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The Design of Ship Screws of O p t i m u m

Diameter for an Unequal Velocity Field'

BY J. D . V A N M A N E N , V I S I T O R / AND L . TROOST, MEMBER^

I N T R O D U C T I O N

This paper deals w i t h the practical results of research work carried out in the Netherlands Ship Model Basin, Wageningen, by the first author, for the Doctor's thesis [1]'* under the supervision of the second author. The underlying theory has been reduced to the minimum amount for a good understanding of the practical issues.

For many years a great number of ship screws have been designed at the Wageningen Model Basin by means of the circulation theory, origi-nally developed in 1919 by Betz and Prandtl [2]. I n 1927 Helmbold [3] derived a condition for minimum energy loss of the "ship and screw" system. This theory holds only for a propeller behind a solid of revolution. For the propeller in homogeneous flow, Helmbold [4] and Lerbs [5] developed practical design methods. Van Lam-meren [6] presented a design method for the wake-adapted screw. I n 1942-45 Lerbs [7] worked out a new condition for minimum energy loss in the combined "solid of revolution and screw" system.

which is conspicuous on account of its elegance and practical applicability. Starting f r o m Lerbs' fundamental equations, Balhan and Van Manen [8] found a condition for minimum system energy loss difl'erent f r o m Lerbs'.

Balhan and Van Manen's condition, as based on Lerbs' (and Helmbold's) pioneer work, w i l l be the starting point of the present method of design for ship screws in a velocity field which is unequal in the radial as well as in t h é circum-ferential direction, as is particularly the case w i t h single screw ships. The method is eminently suited to the design of lightly and mediumly loaded cavitation-free propellers of optimum diameter for commercial and naval vessels, for which the eftect of the race contraction is practi-cally cancelled by the effect of the centrifugal pres-sure gradient in the race. For a theory of heavily loaded open-water propellers, e.g., tug-boat screws, the reader is referred to Lerbs' latest con-tribution [13].

SECTION 1

The propulsion coefficient of the system "ship and screw" is:

N

ehp

php (1)

Lerbs, by making use of a principle of Helmbold's theory, considers this propulsion coefficient to be a function of the screw radius r. He assumes that the cylindrical sections dr of the propeller blades at radius absorb a power dN and deliver a

con-1 Paper presented at the January con-1con-1, con-1932 meeting of the New England Section of T h e Society of N a v a l Architects and Marine Engineers. Received the President's Award for 1952.

2 Naval Architect, Netherlands Ship Model Basin. 3 Professor, Superintendent, Netherlands .Ship Model Basin. J Numbers in brackets refer to references at the end of the paper.

tribution in the thrust dT. This thrust element

dT overcomes a resistance element dR. I f the

propulsive coefficient per ring element is denoted

e', we obtain:

dR-V, d.R-V, d.N

where dFt = tangential force element per ring element.

We denote the mean wake fraction at the ring element w' and the mean thrust deduction frac-tion t':

dT = dR + t'dT and VJ = V,{1 - w') , dT VJ 1 - t'

dF, wr I (3)

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T H E D E S I G N O F S H I P S C R E W S 443

For a frictionless screw (subscripts o for forces;

i for efficiency), we write:

dT, Yl l - t '

et = dF„ 1 - w' (3a)

For obtaining the condition of minimum energy loss (optimum distribution of circulation and thrust over the radius), Balhan and Van Manen now apply Betz' method of adding at radius r a small circulation element AF. The thrust will increase with an amount A(dTo) and the tangential force w i t h an amount A{dF,o). Introducing the notation:

A(dTa)

A(^™)

Zl. 1 t'

1 = K (4)

the ratio K must be independent of the radius

r for minimum energy loss. I f not, i t should be

possible to keep the absorbed energy constant and to increase the thrust output energy (in-creasing the total efficiency and reducing energy loss) by increasing the amount of circulation at a blade element with a large /C-value, taking this amount away f r o m a blade element w i t h a small /C-value. Thus the original distribution of the circulation should not have been optimum.

By application of Munk's displacement theorem and Kutta-Joukowski's Law, we obtain (Fig. 1):

(5) dTo A{dT,) = AV-p(^car - )dr A(rfF,„) ^ A r . p ( T V - f f ) dr C n 2 V . 2 ^ i \ \ d L C u \ Tt/2 C A j r = Trnd Ve.Ve'

F I G . 1.—VELOCITY AND FORCE DIAGRAM OF A FRICTION-LESS PROPELLER BLADE ELEMENT

Our condition of minimum energy loss now reduces to: car - (C„/2) V/ l - t ' K = TT^^T^^reTT;^ • — •:; ; = constant ( I / / + C „ / 2 ) o:r I - w or (Fig. 1): (6) tan^S,- = _ ^ / - t a n /3 (7) From Fig. 1: , ^ dT, VJ I dF,J m'l (C„/2) VJ I - I' VJ + (C„/2) 0.;- 1 - w' Therefore: tan /? 1 - / ' tan I w 1 = K (8) (9)

For a screw behind a solid of revolution, the relative rotative efficiency assumed to be = 1, we obtain, w i t h ideal efficiency of screw in open water = €„,: 1 K = d = Cjti t " l - w I -(10) (11)

Our final condition for minimum energy loss of the "behind" screw reduces to:

- ^ . = i ( f ^ ) ( f ^ ) - .

(12) vSince for the open water screw (homogeneous velocity field), t', w and w' are zero, equation (12) reduces for this condition to the well-known Betz equation:

tan j3

tan fit (13)

which shows the connection between the screw in the "behind" and "open" conditions.

I n Fig. 1 and the above derivations, the mean-ings of the symbols are:

cor = circumferential velocit)' of blade ele-ment at radius r

Vc = mean speed of entrance in screw disk VJ = mean speed of entrance at radius r C„/2 = induced velocity at blade element

be-cause of finite span of wing

C„/2 = component in axial direction of C„/2

C,i/2 = component in tangential direction of Cn/2

dL = lifting force on blade element dTo = thrust component of dL

dF,„ = tangential force component of dL

i3 = hydrodynamical pitch angle (uncor-rected)

0,- = hydraulic pitch angle, corrected for induced velocity

Up to this point, our reasoning is similar to Lerbs' deduction of a condition of minimum energy loss. However, Lerbs does not apply Munk's displacement theorem and obtains:

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By assumption that the variation of /' with radius is only small and that the value of • \ / ( l — / . ' ) / ( ! ~ 0 niay be taken as one, he reduces equation (14) to the simple approximate minimum condition:

t a i L ^ = i J l - ^ (15) tan p Bpijl w'

vSince the values of w' as a function of r can be established by blade wheel measurements behind the model, and the value of w f r o m integration of

w', this formula supplies a base for the design of

propellers of minimum energy loss in a variable wake. I n the same way, to the neglect of the radial variability of thrust deduction our equation

(12) reduces to:

tan/3, 1 /I - w \

t a M = ^ A r ^ ' j ^''^

Both equations (15) and (16) have been applied

in a number of cases, but not always w i t h satis-factory results. A systematic prosecution of these studies w i t h propellers behind a solid of revolution, carried out in the Wageningen cavita-tion tunnel, gave best results for condicavita-tion (16). A t the same time i t was concluded f r o m these tests, that the effect of the radial inequality of the velocity field as produced by the solid of revolution (which was a good representation of the inequality as measured with single-screw ship models) on • the onset and extent of cavitation could be neglected as compared w i t h these characteristics in a homogeneous flow field.

SECTION 2

1. I t was clear, however, that the effect of circumferential inequality of the velocity field should be studied. Owing to this inequality, and consequently to the inequahty of the intake velocity of a blade element performing a circular movement of radius r, the geometric angle of incidence of such a blade element continually changes, particularly in the case of a central screw. I n consequence, both the thrust and the tangential force continually change. W i t h any design method, using circulation theory, the cal-culations are performed w i t h the mean intake velocity over the circumference of the circle con-cerned, consequently, w i t h the thrust and tan-gential force which are in accordance w i t h the mean intake velocity.

2. A study of the radial distribution of the thrust-deduction fraction was also considered to be necessary. This should imply the effect of

neglecting the ((1 — T}/(1 — t)) term in equation (12).

