The design of wake adapted screws and their behaviour behind the ship

(1)

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The Design of Wake-Adapted Screws

and their Behaviour behind the Ship

BY

DR. IR. J. D. VAN MANEN

and

PROF. W. P. A. VAN LAMMEREN

PUBLISHED BY THE INSTITUTION

ELMBANK CRESCENT GLASGOW

IBM

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The Design of Wake-Adapted Screws

and their Behaviour behind the Ship

BY

DR. IR. J. D. VAN MANEN

and

PROF. W. P. A. VAN LAMMEREN

The responsibility for the statements and

opinions expressed in papers and discussions

rests with the individual authors: the Institu-tion as a body merely places them on record.

Copyright by The Institution of Engineers and Shipbuilders

in Scotland, 39 Elmbank Crescent, Glasgow. Original date

of publication is shown under the title of each paper.

Reprinted from the

Transactions of the Institution of Engineers and Shipbuilders in Scotland

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Paper No. 1195

THE DESIGN OF WAKE-ADAPTED SCREWS AND THEIR BEHAVIOUR BEHIND THE SHIP

By Dr. Jr. J. D. VAN MAN EN* and Prof. W. P. A. VAN LAMMEREN*

8th March, 19,55

SYNOPSIS

The paper deals first with the deficiencies in the theory of

wake-adapted ship screws. Next, a method of design for these screws

is discussed which, by starting from the available data of the circula-tion theory, has been developed with the aid of experimental data in such a way that the screw design satisfies the requirements with

regard to correct pitch and cavitation properties. Using this method

of design, a series of 4-bladed wake-adapted screws has been cal-culated. The appropriate design charts are provided, and a

numeri-cal example given.

For readers not acquainted with the fundamentals of the circu-lation theory a brief review of the screw design according to this

theory is given in the Appendix.

INTRODUCTION

No important ship is nowadays built without previous elaborate

model scale experiments on hull and propeller, and owners and builders have become quite familiar with this mode of scientific

research into problems of naval architecture. During the

de-velopment of scientific research, the purely theoretical treatment

of marine hydrodynamics fell into the background. Now,

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424 DESIGN OF WAKE-ADAPTED SCREWS,

however, as in aerodynamics, a start has been made to investigate

the field of unstable flow phenomena, of which very little is known, and the necessity of carrying out more fundamental research in this field is apparent. If it is remembered that the

outcome of experimental research is almost directly dependent on the fundamental theoretical basis on which the experiment is founded, this leads to the conclusion that the recent theoretical work performed by various experimenters in the field of marine propeller design, must be regarded as an important advance ; in particular, the work done by Lerbs,1-3 Ludwieg and Ginze1,4

Strscheletzky,5 and Guilloton,6 may be mentioned here. In

this connection, the manner in which the velocity field of the screw in the open-water condition has been calculated, either with the aid of Biot-Savart's law or by putting the problem

as one of boundary conditions and by trying to find the

appropri-ate solution of Laplace's equation comes to mind. In addition

to these calculations, the introduction of vortex sheets instead of vortex lines in the calculation of the curvature of flow in way of the propeller blade leads one to expect that the theory of

marine propellers in a homogeneous flow may be mathematically

justifiable in the near future.

With regard to the theory of propellers adapted to the unequal velocity field behind the ship, the situation is less favourable. For propellers rotating in a radially unequal velocity field, the Goldstein reduction factors and the induction factors are at

present being calculated in the Netherlands and the United States

respectively. The Goldstein reduction factors are " average "

factors, which determine the decrease in circulation due to the

finite number of blades. They are calculated for optimum

distribution of circulation corresponding to the condition of

minimum energy loss. The induction factors are non-dimensional

quantities of the induced velocity components. They are cal-culated for an arbitrary distribution of circulation.3 When these factors are known, it will also be possible to introduce vortex

sheets instead

of vortex lines

in order to calculate the

curvature or deflection of flow in way of the blade sections of

these propellers.

In the calculation of the influence of the circumferential inequality of the velocity field in way of the propeller in the

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DESIGN OF WAKE-ADAPTED SCREWS 425 behind condition, considerable difficulties are still encountered.

It will probably be necessary to deal with this essentially

un-stable phenomenon in a way analogous to that in which the

phenomena of aerofoil flutter are treated. In addition, the effects

of the ship's movements due to a seaway on the intakevelocities

of the water into the screw disc will probably demand a similar

treatment.

Next, it should be kept in mind that the flow

around the propeller is influenced by the presence of the rudder, which must, therefore, be taken into account in the calculation

of the propeller.

Thus a theory developed for wake-adapted screws can only

be said to be perfect if, by starting from a condition ofminimum energy loss in calculating the CTi or Cpi-X-71 diagram and the Goldstein and induction factors, the following phenomena can betaken into account :

The radial inequality of the velocity field.

The circumferential inequality of the velocity field.

The change in the intake velocities of the water into the screw disc due to the ship's movements in a seaway.

The influence of the rudder on the propeller performance.

Moreover, vortex sheets must be introduced in the calculation of the curvature and deflection of flow in way of the propeller

blade.

Owing to the risk of erosion, close attention must be paid to the choice of profile because it is, in practice, impossible to design a cavitation-free propeller operating in the

circum-ferentially unequal velocity field behind a ship. Therefore

a profile will have to be found that will show little, if any,

erosion with the inevitable cavitation.7 The magnitude of

each section's average angle of attack during one revolution is

also of importance to the solution of this problem. Finally,

an improvement in the present simplified strength calculation for the marine propeller should be aimed at with the aid of the above-mentioned refinements of the theory.

It has been learned from experience that the design of wake-adapted screws according to the modern circulation theory

leads to propellers with too small a pitch. Corrections are

needed to give the screw the correct pitch and render it

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426 DESIGN OF WAITE-ADAPTED SCREWS

unequal. The Authors therefore intend first to consider the

imperfections of the theory for wake-adapted screws more closely, and next to discuss the compromise between theory and empirical data that has been arrived at by the Netherlands Ship Model Basin in order to design a wake-adapted screw which comes closest to both the correct pitch and freedom from cavitation. By means of this compromise a series of 4-bladed

wake-adapted screws will be calculated,8,9 followed by a numer-ical example and concluding remarks.

DEFICIENCIES IN THE THEORY OF WAKE-ADAPTED SCREWS

In designing a wake-adapted screw according to the circulation

theory one has to start from the following data

The power P to be absorbed or the thrust T to be delivered by

the propeller.

The speed of rotation of the propene, per sec.

The screw diameter D and the number of blades z.

The ship speed Vs, the corresponding mean intake velocity vs

of the water into the screw, and the radial distribution of v e' over the screw disc.

The distance ii from the centre line of the propeller shaft to the water surface for calculating the cavitation number ao. From the above data it is possible to calculate the following

dimensionless factors:

The thrust constant:

CT =-- T/[( pv82/2 )7rD 2/4] the power constant :

Cp=75 S.H.P./[( pv 83/2)71)2/4]

and the advance coefficient : A=ve/nnD.

For calculating the frictionless thrust constant and the power constant the following approximate formulae may be used:

Gr1=CT/(1-2e, A)

and Cpi=C01+2e,/3A,),

where ei=the drag-lift coefficient of an aerofoil with infinite aspect ratio; and xi=the advance coefficient corrected for the

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induced velocities. For an explanation of these formulae, refer

to the bibliography. 1,9 Before the characteristic hydrodynamic

quantities of the blade sections can be calculated the idea/ efficiency 7i of the screw must be determined Writing for the frictionless thrust constant per ring element of the screw:

dOri=dTiff( pv e2/2)7rD2/4-j=-27cr pc.( cul2)dr1[( pv.3/2)r.D2/4]

(for the meaning of the symbols see Fig. I),

DESIGN OF WANE-ADAPTED SCREWS 427

dFti _r

Fig. 1.Velocity and force diagram ofa frictionless

propeller blade element. we can, by using the equations:

x=r/R

X/Xi (condition for minimum energy loss according

to Betz for homogeneous flow), and

caj2v,=--[(1r0/74,dxXi/(x2+ x,2) reduce this equation to :

1-7) 3.3 ₁ _x3

[

-

2 _dx.

x2+ X12 74,2 _{(x2± Al2)2}

Integrating over the screw disc, this yields

4(1

[

A2

2 1-1(1 7ivi) o

(2 7)

x2+73,,i2]

. In _A2

With the aid of the equation 712,= i/Gp i the equation for Cr i can

be determined d A V c.0 2 CT Cn.

