Introduction cavitation

Chapter 2

INTRODUCTION

Objective: An introduction of the phe-nomenon of cavitation, its history, occurrenceand effects.

Cavitation is the occurrence of vapor in afluid due to local low pressures which are gen-erated by high local flow velocities.

2.1 Cavitation and boiling

A general description of cavitation is thetransition of fluid into vapor due to local re-duction of the pressure. There are more waysto arrive at a transition of water into vaporand the most common method is cooking ofwater. There the transition of water intovapor is not due to a lower local pressure,but to an increase of the local temperature.The term cavitation is generally reserved forconditions in which the temperature of thebulk fluid is not changed. This distinctionbetween cooking and cavitation is not alwayssharp, because both temperature and pressureeffects can be present at the same time, e.g.in the flow around pipes in heat exchangers.In the case of hydraulics or hydrodynamicsthe bulk temperature during the process ofcavitation is generally isothermal and theoccurrence of vapor regions is due to localpressure reductions which are related with

the local flow velocities. In that case thephenomenon is called cavitation.

Cavitation is the transition of a fluid intovapor. That can be any fluid. An example isthe occurrence of cavitation in blood as mayoccur in artificial heart valves [32] or cavitationin pumps and valves in which all types of fluidsare used. In principle the local pressure in afluid cannot be lower than the vapor pressure.If that occurs vapor will be generated fromthe fluid and the resulting volume increase willonly be checked when the vapor pressure isreached. Reality is often more complex, as willbe discussed in chapter 4.

2.2 Cavitation in water

and other fluids

The pressure at which both vapor and fluidcan be present in a fluid is called the equilib-rium vapor pressure or shortly the vapor pres-sure. The main parameter of the vapor pres-sure is the temperature. For technical pur-poses a ”standard” temperature of 15 degreesCelcius is often used. The vapor pressure ofwater in that condition is 1706 Pa. The datamay vary between different sources. The dif-ferences are small, but may lead to confusion.For application in ship hydrodynamics the In-ternational Towing Tank Conference adopteda line with pressures as given in Appendix C.

5

6 G.Kuiper, Cavitation in Ship Propulsion, January 15, 2010

2.3 Adverse effects of cav-

itation

The main effects of cavitation are adverse ef-fects: noise, erosion, vibrations and disruptionof the flow, which results in loss of lift and in-crease of drag.

Cavitation is known for its violent behavior.That is caused by the fact that vaporizationof water and condensation of vapor are veryfast processes, much faster than the dynamicsof a vapor cavity. As a result the growth andcollapse of a cavity is not slowed down by theseprocesses. The violent behavior of cavitationhas several adverse effects. Because cavitationis part of the flow, it can move rapidly fromregions of low pressure into regions of a higherpressure. This leads to a very rapid collapse.The collapse is so rapid that the local speed ofsound in the fluid is exceeded and shock wavesoccur. The consequence is that cavitation isvery noisy and radiates noise over a wide rangeof frequencies, especially higher frequencies.

Also the local pressure rises very stronglyat collapse, leading to damage of a nearbysurface. This effect is called erosion. Whenlarger amounts of vapor are involved theimplosion of cavitation can cause pressurevariations in the fluid, which lead to vibrationsof the cavitating structure. The majority ofthe adverse effects of cavitation can be relatedwith erosion, noise and vibrations.

Cavitation can also alter the flow. This ise.g. the case on propellers when the cavita-tion becomes extensive. In that case the flowover the blades and the lift of the blades is al-tered by the cavitation and the thrust of thepropeller is strongly reduced. This is the so-called thrust breakdown. In valves cavitationcan also block or choke the flow.

The volume of vapor in cavitation is muchlarger than the volume of the water that hasevaporated. In cases of extensive cavitationthis leads to large volume increases and

decreases when cavitation grows and col-lapses. The volume variations cause pressurefluctuations in the surrounding fluid, withstructural vibrations as a result.

The properties of cavitation and its implo-sion can also be used, as will be mentionedbelow.

