Chapter 4 Physics of Cavitation:CAVITATION INCEPTION

Chapter 4

Physics of cavitation:CAVITATION INCEPTION

Objective: Description of the mechanism the equilibrium vapor pressure or shortly the of cavitation inception, the phenomena which vapor pressure Pv. But is is still possible that influence inception and the definition of the no vapor (cavitation) is present when the inception bucket local pressure is below the vapor pressure. In that case the difference between the vapor pressure and the local; pressure is called the fluid tension. When the pressure is lowered Cavitation inception is important for two the tension will increase and at some pressure reasons. The first reason is that the radiated vaporization will start. The pressure at which noise level of any form of cavitation is an this occurs is the inception pressure and the order of magnitude higher than the noise level difference between the vapor pressure and of a non-cavitating flow. This is used by Navy the inception pressure is called the tensile ships to detect and locate other ships and strength of the fluid. by torpedo’s to home in on the ship. This is the Navy problem of cavitation inception For cavitation inception nuclei are required, and the inception speed of a navy ship is very and these nuclei generally consist of small gas important. bubbles. In this chapter the behavior of nuclei Commercial vessels generally are not inter- in a pressure field is investigated. ested very much in the precise inception of cavitation. Only special ships such as fishery research vessels require a high inception 4.1 Bubble Statics speed. However, for cavitation testing at model scale it is necessary that inception Consider a gas/vapor bubble of radius R in occurs in the properly scaled conditions. a fluid with pressure p. The pressure inside Otherwise no cavitation will appear. When the bubble is equal to the outside pressure p cavitation inception is delayed at model plus the surface tension 2s/R. Here s is the scale, the properties and effects of developed surface tension in N/m. The pressure inside cavitation cannot be investigated properly. the bubble is the sum of the vapor pressure pv This is the scaling problem of cavitation and the partial gas pressure pg. So inception and this problem is occurs in any case when cavitation occurs. 2s K p + − 3 = pv (4.1) Water and vapor are in equilibrium at R R0

17 18 G.Kuiper, Cavitation in Ship Propulsion, June 21, 2012

entiation of the pressure:

∂(p − p ) 0 v = 0 ∂R from which condition it follows that r 3K R = (4.2) crit 2s

The pressure at which this critical radius is reached is taken as the inception pressure pi. This pressure can be found by substitution of the critical radius in eq. 4.1 and is: Figure 4.1: Bubble Radius as a Function of the Outside Pressure 4s pi = pv − (4.3) 3Rcrit In isothermal conditions the gas pressure is The difference between the vapor pressure and inversely proportional to the volume, so the the inception pressure, as expressed in eq. 4.3, gas pressure can be written as K , where K is R3 decreases with increasing critical radius and a measure for the amount of gas in the nucleus. thus with increasing nuclei size in the undis- The relation between the bubble radius R and turbed flow. the pressure p is given in Fig.4.1 for a number This leads to the important conclusion that of values of the amount of gas K. The curve shows a minimum, and when the radius is at that minimum the radius The inception pressure depends on the size increases with increasing pressure. This is of the largest nuclei in the fluid an imaginary situation, which cannot exist. When the pressure reaches the minimum value equilibrium is no longer possible and the bubble will grow so rapidly that inertia This conclusion can be expressed in a non-dimensional way. terms have to be included in the equilibrium The pressure is expressed as the cavitation index equation. The minimum radius at which a stable condition is possible is called the p − p σ = 0 v (4.4) critical radiusand the corresponding pressure 1/2ρV 2 the critical pressure. The critical pressure is often taken as the inception pressure, because where p0 is the pressure in the undisturbed flow at the same at that pressure the nuclei are beginning to location. The surface tension s is expressed as the Weber num- grow rapidly and will become visible. The ber W e critical pressure is indicated with a dot in Fig. 4.1. With increasing bubble size the critical ρV 2r pressure approaches the vapor pressure, but W e = 0 (4.5) for smaller nuclei the critical pressure can be s significantly lower than the vapor pressure. 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 The critical radius can be found from differ- inception condition, so June 21, 2012, Inception of Bubble Cavitation19

2s K p0 − pv + − 3 = 0 R0 R0 2s K pcrit − pv + − 3 = 0 Rcrit Rcrit 1 3K 2 R = ( ) crit 2s

Elimination of Rcrit and p0 and expressing the equations in cavitation index and Weber number results in [27]

