Blade spindle torque and off design behaviour of controllable pitch propellers

ARCHiEF

BLADE SPINDLE TORQUE AND

OFF-DESIGN BEHAVIOUR OF

CONTROLLABLE PITCH

PROPELLERS

C. PRONK

0 9 NO 1988

Delft University of

_Technology

ShIp Hydromechanics Laboratory Mekelweg 2 _{2628 CD DELFT}

BLADE SPINDLE TORQUE AND OFF-DESIGN BEHAVIOUR

OF CONTROLLABLE PITCH PROPELLERS

BLADE SPINDLE TORQUE AND

OFF-DESIGN BEHAVIOUR OF

CONTROLLABLE PITCH

PROPELLERS

PROEFSCH RIFT

TER VERKRLIGING VAN DE GRAAD VAN DOCTOR IN DE

TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE

HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS

PROF.

DR. FA.

KIEVITS,

VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN

DEKANEN TE VERDEDIGEN OP WOENSDAG II JUNI 1980 TE 16.00 UUR

DOOR

CORNELIS PRONK

SCHEEPSBOUWKUNDIG INGENIEUR

GEBOREN OP 15 SEPTEMBER 1945 TE ROTTERDAM

Dit proefschrift is goedgekeurd

door de promotor:

Prof. Dr. Jr. J.D. van Maneni

CONTENTS: Page

Introduction 1

Geometry and centrifugal loads 6

2.1 General

2.2 Coordinates of blade surface in off design pitch condition 2.3 Effects of main parameters on section deformation, analysed

by means of analytic solutions for simplified blade surfaces 2.4 Results of numerical analysis

2.5 Centrifugal forces and moments in arbitrary pitch position

Blade spindle torque measurements ₁₉

3.1 General

3.2 Results of previous measurements

3.3 Experimental technique, procedures and conditions 3.4 Results for static component of spindle torque

Computation methods 57

4.1 General

4.2 Review of existing calculation methods 4.3 Surface vorticity method

4.3.1 _{Considerations for the development of a new} calculation method

4.3.2 Description of the model 4.3.3 Intermediate test results

4.3.4 Comparison of computed and measured spindle torques

Dynamic spindle torque measurements, adaptable pitch propeller 79 5.1 General

5.2 Results of dynamic spindle torque measurements

5.3 _{Application of dynamic spindle torque to investigation of} feasibility of adaptable pitch propeller

Summary and concluding remarks ₉₂

1. INTRODUCTION

Controllable

pitch propellers were considered as "special propul-sion devices" up till about 10 years ago. Only during the last decade the application of controllable pitch propellers is gene-rally accepted and no longer considered as special. Nowadays about 30 percent of all newbuilt seagoing ships are equiped with C.P. propellers.

For this wider application of C.P. propellers several reasons can be pointed out. First of all there is the acceptation based on proper functioning of the earlier C.P. propellers and their strong-ly increased reliability. This has led to a growing number of "classical" applications i.e. for ships with strongly varying operating conditions and for vessels where frequent manoeuvring is required. Next to this, new applications were found by the in-troduction of marine gasturbines which did not possess reversing possibilities. Other developments where the particular features of the C.P. propellers show their benefits and have contributed to expansive use are for example: dynamic positioning with

continu-ously varying thrust demand, auxiliary power generated by shaft driven generators requiring propulsion with constant rotational speed, anchor handling and ice breaking with optimized backing thrust, multi engine configurations driving one single shaft.

Parallel to increased application in the construction a continuous development has taken place. Some of the components characterizing the modern concept of a controllable pitch propeller are: high pressure hydraulic motor for pitch actuating inside the hub, bearing constructions able to withstand the very large centrifugal loads on the blades and having optimal friction behaviour, elec-tronic pitch sensing devices inside the hub with contactless signal transmission from the rotating shaft. For the design of the com-ponents knowledge of the external loading, of which the main part is generated by the blades, is essential.

Therefore, the blade design of C.P. propellers must also be based on interference effects with other components_

Although there are some principal differences with fixed pitch propellers the same design methods are used. Adopting these same methods is correct provided that essential differences with fixed pitch propellers are taken into account. The main differences to be dealt with in designing blades for CA'. propellers are:

larger hub diameters when compared with monobloc propellers, restriction in chordlengths for the higher blade area ratios, restriction of dimensions of blade root sections,

impact

of

off design conditions,

A larger hub diameter is necessary to locate the pitch adjusting mechanism. With the influence of a larger hub mostly the influence on the efficiency of the propeller is meant. By means of linear momentum theory the reduction of efficiency can be estimated and within the practial range of hub diameter ratios this reduction will be limited to a maximum of 2%. Taking into account the effect of different loading distributions for C.P. propellers this re-duction will increase further. In literature even values of up till

5% are mentioned e.g. [Ref. 1]. Bearing in mind that experiments were recently performed showing that different hub shapes can change the efficiency with 3% [Ref. 2] , it can be assumed that the.

influence of larger hub sizes will be restricted when the hub shape is optimized. Unfortunately the theoretical treatment of hub in-fluence is not yet developed far enough and therefore only opti-nitzatiom 'by means of experiments is possible. Although efficiency is very important and should be optimized it is generally accepted that C.P. propellers have lower efficiencies, in design condition.

For complete reversing of the blades a limitation in blade chord lengths is a necessity.

-2-However, the restriction of blade area ratio is only important for a particular class of propellers, namely those propellers which need very large blade area ratios. The consequence of restricted blade area is that C.P. propellers are not applicable in extreme .cases where F.P. propellers are still feasible [Ref. 3]. For the majority of the propellers this restriction does not play an

im-portant role, although an adaption of the radial load distribution is sometimes required.

Part of the external loading is transmitted through the housing of the hub and in some constructions blades are bolted to the blade carriers. These parts of the construction require their space between the blades and limit the dimensions of the blade root

further than the already mentioned restrictions necessary for pitch reversing. For strength reasons these restrictions have to be compensated by increased thickness of the root sections of the CJP. propeller blades. In case of high speed propellers this will am-plify the problems with blade root cavitation [Ref. 4]. In case the blade root cavitation is unavoidable a further increase of huh diameter

will

be required.

The subject of this thesis is not only of importance for the ana-lyses of the performance of the C.P. propeller, but gives also tools for a better design, of which proper functioning in other' conditions is assured. The influence of off design behaviour on the design is found in the following subjects:'

strength of blades and 'hub cavitation erosion

noise radiation and vibration excitation forces efficiency

spindle torque capability of the hub.

-3-for cavitation erosion, noise radiation and efficiency off design conditions of long duration are determinant [Ref. 4 and 5]. For strength such steady state conditions can be determinant but also transient conditions can be dominant. While the spindle torque level during transient conditions is normally having the more important consequences. This is due to the fact that maximum

spindle torques in the majority of the cases occur in the low pitch range where the power absorbtion and the propeller thrust are small. This of course is a condition which occurs during longer periods only by way of exception. During manoeuvring however this condition happens frequently, however as a transition between steady state conditions. A characteristic spindle torque pitch

relation is shown in Figure 1, where the results of a full scale measurement clearly shows the larger spindle torque values in the low pitch interval.

gsma..

1Ppio.

rants1424,-3 6

lialla

OPPSW' 40 PITCH

_{ANGLEN140026111}

(DEGREES)--I,

_{4,..aff>7141-1*}

a'

Figure Y. Full scale measurement results of spindle torque as function of pitch setting.

