physics of cavitation: cavitation inception

Chapter 4

Physics of cavitation:CAVITATIONINCEPTION

Objective: Description of the mechanismof cavitation inception, the phenomena whichinfluence inception and the definition of theinception bucket

Cavitation inception is important for tworeasons. The first reason is that the radiatednoise level of any form of cavitation is anorder of magnitude higher than the noise levelof a non-cavitating flow. This is used by Navyships to detect and locate other ships andby torpedo’s to home in on the ship. Thisis the Navy problem of cavitation inceptionand the inception speed of a navy ship is veryimportant.Commercial vessels generally are not inter-ested very much in the precise inception ofcavitation. Only special ships such as fisheryresearch vessels require a high inceptionspeed. However, for cavitation testing atmodel scale it is necessary that inceptionoccurs in the properly scaled conditions.Otherwise no cavitation will appear. Whencavitation inception is delayed at modelscale, the properties and effects of developedcavitation cannot be investigated properly.This is the scaling problem of cavitationinception and this problem is occurs in anycase when cavitation occurs.

Water and vapor are in equilibrium at

the equilibrium vapor pressure or shortly thevapor pressure Pv. But is is still possible thatno vapor (cavitation) is present when thelocal pressure is below the vapor pressure. Inthat case the difference between the vaporpressure and the local; pressure is called thefluid tension. When the pressure is loweredthe tension will increase and at some pressurevaporization will start. The pressure at whichthis occurs is the inception pressure and thedifference between the vapor pressure andthe inception pressure is called the tensilestrength of the fluid.

For cavitation inception nuclei are required,and these nuclei generally consist of small gasbubbles. In this chapter the behavior of nucleiin a pressure field is investigated.

4.1 Bubble Statics

Consider a gas/vapor bubble of radius R ina fluid with pressure p. The pressure insidethe bubble is equal to the outside pressure pplus the surface tension 2s/R. Here s is thesurface tension in N/m. The pressure insidethe bubble is the sum of the vapor pressure pvand the partial gas pressure pg. So

p+2s

R− K

R30

= pv (4.1)

17

18 G.Kuiper, Cavitation in Ship Propulsion, June 21, 2012

Figure 4.1: Bubble Radius as a Function ofthe Outside Pressure

In isothermal conditions the gas pressure isinversely proportional to the volume, so thegas pressure can be written as K

R3 , where K isa measure for the amount of gas in the nucleus.The relation between the bubble radius R andthe pressure p is given in Fig.4.1 for a numberof values of the amount of gas K.

The curve shows a minimum, and whenthe radius is at that minimum the radiusincreases with increasing pressure. This isan imaginary situation, which cannot exist.When the pressure reaches the minimumvalue equilibrium is no longer possible andthe bubble will grow so rapidly that inertiaterms have to be included in the equilibriumequation. The minimum radius at whicha stable condition is possible is called thecritical radiusand the corresponding pressurethe critical pressure. The critical pressure isoften taken as the inception pressure, becauseat that pressure the nuclei are beginning togrow rapidly and will become visible. Thecritical pressure is indicated with a dot in Fig.4.1. With increasing bubble size the criticalpressure approaches the vapor pressure, butfor smaller nuclei the critical pressure can besignificantly lower than the vapor pressure.

The critical radius can be found from differ-

entiation of the pressure:

∂(p0 − pv)∂R

= 0

from which condition it follows that

Rcrit =

√3K

2s(4.2)

The pressure at which this critical radius isreached is taken as the inception pressure pi.This pressure can be found by substitution ofthe critical radius in eq. 4.1 and is:

pi = pv −4s

3Rcrit

(4.3)

The difference between the vapor pressure andthe inception pressure, as expressed in eq. 4.3,decreases with increasing critical radius andthus with increasing nuclei size in the undis-turbed flow.

This leads to the important conclusion that

The inception pressure depends on the sizeof the largest nuclei in the fluid

This conclusion can be expressed in a non-dimensional way.