.3. Attention should be given to the influence of a rudder behind the screw on the propeller action. I t was known that both the velocity field and the pressure distribution of the screw race are changed by the rudder. The transforma-tion of the velocity field in the propeller disk and the possible improvement of the screw efficiency by a streamlined rudder are phenomeca, which had not yet been examined sufficiently for deriv-ing a theory suitable for practical application.

4. The determination of the optimum diam-eter of a screw acting i n the circumferentially and radially unequal velocity field behind the ship was a problem which had not been solved in a satisfactory way, both through lack of sufficient theoretical foundations and on account of the small number of experimental data.

SECTION 3

This investigation was specially concerned w i t h screw design for the modern single-screw com-mercial ship. For this purpose, the propulsion tests were- carried out w i t h a 20-ft model of a 16j^-knot single-screw cargo ship, the propelling machinery of which develops 9130 shp at 115 rpm.

As to Section 2.1, a pair of fins were fitted to the sohd of revolution, which is 6 f t long and has a diameter of 1 f t , 2 in. I n this way, a double model is obtained of the part of the ship extending between the keel and the center of the shaft.

Three bronze screw models of 16-in. diameter were tested behind this bod3^ Besides, a theo-retical investigation was made into the alteration of the forces on a blade element due to the cir-cumferential inequality.

The principal conclusions of these investiga-tions are as follows:

(a) On theoretical grounds i t is shown that the l i f t of a profile moving linearly at an angle of incidence a is equal to the mean l i f t of the same profile at a periodically changing angle of inci-dence, the mean value of which amounts to a.

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T H E D E S I G N O F S H I P S C R E W S 445

(b) The momentary h f t force of a profile moving linearly, and at the same time oscillating at frequencies such as occur w i t h the screws of single-screw ships, is smaller than the l i f t which should follow f r o m statical calculations. I n addition a difference i n phase between the l i f t force and the angle of incidence occurs.

(c) The method of calculation w i t h the thrust value belonging to the mean inflow velocity at a certain radius gives correct results for the cir-cumferentially unequal velocity field of single-screw ships.

(d) The circumferential inequahty of the velocity field of a single-screw ship always causes momentary sheet cavitation at the blade tips i n those parts of the propeller disk where the intake velocities are small (behind the skegs). No cavitation occurs i n the parts of high intake velocity. As a general conclusion i t can be stated that the radial inequality of the velocity field can be taken into account w i t h the design of a wake-adapted propeller but not the circumferential inequahty, except b y the term (1 — t')/{l t)

in equation (12).

SECTION 4

W i t h regard to Section 2.2, a theoretical solu-tion is proposed for a justifiable radial distribu-tion of the thrust-deducdistribu-tion fracdistribu-tion to be derived. A consideration of the results of re-search in this field by Dickmann [9], Fresenius

[10] and Van Lammeren [11] lead to the con-clusion that the fraction of the thrust deduction resulting f r o m the circumferential inequality of the velocity field mainly consists of frictional and not of potential thrust deduction, opposite to the conditions prevaihng w i t h propellers behind solids of revolution. A relation was found be-tween the value of the local frictional thrust-deduction fraction and the local and mean annular frictional wake fractions, which held good for various single-screw ship forms. Likewise, the value of (1 — — /.) as a function of the radius showed for all ship forms under investiga-tion a similar picture, this value being about 0.8 at r/R = 0.2, 1 at r/R = 0.6 and 1.02 at r/R = I.

Over the outer radii, where the contribution in thrust is greatest, the thrust-deduction fraction is rather constant. From measurements and calculations i t appeared that the following rela-tion gave a good approximarela-tion for this ratio:

so that the condition for minimum energy loss (12) changes into:

tan 13, ^ 1 1 - w 1 ~ t' ^ tan /3 gpf'l - w ' ' 1 - / ~

- i ( f ^ , r (18)

ei,i\l - w'J ^ '

From this condition i t appears that the radial distribution of the thrust-deduction fraction gives rise to a smaller loading of the inner part of the screw blades, hence, of the parts where the thrust-deduction fraction is greatest.

SECTION 5

Concerning section 2.3, the influence of a rudder in the screw race, a blade-wheel measuring instru-ment has been designed, w i t h which i t is possible to measure the tangential velocities in the screw race. These measurements were performed be-hind screws i n open water both w i t h a stream-lined rudder and without rudder. A series of tests w i t h screws, plates and rudders, both i n homogeneous flow and behind a solid of revolution (with and without skegs) was carried out to meas-ure values of Kr, KQ and Bp. I t was shown by experiment, that the tangential velocity losses behind the screw are not transformed b y the rudder into axial velocity losses. The reduction in the tangential velocities by the rudder is as-sociated only w i t h a decrease of the pressure reduction caused by centrifugal force. This virtual pressure increase behind the screw gives an improvement of propulsive efficiency,

amount-ing to about 4 per cent and dependamount-ing only to a small degree on the screw type. The gain in thrust, obtained by the rudder action, appears to be greatest near the boss. B y increasing the pitch near the boss, the gain in thrust will i n -crease; on the other hand the screw efficiency proper will decrease w i t h this deviation f r o m the optimum distribution of circulation. A pro-pulsive action of the rudder does not occur, since the leading edge of the rudder is in the field of high pressure close behind the screw.

For the "screw and rudder" system a new con-dition for minimum energy loss was obtained, f r o m which could be derived that an increase of the loading concentrated very near the boss would slightly favor the efficiency of the system. How-ever, this fact was hardly considered to be of sufficient importance for abandoning the simple condition (18) for design purposes

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SECTION 6

T H E O P T I M U M D I A M E T E R OF PROPELLERS I N T H E " B E H I N D " C O N D I T I O N

1. W i t h powering problems, the selection of the propeller diameter which will give the highest propulsive efficiency is of utmost importance. I n those cases where this diameter is the largest which practical considerations will permit (free-dom in choice of r p m b y use of reduction gears) i t has a bearing on the selection of the optimum number of revolutions. I n the case of ungeared machinery installations (prescribed rpm) the cor-rect adaptation of the propeller diameter to the behind conditions is important.

I f propeller charts were available of systematic screw series, run in radially unequal velocity fields (e.g., generated by gauzes) and with stream-lined rudders of various description behind, this problem for the single-screw ship should be solved at once. The optimum propeller diameter could be selected f r o m such charts as the best solution for the prevailing "behind" conditions, and we should not have to bother w i t h book-keeping quantities like relative rotative efficiency, thrust wake and torque wake fractions. Alas, for the time being, we will have to resort to the usual propeller charts for a homogeneous velocity field, since even the practical apphcation of cir-culation theory to propeller design problems does not give us an answer to the question of optimum diameter. We have to select this f r o m propeller charts before starting a calculation by circulation theory.

Experience has shown that the diameter for optimum open water results as obtained f r o m the charts is most definitely not the diameter for optimum results in the "behind" condition. For many cases we have tested sets of B-series pro-pellers behind ship models. These sets consisted of propeller models of several diameters, designed for equal power absorption and number of revolu-tions (C-series). I n every case a propeller of smaller diameter than that indicated by the B-series design charts proved to give the lowest php and the highest propulsive efficiency. For self-propulsion point of (smooth) ship, this reduction i n diameter ranges f r o m 5 to 8 per cent for fine to f u l l single-screw ships, and is about 3 per cent for wing screws. The reduction is somewhat smaher for higher loading. I t refiects the effect of huh (and rudder) on propulsion.

2. The practical application of this reduction principle requires quite some judgment and experience, since i t depends on type of screw and ship form. I t was, therefore, considered that a more systematic approach to this problem was

necessary. Two systematic screw series have been investigated under the following conditions:

(a) I n open water (homogeneous velocity field).

(b) I n open water (radially unequal velocity field generated by a combination of gauzes).

(c) Behind a ship model without rudder (radially and circumferentiahy unequal velocity field).

(d) (1) Behind a ship model w i t h rudder (at self-propulsion point of smooth ship). (2) Con-dition (d (1)), but w i t h 20 per cent overload. (3) Condition (d (1)), but w i t h 40 per cent over-load, representing about the average service con-dition of an ocean-going vessel.

