumed that the roagation and reflection of waves of relatively low but finite amlitude may be modelled by simle acoustic relations i.e. weak waves are aroximated by sound waves. Since air is far more comressible than water, the acoustic aroximation is valid over only a narrow range of wave ressures for air and over a relatively large range for liquids. For all fluids, non-linearities in the medium become more imortant [5] with increasing incident ressures. When the ressure and density ratios across a wave are non-negligible, the wave leads to article flow that is non-negligible comared to the wave velocity, and that the comression across the wave is small and occurs adiabatically. Hence, the wave travels at a seed higher than the sound seed a. In the fast moving fluid, the wave travels at an increased

35 6 sound seed and its velocity of roagation is increased relative to a stationary observer. Indeed, in the first theoretical study of large amlitude sound waves [44], Poisson derived a relationshi indicating that the velocity of roagation is the sum of the seed of sound and the disturbance (article) velocity roduced by the wave. Shock, weak finiteamlitude, and Mach waves are discussed in the following sections..4. Shock Waves and Rarefaction Waves in Gases.4.. General Equations for Gases The erfect gas law is: = ρrt (.7) Since R o is the universal gas constant, R0 R = (.8) M The first law, equation (.3b), and the second law ( ds = δq /T ) of thermodynamics and the definition of enthaly combine to form the well-known Tds equations [47] from which the following equations relating two arbitrary states (even states on either side of a shock in a gas) are derived [5]: T v s v + ln s = c ln R or T v T s ln s = c ln R (.9) T The secific heat ratio γ of air varies by only about 5 % across the temerature range K and is considered constant. Similarly, c and c v are considered constant [5]. For an ideal gas, the internal energy is a function of temerature only [46, 53]. The following equation is valid for a olytroic gas, such as air at moderate temeratures, and is assumed in most alications [46]: e = cv T = RT /( γ ) = /[ ρ( γ )] (.0) The internal and secific internal energies may be related by [46]: E e + = (.) u.4.. Sound Waves in Gases

36 7 From the definitions of the secific heats c and c v, and the adiabatic exonent γ, for gases obeying the ideal gas law, the adiabatic exonent reduces to the ratio of secific heats [0]. For an ideal gas, equation (.4) and the ideal gas equation (.7) give: ρ γ γ γ RT M T R a = = 0 (.) For an adiabatic (δq = 0) rocess [47], from equation (.5): γ ρ ρ = (.3) Aendix A illustrates the derivation of following useful entroic relations, ) ( S S e a a = γ γ γ (.4) ) ( S S e a a = γ γ ρ ρ (.5) The following isentroic relations are frequently derived from the adiabatic relation (.3) and the sound seed equation (.) [9,44] or simly from (.9) and (.0), letting S = S : = γ γ a a (.6) = γ ρ ρ a a (.7).4.3. Finite, Isentroic Waves in Gases The following equation, which is derived in [48], [46] and [4], relates the article velocity u behind an isentroic (comression or exansion) wave to the ressure ahead of the wave u and the ressure ratio across the wave: ± = γ γ γ a u u for a gas (.8)

37 8 Note that the ositive sign alies for a wave moving in the ositive direction (referred to as a P-wave) and the negative sign is valid for a wave moving in the negative direction (a Q-wave). Substituting the isentroic relation (.6) into (.8), one may obtain the velocity of roagation of a P-wave in a gas: + + = + = + ) ( γ γ γ γ γ γ γ a a u u a u (.9) and the velocity of roagation of a Q-wave in a gas is: + = = ) ( γ γ γ γ γ γ γ a a u u a u (.30) Thus, when the wave is weak, its seed of roagation tends to the seed of sound a and secondly that strong waves may roagate significantly faster than sound waves [4]. The seed of sound, ressure, density and temerature ratios across a wave searating the initial state from the final state are [46, 48]: ) ( a u u a a = γ (.3) ) ( + = γ γ ρ ρ a u u (.3) ) ( + = γ γ γ a u u (.33) ) ( = γ γ a u u T T (.34).4.4. Shock Waves in Gases Using the energy equation (.3b) and equation (.), the Mach number of a shock wave in a gas may be exressed as [45]: γ γ γ ) ( ) / )( ( + + = M S (.35)

38 9.5. Shock Waves and Rarefaction Waves in Liquids While the generation, measurement and study of shock waves in air are relatively well advanced, liquid shock wave research has until recently been relatively unexlored due, in art, to the high sound seed of most liquids [5]..5.. General Equations for Liquids The bulk modulus K is the recirocal of the comressibility β, which is defined by [0]: v = β. (.36) v This equation exresses the change in secific volume, which occurs when a liquid with secific volume v and at a ressure is subjected to a ressure rise of. The values of K and β are, in general, different for isentroic and isothermal rocesses. The bulk modulus of comressibility may be considered constant [0] for the moderate ressures considered in the resent discussion. The modified liquid equation of state, known as Tait s equation, as derived in [54] is: ' ' ρ = ρ n or ρ ' = ρ ' where ' and ' are known as the modified ressures: n (.37) ' = + A W and ' = + A W (.38) Tait s equation may be written as the ressure-density relation for adiabatic comression: + A W = constant ρ n (.39) The constants A W and n are characteristic constants of the medium and are obtained emirically [4]. A W is a weak function of entroy and n is a function of ressure and temerature but may be considered constant for weak to moderate shocks in water and in the range of ressure and temerature under consideration [54]. For the uroses of this reort, it is assumed that A W = 96.3 MPa and n = These values are valid [0] for ressures below 500 MPa.

39 0 The gas dynamics equations may be transformed to describe shock waves in liquids using the modified ressure instead of the ressure and the value n instead of the secific heat ratio γ e.g. equation (.37) is analogous to the equation (.3) for an isentroic flow in air. However, due to the large magnitude of A W, the modified ressure ratio (or shock strength) is close to unity and thus, shock waves in water u to a few hundred bars may be regarded as weak and most waves in water behave like acoustic waves even for shock ressure changes of hundreds of megaascals [55]. This exlains why the isentroic equation (.37) is valid across shocks, for ressures u to 000 MPa [43], although the analogous gas dynamics equation (.3) is not, since shocks in gases can seldom be treated as isentroic. In addition, for ractical cases, the article velocities behind such weak shock waves in liquids are always subsonic [55]. For liquids and other relatively incomressible substances, it is unnecessary to distinguish between the secific heats [5] c and c v and one may write c = c = c v. Then, the entroy values on either side of a shock wave in water may be related by: s T s = c ln (.40) T This equation is a simlification of equation (.9) for air, with density taken as constant for the relatively incomressible substance. Though the equation (.40) is a reasonable aroximation, one should note that entroy changes in liquids are usually negligible. It follows that temerature changes across liquid shock waves are also usually negligible. Itoh [0] has shown that the seed of sound behind an underwater shock wave is greater than the velocity of the shock at the same ressure. This imlies that the strength of a shock wave that roagates in water is more easily attenuated than one in a gas. However, the calculated data of Kirkwood and Richardson as detailed in [55] contradicts these statements and demonstrated that the seed of sound behind an underwater shock is less than the velocity of the shock at the same ressure. This is also true in gas media. In any case, such wave-steeening or overtaking effects are only clearly observable at significantly higher shock ressures due to the high comressibility of liquids.

40 .5.. Sound Waves in Liquids The seed of sound in a liquid may be exressed [4,43,5] as follows: ' a = n (.4) ρ This equation is identical to equation (.), with the secific heat ratio γ and the ressure of the air medium relaced with the adiabatic exonent n and the modified ressure of the liquid medium resectively. Using the one-dimensional wave equation, the sound seed in a liquid may be exressed in terms of the bulk modulus by [0]: K a = (.4) ρ where K is the isentroic bulk modulus K S of the liquid, although the subscrit has been droed since it differs only slightly from the isothermal bulk modulus K T for a fixed temerature and ressure (esecially at low ressures) and can be used interchangeably. The following equations (see derivation in aendix B) are the analogous to equations (.6) and (.7) with γ and substituted with n and : n a n a = (.43) a a = ρ ρ n (.44).5.3. Shock Waves in Liquids Liquid shock waves may be generated using the following techniques, which are discussed in detail in [5]: Underwater exlosions the detonation of submerged exlosive charges Underwater sark ga discharge the raid generation of a high voltage between two electrodes resulting in formation of high-ressure, high temerature lasma in a small volume and subsequently, a shock wave

41 High-ower lasers or oto-electronic sources localised heating in a fluid, which causes exansion and sherical shocks Electromagnetic emitters the acceleration of a late in contact with the liquid by Lorentz forces Liquid or hydrodynamic shock tubes the use of gas shocks to iminge uon and transmit shocks into liquid samles. This is discussed further in [56]. Using the Rankine-Hugoniot relation (.) and the Tait equation (.37), the Mach number of a shock wave in a liquid may be exressed as [57]: n n n M S ) ( ) / )( ( + + = (.45).5.4. Finite Waves in Liquids It may be roved, with a similar derivation to that in [48], [46] and [4] that: ± = n n n a u u for a liquid (.46) where the ositive and negative signs aly for waves moving in the ositive direction (P-waves) and waves moving in the negative direction (Q-waves) resectively. As with gases, it may be shown, using the isentroic relation (.43), that the velocity of roagation of a P-wave in a gas is: + + = + = + ) ( n n n a a u u a u n n n n (.47) and the velocity of roagation of a Q-wave in a gas is: + = = ) ( n n n a a u u a u n n n n (.48) The following equations are analogous to equations (3), (3) and (33) resectively:

42 3 a a n ( u u) ρ ρ = ± (.49) a n ( u u) n = ± a n n ( u u) n = ± a where the ositive signs are for Q-waves and the negative signs are for P-waves. (.50) (.5).6. The Shock Tube A shock tube is a device for generating flows in which shock waves aear in a laboratory environment. A tyical shock tube consists of a closed duct searated into two comartments, a high-ressure driver section and a low ressure driven section, by a diahragm. The diahragm is rutured (instantaneously in the idealised situation), either by ressure difference between the two sections or by ricking of the diahragm with a needle [0], causing transient ressure disturbances: A shock forms in the driven section and a diverging exansion fan roagates into the driver section. The ressure ratio required to generate a shock of strength γ deends on the initial gas temeratures and the roerties of the gases and may be calculated by [48]: = ( n ) a ( ) 4 (.5) ( n4 ) a4 (( n + )( + n ) where ij = i / j, n ( γ + )/( γ ) and the states, and 4 are those in the driven i = i i section, driver section and the region behind the shock wave resectively. This equation is resented in various forms in [48, 56, 44, 49]. The most imortant factors determining shock strength are the diahragm ressure and sound seed ratios 4 and a /a 4. Equation (.5) suggests the use of gases with low molecular weights and high sound seed, such as helium and hydrogen, in the driver section for roducing strong shocks. Figures. and.3 show the variation of the shock strength of the first shock roduced in a shock tube with diahragm ressure ratio and illustrates the tendency of the shock strength to ( n 4 + )

43 4 tend to a value with increasing ressure ratio u to 4 / =. At room temerature, the maximum obtainable shock strengths P MAX (the shock ressure ratio) for each combination of gases in the shock tube are equal to: 574 for H in the driver section and N in the driven section, 3 for He-air and 44 for air-air combinations [56, 44, 48]. Figure.. Required diahragm ressure ratio for generating a shock of strength γ in a shock tube using air in the driven section and air, helium and hydrogen in the driver section. Figure.3. Required diahragm ressure ratio for generating shock of Mach number M S in shock tube with nitrogen in the driven section and helium/nitrogen in the driver section.

44 5.7. Methods of Analysis.7.. The Wave Diagram Figure.4. The wave diagram resulting from diahragm ruture in a shock tube. Shock tubes involve the roagation of ressure waves in ducts. If the cross-sectional dimensions of the duct relative to its length are negligible, the flow may be considered as one-dimensional. A grahical lot reresenting the roagation of ressure waves in such a duct is called a wave diagram and the method of comutation is known as the method of characteristics. While the fundamental artial differential equations for comressible flow are too comlicated to be dealt with directly, in such cases, the method of characteristics allows the state of the fluid, described by two state arameters, and the flow velocity at all times and oints of the duct to be determined. A detailed discussion of wave diagram methods was given by Rudinger [44]. Here, only a few imortant, relevant facts are discussed. Characteristic lines reresent finite waves across which flow variables change discontinuously. They differ from shocks in that they are considered isentroic. A P-wave is the characteristic of a ositive (right-going) wave and a Q-wave is the characteristic of a negative (left-going) wave. P and Q waves may be either exansion or comression waves. One P-wave, one Q-wave and a article ath [44] asses through every oint of a wave diagram. It follows that in cases of adiabatic [43] flows in ducts of constant crosssection, the Riemann variables P and Q remain constant along its resective

45 6 characteristics [44]. P is constant along P-waves and Q is constant across Q-waves while P is constant across Q-waves and Q is constant across P-waves. While any combination of a and u could serve as a flow variable, the following combinations are defined as the Riemann quantities in gases [44] and liquids [43]: P = a + u (in a gas) γ Q = a u (in a gas) γ The sloes of the characteristics are then exressed as [44]: dx dt dx dt P = a + u (in a liquid) (.53) n Q = a u (in a liquid) (.54) n = u + a (for P-waves) (.55) = u a (for Q-waves) (.56) In water, the velocity Uˆ is very small comared with Â, and may usually be neglected in equations (.55) and (.56). The Riemann variables may be defined in more convenient, non-dimensional form, using the relations shown in aendix C: P Aˆ Uˆ = + (in a gas) P = Aˆ + Uˆ (in a liquid) (.57) γ n Q = Aˆ Uˆ γ (in a gas) Q = Aˆ n U ˆ (in a liquid) (.58) An exansion fan is, in reality comosed of an infinite number of Mach waves. However, for the urose of analysis, an exansion fan is reresented by a finite number of diverging characteristics, which searate finite regions of roerties. The Riemann quantities P and Q and the relations (.6), (.7), (.8) and (.44) are sufficient to solve for all wave diagram arameters across isentroic characteristic waves but not adequate when discontinuities such as shock waves and contact surfaces are encountered. However, simle matching relations (such as normal shock tables for shock

46 7 waves and the ressure and velocity comatibility relations for a contact surface) between adjacent regions may be used, rather than the general conservation equations. Normal shock tables, derived from the Rankine-Hugoniot equations are convenient for solving for values across the shock and thus rovide the matching conditions. Usually, only the ressure, density, sound seed and temerature ratios, the entroy change and the Mach numbers M and M are tabulated. These quantities may be calculated from the simle relations in aendix B. However, the quantities most useful for the solution of wave diagram roblems, namely P / A ˆ, Q / A ˆ, Û / Â and S are not usually tabulated but may be calculated as shown in aendix C, which describes shock relations for liquid water. The aths of contact surfaces are easily lotted on a wave diagram as its velocity is the same as the equal velocities on either side of it (refer to section.8.). For the uroses of wave diagram construction, contact surfaces are idealised as discontinuities. The widening of the contact surface by mass diffusion, which occurs when the gases are at rest or moving at constant velocity, as well as the instability and subsequent disintegration of the interface that occurs if the gases are accelerated toward the one of higher density, are neglected..7.. The Pressure-Velocity Diagram Problems involving the one-dimensional interaction of shock and rarefaction waves may be solved grahically using a ressure-velocity diagram. This method is discussed at length in [46]..8. Shock and Rarefaction Wave Interactions Shock and rarefaction wave interactions in a one-dimensional flow are the subject of the resent discussion. The resulting flow field from all such interactions may, in rincile, be obtained by solving the conservation equations of mass, momentum and energy in a one-dimensional flow, using the aroriate boundary conditions.

47 8 One-dimensional interactions can be one of three tyes: Collisions of two waves. Overtaking of one wave by another. Interactions of a wave with walls or contact surfaces. General results of these tyes of interactions are summarised and noted, with reference to ressure-velocity diagrams, in aendix D. The third tye is most relevant to discussion of free surface reflection and is discussed in detail in the following section..8.. Interactions of a wave with walls or contact surfaces Figure.5. Collision of a wave with a contact surface: initial and final conditions. Interactions of a wave with walls or contact surfaces occur frequently. A contact surface is defined as an interface between two or more sections of different fluids or of the same fluid at different states or entroy levels [44]. The boundary conditions across any contact surface are that the ressure and article velocity left behind the transmitted wave must always equal the ressure and article velocity left behind the reflected wave [44]. Uˆ = ˆ (.59) L U R L = R (.60)

48 9 These continuity conditions are, generally, only true at the interface. The fluids on either side of the contact surface may have different densities and temeratures. The most general case of collision of a lane wave with a wall or contact surface, during which the above conditions must be satisfied, is illustrated in figure.6. Thomson [49] has shown, by alying only the equations of conservation of mass and momentum to a control surface enclosing a wave, that the of seed of the wave relative to the fluid ahead of it (w) may be related to the states ahead () and behind () it by: ρ ( = (.6) w u u) where the subscrits and denote the arameters ahead and behind the wave resectively. This can be rearranged giving, where Z is the imedance of the wave, δ = = ρw u u δu Thus, for an acoustic wave, the acoustic imedance is: = Z (.6) Z = ρ a (.63) Extending the concet, the shock imedance is [0]: Z = ρ U S (.64) Since the characteristic imedance Z it is a roortionality constant between the imressed article velocity δu and the ressure δ, it is a measure of the stiffness of a material. At the interface between two materials with equal acoustic imedance, an incident sound wave transmits with no reflected waves. For the collision of weak to moderate waves with a contact surface, the Riemann solution alies [46]. It states that for the collision of a shock with a contact surface, the reflected wave will be a shock for: ( ρ a ) > ( ρa) (.65) T and a rarefaction wave for: T I ( ρ a ) < ( ρa) (.66) I where the subscrit T denotes a roerty of the undisturbed, receiving material into which a wave is transmitted and I denotes a roerty of the undisturbed material through which the incident wave travels. Wave imedances are discussed in detail in [46] and [58].

49 30 Reflection of a wave from a rigid wall, area change discontinuity or free surface may be considered as secial cases of collision of a shock or rarefaction wave and are discussed in the following sections Reflection from a Rigid Wall The collision of a shock wave with an infinitely rigid (no movement is ossible) wall may be considered a secial case of collision of a wave with a contact surface. In this case, the aroriate comatibility condition [55] is that the article velocity behind the reflected wave must be equal to zero (as it is behind the wave that is transmitted through the wall), since the acoustic imedance of the wall is infinity. The reflection of acoustic waves in any fluid and that of finite waves in gas or liquid media from a wall are considered in aendix E Shock or Rarefaction Wave Passing Through a Discontinuous Change in Cross Section An area change discontinuity may be described as a secial tye of contact surface. Disregarding any of unsteady disturbances described in aendix F the strength of a shock assing through a change in cross section is modified and a reflected wave created. When assing through an area enlargement, the transmitted shock will be weaker than the incident wave, since it must comress a greater volume of fluid. When assing through an area reduction, the transmitted shock will be stronger than the incident shock. Methods used to solve these roblems, described by Rudinger [44] and, for acoustic waves, by Parmakian, are discussed in aendix F Free Surface Reflection Consider the secial case of normal incidence and collision of a shock wave with a contact surface, where the incident wave roagates in a liquid and the transmitted wave roagates in a gas. The surface of the water will initially be accelerated uwards. Since the comressibility of water is far greater than that of air (refer to the tables of roerties in aendix G), there is virtually no oosition to motion of the boundary and hence no

50 3 restriction on the normal comonents of velocity [55]. At room temerature, the acoustic imedance of water is about 3500 times that of air (refer to aendix G). Since the acoustic imedance of the liquid is high, the velocity and dislacement of the free surface (i.e. the velocity imarted to the medium) will be low, such that changes in gravitational otential energy are much smaller than the energy of comression [55]. The ressure of the air above the free surface cannot change significantly for all realistic movements of the surface [55] i.e. the accelerated liquid interface cannot transmit a significantly strong shock wave into the air above it, leaving the ressure of the air above the liquid at its initial value. Thus, free surface reflection of a shock requires that the ressure above the surface, and by the comatibility condition (.60), below the surface as well, be unchanged [55]. A reflected rarefaction wave, and not a shock wave, is required to reduce the ressure to its initial value [55,59]. The rofile of this rarefaction wave is the negative or a reflection [55,60] of the shock wave rofile (essentially equal in magnitude or strength but oosite in sign to that of the incident wave). Thus, free surface reflection may be seen as an oosite case to that of rigid wall reflection where the reflected and incident wave are, in the acoustic aroximation, of equal strength. The resultant ressure variation at all deths below the free surface may then be determined by suerosition (simly the algebraic sum) of the ressures in the incident comression and reflected rarefaction waves [36,55,60,6], taking into consideration the aroriate delay times (difference in time of arrival of the waves at the deth under consideration). This suerosition rincile is an acoustic aroximation, which is valid since liquid shock waves are weak as discussed reviously. Consider the illustration of the absolute ressure at some deth below the surface shown in figure.6. As the head of the rarefaction rogresses downwards, it encounters the ressure remaining behind the incident wave some deth below the surface, which has decayed. The negative rarefaction front (the ressure dro) is suerimosed on the later, weaker art of the ositive wave. Thus, the absolute ressure values deend on the rate of

51 3 decay of the incident wave, ath differences and the deth under consideration. The net ressure behind the front may reach negative absolute ressures when the front encounters regions of lower excess ressure [55]. If the incident ressure wave is of sufficient strength, the reflected wave may decrease the ressure, at some deth below the free surface, below the threshold level, causing the liquid to cavitate. The dashed line on figure.6 reresents the net ressure obtained by considering suerosition of the incident and reflected waves but ignoring cavitation. It shows an unrealised state of tension that would occur if the liquid resisted cavitation, while the solid line in fig indicates the net ressure when cavitation occurs [55]. Figure.6. Resultant absolute ressure-time curve at some deth below the free surface (adated from [55]).

52 33 3. Cavitation The cavitation rocess is comlicated by such factors as energy losses involved in the damed oscillations of a cavity, heat conduction, viscosity, comressibility, surface tension, temerature discontinuities at the hase interface [] and transfer of heat and mass (by diffusion of ermanent gas, evaoration of liquid and condensation of vaour) across the bubble wall. The roblem involves, essentially, two hases couled through a moving boundary (the bubble wall) []. 3.. Liquids Under Negative Pressure A liquid may, deending on the ambient conditions, exist as: A subcooled liquid, also known as a comressed liquid or simly as a liquid [5], where the temerature of the liquid is below its saturation temerature and the liquid is under more ressure than it needs to be in order to stay liquid. A saturated liquid [5], that is, a liquid that is at its boiling oint (or saturation temerature), which may coexist with its vaour hase. Liquid in a suercooled state, which is metastable with resect to freezing. Liquid in a suerheated state, which is metastable with resect to boiling (liquidvaour transition). Liquid in a stretched state, i.e. liquid under negative ressure, which is metastable with resect to cavitation (liquid-vaour transition). Liquids may exist as metastable states (the last three mentioned above), which are states outside the stability region of the hase diagram of the substance, for a certain time. The resent discussion will be limited to the last two (suerheated and stretched) metastable states only. The condition of mechanical stability ( P V) < 0 must be fulfilled for both hases on the solid-liquid line, including the metastable section of this line [65]. At metastable states, the condition ( P V) 0 holds. Metastability means that a small can T fluctuation or disturbance of the liquid of sufficient magnitude will cause [7,0,65] the T

53 34 mechanical stability condition to be violated, which results in vaour, exlosive or raid boiling or cavitation in the liquid, which will revert to the ositive vaour ressure. Refer again to the conventional hase diagram for water, figure.. It is known that the rocesses a-b and b-c corresond to those of boiling and cavitation. However, liquid may exist in the ressure-temerature region labelled vaour as a metastable liquid. From the arbitrary liquid state a, one may reach the metastable state by either increasing the temerature above its boiling oint (a to b, where b is a metastable, suerheated liquid state) at constant (ambient) ressure or by decreasing the ressure at constant temerature (a to c, where c is a metastable, stretched liquid state). Therefore, it is ossible to reach liquid states usually associated with stable vaour states, which have lower chemical otential, without the vaour hase develoing [66]. So arbitrary states in the vaour or metastable liquid region, such as b and c in figure., may exist as vaour, suerheated liquids or liquids under negative ressure, deending on the way in which the state was reached. Cavitation results because stretched liquids tend to equilibrate to the vaour state [67]. The binodal is thus, in the case of suerheated and stretched liquid, a boundary between stable and metastable liquid []. Interesting henomena occur only in liquids under negative ressures. However, if one decreases the ressure below absolute zero ressure, which is not a articularly secial oint in fluids, the liquid roerties do not change discontinuously at that oint [68]. Martinás and Imre [69] have shown, using classical thermodynamics, that negative absolute ressure states, though forbidden in gases, are ossible in liquids. For sufficiently large tensions, a liquid may be exected to be ulled aart ; with holes (i.e. a cavitation region) forming [55]. In very ure liquids with small nuclei, cavitation may occur at large negative ressures [70]. A liquid under negative ressure ulls inward on its container as oosed to exerting an outward ressure when under ositive ressure. The intermolecular cohesive forces between the liquid molecules as well as the adhesive forces between the liquid and container allow liquids to withstand negative ressures [,3]. Liquids under negative

54 35 ressures may be found quite widely in confined saces in the human body [4] and in the xylem of trees [3,69]. From the elementary considerations of section. alone, one might exect that cavities form when the local, external, ressure dros to the saturated vaour ressure of the liquid at the current, ambient temerature i.e. the saturated vaour ressure within the bubble []. However, cavitation does not always begin near the vaour ressure: some exeriments reveal deviations that are not reconcilable with this vaour-ressure concet [] e.g. water has been subjected to a tension of -40 MPa before cavitating [66]. However, this required a very small liquid volume (an inclusion in a quartz crystal) to minimise the robability of imurities being resent [66]. Thus, the cavitation threshold is also known as the tensile strength of the liquid, and is not, in general, equal to the vaour ressure [,5], as will be shown, in terms of bubble dynamics, in section 3.3. In summary, if the ressure dros down to, or below, the vaour ressure any minute cavity will grow and if the ressure dros to absolute zero or negative ressures, the rate of bubble growth will increase [] Cavity Dynamics of Bubbles Initiated in a Stretched Medium Hassanein et al. [7] ostulated that a cavity that forms during a negative ressure hase initiates a relaxation shock wave when the stretched medium returns from low density and ressure to normal density and ressure. The shock is a result of the ressure difference across the bubble wall: the ressure in the cavities is equal to aroximately zero while the liquid is at negative ressure. Hassanein et al. concluded, from calculations [7,7], that any cavity that is initiated during a negative ressure hase will continue to grow and not collase when the ressure increases. It will exand freely at a decreasing rate to a value determined by the amount of elastic energy stored in the system. This illustrates a difference between cavity dynamics in a stretched medium and cavitation initiated at ositive ressures, where vaour bubbles collase during an increased ressure hase. The conclusion that bubbles initiated in a stretched medium do not collase, during the ositive ressure hase, was

