A study of rudder action with special reference to single-screw ships

sA STUDY OF RUDDER ACTION WITH

SPECIAL REFERENCE TO

SINGLE-SCREW SHIPS

BY

MAGNE LoTVEIT, Graduate Member

Excerpt from the Transactions of the North East

Coast Institution of Engineers and Shipbuilders,

Vol. 75.

NEWCASTLE UPON TYNE

PUBLISHED BY THE NORTH EAST COAST INSTITUTION

OF ENGINEERS AND SHIPBUILDERS, BOLBEC HALL

-fife

Nbr.

time ifit'

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MADE 4NDP-*YINTiô GREAT

A STUDY OF RUDDER ACTION WITH SPECIAL

REFERENCE

TO

SINGLE-SCREW

SHIPS

By MAGNE LoTVEIT, 'Graduate Member

SYNOPSIS.This paper details the results of some investigations which have been made to clarify the action of a ruddPr working in the propeller slip-stream

of a single-screw ship. As there is no difference, in principle, between a low

aspect-ratio wing and an ordinary streamline balanced rudder, the hydro- or

aero-dynamic properties of rectangular wings having aspect ratios 1.0 < A <20

are first briefly discussed. Scale effects, especially those in the maximum lift

coefficient, are dealt with ; and it is shown how these limit any conclusions which

can be drawn from model tests concerning rudder stalling in the ship itself. It is

further shown how the effective rudder angle of attack and the effective rudder

velocity in relation to the surrounding water are different for a free-running rudder

and a rudder placed behind a hull and propeller. The simple momentum theory

of the propeller has been used to calculate the mean axial velocity of the water

in the slip-stream ; and this velocity is compared with the measured velocity for

one special case. The properties of the propeller slip-stream are considered and

its influence upon the rudder is discussed. The pressure distribution over the

surface of a rudder placed in the slip-stream behind the propeller on a model of a

24,000 tons deadweight tanker has been measured for seven different rudder angles. Based upon these pressure measurements it is shown how the spanwise load distribution for the rudder is different for the rudder to port or to starboard ;

and it is further shown that it is impossible to obtain elliptical spanwise loading

for different rudder angles for a rudder placed in the slip-stream of a propeller

without introduction of twist.

The lift of the rudder has been determined from the pressure distribution and

has been compared with the lift found by direct measurements. Agreement within ±3 per cent is obtained. Furthermore, it is shown that the drag /lift ratio

of the rudder in the slip-stream is considerably higher than for a free-running

rudder due to the unfavourable pressure distribution of the untwisted, symmetrical rudder in the slip-stream.

Finally, the calculation of the rprid,or forces is discussed and it is shown how

these forces may be calculated in the design stage. The calculations have been

carried out for model M 291 and a comparison is made between the predicted

rudder lift and moment and the actual, both for ahead and astern motion. Introduction

THE

upon the forces generated by its rudder.

manceuvring characteristics of a ship do not solely depend

They depend also upon

the hull form and its appendages. The rudder acts more or less

as a triggering device inducing the ship to assume an angle of attack

to the flow, and the forces and moments then generated by the hull

itself together with the rudder forces will determine the manceuvring

, characteristics of the ship. It is clearly difficult to relate the forces and

moments generated by the rudder directly to the ship manceuvring

characteristics, and much investigation must be undertaken before

every detail in connexion with steering and manceuvring is completely

88 STUDY OF RUDDER ACTION WITH SPECIAL REFERENCE To SINGLE-SCREW SHIPS

and, as the greater part. of the merchant fleet of the world consists of

single-screw ships having streamline balanced rudders, a study of this

type of rudder should be of special interest.

Some investigations,

which possibly throw new light on this subject, have recently been

carried out at the Norwegian Ship Model Experiment Tank, Trondheim,

Norway, and are here reported.

The tests were made on a model of

a single-screw tanker of about 24,000 tons deadweight. A full

descrip-tion of the model, its rudder and propeller is given in the Appendix.

The rudder tested is an ordinary streamline balanced rudder of 20 per

cent thickness ratio and aspect ratio 1 .585. All tests have been carried

out with the model restricted to a straight course.

Forces Acting upon a Rudder in Open Water

In principle there is no difference between a deeply submerged streamline rudder and an aerofoil of low aspect ratio. When the rudder is working in a uniform flow, aerodynamic theory can be directly applied to determine the

rudder forces and moments. This theory and the great amount of experimental

data available, are very useful tools in the hands of the rudder designer. The forces acting upon an aerofoil are usually expressed as dimensionless lift, drag,

and moment coefficients denoted by CL, CD, and Cmc/4 as functions of the angle of attack a. For high aspect-ratio wings the lift coefficient curve can be

considered as a linear function of a for values of a less than the stall angle. For symmetrical aerofoils we always have for a = 0, CL = 0; and the slope

of the lift-coefficient curve therefore determines the values of CL for values of a

somewhat less than the stall angle. For low aspect-ratio wings we may as an

approximation consider CL to be a linear function of a for values of a less than

the stall angle and for A> 1 .0 ; and thus for the usual symmetrical rudder

profiles the lift-coefficient curve is approximately determined up to the stall angle if the slope of the lift-coefficient curve is known. As shown later, the scale effects on the stall angle and the maximum lift coefficient are of such magnitude that it is hardly possible to determine the stall angle of a rudder by model tests, and therefore we are limited to consider values of CL,<C1.... The slope of the lift coefficient curve dCLIda is mainly a -function of the

aspect ratio A and the plan form of the wing, and only to a small extent a

function of the thickness and the form of the profile, Fig. 1 shows the variation

of dCL/da with variation of A for rectangular wings according to the ordinary aerofoil theory (Glauert)' and according to the simplified lifting surface theory (Weissinger's method)2 and three spots obtained from actual tests with low

aspect-ratio wings3. The actual spots are situated between the two theoretical

curves, and a curve through these spots gives an empirical relation for the slope

of the lift coefficient curve: For the actual rudder " A " of aspect ratio A = 1 .585 the slope of the lift coefficient curve is, according to the empirical curve in Fig. 1, dCLIda = 0.0384 per degree. This slope is compared with the actual measured values of CL for the ru dder alone in Fig. 2. The tests

suffered from laminar flow, and values of CL > 0.6 were not obtained. The

results are therefore of limited value, but the spots agree quite well with the

straight line determined from the empirical curve in Fig. 1. It should be noted,

however, that the slopes of the lift-coefficient curves obtained from Fig. 1 are, strictly speaking, valid only for a load distribution which corresponds to the load distribution of a rectangular wing in uniform flow. For a rectangular rudder placed near the surface and in the propeller slip-stream or in the wake behind a hull, the flow will not be uniforfir and the spanwise load distribution will not remain the same as for the " open " rudder. This may result in the slope of the lift coefficient curve being slightly altered. If the CD for a wing

STUDY OF RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS 89

with the same profile as the rudder is known, then the CD for the rudder alone may be computed by the reduction formula

CD = CD0 -r (1)

Suffix o indicates A = co

This drag coefficient is of little value if the rudder is placed in the slip-stream behind a propeller. Due to the rotation of the slip-stream there will be, for a non-twisted rudder, some local lift at the different rudder sections, even if the total lift is zero; and due to the variations in this local lift, which varies along

the span, there Will be some induced drag. The magnitude of this induced drag

will depend upon the flow conditions in the slip-stream at the rudder. The

actual drag coefficient for the rudder behind the propeller will therefore be

considerably greater than that obtained from the reduction formula (1), especially

for small rudder angles.

