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The Choice of the Propeller

B y J . D . v a n M a n e n ^

In this paper the four main requirements for a propeller ore dealt with. These require-ments concern efficiency, cavitation, propeller-excited forces and stopping abilities. In a propeller diagram the characteristic efficiency curves for different conditions are ex-plained. A comparison of the optimum efficiencies for various types of propulsors is given, and the applications on a 1 3 0 , 0 0 0 - d w t tanker are considered. Cavitation-in-ception curves both for a specific propeller and f o r systematic propeller series are dis-cussed. Predicted torque and thrust fluctuations, based on model-test d a t a , and the results o f measurements on the full-size ship are compared. Finally a quasi-steady testing technique, developed to analyze difTerent types of stopping maneuvers, is described.

IN this paper an attempt is made to explain i n an

instructive way the results of applied research i n the propulsion of ships. I n particular i t is hoped that i t w i l l be instructive to those who are active i n the field of ship design.

The main requirements for a ship propeller are: 1 H i g h efhciency.

2 M i n h n u m danger of cavitation erosion.

3 M i n i m u m propeller-excited vibratory forces. 4 Good stopping abilities.

5 Favorable interaction w i t h the rudder, to improve maneuverability.

• 6 Dependability—minimum vulnerability. 7 Low i n i t i a l and maintenance costs.

I n the following sections the author has restricted h i m -self to a discussion of the requirements mentioned under points 1 through 4.

There are two important diagrams, w i t h the aid of which insight into nearly every propulsion problem can be obtained. These are:

(a) The diagram giving the relation between the thrust coefRcient KT and torque coefhcient KQ and the advance ratio J of the propulsor is shown i n Fig. 1.

(b) The velocity and force diagram of a screw-blade

element is shown i n Fig. 2.

Results such as those given i n Fig. 1 were obtained f r o m an "open-water test" of a given screw model. I n such tests the screw model is driven f r o m behind. The propulsion motor and the measuring apparatus are housed i n a boat which is a considerable distance behind the screw model and is connected to the carriage of the towing tank. I n this manner the thrust T and the torque Q can be measured for constant values of rotative speed n and varying advance speed Va without the

' P - I ^ T ^

1 L e c t u r e h e l d a t a Seminar of T h e Roj^al I n s t i t u t e of Engineers, D e l f t , T h e N e t h e r l a n d s , 1965.

' A s s i s t a n t D i r e c t o r , N e t h e r l a n d s Ship M o d e l Basin, W a g e n i n -gen, T h e N e t h e r l a n d s .

F i g . 1 R e l a t i o n b e t w e e n t h r u s t c o e f f i c i e n t KT, t o r q u e c o e f f i c i e n t KQ a n d advance c o e f f i c i e n t J o f p r o p e l l e r

influence of the ship that ultimately is to be driven by the propeller.

As a rule, thrust and torque are given i n nondimen-sional f o n n KT = KQ = Q Q (thrust coefficient) (torque coefficient) where p = mass density of f l u i d

ni) = a, measure of rotative speed of screw

= a measure of screw-disk area

These thrust and torque coefficients are plotted as a function of the advance coefficient J , which is the ratio between the speed of advance 7 a and the rotative speed

i:iD.

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dQ <1Q|

F i g . 2 F o r c e a n d v e l o c i t y d i a g r a m o f s c r e w - b l a d e e l e m e n t > t

L j r a d i u s r

J

F i g . 3 i C r - j K g - / d i a g r a m

screw using a diagram such as Fig. 1. I f , for instance, f o r the screw concerned, the speed of advance Va and tlie rotative speed n are Imown, then the thrust and torque can be read off, or, w i t h known Va and Q the rotative speed n can be found. Thus, when two out of the four quantities Va, n, T and Q are given, the other two can be determined f r o m the diagram.

I n Fig. 2 the force and velocity diagram of a screw-blade element at a certain radius r is given. The thrust-ing action of the screw induces velocities i n the fluid. The magnitude of these induced velocities depends on the screw loading. I f the speed of advance Va (or ship speed Fs) is decreased, while the rotative speed is held constant, the screw loading w i l l increase and the induced velocities w i l l increase at a rate proportional to the i n -crease i n l i f t force dL and the effective angle of attack Oti.

The induced velocities c„, which are, to a good ap-proximation, at right angles to the resultant incoming velocity V, can be resolved into axial and tangential components and c,. A t the screw disk the induced velocities are one-half of their ultimate values far behind the screw.

I n the diagram, Fig. 2, the following symbols are i n -dicated:

cor or irnd = tangential speed of blade element at radius r

dDp = profile drag of blade element

- B p

F i g . 4 Bp — S d i a g r a m

F i g . 5 C h a r a c t e r i s t i c curves i n a Bp-S d i a g r a m

dTi, dQi = thrust and torque force of blade element

without influence of profile drag

dT, dQ = thrust and torque force including influence

of profile drag

P = hychodynamic pitch angle uncorrected for

induced velocities

Pi = hydrodynamic pitch angle corrected for

induced velocities

This force and velocity diagram forms the basis for the lifting-hne theory for ship propellers. This theory w i l l be treated i n greater detail i n the section "Cavitation of the Screw-Blade Sections."

This diagram is also helpful when analyzing propulsion problems using quasi-steady considerations, see also sub-sequent sections.

Efficiency of the Propeller

A n important source of data for screw design are the results of open-water tests w i t h systematical screw series. A systematic screw series consists of a number of screw models, i n which only the pitch ratio P/D is varied. AU other characteristic dimensions, including diameter D, number of blades z, blade-area ratio A,/A, blade plan-form, f o r m of blade sections, blade thicknesses and