2.4 Examples of effects of

cavitation

The focus of this course is on ship hydrody-namics and ship propellers. But cavitationoccurs in many other situations, mostly withadverse effects. Cavitation in pumps is relatedto that on ship propellers, and the effects arethe same. First the pump may be damagedat the location of cavitation, but with anincreasing amount of cavitation the efficiencyof the pump decreases. Cavitation can alsooccur in pipes and valves, especially whenthe temperature is high and the differencebetween the mean pressure and the vaporpressure is small. A rattling noise in the tap ofa shower indicates cavitation and eventuallywill cause damage. Extensive cavitationcan reduce the flow through a valve or pipesignificantly.

Cavitation also occurs in dam overflowswhen the overflow accelerates the waterlocally. This may lead to large scale damageof the concrete. Local acceleration of thewater also occurs when gates of sluices areclosed or opened and cavitation may occur,generally resulting in erosion damage.

Cavitation also occurs in small fluid lines,such as in the heads of inkjet printers or dieselfuel injection [46]. The flow velocity in thoselines is limited by cavitation.

Small fluid lines also occur in plants. Water

January 15, 2010, Cavitation 7

in a tree is predominantly sucked from thetop through capillary vessels. This limits theheigth of a tree, because when the pressureat the top is too low, the fluid will cavitateand the flow will stop due to the occurrenceof vapor. In times of drought vapor is indeedpresent in these capillary lines. When sucha plant is watered abruptly it seems thatimplosion of the vapor cavities in those vesselscan cause fatal damage to the capillary linesdue to implosion of the cavities.

Cavitation is also used for specifi purposes.The destructive effect of acoustic pressurewaves focussed on kidney stones is due tocavitation. Cavitation is a main factor inthe acoustic destruction of kidney stones(lithotripsy). The imploding cavities destructthe kidney stone. It is clear that the focussingis very important to avoid damage to tissue.This risk of damage due to cavitation is alsoa limitation in the application of acousticdiagnostics [30]. The destructive propertiesof cavitation are also used to destruct cancercells. This is a way to remove cancer cells ina non-intrusive way. But when the energydensity becomes too high cavitation in thetissue will also cause damage. Medical ap-plications use very small nuclei to transportdrugs. These nuclei are made to cavitate inlocations where the drugs have to be applied[41].Kato proposed cavitation as a means tokill unwanted biological life from e.g. ballastwater of ships[23],[49].

Cavitation erosion is also used in underwa-ter cleaning. A strong fluid jet which hits asurface will cause local cavitation and the im-plosions will remove fouling from the surface.It is also used in rock cutting and in cuttingsteel with high speed jets. Rock cutting indeep sea is notably more difficult due to thelack of cavitation in that environment.

2.5 History

Figure 2.1: The Turbinia, used by CharlesParsons to obtain 30 knots ship speed

Cavitation entered its rle in ship propellersaround 1894, when Charles Parsons treis toestablish a speed record by using a steam tur-bine for ship propulsion. The power of thesteam turbine was enough for a small boat torun at 30 knots or more, but the problem washow to bring this power in the water by a pro-peller. The ship he used was the Turbinia,a 30 meter fast boat [6]. Sir Charles Parsonsfound that the propellers had an excessive slip,which means that they rotated much fasterthan the pitch of the propeller. He suspectedthat ”there appears to be vacuum behind theblades”. He made that visible by designinga small cavitation tunnel of about 60 cm inlength with a 5 cm diameter propeller model,in which tunnel he applied stroboscopic illu-mination of the propeller by using a rotatingmirror with the same rotation rate as the pro-peller.

There it became apparent that the wasan empty space on and behind the propellerblade. The term cavitation was born, althoughBurill mentions in his paper that from un-named sources it seems that R.E.Froude usedthe term cavitation first. After many propellerdesigns it took nine propellers on three shafts


to obtain a speed of more than 32 knots in1897. He showed the possibilities of steamturbine propulsion by running the Turbiniaunannounced on the Spithead Navy Review in1897, which was held on the occasion of queenVictoria’s diamond jubilee. The Turbinia wasfaster than any other ship and easily outranthe Navy’s parol boats.

This was the beginning of a developmentof propeller design and cavitation testing. Alarger cavitation tunnel was built at Wallsendin 1910.

Figure 2.2: The first cavitation tunnel used byCharles Parsons

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[30] Kreider, W., Crum, L., Bailey,M., Mat-ula, T.,Khoklova, V., Sapozhnikov,O.,2006, Acoustic Cavitation and MedicalUltrasound,Sixth International Sympo-sium on Cavitation CAV2006, Wagenin-gen, The Netherlands.