According to the usual two-dimensional theory, the centre of pressure and thus the C,ch.are independent of the aspect ratio for wings in a uniform stream

as long as no separation of flow takes place. This is true for high aspect-ratio

wings, but for low aspect-ratio wings the three-dimensional flow at the ends of

the wing influences the pressure distribution to such an extent that the conclusion

drawn from the two-dimensional theory about the variation of C,,/, with

aspect ratio must be somewhat modified. As far as the Author is aware there

do not exist any simple formulae which take into account the variation of

with aspect ratio and thus permit a prediction of the for wings of any aspect ratio knowing the values of Cnic /4 for a wing of the same section. There are, however, some experimental results available which can illustrate the trend, and Fig 3, which shows this tendency for NACA 0012, has been

prepared from data given in Ref. 3. For A = oo there are only small variations

of with variation of the thickness ratio for symmetrical four-digit NACA

sections and the same will probably also be the case for other values of A. Thus the data given in Fig. 3 may be used as approximations for the four-digit symmetrical NACA sections other than NACA 0012.

Scale Effects

The main purpose of the rudder is to produce a side force, by which it is

possible to control the forces and moments generated by the ship itself in

unsymmetrical flow conditions. It is therefore of great importance to know

the magnitude of the rudder " lift " or side forces at given speed and rudder angle, and especially the magnitude of the maximum lift force and the corres-ponding rudder angle. So long as the flow follows the surface of the rudder profile, the lift force is nearly a linear function of the rudder angle, but as the rudder angle approaches the stalling angle this linearity is lost. The pressure

distribution of the low-pressure side of conventional aerofoil sections at lift

coefficients a little below the maximum is characterized by a negative pressure

peak at a small distance behind the leading edge and by increasing pressure

from this point in the direction of flow to the trailing edge. The reduced energy

water in the boundary layer may then fail to progress against the pressure

gradient. It thus accumulates, and then produces separation of the main flow.

The separation, of course, alters the pressure distribution on the low-pressure side of the rudder and reduces the lift. The resistance to separation is mainly

dependent on the boundary-layer conditions. If the boundary layer contains much energy, such as in a well-developed turbulent boundary layer, it is much more resistant to separation than if its energy content is less as in the laminar

boundary layer. As the maximum lift coefficient is closely related to stalling

and separation, we must expect some variation of the maximum lift coefficient with Reynolds Number, and tests of aerofoils in wind tunnels have shown that

it is so.

The scale effects on the maximum lift coefficient seem to be of somewhat

90 STUDY OF RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS rudders is very seldom less than 15 per cent of the chord and they may therefore

be regarded as thick aerofoils. Investigations carried out in wind tunnels4,6 show

that for such aerofoils the maximum lift coefficient continues to increase with increasing Reynolds Number at least up to = 20. 106. The variation of

the maximum lift coefficient with variation in Reynolds Number for some thick,

symmetrical profiles is shown in Fig. 4 (data from Refs. 4 and 5), and this variation may be considered as typical for thick sections suitable for rudder profiles. The conclusions to be drawn from Fig. 4 may be correct for a smooth aeroplane wing moving through undisturbed air, but not for a rudder working in the wake of a. ship or in the slip-stream of a propeller. First, the actual rudder surface cannot be considered as smooth, and second the water meeting the rudder is turbulent before it reaches the rudder. Both these circumstances

can be important. Wind-tunnel investigations6 show that roughness on the

leading edge of an aerofoil has the effect of reducing the maximum lift coefficient.

Stall studies carried out in a wind tunnel at Reynolds Number = 6. 106 show that aerofoils of the NACA 63series of 12 per cent thickness and less, stall

as a result of abrupt laminar separation of the flow near the leading edge,

whereas 63series aerofoils of 18 per cent thickness stall as a re:sult of gradual

separation of the turbulent layer moving forward from the trailing edge. This shows that we may have laminar separation at quite high Reynolds

Number, at least with thin aerofoils in a non-turbulent, uniform flow. But if the flow is already turbulent when it reaches the leading edge of the profile, as it is with a rudder behind a ship, what will then happen to the separation ?

We might expect the staffing to take place at higher lift coefficients under these

circumstances. Rudder cavitation may also have some effect upon the stalling

conditions of a rudder.

It is difficult, therefore, to predict the maximum lift force and the corres-ponding stall angle for a ship's rudder. Because of serious scale effect, model

tests on stalling are of little use in this connexion ; and a method which may

predict the stall angle on the basis of wind tunnel tests of aerofoils is of doubtful

use if we cannot predict the effective Reynolds Number for the rudder behind

the ship. Any conclusions about rudder stall angle have thus to be based upon

experience with full-scale ship rudders. When the scale effect on maximum lift is so marked and has such serious effects, the question of how far model

tests can be used at all to predict the rudder forces and manceuvring

characteristics for full-scale ships naturally arises. The aerofoil tests',

show, however, that, maximum lift coefficient apart, the scale effects on the

other aerofoil-section characteristics are not so very serious and may be

disregarded in this connexion. Thus it should be possible to study rudder phenomena and rudder forces on the model scale both qualitatively and

quantitatively if the rudder angle does not exceed the stalling angle. Effective Rudder Angle of Attack in the Behind Condition

The forces generated by one particular rudder are mainly determined by the effective angle of attack of the rudder and the speed of the water relative

to the rudder. For an" open " rudder both the angle of attack and the velocity

can be directly determined, but for a rudder behind, a ship the conditions are not so simple. If we disregard the rotation of the propeller slip-stream, the effective rudder angle of attack a may be considered to be a function of the

rudder helm angle 8, the drift angle of the ship q), and the straightening influence

of the hull and appendages other than the rudder e. Fig. 5 illustrates the

conditions.

In 'a manceuvre all the angles 8, tp, and e will change from the instant the rudder is put over until a new state of equilibrium has been reached and the ship is moving in a circle with constant speed. It is the first part which is of

most interest in such a manceuvre because a merchant ship very seldom makes

a complete turn. It takes some time to put the rudder over to the desired

angle, but this time is so short that the ship has not changed heading very much when this angle is reached, and therefore the flow conditions will then be quite

STUDY OF RUDDER ACTION WTTH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS 91

is restricted to a straight course. The rudder forces and rudder pressures for

the initial part of a turn may thus be at least approximately studied on restricted

models. When the drift angle has assumed some particular value, it may be possible to correct for this effect and thus make the conclusions which can be drawn from tests with restricted models valid for a greater part of the turn. When the rudder is forced to move relatively to the hull, forces additional to those due to the rudder angle of attack are generated. These additional forces

increase the torque on the rudder stock of balanced rudders and have to be taken into account when designing steering gears and when determining the

scantlings of the rudder stock. It is impossible to account for this extra torque

by some percentage allowance over what may be called the static rudder torque on the rudder stock since the magnitude of these two torques is quite independent.

The static rudder torque is a function of the position of the rudder stock in relation to the rudder, while the extra torque is a function of the rudder area

and shape, and also of the angular and axial velocities of the surrounding water

in relation to the rudder.

The Effective Velocity for Rudder behind a Hull and Propeller

If, for some particular rudder, the effective angle of attack has been deter-mined for one condition, the dimensionless lift and drag coefficients may be

found. To get from these the actual magnitude of the rudder forces, the

velocity of water in relation to the rudder has to be determined ; and as the

rudder forces are proportional to the square of this velocity, the velocity should

be determined as accurately as possible. It is influenced by the ship wake

and the propeller action, and these influences are oppositely directed. The wake

is a somewhat vague concept. We may get different wake fractions for the same ship, propeller, and speed, dependent on how the wake fraction has been

determined. If the ship speed is v and the velocity of the water in relation to

the ship at a point behind the ship is v0 then the actual local wake fraction for

this point is vv0 and we have,

= v(l w0).

If v0 has been measured for various points in the propeller plane, the mean

wake fraction

w°

may be determined, but if the wake fraction has been determined

using propeller thrust identity wo or torque identity w,,., we get values for the wake fraction which usually differ from w.. Thus wo and wpq may then be called effective wake fractions (as their magnitude is found from propulsion

tests) to distinguish from the actual mean wake fraction calculated from

measurements in the basic flow. For the calculation of rudder forces we are

mainly interested in the local wake fractions, but as these are rarely known, we

have to use the effective wake fractions and we then suppose that v (1 wo)

or v (1 Kim) is the real mean speed of advance of the propeller through the

Wake and denote it by v'.: and as 'wy, or wp.is usually known, the magnitude

of v'0 can be determined.