3 p − p 2 8 2 Figure 4.2: Bubble radius versus pressure for crit v = − √ σW e 8 1 p0 − pv 2 bubbles with a higher gas content 3 3(1 + σW e ) The tensile strength of the water, which is the difference be- 4.2 Gaseous Cavitation tween inception pressure and the vapor pressure, as expressed in the left hand of this equation can also be written in non- The definition of the critical pressure as he in- dimensional terms as ception pressure raises some problems. The ∆σ first problem is that bubbles with a higher σ amount of gas do not become unstable (The where two highest values of K in Fig.4.1). In experiments cavitation is mostly detected visu- p − p ∆σ = crit v ally. This means that the cavity has to have 1/2ρV 2 a certain minimum size to be detected. Let

This gives the ﬁnal non-dimensional form to the inception us assume that this minimum size is 1 mm in pressure: diameter or 500 microns in radius, the end of the scale in Fig.4.1. The bubbles which do not become unstable in Fig.4.1 are detected at the 3 ∆σ 2 8 2 vapor pressure, so inception is called at the = − √ σW e 8 1 vapor pressure. When the amount of gas in- σ 3 3(1 + ) 2 σW e creases the bubbles reach the visible size at a pressure above the vapor pressure. In Fig. 4.2 the two highest gas contents This equation gives the diﬀerence between the vapor pressure of Fig.4.1 are plotted, together with three and the inception pressure in non-dimensional terms. bubbles with a higher gas content. This Figure shows that for K > 10−7 the nuclei This has severe implications for scaling of cavitation become visible at a pressure above the vapor inception. It means that, apart from the cavitation index σ pressure. In that case the observation is called also the Weber number W e has to be the same at model and gaseous cavitation [21], which is not really full scale. So the Weber number is an additional scaling law, cavitation, because the pressure inside the introduced by the surface tension as a new parameter. The bubble is still higher than the vapor pressure. Weber number can also be derived as a scaling law using the There is, however, a range of gas contents Π-theorem as mentioned in [42] (or nuclei sizes) which has no critical radius but which still lead to a correct prediction 20 G.Kuiper, Cavitation in Ship Propulsion, June 21, 2012

of the vapor pressure as the inception pressure. order 105) that the amount of water required to form vapor can generally be neglected In Fig.4.2 the pressure has been extended to atmospheric pressure. This shows that in Assuming the flow to be inviscid and irro- the same fluid at atmospheric pressure nuclei tational the bubble can be formulated as an with a radius above 100 microns (10−4m)will unsteady source in potential flow. Assuming lead to gaseous cavitation. In experimental further that the behavior is isothermal and facilities the mean pressure in the test section that no diffusion takes place through the bub- is often lower, but that does not change the ble wall the unsteady equation of motions of a situation very much, since the curves are very bubble becomes an extension of eq. 4.1 steep. From Fig.3.3 it is found that these nuclei are relatively scarce in test facilities at 3 ∂R ∂2R 2s low air contents, but not completely absent. p + ρwR + ρw + = pv + pg 2 ∂t ∂t2 R An incidental observation of a visible bubble (4.6) on a surface therefore does not necessarily mean inception. To avoid observation of This dynamic terms were first derived gaseous cavitation the water in a test facility by Raleigh(1917). Plesset [53] combined it should not contain too many large bubbles. with the contents of a gas-vapor bubble. In that case all visible bubbles are cavitation This equation is therefore referred to as bubbles with an inside pressure close to the the Raleigh-Plesset equation. To investigate vapor pressure which have reached the critical the dynamical behavior of a bubble we will pressure. consider its radius when subjected to a jump When the observation is more careful, smaller in pressure, as shown in Fig.4.3. bubbles can be observed, e.g. with a radius of 250 microns. This will increase the risk of A nucleus with radius R0 is present in a observing gaseous cavitation. flow at pressure p0 at time t=0. At time t=1 this nucleus enters a negative pressure peak. When the pressure reaches the critical pres- During the period t[1,2] the pressure is still sure equilibrium is no longer possible and the above the critical pressure and the bubble will nuclei will grow dynamically and become visi- grow, but the increase in size is small. ble cavitation bubbles. However, when the low At t=2 the critical pressure is reached and pressure region is very small the time of growth the bubble begins to grow more rapidly while will be very short and even when the nuclei the pressure decreases further. The growth of have reached their critical size, they may still the bubble is such that the gas pressure in the not reach a visible size. So we have to check bubble can be neglected at this stage as well our concept of inception in dynamic circum- as the surface tension. So the pressure inside stances. the bubble is close to the vapor pressure The period t[2,3] is controlled by eq. 4.6. During the period t[3,4] the negative pres- 4.3 Bubble Dynamics sure on the bubble is constant and equal to pv − pmin and the rate of growth of the Vapor is relatively light relative to air. At bubble becomes constant too. This rate of the vapor pressure and at 15 degrees Celcius growth will be considered later. In principle the the vapor density is 0.013kg/m3. The the growth rate of the bubble is unrestricted difference between the density of water and as long as the pressure remains unaffected. In vapor in that condition is so large (a ratio of practice the bubble may grow to such a size June 21, 2012, Inception of Bubble Cavitation21