55 36 attributed to discharging or uloading of the liquid medium, away from the bubble, by the relaxation shock wave initiated at incetion. 3.. Nucleation Knowledge of the local ressure variations alone does not determine when and how cavitation will occur [7]. Cavitation is a comlex rocess where the liquid s hysical state i.e. the nature and content of heterogeneities in the liquid and container walls lay a major role in the nucleation of bubbles, are difficult to redict accurately. The influence of nuclei makes the rediction of when cavitation will occur, i.e. cavitation threshold, difficult [5,7,34]. From the theory of Fisher [40,73], the homogeneous cavitation ressure, for ure water, has been estimated as 3 MPa in water. According to Young [], the tensile strength of water at room temerature has been redicted as aroximately 00 MPa. Other sources have estimated the ultimate tensile stress of water as 5-00 MPa [74,75] and hundreds of atmosheres [76]. Using classical nucleation theory [4], the tensile strength of water near the critical oint has been estimated as 0 MPa. However, such values are unrealistic for most normal liquids [38]. Consider the first stage of cavitation, during which a bubble forms, which is referred to as the nucleation stage. There are two main tyes of nucleation [77]: Heterogeneous nucleation occurs when the bubble formation is influenced by external factors such as imurities or walls [5,66]. In this case, the major weaknesses, where ruture would be most likely to occur, are at sites in liquidsolid interfaces (walls or solid articles). Under ractical circumstances, this is the most common tye occurring. It is difficult to analyse quantitatively as the shaes and sizes of the nucleation sites vary greatly and are random. Homogeneous nucleation occurs only in the absence of imurities, walls, etc. This tye of nucleation is thus an intrinsic roerty of the liquid system. As a result, homogeneous nucleation is simler to describe quantitatively [66]. However, it requires secial conditions and usually takes lace very far from equilibrium

56 37 conditions [77]. In this method, thermal motions within the liquid form temorary, microscoic bubble nuclei that may exand to macroscoic sizes [5]. The above values are based on assumtion of homogeneous liquid. Homogeneous cavitation, however, rarely occurs in nature and in ractical alications [66]. Water, which has been elaborately filtered and re-ressurised to several hundred atmosheres, may cavitate at tensions of -30 MPa [7]. When solid non-wetted nuclei of size 0-5 mm are resent, it is likely that otherwise very ure water will ruture at tensions in the order of megaascals [7]. It is certain that cavitation may occur in untreated water at ressures between the ositive vaour ressure and small negative ressures [7]. These emirical observations imly that nuclei or weak sots exist within liquids i.e. liquids are usually not homogeneous. Now consider untreated water. Liquids that cannot withstand significant tensions may be exected to cavitate when tensile stresses in the order of one atmoshere are reached [78, 55, 6]. Studies of the liquid strengths [6] have showed that such liquids will sustain tensions with a limiting magnitude in the order of the vaour ressure (i.e. close to zero). Cole concluded, artly from an estimate that a negative ressure of - bar will suort bubbles more than one or two microns in size and from observations of cavitation behind waves [55], that oen water and water that has not been articularly well urified is unlikely to be able to sustain tensions greater than - bar. This is verified by the exerimental results of Eldridge et al. [79], which indicated that oen sea water can only sustain a negative ressure of only a few bars before cavitation occurs. Štuka et al. found that the magnitude of the tensile strength of non-degassed water is less than 5 bar [4] Cavitation Nuclei Detailed exerimental studies have shown that even after several urifications, distillation and deionisation, real liquids settled contain microinhomogeneities, existing as microbubbles of free gas, solid articles or their combination, which act as cavitation nuclei [38,6,80]. The comlex initial state of the liquid deends on the hysical nature, size, distribution and concentration of these nuclei and determines the initial dynamics of

57 38 cavitation rocesses [35]. The intrusion of imurities or nuclei revents liquids from sustaining tensions as readily as solids and reduces the tensile strength of a liquid from the high theoretical values listed to the low values encountered in exeriments []. This exlains why cavitation occurs at much lower tensile strengths than is estimated thermodynamically. In the absence of nuclei, a liquid may withstand large negative ressures in the order of the theoretical values [7,66]. However, this necessitates a small volume (to minimise the robability of imurities being resent) of very ure liquid [8]. The many tyes of cavitation nuclei include ermanent bubbles containing undissolved gas or uncondensed vaour and quantum vortices in Helium [,8]. In seawater and other bodies of water exosed to the atmoshere, small vaor bubbles may originate from highenergy articles roduced by cosmic rays or radioactivity. Other forms of nuclei include ionising articles and neutrons [70]. However, not all imurities in a liquid will affect the cavitation rocess or the tensile strength. The main tyes of imurities are distinguished between in the following sections Miscible Liquids and Dissolved Solids The resence of a miscible liquid or dissolved solids does not change the hysical roerties of a liquid (such as viscosity, density and surface tension) significantly enough to affect the growth or collase of cavities [], unless resent in large amounts and thus have little effect on the effective tensile strength of the solvent Non-miscible Liquids and Undissolved Solids All liquids wet solids to some measurable extent but do so imerfectly [,]. For both non-miscible and non-soluble substances, when the bond at the interface between the liquid and the solid or between two dissimilar liquids (i.e. low wettability or hydrohobic) is low, weak sots are resent and vaour bubbles form readily. This is exected as adhesion (wetting) is due to the same intermolecular forces that allow liquids to withstand tensions []. Wetting, surface tension, contact angles and the relation between them is given in []. High wettability between a liquid and a non-miscible liquid or undissolved solid imurity will not necessarily revent that imurity from acting as a

58 39 nucleus and romoting cavitation. Exeriments [] suggest that the rigidity of a liquid is also an influential factor: Cavities may form, even at a hydrohilic surface, if it is too rigid to follow the motion of the liquid under consideration [83]. More detailed research in this area may be worthwhile Dissolved Gas It is virtually imossible to eliminate dissolved gas from a macroscoic liquid volume, even after long eriods of degassing [5]. Exeriments (discussed in chater 4) have shown that ressurised water samles may be more resistant to cavitation than unressurised ones containing undissolved air [5], which may exhibit virtually no tensile strength. These exeriments suggest that, in contrast with undissolved gas, gas that is entirely dissolved in the liquid does not romote cavitation or decrease the tensile strength of the liquid significantly Undissolved Gas and Vaour Bubbles Even if no gas bubbles are visible, submicroscoic gas bubbles may act as nuclei [5,7]. Undissolved free bubbles filled with a mixture of the uncondensed liquid vaour and dissolved gases are the most obvious cavitation nuclei, tyically resent in liquids, to consider. Such bubbles of undissolved gas or uncondensed vaour are already cavities in the liquid and hence, must lower the tensile strength of a liquid. Many exeriments [] in hydrodynamic and acoustic cavitation show that the ressure at which cavitation occurred decreased as the vaour and gas content was reduced i.e. after degassing rocesses. Free gas bubbles, articularly large, visible ones [7], will slowly (rise) float to the surface and escae. A bubble of radius 0µm in water will rise at a rate of about 0.3 mm.s - []. In addition, diffusion of gas out of the bubble into the liquid will occur. Surface tension forces would increase the bubble gas ressure resulting in the gas and vaour content being comletely dissolved and condensed []. Thus, diffusion and surface tension forces will cause a bubble to collase comletely [7]. It has been estimated [84] that an air

59 40 bubble, of radius 0µm, in air-saturated water will take about 7 s to dissolve. Another calculation redicts a time of.5 s [5]. Similarly, vaour should condense comletely []. In addition, free bubbles have been found to ersist, even under ressurisation, although their sizes diminish [5]. Thus, some stabilising mechanism or host must be resent for bubbles containing undissolved gas or uncondensed vaour to exist stably as nuclei in the liquid. Various stabilisation mechanisms have been roosed [,,5]. According to Harvey et al., [85] minute, undissolved gas nuclei ockets may exist in the microscoic and submicroscoic, hydrohobic cracks, crevices or interstices of hydrohobic solids, i.e. container walls or in imerfectly wetted, articles []. They are stabilised because, under such conditions, the surface tension acts to decrease rather than increase the ressure, thereby reventing the gas from dissolving []. Pressurisation may force such traed nuclei to collase by overcoming the surface tension and forcing liquid into the crevices, thereby dissolving the gas []. In other cases, the bubble will not collase because its geometry is such that the gas-liquid interface has a convex curvature viewed from the liquid, such that the surface tension sustains the high ressure [5]. Fox and Herzfeld [,86] roosed that small bubble nuclei are surrounded by monomolecular skins made u of organic imurities, which give the free surfaces of the bubbles sufficient elasticity to withstand high ressure [5], modify the effective surface tension, retard evaoration and act as diffusion barriers, thereby reventing the bubble from dissolving. The model of Harvey et al. is thought to be more satisfactory as it is able to exlain all observed behaviours (e.g. scatter in the cavitation threshold as discussed later in section 4.4) without ostulating imrobable fluid roerties []. Desite this, the organic skin mechanism is now more widely acceted because of studies confirming the existence of small amounts of surface contamination and their ability to generate significant surface effects [5]. Another stabilisation mechanism, involving the ossible roduction of nuclei continuously by cosmic radiation has been suggested [5]. Consider the model of Harvey et al. It is known that cavitation often starts at or near boundaries. This is due to crevices, which are often resent in container, ie or channel

60 4 walls and are able to suort gas nuclei of size in the order of 0-4 mm []. Exerimentally [,87], cavitation also occurs readily within the body of a samle of liquid. Within the body of the liquid, susended, solid articles rovide the crevices for nuclei stabilisation instead of the walls. Ordinary ta water may contain thousands of these solid articles er cubic centimeter []. The size, shae and number of solid articles affects the tensile strength of the liquid []. Susended articles, such as atmosheric and industrial dust, with diameters in the order of 0 µm, may suort gas nuclei of size in the order of µm [] The Statistical Nature of Cavitation Figure 3.. Cavitation robability as a function of Transducer Driving Voltage [40]. Cavitation is essentially a stochastic or intrinsically random henomenon [40] i.e. reeated exeriments under the same conditions do not always result in cavitation [35]. Cauin and Fourmond [40] induced cavitation in Freon ultrasonically, using a iezoelectric transducer, and detected incetion using light scattering, otical imaging and acoustic detection methods. Their study illustrated the statistical nature of cavitation. The cavitation robability was defined as the fraction of nucleation events resulting from ulses of low ressure. Figure 3. is the result of their analysis of the robability of nucleation occurring. Its general trend is similar to that of results of Maris and Konstantinov [88] who used similar methods to roduce cavitation in liquid Helium. The transducer driving voltage was aroximately roortional to the maximum negative ressure roduced at the focus i.e. the amlitude of the negative ressure swing at the

61 4 focus [88]. Thus, it is reasonable to assume that the variation of cavitation robability with the strength of a rarefaction wave in a liquid, rather than transducer driving voltage on the abscissa of the grah, will follow the same trend shown in figure 4.. Isolated cavitation events may be caused by large, random nuclei [39] Sherical Bubble Dynamics The following theory, initially develoed by Rayleigh [89] and later extended by Plesset et al. [90,9] allows analysis of the dynamic behaviour of a bubble in an infinite liquid. Figure 3.. A sherical bubble in an infinite liquid [5]. A bubble is forced to collase by external ressure and surface tension and forced to stay oen by the ressure within it. Thus, a ressure balance across the interface of a erfect gas bubble of radius R yields [,5]: B σ = (3.) R where B is the ressure within the bubble, is the external ressure at infinity and σ is the surface tension, which is the manifestation of inter-molecular forces reventing hole formation [,,5]. If the bubble contains gas and vaour, then B = G + v ( T ) (3.) and from equations (3.) and (3.) v ( σ T ) = R (3.3) G +

62 43 where the bubble ressure is the sum of G, the artial ressure of the gas within the bubble [5] and v, the saturated vaour ressure, which is a function of external temerature only [5]. Consider a bubble in equilibrium in a liquid. In the following, arameters at this initial equilibrium state are reresented using the subscrit i. If the temerature in the bubble is uniform and the bubble is assumed to contain only vaour, then = and from equation (3.), the stability condition is [,,5,77]: Bi vi σ = vi (3.4) Ri Thus, the external liquid ressure must be lower than the vaour ressure vi for the bubble to remain in equilibrium. If is maintained at any value below σ / R, then the excess ressure causing growth will increase and the bubble will no longer be in equilibrium and will exand [,,5,77]. If, the radius of the largest bubble nucleus resent is referred to as the critical radius and denoted by R Ci, then the liquid tensile strength is vi σ = c = (3.5) R Ci Thus if one assumes vi = 0, then the cavitation threshold will be negative c < 0 i.e. the liquid will sustain a negative ressure before cavitating. If one assumes that vi = 339 Pa then c < 339 Pa. This illustrates that, cavitation occurs, in general, at a ressure below the saturated vaour ressure. Winterton [77] modified the stability equation (3.5) to include the contact angle of a crevice for heterogeneous cavitation. However, his equations are not readily alicable due to the need for secification of nucleation site arameters, which vary widely and are difficult to quantify. v i The Rayleigh-Plesset Equation The general form of the Rayleigh-Plesset equation is [5,89,90,9]: B ( t) ( t) d R 3 dr = R + ρ dt dt L 4ν L + R dr dt S + ρ R L (3.6) In the above equation, it is assumed that the bubble contains both vaour and contaminant gas. It is also usually assumed that no areciable mass transfer occurs between the gas and liquid excet by vaorisation and condensation i.e. no diffusion takes lace. Defining

63 44 a reference state, denoted by the subscrit o, at which the bubble radius, external temerature and the artial ressure of the gas within the bubble are R o, T and Go resectively, and assuming that the ideal gas equation are valid during the comression and exansion of the gas, the artial ressure at any time is: 3 T B Ro G = Go (3.7) T R Combining equations (3.), (3.6) and (3.7) yields the following equation, equation (3.8) V ( T ) ( t) V ( TB ) V ( T ) Go T B Ro d R 3 dr 4ν L dr S + + R + + ρ L ρ L ρ = + L T R dt dt R dt ρ L R It is now assumed that the exansion and comression of the gas in the bubble are olytroic rocesses. Thus, in terms of the olytroic constant k, G 3k Ro = Go (3.9) R and the third term in equation (3.8) becomes: 3 ρ Go L T T B Ro R 3k (3.0) The first term in equation (3.8) is referred to as the tension or driving term as it is associated with the external, driving ressure field, while the second term is called the thermal term as it is associated with thermal effects and the first and second terms on the right-hand side of the equation are the inertial terms. Brennen [5] defined the first critical time as the time during a bubble s growth when the order of magnitude of the thermal term increases and aroaches the order of magnitude of the inertial terms. Once this time has elased, the comarative significance of the terms in the equation (3.8) changes, with the thermal term becoming dominant. This first critical time deends on the first term (the tension) and a thermodynamic quantity that in turn deends on the liquid temerature only. The value for water, at temeratures of about 0 C, is in the order of 0 s, as estimated by Brennen [5]. This is much longer than the duration of the growth hase of a cavitation bubble. Thus, bubble growth occurring in water at room temerature is said to be inertially controlled, which simly means that the growth is unhindered by thermal

64 45 effects i.e. the growth does not take long enough for the thermal term to become dominant. Consequently, T B =T and the thermal term in equation 3.8 is neglected, giving V ( T ) ( t) + ρ ρ L Go L R o R 3k d R 3 dr = R + dt dt 4ν + R L dr S + dt ρ R L (3.0) In summary, this equation was develoed on the basis of the following assumtions [5]: The bubble is isolated in an infinite liquid (i.e. unaffected by solid articles, walls and other bubbles). The bubble contains both vaour and contaminant gas. The liquid is a Newtonian fluid. The bubble remains sherical throughout the growth or collase rocess. The liquid density, ρ L and viscosity µ L, as well as the temerature of the liquid T (and hence T B ) are constant and uniform. The bubble contents are homogeneous and the temerature, T B (t), and ressure, B (t), within the bubble are uniform. The olytroic assumtion, equation 3.9, and ideal gas equation aly for the bubble contents. It is assumed that the growth and collase events occur too raidly for significant mass transfer of contaminant gas to occur between the bubble and the liquid i.e. the mass of contaminant gas in the bubble is constant and no diffusion takes lace. The form of Rayleigh s equation above only alies if the seed of the collasing bubble wall, R &, is small comared to the seed of sound in the gas a g. i.e. R & / a << []. g The Resonse of Bubbles to Rarefaction Waves Qualitative analysis using the first integral of the Rayleigh equation has shown [9] that, assuming an instantaneous ressure dro three tyes of solutions are ossible for different values of the ressure at infinity and the nucleus radius R o : Unbounded bubble growth. Asymtotic attainment of a finite size during an infinite time. Periodic oscillations of the nuclei. With the only other ossibility being of invisible bubble ulsations.

65 46 Figure 3.3. Regions of the P-V o lane (adated from [9]). Which region occurs deends on the rarefaction wave strength (/ o ) and the initial radius R o and visible radius R v as illustrated in figure 3.4, where R o and R v are exressed in terms of the non-dimensional arameter V o =(R o /R v ) 3. From this qualitative numerical analysis, Kendrinskii also found that cluster formation may develo in two ways [9]:. In weak rarefaction waves (e.g. waves across which the ressure dros from a value o to a value of about -0. o ) or acoustic fields, the time to attain visible size deends largely on the initial radii R o in the sectrum. The zone is gradually saturated with bubbles, which attain visible size after different time intervals according to their initial size [,9].. In intense rarefaction hases, where the negative ressure may be large, the cavitation zone is instantaneously saturated by bubbles of the same size. All of the bubbles of the whole initial sectrum of nuclei attain detectable size simultaneously. Thus, the bubble density and concentration eaks immediately in the cavitation zone, which is characterised by a uniformity of bubble sizes. These theoretical results have been verified emirically and are acknowledged as exerimental facts [9].

66 47 The oscillation of a regular cavitation bubble is similar to that of a gas shere containing detonation roducts, which results from an underwater exlosion. Such a gas shere may be considered as a large cavitation bubble [0]. In all cases, the inertia of the liquid and the elasticity of the liquid and gas rovide the necessary conditions for an oscillatory system [55]. The oscillation of a gas bubble roceeds as follows: Firstly, during bubble growth, the exansion is sustained by the inertia of the out-flowing liquid [55]. This rocess slows down as mass is transferred from the liquid to the cavity, which becomes increasingly occuied. The dense fluid within must rearrange to accommodate more molecules [4]. Then as the gas ressure within the bubble dros, the external ressure constrains the bubble and causes it to contract. The bubble contracts until the gas comressibility becomes significant, arrests the motion and then reverses it [55]. The oscillations continue with diminishing amlitude until a uniform state is reached [4] Collase of Cavitation Bubbles If an exanding bubble is subjected to a rise in ressure, its growth will be arrested and it will collase violently and ossibly disaear by solution of gases and condensation of vaour []. Xiao et al. [4] simulated the collase of a cavitation bubble, using a Molecular Dynamics aroach, which has been shown to be a better suited to the final stage of collase and to rovide a more realistic descrition than traditional fluid mechanics or classical nucleation theory, which rovides no details of the collase rocess. Simulations showed that bubble collase to an undetectable size is followed by the reviously discussed damed oscillatory behaviour, which may or may not be detectable. It was found that the larger the initial bubble size, the more ronounced the oscillatory nature of the collase and the higher the final temerature will be. According to the simulations, the final temerature was at least (for the smallest bubble) 9 times the critical temerature (about 5800 K for water). Bubbles may either collase while retaining sherical form and then emitting shocks at the instant of rebound or, in the resence of other cavities, walls or free surfaces, in asymmetric form resulting in liquid jets and violent imacts at solid boundaries [93].

67 48 4. Literature Review The resent discussion is a review of studies that have shown methods of studying the effects and behaviour of cavitation rocesses. Tensions and cavitation may be induced in a liquid by static or quasi-static methods (by the Berthelot or centrifugal methods) or by dynamic methods (ulse reflection techniques, tube arrest or shock tube methods). Overton and Trevena [95] and Williams and Williams [96] rovided a good overview of dynamic stressing, while both static and dynamic methods are surveyed in [59] and [76]. Aendix H lists cavitation thresholds determined by static and dynamic exeriments. Berthelot Tubes are a widely used method for subjecting liquids to negative ressures and roducing cavitation. They are discussed, at length, in [,59,75,97-03]. In these exeriments, cavitation is most likely to occur at the walls i.e. by loss of adhesion [73,99,03]. The centrifugal method is discussed, in detail in [,59,04]. It essentially involves a horizontally sinning tube filled with liquid in which tension is generated by centrifugal force []. Dynamic methods are the main focus of this chater. 4.. Pulse Reflection Techniques Pulse reflection techniques make use of the henomenon described in section Such exeriments usually differ in the methods used to roduce the incident comression ulses and in the nature of the free surface. These exeriments may also differ in the method of reflection. The concet of free surface reflection and its associated consequences, as described earlier, need not involve a gas-liquid interface. A similar henomenon will occur when a comression wave roagating in a high imedance medium is reflected at any relatively low imedance boundary [96]. Davies et al. [,05] erformed free surface reflection using a tube with istons at both ends. One was struck by a lead bullet to generate a comression ulse within the tube. The other iston, which was light, acted in as a free surface, reflecting a rarefaction wave when iminged uon by the ulse. The liquid-gas interface in free

68 49 surface reflection exeriments may also be relaced by a low inertia or free late as illustrated by the exeriments of Eldridge et al. [55,79] who used a 0.5 mm thick sheet of cellulose acetate, about 50 mm in diameter and backed by air. A common material used as a flexible membrane in such exeriments is Mylar. The technique of ulse reflection in which the liquid is confined between a solid surface and a flexible membrane, is referred to as the cell method [59] Exlosions The earliest exeriments using ulse reflection techniques were erformed in studies of underwater exlosions in large bodies of water [59,60,6,06,07]. In general, the resulting ulses have shorter rise times than those generated in bullet-iston exeriments (described in section 4..) and similar decay time constants in the order of s while tensile strength values are in the order of megaascals [07]. On a smaller scale, Wilson et al. [59,80,95,08] detonated small (0.g) exlosive charges, resulting in sherical comression waves in water that were reflected at a free surface above the charge. The maximum tension that can be sustained by ordinary settled ta water was estimated as 0.85 MPa while the value for deionised and degassed (by evacuation) water was.5 MPa [59,80,08]. These tensions were determined by taking high-seed hotograhs of the surface effects and relating the liquid article velocity to the maximum tension using the Rankine-Hugoniot equations [59]. Richards et al. [09] ignited an oxyacetylene mixture in gas above a liquid column. The detonation wave was transmitted into the liquid as a comression wave, which was subsequently reflected as a tension ulse at a thin sheet, made of Mylar, which suorted the column at the lower end. The maximum ressure behind the transmitted comression waves was about 7 MPa. The duration of the maximum tension was µs. The tensile strength recorded, for deionised water, was. MPa. Kedrinskii studied the effects of the underwater exlosion of a. g charge of high exlosive, fixed 5 cm below a free surface as detailed in [36,37,38,80,06]. Free surface

69 50 reflection of the shock waves resulted in intense rarefaction waves [80]. The hotograhs taken at high seed, figure 4., showed that the violent cavitation caused salls or layers to became detached from the main cavitation zone. This effect is similar to the wellknown effects of brittle solid fragmentation when a strong shock roagates through the solid and encounters a solid-air interface [38]. Thus, a cavitating liquid exhibits both lastic (foam) and brittle (salls/drolets) behaviour [37,80]. The henomenon of free surface reflection is essentially the same as that of salling of metals [80]. Figure 4.. Cavitation zone develoment due to underwater exlosion of a.g charge near the free surface of a liquid. A mass of bubbles formed by intense cavitation (due to a free surface reflection), may, in effect, became a new free surface for the remaining body of liquid such that the wave reverberating within a layer with two free surfaces [0] Bullet-Piston (B-P) Methods The bullet-iston (B-P) method is the most widely emloyed method of achieving ulse reflection [96]. In such exeriments, the initial comression wave is roduced by imact of a bullet on a iston in contact with the test liquid as exlained in []. Tyically, the rise time and total duration of the incident wave are in the order of 50 and 500 microseconds resectively [59]. Incident shocks with eak ressures of u to 30 MPa may be generated in water using this method [59].

70 5 Couzens and Trevena [] erformed conventional B-P exeriments using a vertically mounted stainless steel tube.4 m in length with an internal diameter of 5.4 mm. The bullet was fired, by a 0.-in. calibre rifle, at a iston below the water column. The resulting liquid ulse had a rise time of aroximately 50 µs and a decay, after the eak ressure, lasting µs. Emirical relations between the viscosities of different liquids and their tensile strength were derived. The relations indicated that liquids with high viscosity are able to withstand greater tensions than those with lower viscosity, as the investigators had exected intuitively. Bull [3], Carlson and Levine [4] and Williams and Williams [96] erformed similar exeriments using various liquids (water, olive oil, syru, etc), glycerol and silicon oils resectively and derived similar emirical relations for these liquids. Bull used a B-P method, similar to that of Couzens and Trevena [], while Williams and Williams [96] used a cylindrical tube (.4 m long, inner diameter of 4 mm). In their B-P method, the lower iston was imacted uon by the bolt of a cattle stun gun instead of a bullet. Carlson and Levine emloyed the cell method and generated incident comression ulses by using a ulsed electron beam generator as described in section 4..3 [59,4]. Sedgewick and Trevena [98] found that olyacrylamide solutions with high viscosity had lower tensile strengths that those with lower viscosities. It was thought that this discreancy might have been due to interactions between olymer molecules. Sedgewick and Trevena [5] erformed B-P exeriments using very similar equiment to that of Couzens and Trevena described above. The study illustrated that boiling and deionisation of a water samle increases its tensile strength. Sedgewick and Trevena [98] later extended these B-P exeriments with the same equiment and found that the dissolution of olymer additives in deionised water did not affect the cavitation threshold of the solvent noticeably. Overton and Trevena [95] erformed B-P exeriments using very similar equiment and water as the test liquid. Pressure traces indicated maximum tensions of about 0.45 MPa. All of these tubes used were vertically mounted and made of stainless steel tube m in length with an internal diameter of 5.4 mm. The bullet was fired, by a 0.-in. calibre rifle, at a iston below the liquid column.

71 5 When using B-P and similar methods, additional factors, such as the length of the iston, have been found to affect the observed cavitation threshold [6]. Overton and Trevena considered a ulse, initiated by imact of a bullet on a iston, that travels through the iston and reaches the iston face where it is artly transmitted into the liquid and artly reflected back into the iston. The reflected wave is reflected again when it reaches the oosite face of the iston. As a series of such internal reflections occur, a succession of comression waves are transmitted into the liquid. It follows that, if the iston is short, then the comression waves will be close together resulting in a higher eak ressure than if the iston is long [95,6] Pulsed Electron Beam Generator Carlson and Henry [59,95,7] roduced stress waves, by deosition of energy, using a ulsed electron beam generator, into a solid, mm thick late made of comosite organic material, adjacent to a liquid samle. These waves were transmitted into the 3 mm thick samle of glycerol, and were almost unaffected at the interface due to the similar acoustic imedances of the late and liquid. The comression waves, which had short durations of about 0. µs, were reflected at a 6 µm thick sheet of Mylar film, which acted as a free surface. The cavitation threshold of glycerol was measured as 60.8 MPa. Carlson and Levine [4] and later, Carlson [8] extended the exeriments using glycerol and mercury as test liquids resectively Illustrative Case Secial consideration is now given to a recent study that yielded imortant information about cavitation: Kedrinskii et al. [35] roduced cavitation and investigated the effects of certain factors on the rocess using the free surface reflection method. Water resulting from a single distillation without any additional urification was tested. The comression waves (tyically with rise times of about.7 µs and of duration in the order of 3-5 µs) were generated by an electromagnetic acoustic emitter as described in [5]. High seed filming showed that there were significant differences between the dynamics and intensity of cavitation near the free surface and at greater deths. When exeriments

72 53 were erformed immediately after the liquid samles were laced, the cavitation bubbles were uniformly distributed over the cavitation zone. However, when a samle was left to settle for one hour before being tested, a dense, localised layer of bubbles formed aroximately 3 mm below the free surface. These differences in cavity formation are deicted in figure 4.. This effect of the time of contact of liquid with the atmoshere was attributed to the layer of liquid near the free surface becoming saturated with air. This robably increased the concentration and initial, equilibrium radii of the cavitation nuclei. Thus, while leaving water to settle allows free bubble nuclei to escae or dissolve as discussed in section 3...4, the amount of cavitation nuclei may actually increase. Figure 4.. Photograhs of cavity clusters that develoed in distilled water samles tested immediately (uer frame) and one hour after (lower frame) being laced into a cuvette [35].