A first approximation to the velocity in the slip-stream may be obtained by

the axial momentum theory. The propeller is then assumed to be an actuator

disc capable of imparting an axial motion to the water and of sustaining a

reactive thrust. It is further assumed that the reactive thrust and the velocity

imparted to the water are uniformly distributed over the propeller disc and that

friction is negligible. If such an actuator disc advances through undisturbed

water of density p with a speed of v0 and the cross-sectional area of the slip-stream is F1, the velocity in the slip-slip-stream far behind the disc is v1, then the thrust T of the actuator disc can be written,

T = Fip vj (v1

v.)

. . ... .

. . ... . . . .. . . .

. ... (2)

If the disc area is F and the velocity of water through it u then the condition of continuity of flow requires that,

92 STUDY OF RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS

V0 -I- v,

and thus we have,

It can be shown that u ₂

+ Vo

(

21,1 )

and D, = D

+ V°

' 2v,

The equation (2) then becomes,

T =

F (Y21 00)

and thus,

v, =

Vvic,

+

2T (4)

Pr

If v0 = v' then equation (4) can be used to determine the mean velocity in the

slip-stream of the propeller and equation (3) can be used to determine the

cross-section area of the slip-stream. To test the validity of equation (4),

measurements of the velocity distribution in the slip-stream of model M 297

without rudder were made. A more detailed description of these measurements and a more complete discussion of the special features of the slip-stream are to

be given (see Ref. 6). Only some of the results will be given here. The

measure-ments have been carried out by means if a simple pitot tube in a plane situated 1 . 16 D behind the propeller plane. The measurements were made over a

series of runs. As it is very difficult to keep the speed of the towing carriage

exactly the same for several runs, it is of advantage to express the results in a dimensionless form. The velocities of water in the slip-stream are expressed as wake fractions w, in relation to the model speed by the equation

= v (1 w,) (5)

where v, = the velocity in the slip-stream and v = the speed of the model. Curves showing the variation of w, in the slip-stream are shown in Fig. 6. Negative values of w mean that the velocities are higher than the ship speed. The unsymmetrical distribution of the velocity is due to the angularity of the

flow which enters the propeller. The model propeller diameter D = 0.259m.,

thrust T = 5 .10kg., wake fraction wp, = 0381 and the speed of the model

v = 1 . 50 m /sec. The mean velocity in the slip-stream is then according to

equation (4) v1 = 1 -66 m. /sec. and the diameter of the slip-stream according to equation (3) D, = 0 .228m. The mean velocity of the real slip-stream'inside

a cylinder concentric with the propeller and of a diameter D, = 0 .228m. can be found by integration of the curves shown in Fig. 6. The mean slip-stream

velocity v', found in this way = 1 77 m. /sec. Thus the mean measured slip-stream velocity v', is somewhat higher than the calculated mean velocity v1., Possible reasons for this difference may be :

1.. The calculated velocity v', is based upon the assumption that there is no

rotation of the slip-stream. This is not true ; and because of this

rotation there is a low-pressure zone at the centre of the slip-stream which reduces the propeller thrust. For this model and propeller the reduction amounts to about 5 per cent of the total thrust.

The measured velocities are more or less the total velocity at each spot

and not solely the axial velocity component. There may be some difference between w, and wpr

For the calculation of rudder forces, we are mainly interested in the total velocity and not only the axial component. It may then be concluded that equation (4) gives values of v, which are some 5-8 per cent lower than the actual mean velocity in the slip-stream some distance behind the propeller

plane. As only half of the total increase of velocity given to the propeller

slip-stream by the propeller has been achieved at the propeller plane, and as each cross-section of the slip-stream behind the propeller must contain the

(3)

STUDY OF RUDDER ACTION WITH' SPECIAL REFERENCE TO SINGLE-SCREW SHIPS 93

same amount of energy, there must be at a point immediately behind the

propeller where the velocity increase is only the half of the total, and an increase

of pressure which corresponds to the other half of the velocity increase. From

the propeller plane and some distance aft there will thus be an acceleration of

the flow and a corresponding drop of pressure as the distance from the propeller

increases. If Avec, is the increase in velocity in the slip-stream far behind the

propeller and Avx is the velocity increase at any distance x from the propeller Av,

plane, the ratio --- varies from 0.5 at the propeller to 1 0 far behind it.

Avo,

A vx

Fig. 7 shows the variation of - -_Av,t,with the distance from the propeller

according to Gutsche7. As the distance from the propeller to the leading edge

of the rudder is seldom greater than 1/2 R, there will be an acceleration of the

flow along the rudder surface and a corresponding drop of pressure in the basic

flow from the leading to the trailing edge. This will, to a certain extent, prevent

separation of flow from the rudder surface at small rudder angles. The mean velocity over the entire rudder surface within the slip-stream will therefore be

less than the mean velocity over a cross-section of the slip-stream further behind

the propeller. The agreement between the actual mean velocity over the part

of the rudder within the slip-stream and the velocity predicted by (4) may

therefore be better than the above comparison shows.

Pressure Distribution Round a Rudder

If the theory for an aerofoil in uniform flow could be directly applied to the

rudder once the inflow velocity to the rudder had been determined, the

calcula-tion of the rudder forces would be quite simple. Experience has shown,

however, that in some cases the conclusions drawn from aerofoil theory are

of limited value to the rudder designer. As the calculation of forces and

moments by the simple aerofoil theory applied to the rudder is based on the

assumption that the pressure distribution over a rudder working in the propeller

slip-stream is the same as for an equal aerofoil in uniform flow, it may be of interest to find out what difference there is between the pressure distribution over the rudder surface in the behind condition and the theoretical pressure distribution for an equal aerofoil in uniform flow. Some measurements of the pressure distribution over a rudder surface for different rudder angles in the

behind condition were therefore made. Hull model M 297 and propeller P 177

were used, and a new rudder of the same form as rudder " A ' was made of usual white metal. Twenty-one brass tubes of 3mm. diameter were cast into

the rudder (see Fig. 8). Seven holes of lmm. dia.meterwere drilled into each tube

perpendicular to the rudder surface (see Fig. 8). The rudder profile was correct

within ± 0 -05rr.jn. and the position of the drilled holes to within ± 0.2mm.

When the rudder had been mounted on the model, the brass tubes of therudder

were connected to a multiple tube manometer. The liquid in the manometer

was ordinary water of specific weight 1,000 kg /m3.

To measure the pressure distribution only the holes in one of seven horizontal

rows had to be connected to the manometer. Thin plastic tape was banded across the holes in each of the other six rows. The thickness of the tape is approximately 0 -15mm. and the breadth 15mm., and it maintained complete watertightness when placed in water for several days. A photograph of the rudder mounted and, with plastic tape banded over all the sevenhorizontal

rows of holes, is shown in Fig. 9. The pressure readings were photographically

recorded. The tests were carried out with a model speed of 1 -50 in. /sec which

corresponds to V I V LLivi, O56 or a ship speed of l368 knots, and the

corresponding revolutions of the model propeller 8 -00 revs. /sec. The propeller

loading then corresponds to the self-propulsion loading for the ship, i.e. the model was towed with a force equal to difference in friction resistance for the model and the ship according to Froude. Turbulence was stimulated on the model by a 1 nam. trip-wire 5 per cent of ksvz, from the bow. The rudder was

94 STUDY OF RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS

completely smooth except for the roughness introduced by the tape and the holes, but this roughness was very small.