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

Figure 4.3: Bubble response to a pressure the last stage of collapse the velocity of the jump bubble wall will exceed the velocity of sound and shock waves are also radiated. that the pressure is increased by the bubble An example of the growth and collapse of a size and this will restrain its growth. For bubble is given in Fig. 4.4[15]. The vapor bub- simplicity we assume no interaction between ble is created by a spark from the left hand the outside pressure and the bubble size. electrode. The bubble collapses on the wall In the period t[4,5] the negative pressure on above the electrode. The first bang is the gen- the bubble will decrease, which slows down eration of the cavity. After reaching its max- the growth of the bubble. At t=5 the vapor imum size the bubble collapses on the wall, pressure is reached (note that the critical and it is clear that the impact is high. After pressure does not play a rôleanymore) and the first collapse a rebound is visible. Note the pressure difference on the bubble becomes the time scale: the pictures were taken with positive. Due to the rate of growth of the a frame rate of 40.000 Hz and the whole se- bubble, there is an overshoot in the radius, quence is 120 pictures or 3 milliseconds. after which the radius will begin to collapse. In the very last stage of collapse any gas in Due to the overshoot the rate of collapse is the bubble will act as a buffer and slow down much greater than the linear rate of growth. the rate of collapse. The maximum pressure When the bubble decreases to roughly its is decreased by the gas, but, depending on the original size the surface tension becomes gas, the temperature increases rapidly. The increasingly important and this will further temperature may increase so much that light increase the rate of collapse. In the end the emission takes place (sonoluminiscence) . The bubble walls will hit each other , as all vapor energy, stored in the gas, makes the bubble disappears. The energy of collapse is then to rebound once or twice.After rebound the converted in a very high local pressure. In single bubble is often fractured into many 22 G.Kuiper, Cavitation in Ship Propulsion, June 21, 2012

smaller ones. 4.4 Bubble Resonance

When the duration of a pressure change is too At this point we are interested in incep- short the response of the bubble will decrease. tion only. At the critical pressure the nucleus A measure for this time scale is the resonance is not visible yet, and the question is when frequency of a gas bubble. The resonance fre- is it? What time does it take to grow to a quency of a spherical gas-vapor bubble with visible size? This can be estimated from an radius R at a pressure p is [51][14] asymptotic solution of eq. 4.6. With a constant under-pressure in region l[3,4] of Fig. 4.3 2 2 ∂ R 1 2s 2s 4µ 1 the acceleration term ∂t2 will vanish. With in- 2 f0 = √ [3γ(p + − pv) − − 2 ] creasing radius the gas pressure pg and the sur- 2πR ρw ρw R R ρ 2s face tension term R will become small, leaving (4.8) only in which γ is the ratio of specific heats (its value is value approximately 1.4) and µ is the dynamic viscosity of water. The effect of vis- 3 ∂R cosity in water is mall. For bubbles larger than ( )2 = p − p (4.7) 10 microns in radius the surface tension be- 2 ∂t v comes small and can be neglected and eq. 4.8 can be simplified to This estimation can be used to estimate the time dt in Fig. 4.3 between the time 1 1 f0 = √ [3γ(p − pv)] 2 (4.9) the critical pressure is reached and the time 2πR ρw the bubble becomes visible. Take e.g. the In most cases the vapor pressure is small rel- nucleus with K = 5 ∗ 10−10 from Fig. 4.1. ative to the pressure p. Then eq.4.9 is reduced It reaches the critical pressure at a radius to of 100 microns and the critical pressure is 700 Pa. When this is also the minimum pressure in the fluid the pressure difference 1 1 f0 = √ (3γp) 2 (4.10) p − pv = 1000 Pa, assuming that the vapor 2πR ρw pressure is 1700 Pa. The bubble still has to The assumptions are mentioned here, be- grow 400 microns in radius to become visible, cause eq. 4.10 is often used without noticing so dr = 4 ∗ 10−4m. From eq. 4.7 it is found −5 the restrictions. that this requires a time dt = 1.5 ∗ 10 sec. When e.g. the pressure in the test sec- Even at a very high flow velocity of 40 m/sec tion of a cavitation tunnel is 0.3 at or 33.000 the bubble will thus be detected at a position Pa, the resonance frequency of a nucleus of which is 0.6 mm downstream of the minimum 100 microns radius is 18.7 kHz. The relevant pressure location. So except for very short length scale can be estimated as half the wave low pressure pulses the critical pressure is a length, so at 2.7 ∗ 10−5m. At a tunnel speed useful inception criterion. of 10 m/sec the length of a low pressure region which is of the order of the resonance fre- The Raleigh- Plesset equation is often used quency is 2.7 ∗ 10−4m. When the length of the in numerical calculations to provide a model low pressure region is 10 times higher, so 2.7 for the phase change of water into vapor when mm, the bubble will remain close to equilib- cavitation occurs. rium. This may require attention sometimes, June 21, 2012, Inception of Bubble Cavitation23

