188

4 CAVITATION AND PRESSURE IMPULSES INDUCED BY

THE PROPELLER** The chief aim in this section is:

• to show how the cavities on a propeller blade working in a wake changes volume during one revolution

• to show how the cavity thereby induces pressure impulses on the surface of the

hull and in the surrounding fluid • to show that cavitation is the most efficient noise generator on board the ship

and that it is possible to study this "generator" in a cavitation tunnel before the ship is build

At the different radii in the propeller plane the inflow velocity to the propeller will vary with the angular position of the blade. The velocity is at a minimum when the blade is in the upper position as indicated in Fig.4.3 and Fig.4.4. If the propeller blade "sees" different velocities, the angle of attack during one revolution will vary as sketched in Fig.4.3. When it moves towards the top position, the angle increases, CL increases, the pressure on the suction side decreases and the cavity finally starts to grow. Fig.4.3 illustrates how the thickness of the cavity increases with increasing angle for a constant external pressure. The volume of the cavity is time dependent and its growth a source of cavitation noise and pressure impulses. The pressure impulses cause vibrations in larger parts of the ship and give cracks in the plating. The following contribute essentially in generation of pressure impulses and noise.

1. Motion and thickness of the blade

2. Motion and thickness of the cavity

3. Increase in cavity volume with time

4. Volume and volume variation with time of the tip vortex

5. Lift of the blade

189

Fig. 4.1 Wake distributions for single-screw ships without working propeller

190

Fig. 4.2 Different types of frames in the afterbody

191

Fig. 4.3 Blade position, wake field and cavitation volume

192

Fig. 4.4 Growth of cavity volume due to ship wake field

193

Fig. 4.5 Predicted full scale wake distribution

194

Fig. 4.6 Cavitation tunnel aft body dummy model

195

Fig. 4.7 Axial wake circumferential distribution

196

Fig. 4.8 Cavitation Sketch, 15 degrees blade position

197

Fig. 4.9 Cavitation sketch, 30 degrees blade position

198

Fig. 4.10 Cavitation sketches, 45 degrees blade position

199

Fig. 4.11 Cavitation sketches, 180 degrees blade position

200

Fig. 4.12 Cavitation sketches, 345 degrees blade position

201

Fig. 4.13 Results from pressure measurements

202

Fig. 4.14 Examples of distortions in propeller cavitation

203

Fig. 4.15 Different types of cavitation developments

204

Fig. 4.16

Fig. 4.17 The pressure signal p as a function of time t at the collapse of a sheet caviy on an oscillting foil

205

Fig. 4.18 Schematic behaviour of a pulsating cavity and the radiated pressure

206

Fig. 4.19 Typical signal p(t) from a cavitating propeller. a) Schematic. b) Example from model test

207

Fig. 4.20 Nuclei content in sea water (from Issay)

208

Fig. 4.21 Non-dimensional noise level

209

Fig. 4.22 Pressure measurement results and conclusions

210

The pressure at an occasional point on the surface of the hull or in the water induced by the propeller is according to Bernoulli’s equation:

⎟⎠⎞

⎜⎝⎛

∂∂

V 21 -

t = p 2

Riφρ

where

t = time

VR = resulting velocity

φ = velocity potential in the point due to different contributions as mentioned above.

The thickness is simulated with sinks and sources on the surface of the propeller blade. This is illustrated by the following expression for the induced velocity Vn normal to the cavity on the blade:

n nn R

(r, ,t) (r, ,t) = + V V s t

τ τϕ ϕ∂ ∂∂ ∂

where

τ = cavity thickness. The first part is identical with an expression already known from the "frozen" condition. The second link is an expression for the dynamics in the process. Returning to the expression for the induced pressure a simplification is desirable: The resulting velocity is then expressed as:

2 2 22

R x y zV = (V + U +U +U) where

Ux , Uy , Uz = propeller induced velocity respectively in the x, y and z-directions.

