chapter 3 physics of cavitation: gas content and nuclei

Chapter 3

Physics of cavitation: GAS CONTENTAND NUCLEI

Objective: General description of thebehavior of gas bubbles which act as cavitationnuclei

The beginning of cavitation is called cavi-tation inception. Inception is a special topic,because in the initial formation of cavitationnuclei are involved and once cavitation occursthese nuclei loose their meaning. In this chap-ter the need of nuclei will be explained andthe presence of nuclei in fluids investigated.

3.1 Cohesive Forces

The initial formation of vapor (cavitationinception) is complicated, because pure watercan withstand much lower pressures than theequilibrium vapor pressure without vaporiza-tion. This is due to the mutual attraction orcohesionof the water molecules. A classicalexperiment dating back to 1950 is that ofLyman J.Briggs [4], a soil physicist who afterhis retirement (at age 77!) did investigatethe tension strength of fluids. He used acapillary Z-shaped tube rotating horizontallyin a centrifuge. This created a calculablepressure in the center of the tube. The waterremained inside the tube as long as no vaporwas formed. The maximum tension (negative

pressure)as a function of temperature wasobtained at 10 degrees Celcius and amountedto 277 bars (27.7 MPa)! Although this valuewas reduced to some 20 bars near the freezingpoint, it remained far away from the equi-librium vapor pressure. Similar experimentswere carried out much earlier by Berthelot(1950) and Dixon(1909), See [5].It requiredcareful attention of cleaning and purificationof the water to obtain these high tensions,and still the theoretical tensile strength ofwater molecules is even much higher [5]. Inpractical circumstances one may concludethat pure water does not cavitate. Itshould be kept in mind that it is very difficultto make ”pure” water [7]. Contaminationsof some sort are unavoidable in practice, andthese contaminations determine the tensilestrength of water.The cohesive forces between water moleculesare much greater than the forces betweenwater molecules and other materials, theadhesive forces. It means that the next weakspot in water is the wall of the containeror the surface of a wing in the low pressureregion. So when a low pressure occurs on asurface, cavitation will start at the surface.However,the tensile strength due to theseadhesive forces between e.g. metals and wateris still orders of magnitude higher than theequilibrium vapor pressure.

9

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

In practice the main mechanism of inceptionis the growth of small gas bubbles called nu-clei,which are present in the water. These nu-clei are small bubbles, filled with air (or othergases) and vapor. The diameter can vary froma few microns to nearly visible bubbles of theorder of 1 mm. That seems queer, becausein standing water at atmospheric conditionsgas bubbles will generally disappear due to twomechanisms: bubble rise and dissolution. Tounderstand the behavior of nuclei it is usefulto describe these two phenomena first.

3.2 Rising bubbles

In standing water a bubble will rise to the sur-face due to the difference between the weightof the bubble and the weight of the displacedwater. This difference causes a buoyancy forceon the bubble. Neglecting the weight of theair in the bubble this force F is equal to theweight of water displaced by the bubble:

F =ρw ∗ π ∗D3

b ∗ g6

, in which ρw is the specific mass of water andDb is the bubble diameter.The subsequent rise velocity is determined bythe drag Db of the bubble. That is a compli-cated matter, but the simplest solution is a so-lution of the Navier-Stokes equation in whichthe kinematic forces are negligible relative tothe viscous forces. The drag of the bubbleis thus completely due to friction along thebubble wall wilt a laminar boundary layer andthis condition is expressed in non-dimensionalterms as a low Reynolds number Reb = VbDb

ν,

in which nu is the kinematic viscosity of wa-ter and Vb is the bubble velocity. In this equa-tion it is also assumed that the bubble remainsspherical, which is applicable for small bubblesat low velocities. The drag of a bubble canthen be written as

Db = 3πµDbVb

in which µ is the dynamic viscosity of water(related to the kinematic viscosity ν by ν = µ

ρ.

The rise velocity of a bubble can then simplybe written by equating the rise force FD withthe bubble drag, which results in Stokes’ law:

Vb =g ∗D2

18 ∗ ν(3.1)

This equation is useful to estimate the timeneeded for a bubble to rise out of the measure-ment volume. Consider e.g. a nucleus of 100microns (100∗10−6 m)in diameter. As we willsee later in chapter 4, such a nucleus is suf-ficient in many cases to bring the cavitationinception pressure close to the vapor pressure.Such a nucleus reaches a rising speed (fromeq. 3.1of

9.81 ∗ 10−8

18 ∗ 10−6= 0.55 ∗ 10−2m/sec

or approximately 5 mm/sec. In a towing tankthe upper layer of 1 meter depth will be freeof such nuclei after a few minutes. Nuclei of10 microns take 100 times as much time, so afew hours.At sea the turbulent motions in the upperlayer of the water are often in sufficientmotion to keep nuclei mixed up forever. Butafter a storm the nuclei content of the upperlayer of the sea can contain much more largernuclei. The storm itself creates these nucleiby breaking waves and the increased wavemotions prevent the rise to the surface or atleast delay it considerably. However, overtime they will rise to the surface.