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hub-Table 1 Transformation of the KT — KQ — J Diagram into 6^,-5 Diagram J KQ Bp 6 J P / D = 0 . 6 0 . 8 1 . 0 1.2 1.4 0 . 5 0 . 8 1.0 1.2 1.4 2 0 0 0 . 6 0 6 0 . 1 8 2 0 . 0 1 0 0 0 . 0 2 3 4 0 , 0 3 9 8 0 . 0 5 1 2 0 . 0 8 4 0 1 8 . 2 2 7 8 3 5 . 3 4 5 . 0 5 2 . 3 2 1 0 0 . 4 8 2 0 . 1 6 1 0 . 0 1 0 8 0 . 0 2 4 4 0 . 0 4 1 0 0 . 0 6 2 4 , 0 , 0 8 5 4 2 1 . 3 3 2 . 1 4 1 . 6 5 1 . 3 5 0 . 0 2 2 0 0 . 4 6 0 0 . 1 4 4 0 . 0 1 1 5 0 . 0 2 5 4 0 . 0 4 2 0 0 . 0 6 3 6 • 0 , 0 8 6 8 2 4 . 7 3 5 . 7 4 7 ^ 3 6 3 . 1 6 7 9 2 3 0 0 . 4 4 0 0 . 1 2 8 0 . 0 1 2 2 0 . 0 2 6 2 0 . 0 4 3 0 0 . 0 6 4 7 0 , 0 8 7 9 2 8 . 4 4 1 . 7 5 3 . 4 6 5 . 5 7 6 . 4 2 4 0 0 . 4 2 2 0 . 1 1 6 0 . 0 1 2 8 0 . 0 2 6 9 0 . 0 4 3 8 0 . 0 6 5 6 0 , 0 8 9 0 3 2 , 4 4 7 0 6 0 . 0 7 3 . 4 8 5 5 2 5 0 0 . 4 0 5 0 . 1 0 4 0 . 0 1 3 4 0 . 0 2 7 6 0 . 0 4 4 6 0 . 0 6 6 5 0 , 0 9 0 0 3 6 . 7 5 2 . 3 6 7 . 0 8 1 . 8 9 5 . 2 2 6 0 0 . 3 9 0 0 . 0 9 5 0 . 0 1 3 8 0 . 0 2 8 2 0 . 0 4 5 3 0 . 0 6 7 2 0 . 0 9 0 9 4 1 . 0 5 8 . 4 7 4 . 1 9 0 . 3 1 0 5 . 0 2 7 0 0 . 3 7 5 0 . 0 8 6 0 . 0 1 4 3 0 . 0 2 8 7 0 . 0 4 6 0 0 . 0 6 8 0 0 . 0 9 1 7 4 5 . 9 6 5 . 1 8 2 . 4 1 0 0 2 1 1 6 . 3 2 8 0 0 . 3 6 2 0 . 0 7 9 0 . 0 1 4 8 0 . 0 2 9 2 0 . 0 4 6 6 0 . 0 6 8 7 0 . 0 9 2 4 6 1 . 0 7 1 . 7 9 0 . 6 1 1 0 . 0 1 2 7 6 2 9 0 0 , 3 4 9 0 . 0 7 2 0 . 0 1 5 2 0 . 0 2 9 7 0 . 0 4 7 2 0 , 0 6 9 3 0 . 0 9 3 2 5 6 . 7 . 7 9 . 3 1 0 0 . 0 1 2 1 . 1 1 4 0 . 5 3 0 0 0 . 3 3 9 0 . 0 6 6 0 . 0 1 5 6 0 . 0 3 0 1 0 . 0 4 7 7 0 . 0 6 9 8 0 . 0 9 3 7 6 2 . 2 8 6 . 4 1 0 8 . 8 1 3 1 . 6 1 5 2 . 5 1 0 1 . 2 7 7 BP = 3 3 . 0 8 \ / M :

diameter ratio ch/D are fixed for tlris series. Tire re,sults of open-water tests for sucli a screw series are given i n the KT-KQ-J diagrams, Fig. 3.

The propeller efficiency }?p can be expressed i n terms of these nondimensional coefficients as follows:

^ TVa ^ K T J _

2TvQn KQ2W

B y interpolation i n the KT-KQ-J diagram of a screw series most problems, which arise when designing or analyzing screw propellers, can be solved.

The most widely encountered design problem is that where the speed of advance of the fluid into the screw disk Va', the power to be absorbed by the screw P and the number of revolutions ?i are given. The diameter D is to be chosen so that the greatest efficiency can be ob-tained. This is done as follows:

" • B y choosing discrete values of the diameter D, the corresponding values of the advance ratio J and the torque coefRcient KQ can be calculated. F r o m the

KT-KQ-J diagram, Fig. 3, the corresponding pitch ratios P/D •and the efficiency r/p can be read off for each diameter chosen. Plotting the values of r]p as a function of the diameter w i l l allow the diameter leading to the optimum efficiency to be chosen.

I n order to simplifj^ this frequent design jDroblem, the KT-KQ-J diagrams can be transformed into another dia-gram, f r o m which the optimum diameter D can be read off directly when the speed of advance Va, the power P and the rotative speed n are given. For this purpose a .design coefficient Bp has been formed f r o m the torque coefficient KQ and the advance ratio J i n such a way that the screw diameter is eliminated:

I n the coefficient Bp, N is the number of revolutions per minute, the power P is i n horsepower and the speed of advance Va is i n knots (1 knot = 0.5144 m/sec = 1.689 fps).

I n the usual diagram, the design coefficient Bp is the base and a new speed ratio 8 is used. This speed ratio is defined as B, NP'/'-S = ND Va 101.27 J = 33.08

in which D = screw diameter i n feet.

The manner i n which tlie KT-KQ-J diagram is trans-formed into the Bp-S diagram is shown i n Table 1.

Fig. 4 gives an example of a Bp-5 diagram for a par-ticular screw series.

I n Fig. 5 some characteristic cui'ves i n the Bp-5 diagram are shown:

(a) Optimum rip for P/D = const. This curve goes through the points where the tangents to the curves of equal efficiency {-qp = const) are horizontal (P/D = const). The optimum »;p-values correspond to the peaks of the i7p-curves i n the KT-KQ-J diagram. Fig. 3.

(b) Optimum -qp for / = const. This curve goes

through the points of contact between the curves of ö = const and' those for -qp = const. These optimum values coincide w i t h those on the envelope of the efficiency curve in the KT-KQ-J diagram, Fig. 3.

(c) Optimum ijp for the most favorable diameter D. This curve connects the points of contact between the curves of ijp = const and their vertical tangents {Bp = const; P , i V and F a are given).

(d) Optimum for the most favorable number of

revolutions N'. This is the locus of the points of contact between the curves of constant efficiency = const) and

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T A N K E R S o,so. 5 0.6,74m a s 6,<2fn 7 CR, D.P Twin Tripl» scr. ocr. 3-3 3-6-3 <0 60 60 70 80 90 100 125 150 F i g . 6 C o m p a r i s o n o f o p t i m u m e f f i c i e n c y values f o r d i f f e r e n t types o f p r o p u l s o r s

curves on whicli P, D and Va have constant values. These cui'ves can be constructed easily starting f r o m a certain vahie of and S (for instance points on the v^-optimum curve f o r v^-optimum diameter) and reducing the rotative speed by, say 10, 20, 30, 40 and 50 percent. The Bp and 5-values w i l l then also be reduced by 10, 20, 30, 40 and 50 percent.