The first systematic series was the existing C-series. I t consists of seven four-bladed propeller models, ranging in diameter f r o m 208 to 256 m m and in pitch ratio f r o m 1.173 to 0.699. The expanded blade area ratio of all screws is about 0.45, the blade thickness fraction is 0.052, the rake 10°. The general characteristics are similar to the B-4 series. The face pitch, however, is radially constant. Depending on the pitch ratio, the inner aerofoil sections are washed back at the trailing edge to meet "gap-eft'ect." The series has ogival sections at the tip. The diameters are partly larger, partly smaller than the open-water optimum diameter from the B-4 propeller charts.

From analysis i t is found that the ogival tip sections provide shock-free entrance (zero angle of incidence) only for the optimum open-water diameter. For the other diameters and equal requirements as to shaft power, speed and revolu-tions, the service condition departs f r o m this maximum open-water efficiency condition. The tip sections then no longer have shock-free en-trance and some sheet cavitation may be ex-pected at these sections. Therefore, the C-series does not take account of cavitation phe-nomena.

For this reason, a second screw series has been designed w i t h the aid of the circulation theory. The power, the ship speed and the number of revolutions for which this series has been de-signed, apply to the modern single-screw cargo ship as quoted above. The series consists of eight models w i t h varying diameters, ranging from 207 to 256 m m . For simplicity of design, the velocity field was assumed to be homogeneous.

W i t h this second series, called OD-series, the above experiments have been conducted. I t

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T H E D E S I G N O F S H I P S C R E W S 447

insures a correct determination of the optimum diameter of a screw designed according to the cir-culation theory and w i t h the application of Karman-Tretrtz profiles near the t i p . For each screw is now tested in its optimum condition. B y the application of K . T . profiles (formed b y two circular arcs) of variable camber near the tips, shock-free entrance of these blade sections is obtained i n service condition. Since this series is designed for freedom of cavitation under service conditions, the expanded blade area ratio is greater than w i t h the C-45 series and amounts to about 0.50. I t has Gutsche's V.W.S. nr 7-12 aerofoil sections f r o m blade root to OAR, K . T . sections f r o m 0.7i? to tip and transitional sections f r o m 0.4 to 0.7i?. According to strength conch-tions, the blade thickness fraction varies f r o m 0.0594 for the smallest to 0.056 for the largest diameter.

3. From the test results we arrive at the following conclusions:

(a) Under all circumstances the efficiency obtained w i t h the C-series is somewhat higher than that of the OD-series. This is due to the following causes: (1) The OD-series was designed for a homogeneous flow field. (2) The O D -series has inner aerofoil profiles without wash-back of trailing edge. (3) The OD-series has a larger blade-area ratio than the C-series. (4) The OD-series is free f r o m cavitation, the C-series is not. However, the qualitative examination of the optimum diameter is not influenced b y this difference i n efficiency.

(b) The thrust-deduction fraction in the "model w i t h rudder" condition increases w i t h increase of screw diameter. This is due to the increasing resistance of the rudder w i t h increase of screw diameter. However, for the condition "without rudder" this fraction appears to be constant or decreases to a small degree w i t h in-crease of screw diameter. This is i n accordance w i t h our former conclusions: over the outer radii, where the contribution in thrust is greatest, the thrust-deduction fraction is radially constant. (c) The wake fraction in the "model w i t h rudder" condition is larger than that in the "model without rudder" condition, however, not to such an extent as would follow f r o m the ef-fective wake determination by thrust identity. For the "model w i t h rudder" condition the ef-fective wake determination by means of open-water test results is fundamentally incorrect.

(d) From all experiments i t appears that the optimum diameters in a radially unequal velocity field are smaller than those in a homogeneous flow. (See Figs. 2 and 3.)

(e) The thrust-deduction fraction is nearly

208 216 224 232 240 248 Diameter i n m m .

F I G . 2.—PROPULSION EFFICIBNCY OF C-SERIES I N VARIOUS CONDITIONS

207 214 221 . 228 235 242 249 256 Diameter in mm.

F I G . 3,^PROPULSION EFFICIENCY OF O D - S E R I E S I N VARIOUS CONDITIONS

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constant w i t h increasing diameters in the "model without rudder" condition. The wake fraction, hence also the hull efficiency (1 - / ) / ( ! - w), decreases w i t h increasing diameter. This means that the smaller diameters reduce the momentum losses due to skin friction more effectuaUy. The curve of the propulsive coefficient will have its maximum at a smaller diameter than the maxi-mum in the open-water efficiency curve.

(f) A t the same time i t appears f r o m the experiments that the improvement of the pro-pulsive action by the rudder is greatest w i t h the smaller diameters. This is in accordance w i t h our former deductions. The greatest improve-ment is reached over the inner radii. The opti-mum diameter for the "model w i t h rudder" condition will, therefore, be smaller than that for the "model without rudder" condition.

(g) Overload propulsion tests behind model show an increase i n optimum diameter compared with the "smooth ship" propulsion point. (See Figs. 2 and 3.)

I t also appears that the graph of propulsion coefficient as a function of screw diameter is less curved at the higher loads. I n service con-dition the choice of a smaller diameter than the required optimum diameter (e.g., for obtaining a low polar inertia moment) is not directly at-tended w i t h a strong decrease in efficiency. This especially applies to the OD-series.

(h) I t appears that the optimum diameters of the OD-series are markedly smaller than those of the C-series. Table 1 gives the reduction in per cent for optimum diameter w i t h both series and for various conditions, relative to the

opti-mum diameter of the C-series i n open water.

C-series - 9 OD-series - 2 - 6 - 1 0 - 1 4 T A B L E 1 Open water

Radially unequal velocity field (behind gauzes) Model without rudder Model w i t h rudder,

self-prop, point of smooth ship

Model with rudder, 40 per cent overload (service cond.)

(i) Since the optimum diameter according to the Bp - 8 diagram is about 2 per cent smaller than that obtained f r o m open-water tests, i f the increase i n wake fraction w i t h decrease of diam-eter is taken into account, we find for the optimum diameter in service condition of the C-series a reduction of 3 per cent and of the OD-series a reduction of 5 per cent. I n these investigations the infiuence of variation of ship f o r m is l e f t out of consideration. As the curve of propulsion coefficient as a function of the diameter for service condition is very smooth, we shall reduce the optimum diameter, determined i n the usual way w i t h the Bp - 8 diagrams of the B-series, by about 5 per cent to obtain the optimum screw diameter, which is calculated by means of the circulation theory and w i t h the application of K . T . profiles near the tips of the blades.

S E C T I O N 7

A M E T H O D FOR C A L C U L A T I O N OF STANDARD SCREW SERIES B Y M E A N S OF T H E CIRCUL.^ T H E O R Y FOR W A K E - A D A P T E D SCREWS

1. As has been stated in the preceding section, the blade sections of a B-series screw have shock-free entrance only at a velocity coefficient at which optimum efficiency for the given load coef-ficient is reached. B y calculation and subse-quent tests in the cavitation tunnel i t is ascertained which zone of the 5 , - 5 diagrams of the B-series warrants shock-free entrance of the blade sections near the t i p . A reduction of the chart optimum diameter unavoidably leads to sheet cavitation in spite of possible improvements i n "behind" efficiency. Profiles w i t h a suitably chosen cam-ber may prevent this f o r m of cavitation (Fig. 4).

The ideal propeller must have at each radius a profile of such camber that at zero angle of

incidence the required l i f t per radius is produced. For this condition the camber-thickness ratio

f„/s in general will decrease f r o m blade tip t o

boss. Ogival sections as apphed w i t h the B -series f r o m 0.7i? to tip never can have altogether shock-free entrance under normal loads. I n this respect the most favorable condition is obtained

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T H E D E S I G N O F S H I P S C R E W S 4 4 9

near the optimum values of Cp, but this condition does not agree w i t h that for optimum propulsion coefficient behind a ship.

I t was therefore endeavored to design w i t h the aid of circulation theory a S3'stematic series of propellers, for which the range of loading for maximum attainable efficiency behind the ship should coincide with best cavitation characteris-tics.

