Three dimensional effects in propeller theory

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IN PROPELLER THEORY

by

OVE W. HÖIBY

NORWEGIAN SHIP MODEL EXPERIMENT TANK PUBLICATION NO.105

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SUMMARY

By adapting wing theory to the propeller blade, a fast and simple method is developed for calculation of the propeller pitch angles. The method provides pitch angles with generally less than 3% deviation from corresponding

calculations by complete lift-ing surface theory.

The lift slope of the propeller blade is determined by the same approach. This determination of the lift slope gives rise to a very simple and reliable method of calculating the off-design performance of a given propeller in open water

conditions.

By further adaption of the theories for wings by Weber and Küchemann, a method of -calculating the three-dimensional pressure distribution -on a propeller blade is

evaluated. To check this method, thé pressure distributiqn on

- a piopeller blade of general design was measured by fitting

23 miniature pressure transducers to the blade., arid good

agreement was found between measured and calculated pressure

distributions. -

-A method to determine the section coordinates of the propeller -blade to give a-specifid pressure distribution is

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-PROBLEMS OF PROPELLER DESIGN AND ANALYSIS 1

OUTLINE OF THEORETICAL METHOD 2

II THEORETICAL ANALYSIS I INTRODUCTION

A. GENERAL ASSUMPTIONS 5

B. PROPELLER DESIGN BY LIFTING LINE METHOD 7

Modification for optimum propellers of Lerbs' induction factor method

Preliminary determination of mean line and pitch 13 angle

C. VELOCITY DISTRIBUTION ON A WING ANALOGOUS TO THE PROPELLER BLADE

Velocity distribution due to the symmetrical thickness distribution

Velocity distribution due to the mean line at incidence

D. CORRECTIONS OF MEAN LINE AND PITCH ANGLE 25 Additional velocities induced by vortices 25 Additional velocities induced by blade

33 thickness

Pitch angle and mean line corrections 38

E. PERFORMANCE ANALYSIS OF A GIVEN PROPELLER 245

General remarks on the performance in off-design conditions

Suggested method for calculating the performance in off-design conditions

114

15

18

'45

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F. PRESSURE DISTRIBUTIONS ON THREE-DIMENSIOÑAL PROPELLER BLADES

The pressure distribution on a known 'mean line at incidence

The pressure distribution on à given thickness distribution

The thickness distribution to give a specified pressure distributioñ

III EXPERIMENTAL WORK

SYMBOLS

'REFERENCES

NUMERICAL METHODS FOR CALCULATION OF VELOCITY DISTRIBUTION

THE VELOCITIES INDUCED ON ONE BLADE BY THE REMAINING BLADES, FROM THE THEORY OF TWO-DIMENSIONAL CASCADES OF BLADES

GENERAL TEST TECHNIQUE 61

PRESSURE MEASUREMENTS ON STATIONARY WING 62

Model and test apparatus 62

Test procedure and results 64

C,. PRESSURE MEASUREMENTS ON PRÖPELLER BLADE 66

Model and test apparatus 66

Test procedure 70

Test results 72

IV DISCUSSION AND CONCLUSIONS

DISCUSSION 81 CONCLUSIONS 84 V APPENDICES 5M 54 56 57 86 91 94 99

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Fig. 1 Prope11e velocity diagram

2 _{Spanwise variation of interpolation}

fúnct ions K and

't 3 Cylindrica4. .cordiiate

44 Ideal ng1e of incidence corrected for loading " page 8 29 42 43 244 51 51 52 52 53 53 60 63 64 65 67 68 69 71

5 Pitch angle corrections du to blade

thicknes

it

"

6 Mean line correction factors due to

loading

ti

7 Open ter performaice, propeller D, PID=O.96O (design)

t,

8 Open water performanc, propeller E,

P7D=i.000 (design)

ti

9 Open water perforiiance, propeller E,

PIDO.800

't

10 Open water performance, propeller E,

P/Dl.l00

't

11 Open water performance, propeller F,

P.ID0.84444 (design)

i,

't ₁₂ _{Open water performance, prppeller G,}

P/D0.782 (design)

13 Calculated thickness distributions 't

't ₁₄₄ _{Calculated thickness distributions} _'t

for stationary wing

t

₁₅ _{Stationary wing mounted in cavitation} _'t

tunnel

t

₁₆ _{Prescribed and measured pressure}

distri-butions on stationary wing 't

it 17

1

Propeller for pressure measurements

Arrangement of pressure transducers

't

19 Propeller blade with lead, grooves and transducers

't

t,. ₂₀ _{Instrumentation diagram for pressure} i, measurements

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Chordwise pressure distrbutïons

for

J0.1O68

Chordwise pressure distributions

for

J02510

Chordwise pressure distributions

for

J0.375O

. Chordwise pressure distributions

for

J=0.'370

Chordwise pressure distributions for

J=0.5930

Spanwise pressure distributions for

_J=O.1068,

Spanwise pressure distributions for

J=O.251O

Spanwise pressure distributions fOr

J=D.3750

Spanwise pressure distributions

for

J0.'37O

Spanwise pressure distributions fOr

J=O.5930

Cascade geometry page 7t Fig.

21

22 " 23 " 2t1 " i' 25 26

"

2.7

"

28

"

29 -30 'I

₃₁

7 5 V t,

".

76 77 78 TV -t, t, t, t,

"

79 79

79.

80 80

103

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A. PROBLEMS OF PROPELLER DESIGN AND ANALYSIS. A common problem in propeller theory is to design a propeller to give a specified performance under given

condi-tions. And conversely to determine the performance of a given propeller operating under specified conditions.

For the first problem, the radial loading distribu-tion is assumed to be given. It is also assumed that the dia-meter, the number of blades, the blade outline and the boss ratio have been decided. A theoretical model is then needed to pro-duce the profile coordinates and the pitch angle giving the de-sired performance.

In the inverse problem, the geometry of the propeller and the operating conditions are assumed to be known. One must then determine the pressure distribution over the blade, the thrust produced and torque required..

The vortex theory of propellers gives two principal methods of obtaining the desired theoretical model, the lifting surface theory and the conventional lifting line theory with correction factors. The theoretical model based on the lifting surface conception gives a more correct approach than the one based on lifting line. However, the complete lifting surface theory is of such a complex nature that even when using fast digital computers, the calculation cost is still too high for practical use. Thus, until faster computers are evaluated, the complete lifting surface theory seems to be best suited for research purposes.

The lifting line theory with correction factors based on lifting surface théory have been in use or some time. Some of the most common correction factors are due to Ludwieg and. Ginzel (l)C,.Lerbs (2'), Cox (3), Pien (14),, Kerwin and Leopold (5),

x)

Numbers in square brackets refer to references at end of paper.

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-2-Cheng (6,7), Nishiya.ina and Sasajima (8) and Morgan, Silovic and Denny (9). The first correction factors were correcting the camber because of _{flow", and in recent years the} correction of pitch angle because of blade thickness have also been considered. _{These correction factors have been established} for definite propellers operating at specified _conditions. _To find the appropriate values of the correction factors, it is necessary to interpolate between the given values. Such a procedure is inconvenient as the data provided for interpolation is in most cases insufficient. Thus it appears favourable to determine the correction factors in such a simple way that they could be calculated for each actual case. This paper describes an attempt to determine the correction factors in a sufficiently Simple manner.

