The role of cavitation in the design of controllable pitch propellers

LIPS

T 95-18

The Role of cavitation

in the Design of

Controllable Pitch

_Propellers

By:. T. y. Beak

Technical Manager

Lipi B.V.

Drunezi, 1995-05-08

Paper prepared for PROPCAV 95

CONTENTS:

1.

_Introduction

2

_{The design process}

Characteristic design features of

_{controllable pitch}

propellers

Trends

_{in vIbrations and noise}

Conclusions

MT 26

LIPS

1.

_Introductjo

In recent years the

_{number of controllable}

pitch propellers

Compared to the number of fixed pitch

propellers has shown

a steady growth, which can be derived from

_{official LRS}

statistics (ret,. 1).

This regular

_{growth is a reflection}

of the reputation

_of

the controllable pitch propeller for its

_inóreased

reliability and

_{reduced costs.}

Apart from this

_{nowadays there exist}

only a few four

_stroke

engines, Which

_{are reversible. Thereby}

for ships with

a

four stroke engine the controllable

_{pitch propeller}

appears

to be a natural choice.

Several reaso

_{for the selection of}

controllable pitch

propellers are:

-

good manoeuvring

_properties

-

constant number of

_{revolutions in view of}

the use of

a

shaft generator

-

_{the availability of full engine}

power for a vide

_range

of ship speeds without

overloading the engine

_(towing

conditions but also in heavy weather)

-

the possibility to

_{use multiple engines}

per shaft.

The application of both

_{fixed and}

controllable pitch

propellers can be found

_{in a high ship}

speed and high

_power

densities (fig.. 1.1,

_{ref. 2).}

The additional degree

_{of freedom obtained}

by changing

_the

propeller pitch complicates

_{the design}

process. Basically

the wide range of

_{inflow Conditions}

makes it more

complicate to control cavitation.

Most shipdesigners

_{have a hiétory in the}

application of

f ixed propellers,

the appearance of

_{more controllable pitch}

propellers requires

an extension of this knowledge. This

paper is not written to

Contribute to the theoretical

development of the control

of cavitation for

_controllable

pitch propeller8 but concentrates on design

_procedures,

LIPS

2.

The design process

For propellers in

_{general the highest}

possïble level of

propeller efficiency must be achieved while vibration

_and

noise thus cavitation are kept at the lowest

_possible

level.

This leads to conflicting

_{boundary conditions,}

as less

cavitation leads to

_{a large blade area ratio,}

but with

lower efficiency and vice versa.

Therefore any propeller design should be

_{a subtile balance}

between several extremes

_{resulting in a}

comprojse

according to the experience of the propeller

_{designer and}

the correct tools at his

_disposition,

In order to overcoe the

boundary liinitationg as muc1 as

possible Lips has developeã

_blade

sections that combine

large cavItation-fr

_{operation with}

good

structal

characteristics and low drag

_{properties. The end result}

is

an optimised design with

_higher

_efficiency.

In addition the propeller design system as

_{used and}

developed by .Lps, consists of a series of

_interactive

design and analysis modules

_{as shown in fig. 2.1.}

Any design can only be

initiated after the design criteria

have been selected. These

_{design criteria}

consist of

information with respect to shiptype, mission

_{profile and}

possible limitations regarding propeller

_diamete.

efficiency. ship speed

_{or any other manoeuvzinq}

requirement

Each of these criteria

_{relevant to the design}

of controllable pitch

_{propellers are discussed}

below.

The shiptvpe generally is

an important factor for

the wake

distribution and

consequential variation in inflow. Sut it

also determines the normal clearance at the

_{propeller. The}

required maximum pressure pulses :

-are related to

the strength of the ship's structure.

The

ission profile is vital information for the desjg

of

a controllable pitch propeller.

The percentage of

time at

which speed the vessel operates indicates the

_{importance of}

generator is present it is important in which

_{conditions a}

constant number of revolutions

_{is used.. in the}

case of tugs

the trade-of f between relevant towing condItions

_{and the}

transit speed should be carefully Considered.

Even more complication exists

_{in the. case}

multiple engine

are connected to a single propeller. In that case the

shipowner can operate the ship at lower

_{shipepeeds on one}

or two engines at very different pitch settings.

J4anoeuvrjng requirements for twin screw ferries normally

lead to high requirements for the bollard ahead

_{and bollard}

astern thrust values for defined

_manoeuvring

condition.

For

instance during a sidevards crabbing

_{manoeuvre one}

propeller generates forward thrust

_{which is}

compensated by

the astern thrust of the

_{other propeller..}

For terries operating at the Nordsea the power

_{used for the}

LIPS

These requirements for the manoeuvring dò

lead to

limitations in the propeller

characteristics, which

_{will be}

explained in the

_following.

For al]. the relevant sailing

Conditions vibration

_{and noise}

limits are present. In most cases the maximum

_power

condition will limit the maximum induced

_{pressure amplitude}

on the hull structure as most newbuilding

_{contacts include}

vibration limits in those conditions. But also

other

relevant sailing condition

_{should be regarded}

as to the

requirements of vibrations especially in view of

_comfort.

In the case of

controllable pitch propellers this can also

imply that a louer limit in power exista in

_{order to avoid}

pressure side cavitation at

_{reduced pitch settings,}

Above

requirements are Considered

_{general design}

information,

_which

_{are used in the design of}

controllable

pitch propellers.

The first step in the design process of the

_{CP propeller as}

given in fig. 2.1 is the design of a propeller

_(generation

of detailed propeller

_geometry).

The propeller design

_{module runs in an interative}

sequence

a number of calculations, whereby

_{a low blade area}

(efficiency) will be maintained as long

_{as possible. The}

propeller design consists of

_{three segments}

related to

power absorption, strength and

_{cavitation properties}

in

design condition.

In the power absorption

segment the mean pitch of

_the

propeller is determined to accomodate the design

_condition

and takes into account the

specified radial pitch

distribution.

The strength calculation

_{segment is based on beam}

theory.

The radiaI, thickness distribution is determined

_{taking into}

account the fatigue load due

_{to varying stress}

levels

durtng a propeller

_{revolution and}

_the

fatigue properties of

the propeller material to be

_{used. The resulting}

thicknes

distribution will-meet the classification requirements.

The cavitation segment determines at each radial

_{station a}

combination of minimum chordlength and camber.

_The

requirements as stipulated by the designer as to suction

side and pressure side cavitation together with

_{a built-in}

criterium against harmful bubble cavitation are met.

