Fundamentals of ship resistance and propulsion Part B: Propulsion

OF SHIP RESISTANCE AND PROPULSION

(Course held before "The Netherlands Society of Engineers and Shipbuilders")

PART B: PROPULSION

by

Dr. Ir.

J. D. VAN MANEN

Head of the Research Department of the N.S.M.B. at Wageningen

Publication No. 132o of the N.S.M.B.

Reprinted from

INTERNATIONAL SHIPBUILDING PROGRESS SHIPBUILDING AND MARINE ENGINEERING MONTHLY

Introduction

Allowances on resistance

Allowances on thrust

Allowances on power

Number of revolutions

Practical application of the systematic screw-series diagrams

Introduction

Velocities induced by a propeller

Condition of minimum energy loss

Determination of the ideal efficiency nvi; the - A - Tin, diagram

Effect of finite number of blades; Goldstein re-duction factors

Effect of profile drag

CONTENTS

PART B. PROPULSION

Section I. Introduction to the theory of propulsion

The product of lift coefficient aand chord length

1 of a blade element

Dimensions of the blade sections

Choice of the type of blade sections The Ludwieg-Ginzel camber correction The friction correction

Numerical example of the design of a screw for a uniform wake

Section VI. Screw design according to the circulation theory for a non-uniform wake

Introduction 50. Design method for wake-adapted screws as adopted

Condition of minimum energy loss by the Netherlands Ship Model Basin

The - 2 - np, diagram; Goldstein reduction

factors and Ludwieg-Ginzel camber corrections

Section VII. Screw series diagrams calculated with the aid of the circulation theory

Introduction 53. Application of the diagrams

Design of a systematic screw series 54. Concluding remarks

Section VIII. Special types of propellers

Propellers in nozzles 58. Contra rotating propellers

Paddle wheels 59. Adjustable pitch propellers

Screw propellers with wide blades 60. Vertical axis propellers

Section IX. Trial and service prediction

Average allowances for the service condition

The trial prediction diagram

Measurements on board ships on trial Measurements on board ships in service

1. Historical review 6. Momentum theory of the screw

2. Propeller types 7. Blade-element theory of the screw

3. Geometry of the screw propeller 8. Introductory remarks on the circulation theory of

4. Definitions the screw

5. Development of screw theory 9. Model tests and laws of comparison

Section III. The velocity field behind the ship - Nominal and effective wake

Wake components 28. Relation between wake and thrust deduction.

Inequality of the velocity field Effect of inequality of the velocity field on thrust

Effect of inequality of the velocity field on screw deduction

efficiency and relative rotative coefficient 29. Effect of the rudder on propulsion Section IV. Cavitation

Model tests in the cavitation tunnel; laws of com- 32. Effect of inequality of the velocity field upon

parison cavitation phenomena of screw propellers

Measurements on profiles; types of cavitation 33. Thrust breakdown due to cavitation 34. Cavitation-erosion theories

Section V. Screw design according to the circulation theory for a uniform wake

Analysis of a screw designed for towing condition Optimum number of revolutions

Pitch corrections Data required for a screw design

Numerical example of the design of an optimum screw

Analysis of a given screw in towing or overload

condition

Section II. Design of screw propellers with the aid of systematic screw-series diagrams Theory of the design

Systematic screw series 15. Blade profiles and pitch distribution

Number of blades 16. Data on wake and thrust deduction

Blade area and blade contour 17. Cavitation and cavitation criteria

Blade thickness and blade edges 18. Strength calculation of screw propellers

Shape of generator line and boss; rake angle;

clearances 57,. 61. 30.

-22,

-10. 11,. 12. 13. 14. .43.

FUNDAMENTALS

OF SHIP RESISTANCE AND PROPULSION

(Course given before "The Netherlands Society of Engineers and Shipbuilders") Publication of the Netherlands Ship Model Basin

Part B. Propulsion

by

Dr. Ir. J. D. VAN MANEN

Head of the Research Department of the N.S.M.B. at Wageningen

SECTION I

INTRODUCTION TO THE THEORY OF PROPULSION

1. Historical review

"When a ship moves through water at a certain speed she meets with resisting forces due to water and air. These resisting forces have to be overcome by a propulsive force. This propulsive force, refer-red to as thrust, is generated by mechanisms called propellers.

Transport over water and propulsion of ships

by oars and sails are probably as old as mankind itself. Mechanical propulsion is of a more recent date, though it is difficult to ascertain what mode of mechanical propulsion may claim the right of being called the oldest. The paddle wheel, however, seems to have the best claim to this right, as is testi-fied by the historical fact that in 1543 Blasco de

Garay, by order of Charles V., installed a steam

engine with a steam boiler in a ship propelled by means of rotating wheels. The results at her trial trip held at Barcelona gave rise to a violent con-troversy, and the inventor removed his engine from the ship without once showing it to anyone. Some

250 years passed before another steamship was

placed in service. This was the Charlotte Dundas,

built by Symington for the trade in the Forth-Clyde Canal. In 1807 Robert Fulton's Famous

Clermont was built for the passenger service on

the Hudson River near New York. The Savannah,

a sailing-vessel with auxiliary propelling power, crossed the Atlantic in 1819. The sea-going paddle steamer maintained her supremacy at sea until about

1850, when she was superseded by the

screw-propelled vessel.

Paddle wheels are far from being the ideal means

of propulsion for sea-going ships. Although the propulsion efficiency of the wheel as compared

with that of other propelling mechanisms is quite

satisfactory, there are many other defects in its

operation which have mitigated against its employ-ment.

The principal drawbacks of paddle wheels are:

1. The variable immersion under different loading

conditions of the ship;

The alternate -rise of the wheels above water level while the ship is rolling causes her to make an irregular course;

The high risk of being damaged during rough weather;

The need for large, heavy machinery installat-ions because of the low number of revolutinstallat-ions of the wheel;

The great influence which the large engine

installation has on the subdivision of the ship; and

The increase of the ship's breadth necessitated by the installation of the wheels.

For trade on inland waterways and lakes, espec-ially in cases of restricted draught in association with the limited depth of water, paddle wheels have held their own for ship types whose draughts vary

only slightly (pleasure boats, tugs). The ease of

manoeuvering a paddle wheel ship to a mooring is one of the reasons which account for this fact.

The invention of the screw propeller is claimed

by various nations. The idea of adopting the

Archimedean screw, which had been employed as

a water pump for many centuries, as a ship pro-peller was patented by Toogood and Hayes in

England in 1661. This propeller seems to have been

of the

jet type consisting mainly of a channel

situated in the centre of the ship in which a plunger or centrifugal pump was installed which pumped

the oncoming flow of water in at the bow and

forced it out

at the stern. The reaction thus

generated caused the thrust which propelled the

ship. At ordinary ship speeds the jet propeller is much less efficient than other types of propellers, and it has never been capable of holding its own

against these types. Its use is restricted to very

special ship types where it is of paramount impor-tance that there should be no projecting parts out-side the hull, i.e., a lifeboat.

The first practical application of the screw

propeller took place at a much later date and is

generally attributed to Colonel Stevens, an Ameri-can who from 1802 to 1804 experimented with a boat 71/4-m-long provided first with one and later with two steam-driven propellers. These propellers

4

Colonel Stevens

Pettit Smith

Ericsson

Fig. 1. Screw design by Colonel Stevens (1802), Josef Ressel (1828), Pettit Smith (1836), Ericsson (1836)

showed a marked resemblance to the type of con-struction used to-day even to the extent of being fitted with adjustable blades (fig. 1). However,

Stevens' views found no acceptance in America

because of the lack of interest in and understand-ing of scientific research work in those days.