[31] Kumar, S., Brennen, C.E., 1996, A Studyof Pressure Pulses generated by TravellingBubble Cavitation, Journal of Fluid Me-chanics Vol 255 pp541.

[32] Hwansung Lee, Tomonori Tsukiya,Akihiko Homma, TadayukiKamimura,Eisuke Tatsumi, YoshiyukiTaenaka, Hisateru Takano, Observationof Cavitation in a Mechanical HeartValve, Fifth International Symposium onCavitation (cav2003),Osaka, Japan.

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[37] Ligtelijn, J.T., van der Kooij, J., Kuiper,G., van Gent,W., 1992, Research onPropeller-Hull Interaction in the Depres-surized Towing Tank, Hydrodynamics,Computations,Model Tests and Reality(Marin ), Elseviers Science Publishers.

[38] Morch E.M., 2000, Paper on Cav2003.

[39] Moerch, K.A. 2009, C avitation Nuclei:Experiment and Theory,Journal of Hy-drodynamics Vol21 p176.

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[41] Ohl, C-D., Arora, M. ,Roy, I., Delius.M., Wolfrum, B.,2003, Drug Delivery Fol-lowing Shock Wave Induced Cavitation ,Fifth International Symposium on Cavi-tation (cav2003),Osaka, Japan.

[42] Plesset, M.S., 1949, The Dynamics ofCavitation Bubbles, ASME Journal ofAppl. Mech. 1949, pp 277-232.

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[45] Terwisga, T.J.C., Fitzsimmons, P.A.,Li,Z., Foeth, E-J. Cavitation Erosion-Areview of Physical Mechanisms and Ero-sion Risk Models, 7th International Sym-posium on Cavitation, CAV2009, AnnArbor, Michigan, U.S.A.

[46] Jin Wang, 2009, Nozzle-geometry-dependent breakup of diesel jets byultrafast x-ray imaging: implication ofin-nozzle cavitation, Seventh Int. Symp.on Cavitation: CAV2009, Ann Arbor,Michigan, U.S.A.

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Appendix A

Air Content of Water

The amount of air dissolved in water α canbe expressed in many ways. The most commonways in literature are

• the gas fraction in weight ratio αw

• the gas fraction in volume ratio αv

• the molecule ratio

• the saturation rate

• the partial pressure of air

A.1 Solubility

Air is a mixture of 21 percent oxygen, 78 per-cent nitrogen and one percent of many othergases, which are often treated as nitrogen. Thespecific mass of gases involved in air are:

Oxygen (O2) 1.429 kg/m3

Nitrogen (N2) 1.2506 kg/m3

Air 1.292 kg/m3

The maximum amount of gas that can bedissolved in water, the solubility), depends onpressure and temperature. It decreases withincreasing temperature and increases with in-creasing pressure. The solubility of oxygen inwater is higher than the solubility of nitrogen.Air dissolved in water contains approximately36 percent oxygen compared to 21 percent inair.The remaining amount can be consideredas Nitrogen. Nuclei which are in equilibrium

with saturated water therefore contain 36 per-cent oxygen. But nuclei which are generatedfrom the air above the water contain 21 per-cent oxygen. Since the ratio between oxygenand nitrogen is not fixed, it is difficult to re-late measurements of dissolved oxygen (by os-mose) to measurements of dissolved air (frome.g a van Slijke apparatus).

The amount of oxygen dissolved in water atatmospheric pressure at 15 degrees Celcius isapproximately 10 ∗ 10−6kg/kg. For nitrogenthis value is about 15 ∗ 10−6, so the solubilityof air in water is the sum of both: 25 ∗ 10−6.Here the dissolved gas contents are expressedas a weigth ratio αw.Air is very light relativeto water and the weight ratio is very small.This ratio is therefore often expressed as partsper million (in weight), which is 106 ∗ αw.