The induced velocities are small relative to V, which is the main velocity. If all U2 links are equal to 0:

2 2 2RV V V Ux= + ⋅ ⋅ If this is used in the expression for pi:

⎟⎠⎞

⎜⎝⎛

∂∂

∂∂

x V -

t = pi

φφρ

211

neglecting 1 / 2 V2 which is constant. Only the variable pressure is of interest, which leads to the following expression for the propeller induced pressure:

) + + + + ( x

V - = p 54321n

i φφφφφϕ

ωρ ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

⋅∂

∂

With reference to the different contributions mentioned earlier.

ϕn = angular position for blade number n The problem is now to determine the source/sink distribution and its dependence of the blade position. Many attempts have been made in order to estimate pi for a given propeller and a given wake distribution. But since the theory still has its shortcomings, model tests and full scale measurements are required to determine the relation between pi and ϕ n. On Fig.4.5 – Fig.4.13 results from such measurements applying pressure cells on a model of an after body in the cavitation tunnel is shown. To keep control with cavitation and pressure impulses is one of the chief tasks in the design of a propeller. Two parameters are of special interest in reduction of pressure impulses. These are:

1. Propeller geometry

2. Wakefield or wake distribution

Different after body shapes (Fig.4.2) give different wake fields. It is not necessarily the absolute mean value of the wake w (r,φ ) that is important. It follows from the expression for Vn that the derivative of the cavity thickness with time is more important. This indicates that strong gradients in w (r, φn) may give large pressure impulses. Closely related to the development of pressure impulses is the generation of noise. Pressure impulses are low frequent while propeller induced noise due to oscillations from single bubbles and clusters of single bubbles covering the entire frequency range. If a cavity on a blade, which passes through a wake peak, is studied different types of cavitation will be observed. The surface may be covered by a silvery area or bubble that passes into a cloud looking sheet or into clusters of bubbles. The cavities may as well roll up in sausage like vortices. This is illustrated in Fig.4.14. In Fig.4.15 the dynamics in the development of cavitation sketched for different alternatives. Common for all cavities is that they in reality are collections of tiny bubbles. One thing is that a cluster of bubbles can increase or decrease in volume. Further each bubble as well may increase and decrease in volume. When a bubble or a cluster of bubbles implodes, the local pressure may increase violently. This sends shockwaves towards the surface of the blade and the pressure is so high that blade material is hammered out and finally eroded. This process is accompanied by noise.

212

A different mechanism is that the cluster of bubbles divides in two parts and induces a jet hitting the surface with very high velocity. Both these processes are illustrated in Fig.4.16. In Fig.4.17 it is illustrated how the pressure signal is when the bubble implodes on the surface of the foil in time dependent flow. It is observed that large pressure oscillations may spring up and last only a few milliseconds. Within a given volume of seawater there are an infinite number of small bubbles filled with gas. Each of them has a radius R(t) that varies with the time t. Due to the variation in volume there is a variation in pressure induced by the bubble at a distance r from the bubble.

r

)R R + R(2R = r 4

Q = t)p(r,2

2.⋅ρ

πρ (Fitzpatrick)

where

R = radius of the bubble

R = velocity of the surface of the bubble

R = acceleration of the bubble surface

Q = volume of the bubble r = distance from the bubble to a given point

This is in linear acoustics and in incompressible flow the classic approach. In Fig.4.18 the behavior of a pulsating cavity that follow this law been studied over time. It is observed how sharp tops originate and how the bubble collapses and arises again. How clusters of such bubbles grow and break off from the main cavity when the propeller blade passes through the wake field is shown on the figures 4.14 and 4.15. When a cluster of such bubbles are oscillating with different phases, violent pressure peaks as already mentioned arise with noise and erosion as a consequence. This is illustrated in Fig.4.18 and Fig.4.19. The following expression for the radius of the cavity is often applied:

(Plesset) )C + (21- = R

23 + R R prel

2relrel σ⋅

where

Rrel = RR

0

, (R0 = Rmax)

σ = cavitation number

2

0p

2

p(x,t) - pC V

ρ= = local pressure coefficient

213

2

vp

2

p(x,t) - pC V

σ ρ+ = , expression for the driving pressure

or difference in pressure between the outside of the bubble (profile) and the vapor pressure pv.