At Triest in 1828 the experimental steamship

La Civetta which was 18-m-long was successfully

propelled by means of a screw which had been designed by the Austrian Josef Ressel (fig. 1).

La Civetta was powered by a 6 h.p. engine and obtained a speed of 6 knots. Unfortunately, the bursting of a copper steam pipe, which injured

several persons on board, brought Ressel's demon-stration to a premature end after a ten minute run. Subsequent attempts by Ressel to arouse interest in his device in France and England, failed because of financial reverses.

The first really practical use of the screw pro-peller was made simultaneously in 1836 by Pettit Smith, an English farmer, and Ericsson, a Swede.

Smith obtained a patent for a design which was

very similar to Ressel's, viz, a screw with one thread and two complete turns (fig. 1 ) . Ericsson obtained his patent for a quite different type consisting of two co-axially contra-rotating rings with circum-ferential blades (fig. 1). Ericsson's propeller created

no interest in England; therefore he moved to

America. In America, and later in France, his

device found wide application.

On the Paddington Canal in London, Smith

carried out demonstrations with a 6-ton boat fitted with a wooden propeller which was actuated by a

6-hp. steam engine. During one of his trial runs

his boat collided with a vessel moored alongside the bank carrying away one half of the long vane of his propeller. This seeming misfortune actually

J. Ressel

Pettit Smith

turned out tobe a stroke of good luck, for with his battered screw he at once attained a considerably higher speed. Smith possessed a highly developed

faculty of observation and applied the results of

his accident to improve his design by decreasing the width of the blades and increasing the number of threads, producing a design quite similar to our modern marine propellers (fig. 2).

In 1839 Smith built the 237-ton Archimedes.

The trials of this large ship in the open sea around

the British Isles were so successful that the in-troduction of the pushing screw in navigation

henceforth was only a matter of time. It gradually superseded the large unwieldy paddle wheel and by 1860 had acquired undisputed mastery of the

sea. Of course, it must be remembered that the evolution of the steam engine towards, higher

revolutions per minute contributed its share to the

development of the screw propeller. The first screw-propelled steamer was the English vessel

Great Britain. This ship made its first passage across

the Atlantic in 1845. In 1861 the keel of the last

sea-going paddle steamer was laid.

At the close of this brief historical survey it is appropriate to quote part of a speech made by

Prof. Troost. [1] He says, "It is interesting to con-sider what sort of people had the most important

share in the evolution of the screw propeller.

Colonel Stevens was a solicitor and treasurer of the State of New Jersey, Ressel was a forest conser-vator, Smith a farmer. Only John Ericsson was a

technical expert and was already well-known as

the designer of the locomotive engine "Novelty",

which, in 1829, he ordered to be built in

com-petition with Stephenson's "Rocket". Though, in the meantime, some changes have taken place in

N1800

1860

Fig. 2. Different evolutionary stages in the history of the screw propeller

1840

f

Fie., 3. The development in power absorbed by the screw ,propeller

the technical world, even to this very day we find

inventors of ship propellers in the most varying callings. My personal experiences in the matter

range from an innkeeper to a monk. The cause of this penomenon is best expressed in the English saying, "Any propeller will propel any ship", and

by Sir William White's well-known statement: "It takes a first-class naval architect to design a

really inefficient screw propeller". A certain chance

of propulsive action must even be attributed to the most primitive conception in this respect

frequently even in the direction of motion intended by the inventor. This has such a stimulating effect on inventive minds that they are sometimes inclined to apply the converse of Sir William's proposition to themselves".

As we have seen, the screw propeller has reigned supreme as the primary method of ship propulsion for more than a century. Of course many problems have remained, the principal of which is the

ever-lasting demand for screw propellers capable of

absorbing higher powers without increased cavita-tion and consequent erosion (Fig. 3). Nevertheless, the screw still has no actual competitor as a means, of propelling ships.

In concluding this introduction perhaps we

should remark on a mode of propulsion with non-stationary action. Anyone who is confronted with the problem of propulsion occasionally asks

him-self the question: Why should not we use the

system practised by fishes, i.e. that of executing

periodically repeated strokes? The answer to this

question is in all probability implied by the fact

that the technique of the employment of rotating

parts, such as shafts, wheels, etc., has, from the

view-point of efficiency of motion, obtained a lead

over nature. A fish, just like a man on a bicycle,

might advance in a more efficient manner with the

aid of ,a screw propeller which would derive its

ay 'Weer Intakes b Water outlets 'Grids Id Propeller blades Worm-gear hansmission, It Propeller box. 'Faigers v, Course of ship

Fig.. 4., The Hotchkiss internal-cone propeller

rotation from strokes carried out with the tail. When taking a retrospective view of nature for

solving our technical problems we should realize, however, that living beings form a totality to which considerations of efficiency can scarcely be applied.

2. Propeller types

The various propelling mechanisms may be divid-ed into three principal groups:

The so called jet propeller which imparts to the water flowing from ahead an impulse directed

aft. A propeller whose construction is based on

this principle is the Hotchkiss internal-cone

propel-ler (fig. 4). [2] This propelpropel-ler has the advantage

of having practically no parts projecting from the ship's hull and the disadvantage of low efficiency.

Propellers which derive their thrust in the

direction of the ship's course principally from the resisting forces on their moving parts. Among these propellers are the paddle wheels rotating around a horizontal shaft. Two types of paddle wheels are used in practice, viz, those having fixed and those having movable blades. The former type has the advantage of simplicity, solid construction, light weight and low cost of maintenance. Its main

dis-advantage is that for high efficiency the wheels

require a large diameter and therefore must neces,

sarily have a low, number of revulutions. This results in general in having to use heavy,,

low-speed engines. The diameter depends on the angle between the blades and their resultant velocity in respect to the water at their entrance into and exit out of the water. In fig. 5 these angles are indicated

by a and j respectively. From a consideration of efficiency it is recommended that these angles be

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6

Direction of rotation

Course of ship

Fig. S. Wheel with fixed blades

small, so that the entrance and exit of the blades

take place as gradually as possible.

Attempts to improve the design of paddle wheels have led to a design with adjustable blades (fig. 6). This permits the angles of entrance and exit to be reduced by means of a link mechanism. The entrance and exit of the blades can now take place gradually and the inevitable energy losses are reduced to a minimum. The revolutions per minute of this type of wheel can be increased above those of wheels

Direction of rotation

Course of schip

Fig. 6. Wheel with adjustable blades vr

-having fixed blades. Disadvantages of the adjust-able blade paddle wheel are its heavier weight and greater vulnerability. The efficiency of wheels with fixed blades may range from NO to 60 per cent, if the ship is without tow; that of wheels with adjust-able blades is considerably higher and sometimes

exceeds that of the screw propeller. [3] [4] [ 5]

The wheels are generally fitted amidships on either

side. They are so placed to minimize effects of

change of trim and pitching. Where ships have to operate in narrow waterways the wheels are fitted at the stern (sternwheelers).

c. Propellers which derive their thrust in the

direction of the ship's run principally from the

lifting forces on their moving parts. To this group belongs the most important type of propellers, viz. the screw propeller. This propeller, in its present form, consists of a boss on which from 2 to 6 but

generally 3

or 4 separate

blades, either fixed,

detachable or movable, are mounted. It derives its

name from its characteristic movement, namely

the combination of a uniform rotating motion

with a uniform progressive movement. Many types

and special dispositions of the screw have been

designed during the course of the years.