The pressure orifices were symmetrically distributed about the centre plan of the rudder and their chordwise distances from the leading edge were as a

percentage of the chord 0, 5, 10, 20, 30, 40, 50, 60, 70, 80, 90. As the pressure

vector always acts perpendicularly to the surface and as the surface of the rudder

is curved, due to the thickness of the rudder, the pressure at every point on the

rudder will, as a rule, have a normal and a chordwise component. The measured

pressure is the total pressure which acts perpendicularly to the rudder surface, and this pressure has then been divided into normal and chordwise pressure. It is usual to express the pressure distribution over an aerofoil as the variation of a non-dimensional pressure coefficient. This presentation is generally the

most convenient, but it involves a knowledge of the reference velocity. In the

propeller slip-stream the velocity changes both in magnitude and direction from

point to point, and it is not easy to choose any particular velocity as reference

velocity. It is also difficult to predict the velocity distribution and use the local

speed as reference velocity for each horizontal section. Thus it is very difficult

to find any simple velocity distribution which could serve as reference velocity

for the pressure distribution on a rudder in the propeller slip-stream.

Further-more, some pressure changes take place in the basic flow behind the propeller.

This involves still more complication ; and it was therefore decided to give the results of the pressure measurements in dimensional form as the distribution of the normal and chordwise pressure components for the model rudder. The

normal pressure distribution of the rudder for the rudder angles 0 degrees, 10 degrees to port, 10 degrees to starboard, 30 degrees to port, and 30 degrees to starboard is shown in Figs. 10-14, and the chordWise pressure distribution is

shown in Fig. 15. Only very slight fairing was necessary to produce the normal

pressure curves, but as there was no pressure orifice between the leading edge

and the orifice situated chordwise 5 per cent of the chord from the leading edge

(which means that the normal distance from this orifice to the leading edge orifice is 29.7 per cent of the maximum thickness of the rudder), the

deter-mination of the chordwise pressure-distribution curves in this area was somewhat difficult, and therefore the form of the chordwise pressure curves in this region

may not he absolutely correct.

Spanwise Load Distribution

All the pressure curves of the type shown in Figs. 10-14 have been integrated

to get the spanwise distribution of the normal force, and the result is shown

in Fig. 13. From this, it is evident that for port rudder it is the upper part of

the rudder which carries the greater part of the load ; and for starboard rudder the greater part of the load is carried by the lower part of the rudder. For zero rudder angle the upper part of the rudder produces a force towards star-board while the lower part produces a force acting in the opposite direction. Thus the rudder acts more or less as a twisted aerofoil in a uniform flow. As the rudder has no geometrical twist, this must be due to the rotation of the

propeller slip-stream. The form of the curve for spanwise loading for zero

rudder angle is more or less repeated for the other rudder angles, but the curves

are bodily moved away from the position of the zero rudder-angle curve. If

some particular geometrical twist distribution had been given to the rudder in the slip-stream, it might have been possible to obtain a rudder which, in the slip-stream, had a spanwise loading, which was quite similar to the spanwise

load distribution of a corresponding non-twisted aerofoil in a flow which

contains approximately the same axial velocity distribution but no rotation. For zero rudder angle there would be no local normal force acting on such a rudder. If the lift of an aerofoil is not constant along the entire span, there will be some induced drag, and this drag force will be greater the more rapidly the magnitude of the lift force changes. Therefore it ought to be possible to reduce the drag force of a rudder on a single-screw ship by giving the rudder

STUDY OF RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS 95

It is well known that the plan form of an aerofoil in a uniform flow which

gives the least induced drag for any finite value of the aspect ratio is the elliptical, which also has an elliptical spanwise load distribution. From this it has been con-cluded that the best plan form of a rudder is the elliptical or some other tapered

plan forms, which give an approximate elliptical loading in uniform flows, but

this conclusion is not correct for the rudder. Both the rotation of the propeller

slip-stream and the uneven velocity distribution of the water in the vicinity of the rudder, influence the load distribution to a greater extent than the rudder plan form. To get an approximately elliptical loading for all rudder angles,

the rudder has to be twisted, and the plan form combined with the twist must be

such that elliptical loading is obtained in the uneven velocity field in which the rudder is going to operate.

Chordwise Distribution of Normal Pressure

If, the pressure distribution for zero angle in Fig. 10 is examined, the pressure

curves for port and starboard side of the rudder cross each other at all sections,

except section D. For an ordinary aerofoil in uniform flow, it is only possible

to obtain this type of pressure distribution (with the pressure curves crossing each other) if the aerofoil camber line has some curvature and a negative angle of attack.

As the mean camber line of the rudder sections is without geometrical

curvature, this type of pressure distribution has been produced by the curved flow in the propeller slip-stream. This special type of pressure distribution persists for some of the sections for rudder angles up to about 20 degrees. It

is typical for an inverted, curved aerofoil that the distance from the leading edge

to the centre of pressure may differ considerably from 25 per cent of the chord and in some cases the apparent centre of pressure may be outside the section. When the angle of attack increases for such an inverted aerofoil, the apparent centre of pressure usually moves towards the 25 per cent station. In Fig. 17

the measured normal pressure distribution over a typical rudder section (Section

B) for different rudder angles is compared with an approximated theoretical pressure distribution for the aerofoil in uniform flow calculated according to the method given in Ref. 9. The reference velocity used is the mean velocity

inside the slip-stream found by integration of Fig. 6. Using this velocity, the normal force coefficients CN have been determined on the basis of the measured

pressure distribution, and the theoretical pressure distribution for these values

of CN calculated. For small rudder angles there is a considerable difference

between the " theoretical " and the measured pressure distribution, but for

higher rudder angles the difference is reduced. Rudder Lift

' The magnitudes of the lift force per unit length of the model rudder for the

different sections and rudder angles are-given in Table I. The values given are

based directly on the pressure measurements.

Table 1Lift in kg. lcm. for the Model Rudder

Rudder Angle . . Section ' _Total lift kg. A B C D E F G 300 to port -244 .324 .261 .225 -206 .225 -234 6-80 20° -164 -240 181 -130 -107 .122 -124 4.20 10° ' -084 -146 -112 -031 -032 .033 -031 1-91 0° -021 -062

.027 --036 --042 --047 .011 0.10

10° to

star-board --021 .015 --048 .118 --118 --121

078 2-00

200

_{057 --101 136 .210 --196 --216 .210 4-38}

30° ,, 127 -199

-238 .273

-301 -330

312 6.90

96 STUDY OF RUDDER ACTION WITH SPECIAL REFERENCE TO.SINGLESCREW SHIPS

Direct measurements of the rudder forces by a special rudder dynamometer

have been carried out under the same conditions as the pressure measurements,

but on the original wooden rudder "A ", which has the same form as the

rudder used for the pressure measurements, only without the tubes and pressure

orifices. The lift curve for the rudder behind the model at a model speed of

1-50 m. /sec. and with the propeller running at 8-00 revs. /sec. is shown in Fig.

18. The total lift force for the rudder calculated from the pressure

measure-Ments for the different rudder angles is marked in Fig. 18 and agrees with the

mean curve through the spots representing the direct measured lift forces

within ± 3-0 per cent. The lift curve for the rudder for the same speed without

propeller is shown for comparison : the propeller action increases the rudder lift by about 150 per cent. The slight curvature of the lift curve (Fig. 18) is more likely to be due to the low aspect ratio of the rudder than to the altered pressure distribution caused by uneven flow in the slip-stream,

but as an

approximation it may be considered as a straight line. There is a small, but definite, difference in the lift when the rudder is to starboard or to port. This

is due to the rotation of the slip-stream. The difference is also found in the

loading curves shown in Fig. 16 and explains the rudder bias.

