Types of cavitation: artificial roughness and inception of sheet cavitation

Chapter 8

Types of cavitation: ARTIFICIALROUGHNESS AND INCEPTION OFSHEET CAVITATION

Objective: Description of the applicationof leading edge roughness and its effects

The ultimate goal of cavitation investi-gations at model scale is the prediction offull scale cavitation behavior. That meansprediction of cavitation behavior at highReynolds numbers with ample nuclei. Theproblems with lack of nuclei and with laminarboundary layers are complications raised bymodel testing. There are two ways to dealwith these complications. The first way is tocontrol the complications and to ”correct”model scale results for the differences betweenmodel and full scale, such as the correctionof the viscous model drag to extrapolate shipresistance to full scale. The second way isto develop model scale techniques to avoidthese complications. For cavitation inceptionthis would mean supply of ample nuclei ofthe proper size and avoidance of laminarboundary layer flow.

8.1 Paint Tests

Avoidance of laminar flow has been the basisof the ITTC recommendation to use a mini-

mum Reynolds number of 2 ∗ 105 on propellersections. In many cases, depending on thepressure distribution, this minimum Reynoldsnumber does not avoid laminar flow. In uni-form flow this can be checked by a ”paint test”,in which a thick layer of ”paint” is applied atthe leading edge of a propeller model (Fig. 8.1)

The ”paint” is not real paint in that it issupposed not to dry rapidly. The propelleris then mounted in a tunnel or a towing tankand run in a certain condition for a short time.The time should be enough for the paint to bespread out by the flow over the blade, as inFig 8.2. In this picture the paint was mixedwith fluorescent dye, so that the very thinpaint layer was better visualized when illumi-nated with ultra-violet light.

The pattern in this figures is caused bytwo forces on the paint particles. The firstforce is the friction force between the paintand the water, which is directed in chordwizedirection. The paint is moving very slowlyrelative to the propeller, and is thereforesubject to a centrifugal force , which isdirected in outward radial direction. Theresult is that the paint will move outward. Ina region with laminar boundary layer flow thefriction force is smaller than in a region withturbulent boundary layer. This results in achange in direction of the paint streaks: in

57

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

Figure 8.1: Application of paint

regions with a turbulent boundary layer thepaint streaks will be close to the tangentialdirection. In regions with laminar boundarylayer flow the streaks will be pointed outward.And in regions with separated flow there willbe no paint at all or the paint streaks willbe directed radially outwards. This makes itpossible to distinguish various boundary layerregimes on a propeller blade.

The pattern shows an abrupt change in thedirection of the paint streaks at a certain ra-dius. Outside that radius the paint streaks arein chordwise direction, which indicates a tur-bulent boundary layer flow from the leadingedge on. This is caused by a laminar sepa-ration bubble at the leading edge. A laminarseparation bubble occurs when the boundarylayer separates near the leading edge due to a

Figure 8.2: Paint test on propeller of Fig. 7.4

strong low pressure peak there. At separationthe boundary layer is still laminar. However,the separated laminar boundary layer is unsta-ble and becomes turbulent soon after separa-tion. The turbulent boundary layer causes theturbulent boundary layer to reattach to thesurface and the boundary layer is thus turbu-lent from there on. A sketch of a short laminarseparation bubble is given in Fig. 8.3.

Paint tests revealed that for most propellersthe criterion as used by the

When the pressure peak becomes strongand the adverse pressure gradient downstreamof the minimum pressure becomes very ad-verse the separation bubble can become longand reattachment takes place in the midchordregion of the section. If no reattachment takesplace the profile is stalled.In some cases the laminar separation bubble

January 15, 2010, Leading Edge Roughness59

Figure 8.3: A short laminar separation bubble

is revealed by the paint test by a thin line ofpaint at the leading edge. Mostly the lami-nar separation bubble is too short to be visible.

Laminar separation is independent of theReynolds number. A confirmation of thefact that laminar separation is involved isthat the radius at which the paint linessuddenly become tangential is independentof variations in the Reynolds number. Onthe other hand this radius is very sensitiveto variations in propeller loading. Thisradius could be predicted by two-dimensionalboundary layer calculations on propellerblade sections ([33]). So the role of laminarseparation has been confirmed in various ways.