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DESIGN OF WAKE-ADAPTED SCREWS

From this derivation, valid for a screw having an infinite

number of blades in a homogeneous flow, the condition of

mini-mum energy loss according to Betz having been satisfied, it appears that

CT i=f

Kramer has calculated this relationship numerically for an infinite number of blades and for some finite numbers of blades. Fig. 2 shows the result of his calculations. A special property of the occurring integrals enables all results to be shown in one

diagram by shifting the abscissae. This transformation is not

exact. The diagram is only accurate for 2-bladed propellers;

the error increases with an increasing number of blades. For

judging this inaccuracy, the values (Jr, have been recalculated

for co and for 74i=0-50, 0.80, 0.95 with different A-values

(see Table I and Fig. 3).

TABLE I

The values in brackets have been read off Kramer's diagram. Considering the importance of determining the value of as accurately as possible at the initial stage of the screw design,

the diagram showing the relationship between CT and

must be calculated for any number of blades that may occur.

In using Kramer's diagram for propellers of single-screw vessels,

it must be realized that incorrect values of may be obtained.

As the condition for minimum energy loss in the unequal velocity

field behind a ship, the one by van Manens and by van Manen and Troost,9 in which the influence of the thrust-deduction distribution over the screw disc is taken into account, has been

chosen, namely: 1

\

1-0'

\

--tv

\ 1-0 /.

co 73p, = 0.50 12,1 = O. 80 CTi =095 0.04 7.637 (7-50) 1228 (1.17) 0.219 (0.21) 0-10 6-590 (6.80) 1-156 (1.10) 0-210 (0.20) 0-20 4-748 .(5.25) 0-999 (1.00) 0-190 (0-18) 0.40 2-334 (2.80) 0696 (0-72) 0-145 (0.15) 0.80 0-746 (0.90) 0-335 (0.35) 0-081 (0-08) 428 A -.

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rummosmomfinomimm

0.001 2 3 4 6 8 0,01 2 3 4 6 8 0,1 2 3 4 6 8 2 3 4 6 8

Fig. DA diagram for a homogeneous flow.

2F

DESIGN OF WAKE-ADAPTED SCREWS 429

80 60 40 30 20 lCj 8 6 4 3 6 4 3 0) 6 4 3 0,01 8 6 Ti 4 3 0,001 a 2 2

(10)

001

Kramer (z, po)

0 recalculated values CZ: co)

002 au 006

A, 0.1 02 0.3 04 0i5 de lie

Fig. 3.Comparison of some recalculated values ofCT i 71, and ).

and the original Kramer diagram.

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c 4pi z=4 00 - 0-0800 0-1423 0-5025 0-1451 0.2493 0.6513 0.2102 0.3536 0.7337 0-2753 0.4562 0.7863 0-3404 0-5579 0-8226 " TABLE III

Fig. 4 shows the curves of yi and Cri for the given values of

71i and X and the corresponding valuesof CT: one curve for the

homogeneous flow according to Kramer, and one curve for the unequal velocity field. In this figure, only the difference in

the value of CTi with ) constant is caused by the difference

between the homogeneous and the unequal velocity fields.

For calculating the change in -1i with CTJ and constant, on

et; . A Cri ". Cri CT/

Q1

Q#1 Kramer 5-711 5.252 6-18 2-098 1-848 2-23 , t 1-107 0-932 1-14 0-678 0.543 0-71 0.451 - 0?340 . 0-47 , x .== 0-2 0.3 0-4 0.5 06 0.7 0.8 0-9 0-95 --q/ = 104255 0-6061 07813' 0-9091 1-000 1-0593 120989 1-1261 1:139 1 1,2038 1.1346 _ 1-0727 1.0222 0-9926 0-9804 10-9785 0-9785 0-978-5' 1 0' Q 0-5122 0-6877 0-8381 0.9293 0-9926 1-0385 1-0753 1-1019 1.1145

filESIGN OF WARE-ADAFTED ;SCREWS 431

he expression for dCT1 will then be as follows:

1Q ^t6i [

""'"

1Q 7)

i x2 x3

dCT, 8

Q2 ylvi2 x2+Ai2 Q2 .19i2 _(x2±A42)2 dx

where

'

Q-/

k

/

i-ty)

1-4/\ I. 1-0 \

if for Q we use the values represented in the Table II (see

'A " item 9 ofbibliography) and calculate the CTi values with given

and i, these Cpi values for the unequal velocity field appear

to be smaller than those for the homogeneous flow (seeTable III).

TABLE II

=

A

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432 DESIGN OF WAKE-ADAPTED SCREWS

account of the unequal velocity field a linear extrapolation has been used, as shown in the example given below:

With constant )4=02102 and Xoc,=0.3536, for the

homo-geneous flow

CT = 1.14 -10,=0-7337

CT i= 0-932 -,1=-0.7720

and for the unequal velocity field

CT i 0.932 _Ni=07337 CT =1.14 71i= [(0.7337/0-7720) x 0.7337] =0.69S. OBO 070 /060 I

homogeneous flow (Kramer)

radially unequal field (MC.)

050

0.1 0.2 03 04 as 08 1.0

CT

20 30 4.0 60 80

Fig. 4.Comparison of the 71,,CT, relationship for given A-values for a

homogeneous flow and for a radially unequal field.

As soon as the calculations of Goldstein factors (y.) for the un-equal velocity field have been completed, it will also be possible

to calculate this diagram for screws with different numbers of

blades.

When once 71,i has been determined, Ai can be calculated with

the condition of minimum energy loss. With given wake

dis-tribution the product of lift coefficient and chord length / of the

blade elements can be determined at any radius of the screw disc: 0/=(47t1)/z)x x sin 13, tan ( 3).

Fig. 1 gives the meaning of the symbols in this formula, and the derivation of the formula is given in existing literature,

(13)

on the subject. To make the formula usable the Goldstein

factors are required. Since, as already mentioned, these fac-tors are known only for homogeneous flow, in designing

wake-adapted screws there is nothing left but to use them as an

approximation for the unequal velocity field.

Under the direction of the Netherlands Ship Model Basin, the following calculations are being carried out by the Mathe-matical Centre at Amsterdam:

Recalculation of the x functions for 3-bladed propellers

in a homogeneous flow. Recalculations of the

diagram for 3-bladed propellers.

Calculation of the Ludwieg-Ginzel camber corrections

for 3-bladed propellers with normal blade contour.

Calculation of the x functions for 4-bladed propellers

in a "normal " radial wake distribution.

Calcula-tion of the CT,- X-76, diagram for 4-bladed

propellers.

(1) Calculation of the Ludwieg-Ginzel camber corrections

for 4-bladed wake-adapted propellers with normal

blade contours.

In carrying out the calculations mentioned under (a), there has been deviation from the method of solution used by Gold-stein and Kramer owing to the intricate construction of some coefficients, which render work on an automatic calculating

apparatus very difficult. In using the method of solution adopted

by the Mathematical Centre the coefficients which occur are composed of e-powers and sine-functions, and a relaxation

method has been used. It has the advantage that calculations

by means of automatic calculating apparatus can be carried

out much more easily. The coefficients for each point in the

field have been calculated on the A.R.R.A., while the relaxation

has been further performed on punched card equipment. In

this way satisfactorily differentiable x-values could be obtained to four decimal places with smaller intervals than have been

done by Goldstein and Kramer.

Table IV shows a

com-parison between the Goldstein factors calculated by Kramer

and by the Mathematical Centre respectively.

From Table IV it appears that the differences between the orig-inal x-values (Kramer) and the recalculated values are very small

(a)

,(b). (e)

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434 DESIGN OF W (KE-ADAPTED SCREWS

TABLE IV

indeed. An advantage of the recalculated x-values is, however, that the calculations of the Ludwieg-Ginzel camber correction, mentioned under (b), have shown much more satisfactory progress, as it appears that the inaccuracy of the factors of camber correction that have been calculated by various authors up to now is chiefly caused by the irregularity of the function:

d

I

dx

where

9-

1 l/D,

/x 2+

and ro=the circulation around the propeller blade.

d

r

In Fig. 5 the values of

-

have been given

for

dx

A=0.400 and a special blade contour, as calculated both with the x-values according to Kramer and with the recalculated

x-values of the Mathematical Centre. This figure clearly shows

that the calculation of the function

_dx

d

--°)

with the aid

of the recalculated x-values gives rise to a more regular

develop-ment of this function. These calculations and those for the

radially unequal velocity field, mentioned under (c) and (d), are progressing and will be published in their entirety in due course. As has already been stated, since the new x-values are not yet available, the x-values for the homogeneous flow according to Kramer, will be used as an approximation for the design of

wake-adapted screws.