The method of designing individual propellers, based upon the results of the circulation theory for wake-adapted screws and using the results of l i f t and drag measurements on profiles, takes up much time and skill, though very accurate and pre-eminently suited for fast merchant ships and warships.

The method, based upon the results of experi-ments carried out w i t h systematically varied screw models in open water, is most frequently applied for designing screws. I t does not require profound theoretical knowledge. Simplicity, quickness and a good result as regards the efficiency of the screw are requirements which are to be fulfilled by this method.

Screws calculated by means of the circulation theory give better results as regards the cavita-tion phenomena and offer possibilities for small efficiency improvements w i t h regard to existing screw series, together w i t h often appreciable savings in weight and polar inertia moment.

A practical combination of both design methods should be of advantage to the naval architect.

2. For the design of a screw by means of the circulation theory, the following data should, in general, be available:

(a) The power php to be absorbed by the pro-peller, or the thrust T to be delivered for the condition concerned.

(b) The number of revolutions of the screw M.

(c) The ship speed F,, for the condition con-cerned and the corresponding mean intake velocity Vc of the water to the screw.

(d) The number of blades z.

(e) The static jDressure at the center of the screw shaft.

I n general, the diameter of the screw will be determined by means of the Bp — 8 diagrams of the B-series. A f t e r this i t is possible to calculate the dimensionless quantities, which form the proper starting points for the screw design b)^ means of the circulation theory:

The velocity coefficient

V V ^ ' e ' c

TTUD ioR The thrust constant

{l/2)pV/-{7r/m'

The power constant

^ 27rQn

( l / 2 ) p F / . ( V 4 ) i ? ' ^

I t is clear, therefore, that for the design of stand-ard screw series by means of the circulation theory, the relation between the velocity coef-ficient and the thrust constant CT or the power constant should be known. As, in general, we determine the screw diameter f r o m the B-series diagrams, i t is possible to derive this relation between X and Cj. from the B-series characteris-tics.

We transfer the curve of optimum 5-5% f r o m the Bp-8 diagram (Fig. 5) to the KT-KQ diagram (Fig. 6) and find curves for optimum KT, KQ and

Cp. {Note: KT = K,; KQ = K„; Ip = n„; Ho = Po = pitch at circumference.) We now estabhsh

for these optimum "behind" diameters the rela-tion between optimum KT and A. Starting f r o m this \-KT relation, we design w i t h circulation theory a standard screw series of which all pro-peffer profiles at all radii have shock-free entrance. Herewith, the shock-free condition is obtained in the range of optimum propulsion coefficient.

We establish the KT-A relation for the series B 4.40-B 4.55 and B 4.70 making use of following transformations:

. _ 101.277 , _ A ^ 8 Kr A — — I — ; A — LT — — T V

0 TT TT A"

For the calculation of the frictionless thrust constant, we make use of the well-known formula:

where e,- = drag-lift ratio, and: 6, = 0.020 for FJF = 0.40 e,- = 0.035 for FJF = 0.55

ei = 0.050 for FJF = 0.70

This formula is of sufficient accuracy, as the i n -fluence of friction on tlu-ust is small. We prefer the use of CT over Cy, since the relation of the latter with X can be obtained only by a process of trial and error. The values of X as well as of CTO are now known and w i t h the use of Kramer's diagrams [12] of circulation theory we obtain values of the ideal efficiency ept.

Since according to Betz' minimum energy loss condition for propellers i n homogeneous flow:

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— — —' — — ö ö d ö ö

a / ° H

where

^ Vc + (C/2) ^ Vc + (C/2) ' wR T7lD

we can define X as well as CTO as functions of Xj. As we might expect, the relation between X and

CTO appears to be independent of blade area

ratio, since this can play no part as soon as we have eliminated the influence of friction f r o m our calculations (Fig. 7). I n Table 2, the funda-mental values of the screw series are equal:

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T H E D E S I G N O F S H I P S C R E W S 4 5 1

\ T

Type B 4 Blades Fa/F = 0.40

s i/ D = 0.045 d n/ D = 0".l67 HQ = Pitch at Blade Tip Ho/D = 0.6-0.8-1,0-1.2-1,4 T 0.9 0.8 0.7 0.6 5 0.4 0.3 0.2 0.1 0 0.5 a. 4 1.5 1.6

F I G . 6 . — K T - K Q DIAGRAM OF B . 4 . 4 0 SCREW SERIES

T A B L E 2 P/D = h X CTO 0 . 5 0 , 1 5 9 2 0 0 8 0 0 0 . 9 5 0 0 . 5 0 2 5 0 . 7 0 , 2 2 2 8 0 T 4 5 1 2 . 1 3 0 0 , 6 5 1 3 0 . 9 0 2 8 6 5 0 . , 2 1 0 2 1 . 1 1 5 0 , 7 3 3 7 1 . 1 0 , 3 5 0 1 0 2 7 5 3 0 . 6 6 6 0 , 7 8 6 3 1 . 3 0 4 1 3 8 0 3 4 0 4 0 . 4 4 0 0 , 8 2 2 6

(Since at all radii we apply shock-free entrance profiles, ai = 0 (see Fig. 1 ) , the pitch angle at

any radius = (3,- and P/D = TTX,-.)

The relation between X, and X appears to be linear for the B.4 series:

X,- = 0 . 9 7 8 X - f 0 . 0 8 1 0

This equation is vahd for those screws, the diam-eter of which is defined by the 5 per cent reduc-tion hne in the Bp-8 diagram. To reproduce these relations in a design chart, we introduce the coefficient:

BT = 2 . 0 3 9

where

n = rps

= mean speed of entrance i n m/sec or ft/sec Q 3 5 0 3 0 0 2 5 0 2 0 0 , 1 5 0 , 1 0 0 , 0 5 Q 5 0 9 T t A i 1,3 V r o

F I G . 7.—RELATION BETWEEN THRUST CONSTANT CTO, VELOCITY CONSTANT X AND PITCH RATIO ?rXi

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TosM = frictionless thrust in kg or lbs for sea

water. This is 1.01 to 1.02 X screw thrust T

Fig. 8 gives the functional relation between X and

BT, f r o m which we can find X for optimum

"be-hind" diameter, and this diameter D f r o m X. 3. Our condition for minimum energy loss for wake-adapted single screws is:

tan Pi tan /3 w 1 ? 1 - t' 1 - / (18)

For the calculation of a series of wake-adapted propellers i t remains for us only to set up dis-tributions of general validity of the factors:

w Ul' and 1 /' 1 1 - w'y/ l - w As to the distribution of (1 — w ) / ( l — w'), we have found that this is practically equal for all normal single-screw ships f r o m 0.5i? to R. From 0.2i? to 0.5i? there occur larger differences in this distribution.

I f we select a standard distribution for (1 — w ) / ( l — TO')) a shock-free entrance can be main-tained in the region of 0.5i? to R, which is the most important region f r o m the cavitation point of view. From 0.2i? to 0.5R we select a radial distribution, which does not vary too much. The distribution obtained w i t h the 16.5 knot cargo ship suits our purpose. I f , for a ship of greater block coefficient, the radial distribution of (1 - TO)/(1 - w') between 0.2i? and 0.5R wih be more variable than the standard distribution, this means that we deviate somewhat f r o m our condition of minimum energy loss. This, how-ever, is not at all serious for loss of efficiency, since only the little thrust producing inner radii are concerned. For a f u l l ship form, the ex-ponent 3/4 should be somewhat smaller, for a very fine ship form somewhat larger. Calcula-tions and experiments for several cases have shown that i t is f u l l y justified to make use of a standard distribution in this formula as a uni-versal relation for the calculation of propellers by circulation theory for all normal single screw sea-going ships (Fig. 9). The standard distribution of (1 - t')/{l - t) « [(1 - w ' ) / ( l - w)]'/* is shown in Fig. 10. From Fig. 1 we obtain tan fi =

Ve'/2Tr7ir. Since X = F,/27rwi? and r/R = x

. « F. Ve' X 1 - w'

From equation (18) we obtain w i t h X,- = X/gpf:

0.30 0 . 2 5 3 4 5 6 7 8 9 1 0 1 5 2 0 5 0 4 0 5 0 6 0 (n in Revs./Sec.,Vein m/Sec.,To in Kg.) I I I I I I I f I I I I 0 4 0 5 0 6 0 . 8 I 1.5 2 3 4 5 6 7 8 ^••"(n i n Revs./Sec.,Vein Ft./Sec.Jo in Lbs.)