The pressure is expressed as the cavitation index

σ =p0 − pv1/2ρV 2

(4.4)

where p0 is the pressure in the undisturbed flow at the same

location. The surface tension s is expressed as the Weber num-

ber We

We =ρV 2r0

s(4.5)

where r0 is the radius of the nuclei in the undisturbed flow.

Eq. 4.1 is valid in the undisturbed flow as well as in the critical

inception condition, so

June 21, 2012, Inception of Bubble Cavitation19

p0 − pv +2s

R0

− K

R30

= 0

pcrit − pv +2s

Rcrit

− K

R3crit

= 0

Rcrit = (3K

2s)

12

Elimination of Rcrit and p0 and expressing the equations

in cavitation index and Weber number results in [27]

pcrit − pvp0 − pv

= −2 8σWe

32

3√

3(1 + 8σWe

)12

The tensile strength of the water, which is the difference be-

tween inception pressure and the vapor pressure, as expressed

in the left hand of this equation can also be written in non-

dimensional terms as

∆σ

σ

where

∆σ =pcrit − pv1/2ρV 2

This gives the final non-dimensional form to the inception

pressure:

∆σ

σ= −

2 8σWe

32

3√

3(1 + 8σWe

)12

This equation gives the difference between the vapor pressure

and the inception pressure in non-dimensional terms.

This has severe implications for scaling of cavitation

inception. It means that, apart from the cavitation index σalso the Weber number We has to be the same at model and

full scale. So the Weber number is an additional scaling law,

introduced by the surface tension as a new parameter. The

Weber number can also be derived as a scaling law using the

Π-theorem as mentioned in [42]

Figure 4.2: Bubble radius versus pressure forbubbles with a higher gas content

4.2 Gaseous Cavitation

The definition of the critical pressure as he in-ception pressure raises some problems. Thefirst problem is that bubbles with a higheramount of gas do not become unstable (Thetwo highest values of K in Fig.4.1). In ex-periments cavitation is mostly detected visu-ally. This means that the cavity has to havea certain minimum size to be detected. Letus assume that this minimum size is 1 mm indiameter or 500 microns in radius, the end ofthe scale in Fig.4.1. The bubbles which do notbecome unstable in Fig.4.1 are detected at thevapor pressure, so inception is called at thevapor pressure. When the amount of gas in-creases the bubbles reach the visible size at apressure above the vapor pressure.

In Fig. 4.2 the two highest gas contentsof Fig.4.1 are plotted, together with threebubbles with a higher gas content. ThisFigure shows that for K > 10−7 the nucleibecome visible at a pressure above the vaporpressure. In that case the observation is calledgaseous cavitation [21], which is not reallycavitation, because the pressure inside thebubble is still higher than the vapor pressure.There is, however, a range of gas contents(or nuclei sizes) which has no critical radiusbut which still lead to a correct prediction


of the vapor pressure as the inception pressure.

In Fig.4.2 the pressure has been extendedto atmospheric pressure. This shows that inthe same fluid at atmospheric pressure nucleiwith a radius above 100 microns (10−4m)willlead to gaseous cavitation. In experimentalfacilities the mean pressure in the test sectionis often lower, but that does not change thesituation very much, since the curves are verysteep. From Fig.3.3 it is found that thesenuclei are relatively scarce in test facilities atlow air contents, but not completely absent.An incidental observation of a visible bubbleon a surface therefore does not necessarilymean inception. To avoid observation ofgaseous cavitation the water in a test facilityshould not contain too many large bubbles.In that case all visible bubbles are cavitationbubbles with an inside pressure close to thevapor pressure which have reached the criticalpressure.When the observation is more careful, smallerbubbles can be observed, e.g. with a radiusof 250 microns. This will increase the risk ofobserving gaseous cavitation.