A.2 The Gas Fraction in

Volume Ratio

The volume of gas dissolved per cubic meterof water depends on temperature and pres-sure. Therefore this volume ratio is expressedin standard conditions of 0 degrees Celcius and1013 mbar (atmospheric conditions). The de-pendency of the volume of water on tempera-ture and pressure is neglected. The volume ofthe dissolved air is then described by the lawof Boyle-Gay-Lussac:

69


p ∗ V ol273 + T

= constant (A.1)

The volume fraction at (p,T) can be relatedto the volume fraction in standard conditions:

αv = αv(p, T )273p

(273 + T )1013(A.2)

The gas fraction in volume ratio is dimen-sionless (m3/m3). Be careful because some-times this is violated by using cm3/l (1000∗αv)or parts per million (ppm) which is 106 ∗ αv.αv is found from αw by:

αv =ρwaterρair

αw (A.3)

in which ρ is the specific mass in kg/m3. At15 deg. Celcius and 1013 mbar pressure thespecific mass of water ρw = 1000kg/m3 andthe specific mass of air is 1.223kg/m3, so forair αv = 813αw.

A.3 The Gas Fraction in

Molecule ratio

The dissolved amount of gas can also be ex-pressed as the ratio in moles(Mol/Mol). Mo-lar masses may be calculated from the atomicweight in combination with the molar massconstant (1 g/mol) so that the molar mass of agas or fluid in grams is the same as the atomicweight.The molar ratio αm is easily found from theweight ratio by

αw = αmM(water)

M(gas)(A.4)

in which M is the molar weight, which is 18for water, 16 for oxygen(O2) and 28 for Nitro-gen (N2). For air a virtual molar weigth can

be defined using the ratio of oxygen and nitro-gen of 21/79 this virtual molar weight of air isabout 29.

A.4 The saturation rate

The saturation rate is the amount of gas in so-lution as a fraction of the maximum amountthat can go in solution in the same conditions.Since the saturation rate is dimensionless. It isindependent of the way in which the dissolvedgas or the solubility is expressed. The satura-tion rate is important because it determines ifand in which direction diffusion will occur ata free surface. The saturation rate varies withtemperature and pressure, mainly because thesolubility of gas changes with these parame-ters.

A.5 The partial pressure

Sometimes the amount of dissolved gas is ex-pressed as the partial pressure of the gas (mbaror even in mm HG). This is based on Henry’slaw, which states that the amount of gas dis-solved in a fluid is proportional to the partialpressure of that gas. In a van Slijke appa-ratus a specific volume of water is taken andsubjected to repeated spraying in near vac-uum conditions (a low pressure decreases thesolubility). This will result in collecting thedissolved in a chamber of specific size. Bymeasuring the pressure in that chamber theamount of dissolved gas is found. Note thatthis pressure is not directly the partial pres-sure. A calibration factor is required whichdepends on the apparatus.

Appendix B

Standard Cavitators

A standard cavitator is a reference bodywhich can be used to compare and calibratecavitation observations and measurements.Its geometry has to be reproduced accuratelyand therefore an axisymmetric headform hasbeen used as a standard cavitatior.

Such an axisymmetric body has been in-vestigated in the context of the ITTC (In-ternational Towing Tank Conference).This isa worldwide conference consisting of towingtanks (and cavitation tunnels) which have thegoal of predicting the hydrodynamic behaviorof ships. To do that model tests and calcula-tions are used. They meet every three yearsto discuss the state of the art and to definecommon problem areas which have to be re-viewed by committees. The ITTC headformhas a flat nose and an elliptical contour [22].Its characteristics are given in Fig B.1.

This headform has been used to comparecavitation inception conditions and cavitationpatterns in a range of test facilities. Theresults showed a wide range of inceptionconditions and also a diversity of cavitationpatterns in virtually the same condition,as illustrated in Fig B.3. This comparisonlead to the investigation of viscous effects oncavitation and cavitation inception.

The simplest conceivable body to investi-gate cavitation is the hemispherical headform.This is an axisymmetric body with a hemis-pere as the leading contour. Its minimumpressure coefficient is -0.74. The hemisperical

Figure B.1: Contour and Pressure Distribu-tion on the ITTC Headform [31]

headform was used to compare inceptionmeasurements in various cavitation tunnels.However, it was realized later on that theboundary layer flow on both the ITTC andon the hemisperical headform was not assimple as the geometry suggested. In mostcases the Reynolds numbers in the investi-gations was such that the boundary layerover the headform remained laminar and thepressure distribution was such that a laminarseparation bubble occurred, in the positionindicated in Fig. B.1. This caused viscouseffects on cavitation inception and made theheadform less suitable as a standard body.Note that the location of laminar separation isindependent of the Reynolds number. Whenthe Reynolds number becomes high transitionto turbulence occurs upstream of the sep-

71


Figure B.2: Contour and Pressure Distribu-tion of the Schiebe body [31]

aration location and separation will disappear.