Rudder Drag

The drag of the rudder consists of pressure and frictional drag. The total

drag has been measured by the rudder dynamometer, but the curve of chordwise

pressure distribution is not accurate enough for calculating the pressuredrag of

the rudder and splitting up the total drag of the rudder into pressure and

frictional drag components, but it is evident that the greater part of the rudder drag in the slip-stream is pressure drag, even for small rudder angles. The

pressure drag is mainly a function of the rudder shape whereas the frictional

drag chiefly depends on the conditions of the rudder surface. The drag /lift ratio

is a dimensionless quantity and if the variations of the drag coefficient with Reynolds Number are neglected, the drag/lift ratio is independent of the test

speed. The curves, forDIL as a function of rudder angle for the rudder behind

the model with and without propeller, are shown in Fig. 19. The tests were carried out at a model speed of 1-50 m. /sec. both with and without propeller.

The velocity of the water relative to the rudder was thus considerably higher in

the case with propeller than without. Provided that the flow is turbulent in

both cases (as it must be, due to the high rudder stall angle), the effect of raising

the speed of water relative to the rudder on the frictional drag is to reduce the D/L ratio, but the D/L ratio-for the rudder behind the model with propeller is considerably higher than without propeller. This rise in D/L for the rudder

behind the propeller must then be due to a corresponding increase of the pressure drag.

It is the drag in the vicinity of zero rudder angle which is of greatest interest since the rudder usually will be in this position during the greater part of a

velyage. The drag of this rudder behind the propeller amounts to about 7-0

per cent of the total model resistance and, if it is possible to reduce the D/L ratio to the same value as for the rudder behind the model without propeller, then the drag force would be reduced to about 4-0 per cent of the total model

resistance. It ought to be possible to alter the shape of the rudder and thus

reduce the drag by this amount, and if the optimum rudder form is found, a still greater reduction may be obtained. The form factors which influence the

pressure drag and can be altered are: introduction of twist into the rudder, reduction of thickness, and changing the thickness distribution. The induced

drag will be reduced due to the load distribution of the rudder being more even

if the rudder has been given the correct twist, but, furthermore, the normal and chordwise pressure distribution of some of the rudder sections may be altered in a favourable way, thus reducing the drag. If there is no breakdown

of the flow at the leading edge of the rudder, a reduction of 'the rudder thickness

will always result in a reduction of the chordwise pressure force and thus in an equal reduction in the drag force. A change in thickness distribution, which

STUDY OF RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS 97

thinner at the fore part, will have a similar effect on the drag force. Both

these changes involve a reduction .of the leading-edge radius, and if theradius. is made too small a breakdown of the flow, which could possibly have been

avoided by a greater leading edge radius, may take place near the leading edge.

If the rudder thickness is reduced to reduce the drag, care must be taken to

avoid flow breakdown. The reduction of the rudder thickness may also have some effect upon the propulsive efficiency and this should be considered together with

the reduction of the drag force. In one particular case where three non-twisted and one twisted rudder have been tested", the best propulsive efficiency was given by the twisted rudder, and then by a rudder of 19 -.3 per cent thickness, but the variation in propulsive efficiency due to the twisting of the rudder and

by varying the rudder thickness from 9-7 to 29-0 per cent of the chord is small. The rudder forces were not measured in this case.

RuddPr Torque

The rudder torque can either be directly measured or calculated on the basis of the measured pressure distribution. For model M 297 self-propelled at a speed of 150 m. /sec. the measured rudder torque or twisting-moment curve

for rudder " A " is shown in Fig. 20. The spots representing the rudder

twisting moments calculated on the basis of the measured pressure distribution

are also shown. The actual values for the twisting moment about the leading

edge of the fudder for the different sections calculated on the basis of the

measured pressure distribution are given in Table 2.

Table 2----Moment about Leading Edge in kgcmIcm. for the Model Rudder

NotePositive moments turn the rudder to port.

The fact that the centre of- pressure for low-aspect-ratio aerofoils moves aft at high angles of attacks may partly explain the nature of the twisting-moment

curve. This movement of the centre of pressure is a function of the deformation

of the normal pressure curves near the ends of the rudder which takes place especially at high angles of attack. This deformation depends on the aspect

ratio and the spanwise load distribution of the rudder. The heavier the loading

-of the section near the ends -of the rudder, the more pronounced will be the deformation of the normal pressure curves in this Vicinity. Thus the moment

coefficient Cmc/4 cannot be regarded as independent of aspect ratio in wings

or rudders. For rudders placed behind a propeller the centre of pressure usually moves further aft than for the same rudder in free-running conditions. If the_ theoretical and the measured pressure distribution for section " B " (Fig. 16) is considered, it is not surprising to find that there is some difference between the actual moment for a rudder in the slip-stream and an equal wing

in uniform flow. This difference may be due either to the rotation of the

slip-stream, the curved flow in the slip-slip-stream, or to the accelerated motion of the water in the slip-stream.

The rotation of the slip-stream has the effect of increasing the loading at

Rudder Angle Section -Calculated total moment about actual A B - C 'D E ' G rudder axis kgcm. 30° to port

1-53 .1-62 1-43 1-20 1-08 1-30 P99 2-70

20°

-89 1-09

-68 -63 -62 .-60 99 +1-27 100 ,' ,, -29 51 -29 -12 23 14 19 +2-68 ,, 00 ' + -03 -07

-06 + -15 + -02 + -01

02 +0-23 100 to star-board 200

+ 14

+ -41

+ 20

+ 60 + -32

+ 77

+1-04+-.-57

+ 31

+ -64 + -28+ -81

+ :45

+1-66 71-65

0-71

, , 30°

+ 94 +1-17 +1-34 +1-55 +1-55 +1-67 +3-49 +4-31

98 STUDY OF RUDDER ACTION wrni SPECIAL REFERENCE TO SINGLE-SCREW SHIPS

one end of the rudder and this is an effect which is quite similar to the effect of reducing the aspect ratio. The curved flow in the slip-stream will alter the form of the normal pressure distribution, and the acceleration of the flow at the rudder will effect a shift of the loading towards the trailing edge of the rudder where the velocities are highest. To account for these effects in a

complete and theoretically correct manner is very difficult, but in practice the effects may be allowed for by shifting the reference axis for the moment co-efficient 2-3 per cent of the chord aft.

Calculation of the Rudder Forces

The calculation of the rudder forces and moments for a rudder operating in

the slip-stream of a propeller behind a hull is clearly very complicated. As far

as the Author is aware no complete theoretical treatment of the subject has yet been given, but many authors have published work concerning the calculation

of rudder forces (Refs. 11, 12, 13, 14, 15, 16, 17, 18, 19, 20). Using these, aero-dynamic theory, and the present experiments, it is now possible for single-screw

ships to give a method for calculating the rudder lift and torque for rudder angles less than the stall, taking into account the influence of the wake and the

slip-stream. The mean slip-stream velocity can be obtained from equation (4) and the diameter of the stream from equation (3). The velocity outside the

slip-stream can be determined by assuming a wake fraction. This wake fraction may be assumed to be a little higher than the propeller wake fraction as the

greater part of the rudder outside the propeller race usually . is situated

above the propeller and thus in a zone with heavy wake. The mean velocity

in relation to the rudder can then be computed. The slope of the lift-coefficient curve can be obtained from the empirical curve Fig. 1, and the rudder lift can be directly calculated straight away. Due to the rotation of the propeller race

and the accelerated and curved flow in the slip-stream, the centre of pressure for

a rudder placed behind a propeller will be situated further aft than for the

rudder in a uniform flow. This shift amounts to about 2-3 per cent of the

chord and must be taken into account to get good agreement between the actual

and the calculated rudder torques. The drag forcesof a rudder placed in the

propeller slip-stream are generally considerably greater than for the rudder in open water especially for small rudder angles; and as the drag force is largely dependent on the form of the load distribution curve, and as this for small rudder angles is

determined chiefly by the rotation and the curved flow of the propeller slip-stream, any accurate calculation of the drag force must allow for these. It is

generally, however, not very important to the ship designer to have exact

know-ledge of the magnitude of the rudder drag forces and therefore no great effort has been made to develop a method by which the rudder drag forces can be determined with complete accuracy.