F I G . 8 . — D I A G R A M FOR DEFINING OPTIMUM DIAMETER FOR FOUR-BLADED SCREWS I N A VARIABLE W A K E

v , / i - w'Y'

c \ l - i v ) (20)

We can now calculate for all pitch-ratios, w i t h the values of X and X; f r o m Table 2 and those of (1 — w ' ) / ( l — w) f r o m Fig. 9, the values of tan fi for all radii. The same apphes to tan fit (Fig. 10).

4. From these values we obtain the product

Cal/D from the formula:

C „ 4 = —('''•AO'sin fiftanifii - fi) (21) JJ z

where

Ca = l i f t coefficient at :v; = r/R

I = chord length of blade element at x K = Goldstein's reduction factor at :\.- [13]

2 . 5 2 . 0 ^ 5 I I 1.0 Q 5

\

\

0.2 0 3 0 4 0 . 5 0 . 6 0 7 0 . 8 0 . 9 1.0 r/R

F I G . 9 . — R A D I A L DISTRIBUTION OF THE RATIO

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T H E D E S I G N O F S H I P S C R E W S 453

0.85

0 . 2 0 3 0 , 4 0 5 0 6 0 , 7 0 8 0 9 1,0

r / R

F I G . 1 0 . — R A D I A L DISTRIBUTION OF THE RATIO

1 - / \l - w ) 1,0 0.9 0,8 0,7 f 0.6 0,5 0.4 0,3 0.2 0.1 0 s - S t p D T D 788 ^ 0 . 565 ^ 0 5 51 <-0.4 43 ,^0.3 44 <^0.1 51 016 2 ^ so.c )79 0.03 9 - ^ 0 0.1 0.2 0.3 04 0,5 06 0.7 0.8 0.9 1.0 r/R F I G . 1 1 . — R A D I A L DISTRIBUTION OF M A X I M U M THICKNESS OF BLADE ELEMENTS

The radial distribution of the maximum thickness 5 of the blade elements can be obtained w i t h Fig. 1 1 . The blade tip thickness ratio Sup/D is mostly taken 0.003. The blade thickness frac-tion for the series Sj/D = 0.052. Since now for each pitch ratio the distribution of Ca-l/D and of

s/D is known, the distribution of Ca-l/s is fixed

and we can define the distribution of l/D (blade contour) i n such a way t h a t the cylindrical

sec-0 . 2 5 0 2 0 0 . 1 5 0.10 0.05 0.30 0 2 5 0 2 0 0.15 0.10 r / R = 02N^ 0 6 0 . 4 " 0 7 ^ 0 8 " ^ ^ . 9 ^ 0 9 5 F a / F = 0 . 3 5 z = 4 1 0 5 0 7 0 9 l . l 1.3 P/D 0 3 5 U O 3 0 Q 0 2 5 0.20 0.15 l . l 1.5 0.9 P/D

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eavitation at a given cavitation index cro =

(po — B)/[Q/2)pVe^]. The cavitation index a = {p — e)/[{l/2)pV'^] for each blade section is

given by the following relation:

sin^ j3 (I - wV " - "°cos^ {p^- f i ) \ Y ^ ' )

W i t h the aid of the diagram, developed by Lerbs, which gives for K . T . profiles the relation between (Cfl-QA, s/l, f / l (camber ratio) and Ap/q (mini-mum pressure ratio) for shock-free entrance [14]

to the cavitation requirements.

This adaptation is not advisable nor necessary for the inner sections. The ^//-ratios should become too large and should cause loss of effi-ciency. We therefore chose s/l ^ 0.20 at 0.2i?. For the definition of the eft'ective camber ratio of the profiles f J l we use the correction factor k for curved flow as given by Ludwieg and Ginzel

[15] and the l i f t correction factor ti for circulation decrease due to friction as obtained f r o m profile measurements in the Wageningen Cavitation Tunnel [16].

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T H E D E S I G N O F S H I P S C R E W S 4 5 5

P / D

F I G . 13.—(CONTINUED) (b)

A f t e r having adapted the blade contour to a series of consecutive cavitation numbers, we easily can define the distributions of l/D belong-ing to the blade area ratios FJF = 0.35, 0.50 and 0.65. These l/D values are given in Fig. 12 as a function of the mean pitch ratio P/D.

5. For completion of the design of the stand-ard series, i t remains for us to calculate the dis-tribution of camber ratio and to carry out a final cavitation calculation. A l l calculations were carried out for 5 pitch ratios and for 3 blade area ratios. The results are given i n F i g . 13 and Fig. 14. I n Fig. 15 the pitch distribution over the radius is given as calculated w i t h the relation

0.2 Q3 04 Q5 06 07 0.8 09 1.0

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SECTION 8

E X P L A N A T I O N OF T H E U S E OF T H E D I A G R A M S

1. W i t h the aid of the diagrams as presented i t is possible to design a cavitation-free, four-bladed, wake-adapted propeller of very good efficiency for a single-screw ship. The data re-quired for this design are:

(a) The thrust T.

(b) The number of revolutions of the screw

n/ sec.

(c) The ship speed (F,) and the mean wake fraction w.

(d) Blade number s = 4.

(e) Static pressure at center of screw shaft. 2. We shall carry out the calculation for an instance

(a) The thrust to be delivered in sea water

T = 62,485 kg.

(b) n = 100/min = 1.067/sec.

(c) F, = 15 knots, w = 0.324, F, = 5.216 m/sec.

(d) z = 4.

(e) Static pressure at center of screw shaft minus vapor pressure (/>o — e) = 16,360 kg/m^.

3. ro„H = l - 0 2 r = 1.02 X 02,485 = 63,760 kg. 1.667 (5.216)^ ^•V63,7m= 15.47 0.1692, so Z? = = 5.885 m . We 5.900 m, and so From Fig. 8: X = Ve/irnD

5.216/(7r X 1.067 X 0.1692) take D = 5.885 + 0.2% = X = 0.1689. Xi = 0.978 X + 0.0810 = 0.2462. The mean pitch ratio is P / Ö = TTX, = 0.7740. 4. Cavitation calculation .v = r/R = 0.8, see Fig. 14: / Ve \-(o'o)o.8( J = Ó'Ö X i\/xy[i + (x,/.v)^] w)] = 0.4185 1 + 2(X.Xi/.v^) • [(1 - w ' ) / ( l

From a strength calculation is obtained: Sj/D = 0.0513. Since the charts are to be worked out for values of Ap/q = a — 20%, there is sufficient margin in the onset of back cavitation for values of St/D somewhat higher than the standard value of 0.052 and a correction of Ap/q wih seldom be necessary. For sJD = 0.062 we may calculate w i t h the greater margin of safety Ap/q = a — 25% and interpolate linearly between a deduction of 20 and 25 per cent. Say we find St/D =

0.0583, then the o--deduction will amount to 20% + 0.63 X 5% = 23.2% and the [Ap/q]o.s value with which to enter Fig. 14 must be:

[Ap/q]o.s = ffo.s - 23.2% = 0.768 X 0.4185 =

0.3214. This will result in a somewhat larger blade area ratio to prevent cavitation than is found i n our case, where no correction is required and [Ap/qU = o-n.s - 20% = 0.8 X 0.4185 = 0.3348. From Fig. 14 we find graphicahy for P/D

= TT.Xi = 0.7740 and [Ap/q]o.s = 0.3348, a blade area ratio Fa/F = 0.421. Since this diagram is based on a hub-diameter ratio d„/D = 0.167, this blade area ratio must be corrected for deviatinghub diameters to obtain the same effective blade area. 5. W i t h the aid of Figs. 12 and 13, we now can define by graphic interpolation for the values Fa/F

= 0.421 and P/D = 0.7740 the radial distribution of the chord length ratio l/D and the camber ratio fo/l of the blade sections. From the point of view of cavitation we use Karman-Trefftz pro-files f r o m the blade tip to 0.6i?. From OAR to the hub we build aerofoil profiles around the skeleton line i n order to obtain highest efficiency. We may use i n principle any good aerofoil form, provided the maximum thickness is not situated at less than 35 per cent of the chord length f r o m the leading edge. Table 3 gives the thickness ordinates of three good aerofoils for 0.2 to 0.47?, w i t h maximum thickness at 0.35/.