When the pressure reaches the critical pres-sure equilibrium is no longer possible and thenuclei will grow dynamically and become visi-ble cavitation bubbles. However, when the lowpressure region is very small the time of growthwill be very short and even when the nucleihave reached their critical size, they may stillnot reach a visible size. So we have to checkour concept of inception in dynamic circum-stances.

4.3 Bubble Dynamics

Vapor is relatively light relative to air. Atthe vapor pressure and at 15 degrees Celciusthe the vapor density is 0.013kg/m3. Thedifference between the density of water andvapor in that condition is so large (a ratio of

order 105) that the amount of water requiredto form vapor can generally be neglected

Assuming the flow to be inviscid and irro-tational the bubble can be formulated as anunsteady source in potential flow. Assumingfurther that the behavior is isothermal andthat no diffusion takes place through the bub-ble wall the unsteady equation of motions of abubble becomes an extension of eq. 4.1

p+ ρwR3

2

∂R

∂t+ ρw

∂2R

∂t2+

2s

R= pv + pg

(4.6)

This dynamic terms were first derivedby Raleigh(1917). Plesset [53] combined itwith the contents of a gas-vapor bubble.This equation is therefore referred to asthe Raleigh-Plesset equation. To investigatethe dynamical behavior of a bubble we willconsider its radius when subjected to a jumpin pressure, as shown in Fig.4.3.

A nucleus with radius R0 is present in aflow at pressure p0 at time t=0. At time t=1this nucleus enters a negative pressure peak.During the period t[1,2] the pressure is stillabove the critical pressure and the bubble willgrow, but the increase in size is small.At t=2 the critical pressure is reached andthe bubble begins to grow more rapidly whilethe pressure decreases further. The growth ofthe bubble is such that the gas pressure in thebubble can be neglected at this stage as wellas the surface tension. So the pressure insidethe bubble is close to the vapor pressure Theperiod t[2,3] is controlled by eq. 4.6.During the period t[3,4] the negative pres-sure on the bubble is constant and equalto pv − pmin and the rate of growth of thebubble becomes constant too. This rate ofgrowth will be considered later. In principlethe growth rate of the bubble is unrestrictedas long as the pressure remains unaffected. Inpractice the bubble may grow to such a size


Figure 4.3: Bubble response to a pressurejump

that the pressure is increased by the bubblesize and this will restrain its growth. Forsimplicity we assume no interaction betweenthe outside pressure and the bubble size.In the period t[4,5] the negative pressure onthe bubble will decrease, which slows downthe growth of the bubble. At t=5 the vaporpressure is reached (note that the criticalpressure does not play a role anymore) andthe pressure difference on the bubble becomespositive. Due to the rate of growth of thebubble, there is an overshoot in the radius,after which the radius will begin to collapse.Due to the overshoot the rate of collapse ismuch greater than the linear rate of growth.When the bubble decreases to roughly itsoriginal size the surface tension becomesincreasingly important and this will furtherincrease the rate of collapse. In the end thebubble walls will hit each other , as all vapordisappears. The energy of collapse is thenconverted in a very high local pressure. In

Figure 4.4: High Speed Video:Implosion ofspark generated vapor bubble. courtesy:MARIN

the last stage of collapse the velocity of thebubble wall will exceed the velocity of soundand shock waves are also radiated.

An example of the growth and collapse of abubble is given in Fig. 4.4[15]. The vapor bub-ble is created by a spark from the left handelectrode. The bubble collapses on the wallabove the electrode. The first bang is the gen-eration of the cavity. After reaching its max-imum size the bubble collapses on the wall,and it is clear that the impact is high. Afterthe first collapse a rebound is visible. Notethe time scale: the pictures were taken witha frame rate of 40.000 Hz and the whole se-quence is 120 pictures or 3 milliseconds.

In the very last stage of collapse any gas inthe bubble will act as a buffer and slow downthe rate of collapse. The maximum pressureis decreased by the gas, but, depending on thegas, the temperature increases rapidly. Thetemperature may increase so much that lightemission takes place (sonoluminiscence) . Theenergy, stored in the gas, makes the bubbleto rebound once or twice.After rebound thesingle bubble is often fractured into many

http://youtu.be/hFZ7BBvn-0g


smaller ones.