because on thin foils as used in ship propellers, The main reason for the limited effect of the length of the low pressure peak at the lead- thermal effects is that the mass of the vapor ing edge is only a few percent of the chord. needed to influence the flow pattern is limited. With a chord length of 10 cm the low pressure Cavitation in water occurs at a low pressure peak may become so short that the nucleus and at this pressure the amount of water to has no time to react to the pressure drop at generate the vapor is very small. Much smaller the leading edge. than when e.g. evaporation is used for cooling (sweating) at atmospheric pressure. In some cases, where the evaporation is very strong, 4.5 Thermal effects thermal effects can become more important [62] Evaporation of water requires energy. Evapo- ration of water lowers the temperature of the surroundings, as is well known by everybody 4.6 Scale Effects who is sweating. The amount of required energy is the evaporation heat (2270 kJ/kg). The inception criterion developed until now To get an impression of the thermal effects of has been the unsteady growth of gas nuclei evaporation, consider a cubical vapor volume and the definition of the inception pressure of 1m3. Creation of a vapor volume of 1m3 is the critical pressure. This means that in requires the evaporation of 10−5 ∗ 1000kg of theory cavitation inception is determined by water, or 10 g of water. When the evaporated the largest nuclei in the flow. When the den- water comes from the six sides of 1m2 each, sity of the largest nuclei is low, it may take the evaporated water thickness is only 0.0017 time to encounter a nucleus. On a propeller millimeter! The amount of heat involved to in uniform flow this time is available, but in evaporate 10 g of water is is 22.7kJ. The spe- non-uniform inflow the time in which the pro- cific heat of water can be taken as 4.2kJ/kgK. peller blade can reach inception is limited. So Now assume that the water temperature is the encounter frequency or the nuclei density is affected over 1mm of water depth. This is an also becoming a parameter in inception. It is amount of 6 kg. The drop in temperature of generally assumed that in inception tests the that amount of water is then 22.7/4.2/6 = 0.9 density of the relevant nuclei is high enough degrees K (or Celcius). Its main effect is the to make the encounter frequency much higher temporary change of the equilibrium vapor than the blade passing frequency. So the def- pressure by 100 Pa. inition of the inception pressure is the critical pressure of the largest nuclei which become In the above example the critical assump- unstable and which are available in sufficient tion is the thickness of the layer which is cooled quantity.// by the evaporation. There will be a temper- Cavitation tests on ship propellers are ature gradient and the minimum temperature generally done at model scale. That means will be at the evaporating surface. For a short that scaling rules have to be maintained. time the temperature can therefore be much These rules can be defined as non-dimensional lower than the average temperature in a thin parameters which have to be maintained at fluid layer. This is especially important when model and at full scale. Examples are the the time scale is small, such as in acoustic cav- Reynolds number, the Froude number, the itation. In water the thermal effects are often cavitation index and the Weber number. small, but in other fluids they can be important. Visible cavitation bubbles have an internal 24 G.Kuiper, Cavitation in Ship Propulsion, June 21, 2012