73 54 The light scattering method showed that the cavitation zone reached a state where aroximately 50 bubbles er cubic centimeter were resent with bubble diameters ranging from 70 to 70 µm (average ~30). Using light absortion and caacitance methods simultaneously, the threshold of cavitation incetion of distilled water was determined as between.7 and -.9 MPa. Comression ulses with ressures of about.9 MPa had to be generated in order to roduce cavitation Intense Cavitation Figure 4.3. Sequence of x-ray frames (time between frames is 00 µs) of cavitation zone dynamics resulting from shock wave reflection at a free surface. Under intense tensile stresses, cavitation is characterised by ractically unbounded, inertial growth of bubbles [38]. On a larger scale, intense cavitation, or cavitation destruction, involves large scale change of state by unlimited develoment of clusters of bubbles, until the liquid is transformed into a foam tye structure [36], followed by (disintegration) fracture into cavitating fragments [80] and then transition into a gas drolet hase state [36]. A comlete inversion of the two hase state of liquids occurs [7,06]: The initial liquid medium, containing a disersed gas hase (gas bubble nuclei) is converted, ultimately, into a gas medium containing the disersed liquid hase (drolets). A suitable mathematical model (the frozen field model), for describing the late stages of intense cavitation resulting from unbounded growth of bubbles, has only

74 55 recently been develoed [9,0]. Kedrinskii erformed ulse-reflection exeriments using shock tubes to illustrate intense cavitation. In one case, a shock wave was generated in the liquid by the imact of a iston [38]. The use of three x-ray aaratus, triggered by ressure gauge signals, with different time delays, allowed ultra-high-seed x-ray ictures to be obtained. Comuter rocessing allowed the internal structure of the oaque cavitation zone to be determined as shown in figure 4.3. Kedrinskii et al. [37,38,] erformed novel exeriments to investigate the inversion of two-hase liquid to states (transition from a bubbly liquid to a gas drolet structure). A dro of distilled water, only about cm wide [37,38], or a dro of viscous liquid, such as a gum-resin-acetone solution [] was laced on the diahragm of an electromagnetic shock tube. The shock tube generated a 5-6 MPa short (3-4 µs long) shock wave, which imacted the diahragm. The shock waves were transmitted into and reflected from the free surface of the dro, resulting in negative ressure and cavitation (mainly in the central region of the dro). The liquid underwent a rocess of transformation from liquid to a foam or mesh tye structure as shown in figure 4.4. Figure 4.4. Process of Intense Cavitation in a Liquid Dro for times 0, 50,60,70 µs [].

75 Tube Arrest Methods Tube arrest or cavitation tube methods involve imact resulting in downward acceleration of the solid boundary at the bottom of the column. The inertia of the liquid caused it to continue its uward motion, resulting in a tension wave. The wave was first thought to originate at the bottom of the column and travel uwards [87,95,,3,4] because cavitation usually occurs near the base [5]. However, studies [5,6] have shown that, in fact, the tension ulse originates from the to of the liquid column and travels downwards as assumed in [7]. Cavitation tends to aear at the base because the tension at the base increases momentarily after the ulse is reflected, resulting in cavitation in the vicinity as confirmed by hotograhs [5]. The tension wave arises from continuing uward motion of the free surface [5]. Chesterton [87,95] erformed exeriments using a tube arrest method. His aaratus consisted of a vertically mounted glass tube, half-filled with water. The tube was ulled down against tensioned suorts and then released. This caused it to accelerate uwards, over about 50 mm, before being stoed by a rubber buffer. At this buffer, the tube accelerated downwards at a raid rate. High-seed hotograhs showed cycles of cavitation growth and collase. Overton and Trevena [95], and, more recently, Williams et al. [6,7] erformed similar exeriments. Chesterman [87], Overton and Trevena [95] and [7] used glass, Persex and olycarbonate tubes, with similar internal diameters of 5.4,.5 and mm, resectively. All of the tubes were m long. In some exeriments, Chesterman used a square (5 5 mm) Persex tube [87]. Chesterton [87] could vary the tube velocity at imact between and 6 m.s -. He tested distilled water and acetone solutions. He found that reeated stressing caused a decrease in tensile strength: In his tube arrest exeriments, initial reeated tests (referred to as riming ) were required before cavitation occurred in acetone solutions and carbon tetrachloride, while the number of cavities that formed aarently increased with the number of tests erformed. Chesterman showed that a single, isolated bubble could be roduced and analysed. This was achieved, firstly, by suitable control of the number and tye of riming tests run and, secondly, by close-u hotograhy and secial negative

76 57 rocessing. This illustrates the value of the tube arrest method in that a single cavity can be studied. Overton et al. [95,] also used water as the test liquid. By varying the height through which the tube traveled before being arrested, they varied the magnitude of the tension ulse generated. Overton and Trevena measured tension ulses in the order of -0.5 MPa [95]. High-seed hotograhy showed that a small cluster of bubbles formed - cm above the base of the tube. Overton et al. later extended this method [] and investigated other liquids. Williams et al. [4-7] used water that had been subjected to filtration and nuclear grade deionisation rocesses. The full exerimental details regarding the exerimental equiment and methods of Williams et al. are given in [4-7] (reference [8] details older equiment). The imact velocity of the tube was about 5 m.s - in [6]. Earlier work [3,4] on the roblem was focused on exlaining effects of secondary ressures waves, which limited the usefulness of the method [3], as discussed later. More recent work [5,6] was more focused on the study of the effect of cavitation bubble collase and may contribute significantly to the assessment of the safety and effectiveness of lowfrequency ultrasound. One conclusion of this recent work was that shocks resulting from cavitation bubble collase and rebound may be reflected at a free surface, resulting in a tension wave that may cause further cavitation [6] as suorted by hotograhic records. The characteristic times of these waves are similar to those arising in biomedical alications. Williams et al. [4] built a searate rig and confirmed that tensions ulses could roagate further cavitation, the extent of which was also investigated. In other exeriments [5,6], an air bubble was introduced through a metered syringe, such that it rested below the free surface. The tube arrest method was subsequently emloyed, thereby generating cavitation at the lower end of the tube. When those bubbles collased, a shock wave was initiated, which then traveled uwards and iminged uon the bubble near the surface. Collase of this bubble roduced a liquid jet in the direction of the free surface. Such exeriments have been erformed in water [6], and motor

77 58 lubricants [5]. The results were confirmed by video images and ressure records [5,6]. In water, a liquid jet roduced in this way extended about cm above the free surface. In the lubricants, the liquid jet ierced a latex membrane stretched across the to of the tube. This collase behaviour near a free surface is similar to the that of bubbles near flexible membranes or solid surfaces coated with an elastomer (see references in [5]) and thus warrants further consideration. Williams et al. also used the tube arrest method to find the tensile strength of water [6, 7]. Tension ulses with stressing rates of about 0 bar.µs - were roduced. Sometimes, one or more isolated cavities were visible at the base of the tube. On other occasions, bubbles formed both within the body of the liquid and at the tube walls and base. The exerimenters used ressure transducers indirectly, by determining the seed of the initial tension ulse, from which the associated ressure changes were calculated. The calculation method is detailed in [5,6,7]. The mean value yielded a value of 6.5 MPa for the tensile strength of water. Lackmé [59,95,9] created an uward-travelling exansion ulse by fixing a metal late to the bottom of a vertical bar, which suorted a column of liquid above it, and droing a weight to fall on the late. This method is similar to the tube-arrest method. The column of liquid was contained in a tube with an internal diameter of cm and a length of 40 mm. The duration of the tension ulses were about 00 µs and could be varied by changing the height of the fall of the weight. The water column could be subjected to tension waves of u to -0.5 MPa. This method is relatively unexlored. Another exeriment involving downward acceleration of a tube was erformed by Hansson et al. [30]. They used a rectangular ( 8 mm) tube, suorted by a sring. Imact of a mass on the uer end of the tube resulted in downward acceleration for a few hundred microseconds. For large accelerations, cavitation occurs, as illustrated in figure 4.5. This case is analysed theoretically in [36,37,38,80,9,06,30].

78 59 Figure 4.5. Cavitation develoment in a tube due to downward acceleration of m.s -. Time between each icture is effectively 50 µs [30] Hydrodynamic Shock Tube Methods Fujikawa and Akamatsu [93,94] emloyed a more simle method to study cavitation. Their equiment consisted of a hydrodynamic shock tube made u of a.8 m long, gasfilled driven section above a. m long driver section, filled with water to a height of m. Their aim was to observe bubble growth and collase (due to shocks) and deendencies on the initial gas ressure Go within the bubbles and the distance from the from tube walls to the bubbles and to investigate the mechanism of violent imact leading to damage to walls. The exansion fan iminged on the free surface of the liquid and was transmitted into it, thereby initiating bubble growth. The rarefaction wave transmitted into the water was also reflected at the bottom, increasing the magnitude of the tension further. The exansion waves were subsequently reflected from the free surface as downwardtravelling shocks, which initiated bubble collase. The driver section was filled with a helium-air (ratio of 60 to 40) mixture ressurised to.64 atm while the driven section was left at atmosheric ressure (air at atm). This allowed a small negative absolute ressure of 0.46 atm to be generated in the liquid behind the exansion wave reflected from the bottom of the tube. Hydrogen-filled

79 60 bubbles, with radii R o equal to 0.4 mm, were generated by electrolysis, using latinum electrodes, to make collase henomena more readily visible (refer to [93] for details). The initiation of the bubbles was synchronised with the electro-magnetically actuated diahragm burst mechanism such that the exansion waves and bubbles reached the test section at the same time. The bubbles exanded and reached a size of R = mm. Pressure traces showed that the jet imingement was undetectable while the violent imacts due to shock waves from rebound were detectable regardless of whether the collasing bubble was sherical or asymmetric. The comression ulses from collasing cavities had durations of about µs. During exlosive erution of a volcano through a vent, magma, which has high gas content, roagates towards the earth s surface while dissolved gases contained therein come out of solution. Once the magma reaches the earth s surface, the degassing may become exlosive, with the final result being total fragmentation into discrete articles. Exlosive volcanic erutions (of liquid magma, ash, etc.) are often driven largely due to raid fragmentation [3], which is often a major factor in determining the intensity of the erution [3]. The rocess is essentially a wave rocess where an exansion fan, centred at the vent, roagates through the magma, accelerating it towards the surface. Sturtevant et al. [33] erformed exeriments using a shock tube technique in which the magma was simulated by suerheated, volatile refrigerant R and R4 liquids. The liquid was laced in the high-ressure section driver section. The refrigerant R was ressurised to 5.69 bar, which is slightly above the saturation ressure at 0 C. The driven section was evacuated. When the diahragm was rutured, raid decomression occurred as tensile stresses were alied. Figure 4.6 is a hotograh of the refrigerant R liquid in the test section during an exeriment. The liquid level was initially at the 8cm mark. The hotograh shows that the combination of suerheat and low ressure had caused the liquid to cavitate or boil intensely forming a two-hase bubble mass above the liquid (aears dark due to backlighting). Such an exansion wave, which results in exlosive vaorisation, is referred to as an evaoration wave.

80 6 Figure 4.6. Side view of test section, showing the result of roagation of a evaoration wave through a suerheated refrigerant samle [33]. Alidibirov and Dingwell [3] used the same exlosive degassing technique, but erformed the exeriments on actual magma samles at high temeratures Ultrasonic and Wave-Focussing Methods Ultrasonic methods usually involve a transducer that asses a high-frequency wave through a liquid, thereby subjecting it to a comression and tension during the ositive and negative half-eriods of each ressure cycle resectively. For sufficiently high wave amlitudes, a bubble will grow during negative half-cycles and collase during the comressional half-cycles [59]. These methods are not discussed in detail. It has been suggested that these methods are not strictly relevant in measuring cavitation thresholds due to the effects of rectified diffusion, which makes comarison with other methods difficult [76]. Rectified diffusion imlies that eriodic ressure variations will, in the long term, encourage gas to diffuse into the bubble and should decrease the observed tensile strength [34]. Ultrasonic methods generally yield low cavitation threshold values ranging from about - bar (gassy water) to -0 bar (degassed water) [07].

81 6 Negative ressures and/or cavitation may be achieved in liquids by focusing sound waves generated by a hemisherical ultrasonic transducer. Tensions occur during the negative art of the ressure swing at the acoustic focus. Imortantly, such exeriments usually roduce homogeneous cavitation as the acoustic focus is far from walls. Exeriments of this nature are discussed in [34,40,88,35,36]. For examle, Arora et al. [34] used a iezoelectric shock wave generator (a modified version of a commercial extracororeal lithotriter) to focus strong shock waves at the liquid samle at its acoustic focus. The ressure record at the focus showed a shock with a eak ressure of about 4 MPa, followed by a large negative ressure hase, where absolute negative ressures as low as -7 MPa were roduced in filtered, deionised and degassed (the concentration of oxygen imurities was 3.3 mg er litre of liquid) water. Müller [54] generated weak sherical shock waves in water by underwater sark discharge. The shock waves were focussed when reflected at ellisoidal reflectors resulting in ressure rises of u to 30 MPa at the focal sot. In every exeriment, shadowgrah images showed a diffracted wave, generated at the reflector edge and small cavitation bubbles that aeared in the focus region. On collasing a few microseconds later small, intensive, sherical shocks were generated and micro liquid jets formed in the direction of boundaries. In the focal region a comlete chain of bubbles collased almost simultaneously, forming a new ressure distribution and an enveloe of shock fronts, which were visible on the shadowgrah image, as a dark ring. Such focussed shocks, roduced by half-ellisoid reflectors, are alied in the medical field, in fracturing kidney stones. Using the same exerimental method and soft reflectors (made of olyurethane foam), which act like a water-air free surface due to their very low acoustic imedance, Müller generated converging exansion waves, thereby roducing negative ressures u to 9 MPa in deionised, degassed water before cavitation occurred. A study by Bailey et al. [7] also shows the effect of such soft reflectors. They used a Dornier HM-3 tye lithotriter with rigid and ressure-release (i.e. soft) reflectors. The intensity and destructive otential of cavitation roduced was measured by the sizes of

82 63 its in aluminium foil detectors and by the amlitude of shocks emitted at collase. Cavitation was detected by assive acoustic and high-seed hotograhy methods. The ressure release reflectors ractically eliminated tissue damage in ig kidneys, which were evident in exeriments using the rigid reflectors. However, the ulses roduced by the ressure-release reflectors were ineffective in stone comminution. These effects were attributed to cavitation of relatively low intensity occurring due to stifling of bubble growth by the ositive ressure eak following the negative ressure. In contrast, the rigid reflector waveforms caused cavitation of duration 50 times that resulting from the ressure release reflectors. The cavitation intensity was at least 3 times larger Gas Nuclei Studies Studies have shown the effects of ressurisation, evacuation and reeated stressing on the tensile strength of liquid samles The Effect of Precomression As mentioned earlier, studies have been made to comare the characteristics of ressurised and unressurised samles of water. Harvey et al. [85,37-39], Kna [40] and Briggs [4] subjected samles of water to high hydrostatic ressure and then determined their tensile strengths using various methods. Firstly, they measured the saturation temerature of the liquid at atmosheric ressure and the corresonding saturation ressure. The effective tensile strength was calculated as the difference between the saturation ressure corresonding to the reviously measured saturation temerature (from saturated steam tables) and the saturation ressure corresonding to the local atmosheric temerature. Harvey neglected the latter ressure as it is close to zero. This rocess was not true cavitation, but rather boiling, as it was caused by temerature increase and not ressure decrease. Kenrick et al. [4] and Winterton [77] have erformed similar suerheating exeriments. In all of the exeriments of Harvey et al. [37-9], unressurised water samles, containing undissolved air, boiled within a few degrees of the saturation temerature at

83 64 the relevant atmosheric ressure. This corresonds to cavitation occurring at or near the vaour ressure and imlies that the samles were found to be unable to sustain significant tensions. However, after being ressurised, all samles boiled at considerably higher temeratures when tested at the same ressures (corresonding to higher effective negative ressures). Calculations showed that re-ressurised liquid samles exhibited substantial, effective tensile strengths. Since ressurisation could not have affected the total gas content of the system, the high-ressure treatment must have forced the initial gas bubbles into solution []. This effect increased with the level of ressurisation but aeared to reach an uer limit of about 3-0 MPa, above which the tensile strength could not be increased. Briggs achieved similar results. Trevena [43] and Winterton [77] discussed the effect of recomression in detail. Harvey et al., Briggs and, more recently, Kna and Winterton found that even when samles were treated and handled in the same way throughout, the observed boiling oint showed wide scatter. In Kna s exeriments on ressurised samles, the boiling oint values varied from 7-7ºC, corresonding to maximum tensions of about -.5 bar and -.6 MPa and resectively. Harvey measured a maximum temerature corresonding to a tensile strength of.6 MPa while Briggs (64-67ºC) calculated values of between -4.9 and -5. MPa using water, contained in tubes with bore diameters in the range mm, which had reviously been boiled. Such scatter in fact, validates the stabilisation model of Harvey et al. [85], which imlies that the hysical characteristics of the hosts (interstices) are uncontrolled, vary widely and resond differently to ressurisation [] such that ressurisation reduces the size of nuclei but may not remove all of them. Thus, the model imlies that observed boiling oints and tensile strengths are exected to show scatter since they are determined by the nuclei remaining after ressurisation, which is a random function of the heterogeneities in the liquid e.g. If one article has, by chance, a satisfactorily shaed crack, it may act as a nucleus that will resist dissolution of free gas, even after ressurisation []. Variations, in the contact angles of interstices, of even a few degrees can significantly affect the ressure at which nucleation occurs [77]. The scatter of the exerimental results rovides further verification of the model of Harvey et al. over the organic skin model, which redicts more regular characteristics [].

84 65 Harvey [39] also roduced true cavitation dynamically, by raid withdrawal of a square ended rod from a liquid. The rod was inserted into a liquid samle and ressurised to remove free gas from its surface before being withdrawn at seeds of u to about 37 m.s - by means of a sring bow. The unressurised samles showed cavitation at the end of the rod during withdrawal while the treated samles did not. Kna [] made the same conclusion from exeriments involving hydrodynamic cavitation in a Venturi tube. Iyengar and Richardson [44] and Hayward [45,46] have also shown the effect of recomression. The former used a transducer to initiate cavitation in degassed water, ta water and seawater. The voltage across the transducer required to induce cavitation was found to increase (i.e. the tensile strength increased) when ressures of 7 and 0 bar were alied to the liquid samles [44] until an uer limit of maximum tensile strength was reached after about 5 and 60 s resectively. Winterton [77] built three rigs for boiling oint and cavitation threshold measurements in distilled water. In one rig, the lower end of a vertical tube was exosed to a reservoir and a vacuum um, allowing tensions of u to 0.74 bar to be obtained at the to of the tube, 7.5 m above the water level in the reservoir. In one series of tests, deactivation ressures of u to 5 bar for 30 minutes yielded no effect, while other tests showed increased boiling oints or cavitation thresholds with greater recomression until uer limits aroached asymtotically. In one set of exeriments, the boiling oint aeared to decrease with increasing dissolved air content. This is at variance with the earlier statements that dissolved gas content does not affect cavitation thresholds. Thus, either dissolved gas does actually have an effect on tensile strength, or the manner in which the dissolved gas content was increased (by exosing the liquid to gas in a reservoir) may have introduced undissolved gas content The Effect of Evacuation Rather than converting undissolved gas bubbles into a form (dissolved) less likely to act as cavitation nuclei, undissolved gas may be removed comletely from a liquid. Overton et al. [] studied the effect of reducing the amount of entrained gas, traed in crevices at solid boundaries, by evacuating the liquid using a vacuum um before transferring it

85 66 to the test chamber of the aforementioned tube arrest aaratus. This degassing was found to raise the cavitation threshold, measured under dynamic conditions, by a factor of about (refer to aendix H, table H.). A better alternative would have been to evacuate the tube with the liquid in situ as some air is entrained during filling []. Jones et al. [47] and Overton et al. [48] studied the effects of degassing, using Berthelot tube methods. Jones et al. showed that the tensile strength of distilled water (recorded as 3.5 MPa) could be increased to 4.7 MPa by evacuating samles rior to testing. Overton et al. used deionised water as the test liquid. Samles were degassed, by reeated evacuation, in a searate, sealed vessel before being transferred to a Berthelot tube. Between degassing and sealing of the Berthelot tube, the liquid was exosed to the atmoshere for less than minutes to minimise air content. The breaking tensions of the samles varied greatly between -3.0 and -6.9 MPa although most of samles cavitated in the uer range of -4.6 to 6.9 MPa Reeated Cavitation in a Single Samle Ohde and Tanzawa [49] studied the effects of reeated cavitation in water-filled Berthelot tubes. The cavitation strength of degassed water, though widely scattered, was found to increase gradually with the number times it was stressed. It was also found that degassing of the metal container was needed to achieve negative ressures below 0 MPa (-7 MPa was achieved in water, but only after several thousands stressings showing widely scattered negative ressures). Again, the wide scatter was attributed, largely, to uncontrollable randomness of gaseous nuclei on the tube wall. The effect of reeated cavitation has also been studied by Sedgewick and Trevena [98,5]. They found, using Berthelot tube methods, that olymer solutions had to be recycled a few times before being able to sustain any tension [98]. In addition, as tests, using the B-P method, were reeated, the tensile strengths of water and olymer solutions rose by 0 [5] to 30% [98] resectively before levelling off at final values. In [5], tests were reeated every 3 minutes, the minimum for their aaratus [43]. After about reeated tests, the tensile strength of a samle of ordinary ta water increased from 9.0 to.0 bar. When this samle was left to settle for 4 hours and tested again, its tensile strength was aroximately 9.8 bar [5]. The rogressively higher tensile strength recorded as the

86 67 liquid is reeatedly tested is attributed to the exhaustion of cavitation nuclei on the tube wall [49] i.e. some of the bubble nuclei in the water are removed (rise and disaear into the atmoshere) in each successive test [5]. The influence of cavitation history has also been studied in [,8] and is discussed further in [43]. Overton et al. [] erformed tube arrest exeriments on many liquids, including seawater, deionised water, ta water and Jet A- fuel. Many series of tests were erformed, with intervals between each test varied between 5 seconds and 5 minutes. Since the cavitation threshold is deendant on the time interval, it is also deendant on the frequency of testing: a higher frequency of stressing (i.e. shorter time interval between each shot) leads to a lower tensile strength because it gives less time for the nuclei to escae comletely []. Alternatively, as the frequency increases, the recovery between shots decreases [43] The Initial State of Samles of Liquid Water Various studies of the initial state of liquid samles have been erformed. These include the investigations of [50-54] and [38]. Exerimentally determined and tyical values of the radii of free gas bubbles, the initial number of bubbles er unit volume and the volume concentration of the gas hase in treated and untreated water samles is shown in aendix I Solid Nuclei Studies Parametric studies of relations between tensile strength and the characteristics of nucleation sites in solid articles and walls are difficult because natural articles are of irregular, random shae [39]. To investigate the role of the surface structure and article size of solid nuclei on cavitation incetion, while eliminating doubts concerning the size, shae and texture of solid nuclei, Arora et al. [34] and Marschall et al. [39] deliberately seeded samles of water with sherical articles with known characteristics. Marschall used a circulating water system, with a vortex-flow nozzle, to study the cavitation rocess, while Arora et al. used a shock wave generator (as mentioned in section 4.4). Marschall et al. [39] seeded filtered, degassed (to about 0. bar) ta water, which had initial article sizes of less than µm (larger articles were filtered out), with solid balls

87 68 with radii between.5 and 38 µm. They detected cavitation by using an acoustic and a light scattering method. Seeding of the liquid with the smallest sheres of radius.5 µm did not result in a measurable decrease in the tensile strength even though they were hydrohobic and distinctly larger than the natural nuclei remaining after filtration. It was concluded that a liquid containing near sherical solid nuclei has a greater tensile strength than one containing natural articles of the same size i.e. the irregular shae of natural articles contributes significantly to their role as nuclei. The addition of sheres much larger than the natural nuclei (0-40 µm in size) reduced the tensile strength by only between one and two thirds of that measured for the unseeded water. The fact that the cavities that develoed on these larger balls were much smaller than the balls themselves validates the hyothesis that the suitability of articles as cavitation nuclei deends on their fine scale surface. Since Marshall et al. [39] found that even very smooth articles can reduce the tensile strength of a liquid, the exeriments confirmed the effect of the size of solid nuclei [34]. Arora et al. used filtered, deionised and degassed water (initial maximum bubble radius was thought to be in the order of 50 nm) as their test fluid. The liquid was laced in a flask and seeded with hydrohilic, olystyrol articles, with radii between 30 and 50 µm, and density of.07 kg.m -3. Initials test using globally sherical articles were found to yield no noticeable effect on the tensile stress and, in fact, no cavity formed on them, even for the highest device settings. This is in conflict with the findings of Marschall et al. [39]. Pictures of exlosive bubble growth on article surfaces, which was found to occur at tensions of about.8 MPa, were catured using a sensitive camera, equied with a long distance microscoe. They showed that the growing bubbles accelerated the solid articles, on which they formed, into translatory motion. This effect might be alied in the acceleration of microscoic and nanoscoic articles as noted in section.4. Free bubbles may also be deliberately introduced into the liquid by electric sarks, electrolysis or by energy deosition using a laser [93]. Electrons, which may act as

88 69 cavitation nuclei, may be deliberately introduced into a liquid by a radioactive source and discharged through a metal ti. This is useful for studying heterogeneous cavitation [88] Cavitation Thresholds Aendix H rovides a summary of many cavitation threshold values determined by different methods. It is evident that, in general, the exerimental values and the theoretical values in section 3. differ greatly. Threshold values close to the homogeneous nucleation ressure have only been obtained in inclusions in minerals such as quartz, calcite, fluorite due to the small volumes involved, which minimises the robability of nuclei being resent. For examle, water was suerheated u to more than 300 C in inclusions, but only u to about 70 C in laboratory tubes (refer to table H.5). Discreancies between values from dynamic and static exeriments are evident e.g. Sedgewick and Trevena demonstrated tensile strengths of deionised water samles of.0 MPa and.3 MPa (mean value) using dynamic (B-P) and static (Berthelot tubes) methods resectively [98,5]. The tables in aendix H show that the cavitation thresholds determined using static methods are, in general, greater than those determined using dynamic methods, for liquids of comarable urity. The differences in threshold values may be attributed, largely, to variations in the urity of test liquids and their containers, differences in the rate of stressing and, in the case of, ulse reflection exeriments, the nature of free surfaces. The first factor has been discussed in this chater and accounts for the wide scatter, as exlained earlier (sections 3.. and 4.5.). The threshold data is essentially different and deends on both the state of the liquid and the dynamic of tensile loading Liquid Purity Firstly, it should be noted that it is highly unlikely that different investigators will utilise liquid samles urified (distilled, filtered and/or deionised) to the same degree [35]. The effect of liquid urity is illustrated in the summary of threshold values obtained by centrifugal stressing methods, table H.4: Threshold values obtained for degassed/distilled

89 70 water are an order of magnitude greater than those obtained for ta water. The difference aears less marked in other cases. The significantly higher threshold values (relative to values from similar exeriments) obtained by Williams et al. [5-7] were obtained using heavily urified water. Samles were rocessed by a two-stage, reverse osmosis, ionexchange urification system with a activated carbon filter for organics, a nuclear grade deionisation stage that reduced trace metal contamination to under art er billion levels and a final 0. µm membrane filter [96, 4-7]. Photograhic evidence and tension wave velocity measurements indicated that cavitation occurred within the body of the liquid in these exeriments. This accounts, artly, for the relatively high cavitation threshold determined in these exeriments [7] Rate of Stressing Firstly, it should be noted that all stages of cavitation are characterised by the relaxation of tensile stresses [38]: The aearance of clusters of bubbles interruts the continuity of the liquid hase [] i.e. the homogeneity of the liquid is altered. Thus, in acoustic cavitation the rarefaction waves are quickly absorbed [80] or extinguished [55] by the gas-filled bubbles (note again that the ressure of a gas may never be below absolute zero) while further ressure decrease is revented [55]. This exlains why the tension due to rarefaction waves from free surface shock wave reflection, as illustrated in figure.6, is not reached: the negative ressure hase is immediately distorted [35] and the tension is relieved [7]. Thus, cavitation alters the alied ressure field that caused it [06]. The rofile of a rarefaction wave is continuous and eaks after a finite time denoted by t *. The stress rate affects the observed tensile strength as during the rise time (the eriod t t * ) of the front, the volume concentration of free gas increases by several orders of magnitude i.e. the state of the medium is changed significantly [36,60]. As exlained above, the alied stress field itself is changed by the develoing cavitation, resulting in a lower observed tensile strength. If the rise time is instantaneous i.e. the negative ressure is alied instantaneously or for a time during which the cavitation nuclei have not exanded areciably (t 0), then the cavitation threshold is unaffected during the rise time and may reach its maximum value. Once it has reached that maximum value,

90 7 cavitation occurs and the tension is relieved. Thus, threshold discreancies may be due to the rate of stressing of a liquid, which has been found to be an imortant factor in determining cavitation threshold: For a given liquid, the cavitation threshold increases with increasing stressing rate (i.e. steeer ressure change) [35,96,6,7] e.g. Kedrinskii et al. [35] found that increasing the length of the ressure ulse by a factor of led to a decrease of the observed cavitation threshold by a factor of about 4. Exeriments may be classified as either static or quasi-static (Berthelot tube, suerheating and centrifugal methods) or dynamic (ulse reflection, tube-arrest and ultrasonic methods). In B-P exeriments, the incident comression and reflected exansion waves tyically have durations in the order of a millisecond and have stress rates of u to atm.µs - [76]. The high rate of stressing (0 bar.µs - ) accounts, artly, for the relatively high tensile strength value (shown in table H. determined in the tube-arrest exeriments of Williams et al. [7]. Turning attention to the threshold values for Mercury shown in table H.9, the tensile strength value determined by Carlson [8] is higher than that of Williams et al. [5] due to the much higher stressing rate (06 bar.µs - ) involved in his exeriments (achieved using the ulsed electron beam generator method). Williams et al. used the tube arrest method, which resulted in stressing rates of about bar.µs - and led to an observed threshold value of 300 MPa, while Carlson achieved a much higher value of 900 MPa, which is comarable to the homogeneous value 30 MPa [7]. In B-P exeriments, in order to achieve higher eak ressure in the liquid, bullets of greater mass or greater velocity at imact should be used. It has been shown that the use of bullets with greater momentum results in higher rates of stressing of the comression wave in the liquid (and hence that of the exansion wave that results from free surface reflection), which in turn affect the observed cavitation threshold [6]. The conclusion, based on the arguments above, is that there are no established critical values of the critical tensile stresses causing cavitation in a real liquid, even if the gas content of the liquid is fixed [60].