At this point we are interested in incep-tion only. At the critical pressure the nucleusis not visible yet, and the question is whenis it? What time does it take to grow to avisible size? This can be estimated from anasymptotic solution of eq. 4.6. With a con-stant under-pressure in region l[3,4] of Fig. 4.3the acceleration term ∂2R

∂t2will vanish. With in-

creasing radius the gas pressure pg and the sur-face tension term 2s

Rwill become small, leaving

only

3

2(∂R

∂t)2 = p− pv (4.7)

This estimation can be used to estimatethe time dt in Fig. 4.3 between the timethe critical pressure is reached and the timethe bubble becomes visible. Take e.g. thenucleus with K = 5 ∗ 10−10 from Fig. 4.1.It reaches the critical pressure at a radiusof 100 microns and the critical pressure is700 Pa. When this is also the minimumpressure in the fluid the pressure differencep − pv = 1000 Pa, assuming that the vaporpressure is 1700 Pa. The bubble still has togrow 400 microns in radius to become visible,so dr = 4 ∗ 10−4m. From eq. 4.7 it is foundthat this requires a time dt = 1.5 ∗ 10−5 sec.Even at a very high flow velocity of 40 m/secthe bubble will thus be detected at a positionwhich is 0.6 mm downstream of the minimumpressure location. So except for very shortlow pressure pulses the critical pressure is auseful inception criterion.

The Raleigh- Plesset equation is often usedin numerical calculations to provide a modelfor the phase change of water into vapor whencavitation occurs.

4.4 Bubble Resonance

When the duration of a pressure change is tooshort the response of the bubble will decrease.A measure for this time scale is the resonancefrequency of a gas bubble. The resonance fre-quency of a spherical gas-vapor bubble withradius R at a pressure p is [51][14]

f0 =1

2πR√ρw

[3γ(p+2s

ρw− pv)−

2s

R− 4µ2

R2ρ]12

(4.8)

in which γ is the ratio of specific heats (itsvalue is value approximately 1.4) and µ is thedynamic viscosity of water. The effect of vis-cosity in water is mall. For bubbles larger than10 microns in radius the surface tension be-comes small and can be neglected and eq. 4.8can be simplified to

f0 =1

2πR√ρw

[3γ(p− pv)]12 (4.9)

In most cases the vapor pressure is small rel-ative to the pressure p. Then eq.4.9 is reducedto

f0 =1

2πR√ρw

(3γp)12 (4.10)

The assumptions are mentioned here, be-cause eq. 4.10 is often used without noticingthe restrictions.

When e.g. the pressure in the test sec-tion of a cavitation tunnel is 0.3 at or 33.000Pa, the resonance frequency of a nucleus of100 microns radius is 18.7 kHz. The relevantlength scale can be estimated as half the wavelength, so at 2.7 ∗ 10−5m. At a tunnel speedof 10 m/sec the length of a low pressure re-gion which is of the order of the resonance fre-quency is 2.7∗10−4m. When the length of thelow pressure region is 10 times higher, so 2.7mm, the bubble will remain close to equilib-rium. This may require attention sometimes,


because on thin foils as used in ship propellers,the length of the low pressure peak at the lead-ing edge is only a few percent of the chord.With a chord length of 10 cm the low pressurepeak may become so short that the nucleushas no time to react to the pressure drop atthe leading edge.