To better illustrate the use of the correction factors, a lifting, line theory is needeto which these can be applied. The indubtion factor method of Lerbs

(lo)

is found to be the most advanced lifting line theory at present. _{As this theory} also is well suited for ap1ication of the 'correction factors, it was chosen to form the basis for the corrections.

B. OUTLINE OF THEORETICAL THOD.

When tle correc-ion factors are to be determined in a relatively simple manner, it might be advantageous to consider the related and well established fields of wing and _cascade

'aerodynamics. 'In these fields the theories of the lifting surface first occured, and here the theories might have reached a more advanced phase, resulting in more direct o±' "short cut" methods. _{Such methods are for instance the theories given by} Küchemann and Weber (ii), (12), (13), (lL), (15) and (16) for calculation of the velocity distribution on swept wings of finite aspect ratio.

-±n the inner regions

_of

_{the propeller blades, and} particularly for prope11es with many blades, there is a notice-able interaction between the blades. This interaction effect can to,some extent' be determined from two-dimensional cascade theory. _{A cascade can be calculated in two ways, either by the}

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use of conformal transformations, or. by the method of singularities.

As the latter method is best suited or numerical calculation, and as the. basic method of the lifting line, is also based on singulari-ties, the singularity method of Scholz (17), Schlichting (18) and Pollard and Wordsworth (19) were chosen. These methods give

the velocities induced on one blade from the remaining blades due

to blade thickness in two-dimensional flow.

The blade loading and blade thickness induce velocities normal to the chordline. These induced velocities have the effect

of changing the apparent camber and pitch angle. To obtain the prescribed loading, the camber and pitch angle must therefore be corrected. This correction is approximated from the indúced

velocities calculated by cascade theory, and from a more direct approach for the corrections due to loading.

In recent years, heavily loaded propellers have been

much in use, creating an increasing interest in the cavitation

problem. Closely related to the problem of cavitation, is the

problem of determining the velocity or pressure distribution on

the propeller blade. If a complete lifting surface theory is

not applied, the usual way to calculate the pressure distribution

is by the plain two-dimensional distribution for the actual mean line and thickness form. For three-dimensional swept wings, however, there exist methods which take into account three-dimensional flow. In

this

paper an attempt is madeto modify

one of these methods for application to the propeller blade. The original theory is described by Küchemann and Weber in refs.

(12) , (13) and (114). They have shown that the effects of the

three-dimensional approach are quite noticeable in the tip and

centre regions for swept wings. The trend is to give suction peaks at the leading edge in the tip region. The method of

Küchemann and Weber seems to make it possible to determine the pressure distribution over the propeller blade, and to check the modification of this theory, an experiment was performed. A

model propeller was fitted with 23 miniature pressure transducers in one blade to measure the actual pressure distribution over the blade.

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In ref. (ii), Küchemnn gives a method to determine

the sectional lift slope of a wing of arbitrary planform. . The

sectional lift slope occurs whn only the bound vortices are

taken into account. _{Knowing this sectionaljift slope, the lift} coefficient and the induced velocity, angle cán be determined in off-design conditions.. _{And as will be shown in part II, section}

E, the inverse problem in propeller theory then can be handled

in a simple way.

So far, it has been usual to select the sections because of their favourable pressure distributions, determined by two-dimensional theory. As mentioned in the preceding paragraph, the

two-dimensional pressure distributions can have remarkable deviations from the actual distributions, especially in the tip region where the cavitation danger is greatest. Thus it seems advisable to calculate the profile coordinates for each station on the actual propeller blade. The coordinates of the mean line are given by the lifting line calculations with the correction factors included. When a desirable pressure distribution is prescribed, the difference between the desired distribution and the pressure distribution from the mean line at the actual angle of incidence, must be given by -the thickness distribution. With a modification of one of Weber's theories (13), it is possible to determine a thickness distribution giving a prescribed pressure distribution. And it is then possible to prescribe à favourable pressure distribution and calculate the profile coordinates to give this pressure distribution, even in the tip region.

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A. GENERAL ASSUMPTIONS.

In the mathematical model used for the theoretical analysis, the bound vortices, generating the pressure dis-continuity across the blade, are either assumed to be distri-buted over the blade or to be concentrated in a lifting line.

The continuity of vorticity requires free vortices to be shed from the b6und vortices downstream along helical paths. Blade thickness is représented by a distribution of sources and

sinks over theblade. Based on -this mathematical model, the following assumptions are made:

The fluid is inviscid, incompressible, infinite in extent and free of cavitation. The free-stream velocity is, axially directed, and the magñitude is a function of radius

only.

The effects of slipstream contract-ion and centri-fugal force on the shape of the. free vortices are ignored.

Then each of the free vortices has a constant diamete±' and a constant pitch in the, downstream direction. In the radial

direction, however, the pitch may vary.

- The propeller has Z' symmetrically spaced identical

blades. The boss is assumed to be small enough to be

neglected. This is not particularly realistic, but it is not

- considered worthwile trying to simulate the boss by sources

and sinks. The propeller blades begin at a radius rb corre-sponding to-the boss radius of the actual propeller. The lift is therefore required to be zero at the inner and outer extremities of the blade which are regarded as free ends.

Blade rake is not considered.

-The disturbance to the flow caused by the propeller

- is assumed to be small.

This requires that the blades are thin, and that the perturbations from camber and incidence of

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-6-In conventional lifting line propeller theories, the propeller blade is-treat-ed as a straight wing of large aspect ratio, the difference being in the calculation of the velocities

induced by the helicoidal trailing vortices of the propeller blade. The effect of the helicoidal form of the trailing vortices is taken into account by a factor, for instance the

induction factor, applied to the general downwash equation for a wing of large aspect ratio. In the induction factors calculated by Lerbs, the mutual interaction between the trailing vortices of neighbouring blades is also included. The blades are considered to be infïnitely thin, as the blade thickness is ignored.

However, the propeller blades in general use today, characterized by skewed, broad blades of finite thickness, are not matching this conception particularly well. It is therefore suggested to replace the propeller blade by the following analogous wing.

To take account of the skewback of the blade, the analogous wing is considered to have sweep back. As the aspect ratio of most propeller blades are in the region 0.5 to 5.0, the analogous wing is considered to have this medium aspect ratio instead of- the large aspect ratio generally used. T}ie

span of the wing is equal to the length of the blade from boss to blade tip, and the aspect ratio of the wing is given as

b2_ Z.(l-rb)2

S

e

b and S are span and area respectively of a general wing, Z is the number of blades, rb the non-dimensional boss radius and Ae/Ao the expanded blade area ratio. The effect of the form of the trailing vortices .is still accounted for by the induction factqrs of Lerbs. The thickness of the propeller blade is considered by giving the analogous wing the corresponding thickness. The wing is moving through the fluid describiñg a helicoidal path. The sweep angle is measured between the midchord line of, the expanded blade, and the radius through the intersection of mid-chord line with boss.

if not anyhing else is specified, the x-axi.s is directed along the chord line of the blade section, with origin

(13)

at the leading edge, and the z-axis normal to the chord line, positive on the suction side of the blade. The chord line is defined as the straight line between the ends of the mean line. The x- and z-coordina-tes are made non-dimensional with the

actual blade chord c. Radial distances are given by the non-dimensional r. The velcity components in x- and z-directions will be denoted u and w respectively for induced velocities, and capital letters for free stream velocity components.