91 Transition Layer Some threshold values for water, determined by Berthelot tube, suerheating, centrifugal and tube-arrest methods, are in reasonable agreement with theoretical estimates in the order of tens of megaascals [07,55]. However, in the case of free surface reflection of comression ulses with rise times in the order of a millisecond (i.e. B-P exeriments excet that of Williams [96]), the highest tensions recorded were less than -4 with no other values exceeding -.5 MPa [59,]. In Berthelot tube exeriments, the tension develos over a eriod of several minutes, while in dynamic exeriments: stress develos over tyically, 0-6 to 0-3 s [96,56]. Thus, the rate of stressing cannot account for the low values recorded in ulse reflection studies since the stressing rate is lower in the Berthelot tube case. The aarent discreancies have been resolved, artly, by considering the nature of free surface reflection [07]. In reality, a free surface or liquid-vaour interface is not of zero thickness. A transition layer, of finite thickness, exists, at a free surface, between the liquid and vaour as discussed in [7,57]. The density decreases with height within the transition layer [07,7,34], as confirmed by x-ray studies [7]. The characteristic thickness of the layer is in the order of 3 intermolecular distances of water surfaces [57]. A ulse can be reflected u and down many times, within the layer, but with decreasing amlitude [34]. This transition layer is thought to be both a source of nucleation for vaour bubbles and is, imortantly, a major factor in making the free surface an imerfect reflector [34]. Attenuation and disersion of ulses within the transition layer results in anomalously low measurements of their amlitude (and hence the cavitation threshold), after reflection and below the layer [6-7]. Thus, ressure transducers cannot measure the tensile strength in such ulse reflection exeriments [07,34,55] directly. The effect of the transition layer on incident ulses was illustrated in exeriments by Sedgewick and Trevena [74,07,5,34] and Couzens and Trevena [-,07]. They found that a ressure ulse is not reflected as a mirror image tension ulse even if the eak ressure is so low that cavitation does not occur and distort the ressure rofile. The eak tension is significantly less than the eak ositive ressure and is accomanied by distortion of the reflected ulse rofile, which was sread or drawn out [07,34].

92 73 Sedgewick and Trevena investigated if the effect of the layer was influenced by the resence of ions or gas and found that the ratio of eak tension to eak ositive ressure for ta water was 0.34 while the ratio was 0.36 for deionised water and 0.50 for both boiled ta water and boiled deionised water [74,07]. Thus, the resence of ions may affect the ratio slightly while the resence of dissolved gas definitely and substantially affects the ratio. Couzens and Trevena [07, -] found that the ratio of the magnitude of the eak tension to the eak ositive ressure was about 0.5 for boiled, deionised water over a fairly wide range of incident wave eak ressures. The existence of similar transition layers at solid-liquid interfaces may artly account for cavitation at solid walls as well as the relatively low tensile strength values recorded in ulse reflection exeriments using flexible membranes [7]. However, the high tensile strength value obtained by Marston and Unger [58], who used a Mylar membrane for ulse reflection, suggests that solid membranes behave as erfect reflectors of comressive ulses [56]. Thus, the low values obtained by Richards et al. [09] in similar ulse reflection exeriments involving Mylar membranes cannot be attributed to transition layers but are, instead, due to the low acquisition rates and rise times of transducers used in their exeriments [56] Measurement of Tensile Strengths Other differences between threshold data may be attributed to different ressure and force measuring equiment used [,60,96]. Another imortant factor has recently been considered. Tensile strength measurements may be underestimated in some cases, due to inadequacies of ressure transducers as shown in [96,4,56]. The transduction equiment used by various researchers [95,09,,5,] had an equivalent samling rate of less than 0 khz and included a transducer with a rise-time of aroximately 6 µs. The ressure records resulting from such equiment were deemed inadequate and may underestimate the tensile strength of liquids by as much a factor of 3 [4]. Williams et al. showed that by overcoming the adequacies, low values of tensile strength could be reconciled with higher values of u to about 60 bar [96,4].

93 Overlaing of Tension Pulses The last factor suggested as a ossible cause of discreancies in the tensile strength is the effect of overlaing of the tension ulse resulting from free surface reflection with the tension ulse reflected from the base of the liquid column, which results in cavitation. It is known that cavitation regions may affect the seeds of waves assing through them such that they may aear to roagate at low seeds tending to the sound seed in water vaour (aroximately 494 m.s - ) as oosed to the high seeds associated with water [96,4-7,56]. This accounts for some low measured values of the tensile strengths.

94 75 5. Exerimental Facilities and Methods Two shock tubes were designed and built for the urose of achieving low and ossibly negative absolute ressures and consequently roducing visible cavitation in a liquid. The exerimental equiment, and their design, are described in the resent sections 5. and 5., while the rocedures of oeration are detailed in section 5.3. The first shock tube was designed to generate flows in which strong exansion waves roagate directly towards and iminge onto a column of water. This shock tube was designed such that the Mach number of a shock wave resulting from diahragm burst was to be about 3. In the following, it will be referred to as the Mach 3 shock tube. The second shock tube was designed to generate shock waves, with conventional gas driver and driven sections, which would iminge on and transmit into a column of water. The transmitted shocks were intended to reflect as exansion waves at the free surface of the liquid, thereby lowering ressures below the ambient, initial value and ossibly causing cavitation. In the design case, a shock waves roduced in the gas driven section resulting from diahragm burst were intended to have Mach numbers of about and thus, in the following, it will be referred to as the Mach shock tube. 5.. The Mach 3 Shock Tube The first shock tube is vertically mounted and clamed, at one lace, to a wall, as illustrated in figure 5. and the assembly drawing (aendix J), of the tube. As shown, the driver section is situated below the driven section. Circular steel tubes were used for the driven section and art of the driver section of the tube since they are relatively easy to obtain and rigid and do not introduce stress concentration areas as encountered in square tubes. However, circular tubes are not ideal for accommodating windows (essential in most cavitation studies) and a transition section from circular to square cross-section, which leads to attenuation or modification of waves assing through it [93], is usually required. Instead, a transarent tube was chosen to

95 76 facilitate otical observation of the test section. Polycarbonate was chosen for its high strength, low water absortion (0. % after 4 hours and 0.35 % after equilibrium is reached). Polycarbonate tubes are not as easily obtainable in small quantities as sheets. Polycarbonate is also known by the trade name Makrolon and the tube obtained was a general urose grade Makrolon 303. This material has yield and ultimate tensile strengths of 65 and 70 MPa resectively [60]. Figure 5.. Photograh of the Mach 3 shock tube.

96 77 The lower, driver section consists of a flange welded to a ie with a collar on its other end. The olycarbonate tube, sulied by Seal Cool Industries cc., is tightly clamed between the collar and the base of the shock tube bottom by the flange arrangement at the bottom. Rings, cut from rubber sheets, were clamed at the ends of the olycarbonate tube to revent leakage. Figure 5.. Photograh of the Mach 3 shock tube test (driver) section. The uer, driven section consists of a.35 m long stainless steel tube welded onto a flange. The chamber was designed to allow sufficient time for the exansion waves to travel through the water column and be reflected back uwards through the column before the shock interfered with the low ressures. A single olyurethane diahragm of thickness 5 microns was used in each test. A diahragm and a gasket were clamed between the two flanges of the driver and driven sections.

97 78 The driver section was designed for adequate static strength, corresonding to the maximum design ressure, while assuming a large safety factor to withstand transient loads, [48] and to withstand the high ressure behind the reflected shock. The olycarbonate tube, the most likely source of failure, could be considered as a thin-walled vessel making the well-known hoo stress relation σ = r/t valid [6]. The tube was designed to withstand a ressure of MPa with a safety factor of aroximately 6 on the yield strength. In small shock tubes, such as this one, the tube and its suorts are not subjected to significant axial, imulsive loads following the diahragm burst [48]. Thus, the claming mechanism was only designed, using standard rocedure for offset bolt wall mountings [6], to suort the static loads of the structure Instrumentation Three PCB (model number 3A) ressure transducers were installed: two in the shock tube walls of the driver and driven sections and one mounted axially into the bottom of the tube. The iezoelectric ressure transducers used were sulied by PCB Piezotronics, Inc. and are designed secifically for use in alications requiring high frequency, near non-resonant resonse such as shock tubes [6,63]. The transducers consist of quartz slides, an insulator and a metal housing. When the ressure is alied to the quartz crystal sensing elements, they roduce an electric charge. By converting the charge to a voltage, the ressure alied to the transducer is known. The sensors were installed into convenient M0 thread adator housings, which diminished the need for recision machining. A transducer mounting ort is illustrated in aendix J. Dimensions and instructions for rearing orts are given in installation drawings in [63]. When considerable leaking occurred through the ga between the housing and transducer, they were effectively sealed using commercially available silicon sealant. The transducer secifications are given in aendix K. The transducers were connected to the signal conditioners by general urose, white, coaxial Teflon cable assemblies (series 00) and low-noise, blue, coaxial, Teflon cable assemblies (series 003) of various lengths with 0-3 lugs on either end. The signal outut from the signal conditioners was fed

98 79 into a high-seed digital oscilloscoe. Throughout testing, two oscilloscoes, a Yokogawa DL708E and a Yokogawa DL00A, were used. During testing, a ressure gauge calibrated from 0 to 000 kpa, rovided by Kerr Valve and Industrial Sulies (Pty) Ltd, and a gauge calibrated from 00 to 400 kpa, sulied by Wika Instruments (Pty) Ltd, were used. The low ressures at the vacuum um outlet and in the driven section were measured using two barometrically comensated, Seedivac (tye C.G.3) vacuum gauges with dials calibrated 0-40 Torr and 0-00 Torr, resectively, which read absolute vacuum ressures High ressure and Vacuum Sulies The driver section high ressure suly was rovided by the laboratory comressor (0-80 si gauge), which was charged to about 0 MPa before each series of tests. A Seedivac High Vacuum Pum manufactured by Edwards High Vacuum Ltd. was used to reduce the ressure in the driven section. The um was a single-stage model (Model number ISC4508, Serial number 3688). Pressures as low as 5 mm Hg could be obtained at the vacuum um outlet Flow Visualisation A Reflecta Flectalux GLX006 lam, mounted on a triod, was used to illuminate the test section. Pictures were recorded using a high-seed camera that was manually triggered. An EG&G Reticon camera, fitted with a.8/35 Meyer-Otik Görlitz Orester lens (Serial no ), was used. The camera s maximum cature rate was 000 frames / second. 5.. The Mach Shock Tube The second shock tube is also vertically mounted. It was constructed by modifying the shock tube designed and built by Karnovsky [4] for shock focusing exeriments. It consists of a gas driver section at the bottom, a gas driven section in the middle and a

99 80 water filled test section at the to. The tube extends across two floors as illustrated in figure 5.3. The gas driver and driven sections are made of mild steel and has been tested at a ressure of 700 kpa. The driven section used by Karnovsky [4] was modified by adding a diahragm bursting mechanism (refer to the the drawing of the modified gas driven section in aendix J). In addition, a ortion of ie was cut off the to and a new flange added to accommodate the water-filled test section. A hole had been drilled into the wall, which served as a vent, allowing the high ressure in the shock tube to normalise to atmosheric levels after tests [4]. The driven section was held in lace by a clam bolted onto the adjacent wall and a large steel disc resting on the floor above it. A gasket was clamed in lace, below the diahragms, between the driver and driven section flanges. The water-filled test section is made of stainless steel and was designed to accommodate olycarbonate (sulied by Plastic World (Pty) Ltd.) windows, for otical observation of cavitation, near the to. This section was clamed to the floor to revent longitudinal and lateral movement of the uer ortion of the tube above the bellows units. In order to allow the liquid shock wave to become fully formed before reaching the free surface, the uer section was designed to be as long as ossible. The length of the ie used, limited by sace constraints, was. m while the wall thickness was aroximately 4 mm (the inner and outer diameters were equal to 5.5 and 60.3 mm resectively, as secified by the manufacturers). As the initial ressure in the tube was small and equal only to the hydrostatic ressure, it was not designed for static strength. It was designed to withstand a maximum shock wave ressure of.5 MPa assuming a safety factor of about 0 on the yield stress. The windows were incororated by cutting through the to ortion and fitting two Polycarbonate sheets in lace of the tube walls. Each sheet was fixed in lace by M4 bolts, which were intended to ensure adequate ressure along the window edges to ffffffffffffffffffffffffffffffffff

100 Figure 5.3. Photograhs of the Mach shock tube. 8

101 8 revent leakage. Assuming a safety factor greater than 5, the windows and associated bolts were designed to withstand shock ressures of u to.5 MPa Instrumentation Four PCB3A ressure transducers were mounted along the tube wall. All transducers and associated ower sulies and oscilloscoes used were the same as those used for exeriments using the Mach 3 shock tube. Again, their secifications are given in aendix K. The same kpa ressure gauge mentioned above was used throughout testing The High Pressure Suly A Helium cylinder, sulied by African Oxygen Limited (AFROX), was used as the highressure suly for the driver section. The cylinder contained instrument grade helium with total imurities less than 7 vm (arts er million by volume) ressurised to 0 MPa. It was connected to the driver section by a regulator The Liquid Gas Interface A sheet of water resistant material was used as an interface between the gas driven air section and the column of water in the test section above it. Different materials, namely, stainless steel, olycarbonate (sulied by Plastic World Pty. Ltd.), silicon rubber and insertion rubber (sulied by TRUCO or Transvaal Rubber Comany Pty. Ltd.), were used as the interface. It was clamed in lace between two single axial bellows units sulied by Sirax Sarco (Pty) Ltd.. The bellows (Product code SAF ) have a nominal diameter of 50 mm and were welded to standard BS4505 stainless steel flanges. Such a bellows unit is usually used as an exansion joint i.e. to absorb dimensional changes for reventing ie strains from becoming too large. In this case, the bellows were intended to allow the interface to move when struck by the gas shock wave and so transmit shock waves into the liquid Diahragm Bursting Mechanism

102 83 The diahragm bursting mechanism is illustrated in figure 5.4. The needle is ulled away from the diahragm and held in lace by a catch. It is released when the driver section is fully ressurised and diahragm is to be burst. It was found that the bursting mechanism erformed adequately over the entire range of testing ressures. Figure 5.4. Photograh of the diahragm bursting mechanism on Mach shock tube Exerimental Procedure The Mach 3 Shock Tube The following rearation rocedure was done before each series of tests:. Switch the PCB line ower sulies on at least an hour before commencing a test to allow them to stabilise.. Switch on the oscilloscoe and check the connections between the transducers, ower sulies and the oscilloscoe. 3. Check that the vacuum um has sufficient oil and relenish if necessary.

103 84 4. Cut diahragms such that they will comfortably searate the driver and driven section but allow sufficient sace for the bolts. 5. Place the camera aroximately 5 to 6 m away from the tube, with the lens focussed horizontally, at a ortion of the test section. 6. Note the atmosheric ressure Before each test: 7. Fill the test (driver) section, from the to of the assembled driver section, with aroximately 500 ml of water. 8. Insert a single 5 micron diahragm between the comression and exansion chambers above the gasket and tighten the bolts. 9. Ensure that a valve between the comressor suly and driver section and the vent valve, are closed. Oen the valve between the vacuum um and driven section. 0. Switch the vacuum um on and allow it to run (letting the gases achieve thermal equilibrium before burst) until the required driven section ressure obtained. Close the valve between the vacuum um and driven section.. Set the required arameters on the oscilloscoe.. Start the camera cature rogram and choose the acquire otion (the rogram then waits for the user to ress a key to trigger the caturing rocess). Then, the following stes were comleted during the exeriment 3. The comression chamber was charged to a ressure of about 7.5 bar by oening the valves from the high ressure suly. This is done slowly enough to allow observation of the burst ressure and to allow the gases to achieve thermal equilibrium. 4. When the ressure gauge of the driver section reaches a value of aroximately between 6 and 7 bar, ress the trigger key on the keyboard, while the ressure rises. This has usually roved to be sufficient for caturing the events within the test section when a frame rate of 000 frames er second is used. 5. Observe the driver ressure at diahragm burst After each test, comlete the following stes:

104 85 6. Oen the vent above the driven section to relieve the high ressure within the tube. 7. Follow the romts on the comuter monitor to view and save the catured frames. 8. Save the oscilloscoe traces to floy disc, the hard drive or rint the results directly from the oscilloscoe. 9. Process the data, using a suitable rogram such as Micrsoft Excel. The quartz crystals of a iezoelectric ressure sensor generate a charge when ressure is alied. The charge eventually leaks to zero, even though the electrical insulation resistance is fairly large, and thus the sensors may only measure dynamic ressure changes. Consequently, oscilloscoe readings had to be adjusted to the known ressures in the driver and driven sections (read off the gauges) after being converted to ressures using the calibration gauge factors shown in aendix K, section K.. In order to eliminate extraneous effects and thereby increase reroducibility, care must be taken, throughout the exeriment, to kee all transducer wires still. The ressure transducer in the to (driven) chamber was connected to channel of the oscilloscoe. The volts er division was set to 500 mv/div. Nominally, the trigger level was set to about -50 mv based on the middle transducer i.e. data acquisition commenced once the ressure at the transducer, exosed to air above the water column, droed slightly. The uer and lower ressure transducers mounted in the bottom chamber walls were connected to channels and 3 and set to V/div and V/div resectively. The time er division (for all channels) was set to ms/div The Mach Shock Tube The exerimental rocedure to be comleted before, during and after tests using the Mach shock tube is similar to that of the Mach 3 facility above. It differs in that no vacuum um is used and that the following additional stes aly:

105 86 Before each series of tests,. Clam a late or sheet of aroriate size, to serve as an interface between the liquid above and the air below, between the bellows units. Before each test, the uer art of the tube is reared as follows:. After ensuring that the drainage valve is closed, fill the uer test section of the tube, from the to, such that the free surface of the water is visible through the windows and the uermost transducer is submerged. The gas driver and driven sections are reared as follows: 3. Unbolt the driver section. 4. Retract the diahragm bursting mechanism needle and fix it in lace, using the catch. 5. Insert two 5 micron thick diahragms between the comression and exansion chambers, above the gasket, and bolt the sections together. 6. Ensure that the vent valve is closed and that the valve between the driver section and the gauge of that section is oen. During the exeriment, 7. Pressurise the driver section slowly, allowing the gas to achieve thermal equilibrium before burst, until the required ressure is reached. Close the highressure suly valves. 8. Close the valve between the driver section and the gauge of that section, simly to rotect it from transient ressures and moisture. 9. Release the diahragm bursting needle. Four ressure transducers were connected to the oscilloscoe. The volts er division was set to or V/div. The trigger level was set to about +00 mv (for a slight ressure rise) based on the transducer in the air driven section. The time er division (for all channels) was usually set to or ms/div.

106 87 6. Exerimental Observations and Results In this section, the results of tests, in the form of ressure and hotograhic records is noted. More detailed discussion of these observations is reserved for the next chaters. 6.. The Mach 3 Shock Tube 6... Pressure Transducer Records Adjusted Absolute Pressure (kpa) Time (ms) 3 Figure 6.. Pressure trace from test using the Mach 3 shock tube. Samling rate MHz, 0000 data oints. (43C) Figure 6. is a tyical ressure trace recorded during a test. It shows the absolute ressures at the two transducers in the driver section, adjusted to the ressure before diahragm burst. The atmosheric ressure during this test was 8.5 bar. A negative value of the absolute ressure of 00 kpa is recorded at the transducer mounted axially in the bottom of the tube, referred to in the following as transducer 3. The middle transducer, in the following referred to as transducer, recorded a minimum ressure of

107 88 bar. A very noticeable feature of the grah is the stationary oint on the ressure record of transducer, which occurs when the local ressure has droed to about 470 kpa. This is the oint where the exansion wave, reflected from the to of the water surface arrives back at transducer. The ressure variations from the transducer in the driven section were of no interest and no ressure records are shown here. Transducer 3 shows that the liquid at the base of the tube sustained an absolute negative ressure for aroximately 3 ms. Then the ressure at the base of the liquid column rises to the same value as the air above it before droing again, to a value of about 30 kpa. This ressure rise is attributed to the reflection of the exansion waves at the free surface, which results in downward-travelling comression waves. A shock wave, with a strength γ of aroximately, is recorded at transducer after aroximately 7.3 ms. In addition, at aroximately 8.7 ms, the ressure at transducer 3 rises abrutly, reaching a eak ressure of about 6.5 bar. This is followed by oscillating ressure variations above and below a ressure of about 3.5 bar. Similar ressure records are shown in aendix L. These results were recorded from tests under the same conditions but show the ressure variations with different time scales to illustrate the overall trend of the ressure variations. In all exeriments, the ressure records aeared fairly similar to figure 6.. The maximum negative ressure in the liquid varied within the range 90 to 50 kpa. The most significant differences occurred between records of the ressure at transducer 3 when the ressure increased abrutly (e.g. the first ressure rise between 4 and 7 milliseconds and the second ressure rise at 9.7 ms on figure 6.). Figures L. shows the first ressure rise followed by distinct ressure ulsations that roceed with decreasing amlitude. Figure L. shows a less regular ressure variation. The ressure oscillations are also visible from closer insection of figure 6.. Note that, in all exeriments conducted, the ressure at transducer 3 immediately after the first ressure rise was aroximately equal to the ressure, at the same time, at transducer. In all tests, the second, more abrut, ressure rise aeared similar to the corresonding event shown in figure 6., although the eak ressures varied between about 350 and 700 kpa.

108 Photograhic Records Aendices M, N and O show sequences of hotograhs catured during exeriments at a rate of 000 frames er second. The ictures resented in aendix M corresond to the same test corresonding to figure 6. and the discussion above. No ressure waves were visible in the hotograhs. Large ressure gradients (e.g. shock waves) may be visible when simle direct hotograhy, with no otical arrangements, is used if the subject is suitably illuminated [55]. In this case, however, the ressure gradient involved here may be too low to cause visible effects. In addition, due to the high sound seed in water, any wave will travel the distance from the free surface to the tube base and back in less than half a millisecond. Thus, the hotograhs, searated by millisecond intervals, cannot be exected to consistently cature the waves if they are visible. A wave could not be visible in consecutive frames and may not be catured at all. Figure 6.. Cavitation observed at the bottom of the Mach 3 shock tube (aendix M, frame 4).

109 90 In the first three frames, no bubbles are clearly visible. The absolute ressure in the test section at the corresonding times aroaches 7.93 bar. It aears as if cavitation occurs both within the body of the liquid and on the walls of the tube: The next frames illustrate saturation of the test section with bubbles of varying sizes. In frame 4, reroduced in figure 6., a large bubble with a diameter of aroximately 5 mm is observed. The relatively large sizes an number of the bubbles suggest that the favourable nucleation sites are abundant. Evidence is resent of the oscillatory nature of the cavitating bubbles: In the three frames following frame 4, the number and sizes of visible bubbles decreases rogressively. The largest bubble in frame 4 disaears from view in frames 5-7. In frames 8 and 9, bubble growth has occurred throughout the region of the tube under consideration. The intensity of the cavitation aears to have increaseed from the revious frames. A articularly large bubble, with a diameter of about 4.8 mm (estimated from enlarged ictures of these frames), is resent in the bottom left hand corner of the frames. In frames 0 and, very few of the bubbles are clearly visible. Then, from frame, it is evident that another growth eriod occurs. The intensity of cavitation aears to decrease from frame 3 to 5. In frame 6, some bubbles aear to have exanded again. Over the next frames, the bubbles disaear again. Enlarged images of the frames showed that growth occurred at frame, although this is not clearly visible here. It is evident that the intensity of cavitation decreased as the oscillations rogressed. This imlies daming of the bubble growth and collase cylces. The frames that show markedly greater cavitation intensity than the adjacent frames before and after them are judged to be frames 4, 8,, 6 and. This seems to suggest that the eriod of oscillation of the bubbles is about 4 ms. However, this statement is based on rather inaccurate data: The eriod of oscillation may be smaller than the time interval between each camera frame ( ms). In addition, from the trend shown and bearing in mind the camera s relatively slow frame rate, it is reasonable to assume that the cavitation intensity decreases throughout the sequence of frames, even though the number of visible bubbles aears greater in frames 8 than in frame 4.