214

Scaling of Measured Noise due to Cavitation The chief intention in this section is • to show how propeller noise is measured in a cavitation tunnel and later is scaled to

full scale noise In a cavitation tunnel or in full scale, it is possible to measure the sound pressure. If the mean pressure is taken over the bandwidth a ⋅ f as in Fig.4.19b, the sound pressure on the ship model is expressed as:

)(r/DV )f a ,f(p

m

2mm

mmm

⋅=⋅

ρ

Where:

Vm = characteristic velocity of the model

r = distance from the noise source to a given point

D = characteristic dimension like the propeller diameter The measured sound pressure in model pm is scaled to the following sound pressure in full-scale:

)( )VV(

DD

rr f) a,f(p = p

m

s2

m

s

m

s

s

mmms ρ

ρ⋅⋅⋅⋅⋅

If a bubble is growing due to the driving pressure ∆p, the growth is described by:

t )p(32 R(t) 0.5 ⋅

∆=

ρ

where t is the growing time. It has a collapse time given by

)p

( D k = T 0.5 c ∆

ρ

pm ( fm )

fm ∆ f = a f

f

215

The frequency in model and in full-scale is:

2

2

2ss s

m ms m

2ms sm s m m

Vf T D = = f T D V

ρσρ

ρρ σ⋅

nn 1 =

ff

m

s

m

s

λ

where

λ = scale

n = number of revolutions

A requirement for the scaling made above is that nDV = J is equal for model and full scale.

It is also assumed that: σm=σS It is now possible to scale both sound pressure and corresponding frequency: The sound pressure is normally presented as "root mean square"-values (r-m-s-value) Dimensionless the pressure is expressed as:

2 2rms

pp = K n Dρ ⋅ ⋅

LP is the basic measure for sound pressure and is defined as follows:

logp0

pL 20 dBp

⎛ ⎞= ⋅ ⎜ ⎟

⎝ ⎠

where p is the pressure at a given point and p0 is the reference pressure. When two levels are compared this is done in the following way:

12 1

2

logp ppL = L L = 20 p

⎛ ⎞∆ − ⋅ ⎜ ⎟

⎝ ⎠

In Fig.4.21 we have shown results from measurements of noise from a cavitating propeller. The propeller was working in a cavitation tunnel behind a model of the after body. Frequency and pressure is scaled to full scale and the result compared with full-scale measurements. The measurements are made with a certain content of air dissolved in the water. In the actual case, the air contents is 40 % . This means that:

400 = s

.αα

216

where

α = air contents

αs = air contents when the water is saturated with air. It is important to keep the air contents and the nuclei content at a constant level because dissolved air and gas are the origins of the cavitation bubbles. See Fig.4.20.

If the water is “dead” or s

αα

low, it will be difficult to generate cavitation at all. The higher

s

αα

the earlier the cavitation starts.

The air contents and density of nuclei should therefore be equal in model and full scale. The distribution of bubble sizes should also be identical. Some laboratories do only control the air contents. It is clear from Fig.4.21 that the ratio between velocity and number of revolutions is approximately constant.

V( ms

) 4.5 6.0 7.5

n( rs

) 25.7 35.4 46.2

Vn

0.1751 0.1695 0.1623

This means that the most important model law is fulfilled. Because we have the same J value in all cases, KT and CL (r) are equal for all numbers of revolution. Consequently, extent of cavitation and noise should be equal, which the scaled sound pressures also indicate. The sound pressure is made dimensionless by dividing with ρ n2 D2