The design of the blade section is done by

_{a variation of}

camber to chord ratios. The

cavitation inception lines of

the blade sections, also

_{called cavitation buckets,}

are

shifted drastically dependent

on the maximum camber to

chord ratio.

The lift of the section and thus the thrust of the

propeller is dependent on the power, the 'number

_of

revolutions, the entrance speed,

_{the wake and the 'pitöh}

distribution. The optimum blade camber to chord ratio

_in

LIPS

A design module is always

_{limited to one or two conditions.}

Therefore at this point the Propeller

_{designer has to}

consider if the designed

_{propeller meets ail}

requirements.

Acceptance of the design depends on the

_{calculated resuits}

and on the quantity

and reliablity of the available

information.

With the available

_{blade geometry the}

cavitation properjes

of the propeller working

_{in its wakefield}

are deterrnjne

_by

the cavitation analysis module for any given

_{combination of}

shipepeed, number of propeller revólutjon,

_absorbe

_thrust

to be delivereti and

propeller eubmergece, this

_{way also}

otf-desig

_{condition can be considered. At fixed}

angular

blade positions the loading on each blade is calcuiate.

_In

the figure the analysis

_{resulta regardjg}

suction side an

pressure side cavitation is given for all blade

_sections

and all angular positions

_{considered. The}

experience shows

that the calculated

method compares well with cavitation

test results Therefore

_{a reliable design}

can be made

without testing.

Fig. 2.2. shows

_{a comparison between the}

calculated

cavitation pattern

_{compared to a full scale}

observation of

a controllaj,le pitch Propeller for a container

_ship.

When the cavitation

performance of the propeller fulfills

the requirements for the respective

_{operationai cOnditi}

other design criteria have to be fulfilled.

These relate to

the pressure amplitudes

_{and strength requirements.}

Pressure

pulses are calculated

numerically with a combination of

lifting line and lifting surface analysis

_{methods (ref. 3)}

and compared with

_{a maximum, allowed level.}

Fig. 2.3. shows

calculation as compared to Ï7 other institutes

_{as compared}

to in the 18th ITTC in

_{1987. In certain cases this should}

change design properties and lead for instance

_{to an}

increased skew distribution

or increased blade area ratib.

Obviously in the design of

controllable pitch propellers

the loading of the hub

_{has to be considered.}

This concerns

the actuating forces,, the loading of parts

_{of the hub such}

as blade bol-t8, blade

flange-andother-ìi-at.-s of

the

controllable pitch propeller.

Therefore the design of the

blade and the hub are strongly

related and can not be

separated.

Based on the calculated

pressure distribution in the wake a

finite element calculation is carried out

_{and evaluated}

using fatigue

_{properties of cast material.}

Optimisation of the propeller geometry in the final

_stage

of the design is done by

iterative change of the geometry

such as': (consider diameter

and blade number fixed.)

-

_{chordiength distributions}

-

pitch distribution

-

_{maximum camber distribution}

-

_{skew distribution}

-

_{rake distribution}

In order to investigate in what way the designer

_{can obtain}

the best possible

_{compromise a number of items}

are

considered fixed

or constant.

LIPS

For instance the

_{propeller diameter}

shou]d be selected

_for

high

efficiency and clearance.

During the more detailed

optimisations it is impractical

_{to vary this as well.}

This

also holds for the blade

_{number, Which}

normally is selected

for avoidance of

_{resonance in the}

superstructUr

Additionally the type of

_{blade section used}

(a llaca 0.8

modified mean]jne and some fixed Naca thickness

distribution) is

assuine

constant during the iteration. As

a last simplification it is assumed that the blade

thickness used is

_{sufficient to}

accommodate small changes

in blade shapes.

_{What is left is the}

chord distribution,

the pitch distribution,

_the

axiint

_{camber distribution and}

the skew distribution.

In the regulaz- design

_{process a selection}

of both pitch

distributions and chordiengtj

distributions is varje

_in

order to find the

_{proper cavitation behaviour}

with given

wake distribution and operating conditions.

The

_blade

camber is varied until

_{a proper margin against}

pressure

side cavitation is

_{obtained. Here also other operatjg}

Conditions have to be

considered. As to the skew

distributions, large skew

_{values are normally applied i}

those cases were the

_{pressure pulses are playing}

a decisive

role. For instance in the case of modern tankers

_the

pressure pulses are so small due to the large tip

_clearance

that a moderate skew is

sufficient. Decisive is the

efficiency in behind

condition whereas the minimum blade

area depends on the risk of

erosive cavitation in the

ballast condition. For

_{large container}

vessels the level of

the pressure pulses is

_{a fixed limit. As}

a result the skew

angle will be larger.

_{it Should be noted}

however that this

should be properly

_{balanced against the}

pitch distribution.

Figure 2.4. shows the calculated cavity volumes

_{for given}

propeller geometry but with increased skew

_{distribution.}

It is known that an increased skew leads

_{to a larger}

loading at the tip. This is reflected in

_cavitation

behaviour and the predicted

pressure puises. Whereas

_a

large skew angle reduces the pressure pulses

_{as a result of}

the smoother leading edge shape this effect

_ispartiy

compensated by the increased tip

loading. Therefore,

increased skew should be balanóed by proper tip

_Unloading

in order to achieve the combined effect of low pressure

pulses and high. efficiency.

In figure 2.5. results are shown for systématic

variations

in blade contour, pitch

distribution and maximum blade

camber. Changes are arbitrarily

selected but show the

effectiveness in reduction of pressure puises at the

_cost

of efficiency. Based such a variation the

_{designer can}

select the most promising option, e.g. the

_maximum

efficiency with the smallest increase in

_{pressure puises.}

In this case the blade camber is the most

_{effective for the}

control of cavitation and pressure pulses. However,

_as

usual in this sort of optimization processes better

cavitation properties lead to a lower efficiency.

Therefore

given a maximum

pressure ampiitude the best efficiency can

be obtained.

LIPS

3.

_{Characteristic desjqn features of}

controllable pitct

prone 1. 1ers

In section 2 the general chracteristj.cs of the design

process are explained. In this section

_{characteristic}

differences between controllable

_{and fixed pitch propellers}

are discussed.

For fixed pitch propellers the blade length and thickness

distribution at the hub can be freely

_{optialsed. The}

maximua thickness is determined from

_{the strength}

rerquirement whereas the chordìength should be long enough

to avoid bubble cavitation. Special care has to be given to

the faizing of the root fillets.