110 9 Cavitational activity was not limited to the lower section of the tube. Aendices N and O show sequences of hotograhs from other tests, erformed using the same method, but with the camera focussed on the middle and uer segments of the test section resectively. Referring to aendix N, it it observed that no cavitation is clearly visible in the first 3 or 4 frames. Then, cavition, beginning at the lower section, is visible. The cavitation aears to intensify in frame 5, with a large bubble (diameter aroximately 5.4 mm) forming in the lower region. Cavitation intensity aears to decrease markedly in frames 6-8. Frame 9, reroduced in figure 6.3 illustrates that cavitation then develos intensely. Many bubbles, of varying sizes and reasonably close together, form over the entire middle section. The rest of this record shows successive bubble growth and collase hases. The intensity of cavitation aears to reach rogressively decreasing maximums. Frames 3 and 7 show maximums of the cavitation intensities, which again suggests a eriod of oscillation of 4 ms. Figure 6.3. Cavitation observed at the middle section of the Mach 3 shock tube (aendix N, frame 9).

111 9 Aendix O aears to illustrate different behaviour. In the first frame, two bubbles are barely visible near the free surface. In the next four frames, a few isolated bubbles form beneath the free surface. The next frames show slight, initial disturbance of the free surface, which corresonds to the assage of the rarefaction wave through the the liquid column. The free surface aears relatively undisturbed from its initial osition. In frames 6 and 7, the free surface begins to assume a corrugated shae as a result of many bubbles forming and coalescing in the region. Figure 6.4 is an enlargement of frame 8 and illustrates the nature of the free surface region after the assage of the wave. The initial level of the liquid column is lowered as the vaour mass forms above it. The rest of the ictures show vaorisation of the to layer of the liquid while the isolated bubbles below the free surface ersist. Figure 6.4. Cavitation observed at the uer section of the Mach 3 shock tube (aendix O, frame 8). 6.. The Mach Shock Tube Initially, difficulties in roducing a liquid shock were encountered: As mentioned earlier (section 5..3), a number of different interfaces between the air and water sections of the tube were tried. In early attemts, a single sheet of olycarbonate (6 and 8 mm thick) or

112 93 stainless steel was clamed between the bellows. However, the bellows did not rovide enough movement for the interface to transmit a shock into the water. Later, a mm silicon rubber sheet roved too weak and broke after just one comleted test. Finally, insertion rubber sheet was used. Since mm thick sheet was found to break after only 3 or 4 tests, a 4mm thick was used instead. Use of the thick insertion rubber resulted in comression waves in the water of similar magnitude to those obtained using silicon rubber and thin insertion rubber. Consequently, the thick insertion rubber was chosen, as it did not break, even after more than 30 tests Static Pressure (bar) Time (ms) Figure 6.5. Pressure trace from test using the Mach 3 shock tube. Samling rate MHz, 0000 data oints. Driver section filled with helium at bar. Trigger level +0.V, source CH-, osition 4V, delay 0. Figure 6.5 shows a tyical set of ressure traces recorded. The absolute ressure during this test was 0.83 bar. In the following, the transducer mounted in the wall of the airfilled driven section will be referred to as transducer, while the first, second and third

113 94 transducers encountered by an uward-travelling ulse in the liquid section are referred to as transducer, 3 and 4 resectively. The ressure behind the air shock wave, recorded at the first transducer, transducer, at 5.0 ms, is aroximately.5 bar. This is followed by another shock at 6. ms. These two shocks corresond to the uward-travelling incident wave and downward-travelling reflected waves. The strengths of the incident and reflected waves are as 4.0 and. resectively. The reflected shock is re-reflected at the bottom of the shock tube and then travels u the tube. It crosses transducer again and is reflected from the interface again: the second incident and reflected waves occur at 3.8 and 5. ms. The ressure trace for transducer shows that a shock wave of ressure 6. bar was transmitted into the water column. This corresonds to a shock strength of about 7.3. From closer insection (shortening the time scale), it was evident that this shock wave had a measurable rise time of aroximately 60 µs. The ulse is not a strictly discontinuous shock wave. The initial ressure rise at this transducer is followed by a dro and subsequent rise in ressure. Then the ressure dros raidly, reaching a minimum value of 33.5 bar (static ressure) at 0 ms. It aears that weaker comression waves are exerienced at transducer during the time interval between 9.5 and 5 ms. Then, the attern aears to reeat, with lower amlitude waves, from 5. ms. As stated earlier, this reeating ressure variation is due to the re-reflection of the gas shock wave at the lower end of the tube, which results in another comression ulse that is transmitted into the liquid. The ressure record of transducer 4, 60 mm below the free surface, shows unexected results: the main comression wave aears to have decayed to a much weaker wave behind which the ressure was only.5 bar (an almost three-fold decrease in shock strength to a value of.8). In addition, the comression wave seems to have been receded by a low amlitude transient ressure disturbance between 7 and 8 ms. After the main comression wave, a negative static ressure occurs for about ms. However, this ressure is not a tension since it is equivalent to an absolute ressure of about 0.7 kpa.

114 95 Since this value is well above the vaour ressure, no cavitation is exected. Between 4 and 8 ms, slight ressure increases occur. Again, these were due to the waves resulting from transmission of a second shock from the gas section into the liquid. Pressure records from transducer 3 were not included in figure 6.5 but follow the same trends as the records from transducer 4 excet that the ressures involved are higher. The magnitudes of the ressures at transducer 3 are aroximately halfway between those of transducer and 4. This indicates that the attenuation of the comression wave is cumulative and aroximately roortional to the distance travelled. Aendix P shows additional ressure records from other tests. Figure P. shows ressure records from a test under the same conditions as figure 6.5. Figure P. differs from figure 6.5 in some resects. Firstly, the strength of the first incident gas shock wave is weaker. It follows that the reflected gas shock and the transmitted liquid shock are weaker. In addition, the unexlained comression waves, between the first and second main waves transmitted from the gas section, result in much greater ressure increases. In other similar tests, these unexlained increases in ressure were less than those shown in figure P.. In figure P., most of the comression ulses after the main wave seem to be of similar strength. Otherwise, ressure records of figure P. resembled those shown in figure 6.5. For all tests using a bar driver section, no negative ressures, or even ressures near the vaour ressure, were recorded at either of the transducers. Figures P., P.3 and P.4 show ressure records from tests in which the driver section was filled with helium and ressurised to 8 bar before bursting the diahragm. These records show inconsistencies and significant deviations from figures 6.5 and P.. Consider the records from transducer in figures P., P.3 and P.4. Firstly, the uward-travelling gas shock waves, incident on the interface, aear to not be fully formed i.e. The ressure rises twice before a larger ressure rise, corresonding to the reflected wave from the interface, is registered. Secondly, the gas shock strength varies. The strengths of the incident wave vary between 3.3 and 4. while the reflected wave strength is close about.0 in all three cases. The static ressure behind the reflected shock is similar in figures

115 96 P.3 and P.4 (6.0 and 5.8 bar resectively); the value in figure P. is only 4.7 bar. Again, these ressure waves are reeated, when the gas shock travels the length of the gas section of the tube, is reflected and reaches the interface again. The ressure records for transducer in figures P., P.3 and P.4 show that for an 8 bar driver ressure, the transmitted shock is actually greater than the ressure behind the reflected gas shock. The ressure records for transducers and 3 show that after the assage of initial comression waves, the static ressure dros raidly to a negative value of between - and - bar. This corresonds to negative absolute ressures with magnitudes slightly below bar. During these low-ressure hases, high ressure sikes, u to ressures of about MPa are recorded at these transducers. It should be noted that these abrut changes in ressure occur before the second comression ulse is transmitted into the liquid section. The eak ressures vary with a maximum value of about 0 bar. Consider lastly, the ressure records of transducer 4 situated 60 mm below the free surface in figures P., P. and P.3. The initial comression wave is attenuated markedly. In all cases, the ressure rise is followed by a low-ressure hase lasting about ms, as recorded in the exeriments using driver ressures of bar. In all tests using the 8 bar driver ressure, the lowest static ressure, recorded during this hase (see figure P.3), was aroximately 0.5 bar. This corresonds to an absolute ressure of about 0.35 bar (the atmosheric ressure was 0.85 bar), which is an order of magnitude greater than the vaour ressure of water at 0 C.

116 97 7. Theoretical Analysis 7.. The Mach 3 Shock Tube The exeriments using the Mach 3 shock tube showed that cavitation was induced in the liquid. In this section, an attemt is made to exlain the results theoretically Wave Diagram KASIMIR (Coyright 993 by Stoßwellenlabor, RWTH Aachen University) is a rogram used to simulate the hysical henomena inside one-dimensional shock tubes. It calculates the wave diagram, including information about the state variables at all oints and the shock, comression and exansion waves and, dislaying it on the screen. Inuts to the rogram include wall tyes (rigid for the current alication), driver and driven section ressures and the exansion fan ressure ratio, which is, nominally, the ressure ratio across each characteristic in the exansion fan. For some combinations of inuts, the rogram would fail and generate an error message. In such cases, it was necessary to calculate the wave diagram using established methods. In the following, the built in real gas model for air (an air model consisting of nine comonents) was used. The results were found to be almost identical to results obtained using the ideal air model. Figure 7. is the wave diagram for the first shock tube with the driver section ressurised to 7.93 bar and the driven section evacuated to a ressure of 0.07 bar (both ressures are absolute). The lower, test section containing the liquid column is reresented by the ositive ositions from 0 to 0.49 m while the osition values from 0 to.350 m corresond to the driven section. Thus, note that uward velocities are indicated as negative and downward velocities as ositive. The initial temerature of the gases (states A and B) and water (state W ) before diahragm ruture is usually room temerature [48] and taken as 0ºC. Thus, the reference value of the seed of sound a 0 was m.s - for all gas regions of the wave

117 98 diagram and 48 m.s - for all water regions. Entroy changes in the liquid were neglected. The analysis neglects losses caused by the ruture of the diahragm as well as the influence of the boundary layer. All ressures are absolute values. Bold lines reresent shock waves. Figure 7.. Wave Diagram of the Mach 3 Shock Tube (driver section: air at 7.93 bar, driven section: air at 0.07 bar). Note that the waves resulting from uward-travelling exansion waves that collide with the free surface (i.e. waves transmitted from the liquid to the gas) may be neglected for the same reasons that the gas above the free surface reflection of a shock remains unchanged (refer to chater ). The wave diagram redicts ressure variations, at the transducers, which are in satisfactory agreement with the exerimentally recorded values. The values of the fluid roerties at the above regions are summarised in aendix Q, table Q.. It may be seen that the ressure in region W 3 (behind the exansion wave

118 99 reflected off the bottom of the shock tube) is slightly negative ( bar). Since a gas cannot sustain any negative ressure [69], the comatibility condition used to construct the wave diagram, namely that the ressures on either side of the interface are equal, cannot be satisfied. Consequently, the method of characteristics becomes ineffective for solving the regions that follow the collision of the exansion wave with the free surface. The unlabelled regions in figure 7. are the unsolved regions Pressure-Velocity Diagrams Since the wave diagram rocedure was of no use once a negative absolute ressure value was reached and could thus not yield values for the lowest ressure reached in the liquid (an imortant arameter affecting cavitation), a different aroach was required. The collision of the exansion fan with the liquid surface is illustrated grahically in the ressure velocity or -v diagram, figure 7.. Again, the convention, in this case, is that negative and ositive signs corresond to uward and downwards velocities resectively. The curves labelled i, r and t reresent the loci of states, which may be reached by a oint on them an incident, reflected or transmitted exansion wave roagating in the relevant medium. Note that the transmitted wave has a ositive sloe but aears vertical because it roagates in water, which has a relatively high acoustic imedance. Curves for reflected waves starting from any ressure on the incident wave curve may be lotted. However, only selected loci, starting from ressures of 5.569, 3.9,.746,.99,.355 and 0.95 bar, are lotted here. These are the ressures in the regions P, O, N, M, L and K behind the six characteristics on the wave diagram closest to the liquid column. Using the same naming conventions as in chater, the initial state of the air and water are denoted by () and (5) resectively. These states are coincident on the -v diagram because of the comatibility conditions on the ressures and velocities across the contact surface. The incident exansion wave causes the air of state (), above the liquid samle, at rest and a ressure of 7.93 bar, to be exanded to the state (4) and to acquire an uward velocity. The reflected wave causes the ressure to dro further and the magnitude of the velocity to decrease until a state () is reached. The water at state (5) is exanded and

119 00 accelerated uwards by the transmitted wave to a state (3), which by comatibility has the same ressure and velocity as (). Thus, these states are coincident. Figure 7.. Pressure-velocity diagram for the collision of the exansion waves with the gas-liquid interface (initial driver ressure 7.93 bar). As the lowest negative ressure obtained in the water is required, the lowest ressure behind the downward-travelling, incident waves,.99 bar (the state M on the wave diagram), was chosen as the starting oint for the reflected wave curve. Thus, the ressure at the state (,3) is, from closer insection of the diagram, aroximately 0.33 bar, while the velocity of the air and water is aroximately 0.5 m.s - uwards. In summary, the analysis redicts that the ressure of the air, initially at 7.93 bar, is reduced (by 6.00 bar) to.99 bar and then (by.60 bar) to 0.33 bar by the incident and reflected waves resectively. The ressure of the water is reduced, by the transmitted wave, to 0.33 bar (a ressure dro of 7.6 bar).

120 0 The ressure in the liquid is reduced further when the transmitted wave collides with the bottom of the tube, which is assumed to be a rigid wall. For this second interaction, another ressure-velocity diagram was constructed. The diagram, resented in figure 7.3, simly shows the acoustic doubling effect discussed in aendix E i.e. the incident and reflected rarefaction waves are of equal strength. Thus, since the ressure dro caused by the incident wave was about 7.6 bar, the resulting, minimum absolute ressure in the liquid is aroximately 7.7 bar. Grahically, this ressure could be obtained by extraolating the curve, on figure 7.3, which corresonds to the reflected exansion wave starting at 33 kpa until it intersects the curve of the transmitted exansion wave. Figure 7.3. Pressure-velocity diagram for the collision of the exansion waves with the base of the Mach 3 shock tube (initial driver ressure 7.93 bar). Although the analysis u to this oint rovides a lausible and reresentative maximum negative ressure value, it neglects three imortant factors:. The reflection of the exansion waves, at the free surface, results in downwardtravelling comression waves, which raise the ressure. In similar work [93],

121 0 these waves were urosely used to initiate bubble collase. These comression waves result in the rise in ressure that follows the initial tension in the liquid, as shown on ressure records from transducer 3: figures 6., L. and L... Secondly, consider the strong, downward-roagating shock wave that results from the reflection of the initial shock, formed at diahragm burst, at the uer wall. This incident gas shock iminges on the free surface and increases the liquid ressure before all of the downward-roagating waves in the exansion fan have entered the liquid. The rise in ressure caused by this shock, and the shock resulting from its reflection at the base of the tube, would certainly increase the ressure well above the atmosheric ressure, as well as cavitation bubble collase. This was confirmed by the ressure records (refer to figure 6.): The abrut ressure increase at transducer 3 occurred follows a shock exerienced at transducer. The ressure records showed adequate agreement with the wave diagram, figure 7.. However, the shock strength, from figure 6., is only between and.5 while the strength of the shock, which is weakened by the exansion wave system it travels through, was estimated as between 6 and 0. This value was determined by treating the free surface of the liquid as a rigid wall. 3. Thirdly, some of the uward-travelling exansion waves would be curved on the wave diagram (towards the liquid column), due to the their interaction with exansion waves reflected from the liquid surface, and iminge uon the liquid before the strong shock, thereby reducing the ressure further. The arameters of these waves could not be calculated accurately using the wave diagram method because, again, the negative ressure in the liquid limited its effectiveness. Thus, the magnitude of the maximum negative ressure may be lower (due to the strong shock) or greater (due to curved waves entering the liquid) than 7.7 bar, as found by combining the wave diagram and -v diagram methods. A further analysis of the strong shock was carried out. The limiting value of negative ressure in the liquid must be that resulting from transmission of the characteristic waves that imact the free surface at the same time as the strong shock wave. Only the waves in the exansion fan that arrive at the free surface before or at the same time as the strong

122 03 shock, were considered. They were treated as a single wave. The low ressure behind the single incident wave was then easily calculated as.7 bar using an extended wave diagram, including times u to 0 ms and a very long driver section to neglect the effect of the exansion waves reflected from the free surface. The characteristic between regions M and N on figure 7. has a very slight gradient and would thus not reach the liquid surface before the strong shock wave, while the characteristic between N and O reaches the liquid well before the shock. Thus, the value of ressure obtained seems reasonable as it is close to the value at N,.746 bar. Then, the interaction of this single wave with the free surface (transmission and reflection) was solved on close insection of the same -v diagram, figure 7.. This yielded a value of 0.77 bar for the ressure of the liquid behind the transmitted exansion fan. Again, when the exansion fan is reflected at the base of the tube, the dro in ressure (-7.6 bar) caused by the exansion wave is doubled. Thus, the maximum tension to which the liquid may be subjected, in this method is 6.39 bar. This minimum ressure is the absolute limiting value that is aroached. It is not sustained for an areciable length of time before the ressure increases. This correction, from the value of 7.7 bar, accounts for the effect of the strong shock, but the magnitude of the actual tension imosed on the liquid may be greater or lower due to curved exansion waves waves that may enter the liquid and the comression waves from the free surface, resectively. It should be obvious that the magnitude of this negative ressure exceeds the value encountered in the exeriments, where maximum tensions of about bar were measured. It is, thus concluded that cavitation occurred before this value was reached, and revented further ressure decrease. The high number of large bubbles observed (e.g. figure 6. and 6.3) and intense cavitation (figure 6.4) aear to account for this Bubble Dynamics Simulations The Rayleigh-Plesset differential equation, equation (3.0), including viscosity effects, was modelled using Simulink (Coyright by the The Mathworks, Inc ), which is integrated into MATLAB. The model is shown in figure 7.4. The inuts

123 04 required for the model were the initial bubble nucleus radius R o, the initial rate of growth R & o and the ressure field (t). The main difficulty encountered in these simulations was associated with the latter: It was shown, in sections 7.. and 7.., that determination of the ressure variation imosed on the liquid involves consideration of many factors. Figure 7.4. Matlab Simulink model of the Raleigh-Plesset differential equation. The shaded block in the far-right section of the figure 7.4 is the ressure variation inut. The ambient ressure variation with time could be reresented by various blocks such as a ste-function and ram-function. The lowest shaded block is the nucleus radius R o. The first and second shaded blocks from the left are integrator blocks, which require, as inuts, the initial rate of change of bubble radius and initial bubble radius, resectively. Figures 7.5, 7.6 and 7.7 show results of a simulation of bubble growth using the Simulink model. The ressure dro due to the exansion waves is reresented by simly a stefunction inut, occurring at the time equal to zero, with the initial and final ressures equal to 7.93 and bar resectively. This aroximates the ressure variations recorded by transducer 3 mounted at the base of the Mach 3 shock tube.

124 05 Figure 7.5. Variation of radius with time for growth of bubble (R o 0.5 mm) subjected to ste-function ressure dro from 7.93 to bar at time zero. Figure 7.6. Variation of bubble wall velocity with time for growth of bubble (R o 0.5 mm) subjected to ste-function ressure dro from 7.93 to bar at time zero.

125 06 Figure 7.7. Variation of bubble wall acceleration with time for growth of bubble (R o 0.5 mm) subjected to ste-function ressure dro from 7.93 to bar at time zero. The initial values R and o R & o in the above simulation were equal to 0.5 mm and 0 resectively. The figures show that the bubble wall accelerates to a maximum velocity of 3.98 m.s - before stabilising at a seed of about 8.6 m.s -. Figure 7.5 shows that the bubble wall exands without bound. The radius reaches a rather unrealistic value of 7.3 mm after ms. Aendix R shows additional simulations for the same inut ressure but widely different nucleus sizes, in adiabatic and isothermal growth. The general trend of the results here and in aendix R (figures R.-R.7 and tables R. and R.) is the same. The bubble wall velocity rises raidly and reaches a eak value denoted by R & MAX after a time t V. The velocity then aroaches a constant velocity, R & FINAL, asymtotically. Table R. and R. show that this final rate of exansion is relatively constant. The velocity can be estimated quite accurately using the equation * R & FINAL = [⅔( V - ) / ρ L ] ½, where V, * and ρ L are the vaour ressure, the final ressure after the ressure dro and the liquid density resectively. R & MAX increases

126 07 slightly with increasing initial bubble radius R o, while the time taken to exand to this maximum growth rate increases significantly: the growth of the bubble of radius µm u to a eak rate of about 3 m.s - is indiscernible from figure R.5, even though the time scale has been shortened. In addition, the time taken in attaining visible size t V decreases markedly with increasing R o, while the radii at fixed times increase. Thus, the model redicts that, in the exeriments, the bubble nuclei do not aear simultaneously. The sizes of the bubbles when they become visible will vary according to their initial sizes. This is in agreement with the exerimental evidence of figures 6. and 6.3 and aendices M and N. Figure 7.8. Regions of the P-V o lane (initial ressure 7.93 bar, surface tension J.m - ). It is also evident that for the ressure variation described above, the model redicts unbounded growth, with no eriodic oscillations, for all nuclei sizes continued. Figure 7.8 deicts the resonse of a bubble to a rarefaction wave, in the P-V o lane. The curves were lotted using the theory exlained in section 3.3. and [9]. Interestingly, the lot shows that almost any negative ressure will lead to unbounded growth. Figures R.8, R.9 and

127 08 R.0 illustrate a tyical case of eriodic, oscillating growth of a bubble (R o = 0 µm) subjected to a ressure dro, from 7.93 bar to the vaour ressure 339 Pa, at time zero. The bubble oscillates with diminishing amlitude and eriod. Figure 7.9. Variation of radius with time for collase of a bubble (R o 0.5 mm) subjected to ste-function ressure rise from - to bar at time zero. R(0)= mm, R & (0)=8.3 m.s -. The analysis of bubble growth has shown, unequivocally, that no bubble oscillations will occur, in the test liquid of the Mach 3 shock tube. However, hotograhic evidence (aendices M and N) roves the contrary. It has been found that such oscillations may have been caused, at bubble collase, by ressure increases due to comression waves. As discussed in section 6.. and 7.., two ressure increases are exerienced at the bottom transducer. Firstly, the ressure rises from to bar. This was attributed to reflection of the exansion waves from the free surface. Later, the ressure increases to u to more than 6 bar. Figures 7.9, 7.0 and 7. are results of simulations of the resonse of a bubble, with an initial radius R o of mm, subjected to a ressure rise from bar to some value *, at a time zero. At this instant, the bubble radius R(0) is 5 mm, while the bubble wall velocity R & (0) is 8.3 m.s -. The lots show oscillations with

128 09 aroximately constant eriod and slightly diminishing amlitude. Thus, the bubble collase is followed by cycles of rebound and collase. Figure 7. shows that the maximum bubble wall acceleration, at the instants of minimum bubble radius, is almost than m.s -. These corresonds to the instants of bubble rebound. The acceleration of the liquid surrounding the bubble has the same high velocity. This illustrates the otential for significant shock waves to be radiated from the bubbles at collase. Figure 7.0. Variation of bubble wall velocity with time for collase of bubble (R o 0.5 mm) subjected to ressure rise from - to bar at t = 0. R(0)= mm, R & (0)=8.3 m.s -. Aendix R (figures R.-8) shows similar results for different values of the arameters. Comarison of figures 7.9 and R. shows that the frequency of oscillation decreases with decreasing radius R(0). Both figures show oscillations with near constant eriods over 5 ms. However, figures M.7 and M.8 show oscillations with rogressively increasing eriods. Such oscillations roceeded until a singularity occurred at the instant of rebound of the bubble after collase from a large size.

129 0 Figure 7.. Variation of bubble wall acceleration with time for growth of bubble (R o 0.5 mm) subjected to ressure rise from - to bar at t = 0. R(0)= mm, R & (0)=8.3 m.s -. Now consider figures 7.9, R. and R.4. In figures R. and R.4, the amlitude increases rogressively as oscillations increase, while the amlitude decreases slightly. This imlies that an increase in the radius of the bubble R(0) before collase or an increase in the ressure * (i.e. a stronger comression wave) results in rogressively increasing amlitude and hence, in effect, more intense rebounds from minimum size. The effect of R(0) on the intensity of the rebound is in agreement with the molecular dynamics simulations [4] (refer to section 3.4). The effect of * is hysically realisable: An increased ressure on the bubble causes greater comression of its contents, resulting in more intense collase and rebound. Comarison of figure 7.9 with figure R.7 and of figure R. with figure R.8 illustrates that a smaller nucleus radius, R o, also results in more intense rebound. This is understandable since if equation (3.3) is alied for the initial nucleus in equilibrium, then the initial gas ressure Gi will be larger for smaller values of R o (assuming the same values for, v and σ). From equation (3.9), this results in higher gas ressure at the instant of rebound.

130 In figure 7.9, the eriod of oscillation is aroximately 0.9 ms. In the case of figure R., the eriod is below ms for the first 30 ms, while for the case of figure R.4, the eriod increases from 0. ms to about ms over the first 0 ms. In figures R.7 and R.8, the eriod reaches values between 4 and 8 ms. From this analysis of bubble collase, it is concluded that bubbles of different sizes may show very different collase behaviour, even under the same ressure field e.g. comare figures 7.9 and R.. The analysis suggests bubble oscillations with eriods in the order of tenths of milliseconds initially and ossibly in the order of milliseconds for later times. Thus, the bubble behaviour at collase caused by the comression waves in the Mach 3 shock tube may account for the observed oscillations detailed in section Comutational Fluid Mechanics Modelling of cavitation using CFD is a comlex task. Density ratios between the hases in such multi-fluid roblems involving air and water are high (in the order of 000). The codes adoted by most commercial CFD ackages have only recently been develoed and little information on their alications is available [64]. The relevant methods are discussed briefly in aendix S. It was found that none of the available CFD ackages could simulate cavitation while considering the liquid, vaour and non-condensed gas to be comressible at the same time. 7.. The Mach Shock Tube Since exeriments showed unexected results, namely aarent decay of shock waves to low values, various factors were analysed to determine the cause of the discreancies. First, the wave diagram of the tube was constructed to ascertain if the unexlained ressure variation or the low shock strengths measured below the free surface may have been caused by any extraneous waves or if any unforeseen interactions took lace. Other factors are discussed in sections 7.., 7..3 and 7..4.

131 7... Wave Diagram Figure 7. is the wave diagram for the second shock tube with the driver section ressurised to bar and the driven section at atmosheric ressure 0.83 bar. The liquid section is filled with water to a level of 60 mm above the transducer 4. In KASIMIR, the built-in ideal gas models for air and helium were used. The characteristics and related state variables in the uer, water-filled section of the tube were calculated manually. The bold lines reresent shock waves. The values of the fluid roerties corresonding to the labeled regions of the wave diagram are summarised in aendix Q, table Q.. For clarity, a ortion of the gas section, from about 5 ms onwards and the resultant, transmitted liquid waves, are omitted. In addition, the waves in the liquid, reflected from the gas-liquid interface were not lotted. The comression and exansion waves are exected to be reflected as exansion and comression waves resectively. Firstly, one can see that the deth of the uermost transducer below the free surface is large enough to ensure that the incident and reflected waves should be distinguishable in ressure traces recorded there: The incident and reflected waves travel the distance from the transducer to the free surface and back again in aroximately 80 µs. Since this value is much larger than the rise time of the transducer, the ressure waves should be resolved on ressure records. The motion of the free surface was also analysed to determine any effects the acceleration of the free surface might have on the readings of the uer transducer, which was close to the free surface. The initial velocity of the liquid at the free surface was found to be less than 0.5 m.s -. Thus, over the time interval between the the arrival of the incident wave at transducer 4 and that of the reflected exansion wave, the free surface moves uwards less than 0 µm. It is concluded, from this insignificant value, that the motion of the free surface does not affect the transducer readings.