The screw propeller is usually constructed as a pushing screw and fitted as low as possible in way

of the stern (fig. 7). In the

case of sea-going

vessels the screw should have a diameter such that

when the ship is in her fully load condition the

propeller is sufficiently submerged so that the

phenomena of air-drawing and racing of the pro-peller during pitching is avoided as much as pos-. siblepos-. A rough rule of thumb which may be used

in the preliminary design stage of single-screw

ships is to set the screw diameter equal to 0.7 the draught.

In certain types of river and canal vessels it may be that because of design requirements the screw diameter is too large for the propeller to be fitted

entirely below the water level. In such cases the propeller is enclosed in a tunnel. This tunnel is constructed by extending the aft part of the hull so as to bulge over the entire screw or by fitting

a counter plate in way of the propeller.

The choice of the number of screws for a ship depends on many factors, such as the power, di-mensions and type of engines and the ship's draught. Ships are generally built as single-screw or as twin-screw vessels, but triple- or quadruple-twin-screws are sometimes used.

Ferry boats are often propelled by a

tractor screw at the bow and a pushing screw at the stern.

'When the ship's course is reversed, the screws change functions. The fore and after parts of such a ship are generally designed symmetrically.

Ships which must have a combination of satis-factory towing performance and high free running

speed (sea-going tugs, trawlers etc.) are often

fitted with adjustable-blade propellers.

In addition to the propeller types mentioned

above there are several types of propellers which are use. 'ill in special applications.

Fig. 8. Arrangement of a screw with nozzle

A conventional propeller fitted in a nozzle is

very useful for tow boats of restricted draught (fig. 8). The tandem propeller consisting of two

propellers mounted on a single shaft with the

pro-pellers rotating either in the same or in opposite

directions is a type which is only employed for very special purposes. Finally the Voith-Schneider

pro-peller is a propelling mechanism of an entirely

different construction designed in accordance with the aerofoil principle [6].

In contrast to the screw propeller, the

Voith-Schneider propeller must be classed as an nonsta-tionary propeller. It is the only propeller used in engineering whose design has successfully incor-porated a system of propulsion adapted from the natural life of birds and fishes based on the

non-stationary aerofoil principle.

The Voith-Schneider propeller has a large disc fitted in a flat portion of the ship's hull. This disc

carries

a number of

vertical blades resembling

spade-shaped rudders. The construction of the pro-peller is such that the perpendiculars on the blades constantly pass through an eccentric point as the blades rotate around a vertical axis. Because of this oscillating movement of the blades the total thrust acts in one direction. The resultant direction of the

8

Fig. 10. Helicoidal surface with constant pitch

thrust can be altered by changing the disposition of the blades, thus allowing the ship to be propelled straight ahead, astern or sideways (fig. 9). Where high manoeuverability and restricted draught are required this propeller can be employed success-fully. Its disadvantages are: its complicated struc-ture, its high weight, and its vulnerability as com-pared with the screw propeller. Because of higher mechanical losses, its efficiency is lower than that of the screw propeller. The Voith-Schneider pro-peller has rendered good service in practice; and there is already a large number of ships equipped with this type of propeller, particularly for naviga-tion on rivers and lakes.

Of all the propellers discussed above the screw propeller is the most important. The present course will deal principally with propulsion by this pro-peller. The other propellers will be discussed fully in a separate section.

3. Geometry of the screw propeller

When one stands astern of the ship and looks for-ward, the surface of the propeller blade which is

seen, is called the face or the high-pressure side,

and the opposite surface is called the back or the low-pressure side of the blade. The high-pressure side in its simplest shape is a helicoidal surface, which can be defined as the surface formed by a straight line rotating at a constant velocity around an axis through one of its extremities and

simul-Fig. 12. Circumferentially constant and variable pitch

HAI

"v4

Fig. 11. Helicoidal surface with radially variable pitch

taneously moving along this axis

at a uniform

velocity (fig. 10). The axial distance covered per revolution is called the pitch.

In the case in question, the pitch is constant for all the various radii of the propeller but the pitch may also be radially variable. Fig. 11 shows a heli-coidal surface with radially variable pitch. Here,

the pitch at the boss of the propeller, H, is less

than that at the tip, Ho. In this case the pitch

distribution is linear,

but the curve of pitch

distribution, AB, may, assume any possible shape.

With a non-linear variable pitch the generator is

no longer a line but a curve.

When the areas BCD, B'C'D' etc. are developed into a flat surface the lines BD, B"D" etc. always appear to be straight if the helicoidal surface of the propeller has a constant or radially variable pitch.

If this is not the case, the pitch is

also circum-ferentially variable (fig. 12).

The low-pressure side of the screw, is in contrast

with the high-pressure side, not a true helicoidal

surface. The pitch of the low-pressure

side is

circumferentially variable (curve POR in fig. 10). The actual or virtual pitch, Hy, of the screw is

a mean of the pitch of the high-pressure and of

the low-pressure sides (fig. 13).

Fig. 13. Nominal and virtual pitch

Ir. projection

The pitch of the screw, also called the nominal

pitch or face pitch, is generally understood to be

the pitch of the high-pressure side. The use of the nominal pitch is advantageous in that it is indepen-dent of the shape of the blade, the thickness of the

blade, and the shape of the blade sections (screw-blade sections). In addition the use of the nominal pitch facilitates the drawing and the construction of the screw and its measurement and inspection. Drawings of the screw propeller are made using various projections by which the blade surface is developed in a plane. The projections are derived by the intersection of concentric circular cylinders with the propeller blade (fig. 10). Fig. 14 represents a simple screw plan and shows the various charac-teristic names of the parts of the screw.

The area of the projection of the screw blade in the direction of the propeller shaft is the projected

blade area (F)...

If the screw blade is rotated into the plane of the

drawing (each blade section, therefore, rotating through a different angle) the developed blade

area is obtained. If next the curved

cylindrical

sections are expanded, the expanded blade area

(Fa) is obtained. The area of the screw disc (dia-;peter = screw diameter) is the screw-disc area (F). The pitch of the screw is usually expressed as the

ratio of the pitch to the screw diameter, H/D.

With a radially variable pitch the pitch distribution is indicated in the drawing.

Fig.. 14. Various projectio s of the screw

Fig. 15:., Profile with camber tine

For a

proper understanding of the

factors

determining the shape of the blade sections, we

must first know how a section is formed (fig. 15). On either side of a given centre or camber line

(the dotted line in fig. 15) a uniform thickness' distribution throughout the length of the section is plotted. The camber line, is therefore the line

which can be drawn through the mid points of the thicknesses of the blade profile. The shape of the camber line and the distribution of the thicknesses

of the profile over the length of the profile

de-termine the shape of the profile.

Fig. 16 shows some optimum shapes of blade

sections.