For controllable pitch propellers the situation is much

more complicated. The simple requirement to be able to

operate at negative pitch, limits the aaximu

_chord1engu,

especially at the root. Again the maximum thickness follows

from strength requirements.

The freedom jn choice of design parameters at the hub is

fairly limited to a local change in

_{pitch and camber..}

Apart from the hub the largest difference between fixed and

controllable pitch propellers is obviously

_{the additions].}

degree of freedom which is obtained by

_changing

_{the pitch.}

This has consequences for the local inflow angles and the

blade section shape. This deformation of the original

design geometry is defined as pitch deflection and biade

section distortion. Basically the effect is caused by the

fact that after significant reduction in the pitch the

cilindrica]. cross sections connect sections, which at the

original design pitch.were located at different radii. This

is further. illustrated in fig.

3.1.

Notj only the blade section shape changes from character.

Also the operating condition changes.

_{For instance at}

constant number of revolutions the local cavitation of the

section remains constant while the lift

_{coefficient is}

reduced. As a result operation closer to the pressure aide

cavitation is therefore unavoidable. This is

_{also shown in}

an open water diagram for a controllable pitch propeller

where the lower limit against_pressur&sje

_{cavitation-has}

been

shown

(fig. 3.2).

Normally pressure side cavitation is avoided in order to

avoid cavitation erosion. Pressure side cavitation is found

to be erosive which is relatid to the type of pressure

distribution on the blade. Especially the large pressure

gradient close to the leading edge causes a type of

implosion on the blade which is erosive.

This effect of pressure side cavitation has been known for

a very long time.

In

_{recent years, especially as a result}

of the application of resiliently mounted engine

_{noise and}

vibration problems

have

_{arisen as a result of pressure side}

cavitation. Examples of several of those cases have been

given by Lepeix (4]. Lepeix describes two problems, one on

a cruise vessel (Sovereign of tite Seas 10.2 MV,

_{4.9 ii}

LIPS

The normal

_{operating number of}

revolutions is 135 RPM.

_The

minimum number of

_{revolutions was set at 125}

_{RPM! This}

value is to avoid

any

resonance of the engines

_{on their}

elastic mountings.

As

_a

result the mean pitch value at

_{low speed vas. then}

quite low. Moreover

the limited excitation

_{at full speed}

was partly obtained by a Pronounced tip unloading

_{with a}

strong decrease of the pitch at the top of the blades. By

using. these propellers with a

ow mean pitch complicated

flow phenomena occur when the main part of the blades

produces positive thrust and the. outer area of the blades

produced negative thrust.

Fig. 3.3 shows three

_{pitch distributions (31}

_{degrees), one}

for the free running design condition for zero thrust

_and

at 6 degrees

_{pitch. The free running pitch shows a normai}

distribution and thereby radial load. At zero thrust the

outer negative thrust is compensated by the inner positive

thrust

_{It is therefore understandable that at any}

intermediate

condition

_{the pitch at the tip becomes.}

negative. The importance of these

intermediate. conditiòns

has been illustrated by the noise. and vibration problems,

which have been. encountered in

practice. There has not been

a proper description of the cavitation phenomena in

technical literature.

In the following a first attempt is presented.

In the intermediate: pitch

_conditions

_{of fig. 3 3 cavitation}

observations

_{in a large size cavitation tunnel with axial}

and tangential simulated wake showed

_{(f ig. 3.4);}

-

_{Large amount of vortex type pressure side cavitation}

from 0.7 radius and upwards,.

-

_{At the blade: tip a secondary tip vortex occurréd with}

lower strength.

-

_{Looking from aft the two vortices are not contracted}

but the move outwards, such

that effectively the tip

clearance between the vortex and the hull is reduced.

-

_{The two vortices are counterrotating and therefo}

show a strong interaction.. This

_{interaction causes the}

individual vortices tO become unstable.

The consequences of such behaviour is shown in fig. 3.5.

instead of clear blade rate frequencies,

_{broad band}

pressure pulses are generated at 3 to 6 times blade

frequency.

Por the normal design optimisation

the avoidance of

unstable pressure side cavitation is of vital importance in

the condition which regularly occur in the sailing profile

f the ship.

Although it is possible to theoretically predict the

unstable behaviour of interacting tip vortices an

engineering approach has been developed

_{which, gives a good}

estimate of the Unstable pressure side cavitation.

Then

_{the operational limits of the controllable pitch}

propeller can be outlined as indicated in fig. 3.6. At high

number of revolutions a lower

limit in

_{pitch exist below}

LIPS

4.

Trends in vibrations and noise

Proa the foregoing it is clear what the limitations are in

the use of controllable pitch propellers. At maximum (or

high) power the level of pressure amplitudes should

_be

correlated with the vibratrion limits at multiple blade

frequencies.

Not much more than five years ago the levels of blade rate

amplitudes vere in the order of 3 to 6 kpa. Nowadays, in

spite of the increase in power, by careful optimisation of

the propeller geometry, values of 1 to 2 kpa have been

reached. These values bave been measured on full scale

ships such as high powered cruise vessels (twin screw) and

container vessels (single screw).

It should further be remarked that in spite of the strong

reduction in blade rate components the higher order

components cannot be affected that much. The level of these

components depends on the wake, distribution rather than

_on

the propeller geometry.

During recent experiments with several designs for a high

powered container vessel it become apparent that blade rate

pressure ampiitùdes vary with about 30 percent. The levels

of 2nd and 3rd blade rate pressure amplitudes however

varied with only 10 perceñt in a now consistent way.

A second point should be clear from the observations made

in section 3. At low pitch settings the level of excitation

is not the problem. Rather it is the broad band character

of the excitation which creates local resonance and noise.

Prom the acoustical point of view the situation is even

more complicated.

The shipyards have adjusted the constructions of their new

ships to take advántage of the lower levels of pressure

amplitudes. In combination with more sophisticated design

tools (such as finite element techniques) the lower blade

rate pressure amplitudes lead to a lighter and weaker

structure.

From acoustical point of view it is then obvious that the

properties are worse than before. This is especially

troublesome as the control of noise is difficult as:

It is as yet not comion practice to use experiments t

predict noise levels from propellers.

Prediction methods either based on numerical or

statistical methods sometimes give quite different

results.