132 3 Figure 7. Figure 7.. Wave diagram of the Mach shock tube.

133 4 Other wave diagrams were constructed for different driver-section ressures (the drivensection ressure was 0.83 bar in all cases). In the case of the driver-section filled with helium at 8 bar, the wave diagram was similar to figure 7. although the time interval between the arrival of the shock and that of the head of the exansion fan was less. This is exected as the lower diahragm ressure ratio should reduce the shock seed markedly but not affect the head of the exansion fan much. Figure 7. and its associated arameters shown in table Q. involve certain assumtions that differ from the exeriments. Firstly, table Q. indicates that the ressure behind the gas shock reflected from the gas-liquid interface is over 7 bar. This value is much higher than the exerimental values because the interface was assumed to behave like a rigid wall and that reflection that occurred at it was erfect or hard reflection, which corresonds to a reflection factor equal to. The relatively low density of the rubber means that the reflection factor is lower than one [43]. Kosing assumed a value of 0.35 for lastic [43]. In the resent case, such a value of the reflection factor would result in more reasonable estimates below 0 bar, for the ressure behind the reflected gas shock. Secondly, when the gas waves iminged uon the interface, transmitted waves were assumed to form instantaneously. This aroximation is justified as the low inertia of the interface should not delay the transmission of waves significantly and aeared to be in agreement with the ressure records. The following sections imrove on the redictions of the wave diagram by analysing the effects of other factors The Area Change A discontinuous area change exists between the ie section and the otical observation section. This was incororated to facilitate the windows. Figure 7. shows reflected exansion waves where waves ass through the area change. For clarity these reflections are not shown for the downward travelling waves, though it should be understood that those waves are also strengthened as they ass through an area reduction. The quasi-steady analysis, described by the comrehensive rocedure of Rudinger [44] was alied to determine the loss of normal shock strength across the area change

134 5 between the ie section and the otical observation section. The steady flow attern was exected to be reached quickly since the change in cross-section is rather small [46]. Referring again to aendix F, figure F., the standing shocks in cases (B) and (C) and the shocks that are swet downstream in cases (C) and (E) are caused by the local article or flow seeds being greater than the wave seeds [46]. This condition is unlikely for waves in water due to the high acoustic imedance of the medium and the low flow seeds. Thus, the condition that would result in the resent case is case (A). In any case, the waves swet downstream in a liquid medium would be ractically coincident on the wave diagram due to their similar seeds. The cross-sectional areas below and above the area discontinuity are (A ) 06 mm and (A ) 709 mm and resectively. Thus, the uward-travelling waves ass through a discontinuity where the area increases by.8 %. The quasi-steady analysis redicted that for an incident shock wave of / =6 in the ie-section, the ressure ratio across the shock wave that is transmitted into the otical observation section will Due to the near acoustic behaviour of shock waves in liquids, simler methods than that of Rudinger may be emloyed. The waterhammer analysis of Parmakian [6] was used for the resent case. The author found that the maximum error between these methods, for the current case was aroximately 3%. In another aroximate method, the Chester-Chisnell-Whitham Channel Formula, the transmitted shock strength is determined analytically by considering the rimary effects of the area change on the wave but neglecting secondary effects of overtaking disturbances [65]. This method show good agreement with the quasi-steady analysis for weak shocks and small area ratios, as shown in figure 7.3. In [65], the one-dimensional equations of unsteady motion were solved numerically for gases over a wide range of incident shock strengths and area ratios. The results also showed good agreement with the methods described above for weak shocks and small area ratios. The methods of Rudinger, Parmakian and the Chester-Chisnell-Whitham formula are suited to cases where area changes are gradual or small [46,65,66]. It is thus

135 6 necessary to discern whether the area change involved in the resent discussion are small. Rudinger [44] stated that for the quasi-steady analysis to accurately describe the interaction of a shock with an area change, the condition ln(a /A ) 0. must hold. The area ratio under consideration,.8 results in ln(a /A ) 0. and thus, the accuracy of the results of the analysis here is uncertain. Figure 7.3. Results from Chester-Chisnell-Whitham analysis (solid lines) and the quasisteady analysis (dashed lines) for the de-amlification of a shock with initial Mach number M Si (adjusted from [65]). The results of the quasi-steady analysis, though conceivable, do not, generally, reresent accurate, -D flow. Parts of the flow field may never aroach a -D flow. While some two-dimensional flow results aroach the resective -D results after sufficiently long times, some are genuinely -D and cannot be reduced to a one-dimensional equivalent [66]. The real flow attern at such an interaction is more comlex with two-dimensional effects as illustrated in figure 7.4 from simulations using -D numerical code [46]. The figure shows a strong, ste-rofile shock after assing through a discontinuity with a

136 7 large area ratio of 0. The working fluid is air. The absolute ressure behind the transmitted shock may be exected to be about 40% that behind the incident shock. The simulation, though very different from the resent discussion of the shock in the Mach shock tube, illustrates the two-dimensional effects that may be resent in it. Since the incident shocks in the Mach 3 shock tube case are effectively much weaker and ass through a discontinuity with a much smaller area ratio, the ratio of the transmitted wave strength to that of the incident wave may be exected to be much greater than 40 %. Since ressure traces show that this is not true, it is doubtful that the area discontinuity alone is the cause of the decayed shock. Figure 7.4. Results of simulation of roagation of shock wave (M s = 3) through a discontinuity having an area enlargement ratio of /0 at the instant 0. ms [46]. The ressure records show that the first transmitted liquid shock becomes rogressively weaker, even before reaching the area discontinuity. Thus, other factors must have contributed to the decrease in shock strength.

137 The Motion of the Interface The wave diagram, figure 7., imlies that the initial shock wave has a ste-rofile although the ressure records showed that the ressure decreases after the eak value is reached. In addition, ressure records showed that the ratio of the transmitted shock strength to that of the incident shock varied for different incident shock strengths. Thus, it is not certain how the strength of the transmitted shock and exansion waves were related to that of the incident waves in the gas and further analysis was erformed to describe the nature of the transmission of the shock. It was thought that the ressure recorded at the transducer just below the surface was smaller than redicted by theory due to extraneous waves originating from the motion of the rubber interface. If the late were to accelerate downwards (or if its motion decelerates while moving uwards), exansion waves would be emitted uwards into the water column. These waves might be exected to overtake and weaken the shock wave, causing the aarent decay noted section 6.. The insertion rubber sheet, which acted as the air-water interface, was treated as an isotroic late and as such, resists bending and has comlex internal stresses, as oosed to a membrane, which has little stiffness (rubber roerties are listed in aendix G). The actual occurrence of transmission and internal reflection of elastic waves within the late will be neglected (this imlies infinite velocities of waves in the late). The late was assumed to not move areciably, such that the vertical dislacement of the late, over its whole area, could be assumed equal. The analysis is based on the theory of Taylor [67], who exlored the behaviour of a nonrigid late of finite size and derived, with simlifying assumtions, an equation that describes the motion of the late in terms of the incident ulse arameters. The vertical dislacement of the late may be exressed by the equations derived in aendix T. As described in aendix T, the author found that, including the effect of the column of liquid above the late, which served to stiffen the late against vibration; the effective

138 9 natural frequency of vibration of the interface ω n in the first mode was found to be 9745 rad.s -. Using the values given in aendix T, the motion of the interface was calculated and the results are shown in figure 7.5. It was found that the late accelerated for aroximately one millisecond before reaching a constant velocity, which was maintained until the late started decelerating aroximately 4 ms after imact of the wave. This means that uward-travelling exansion waves could only be generated from the interface surface 4 ms after the shock wave was generated. Such waves would be too late to catch u and attenuate the shock wave or interfere with the shock and reflected exansion registered at the uermost ressure transducer, near the free surface. In any case, it is unlikely that any waves from the interface could catch u with the shock due to the acoustic nature of all waves in water i.e. the shock and exansion waves travel at a seed very close to the sound seed. Thus, it was concluded that extraneous effects from the interface did not attenuate the shock wave or interfere with the ressure readings at the uermost transducer. Figure 7.5. Results of analysis of the liquid-air interface using Taylor s theory [67].

139 The Effect of Pie Elasticity For a shock wave roagating in a tube, elasticity of that ie may attenuate the shock strength in the same way as an area increase does. The reduction of shock strength due to ie stretch is a relatively unexlored effect Streeter s Method The only known analysis of the effect of ie elasticity on comressible flows is that of Streeter [68] who exressed the rate of area increase as a function of the modulus of elasticity, diameter and thickness of the ie: da D 3 π d = dt 4 t E dt For small time increments ( dt t ), (7.) A = D 4t E 3 π (7.) So, the change in area from its initial area, at any oint, is a function of the excess ressure at that oint. For discontinuous waves, the area is assumed, as is usually done in waterhammer roblems [6], to change instantaneously as imlied by equation (7.). The effect of ie wall elasticity on sound seed may be comuted from the following equation [68,69]: a = K / ρ + ( K / E)( D / t ) Y (7.3) It is evident that if the modulus of elasticity were aroximated to infinity, the sound seed would reduce to equation (.4). Taking the density as 000 kg.m -3, and the bulk modulus as N.m -, the seed of sound redicted by equation (7.3) is 375 m.s - comared to the value of 48 m.s - obtained from equation (.4). Streeter [68] divided the ie into a number of ortions of equal length, and since the magnitudes of the gradients of the characteristics were all equal, the wave diagram was divided into a grid. Interior oints were solved for using the two characteristic equations in finite difference form. At the ends, the two unknowns were solved by two relations,

140 one rovided by the characteristic relation of the characteristic, which assed through it, and the other from the external boundary condition, which must be secified. Attemts to calculate the effects of ie stretch using this analysis and variations thereof did not yield any lausible results. It was concluded that this rocedure could not be alied in the resent case as the shock wave travelling u the ie, with decreasing strength, resented a moving boundary condition Method of Characteristics Consider figure 7.6, the wave diagram of the liquid-filled tube section. The region of the wave diagram of the water column behind the shock was divided into regions (labelled O,R,S,T,U,V,W,X,Y,Z) searated by a family of characteristics. Each exansion wave in the family imlies a weakening of the uwards-travelling shock wave due to ie stretch. In reality, there would be an infinite number of reflected exansion waves between the waves, but since the wave diagram method can only be alied for a finite number of waves, the infinite number of exansion waves was reresented by nine characteristics. For the uroses of ascertaining the effect of ie stretch, only the ortion of the tube from the region U uwards was considered. This allowed a known value of Q to be established since the ressure in region U corresonds to that recorded by transducer, behind the shock. Then, the value of Q in region U was calculated from the known shock strength and shock tables. Firstly, Q was made to vary uniformly from the value at U to the value behind the last characteristic. The latter was taken as the atmosheric conditions as the comatibility condition requires that the ressures on either side of the free surface remain unchanged. However, this roved to be incorrect since it caused the ressure to vary almost uniformly from the first wave (i.e. the ressure at U) to the last (atmosheric ressure). The correct method would have been to choose the values of Q at the characteristics such that they varied from the value at U to the value at Z, since the characteristics between Z and J and between K and M are not caused by ie stretch, but rather by the area discontinuity and free surface resectively. However, the value of Q at Z could not be determined since the strength of the last rarefaction wave resulting from

141 the free surface reflection could not be established. Thus, no results could be obtained from this method. Figure 7.6. Wave diagram of the uer, liquid section of the Mach shock tube, including the effects of ie stretch Elementary Calculation A more simle method, based on equation (7.) and the definition of the bulk modulus (.36) was used after Streeter s method and the adatation of the method of characteristics roved ineffective. The method is, essentially, a calculation of the dro in ressure of the liquid behind the shock wave due to ie stretch. The reduction in ressure is attributed directly to the change in the total volume occuied by the mass of comressed liquid behind the shock.

142 3 The transducers are located 0.83,.48 and.86 m above the interface. When the shock is at each of the transducers, the area change is calculated using equation (7.), where the initial area A i and initial ressure i are the same reference area and ressure for each calculation. Since the area change is roortional to the overressure, V V = V V = V V 3 3 (7.4) In the most general way, the constant in the sound seed equation (7.3), Y, may be calculated by 5 Y = K + ν) + K ( ) for a ie anchored at the ustream end only, ( ν 4 Y = K( + ν) + K ( ) for a ie anchored against longitudinal movement and ν = K( + ν ) K, or alternatively Y = µ Y +, for a ie with exansion joints throughout its length where K = t / D and K /( ) = D D + t. The exressions for a ie with exansion joints throughout its length reduce to aroximately. The ie of the Mach shock tube, with an exansion joint on one end and a free end, is assumed to have a value of Y of aroximately. From the definition of the bulk modulus of comressibility, the change in ressure can be exressed in terms of finite, non-secific volumes: v (/ ρ f) (/ ρ i) i K = ρ = = K v i β / ρ i ρ f M V ( / i) = K (7.5) M V ( / f) Since the resent case involves a constant mass of water at each transducer, V f = K = [ C] K V where i C = V (7.6) V i However, since both V and V i are a function of the length of tube considered, C is constant for all sections considered. Thus, C T = CT = CT3 = V / Vi = A/ Ai = Since the ie inner diameter is 5.5 mm, the initial area is A i = 64.8 mm

143 4 For a shock ressure of 6 bar, the change in area is (5.5 0 ) π(6 0) A = = 0.09 mm 3 9 4( )(00 0) Thus, the change in ressure (from the nominal value of 6 bar) felt the transducers is T T T3 = = = 946 Pa In conclusion, this method redicts that the effect of ie stretch is to lower the shock ressure from 6 to 5.08 bar (a 5% decrease in ressure). However, this elementary method does not imly a cumulative effect i.e. the strength of the shock wave does not decrease rogressively as the wave roagates u the tube, rather, its strength is smaller than its original value (at the bottom end), but constant along the ie length. The magnitude of the estimated ressure dro is too small to comletely account for the decay of the shock evident from the ressure traces.

144 5 8. Discussion 8.. The Mach 3 Shock Tube The direct method of creating tensions in liquids, using a hydrodynamic shock tube, is a relatively unexlored aroach, making comarison difficult. The only known, similar study using water is that of Fujikawa and Akamatsu [93]. Their studies differ from the exeriments using the Mach 3 shock tube in that their driven section was left at atmosheric ressure and not evacuated. In addition, their driver section was ressurised to only about.64 bar as oosed to the corresonding value of 7.93 bar in the Mach 3 shock tube. The effect of the far greater diahragm ressure ratio in the Mach 3 shock tube was that the exansion fan was substantially more sread out i.e. some of the characteristics (refer to the wave diagram, figure 7.) in the exansion fan reached the driven-section of the tube. This did not occur in the tube of Fujikawa and Akamatsu. The theoretical analysis of chater 8 also showed that greater negative ressures, of u to 6.39 bar, could be achieved in the Mach 3 shock tube than in the tube of Fujikawa and Akamatsu who obtained tension of less than 0.5 bar). The longitudinal dimensions of their tube was also significantly larger e.g. Their liquid column is m in height comared to the height of the column, about 300 mm, used in the Mach 3 shock tube. This allowed Fujikawa and Akamatsu to observe greater temoral resolution in ressure traces and high-seed camera ictures. It should be noted that the magnitude of the maximum negative ressure that the test liquid, in the Mach 3 shock tube, is subjected to could be increased, from the estimate 6.39 bar, to greater values by lengthening the driven section, thereby delaying the increase in the liquid ressure caused by the shock that is reflected from the uer end of the tube. The main conclusion made from exeriments using the Mach 3 shock tube, as evidenced by ressure transducer and hotograhic records, was that cavitation, induced by rarefaction waves, was successfully demonstrated in ta water. According to the ressure records from transducer 3, at the bottom of the tube, and the hotograhic records, cavitation occurred at a ressure of bar. This inability to withstand more substantial

145 6 negative ressures is consistent with the values of Cole [55] and Eldridge [79] for untreated water (refer to section 3.). The likelihood of higher tensions occurring at the tube walls and at the liquid-gas interfaces was reduced further by the resence of free gas and other imurities [55]. Exerimentally, it is difficult to distinguish homogeneous nucleation from heterogeneous nucleation that occurs on small, invisible solid articles within the body of the liquid [5]. However, the low tensions achieved indicate that heterogeneous nucleation occurred. It is difficult to ascertain, from the hotograhs, if the nucleation occurs at solid boundaries or within the body of the liquid. Cavitation occurred throughout the height of the liquid column. However, the formation of cavities at the to of the column aeared different to that elsewhere. The more intense cavitation at the to is attributed to the saturation of the to layer of the liquid with suitably large gas nuclei as exerienced by Besov et al. [35]. The corrugated surface of the modified free surface showed that bubbles had formed and coalesced such that the to layer of liquid was totally evaorated. The high-seed camera available was, resumably, not fast enough to accurately cature the oscillating bubble or other effects, such as sonoluminescence as discussed in the following. Nevertheless, the eriod of oscillation of the ulsating bubbles, suggested by the exerimental hotograhic records, aeared to be aroximately 4 ms. In addition, the oscillations of a bubble do not, in general, have a constant eriod [,,95,4,56,70,7]. Thus, in the resent case, the oint-by-oint (i.e. constant frame rate) hotograhic method used may not accurately demonstrate the irregular bubble oscillations []. The bubble simulations (section 7..3) using the Raleigh-Plesset equation showed that no oscillations occurred during the growth hase of all nuclei sizes considered. The oscillations were, instead, thought to have occurred at collase. In all exeriments, the collase may have been initiated by ressure increases. The ressure increased to ositive ressures less than 3 ms after the initial negative ressure was reached. The simulations redicted oscillations at bubble collase of varying amlitude and frequency with eriods of oscillations varying in the order of tenths of milliseconds or milliseconds. The eriod deended significantly on arameters such as the size when

146 7 the ressure increases. The hotograhic records suggested more regular characteristics, with all bubbles reaching maximum size and disaearing almost simultaneously. Pressure records also aear to indicate bubble oscillations. At each increase in ressure from negative to ositive values, oscillations were recorded at the submerged transducer at the bottom of the tube. These ulsations were evident in all ressure records although the eak ressures and trends of the ulsations varied to some extent. These variations suggest cavitational effects. Secondary ressure ulses have been recorded and observed in tube-arrest exeriments [87,95,,4-6] and B-P exeriments [,70] and analysed, in detail, in [95,3]. In such exeriments, the ressure ulsations have been found to corresond to the oscillations of the bubbles. In all ressure records from the lowest transducer in the Mach 3 shock tube, the ressure ulsations aeared to have eriods of less than half a millisecond. Thus, like the simulations, the ressure records suggest oscillation eriods smaller than that showed by the exerimental, hotograhic records. Furthermore, these simulations redicted that bubbles of various sizes grew to sizes of u to about 0 mm (R 0 mm), ms after the minimum ressure of - bar was reached. Then, ms after the minimum ressure of - bar was reached, the bubbles were redicted to have exanded to about 36 mm (R 8 mm). The simulations redict that sizes of this order of magnitude should be visible since the negative ressure is sustained for comarable eriods of time. The maximum bubble radii suggested by hotograhic records were about.5 mm (refer to section 6.), an order of magnitude lower than the theoretical estimates. Possible reasons for the discreancies between the large redicted bubble sizes and those suggested by the hotograhic records are given in the following. Firstly, note the relatively slow frame rate of the camera used: Assuming that the bubble wall exands at the asymtotic rate R & FINAL equal to 8.6 m.s -, the diameter of the bubble under consideration will increase substantially, by about 6.5 mm, over ms, the time between each hotograhic frame. This illustrates that the hotograhic records may significantly

147 8 underestimate the maximum bubble sizes by missing the corresonding instants in time. To cature the bubble growth such that the maximum observed sizes of bubbles are consistently recorded within a millimetre of the actual maximum values, a camera with a maximum frame rate in the order of tens of thousands of frames er second is needed. Simulations showed that bubble collase involves even greater bubble wall velocities. Secondly, it was thought that the rate of evaoration, which is finite, may have been too low to kee u with the exansion of the bubbles and consequently, that the velocities of the bubble walls had been limited to values below the rates redicted by the bubble dynamics analysis (which had not included such factors). It was thought that this may have exlained why the radii of the exerimentally observed bubbles, one or two milliseconds after the minimum ressure was reached, were smaller than the theoretically redicted values. From kinetic theory, it has been found [93] that the finite rate of evaoration m& is: αv RT m& = = αρv = ρvvd (8.) πrt π where m& is the mass of liquid that is evaorated or gas that is condensed er unit time, α is the accommodation coefficient of the liquid, V D is the velocity associated with the rate of evaoration or condensation and V, ρ V and T are the ressure, density and temerature of the vaour for which the ideal gas equation is assumed to aly. Assuming a value of one for the accommodation coefficient, the value of V D in the resent case is about 0 m.s -. Since this value is much greater than the maximum and final velocities of the bubble wall, it is concluded that the evaoration of the liquid occurs raidly enough too kee u with the rate of exansion of the bubble. Now, the validity of the Rayleigh-Plesset model is considered. The assumtions of the Rayleigh-Plesset equation (3.8) were listed in section The assumtions that the bubbles were not affected by other bubbles, or solid articles or walls and that the bubbles remained sherical throughout are doubtful. The assumtion that the liquid density is constant is justified, as liquid comressibility does not affect the bubble dynamics significantly. Its main effect is in the formation of shock waves at rebound after

148 9 collase. While the effect of ressure on the surface tension is negligible, the temerature affects the liquid s surface tension significantly [0]. However, since thermal effects are neglected, the liquid viscosity may be considered constant. The assumtion of negligible thermal effects during bubble growth was justified in section However, at the final stage of bubble collase, temerature effects are always significant due to the high comression of the non-condensed gas within the bubble. In addition, it should be noted that since bubble collase usually becomes very raid, there is insufficient time (collase occurs in microseconds) for areciable heat transfer to occur and the gas behaves adiabatically rather than isothermally i.e. to assume k =.4, k [5]. This was assumed in most of the simulations. Finally, it was assumed that the mass of gas (as oosed to the vaour of the liquid) within the bubble remained constant. This is imlied by the equations (3.7), (3.9) and (3.0). The effect of non-condensable gases is to cushion the collase of a bubble, thereby reducing the high ressures and temeratures that would occur at the final stage of collase. The limited, finite rate of condensation of vaour is the same as the evaoration rate calculated above and thus, V D = 0 m.s -. Since the condensation rocess occurs during bubble collase, consider the figures in aendix R for the collase of different bubbles subjected to ressure increases, as well as figure 7.0. In most cases, the radial velocity of the bubble wall reached negative values with magnitudes greater than 0 m.s -. In addition, in aendix M, a large bubble with a radius of about.5 mm was visible in frame 9 but ractically invisible in frame. Assuming that the radius of the bubble is 0.5 mm in frame, the hotograhs suggest a bubble wall collase velocity of at least 000 m.s -. Thus, it is concluded that, during collase, some bubbles collase too quickly for the vaour within the bubble to condense. In these cases, the noncondensable vaour then acts more as though it was non-condensable gas and thus, the mass of gas effectively increases. A buildu of non-condensable gas near the bubble wall may form a barrier through which vaour must diffuse if it is to condense on the interface [5]. This may effectively slow the finite rate of condensation. In addition, if the effect of the increase in gas content (i.e. the additional cushioning of the collasing bubble) were

149 30 included in the simulations, the results would have shown, resumably, more significantly damed oscillations. Consider that hotograhic records showed that a few bubbles were visible well after the absolute ressure had increased to ositive values. This is visible from the last frames of aendices M and N. In the resent case, the relaxation shock wave ostulated by Hassanein et al. [7] and discussed in section 3.. may be exected to be weak due to the low magnitude of the negative ressure. It follows that the bubbles are not exected to ersist for very long when the ressure increases i.e. the liquid is not discharged to a great extent from the bubble, which is thus exected to collase during the ositive ressure. The resence of bubbles after the ressure has increased may be attributable to noncondensable remaining in some of the bubbles [5], rather than relaxation shock waves. There exists a fixed ressure ratio, that deends on the tube cross-section and the method of claming, for any diahragm or combination of diahragms, which may be referred to as the natural bursting ressure ratio [48,7]. The diahragm will ruture neatly and consistently if it is ricked and bursts when the ressure ratio is above this value [7]. Otherwise, messy flow and inconsistent ruture and hence shocks of different strengths, which are lower than is redicted by theory, for the same ressure ratio) result [7]. Theoretically, for the diahragm ressure ratio of 7.93/0.07, i.e. aroximately 93, the theory redicts a shock strength of about 9. Pressure traces from the transducer embedded in the tube wall of the driven section (transducer ) showed that the shock wave was not fully develoed. A shock generated using such a high ressure-ratio will only become fully formed after travelling a distance equivalent to 3 ie diameters from the diahragm station [48]. Since transducer was 609 mm, or about ie diameters, from the diahragm station, the shock is exected to be fully formed. The formation distance may be larger in the case of the Mach 3 shock tube due to the fact that the diahragm was not comletely removed: In each test, the diahragm remained in one iece although the high ressure broke a hole through it. The remaining material robably disturbed the transient flow, thereby delaying the formation of the shock, even though the diahragm was obviously at the natural burst ressure ratio. This is in contrast to findings [48] that the

150 3 closest agreement between the observed and theoretical shock strength occurs when the diahragm shattered and left little material at the diahragm station. This corresonded to when the diahragm was near its natural bursting ressure ratio. This was not true in the case of the Mach 3 shock tube. The formation decrement may deend, not only on whether the diahragm ressure ratio was above or below the natural bursting ressure ratio, but also on the brittle or lastic behaviour of the diahragm material. In section 7.., the strong, downward-roagating shock that resulted from reflection of the first shock generated in the tube at the uer tube wall was considered. The strength of this shock, which iminges on the liquid column, was estimated as between 6 and 0. However, the ressure record for transducer, figure 6., shows that the shock strength was about. The fact that the initial shock ressure was lower than the value redicted theoretically is artly attributed to boundary layer effects. For the relatively strong air shock strengths involved, this is exected, with the theoretical value only reached behind the shock front, where the ressure increases [7]. Now consider how some of the factors in section 4.7 affect the Mach 3 shock tube. The rate of stressing caused by the Mach 3 shock tube is relatively low: The time taken for the ressure at transducer 3, at the base of the tube, to dro from 7.93 to bar was aroximately.33 ms. This value was consistently reeatable and varied within 0.04 ms of the mean. It follows that the stressing rate was aroximately 6.7 bar.ms -. The conventional bullet-iston exeriments listed in aendix H, table H. (excluding that of Williams and Williams who used a cattle stun gun to imact the iston) roduced ulses with similar stress rates and times elased in decreasing the ressure to the cavitation threshold. These values were aroximately bar.µs - [43] and. µs resectively. The other exeriments listed in table H. involved higher stressing rates and lower times elased during the ressure dro. Where ossible, these values are resented in the tables. In the tube-arrest exeriments of Williams et al. [97], the stressing rate of the exansion wave was about 0 bar.µs -. Thus, the stressing rates of the exansion waves roduced in the liquid in the Mach 3 shock tube were markedly lower than those of other exeriments and artly accounts for the relatively low exerimental tensile strength achieved: The