Fig. 17 represents a complete screw plan, the

set-back of the blade sections being taken into

con-sideration in all projections.,

The construction of the various projections is, evident from the figure. The shaded portion shown in the third projection is not a section, but denotes the radial distribution of maximum thicknesses of the blade sections. The first projection is generally omitted from the screw plan. From the third

pro-camber tam bentme

nose Inv

10

asR

0:2A

Fig. 16. Optimum shapes of blade sections

jection it is obvious that the screw blades are

in-clined to the vertical. This is done to obtain reason-able clearances between screw and screw aperture. The angle e, showing the degree of inclination of the blades,, is called the rake.

4: Definitions

The propulsive coefficient, which is a measure of the efficiency of propulsion, is:

eo

E.H.P. Wo Vs.

S.H.P. M .

The propulsive coefficient is defined as the ratio

between the towing power and the propelling

power. The towing power, E.H.P. is the power

re-quired to tow a ship with a resistance, Wo, at a certain speed, V.

The propelling power S.H.P. is the power sup-plied to the propeller to propel the ship at a speed Vs.

The towing power E.H.P. is expressed as the

'effective horse power which must be supplied:

E.H.P. =

Wo

75

where:

= the ship resistance in kg at a ship speed

V,; and

V, the ship speed in m/sec.

The propelling power is expressed as the horse

power supplied to the shaft or shafts of the

propeller. The horse power supplied to the

pro-peller is indicated by S.H.P.

271 M n

!Fig. t7.!.. Complete screw plan with expanded cylindrical sections

S.H.P. supplied to the propeller is less than the power of the propelling machinery, S.H.P.. (the

so-called "brake" horse power).

As a result of the mutual interference of ship

and propeller, the propulsive coefficient $0 generally is not equal to the efficiency of the screw propeller

alone. II

S . ve

M. n

where:

S = the thrust of the screw in kg; and

ve = the speed of advance (speed of translation) of the screw in respect of the surrounding

water in m/sec.

This speed, vi, therefore, is the relative velociti

of the water particles behind the ship in way of

the propeller. Owing to

the widening of the

streamlines in way of the stern, the friction along

the ship's hull, and the wave system of the ship, this intake velocity of the water into the screwr

v is generally less than the ship speed V3.

The absolute velocity of advance, which is the difference between the ship speed Vs and the intake

velocity ve

Vv

Vs Ve

is called wake speed.

It is common practice' to express the in. relation to the ship speed.

Vs Ve

₌

=

_=

V, Vs

ye = V,

w) where 11, = the wake fraction.

The thrust S of the propeller generally exceeds the ship resistance Wo, which the propeller has to overcome at a speed Vs. This difference between the thrust S and the ship resistance 1170 is in part due to the screw accelerating the water in way of'

the stern, which causes an increase in frictional resistance and is in part caused by the fact that the screw operates in the potential velocity field' in way of the stern. In addition, the stern-wale system of the ship may, in special cases, be in-1 fluenced by the propeller; and this in turn may

cause 2 change in the wave-making resistance.

Li -0011MINI, T ( '-'-e

Alm .

116. E1111111111M1111M1 111117 .---11113.1111111=M111111 Efe:

7iii

1-11

--Fr --- .0 ..--. ..,;.!, 7::

EvAmeatam.,

_

,

. . 1 12 _.55.-'W

4.

=

75 where:

M = torque in kgm at the propeller; and

n number of revolutions per sec.

Because of friction losses in bearings, thrust blocks, stuffing boxes and transmission gear, the.

voj tc Ion Dr0,eCtiOn 0100e. seCIlons,

wake speed v. V, 2.7 n . S.H.P. 3" (1

The difference in thrust, S, and ship resistance,

Wo, is called "thrust deduction". This force is

generally expressed as a fraction 79 of the thrust.

S = S Wo

, 1Fo

S

=

S w (-)

where 0 = the thrust deduction factor.

The propulsive coefficient So can now be divided as follows:

= S (1

0) S vs

Wo . Vs M'

27r M n 27t M' . n S M Wo $0

The first term represents the efficiency, lip, of the propeller alone (open water condition), which yields the thrust S required at the intake velocity, and R.P.M., n. The corresponding open water

torque, M', usually differs from the torque, M,

of the screw operating behind

the ship. This phenomenon occurs as a result of the difference in the conditions of the screw behind the ship and in open water (see section III, § 27), caused by such factors as turbulence, inequality of the flow field behind the ship, and presence of the rudder.

The factor

M'

m

is called the relative rotative efficiency.

The second term on the right-hand side of the

equation denotes the ratio of the effective towing power W5 . V,5 and the propulsive efficiency S . v, of the screw.

This ratio may be written as follows:

Wo V_

'0

S V, 1

This coefficient, which is determined by the components of wake and thrust deduction, is a

measure of the mutual interference of propeller

and ship and is termed hull efficiency.

The propulsive coefficient o now becomes:

If a propeller moved forward through the water

as it rotates with the same action as a screw in a nut, in one revolution it would move forward a

distance equal in magnitude to its pitch.

However, because the water is accelerated aft,

the propeller actually moves forward in one

re-volution somewhat less than its pitch. This diffe-rence in forward motion is referred to as slip.

With a screw having constant face pitch the

slip velocity is defined as the difference between

the pitch velocity n . H (H being the pitch of the

screw, and n, the number of revolutions per second)

Fig. 18. Relation between pitch velocity nH, speed of advance and slip velocity nH-v

and the speed of advance of the screw, v, (fig. 18). Slip is then defined by the equation:

nH ve v,

sw = _nH

= 1

_nH

Since, in this equation, ve is the speed of advance of the screw in respect to the surrounding water, this slip is called the true slip.

If the wake speed is unknown, slip is often based on the ship speed. This is the apparent slip.

nH Vs

ss =

= 1

nH

nH

The ratio between the two slip values and the

wake fraction is represented in fig. 19 and in the following equation:

1

₌

ve

1-55

Vs

It is customary to determine the slip values using

the face pitch of the screw (nominal pitch). In

this case the term nominal slip value is used. If the

slip values are based on the virtual pitch of the

screw, the term virtual slip value is employed.

5. Development of screw theory

In the

first stage of development of screw

theory, the screw action was explained in a primi-tive way by means of the screw and nut principle.

nH

vs

'ON, 5,nH

ye sw.n H

Fig. 19. Relation between real slip, apparent slip and wake speed

. .

1

=

.

-12

According to this theory the efficiency of the

screw is:

S .

Sw

11P

= S

. nH

With a slip value of sa, = 0, such a definition

results in an efficiency of ?hi = 1.

Later, the screw theory principally developed

along two lines, which led to the adoption of the

momentum theory and the blade-element theory

respectively [7].

The momentum theory explains the origin of the thrust of the screw entirely from the change of momentum caused by the screw in the sur-rounding fluid. It shows more clearly than the screw and nut theory that the efficiency of the

screw without friction is dependent on the screw loading. This momentum theory gives no indication

as to the shape of the screw.

The blade-element theory determines the forces

on each blade element. By integration the total thrust delivered can be found. This theory gave indications as to the shape of the screw, but led to untenable conclusions with regard to the

ef-ficiency.

With the aid of the vortex theory the relation between the forces acting on the blade elements and the changes of momentum occurring in the

surrounding fluid may be derived. 6. Momentum theory of the screw

The screw imparts an increase in pressure, A p

= pi

-

pi (fig. 20) to the fluid flowing through the screw disc F.