In fig. 4.1. two predictions are compared tot the noise

levels induced by the same propeller. The numerical method

is based on unsteady lifting surface theory modelling the

cavitation volume as a monopole. The statistical prediction

is based upon the method of de Bruijn et al [5] and

includes a correction of 10 dB to account for a low noise

design. Large deviations are present. in this case the

higher level is expected to be more realistiö given the

simplifications in the physical model

The combination of

limitations in prediction methods and the trend of building

shipsmore critical frOm acoustical point of view makes it

necessary to develop a consistent design methodology from

acoustical point of view.

S..

0$

LIPS

The combinatjo of the limitations in

_{propeller noise}

prediction methods and

the

_{building of ships,}

which are

acoustical more critical.,

makes it necessary to dèvelop

_a

consistent design

_{methodology from}

LIPS

5.

_Conclusions

-

The number of controllable pitch

_{propellers compared}

to fixed pitch propellers is still

_growing.

In the design several more

_{complications ara present}

then for fixed pitch propellers such as

* limited chord length in

_{the root area.}

*

_{many off design conditions to be}

considered.

-

_{Especially at Iou pitch settings the occurence of}

instable pressure side cavitation

_{should be avoided in}

normai operating conditions.

-

The cavitation

phenomena

_{in such conditions are}

dynamic of character and creates

_broadband

_excitation.

-

_{Blade rate pressure amplitudes at maximum power}

can be

controlled succesfully. Absolute low pressure

amplitudes are measured at full scale nowadays

(1 - 2 kpa).

The

_{shipyards have take: n advantage of the lower}

pressure puises by reducing the strength

_{of the ship}

structure. A negative consequence

_{of this is that also}

the

acoustical properties become worse.

-

_{The development of consistent acoustical}

_design

approach is

necessary to avoid future problems.

LIPS

Beferences:

J.S. Canton: "Propeller

_{Service Experience".}

7th Lips Propeller Symposium

_1989.

H. v.d. Vorst: "High-speed

_{Propellers or waterjets?}

Un

update for owners and

_designers".

Paper presented at the Cruise and Ferry

_{Conference, London,}

1995.

T. y. Beek, H. v.d.

Vorst: "On the validation of an

unsteady lifting surface method for propeller

_design

analysis".

SNANE Propeller/Shafting

_{'94 Symposium, Virginia}

Beach,

1994.

R. Lepeix: "Propeller induced excitations

_{and responses}

_on

large passenger vessels in transient

_conditions".

INSDC '91, Proceedings Vol. I, Robe, 1991.

A. de Br-uijn, W. Moelker,

_{F.G.J. Absil: "Prediction}

method

for the acoustical

_{source strength of propeller}

cavitation".

'p

_LIPS

2.5--i i i

II_:

0.5

0.0

o

A

10

20

₃₀

SHIP SPEED EKNOTSJ

Fig. 1.1. Power density (

power divided by propeller disk

_area)

for fixed and controllable pitch propellers versus

j

ship speed.

40

SMALL PROPELLERS

A

CONTROLLABLE PITCH

a

FIXED PITCH (LARGE)

a

L

a

LA

L L

L.

L.

A L&$

àt.

a

a

.

.* $.

4A4

_.

_.

L

.

I

aa

.

L L

a.

.

.

AA &-.

Â.

L.

.

_a

L

a

I

IP

t

I I I I I I

2.01.5

-.

a

.

a

s..

_LIPS

PROPELLER SPECIAL/ST

IFWALJES/Thy

_J

LIPS

Fig. 212

_{Model scale (upper) and full scale (lower) observation}

compared with calculations for a container ship with a

LIPS

0.5

Q03

x/R

ca vita/in g

-as

---OS

xfR

-05

non-ca vitciting

Fig. 2.3. Calculated

_{pressure pulses for ITTC 87 propeller}

compared to other predictions.

A - maximum calculated level

B - mean value, of 16 calculations

C - Lips calculation

D - minimum caculated level

sp

_LIPS

7.0

6.0

w 5.0

4.0

3.0

0 2.0

1.0

30 DEGR. SKEW

44 DEOR. SKEW

w

DEGR. SKEW

0.5

0.4

o

o

0.1

>

w

0.0

-0.2

-0.3>

-0.4

0.0

_-0.5

-60-40-20 0 20 40 60 80

_100120140

ANGULAR 'POSITION (DEGREES)

Fig. 2.4. Ef fect of skew angle on calculated cavity volume and

cavity volume velocity.

LIPS

Fig. 2.5. Effect of geometry changes on efficiency and pressure

pulses ( O - camber, O - pitch disribution,

ORIGINAL AND DEFORMED PROFILE

ON

.90 RADIUS. DF!

_{= -24.5 DEC}

ORIGINAL AND

_{DEFORMED PROFILE}

ON

_{.70 RADIUS,}

DF! = -24.5 DEC

OR I G I NAL AND-DEFORMED--P-ROF I L E

ON

.50 RADIUS. DFI = -24.5 DEG

LOADCOEFFICIENT KT

LIPS

Fig. 3.2. Open water diagram

_{of controllable pitch propeller}

with operating lines and

_{pressure aide oavitatjà}

Hub

Tip

0.8

0,6

0.4

pitch distribution at design pitch

- .

positive thrust

pitch diameter ratio

0,2

0,96

negative thrust

positive thrust

i 1.06 1,1 . 1.15.. 1,2 1,25 _1i3 1.35 1.4

pitch distribution at dead slow condition

.350 400

0,8

06

Positive thrust

-__.,

negative pitäh

positive pitch

0,2-

-600 -400 -300

200

100 Ó

Fig. 3.3. Pitch distribution

_of

_{C? propeller at various}

conditions.

04

.100 _{.200 .300 .400} .500

Hub

Tip

0,2 -.150 I

positivé pitch

negative pitch

t -10O -60 0 .50 .100 .150 .200 .250

pitch distribution

at zero pitch

condition

.300

-4

_{negative thrust}

Tip

0,8 0,6 0,4

Hub

'p LIPS

-s

ò

Fig. 3.4. Cavitation

_{observations at}

extremely low pitch

settings (6 degrees).

LIPS

Fig. 3.5. Characteristic

_{pressure aIplitudeg at extrinely}

low

LIPS

h?

Ship speed

14

Upper limit of suction side cavitation

A

"J,

ff1414

44

Lower limit of

pressure side cavitation

AI

Lower

limit

for unstable pressure side cavitation

I I I

rl!