151 3 lowest magnitude of the maximum sustainable tension of untreated water, obtained from exeriments involving tension ulses, listed in aendix H is 0.85 MPa or 8.5 bar, which is several times larger than the value found here. Consider the measurement system used. The work by Marston and Unger [58], Williams and Williams [96] and Boteler and Sutherland [73] redicted similar, relatively high values for the tensile strength. The similarity between these exeriments is that they did not use ressure transducers to measure negative ressures directly. Instead, Marston and Unger and Boteler and Sutherland estimated the tensions by recording the dislacement and velocity of the membrane that acted as the free surface by interferometric means and relating the article velocity to the shock strength and tension. Williams and Williams determined the tensile strength value by detecting cavitation by using transducers to record the time between arrival of the comression ulse from the iston and the secondary comression ulse emitted from bubbles at rebound and determining which arameters result in suressed cavitational activity (i.e. no secondary ressure ulses) by extraolation as detailed in [96]. In addition, these exerimenters used distinctive methods of roducing comression ulses that resulted in higher stress rates and contributed to their high tensile strength values. The method used in the resent case, was to measure the ressures and tensions directly with PCB 3A transducers. The rise time of these transducer was less than µs and natural frequency of these transducers were greater than 500 khz. These erformance measures, as well as the samling rate of the transduction system equal or better those of transducers used by many other researchers in this field [56,3-7]. Williams et al. [4] showed the effect of slower transducers, which underestimate the magnitude of secondary ressure ulses (refer to section 4.7.4). The study validated the use of transducers such as the PCB 3A transducer. However, shock waves in water will have rise times of the order of nanoseconds and a short ulse length of only a few microseconds, which requires measurement systems with great temoral and satial resolution [5,54]. A transducer or hydrohone with a smaller ressure sensing element may be installed for imroved satial resolution. Such micro-ressure hydrohones, which have very short rise times include the Müller-Platte-Gauge, as described in [4,5,54] and the KP-36 transducer

152 33 (Ktech Cor., USA) used by Williams and Williams [96], which have rise time of about 50 and 66 ns resectively and is ideal for the measurement of cavitation effects [4,96]. However, the liquid ulses generated in the Mach 3 and Mach shock tubes are not discontinuous shocks and aear to have rise and fall times in the order of tenths of milliseconds for which the PCB transducers should be adequate. Now, note that the hotograhic records showed very different cavitational behaviour occurring near the to of the liquid column than in the lower regions: Only a few bubbles were visible below the vaorising to layer of liquid (refer to figure 6.), while many bubbles were visible throughout the lower regions (figures 6.3 and 6.4). In other words, the to layer of liquid shows intense cavitation, while the liquid directly below it aeared to exerience very little cavitation. The exlanation for these observations is firstly that the to layer of liquid had a lower, local tensile strength than the regions below it (due to its saturation with vaour nuclei), and cavitated intensely. Secondly, it is believed that the cavitation at the lower regions of the liquid column extinguished the exansion waves resulting from reflection of the transmitted exansion waves, at the base of the tube. This caused the ressure in the region just below the free surface to dro less than at the bottom of the liquid column and consequently resulted in the lower intensity of cavitation there. This could only be confirmed by installing a ressure transducer through the olycarbonate tube, just below the free surface. One further factor should be mentioned with regards to cavitation. Within limitations of the otical method of detection used, the visible bubble radius R V was chosen as 0.5 mm i.e. bubbles mm in size could be discerned from hotograhs. Bubbles were assumed to be fully develoed once they have reached this size. Kedrinskii [60] chose R V equal to 0. mm. 8.. The Mach Shock Tube The Mach hydrodynamic shock tube demonstrated the use of conventional air shock tubes to roduce liquid comression waves. However, the exeriments did not roduce cavitation by free surface reflection. The factors resonsible for this failure to roduce

153 34 visible cavitation are considered in this section. Firstly, consider the rofiles of the comression ulses roduced. It should be noted that the comression ulses recorded at the lowest transducer in the liquid section of the tube, transducer, resulted in eak, local static ressures of u to about 0 bar. Consider the durations of the ulses near the free surface. At transducer 4, the static ressures decreased to zero (i.e. absolute ressure droed to atmosheric ressure) after several hundreds to thousands of microseconds. Consider the reresentative case of the tyical ressure records of figure 6.5. The ressure record for transducer 4, which lies within the length of the window section, can now be understood: the tension ulse, from the free surface reaches the transducer in only 40 µs. Thus, since this is much smaller than the duration of the wave, the ulse reaches the transducer when the ressure there has not droed much from the eak value. Now, consider transducer. The wave diagram for a gas driver ressure of bar, figure 7., shows that aroximately 0.7 ms after the lower, submerged transducer exeriences the initial, transmitted comression wave, exansion waves, transmitted from the gas section, reach it. Then, less than half a millisecond after the tail of this exansion fan asses the transducer, the first comression wave, of a family that results from free surface reflection of the exansion fan, asses. The dro and subsequent rise following the eak ressure, caused by the families of exansion and of comression waves is evident from the ressure records from transducer on figure 6.5 (between the times marked 6 and 8 ms). Then, the ressure dros again due to the exansion wave that results from free surface reflection of the initial comression wave reaches the transducer. Now, an exlanation for the aarent ressure rise that occurs at the time marked 0 ms is roosed. It is believed that the ressure rise is actually the remaining ressure left behind by the first comression wave. Figure 8. shows the recorded ressure variation at transducer (dark line) with the roosed rofile (light line) of the incident comression wave. It shows that exansion waves transmitted into the liquid after the comression wave and the wave resulting from free surface reflection of the relatively strong comression wave suerimosed onto the ressure at transducer (due to the first comression wave) taking into account the aroriate delay times. This could exlain

154 35 the unexected ressure variations there. The time intervals between the arrival different waves at the transducer corresond, adequately, with the wave diagram. Figure 8.. Suggested form of the ressure rofile of the first comression wave transmitted into the liquid (transducer ). The considerations above exlain why no absolute negative ressures are imosed on the liquid in the case of a bar driver ressure. The lowest absolute ressure, which was recorded at transducer 4, 60 mm below the free surface, was an order of magnitude higher than the vaour ressure and no cavitation was observed through the windows. As noted in section 6., the ressure records for an 8 bar driver ressure aeared different. The incident liquid comression wave aeared more difficult to characterise. Again, the ressure records showed that the liquid at transducers and 3 sustained a negative static ressure of between and bar for a few milliseconds. These corresond to absolute negative ressures of between about 0. and.75. Then, at various times, the ressure rose abrutly to relatively high values. For most of the tests, two of these ressure ulses could be distinguished from the records of transducer while one could usually be distinguished from those of transducer 3. It should be noted that these increases in ressure were recorded before the next comression wave transmitted from the gas section had arrived. The variability of the amlitude of the ressure rises suggests that cavitation occurred during the negative ressure hases at

155 36 these transducer: The collase and rebound of many bubbles was though to have emitted shock waves. From the wave diagram, it was seen that the ressure rises that caused bubble collase were due to:. The downward-travelling comression waves resulting from free surface reflection of the exansion fan and. The uward-travelling comression waves that result from reflection, at the gasliquid interface at the base of the liquid column, of the exansion wave that results from free surface reflection of the first liquid comression wave. However, since the liquid at these stations could not be observed since they were situated well below the windows, the cavitation could not be confirmed hotograhically. The fact the maximum tension value reached was close to the value of bar, found in the Mach 3 shock tube and since the working fluid in both cases was settled ta water reinforce the belief that the negative ressures resulted in cavitation. The wave diagram for an 8 bar driver ressure showed no major differences from the bar driver case. The most significant difference was that, in the lower, gas section of the tube, the time between the arrival of the head of the exansion fan and the shock wave was less than in the bar case. This is understandable since the lower diahragm ressure ratio should affect the shock seed much more than the seed of the head of the exansion fan. Consider transducers and 3: The dro in ressure from the ositive value, after the eak is reached, to the negative ressure is attributed to the exansion wave from free surface reflection of the first liquid comression wave. In the case of the 8 bar driver ressure, like that of the bar driver ressure, no absolute negative ressures were imosed on the liquid at transducer 4. This was again attributed to the transducer being too close to the free surface and the tension ulse, resulting from free surface reflection, being suerimosed on a ositive ressure close to the eak value of the incident comression wave. From all of the ressure records, it may be concluded that for a shorter ulse length or greater transducer deth, one could imose a maximum absolute negative ressure of

156 37 aroximately 0.64 or 0.7 bar for a or 8 bar driver ressure resectively. These maximum tensions are less than the tension that the Mach 3 shock tube is caable of imosing. One could theoretically subject the liquid to greater tensions if the first liquid comression ulses had not been attenuated during their roagation u the ie, because a weakened comression wave would result in an equally weak exansion wave when reflected from the free surface. The attenuation was evident in all tests, regardless of the driver ressure. Such attenuation of a comression wave travelling through a water-filled tube has been found to occur in other ulse reflection exeriments involving water as the working fluid [95,98,6,74]. Overton and Trevena [6] found that the smaller the rise time of the ulse, the less attenuation occurs. This was attributed to the dissiative forces having less time to act for these higher stress rate. Earlier, they found that tension ulses also suffer attenuation when travelling through a liquid [95,6]. Couzens found that the ressure behind a ulse travelling in boiled, deionised water would decrease by about 0i% for every metre travelled. No other ublished work on the characteristics of shock attenuation in water was found. None of the exerimenters described or analysed the cause of this attenuation. It is evident, from aendix P (figures P.-P.4), that the attenuation of the uwardtravelling comression wave is greater from transducer to 3 than from 3 to 4. Since these distances are equal, this indicates that either the attenuation of the ulse is a cumulative rocess or, alternatively, the discontinuous change in area, between transducers 3 and 4 contributed significantly to the attenuation. It was shown that the change of area needed to incororate the observation windows would not affect the liquid comression wave to the extent that ressure traces showed. The analysis showed that if the incident shock has strength of 6, it would be attenuated by about 8i% to strength of The attenuation must be substantially less than 40i%. The aarent decay of the uward-travelling liquid comression wave could also not be exlained by the effects of ie wall elasticity: Simle calculations redicted that for a

157 38 shock with a eak ressure of 6 bar, the ressure would be attenuated to 5.08 bar. This corresonds to a decrease in shock strength of about 3i%. In addition, the alication of Taylor s method redicted that no waves from the gas-liquid interface at the base of the liquid column would overtake and attenuate the shock. Other factors that may contribute to attenuation of a moving shock are boundary layer and transverse wave effects. A boundary layer generates moving comression and exansion waves, which alter the roerties in the region behind the shock, which becomes non-uniform, thereby decreasing the shock strength while the bulging of the diahragm of the shock tube leads to the generation of a curved shock and, consequently, transverse waves [48]. The high slenderness ratio of the tube is exected to contribute to shock attenuation by increasing the effects of transverse wave reflections, the boundary layer and viscous dissiation [48,7]. The surface roughness and imerfections on the inside of the tube, which had not undergone any secial surface finishing treatments, as well as the relatively high gas driver ressure leads to the growth of the boundary layer, which restricts the flow significantly, resulting in shock attenuation [7]. However, because of the high acoustic imedance of water, article seeds behind the shock are low. This, as well as the near acoustic behaviour of waves in water, minimises the effect of the boundary layer. It was thought that the liquid shock waves roduced might have become unstable and slit into two, weaker waves moving in the same or oosite direction as detailed in [0]. However, the equation of state of water is convex at all temeratures i.e. the value of the fundamental derivative G, a third derivative of the internal energy, is greater than zero. Secondly, its adiabatic constant k (= γ) is greater than Γ, the Grüneisen coefficient (a second derivative of energy). These conditions guarantee shock stability and indicated that shocks travelling in water will not become unstable and slit u. Over many tests, the incident gas shock strength varied quite widely. In addition, the initial ressure behind the gas shock was lower than that redicted by theory. This decrease in shock strength may be attributed to distance attenuation and formation

158 39 decrement described above [48,7]. To obtain more consistent shock strengths, the diahragms must be more carefully selected as detailed in [7]. An imortant factor to consider, again, is the rate of stressing of the waves generated in the Mach shock tube. Consider the ressure records for a bar driver ressure, figures 6.5 and P.. At transducer, the lowest submerged transducer, the rise time of the shock varied between aroximately 60 and 90 µs. Near the free surface, the rise time of the shock varied between about 80 and 360 µs. For 8 bar driver ressures, the rise times of the comression wave at transducer and 4 varied from.-.3 ms and from ms resectively. Now, consider the effect of the transition layer in the case of the Mach shock tube. A characteristic time of a transition layer of thickness z o is the time taken by a ulse, roagating at the sound seed of the liquid c l, to travel u and down the layer. It is equal to z o /c l [34]. If the characteristic time was very much smaller than the time constant of the incident wave, transition layer will not affect its reflection at the free surface. However, if the magnitudes of the characteristic time and incident wave time constant are comarable, the effect of the transition layer will be to lower the eak tension and sread out of the ulse [34]. The time constants of the comression ulses in the exeriments of Sedgewick and Trevena, who found that the free surface reflected was not a mirror image of the incident comression wave, were in the order 0-4 s i.e. tenths of milliseconds [5,34]. Since these ulses were seriously distorted on reflection, Temerley and Trevena [34] inferred that the characteristic time delay introduced by such a free surface is of the same order. For and 8 bar driver ressures, the time constants of the liquid comression waves near the free surface, generated in the Mach shock tube, was thus an order of magnitude higher and of the same order of magnitude as the characteristic time resectively. Thus the transition layer is likely to have had an effect on the reflected tensions. Again, since the waves involved were not discontinuous and had rise times in the order of tens and hundreds of microseconds, the transducers were judged to have adequately measured the ressures behind them. Overlaing of tension ulses

159 40 9. Conclusions Samles of water were forced to cavitate by generating tension in the liquid, using the Mach 3 hydrodynamic shock tube. Tensions of greater magnitude than about 6 or 7 bar could be imosed on the test liquid if the length of the driven section is increased. Bubble dynamics simulations showed reasonable, qualitative agreement: the bubbles generally grew during the negative ressure hase and exhibited oscillatory growth and collase when the ressure increased to ositive values, due to comression waves from the free surface and from the uer end of the tube. Pressure records showed secondary ressure ulsations, confirming the oscillatory nature of the collase at each rise in ressure. More quantitative comarison of theory and exeriments would require a high-seed camera with a higher frame rate. Pulse reflection exeriments were not successful in roducing cavitation. The test facility was similar to bullet-iston equiment, excet that the comression wave was generated by a conventional gas shock tube searated by a flexible sheet. Liquid comression waves with eak static ressures of u to about 0 bar were roduced. However, no negative ressures or cavitation was observed through the otical section of the tube. Pressure records suggested that maximum negative ressures, similar to those obtained in the Mach 3 shock tube (~ - bar), and cavitation were generated for lower driver ressures. However, this could not be confirmed as these events occurred below the windows and were not visible. The fact that no absolute negative ressures or even ressures below the vaour ressure were exerienced was attributed to the combination of two factors: the ulse duration and the osition of the free surface with resect to the windows and the transducer near the free surface. The comression ulse length, which could not be redicted theoretically before design, was, in all tests, greater than 000 µs. The free surface was not high enough above the transducer for the eak, reflected tension ulse to be suerimosed on the lower ressure behind the incident comression wave. In addition, the transition layer should, in the case of the Mach shock tube, have an effect in lowering the eak, reflected tension value.

160 4 0. Recommendations 0.. Imrovements of the Mach 3 Shock Tube Rig Various means may be ursued for demonstrating more intense cavitation. One may seed the liquid with solid articles or free bubbles. As stated earlier, it would be ossible to lengthen the driven section such that the arrival of the strong shock wave at the test section and subsequent collase caused by it would be delayed. Also, the water can be heated to a temerature near the boiling oint to further romote bubble growth []. To reduce the intensity of cavitation in the tube, the liquid and containers must be urified more. To reduce the amount of free gas entrained in the test liquid, the lower section of the tube should be modified, allowing the tube to be filled the through the bottom [54]. The rig could also be connected to an absortion system for absorbing any gas bubbles, while the dissolved gas content could be controlled by an air-content control system []. To generate large liquid tensions, heterogeneous nucleation must be avoided. For this urose, secial equiment is needed [8] while reliminary washing of exerimental equiment is imortant [35]: The samle and its container must be elaborately and thoroughly cleaned as chemical cleaning agents in water may alter the microbubble size distribution, and thereby considerably affect cavitation in the liquid samles [,35,86]. All surfaces would need to be ground down and a totally re-designed rig, with a test section designed to contain a smaller volume of liquid, to minimise imurities, would robably be needed. Using a commercial gas cylinder, having a very low dew oint, would minimise the effects of water condensation [48]. In light of the foregoing, it is obvious that a camera with a higher frame rate would be beneficial to the studies. In addition, a hydrohone could be installed into the tube to ensure that raid changes are recorded accurately (note that the measurement of some shocks emitted at bubble collase may be ractically discontinuous and require a measurement system including a transducer with a faster resonse.

161 4 0.. Imrovements of the Mach Shock Tube Rig A material with lower inertia than insertion rubber may be used to act as the gas-liquid interface. One benefit of this may be in generating comression waves with shorter rise times (by accelerating the liquid more raidly), thereby minimising the effect of the transition layer. Another effect might be in generating liquid comression ulses with shorter durations, resulting in cavitation within the length of the observation section of the tube. On the other hand, one may alter the tube length: The ressure records for a gas driver ressures of 8 bar showed negative absolute ressures and suggested that cavitation occurred at the lower art of the tube, at transducer. One could shorten the uer, liquid-filled section such that the transarent observation section is at this lower art of the tube. However, one would then have to ensure that unwanted overlaing of waves, near the base, does not occur. Alternatively the tube could be of modular structure such that is would be adjustable. Shortening the liquid section of the tube would also result in less attenuation of comression ulses and thus, stronger tension waves. Alternatively, one could construct a transition section to make the change in cross-sectional area between the window section and the ie more gradual. [93] Further Studies While Shock Wave Lithotrisy or SWL is commonly and effectively used to break kidney stones and for treating other uncomlicated, uer urinary tract calculi, it is currently ineffective in more comlicated cases, such as that of urethral stones [7]. While cavitation lays a significant role in stone breakage in SWL, exeriments have shown that cavitation is definitely linked with SWL-induced tissue damage [7]. For the develoment of SWL and imrovement of lithotrisy rocedures and techniques, the cavitation events must be controlled. Thus, further research on the arameters affecting and effects of cavitation as well as the recise mechanism of cavitation and stone breakage, which are not comletely understood, would be worthwhile.

162 43 In articular, cavitation that results behind focussed sound waves using soft or ressurerelease reflectors, as observed by [7,54], deserves further attention. It is articularly interesting as it may find alication in medical rocedures where conventional shock wave lithotrisy or SWL is ineffective. Müller first suggested the ossible alication of the soft focussing technique in maniulating human body tissue (which has a similar acoustic imedance to water) such as cancer cells, where focussed shocks are also ineffective [54]. He ostulated that the combination of the tensile stresses and shocks from bubbles collase may attack soft materials. The technique could also be used for more comlicated cases of stone breakage, as mentioned above [7]. The effect of using non-rigid or soft reflectors is, effectively, to alter the ressure variation at the focus such that the negative ressure recedes the ositive eak. The effects of soft reflectors are not yet comletely understood and the studies of Müller and Bailey et al. are the only known exeriments that involve them. As discussed in section 4.4, the exeriments of Bailey et al. [7] showed that it may be safer than using conventional reflectors and imrove SWL alications although it aears to be less effective in stone comminution (breakage) [7]. Such focussing exeriments could easily be erformed using the Mach shock tube rig, which could be modified again to the focussing setu described and used by Karnovsky [4]. In his exeriments using hard reflectors, absolute negative ressures and what aeared to be cavitation clouds were roduced in water. As stated earlier, an advantage of focussing of waves is that one can roduce true homogeneous nucleation away from solid boundaries. Further studies may be worthwhile in the field of heterogeneous nucleation. Adequate techniques for resolving and characterising the size and nature of the total sectrum of nuclei have yet to be develoed [80]. Exeriments with this goal may involve the deliberate seeding of a test liquid with solid articles of known characteristics such as those of Arora et al. [34] and Marschall et al. [39]. Indeed, the characterisation of nuclei may be one of the least develoed fields related to cavitation. Consensus has not been reached on the mechanism of stability of free gas nuclei. Until research yields a better

163 44 understanding of nuclei, exact redictions of the distribution and size of nuclei and imroved heterogeneous nucleation models resent in water samles cannot be made [7]. The Mach 3 or Mach shock tube or modified versions of them could be emloyed for such studies.

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179 60 8. Reynolds, O. On the Internal Cohesion of Liquids and the Susension of a Column of Mercury to a height of More than Double that of the Barometer. Mem. Manchester Lit. Phil. Society, Vol. 7, 3d ser.,. -9, Kna, R. T. Cavitation and Nuclei. Transactions of the ASME, Vol. 80, 958, Zheng, Q., Durben, D. J., Wolf, G. H., Angell, C. A. Liquids at Large Negative Pressure: Water at the Homogeneous Nucleation Limit. Science, Vol. 54, 99, Roedder, E. Metastable Suerheated Ice in Liquid Water Inclusions Under High Negative Pressures. Science, Vol. 55, 967, Henderson, S. J., Seedy, R. J. Temerature of Maximum Density of Water at Negative Pressure. Journal of Physics and Chemistry, Vol. 9, Greensan, M., Tschiegg, C. E. Radiation-Induced Acoustic Cavitation; Aaratus and Some Results. Journal of Research of the National Bureau of Standards, Section C, Vol. 7, 967, Galloway, W. J. An Exerimental Study of Acoustically Induced Cavitation in Liquids. Journal of the Acoustical Society of America, Vol. 6, Willard, G. W. Ultrasonically Induced Cavitation in Water: A Ste-by-Ste. Process Journal of the Acoustical Society of America, Vol. 5, 953, Briggs, L. J. The Limiting Negative Pressure of Mercury in Pyrex glass. Journal of Alied Physics, Vol. 4, Carlson, G. A. Dynamic Tensile Strength of Mercury. Journal of Alied Physics, Vol. 46, 975, Besov, A. S., Bichenkov, E. I., Kedrinskii, V. K., Palchikov, E. I. Investigation of Initial Stage of Cavitation by Diffraction Otic Method. In: Pichal, M. Proceedings of the IUTAM International Symosium on Otical Methods in Dynamics of Fluids and Solids. Pichal, M. (Ed.), Sringer-Verlag, Berlin, Heidelberg, 984, Franc, J. P. Partial Cavity Instabilities and Re-Entrant Jet. Proceedings of the Fourth International Symosium on Cavitation, Pasadena, California, U.S.A., 0-3 June, 00, Cav00: lecture 00. (6) 94. Fluent, Inc. FLUENT 6.. User Guide. Section.5.3.

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181 6 Aendix A Entroic Relations for Arbitrary Gas States The relations between ressure, density and sound seed in ideal gases will now be derived. Multilying equation (.9) by /c (γ-) and defining the non-dimensional form of the secific entroy s as S = s/c (γ-), γ γ ) / ln( ) / ln( T T S S = = γ γ ) / ( ) / ( ln T T S S (A.) = ) / ( ) / ln( ) ( T T S S γ γ γ (A.) This may be rearranged to give: ) / ( ) / ( ( ) T T e S S = γ γ γ Thus, ) ( ) ( ) ( S S S S S S e a a e R a a R e T T = = = γ γ γ γ γ γ γ γ γ γ γ ) ( S S e a a = γ γ γ (A.3) Similarly, using the ideal gas equation (.7) and the sound seed equation (.), ) ( S S e a a = γ γ ρ ρ (A.3) (.9)

182 63 Aendix B Common Relations Across a Liquid Shock Wave Here, exressions for the ratios of the Mach number, ressure, density, sound seed and article velocity behind a liquid shock wave to those roerties ahead of the shock, are derived. These ratios are the most commonly tabulated arameters in shock tables. A similar derivation, for air, is given in [45]. The conservation of mass equation is u u ρ ρ = (.6b) a M a M ρ ρ = Using the sound seed equation (.4), / / = ρ ρ ρ ρ n M n M / / ρ ρ M M = Using Tait s equation, n M M / ) / ( = ρ ρ n M M / ) / ( = n M M / / = Thus, n n M M + = (B.) Conservation of Momentum: u u ρ ρ + = + (.7b) + = + ρ ρ ρ ρ n M n M n M B n M B + + = + +

183 64 ) ( ) ( nm nm + = + nm nm + + = (B.) Combining equations (B.) and (B.), The ustream and downstream Mach numbers may be related by: n n nm nm M M = (B.3) This equation has two roots: M M = which corresonds to a Mach wave (B.4) n n nm nm M M = which corresonds to a shock wave. (B.5) Equation was used to draw u shock tables for normal shock waves in liquids. Generally, the article seed ahead of the shock wave is assumed to be zero. Thus, M =M S. It follows that the shock table will include all values where M 0 (M <0 corresonds to rarefaction shocks) and M M. The density ratio across the shock wave, also included in shock tables can be calculated directly from the ressure ratio across the shock and Tait s equation: n = ρ ρ (B.6) The sound seed ratio across the shock is determined as follows: From equation (.4): ρ ρ ρ ρ n n a a = = Using Tait s equation (.37), n a a / = Thus,

184 65 n n a a = (B.7) or = n a a ρ ρ (B.8) These equations are analogous to the isentroic relations (.6) and (.7) and, like Tait s equation, is valid for liquids at ressures u to 500 MPa. From the equation of continuity, the article velocity ratio across a shock wave is: ρ ρ = u u (B.9)

185 66 Aendix C Wave Diagram Considerations C.. Non-dimensional Forms of the State Variables Wave diagram rocedures are greatly facilitated by the use of non-dimensional quantities. The following non-dimensional forms of the seed of sound, article velocity and secific entroy are defined as [44]: ˆ a A = (C.) a 0 ˆ u U = (C.) a 0 s S = (where S 0 = s0 = 0 ) (C.3) ( γ ) C P C.. Method of Characteristics From the fundamental differential equations (conservation mass and momentum equations), the following characteristic relations, derived by Rudinger [44], allow systematic solution of nonsteady-flow roblems. δ γ 3 + P A A S DS γ γs AU ˆ ˆ ln δ Aˆ ln Aˆ + γ Aˆ Aˆ = I Ψ e (C.4) δτ ξ τ δτ δ γ 3 Q A A S DS γ γs AU ˆ ˆ ln δ Aˆ ln Aˆ ( γ ) Aˆ Aˆ = + + I Ψ e (C.5) δτ ξ τ δτ where I is the non-dimensional form of f (the sum of body and dissiative forces er ( ) Dτ Dτ unit mass) = I and Ψ is the non-dimensional form of ψ (the mass flow fl 0 / a 0 removed er unit length) Ψ = L a ψ Aˆ. 0 0 / γ 0 It is evident from these equations that changes in the arameters (P and Q, S) may be calculated by eliminating negligible terms and multilying both sides of the equations by the differential time δτ.