If the pressure distribution over the screw disc is assumed to be constant, the force exerted by the screw on the fluid or the reaction force equivalent

to it, can be represented by the equation:

S = Ap F

This reaction force is called the thrust. Further-more, according to the momentum law, reaction force is equal to the change of momentum,

S = e . 0

. C,,

where:

0 = the volume of fluid flowing through

The magnitude of this velocity, v1, generally

differs from the speed of advance ye of the screw

(ve is the relative velocity of the flow at a very

large distance forward of the open-water screw).

The relation between v1, ve and ca can be derived

with the aid of Bernoulli's equation. For the flow aft of the screw we have:

h, (ye

±

ca)2

+ Po = O

V12

+ pi'

and for the flow forward of the screw:

leVe2 + Po = 1-Q -v12

+ pi

Combining these two equations, we obtain:

ie ca (2 V, + Ca) = pi'

pi = A p

S

0

=

=

= Q . v1 . ca or: Vl _{= ye} 1 2C0

From the above remarks on momentum it appears

that the increase of velocity in way of the screw

amounts to half the total increase of velocity. The velocity and pressure changes are shown diagram-matically in fig. 20.

The increase of velocity is associated with a

contraction of the radius. Forward of the screw

the pressure falls with the increasing velocity, which

Ye Ye Po Po AP = P.- P _F P; P2= Po V1 ye P; Po p,

the screw disc per unit time;

e _= the density of the fluid; and ca = the imparted axiallyvelocity change.

With a constant pressure distribution over the

screw disc, is also constant over the screw disc.

The volume of water 0 flowing through the screw

disc is determined by the screw-disc area F and

Ve C.

vi vo +co

2

the relative water velocity v1 in way of the screw (velocity of the fluid in respect of the screw disc);

hence: _{Fig. 20.} Screw-race contraction; velocity and pressure changes in

0 = F . vi

the screw race

.

is in agreement with Bernoulli's theorem. After the

jump in pressure in the screw the pressure falls

again, until at a distance very far behind the screw the pressure in the undisturbed flow Po has been attained once more. The velocity in the screw race has then attained the value:

V2 = ye ±

With the aid of the equation derived above the following expression for the thrust can be found:

S=-Q.Q. ce=e.F.yt.ca=

= e . F (ye + ice) ca

where:

e . F (ye + Ica) = the fluid mass flowing through the screw disc per unit time.

According to this theory the ideal efficiency of the screw, i.e. the efficiency without the influence of the fluid friction, is:

S ve ye

Pi =

. v, ca

where:

S. ve = the effective power delivered by the

screw; and

S. vt the power supplied to the screw.

From this equation it follows that the efficiency increases if the velocity change c, decreases or the speed of advance V,, hence the mass of fluid flowing through the screw per unit time, increases.

The ideal screw, therefore, is one which gives an acceleration as low as possible to a fluid mass as large as possible.

The non-dimensional thrust constant is defined as follows:

C =

Si _{io ye'} . F (2 F (ye ± lca) cc ve2 . F 2 (1 Ve 2 re)

The efficiency expressed in this thrust constant then becomes:

ve . 2

V0 Ca

From this equation it is

evident that the

ef-ficiency is higher in proportion as the screw loading (Cu) is lower, and therefore, the disc area of the screw larger. It also appears that the ideal efficiency

= 1

if the thrust S = 0 and, the thrust

constant C., 0.

In a later stage of the development of screw theory allowance was made for the influence of the rotation in the screw race on the efficiency

[8]. For a rotating movement a momentum

equa-n H

Fig. 21 Diagram of velocities for a screw-blade element with the

influence of induced velocities

non can also be derived, analogous to a movement of translation. For a screw-blade element situated at

a radius r, with a width dr and an area dF, and

rotating at a uniform angular velocity the

tangential force becomes, according to the momen-tum law:

dT = e

. dO . c,,

where:

dQ = dF

. v1 = dF (v, ic,i); and

= the tangentially imparted velocity change. The ideal efficiency will then be:

dS ve . dO . . ye _{1.), Ca}

= dT

. o)r e . dQ (or co r Cu

From the diagram of velocities (fig. 21) of a

screw-blade element, which will be more fully dealt

with in our discussion of the vortex theory, it

follows that

Ca wr

cu Ve ± ica

so that

np. = (ye

_{V, ±} ) ( (or _(or l-cu)

From this equation it will be seen that, owing to

the correction for the rotation in the screw race, the ideal efficiency decreases. This efficiency should also be corrected for the pressure resistance caused

by centrifugal force, which results from the

ro-tation in the screw race.

7. Blade-element theory of the screw

According to this theory the screw blade is

divided into a number of elements [9, 10]. For each blade element the forces which are set up are calcu-lated. These forces are dependent on the magnitude

of the relative velocity V i.e., the velocity at which

the fluid flows along the blade element on the Ca

+

=

. . :=

+

1 r 2 2 .

+

e Ca .

14

Cn

2 2

Fig. 22. Diagram of velocities and forces for a screw-blade element without the influence of induced velocities

angle a, at which the blade element meets the flow, and on the area of the blade element. Fig. 22 shows

the diagram of velocities and forces of a

screw-blade element. For a screw-blade element situated on a radius r, with a length 1 and a width dr, the relative velocity V is the resultant of the speed of advance ve and the rotational speed cor. The blade element meets the flow at a small angle a. A lifting force dA is set up, perpendicular to the direction of the

relative velocity and a resisting force dW in the

direction of this velocity. The components dA and dW, combined, yield a force dP, which, resolved in

the direction of translation and a direction per-pendicular to it, yields the components of thrust

and torque dS and dT.

The total thrust developed by a screw with z

blades can be found by integration:

S = z fdS. dr = z f (dA . cos

dW . sing) dr

The torque required for this follows from:

M = z f dT . r . dr = z f (dA .

sin 13 +

0

dW . cos

g) r. dr

The lifting and resisting forces were determined

by Froude with the aid of experiments with flat plates which were moved forward through the

water at a definite angle of attack.

The neglect of the changes in the velocity of the water in way of the screw must be considered as one of the chief defects of this theory and one of the reasons why it has never been possible to obtain

satisfactory agreement between experiment and

blade element theory calculations.

8. Introductory remarks on the vortex theay of

the screw

The line integral of a flow field along a closed

curve (at a definite instant in time t) is termed circulation T.

v. ds = P

Fig. 23. Magnus effect

A line integral represents the integration of the

product of a path element ds and the component

of velocity vs in the direction of this path element.

In hydrodynamics two kinds of flow fields are

distinguished, viz.:

vortex-free fields, for which the circulation I' is zero for any given closed curve; vortex fields, for which the circulation I' is not zero for any given closed curve.

An infinitely long circular cylinder has no lift in an ideal flow. It appears, however, that if through rotation this cylinder causes a vortex flow or

circu-lation, a lifting force is created (Magnus effect). The Flettner rotor, formerly employed on ships,

was based on this principle. The superposition of a

streamline flow and an eddying flow (fig. 23)

results in an asymmetrical flow field. By applying Bernoulli's equation it is found that the pressure at P is lower than that at O. A lifting force A is

therefore caused perpendicular to the direction of

Fig. 24. Fluid particle in eddy flow

. fl .

.

flow. The curve of the streamlines and the position

of the stagnation points

S1 and S,, where the

velocity is

zero, depend on the strength of the

circulation.

The vortex flow is characterized by a definite

relation between the velocity v and the distance r from the circular streamline concerned to the centre

of the cylinder. In this case the velocity along a

streamline is constant and inversely proportional to the radius r.