Number of revolutions

Fig. 3.6. Example of recommended operational limits of a

controllable pitch propeller.

LIPS

120

100

60

60

40

20

dB re 1O'm'/sec f i 4 1

i W

.-3---1.---J-_i ìi4

$

4 IS 700

100

Fie uncy (113-octaves) e

Fig. 4.1. Comparison öf a numerical prediction of the noise

level compared to a statistical.

A -

statistical average

B -

statistical average with lo .dB correction for low

noise design

C -

numerical prediction based on Calculated

cavitation volume

.ight 1 by

sain-be

tion

Design aspects of efficient marine propellers

T. van Reek, R. Verbeek, Research & Development Department Lips B.V., Drunen, The Netherlands

I Introduction

For the design of fixed and controllábie pitch propellers efficiency is the main target. Dùe to

boundary conditions.the maximum attainable efficiency is Limited, For instance if the propeller could

operate without a ship the diameter is not limited and very high efficiencies can be reached. The presence of the hull restricts the diameter and implies inhomogeneous inflow velocities. The finite draught of the ship causes that cavitation behaviour and pressure fluctuations induced by the pro peller on the ship's structure, are additional boundary conditions to be dealt with. AU these factors by Itself have their influence on the efficiency of the propeller. Maximizing the efficiency, while

taking into account these boundary conditions, is difficùlt and the designer has to use sophisticated design tools.

In this paper the basic parameters and their consequences for the efficiency are discussed. The influence of propeller-hull interaction, number of blades, blade area ratio and propeller load are described. The results of cavitation and pressure fluctuation requirements on the blade design are

discussed. Effectsof skew and its consequences for the blade stresses areshown.

2 The main parameter effecting the propeller efficiency 2.1 The efficiency of a single propeller without presence of the ship

The basic action of the propeller is to accelerate the flow through the propeller disk so that an

increase In momentum is generated and a thrust force results.

With the aid of simple momentum theory a formula can bederived for the efficiency under the assumption that there is no friction and that there are no rotational losses:

2 where CT =

-p VD2

T - thrust

Ve -entrancevelocity D - diameter p - specific density CT - thrust coefficient

For a given ship and shlpspeed the required thrust is known and therefore the efficiency of the

propeller increases with increasing propeller diameter (Fig. 1). For a more refined theory taking Into account the propeller r tatlonal losses the following relation can be found [1 J:

4(1 - noi)

X2 X2 ¡X2 +

CT=.

I +(l noi)22 (2

0)-h

₂ (2)

Jahrbuch der Schlifbautechnischen Gesellschaft 77 (1983)

lt

106 Jahrbuch der Schtffbautechnlschen Gesellschaft 77(1983)

B 2

uloh?oI

_{1+2/3 e/X}

0.8 TI (CT) 0.7-u

1?ol ideal open water efficiency

n - rotational speed

The open water efficiency thus calculated Includes rotational losses for the propeller with Infinite

blade number without friction, but is only valid for one radial circulation distribution. The result is

plotted in Fig. 2 which clearly shows that the rotational losses increase with the advance ratio J. So far the propeller efficiency Is a function of the CT and the J-value of the propeller. An estimate of the influence of the viscous drag on the efficiency is given by:

I 2eX1

with

and = drag lift ratio of the propeller section profile.

- 0.L

-.-..---,----D

0.2 OE 0.6 0.8 1.0

Advance ratio T

Fig. 2. Rotational and frictional losses as function of the advance ratio

(3) 1.0 Ui 0.8 _0.5-

_.02

0.6 ¿i Friction C w u 0.L .0.1w 0 1 2 3 ( 5 Thrust coefficient

Fig. 1. ideal efficiency as function of thrust coefficient

V i with X J

vnD

ir 0.8 0.7 : ° w 0.5 0.30 alo ais 0.20 0.25 Ao/Ao z

¿ n 'n o .2 -u .1 nite ut is nate (3)

Design aspecusof efficient marinepropellers 107

FIgure 2 shows that e has a distinct influence on the efficiency. More sophisticated calculations

can be made according lifting line or lifting surface theories. Then also the effect of a finite blade

number and the effect of the blade area ratio are taken Into account.

The approach has been completely theoretically thus far. For practical applications one often uses experimental results of propeller series, for instance Wageningen B-series, where certain parameters such as propeller pitch and blade area ratio are systematically varied. With the aid of these measured

open water characteristics one can choose for given thrust and diameter the optimum number of

revolutions and determine the correspondIng efficiency. When these resultingefficlencies are plotted, as a function of the blade area ratio per propeller blade (A0/A0/z), it appears that there is an

opti-mum blade area ratio for a given thrust coefficient (FIg.3). This figure is within limits valid for all

B-series propellers. The physical meaning behind this optimumcan be explainedby known

character-istics of wings with a low aspect ratio. Each wing in a potential flow (no friction) has alift L and a

drag D. This drag force is Induced by the circulation distribution around the wing:

C0=k-

(4)

where CD - induced dragcoefficient (D/(b . s 1/2 p V2))

CL - lilt coefficient (L/(b . l/2p V2))

b - mean chord ofthe wing

sspan of the wing

A - aspect ratio

k - factor depending on circulationdistributlon and wingplatform

0.6 0.5 0.2 0.1 9 4) 0.6 02 o 01. 5)

Figs. 4 and 5. Propeller open-water efficiency as a function of blade roughness

10KO R6.0-10

\\1

'

\

IW

K:400j,m 125 85

I

K:1SIJm

°°

R:60- i0 01. 0.6

I

08 10 06J- 10 08

108 Jahrbuch der SchUTbautechnlschen Gesellschaft 77 (1983)

A measure for the aspect ratio of a propeller blade is given by:

D2 z

A0/A01z D2 AJÀO

(5)

Thus for a given thrust (lift) and AeIA0 the aspect ratio increases with the number of blades.

In Fig. 3 both viscous friction due to the blade area ratio and induced drag depending on the

aspect ratio play a role. With constant aspect ratio an increase of the blade number causes an increase in blade area and therefore the viscous friction. This decreases the efficiency. The lift coefficient CL is proportional to T/Ae, and therefore the induced drag is proportional to

C01 (6)

For constant aspect ratio and constant thrust an increasing blade number decreases therefore the induced drag. Apparently both effects compensate each other.