Returning to propeller design, the Authors would, in conclusion,

observe that the relationship between pi and is determined by

the condition of minimum energy loss:

tan pi 1 ---tV

1-6 \

tan p

f k 1-'4

1-0'

= 0.400 z= 3 homogeneous flow x=r/R 0.1 02 0.3 0.4 0.5 0.6 0.7 0-8 0.9 1.0 x Kramer 1.832 1.225 1.013 0.914 0.852 0.795 0.724 0-624 0-465 0 M.C. 1.8264 1.2138 1.0076 0-9107 0.8493 0.7925 0.7225 0.6226 0.4644 0

(15)

The Netherlands Ship Model Basin uses the condition derived in

the-paper9 by van Manen and Troost, namely,

tank

1 1,11

1q)

1

1tV

tan (3

\

k

1-4/ /

/

where

\*

/1-0

k

1-0 cc

k ie' ).

02 03 04 05

Fig. 5.Results of the calculation of the function d 022\

dx\ cp

for Ai=0.400 with the x-values according to Kramer and with

recalculated x-values of the Mathematical Centre.

With known 1 distribution and blade-element thickness

distribution, the cavitation calculation for the various blade sections can be started. In the recent section, dealing with the imperfections of the design of wake-adapted screws, this part

of the calculations does not require special notice. When,

however, with the aid of a cavitation calculation the shape of

I / / /

/

Mathernattsch Kramer i Centrum It-values I i with 06 0.7 0.8 09 1.0 '215' '014 /

-I__

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436 DESIGN OF WARE-ADAPTED SCREWS

the blade sections has been determined, the Ludwieg-Ginzel

camber correction has still to be calculated.

The calculations that are now being carried out to obtain

accurate camber correction factors have already been mentioned. Nowadays, the k-values given in Fig. 6 are being used for

design-ing marine propellers accorddesign-ing to the circulation theory. These k-values have been adopted on the basis of various values published, and by making use of the fact that for a blade area

ratio Fa/F=0 the value of k must be 1.

Strictly speaking, these k-values naturally only hold true for a homogeneous flow. If a theoretically derived cavitation chart (Karman-Trefftz profiles) is used, the friction correction factors t.t will also have

to be known. As a matter of fact, the influence of friction shows

itself as a decrease in circulation. The lift coefficient occurring in a viscous flow is less than the value calculated with the potential theory. The ratio of the experimentally determined

and the theoretically calculated lift coefficient is called the friction correction factor tl.. Lerhs gave values of v. as a function of 811,

the thickness : length ratio of a blade element (Fig. 7). Accord-ing to Lerbs this friction correction only gives rise to a camber

correction.

Hill'° ascribes to v. a constant value of 0-8 and interprets this friction correction as a camber and a pitch-angle correction.

During his profile measurements, Balboni.' found that the

camber ratio of the sections also has a considerable influence

on the values of p.. Balhan further derived that 50 per cent, of

the friction correction should be applied as a camber correction and 50 per cent. as a pitch-angle correction, so that shock-free entrance of the blade sections is theoretically maintained.

Summing up, the deficiencies in the theory of wake-adapted

screws are therefore:

The 74,-values, which are known for the homogeneous flow,

are too high for the radially unequal velocity field.

The Goldstein factors (x) are not yet known for the radially

and circumferentially unequal velocity field. As an

approxi-mation, the x-values for the homogeneous flow are used

in this case.

The Ludwieg-Ginzel camber correction is not known for the radially and circumferentially unequal velocity field. The

If

(17)

-r?

I.

0.

1 02 03 04 05

Jb

Fig. 6(4Camber correction k for z=4; FalF=0-35,j'.

t. 4

,

S z .4 ' _ , 0 3 R FPO" .0.35 Hub diameter.0,180 I 0 5 R R -, 1 -. 0.9' R -...__ I.6 d _ -,2 2 a 4 rair .0.50 _ i , a 03 R

Hub diameter OAS 0

1 1 a78 -... I 0. R ___,..._ . -", -1

DESIGN OF WAKE-ADAPTED SCREWS. 437

0 OA Xi 02 03 4 05 06

Fig. 5(b)Camber correction k for z=4; FalF=0-50.

1.0

.4

02

07

(18)

438 _{DESIGN OF WAKE-ADAPTED SCREWS}

odie

k-values for the homogeneous flow

_{can again be used}

as an approximation (see Fig. 6).

The magnitude of the friction

_{correction factor} [.L

is not

accurately known for blade sections commonly used in

marine propellers.

From the above it is evident that

_{a wake-adapted screw,} designed with the aid of the available data, can only be an approximation to the ideal. Only with the aid of experimental

data will it be possible to devise a method of design that

will give a propeller with the

_{correct pitch and minimum}

cavitation.

THE METHOD OF DESIGNING WAKE-ADAPTED SCREWS AS ADOPTED BY THE NETHERLANDS SHIP MODEL BASIN

Many experiments have shown that the use of the values of

x and k valid for the homogeneous flow, leads to too

small a pitch ratio when used in the _{design of wake-adapted}

propellers for single-screw ships. _{In many instances,} calcula-tions of the virtual pitch for a wake-adapted propeller _designed

according to this method show _{a difference of approximately}

10 per cent. in pitch from the corresponding _{standard series}

screw. _{It is not advisable to eliminate this difference}

_{as a}

pitch-correction at the end of the screw design from the point of

view of shock-free entrance of the blade sections. It is preferable

to correct the 7) _{value, and this correction is thus made in the}

very first stage of the design.

Systematically varied experiments have shown that the _for

the homogeneous flow has to be corrected by approximately

one-third of the percentage difference between the pitch_{of the face at}

0-7R of the wake-adapted propeller (ra) and theB-series propeller

(P0.7/D) with which it is compared. In most cases this amounts to a reduction of about 3 to 4 per cent. of the -ii-value. _With

given values of CTi and X the calculation of r _{the radially}

unequal, and for the homogeneous velocity field,

_{results in a}

fairly similar difference (Fig. 8).

With the aid of the above-mentioned experimental and

theoret-ical results it is now possible to give a mean correction as a function of the advance coefficient x for wake-adapted screws,

(19)

3 1.0 09 '09 0.7 16- 5/I

Fig. 7.The friction-correction factor

-DESIGN OF WAKE-ADAPTED SCREWS

0.6 0 ckos 0fb. 020 -6.26 439' , 0.65 !

III

i.z JO 0 3 R = Hub diameter .01 SO 10. . ₁₁₁₁₁₁ DB 1-I coA 1 ag ii i 02 -.. Lerbs ris=0.450 ris= 0.700

P.-hilt.

ir

A

r

:halechner ,, 1 , 02 03 4 e 4

Fig. 6(c).Camber correction k for Fal,F=0)65.

056

21 05

z--=-4;

(20)

96 0.40 97 030 98 0.20 99 0.10 ₁₀₀ 0 0.0 862 IIp (rad. un field) C-(homogeneous flow) 005 010 A 015 020 025 030 0.35 0.40

Fig. 8.The 7)5-c0rrection for a radially unequal field. Relationship

between

the

velocity

constants X and Xi for screws with optimum diameters.

-8

4

(21)

whose optimum diameter can be chosen. This correction can

be applied to the relationship' between and xi, namely:

0-978x+ 0-0810.

After correction of 7) this relationship becomes

xi= X+0-0862.

This equation is valid for screws whose diameter is defined by the 5 per cent. reduction line of the Bp-8 diagram. To reproduce these relationships in a design chart the Authors introduce the

coefficient :

BT(nIve2)-VTi st,

where n =rev. per sec.;

ve=mean speed of advance in m per sec. or ft. per sec.

Tiscat=--frictionless thrust in kg. or lb. for sea water. This is

from 1-01 to F02 x thrust T on the screw.

Fig. 9 gives the functional relationship between B., and -A. From this diagram can be found X for the optimum diameter of the screw behind the ship, and this diameter D from x.

According to experience gained at the Netherlands Ship

Model Basin, wake-adapted propellers designed with 74, corrected

for a radial flow, or with the newly derived A-? formula and the values of x and k valid for the homogeneous flow, have correct pitch behind the ship.

The second necessary condition of the screw design is that of

shock-free entrance. This requirement can be met by a special

interpretation of the friction

correction if Kirman-Trefftz

profiles are used. Systematically varied experiments have

resulted in this correction being made as follows :

The friction correction factor will be equal to 0-75.

Of the decrease in

circulation due to friction phenomena

35 per cent, must be compensated by a pitch-angle correction,

and 65 per cent. by a camber correction. The experiments,

therefore, give rise to a compromise being made between Balhan's

theoretically derived interpretation of the friction correction and the conventional interpretation adopted by Lerbs. In fact, according to Balhan's interpretation the pitch-angle correction and camber correction should neutralize equal parts of the decrease of circulation due to the friction phenomena, whereas

(22)

according to Lerbs, the friction correction is exclusively a camber

correction. The friction corrections can now be calculated by

the following formulae:

A« in rad.=0.7[(1 )hl] (11k) (El), fol1=-[(1.3+0.7[.)/2v.] (1/k)(E1),

where p.=-0.75 (see Fig. 7).