T A B L E 3 . — D I S T A N C E OF O R D I N A T E S TO L O C A T I O N OF M A X I M U M T H I C K N E S S ( L . M . T )

' From L . M . T . to trailing edge

r/R 80% 60% 40% 20%,

0.2 35.15 61.75 81.45 94.90 0.3 38.75 65.80 85.10 96.80 0.4 41.50 68.75 86.55 97.00

-From L . M . T . to leading

edge-r/R 20% 40% 60% 80% 90% 95%,

0.2 97.70 89.65 74.95 51.95 35.55 24.65 0.3 97.70 90.05 75.80 53.95 37.45 26.75 0.4 97.50 90.10 76.60 55.70 38.85 27.30

The thicknesses are given as percentages of maximum thickness. The distribution of maxi-mum blade thickness over the radius is given by Fig. 11 and the pitch distribution by Fig. 15. The generator hne can be placed at 0.55/ f r o m the leading edge of section 0.2R

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T H E D E S I G N O F S H I P S C R E W S 4 5 7

SECTION 9

STANDARD SCREW SERIES FOR Z = 3

I n general, 3-bladed screws are designed for the wing position. The velocity field is much less variable than for center screws and f r o m experience we know that wing screws may be designed for a homogeneous velocity field.

I n analogy w i t h the 4-bladed series, the X —

CT, relation has been defined with the aid of the

Wageningen B-3 series. The reduction for opti-mum diameter according to the Bj,-b diagram is 3 per cent. The results of the calculations are

presented i n Figs. 16, 17, 18 and 19. The blade thickness fraction s,/D for this series is 0.055. The distribution of maximum thicknesses of the blade elements is to be taken f r o m Fig. 11. The radial pitch distribution according to the for-mula :

P/D = 7r.%--tan fii = TTX;

is constant i n this case, because the design was set up for a homogeneous velocity field.

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0 . 2 5 0 2 0 0 . 1 5 0.10 0 , 4 0 035 r/R = 0 , 7 , 0 . 6 • ^ 0 . 4 ^ 0 . 9 0 2 - ^ _ £ 0 9 5 Fa/F z = 0 . 3 5 = 3 Q 030 0 , 0 5 0 , 2 5 0 . 2 0

F I G . 1 8 . — C A M B E R RATIO FOR THRBE-BLADED SCREWS

0 . 5 0 045 0 4 0 0 3 5 0 3 0 0 . 2 5 r / R = 0 . 7 ^ = • 0 . 9 5 Fa/F =0.65 z = 3 0 . 5 0 . 7 0 . 9 P / D 0 , 0 4 0 , 0 5 , 0 . 0 2 0.01 r / R = Q 2 0 . 9 5 ^ ^ 0 , 4 ^ 0 9 _ C ^ ^ ^ 6 0 . 8 Fa/F z = 0 . 6 5 = 3 Q 5 07 09 P / D 1.3

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T H E D E S I G N O F S H I P S C R E W S 4 5 9

S E C T I O N 10

G E N E R A L R E M A R K S I n case that the screw diameter available f r o m

practical conditions and given rpm is restricted to a value below the optimum, the design method as presented above cannot be used. We then have to resort to an "individual" circulation theory calculation or to propeller design charts of the B-8 type.

I n case that the optimum diameter is the largest which practical considerations will permit, and w i t h reduction gears the revolutions can be chosen at will, these can be found by trial and error from the BT-\ diagrams.

We can obtain thrust T f r o m ehp and /. I f no ehp is available, we can obtain an estimate of

T f r o m the formula:

„ ^ php X X r.r.e. F . ( l - w)

taking Bp f r o m Bp-8 chart and relative rotative efficiency for service condition = 1.04 for single-screw ships, = 1.00 for twin-single-screw ships.

I t must be noted that Fig. 16 contains, i n addition to the BT-\ curve for homogeneous flow and z = 3, curves for z = 2, 4 and 5. Calcula-tions for these blade numbers are under way, b u t cannot yet be released.

I t is recommended to improve the design by making use of Walchner's [17] modification of K . T . circular arc profiles, consisting of a rounded and lifted nose. These modified profiles have

Fa/F = = 0 . 3 5 ^ Fa/F= = 0 5 0 ^ Fa/F = Q65^^ si/D = z 0.055 = 3 0.5 07 0.9 M 1.3 P/D

F I G . 1 9 . — C A V I T A T I O N DIAGRAM FOR THREE-BLADED SCREWS

a wider range of angle for shock-free entrance and therefore an extra margin of safety against cavitation i n a circumferentially unequal velocity field. The coordinates of these profiles are given in Table 4. T A B L E 4 .%•// 0 2.5 5 7.5 10 15 20 30 y./d UA 35.1 45.1 52.9 59.9 71.0 79.5 92.0 yjd 14.4 4.7 3.2 2.2 1.5 0.3 0 0 x/l 40 50 60 70 80 90 100 y,/d 98.5100 96.6 85.6 67.6 41.4 3.0 yjd 0 0 0 0 0 0 0

x/l = percentage of chord length / from leading edge.

xb/d = ordinates of baclc of profile in per cent of maximum

thick-ness at 1/2.

xf/d = Oldinates of face of profile.

R E F E R E N C E S

[1] Van Manen, J. D . , Doctor's Thesis 1951, Delft. Publication No. 100 of the Netherlands Ship Model Basin.

[2] Betz, A., "Schraubenpropeller m i t gering-stem Energieverlust," m i t Nachschrift von L . Prandtl-Nachr. K ö n . Ges. der Wiss, zu Göttingen. M a t h . Phys. Klasse, 1919, p. 193.

[3] Helmbold, H . B., "Nachstromschrau-ben," W . R. H . , 1927, p. 528.

Helmbold, H . B., "Ueber den Vortriebs-wirkungsgrad," W. R. H . , 1928, p. 151.

[4] Helmbold, H . B., "Die Betz-Prandtlsche Wirbeltheorie der Treibschraube und ihre Aus-gestaltung zum technischen Berechnungsver-fahren," W. R. H . , 1926, p. 565.

[5] Lerbs, H . W., "Kurbentafeln zur Berech-nung starkbelasteter Freifahrtschrauben nach der Tragflügeltheorie," W. R. H . , 1933, p. 29.

[6] Van Lammeren, Troost and Koning, "Resistance, Propulsion and Steering of Ships," H . Stam, Haarlem, 1948.

[7] Lerbs, H . W., "Bemerkungen zur Theorie und zum Entwurf von Nachstromschrauben."

[8] Balhan, J., and Van Manen, J. D . , Publ. No. 88 N.S.M.B., 1950.

[9] Dickmann, H . , "Wechselwirkung zwis-chen Propeller und Schiff unter besonderer Berücksichtigung des Welleneinflusses," Jahrbuch S. T . G. Berlin, 1939, p. 234.

[10] Fresenius, R., "Das grundsatzliche Wesen der Wechselwirkung zwischen Schiffskörper und Propeller," Schiffbau, 1921, p. 257.

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[11] Van Lammeren, W. P. A . Doctor's Thesis, 1938, Delft. Publication No. 32 N . S. M . B .

[12] H i h , J. G., "The Design of Propellers,"

Transactions S N A M E , Vol. 57 (1949), p. 143,

Figs. 24 and 25.

[13] Lerbs, H . W., " A n Approximate Theory of Heavily Loaded, Free-Running PropeUers i n the Optimum Condition," Transactions S N A M E , Vol. 58 (1950), p. 137, Figs. 2 and 3.