-4-

-5-In both steady state and transient off design condition all parameter values are entirely depending on the control system once the design of ship and propeller is fixed. In case undesired parameter values occur during transient condition this fact can be used to limit possible negative effects of off design condition on the propeller itself. Then provisions in the control system can be made to avoid undesired conditions. Nevertheless for the majority

of the designs this is simply impossible, since response time during transients is normally limited. This means that a thorough investigation of off design behaviour of a C.P. propeller is necessary to establish the reliability of the propeller.

In this thesis both experimental and theoretical developments will be treated in order to improve the tools available for off design analyses. In Chapter 3 the results of model experiments are treated.

In particular the influence of cavitation on blade spindle torque, which up till now was only known qualitatively, will be thouroughly investigated. In addition the effect of propeller geometry on blade spindle torque will be shown by means of systematic variations of the main parameters. Since the experiments were performed in so called "behind condition" fluctuations of blade spindle torque during a propeller revolution could also be measured even in cavi-tating condition. In Chapter 5 these results are presented. They are used to make theoretical analyses of the possibility to change the blade pitch during a revolution. A propeller with this feature is called an adaptable pitch propeller. The purpose of this propeller is to solve the ship vibration problem at the source namely to

react on the varying inflow velocities in the wake field. In Chapter 2 considerations are given which contribute to a better understanding of the experimental results. Finally in Chapter 4 a theory is given for the calculation of blade spindle torques. This lifting body theory has no restraints with regard to incorporating non linear effects.

2. GEOMETRY AND CENTRIFUGAL LOADS

2.1 General

When the blades of a C.P. propeller are rotated to change their pitch the dimensions of the blades do not change since the blades are rigid. Yet is this respect often the motion section deformation due to pitch change is used. For understanding of hydrodynamic propeller behaviour in theoretical calculations by means of a

lifting line model and in cavitation analyses the geometrical and hydrodynamic characteristics of cross sections between the blades and cilinders coaxial with the propeller shaft play an important

role. Such a cross section with one and the same cilinder will have a different shape for different pitch angles and this difference is referred to as section shape deformation.

In earlier publications [6] [7] [8] it has been pointed out that the sections of the propeller will be distorted due to blade ro-tation and typical Sshaped sections were shown. For design and analyses purposes insight into the influence of the various para-meters on deformation is of importance. Of course, when rigorous numerical computation methods are used coordinate rotations can easily be incorporated without separate geometrical analysis.

Due to the complicated geometry of the propeller a direct analy-tical approach is impossible. Therefore numerical methods are used to calculate the off design geometry and with these methods the effect of parameter variations can be computed. Next to the numeri-cal solution it is possible to analyse blade distortion analyti-cally by means of simple blade models. By this analytical approach a better understanding of the phenomena can be obtained and the benefit will be shown in analysing experimental results and judging

theoretical methods. Since the centrifugal loads entirely depend on the blade geometry the forces and moments in off design condition can be derived directly from their values in the designposition.

-6-X

Figure 2_ Coordinate system

The blade surface in this coordinate system (see Figure 2) are given, in, parameter presentation by the following, formulae:

yo + s sins

4

yf(xl) coss cost - sins y(x1)

Y:t(xl)+ a

(1 +

(4(x1)) a)

2.2 Coordinates of blade surface in off design: pitch, condition.

In the conventional way the geometry of the propeller is described by the two dimensional section

offsets

of camberline and thickness distribution in a developed cilinder plane at various radii. Rotation over the particular pitch angle and translation over the distances corresponding to skew and rake give the ordinates., in a. cilindrical coordinate system.

-7-= (x1

rono d(x1 + s) cos* - yf(xl) sit*

sin* + cos* yf(x1)

T Yt()

' 2

41 (Yf(x1)) )

where':

x1 = parameter, helicoidal distance at cilinder r = rip to midchord point of section.

= skew defined as helicoidal distance at r = r cildnder between midchord point of section and (0,0,r0)'.t

0 = pitch angle at cilinder r = ro.

Yf camber ordinate at cilinder r =

Yf = slope of camberline at citinder r = ro.

Yt = thickness ordinate at cilinder r = r.

A rake at cilinder r

The index o refers to the coordinates of the surface in design condition. The upper signs hold for the back side and the lower signs hold for the face side of the blade. The meaning of the parameters is illustrated in Figure 3. The generator (z-axis) line has been selected perpendicular to the shaft and is lying in the plane through, shaft and spindle axis.

-8-= / + (2.2) s = ro. = = ro.

Figure 3., Coordinates of two dimensional sections in developed ctlinder plane.

Assuming that propeller shaft and spindle axis lie in the same plane making an angle 6, the coordinates of the blade surfaces after rotation over an angle AO are given as function of the original coordinates by:

sin a = ro cosa0sim5 SiMAS + ro sinao cosa. + yo cosS sin.. .1(2.37

r cos a = ro cosao (cOS26 + sin26 cos110)

-ro sina simo sinta +

+ yo sind cos6(1-cosAg) 01

24i

= r

cosao sind cos6(1-cosAfl + ro sima cosd +

0

+ yo(sin26 + costs, -

sin2S cosA0)

-9-r

-10-The coordinates of the blade surface in off design condition are then given by equations (2.1) - (2.5).

A numerical solution can be obtained by starting from coordinates in design position. For each selected value of ao a number of values

ro is taken. With an iterative process, e.g. Wegsteins procedure,

the parameter x/ can be obtained from equation (2.2). The yo, r and a are computed directly from (2.1), (2.3) and (2.4).

This means that r0, yo and a are known as tabulated functions of r for each

txo. The desired values of section ordinates at a radius

follow from parabolic interpolation in these tabulated functions and from inserting the values in equation (2.5) for each ao. A flow

diagram of the computerprogram is given in Figure 4.

-.3 Effects of main parameters on section deformation, analysed by means of analytic solutions for simplified blade surfaces.

The deviation from the original section shape, when rotating the blade from ahead design pitch position to "zero pitch", will have the same character, but have opposite signs, when compared with the deviation starting from "zero pitch" and then increasing the pitch with the same angle. This can be used to analyse the shape defor-mation, since the most simple blade surface is that of a flat blade with zero pitch, thickness, camber and rake. Then holds yo = 0. Defining u = ra and using the transformation formulae the section in the developed cilinder plane can be described by:

I = cos6 sin

uT-tp4 -(cos sins tgA0) tgS (1 - cosA(0 (2.6)

(tg2 + cost)

For a spindle axis perpendicular to the shaft this shape gets further simplified to:

y . u

j loop

read: geometry of original propeller pitch change

compute maximum value of ao

ao = a

-

decrement

select array r0

iteration proces for

x1. i loop

compute

y0, a.,

r.

F

select aray for desired radii r.

interpolate in tabulated function r (r.), y (r.), a(r)

J J

compute yj

off design geometry

ANGLE OF ROTATION .7 .6 .5 .4 PITCH ANGLE

AFT SKEWED

Figure 5. Section distortion as function of skew.

It can be seen from this Figure that, for reduction of pitch, a forward skew induces negative cambers with the nature of a sine function and causes less pitch reduction than corresponding with the angle of rotation. Backward skewed blades will, for the same blade rotation, show positive induced cambers, while also the pitch reduction shows a pitch lag. For increasing skew and increasing chord length both effects - induced camber and pitch lag - are amplified. In Reference [9] these effects have been utilized to

reduce the cavitation extent in off design condition. For a flat bladed thruster of which the design pitch is actually zero the planform was changed such that, in the operating condition with

increased pitch, the pitch lag caused a reduction of blade tip

loading.