To illustrate the method proposed, the calculation for model M 297 with rudder " A " and propeller P 177 at a speed of 1 .50 m. /sec. is shown 'below.

The particulars for the model rudder and propeller are given in the Appendix.

For a model speed v = 1.500 m. /sec. and a wake fraction win = 0.381, we have v, = v (1 win) = l500 . 0-619 = 0-929 m./sec.

The thrust T = 5.10 kg. and the mean velocity in the slip-stream,

V1 'VI

2T 2T

The diameter of the slip-stream,

/vi

D, = D

0.228m.

... . . ... (3)

2v,

1.66 m. /sec.

STUDY OF RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS 99.

Assumed wake fraction outside the slip-stream w = 0.400. The mean velocity over the rudder surface becomes, since

Div, + (1- w) v (b - Di)

0.228 x 166 + 0.6 x 1.50 x (0.282 - 0.228)

=

_{0 282} - 151 m./sec.

b = height of rudder

The slope of the lift-coefficient curve (from Fig. 1, empirical curve) is given by

dCLIda 0.0384 per degree and then the lift-force curve is taken as a straight

line with slope 0.219 kg. /degree. This slope is valid for rudder angles less than the stall angle. For the ship according to Fig. 4, the stall may not take place until values of CL, above 150 are leached. The mechanism of stalling

is, however, not completely explained, and therefore any prediction of the stall

angle or the maximum lift coefficient for a full-scale rudder must be doubtful.

The moment coefficient for the rudder can be obtained from Fig. 3

by interpolation, and since the centre of pressure for a rudder in the behind condition shifts about 2-3 per cent of the chord aft in relation to the position for the open-water rudder, this may be taken into account by using a reference axis 27 -5 per cent of the chord from the leading edge instead of 25 per cent. 'The actual rudder axis is situated 28.1 per .cent of the chord from the leading

edge and the correction for this distance between the actual and the reference

axis then becomes L x 0-006c. The calculation of the rudder moment is

Shown in Table 3.

Table 3-Calculation of Rudder Moment for Model Rudder

(1) From Fig. 3. (3) Reference tuns 27.5 per cent of chord from leading edge.

The calculated moments or torques are valid only for rudder angles less than the stall angle. For rudder angles higher than the stall angle the moment

coefficient varies from about -0.10 to -0.20. The predicted magnitude of

the rudder lift and moment is compared with the actual measured values in Fig. 21 and the agreement is, as shown, satisfactory for most usual purposes. But it must be remarked that the calculated and observed rudder moments are static rudder moments for the rudder fixed in one particular angular position.

For designing steering gears the additional moment necessary to "give the rudder

the desired angular speed must be added to the static rudder moment to get the magnitude of the moment which the steering gear has to overcome. This

additional moment for balanced rudders can be of considerable magnitude and carmbt be accounted for by a percentage addition to the static moment-.

It

must be treated separately.

Torque on Rudder Stock when going Astern

It is importSnt for the designer to know the magnitude of the rudder torque when going astern,, as this very often exceeds the ahead torque.

The main difficulty in predicting the astern torque is to determine the astern

L

-CL .

c.

(3) e)

Moment about

reference axis x 0.006c Ma°cmtueant axatut

C4 0.0384

02

52°

+0l3

-1.35 kgcm -0.13 kgcm -148 kgcm 0.4 10.4° +.012 -1-25 -0.25 -1-50 O'6 15.6° +.003 -0.31 -0.38

-0.69

0.8 20.8° -.011 +1.14 -0.50 +0.64 1.0 26.00 - 029 +3.00 -0.62 +2.38 1.2 31.20

-.049

+5-10 -0.75 +4-35 ,

100 STUDY OF RUDDER ACTION wan SPECIAL REFERENCE TO SINGLE-SCREW SHIPS

speed. It is outside the scope of this paper to discuss in detail the factors

influencing backing speeds, but it may be pointed out that for the same propeller revolutions the astern power is considerably less than the ahead. This is mainly

due to the reduced astern wake and the reduced effective pitch of a propeller with fixed blades when reversed. Tests carried out with model M 297 and propeller P 177 show that this model, for the same revolutions, had an astern speed of about two-thirds the ahead. We can thus conclude that forthis type of ship the maximum speed astern seldom exceeds two-thirds of full speed ahead.

Information on wake and speeds induced at the rudder by the propeller for astern motion is very scarce, but as an approximation it may be assumed that the wake and 'speeds induced by the propeller cancel each other and that the water velocity in relation to the rudder is equal to the astern speed of the ship.

Denoting the distance from the after end of the rudder to the centre of the

rudder stock by e and the distance from the after end of the rudder to the

centre of pressure by a then the rudder torque is

M= N (e

a) (8)

As an approximation we may use

M L(ea)

' (9)

and a is for astern motion approximately 25 per cent of the chord c.

The slope of the lift coefficient curve tiCLIda may be taken from Fig. 1, and

then we have

L,

M =

dC x x S vs x (e a) (10)

Equation (10) has been used to predict the torque of rudder "A "behind

model M 297 with propeller P 177 when running astern, and the results are compared with the actual observed values for two different speeds in Fig. 22.

The tests were carried out with the model self-propelled, less the Froude friction

correction. It is evident from the figure that equation (10) can be used to

predict the rudder moment approximately when going astern, and as the

prediction of the astern speed in most cases must be very uncertain any

refine-ment of the method seems unnecessary.

Acknowledgments

This work has been carried out dining a period when the Author was a

research student at the Technical University of Norway. He wishes to express his gratitude to the Council of Professors at the University for awarding him a scholarship, and to those of the staff at the Norwegian Ship Model Experiment

Tank and the Shipbuilding Section at the University who have taken part in

'STUDY OF RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS 101

APPENDIX

Details of Model, Propeller and Rudder used for Tests

i'v1 297 used for the rudder tests is a wooden model of a 24,000 tons deadweight

tanker. The model scale is 1 : 22, which in this case means an exceptionally

large model. The model _was originally designed for a special purpose, but as

it is an advantage to use a large model for rudder tests, it was decided to use

this existing model again. During all the tests the model was fitted with a 1.0

mm. trip-wire, 5 per cent of the load-water-line length from the forward

perpendicular. ' The main particulars for the model and the corresponding

24,000 tons deadweight ship are given in Table 4. Table 4Data for Model M 297

Ship. 1: 22 1835m. 177 0m. 24.4m. 9 6m. 0.11m. 32,030m.a 33,087 tons*

Scale model to ship Length of water-line

Length between perpendiculars ..

Breadth moulded .. Draught moulded..

'Rise of floor ..,

Volume displacement moulded .. Weight displacement

Lt.wiJa

.. Block coefficient (on LLWL) Prismatic coefficient (on LL.wz.)

Load water-line coefficient ..

-Centre of buoyancy forward of ..

Half angle of entrance ..

I ton = 1,000 kg.

The body plan and the

Fig. 23.

The model propeller, P

.drawing of the propeller.

LLWL Lpp CB Cp CLWL LLB/B 2 1 /2ce 7-520 2.542 -745 756 0.834 Model 8 341m. 8045m. 1-'109m. 0 436m. 0.005m. 3 -008m.a 3,085 7]g. 329% of LLWL 32°

bow and stem contours of the model are shown in 177, is of the usual white-metal. Fig. 24 shows a The main particulars are given in Table 5.

Model 259.1nun. 46-9nun. 4 12-5° 792

.Expanded blade-area ratio F alF -613

The stern arrangement with propeller and rudder is shown in Fig. 25. The

rudder used for the lift, drag, and moment measurements, is made of teak ; and as the surface has been lacquered and polished it can be considered as .completely smooth. The data for the rudder are given in Table 6.

Table 5Data for Propeller, P 177

Ship

'Propeller diameter D 5-700m.

Boss diameter .. 1032m.