pressure which is close to the vapor pressure, small nuclei of the order of R=0.1 mm present since the gas content has become negligible So in sea water (Fig. 3.3)at about atmospheric the parameter which has to be maintained for pressure. The critical radius of such nuclei will proper scaling of cavitation is the cavitation be much larger than the initial radius at at- index (equation 4.4). Cavitation inception is mospheric pressure, because at full scale the not determined by the vapor pressure, but by difference between the vapor pressure and the the critical pressure of the largest nuclei, or by initial pressure is large (which is of the order 5 the pressure difference pcrit − pv (eq. 4.3). In of 10 Pa.). From eqs.4.1 and 4.2 it can be non dimensional terms this means that a new found that the critical radius of such nuclei is scaling parameter ∆σ is introduced, which has larger than 0.1 mm. The value of pcrit − pv to be maintained at model scale and full scale: from eq.4.3 is then 100 Pa. From Fig. 4.1 it p − p can be seen that this is relatively close to the ∆σ = crit v (4.11) 1/2ρV 2 vapor pressure (about 1700 Pa at 15 degrees Celcius). So at full scale the inception pressure Using eq.4.3 it can be derived that when is always close to the vapor pressure. eq.4.11 is the same at model and ship, this means that To maintain ∆σ at model scale at a scale 2 ratio of say 25, this requires nuclei with a crit- Rcrit(model) V = s ical radius larger than 25 ∗ 0.1mm or 2.5 mm. R (ship) V 2 crit m Such nuclei will rise relatively fast and will ob- When the Froude number is maintained in scure visibility when present in sufficient num- case of a free surface this ratio is equal to the bers. In practice the largest nuclei in a model scale ratio, which in model testing generally is tt will be smaller. But from Fig. 4.1 it can between 10 and 30. It means that the critical also be seen that when the critical pressure ap- radius of nuclei at model scale have to be larger proaches the vapor pressure, the curve of crit- than at full scale by approximately the scale ical pressures approaches the vapor pressure ratio. asymptotically. This means that deviations of Apart from this requirement also the geo- the critical radius have little effect on the in- metrical distribution of the nuclei should be ception pressure. So when at model scale the scaled properly at model scale. That means required nuclei size is not reached, the error that at model scale there should be the same in inception pressure will be small, provided amount of bubbles on the model propeller that the inception pressure is close to the va- blade as on the ship. This means that the por pressure. number density of the nuclei at model scale The conclusion therefore is that proper scal- has to be increased by the third power of the ing of bubble inception at model scale is not scale ratio! possible. To obtain a good approximation of The combination of both requirements full scale inception at model scale the nuclei would lead to a very high free air content, a size should be such that the inception pres- ”soda pop tunnel”.The transparency would be sure (critical pressure) is close to the vapor such that visual observations were impossible pressure. So the nuclei should be as large as and the fluid would not be incompress- possible without harming visibility. On the ible anymore. So it is impossible to maintain other hand the maximum size of the nuclei at the scaling laws of bubble cavitation inception. model scale should not be so large that the nuclei become visible before the vapor pressure is To evaluate this problem consider the value reached. In that case gaseous cavitation will of ∆σ at full scale. There are generally ample be observed, and the inception pressure at full June 21, 2012, Inception of Bubble Cavitation25

cidentally and that the number density of the nuclei is not suﬃcient. The inception pressure is much lower than during the observation in Fig 4.5.