-12-Depending on the skew of the blade, the position of the blade in this plane varies and, consequently, both the shape and orientation of the section vary (See Figure 5).

FORWARD SKEWED

.4 3 .2 .1 .1 2 3 .4 ANGLE OF GLE \ROTATION PITCH .5 .8 / CENTRE OF ROTATION .2 .3 .4

-13-The effect of rake on section distortion can be illustrated by rotating a plane yo =

ro cosao tgc. The equation of the plane after rotation with an angle AG, will be:

= sin Li tgA4) + cos toe

r

cos*

It follows from this formula that forward rake increases the dis-tortion effects for positive values of u and decreases the effects for negative u values. This means that for highly backward skewed propellers the application of forward rake has a compensating effect on the section deformation. This raking forward is often applied for highly backward skewed propellers of both the fixed and controllable pitch type. Normally, rake is applied to increase the clearance with respect to the rudderpost and to reduce tension stresses at the pitch side of the blade root. As shown above an additional advantage is effecting the blade spindle torques. Bearing in mind that sections at the lower radii mostly have low skew values, it can be seen that rake hardly influences the pitch lag but only induces camber at these lower radii.

A comparison of the distorted planes as given by equations (2.6), (2.7) and (2.8) is made in Figure 6. The data in this Figure refer to an angle of rotation of 25 degrees. A forward rake of 10 degrees and a spindle axis inclination of 10 degrees aftward have been chosen, such that in both cases the blade is in front of the spindle axis. It can clearly be seen that spindle axis inclination hardly effects the section deformation and that the influence of forward rake is more pronounced. Occasionally spindle axis inclination is used for constructional reasons.

Insight in the effect of thickness and camber can be obtained by rotating a plane

yo = ao and comparing the shape after rotation with that of the plane yo = 0 after rotation.

0.5

100 FORWARD RAKE 0° SPINDLE AXIS

0.4 0° RAKE 00 SPINDLE AXIS

RAKE 10° AFTWARD SPINDLE At,,'

0.32

/7

0.2

//'

0.0

//

/

77

0.4 0.52 -1.0 -0.75 -0.50 -0.25

-14-7.

a = sin Li

tgo

+ 1 r cosA.

So, a constant thickness will have no influence on the deformation, since the second term in this equation only gives translation in axial direction.

0 0.25 0.50 0.75

u/R

Figure 6. Comparison of section deformation due to rake and spindle axis inclination.

For the rotated plane yields: 0.04 0.16 0.28 1.0 (2.9) INCLINATION

2.4 Results of numerical analysis.

Although influences of some parameters can be investigated by means of analytical solutions for simplified blade surfaces, more detailed parametric studies can only be performed when using the numerical method.

As was shown in the previous paragraph, it can be concluded from simple blade theory that effects of thickness on deformation are small. This leads to the often used method, in which only the dis-tortion of the nose-tail line is taken into account, while the original thickness coordinates are added to the transformed nose-tail line on the new radius. Points lying on the same radius after rotation originally come from different radii. Even points at face and backside come from different radii. Although the different original radii lie rather close together, at the edges, where large thickness gradients excist, differences in edge form occur when the approximate method is used. Results of numerical analysis demon-strate that the influence of the approximate method on the leading edge shape can be neglected for normal applications. In this respect it should be borne in mind that cavitation inception calculations are extremely sensitive to proper description of section nose shapes, but even then approximations as mentioned above are acceptable.

Another consequence of the fact that points arrive from different radii is the influence of pitch distribution on section defor-mation. For the example propeller the effect of two different pitch distributions have been investigated: the original pitch distri-ribution and a constant pitch distdistri-ribution equal to the pitch at 0.7 radius. The sections at 0.7 radius after a 30 degrees rotation are shown in Figure 7. Although the pitch at this radius for both propellers was the same, after rotation a difference in pitch angle of about 1 degree results.

-15-

-16-CONSTANT PITCH

VARIABLE

PITCH

Figure 7. Section shapes after rotation for constant and variable

pitch.

Centrifugal

forces and moments in arbitrary pitch position. The centrifugal forces and moments for the blade in arbitrary position are given by:

F

p0)2 fif

xdydz F = 0 Y Fz .

pw2 fjf

zdxdydz Qx . pw2

1ff

yzdxdydz Qy . 0 Qz =

pw2 iff

xydxdydz

In which are: p = specific density

STELLINGENI

Eltj de bepaling, van verstelkrachten voor verstelbare schroeven dient met de invloed van cavitatie rekening te warden gehouden.

Verdergaande ontOkkeling van schroefontwerp- en, analysemethoden Neeft meer kans van slagen wanneer men daarbij uitgaat van die methoden die

gebaseerd, zijn op modellen, testaande uit discrete wervels.

[en veelbelovende remedie ter vermindering van scheepstrillingen is het aanpassen van de stand van de bladen van een schroef tijdens een omwente-ling, aan de aanstroomrichting in het volgstroomveld.

IV

Vele bekende, doch niet of nauwelijks toegepaste oplossingen ter vermindering van het brandstofverbruik aan board van schepem, kunnen zonder veel moeite en op korte termijn met succes warden aangewend.

'V

Het uitvoeren van weerstandsproeven Voor het onderzoek naar de vorm, van het achterschip is onjuist.

VI

Aangezien cavitatie een dynamisch verschijnsel, is en, de momenteel bestaande. rekenprogramma's voor cavitatie-analyse dit dynamisch effect niet voldoende in rekening brengen, dienen cavitatieproeven beschouwd te worden als een onderdeel van het ontwerpproces en niet als een afnametest van een schroef-ontwerp.

VII

Het kiezen van de spoed van de verstel bare schroef als onafhankelijke regelgrootheid berust slechts op historische gronden en niet op een, juiste analyse van het voortstuwingsproces.

VIII

Het behalen van extreem hoge reductieverhoudingen met behulp van een hydrostatische-extrusiepers berust niet op een hoog rendement van het proces maar op de grate hoeveelheid energie die wordt toegevoerd.

IX

Het onbeheerste proces van stolling en afkoeling van grate gietstukken kan warden verbeterd door het vooraf analyseren van de warmtetransporten of door temperatuursmetingen en gerichte koeling tijdens het proces.

X

Organisatieschema's en functieomschrijvingen geven de werkelijkheid niet weer.

XI

Met het ontwerpen van een geintegreerd scheepsbesturingssysteem en het afstellen hiervan door middel van een manoeuvreersimulator bewijst men het aloude gezegde: "De beste stuurlui staan aan wal".

XII

Hoewel het een kunst is een goede schroef te ontwerpen, is het ontwerpen van een schroef nog geen kunst en komt dus niet in aanmerking voor een subsidieregeling.