[14] Lerbs, H . W., 1. c. (13), Figs. 18 and 19. [15] Strassl, H . , "Camber Correction for Screw Propellers," M A P Voelkenrode, Ref. M A P -VG90-T, 1945.

[16] Balhan, J., Doctor's Thesis, 1951, D e l f t Publication No. 99, N . S . M . B .

[17] Walchner, O., "Report on Profile Meas-urements," Report of the K.W.-Institute f ü r Strömungsforschung, Göttingen, 1934.

DISCUSSION

PROFESSOR M A R T I N A . A B K O W I T Z , Member:

The Society is greatly indebted to Professor Troost for a most .stimulating and informative paper, and I am personally happy that the mem-bers of the New England Section were given opportunity of partaking of the wealth which we at M . I . T . have been almost mercilessly exploiting all term. This wealth, is, of course, the knowledge, experience and ingenuity i n the field of naval architecture that Professor Troost possesses and which he has lavishly and willingly shared with us. This technical wealth is weh amplified by a wealth of personal qualities which those who have had the privilege of knowing h i m have enjoyed.

The paper is a very timely one and tackles a very difficult problem; i.e., how to get something beneficial out of the apparent chaos t h a t exists in the velocity field at the stern of the ship. Professor Troost's attack of the problem is expert. He shows us that the velocity field, when properly studied, is not as chaotic as i t originally seemed and in fact for a single-screw ship has quite a definite pattern. A regular radial distribution of wake and thrust deduction and a circumferential variation of these quantities can be resolved b y their average effect. Advan-tage is taken of this procedure to produce a more efficient ship-propeller unit.

The manner i n which the problem is attacked and solved is well worth noting and admiring. First, the most recent theoretical contributions to the subject are analyzed and used, when the extent of theoretical analysis is reached, detailed and well-planned experiments have been carried out to fill the gap in the necessary information desired and evaluated. The results first evolve i n what may be called mathematical form but are i m -mediately Jnterpreted in terms of diagrams and design procedure. The train of events are all

there—theory, experiment, solution, design. I t is hoped that in his talk or in his reply to the dis-cussion the author will extend the train one point further—to practice. I am sure the author has interesting information about these propellers in practice. I should like to ask the author to discuss the problems or the lack of problems pre-sented in the manufacturing of a large propeller with K a r m a n - T r e f f t z profiles.

Searching for, evaluating and using all the latest theoretical and experimental developments is nothing new w i t h Professor Troost—for example, stimulating turbulence on large models has been standard practice at the Wageningen Tank since 1937—this practice only recently has been adopted by the m a j o r i t y of the other tanks. The present plans of the Wageningen tank to conduct an elaborate and expensive series of experiments on models and f u l l size ships, as revealed at the recent International Conference of Ship Tank Superintendents, is another example of the author's desire to get the necessary informa-tion beyond the realm where theoretical con-siderations have reached their present l i m i t .

The contribution of the author in the presenta-tion of this paper is one which I am sure is wel-comed by all.

M R . N . J . BRAZELL, Member: I agree w i t h the authors to the effect that the optimum pro-peller diameter is generally less in the "behind" condition than the optimum diameter established f r o m "open-water" Basic Series propeller charts. However, this concurrence applies only to pro-pellers when operating in the non-cavitating field.

Contemplate, if you will please, the plot of the conventional 13 orifice p i l o t tube wake survey i n way of the propeller disk of a single-screw ship. There you will perceive wide variations in the

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T H E D E S I G N O F S H I P S C R E W S 4 6 1

magnitude of the fore and a f t wake currents throughout the propeller disk and also non-axial flow stream-lines. Remember, a helicoidal screw propeher is essentially an axial flow reaction thrust instrument.

Propeller action in the "behind" position smooths out the variable wake and non-axial flow patterns. Consequently there occurs a greater relative induced velocity (forward of the propeller disk) than obtained during the operation of the propeller in the "open-water" test. Therefore, in the "behind" position, the virtual or effective diameter of the propeller is increased. This situa-tion will be evidenced by a relatively greater diameter i n the Vena-Contracta of the propeller slipstream. To illustrate: a similar augment in the virtual propeher diameter occurs when em-ploying the well-known K o r t nozzle. The non-axial flow in way of some steeply raked destroyer shafts is the cause of the negative wake fractions (up to 2 per cent) which have been reported in some self-propelled model tests.

When the propeller diameter is reduced a cer-tain amount, then the propeller pitch also simul-taneously must be increased by some two-thirds of the diameter reduction. Experience indicates that the propeller w i t h the higher pitch is a better instrument than the lesser pitch propeller to smooth out the variable wake and non-axial flow. A n inspection of the velocity diagram. Fig. 1, also will lend theoretical support to the conclusion of experience. (Note the lesser angular change in the direction of the resultant velocity, due to vary-ing the speed of advance, when the geometric angle of the hydrofoil has been increased.) Re-member, too, that the maximum effect on a rotat-ing propeller, due to wake or non-axial flow, is positioned some 30 degrees of angular arc prior to coincidence w i t h the non-uniform flow.

Now, consider the flow in way of a cavitating propeller. Cavitation is a vaporization process. The ambient water around the hydrofoil is not only of lesser density but the flow departs f r o m the original stream-line character. I have found the inception of cavitation to be substantially delayed on destroyer propellers by using a propeller diam-eter 2 V 2 per cent larger (not less) than the opti-mum diameter obtained from propeller Basic Series charts. Also, the larger propeller is subject to a lesser loss i n efficiency when operating deep in the cavitation fleld. M a n y of the propehers on our high-speed P T boats t u r n at 2800 rpm. The optimum propeller diameter for this P T boat ap-plication was found to be some 10 per cent larger than predicted by the Basic Series charts. I t therefore appears that a constant correction factor cannot be apphed universally to the propeller

diameter which is obtained from Basic Series charts.

This paper embodies a wealth of fine material. The practical aspects of the information are particularly commendable.

D R . H . W . LERBS,^ Visitor: The writer regrets very much that the copy of this paper was received too late to prepare a thorough comment, which corresponds to the great merits of this paper. The remarks will be restricted to equations (5) and (21) of the paper.

The writer cannot see in which way equations (5) follow f r o m Munk's displacement theorem rigorously since, by this theorem, the elementary circulation AF is applied to the ultimate wake whereas equations (5) are written i n terms of the flow at the disk. To relate the flow in the u l t i -mate wake to that at the disk an approximation is necessary, viz., that second and higher powers of the induced velocity components can be neglected (by which neglect the medium loaded propeller is defined). These are just the approximations which the writer had applied when deducing the minimum condition equation (14). For this reason, the writer is not entirely convinced that the theoretical basis of the condition (12) which follows f r o m (5) is more accurate than that of the condition (14).

The second remark refers to the configuration of the vortex sheets behind a wake-adapted propeller in connection w i t h equation (21). This equation is essential since i t determines the design of the propeller i n such a way that the circulation distri-bution which follows from the minimum condition is realized in a radially varying wake. W i t h i n the deduction of this equation, the condition of normality is involved, viz., that the resultant of the components CJÏ and C„/Ï be normal to the re-sultant relative velocity (Fig. 1). This condition is satisfied only if the vortex sheets are of a true helical shape which is, in general, not the case for a wake-adapted propeller. Secondly, the Gold-stein function K is related to free-running opti-mum propellers, i.e., to vortex sheets of a true helical shape w i t h a definite circulation distribu-tion which differs f r o m that of an optimum wake propeller. Introducing both the condition of normality and the Goldstein function into the re-lation for {cJ), i t can not be expected that the circulation distribution which is required by the minimum condition of the wake propeller w i l l be realized by the calculated quantity {cJ). From this reasoning the writer has developed recently a more general method for wake-adapted propellers within which the radial variation of both the pitch

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of the vortex sheets and of the cireulation is taken into account. This becomes possible when basing the method on the velocity field of helical vortex hnes which are the elements of the vortex sheets. W i t h i n such a method the induced velocity com-ponents follow independently without being re-lated by a condition introduced from the begin-ning. Further, the Goldstein function appears as a special case of more general functions, the so-called induction functions, which have been cal-culated numerically. B y means of these func-tions i t becomes possible to design a wake pro-peher, i.e., to calculate the product (£„//) such that the prescribed circulation distribution is realized. The only assumption which is necessary is that the circulation distribution can be represented by a Fourier series which assumption, however, does not restrict the practical application. Therefore, in the opinion of the writer, i t is premature to draw conclusions w i t h respect to a minimum con-dition for a wake-adapted propeller before tests have been carried out w i t h propellers which are designed f r o m the aforementioned induction func-tions.

