Chapter 5

Types of Cavitation: BUBBLECAVITATION

Objective: Description of the appearanceand behavior of bubble cavitation

When observed in white light cavitation isonly a hazy blur on a surface. As is often thecase (see the work of Parsons in chapter 2)good observations in an adequate test setupopened new perspectives. This was the casewith the first high speed observations in 1948by Knapp and Hollander [26]. They investi-gated a headform, which is a cylindrical bodywith a certain contour. The contour they in-vestigated was a hemispere, which at that timewas used as a standard body for the compar-ison of cavitation and cavitation inception invarious facilities.

Knapp and Hollander observed cavitation inthe low pressure region of such a headform.Instead of stroboscopic illumination, which re-sulted in a series of unrelated observations ofbubble cavitation (Fig. 5.1 [19]), they applieda rotating mirror in combination with a filmcamera to observe the cavitation. This wasthe beginning of high speed observations. Asa result they could observe the growth andcollapse of bubble cavitation on the headform(Fig 5.2).

This made it possible to plot the bub-ble radius over time. Knapp and Hollandershowed that the bubble growth could accu-rately be described by the Raleigh Plesset

Figure 5.1: Observations of Bubble Cavitationon a Headform. From [27]

equation (eq.4.6), as shown in Fig. 5.3.

When a nucleus grows into a bubble cav-ity the effect of the surface tension becomesnegligible, so the bubble motion is governedby inertia. From Fig. 5.3 it can be seen thatthe overshoot of the bubble radius is consid-

29

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

Figure 5.2: High Speed Observations of Bub-ble Cavitation on a Headform. From [27]

Figure 5.3: Pressure and Bubble Response ona Hemispherical Headform

erable: the bubble reaches its maximum sizewhen the pressure has already returned to theundisturbed pressure. The potential energystored in the cavitation bubble is then con-

Figure 5.4: Bubble Cavitation on a 2D foil

verted into kinetic energy during the collapsephase. This was used by Raleigh to formulatein 1917 the collapse time of a bubble for a bub-ble with radiusRmax and zero internal pressure(the vapor pressure is neglected, as well as thegas pressure and the surface tension):

t = 0.91468Rmax

√ρwp∞

(5.1)

The collapse time is generally very short.To get an idea: a cavitation bubble of 1cm (0.01m) in radius which returns fromthe vapor pressure to atmospheric pressure(p∞ = 105Pa) takes only 1 millisecond tocollapse! However, on a propeller blade ofa model propeller at a fluid velocity of 10m/sec this bubble will still move over 1 cmin distance during collapse, which is notnegligible.

5.1 Bubble cavitation on a

foil

On a two-dimensional foil the appearance ofbubble cavitation can be observed in more de-tail. An example is Fig. 5.4.

In this situation the bubble cavities reacha significant size, which indicates that the nu-clei content is not very high. It is also observedthat the bubble cavities are no longer spheri-cal. The streamwise pressure gradient causesa flat part on the bubbles. Although not seen

January 15, 2010, Bubble Cavitation31

Figure 5.5: Bubble Cavitation on a PropellerModel

in this picture, the bubbles will also attach tothe surface and become non-sperical near thewall. The shear in the boundary layer causesa distortion of the cavity, which is apparent bythe small twin ”horns” which are directed up-stream. At inception this deformation is notyet present, so for nuclei the assumption ofspherical bubbles can be used.

5.2 Bubble cavitation on a

model propeller

The natural type of cavitation following fromexpanding nuclei is bubble cavitation. Whenthe nuclei grow to visible size and larger due toa low external pressure the gas content insidethe bubble as well as the surface pressure onthe bubble wall become insignificant and thepressure inside the bubble will be close to thevapor pressure. Fig. 5.5 shows bubble cavita-tion on the model of a ship propeller.

The origin of each bubble in Fig.5.5is a nu-cleus which grows with decreasing pressure. Itshould be kept in mind that when the bubbles

Figure 5.6: Schematical Pressure Distributionon a Foil with Bubble Cavitation

become visible the partial pressure of the gasis already negligible and that the bubbles canbe considered as filled with vapor at the va-por pressure. The pressure distribution on theblade sections of Fig.5.5 is such that there is amore or less gradual decrease of the pressurefrom the leading edge, with a minimum pres-sure between 30 and 80 percent of the chord,after which the pressure increases again to theundisturbed pressure, as sketched in Fig.5.6.The essential element of the formation of bub-ble cavitation is that the bubbles have time togrow without causing flow separation. The re-sult is that the cavitation bubbles move withthe flow. Therefore this type of cavitation isalso called traveling bubble cavitation.