4.5 Thermal effects

Evaporation of water requires energy. Evapo-ration of water lowers the temperature of thesurroundings, as is well known by everybodywho is sweating. The amount of requiredenergy is the evaporation heat (2270 kJ/kg).To get an impression of the thermal effects ofevaporation, consider a cubical vapor volumeof 1m3. Creation of a vapor volume of 1m3

requires the evaporation of 10−5 ∗ 1000kg ofwater, or 10 g of water. When the evaporatedwater comes from the six sides of 1m2 each,the evaporated water thickness is only 0.0017millimeter! The amount of heat involved toevaporate 10 g of water is is 22.7kJ . The spe-cific heat of water can be taken as 4.2kJ/kgK.Now assume that the water temperature isaffected over 1mm of water depth. This is anamount of 6 kg. The drop in temperature ofthat amount of water is then 22.7/4.2/6 = 0.9degrees K (or Celcius). Its main effect is thetemporary change of the equilibrium vaporpressure by 100 Pa.

In the above example the critical assump-tion is the thickness of the layer which is cooledby the evaporation. There will be a temper-ature gradient and the minimum temperaturewill be at the evaporating surface. For a shorttime the temperature can therefore be muchlower than the average temperature in a thinfluid layer. This is especially important whenthe time scale is small, such as in acoustic cav-itation. In water the thermal effects are oftensmall, but in other fluids they can be impor-tant.

The main reason for the limited effect ofthermal effects is that the mass of the vaporneeded to influence the flow pattern is limited.Cavitation in water occurs at a low pressureand at this pressure the amount of water togenerate the vapor is very small. Much smallerthan when e.g. evaporation is used for cooling(sweating) at atmospheric pressure. In somecases, where the evaporation is very strong,thermal effects can become more important[62]

4.6 Scale Effects

The inception criterion developed until nowhas been the unsteady growth of gas nucleiand the definition of the inception pressureis the critical pressure. This means that intheory cavitation inception is determined bythe largest nuclei in the flow. When the den-sity of the largest nuclei is low, it may taketime to encounter a nucleus. On a propellerin uniform flow this time is available, but innon-uniform inflow the time in which the pro-peller blade can reach inception is limited. Sothe encounter frequency or the nuclei density isalso becoming a parameter in inception. It isgenerally assumed that in inception tests thedensity of the relevant nuclei is high enoughto make the encounter frequency much higherthan the blade passing frequency. So the def-inition of the inception pressure is the criti-cal pressure of the largest nuclei which becomeunstable and which are available in sufficientquantity.//

Cavitation tests on ship propellers aregenerally done at model scale. That meansthat scaling rules have to be maintained.These rules can be defined as non-dimensionalparameters which have to be maintained atmodel and at full scale. Examples are theReynolds number, the Froude number, thecavitation index and the Weber number.

Visible cavitation bubbles have an internal


pressure which is close to the vapor pressure,since the gas content has become negligible Sothe parameter which has to be maintained forproper scaling of cavitation is the cavitationindex (equation 4.4). Cavitation inception isnot determined by the vapor pressure, but bythe critical pressure of the largest nuclei, or bythe pressure difference pcrit − pv (eq. 4.3). Innon dimensional terms this means that a newscaling parameter ∆σ is introduced, which hasto be maintained at model scale and full scale:

∆σ =pcrit − pv1/2ρV 2

(4.11)

Using eq.4.3 it can be derived that wheneq.4.11 is the same at model and ship, thismeans that

Rcrit(model)

Rcrit(ship)=V 2s

V 2m

When the Froude number is maintained incase of a free surface this ratio is equal to thescale ratio, which in model testing generally isbetween 10 and 30. It means that the criticalradius of nuclei at model scale have to be largerthan at full scale by approximately the scaleratio.

Apart from this requirement also the geo-metrical distribution of the nuclei should bescaled properly at model scale. That meansthat at model scale there should be the sameamount of bubbles on the model propellerblade as on the ship. This means that thenumber density of the nuclei at model scalehas to be increased by the third power of thescale ratio!

The combination of both requirementswould lead to a very high free air content, a”soda pop tunnel”.The transparency would besuch that visual observations were impossibleand the fluid would not be incompress-ible anymore. So it is impossible to maintainthe scaling laws of bubble cavitation inception.