Fig 8.2 shows that inside of the radius oflaminar separation there is a significant areaof laminar flow. Transition to turbulenceis indicated by a rather corner where theoutward directed paint streaks change intotangential lines. In Fig. 8.2 the boundarylayer in the low pressure region near theleading edge at inner radii is fully laminar.At the root the flow is close to separation.The friction force becomes very low there,resulting in a collection of paint in that region.The direction of the paint streaks is not yetfully radial, indicating that no separation

Figure 8.4: Paint pattern on the pressure sideof a conventional propeller

occured yet. This is confirmed by the factthat there are paint streaks downstream ofthat region with thick paint.

The transition is not always as sharp as inthe previous pictures. Especially at the pres-sure side the direction of the paint streakschanges more gradually. So the transition re-gion is much longer in that case (Fig 8.4).

An increase of the Reynolds number inpropeller testing is not very effective. Thelocation of transition depends more on thepressure distribution than on the Reynoldsnumber. When the local pressure gradient isfavorable, it is very difficult to generate tran-sition. A paint test at a very high Reynolds


number is shown in Fig shown in Fig. 7.5.The result is not that the transition locationhas moved much towards the leading edge.However, turbulent streaks occur instead inthe region of the laminar boundary layer.This is due to the fact that the high Reynoldsnumber makes the boundary layer thinner andthus more susceptible to surface irregularities.The turbulent streaks originate at irregularspots on the surface of the propeller. This canalso be simulated with larger irregularities onthe surface and application of leading edgeroughness is therefore a logical simulation ofa high Reynolds number.

8.2 Turbulence Stimula-

tors

Artificial stimulation of turbulence is a wellknown way of creating a turbulent boundarylayer. It is applied in airplanes to preventseparation, generally by small triangles at anangle with the flow. A turbulent boundarylayer is also stimulated at the bow of shipmodels, where trip wires or studs (cylindersvertical to the surface) are used. To under-stand the effect of turbulence stimulators themechanism of transition has to be understoodin more detail, although it should be kept inmind that this description is very simplifiedand many details are still unknown.

Transition to turbulence occurs in thelaminar boundary layer after at some pointthe boundary layer becomes unstable. Thisinstability means that small disturbancesin the velocity (e.g. due to turbulence) aremagnified. After some distance the magni-fication is so large that the boundary layerbecomes turbulent. The region between thelocation where the boundary layer becomesunstable and the location of fully developedturbulence is the transition region. In the

transition region only a specific bandwidthof disturbances is amplified. Al disturbancesoutside this bandwidth are damped out andwill disappear. As a result the transitionregion contains waves of velocity fluctuations,the so called ”Tollmien-Schlichting” waves.The magnification of disturbances in a lam-inar boundary layer is very local, leading toso-called turbulent bursts in the boundarylayer.

At a small distance downstream of a singleroughness element the shape of the velocitiesin the boundary layer is similar to the bound-ary layer without roughness. But the fluctu-ations of the velocity are increased. So themechanism of turbulence stimulation seems tobe through the enhanced instability [25].

The effects of roughness on flat foils hasbeen studied extensively, but mostly with thefocus on maximum lift or stall. However, incase of cavitation inception there is an impor-tant additional requirement: the local meanpressure should not be affected. It means thestimulator should be small. Investigations ofa trip wire [15], [2] were not very satisfactory,because the trip wire was causing a separatedregion which was Reynolds dependent and de-pendent on the size of the trip wire. So the in-ception conditions were too dependent on thelocation and the size of the trip wire. It wasalready found by Klebanov et al [25] that threedimensional roughness was less dependent onthe Reynolds number and caused transitiondepending on the roughness Reynolds number

Reθ =vkk

ν(8.1)

where vk is the velocity at the top of the rough-ness in the absence of roughness and k is theheight of the roughness. The value of theroughness Reynolds number causing transitionto turbulence was between 500 and 750. Thesenumbers were obtained on a flat plate, so with-out pressure gradient.

Distributed carborundum particles proved


Figure 8.5: Leading edge roughness on a pro-peller model

to be more successful in cavitation inceptiontests in the Depressurized Towing Tank [33],[34] and this technique will be discussed inmore detail. In case of leading edge roughnessturbulence stimulation is done by distributedroughness particles which are glued to thesurface over a certain length near the leadingedge of a foil.