3.3 Dissolving bubbles

Another mechanism for gas bubbles to disap-pear is dissolution. In that case the gas bub-bles disappear because the air dissolves in thesurrounding water due to diffusion and the va-por in the bubble condensates because of thesurface tension.

January 15, 2010, Nuclei11

Water can absorb a lot of air and it is veryimportant to distinguish between dissolvedair and free air. Free air is present in the formof bubbles. The majority of the air in water isdissolved and dissolved air has no significanteffect on the cohesive forces in the water.At 15 degrees Celcius in atmospheric condi-tions the amount of air that can be dissolvedin one cubic meter of water is about 25-30g(25-30 liters in atmospheric conditions).This is expressed as the gas concentrationC = 3 ∗ 10−2kg/m3. In that case the wateris saturated. In that condition there is nonet transfer of air molecules through the freesurface. The air content of water is oftenexpressed as the fraction of the maximumamount of gas that can be absorbed: thesaturation rate. For the calculation of theair content in in various units see AppendixA. Note that the saturation rate can beexceeded considerably without gas comingout of solution immediately. In that case thewater is oversaturated or supersaturated andthis can amount to two or three times thesaturation rate.

As mentioned bubbles in water contain airand vapor. Dalton’s law states that the totalpressure exerted by a gaseous mixture is equalto the sum of the partial pressures of eachindividual component in a gas mixture. Thisempirical law was observed by John Dalton in1801. So the pressure inside of a bubble is thesum of the partial pressures of air and vapor.The partial pressure of vapor is the equi-librium vapor pressure. In many conditionsthe vapor pressure can be neglected and abubble can be considered as a pure gas bubble.

Transport of gas through the bubble wallis governed by Henry’s law (formulated byWilliam Henry in 1803), which states thatat a constant temperature, the amount of agiven gas dissolved in a given type and volumeof liquid is directly proportional to the partial

pressure of that gas in equilibrium withthat liquid. The coefficient is called Henry’sconstant, but this ”constant” is temperaturedependent.When the water is not saturated, there willbe a mass flow through the bubble wall andthe bubble will disappear because the freeair goes into solution. Even when the wateris saturated, small bubble will disappearbecause the pressure inside a bubble is higherthan in the surrounding fluid due to thesurface tension, as will be discussed later.

Contrary to evaporation, diffusion is arelatively slow process. Evaporation takesplace at the surface of a bubble only, butdiffusion extends over a much thicker layerof fluid around the bubble. At the bubblewall the saturation rate of the water willimmediately be such that the partial pressureof the dissolved air is equal to the partialpressure of the air in the bubble. When thesaturation rate of the bulk of the water isdifferent, air molecules will migrate to orfrom the bubble. Over a certain distance aconcentration gradient will form.

The mass flow m in kg/m3 through the wallof a bubble with radius R can be written as

dm

dt= 4πR2Dg(

dCgdr

)R (3.2)

in which Dg is a diffusion coefficient(m2/sec) and Cg is the gas concentration in

(kg/m3). The gradient (∂Cg

∂r)R is the gradient

of the gas concentration at the bubble wall.When the vapor is neglected and the gas den-sity remains the same the mass change is

dm

dt= ρg

d(4/3πR3)

dt

and from eq.3.2 the bubble radius changes as

dR

dt=Dg

ρg(dCgdr

)R (3.3)


Eppstein and Plesset [9] gave an approxi-mate solution for the concentration gradient:

(dCgdr

)R = (1

R+

1

(πDgt)1/2

)(Cg∞ − CgR)

(3.4)

in which the length (πDgt)1/2 can be re-

garded as a diffusion boundary layer thickness.For small bubbles this thickness is much largerthan the radius R, so eq. 3.4 can be writtenas

(dCgdr

)R = ((Cg∞ − CgR)

R

and eq.3.3 can be integrated and written as

R2 = R02 +

2Dg(Cg∞ − CgR)

ρgt (3.5)

The time to go completely into solution iswhen R=0 in eq.3.5 and when Cg∞ − CgR isnegative:

t =R0

2ρg2D(CgR − Cg∞)

(3.6)

The diffraction coefficient Dg is close to2 ∗ 10−9m2/sec.