The four typical curves coincide at one point i n the l e f t upper corner of the diagram. This point corresponds to the maximum of the envelope of the efficiency curves in the open-water characteristics, Fig. 3.

I f both the screw diameter D and the rotative speed are free to be chosen for a given power P and speed of advance Va, the optimum propeller w i l l correspond to this optimum point. As a rule, however, either the resulting diameter is too large or the resulting number, of revolutions too small f o r practical purposes.

I t is of great importance that the propeller designer makes himself thoroughly familiar w i t h one type of design chart. I t is advised, therefore, that the propeller de-signer restricts himself to the open-water characteristics (KT-KQ-J diagrams) and the B^-S diagrams when making use of systematic screw-series data. I f necessary, the open-water characteristics can be transformed i n such a maimer that the required calculations f o r any given de-sign requirements are reduced to a bare minimum. Ex-amples are the Bu-S diagrams f o r given values of the speed Va, r p m N and thrust T and the ti-a diagrams for the construction of the tow-force diagram of a tug [1].^ The advantage of the reduced amount of calculations when using a special diagram does not outweigh the dis-advantages arising f r o m the use of an unfamiliar diagram. I n Table 2 a survey of the systematic screw series tested by the N S M B is given. Results are available i n the four forms discussed previously [1-3].

2 N u m b e r s i n brackets designate References a t end of paper.

F i g . 7 P r o p u l s i v e c o e f f i c i e n t s f o r a 1 3 0 , 0 0 0 - t o n t a n k e r

Table 2 Survey of Wageningen B-Series

N U M B E R OF B L A D E S BLADE A R E A - R A T I O 2 0 . 3 0 0 , 3 8 3 0 . 3 5 0 . 5 0 0 . 6 5 0 . 8 0 i 0 . 4 0 0 . 5 5 0 . 7 0 0 . 8 5 1.00 5 0 . 4 5 0 . 6 0 0 . 7 5 1 . 0 5 B 0 . 5 0 0 . 6 5 0 . 8 0 7 0 . 5 5 0 . 7 0 0 . 8 5

Open-water tests have been performed w i t h systematic series f o r types of propellers other then the conventional screw. Some of those results are summarized i n Fig. 6. I n this figure curves are given showing the liighest obtainable efficiency rjj, f o r dift'erent types of propellers as a function of Bp. A t the top of the figure the ranges of .Bp-values typical for different slrip types are indicated. The lightly loaded screws of fast slnps are at the left-hand side, whUe the heavily loaded propellers of towing vessels are at the right. Such a chagram can give a quick indication which type of propeller will give the best efficiency for a given type of ship.

I n particular, i t can be seen f r o m the diagram t h a t propellers i n nozzles are to be reconunended f o r heavy loadings, such as occur i n tugs, trawlers and large tankers. For heavy towing a long nozzle w i t h a chord-diameter. ratio of 0.83 is preferable to a short nozzle w i t h a chord-diameter ratio of 0.50 [4].

For fast ships, contrarotating propellers appear to give a higher efficiency than conventional screws, such as the B-4.70 or broad-bladed screws such as the Gawn 3¬ 110.

I n f o r m a t i o n on the optimum efficiency of f u l l y cavitat-ing and vertical-axis propellers has been included i n the diagram [5], [6].

Fig. 7 gives the results of calculations of the propeUer for a tanker w i t h a deadweight of 130,000 tons, a ship

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Table 3 Required Power for a Tanker D i s p l a c e m e n t v o l u m e = 156,000 m^; deadweight = 130,000 tons R P M = 100; 7 . = 15.8 k n o t s ; 100 percent S H P = 27,720 h p . D = 7.20 m 7.60 m 8.00 m C o n v e n t i o n a l screw z = 4, percent. . . . 1 0 4 100 96 z = 5, percent. ...103 100 98 z = 6, percent. ...103 100 100 z = 7, percent. . . . 1 0 2 101 103 C o u n t e r r o t a t i n g propellers 2 = 4, 5; D = 6.57 - 5.78 m . 9 5 % P r o p e l l e r w i t h nozzle D = 7.00 m

Resistance increase due to 8 9 % H o g n e r stern and nozzle a t t a c h m e n t

. . . . 6 p e r c e n t T w i n - s c r e w ship

z = 3 - 3; D = 7.47 m

Resistance increase due to bossings 1 0 0 % . . . . 5 p e r c e n t

T r i p l e - s c r e w ship g = 3 - 5 - 3; D = 6.74 - 6.42 m

Resistance increase due to bossings 9 4 % . . . . 4 percent

speed of 15.8 laiots and an installed power of 27,720 shp. The rotative speed of the propeller has been fixed at 100

rpm.. The propulsive coefficient, which is a measure of

the efiiciency of the propeller including the hydrodynamic interaction of the ship's hull and the propeller, is plotted vertically.

A t the left-hand side of the diagram results are given for conventional ship propellers w i t h 4, 5, 6 and 7 blades. The calculations have been carried out f o r tluee different diameters: D = 7.20, 7.60 and 8.00 ni. I n order to give a complete survey of the potentialities of a conventional screw propeller i t is necessary to carry out calculations for a range of number of blades, diameters and rotative speeds. I n tins diagram the rotative speed has been fixed at 100 r p m and i t is obvious that for the higher number of blades (6 and 7) a screw diameter of 8.00 m is too large; the highest efficiency is found for a diameter of 7.60 m . For the lower number of blades the efficiency continues to increase w i t h increasing diameter.

I t might be the question if a diameter of 8.00 m gives already the optimum for a 4-bladed propeller f o r this slnp. For lower rotative speeds, f o r instance 80 r p m for turbine-driven ships, these tendencies are intensified.

T o the right, results are given for a set of contrarotating propellers (CR) and a propeller i n a nozzle (DP = ducted propeller). Results are also given for a twin-screw arrangement of two 3-bladed propellers and a triple-screw slup w i t h two 3-bladed and one 5-bladed pro-peller. The increase i n resistance (EHP) due to the nozzle attachment or due to the brackets for the t w i n -screw or triple--screw configuration are specified to be 6, 5 and 4 percent, respectively. For each configuration the optimum diameter, at 100 rpm, is indicated.

F i g . 8 A r r a n g e m e n t o f p r o p e l l e r i n nozzle f o r a t a n k e r m o d e l

. I n Table 3 the required power for each case is given, using 27,720 shp as 100 percent.

The large saving i n SHP which can be obtained using a shrouded propeller should be noticed.

As tanker sizes continue to increase, the advantage of shi-ouded propellers w i l l become greater. I n Fig. 8, the installation of a propeller i n a clear-plastic shroud on the afterbody of a tanlcer model is shown.