To evaluate this problem consider the valueof ∆σ at full scale. There are generally ample

small nuclei of the order of R=0.1 mm presentin sea water (Fig. 3.3)at about atmosphericpressure. The critical radius of such nuclei willbe much larger than the initial radius at at-mospheric pressure, because at full scale thedifference between the vapor pressure and theinitial pressure is large (which is of the orderof 105 Pa.). From eqs.4.1 and 4.2 it can befound that the critical radius of such nuclei islarger than 0.1 mm. The value of pcrit − pvfrom eq.4.3 is then 100 Pa. From Fig. 4.1 itcan be seen that this is relatively close to thevapor pressure (about 1700 Pa at 15 degreesCelcius). So at full scale the inception pressureis always close to the vapor pressure.

To maintain ∆σ at model scale at a scaleratio of say 25, this requires nuclei with a crit-ical radius larger than 25 ∗ 0.1mm or 2.5 mm.Such nuclei will rise relatively fast and will ob-scure visibility when present in sufficient num-bers. In practice the largest nuclei in a modeltt will be smaller. But from Fig. 4.1 it canalso be seen that when the critical pressure ap-proaches the vapor pressure, the curve of crit-ical pressures approaches the vapor pressureasymptotically. This means that deviations ofthe critical radius have little effect on the in-ception pressure. So when at model scale therequired nuclei size is not reached, the errorin inception pressure will be small, providedthat the inception pressure is close to the va-por pressure.The conclusion therefore is that proper scal-ing of bubble inception at model scale is notpossible. To obtain a good approximation offull scale inception at model scale the nucleisize should be such that the inception pres-sure (critical pressure) is close to the vaporpressure. So the nuclei should be as large aspossible without harming visibility. On theother hand the maximum size of the nuclei atmodel scale should not be so large that the nu-clei become visible before the vapor pressure isreached. In that case gaseous cavitation willbe observed, and the inception pressure at full


Figure 4.5: Bubble cavitation due to a singlenucleus

scale will not be predicted correctly.

Note that for proper scaling of cavitationand its dynamics the cavitation index is stillthe proper index, because the pressure insidethe cavitation is the vapor pressure. It isnot correct to base the cavitation propertieson σ/σi, as is sometimes done in literature.When inception has occurred the inceptionpressure does no longer affect the cavity sizeor dynamics, except for the very short timethe cavity grows from inception to its dynamicequilibrium (see section 4.3).

At model scale the nuclei density of thelargest nuclei will be smaller than at fullscale,which has an effect on the appearance ofbubble cavitation. When very few nuclei areavailable which can reach the critical size, in-ception will occur only incidentally, dependingon the passage of an incidental nucleus. Sucha single nucleus will grow rapidly because it isunstable and the minimum pressure is far be-low the vapor pressure. Since no other nucleiare growing around it,its surrounding pressurewill not be affected until it has grown to a sub-stantial size. An example is shown in Fig. 4.5.

Such large incidental bubble cavities indi-cate that the critical radius is only reached in-

cidentally and that the number density of thenuclei is not sufficient. The inception pressureis much lower than during the observation inFig 4.5.

4.7 Acoustic Inception

Up to now e have focussed on visual detectionof cavitation inception. As indicated inFig. 4.3 the collapse of a cavitation bubbleis not symmetrical with its inception. Afterbecoming unstable the nucleus grows andreaches a maximum value, after which itcollapses. This collapse generates noise, dom-inated by one or several pressure spikes. Inthe frequency domain this means a significantincrease of the high frequencies. The spike orthe increase in high frequency noise can beused to detect cavitation. In that case it iscalled acoustic inception.

There may be differences between acousticinception and visual inception. When the nu-clei are small and the low pressure peak is nar-row, the nuclei may become unstable but mayremain invisible. In that case acoustic incep-tion is detected first. In a cavitation tunnelthis is the dominant situation. However, fromFig. 4.2 it was observed that larger nuclei cangrow to a visible size without becoming un-stable. In such a situation the bubble returnsto its approximate original size without a vio-lent collapse and thus without being detectedacoustically. Stricly speaking this is gaseouscavitation.