An example can give a rough idea abouttime scales. Consider a bubble with a diame-ter of 100 microns (R = 0.5 ∗ 10−5m) in waterat atmospheric pressure with a saturationrate of 50%. The saturation rate of thewater at the bubble wall will be 100 %. Theconcentration gradient is then estimated as1.5 ∗ 10−2kg/m3 and with ρg = 1 this means

that(Cg∞−CgR)

ρg= −1.5 ∗ 10−2. From eq.3.6

it follows that the time for the bubble todissolve is 41 seconds. When submerged at1 m below the surface, such a bubble willnot reach the surface (at 5mm/sec), but gointo solution. At a time t = 41 the term

(πDt)1/2 is 5 ∗ 10−4, which is still an ordersof magnitude larger than the initial radiusR0 = 5−5. It is therefore justified to neglectthis term in eq.3.4.

Surface tension will reduce this time for abubble to dissolve, but these approximationscan be used to determine a time scale. It fol-lows that the diffusion rate is low when com-pared to the time scale of bubble dynamics ina flow. However, when the existence of gasbubbles in a fluid is considered, small bubbleswill tend to go into solution.

3.4 Mechanisms to sustain

nuclei

From the foregoing it is still unexplainedwhy there are so many nuclei in water.One mechanism that can keep nuclei in thewater without rising to the surface is thecombination of gas and solid particles. Forthat mechanism we have to consider thesurface tension of a fluid. The difference inbond between cohesion between the watermolecules and adhesion between water andair molecules causes surface tension. Thecohesion attracts the molecules at a free sur-face and prevent these molecules to mix withthe air. The result is also a tension on thesurface molecules, which curves the surface inthe neighborhood of a solid wall. The surfacetension makes that a small gas bubble tendsto be spherical. The surface tension increasesthe pressure inside a spherical gas bubblewith an amount of 2s/Rb Pa, in which s isthe surface tension and Rb the bubble radius.The surface tension of water in contact withair at 15 degrees Celcius is 0.075 Nm andthe sensitivity to the temperature is not verystrong.

Consider again a nucleus of 0.1mm diame-ter in a fluid of 1 bar (105Pa). The pressure


Figure 3.1: Contact Angle

inside the bubble is higher than the surround-ing pressure due to the surface tension. Thedifference is 0.15/10−4 = 1500Pa. Note thatthis is of the same magnitude as the vaporpressure. When the bubble is only 10 micronsin diameter the surface tension increases thepressure with 15000 Pa!. The surface tensionwill increase the diffusion of gas into the fluidand thus decrease the time required to go intosolution and this is increasing with decreasingbubble size.

Surface tension also causes a contact angleat the interface of water-gas-solid. This con-tact angle depends on the molecular forces inthe water, the gas (air) and the material. Twotypes of material can be distinguished. On hy-drophobic surfaces the contact angle is largerthan 90 degrees, and air bubbles have the ten-dency to stick to the surface. In hydrophilicsurfaces the contact angle is smaller than 90degrees and air bubbles have the tendency todetach from the surface (Fig. 3.1).

When the surface is not flat but has theshape of a crevice, the schematic model is asin Fig. 3.2.

The material in Fig.3.2 is hydrophobic. Thepressure inside the gas bubble will be lowerthan in the water due to the surface tension.When the water is not saturated an equilib-rium is possible where no net gas diffusiontakes place and the bubble is stable. The den-sity of the particle and bubble together canin principle be equal to water and the nucleusis stable. Note that this model was first usedby Edmund Newton Harvey [16], a biologist

Figure 3.2: Gas Pocket in a Crevice

working on bioluminescence and interested innuclei in blood.

Diffusion can also be prevented by a molec-ular skin on the surface of the nucleus, as wasalready assumed by Fox and Herzfeld in 1964[14]. A variation on this assumption was usedby Yount [50], who assumes a skin of surfaceactive molecules that make the nucleus stable.Morch [38],[39] showed that gas could not onlybe present in crevices, as Harvey assumed, butalso in slightly concave surfaces on solid parti-cles and that the effective radius of these thingas remains could be larger than the size ofthe solid particle.

Experiments have shown that solid particlesare important for the tensile strength of water.Filtering of water delays cavitation inceptionand increases the tensile strength of the water[24]. But also pre-pressurization can stronglyincrease the tensile strength of water by driv-ing the nuclei attached to the solid particlesinto solution before the low pressure occurs.Nuclei with an organic skin seem to be crushedat some pressure and thus can go into solutionalso.

3.5 Mechanisms to gener-

ate nuclei

As discussed in previous sections, diffusion andgravity will remove nuclei from water. How-


ever, in general there are still ample nuclei inregular tap water and also in sea water. Atsea, the breaking of waves is a major sourceof bubbles in the water and the mixing effectof these waves and the related turbulence willprevent small nuclei to rise to the surface. Bi-ological life in the water can also produce freegases and in combination with biological mat-ter it can be a source of nuclei. Actually thetensile strength of sea water in prototype con-ditions is close to zero and cavitation inceptionwill thus take place at the vapor pressure. //

In test facilities this is different. First of allthe amount of organic or solid matter may beless in clean test facilities, but also the sizeof the nuclei required for inception is larger.And these larger nuclei are not always avail-able in sufficient numbers. So in test facilitieslike cavitation tunnels and depressurized tow-ing tanks the nuclei content is always a pointof concern. Locations where nuclei are formedare low pressure regions in a flow, especiallywhen sharp edges cause local flow separation.These edges may occur on a very small scale,such as on surface roughness.