For the very big. tanlcer the contrarotating propeller leads to a power reduction of 5 percent compared to conventional screws. The advantage of this type of propulsion for large tanlcers lies primarily i n the smaller allowable screw diameters and thus a possible reduction i n the danger of vibration caused by the fluctuating propeller-force field. As shown by Fig. 6, the centra-rotating' propeller may become a serious competitor of the conventional screw propeller for fast cargo ships [7].

Cavlfatlon of the Screw-Blade Sections

The lifting-line theory for ship screws gives an idea of the induced velocities, pressures and forces along the different radii of the screw blades. This information forms the basis of considerations of cavitation and strength.

Fig. 9 gives schematically the screw w i t h a bound vortex or circulation T at the radius r and trailing heli-coidal vortices. The circulation r of a screw-blade sec-tion is defined as the line integral of the flow field along a closed curve around this profile. This line integral is the integration of the product of a line element ds and the component of velocity tangent to this line element. B y clioosing the closed curve along which the line integral is to be evaluated, i n the manner indicated i n Fig. 9, a relation can be derived easily between the circulation r of a screw-blade section and the tangential-induced

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1.00 G ' 2 i r r c t . I zT F i g . 9 R e l a t i o n b e t w e e n c i r c u l a t i o n r a n d t a n g e n t i a l i n d u c e d v e l o c i t y c, 0.50 —r r 6

/

2' f "7 / \y— \ o . l 6 l/l .0.04 0.07 * -1' LL 6 ° [ _ l' 0.5 . 1.0 S T E A O y V O R T I C E S F i g . 1 1 CL — t/l — Apll d i a g r a m for c i r c u l a r - a r c p r o f i l e s F i g . 10 O o s t e r v e l d ' s h y d r o d y n a m i c m o d e l for s c r e w blade as v o r t e x sheet

velocity c, at the same radius r. The curve shown i n Fig. 9 consists of a coaxial cylinder, of radius cut open along a line paraUel to the axis. Far forward of the propeller C( = 0, so that that part of the boundary has zero contribution to the line integral. The two longi-tudinal boundaries make equal but opposite contributions and, hence, cancel each other. Far aft, the contribution is 27rrc,. When the axial cylinder is flattened out, as at the right-hand side of Fig. 9, the chosen,boundary sur-rounds the bound vortices of the 2-screw blades and thus, according to Stokes' law, we find

zV = 27rrc,

For a finite number of blades, say 4 or 5, the value c, for the tangential induced velocity far a f t the screw w i l l occur only at the helicoidal vortex sheets. Between these vortex sheets the tangential induced velocities w i l l be less. On a circle with radius r the tangential induced velocities w i l l have an undulating magnitude w i t h a maxi-mum Ci at the free vortices and a minimaxi-mum i n between. The ratio between the mean and the maximum induced tangential velocity is given by the Goldstein reduction factor: .o.eo VO.70 - -•050 0.02 0.04 0.06 0.08 0.10 0.12 OM Vl 0.16 0.1S F i g . 12 R e l a t i o n b e t w e e n l i f t c o e f f i c i e n t d , r a t i o t/l a n d c a v i t a t i o n n u m b e r a, r e s p e c t i v e l y , pressure c o e f f i c i e n t Ap/q f o r c i r c u l a r - a r c p r o f i l e s

I t follows for 23ractical screws that for the circulation around the screw blade we have

ZT = 27r)'Cimeaii = 27r?'f<C(

The product of the l i f t coefficient and chord I of a blade section is a basic parameter i n the calculation of the cavitation properties of the blade section. According to the law of Kutta-Joukowski the l i f t i n g force of a screw-blade element, see Fig. 2, is

dL = pVTdr

where V is the resultant velocity of the screw-blade sec-tion. B y definition

dL = CLipVHdr

and hence

Cd = 2 r

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Mcr NACA 6 - S E R I E S D E S I G N C L - 0 — -T' t/l =0.0

"^^^^

u . u a / / / 0 . 1 2 / / / 0 1 5 / / / / o . i e / / ' 0 . 2 1 / -04 -0,2 0 02 0,4 0,6 0,6 1.0 _ C L F i g . 13 R e l a t i o n b e t w e e n c r i t i c a l M a c h n u m b e r a n d pressure coefficient l\plq

where .1; = nondimensional raduis =

W i t h the aid of Fig. 2 a geometric relation can be derived between \ctlY and the hydrodynamic pitch angles /? and jSj

^ ' = sin /?.• tan (^^ - /3)

For a known velocity distribution the forces on the screw-blade element can now be calculated easily.

Tills paper will be restricted to these elementary re-marks based on the steady lifting-line theory for ship propellers. Further details can be found i n the available literature, such as [3]. For a clear insight into the pro-peller action, the construction of more complicated hy-drodynamic models may often be necessary. Theories have been developed i n wliich the screw blades are re-garded as vortex sheets, both for steady and unsteady phenomena. Fig. 10 gives Oosterveld's hydrodynamic model f o r the screw blade as a vortex sheet. The addi-tional unsteady vortices are created when the screw passes a region of low speed of advance, such as occurs when the blade passes the stern post.

The creation of l i f t at the different radii of the screw blade can be accomphshed by camber and b y angle of attack. The importance of the distribution of l i f t be-tween these two f o r good cavitation properties can be determined f r o m :

(a) Characteristics for two-dimensional profiles giv-ing the relation between the l i f t coefficient CL, the angle of attack a and the cavitation number cr, or the pressure coefficient

t^vll)

see Figs. 11, 12 and 13.

(&) Cavitation-inception curves such as a <S-KT dia-gram f o r a ship screw, Fig. 14.

(c) Cavitation-inception curves superimposed on a

Bp-h screw-series diagram, Fig. 15.

I n Fig, 11 the maximum pressure coefficient Ap/g = (F^ma^ — V^)/V^ is given as a function of the l i f t coefficient

C i for four circular-arc profiles w i t h thickness-chord ratio of 0.04, 0.07, 0.1 l a n d 0.15.

Three areas can be discerned:

(a) A n area w i t h low Ci-values, where a small change

— T H R U S T C O E F F I C I E N T K^

F i g . 1 4 C h a r a c t e r i s t i c curves i n a a-Kr d i a g r a m

in CL (and, hence, i n angle of attack a) gives a large change i n Ap/q (pressure-side cavitation).

(6) A n area where the lowest Ap/g-values are reached and a change i n C L gives almost no change i n Ap/q (the area of shoclc-free entrance).

(c) A n area of high Ct-values where a small change in CL corresponds to a large change i n Ap/q (suction-side cavitation).