These remarks are not intended to restrict the value of the paper. The authors have clearly shown the potentialities of the vortex theory in the case of the complicated problem of the interaction between propeller and hull and have deduced f r o m their investigations practical design methods on the basis of the vortex theory. The writer wants to congratulate both Dr. Van Manen and Profes-sor Troost for this valuable contribution.

M R . E D W A R D V. L E W I S , Member: The work of

Professor Troost and his associates at the Nether-lands Ship Model Basin has been of great value to all naval architects through the years. A t the Experimental Towing Tank this work has been most helpful in connection w i t h our studies of propulsion scale effect and i n practical propeller design problems. The present paper is again valuable on both these counts, and i t is a pleasure to record our appreciation to the authors for mak-ing i t available.

A few general comments can be made at this time, although further study will be needed for the real appreciation of all aspects of the paper. A ' -though the B-series has become a standard ref-erence in designing modern propellers, i t is gra-t i f y i n g gra-to know gra-thagra-t gra-the circulagra-tion gra-theory is being applied to improve upon i t , particularly i n ref-erence to cavitation characteristics. The design charts presented should be of great value in many cases by making possible the design of propellers in accordance w i t h the circulation theory without the need for going through the

elaborate calculations usually involved.

A particularly significant statement is the con-cluding one indicating that a modified Karman-Trefftz section is under investigation, having rounded and lifted leading edge. The section used i n the paper (Fig. 4) would seem to have a serious disadvantage i n its sharp leading edge, providing "shockless entry" only at exactly 0 angle of attack. As pointed out on page 5, item (d), circumferential variation i n wake velocity causes momentary cavitation which certainly would be reduced by a rounded leading edge. Of even more importance is the fact that fouling and weather effects cause a merchant ship propeller tD operate through a wide range of propeller loadings. Hence, the angle of incidence will change and an adaptable type of section is badly needed.

The information on radial variation of thrust deduction is helpful. I t would be interesting to know how this quantity was measured experi-mentahy.

I t is believed that the theoretical conclusion on page 5, item (a), that the mean l i f t of a profile w i t h fluctuating angle of incidence is equal to the l i f t at the l i f t at the mean angle is not borne out by some wind tunnel tests which have been reported. We will look up these references and forward them at a later date, since they have a bearing on the question of the effect of circumferential variation in velocity.

A particularly important point is the conclusion that the optimum diameter in the behind condi-tion is about 5% less than i n open water, although the effect of overload due to added resistance has an opposite trend. I t might be well to point out that the conclusion regarding reducing the di-ameter for optimum efficiency does not mean, for example, that typical American cargo ships neces-sarily should have new propellers fitted. For, as the authors state, this conclusion applies only to cases where the r p m are limited, which is typical of European design problems where the Diesel engine is quite generally used. W i t h typical geared tur-bine installations on the other hand, the diameter usually is governed by the practical consideration of propeller aperture, immersion, etc., and is there-fore quite commonly less than that for maximum efficiency.

For example, using the B4.40 chart applied to a C3 cargo ship i t is found that for 85 r p m at 8500 shp the optimum diameter is about 22 f t 9 in. The actual diameter, which was felt to be the maximum practicable, is 21 f t 8 i n . (The chart also shows that increasing the r p m at this diameter would decrease the efficiency slightly.) Thus for typical design problems in this country, the auth-ors' data should be applied from the point of view

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T H E D E S I G N O F S H I P S C R E W S 4 6 3

of optimum r p m for a given diameter, as men-tioned under General Remarks, Page 459.

I t should be emphasized that accepting pro-peller diameters less than the absolute maximum practicable almost always restricts the possible open water efficiency obtainable, no matter what refinements may be made in detail design. This is clearly seen when ideal efficiency in the basic momentum theory is plotted against thrust load coefficient, indicating that i t is a direct function of (diameter) 2. Hence, i t is felt that the complete picture is obtained only by realizing that every effort should be made to reduce r p m to take ad-vantage of larger diameters. There can be little doubt that the gains f r o m increasing diameter

with reduction in rpm will more than balance the

losses f r o m unequal velocity distribution i n the wake. However, the authors have shown that the latter considerations will indicate a somewhat higher optimum r p m than would otherwise be selected.

We hope that this valuable work will be con-tinued, since i t leads to a better understanding of propeller efficiency in the condition which is of ultimate importance—that is, behind the ship.

PROFESSOR F R A N K M . L E W I S , Vice-President:

The Section is to be congratulated upon having obtained this interesting paper which, I believe, to be among the most valuable which have been con-tributed to the Society. The authors have under-taken a formidable task in attempting to formu-late the conditions which fix the size and other characteristics of the optimum propeller, operat-ing in a non-uniform velocity field. I will con-fess to a feeling of considerable uneasiness even i n considering the problem f r o m the theoretical viewpoint. One finds oneself asking questions regarding the validity of fundamental hydrody-namic laws, such as Munks displacement theorem or the Kutta-Joukowski law, when they are applied to a potential flow which is superposed on another flow, non-potential in character. As evidence of the pitfalls which may exist we note that Lerbs obtains equation (15) for the optimum condition and Van Manen and Troost obtain (16) quite a different expression. I have not studied the prob-lem sufficiently to f o r m any opinion as to which of these expressions is the correct one.

Fortunately, f r o m the very nature of an opti-mum, i t is not necessary that the exact optimum conditions be found. There exists, for example, an optimum radial pitch distribution b u t for very considerable departures f r o m this condition the efficiency will be affected very little.

I n Section 4 the authors note a radial variation of the thrust ratio l—t' and state that this obtained

theoretically and experimentally. I would like to know just what is meant by this and how one goes about measuring the thrust ratio associated w i t h a particular radius.

The authors find that the optimum propeller diameter, for a non-uniform wake condition, is smaller by the order of 5% than the optimum diam-eter as ddiam-etermined by the optimum diamdiam-eter line on a series of open-water test charts. The reason for this is obvious. However, i t should be pointed out that an optimum diameter, as applied behind a ship, still can be obtained if one properly can as-sess the thrust and flow ratios and the resistance thrust ratio correctly. To do this, a series of diam-eters is assumed and for each diameter ƒ„ ƒ, and

y are assigned suitable values. Propeller

character-istics for each diameter are obtained and the opti-mum is obtained from a plot of the results.

From this viewpoint i t would add greatly to the value of the paper if tables or curves of ƒ, and y, as well as the propeller characteristics, for the series tests w i t h varying diameter, were appended.

I n American practice w i t h geared turbines the propeller diameters fitted are usually the largest which are practicable. W i t h the conclusions of this paper in mind a re-examination of some of these vessels, to determine whether or not exces-sively large propellers have been fitted, might be i n order.

I n practice, one wih not want to fit the propeller of theoretically optimum diameter but one slighter smaller, if cavitation conditions will per-mit. The authors i n Section 10 remark that their design method does not apply i n this case. I speculate if this statement is strictly correct or if some slight modification of the design procedure will not make i t apply to almost optimum screws.

M R . A . J . T A C H M I N D J I , Associate Member: The paper contains a great wealth of information on a subject which is refiected by its general lack of knowledge and the great need for experimental work of a fundamental nature. The introduction of the wake varition across the propeller plane raises a number of points w i t h regard to propeller theory. The presence of wake being a result of frictional forces, i t is doubtful whether the ideal or frictionless propeller theory still can be used. The development of the optimum condition accounting for the wake variation at least in the radial d i -rection (the circumferential variation may or may not exist as w' is taken to be the mean) can be ex-tended to include the effect of drag. Consider-ing equation (3):

^ dT- VJ{\ - t') dFT-co-r{l w"

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