3.6 Nuclei measurements

It is not easy to measure the size distributionof nuclei. These techniques will be discussedlater. Measurements indicate that there aremany small nuclei in water. Gates [15] com-piled such measurements in his thesis andshown in Fig. 3.3. The size range of the nucleimeasured is from 5 microns to 500 microns.The number of nuclei is expressed as the num-ber density Nd, which is the number of nucleiin a small small size range divided by the sizerange. Its unit is therefore m−4.The range of number densities is between 1010

and 1014. It helps to get a feeling for thosenumbers. When the number density is 1012

and we consider nuclei in the range of 50-100microns (a size range of 50∗10∗−6m) it meansthat there are 50 ∗ 10−6 ∗ 1012 = 50 ∗ 106 nu-

Figure 3.3: Nuclei densities

clei per cubic meter or, more imaginable, 50nuclei per cubic centimeter. The average dis-tance between the nuclei is then 2.5mm.Fig. 3.3 also shows that this amount can eas-ily vary with a factor 100 depending on thecircumstances. Even when there are 50 ∗ 106

nuclei of in average 75 microns in diameter in1m3 of water, this means that there is a volumeof 1.1 ∗ 10−5m3 of free gas per m3 of water. Inatmospheric conditions this is also 1.1∗10−5kgand this is the free air content in 1m3. This isstill only a third of the dissolved air content ofC = 3 ∗ 10−2kg/m3. In general the amount offree air will be (much) smaller then the amountof dissolved air. So when there is a mechanismto bring air out of solution and create bubbles,this will not immediately deplete the dissolvedair content. It is also important to note thatthe dissolved air content has little or no effecton cavitation inception.

From Fig.3.3 it can be seen that the nucleiare predominantly small. These small nucleican easily be transported through the fluid by


currents and water motions such as waves. Butthey would still dissolve over time . This seemsto be significantly reduced by the presence ofan organic skin on the surface of a nucleus.When the nuclei dissolve, the reduction of theradius makes the skin thicker and this ulti-mately stops the diffusion, making the nucleistable.Since in sea water ample nuclei are availablenot much attention has been given to their ori-gin and presence. However, for model tests therequired nuclei size is larger, as explained inchapter refchincept, and these nuclei are oftenscarce or not available in test facilities. Thismakes that nuclei distributions dominate theproblem of scaling of cavitation inception andthe mechanisms of dissolving and rising nucleihave to be considered when test facilities areused.

3.7 Bubble Screening

A phenomenon which is often observed isthat cavitation does not take place in theminimum pressure location with a minimumpressure significantly below the vapor pres-sure. This occurs even in conditions withample free nuclei in the incoming flow. Sincethis is observed spedifically in cases whenthe boundary layer is still laminar in hteminimum pressure region, it was suspectedthat this delay in inception was caused bya local lack of nuclei. It means that theincoming flow contains enough nuclei , butthat they are inhibited to enter the lowpressure region. The interpretation of whathappens is only tentative because it is verydifficult to verify experimentally. A possibleexplanation is The common explanation isthat bubble screening occurs. In a pressuregradient nuclei experience a force, similar tothe rising force in a gravitational field. Nearthe leading edge of a foil there is a strongpressure gradient along the foil surface, butalso perpendicular to the wall outside the

Figure 3.4: Path of a Nucleus in the PressureField near the Leading Edge of a Foil

boundary layer (the pressure in a thin bound-ary layer is constant perpendicular to thewall). This pressure gradient perpendicularto the wall is is especially strong near thestagnation point and, in opposite direction,near the minimum pressure location. Nearthe stagnation point the pressure gradientwill move the nuclei away from the wall. Inthe minimum pressure region the nuclei aremoved towards the wall. As a result the pathof the nuclei will not follow the streamlines,but follow a path away from the wall, assketched in Fig 3.4. The path of the nucleimay thus avoid the minimum pressure closeto the wall. The pressure gradient thus has ascreening effect and this effect will be greaterwith increasing size of the nuclei.Calculations of the path of nuclei were firstmade by Johnsson and Hsieh [22] in 1966,assuming a certain drag coefficient of nucleiand assuming that they remained spherical.

Although bubble screening is a plausible hy-pothesis, experimental evidence is difficult toobtain.

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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

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

chapter 3 physics of cavitation: gas content and nuclei

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