B. PROPELLER DESIGN BY LIFTING LINE METHOD. 1. Modification for optimum propellers of Lerbs' induction factor method.

To illustrate the use of the correction factors, these will be applied to the well-known induction factor method presented by Lerbs in ref. (lo). The correction factors will here be applied to the design of optimum propellers only, but they can also be applied to non-optimum and to wake adapted

propelle' desins in a similar way.

The induced axial and tangential velocities for a propeller blade of large aspect ratio is given by Lerbs as

(2) (3) 'a -a rb dy(r') a dr' r-r' 1 w t , f dy(r') -j dr' r a rb

Here _a and i are the axial and tangential induction factors giving the relation between the velocities induced by a helicoidal and a straight trailing vortex. tn the two expressions (2) and

(3), y is the non-dimensional circulation given by y

- ______ The meaning of the other symbols will be seen from fig. 1. a For an optimum propeller, the resultant w of Wa and w is normal to the resultant velocity V (fig. 1). Then 1a and i are replaced by their resultant i for the direction normal to V.

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-8-Fig. 1. Propeller velocity diagram.

As the induced angle is small, equations (2) and (3) cari be replaced by i r dy(r') _..!_..Tdr' VT

T

_J dr' r-r rb

From fig. i it is seen that u. = then

10 1

i

sine

f dy(r').

-T

J dr' r-r rb

Equation (5) is then the first relation between the angle ß and

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To treat the singular integral, a new variable is introduced. c is zero for r = rb and ir for r 1. 4 is given by

r = _{(l+rb)_(l_rb)cos}

The non-dimensional circulation y is represented by a Fourier series. The circulation is assumed to be zero at the boss and the blade tip, then y is replaced by

y

'sin

The resultant induction factor i is a function of c and ',

apart from the number of blades Z and the pitch of the trailing vortices

..

As proposed by Schubert (20) , the inductidn factor is resolved, with respect to , into an even Fourier series:

i(cp,4') =

I()cosm'

m0

Inserting from equations (6), (7) and (8) in equation (5):

il

fly cosfl

-

"b

_I

co'OS

sine ç Cl-rb)

_n1

- 2r)

_nL

_JO

'm{

fcos(n-m)' cos -cos 4 ir

cosn' cosm4,

ny

I ()

n

_m0

m j cos'-cO5c1' o

(16)

lo

-The singular integral of equation (5) is thus replaced by two singular integrals of which the principal values are known from Glauert (21): TT

(lo)

J

cosn,

d' -o

This relation, however, is only valid for n > O, thus one must distinguish between the two cases n > in and n < m. Then the sum of the two integrals inside the square brackets will be:

2rr

n > m: sinn COsm

n<m

271

sinm cosn

As proposed by Lerbs, equation (9) can then be written

7rsin

(li)

sin1_rb)

_nl

n[s

_mU

Im)co8m

+ cosn

I ()sinm)

in=n+1 n

For an optimum propeller, the following relation exists

(12) tan constant

r

As the inductjon factor is a function of

., a value of must

first be assumed, and using equation (12), the distribution

of over the radius is given. By considering only the first p terms in the sums of equation (li), and applying this equation at p different stations over the radius, a set of p equations is obtained for determining the Fourier coefficients of y for the assumed valua of ..

i

Siflfl

(17)

required to determine both y and Applying the well-known relation

L pVr

in.axial and tangential directions, the moment and thrust for a propeller working i-n a non-viscous fluid will be

i KQ _nzDS r3ZÀ2

J

y(r)r(l+)dr

rb i T ir3ZX2 f r w T pn D 2 j y(r)( -a

For an optimum propeller, it is seen from fig. 1 that and can be expressed as 2 w r2 V ax' a = V

_2+X2(l+)2.

a a a w (16)

cos3

sin = Substituting and (14): (17) KQ = from equations 1T3zx2 1

(15) and (16) into equations (13)

-4 f (r)r[l+Ç-r2 dr -

a r2+A2(1+)2J

b a i (18) _KT

2z2

-

(l+-)A

J

Y(r)[

r2+A2(1+f)2Ìx' rb - a

(18)

When a prope1le _{is to be designed, either the mcment} coefficient KQ

viscosity of a (13) will give

If the lift coefficient is determined, alternatives to equations (17) and (18) can be written down by _considering or the thrust coefficient KT is given. As the real fluid affects KQ more than

T' equation

a better estimate fòr the real fluid.

Here CD is the drag coefficient given by the empirical expression

CD 0.0085 + 3(-ß.)2

and the resultant velocity V calculated from

Cosa.

V-V

_- _{a sine}

The additional information requiréd to determine both y and is then obtained either from equations (17) and (18) or from equations (19) and (20).

The following calculation procedure is then used. A value of the constant in equation (12) is assumed. With known from (12), the induction factors are calculated and the Fourier coefficients Lm( determined. By applying equation

(11) at p radial statins, a system of p equations is _obtained

to determine the p Fourier coefficients of y. _{The value of y} determined in this way is then inserted in one of the equations (17), (18), (19) or (20). The ??dummyt? velocity w is related to the assumed constant of .auation (12) by the expression en

from fig. 1, - 12 fig.

1:

1 (19) KQ

_{.V2cr(CL sin.0 cos.)dr}

i D i 8n2D _J rb 1 (20) z _{V2c(CL cos8l_CD sin.)dr} - 4ri2D3 rb

(19)

.v +w4

constant - 27rnR

KQ or is then calculated from one of the equations (17) -(20) and this value compared to the stipulated value. If the

new value is too far from the desired, anew constant is thên

found from the expression.

KT-K

New constant = previous constant x (1+ D T)

SKTD

and the whole procedure done over again. This itération is carried on until the calculated value of is within a certain

limit from the stipulated KT Relation (23) is purely empirical,

it was first proposedby Eckhardt and Morgan (22), and has proved

to give good convergence of the iteration. KTD in (23) is the stipulated thrust coefficient.

2. Preliminary determinatior of mean line and pitch angle.

As the circulation y is now detérmined, the product

of lift coeffïcjent and chord is obtained from the definitions

L pvr _PV2CLC

so that

ci

R

(2)

CLc = 2irD y

lo

Generally, the blade form is stipulated before the calculation starts, with eventual modifications to improve the safety margin against cavitation. This means that the chord c in equation (2) is determined, and the lift coefficient canthen be calculated.

The preliminary coordinates of the.mean line, which in two-dimensional flow would give this lift coefficient, can now be calculated if a standard mean line is used, e.g. the MACA a-series mean lines. The ordinates of these mean lines, z, are

(20)

arid (25) (l_x)2_ _{(a-x.) 2)-xlnx+g-hx}} where g - ia(a2(1n1 a

1))

h

!(l_a)2ln(l-a)

- (l_a)2)+g

Here a gives the fraction of the chord over which the velocity distribution is constant, and CL is, the lift coefficient at ideal angle of incidence. As this relation is for two-dimensional aerofoils, these mean line ordinates must be corrected by the

camber correction k evaluated in,section D 3 to obtain the desired c

lift coefficient.