Dissertatie C. Pronk Delft, 11 juni 1980

When these forces and moments are made dimensionless in' the form which is usual for propeller forces and moments by dividing through pn2D4 and pn2D5, the force and moment coefficients become:

2 Sx t: 4n 2' Sz = = _{4%2 -Ye} = 4%2 Cxy

-5-D 'Where: D Sx'Sz C ,C xy yz = propeller diameter = static moment =- centrifugal moment-;,

For rotation around the spindle axis the following transformation formulae hold for static and centrifugal moments:

S _{= Sx} cosAs sinAs Ya Sz = Sz or ... (2.11N Cy

_{0 0}

z cosAcP + C SlMA(1) Z X

_{0 0}

C C lx - ry xy xoyocos2A$ + 2 _sin2As Where: IX

fff

x2dxdydz

fff

y2dxdydz.

and the suffix 0 refers to the, condition before rotation.. CF CFz (2.10) -C yz = = = =

-18-From equations (2.10) and (2.11) it follows that the dimensionless forces and moments in off design condition can be expressed in the ordinates of the blade in design condition:

472 CFx (Sx COSAct, S sinAO D Yo 472 C = S F z z D o 472

,,

Y CQx - , ku cost+ + C sinao) DJ

0 0

Z

00

x 472 I I x - y o Cr, = IT (Cx cos2A0 + o 2 sin2a4)) 'z "

0'0

(2.12)

So the centrifugal forces and moments in any off-design condition can be obtained by calculating eight volume integrals for the blade

in design pitch condition. With the same dimensionless coefficients the centrifugal spindle torque can be calculated also for an in-gifted spindle axis. The spindle torque coefficient will then be:

I - I 4n2

,r

Xo 2 Yo n = ku cos2no + sin2A0)cos5 D' NoYo

-BLADE SPINDLE TORQUE MEASUREMENTS

3.1 General

Whereas numerous data on thrust and shaft torque values are avail-able, experimental information on blade spindle torques is limited. Before the construction of the larger propellers is started, it is standard practice to verify their predicted performance by means of model tests. The strong desire for verification is due to the strict quantitive requirements, such as contract speed and proper power absorbtion, of which fulfilment is tested during trials. Requirements regarding spindle torques are only qualitative, namely adjustability. For this reason model experiments are only performed by exception. Full scale experimental data are _{even more scarce.} This can be explained by the restricted value of the _recordings. As can be seen from Figure 1 _{torques including all frictional} contributions can be recorded without the possibility of an

acct.,-rate separation into components.

Although before 1960 in several publications measurements of blade spindle torques are treated, the results can not be considered as

valuable design information, but have to be regarded as incidental information of the qualitative kind. The first attempt for syste-matic measurements on model scale was succesfully performed _{in 1962} by Bossow [Ref. 10]. Over the entire range of pitch angles the spindle torque coefficient was measured for blade _{area ratios of} 0.3 to 0.9, in open water condition, at atmospheric _{pressure and at} positive advance ratios. The measuring device _{was a mechanical one} based on springs. With this device 900 measurements _{were performed.}

The next step in spindle torque measurements _{was the introduction} of straingages, while splitting up mechanically the spindle torque from the other moments and forces. With such _{a device tests were} performed at DTMB to investigate the effect of spindle axis loca-tion and type of secloca-tions [Ref. 11].

-20-Further improvement of the measuring equipment offered the possi-bility to investigate more parameters. In Reference [12] results at different Reynolds numbers and cavitation numbers show that both these parameters have effects which must not be neglected. Only

later on it was found that the dependency on Reynolds number resulted from the test set up in a cavitation tunnel. With tests carried out in a deepwater basin no effect of Reynolds number was measured [13], [14].

In the meantime also at the NSMB a device had been developed to measure spindle torques under cavitating conditions [Ref. 15]. Although no systematic measurements were done, this instrument was

fully operational and was used in five commercial applications. Nevertheless it was felt that there still were some drawbacks on essential points. In 1974 on the authors request the NSMB developed new instrumentation for measuring spindle torques. The equipment was based on the principles of the already operational six compo-nent balance for measuring shaft forces and moments. The descrip-tion of this five component dynamometer will be given in paragraph

3.3.

From the above it can be concluded that measuring spindle torques has always been difficult. Only recently at various institutes simultaneously so called 5 components dynamometers for propeller blades were developed [Ref. 16, 17], with which accurate measure-ments on model scale are possible.

3.2 Results of previous measurements

From the measurements performed with the DTNSRDC spindle torque equipment only the more recent experiments are considered as

valuable. See Ref. [14]. These experiments were carried out as open water tests and contrary to the earlier experiments not in a tunnel. Unfortunately the six tested propellers, although covering a large range of blade area ratios and skew distributions, do not have systematically varied particulars.

However, from the results some important conclusions can be drawn. The highest absolute values of the spindle torque coefficient occur

at very large advance ratios and at advance ratios corresponding to nearly bollard condition. Since the very high advance ratios are only possible in those conditions where the rotational speed is low, these extreme values of spindle torque coefficient do not result in high absolute values of the spindle torque, for the spindle torque is proportional to the rotational speed squared. This restricts the important area to the interval of advance ratios around bollard condition, which is in full agreement with full scale experience and is also mentioned in Reference [17]. The results of Reference [14] for zero advance ratio are summarized in Figure 8. -21-4496 4575 4536 4572 4517 4535 nmimm 4402

,/

...---

,/

. 0

/

--,

...

,./'.

,7

"?.

---. --;;'4.'""

...

A..

---...',* /..

..//

.1 11.4*.

,... / /

..\,...

X.1

...,

/

0 0.5 1.0 15 PITCH RATIO

Figure 8. Spindle torques as function of pitch setting for bollard condition.

/

.

=22-The most dominant effect occurs. due to skew. Both. skew back of radii and balanced skew back - forward skew of inner radii and backward skew of outer radii - reduce the spindle torque in design condition. In addition the gradient of spindle torque with advance ratio is reduced and gets an opposite direction for the larger skew values. The consequence is a much stronger negative spindle torque in the low pitch low advance ratio range, which makes required hub dimensions larger. In this condition balancing of the skew has not much influence on the spindle torque level.

The experimental results of Rusetskiy [6] cannot be compared directly with those of Bossow [10] since the blade number is differing.. Other propeller parameters differ only slightly so that use of both series of tests could give an indication of the effect of blade number. Comparison of the results for the same blade area shows curves of the same shape. However, the values for the lower blade. number are higher due to blade interference effects, Figure 9.

v,

0 02 04 0,6 OB COP Xi

ADVANCE RATIO

Figure 9. Comparison of spindle torques for different blade numbers where blades are comparable.

IN

1 1 2 BLADES BLADES 3

in_

_ill

Ii

I I

EN

_a

1 , A

\

II

I

,,

\ .

. all

-23-From Rusetskii's test results it follows that the design pitch angle has an influence on magnitude of spindle torques in off design pitch position. The most important effect is the increasing value at zero advance ratio and low pitch values with increasing design pitch ratio (Figure 10). This must be explained by section shape deformation.

A A

-..

Y

0.5 0.6 0.7 0.8 0.9 1.0 1.1

DESIGN PITCH RATIO P/D

Figure 10. Spindle torques at P/D = 0.2 and J = (las function of design pitch ratio.

From the results of the experiments carried out by Bossow the following conclusions can be drawn in addition to conclusions already mentioned in Reference [10].

1 . 3 1.11 1.J0 0.9 0 . 8 0.7

-24-For the design pitch ratio (i.c. P/D = 1.0) the spindle torque value over the range from small advance ratios to 1.0 is practically proportional to the blade area ratio. This also means that the advance ratio where the spindle torque is zero

is independent of the blade area ratio. This can only be the case when the pressure distributions are such that lift

applies at midchord as occurs with shockfree entry.