0,30 425 420 0,1 0,4 Q5 0,6 0,7 0,8 1,0 1$ 2 BT (n in r ps, vein ft/sec, Ti In lbs)

Fig. 9.Diagram for defining the optimum diameter for

4-bladed wake-adapted screws.

SERIES OF WAKE-ADAPTED SCREWS CALCULATED ACCORDING

TO THE NEW METHOD

With the aid of the newly derived relationship between Ai and A: Ai= X+ 0.0862

\

Diagram for optimum = .1- 2 'e-.= ve =A+ 0,0862 defining diameter Ti salt z= 4 BT 1 XI X 3 4 B 5 6 7 8 9 10 15 20 30 40 50 60 (n in r s In m/sec ,Iiirc )

(23)

fir

9

DESIGN OF W.A.KE-ADAPTED 'SCREWS - 443

it is possible to calculate a series of 4-bladed propellers.9 The

calculations have been carried out and the results collected in Tables VX and in Figs. 10-14. In these calculations, the neW interpretation of the friction correction has also been adopted, namely 35 per cent. of the correction as a pitch-angle correction

and 65 per cent, as a camber correction. In contrast to the

pub-lished series9

the strength calculation has been carried out

.--.0..., _ _ 0 6 0.7 0.8 i0 4 _ I 2 1.3 --=. z g4

wake adapted screw series

_

035 FafF7- 0 50.

Fig.. 10:Cavitation diagram for 4-bladed wake-adapted screws:

_

according to Romsom's data. 1 3 This method of strength

calcula-tion has the advantage of providing values for the blade-element thickness at 0-6 R together with values for the thickness at 0.2 R

(Fig. 13). The propeller dealt with in the numerical example

will also be designed with the aid of these design charts for

wake-adapted screws. Since the radial wake distribution of the ship

-to which this propeller belongs is of an extreme character, it will also be possible to judge the validity of the use of a standardized mean radial wake distribution in designing

wake-adapted screws. 8 0 60 4 30 .20 fo 6 ;. 4 f51 3

b2

(24)

=N. 0.30 01 050 0.1 ;."

,DESIGN OF WARE- A.DAPTED SCREWS.

1 co 1.27. r I _ rill a 0.3 0.4 0.5 0.6 . Si ' -L.4 Fa/F .,. 0.35

wake adapter: screw serIes

. ...--e 1 I 1 . r -rili -1 1 -0.3 0.4 ill 0.7

\

0.8 ---1 i . 095 . _ z =4 Fair a 0.50

Wake adapted screw series

____,,______-____-.. .. -1 I, t . I eilka0,5--, ... 0.6 0.7 a 2 QL,.., 095 1 z-4 Fair .0.65

.

Pip rew 0 13

05 P/D 0- 07 .9 11 3

05 P/0 07 09 1

Fig. 11.kade contour for 4-bladed wake-adapted screws.

020 YD. 0,1 0.30 Is 02 (c) 03 Ills 0.2 040 04 0.3 141 (a)

(25)

' 0,0 0:0 '0. 0,0 0,02 0, 0,5 0,1 p 0,9 'II 1,3 00,5 _0,7 i 0,9 Po

- -- Fig. 12.Camber ratio for 4-bladed wake-adapted screws..

15.5 vv. a It 4.7 1_11 7., li; . i 03 - . . 04 wall

/all

OM.--propPir

0 0.5 0 6 0.7 PP rr-":-.. ---- -_ i - -. -Fa/F=0.35 _ z=4 i .

-wake adapted screw series I

0---1.---.. _ rift'--.0.2 - -- - 11._ e a, 03 . 16 04 1 05 14 0,95 0,6 06

I

,. 0,9 rafr .0,50 z=4 _ -_ -.

wake adapted screw series 1 =

..___ 1

-

---p.' r/R. 0,2 I , 03 04 0 0.7 Fa/F =0,65 z =4 ' , .-. El 0.8

wake adapted screw series v.

4;1 0,5 0z7 08 2Cr .w.` Art Wr0.4,1, wv. (b) 0, 0. 0,08 0,0 (.0 0,04 0.02 0,5 11 10 _r/,

(26)

003 0.0 0.01 elf. t 0,8 02 0.3 x.T/R 04 as a. 0,7 0.8 09

Fig. 14.Radial distribution of pitch ratio for wake-adapted screws.

strat ht fatrod In part straight

N".11111111111111111111

.1

oil iiiall 0.35 . 65 _

11 ,

z .t

wake adapted sum ries

02 03 03 Q. 0.5 09 07 0.9 0.9

Fig. 13.Radial distribution ofmaximum thickness of blade elements.

0.0 004 -1,3 I

(27)

-DESIGN OF WAKE-ADAPTED SCREWS 447 DESIGN No. 1

NUMERICAL EXAMPLE OF THE DESIGN OF A WAKE-ADAPTED SCREW FOR A GIVEN THRUST

The data required for this design of a 4-bladed propeller for

a single-screw cargo ship are: Ship speed V8=16-73 knots Wake fraction 4,=0.348

v8=16-73 (1-0.348) 0-5144=5.612 m. per sec.

Number of revolutions n=100.8 (100-2.2)/100=-98.6 r.p.m.

(corrected for the scale effect on the wake). Thrust in sea water= 110,300 kg.

Immersion of shaft centre line including the wave height= 7-570 m.

Determination of x, D and 'Av. The frictionless thrust for sea water

Ti =- 102 Tsalt=1.02 x 110,300=112,500 kg.

13T= (nit, 69-VT_{i sot=1.643-V112,500/31.49=17.5.}

From Fig. 9: X= veRrnD=-0.1581

D=6-880 m.

Xi= )40.0862=0.2443.

Calculation of the Static Pressure at Shaft Centre Line. Immersion of shaft centre line

=7570m.

Hydrostatic pressure in sea water

=7.570 x 1,025= 7,759 kg. per sq. m. Atmospheric pressurevapour pressure e=10,100 kg. per sq. m. Static pressure at shaft centre line e

=p0e=17,859 kg. per sq. m.

The numerical calculation of of the cavitation number a,

of the blade sections and of the pitch distribution is given in Tables V-XV:

(28)

41-8 DESIGN OF WAKE-ADAPTED SCREWS

TABLE VIII

ao a.a =1.25(Ap/q)0.8 (V/va)2

TABLE V

RESULTS OF THE NUMERICAL CALCULATIONS OF A WAKE-ADAPTED SCREW SERIES

TABLE VI

1-'

1 - 0 P/D= 0.5 0.7 0-9 1-1 1-3 1 - 0' 0.2 0255 1-2038 0-1587 0-2199 0-2641 02901 03016 0-3 06091 1-1346 0-1181 0.1625 0-1967 0-2194 0-2310 04 0.7813 1-0727 0.0937 0-1258 0.1489 0.1641 0-1691 0-5 0.9091 1-0221 0-0793 0.1041 0-1202 01285 0-1310 0-6 1-0000 0-9925 0.0676 0-0869 0.0976 0-1017 0-1009 0-7 1-0592 0-9804 0-0579 0.0709 0-0773 0.0784 0-0756 0-8 1-0989 0-9785 0-0474 0-0559 0.0587 0-0572 0-0530 09 1.1261 0-9785 00361 0-0396 0-0394 0-0365 00325 0-95 1.1390 0.9785 0-0271 0-0279 0-0271 0-0246 0-0214 X Fa/:F=O-35 8/D FalF =0.50 8/D Fa/F=065 8ID 02 0.0423 0.0423 0-0423 0.3 603872 0-03724 0-03650 0-4 0.03520 0-03218 0-03070 0-5 0-03102 0-02716 002520 0-6 0-02598 0-02234 0-02058 07 0-02036 0.01763 0-01631 0-8 0-01474 0-01292 0-01204 0-9 0-00912 0-00821 000777 0-95 0-00630 0-00586 0-00564 Fa/F P/D = 0.5 0-7 0-9 1-1 1-3 035 75-0149 24-1243 12-1344 7-3430 4.8922 050 41.2582 13-2068 6.6478 4.0076 26604 0.65 27-6055 8-9366 4.4745 2.7148 1-8033

1 -=

(29)