5.2.1 The extent of bubble cav-itation

In Fig.5.5 it can be seen that the maximumsize of the bubbles occurs at the trailing edgeof the cavitating region. This location is down-stream of the minimum pressure location be-cause of the overshoot of the bubble radius andbecause of the interaction between the bubblesand the pressure distribution. After reachingthe maximum size the bubbles are in a muchhigher pressure than equilibrium and a rapidcollapse will follow. The chordwise extent ofbubble cavitation is therefore greater than thearea on the propeller at which the pressure is

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

Figure 5.7: Estimate of cavity length from thenon-cavitating pressure distribution

below the vapor pressure! Limited cavitationhas no drastic effect on the lift of a section.Using this a very rough estimate of the cavityextent can be found from the non-cavitatingpressure distribution and the cavitation index.The pressure increase due to cavitation has tobe compensated by a pressure decrease down-stream, as shown by the red area in Fig. 5.7.

5.2.2 Relation of bubble cavita-tion with nuclei distribu-tion

When a nucleus is in a region with a pressurelower than the vapor pressure, it will grow in-definitely. The mechanism to stop the growthis interaction between the bubble and the sur-rounding pressure field. The bubble growthincreases the surrounding pressure and the re-sult will be that the pressure around the bub-ble cavities rises until the vapor pressure isreached. As can be seen in Fig.5.5 the bub-bles coalesce and individual bubbles cannot bedistinguished anymore. Efforts to relate the

nuclei density with the amount of cavitationbubbles are fruitless, because when the largestnuclei begin to affect the flow, the smaller willnot grow anymore or even do not become un-stable. The number of cavitation bubbles inbubble cavitation only reflects a band of thelargest available nuclei. The width of thatband is very difficult to determine.

5.3 Inception of bubble

cavitation

The definition of inception of bubble cavita-tion has been discussed in detail in section 4.The conditions of the flow should be such thatno gaseous cavitation is present by limitingthe number and size of large nuclei in theinflow. In general there is more risk of a lackof nuclei, so that inception is detected toolate. The fact that bubble cavitation has thereputation of being erosive may be due to thefact that it was detected too late at modelscale.

A lack of nuclei density can be detected byobserving the size of nuclei at cavitation in-ception. When nuclei are lacking the mini-mum pressure on the blade sections will bebelow the vapor pressure. When a single nu-cleus becomes unstable at that pressure it willgrow rapidly and relatively unrestricted to asignificant size (Fig. 4.5). Then it can be con-cluded that the inception pressure is below thevapor pressure. When the cavitation bubblesare small at inception there are enough nu-clei present. That does not yet mean that thenuclei are large enough. The inception pres-sure of small nuclei may still be much lowerthan the vapor pressure, even when there areenough of them. However, in general the nu-clei density will decrease with increasing nucleisize. So at inception there will be less bubblesthan at a lower pressure. Proper inception iscalled when a few bubbles are visible just be-

January 15, 2010, Bubble Cavitation33

fore inception and when the size of these bub-bles is small.

5.4 The appearance of

bubble cavitation

When large nuclei are scarce the amount ofbubble cavities is small. In that case the in-dividual cavities can grow to a large size un-til their effect on the pressure is felt. Whenmany nuclei are present the same effect on thepressure is felt by many smaller nuclei and themaximum size of the cavities will thus remainsmaller. In the first three figures of Fig.5.8 theamount of nuclei in the incoming flow is grad-ually increased. The observations are made inuniform inflow and the increase in nuclei con-tent is obtained by electrolysis upstream of thepropeller ([34].

The number of bubble cavities increaseswhen nuclei are generated, but the maximumsize is not so much different yet. When the(local) nuclei density is drastically increased,as is the case in the fourth picture of Fig. 5.8,the maximum size decreases. The increase innuclei in this fourth picture has been obtainedby leading edge roughness, as will be discussedlater. The change in appearence of bubblecavitation with incresing number of nuclei canbe extrapolated to full scale. At full scale anabundance of nuclei is available and the maxi-mum size of the individual bubble cavities willbe small. An observation of bubbly cavitationat full scale is given in Fig. 5.9 and the fullscale bubble cavitation is a fine mist of verysmall cavitation bubbles.