In the conventional lifcing line approach, the pitch angle is obtained by adding the two-dimensional ideal angle of incidence of. the sèction to ß. The ideal angle of incidence is however influenced by blade loading and thickness, and will be evaluated for the three-dimensional propeller blade in section D 3.

C. VELOCITY DISTRIBUTION ON A WING ANALOGOUS TO THE PROPELLER BLADE.

In this problem, the wing geometry is considered to be known. And as usual in linearïzed theories, it is assumed that th wing sections can be separated in a mean line at incidence and a symmetrical thickness distribution. The velocity distribution on t wing has then two contributions, from the symmetrical thickness

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(26)

distribution, and from the mean line at incidence.

1. Velocity distribution due to the srmmetrica1 thickness distribution.

The thickness distribution z(x) is represented

by a distribution of sources and sinks of strength q(x). This strength is determined from the condition that the,

surface is a streamline, and in linearized theory, this can be written as

dz(x)

q(x).

t

2U dx

At the chord line of a two-dimensional, straight wing, this source and sink distribution induces a velocity uqs along the chord

-- f..., \

U f

LL&tA '

dx'

Uq5(XO)

J dx' x-x'

For a sheared, swept wing of infinite aspect ratio and sweep angle !, this relation is modified to

'r dz (x')

Ui

t dx

u (x,O) = cosA ,

.,

qs Ir j dx x-x

At the centre section of a swept wing, the thickness distri-bution is replaced by a distridistri-bution of kinked source lines, as shown by iKüchemann and Weber in ref. (lt). The strength of the source distribution is again approximated by

dz(x)

q(x) 2U dx

as the normal velocity induced by a continuous distribution of kinked source lines is still equal to q(x)I2. At the

centre line, Küchemann and Weber have shown that an additional term must be included to account for the centre line effect. This term is given as

(22)

16

-where f(A) again is given a

1 .l+sinA f(A) = 1nsl_sjflA)

The total velocity induced along the chord, at the centre

line, by the thickness is then

u (x,0)

1 dzt(x') d dz (x)

g

- cosA

J

xx'

cosAf(A)

In ref. (114), Küchemann and Weber have shown how to interpolate between the two limiting cases of' expressions

(28) and (31), forswept wings of constant chord and square

cut tips. It is also pointed out that the tip of such a swept back wing may be treated similar to the centre of a swept forward wing. The spanwise extents of the centre and tip regions are given by mainly empirical interpolation functions

T and K. The centre section is characterized

by the kink in the source lines and the spanwise bound

vortices.. A general propeller blade with skew back is, however, assumed to have smoothly curved source lines and spanwise bound vortices without any kink. An effect similar to the centre line effect is therefore not expected to be found on a propeller blade. The wing analogous to the propeller blade will not have a constant chord, and the wing tip corresponding to the blade tip will not be square cut. In fact, this wing tip will generally have a shape more similar to the tip shape evaluated to eliminate the tip effect. ,From fig. 20 of ref. (114) it is seen that the tip effect is greatly reduced when the tip, leading edge is curved, and fig. 26 of the sane reference gives a tip shape to eliminate the tip effect. From these two figures, it is assumed that the tip effects at the blade tip will be negligible.

At the blade root, however, the shape is similar

to the square cut tips considered in ref. (114). At this end,

however, the boss is situated. The boss may be considered

to act like a kind of small reflection plate, thus tending

to increase the value of

'T and K. If the boss was replaced

by an infinite reflection plate, the value of 1T and K would be 1.0. However, the boss.cannot by any means replace the

(23)

infIni-te reflection -plate, thus -the value of UT and K

would-be well would-below 1.0. Alsò, the pressure fields of the

neigh-bouring blades will counteract the reflection effect of the

boss, so that the value of and K is assumed to be in the region 0.7 to 0.75. For the sake of simplicity, the effect

of the boss is neglected,, i.e. UT:KO.7

at the blade

root. in the middle part of the blade, where the tip effects are not observable, liT has the value zero and K equals 1.-O. This -part of the blade is thus treated as the sheared part of the swept wing. The values of UT and K for intermediate

spanwise stat-ions are reproduced from ref. (i'+) in fig. 2.

The general expression for the induced velocity at any spanwise station can then be written

u (x,O)

f(-A) = -f(Î).

By introducing the parameters Sl(x) and S2(x), defined as

(33) SÎ(x) = } dz(x') dx' j dx' x-x'

o

dz(x)

(3i4)

S2x)

_dx

expression (32) can be written as

u (x,O) g

--- KCOSAS1(X) + UTcOfMS2(x)

By using the numerical method of Weber (13), the parameters

- Sl(x) and S2(x) can be given as the finite sums

(36) Sl(x)

-1

dzt(x') dx'

dztx)

(32) as

-KcosA

_-

+

--U j dx' x-x' o liTcos4f(A) dx

(24)

in

0.8 0.6 0.2 0.0 T (38)

w(xO)

-

J y(x'),

0.1 0.2 0.3 0./+ 0.5 0.6

0.7 nc

0.8

Fig. 2. Spanwise variation

of

_{interpolation functions} _K

and UT.

(37) S2(x)

N(2)

Here and _{are given in appendix A.} _u _{¿md y are} any integer 1 NT and N is an arbitrary even number.

x and x are the x-values corresponding to _u and according to x

2. Velocity distribution due to the mean line at incidence.

If a two-dimensional, straight wing of negligible thickness is represented by a vortex sheet of local strength y(x), the normal induced velocity at the point

_{xon the chord}

line is given as

(25)

For the centre line of a swept back wing of sweep angle A, Küchemann (ii) has shown that an additional term must be

included to account for the effect of the kinked spanwise bound vortices. Then the expression fär the centre line can be written

(39)

w5(xO)

-+i:

J y(x'),

-tanAy(x)

The reasoning on the centre and tip effects in the preceding section applies here as well, so expression (39) will only be applied in the tip behaving region of the blade root. As the sweep angle shoul be reversed when applying centre

line relations in the tip region, the actual expression for the boss end of the wing can be written

1 1 r _(x,)dx

(0)

w (x,O)

Q.7-j

y +

tanACx)}

ys

Using the interpolation functions _T and K of the preceding

section, the general expression for any spanwise station can be written

K r

(L1) w15(x,O)

---- j

y(x'), +

tanAy(x)

From the boundary condition,

dz Cx) w (x,O) s ys dx - U so that dz (x) (L12) s K

y(x').

dx' + dx U ' o

This is an integral equation for the vortex strength y(x)

of a mean line given by z5Cx) on the swept analogous wing. Carleman (2k) has given the solution of such singular integral equations, so the solution of equation CL2) can be written as

(26)

with 20 -y(x) TtanA dz5(x) K (1_x)p . Ti

_-

_KZ+UT2tan2A _dx _KZ+UTztanZA _x 1

dz5(')

)' )P dx +

C (lX)P

dxT

Tr

1-x X where i tan A-ic P i- arccos p2tan2A+ic'

and C is an arbitrary constant. _{C is determined from the}

condition that y(1)=0, and Weber (13) has shown that it is necessary and sufficient to take C=0 to obtain y(1)0.