I n Fig. 12 the data of Fig. 11 have been replotted i n a manner, more instructive to the screw designer. This figure indicates, for a certain value of a (or Ap/q) the thickness-chord ratio t/l where a maximum variation i n the l i f t coefficient CL can be sustained without cavitation. The point at the extreme right-hand side of the GL-I/I loop for a given Ap/g-value is the point where a is about 0 deg and where shock-free entrance changes into pres-sure-side cavitation. For calculation of propeller cavita-tion according to steady lifting-line theory, a margin of safety against cavitation is generally used i n the calcula-tions, for instance Ap/q = a — 20 percent.

Fig. 12 shows that this reduction i n cavitation number affects only the determination of the optimum thickness-chord ratio t/l i n order to obtain a maximum CL varia-tion at a given Ap/q.

Experimental and theoretical data, such as given in Figs. 11 and 12, are scarce. A n extra source for i n -formation, however, is given b y the calculations for the inception of supersonic phenomena i n aerodynamics [8]. There is a simple relation between the maximum pressure coefficient Ap/q of a profile and the critical M a c h num-ber, as is shown i n Fig. 13. The critical M a c h number is the ratio between t h a t advance velocity V, when at some point on the profile the velocity of sound c has been reached, and that velocity of sound:

Fig. 13 gives an example of the relation between the crikcal M a c h number M „ and the l i f t coefficient Cr, of a

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F i g . 15 C a v i t a t i o n - i n c e p t i o n curves i n a Bp-d d i a g r a m

P-f (1.6„

F i g . 16 A x i a l w a k e d i s t r i b u t i o n b e h i n d s i n g l e - s c r e w s h i p m o d e l

profile m t h o u t camber but w i t h six thickness-chord ratios [8].

For the investigation of screw models i n the cavitation tunnel i t is important to show the different results i n a characteristic diagram. Fig. 14 gives, i n an instructive manner, the onset of the different types of cavitation. Such a diagram can be made f o r every screw model by systematically varying the cavitation number ir at cer-tam values of the thrust coefhcient KT. From such a diagram i t can be ascertained i f the screw model is too near the inception of pressure-side cavitation at the design-condition.' B y reducing the camber somewhat and compensating for this camber reduction by a p i t c h increase, the curves f o r the onset of pressure-side and suction-side cavitation can be shifted to the left, making pressure-side cavitation less likely.

Tests i n the cavitation tunnel w i t h systematically varied screw series can give data about the onset of pres-sure-side or suction-side cavitation. B u r r i l l [9] has systematically tested some screw series and has indicated the hues f o r cavitation inception i n a Bp-5 diagram, see. Fig. 15. . From this diagram i t is evident that the curve

for optimum diameter for a given power P, rotative speed A'' and speed Va, lies i n the region of suction-side cavita-tion. B y a slight change i n camber of the blade sections the area more favorable f r o m the viewpoint of cavitation can be moved toward the position of the D„pt-curve.

Vibration Generated by Propeller

Usually the variations of the flow field at the screw can be split up into two components, i.e.:

(a) The radial variation, especially of the axial

veloci-ties. This variation does not lead to unsteady phe-nomena at the screw. A propeller w o r k i n g i n such a velocity field has a steady flow and force pattern. Moie-over, the propeller can be adjusted to this radially non-uniform flow by an appropriate distribution of pitch and camber, and optimum efficiency and cavitation proper-ties may be-expected i n such cases.

(b) The chcumferential (at a given radius) variation

of both axial .and tangential velocities. This nonuni-f o r m i t y is the origin ononuni-f the periodically nonuni-fluctuating nonuni-force pattern and the unsteady pressure distributions along the blade chords, and determines the. dynamic pressure fluctuations induced by the propeller on the stern.

I n Fig. 16 an example is given of the wake distribution behind a single-screw slup model. I n this figure only the axial velocity component is given i n the f o r m of the local wake fraction iv = ( 7 . - Va)/V„ where 7 . is the ship speed.

Although often attempted [10], the expermrental deterinination of the tangential-velocity components usually meets many difficulties. The 5-holed spherical pitot tube of Van der Hegge-Zijnen still gives no consistent results for velocities below 1 m/sec. These tangential-velocity components are much smaller than the axial components but for an accurate theoretical analysis of the unsteady phenomena at the screw propeller the cir-cumferential inequality caused b y the tangential veloci-ties may not be neglected i n the future.

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KT-KQ-, - 7 0 • E R A T E L Y U - S H A P E D 8 0 9 0 U - S H A P E D S E C T I O N 6 0 F I G U R E S I N D I C A T E T H E B L A D E A N S L E S I N D E G R E E S F i g . 17 T h r u s t e c c e n t r i c i t y c a l c u l a t e d b y Stuntz, P i e n , H i n t e r -t h a n , a n d F i c k e n

/ diagram as proposed by Scliuster [11 ] may give a qualitative picture of the forces generated by a screw i n a c h -cumferentially nonuniform flow field.

For every 5 or 10 deg of the circumference an instan-taneous exannnation of the blade is made. The axial wake velocities are regarded as constant at each blade position. W i t h the aid of the open-water characteristics {KT-KQ-J diagram) of the particular screw, the time history of the thrust and torque can be found. The path of the center of the thrust w i l l be symmetrical w i t h re-gard to the longitudinal centerplane of the ship when the tangential wake velocities are neglected. This path Avill be swept 2-times every revolution for a 2-bladed propeller. Usually the region of maximum wake velocity above the propeller axis w i l l be broader (thicker) than that below the propeller shaft. The closed path on which the center of tlu'ust is moving w i l l lie mostly above the propeller axis.

When the tangential wake velocities are included, the rotative speed of the screw blades w i l l be smaller when entering the peak of the wake and larger when leaving this peak. This w i l l cause a shift of the patli of the center of thrust to starboard for a screw that rotates clockwise and to port for a screw that rotates counterclockwise. The shape of the sections i n the ships afterbody has a pronounced influence On the position and f o r m of this path, Fig. 17.

Because of this eccentric position of the tln-ust, hori-zontal and vertical bending moments are created i n the propeller shaft.

F i g . 18 C o o r d i n a t e s , elastic d e f o r m a t i o n s a n d f o r c e s

Looking more closely at the variation of torque, i t is obvious that a dynamic-force pattern is created i n the propeller shaft i n way of the propeller because of the circumferential inequality of the wake and, hence, the torque-generating force. These horizontal and vertical transverse forces and bending moments" have to be absorbed mainly b y the sterntube and the sternpost.