Number of blades

Rake ..

Pitch ratio at rIR = 07.. P

102 STUDY OF RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS

Table 6Data for Rudder

Ship Model Rudder height b . . 6.204m. 282mm. Rudder chord c .. 3.916m. 178mm. Rudder thickness .. 0.783m. 35.6mm. Rudder area 24.293m.2 0-0502m.2 Aspect ratio 1.584 Thickness ratio .. .20

Distance-from leading edge to centre of rudder

stock as percentage of chord .. 28-1

Rudder profile NACA 0020

Rudder area

app

x d) 0.0143

The rudder used for the pressure measurements is made of white-metal and has the same form as the wooden rudder.

STUDY OF RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS 103 SYMBOLS A Aspect ratio = Breadth moulded CD Drag coefficient

y

CL -- Lift coefficient P -2- av N

CN Normal force coefficient

Mat

Cnic, 4 Moment coefficient

Sc

D Drag force kg.

D Propeller diameter m.

DI Diameter of slip-stream M.

F Propeller disk area in.'

F1 Cross-section area of slip-stream in.'

' L Lift force

kg-Lpp Length between perpendiculars m.

LwL Length of waterline m. or ft.

M Moment about any reference axis kgm. (kgcm.)

Me/2 Moment about quarter chord kgm. (kgcm.)

N Normal force kg. P Pressure kg. /cm.' Propeller radius m. L v R, = Reynolds Number S Rudder area m.2. T Thrust kg.

V Ship speed knots

a Distance from after end of rudder to centre of pressure

when running astern m. (cm.)

b Span of wing or height of rudder m. (cm.)

c Chord of wing or rudder m. (cm.)

e Distance from after end of rudder to centre of rudder

axis when running astern m. (cm.)

g Acceleration due to gravity m. sec.-'

104- STUDY OF RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS

Revolutions per second sec.-1

Thickness ratio of rudder Thrust deduction factor

Model speed m. sec.-1

ro = v (1w0) Nominal axial velocity in the wake m. sec.-1

v (1 - wl) Axial velocity in the slip-stream 'm. sec.-1

Mean velocity over the rudder surface M. sec.-1

wo Nominal wake fraction (Taylor)

Wpi Effective wake fraction (thrust identity)

P4 Effective wake fraction (moment identity)

w,

= I

" Wake " fraction in the slip-stream

a Angle of attack degree

8 Rudder deflexion angle degree

Straightening effect of hull and appendages degree

Drift angle degree

.Specific weight of water kg. m.-a

Density of water kg. sec.2m.-4

STUDY OF RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW.SHIPS 105

REFERENCES

GLAUERT, H. "The Elements of Aerofoil and Airscrew Theory," Second

Edition, eambridge University Press, 1948.

DE YouNo, Y., and HARPER, C. W., Theoretical Symmetrical Span Loading at Subsonic Speeds for Wings having Arbitrary Plan Form,"

NACA Rep. 921, 1948.

JONES, GEORGE W., "Investigations of the Effects of Variations in the Reynolds Number between 04 x 10° and 30 x 108 on the Low Speed Characteristics of Three Low Aspect Ratio Symmetrical Wings with

Rectangular Plan Form," NACA RML 52 G 18, 1952.

JACOBS, E. N., and SHERmAN, A., "Airfoil Section Characteristics as

affected by Variation of the Reynolds Number," NACA Rep. 586, 1937.

LoFrrN, L. K., and BURSNALL, W. J., "The Effects of Variations in Reynolds

Number between 30 x 10° and 250 x 10° upon the Aerodynamic Characteristics of a Number of NACA 6-Series Airfoil Sections,"

NACA Rep. 964, 1950.

LoTvErr, M. "On the Working Conditions of a Propeller behind a Ship

and its Relation to the Pressure and Velocity Changes in the Slip-stream." (To be published).

GUTSCHE, F., "Die Induktion der aidalen Strahlzusatzgeschwindigkeit in der Umgebung der Schr-aubenebene," Scluffstechnik 1955, Heft 12 /13.

MANDEL, P., "Some Hydrodynamic Aspects of Appendage Design,"

S.N.A.M.E., 1953.

AsBorr and VAN DOENHOFT, "Theory of Wing Sections," McGraw-Hill, New York, 1949.

HARVAII), S. AA., Medstromskoefficientenes afhengighed af Rorform,

Trim og Hwkbolge," Medd. Staten.s Skeppsprovningsanstalt, Goteborg, Nr. 13, 1949.

SCHOENHERR, K. E., "Steering," Principles of Naval Architecture, Vol. II, New York, 1949.

Fisanat, " Berechnung der Ruderkraft,7 Werft Reederei Hafen, 1938. DANCKWARDT, E., " Berechnung der Ruderlcraft mid des Rudermoments,"

Schiffbautechnik, Heft 1, 1954.

JAEGER, H.

E.' "Approximate Calculation of Rudder Torque and Rudder Pressures," Mt. Sbldg. Frog., No. 10, 1955.

KnvosBrrA, M., and OKADA, S., "On the Twisting Moment acting upon

a Ship's Rudder Stock," Int. Asbldg. Frog. No. 9, 1955.

KUCHARSKI, W., " Neuere Geschichtspunkte fur den Entwurf von

Schiffsrudern," SBTG, Band 32; 1931.

GOVER, S.

C.,

and OLSON, R. C., "A Method for Predicting the Torque of Semibalanced Centreline Rudders on Multiple-Screw Ships,"

David W. Taylor Model Basin Report No. 915, Nov. 1954. GAWN, R. W. L., " Steering,Experiments, Part I," I.N.A., 1943.

ROMAHN, K. and THiplE, H., " Zur Wahl der Balancellache von Rudern im Propellerstrahl," Schiffstechnik, Heft 21, 1957.

BRARD, R. " Manceuvring of Ships in Deep Water, in Shallow Waters and in Canals," S.N.A.M.E., 1951.

106 STUDY OF RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS

Fig. 2Predicted and Measured Values of CI,for

_kidder A F (2) 18 PEDIE8BIJI/Eaff_RE5... 16 EMPIRICAL CURVy'

SIMPLIFIED. LIFTING SURFACE THEORY RE

o FROM_{keit (31}

60'

PREDICTED FROM EMPIRICAL CURVE FIG. 1.

2

-0 tO 20. 30 40 SO

ASPECT RATIO A

Fig. 1Slope of Lill Coefficient Curve as Function of Aspect Ratio A

for Rectangular Wings

0 FROM TESTS WITH RUDDER A REYNOLDS NUMBER 0.35,10

.04

02

J

2.0

x

Am ta : IiJ 0 C.3 1.0 g4.

t .5

10 .02 .04 -.08

STUDY OP RUDDER ACTION yam SPECIAL REFERENCE TO SINGLE-SCREW SKIPS ₁₀₇

.06 .2 !MCA .001R 30-P104 60 12

Fig. 3Cmc14 as Function of CI, for Three Low Aspect Ratio Wings

4.1

1

EF C 11 . IIREF.( 'A 63-016 F. MT A a3B A ..1.0 REYNOLDS NUMBER

Fig. 4Variation of Maximum Lift Coefficient with Reynolds Number for Some

Thick Aerofoils

A 20

tos

108 STUDY. OF RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREWI'SHIPS

.10

Fig. 5Effective Rudder Angle for Rudder behind Hull during Turn

_

-.20 CONTOURS FOR CONSTANT W,

Fig: 6Velocity Distribution in Slip-stream behind M 297

1.0

0.5o

. STUDY OF RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS 109

v.

