Appendix A

Air Content of Water

The amount of air dissolved in water α can with saturated water therefore contain 36 per-
be expressed in many ways. The most common cent oxygen. But nuclei which are generated
ways in literature are from the air above the water contain 21 per-
cent oxygen. Since the ratio between oxygen
• the gas fraction in weight ratio αw and nitrogen is not fixed, it is difficult to re-
late measurements of dissolved oxygen (by os-
• the gas fraction in volume ratio αv
mose) to measurements of dissolved air (from
• the molecule ratio e.g a van Slijke apparatus).
The amount of oxygen dissolved in water at
• the saturation rate
atmospheric pressure at 15 degrees Celcius is
• the partial pressure of air approximately 10 ∗ 10− 6kg/kg. For nitrogen
this value is about 15 ∗ 10− 6, so the solubility
of air in water is the sum of both: 25 ∗ 10− 6.
A.1 Solubility Here the dissolved gas contents are expressed
as a weigth ratio αw .Air is very light relative
Air is a mixture of 21 percent oxygen, 78 per- to water and the weight ratio is very small.
cent nitrogen and one percent of many other This ratio is therefore often expressed as parts
gases, which are often treated as nitrogen. The per million (in weight), which is 106 ∗ αw .
specific mass of gases involved in air are:

Oxygen (O2 ) 1.429 kg/m3

Nitrogen (N2 ) 1.2506 kg/m3
Air 1.292 kg/m3 A.2 The Gas Fraction in
Volume Ratio
The maximum amount of gas that can be
dissolved in water, the solubility), depends on The volume of gas dissolved per cubic meter
pressure and temperature. It decreases with of water depends on temperature and pres-
increasing temperature and increases with in- sure. Therefore this volume ratio is expressed
creasing pressure. The solubility of oxygen in in standard conditions of 0 degrees Celcius and
water is higher than the solubility of nitrogen. 1013 mbar (atmospheric conditions). The de-
Air dissolved in water contains approximately pendency of the volume of water on tempera-
36 percent oxygen compared to 21 percent in ture and pressure is neglected. The volume of
air.The remaining amount can be considered the dissolved air is then described by the law
as Nitrogen. Nuclei which are in equilibrium of Boyle-Gay-Lussac:

89
90 G.Kuiper, Cavitation in Ship Propulsion, June 21, 2012

be defined using the ratio of oxygen and nitro-

p ∗ V ol gen of 21/79 this virtual molar weight of air is
= constant (A.1) about 29.
273 + T
The volume fraction at (p,T) can be related
to the volume fraction in standard conditions: A.4 The saturation rate
273p The saturation rate is the amount of gas in so-
αv = αv (p, T ) (A.2) lution as a fraction of the maximum amount
(273 + T )1013
that can go in solution in the same conditions.
The gas fraction in volume ratio is dimen- Since the saturation rate is dimensionless. It is
sionless (m3 /m3 ). Be careful because some- independent of the way in which the dissolved
times this is violated by using cm3 /l (1000∗αv ) gas or the solubility is expressed. The satura-
or parts per million (ppm) which is 106 ∗ αv . tion rate is important because it determines if
αv is found from αw by: and in which direction diffusion will occur at
a free surface. The saturation rate varies with
ρwater temperature and pressure, mainly because the
αv = αw (A.3) solubility of gas changes with these parame-
ρair
ters.

in which ρ is the specific mass in kg/m3 . At

15 deg. Celcius and 1013 mbar pressure the
A.5 The partial pressure
specific mass of water ρw = 1000kg/m3 and Sometimes the amount of dissolved gas is ex-
the specific mass of air is 1.223kg/m3 , so for pressed as the partial pressure of the gas (mbar
air αv = 813αw . or even in mm HG). This is based on Henry’s
law, which states that the amount of gas dis-
A.3 The Gas Fraction in solved in a fluid is proportional to the partial
pressure of that gas. In a van Slijke appa-
Molecule ratio ratus a specific volume of water is taken and
subjected to repeated spraying in near vac-
The dissolved amount of gas can also be ex- uum conditions (a low pressure decreases the
pressed as the ratio in moles(Mol/Mol). Mo- solubility). This will result in collecting the
lar masses may be calculated from the atomic dissolved in a chamber of specific size. By
weight in combination with the molar mass measuring the pressure in that chamber the
constant (1 g/mol) so that the molar mass of a amount of dissolved gas is found. Note that
gas or fluid in grams is the same as the atomic this pressure is not directly the partial pres-
weight. sure. A calibration factor is required which
The molar ratio αm is easily found from the depends on the apparatus.
weight ratio by

M( water)
αw = αm (A.4)
M( gas)
in which M is the molar weight, which is 18
for water, 16 for oxygen(O2 ) and 28 for Nitro-
gen (N2 ). For air a virtual molar weigth can
Appendix B