This relation can be explained as follows (fig. 24):

According to Bernoulli's theorem,

p iQ v2 = p

dp +1(2 (v + dv)2

hence,

dp ± 2 .

v. dv = 0

or

dp =

. v. dv

[(dv)2 may be considered so small as to be

negligible.]

The centrifugal force exerted on a fluid element

is:

K = (9 . dx . dy. . dr)

.V2 For equilibrium, dx dy dp or dv dr

+

= 0 yields

Except for their cores, whirlwinds and water

spouts are vortex flows occurring in nature. An aerofoil causes vortex flow or circulation on account of its asymmetrical shape. On the under-side of an aerofoil, the high-pressure under-side, there is as a rule an increased pressure p, hence, according

to Bernoulli's theorem, a low velocity. On the upper side of the aerofoil, the lowpressure side,

there is, as a rule, a reduced pressure pz, hence, a high velocity. The relative flow along an aerofoil can be imagined to be composed of a streamline flow of a velocity V, and a vortex flow around the

aerofoil of a velocity v' (fig. 25). According to

v2. dr

=

Q. v. dv

V r = constant

Fig. 25. Circulation around an aerof oil section

Bernoulli's theorem we have:

Pd + 1 (V

v')2 = p, + e (17 + v')2

zip = Pd

p, = 2 t) V. V' and dv = 2v

where Av = the difference in velocity under and above the aerofoil.

The lift dA of an aerofoil element of length 1

and span dx will then be:

dA -= dx f4p.dyo V dx f zlv dy

0

Now,

fAv. .dy=

. ds = F (the circulation)

hence,

dA=t2.V.F.dx

This equation, which is of fundamental im-portance for the calculation of the lift to be

ex-pected with a given aerofoil, is known as Kutta-Joukowsky's theorem.

If it is desired to calculate the lift dA for a given

aerofoil, the circulation I' has to be known. For

determining the circulation, some more intimate acquaintance with aerofoil phenomena is essential.

Fig. 26a and b. Starting vortex and circulation If an infinitely wide section (two-dimensional

flow field) is accelerated

to a velocity V, the

circulation will not instantaneously develop. At

first a flow as shown in fig. 26a will occur in which the after stagnation point S2 does not coincide with the trailing edge of the blade section. A flow around

this trailing edge of the blade section will take

place. Theoretically an infinitely high flow velocity

would occur in way of this sharp trailing edge.

This is in reality hardly to be expected. Owing to the high pressure at the back stagnation point S2, the fluid flowing around the trailing edge is locally forced away, during which a free vortex disengages

itself from the boundary layer around the blade

section. Similar vortices are set up during the take-off of aeroplanes. On this account they are called starting or initial vortices.

. . . . . v r . . .

16

The splitting off of the starting vortex, the

creation of the circulation, and the resultant

alteration of the streamline pattern result in a

shifting of the back stagnation point to the trailing edge of the blade section (fig. 26b).

The strength of the starting vortex and of the

circulation increases until the back stagation point coincides with the sharp trailing edge of the blade section. At the back of the section a smooth flow from this section will then be found to take place at finite velocity. The starting vortex is carried off with the main flow.

The aerofoil-shaped section is the generator or nucleus of the circulation. When the profile chord of the aerofoil is reduced to zero, the aerofoil sur-face changes into a line, the so-called lifting line. This lifting line, which forms the core of the

circu-lation, is called a vortex, in this case a lifting or

bound vortex.

In the case of an aerofoil with infinite span the circulation is constant for each point of the trans-vei-se axis, but this can no longer be the case with an aerofoil of finite span. With an aerofoil of finite span, a flow around the aerofoil tips occurs from

the increased-pressure area under the aerofoil to

the reduced-pressure area above the aerofoil. The result is that vortex flows are created at the aerof-oil tips. Such a flow, after passing the aerofoil,

remains behind in the fluid (air) and forms the

tip vortices. These tip vortices are free vortices, i.e.,

the core of the vortex contains no solid bodies.

Free vortices in contrast with bound vortices, pro-duce no forces.

Helmholz' theorem says that in an ideal fluid a vortex can neither be created nor disappear. With

an aerofoil of infinite span it is evident that this theorem is satisfied. The lifting vortex extends in-finitely in both directions. With an aerofoil-shaped section of finite span the bound vortices around the

section, the trailing tip vortices and the starting

vortex form a closed whole (fig. 27), so that this phenomenon too is in agreement with Helmholz' theorem. The tip vortices, therefore, are the conti-nuation of the bound vortex, whereas the starting vortex makes the vortex system one closed whole.

When the profile chord or an aerofoil is not

reduced to zero, it is no longer possible to replace the aerofoil by one lifting vortex. In this case the

TIT ,NO VOMIT X

r

STAPlING TORTE

Fig. 27. Closed vortex system of aerofoil of finite span

Fig. 28. Vortex system of blade section which has been replaced by a vortex sheet

aerofoil is replaced by a series of lifting vortices, whose combined strength is equal to the circulation. With a section of finite span the vortex system is then as shown in fig. 28.

From fig. 27 it is evident that in the wake of the aerofoil downward velocities are induced by the vortex system. If the magnitude of these

in-duced velocities far behind the aerofoil are indicated by c,, the induced velocity in way of the aerofoil will be icn; since at a point far behind the aerofoil it may be said that the free vortices, which princi-pally induce the velocity, extend infinitely far in both directions, whereas at a point on the aerofoil the free vortices extend infinitely far on one side only.

In summing up all these facts for aerofoils of in-finite and in-finite spans we may say:

For an aerofoil of infinite span the strength of

the circulation or vortex across its width is constant (two-dimensional flow). In an ideal

fluid this aerofoil has lift only given by

dA=9..V.F.dx

and no drag.

For an aerofoil of finite span the circulation

decreases towards the aerofoil tips

(three-dimensional flow). Trailing free vortices are

created which induce downward velocities.

These free vortices represent a loss of energy. The aerofoil of finite span, therefore, has both

lifting and resisting forces in an ideal fluid.

This resistance is called the induced resistance which is

a result of the finite span of the

aerofoil.

Fig. 29. Relation between the effective and the geometrical angle of attack

Owing to the occurrence of the induced veloc- 1,6 0,32

ities, the angle of attack and, consequently, the lift

decreases (fig. 29). An aerofoil of infinite span,

therefore, has, at a smaller angle of attack, the same lift as an aerofoil of finite span.

The ratio between the angle of attack ai for an aerofoil of infinite span (effective angle of attack)

and the angle of attack a for an aerofoil of finite

aspect ratio (geometric angle of attack) is,

where:

F = 1

. dx or 1 . b respectively (area of the element under consideration)

The coefficient of resistance or drag coefficient,

=

. V' . F (finite span)

and

(infinite span)

where the total resistance W = profile resistance

± induced resistance.

The quality of an aerofoil is represented by the drag-lift ratio.

1rP CP

A

In fig. 30 the results of experiments with aero-foils in a wind tunnel are shown diagrammatically.

In this connection the following observations

must be made:

For small angles of attack the lift coefficient

is directly proportional to the angle of attack ai . cc 2 7E ai

The angle ai at which the lift A

= 0,

is not

zero but

negative. The magnitude of this negative zero-lift angle is dependent on the

shape of the aerofoil.