For positive advance ratios the spindle torque is linear with the pitch ratio, although the gradient strongly depends on the blade area ratio.

For zero advance ratio, where we normally expect the highest spindle torques for low pitch values, the linearity with blade area no longer holds. This must be explained by the section deformation being larger for the larger blade area ratios. For the same reason we see that the pitch angle at which the spindle torque changes sign is considerable larger for the larger blade area ratios, where the spindle torque is already nega-tive and where the thrust did not yet reverse.

3.3 Experimental technique, procedures and conditions.

In order to obtain results of spindle torque measurements on model scale, which can be properly scaled to full scale values and which

incorporate all important effects of physical phenomena, a 5 compo-nent dynamometer based on strain gages has been developed for use in the depressurized towing tank. Using this instrumentation the following advantages are achieved:

Proper scaling of cavitation phenomena in the depressurized towing tank without earlier observed wall effects in a cavi-tation tunnel.

Comprising the effect of wake behind a ship model to obtain the correct radial load distribution. The spindle torque value is much more sensitive for radial variations in

Av,

F.15-Figure

Spindle

0.5

CALCULATED

---- MEASURED

-26-Measuring the variations in spindle torque during a revo-lution of the propeller in the wakefield.

Performing spindle torque measurements under the same con-ditions as those during propulsion tests or even performing both tests simultaneously and therefore using the same scale.

The design requirements of the spindle torque measuring device were:

sufficient sensitivity to measure one blade only no interaction with other forces and moments

sufficient stiffness such that natural frequency is at least 5 times higher than frequency of exciting forces.

Figure 11 shows the construction of the measuring device. Static calibration tests showed that interaction of spindle torque with other forces and moments only influenced the output signal with + 1% of the maximum value. Dynamic calibration can be performed by measuring the centrifugal torque during propeller rotation in air. This centrifugal torque can be calculated as accurately as desired as was shown in paragraph 2.5. An example of dynamic calibration is given in Figure 12.

0.5

PITCH RATIO P/D

1.5

Figure 12. Dynamic calibration of spindle torque measuring device. In

-27

The signal transfer from the measuring device requires that the shaft is driven from astern as is the normal mounting during 'measurements of dynamic forces and, moments generated by the

pro-peller. A dummy shafting generates its part of the wakefield.

Scaling the velocity, rotational' speed and static air pressure occur according Froude's law in order to obtain the correct cavitation number in any point of the propeller disk. As usual

in

cavitation tests Reynolds law of similarity is not fulfilled and this introduces scale effects in the wakefield. The effect of Reynolds number related to the propeller has been investigated, The results of this investigation reveal that no scale effect has

to be expected by not obeying the scaling law for friction pheno-mena on the propeller blades. The tests conditions have been selected such that desired combinations of cavitation number and advance ratio were obtained in a steady state condition, without taking care of translation and rotational equilibrium. This means that the test results represent transient conditions simulated in a quasi steady way. The justification for neglecting hull accele-ration effects has been given in Reference [18]. Although inertia effects in' shaft rotation differ in order of magnitude from trans-lation inertia effects also a quasi steady approach can be accepted for rotational transients,, see discussion to Reference [19].,

The particulars of the tested propellers are given in Table 1 and the planforms of the blades are shown in Figure 13. The propellers have been tested in behind condition, therefore the wakefield data and stern arrangement are shown in Figures 14 and 15.

All propellers are non-raked.

Type of camberdistribution: NACA a=0.8

Type of thicknessdistribution: elliptical.

Table 1.

Particulars of tested propellers.

Rip C/0 TIC F/C 5/C RADIUS A B C 0 E 4

BCD

C A B C 0 E A B C D E A 8 C D C 1. .975 .95 .9 .85 .8 .7 .6 .5 .4 .35 Hut. 1,214 I 1,214 1,214 1,214 1,214 1.214 1,214 1 1,214 1,214 1,214 0 .323 .409 .494 .535 .550 .535 .481 .407 .331 .293 .265 0 .323 .409 .494 .535 .550 .535 .481 .407 .331 .293 .265 0 .323 .409 .494 .535 .550 .535 .481 .407 .331 .293 .265 0 .291 .368 .444 .481 .495 .482 .433 .366 .298 .263 .238 0 .259 .327 395 .428 .443 .428 .385 .325 .265 .234 .212 .0031 .0037 .0041 .0054 .0069 .0085 .0126 .0173 .0233 .0321 .0385 .0444 .0031 .0037 .0041 .0054 .0069 .0085 .0126 .0173 .0233 .0321 .0385 .0444 .0031 .0037 .0041 .0054 .0069 .0085 .0126 .0173 .0233 .0321 .0385 .0444 .0031 .0037 .0041 .0054 .0069 .0085 .0126 .0173 .0233 .0321 .0385 .0444 .0031 .0037 .0041 .0054 .0069 .0085 .0126 .0173 .0233 .0321 .0385 .0444 .0025 .0030 .0034 .0039 .0044 .0047 .0052 .0055 .0053 .0043 .0033 .0 .0025 .0030 .0034 .0039 .0044 .0047 .0052 .0055 .0053 .0043 .0033 .0 .0025 .0030 .0034 .0039 .0044 .0047 .0052 .0055 .0053 .0043 .0033 .0 .0025 .0030 .0034 .0039 .0044 .0047 .0052 .0055 .0053 .0043 .0033 .0 .0025 .0030 .0034 .0039 .0044 .0047 .0052 .0055 .0053 .0043 .0033 .0 .0897 .0897 .0831 .0771 .0713 .0660 .0566 .0484 .0431 .0387 .0368 .0355 .0 .D .0 .0 .0 .0 .0 .0 .0 .0 .0 .0 .060L .0406 .0265 .0140 .0063 .0 _{-.G00E -.0065 -.0016} .0173 .0269 .0355 .019: .0897 .0E31 .0771 .0713 .066: .0566 .0484 .043: .03E .0358 .035.0 ....E.97 .039) .0331 .077: .0713 .0663 .356: -1.484 .0431 .0391 .0365 .5350

-29-EXPANDED BLADE OUTLINE PROPELLER MODEL A (----), PROPELLER MODEL B AND PROPELLER MODEL C

EXPANDED BLADE OUTLINE PROPELLER MODEL A (---), PROPELLER MODEL D AND PROPELLER MODEL E

Figure 13. Planforms of tested propellers.

_

\

\

\

cc

1

)

/

XMIEI

-if

1.0

-L

-180

-100

-30-0 ANGULAR POSITION TANGENTIAL COMPONENTS

Figure 14.

Wakefield data.

100 180

1.08

0 97

0 75

0 65

0.43

r/R=0.32

0.86

r/R=0.97

Figure 15. Stern arrangement.

-31-

-32-3.4 Results of measurements of static components of spindle torque.

All results presented in this paragraph are based on processed spindle torque values. The processing of the measured data consisted of harmonic analysis, subtraction of measured centrifugal moments, correction with dynamic calibration data and least square regression. The parameters are expressed in non dimensional coefficients

defined as:

Q,

hydrodynamic spindle torque coefficient K -OS pn205

advance ratio

cavitation numbern

-ipn2D2

where: Po = static pressure e = vapour pressure

h = shaft submergence

n rotational speed Vs = shipspeed

Qs spindle torque, positive values tend to enlarge the pitch.