Figure 5.8: Bubble cavitation with increasingnuclei content

Figure 5.9: Bubble Cavitation on a Propellerat Full Scale

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

5.5 Effects of leading edge

roughness on incep-

tion of bubble cavita-

tion

Inception of bubble cavitation is controlled bythe nuclei spectrum in the bulk flow. However,roughness elements may also serve as nucleigenerators. This is because cavitation has theproperty of absorbing dissolved gases and af-ter implosion these gases remain as free gasnuclei nuclei in the flow, as can be observed inFig. 8.8on page 63. Downstream of the spotsa stream of gas bubbles can sometimes be ob-served. (In fact such a bubble is a tiny sheetcavity as will be discussed in chapter 6). Ex-perience is that even when the roughness el-ements are in regions with a mean pressurehigher than the vapor pressure, still nuclei aregenerated by roughness. The precise mecha-nism and required conditions are not yet clear,but it seems that micro cavitation occurs inlow pressure locations in the roughened re-gion, as sketched in Fig. 5.10. Since cavita-tion causes diffusion of dissolved gas into themicro-cavity, a small gas bubble is created bythe micro cavity. In this mechanism nuclei inthe bulk flow are still necessary for inceptionof micro-cavitation, but the size of the nucleirequired is much smaller then for bubble cav-itation in the free flow.A mechanism which may enhance the presenceof nuclei near the turbulent boundary layer isturbulent mixing, which may reduce the effectof bubble screening. These phenomena stillhave to be investigated further. The effects ofleading edge roughness on bubble cavitationcan be very significant, as shown in Fig. 5.8.The first three pictures show the effect of elec-trolysis, so of an increasing amount and size ofbulk nuclei. The last picture shows the addi-tional effect of leading edge roughness. In thiscase the leading edge pressure is above the va-por pressure and there is no cavitation on the

Figure 5.10: Generation of nuclei in a bound-ary layer with roughness

leading edge roughness. Still the roughnessparticles generate an abundant amount of nu-clei which alter the appearance of the bubblecavitation at midchord completely.

5.6 Erosiveness of Bubble

Cavitation

The question is if it is necessary to scalethe amount of bubbles in model tests. Areason would be to scale the erosivenessproperly. But erosiveness is not measuredup to now, apart from techniques such asthe paint technique (see chapter ??. Theamount of energy in the collapse of a singlebubble and of a spherical cloud of bubbleswith the same radius is about the same. Soeven when the number of bubbles in bubblecavitation is different from full scale, theerosion properties may still scale properly.The noise properties are more difficult to scalebecause of interaction between the bubbles.

Bubble cavitation has a longstandingreputation of being erosive in all cases andconsequently propellers are generally de-signed with a certain margin against bubblecavitation. A consequence of that is thatit is difficult to find full scale observations

January 15, 2010, Bubble Cavitation35

of bubble cavitation.however, the reasonwhy bubble cavitation is more erosive thane.g. sheet cavitation is not obvious. Navypropellers designed for maximum inceptionspeed often have bubble cavitation at fullpower. But erosion problems have not yetbeen encountered there, which may partlybe due to the fact that Navy ships do notoften run at full power for a long time. Thisindicates that it is not necessarily the bubblecavitation which is erosive, but that someadditional properties are required.

One of the reasons that bubble cavitationhas a reputation of erosiveness may be its latedetection. The amount of nuclei at model scaleis nearly always too low and when bubble cav-itation is observed at model scale it may havealready a significant size at full scale.

A more physical explanation of the ero-siveness of bubble cavitation is the locationwhere it normally appears: in the middle ofthe blade sections at inner radii, as shownin Fig. 5.5. Similar to sheet cavitation thislocation of cavitation becomes more violentwhen the propeller operates in a wake. Thedynamic behavior of bubble cavitation can besuch that when the area of bubble cavitationis reducing during a blade revolution, theamount of bubbles collapsing at the sametime increases, while the pressure outside thecollapsing bubbles increases also. Similar tosheet cavitation this can lead to strong erosion.