4.7 Acoustic Inception

Up to now e have focussed on visual detection of cavitation inception. As indicated in Fig. 4.3 the collapse of a cavitation bubble is not symmetrical with its inception. After becoming unstable the nucleus grows and Figure 4.5: Bubble cavitation due to a single reaches a maximum value, after which it nucleus collapses. This collapse generates noise, dom- inated by one or several pressure spikes. In the frequency domain this means a significant scale will not be predicted correctly. increase of the high frequencies. The spike or Note that for proper scaling of cavitation the increase in high frequency noise can be and its dynamics the cavitation index is still used to detect cavitation. In that case it is the proper index, because the pressure inside called acoustic inception. the cavitation is the vapor pressure. It is not correct to base the cavitation properties There may be differences between acoustic on σ/σi, as is sometimes done in literature. inception and visual inception. When the nu- When inception has occurred the inception clei are small and the low pressure peak is nar- pressure does no longer affect the cavity size row, the nuclei may become unstable but may or dynamics, except for the very short time remain invisible. In that case acoustic incep- the cavity grows from inception to its dynamic tion is detected first. In a cavitation tunnel equilibrium (see section 4.3). this is the dominant situation. However, from Fig. 4.2 it was observed that larger nuclei can At model scale the nuclei density of the grow to a visible size without becoming un- largest nuclei will be smaller than at full stable. In such a situation the bubble returns scale,which has an effect on the appearance of to its approximate original size without a vio- bubble cavitation. When very few nuclei are lent collapse and thus without being detected available which can reach the critical size, in- acoustically. Stricly speaking this is gaseous ception will occur only incidentally, depending cavitation. on the passage of an incidental nucleus. Such The response of a nucleus to a pressure a single nucleus will grow rapidly because it is change has also been investigated extensively unstable and the minimum pressure is far be- when the pressure change is periodical. The low the vapor pressure. Since no other nuclei pressure field is then defined by an acoustic are growing around it,its surrounding pressure pressure field: a mean pressure pmean, a will not be affected until it has grown to a sub- pressure amplitude pa and a frequency f. stantial size. An example is shown in Fig. 4.5. Vaporous bubble growth (inception) of nuclei may occur when the minimum pressure Such large incidental bubble cavities indi- pmean − pa is lower than the vapor pressure, cate that the critical radius is only reached in- even when the mean pressure is higher than 26 G.Kuiper, Cavitation in Ship Propulsion, June 21, 2012

the vapor pressure. However, when the noise or machinery noise considerably. A cri- minimum pressure is only slightly lower than terion for Navy ship is therefore the inception the vapor pressure the nuclei will respond by speed, which is the speed at which the first a periodical motion. In that case there is no cavitation occurs. Inception measurements on implosion and no acoustical cavitation will Navy ships are therefore always done acous- be detected. When the minimum pressure tically. The determination of the inception decreases below a certain limit value below speed of a Navy ship at full scale is still diffi- the vapor pressure the response of the nuclei cult and is often not very repeatable. It may changes and the nuclei will collapse period- also change considerably in time and certainly ically. From experiments and calculations with seastate, wind and fouling. At model it is found that the critical pressure from scale the determination of cavitation inception eq. 4.1 can also be used with good accuracy exhibits a lot of scatter due to flow parameters for inception of acoustic cavitation [[51],[14]]. such as nuclei, but also because visual inception is difficult to see. Acoustic inception is not well defined either, because cavitation inception starts with a single incidental implo- 4.8 Inception measure- sion and the frequency of the implosions in- ments creases gradually and a certain criterion has to be used. . The definition of cavitation inception as 4.8.2 Inception measurements given until now is from the instability of a sin- for comparative purposes gle bubble. Even then the distinction between gaseous cavitation and inception cannot be Cavitation inception is not very important for made without knowledge of the nuclei content commercial propellers, except in some specific in the flow. When a nuclei distribution is cases where a very low noise level is required involved, the nuclei density distribution is (research vessels), which does not allow any another factor, determining the frequency cavitation. The noise level generated by a of cavitation inception. Also the accuracy small amount of cavitation (incipient cavita- of observation (what is defined as a visible tion) is not high and in most cases can be ig- bubble) is a factor in cavitation inception. nored. Cavitation inception is still often mea- The determination of cavitation inception is sured at model scale, as a criterion for the op- therefore difficult and the accuracy of the erational conditions or for comparison between observed inception pressure of a propeller is different propellers. This is only because there not high. is no other more repeatable criterion available. It should be kept in mind, however, that cavitation inception curves are more inaccurate than is often suggested. Inception conditions 4.8.1 Inception speed on Navy will be discussed with the various types of cav- ships itation. The inception pressure is only important for Navy ships, where the acoustical detection of 4.9 The inception bucket the ship depends on the occurrence of the first small cavitation event. The noise level of such The inception properties of a propeller are an event can be such that it exceeds the flow expressed in an ”inception diagram”. This di- June 21, 2012, Inception of Bubble Cavitation27

Figure 4.6: Example of an inception diagram of a propeller agram gives the inception pressures at various loadings of the propeller. The loading is generally expressed in the non-dimensional thrust coeﬃcient Kt, or in the non-dimensional advance ratio J. An example of such a diagram is given in Fig. 4.6.

The corresponding curves of sheet cavitation have been marked red in Fig. 4.6. Conditions outside the V-shaped curves will have sheet cavitation. Conditions inside the V-shaped curve will be free of sheet cavitation. Because the shape of the curves generally is V- or U- shaped, such an inception curve is called an inception bucket. To be free of any type of cavitation the conditions must be inside the innermost bucket.