The forces and moments acting on the propeller work-ing i n the flow field behind the ship can be divided into six components:

Axial. Tlu'ust and torque. . .

Transverse. A transverse force, because of the

circumferential inequality of the torque force (unbalance of torque); a vertical bending moment due to the thrust eccentricity.

Vertical. A vertical force and a horizontal bending

moment f o r the same reasons as stated i n the foregoing, see Fig. 18.

The experimental determination of the thrust and torque fluctuations of a screw model behind a model of a single-screw ship was carried out successfully for the first time by K r o h n and Wereldsma [12]. They carried out their measurements of the hydi-odynamic forces created by the propeller using a measuring shaft of very great stiffness.

M a n y systematic and individual experiments have been done using the'^apparatus of K r o h n and Wereldsma [13,14]. The systematic experiments give information about the influence of number of blades, the shape of the afterbody and the position of the propeher shaft.

The frequencies of the periodic force fluctuations due to the propeller running i n the flow field behind the ship, will be equal to the number of revolutions of the pro-peller times the number of blades (the blade frequency) or a multiple thereof.

Regarding the influence of the number of blades, the characteristic difference between propellers w i t h even and odd numbers of blades must be mentioned.

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TORQUE VARIATIONS VERTICAL BENDIN6 MOMENT (prop.U«-VrtigN Mctud«d) ••6 A 0 . \

A

--5 90° THRUST VARIATIONS

A • A

i A/lA 111 A/ A /,

90° 180° 270° Propeller position 6 Z-i Z . 5 Z= 6 90° 180° 270° PropeUer position 8 360° F i g . 19 E i f e c t o f n u m b e r o f blades o n d y n a m i c p r o p e l l e r f o r c e s , e x c i t e d i n " b e h i n d ' c o n d i t i o n

Table 4 Formulas Approximating Transverse Forces and Bending Moments, Excited by Propeller ( ] 5 - K n o t Tanker)

FORMULAE APPROXIMATING THE TRANSVERSE FORCES AND BENDING MOMENTS E X C I T E D B Y A PROPELLER ( 15KN0TS TANKER )

1) HORIZONTAL TRANSVERSE FORCE

2) VERTICAL TRANSVERSE FORCE

3) HORIZONTAL BENDING MOMENT

4 ) VERTICAL BENDING MOMENT

Tz gem. Fv. 0.7 D T2 gem. F2 gem.0.7D F2 gem.O.TD = 0 . 1 2 . 0.073 S I N ( 4 p t 8 0 ° ) t. Z • A ) • 0.12 . 0 . 1 5 0 SIN ( 5 p . 1 0 1 ° ) ( Z . 5 ) = 0 . 0 6 . 0 . 0 7 6 SIN ( 4 p . 1 2 6 ° ) ( 2 = 4 ) = 0 . 0 6 t 0 . 1 3 0 SIN ( 5 p . 2 0 1 ° ) ( Z = 5 ) = 0 . 0 0 7 . 0 . 0 0 8 S I N ( 4 0 + 1 4 7 ° ) ( 2 = 4 ) = 0 . 0 0 7 » 0 . 0 1 9 SIN ( 5 p - . 1 5 9 ° ) ( Z .= 5 ) t = 0 0 3 2 * 0 . 0 1 2 S I N ( 4 p . 1 3 0 ° ) ( Z = 4 ) = 0 0 3 2 . 0 0 5 3 S I N ( 5 p . 1 5 5 ° ) ( Z = 5 )

For a screw propeller w i t h an even number of blades, the fluctuating forces of two opposite blades w i l l give rise to a larger total thrust and torque amplitude because two blades pass simultaneously the stern and its associated peaks i n wake velocities. The transverse force and bending moment of one blade w i l l be compensated more or less b y those of the opposite one.

For propellers w i t h an odd number of blades, the blades w i l l pass, alternatingly, the upper and lower wake peak. The total thrust and torque fluctuations wiU thus be smaller than for an even-bladed propeller. ' For an odd number of blades the transverse forces and bending moments, the favorable mutual compensation experi-enced by the even-bladed propeller wih not occur. Fig. 19

gives an illustration of results of measurements on 4-, 5-and 6-bladed screw models i n the wake of a ship model.

A statistical investigation of the experimental data on thrust and torque fluctuations of some 40 different ship models, tested at the N S M B , leads to the following conclusions:

1 No systematic relation can be found to ëxist be-tween the amplitudes of the force fluctuations and the principal ship-shape parameters such as block coefhcient, prismatic coefhcient and screw diameter-ship length ratio. 2 For prismatic coefhcients of the afterbody . be-tween 0.73 and 0.79 i t was ascertained for 4-blade<i pro-pellers t h a t w i t h a probability of about 80 percent the following results wiU be obtained: The amplitude of the

(11)

, E , C , . F , H y d r o d y n a m i c m o m e n t of i n e r t i a H y d r o d y n a m i c t o r s i o n a l d a m p i n g H y d r o d y n a m i c m a s s H y d r o d y n a m i c a x i a l d a m p i n g A c c e l e r a t i o n c o u p l i n g V e l o c i t y c o u p l i n g symbol 7.6 • 1 0 ' l^gm s e c 0 . 8 S 10 k g m s e c Kg s e c 1.4.10' k g s e c 2 0 . 2 3 l(g s e c F i g . 2 0 Scheme o f c o u p l e d d i f i f e r e n t i a l e q u a t i o n s o f s c r e w shaft-t h r u s shaft-t b l o c k sysshaft-tem

first harmonic* of the torque fluctuation wiU be 63^ per-cent of the mean total torque, the amplitude of the first harmonic of the tlirust fluctuations w i l l be 10 percent of the mean total thrust, and the amphtudes of the higher harmonics w ü l be substantiaUy lower.

And lilcewise for a 5-bladed propeller: The amplitudes of the first and second harmonics of the torque fluctua-tions w ü l be, respectively, lYi and 1 percent of the total torque, and the amphtudes of the first and second har-monics of the thrust fluctuations will be, respectively, 2 and V/2 percent of the total thrust.

Deviations f r o m these indications larger than 2 per-cent absolute do not occur. •

3 Fine-ended vessels, which includes most fast ships, can have substantially greater force fluctuations.

I n Table 4 a review of the formulas which appioxi-mate the transverse forces and moments generated by a 4- bladed and a 5-bladed propeUer behind a 15-knot tanlcer is given. .^For the loading of the shaft i n the ver-tical directipn besides the hydrodynamic forces the weight of the propéller has to be taken into account.