Obviously the results can be presented in two ways: in the usual K053 diagrams and, for the illustration of cavitation effects, in the KQS -an, diagrams.

The curves showing Ku as function of an are in principle S-shaped curves. For larger cavitation numbers no cavitation occurs and there will be no effect of cavitation number on the spindle torque values. For low cavitation numbers the blade will fully cavitate at

one side and also no effect of further reduction of cavitation number can be noticed. A further reduction of static pressure would

cause cavitation on both sides of the blades simultaneously, but such conditions are not realistic.

=

-33-So the KQS-an curves show two horizontal lines with a transition curve in between representing the condition from cavitation inception to full cavitation. During the tests it was observed that the

difference in spindle torque for non cavitating and fully cavitating blades is large and can be of the same magnitude as the spindle torque value itself. This means that the use of spindle torque

values obtained from tests under atmosferic condition for engineering purposes can be unrealistic. Only for the small pitch angles of this series of propellers the influence of cavitation number appeared to be minor, which can be explained from the fact that in these conditions no cavitation was observed. However this does not mean that the same will hold for other types of propellers, in particular not for those with a strong pitch reduction towards the tip where cavitation can occur for the same range of cavitation numbers.

The tests confirmed the effect of blade area ratio and skew on spindle torque as already mentioned in the previous paragraph. The influence of cavitation does not reduce any of these effects. The influence of blade area ratio and skew in cavitating condition can mainly be explained by arm effects i.e. the under pressure peak due

to angle of attack has a weighing factor in the pressure integration depending on blade area ratio and skew distribution. While of

course the earlier mentioned section deformation, which strongly depends on blade area and skew, causes in cavitating conditions the

same effects as in non cavitating conditions. From Figure 16 it can be seen that the gradient of Ko for non skewed propeller B is larger than for the skewed propeller A. Cavitation reduces this effect. Furthermore increasing skew back to moderate values reduces the blade spindle torque for large positive and negative pitch angles in bollard condition. Balanced skew (propeller C) gives a reduction in spindle torque for positive pitch angles, however for the neutral pitch interval hardly any effect was observed.

0. 0. 0 0. 0. -2.

-34-liallilli

A

KM

,n = 4 =

-

On 1.5

FM

-

Ka

rI4

NI

_

____ Irv,

Ni

E _DESIGN

i

LPITCH

... ftuAlinilli

.4

--110v4111

.." B

.111

P ITCH 5°

11.11111.E

B

P'

.2 .4 .6 .8 1.0

12

1.4 ADVANCE RATIO J

Figure 16. Spindle torque coefficient as function of advance ratio for propellers A-C, without cavitation and for an = 1.5.

1 0. 21 -35L , . 0 0.2 0.4

0.6

0%8 a 0 1.2 1,4/ ADVANCE RATIO

Figure 17.

_{Spindle torque coefficient as function of}

_advance

ratio for propellers D and E,, without cavitation

and for an

1.5.

DESIGN' PITCH on

=4

----

76 '

/' 5

"111111

hi

_

_1.14

taMIN

____

IF--I

Tr

I

, z

//

/

. r PI CH 45.o -411111"111111111"-

la

alla

..."

7a

(

/

Di

ilaWPlanpral

,,--eadigelria

WPM

1E ITCH 59 I, I

IIII

1.2 0.8 0 0.4 0.8 -1.2 =

--from Figure 17 it appears that Blade area reduction gives the same trends as moderate skewing backward.! The reduction due to reduced blade area for negative pitch angles im cavitating condition causes that values of spindle torque coefficient in, neutral pitch position are even higher and thus neutral pitch conditions become determinant in analysing spindle capability. For the neutral pitch interval the spindle torque coefficient for smaller blade areas is definately influenced by cavitation. In connection with this it is useful to point out that the spindle torque coefficient itself is not entirely decisive for the absolute spindle torque level. From the definitions of KQS and

an can be derived that for a particular propeller

QL = comst. . Therefore in Figure 18. for propeller B the

an K

relation between and advance ratio is given. This figure

an

demonstrates that, although cavitation reduces the spindle torque coefficient, the highest spindle torques occur in cavitating condition.

ADVANCE RATIO J

0

Oh.2 0 . 4 0 .6 0.8 1.0

WITHOUT CAVITATION

an = 1.5,

figure, 18. Kos/an as function' of Js for propeller FK., -36-0 _50 -0.4 -15 -0.8 -5o -1.2

When comparing the cavitation patterns at high positive and negative pitch angles it shows that for positive pitch cavitation extends from lower inner radii towards the tip while for negative pitch angles cavitation starts only at the outer radii. This results for negative pitch angles in an influence of

An on KQS also for the range of low cavitation numbers, where in this range for positive pitch the in-' fluence of

ani disappears due to full cavitation. See Figure 19, 21

and corresponding photographs. At the small pitch angle of 5 degrees for the lowest cavitation number investigated slight cavitation started in bollard condition with no effect on spindle torque., From these results (Figure 20) it can be seen that in this condition, skew has a strong effect, where skew back causes an increase in spindle torque.

A somewhat smaller but for practical purposes still considerable reduction in spindle torque can, be obtained by reduction of the blade area ratio. The balanced skew distribution of propeller C causes some increase in spindle torque but gives a considerable improvement as compared with the skew, distribution of propeller A. This can be explained by the influence of section deformation,, which is more extreme for the backward skewed, propeller, causing larger negative spindle torques,

For the same pitch angle and increasing advance' ratio' (Figures 23 and 26) pressure side cavitation becomes important and the propellers, with the smaller blade areas appear to be much more sensitive for cavitation effects. Apart from the effect of blade area on cavitation', the pitch lag and negative leading edge camber due to section

deformation, which are higher for the larger area, result in smaller angles of attack thus in less cavitation%

-37-7

r&_2-1 2' 5 6 J

-38-. 9

-A

414.

kt\

-;"-3 8

1.6 1.2 0.8 0.4

-1 2

-2.0

0 0.8 1.6

-39-2.4 3.2 CAVITATION NUMBER on

Figure 19. Hydrodynamic spindle torque as function of cavitation number for P/D = 1.214 and J 0.

4.0 4.8 5.6 6.4

44.

04

12 5

p

li

0,1,1111111 14 0 E 13 16 =

_

10

It

g, 11

14

1 12

vas-t3

C

-11 16

IS

-40-L>

-41-0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4

CAVITATION NUMBER

0n

Figure 20. Hydrodynamic spindle torque as function of cavitation number for P/D - .1924 and J = 0.

1.6 1 2 0.8 0.4 ^ A

-2.0

I 1 4 C 2

a 1 2.

-42-2 , I , , I I

1

, I I , 1

i

41 __, 8 NN.%%%"-., _ I

Il

2 6 frm

't

D f21 C ₂₀ If ----...---,B 17 I 0 m

TL

4.8

16

2.4 3.2 14.0 4.8 516 6\4 CAVITATION NUMBER 0n

Figure 21.. Hydrodynamic spindle torque as function of cavitation number for P/D = .5890 and ml =

0-6

0.8

B

AP" 2 I

I.

I CJ CO CJ

0.4 .--1 L.) rTh 1.6 1.2 0.8 -0.8 1.6 2.° -44-_ 25, 24 23 14 B 26 0 29/A 31 34 33

i

32 so. D E 37 36 .15 _ 0 0.8 1.6 2.4

32

4.0

48

5.6

64

CAVITATION NUMBER 0n

Figure 22. Hydrodynamic spindle torque as function of cavitation number for P/D = 1.214 and J .7.