For the centre line,

T1' icl and as (3) is

evaluated for tip regions, the sweep angle A must be reversed

for the centre line, i.è.

-tanA dz5(x) - i

2U cent-re l+tan2fl dx + 1+tan2fl x

dz(x')

X'

)P dx' +

Jdx'

x-x' 1-x x o dz (x) = -sinhcosA S - +

cos2A(.)P

dx x X . j dz5(x')

X'

_)P ₊ centre 1 - arccos tan2A-1 tan2 A + 1 =

i - .r-2(-A))

= i --

2-irf2

(27)

For the sheared wing, =0, Kl, i.e.'

(2)sheared

(i)P1J dZI)(I;.T)p

+

with

sheared = arccos(-l) =

It is seen that for these two liming cases, which it is possible to check, the expressions are identical to the corresponding expressions given by equations (3.5) and (2.7) of ref. (13).

These expressions for the chordwise vortex distri---butioñ are.originally derived for wings of infinite aspect

ratio. Küchemann and Weber

(114),

however, found experimentally, that for aspect ratios of 1 and 2 for A = 530, the centre

and tip effects were not changed from the values, obtained for infinite aspect ratio. It is, therefore suggested to use expression (43) for propellers which do not have exceptionally. broad blades.

Equation (4-3) cannot be solved explicitly for any given mean line, so a numerical method developed by Weber (13) will be used. An outline

of

this method is given in appendix A. As shown in 'this appendix, the loading can be written as a finite sum containing fixed coefficients and the slopes of the mean line at definite stations, so that

-

y(x)

N (4)fdZ5(X))

2U -

1iv

dx

The coefficient

s7

is calculated in appendix A to be

-(14) K x (1-x ) p (1)

(147)

_sv

-

K2+T2tan2A{v(1_XM)} (x-x)s

(28)

for p N, and (P8) where 22 -(p) K _f

i

5Nv K2+1iTztàn2AtanhTp s PTtanA K l-x N-1 X

K2+pTZtafl2A(

x V)P (__...)P (x -x )s (1) V

TJl

XlJ 11 V UV VV

-

K2+P2tan2A

for p = . The coefficient is given in appendix A by equation (Al).

Generally the ordinates and not the slopes of the mean line are given, therefore Weber has shown how the slope car. be approximated from the ordinates.

dz N-1 s (5)

(z5)

(5) 2(-l) s -ATT

cos(L)

- cos(--)

Xii

(Ot(-)

5AA - AIT s in (-fl--) (5) L(-fl 5pN

i+cos(4)

Then from expressions (P6) and (P8)

(P9)

y(x)

= for X p, N for X ii for X = N.

pl

XV sinlip

(29)

with N (6)

(L)

(5) s s Av p A=l

If the mean line nakes an angle of incidence c with the resultant inflow velocity V, the velocity

distri-bution at the tip is given as

VL(x) _2U

V

dz(x)

2

[i+(

S ) 1

dx

and at the sheared part as

y(x) . l-x.

VL(x)

cos(coSA±_)±KSlflc(_)P(l+S3))

cosA

V - dz5(x)

[i+(

dx /cosfl)2}5

The general velocity distribution for any spanwise station can then approximately he written as

x)

(so) VL(x) cosa(cos(l_PT)A± U_)±Ks1n

cos(l_K)A(j)+S3)

cOs(l-pT)A

V dz (x)

[i+(

dx /cos(l_UT)A)2]5

Here VL(x) is the local velocity on the mean line, is given by expression (P3) or the numerical solution of expression (P6).

dz (x)

[l+(

)2J is the so-called Riegels factor, included to prevent the velocity going to infinity at the leading edge. The upper and lower signs correspond to suction and pressure surfaces respectively. The term

(30)

- 2'

-gives the additional ve1ocity due to incIdence, _{where S3(x) is}

a parameter included toaccount for the finite thickness of the

wing. _{S3(x) is given by Weber (12) as}

i

dz(x')

2zt(x') )dx'

S3(x)

{

d'

l-(l-2x')2J-xt

S3(x) is also approximated by a sum. of the products of the

thickness ordinates zt(x) and certain coefficients

independent of the section shape

N-1 S3(x)

TJl

s3.3(zx))

+SNV,/:

where p is the radius of curvature at the leading edge. The coefficïent is given by Weber as

(53)

and

(3) (1)

2 i(i)

i

- S11,

N sinO1 cosO -cosO

(3.) (-l)'-1 i

- N _l-cosO for ii=N.

for N

and

iv =

(1)

(31)

D. CORRECTIONS OF MEAN LINE AND PITCH ANGLE. In- the inner regions of the propeller blades, and especially for propellers with many blades, there is a notice-able interaction effect between the blades. This interaction effect is due to the velocities induced by the camber and incidence and by the blade thickness of all the blades. The velocities induced normal to the chordline give rise to

incidence and camber corrections, wh-ile the chordwise components of the induced velocities are included in the calculation of the pressure distribution over the blade. The effects of camber and incidence are given by the vortices, whereas the blade

thickness is represented by a distribution of sources and sinks. The total induced velocity is considered to contain the induced velocities obtained by a lifting line approach, and additional induced velocities due to blade plan-form and neigh-bouring blades. The additional induced velocities arising from the vortices and from the sources are treated separately, while the incidence and camber corrections are considered in the last section.

1. Additional velocities induced by vortices. The vortices are separated into bound and trailing vortices. In lifting line theories, tfle bound vortices are assumed to be concentrated in a "lifting line". This is a fair assumption when the propeller blade is long and narrow, i.e. high aspect ratio, or when the velocity is to be deter-mined at points at some distance from the lifting line. The induced velocity calculation for ti-te general propeller blade, however, must consider the effects of blade plan-form. In ref. (11), Küchemann provides a method for calculating the induced velocity on wings of. any aspect ratio.

The. bound vorticés are resolved into spanwise and chordwise vortex components. Behind the wing, the chordwise vortices are continued by the trailing vortices. For large aspect ratios, the chordwise vortices are considered to be so small compared with the spanwise vortices that the former can be neglected. The induced incidence is then created only by

(32)

(5)

26

-the trailing vortices, and for a wing moving along a straight path, this induced incidence is given, by the well-known relation

sin _f dy(r') dr'

o.1

-

_J' dr'

F'

rb

The subscript o is used to indicate parameters for large aspect

ratios. As the aspect ratio decreases, the effect of the chord-wise vortex components increases, so that the induced incidence is increasing. For wings of very small aspect ratio, Jones (26)

has shown that o.. in the limit A O. To treat wings of

aspect ratios between the limiting cases above, i<ichemann introduces the tTdownwash factorT' w. Then the mean induced 'incidence from the chordwise and trailing vortices, can be

expressed as a multiple of its value, o.., at large aspect. ratios Then,, from the above expression

(5L)

. wo.. wsin$

I dy(r')

dr

1

_lo

2 J dr'

T

rb

This..expression' is recognized from the propeller theory with exdeption.of the downwash factor'w, and the induction factor in, the expression from propeller theor'. And as the propeller blade is assumed' to have the sam induced incidence a the

analogous wing moving in the same manner, the downwash factor w is taken as the correction factor for finite aspect r.atio to the induded' incidence of the propeller blade..