Standard Cavitators

A standard cavitator is a reference body

which can be used to compare and calibrate
cavitation observations and measurements.
Its geometry has to be reproduced accurately
and therefore an axisymmetric headform has
been 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 is
a worldwide conference consisting of towing
tanks (and cavitation tunnels) which have the
goal of predicting the hydrodynamic behavior
of ships. To do that model tests and calcula- Figure B.1: Contour and Pressure Distribu-
tions are used. They meet every three years tion on the ITTC Headform [38]
to discuss the state of the art and to define
common problem areas which have to be re-
viewed by committees. The ITTC headform headform was used to compare inception
has a flat nose and an elliptical contour [27]. measurements in various cavitation tunnels.
Its characteristics are given in Fig B.1. However, it was realized later on that the
This headform has been used to compare boundary layer flow on both the ITTC and
cavitation inception conditions and cavitation on the hemisperical headform was not as
patterns in a range of test facilities. The simple as the geometry suggested. In most
results showed a wide range of inception cases the Reynolds numbers in the investi-
conditions and also a diversity of cavitation gations was such that the boundary layer
patterns in virtually the same condition, over the headform remained laminar and the
as illustrated in Fig B.3. This comparison pressure distribution was such that a laminar
lead to the investigation of viscous effects on separation bubble occurred, in the position
cavitation and cavitation inception. indicated in Fig. B.1. This caused viscous
effects on cavitation inception and made the
The simplest conceivable body to investi- headform less suitable as a standard body.
gate cavitation is the hemispherical headform. Note that the location of laminar separation is
This is an axisymmetric body with a hemis- independent of the Reynolds number. When
pere as the leading contour. Its minimum the Reynolds number becomes high transition
pressure coefficient is -0.74. The hemisperical to turbulence occurs upstream of the sep-

91
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]) and
this headform bears his name ever since.
The contour and pressure distribution on
the Schiebe headform are given in Fig. B.2.
This headform has no laminar separation and
transition to a turbulent boundary layer will
occur at a location which depends on the
Reynolds number.

Many other headform shapes have been

investigated with different minimum pres-
sure coefficients and pressure recovery
gradients.(e.g.[24])
 June 21, 2012, Standard Cavitators93

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

source:ITTC
94 G.Kuiper, Cavitation on Ship Propellers, June 21, 2012
Appendix C

Tables

T pv
Celcius N/m2
0 608.012
2 706.078
4 813.951
6 932
8 1069
10 1226
12 1402
14 1598
15 1706
16 1814
18 2059
20 2334
22 2638
24 2981
26 3364
28 3785
30 4236
32 4756
34 5315
36 5943
38 6619
40 7375

Table C.1: Vapor pressure of Water.

95
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 1.82844
1 1.72701 1.76915
2 1.67040 1.71306
3 1.61665 1.65988
4 1.56557 1.60940
5 1.51698 1.56142
6 1.47070 1.51584
7 1.42667 1.47242
8 1.38471 1.43102
9 1.34463 1.39152
10 1.30641 1.35383
11 1.26988 1.31773
12 1.23495 1.28324
13 1.20159 1.25028
14 1.16964 1.21862
15 1.13902 1.18831
16 1.10966 1.15916
17 1.08155 1.13125
18 1.05456 1.10438
19 1.02865 1.07854
20 1.00374 1.05372
21 0.97984 1.02981
22 0.95682 1.00678
23 0.93471 0.98457
24 0.91340 0.96315
25 0.89292 0.94252
26 0.87313 0.92255
27 0.85409 0.90331
28 0.83572 0.88470
29 0.81798 0.86671
30 0.80091 0.84931

Table C.2: Kinematic viscosities adopted by

the ITTC in 1963
 June 21, 2012, Tables 97

Rn Cf × 103
1 × 105 8.333
2 6.882
3 6.203
4 5.780
5 5.482
6 5.254
7 5.073
8 4.923
9 4.797
1 × 106 4.688
2 4.054
3 3.741
4 3.541
5 3.397
6 3.285
7 3.195
8 3.120
9 3.056
1 × 107 3.000
2 2.669
4 2.390
6 2.246
8 2.162
1 × 108 2.083
2 1.889
4 1.721
6 1.632
8 1.574
1 × 109 1.531
2 1.407
4 1.298
6 1.240
8 1.201
1 × 1010 1.17x

Table C.3: Friction coefficients according to

the ITTC57extrapolator.
98 G.Kuiper, Cavitation on Ship Propellers, June 21, 2012

Temp. density density

deg. C. fresh water salt water
kg/m3 kg/m3
0 999.8 1028.0
1 999.8 1027.9
2 999.9 1027.8
3 999.9 1027.8
4 999.9 1027.7
5 999.9 1027.6
6 999.9 1027.4
7 999.8 1027.3
8 999.8 1027.1
9 999.7 1027.0
10 999.6 1026.9
11 999.5 1026.7
12 999.4 1026.6
13 999.3 1026.3
14 999.1 1026.1
15 999.0 1025.9
16 998.9 1025.7
17 998.7 1025.4
18 998.5 1025.2
19 998.3 1025.0
20 998.1 1024.7
21 997.9 1024.4
22 997.7 1024.1
23 997.4 1023.8
24 997.2 1023.5
25 996.9 1023.2
26 996.7 1022.9
27 996.4 1022.6
28 996.2 1022.3
29 995.9 1022.0
30 995.6 1021.7

Table C.4: Densities as adopted by the ITTC

in 1963.
Appendix D

Nomenclature
kg
ρ density of water m3
See TableC.4
3
Cg gas concentration kg/m see Appendix A
Dg diffusion coefficient m2 /sec representative value 2 ∗ 109
D diameter m
Fd drag N
m
g acceleration due to gravity sec2
Taken as 9.81
−
Nd number density of nuclei m 4
pg gas pressure f racN m2
N
m2
pv equilibrium vapor pressure
R radius m
V.L
Rn Reynolds number ν in which V and L are a velocity and a length scale
kg
µ dynamic viscosity of water m∗sec
m2
ν kinematic viscosity of water sec
(ν = µρ )See Table C.2
s surface tension Nm for water 0.075

99