The response of a nucleus to a pressurechange has also been investigated extensivelywhen the pressure change is periodical. Thepressure field is then defined by an acousticpressure field: a mean pressure pmean, apressure amplitude pa and a frequency f .Vaporous bubble growth (inception) of nucleimay occur when the minimum pressurepmean − pa is lower than the vapor pressure,even when the mean pressure is higher than


the vapor pressure. However, when theminimum pressure is only slightly lower thanthe vapor pressure the nuclei will respond bya periodical motion. In that case there is noimplosion and no acoustical cavitation willbe detected. When the minimum pressuredecreases below a certain limit value belowthe vapor pressure the response of the nucleichanges and the nuclei will collapse period-ically. From experiments and calculationsit is found that the critical pressure fromeq. 4.1 can also be used with good accuracyfor inception of acoustic cavitation [[51],[14]].

4.8 Inception measure-

ments

.

The definition of cavitation inception asgiven until now is from the instability of a sin-gle bubble. Even then the distinction betweengaseous cavitation and inception cannot bemade without knowledge of the nuclei contentin the flow. When a nuclei distribution isinvolved, the nuclei density distribution isanother factor, determining the frequencyof cavitation inception. Also the accuracyof observation (what is defined as a visiblebubble) is a factor in cavitation inception.The determination of cavitation inception istherefore difficult and the accuracy of theobserved inception pressure of a propeller isnot high.

4.8.1 Inception speed on Navyships

The inception pressure is only important forNavy ships, where the acoustical detection ofthe ship depends on the occurrence of the firstsmall cavitation event. The noise level of suchan event can be such that it exceeds the flow

noise or machinery noise considerably. A cri-terion for Navy ship is therefore the inceptionspeed, which is the speed at which the firstcavitation occurs. Inception measurements onNavy ships are therefore always done acous-tically. The determination of the inceptionspeed of a Navy ship at full scale is still diffi-cult and is often not very repeatable. It mayalso change considerably in time and certainlywith seastate, wind and fouling. At modelscale the determination of cavitation inceptionexhibits a lot of scatter due to flow parameterssuch as nuclei, but also because visual incep-tion is difficult to see. Acoustic inception isnot well defined either, because cavitation in-ception starts with a single incidental implo-sion and the frequency of the implosions in-creases gradually and a certain criterion hasto be used.

4.8.2 Inception measurementsfor comparative purposes

Cavitation inception is not very important forcommercial propellers, except in some specificcases where a very low noise level is required(research vessels), which does not allow anycavitation. The noise level generated by asmall amount of cavitation (incipient cavita-tion) is not high and in most cases can be ig-nored. Cavitation inception is still often mea-sured at model scale, as a criterion for the op-erational conditions or for comparison betweendifferent propellers. This is only because thereis no other more repeatable criterion available.It should be kept in mind, however, that cav-itation inception curves are more inaccuratethan is often suggested. Inception conditionswill be discussed with the various types of cav-itation.

4.9 The inception bucket

The inception properties of a propeller areexpressed in an ”inception diagram”. This di-


Figure 4.6: Example of an inception diagramof a propeller

agram gives the inception pressures at variousloadings of the propeller. The loading is gen-erally expressed in the non-dimensional thrustcoefficient Kt, or in the non-dimensionaladvance ratio J . An example of such adiagram is given in Fig. 4.6.

The corresponding curves of sheet cavitationhave been marked red in Fig. 4.6. Conditionsoutside the V-shaped curves will have sheetcavitation. Conditions inside the V-shapedcurve will be free of sheet cavitation. Becausethe shape of the curves generally is V- or U-shaped, such an inception curve is called aninception bucket. To be free of any type ofcavitation the conditions must be inside theinnermost bucket.

physics of cavitation: cavitation inception

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