The mean value of the propeller-generated transverse forces can be neglected compared to the propeller weight f r o m a viewpoint of static' shaft loading. The static bending moment l ü t s the propeller up and reduces the deflection of the. sterntube. Reckoning has to be lieH, however, w i t h a large beiidihg moment i n the shaft i n way of the screw plane. ... ' '

, Comparing the dynaniic" behavior of a 4-bladed and 5- bladed propeller, i t i s hqted t h a t :

r (a) The fluctuations in, the transverse, force of the

5-Table 5 Stopping Maneuvers for a 100,000-dwt Tanl<erfor Headreacli of 4 l<m (2.5 miles)

Initial speed kn 10.1 11.7 13.3 14.6 Speed at which tugs assist

effectively In braking and keep

the ship on course kn 7 8 9 10 Reach at, 20 R P M . ahead

before tug assistance becomes

effective km 3.1 2.9 2.6 2,3 Reach at 60 R.RM. astern

and 40 tons extra braking force

exerted by tugs km 0.9 1.1 1.4 1.7 Head reach km 4.0 4.0 4.0 4.0

* T h e first h a r m o n i c has t h e blade frequency.'

bladed propeller (although unimportant) are twice as large as those for the 4-bladed propeller.

(&) The fluctuations i n bending moment are much higher for a 5-bladed than for a 4-bladed propeller.

Realizing that the ship designer generally has at hand effective and relatively cheap means of avoiding axial shaft vibrations (torque and thrust) and that he has to reduce the excitation of horizontal huU vibrations to a minimum, the 5-bladed propeUer is to be regarded as an unfavorable propeUer compared to a 4-bladed one.

From results of recent systematic tests w i t h , among others, the "Wageningen B-Series i t could be deduced that a 6-bladed propeller has about 3 percent less efhciency than the comparable 4-bladed propeUer [4]. The smaller screw diameter, the larger screw clearances and the very favorable pattern of the fluctuating forces (see Fig. 19) are, however, distinct advantages, justifying the applica-tion of 6-bladed propeUers for single-screw ships.

The propeller shaft and the ship's stern are not i n -finitely stift'. Thus, because of the described force patterns, elastic deformations wül occur. •

The torsion and the axial displacement of the screw owing to the elastic shaft give rise to hydrodynamic coupling between the axial dynamic screw forces (thrust) and the dynamic torsion forces (torque).

The deflections of the propeUer shaft due to the bending moments create gyroscopic phenomena at the propeller.

A certain volume of water follows the unsteady move-ments of the screw blades, m a n ü e s t i n g itself as an added mass. • •

The unsteady character of the screw loading wUl i n -duce i n the screw race helicoidal traUing vortex patterns, varying periodically m strength, see Fig. 10. The energy, carried away by this vortex system causes hydrodynamic damping. AU tlie hydrodynamic quantities of the screw as a source of vibration are summed' up now. I f i t were possible to calculate or determine experimentally these hydrodynamic quantities, then i t would be possible to predict the expected stresses i n the stern construction and in the' propeller shaft resulting f r o m the unsteady forces of itlïê'-ship propeller.

- - - lïjg. 20 gives the scheme of coupled differential equa-tions of the screw-sliaft-thrustblock system.

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P I C K - U P FOR A X I A L MOTIONS Fig. 2 1 P r o p e l l e r e x c i t e r f o r d e t e r m i n a t i o n o f c o e f f i c i e n t s f o r h y d r o d y n a m i c mass a n d d a m p i n g a n d h y d r o d y n a m i c c o u p l i n g b e t w e e n t h r u s t a n d t o r q u e v i b r a t i o n s P r o p e l l e r p o s i t i o n 0 • M £ A 5 U R t D - F U L L S I Z E S H I P • P R E D I C T I O N D E R I V E D F R O M M O Ü E L T E S T - R E S U L T S

Fig. 2 2 C o r r e l a t i o n o f measurements o n f u l l size a n d p r e d i c t i o n o f t o r q u e a n d t h r u s t v a r i a t i o n s , based o n m o d e l - t e s t r e s u l t s

The hydrodynamic mass and damping and the hydro-dynamic coupling between thrust and torque vibrations have to be determined for the prediction of the stresses in a given shaft configuration. Wereldsma has developed a propeUer exciter, Fig. 21, to evaluate the coefficients appearing i n the left-hand side of the equations. W i t h this exciter a given axial or torsional vibration can be i m -posed on a model screw at a certain load KT or KQ and at a certain advance ratio J. I n the column at the right i n

Fig. 20 the values for a model of a specific single-screw cargo ship, as measured by Wereldsma, are given [15].

The results of Wereldsma's prediction of the torque and thrust fluctuations based on model-test data and the results of measurements on the full-size ship are compared in Fig. 22. The good correlation between prediction and measurement indicates that a new area of ship-model testing has been opened as a service for the shipbuUding and ship operating industry.

Stopping of Ships

W i t h the aid of the quasi-steady velocities and forces diagram acting on a screw-blade section an insight can be gained into the force pattern around the screw during

stopping, Fig. 23. -' When the rotative speed of the screw is reduced the

angle of attack, and, hence, the thrust, will decrease f r o m that at full-ahead power (phase 1). A t about 70 per-cent of the normal ahead number of revolutions the thrust will become zero and the screw turns freely (phase 2).

As the R P I \ I is further reduced, a negative angle of attack wiU result i n negative lUt and thrust. A further reduction i n RPJM leads to such large negative angle of attack that flow separation on the screw blades wiU occur w i t h an accompanied loss i n l i f t . This separation starts at about 30 percent of the R P M ahead (phase 3).

The decreasing l i f t causes a decrease i n braking force

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Fig. 23 Relation between thrust and R P M at a constant ship speed. Force and velocity diagrams f o r blade element of screw

dt ds dt ds

ds= m .J^ dv

Vi • S . — / - d v

Vv,

in which S= Head reach rn A= Displacement ton g = Gravitational acceleration m.sec"

V j = Initial speed m.sec" V( = Terminal speed m.sec"

K= Braking force ton

Fig. 24 Integral f o r calculation of headreach of ships

until such time as the separation or profile drag become large enough to predominate and the braking force again increases (phase 4).

The continued increase of profile drag w i t h the astern operation of the propeller w i l l further increase the brak-ing force (phase 5). A t a high number of astern revolu-tions the probability of cavitation and of drawing air into the propeller increases. The occurrence of one of these may cause a decrease m braking force.