These considerations have lead to a reassess-ment of the erosion danger of bubble cavita-tion. It seems that steady bubble cavitationas such has no strong erosive properties. It isthe dynamic behavior, leading to a collectivefocussed collapse, which seem responsible forerosion. It is therefore important to observethat on model tests. This requires a properamount of nuclei.

Bibliography

[1] Arakeri,V.H., Acosta, A.J., 1979, Vis-cous Effects in the Inception of Cavita-tion, A.S.M.E. Int. Symposium on Cavi-tation, New York, USA.

[2] Arndt, R.E.A., 1976, Cavitation on ModelPropellers with Boundary Layer Trips,A.S.M.E.Conference on Polyphase Flow,New Orleans, USA.

[3] Arndt, R.E.A., Ellis, C.R., Paul, S. 1995,Preliminary Investigation of the Use ofAir Injection to mitigate Cavitation Ero-sion, A.S.M.E. Journal of Fluids Engi-neering, Vol.117.

[4] Briggs, L.J., 1950 Limiting negativepressure of water., J. Applied Phys.21:721722.

[5] Brennen, C.E., 1995 Cavitation and Bub-ble Dynamics, Oxford University Press,ISBN 0-19-509409-3. Also available on in-ternet.

[6] Burrill, L.C., 1951, Sir Charles Parsonsand Cavitation, Transactions of the Insti-tute of Marine Engineers.

[7] Borkent, B.M., 2009, Interfacial Phe-nomena in Micro- and Nanofluidics:nanobubbles,cavitation, and wetting, Thi-sis Twente University, The Netherlands.

[8] Coutier-Delgosha, O., Devillers, J-P.,Leriche, M., Pichon, T., 2005, emphEf-fect of Roughness on the Dynamics of Un-steady Cavitation, Journal of Fluids Eng., vol 127.

[9] Epstein, P.S., Plesset, M.S. 1950 On thestability of Gas Bubbles in Liquis-GasSolutions, Journal of Chemical Physics,Vol.18., pp1505-1509.

[10] , Feindt, E.G., 1956,Untersuchungenuber die Abhangigkeit des UmschlagesLaminar-Turbulent von der Oberflachen-rauhigkeit und der Druckverteilung,Jahrbuch STG, Bd.50, pp180-205

[11] Flynn, H.G., 1964, Physics ofAcoustic Cavitation, W.P.Mason ed.Vol1,PartB,Academic Press New York-London.

[12] Foeth, E-J., Kuiper, G. 2004, Ex-ploratory experiments to determine flowand structure borne noise of erosivecavity implosions, A¿S¿M¿E¿ FluidsEng. Summer Conference, HT-FED2004-56789, Charlotte, NC. USA.

[13] Foeth, E-J., The Structure of Three-Dimensional Sheet Cavitation, ThesisTechnical University Delft.

[14] Fox, F.E., Herzfeld, K.F., 1954 Gas Bub-bles with Organic Skin as Cavitation Nu-clei, J.Acoust. Soc. Am. Vol.26, pp 984-989.

[15] Gates, E.M., 1977 ( The influence of Free-Stream Turbulence, Free Stream NucleiPopulations and a Drag-reducing Poly-mer on Cavitation Inception on Two Ax-isymmetric Bodies California Institute ofTechnology, Rep. No. Eng 183-2.

65

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

[16] Harvey, E.N., McElroy, W.D., Whiteley,A.H., 1947 On Cavity Formation in Wa-ter, Journal of Applied Physics, Vol.18,pp162-172.

[17] Hoekstra, M., Vaz, G. The Paartial Cav-ity on a @D Foil Revisited, 7th Int. Con-ference on Cavitation CAV2009, Ann Ar-bor, Michigan, U.S.A.

[18] Holl, J.W., 1960, An Effect of Air Con-tent on the Occurrence of Cavitation,Trans. A.S.M.E., Journal of Basic Eng.Vol.82, pp941-946.

[19] Holl, J.W., Carrol, J.A., 1981, Obser-vations of the Various Types of Lim-ited Cavitation on Axisymmetric Bodies,PUB A.S.M.E. Journal of Fluids Eng.,Vol103,pp415-423

[20] Huang, T.T., 1981, Cavitation InceptionObservations on Six Axisymmetric Head-forms, A.S.M.E. Journal of Fluids Engi-neering, Vol 103,PP273-278

[21] ITTC, 1978,Proceedings 15th ITTC, TheHaque, Report of the Performance Com-mittee.