8.3 Application of leading

edge roughness

The first requirement in the application of tur-bulence stimulator is that the mean pressureover the blade section is not affected. This isa relative statement, because the difference inReynolds number between model and full scalealso causes differences in the mean pressure,as does the occurrence of a laminar separationbubble. In fact a laminar separation bubblecan also be considered as a turbulence stimu-lator.It means that the size of the roughness hasto be small. There is insufficient knowledgeabout the precise mechanism of roughness ele-ments in a boundary layer, but an experimen-

tally determined criterion is that the rough-ness Reynolds number should exceed 120 to beeffective [10]. The roughness Reynolds numberis defined as

V ∗ kν

(8.2)

in which V is the fluid velocity outside theboundary layer at the location of the rough-ness and k is the height of the roughnesselements. At a typical velocity of 10 m/secthis leads to a minimum roughness heightof 12 microns. For roughness to be effectiveLigtelijn [37]found a minimum roughnessReynolds number of 300. In practice the sizeof the roughness elements used in a cavitationtunnel (with a high Reynold number) theroughness size can be taken as 30 microns. Ina Depressurized Towing Tank the roughnesssize is taken 60 microns.

The carborundum size and the roughnessheight are not necessarily the same. Rough-ness particles have to be attached to the foilsurface and this is generally done by someglue. It is important that the glue is notsmoothing the area between the roughnesselements. The best way to avoid that isto take watery thin glue and to keep theroughness particles at some distance of eachother. As a rule of thumb the coverage of theroughness elements should not exceed 50 per-cent of the area. An example of leading edgeroughness properly applied is given in Fig. 8.6.

Application of small roughness elementsis often difficult, because the particles tendto clog together. A way to apply distributedroughness elements is to blow air with parti-cles over an area with freshly applied glue. Toavoid a disturbance in the mean pressure theglue should be very thin.

As roughness elements carborundum parti-cles are often used. These particles, also used


Figure 8.6: Distribution of roughness particlesin leading edge roughness

in grinding paper, have sharp contours, whichmake them effective for cavitation inception.An alternative is spherical balls, but theseare less effective in generating cavitationinception. Other turbulence stimulators likezigzag paper have the same drawback as atrip wire: a rather sharp beginning and end,causing local separation and a disturbance inthe mean pressure.

The length of application of leading edgeroughness depends on the pressure distribu-tion. In general it can be stated that it is verydifficult to cause transition in a strongly fa-vorable pressure gradient, as occurs upstreamof a minimum pressure location. So the ex-tent of the roughness should cover the mini-mum pressure location, in order to generatetransition shortly downstream of this location.On ship propeller models the location of theminimum pressure is very close to the lead-ing edge. In that case coverage of the leadingedge roughness over some 2% of the chord isenough. A further extension of the coverage isundesirable because of the increasing effect ofthe roughness on sectional drag, but at innerradii the leading edge is thicker and the min-imum pressure further away from the leadingedge. There the extent of the roughness hasto be larger.

8.4 Cavitation inception

on roughness elements

Cavitation inception on roughness elements ina low pressure peak occurs typically by smallspots of cavitation, as shown in Fig. 8.7 (A foilat 5 degrees angle of attack).In this figure theangle of attack is not high and the pressure atthe leading edge is low over the whole rough-ened region. The small spots are evidentlycaused by the roughness elements. Their sizeexceeds already the size of the roughness ele-ments, however, which means that the meanpressure outside the boundary layer is closeto the vapor pressure. When the angle of at-tack is higher and the low pressure peak isclose to the leading edge, the roughened re-gion is longer than the low pressure peak. Thiswill result in the inception of cavitation spotsat the leading edge, as shown in Fig. 8.8(thesame foil at 8 degrees angle of attack). Whenthe pressure in Fig. 8.8 was increased gradu-ally the spots would become smaller, but longbefore the length approached zero they wouldsuddenly disappear. From that we may con-clude that local separation at a roughness el-ement induces inception. The inception con-dition will be when the reattachment pressurereaches the vapor pressure. The mean pressureat the inception location outside the boundarylayer will then also be close to the vapor pres-sure, when the carborundum grains are evenlydistributed. It explains why sharp carborun-dum particles are more suitable for stimulationof cavitation inception, because the flow sepa-rates more readily.The problem remains when to call inception.