217

5 LITERATURE Bark, G. ”On the Mechanisms of Propeller Cavitation Noise”, Division of Mechanics, Charmer’s University of Technology, S - 41296 Gothenburg 1985. Breslin, J.P and Andersen, P,”Hydrodynamics of Ship Propellers” Cambridge University Press 1994. Carlton, J. S "Marine Propellers & Propulsion", Bitterroot-Heinemann 1994. Dickmann H.E, Weissinger, J "Beitrag zur Theorie Optimaler Düsenschrauben Kortdüsen" Jahrbuch der STG bd 49,1955 Dyne, G "A Method for the Design of Ducted Propellers in a Uniform Flow", Meddelanden från STATENS SKEPPSPROVNINGSANSTALT Nr 62.1967 Dyne, G "An Experimental Verification of a Design Method for Ducted Propellers", Meddelanden från STATENS SKEPPSPROVNINGSANSTALT Nr 63. 1968 English, J.W Rowe S.J” Some Aspects of Ducted Propeller Propulsion" Symposium of Ducted Propellers", The Royal Institution of Naval Architects London Summer 1973. Gibson, I S,"Theory and Numerical Analysis of Single and Multi-Element Nozzle Propellers" Report No.LR-579 TU Delft, February 1989. Isay W. H, ”Kavitation” Schiffahrts - Verlag Hansa 1989. Küchemann, D.Weber, J "Aerodynamics of Propulsion", Publications in Aeronautical Science, Mc Graw - Hill Publishing Company LTD LONDON Masilge, Ch.1991,"Konzeption und Analyse eines integrierten Strahlbetriebes mit einem rotationssymmetrischen Grenzschichteinlauf", Doctor Thesis, Technische Universität Berlin, Germany, D83 Minsaas, K.J, 1993,"Flow Studies with a Pitot Inlet in a Cavitation Tunnel", Contribution to the 20 The ITTC Workshop on Water jets. San Francisco, U.S.A, Sept.20. Minsaas, K. J and Lehn, E "Hydrodynamic Characteristics of Rotatable Thrusters". NSFI Report R-69.78, 1978. Minsaas, K.J, Jacobsen G.M and Okamoto H “The Design of Large Ducted Propellers for Optimum Efficiency and Maneuverability". Symposium of Ducted Propellers, The Royal Institution of Naval Architects London Summer 1973. Steen and Minsaas, Fast 95, Lübeck - Travemünde 1995. Svensson, R, 1994,"Waterjet Propulsion- Experience from High Powered Installations" International Symposium on Water jet Propulsion - Latest Developments RINA, London, U.K.No.3

218

van Manen, J.D " Recent Research on Propellers in Nozzles" Journal of Ship Research, vol.1, no.2, 13-46. 1957. Terwisga, T.van1993"A Theoretical Model for the Powering Characteristics of Water jet-Hull Systems" Proceedings of FAST 93" S.N.A.J. ITTC 1996 Proceedings of the 21st International Towing Tank Conference Trondheim, Norway September 15-21, 1996.

219

6 SYMBOLS General

V = ship speed UA = axial induced velocity in the propeller jet A0 = disk area or cross section area of the propeller jet Ap = propeller disk area or projected blade area