-, et 0-2 0.3 0.4 0.5 0-6 0.7 0.8 0.9 0.95 P/D = 0.5 ?/D=0-7 0.7 P/D = 0.9 F/F =0.50 F0/F065 Fa/F= 0.35 Fa/F = 0.50 F/F--= 0.65 i y./F.--- 0;-35 F/F = 0.50 1/D P // D

/JD "

IlD 11D 11D , ! 11D 0.235, !'-'7 0.264: ' 0.212 : 0.235 0.264 0.212 , 0.235 ..., 0.241 t '" 0.283 , 0.201 ..-0.242 0.285 0202. 0.243 0-246 0-300 i; ' 0.191 0-249 0.305 0.193 i 0.251 0.249 ' 0.318 ' 0.181 0.253 0.325 0.183 0.257 0.254, 0.334 0.171 " -, 0.259 0.341 0.173 0.263 0.253 ,r11' 0.349 ' . ", 0.161 0-256 0.354 -1 0.162 , 0-260 0.245' 0.348 ' 0.148 0.244 0.344 0.146 .., 0,242 0.211 . " . 0.303 0.120 0204. 0.290 0.116 '., 0.196 0.166 ". 1 0.244 0.088 . 0.160 . . 0.229 0.090 c 0-.150 .. ., r -) -P/D = 0.5 P/D - 0.7 13/13

I=0'9

,, P/D = 1.1 , -- F/F =0.35 F0/F=050 FafF =0.65 Fa/ F = 0.35 Fa/F= 0.50 N'a/le = 0.65 Fa/F =0.35 14'0/ F,= 0-50 F0/1,' -0.65 i F/F = 0.35 0'0/F=0-50 s -foll foil foil foil , . foil . foil foil foll ,. -foil foil -foil 0.2 ' 0.04617 0.04160 0.03718 ' 0 : ,06599 0.05987 : 0.05352 0.08034 0.07382 0.06753 0.09076 ' 0.08413 0.3 0.03955 0.03381 0.02953 0.05404 0.04864 . 0.04223 0.06704 0.05970 0.05263 0.07632 0.06723

0.40.03525

0.02942 0.02515 0-04769 0-04042 0.03460 0.05681 0.04870 0.04191 0.06350' 0.05441 1,-,0.5 0.03110 0.02735 0.02258 0-04550 0.03634 0.03036 0-05273 0.04246 0.03a55 , 0.05670 0.04584 0.6 ' 0.03357 0.02532 0.02055 0.04380 . 0.03307 0.02700 0.04963 0.03751 0.03050 0.05189 0.03929 4P .., 0.7; , , 0.03326 002448 0.01935 0.04174 0.03053 0.02418 0.04636 0.03368 0-02722 4 '1).04760' (1.03460 16e. 0.8 , r 0.03373 0-02421 0.0191 b 0.04092 0.02918 0.02363 0.04361 0.03156 0.02585 0-04345-. 0.03132 4 0-9 .. ., 0.03761 0-02913 0.02354 1 0-04340 0.03219 0.02734 0,04499 0.03340 0.02891 ., 0.04337 0.03291 -. ' !-I ' 0.95 0.04463 0.03612 , 003132, ' 0-05201 0.03918 0.03516 005022 _ 0.04105 0.03756 :.. ' 0-04808 0.04004 , ., v., Pp) 20.7 0 % . Fai F --- 0,35 F0/F=0-50 0.50 ..Fa/F = 0-66 ' Fa/F=0.35 . Fa/F = 0.50 Ft.i/F = 0.6-5 P/D P/D P/D P/D P/13 P/D x 70,, lai 70,., IS'Ai Tai raj ; 0.2 0.8470 0.8454 0.8438 ..7_, 0.8520' ' -' " a -' ., 1, 0.3 ' 0.8991 . 0-8965 0.8946 0-9014 ', 0.4 0.9518 0.9485 0.9462 0.9530 0.5 1.0012 0.9967 -0.9936 1.0016 '''

0.6 '1-0337

1.0273 1.0236 1-0333 . ) , 0.7 1.0499 ' 1.0426 1.0376 1-0476 . 1 .. ,., 0.8 1.0563 1.0468 1.0418 1.0523 0.9 1-0646' 1-0551 1.0486 1-0578

0.951.0751

-` ' 1-0651 1.0597 1.0670 ,

TABLE VII _{TABLE IX} TABLE X

_ F0/F=0-Ss 1"0./F=0-35 Fa/F= 0.50 1/D .'

//D 1. //D

0.264 0.212 0.235 0.287' 0.203 0,244 0.309 0.195 0.253 0.331 0-185 0.261 ..., 0.349 0-176 , 0.267 , 0.358 0.162 0.263 0.341 0.145 .. 1 0.241 0.276 0-112 yl 0-188 0.215 0-086 ' 01142 t n d , c

3.

III 4 v 1 v v w v. I I ' 3 1 4, , r 1 . P/13=0.9 P/D-_,--,1:1 P/D--- 1.3 Fa/F = 0.35 Fa/F= 0.50 1 Fa/F= 0.65 Fa/F =0.35 F0/F0.50 Fa/ F = 0.65 Fa/F =0.35 Fa/F = 0.50 0/F=065 P/D _ P/D P/D P/D P/0 P/D Pp) P/D P/D --7.---)Ai re.Ai . lai .' 72Ai =Ai Tai TrAi TrXi TrXi °

[TABLES VII, IX. X

Falb' --= 0.65 foil '. . 0.07665 0.06089 0-04718 1).03852 0.03249 0.02795 0.02645 0.02938 0.037cm 10/F= 0.35 foil 0.09545 0.08092 ', 0.06875 0.05935 0.05190 0.04631 0.04125 0.04087 T 0104477 P/D = 1.3 Fa/F =0.50 foll 0.08975 '0.07009 0.05688 0-04690 . 0.03997 0.03381 0.03010 0.03193 0.03753 Fa/F= 0.65 foll ' 0.08171 0.06524 .. 0.04952 . 0.03977 0.03270 0.02753 0.02622 0.0283 0 0-03556, 0-8500 0.8994 0-9498 0.9968 1.0269 .. 1.0401 1.0436 1-0485 1.0559 ' ₄ 0.8479 0.8970 0.9247 0.9704 0-9994 1.0116 1.0152 1.0204 1.0278 0.8572 0-9047 0.9544 1-0018 1.0323 1.0458 1-0486 1.0520 1.0571 003551 ' 0.9019 0.9511 0.9972 1.0262 1.0360 1.0413 1.0443 1.0507 0.8530 0.8996 0.9486 0.9941 1.0224 1.0351 1.0378 1.0412 1.0482 0,8623 0.9075 0.9572 1.0010 1.0305 1.0429 1-0449 1.0466 1.0505 4.f" 0.8600 0.9044 0.9540 0.9967 1-0249 1.0366 1.0385 1-0406 _1-0457 ' 0.8573 0.9021 0.9513 0.9940 1-0219 1.0337 1-0359 1.0388 1.0441 '-. 0.8668 0.9098 0.9572 1.0012 1.0291 10405 PO416 1.0429 1-0458 0.8646 . 0.9059 _0.9528 _0.9964 1.0242 1-0349 1.0363 1.0383 1.0419 0.8615 0.9042 0.9501 0.9936 1.0211 1.0321 1.0339 1.0365 1-0409 Fa/F = 0.65 op 0,264 0.288 0.313 0-336 0.355 0.363 0.338 0-263 0.200 F0/13- =0-35 //D 0.212 ). 0.204 0.197 0,187 0.176 0.163 0.144 0-108 0.081 P/D 1-3 F0/F= 0.50. 1/D 0-235 _0.245 , 0.255 0.265 0-271 -0.266 0.240 0.180 0-134

-

_{Fa/F =0.65} 1/D 0.264 0.290 0.317 0.342 0-363 0-368 0-334 0.250 0.186 Fa/F = 0.35 , qD 0.212 0.200 41.189 '0.178 0-170 0.160 0-149 0.124 0.099 4. 0.5

(30)

,r

TABLE

Xi

.,... UALOULATION OF r, at DISTRIBUTION FOR A VVAKE-ADAPTED $OREW°

r V

Q

1-6106 1.3249 1-0637 0.8538 , 0-6705 0-5052 0-3531 0.2338 0:1658 tan pi 1 i\

1,

1-4, y.

tan_

0-7256 1 tan p 5 0.647 (1

-0-1'

tan p _ 1.1215 (-gig 6

f tan (pi -13),= (tan Pi-tan p)/(1-1- efill pi tan p).

_

.,,

8

x-values from Fig. 16.

,:

:V

9

Z.:, /-.=-(47rDiz)(x x) sin p, tan (pi -p).