Looking very closely to the first minute cavi-tation spots on the roughness elements makesthat the inception pressure depends too muchon the local roughness element. Such spotsmay occur locally up to high pressures. How-ever, in such cases the spots are not very sensi-tive to pressure variations. They persist up tohigher pressures. When the pressure is lowered


Figure 8.7: Cavitation inception on roughnesselements

Figure 8.8: Cavitation inception on roughnesselements

the inception pressure is therefore called whenthe spots begin to grow significantly with de-creasing pressure. Fig. 8.8 is a situation whichis close to the inception pressure. This pictureillustrates that it is not useful to detect cav-itation inception on the first tiny spot whichoccurs, because this will probably depend onlyon the size of the local roughness element.It will be clear that acoustic inception onroughness elements can be considerable earlierthan visual inception.

8.5 Effects of leading edge

roughness on propeller

performance

The main effect on propulsion of a laminarboundary layer on propeller sections will bea reduction of the sectional drag. This willreduce both torque and thrust, but especiallythe thrust. The effect will be a slight increaseof the efficiency of the propeller. When leadingedge roughness is applied two effects are intro-duced: -restoration of the turbulent boundarylayer, resulting in a restoration of the turbu-lent drag of the blad sections -additional dragof the roughness elements. The last effect can-not be distinguished from the first effect. How-ever, since the area of roughness is only verysmall the effect is expected to be small.

This turns out to be a simplification ofthe real flow around blade sections at modelscale. Measurements of propeller thrust andtorque often show that the torque increaseswith roughness and the efficiency drops witha few percent. However,the viscous effects ontorque and thrust of a propeller model maybe more complicated. Sometimes the sectionalthrust is affected by the laminar flow, leadingto a change in thrust. This may be in bothdirections, and the relation with the pressuredistribution is not yet fully clear.

Sometimes the thrust increases with the ap-plication of roughness. This is especially thecase when the blade loading is at the trailingedge, so when there is a strong pressure gradi-ent at the suction side near the trailing edge.In such a case the flow an the smooth bladeseparates earlier than on the roughened blade,resulting in an increase in sectional lift coef-ficient when the boundary layer is turbulent.Then the efficiency increases slightly with theapplication of roughness. For the extrapola-tion to full scale is can be expected that theroughened performance is representative forfull scale, eventually after some correction forReynolds number effects has been applied to


the sectional drag (see ITTC 1978, [21]).

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Appendix A

Air Content of Water

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

• the gas fraction in weight ratio αw

• the gas fraction in volume ratio αv

• the molecule ratio

• the saturation rate

• the partial pressure of air

A.1 Solubility

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

Oxygen (O2) 1.429 kg/m3

Nitrogen (N2) 1.2506 kg/m3

Air 1.292 kg/m3

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

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

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

A.2 The Gas Fraction in

Volume Ratio

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

69


p ∗ V ol273 + T

= constant (A.1)

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

αv = αv(p, T )273p

(273 + T )1013(A.2)

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

αv =ρwaterρair

αw (A.3)

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

A.3 The Gas Fraction in

Molecule ratio

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

αw = αmM(water)

M(gas)(A.4)

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

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

A.4 The saturation rate

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

A.5 The partial pressure

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

Appendix B

Standard Cavitators

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

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

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

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

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

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

71


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

aration location and separation will disappear.

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

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

January 15, 2010, Standard Cavitators73

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

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

Appendix C

Tables

T pvCelcius N/m2

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

Table C.1: Vapor pressure of Water.

75


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

m2/sec× 106 m2/sec× 106

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

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

January 15, 2010, Tables 77

Rn Cf × 103

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

Table C.3: Friction coefficients according tothe ITTC57extrapolator.


Temp. density densitydeg. C. fresh water salt water

kg/m3 kg/m3

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

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

Appendix D

Nomenclature

ρ density of water kgm3 See TableC.4

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

D diameter mFd drag Ng acceleration due to gravity m

sec2Taken as 9.81

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

fracNm2

pv equilibrium vapor pressureR radius m

µ dynamic viscosity of water kgm∗sec

ν kinematic viscosity of water m2

sec(ν = µ

ρ)See Table C.2

s surface tension Nm for water 0.075

79

Documents

Types of cavitation: artificial roughness and inception of sheet cavitation