D = propeller diameter

L = length of the duct

Ducted Propellers

2 21 2 p∆p V Vρ ⎡ ⎤= ⋅ −⎣ ⎦ = pressure drop in front of the propeller disk

( )2 22 2 A pp V U Vρ ⎡ ⎤∆ = ⋅ + −⎣ ⎦ = pressure jump behind the propeller disk

( ) AAppp UUVAppAT ⋅⎟⎠⎞

⎜⎝⎛ +=∆+∆=

21

21 ρ = propeller thrust

δ++= Ap UVV21 = ( )0

Ap

A V UA

+ = total velocity through the propeller disk

15.000 −⎟⎟⎠

⎞⎜⎜⎝

⎛−⋅+=

p

A

p AA

VU

AA

Vδ = velocity through the propeller disk induced

by the duct

AA

p UUVAT ⎟⎠⎞

⎜⎝⎛ ++= δρ

20 = total thrust of the ducted propeller

( )2 2

2 2A

p AUP A V V U Vρ δ⎡ ⎤ ⎡ ⎤= + + + −⎢ ⎥ ⎣ ⎦⎣ ⎦

= change in energy flux

220

02

2 A

VT UPV

η = ⋅ =+

= ideal efficiency

2

2

pTp

p

TC

V Aρ=⋅ ⋅

= thrust loading of the propeller

11 −+= TpA C

VU = induced axial velocity in the propeller jet

0

21 1 TC

ητ

=+ + ⋅

= ideal efficiency of ducted propeller

VVU

VU

TT

A

A

p

δτ

++

+==

21

21

0

= ideal ratio between propeller thrust and total thrust

Tp

Tp

C

C

+

++=

12

11τ = ideal ratio between propeller thrust and total thrust

CD f = resistance coefficient of the duct based on wetted surface

AUV21

+

Ducted Propellers according to Van Manen

( ) 2.51

p

S

RPM HPBV w

⋅=

⋅ −⎡ ⎤⎣ ⎦

( )1S

D RPMV w

δ ⋅=

⋅ −

RPM = number of revolutions pr minute

221

sV = ship speed in knots

w = wake

D = propeller diameter in feet

HP = delivered power in horsepower

0

202

vp p gh

V

ρσ ρ− +

= =cavitation number

0p = atmospheric pressure

vp = pressure of saturated vapor

h = distance to the surface

( )0 1SV V w= ⋅ −

w = wake

sV = ship speed in m/s

Tunnel Thrusters

3

2 p aP A Uρ= ⋅ ⋅ = power delivered to the water in the jet

2

0 p aT A Uρ= ⋅ ⋅ = ideal thrust

31

0 2⎥⎥⎦

⎤

⎢⎢⎣

⎡==

ppa A

PA

TU

ρρ= ideal induced velocity in the jet

222

( )01

2 2 3D

T N

D Pη =

⎡ ⎤⋅⎣ ⎦

= factor for thrust gain

rotζ = pressure losses due to rotational energy in the jet

dragζ = pressure losses due to viscous flow on the blades

( )1

2prot drag

ητ ζ ζ

=⋅ + +

Cavitation

0

2 2 2

2

vp p g h

n D

ρσ ρ π

− + ⋅ ⋅= = cavitation number

Water jet

Qj = flowrate

Aj = outlet area

0V = velocity of the craft

Mj =ρ Qj Vj = momentum flux at station j (N)

2( )2j

j

QEA

ρ= =energy flux (W)

1V = inlet velocity

( )0 0j j jM M M Q V Vρ∆ = − = ⋅ ⋅ − = total change in momentum flux (N)

223

Suction Force and Resistance of the Inlet Lip for an Immersed Inlet

CDpr = 21

0

(1 )VV

− = resistance of a sharp edged inlet

( )2

2 10 1

0

12S n N

VF p p dA V AV

ρ ⎡ ⎤= − = − ⋅ ⋅ ⋅ −⎢ ⎥

⎣ ⎦∫ = ideal suction force

21

2 00 1

(1 )

2

SDS

F VCVV Aρ= = − −

Flush Inlet

( ) ( ) ( )1 0x p fru z V u z u z= + ∆ + ∆ = velocity near the inlet

)(zup∆ = change in velocity due to potential flow

)(zu fr∆ = change in velocity due to frictional flow along the hull

1 10

( ) ( )z

xQ u z b z dz= ⋅ ⋅∫ = volume flow rate at z in the boundary layer

Q bal = volume flow rate inside boundary layer.

Qj = volume flow rate of the jet

)(1 zVE =local energy velocity at station j:

( ) ( ) ( )2 21 1 1 02 2E xV z u z p pρ ρ

⋅ = ⋅ + −

1p = local static pressure at station 1.

0p = ambient pressure in undisturbed flow.

)(1 zu x = measured velocity near the inlet

δm = 0.37· x·RN -1/5 (Prandtl) = boundary layer thickness of the model

170.16s Nx Rδ −= ⋅ ⋅ (Wieghardt) = boundary layer thickness of the ship

224

x = flow length RN = Reynolds number based on flow length in front of the inlet.