[22] Johnsson, C.A., Hsieh,T., 1966, The In-fluence of Trajectories of Gas Nuclei onCavitation Inception, 6th Symposium onNaval Hydrodynamics, Washington D.C.,pp163-178.

[23] Takagi,K., Kato,H.,Kato,D.,Sugimoto,A., 2006, Destruction of Plankton byTwo-Dimensional Cavitating Jet, SixthInternational Symposium on CavitationCAV2006, Wageningen, The Netherlands.

[24] Keller, A.P., 1974 Investigations Con-cerning Scale Effects of the Inception ofCavitation, Proc. I.Mech.Eng. Conferenceon Cavitation, Edinburgh, 109-117.

[25] KLEBANOFF, P.S., SCHUBAUER,G.B., TIDSTROM,K.D., 1955, Measure-ments of the Effect of Two-dimensionaland Three-dimensional Roughness Ele-ments on Boundary Layer Transition, J.Aeron. Sciences

[26] Knapp, R.T., Hollander, A., 1948, Lab-oratory Investigations of the Mechanismof Cavitation, Trans. A.S.M.E., Vol.70,pp.419-435.

[27] Knapp, R.T., Daily, J.W., Hammitt,F.G., 1970, C avitation, New York :McGraw-Hill.

[28] Knapp, R.T., Hollander, A., 1948,Laboratory Investigations of the Mecha-nism of Cavitation, Trans. A.S.M.E., Vol70, pp419-435.

[29] Korkut, E., Atlar, M., 2000, On the Im-portance of Effect of Turbulence in Cav-itation Inception Tests of Marine Pro-pellers, Proceedings of Royal Society ofLondon A: Mathematical, Physical andEngineering Sciences, Vol.458 , pp.29-48.

[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.

January 15, 2010, Leading Edge Roughness67

[33] Kuiper, G., 1978, Scale Effects on Pro-peller Cavitation, 12th Symposium onNaval Hydrodynamics, Washington D.C.,USA.

[34] Kuiper, G. 1981, Cavitation Inception onShip Propeller Models, Thesis TechnicalUniversity Delft.

[35] Kuiper, G. 2008, Fundamentals of ShipResistance and Propulsion, Course Lec-tures Technical University Delft.

[36] Landa, E.R., Nimmo, J.R.,2003 The Lifeand Scientific Contributions of Lyman J.Briggs, Journal of the Soil Science Soci-ety of America, Vol 67 no 3, pp 681-693.http://soil.scijournals.org/cgi/reprint/67/3/681

[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.

[40] Neppiras, E.A., Noltink, B.E., 1951, Cav-itation produced by Ultrasonics, Proc.Phys. Soc. London, pp 1032-1038.

[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.

[43] Schiebe, F.R., 1972, Measurement of theCavitation Susceptibility of Water using

Standard Bodies, St. Anthony Fall Hy-draulic Lab, Univ. of Minnesota, report118.

[44] Schlichting, H., 1968, Boundary LayerTheory, McGraw-Hill, 6th edition.

[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.

[47] Watanabe, S., Furukawa, A., Yoshida, Y.,Tsujimoto, Y., 2009, Analytical investiga-tions of thermodynamic effect on cavita-tion characteristics of sheet and tip leak-age vortex cavitation, Seventh Int. Symp.on Cavitation: CAV2009, Ann Arbor,Michigan, USA.

[48] Williams,M., Kawakami, E., Amromin,E., Arndt,R. Effects of Surface Charac-teristics on Hydrofoil Cavitation, SeventhInt. Symp. on Cavitation: CAV2009, AnnArbor, Michigan, USA.

[49] Yoshimura,T., Kubota, S., Seo, T.,Sato,K., 2009, Development of Bal-last Water Treatment Technology byMechanochemical Cavitations,SeventhInt. Symp. on Cavitation: CAV2009,Ann Arbor, Michigan, USA.

[50] Yount, D.E. 1979, Skins of Varying Per-meability: A Stabilization MechanismFor Gas Cavitation Nuclei,A. ACCOUST.SOC. AM. 65(6).

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

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

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

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

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

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

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.

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

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