LXV

Fig. 7 Variation of Av,,uion with

the Distance from the Propeller Plane according to Gi_4tsche (Ref. 7)

8Rudder used for Pressure Measurements

' 178 i 128 1 i.k 5° ;1 , 1 II 1-1 Er 1 1; - III I. , ,i,, 10 11 .11

,

11,- il _ill1,1 1.1 1 4 L'i

1,4- 1.1H 17110 riIII ,-,1.1 r,HI ,L.: II `Isl-1,1o'i[ 111'1 1 1I IA 111 III III I:I II 11 11; 1111 I;', 111 ,,

'I I.I. 111! 1,1: ..1. 11 1,.; ,1,1 1,1 1:11 ° 41 p.., itlIF.LAL-1 ₀

IA AI

.11, '' :I

Irl Il 11 III 'I -Ill 'Lrr--I II I IdI

1 I, 1.1 H I 1,1 II; , III 11 I

, 11

11 I III 1-il-f'

-, 11

, 1.1 , I j'll Ill III lit , P I ,I,li

I.; I i., ij j j, p I:, ,, ,,JI I II ,7 I ,J1 1 it ',I "1,1 1 1 III W 6 it III III 1711 II I H il I.1 il III rl 111 II I 111 11, 1, II III la 1 ,1 ii 4 11 u iii l'i 11, II I :11, 1 il Ili 1, 111, I lil II 1 1 1 ii,, 0 1, III / I, 1,, ,i 1 ', 1 hl 11! 111 .11. .1, 111

FIP'4"

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111"01.

111 ,e, T1_II IT11

1 , ll, 11 "i I ill iii ill 11 ,,,

1 1,1 Id 1, i.1 1,1 ih Ili ill,

1,1, 1.1 , !'.! Iii !!!,I I! !III

. '

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r' II 1 111 di III 11I

11 1,1

11 1, 1 II, III 11 i, ,i ill 1I1

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, 1 1 1,1 11, 11, 1411 ,i, 1,1 ,,i1 Iv! I p. .., ,1 .4i , .g g '4 14 6 _451_6 Id 6' _-Ic , 178 178 178 I; 178 ;178 ; 178 '78 ,, 178 ; 178 89 171; 0.5 10 1.5 20 0.9 0.8 0.7 0.6

110 STUDY OF RUDDER ACTIONwrmSPECIAL REFERENCE 10 SING LE-SCREW SHIPS

Fig. 9Rudder mutinied on the Model with Plastic Tape banded across the Holes

PRCSSURECURVES FOR PORI SIDE Or RUDDER PRESSURECURVES FOR STARBOARD SIDE or RUDDER

Fig. 10Distribution of Normal Pressure for Rudder Amidship

<TO. 2

r\

20r\

- I .1.1

,\ I

c

WyWDY OF:RUDDER APTION.Wrni,spEciAL RE.FERENCE-TO.S.INGLE7SCREW SHIPS 11L

PRESSURECURVBS FOR foRi SIDE V RUDDER

PRESOI/REOLIRVES FOR STARBOARD SIDE OFRUDDER. .

Fig. 11Disfribittion of Normal: Pressure, Rudder 100 to POrt

Top OF RUDDER

RUDDER 10° TO STARBOARD

PRESSURECUR YES FOR PORT SIDE OF RUDDER PRESSURECURI/ES FOR STARBOARD SIDE OF RUDDER

STUDY OF RUDDER 'ACTION WITH SPECIAL REFERENCE TO SINGLE4CREW SHIPS

20

I I I I I

/

I ----1---L,I / \./ 1

r i,1

' 1 1

u\

. 1

i

...

,/

l./

.

,...

-- PRESSURECURVP FOR PORT SIDE OF WOOD?

---- PRESSUREOJRVES FOR STARBOARD SIDE. or RuriDER.

U _{RUDDER 300 TO PORT}

STUDY OF .RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS 113 1"\\

\

\

\

;

\

r\s.

\

_\

_{TOP o r Room}

\

\

RUDDER 300 TO STARBOARD

PRESSURECURVES FOR PORT SIDE or RUDDER PRESSURECURVES FOR STARBOARD SIDE OF RUDDER

f4 .StLibiOF 'ilinibER Actiot4 Writ ittAktisitt-tTO:sit4aLE4dliktIrstitrig

SCALE FOR CHORD-WISE PRESSURE I.

1 T I IttliiliIIIIIIIIIIII li I " I

. 0 90 90 40 SO ao kg/,...lo.

STUDY OF' RUDDER ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS 115

Fig. 15Distribution of the Chordwise Pressure Component (continued)

PROPELLER ;MIRE

OF RUDICA

SCALE FOR SPA//WISE DISTRIBUTION OF RUDDER NORMAL FORCE ROA,

11141111goil,,i,ly.9,11L1/41111

.

Fig. 16Spanwise Distribution of the Normal Force for Different

Rudder Angles

SALE FOR NORMS FRESSURS COEFFICIENI 1111

1

210

1

SECTION

73-Fig. 17Pressure Distribution for Rudder Section in

7.0

SO

4.0

, 10

STUDY OF RUDDER.ACTION WITH SPECIAL REFERENCE TO SINGLE-SCREW SHIPS 117

RUDDER TO STARBOARD WITH PROPELLER 60 RUDDER TO PORT WITH PROPELLER o CALCULATED FROM PRESSURE MEASUREMENTS WITHOUT PROPELLER 10° 30° 40° 500 RUDDER ANGLE

Fig. 18Rudder Lift Force calculated from Pressure Measurements compared

ro STARBOARD LC WITH PROPELLER , WITH PROPELLER O. N. \N...

/-WITHOUT PROPELLER

- ...--.../..-\WITHOUT ,...,./. PROPELLER 1 2,,, 2!)., 10° 10° 20° 30° 40° 50' TO PORT

Fig. 19Drag/Lift Ratio for Rudder behind

_

RUDDER TO PORT

Fig. 20Rudder- Moment calculated from Pressure Measurements compared

with Moment found by Direct Measurements

Li] RUDDER TO STARBOARD co 0 0.

,

tn 0 _ Z o 5-0 /c.)

/

f

_/

111

11111111M11111W111:-,

2 44Zo

WITHOUT PROPELLER WITH PROPELLER

_.... 410

MI

.-1.1 -c) FROM

MEA SUR M ENT'S

50 0 CALCULATED. o Z Ion PRESSURE 30° 41

7

4.

3.

120. STUDY OF RUDDER ACTION WITH .SPECIAL REFERENCE TO 'SINGLE-SCREW SHIPS

I 1

.

. 0 , 0 . 7 +OBSERVED, . OBSERVED, RUDDER TO RUDDER TO PORT STARBOARD . . I I -. . . . 7 PREDICTED LIFT . I I I ' . 0 41 81 cn dici

o

II

21

g

221 7 7

1

psi 011 I .

.

J

q

. -7 I PREDICTED MOMENT . I

'

. . 0 10° 20° 30° 40° 50° RUDDER ANGLE

MODEL 297 RUNNING ASTERN WITH PROPELLER 177

TO PORT

40°

30°

30 30

Fig. 22Predicted and Measured Rudder Moment for Model going Astern

40°

PITCH RATIO. PD too Pk. 24Propeller P 177 -857 ,,,,,,, 95 84 .../ Age, .90 CtA 9 1

/

4

A .74 792 .8 v /V

XPANDED BLADE AREA,Fa

11

Vil

.w.,.... 1f.%.

./,.../.:,

av .Z..., 10 .55 k _

It

,...,44, /IV /.. z ,,,,,,,,

..4

/1.: , ,,,-A ....

/

1 _-1---- \ 1 .16L

1.7Th

DATA FOR MODEL PROPFI I ER lAP. 177)

.

90

100 144

0/A ME TER

25S1IMM

PITCHYARYIN6 AS SHOWN RAKE:

12.5'

EXPANDED AREA RATIO, Fo/F. 0613 NUMBER OF BLADES:

4

-DIRECTION OF TuRNING. RIGHT HAND

0 10 20 30 '40 50 60 70 80

11111,1[1,1111i

IIIII

SCALE

A

czt

178.0

-IVACA 0020

Fig. 25Stern Arrangement of M297

.