0 0

ci.

I A

AVA

LA III

Fig. 30. Relation between the lift and drag coefficients Ca and Cp and the angle of attack of a profile of infinite spin The drag coefficient is fairly constant for small angles of attack.

The drag-lift ratio is a minimum with a small positive angle of attack.

9. Model tests and laws of comparison

In research experiments to investigate propulsion by means of model tests, the laws of comparison which were drawn up for the experimental research into ship resistance are applicable. For conducting these model experiments the following four princi-ple types of model tests are distinguished:

the open-water screw test;

the model self-propulsion test with the

com-bination

ship model + screw

at different

speeds;

the overload test with the combination ship

model + screw at a constant speed but differ-ent tow-rope forces; and

the screw test in the cavitation tunnel. a. The open-water screw test

Open-water screw tests are carried out by the Netherlands Ship Model Basin with the aid of a

a = a

V

Cr'

(in radians)

In a viscous fluid an aerofoil of infinite span has

a profile resistance consisting of a frictional re-sistance and, possibly, a pressure resistance. The frictional resistance is dependent on the length of

the aerofoil (Reynolds' number) and the

rough-ness of the surface. The pressure resistance depends

on the thickness-length ratio, the shape of the

aerofoil, and the angle of attack.

The total resistance of an aerofoil of finite span, consists of the induced resistance and the profile

resistance.

The quantities dealt with are definable

non-dimensionally as follows:

The lift coefficient of an aerofoil,

1,2 024

-8

0 8 16 wp

P =

. V2 . F A 4.(f = 21g. V2

. F

0,8 0,16

Q4 us

. . . Wp 4.a c.. d. d,

18

Fig. 31. Diagrammatic representation of a boat for carrying out open-water screw tests

self-recording dynamometer placed in a fine

wooden boat, especially

built for

this

type of

experiment, with a shaft tube protruding forward. The screw moves in front of the wooden boat in a homogeneous velocity field which is left

undis-turbed by the potential flow of the boat (fig. 31). Newton's general law of similitude, according to which the specific forces (such as thrust and torque constants) on the model and the full-scale object are similar, may be applied if the following con-ditions are complied with:

geometrical similarity; kinematic similarity; and dynamic similarity.

Geometrical similarity is complied with as the model is the same shape as the actual object to a reduced scale.

Kinematic similarity is complied with if the

velocities at corresponding points of the model and the true object have the same direction.

The ratio of the speed of advance ve and the

circumferential velocity .7 nD must, therefore, be the same for the screw model and the actual screw

(the values for the full-size screw are accented).

nD n'D'

The ratio vIs called the advance coefficient

n .D

A, so that .A = A', and, since

v e H

A=

= ( 1 - s)

.

H/D

n . D

nH D

similitude of the advance coefficient also means similitude of the slip

If similitude of the advance coefficient is

complied with, both the speed of advance and the number of revolutions can still be chosen at random with open-water screw tests.

For dynamic similarity Froude's and Reynolds' laws must be satisfied.

Froude number is defined for a screw as follows:

nD

Fr=

-=

n . VD/g

g . D

As a characteristic length the screw diameter D

is chosen, and for the speed, the circumferential

velocity a nD.

For similitude of Froude number the following

equations apply to the screw model and the

full-size screw: nD n'D'

g D

1/g or ' VT.D'

= "

n = n

According to Froude's law, therefore, the screw model should be tested at a number of revolutions

which equals the number of revolutions of the ship screw multiplied by the square root of the

model scale.

According to Froude the same ratio then holds for the speed of advance v e as for the ship speed,

V, in the resistance experiment:

nD = n' D'

lie Ve

of

V, = V = V,

,

n D

1

n' D'

V a Reynolds' number can be defined as follows [11] :

Re = c

. _10,7

n D where:

= A2 + (0.7 a )2

According to Reynolds' law the number of

revolutions would have to comply with the

following equation for the open-water screw test:

10,7 . nD 07 . n'D' V' or, if v v' n n

,D'

D 10,7 0,, n'. a2

The thrust constant (specific thrust) is

gener-ally defined as follows:

Fr = Fr' or

Km =

Ks =

_{(nD)2 D2 e D4 n2}

The torque constant:

e (nD)2 D2 D

. D5 . n2 Ve nD 1:e n'D' of a. c. . D' c . . . .

0,8 0.6 0,4 0.2 a 1.

The values of thrust, torque, the number of

revolutions per second, 'and the speed of advance,

recorded during the measuring runs, are plotted

as K, and K111 values against A (fig. 32).

As is the case with model experiments for investigating ship resistance it is obvious that with the open-water screw test, Froude's and Reynolds' laws cannot be satisfied simultaneously (similitude

of advance coefficient or slip always must be

established).

Experiments carried out with open-water screw models during which the depth of the screw under the water level was varied, have shown that

notice-able surface waves will no longer occur if the

distance from centre of screw shaft to water sur-face is equal to or greater than the screw diameter [12]. From this it follows that if this condition is

complied with, Froude's law can be left out of

account for the open-water screw test.

In the case of open-water tests with screw models it is almost impracticable to comply with Reynolds' law. As with model experiments for investigating

ship resistance care should be taken, by an

ap-propriate selection of the model scale and number

of revolutions, that Reynolds' number does not fall below a certain critical limiting value, since below this value the measurements will be

in-fluenced by laminar-flow phenomena. The results of open-water screw tests are an important aid in the calculation of screws (see section II). They form the basis on which the interaction between

ship and screw has been analysed.

b. The model self-propulsion test

The combination of screw and ship model renders any deviation from Froude's law impossible. For a

self-propulsion test the model scale should be

chosen so that for the ship model as well as for the screw model the critical value of Reynolds' number is exceeded. The development, of the methods of generating turbulent flow artificially now permits satisfactory resistance tests to be carried out with

models of 1.50 m length [13, 14]. With a ship

model of this size, however, it is still impossible to carry out a proper self-propulsion test. According to the experience gained by the Netherlands ship Model Basin, the minimum length of ship models

with which a reliable self-propulsion test can be

carried out is approximately 2.50 m.

If the model scale a has been fixed, model speed and number of revolutions of the screw model will follow from:

vs

V?, 0 del _v7e

where:

Wir = frictional resistance of the model;

Wf: = frictional resistance of the ship;

1 s ,1 A P ' Km . 2 71 I ve Ks = M sn = 1 -05,2 ve n 0

It

i

n H AI 10 Km

IIIII.

A

--4- 0.2 A 0,4 06

Emil

08 1,0 0.8 0,6 0,4 0,2-

-.--

0 sn

Fig. 32. Characteristics of a screw in Wren water; and the K, - Km - A relation

rt -= N/cx

where:

Vtn cni r. I = model speed in m/sec; = ship speed in m/sec;

n number of rev/sec of the screw model;

and

ii'= number of rev/sec of the ship screw.

It is the practice in carrying out a self propulsion test to give the screw model a constant number of revolutions during a measuring run. The speed of the ship model is accelerated by the towing carriage to the model speed to be expected at this number of revolutions. The ship model is afterwards pro-pelled by its own screw; the towing carriage keeps running over the model at the same speed. During

the measuring run, thrust, torque, number of revolutions of the screw, and model speed are

recorded.