From these results it can be concluded that for a given number of blades the blade area ratio can be chosen such that the efficiency is optimum. However, in normal design practice other constraints

have to be taken into account. To optimize the cavitation behaviour of the propeller with regard to

cavitation erosion and pressure fluctuations, the blade area ratio often must be larger than a certain minimum. Therefore the point of optimum efficiency can not always be reached.

Another effect importan t for the efficiency of the propeller Is the roughness of the propeller

blade. The roughness of the blade surface is strongly depending on the quality of the grinding and on

the time spent in environmental conditions. Figure 4 shows some typical values for the centre line

average roughness (Ra) of propellers with different service condition.

New propeller Class I (ISO 484/1)< 6 pm Ra

Class S (ISO 484/1)< 3pm Ra

Propeller 12 to 24 months in service 20 pm Ra Figure 4 typical values for propeller roughness.

At increasing roughness of the propeller blade the skin friction and therefore the drag of the blade increases. This increment of the skin friction has been investigated bySchlichting

(6 and

Mkuradse

andPrandtlwith the aid of sand roughness tests for plates.

This has resulted in the following empirical relation for the increase in drag coefficient of a blade section

¡ c

2 .1(1.89 + 1.62 log - 0.455 (log (7) with

R - local Reynolds number ()

c - chordlength

V - inflow velocity s' - kinematic viscosity kg - equivalent sand roughness

To estimate the effects of section drag upon the propeller characteristics the equivalent blade section approach ofLerbs(4]may be used. The equivalent sand roughness has-tolei'elased to the

actual propeller roughness such that realistic changes in friction can be calculated. The usual method for circumventing this is to measure the drag of the rough surface at laboratojy scale,compare it with that predicted by equation (7) and thereby determine an equivalent sandgraln size. However, some

discrepancies exist, the Prandtl.Schllchtingformula applies to surfaces having roughness elements which are all geometrically similar (characterized by one parameter). This is not usually the case for surfaces produced in an Industrial process, which have textures different from that of unIform sand.

The second reason follows from the fact that the first part of equation (7) does not involve the

Reynolds number. For industrial surfaces the variation of the roughness is such that this difference should be taken into account. For a more theoretical treatment reference (5] can be useful.

To Illustrate the effect of roughness upon efficiency, results of measurements with a B5.75 type propeller with different roughnesses are given in FIg. 5 13]. The results show that, at these Reynolds

he er ne lin Q0 de se de (7) de he od .th ne its or id. he ce pe .ds

Design aspects of efficient.marine propellers 109

numbers, increasing k from 15 to SOpm Ra decreases the efficiency with about 5 percent. Although this difference is expected to be smaller on full scale, the lossin efficiency will be large enough to pay for a repolishing of the propeller blades.

2.2 Propeller hull Interaction

se In the preceding section the Influence of several parameters on the efficiency have beenexplained.

Is In reality the vicinity of the ship's hull changes the efficiency due to propeller-hull Interaction. The

analysis of the propulsion factors by means of the well.known thrust identity method gives values for propeller-hull Interaction in behind condition. The total propulsive efficiency Is written as:

R TV. (9) an

1t

with:

_02irQn

(10)

(Il)

fl Ve

All the factors involved have a clear physical meaning. The Taylor wake fraction w = i

--describes the decrease in entrance velocity at the propeller. The thrust deduction coefficient t account for the increase in resistance due to the propeller suction. The relative rotative efficency '1R accounts for the difference in torque between open water and behind condition. In practical cases '1H ranges

from 0.90 to 1.25 and sg 0.95 + 1.05.

Placing the propeller infinitely far from the ship both t and w become zero and 'j equals 1,

there-fore

'ID = 'Io

When bringing the propeller closer to the ship two things happen:

- the thrust coefficient of the propeller Increases due todeceleration ofthe:water, thus the open

waterefficiency decreases

- the huflefficiency becomesiarger than one (in most cases).

The influence of 'IRi being a value close to unity is for the time being neglected.

To illustrate the influence of the factors w and t on the propulsive efficiency formula (1) for the Ideal efficiency can be used:

I +./i

+CT

2

(12)

in whichCTis the thrust coefficient in behind condition:

(8)

T

CT

-p V D2

The last equatloncan be written as:

CT1

V2.ED2 (I

.w)2(]

_t)CT,(l

w)2(1 t)

2p 54

R 1 1

(13)

in Fig. 6 the resulting total efficiency according to formula (13) is given for constant values of w and t. In Fig. 7 the total efficiency is given as a function of wake fraction for a given value of

110 jahrbuch der Schiffbautechnischen Gesellschalt 77(1983) 1.1 1.0 08 g 0.7 u 0.6 0.5

L

w:0 (:0 w:O.2 (:0.1

/ :0

:0.1 / w:0.2 1:0.2 w:0 (:0.2 0.3 0 1 2 3 1. 5 6 7

Thrust coeffiienl iCT:R/(1/2V F) FIg. 6. Total eFficiency influenced by wake and thrust

deduction 8 0.9 0.8 0.7 0.6 0.5 a, F 0.3 0.2

al

FIg. 7. Total efficiency, variation with wakerfraction

(CT 3)

CT, (

3) for a Fange of constant values of tH and corresponding t. lt Is not always favourable to

strive at a high hull efficiency as illustrated by the points marked A and B In the figure. In the wake fraction three components can be distinguished:

- the potential wake due to potential flow,

.-. _{the viscous wake due to viscous flosv,}

- the wave wake due to the orbital motion ofthe water particles.

Of these three the viscous wake is the largest and thus the most important.

Much effort is put into the prediction of w by describing the viscous wake . Harvald [21 for instance used model tests to show the dependency ofthe wake fraction upon:

- the breadth to length ratio of the ship,

- the propeller diameter to shlpiength ratio (D/L),

- the fullness of the ship (C8),

- height above the keel to draught ratio (E/d),

- extreme variations of frame shape.

The results are shown In Fig. 8. The main relations can all be explained from the behaviour of the

viscous wake. For instance a short ship (SIL high) has a larger viscous wake than a long one. Also

when for a given ship the propeller is enlarged the propeller in relation to the boundary layer thick. ness increases and the wake fraction decreases (see propeller diameter correction in Fig. 8).

The underlying assumption with the thrust identity method Is that the propeller thrust Is described _by the KT.ÇUTy if theopen water diagram. From-the measured thrust the-KT '«

the wake fraction is computed with the aid of Fig. 9. Only if the KT-CurVe is approximated by a

straight line and the number of revolutions is identical behind and far from the ship, it can be shown that a linear relation exists between the thrust deduction coefficient and the wake fraction. This is in accordance with the well-known formuia as given by Taylor

t=O.5+0.7w.