The downwa.s'h factor w is related to another parameter, n, by w . 2n. The paraieter n is characterizing the chordw.ise

load distribution on a flat plate, and for a general wing n is given by Küchemann as'

A

(33)

The parameters involved for the determination of the ddrrection

factor are then the aspect ratio A, the two-dimensional lift

slope ao, the sweep angle A and the intrpolation function The two-dimensional lift slope ao is determined by

a k(1e)21T

Here k gives the lift reduction due to the boundary layex, for practical purposes it may have a value of 0.91. e gives the effect of thickness, and can be approximated by

e = O.8

The velocities induced normal to the chord line by the trailing vortices of the neighbouring blades, are included in the induction factors calculated by Lerbs (io). From expression (5L) it is then seen that the additional normai

induced velocity, due to the trailing and chordwise bound vortices,, cari be approximated as Aw (r) (56) yT f

d'y(r')

idr' U 2. j dr'

F'

rb

The downwash factor, w, is equal to 2n, where n is given by (56). The normai velocity induced by the spanwise vortices of a singie biade. was given in section 02 by expression (41):

w.5(x0)

-J

y(x')-4+-tanAy(x)

Thechordwise vortex distribution y(x) was determined from the boundary condition, i.e. the vortex distriBution was adjutèd

so that the velocity normai to the given preliminary two-dimensionai mean line vanishes. Then for a two-dimensiol)a

w _(x,O) dz5(X

(34)

In section: C2 it was shown how the velocity distribution on a single blade .was.determined. _{By intgrating over the chord} this resulting pressure difference due to the preliminary two-dimensional mean line,, a value of the _lift _{oefiicient i.} _found. As the blade is not two-dimensional, this lift.coefficient must

be multiplied with a factor ki to obtain, _{the desired value}

determined by lifting line calculations. _{Then it is assumed} that the mean line.ordinates shou-ld be multiplied with.the same factor k to obtain the desired lift coefficient. _{Thus the}

cl

normal velocity for the three-dimensional .sinle blade _{is given} by

w (x,O) dz (x)

(57) _{= k}

U cl dx

dz (x)

where :: is the slope of the mèan line determined by

two-dimensional relations from the desired lift coefficient.

When determining the velocities induced by the spanwise bound vortices of the remaining blades, the lifting line _method will be used. _{In the general lifting line approaòh, ali the} bound vortices of one blade are considered to be consentrated in the lifting line. _{When determining the velocity induced by} a blade, at a point a chordlength or moré àway from the blade, a good approximation is obtained by replacing the blade by its lifting line. By using this lifting line conception,a good

aproxirnation cah be expected in the outer regions of the blades, whereas the accuracy will decrease when approaching the boss. Each blade. will now be re1aced by its lift{ng line going _through the quarterchords of the blade.

Introducing a cylindrical coordinate system, fig. 3, the velocity _{induced at a point (a,r,O) by a unit length of} the bound vortex of unit strength at is given by Biot-Savart's law as

(58)

£xS

4iTjSj3

28

-Here ' is a unit vector iñ the radial direction, and the vector

(35)

Y

Fig. 3. Cylindrical coordinate system.

Kerwin (27) has evaluated expressions for this induced velocity , taking into consideration a complete lifting surface.

From fig. 3 the components of are seen to be

(a-e, rcos8-pcos, rsine-psinc)

whIle the components of a radial unit vector are

(0, cos, sin)

Introducing for a while a Cartesian coordinate system, with the x-axis in the axial direction and the y- and z-axes in the plane of rotation. The velocities induced along these axes are

denoted Vx' v' and v' respectively, and from equation (58) they are seen to be

y -

-rsin(-O)

(36)

V a 30

-(a-)sin

a_)2+r2p2_2rpcos(_e))3/'2

-(a-)cosc -. Z

_{a_)2+r2+p2_2rcos(_O)) 3/2}

The axial, tangential and radial induced velocities are then

-rsin(

c-O)

(a_)2+r2+2_2rpcos(_8))2

v'cosO-v'sinO

(a-)cos(--O)

((a_)2r2+p2_2rpcos(_O)J3/2

vr'

v'sinO+v'cosO

(a-)sin(-O)

((a_)2r2+p2_2rpcos(_0)3/2

The components parallel and normal to the chordline at the point (a,r,O) are then

À(r)v '+rv

(61)

u':

a

t

Yc

/r+Ar)

-rX(r)sin( -O)-r(a- )cos(-O)

/2+2(r ((a_)2+r2+p2_2rpcôs(0))

3/2

(6 2) w

Yc

/r+À(r)

-rv '+A(r)v

(37)

r2sin(ct-8)-À(r)(a-)cos(0)

/r2+x2r((a_)2+r2+p2_2rpcos(_e))3/2

The axial distances a and can be written

a = À(r)O and À(p)4.

The angle between the blades, 6k' is given as

2n(k-l)

z

If the unit bound vortex is situated on the k' th blade, so that the angle is increased by the angle 6k' the induced velocities can be wrïtten (63) c

/r2+A2(r)((A(r)e_A(p))2r2+P2_2rPc0S(_ok))3/2

r2sin(_O+ôk)_À(r)(À(r)0_À(P))c05(_8+6k)

(6k)

(r)((A(r)8-À(p)) 2+r2+p22rpcos(O+6

In most cases, X is constant, or fairly constant, over the radius. Then the following approximation can be made

X(r)O-A(p) -X(r)(-e).

The new variable

1i-e

is now introduced.

To obtain the velocities induced at a point (a,r,O) on a blade, by the bound vortices of all the other blades, the induced velocities must be summed over the remaining blades.

Also the expresion5, (63) and (6'4), for a unit bound vortex, must be multiplied by the actual strength of the bound vortex,

Np), and integrated over the radius

(38)

32 -Z

-rin(+5)+rpcos(pi.5 )

-

/r2+À2(r)f2

( 2Cr 2+r2+p2 2rpcos

+ó))312

r(P)dP C i r2sin(p+Sk)+A2(r)pcos(p+t5k) r(p)d V

_L/2+À2(r)J

_V

r(p) gives the circulation distribution _{over the}

radius, and if this function is approximated by a relatively

simple function, it is seen that the above approximaté expressions fór u and w can be solved analytically. When replacing

r(p) by a polynomial, at least a fourth order polynomial is needed to give a satisfying representation of r(p); but then, the analytical expressions are so complicated that a numerical solution, using Simpson's rule for the radial integration, seems tô be much better for practical purposes.

However, the induced velocities àre wanted relative

to the chordwise component of the resultant velocity, U, instead

of to V . From fig. 1 it is seen that

a

CO S.

v=v-

_asine

The angle between the chord line and the resultant velocity is

considered to be so small that U V. Thus

sinß U - V

COS.

a w w

IC

sin

U - V òoscx. a i

Now all the additional ormal induced velocities arising from

(39)

w (x) w (r) w (x,O) w (69) Y yT ys + p -

u

u u J

dy(r')

J rb O

where is given by expressions (68) and (66).