Momentum Flux and Change of Momentum Flux

( )27 7 7 7 0 7xM u dA p p dAρ= ⋅ ⋅ + − ⋅∫ ∫ = momentum flux at the outlet and at station 7

7j

jj

QM Q

Aρ= ⋅ ⋅ = momentum flux at the outlet and at station 7

17 MMM −=∆ = change of momentum flux between out and inlet

M1 = momentum flux at the inlet or point 1

Energy Flux and Loss Coefficients

( ) ( ) ( ) ( )2 21 1 1 1 02 2E x j blE b z V z u z dz V Q Qρ ρ

= ⋅ ⋅ ⋅ ⋅ + ⋅ ⋅ −∫ = energy flux at the inlet or

at point 1 (W)

blQ = volume flowrate inside boundary layer

jQ = volume flowrate of jet

b1 (z ) = width of the suction area at z

( ) ( ) ( ) ( )( )0772

772

7772 prprururV xE −++= Φ ρ

= energy velocity at point 7

( ) ( )∫ Φ−=− 7

7

277

07 drr

rUprp ρ = pressure at point 7and at radius r7

225

( )27 7 7 7 72 E xE V r u dAρ

= ⋅ ⋅ ⋅∫ = energy flux at the outlet (W)

2

772

QE QA

ρ ⎛ ⎞= ⎜ ⎟

⎝ ⎠= energy flux at the outlet

17 EEPJSE −= = change in energy flux between station 7 and station 1(W)

7

7557 E

EE −=ζ = outlet or diffusor loss coefficient

0

3113 E

EE −=ζ = inlet loss coefficient

20 02 jE Q Vρ

= = energy flux at point 0 (W)

2

3 3 03

12

JJ

QE Q p pA

ρ⎡ ⎤⎛ ⎞⎢ ⎥= ⋅ ⋅ + −⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦

= energy flux at point 3 (W)

Power and Total Head

h j jP g Q hρ= ⋅ ⋅ ⋅ = lifting power (W)

0 13 57 7PE JSE hP P P E Eζ ζ= + + ⋅ + ⋅ = ρ g Qj H35 = effective pump power (W )

( )35 7 57 1 0 131 1 j

J

H E E E hg Q

ζ ζρ

= ⋅ + − + ⋅ +⎡ ⎤⎣ ⎦⋅ ⋅= total head (m)

Cavitation and Impeller Characteristics

43

35H

QRPMnq = = specific number of revolution

226

43

h

QRPMnqs

∆= = suction number

0 130

1vj

p ph E hg g Q

ζρ ρ

− −∆ = + −

⋅ ⋅ ⋅= suction head

3QQC

n D=

⋅= torque coefficient

( )35

2HHC g

n D= ⋅

⋅= pressure coefficient

Q

HQp K

CCπ

η2

= = pump efficiency

2 5QMKn Dρ

=⋅ ⋅

= torque coefficient

M = torque

2 4TTK

n Dρ=

⋅ ⋅= thrust coefficient

35p

D

g Q HP

ρη ⋅ ⋅ ⋅= = pump efficiency

:

PESD

p inst

PPη η

=⋅

= power delivered to the shaft (W)

ηinst = installation efficiency ( 0. 975 - 0. 990 ).

Propeller Induced Pressure Impulses

2Ri

1 = - p Vt 2φρ ∂⎛ ⎞⋅ ⎜ ⎟∂⎝ ⎠

= pressure induced on the hull from the propeller

t = time

φ = velocity potential due to different contributions.

227

tt),(r,

+ s

t),(r,V = V nn

sn ∂∂

∂∂ ϕτϕτ

= expression for induced velocity Vn normal to the

cavity and the blade:

τ = blade thickness.

2 22( )R x y z = V U U UV + + + = resulting velocity

Ux , Uy , Uz = propeller induced velocity respectively in the x, y and z-direction.

i = - V pt xφ φρ ∂ ∂⎛ ⎞⋅ ⎜ ⎟∂ ∂⎝ ⎠

=propeller induced pressure

i 1 2 3 4 5n

= - V ( + + + + )px

ρ ω φ φ φ φ φϕ

⎛ ⎞∂ ∂⋅ ⋅ ⋅⎜ ⎟∂ ∂⎝ ⎠

= propeller induced pressure

ϕn = angular position for blade number n

2.2Q (2R R + R R)p(r,t) = =

4 r rρ ρ

π⋅ ⋅ ⋅ = pressure induced by a bubble at a distance r

R = bubble radius

R = velocity of the bubble surface

R = Acceleration of the bubble surface

Q = volume of the bubble r = distance from the bubble to a given point

Rrel = RR

0

= radius of the cavity

R0 = Rmax

(Plesset) )C + (21- = R

23 + R R prel

2relrel σ⋅

σ = cavitation number

228

0p 2

p(x,t) - pC/2 Vρ

= = local pressure

229

Sound pressure

)(r/DV )f a ,f(p

m

2mm

mmm

⋅=⋅

ρ =sound pressure in model

Vm = characteristic velocity

r = distance from the noise source to a given point

D = characteristic dimension (propeller diameter)