186 67 The fundamental differential equations used to derive the characteristic relation do not aly across discontinuities such as shock waves, which effectively searate a wave diagram into two arts. Each art must be treated searately and aroriately matched along the shock boundary and for the shock tube case the regions on either side of the shock may be treated as isentroic. The terms involving entroy in equations (C.4) and (C.5) may then be equated to zero. For the isentroic regions on either side of the shock, δ P + δτ ˆ ˆ ln A ˆ ln A = AU A + I ΨAˆ ξ τ γ 3 γ (C.6) δ Q ˆ ˆ ln A ˆ ln A = AU A + I ΨAˆ δτ ξ τ γ 3 γ (C.7) These exressions may be further simlified since, in most alications, the mass flow removed is usually insignificant and since, in gases, the gravitational body force may be neglected, as shown by Rudinger [44]. Neglecting these terms yields the following equations, which show that cross-sectional area variations are the only remaining factor influencing the values of P and Q. δ + P A A = AU ˆ ˆ ln Aˆ ln (C.8) δτ ξ τ δ Q A A = AU ˆ ˆ ln Aˆ ln (C.9) δτ ξ τ The first terms of these equations are only used when the cross-section of the duct is not uniform while the second terms are only used when the tube area is variable with time. C.3. Other Useful Quantities for Wave Diagram Construction In this section, the arameters that are articularly useful in tthe construction of wave diagrams (when tabulated in shock tables), namely P / A ˆ, Q / A ˆ, Û / Â and S, are exressed in terms of more well-known and commonly tabulated terms. From the definition of the characteristic quantity P, P Aˆ = Aˆ Aˆ γ + Uˆ Aˆ Aˆ γ + Uˆ

187 68 ( ) ( ) + = U U A A A ˆ ˆ ˆ ˆ ˆ γ A U U A A ˆ ˆ ˆ ˆ ˆ + = γ A U U U U A A ˆ ˆ ˆ ˆ ˆ ˆ ˆ + = γ A U U U A A ˆ ˆ ˆ ˆ ˆ ˆ + = γ (C.0) The quantities Â, A ˆ and M S are always ositive, but U ˆ and Û may be ositive (for Q- shocks) or negative (for P-shocks). Thus, For a P-shock: S M U U A A A P + = ˆ ˆ ˆ ˆ ˆ γ (C.) For a Q-shock: S M U U A A A P = ˆ ˆ ˆ ˆ ˆ γ (C.) Similarly, from the definition of the characteristic quantity Q, For a P-shock: S M U U A A A Q = ˆ ˆ ˆ ˆ ˆ γ (C.3) For a Q-shock: S M U U A A A Q + = ˆ ˆ ˆ ˆ ˆ γ (C.4) The arameter A U / ˆ may be calculated from as follows: ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ = = = a a M A A M A U A A A U A U (C.5) The equations above aly for shock waves in any medium. For gases, which are treated as non-isentroic, the non-dimensional entroy changes may be exressed in terms of the shock Mach number as follows: Using the ideal gas equation, equation (A.) becomes: ( ) ) / )( / ln( ) ( + = γ γ γ γ T T S S and using the Rankine-Hugoniot relation (.) and equation (.0) gives [0,44]: = ) ( ) ( ln ) ( γ γ γ γ γ γ γ γ γ S S S M M M S S (C.6)

188 69 Aendix D One-Dimensional Interaction Problems The collision of a shock or rarefaction wave with a wall or contact surface is discussed in section.8.. The other tyes of one-dimensional wave interactions are discussed here. Figure D.. Collision of waves: Initial and final Conditions Collision roblems involve two waves roagating towards each other through a state (denoted by 0), as illustrated in figure D.. The state 0 and the states behind each wave (states and ) are usually known. The unknown state 3 then rescribes the strengths of both waves (waves c and d) resulting from the collision. The loci of all states that may be connected through a shock or rarefaction to the states and may be lotted on a (,u) diagram. The state 3 formed after the collision must lie on both curves and is found at their intersection. However, only one of the final states is hysically ossible. For two shock or rarefaction waves colliding head on, only the state (b), with both of the resulting waves being shock or rarefaction waves resectively, is ossible [46]. Generally, weak shocks are always weakened by the collision [44]. If the initial wave, travelling from left to right, is a shock and the other initial wave is a rarefaction wave, only the state (b) is ossible (with the shock travelling

189 70 left to right and the rarefaction wave travelling from right to left). The result will be a decrease or increase of ressure, deending on the strength of the waves [44]. Figure D..Overtaking of one wave by another: Initial and final Conditions Overtaking roblems, as illustrated, generally, in figure D., involve either a shock overtaking a shock or rarefaction wave, or a rarefaction wave overtaking a shock. For a shock overtaking another shock, the transmitted wave is always a shock, while the reflected wave can be a shock, rarefaction or Mach wave [46]. When a rarefaction wave overtakes a shock, the shock strength is attenuated (it may attenuate the shock to a Mach wave, resulting in a transmitted rarefaction wave). The reflected waves may be comression, Mach or exansion waves [46]. When a shock overtakes a rarefaction wave, the resulting wave attern deends on the strength of the shock. The reflected wave may be a shock, rarefaction or Mach wave. The transmitted wave will be a rarefaction wave, for weak overtaking shocks, and a shock, for stronger overtaking waves [46]. When shock waves of different strength collide or when one shock overtakes another, the entroy rise of the fluid deends on which of the shocks is crossed first. As a result, the article ath through the intersection oint, where the interaction occurs, becomes a contact surface searating regions of different entroy as illustrated in figures D. and D.. In a liquid medium, since entroy changes may usually be neglected, one may disregard this contact surface.

190 7 Aendix E Collision and Reflection of Waves from a Rigid wall E.. Acoustic Waves The resent discussion is strictly limited to reflection at infinitely rigid walls. In the case of acoustic waves, the reflection from an infinite rigid body results in a ressure change at the surface, which is twice the incident ressure [55]. Denoting the ressures behind the incident, reflected and transmitted waves as i, r and t resectively, = = (E.) t r i E.. One-dimensional Wall Reflection of Finite Waves in Air The difference between the acoustic condition (E.) and the real reflection of finite waves in gases may be very large. When a shock roagating in air is reflected at a rigid wall, the ressure ratio across the reflected shock r / i may be exressed in terms of the incident shock wave Mach number M i and the secific heat ratio behind the incident shock as follows γ i : r i = ( γ i ) + + i M (3 γ i i M ( γ ) i ) (E.) E.3. One-dimensional Wall Reflection of Finite Waves In Water Since shock waves in water are regarded as weak, reflection of these waves obeys rules close to those of acoustic theory [54]. Even for relatively large ressures, the deartures from acoustic ressure doubling are not very large [55]. In the secial case of reflection of a lane shock from a rigid wall when the boundary is normal to the incident wave, Cole [55] derived the following exression for the ratio of reflected to incident ressures using the equations of conservation of mass and momentum for the incident and reflected waves,

191 7 ) / ( ) / ( = ρ ρ ρ ρ ρ ρ r i r i r (E.3) where the subscrit 0 imlies roerties ahead of the incident shock. Also, using Tait s relation (.37) for comression for the incident, reflected and transmitted waves, ) / ( ) / ( = n i n r i r ρ ρ ρ ρ (E.4) Equations (E.3) and (E.4) may be solved simultaneously for the ressure and density in terms of the secified values of the arameters behind the incident wave. The ressure ratio on the left hand side of equations (E.3) and (E.4) is always greater than and aroaches this value for weak shocks.

192 73 Aendix F Wave Interaction with Discontinuous Change in Cross-Section Two methods, used to solve roblems involving roagation of a shock wave through a discontinuous change in cross-section, are discussed in the following sections. First, the wave diagram or quasi-steady method, described in [44,46,65,66], is considered. Then, the waterhammer method, described in [6], is discussed briefly. For the similar roblem of roagation of an exansion or isentroic wave through a discontinuous change in cross section, the analysis of Parmakian or a simlified adatation of the quasisteady method, as shown in [44,65] may be alied. F.. The Quasi-Steady Method Figure F.. Possible steady flow wave atterns for the assage of a shock wave through a small area enlargement [65]. When a shock wave asses through a discontinuous change in cross section, it causes unsteady and generally two-dimensional, local flow disturbances at the discontinuity. These transients include transmitted disturbances, which overtake and change the transmitted shock and reflected disturbances, which may form a shock or rarefaction wave moving ustream [46]. The actual flow attern may, after a certain eriod has elased and once all significant, unsteady disturbances and interactions have subsided,

193 74 reach an asymtotic steady flow attern as shown by Greatrix and Gottlieb [65]. Quasisteady analysis yields an aroximate solution, obtained by neglecting all twodimensional and unsteady wave henomena [65,66]. In the most general case, in the case of the assage of a shock wave (S i ) through an area enlargement, 5 unique (quasi-steady flow at late times) wave atterns are ossible [65,66]. For discontinuous cross-sectional area changes in gas flows, it is generally not ossible to determine which one occurs [46]. In all cases, a shock (S t ) is transmitted. F.. The Analysis of Parmakian When incident waves are weak, a simler method than that described by Rudinger may be alied to the roblem of a shock wave assing through a change in cross-section. Parmakian [6] detailed a simle method, which allow the ressure heads of the incident and transmitted wave to be related by simle functions of the area and sound seed on either side of the change in cross section. This method is valid, in the case under consideration in chater 7, due to the near acoustic behaviour of waves in water. When a wave is incident at a change in cross-sectional area, the ressure heads of the incident and transmitted wave are related by: H = sh (F.) while the heads of the incident and reflected waves are related by: f = rh (F.) The transmission coefficient s and the reflection coefficient r are defined as: s r A / a A / a + A = (F.3) A / a A / a + / a A / a A / a = (F.4) and may related to one another by s r =, which follows from the comatibility condition that the ressure behind the reflected and transmitted waves must be equal ( H f = H ). A wave reflection occurs at every change in ie area.

194 75 Figure F.. Incident (F ), reflected (F ) and transmitted (f ) waves in the interaction of a wave with an area enlargement discontinuity [6].

195 76 Aendix G Physical and Thermodynamic Proerties of Substances In this section, the hysical and thermodynamic roerties of various substances relevant to the exerimental facilities, namely water, air, helium, olycarbonate and insertion rubber, are discussed. It is imlied that these values are assumed throughout this thesis. G.. Proerties of Liquid Water For any cavitation exeriment, the knowledge of the liquid roerties is of major imortance []. Water, the most universal of all liquids, is of rincial imortance in nature, chemical and technological alications and the number of devices utilising it as the working fluid greatly exceed all others []. Water is commonly involved in cavitation studies and is also the most common liquid used in shock wave research [5]. While its transarency allows direct visual and hotograhic detection of cavitation, water does not transmit light waves erfectly. Attenuation of light intensity occurs mainly due to imurity content [,55]. Table G.. Proerties of liquid water. Proerty Magnitude Units Comments Molecular Weight M 8.0 kg./kmol [5] Density ρ 000 kg.m -3 At 4 C and atm [0,5]. Within % of exact value in the temerature range 0-50 C. Sound Seed a 48 m.s - Calculated. Bulk Modulus K Pa Accurate for ressures below one megaascal [0]. Isentroic Comressibility β S Isothermal Comressibility at low ressures/temeratures Pa - Accurate for ressures below one megaascal [0].

196 77 Table G.. Proerties of liquid water: Continued. Proerty Magnitude Units Comments At 0 C and atm. [Efunda] Surface Tension S N.m - This quantity varies significantly with temerature but not ressure [0]. Kinematic Viscosity ν 0.00 cm /s 0-6 M /s At 0 C and atm [0]. Adiabatic Exonent γ Obtained emirically [0]. Characteristic Proerty A W Pa Obtained emirically [0]. Acoustic Imedance Z kg/m s At 0 C, and atm. Vaour Pressure V 339 Pa bar At 0 C [5]. Boiling Point T sat 00 C At atm [3]. Secific Latent Heat of Vaorisation 60 kj/kg At boiling oint of: 00 C at atm. At 7 C, and atm. Within Secific Heat c 4.79 kj/kg.k 0.9% of exact value from 0-00 C [5]. Gruneisen Coefficient Γ [0] Table. Fundamental Derivative G At 30 C, and atm [0].T.3 G.. Proerties of Air Table G.. Proerties of air (equivalent). Proerty Magnitude SI Units Comments Gas constant R 87 J/kg.K [45,5] Density ρ.05 kg.m -3 [5] Molecular weight M 8.97 kg.kmol - [5] Sound seed a m.s - [47] Isentroic Comressibility β S Pa - [55] Secific Heat c.005 kj/kg.k at 300K (air) [5]. Secific Heat c v 0.78 kj/kg. K at 300K (air) [5].

197 78 Table G.. Proerties of air (equivalent): Continued. Proerty Magnitude SI Units Comments Secific Heat Ratio γ.4 -- At 300 K [5]. Acoustic Imedance Z 430 kg/m s At 5 C, and atm. Dynamic Viscosity µ kg/m.s [56] N.s.m - Kinematic Viscosity ν m /s G.3. Proerties of Helium Table G.3. Proerties of helium gas. Proerty Magnitude SI Units Comments Gas Constant R 077 J/kg.K [45,5] Density ρ kg/m 3 Molecular weight M kg.kmol - [5] Sound Seed a 007 m.s - Isentroic Comressibility β S? Pa - Secific Heat Ratio γ [56] Acoustic Imedance Z kg/m s At 0 C, and atm. Dynamic Viscosity µ kg/m.s -- Kinematic Viscosity ν m /s -- G.4. Proerties of Polycarbonate, Grade: Makrolite 303. Polycarbonate tubes are resistant to liquids: When submerged in water, at room temerature, the absortion is only 0. % after 4 hours and only 0.30 % after equilibrium is reached. Polycarbonate is not be damaged by frequent, intermittent contact with hot water [60]. Polycarbonate is about 50 times stronger than glass, has good imact strength and is easily machinable. The roerties given in table G.4 are for Makrolite 303, which is a general-urose grade of olcarbonate.

198 79 Table G.4. Proerties of Polycarbonate (Grade: Makrolite 303). Proerty Magnitude Units Comments Density ρ ~00, (50-570) kg.m -3 [60] Tensile Yield Strength 65 MPa [60] Tensile Ultimate Strength 70 MPa [60] Comressive Yield Strength 76 MPa [60] Otical Refractive Index [60] Modulus of Elasticity ~. GPa [60] Coefficient of Thermal Exansion K - [60] G.5. Proerties of Insertion Rubber Insertion rubber was used to suort the liquid column in the Mach 3 shock tube. Since the roerties if rubber vary widely, only a few quantities are given here definitively. Table G.5. Proerties of insertion rubber. Proerty Magnitude Units Comments Density ρ 440 kg.m -3 From corresondance Tensile Strength 3.60 MPa with manufacturers. Poisson s Ratio ~

199 80 Aendix H Summary of Exerimental Cavitation Thresholds Values In this section, various cavitation threshold values are resented. They are classified as ulse reflection, tube-arrest, Berthelot tube, centrifugal method, suerheating, inclusion, ultrasonic and other exeriments. In most of these studies, the data is associated with visible cavities [80]. All tension waves are for water excet the last section, which lists a few values for mercury. Table H.. Tensile strength values recorded from ulse reflection exeriments. Marston and Unger [58] * Boteler and Sutherland [73] Williams and Williams [96] Williams and Williams [96] Magnitude of Peak Tension Recorded (MPa) -0- to -8.7± to to -0. Comments (Liquid Quality/ Comression Pulse Parameters) Degassed and distilled water. Comressive ulse duration.7µs and amlitude ~0 MPa. Trile-distilled, trile filtered (removed articles larger than 0.- µm), de-ionised, and degassed water. Baker-analysed, high urity, low conductivity, with measured H of 7. Initial liquid comression ulse duration:.07±0.5 µs. Degassed, deionised, filtered water. Comressive ulse rise time between 50 and 00 µs. Degassed, deionised, filtered water. Comressive ulse rise time between 50 and 00 µs. Source of Comression Pulses Imactor late imacted on buffer in contact with a liquid. (Flexible Mylar membrane). Imactor late imacted on buffer in contact with liquid. (Flexible 5 µm-thick aluminised Mylar membrane). Exeriments erformed initially at 4 C. Cell configuration. Particle velocity measurements at the water-air free surface. B-P (cattle stun gun). Tensions recorded by a transducer with a rise time of ~ 66 ns. B-P (cattle stun gun). Tensions determined artly by detecting bubble activity as detailed in [96].

200 8 Table H.. Tensile strength values recorded from ulse reflection exeriments: Continued Besov et al. [35] Magnitude of Peak Tension Recorded (MPa).7 Comments (Liquid Quality/ Comression Pulse Parameters) Distilled water. Comression ulse eak ressure =.9 MPa. Stressing rate 9. bar.µs -. Source of Comression Pulses Electromagnetic acoustic emitter. Brown [60] -.0- to 3.7 Aged ta water. B-P (Bullet-iston). Bull [3] * -.6 to -.5 Couzens and Trevena [] Couzens and Trevena [] Sedgewick and Trevena [5] Davies et al. [05][4]* K Wilson [08] * Crum and Fowlkes [75] Richards et al. [09]* -.5 Untreated (ta) water. Stressing rate atm.µs - [43]. Degassed (boiled), deionised water Ordinary untreated ta water. B-P and underwater exlosions. B-P. -.0 Deionised water. B-P. 0.9 Ordinary ta water. B-P. -.0 Deionised water. B-P. -.5 Boiled ta water. B-P Boiled deionised water. B-P. -.0 Degassed and distilled water..5 Deionised and degassed (evacuated) Settled, untreated (ta) water. B-P. In some exeriments, a light iston acted as a free surface, while in others, the ulse was reflected at a true free surface. Underwater exlosion from detonation of 0. g exlosive charge. -. Distilled water. Pulses of ultrasound. -. Deionised water. Shock tube. Exlosion in air resulted in shock in liquid. (flexible Mylar membrane for free surface). B-P

201 8 The exeriments listed in table H. involved free surface reflection, as oosed to reflection at a low imedance medium, unless otherwise secified. Table H.. Tensile strength values recorded from tube-arrest exeriments. Williams et al. [4] Overton, Williams and Trevena [] Magnitude of Tension Recorded (MPa) 6.5 MPa Comments (Liquid Quality/ Stressing rate) Degassed, deionised, filtered water. Stress rate 0 bar.µs atm Sea-water (degassed).87 atm Deionised water (degassed).76 atm Fresh ta water (degassed) 0.94 atm Deionised water (saturated) 0.86 atm Fresh ta water (saturated) 0.8 atm Sea-water (saturated) Method Mean value. Ater reeated testing at a frequency of er minute. Table H.3. Tensile strength values recorded from Berthelot tube exeriments. Magnitude of Tension Recorded (MPa) Comments (Liquid Quality) Method Henderson & Seedy [76] -6 Distilled water Glass Berthelot tubes. Vincent [77] [55] -5.7 Degassed water Glass Berthelot tube. Dixon [78] [5] -5 to 5 Degassed water Glass Berthelot tube. Berthelot [ 97] [4] -5 Degassed water Glass Berthelot tube. Jones et al. [47] -4.7 Degassed, distilled water Steel Berthelot tube. Jones et al. [47] -3.5 Distilled water Steel Berthelot tube. Meyer [79] [38] -3.4 Degassed water Glass Berthelot tube. Rees and Trevena [80] [ Degassed water Steel Berthelot tube. Sedgewick & Trevena [98] Minimum: -.38 Maximum: -.8 Mean value: -.3 ± 0.09 Deionised water 6 reeated tests. The first 5 showed no cavitation.

202 83 Differences between cavitation thresholds in table H.3 are artly attributed to cleaning and degassing rocedures, tube materials (it aears that steel tubes, in general, have more nucleation sites) and assumtions made in calculating tensions []. Table H.4. Tensile strength values recorded from centrifugal stressing exeriments. Briggs [04] Strube and Lauterborn [8] * Magnitude of Tension Recorded (MPa) -8. at 0ºC -6.3 at 0ºC -.0 at 50ºC Comments (Liquid Quality) Degassed water -7.5 Degassed water Reynolds [8] (K: [44]) Ta water Temerley and Chambers [99] (K: [5]) -0. to Ta water Method Centrifugally induced tension was alied to a water column. The following cavitation threshold values, shown in table H.5, were inferred from the boiling oints of samles of water. All of these exeriments involved re-ressurisation or small liquid volumes to allow significant effective tensions to be sustained. Table H.5. Tensile strength values recorded from suerheating exeriments. Kendrick et al. [4] Harvey et al. [37] Magnitude of Tension Recorded (MPa) 70ºC (-5.5 MPa) 0-06ºC (-.6-.8) Kna [,83] -0.5 to -.6 Comments Tension sustained for 5 seconds. Initially ressurised water. Determined effective tensile strength from boiling oints. Pressurised ta water, in Pyrex tubes. Liquid volume: ~300 cm 3. Method Thin-walled caillary tube, oen to atmoshere. After ressurising samles at 00 MPa for 5-30 minutes. Determined effective tensile strength from boiling oints.

203 84 Table H.5. Tensile strength values recorded from suerheating exeriments: Continued. Briggs [4] Skriov [8] [I.4: 7] Zheng et al. [8] Magnitude of Tension Recorded (Ma) 64-67ºC (-4.9 to -5. MPa) 300ºC (~ MPa) 307ºC (~ MPa) Comments Sustained for a few seconds before boiling exlosively. Initially ressurised, distilled, boiled water. Liquid volume: ~0.005 cm 3. Water in inclusions in crystals. Water in inclusions in crystals. Method Thin-walled caillary tube. Bore Diameter less than 0.5 mm, oen to atmoshere. Pulsed heating. Determined by isochoric cleaning. Table H.6. Tensile strength values recorded from inclusion exeriments. Magnitude of Tension Recorded (MPa) Comments Method Zheng et al. [84] Water. In inclusions within Quartz crystals. Roedder [85] -80 Water. Henderson and Seedy [86] [VI.] At least Measured density and temerature of water at this tension. In inclusions within Quartz crystals. Extraolations of melting oint vs. ressure line. [85] In small-volume caillaries using the Berthelot method. Refer to [8] for further discussions these, and other related exeriments and the methods used in inclusion studies. Even greater tensions are ossible in solutions. The exeriments summarised in table H.7 involve ultrasonic cavitation. The values illustrate great differences (order of magnitude discreancies) in cavitation threshold values, even for water samles of comarable urity.

204 85 Table H.7. Tensile strength values recorded from ultrasonic exeriments. Magnitude of Tension Recorded (MPa) Comments (Liquid Quality) Greensan & Tschiegg [87] -6 to Suitably clean water Galloway [88] -0 Degassed water Willard [89,07] - Degassed water Willard [89,07] -0. Gassy water Method Cavitation was induced ultrasonically. Table H.8. Tensile strength values recorded from other exeriments. Magnitude of Tension Recorded (MPa) Comments Müller [54] -7 to -9 Wave Focussing. Kna [83] Hydrodynamic cavitation Venturi tube. Method Harvey et al. [39] No value was determined Pressurised water Raid withdrawal of a submerged rod. Table H.9. Values of the tensile strength of Mercury. Magnitude of Tension Recorded (MPa) Comments Method Briggs, L. J. [90] 4.5 In a Pyrex glass container Static method. Williams et al. [5] 300 Stressing rate: bar.µs - Pulsed dynamic Tube arrest method. Carlson, G. A. [9] 900 Stressing rate: 06 bar.µs - stressing: Dynamic method: Pulse reflection. Pulsed stressing by a ulsed electron beam generator.

205 86 Aendix I The Initial State of Samles of Liquid Water In this section, exerimental data obtained by various researchers, with regards to the initial imurity content in samles of water, is summarised. The summary is intended to give a good idea of the size of cavitation nuclei. The main arameters are the radii R o of bubbles containing undissolved gas and uncondensed vaour, the bubble number density N o (the number of bubble nuclei er unit volume) and the volume concentration of the vaour/gas hase k o. Table I.. Summary of exerimentally obtained values of the range of bubble radii R o. Source Radii R o (µm) Comments Method Hammit et al. Acoustic and other diagnostic 3-6 Settled water. [5] methods. Gavrilov 0.5 [50] Besov et al. [9] Kedrinskii Water which has been left standing. 50 Relatively fresh water (average.5) Distilled (once) water. Acoustic and other diagnostic methods. Light scattering method and shock tube method. Detailed in [35,38]. 4 (maximum Fresh distilled water. Detailed in [53]. in sectrum) [80,53] 0.85 (maximum Settled distilled water. Detailed in [53]. in sectrum) Table I.. Summary of exerimentally obtained values of bubble number density N o. Source Density N o (cm -3 ) Comments Method Hammit et al. [5] -00 bubbles.cm -3 Settled water. Gavrilov [50] ~ bubbles.cm -3 Settled water. Besov et al. [ 94, 84] Microheterogeneities: Microbubbles Distilled (once) water Acoustic and other diagnostic methods. Acoustic and other diagnostic methods. Light scattering method and shock tube method. Detailed in [35,38].

206 87 Figure I.. Histograms of nuclei number densities N o in untreated, degassed and filtered ta water [5,54]. The data of Keller, shown in figure I., is in order of magnitude agreement with the data for non-distilled water shown in tables I. and I.. Most data is for gas nuclei only [38], although the data of Besov et al. [5] accounted for both microheterogeneities (including solid articles) and undissolved gas bubbles i.e microheterogeneities of submicron size and undissolved gas bubbles, with radii of about.5 µm are resent in a cubic centimetre of distilled water. Kedrinskii [80] estimated the total density of microheterogeneities, resent in fresh distilled water, from the exerimental sectrum data of Hammit et al., as er cubic centimetre. Gavrilov [50] found that the initial concentration k o, in settled water, ranges from while, according to Kedrinskii [60], k o ranges from 0-9 to 0-8 for relatively fresh water and from 0 - to 0-0 for water which has been left standing.

207 88 Aendix J Assembly Drawings of the Mach 3 and Mach Hydrodynamic Shock Tubes

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211 9 Aendix K Transducer Secifications and Gauge Factors K.. Transducers Secifications The PCB 3A transducers are of the ICP (Integrated Circuit Piezoelectric) tye i.e. a voltage mode-tye sensor featuring built-in microelectronic amlifiers, which convert the high-imedance charge into a low-imedance voltage outut [6,63]. ICP sensors are generally not affected by moisture [63]. Figure K.. Drawing of the PCB transducer, model 3A [63]. Table K.. PCB 3A Pressure Transducer Performance [6,63]. Performance SI units Measurement Range: (for ± 5V outut) 379 kpa Measurement Range: (for ± 0V outut) 760 kpa Aroximate Sensitivity: (±5%) 5 mv.si - (3.6 mv.kpa - ) Low Frequency Resonse: (-5%) 0.5 Hz Resonant Frequency: 500 khz Electrical Connector: 0-3 Coaxial Jack Resolution 0.0 kpa Maximum Pressure 6900 kpa Rise time µs Linearity <% Full Scale.

212 93 K.. Transducer Gauge Factors Five transducers were used throughout the exeriments using both shock tubes. Here, their gauge factors, calculated from calibration certificates rovided by PCB Piezotronics, Inc., are given, Transducer # 064: Sensitivity = 3.63 Gauge factor = Transducer # 060: Sensitivity = 6.34 Gauge factor = Transducer # 955: Sensitivity = 8.98 mv. si = 3.47 kpa. mv mv. si = 3.80 kpa. mv mv. si = 4.03 Gauge factor = kpa. mv Transducer # 7345: Sensitivity = 8.53 Gauge factor = Transducer # 7938: Sensitivity = 6.30 Gauge factor = 0.68 mv. si = 4.38 kpa. mv mv. si = 3.84 kpa. mv. mv. kpa mv. kpa mv. kpa mv. kpa mv. kpa

213 94 Aendix L Mach 3 Shock Tube: Additional Pressure Records

214 95 Adjusted Absolute Pressure (kpa) Time (ms) 3 Figure L.. Pressure trace from test using the Mach 3 shock tube. Samling rate MHz, 0000 data oints. Adjusted Absolute Pressure (kpa) Time (ms) 3 Figure L.. Pressure trace from test using the Mach 3 shock tube. Samling rate MHz, 0000 data oints.

215 96 Aendix M Mach 3 Shock Tube: Photograhic Records of the Lower Section of the Tube

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220 0 Aendix N Mach 3 Shock Tube: Photograhic Records of the Middle Section of the Tube

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225 06 Aendix O Mach 3 Shock Tube: Photograhic Records of the Free Surface of the Liquid Column

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229 0 9 0 Note that in the following, ictures are searated by two-millisecond intervals for brevity. 4

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