The typical S-characteristic, describing the thrust between 100 percent R P M ahead and 100 percent R P M astern operation, was described for the first time by Thau [16]. For the propeUer alone, tins curve can be determined for quasi-steady operation, when the open-water screw characteristics are available for the ahead and astern running condition [17]. These quasi-steady con-siderations are at the same time the basis for a method for calculating the headreach. I n Fig. 24 an integral is derived calculating the headreach f r o m the basic law of dynamics, that

Force = mass X acceleration

15 ol 1 1 » l i a m supply s h u t d o w n s c r e w turning s t a c k uU p o w a r a s t « m ' J s = - 5 0 1 s c r s w 5topp«d t u g s m a i t t f a s t 1 1 t u g l m i d i f l i t i l o w t y l i t i r n . R P M O F 1 T H E P R O P E t L E R L i S P E E D O F S H I P S P E E D 14-=iEDUCTIOf 9 knots 1

9-6 knots 6-i kno :s

1 1 1

0 1 2 3 i 5 e 7 8

— H E A D R E A C H IN k m

Fig. 25 Headreach for a 100,000-dwt tanker; initial speed 14 knots

The hydrodynamic added mass has been taken into ac-count by the use of a factor 1.05.

The values of the ratio V/K can be determined f r o m a model test for each speed V at different rotative speeds. For a given ship displacement the braking force K can be calculated for any combination of speed V and the rota-tive speed, and the integral for the headreach can be determined.

The ratio between displacement and power A/SHP, or as written i n the integral of Fig. 24, the ratio A/K, is very important for the length of the headreach. A large displacement propelled by a relatively small power w i l l give a long headreach (large value of the ratio A/SHP, tankers). A low value for the ratio A/SHP, as for i n -stance for destroyers and tugboats, wUl give a very short headreach.

Analysis for a maximum allowable headreach of 4 k m (2.5 miles) were made for a 100,000-dwt tanker at differ-ent initial speeds, w i t h the assumption that tugs would assist i n the stopping maneuver. Table 5 is a review of tins analysis. The only possible maneuver is the one starting f r o m an initial speed of 10.1 knots. When brak-ing f r o m 10.1 to 7 knots w i t h a rotative speed of 20 rpm, the distance covered is 3.1 k m (2 miles). A t 7 knots the tugs take hold and exert an extra braking force of 40 tons. The rotative speed becomes 50 r p m astern and the tanker stops after another 0.9 k m (0.6 mUe). Operations re-quu'ing tugs to make fast at speeds greater than 7 Imots must be considered a very risky undertaking. For a maximum allowable headreach of 4 k m the initial speed of the 100,000-ton tanlcer may, hence, not exceed 10.1 knots.

These conclusions are based on model tests. I t may be possible, due to a conservative interpretation of the scale effect, that these results are somewhat pessimistic. Owing to the lack of sufficient data f r o m full-scale tests, a correction of the data i n Table 5 for scale eft'ects is not possible.

I n Fig. 25 the calculation of stopping of a 100,000-ton tanlcer is given for an initial speed of 14 knots. The different phases into which the whole maneuver can be divided are indicated. A n essential part of the maneuver

(14)

is tlie fact that the screw is to be stopped when the speed reaches 6 Icnots and the tugs malce fast. Otherwise the vessel will loose steerageway.

Finally, i t may be noted that data and testing methods as treated i n this review are important resom-ces needed in the choice of the type of propeller and the determina-tion of its dimensions for a given applicadetermina-tion.

References

1 L . Troost, "Open Water Test Series w i t h Modern Propeller Forms," Trans. NECI, 1950^51.

2 W. P. A. van Lammeren, L . Troost, and J. G. Koning, Resistance, Propulsion and Steering of Ships, The Technical Publishing Company, H . Stam, Haarlem, 1948.

3 J. D . van Manen, "Fundamentals of Ship Resist-ance and Propulsion, Part B , Propulsion," International

Shipbuilding Progress, 1957.

4 J. D . van Manen, " A Review of Research A c t i v i -ties at the Netherlands Ship Model Basin," hitemational

Shipbuilding Progress, 1963.

5 A. J. Tachmindji and W. B . Morgan, "The Design and Estimated Performance of a Series of Supercavitating Propellers," Second Symposium on Naval Hydrody-namics, Washington, 1958 (1960).

6 .J. D . van Manen, "Ergebnisse systematischer Versuche m i t Propellern m i t annahernd senla-echt stehender Achse," Jahrbuch STG, 1963; Schip en Werf, 1964.

7 J. B . Hadler, W. B . Morgan, and K . A . Meyers, "Advanced Propeller Propulsion for High-Powered Single-Screw Ships," Trans. SNAME, vol; 72, 1964, pp. 231-293.

8 I . H . Abbott, A. E. von Doenhoff, and L . S. Stivers, "Summary of A i r f o i l Data," N A C A Report 824, 1945.

9 L . C. B u r r i l l and A. Emerson, 'Propeller

Cavita-tion: Further Tests on 16 in. Propeller Models i n the King's College Cavitation Tunnel," Trans. NECI, 1962¬ 63; International Shipbuilding.Progress, 1963.

10 J. D . van Manen, " D u r c h die Schraube erregte Schiffsschwingungen," Schiffstechnik, 1965; Schip en

Werf, 1965.

11 S. Schuster, "Propeller i n Non-Uniform Wake-Collection of Existing Work," Tenth I T T G , London, England, 1963, Report of Propulsion Committee, A p -pendix 7.

12 J. K r o h n and R. Wereldsma, "Comparative Model Tests on Dynamic Propeller Forces," International

Ship-building Progress, 1960.

13 (a) J. Krohn, "Üeber den Einflusz der Propeller-belastung bei verschiedener Hinterschiffsform auf die Schub- und Drehmomentschwankungen am ftlodell,"

Schiff und Haf en, 1958.

(b) J. Krohn, "Üeber den Einflusz des

Propeller-durchmessers auf die Schub- und Drehmomentschwank-ungen am Modell," Schiff stechnik, 1959.

14 J. D . van Manen and R. Wereldsma, "Propeller Excited Vibratory Foi'ces i n the Shaft of a Single Screw Tanker," Ijitemational Shipbuilding Progress, 1960.

15 (a) R. Wereldsma, "Dynamic Behaviour of Ship Propehers," Doctor's Thesis, Technological University, Delft, 1965; Pubhcation No. 255 of the N SJM B .

(6) R. Wereldsma, "Experiments on Vibrating Propeller Models," I hitemational Shipbuilding Progress, 1965.

16 W . E. Thau, "PropeUers and PropeUing M a -chinery. Maneuvering Characteristics During Stopping and Reversing," Trans. SNAME, 1937.

17 H . F. Nordstrom, "Screw Propeller Character-istics," Meddelanden Statens Skeppsprovningsanstalt No. 9, .1948.

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