2. Additional velocities induced by blade thickness. Approximate expressions for the velocities induced by the blade thickness are obtained from the two-dimensional theory for aerofoils in cascade. The theory based on the singularity method, e.g. ref. (17), (18) and (19), gives the velocities

induced on one aerofoil by the remaining aerofoils. An outline of the actual parts of the theory, following the representation of Pollard and Wordsworth (19) is given in appendix B.

The main differences between the two-dimensional

cascade of aerofoils and the arrangement of the propeller blades are: For the propeller, there is a defined number of blades, rarely exceeding seven or eight, whereas the two-dimensional cascade has an infinite number of blades. The propeller blades are not two-dimensional, and they make an angle with each other depending on the number of blades. Another difference is that the velocity induced normal to the chordline, is only due to the neighbouring blades for the two-dimensional cascade, whereas a twisted propeller blade has normal velocities induced by

sourcesand sinks on the same blade. To compensate, to some extent, for these differences in the geometrical arrangement, the two-dimensional cascade theory will be somewhat modified. First, the number of blades contributing to the induced velocity will be reduced from the infinite number of the two-dimensional cascade theory, to the actual number of blades. This means that the so-called cascade influence factors (see appendix B), must be modified. The second modification is to compensate for

(40)

the finite aspeót ratio of the propeller blades. As there is a close rlation between the velocity -distribution and the lift coefficient, it is proposed to use the ratio of the local lift slope to the'two-dimensional lift slope a as a correction factor applied to the velocity distribution on the propeller blades. The propeller blades are not parallel, as for the two-dimensional cascade., but are making an angle with each other depending on the number of blades. Thus the

third modification is an attempt to compensate for this from

geometrical considerations.

a. Modification of cascade influence factors. From expressions (A15) and (A2-0) in appendix B, it is seen that the complex parameter F may be written as

-

31t

-(70) F

x-x)

= n--114) se - ±11) e s

--

r

where the number of the blades n is going from -

to +.

The complex parameter F. is related to the cascade influence factors R and I by

F R + iI

In expression (70), 14j is the cascade stagger angle, and s is

the- cascade spacing. When n goes from - to +, the sum of

(70) can be written as

L

:'

- coth(' e')

For a three-bladed propeller, n should go from -1 to +1, and as the number of remaining blades must be an even number, n is taken to go from -1 to +1 also for a four-bladed propeller. For five änd six-bladed propellers n is going from -2 to +2, and for seven and eight-bladed propellers, n is going from

(41)

ZT Z for Z 2,'-t,6,8

Z?

Zl forZ3,5,7

where Z is the number of blades of the propeller, the éxpression for F can be written

t

ZY2(_)2+n2cos2P

2(<X)

S F r S s ZY2 n2sin2ilj .2

X-X'2

+1 ()

r s

_XX')+4+222()

nl

(----so that the cascade influence factors are modified to

- , Z'/2

(X-X')22214

2(xx)

S R - r

n1

s and Z'/2 n2sin2iP 2( - r s s

b. Modification for finite aspect ratio.

A correction.factor is to be applied to the velocity distributions from the two-dimensional cascade theory, to account for the finite aspect ratio of the propeller blades. The local 11f-t slope is obtained by dividing the lift coefficient with the geometrical angle of incidence, measured from the zero lift

line. For a general propeller, the effective angle of Incidence will only make a small fraction of the total geometrical angle. For the approximate method used here, it is therefore assumed to be sufficient to consider only the induced incidence, so that

3CLCL

_0L

(42)

s

36

-The two-dimensional lift s'ope _{ao was given in section bi.} The proposed correction fadtor for finite aspect ratio is then

(7L)

k1 L

ao - B)ao

c. Modification for the blades not being parallel to each other.

The propeller blades are not parallel to each other, like the aerofojis of the two-dimensional cascade. Thus when considering the root region of the blades, _{the remaining parts} of the blades will have larger cascade spacings _{and stagger} angles. _{And similarly for the tip region, where the remaining} parts will have smaller casbade spacings and _{stagger angles.} This effect of the angle between the blades is _{not expected to} be found at mid semi-span, where only the effect _{of tapered} blade thickness is expected. _{It is then suggested to use the} values of the cascade spacing and stagger angle for a spanwise station somewhat nearer mid semi-span than the actual station. The new spanwise station may then be given

_{as r'O.B25rO.lÔ5,}

where the, values of the coefficients are empirical. The

cascade spacing s was originally used _{so that the circumferential} distance between the blades is given by

D r

- Z chord

This expression is now modified to

(7b) s

.

chord(O825"F0J05)

when not considering the radial variation of the chord. The cascade stagger angle i is given by

r/2

(43)

A.

'p = 7TL2-ß. ir/2-arctan(-.)

This expresion is now modified to

A.

= ir/2-arctan(08250105)

where is the induced advance -coefficient determined from the lifting line calculations of section Bl.

The two-dimensional cascade theory, outlined in appendix B-, gives the following expression for the velocities induced on one blade by the remaining blades:

u-iw

f

(q(x')+iy(x'))(R('p,s,x')+iI('p,s,x')Jdx'

Here u_e and w are the velocities induced parallel and

C

normal to the chordline; q(x') and y(x') are the chordwise source and vortex distributions for the cascade; and

R('p,s,x') and I(P,s,x') the cascade influence factors _which are functions of the cascade geometry, and are given by expressions (72) and (73) in their modified versions. In

this section, however, only the velocities induced by the sources and sinks will be considered, thus expression (77) reduces to

u -iw ,.., r,.. , - ,

Here subscript

q means "due to thickness".

From the boundary.conditions, as indicated in appendix B, the strength of the sources and sinks are given by

dz(x)

q(x) 20J+u) dx

which is approximated here by

dz(x)

(44)

38

-From expression (78.), when introducing the correction factor

k _{for finite aspect ratio, the chordwise and normal induced}

velocities can be written

k1

_d2(x')

dx' w q0 ki r

dz(x')

.-fl- J dx' I(T,s,x')dx!

As mentioned in Section Cl, Weber (12) has provided a method for numerical calculation of the Slope when h? ordinates of the thickness distribution are given. This method is now used to determine the strength of the sources and sinks according to expression (79).

To determine the correction due to blade thickness, however, the thickness distribution must first. be determined..

This distribution must satisfy the minimum thickness rquire_ ments from strength calculations, and shouidgive a total

pressure distribution with the required safety margin against cavitation. ,.

3. Pitch angle and mean line corrections.

The vortices induce the normal velocities given by expression (69) in additthñ to those obtained by a liftiñg line calculation. The chordwise velocity induced by the chordwise bound and trailing vortices, will either vanish completely or

be very small. The spänwise boundvortices induce a chordwise

velocity equal to ±y(x), w}iere the uer ând lower Signs

correspond to the suction and pressure sides respectively. The chordwise and normal velocities induced by the blade thickness of neighbouring blades are given by expreSsiors (80) and (81).