To avoid laminar separation another head-form was developed by Schiebe ([43]) andthis headform bears his name ever since.The contour and pressure distribution onthe Schiebe headform are given in Fig. B.2.This headform has no laminar separation andtransition to a turbulent boundary layer willoccur at a location which depends on theReynolds number.

Many other headform shapes have beeninvestigated with different minimum pres-sure coefficients and pressure recoverygradients.(e.g.[20])

January 15, 2010, Standard Cavitators73

Figure B.3: Comparative measurements of cavitation inception on the ITTC headformsource:ITTC

74 G.Kuiper, Cavitation on Ship Propellers, January 15, 2010

Appendix C

Tables

T pvCelcius N/m2

0 608.0122 706.0784 813.9516 9328 106910 122612 140214 159815 170616 181418 205920 233422 263824 298126 336428 378530 423632 475634 531536 594338 661940 7375

Table C.1: Vapor pressure of Water.

75


Temp. kinem. visc. kinem. visc.deg. C. fresh water salt water

m2/sec× 106 m2/sec× 106

0 1.78667 1.828441 1.72701 1.769152 1.67040 1.713063 1.61665 1.659884 1.56557 1.609405 1.51698 1.561426 1.47070 1.515847 1.42667 1.472428 1.38471 1.431029 1.34463 1.3915210 1.30641 1.3538311 1.26988 1.3177312 1.23495 1.2832413 1.20159 1.2502814 1.16964 1.2186215 1.13902 1.1883116 1.10966 1.1591617 1.08155 1.1312518 1.05456 1.1043819 1.02865 1.0785420 1.00374 1.0537221 0.97984 1.0298122 0.95682 1.0067823 0.93471 0.9845724 0.91340 0.9631525 0.89292 0.9425226 0.87313 0.9225527 0.85409 0.9033128 0.83572 0.8847029 0.81798 0.8667130 0.80091 0.84931

Table C.2: Kinematic viscosities adopted bythe ITTC in 1963

January 15, 2010, Tables 77

Rn Cf × 103

1× 105 8.3332 6.8823 6.2034 5.7805 5.4826 5.2547 5.0738 4.9239 4.7971× 106 4.6882 4.0543 3.7414 3.5415 3.3976 3.2857 3.1958 3.1209 3.0561× 107 3.0002 2.6694 2.3906 2.2468 2.1621× 108 2.0832 1.8894 1.7216 1.6328 1.5741× 109 1.5312 1.4074 1.2986 1.2408 1.2011× 1010 1.17x

Table C.3: Friction coefficients according tothe ITTC57extrapolator.


Temp. density densitydeg. C. fresh water salt water

kg/m3 kg/m3

0 999.8 1028.01 999.8 1027.92 999.9 1027.83 999.9 1027.84 999.9 1027.75 999.9 1027.66 999.9 1027.47 999.8 1027.38 999.8 1027.19 999.7 1027.010 999.6 1026.911 999.5 1026.712 999.4 1026.613 999.3 1026.314 999.1 1026.115 999.0 1025.916 998.9 1025.717 998.7 1025.418 998.5 1025.219 998.3 1025.020 998.1 1024.721 997.9 1024.422 997.7 1024.123 997.4 1023.824 997.2 1023.525 996.9 1023.226 996.7 1022.927 996.4 1022.628 996.2 1022.329 995.9 1022.030 995.6 1021.7

Table C.4: Densities as adopted by the ITTCin 1963.

Appendix D

Nomenclature

ρ density of water kgm3 See TableC.4

Cg gas concentration kg/m3 see Appendix ADg diffusion coefficient m2/sec representative value 2 ∗ 109

D diameter mFd drag Ng acceleration due to gravity m

sec2Taken as 9.81

Nd number density of nuclei m−4pg gas pressure fracNm2

fracNm2

pv equilibrium vapor pressureR radius m

µ dynamic viscosity of water kgm∗sec

ν kinematic viscosity of water m2

sec(ν = µ

ρ)See Table C.2

s surface tension Nm for water 0.075

79

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Introduction cavitation