:I N L lif ma: u . ur, x 0-2 0,3 _.0:4 , 0-5 0-6 0-7 0.8 0.9 9.95 . ,e .-: tap p '2

- 1-4,'

, 0.244 0-312 0-396 0.492 0.588 0-682

'

0.776 0.820 ' _-0.829 ,, 1 Vs 3 ' _4, tan f3 (-1 -V)l . 0.2957 0.3472-0.2520 0.4175 0.2399 -0.4992 0,2385 0.5875 0-2375 0.6715 0.2363 .0:7505 0-2351 , 0-8268 0-2208 0-8617 , 0-2116 ' - 0.8688 , 4: .. 1-4' 8.606

---5

tank 0:9551 0.6815 0.5390 0-4552 0-3967 0.3531 03188 0.2873 _0.2731 1 -4-, -1

0-9423

-6 tan (pi =p) 0-5142 0-3665 0.2649 0-1956 0-1455 0.1078 0-0779 0.0626 00582 -,1 N . -A .? 7 sink 06907 0.5632 0.4748 0-4143 0.3687 0:3330 03038 0.2762 0-2635. ₅ t,.r. '-' , 8 x 1-049 0990 0.979 0.976 0.964 0.930, 0.865 0.695 0.526

t

1 -. ,.-naD .

-x 7.1.643.6.88 x x

:

0

-1 0 3

(31)

TABLE XII

CALCULATION OF THE CAVITATION NUMBER

a

10

(x

in kg. per sq. In; y =specific weight of sea water= 1025 kg.

per cu. m. 14

V=v.' cos (-)/sin

15 p -= 104.5 kg/m-4 . sec.2 16

130- e= static pressure at centre of screw shaft-- vapour pressure.

17 =p0-e- (xR) y/ipV2= (10)/(15). 1 10 11 12 13 14 15 16 17 18 x (x R)y

cos(-)

sin f3 re' V 12P2V27 1)0- e -xRy a a-20% 02 705 0.8893 02835 2.100 6 .587 6 17154 7.5670 60536 0.3 1058 09389 0 -2444 2.685 10.316 5560 16801 3.0220 2.4176 0.4 0.5 1410 1763 0.9667 0.9814 0.2333 0.2320 3-408 4.234 14.120 17.915 10418 16770 16449 16096 15789 0.9598 . 12631 0.7678 0.6 2116 0.9896 0.2311 5.060 21.670 24536 15743 0.6416 0.5133 07 2468 0.9942 02300 5869 25.376 33656 15391 04573 03658 08 0.9 0.95 2821 3173 3350 0.9970 0.9980 09983 0.2288 0.2156 0.2070 6.678 7.057 7134 29.095 32.672 34 .411 44231 55775 61870 15038 14686 14509 0)..2343030 0 .2345 02720 02106 0.1876 R)y

'

J. 16801 .

(32)

-1 x, 9 19 20 18 a-20% =Aplq 21 811 for Aplq 22 1 for Ai)lq 23 definite 24 811 25 02 1.6106 331.3 6.0536 0.220 1505.9 15059 0.22 1.0698 0.3 13249 275.0 2.4176 1622-0. 0.1696 08168 04 10637 2325 1.2631 1738.0 0.1338 0.6128 05 08538 1943. 0.7678 1832.0 01061 0.4667 06 06705 1583 4232 0.5133 0.890 1778.7 1891-0 00837 03543 07 0-5052 1240 4073 03658 0.662 1873.1 1898.0 00653 0.2661 08 0.3531 895 4034 02720 0.500 17900 1790.0 0.0500 02017 09 0.2338 554 4247 0.2106 0383 14387 1461-0 0.0379 0.1612 095 01658 38.1 4.357 0.1876 0.339 11239 11200 0.0340 0.1482 19

8-distribution as determined from a strength

calculation according to Romsom.

23

adjusted to 1 for Aplq and 1 at 0.2R.

21 from Fig. 18. 24 from 19 and 23. 22 from 19 and 21. 25 from 20 and 24. TABLE XIII

CALCULATION OF SECTIONS (DETERMINATION OF THE

BLADE-ELEMENT LENGTHS) . -1 -.. 4

(33)

TABLE XIV

CALCULATION OF SECTIONS (DETERMINATION

OF THE CAMBER-CHORD LENGTH

RATIO) 0 20 from Fig. 17. w 27

camber correction according to Ludwieg

and a itizel (Fig. 6).

29 f0ll---(1/k)(f11)(1.3+ 0.71)/212; p.=0.75 (friction correction factor) 30 from 24 and 29. 1 24 25 26 27 28 29 30 31 t:1 t.i ta x sll a fll k (11k)(fil) foil 418 x 0.2 022 10698 0.0638 1.190 00536 005623 02965 o 0.3 01696 08168 00515 1105 0.0475 0.05781 03409 04 0.1338 06128 0.0413 1.025 00403 0,04905 0.3666 'el I. 05 0.1061 04667 00324 0947 0.0342 0-04162 0.3923 PI 06 0.0837 03543 00254 0.870 0.0292 0-03554 0.4246 t=1 11.-07 0.0653 02661 0.0195 0800 00244 002969 0.4546 -5.6 t:1 08 00500 0.2017 00160 0.705 00213 0.02592 05184 +1.7 0.9 00379 0.1612 0.0122 0.550 0.0221 0.02690 0.7097 +11.5 g 0.95 0.0340 0-1482 0.0113 0.412 0.0273 003322 0.0770 -J-181 tl u, r

(34)

4.

TABLE XV

1

33

Actrad.---(1/k)Cfp)x 0-7(1

=075 (friction correction factor).

34 Ace ---Aocrad. x 57.296,

'

35 q)=-Pi*-F Ace. 37 P/D-=nx tan 7 -3 5

tan;

0 j3:1 Pi° 33 -Aa rad. CALCULATION OF PITCII ,34 35 Aa° 9 36 tan e.p ' 37 _P/D ' 38 P/D faired 39 - Y 0.2 0.9551 43.683 0.01250 0.71162 44.399 0.9792 0h6153 06177 4260 0.3 0.6815 34.274 0.01108 0.635 34.909 0.6978 0.6;377 0.6552 4519' 04 0.5390 28.326 0.00940 0.539 ..-28.865 0.5512 0.6927 0.6930 4780, -0.5 0.4552 24.477 0900798 0.457 24.934 0.4649 07303 0.7292 5029 0.6 0.3967 21.636 0.00681 0.390 22-026 0-4046 0.7626 07625 5268 0.7 0.3531 19.449 0.00569 0.326 19.775 0.3595 0,7906 0.7906 5452 0.8 0.3188 17.684 0.00497 0.285 17.969 0-3243 0.8151 0.8132 5609 ,0-9 0.2873 16.031 0.00516 0.296 16.327 0.2929 0,8282 0.8290 5717 0.95 0.2731 15.275 0-00637 0.365 15.640 02800 0H8355 0.8355 .5762 cp. 1 x

(35)

100

Vil

=

0.2

1

COMPRESSION STRESS. centrifugal stress alto turfed.

-ass.Pme4 Ci ICE ) 40 COONS n = r p 0 -,.. function M of rake where . s? I angle z = number of blades

1,: function of Paa/C1 _{Ce= 225 Po.i0}

Po., / 0 30

04

, = thickness In cm I = chord Ill rn

011110111*

,.:...;..00.,....

,.'-'00:1ffilIN

,

4° a° 12° 6°

----.- rake ang e in degrees

---1

1

I

COMPRESSION STRESS, DUE TO CENTRIFUGAL FORCE

11411

Go. 02.1?(A,C-0.58) WPM: n. r DM too D diameter in rn A

=function of Clisn,and rake.

of PnsiO unction ii...0 -f

Illi

III

im,...

RE

....

II

, IR = where: _ n = r COMPRESSION OA STRESS,centrifugel stress Ct [C2+4-) excluded. 411

4

0 _A

, . function

,

n 0095 sT. I pm of rake angle z = number of blades

pi 4

C,= f unction of P0,,I0 Co. 0.66 NOD _ s = thicknes in cm , . hp,' length in m i. 8i

0,irA,IM

adoloilaidi

--A

--.0

...,,,,.wr-- ,...,,,,.wr--,....01

i

4° 8° 12° 16°

-N.

rake angle in degrees

COMPRESSION STRESS, DUE

TO CENTRIFUGAL FORCE 6,. n. D. (P. C -0,346) where' rpm 100

D=propel ler d Lam. In

m

5.=

tune Hon of 0/soondra.angle

G .-__ 0.5 06 0.7 08 09 10 1.1 12 13 14 0.5 0.6 0.7 0.8 09 10 1.1 12 13 14 P/0 at rig .0 2 pip at rf R 0.6

Fig. 15.-Diagrant for the calculation of the strength of propeller blades according to Ronisotn.