)( )VV(

DD

rr f) a,f(p = p

m

s2

m

s

m

s

s

mmms ρ

ρ⋅⋅⋅⋅⋅ = sound pressure in full-scale

2

2

2ss s

m ms m

2ms sm s m m

Vf T D = = f T D V

ρσρ

ρρ σ⋅ = ratio between full-scale and model scale

frequency

logp0

pL = 20 dBp

⎛ ⎞⋅ ⎜ ⎟

⎝ ⎠= measure for sound pressure

p = pressure at a given point p0 = reference pressure.

2 12

log 1p p

pL = L - L = 20 p

⎛ ⎞∆ ⋅ ⎜ ⎟

⎝ ⎠= expression for the relationship between p1 and p2

α = air contents

αs = air contents near the saturation point.

230

7 INDEX advance ratio, 119 air content, 217 aspect ratio, 2 atmospheric pressure, 26, 50, 222 azipod, 122 blade shape, 26 bow thruster, 39, 130 bubble, 212, 215 bubbleradius, 213, 229 cavitation, 2, 77, 82 cavitation noise, 189 cavitation number, 26, 41 cavitation tunnel, 26, 212 cavity, 189, 213 change of momentum, 12 change of volume, 189 contra rotating propeller, 14, 28, 94, 96, 121 contraction, 38, 39 Dickmann and Weissinger, 12 drag, 39, 223 duct induced velocities, 13 duct length, 11, 15, 28, 220 duct thrust, 14, 27, 77 ducted propeller, 11, 12, 27, 28, 37, 117 efficiency, 10, 11, 26, 28, 38, 105, 117, 119, 128 elliptical blade shape, 26 fins, 2 free vortex, 2 gain in thrust, 38, 40 house, 94, 105, 107, 109, 117, 118, 119, 120, 128 hydrofoil boat, 2 immersion, 82 induced drag, 2 induced velocity, 49 interference, 77, 105, 106, 107 internal losses, 38 Kaplan, 26, 40, 77 lift, 2 limited immersion, 2 model test, 12, 15, 27, 117, 121, 128, 212

momentum, 3, 49 noise, 41, 49, 130, 212, 213, 216, 217 noisesource, 189, 215, 231 number of revs, 41, 77, 105, 128, 216, 217 open water diagram, 26, 105, 120 optimum propeller diameter, 26 power, 14, 15, 36, 37, 38, 49, 51, 77, 96, 117, 120,

121, 222 pressure distribution, 2, 94, 189, 212 profile drag, 2 propeller arrangement, 94 propeller disk area, 11, 107 propeller induced velocity, 211, 229 propeller loading, 80 propeller power, 80, 117 propeller thrust, 10, 11, 14, 15, 27, 39, 77, 107, 117,

128 propeller torque, 36, 77, 128 propulsor, 95, 117, 119, 121 pulling propeller, 94, 95, 107, 117, 128 pump efficiency, 38 pushing propeller, 95, 128 relative rotative efficiency, 28, 119 resistance, 11, 26, 39, 49, 94, 105, 108, 118, 119,

128 ring vortex, 13 ring vortex cylinder, 13 saturation pressure, 26, 222 span, 2 tandem propeller, 95 thruster, 37, 40, 77, 117, 122 time-varying lift, 2 total efficiency, 11 total thrust, 10, 27, 39, 105, 120, 128 tunnel thruster, 10 ventilation, 2 viscous losses, 39, 223 Voith Schneider, 28 vortices, 2 wake, 15, 26, 28, 49, 50, 94, 119, 120, 121, 130,

222 wake field, 120