(14)

A very Important aspect of propeller-hull interaction is the dependency on the propeller diameter

in relation to the ship's length. A well-known procedure to reduce the fuel consumption of existing

ships is to reduce the shipspeed by several knots and to replace the existing propeller by a larger one.

This has been utilized for several large tankers as with these ships the tip clearance of the original

propeller Is large enough to allow for an increase in propeller diameter.

0h 0 0.1 0.2 0.3

sg e, al 0.50.

05

0.40 0.35 030 0;25 0.20 0.15 0.05 Q -0.05 0.05 o -0.05 0.5 0.6 0.7

ce-0.02 0.03 0.04 0.05 0.06 0.07

D'I-0.? 0.3 0 0.5 0.6 0.7

Fig..8. Diagram for thedetermination of thewake

coefflcien of single screw ships

Under the assumption that the thrust decreases with the ship's velocity squared an Increasing diameterleads to a decrease in thrust coefficient:

T

CT =

p V D2

De5ignaspects of efflclentmarine propellers

Ill

Advanceraflo J

Fig. 9. The thrust Identity method.

A)T R;.) = VJnD. B)T= R/(I - Q;.) = V5(l - w)nD

l)

With the definitions according to (10) and (13) and assuming that a Is onlya function of CT this

er can be written as:

aD

òi70 ¡ 2 2 ôw I

öt

I

òw

at

?H CT

w)W5(1 t)

(16) 38 to U-Frcme V-Frame 0.5 0.6 0.7

CB-ce he so k-le

a The open water effldlerscy then increases according to equation (1). The variation of1?D with the

diameter is then given by:

in

_ò0

112 Jahrbuch der Schiffbautechnlschen Geseilschaft77 (1983)

A

Fig. 10. Propeller design conditions; i homogeneous flow; Infinite draught; 2 = propeUer behind chip;Infinito draught; 3 propeller behind chip; finite draught

From Harvalds diagram (FIg. 8) one can deduce that the value of ô w/ö D varies from - 3/L

to - 6/L, where L is the ship's length. The value of ô t/ò D is considered small.

This equation describes that the increase in the total efficiency due to the increase In open water efficiency Is partly cancelled by the reduction in hull efficiency.

2.3 Effect of design crîtèria on propeller efficiency

In the preceding sections only overall effects are treated. For actual propeller design the local

inflow velocities In the wakefield and the cavitation behaviour have to be considered.

The final efficiency of the design will be lower than global considerations indicate. To show that this difference originates from three principal different causes, the following design exercise Is made. In Fig. 10 three situations are shown:

- propeller in homogeneousflow, infinite draught,

- propeller behind ship, infinitedraught,

- propeller behind ship,finite draught.

If no restrictions are present the propeller operates in a homogeneous flow at infinite draught so that no cavitation or any other related problems exist.

For given propeller diameter the blade area ratio can be chosen such that from Fig. 3 the effi-ciency Is highest. lt is also evident that the diametershould be taken as large as possible, as can be

0.80 0.70 0.60 0.50 0.60 1 2 3 Design condition

Fig. Il. Influence of non.unlform flow and cavitation restrictions upontotal efficiency. Design example for

Pe= 147O0kW;N87.5rpm;Z5;V5 18.4kn.

site

0.

0 05 10 1.5 20 25 30 35 O 1.5 50 55 60

Thrusi coelticleni

Fig. 12. Emelency of actualdeslgns compared with B-series optimumand idealefficiency

shown by eq. (1). The presence of the ship as a second step in this process geometrically restricts the diameter. Also the inflow velocities In front of the propeller are inhomogeneous. Now the propeller. hull interaction plays a role. From section 2.2 ¡t will be clear that this has effect on the overall effi-ciencyof the propeller The design is further complicated because fluctuating forcesin blade hub and shaft are to be considered. The effects of cavitatIon can only then play a role when the finite draught of the ship Is takenintoaccount. Then duringone revolution a certain amount of cavitation can exist which may lead to:

- cavitation erosion - cavitation noise,

- pressure fluctuations on the ship's structure.

The design must be such that noeroslon occurs and that cavitation, noise and pressure fluctuation levels are acceptable. Erosion and noise can be controlled by controlling the type and the extent of the cavitation. The pressure fluctuations which are dominated by the cavitation, as will be shown in the next section, can be controlled by changing the pitch and or skew distribution.

All forementloned restrictions on the propeller design decrease the efficiency of the propeller. This is Illustrated by a specific design example (Fig. 11) where three different propeller designs are com-pared. In this case the step from a homogeneous wakefield to a non-uniform wakefield costs 2 per-cent efficiency. The increase in blade area ratio, necessary to avoid erosion and vibration problems,

Design aspects of efficient marine propellers 113

¡deal elf It ency

4 _{B. L-55 propeller serie5}

I.

-1 O I 2 3 L 5 6 7 ß a io

il

12

Difference in eli ciency (i-ri-IOß

Fig.13.Percentage difference in efficiency compared with 8-series optimum

ter al de. 0. 0. ì. g o.'

0

0. D o. 0. so ffi-be £00 300 200 loo

it

114 Jahrbuch der Schifibautechnischen GeseUschaft 77 (1983)

costs an additional 3 percent. It can be concluded that the ideal efficiency, as given by equation (1), cannot be reached due to

- rotational losses, - frictional tosses,

- non homogeneous inflow,

- losses due o cavitation and vibration requirements.

In Fig. 12 results of Lips' designs made during the last two years have been plotted and compared

with equation (I).

Also indicated is the result for optimum B4-55 propellers. Both non-uniform inflow and different blade area ratio are the reason for the differences between B-series optimum and final design results.

To show the seriousness of this loss in Fig. 13 the differences are given as losses in horsepowers.

2.4 Method to improve propulsion efficiency of C.P. propeller equippedships

The difference between fixed and controllable pitch propellers Is the freedom of pitch variation of the latter. The degree of freedom can be used to optimize the fuel consumption of the engine. For a certain sea state and shipspeed the thrust to propel the ship is constant. Several combinations of pro-peller pitch and propro-peller speed can be selected to deliver this thrust.

A specially programmed micro processor can be used for automatic selection of the proper com-bination of pitch and rpm such that the fuel consumption of the engine is minimal. A further advan-tage of such a system is that adaption to changes in the motor characteristicsand the ship's resistance, due to fouling, is automatically achieved.

