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:

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

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 [27].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 [38]

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-

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

Figure B.2: Contour and Pressure Distribu-tion of the Schiebe body [38]

aration location and separation will disappear.

To avoid laminar separation another head-form was developed by Schiebe ([56]) 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.[24])

June 21, 2012, Standard Cavitators93

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

94 G.Kuiper, Cavitation on Ship Propellers, June 21, 2012

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.

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96 G.Kuiper, Cavitation on Ship Propellers, June 21, 2012

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

June 21, 2012, Tables 97

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.

98 G.Kuiper, Cavitation on Ship Propellers, June 21, 2012

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

Nm2

pv equilibrium vapor pressureR radius mRn Reynolds number V.L

νin which V and L are a velocity and a length scale

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

99