0 = 2: II I I

29

-

45-30 31

23 24 25

-46-'Ike

-AN

32 33 34

CD 0 1.2

08

0.4 -2 0

-47-_ _ _ 46

.

.i...;640 43 45 ..39 .42

_____-38

i _41

-0

r,i----_______r____________p

48

---"---44

-47

_ 1 1 II 1 i i , , 0

08

_{1.6 2.4 3.2 4.0}

₄₈

_{5.6 6.4} CAVITATION NUMBER on

Figure 23.

_{Hydrodynamic spindle torque as function of cavitation}

number for P/D

=

.1924 and J

.505.

I

_I

2

41

-48-4111 "

42

#040/NN

4110,1111\11k 43

4,41

38 39 40 44 45 46

sitiOttL.1

T-50

-49-440

51 47 _{48 49}

U-4.1

>-2

1.6 1.2

08

o 0.4

0 -0.4 1 2 1.6

-2 0

0.8 1.6 2.4

-50--D

CAVITATION NUMBER an

Figure 24.

Hydrodynamic spindle torque as function of cavitation

number for P/D

-.5890 and J

.337.

3.2 4.0 4.8 5.6 6.4

--0

F 0.8

6

4 10 Ls Ls LaJ 0.4 Cr LC Li

;

CC CI

=

-2.0

LI 50

1

Dr Or.8 1.6 I 2.4

32

'CAVITATION NUMBER an

Figure 25.

Hydrodynamic spindle. torque As function of cavitation

number for ,P/D

1.214 and J = 1.35.

4.0 94 8

64

0.

-0.

-1.2

-51-5.6 = 2

'. ,..,..

I

-

'I*

ort

411

55

otikk

-52-56

f44.1

57 52 53 54 58 59 60

-1 . 1JJ 0. 1.6

-2 0

=53-I \. I 1 i i I 1 1 I , 1 1 1 1 I I _ __

111116111111ft

54 1 5 31 i

1-re

5 2,7 E E II II li 1 I

)5

1 1 62 6 , , I , , ,

08

1.6

24

3.2 4.0

48

5.6

64

tWITATIONINUMBER o

Figure 26. Hydrodynamic spindle torque as function of cavitation number for P/D = .1924 and J = 1.011.

0

58

Of,

0.8

LU

CAVITATION NUMBER

0n

Figure 27. Hydrodynamic spindle torque as function of cavitation number for P/D = .5890 and J = .674.

cc c,

-0.4

L, F.

-08

cc

-1.2

-1.6

-2.0

1.6

2.4

3.2

0.8

4.0

4.8

5.6

6.4

I I

0

0 a.6 0 . 4

-56-For the design pitch ratio the extent of cavitation for advance ratio Js 03 is less than for bollard condition, which results

id

a smaller gradient of the KQS-an curves. See figure 22 and

corres-ponding photographs. Therefore the curves do not yet start running; horizontally within the investigated range of 'low values of an. Small increases of advance ratio from zero do not show a significant change in cavitation effects for negative pitch angles, Figure 24, 21, A similar phenomenon can be observed for the 5 degree pitch setting. See Figure 23, 20. A further increase in advance ratio for the design pitch angle causes the spindle torques for the different propellers to get values which are closer together. This is due to the smaller loading where under pressure peaks become less important.,

'It can also be seen from Figure 25 that the influence of cavitation already vanishes at lower cavitation numbers. In Figure 26 is shown; that with further increasing advance ratio for pitch angles of 5 de-grees, the effect of cavitation can, be noticed, in particular for the smaller blade areas. Although, this effect only starts at,

relatively low cavitation numbers. This means that in this condition; face side cavitation becomes important. Face side cavitation is of even more importance for the higher advance ratios and negative pitch angles. Figure 27. In this condition the face side almost fully cavitates and therefore the influence of cavitation number on spindle torque is small for the smaller blade areas. From the same Figure can be noticed that in this condition the spindle torque of the propeller with the balanced skew is reduced by cavitation, while the values for the non skewed propeller are not affected.

4. COMPUTATION METHODS

4.1 General

Off design behaviour can in principle be analysed by means of the same methods as used for analysis in design condition. Only adap-tion of input for geometry in off design pitch condiadap-tion and inflow velocities is required. However, all of the propeller analysis methods contain assumptions and simplifications, which are more or

less justified in design condition but may not be acceptable in other conditions. Investigations of one of the International Towing Tank Conference committees where the results of a number of

opera-tional analysis methods have been compared for the design pitch condition - the calculation were performed for a fixed pitch propeller - show large deviations in pressure distributions of the cross sections for the various methods, Reference [20]. It can be assumed that for off design pitch conditions these deviations will be strongly amplified.

For the computation methods of off design behaviour the accuracy of the calculation of the hydrodynamic spindle torque is a good yard-stick. Since for hydrodynamic spindle torque calculations the proper chordwise load distribution must be taken into account. It can be assumed that with a good result of the first moment of the chordwise load distribution the integral of this distribution will also be correct. In other words, if the hydrodynamic spindle torque is correctly computed the thrust and torque must also be accurate. Therefore we will concentrate on the hydrodynamic spindle torque calculation. First a summary will be given of the existing methods

for spindle torque calculations.

Of course any method which gives the cavitation characteristics of a propeller can be extended with a moment integration of the known pressures. However, the discussion will be limited to those methods only which were developed primarily for spindle torque analysis.

-57-

-58-Next, a new method will be developed. This lifting body method, does not have the limiting assumptions of the !existing methods..

42 Review of existing calculation methods.

One of the first methods was given in 1959 by Van der Voorde in Reference [21]. The at that time usual methods - i.e. circulation reduction factors and camber corrections - were used to determine sectional angles of attack and lift coefficients with effective camber ratios. The spindle torque was divided in a part depending on the moment around the aerodynamic centre of each section, where the section is supposed to be expanded, and a part depending on the lift working in the aerodynamic centre. The use was limited to near design conditions. A similar approach was followed in 1961 by Boswell, Reference [22], where the effect of drift was introduced but section deformation in off design condition still only men-tioned. Calculations for off design conditions were made by Bossow,, Reference [10], and Klaassen and Arnoldus, Reference [7], with basically the same method. Parallel to these investigations a method was developed by Rusetskiy whose earlier work is summarized in Reference [6]. In this method section deformation was taken into account to determine the sectional moment coefficients, although only symmetrical sections with respect to the spindle axis are

treated by means of thin airfoil theory. For skew effects a

"compensation factor" is used. Another correction is made for using, moment coefficients of developed sections instead of cilindricaT

sections. The angle of attack and the velocity at each section was obtained by taking into account the slipstream deformation which is, depending on, the operating condition, converging or diverging. The more advanced and most recent development in this field was

done by Scott Tsao, Reference [8]. The deformation of the camber line is determined at a number of chordwise stations. The corres-ponding pressures are obtained by means of a vortex lattice method

where the vortex strength of the bounded vorticity is split up in z number of modes, allowing to use a mode collocation method. The free vorticity can be handled in a non linear way.