14 12 10 6 4 0 (,) 24 22 20 18 16 10 .14 7 .12 .10 5 .08 4 1.06 3 1.04 1.02 1 1 00 (b) 1.14 .12 1.10 108 1.06 1.04 102 100 0.7 f 05 .5 8 4 , e. '0 1 I -I

(36)

.2

t

1

'

" Fig. 16.-Goldstein factor x for z1.7 homogeneous flow

I 7 4,4

,

& _ . i=4 _ e x=42 , . _

...

1Fi.001111111111 111.11111111

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(37)

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(38)

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(39)

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Fig. 19.Comparison of the blade-contour, the camber ratio and the pitch ratio of a 4-bladed wake-adapted screw numerically calculated and de-signed according to the diagrams for 4-bladed wake-adapted screw series.

020 I'D 0.10 0.90 0.90 FY0 0.70 060

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(40)

3 i e r n Via ,

bESIGN OF WAKE-ADAPTED SCREWS)" 459

DESIGN No. 2

DESIGN OF WAKE-ADAPTED SCREW FOR GIVEN THRUST

ACCORDING TO THE DIAGRAMS FOR

4-BLADED-WAKE-ADAPTED SCREW SERIES

The data required for this design are the same as in design

No. 1, that is,

Ship speed V8=16.73 knots,

Wake fraction 0348

v8=16.73 (1-0-348) 0.5144=5.612 m. per sec.

Number of revolutions n=100. 8(100 2.2)/100=98,6r.P:m.

(corrected for the scale effect on the wake).

Thrust in sea water =--110,300 kg.

Immersion of shaft centre line including the wave height =7.570

The determination of D and 'Ai is the same as hi design go8.1.; It is found that:

A=0.1581; D=6.880 m. ; Xj=0-2443 and 11/13=7cXi=0.7675;

The cavitation number do 42,8 is calculated asfollows.

poe-0S R y

17,859-04 D y

y.pve

Go 0.8 2 . 04.5 (5-612)2

1:38.

=

From the strength calculation 80.2/D=0.0481and so, D=0.0230

7 is obtained (see design No. 1).

The charts are to be worked out for values of

. 0' 8=1.25(Aplq)(V/Ve) 2 and 80.8/D=0-0129

for the expected Fa/F ratio of 0.50 (see Fig. 13). There is sufficient

margin in the onset of back cavitation for values of .90.8/D some-what higher than the standard value of 0.0129. It is therefore

not necessary to correct ao 0.8 for the different thickness at 0-8 R.

- From Fig. 10 we find for P/D=---nXi=0.7675 and cra 0.8=9.138

the blade area ratio Fa/F=0.523. With theaid of Figs. 11 and 12,, we can now define by interpolation for the values of Fa/F= 0.523 and P/D=0.7675 the radial distribution of the chord: length ratio //D and the camber ratio foil of the blade sections..

With the aid of Table X we determine the distribution of the

pitch.-In Fig. 19 the values of //D, foll and P/D ofdesigns Nos. 1 and 2 are compared. This figure clearly shows the difference in the

k

" 9.

(41)

distribution of //D, foil and P/D, which is caused by the applica-tion of a standard mean radial wake distribuapplica-tion. _These differ-ences will usually be considerably less, as the radial wake

dis-tribution of design No.

1 is of a very extreme character.

Even with these extreme differences, however, it is not expected

that the efficiency of the propeller constructed in _accordance

with design No. 2 (standard wake distribution) will differ

appreci-ably from that of the propeller of design No. 1 (measuredwake

distribution).

_{Also the propeller of design No. 2 can be}

expected to be free from cavitation in the measured wake field. The differences that appear in the intake velocities at the blade tips are still too small to result in a deviation from shock-free

entrance. If an optimum screw diameter _{can be chosen, the}

method of design which utilizes the charts for series of wake-adapted screws is recommended, because of the saving in calcu-lations, less risk of making errors, and greater simplicity.

CONCLUDING REMARKS

The method of design with the charts of the wake-adapted

screw series (Figs. 9-14) cannot be used where the screw diameter is limited from practical considerations and the r.p.m. is restricted

to a value below the optimum found in Fig. 9. In addition the

newly derived formula

Ai= x-1-0-0862

is no longer valid in this case. _{Resort must then be made to an}

" individual" circulation theory calculation. The correct 7),

value can be determined with the aid of the 71 value from Fig. 2,

corrected by one-third of the percentage difference of the

pitch of the face at 0-7 R of the wake-adapted screw and that of the B series" propeller with which it is compared.

In the numerical examples of calculation a given thrust has

been taken as a starting point. In many instances, however,

the thrust to be delivered is unknown, whereas the available propeller horse-power is known at the early stage of design. The calculation of the thrust T with the aid of the formula is

recommended:

T=p.h.p. x xlrx 75/[173(1-4)) x 0-5144]

where 74, can be determined by means of the B series diagrams

(42)

and 7;7. has a value of 1-04 for single-screw ships. This formula

is at least as accurate as the process of iteration which has to be carried out if, in the case of an individual circulation theory design, the available power, p.h.p., has to be taken as a starting point, and, as a matter of fact, ei and the coefficient

have to be estimated. The use of the above formula, moreover,

saves many calculations.

With regard to the choice of the propeller blade sections, the calculations carried out by means of the circulation theory pro-vide a decisive answer as to the camber ratio foil at a blade-thickness ratio sll, which has been determined by means of a strength calculation. This camber ratio, f0/l, is only correct for the parabolic camber line of the Karman-Trefftz profiles. Such a blade-thickness distribution for this camber line should

therefore be chosen as this allows as large a variation as possible

in the angle of incidence and thus ensures the shock-free

entrance.9 In this way, due to the circumferential inequality of

the velocity field, the blade section will be subjected to cavitation in as small a sector as possible.

Investigations to determine the shape of blade

section

showing the least dangerous type of cavitation in the

circumferentially unequal velocity field, are still in progress. Recent development" in cavitation tunnel test technique, by which an unequal velocity field such as behind a model can be adjusted, offers many new possibilities of approaching the

cavitation-erosion problem.

A camber line favourable from a hydrodynamic point of view is the N.A.C.A. a=1-00 camber line" which is different from the

above-mentioned parabolic camber line. Experiments carried

out by the N.S.M.B. with blade sections having the N.A.C.A.

a=1-00 camber line, in screw models of about 0-24 m. in

diam , have not led to the expected results. This must chiefly

be attributed to the fact that with a screw model of only

0-24 m. in diam it is not very easy to produce the small

radius of curvature of the N.A.C.A. a=1-00 camber line close to the leading and trailing edges of the blade section. The

favour-able properties of the N.A.C.A. a=1-00 camber line may possibly be demonstrated by experiments with larger models.

In addition, attention must be directed to the magnitude of the radii at the leading and trailing edges of the blade sections.

2n

(43)

For propellers designed to the circulation theory, designers

some-times like to construct the sections as close as possible to the theoretical sections, which have no rounded edges: this results

in very small edge radii. It has been proved, however, that due

to these small radii, the thin leading and trailing edges of the propeller blade are too weak. The edges will be deformed,

with corresponding change in pitch, and erosion due to cavitation

occurs. For the occurrence of these troublesome phenomena,

the theory is often blamed and to avoid them, Table XVI gives the edge radii that have proved adequate in practice.

TABLE XVI

Thickness at blade tip0.004 D,ifD<3,000mm.

Thickness at blade tip 0.0035 D, ifD>3,000 mm. Edge radius at 0.2 R, 0.002-0.00175 D for D 1-7 na?

In conclusion it is worth noticing that the type of propeller with cambered blade sections as dealt with in the presentpaper

has a relatively high efficiency if the propeller is overloaded. On the other hand the braking power of such a propeller is not

as great as that of a propeller with ogival sections.

APPENDIX

BRIEF REVIEW OF SCREW DESIGN ACCORDING TO

THE CIRCULATION THEORY

Due to its sucking action, a marine propeller createsa region

of reduced pressure at its back and a region of increased

pressure at its face. These changes in pressure are coupled

with changes in the velocity of the water. The changes in

velocity which the propeller imparts on the water of necessity

represent a loss of energy. These energy losses are divided into

axial losses, which are due to the change in the velocity (ca) of the water in the direction of the propeller shaft, and the tangential losses caused by the change in the velocity (co) of the water in the direction of rotation of the propeller. It will

be evident that the distribution of these velocity changes, co

and co, over the screw disc, is determined by the distribution of

D, m. 6-7 5-6 4-5 3-4 2-3 1-2 0-1

Edge radius at 0.95 R, mm. 4 3-5 3 2.5 2 1.5 1.25

(44)

-DESIGN OF WAKE-ADAPTED SCREWS 463 the screw loading over the blade, since the portion of the blade delivering the greater part of the thrust will impart the greatest

velocity changes to the water.