This process was simulated on a computer for a 30 000 tdw container ship.For a certain sea state

and thrust an additional 3% decrease in fuel consumption was achieved by the proper selection of pitch and rpm. With the control system an optimum adaption of propeller, shipand engine

character-istics can be achieved. Changes in charactercharacter-istics due to fouling etc. are automatically coped with and

the total propulsive efficiency is continuously optimized. The same system can also be used to

optimize other criteria such as noise.

3 Implications of cavitation and pressure fluctuation requirements onblade design

3.1 Introduction

The primary task for a propeller is to deliver a required thrust with a good efficiency. It has been shown that, given the wakefield, the number of blades, the operational conditions and the loading distribution on the propeller, the efficiency mainly depends on the blade area ratio. The choice of the

blade area ratio and the loading distribution however, Is limited by the requirement that a good

cavitation performance must be achieved and that the propeller induced pressure fluctuations are kept within reasonable limits.

Basic requirements to achievè acceptable cavitation performance of the propeller are: - Face-cavitation-should-be-aVOIded under all-condltions

-- Back cavitation la allowable provided that the extent and type will not lead to cavitation

erosion.

The way cavitation requirements are incorporated in the overall hydrodynamicdesign procedure is described elswhere (see fi. [7]) and will not be repeated here.

The requirement that reasonable pressure fluctuations are to be met, influences the blade design also. Part of the Induced pressures are due to the pulsating cavities on the propeller blades. Limita-tions in allowable pressure fluctuaLimita-tions not only restrict the final blade area ratio in order to keep the extent of the cavitation limited but also affects other design parameters as skew, tipoff-loading and chordlength distribution. The effect of pressure fluctuations restrictions on the blade design will be dealt with in more detall in the next sections. Attention will also be given on the strength aspects of the propeller.

Design aspectsofelflcient muinepropeIIers I I 5

3 .2 Propeller induced pressure fluctuations

The development ofhighly poweredships has led tohlghly loaded propeller blades. The bladescan be designed in such a way that the propeller operates free of cavitation during most of its passage in the wake. The variation in inflow velocity in the wake peak however, leads to a rapid increase and decrease ofcavities on the propeller blades.

This growth and collapse of the cavities gives rise to large fluctuations of the pressure in the fluid surrounding the propeller. The fluctuating pressure field round the propeller leads to excitation forces on the ship's hull which can act as an impoítant sotirce of noise and vibration Inboard the ship. For environmental and structural reasons these forces must be kept to a minimum.

Threeeffects contribute to the propeller induced pressures.

- Pressuresinduced by the rotatingnon-cavltating,non-loaded propeller,

- pressuresinduced by the loading of the propeller,

- pressuresinduced by the rapid growth and collapse ofcavities on the blade.

The pressures induced by the thickness and loading effect have a more or Iesssinusoidal character with a piedominant blade frequent component. Phase differences over the aft body are large and the pressures are decreasing rapidly with Increasing distance from the propeller. The resulting excitation forces.are therefore relatively low.

The characterIstics of the pressures induced by the pulsating cavities on the blade are different.

Not only blade frequent components are induced but also higher order componentscan reach significant levels.

The decay of the pressures with increasing distance is less rapid, so a large part of the aft body ofthe ship vlll be influenced;

The phase differences over the aft body are small.

Though the pressure amplitüdes Induced by the pulsating cavities can have the same magnitude as

those Induced by the loading and thickness effects, the resulting excitation force due to cavitation can reach several times the value of the excitation force resulting from the thickness and loading effect. This is mainly due to the approximately constant phase angle of the cavity pressure signal

induced on the aft body. Therefore in thedesign of the propeller care has to be given that thecavita-lion Induced pressures are kept within reasonable limits.

Several methods exist to predict the propeller induced pressure loads, reaching from simple ones (see f.i. 18]), based on a statistical analysis of full scale and tank results to the more elaborate ones (see f.i. 19]), requiring large amounts of computer time and only suitable for analysis purposes in the finalstage of the propeller design.

A good compromise between the two Is the method developed by several authors ([10, Il, 12])

and based on the linearized cavity theory ofGeursi [13,

l4]and

Geurst andVerbrugli [15].

In the calculation procedure the propeller load is represented by rotating pressure dipolesand the

non-loaded, non-cavitating propeller by rotating sources and sinks. The pulsating cavities are also

represented by rotating sources and sinks, though in this case the source strength will also depend on time.

Only attention will be given to the contribution of the cavitation induced pressures. Detailed

information of the completecalculation procedure can be found in literature (see [12]).

The method is based on the calcUlation of the potential flow around the propeller. ignoring the vorticity in the-infiowfleld-ofthe-propell -and:ässumlngthärall perturbation velocities are small in comparison with the ship speed, V5, the equatIon ofBernoulliin any fieidpoint P can be written as

Pj = - +

_{pV5 H}

(17)

where P1 - induced fluctuating pressUre

p - density - potential t - time

V5 - shlp;speed

x - x.coordinate(see Fig. 14 for definition of àxis) d mt tS.

of

a o-n. n-e, te r-re a-te

f

T:

116 Jahrbuch der Schiffbautechnlschen Gesellschaft 77 (1983)

z Fig. 14. DefinitIon sketch

In equation (17) constant terms and higher order terms in the perturbation velocities are neglected and the velocity in point P is assumed to be V5. When the sources and sinks representing the cavity rotate steadily with speed w, equation (17) can be written as

(18)

with w - rotational speed

- propeller position

Consider now a cavity on a blade section at radius r (see Fig. 15). From the thin section theory

one can derive an expression for the source density m representing the cavity thickness r.

m=Ui+-F

(19)

wIth U5 - local inflow velocity x5 - local ordinate

r - cavity thickness

The cavity thickness distribution along the chord can be calculated when the local inflow velocity,

Incidence angle, the camber distribution of the blade section and the local cavitation number are

given (see 113, 14]). The potential associated with the source density is

_J...

J J

(20)

propeller surface

where d - distance from point of blade to point P in free space m - source density of the cavity

and the surface integral is taken over that part of the propeller where cavitation occurs. The Induced pressure is given by (18):

PCSV

=+!__fIl!!clA

_{a'yJJ d}

pV5

a (Çm

Fig. 15. CavIty on binde element with radius r.

T -local cavity thickness

X3 -local cordinate