4.3 Surface vorticity method.

4.3.1 Consideration for the development of a new calculation method.

As follows from the previous paragraph a number of calculation methods for spindle torque determination exist. All of them do not only have drawbacks but also fail if an accurate prediction is required. In this paragraph the considerations will be given which are the basis for a new calculation method. At the same time argu-ments will be formulated why existing methods can or will not be adapted in order to improve the prediction of spindle torques.

Since the chordwise load distribution is necessary to determine moments, it is evident that a lifting line representation of the propeller blade is insufficient. Even when so called lifting line lifting surface correction factors are used. Reliable answers are only obtained for the restricted conditions under which the correc-tion factors have been determined, which are mostly: particular load distributions and average light loading. The adaption of existing lifting surface methods can be considered. The fact that

off design conditions include heavy loading leads to the conclusion that non-linear effects must be taken into account, for high

perturbation velocities cannot be neglected with respect to low advance speeds. Modification of an existing linear lifting surface method has been performed by van Gent [23]. The induced velocities due to non linear effects are implied in the boundary condition of the real geometry of the lifting surface. However, the induced velocities of the linearised model are calculated for a surface with constant pitch. Although a correction is made for this

discrepancy, it can be assumed that for controllable pitch propellers after pitch adjustments the radial pitch distribution is far from constant and this correction will therefore be insufficient. Modification of this method with respect to extreme radial pitch distributions leads to a complete review of the mathematical treat-ment of the integral equation as reported in [24].

-59-

-60-Although the radial pitch distribution is taken into account for the contribution of the inflow velocity to the boundary condition

in the method developed by Tsakonas et al. [25], in the effect of the induced velocity the radial pitch distribution lacks too. This has of course more effect than on the radial boundary con-dition only. That both methods are based on the same kind of boun-dary conditions can be attributed to the fact that the original developing was done at the time that numerical possibilities of nowadays computers were not so far developed and consequently the analytical treatment of singularities in the integral equation led to restrictions. To overcome this a method based on such a numerical approach was started by Kerwin which resulted in a lifting surface design method [26, 27].

An inverse solution, computing pressure distribution for given geometry, based on the discrete vortex distribution over a lifting surface was given by van Gunsteren and Pronk [4], where Kerwins design program was used in an iterative procedure. Since 6 iteration steps were used, this program is not practical for daily use and

therefore extension of this program for non linear effects is not logical. Further development of this vortex lattice method for analysis purposes has been undertaken by Tsao as mentioned above. An extension to instationary lifting surface calculation based on

vortex lattices is feasible as proved by Frydenland and Kerwin [28]. So it can be assumed that the numerical approach is offering advan-tages over a basically analytical approach.

In the discrete vortex representation the boundary conditions for non linear effects can easily be incorporated without the necessity of solving additional singularity problems. Of course, it should be wondered if the accuracy of results is sufficient when discrete vortices are used instead of continuous vorticity distributions in either tabulated form, as in use for two dimensional problems, or in polynomial form, which is in use for three dimensional models, e.g. representing the distribution by a number of modes.

SURFACE VORTICITY

---

6 TERM POLYNOMIAL

-61-The accuracy of the result depends on the proper distributing and can be compared to the accuracy when presenting the load dis-tribution in a series representation as is usual for the analytical solutions. For controllable pitch propellers in off design condition two factors are of importance in this respect. Due to the distortion and the high section loading a series representations of the pressure distribution can give a considerable discrepancy with the actual distribution.

In Figure 28 a pressure distribution represented by means of a poly-nomial distribution is compared with a distribution obtained by continuous surface vorticity method. Considerable differences in pressure difference occur.

NON DIMENSIONAL CHORDWISE POSITION

Figure 28. Pressure distribution computed by means of polynominal and continuous vortex distribution.

-62-From this it can be concluded that replacing the loading by a number of modes offers no advantages with respect to accuracy, since the number of modes or terms in a series development for these types of loading is comparable to the number of discrete vortices which is required to obtain sufficient accuracy as

will

be shown later.

In paragraph 2.4 it has been shown that thickness effect on section deformation plays a practical role at the edges. However, superposi-tion of distorted camber line and thickness is acceptable in most cases. This means that thickness effects can be treated for each pitch setting quantitatively in the same way. Only changing the influence coefficients of the singularities by rotation of the coor-dinate system is necessary.

In Reference [29] it has been shown that thickness effects can be sufficiently accurately represented by discrete vortices. In combination with the discrete vortices which represent the loading a total representation of the blade by discrete vortices on the blade surface is possible. This means that for each pitch setting a set of discrete vortices must be determined and that there is no need for separating the thickness representation. Only those cases where improved accuracy is desired and a surface source distribution is introduced to represent a part of the thickness effect this source distribution can be determined for the design pitch condition and its strength can be applied for other pitch

settings as well.

From the above it can be analysed that a large number of discrete vortices at the surface is necessary to determine the proper pressure distribution. Consequently, this leads to a set of

algebraic equations, which replace the original integral equation, whereby matrix dimensions ask for very large computer storage capabilities. Routines for solving such large sets of equations are available - using external storage capacity - when the core capacity itself is insufficient.

4.3.2 Description of the model.

The vortex model consists of straight vortex elements which connect nodal points. These points lie on radial cross sections at intervals which have the same dimensionless distribution along the chord for each radius. The number of radii is based on the usual number of radii in propeller design and analysis calculations. Coordinates of nodal points in both radial and chordwise direction can be distributed linear or according the wellknown Glauert dis-tribution. The number of chordwise intervals has been determined by analysis of two dimensional airfoils which were represented by surface vorticity. From this analysis of which some results are shown in Figure 29 it followed that 16 discrete vortices were sufficient for obtaining a proper chordwise pressure distribution. The control points where the Neuman boundary condition has to be

fulfilled are taken midway between radial and chordwise adjacent elements and are determined in the developed expanded plane. The position of the control points followed from analysis as reported

in [29 and 30].

Although the present physical model is different from the two dimensional one used in the analysis, it is assumed that the effect of using discrete vortices on the numerical accuracy is comparable. These differences between this three dimensional and the two di-mensional model which is used in combination with lifting line methods are:

the vortex elements are of finite length for the three dimen-sional approach.

vortices are not perpendicular to the cilindrical surface in which the two dimensional section is situated, while for the two dimensional approach the infinite vortices are perpendi-cular to the plane in which the section lies.

for the two dimensional approach the distance between vortex and control point is measured along the arc over the cilindri-cal surface, while distances between control point and points on the vortex elements are real distances.

-63-

-64-r-.3

(IA OLLIM A110013A

Figure 29. Velocity distribution for various numbers of control points with continuous vortex distribution.

r FREE'

'IT BOUND

- FREE

-65-In a two dimensional problem it has been proved that surface vorticity, either continuous or discrete, is sufficient to re-present the section. In three dimensional flow the vorticity

representing the thickness has to be closed in itself, since free vorticity is only possible when external forces are exerted. This closure has to be obtained within the surface when complications like imaging, associated with vortices in the internal region of the blade, are to be avoided. The system of bound vortex filaments in combination with free vorticity and closed vorticity surrounding panels is identical to a system of bound and free vortices only. See Figure 30. However, coefficient matrices for both systems are different. The latter system has been used for reasons of simpli-city, while the required storage capacity is reduced. The resulting grid of bound and free vortex elements on the blade is shown in Figure 31.

r FREE

7 PANEL

- r

PANEL

Figure 30. Identical vortex systems.

F BOUND

+ r

PANEL

7BOUND + F PANEL