The efficiency of the propeller is determined by the energy

losses due to the above-mentioned velocity changes. A maximum

efficiency can only be attained if the velocity changes and cu

are distributed over the screw disc so that a minimum loss of

energy is achieved.

With the aid of hydrodynamic calculation methods the velocity

changes imparted by the propeller on the surrounding water

have been determined with success.

At the same time the

condition that must be satisfied by the distribution of the

velocity changes in the screw race in order to obtain a minimum loss of energy has been established. This condition, generally termed the condition of minimum energy loss, also determines the thrust distribution over the propeller blade.

A difficulty experienced in calculating the velocity field brought

about by the propeller is caused by the finite number of blades of the propeller. Due to this fact, the circumferential velocity changes, that is, along a circle of fixed radius r, are fluctuating, namely, a maximum in way of the propeller blades and a mini-mum between the propeller blades. The mean velocity changes occurring at any radius of the screw disc can be calculated by means of the Goldstein reduction factors x, which indicate the ratio between the maximum and the mean velocity changes.

Using the results of these calculations a diagram can be prepared

where the thrust constant CT, and the ideal efficiency (the

term " ideal " implies that friction has been neglected) have been plotted on the basis of the advance coefficient A=v8hraD

(to be compared with the slip of the propeller).

This diagram may, in fact, be compared with an open-water

screw diagram (KT-KQ-14,-A) of a standard series of propellers.

As is the case with the 1, which can be read directly from such

a KT-KQ-11.-A diagram with given KT and A, the ideal

efficiency 76, can be read directly from the CT,-1,-A diagram with given Cr, and x.

When the design of a propeller is started, the data should therefore be known for calculating the thrust constant CT, and

advance coefficient :

(45)

The condition of minimum energy loss enables the velocity changes in way of the propeller to be calculated for a particular

radius r with the known values of 7) and ( v e/nD)(1 - 1

-In the ratio (1 0/(1-4,), d? represents the mean wake fraction over the screw disc and the mean wake fraction at a radius r.

The distribution of forces along the propeller blade has, however,

been directly coupled to this distribution of velocity

changes.

With the velocity field calculated, the distribution of the

pro-duct of the lift coefficient and the length 1 of the propeller

blade at radiusr can be determined directly by means of

hydro-dynamic laws. This product cl is a direct measure of the force

acting at a radius r of the propeller. When once this product al is known, determining dimensions, shape and pitch angle of

the blade sections can be started.

For calculating these blade sections, measurements of the pressure distribution over the blade sections suitable for marine

propellers should be available. In this connection should be

mentioned the blade-section measurements carried

out by

Gutsche, various reports of the National Advisory Committee for Aeronautics, and theoretically derived pressure minima

Aplq on Karman-Trefftz profiles.

With the aid of these data, the product %1 and the radial distribution of the maximum thickness of the propeller blade obtained from strength calculations, the dimensions of the blade

sections can be determined This is done in such a way that

the pressure minima Aplq, known from the blade-section

measurements, do not exceed the cavitation numbers c, which are

dependent on the external conditions. By the external conditions is meant, in this instance, the static pressure at the centre line of the propeller shaft and the velocity field in way of the propeller. If Aplq< a, cavitation will be avoided.

After the dimensions of the blade section have been determined

and the velocity changes in way of the propeller are known, we have to correct the blade sections for a curvature of flow,

since a ship screw has wide blades. This curvature of flow is

caused by the distribution of the velocity changes over the

chord-length of the blade elements (k correction).

Finally the pitch angle can be calculated at any radius of the

(46)

LIST OF MAIN SYMBOLS

Br =coefficient for defining the optimum

diameter.

Cr =power loading coefficient.

i --power loading coefficient (frictionless).

CT =thrust loading coefficient.

i =thrust loading coefficient (frictionless).

cu =tangential component of induced

velocity.

D= 2R -=screw diameter.

=effective camber.

=geometric camber.

fll foil =camber ratio.

Fa/F =blade area ratio.

=camber correction according to Ludwig and Ginzel.

1 =chord length of blade element.

=number of revolutions per sec.

=power. =pitch.

P/D =pitch ratio.

p0e

=static pressure at shaft centre line

vapour pressure.

Aplq =minimum pressure ratio._

=radius.

=maximum thickness of the blade

ele-ment.

811 =thickness ratio.

=thrust.

T i salt =frictionless thrust for sea water.

Vg =speed of advance of the propeller.

=mean intake velocity over a radius.

Vs =ship speed.

-P

(47)

x=r/R

=non-dimensional radial co-ordinate.

=number of blades.

=pitch angle correction.

e - 0.8 Ry

cIoo-s

ipv,2 cavitation number at 0-8 R.

(3 =hydrodynamic pitch angle

(uncor-rected).

=hydrodynamic pitch angle, corrected

for induced velocity. =specific weight.

=circulation around the aerofoil (blade

element).

=drag : lift ratio for an aerofoil of infinite

aspect ratio.

lift coefficient.

12, =screw efficiency in open water.

=ideal screw efficiency (frictionless).

=relative rotative efficiency of the screw.

0 _{=mean thrust deduction fraction over the}

screw disc.

0' _{=mean thrust deduction fraction over a}

radius.

X = v

-=Goldstein reduction factor for the circu-lation.

=advance coefficient.

=advance coefficient with the influence of the induced velocities.

p0e

=mass density of fluid.

=cavitation number at shaft centre line.

pv 2

cP=N+ =geometric pitch angle.

4, =mean wake fraction over the screw disc.

=mean wake fraction over a radius.

or =circumferential velocity at radius r,

-e nD

(48)

BIBLIOGRAPHY

" Theorie mid Entwurf freifahrender Schiffssehrauben," by H. W.

Lerbs. 1945. Hambiugischen Sehiffbau Versuchsanstalt,

Hamburg.

"An Approximate Theory of Heavily Loaded, Free-running

Propel-lers in the Optimum Condition," by H. W. Lerbs. Trans. Soc.

Naval Arch. & Mar. Eng., 1950, vol. 58, P. 137.

"Moderately Loaded Propellers with a Finite Number of Blades and

an Arbitrary Distribution of Circulation," by H. W. Lerbs. ibid,

1952, vol. 60, p. 73.

" Zur Theorie der Breitblattschraube," by H. Ludwieg and I. Ginzel. 1944. Deutsche Luftfahrtforsch. Untersuch. und Mitt., No. 3097. " Hydrodynarnische Grundlagen zur Beresehnung der

Schiffsschar-uben," by M. Strscheletzky. 1950. Verlag G. Braun, Karlsruhe.

"The Calculation of Ship Screws," by R. Guilloton. Trans. Inst.

Naval Arch., 1949, vol. 91, p. 1.

" Enige Beschouwingen over het Cavitatie-Erosie Vraagstuk bij

Scheepssehroeven," by J. Balhan. Publ. No. 104. Netherlands Ship Model Basin, Wageningen; De Ingenieur, 1953, vol. 65. Nos. 6 and 7.

"The Cavitation-Erosion of Ships' Propellers," by J. Balhan.

Engineers' Digest, 1953, vol. 14. p. 163.

"Influence of the Non-Uniformity of the Wake on the Design of Ship

Propellers," by J. D. van Manen. 1951. Publ. No. 100.

Nether-lands Ship Model Basin, Wageningen.

" The Design of Ship Screws of Optimum Diameter for an Unequal

Velocity Field," by J. D. van Manen and L. Troost. Trans. Soc.

Naval Arch. & Mar. Eng., 1952, vol. 60, p. 442.

The Design of Propellers," by J. G. Hill. ibid, 1949, vol. 57, p. 143. " Metingen aan Enige bij Scheepsschroeven Gebruikelijke Profielen

in Vlakke Stroming met en Zonder Cavitatie," by J. Balhan. 1951.

Publ. No. 99. Netherlands Ship Model Basin, Wageningen.

" Open Water Test Series with Modern Propeller Forms," by L.

Troost. Trans. N.E. Coast Inst. Eng. & Shipbldrs., 1950-51,

vol. 67, p. 89.

Resistance Propulsion and Steering of Ships," by W. P. A. van

Lammeren, L. Troost and J. G. Koning. 1948. H. Stain,

Haarlem.

"Propeller Strength Calculation," by J. A. Romsom. Marine Eng. & Naval Arch., 1952, vol. 75, pp. 51, 117.

411) (5) (8) (11) -,(1a) .