On the Continent of Europe, in America and

in Japan the self-propulsion test generally consists in carrying out a number of measuring runs within a specified speed range, during which for each speed a corresponding tow-rope force Ra, acting in the direction of motion, is provided to compensate for the relative difference in the frictional resistance

between model and ship.

This tow-rope force or friction correction Ra is:

W'1,

a3

or

20 W7 r

=

fr A. .

Q.

specific frictional the model; W

fr =

specific frictional A 17,2 . S23

the ship; and

= wetted area of the model.

The friction correction Ra is, therefore, de-pendent upon the choice of specific coefficients of frictional resistance (Froude, Telfer, Schoenherr, Hughes, Lap-Troost).

The values of torque, thrust, number of

revo-lutions, and speed can be determined for the actual ship by multiplying the measured model values by a4, a3, 1/N,/a, and \/a respectively.

Allowances must be made on the "tank" values

found for the trial and the service condition, on account of the roughness of the ship's skin, the

increased ship resistance in a seaway, the wind

resistance, and the resulting fall in screw efficiency

caused by the heavier screw loading. These

al-lowances will be discussed more fully in section IX.

c. The overload test

In England the self-propulsion test consists in

carrying out a number of measuring runs at

a constant speed (generally corresponding with the service speed) but with variable screw loading (the

so-called overload

test). The screw

loading is

varied by means of the towing force (R0'). The condition in which the towing force is equal to the friction correction (R,), is called the

"self-propulsion point of the ship" (tank condition),

the towing force R,' = R, working in the direction of the motion. The condition in which R,' -= 0 is

termed the "self-propulsion point of the model".

This method of carrying out the self-propulsion test (overload test) has the advantage that accurate values are obtained for the influence of the

over-load on the screw efficiency and the number of revolutions. For a prediction of trial and service

performances within a certain speed range several overload tests, are necessary.

The best way of investigating experimentally ship propulsion in the model basin is to supplement a self-propulsion test within a specified speed range by an overload test at the trial or the service speed.

d. The screw test in the cavitation tunnel

For making predictions in the design of screws

about the occurrence of cavitation and the

be-haviour of the screw under cavitating conditions,

a model test in the cavitation tunnel is required.

Such a model test can be compared directly with an open-water screw test, provided, however, that the static pressure in the water conforms to a model

law, viz., the law of similarity of the cavitation

number. This subject will be discussed more fully in sub section 8 of sections II and in section V. References

Troost, L.: "Proefschepen, modelproeven en coordinatie". In-augurale rede. Delft, 8 mei 1946.

Hotchkiss, D. V.: "The Hotchkiss internal cone propeller". The Shipbuilder 1931. p. 180.

Gebers, F.: "Das Schaufelrad in Modellversuch". Springer Ver-lag, Wien 1952.

vapid', H.: "Paddle Wheels". Part I, The Inst. of Eng. and

Shipb. in Scotland, 1955.

Krappinger, 0.: "Schaufelradberechnung". Schifftechnik 1954. Mueller, H. F.: Recent developments in the design and ap-plication of the vertical axis propeller. S.N.A.M.E. 1955. Todd, F. H.: "The Fundamentals of Ship Propulsion". Trans-actions of the Institute of Marine Engineers Vol. LVIII, No. 2, 1946, p. 23.

Betz, A.: "Eine Erweiterung der Schraubenstrahltheorie". Zeit-schrift fur Flugtechnik und Motorluftschiffahrt. 1920, p. 105.

Froude, W.: "On the elementary relation between pitch, slip and propulsive efficiency". Transactions of the Institution of Naval Architects 1878, p. 265.

Taylor, D. W.: "Resistance of ships and screw propulsion''. New-York 1893.

"The Choice of Suitable Reynolds Number for Model Propeller Experiments". Fifth International Conference of Ship Tank Superintendents. London 14-17 November, 1948.

Kempf, G.: "Immersion of propellers". Transaction of the North East-Coast Institution of Engineers and Shipbuilders, 1933/34,

p. 225.

Hughes, G. and Allan, I. T.: "Turbulence on Stimulation on Ship Models". S.N.A.M.E. 1951.

Nordstriim, H. F. and Edstrand, H.: "Modeltests with turbulence producing devices 1951". No. 18, Publikatie van Starens Shippsprovningsanstalt, Goteborg. resistance of resistance of 2 f .. I I 12. 14.

SECTION II

DESIGN OF SCREW PROPELLERS WITH THE AID OF SYSTEMATIC SCREW SERIES DIAGRAMS'

Theory of design

10. Systematic screw series

An important method of screw design is that

which is based on the results of open-water tests with systematically varied series of screw models. These screw series comprise models whose charac-teristic screw dimensions, such as pitch ratio H/D,

number of blades z, blade-area ratio Fa/F, blade

outline, shape of blade sections, and blade thick-nesses, are systematically varied.

The best-known screw series are those designed by Froude, Schaffran, Taylor, Schoenherr, Gawn

and Troost [15]. In this course we shall confine

ourselves to the B screw series of the Netherlands Ship Model Basin.

Before we enter into further

details, special

attention should be paid to the characteristics of

the screws in open water. As has already been said in Section I, it is customary to reproduce the results of an open-water screw test in a diagram in which the thrust constant K, and the torque constant Km have been plotted against the advance coefficient A

(fig. 32).

K, ;

ve

e . D4 . n2' e . D5 . n . D

The screw 'efficiency can be expressed in terms of these non-dimensional quantities as follows:

S ve K, A

r _{2 OT Mn Km. 2 7r}

In general, the nominal slip sn is also indicated in the open-water screw diagram. The relationship between the nominal slip sa and the advance co-efficient A is:

sn = 1

_nH

ve

= 1

_{nD H}

ve D

_H/D

A

where H = the face or nominal pitch.

In fig. 32 the left side of the diagram denotes

the area of high screw loading (towing). The

condition A = 0, or 1100 per cent, slip,, represents the "bollard pull" condition.

The right side of the diagram denotes the area

of low screw loading and low slip values. Screws of ships of very high speed with little slip have these

characteristics. Within this "low-slip" area the

efficiency reaches a maximum and then rapidly

decreases to zero. This rapid decrease of efficiency

is caused by the fact that with the very low slip

values the frictional resistance forces exerted on the blade sections dominate the lifting forces.

With a nominal slip s, = 0 or, since s. = for the condition A = H/D,, the thrust H/D

and torque constants still have positive values. 'The' reason for this is that, with a nominal slip sn = 0,

the angles of attack at which the blade sections

meet the flow are about 0°. From fig. 30 it will be

evident that with ai = 0 the profile still has a

positive lift value.

The virtual pitch 11, (a mean pitch of the face

and the back of the screw) is the pitch associated with the zero-lift lines of the blade sections

con-cerned. The thrust constant K, = 0 if the virtual

slip se = 0..

The virtual pitch Hi, can now be simply calcu-lated:

V e

1 := 1

n Hy/D

if sy = 0'; hence, if K, = 10, then Ht/D = A.

If the characteristics of a screw in open water

are given, the virtual pitch ratio can be read

directly at the value K, = 0.

A systematic screw series is formed by a number

of screw models (five or six) of which only the

pitch ratio H/D is varied. All other characteristic screw dimensions, such as diameter D, number of blades z, blade area ratio Fa/F, blade outline, shape of the blade sections, blade thicknesses, and

boss-diameter ratio dn/D, are the same. With ,such a

screw series, open-water screw tests are carried out, A

=

. .

K,

.

= 1