When neglecting the induced velocities obtained by a lifting line approach, the velocitis induced normal and

(45)

or

parallel to the chord are given as

_(W_l)n8fdY:)

_{JYxtx:+tanAYx}

2 _j

dzx

t(i4,s,x')dxt and i i _t u dz (x') , dz (x ) J

-'-'

J ' R(4.i,s,x')dxt

Here E and

are given by expressions (68) and (67) respebtively.

If the resultant inflow V has to make an angle with the chordline, the boundary coñdition can now be

written

W+w(x) dz5(x) U+u(x) - dx

tane

dz(x.) u(x) dx i t_J i.e. x X

w(x')

(8'4)

z(x)

=

tanct

.L 1u1x') dxt

To determine a, z5(x) is taken to be zero and the integrations

are performed over the whole chord, i.e.

(85) tanco

- J

wx)d

The mean line ordinates of (84) are related to the ordinates of the mean line determined from two-dimensional

(46)

140

-relations. Then a camber correction factor k is defined as

the maximum camber obtained by f84) divided by the maximum

two-dimensional camber. Ana to reduce the amount of calculations, this correction factor kc is used for all chordwise stations.

This camber correction factor _{is then given as}

X

w (x '

i r 1 _dx'

f

U

(86) _k

= Z(X2D

tana J

1u(x')4

i+2.

dx'

where x _{is the non-dimensional chordwise station of maximum}

max

camber.

The determination of the mean line ordinates, and

thus also k, cannot be very accurate by this method. _The main reason for this is that the chordwise variation of the

normal velocity induced by the chordwise bound vortices

cannot be determined, as this method gives only the average value

over the chord, and is thus not particularly well suited when

the velocity is to be integrated over frations of the chord. When determining a', the integrations are performed over the

whole chord, and this method should therefore be better

suited for this calculation.

In ref. (9), Morgan, Silovic and Denny have

calculated the angle a" due to the effects of loading and

blade thickness separately. They used a complete lifting surface theory and the method of Kerwin and Leopold (s) for the thickness effect. Similar separate calculations were performed by the present method, and the results for three typical

propellers are compared in figs. 14 and 5. The mean line correction factors due to loading are compared in fig. 6. Paritculars of the three typical propellers are given in table 1.

When designing a propeller, the pitch angle ,

which the chord line makes with the plane of rotation, should

be taken as

(47)

Fiere ß. is determined in section Bi and c is the corrected

i

ideal angle of incidence given by (85). The mean, line ordinates are determined from expression (25), with the correction factor kc included.

TABLE 1. Particulars of tested propellers.

Propeller Number of blades Blad9 area ratio Boss

rato

1 Skew back angle Mean line Thickness distr Design J Design K-T A 4 0.550 0.2000 0.000 a-0.8

mA6&

0.2000 B 5 0.750 0.2000 0.245 a0.8

mA66

0.2000 C 6 0.750 0.2000 0.367 aO.8

mA66

0.2000 D 3 0.547 0.1760 0.138 a0.8 NACA 16 0.755 0.1205 E 4 0.450 0.2655 0.106 a:0.8 NACA 16 0.642 0.2020 F 5 0.645 0.2100 0.116 a0.8 NACA 16 0.550 0.1920 G 6 0.595 0.1760 0.165 aO.8 NACA 16 0.430 0.2050

(48)

.0.03 0.02 0. 01 0.00 0.03 0.02 0.01 0.01 0.00 - 42 -0.00 .0.2 0.3 0 4 .0 5 0.6

Propeller B: Z5, Ae/AûrO7S l4 skew.

07

0. 8 0.9 r 1.0 I I I Liting surface ref. (9) calculation, -i f Lifting surface ref. (9) method calculation, Present

-I I- .1 Lifting surface calóuiatiori, ref.. (9) Present method 0.2 1)3

04 ₀₅

0.6

07

08

0.9 r 1.0

Propeller A:

ZLt, AeIAü055

no skew.

0.2 0.3 0 4 0 5 0.6 0.7 0. 8 0.9 r 1.0

Propeller C: Z6, A IA

_e0

0.75, 21° skew.

Fig. 4. Idea! angle of incidence corrected for loading. 0. 03

a

(49)

0.03 0.02 0. 0.03 T 0.02 0.01 0.01 0.00 0.00 0.2 0.3 0 .... 0 5 0 6 0 7

Prpel1.

A: ZU, A/A0O.55- no skew.

0.00

0.2

03

0 L

05

06 Propeller B: Z5Ae/A0r075

ILf° skew.

07

08

0.8 0 9 r 1.0 0 9 r 1.0 ¡ J I

-Complete source and

sink calo. , ref.

Present method

-I I

I

-Complete source and

sink calc. , ref. 19) Present method

I I .1

0.2. 0 3

014. OS

06

07

08

0.9 r 1.0

Propeller C: Z6, A/A0z0.75, 210 5kew.

Fig. 5. Pi.tbngie cth'reotion due to blade thickness.

0.03 AOT

(50)

1.5 1.5 -

44

-Lifting surface calculation, ref. (9)

-- Present method

1.0 1.0 0.2 0.3 0 4 0 5 0.6 0.7

Propeller C: Z=6, Ae/AoC75

210 skew.

Lifting surface calculation, ref. (9)

-- Present method

Fig. 6. Mean line correction factors due to loading.

0.2 2.5 k C 2.0 1.5 1.0

03

04

05

0.6 0.7

08

0 9 r 1.0

Propeller A: Z=4, Ae/Aü=OSS no skew.

I I Lifting surface calculation, ref. (9) Present method

-0.2 2.5 k 0.3 0 4 0.5

Propeller B: Z5, Ae/A0=O7S

0.6 0.7 14° skew.

08

0.9 r 1.0 I I

08

0.9 r 1.0 2.5 k C 2.0 C 2.0

(51)

conditions.

In the performance problem, the geometry of the propeller and the working conditions are given, and the performance is to be determined. The general way to solve this problem is to evaluate two independent relations between the two unknown parameters y and B, e.g. as used in

references (28) and (29)

CLc

±-r<sin.tan(ß1-8)

when using Goldstein's K-function, and the second relation is

dC

CL

-Lerbs (10) has replaced (88) by a relation obtained from

fig. 1: tan (1 + )/( -Here - and Va

and as both

functions of is necessary wt

- are calculated by use of the induction factors,

Va

the induction factors and the K-function are the unknown angle 8, an iteration procedure

The convergence of this iteration then, to a great extent, determines the usefulness of the theory.

As the approach of Lerbs to the performance problem is rather similar to the lifting line design method, an attempt was made to apply the correction factors to this method. For

the performance calculation, however, relation (12) giving the distribution of over the radius for the optimum propeller,

cannot be used. The induction fadtor method gave such induced velocities in the blade root and tip regions that the

convergence of the iteration quite disappeared. It can be

mentioned that a slow convergence was obtained by "smoothing out" the radial distribution of the induced velocities.

This Ttsmoot.hing out" was obtained by fitting a second or

third order polynomial by the method of least squares, and giving the values in the root and tip regions much lower

(52)

46

-weights than the values in the middle of the blade. This

method could of course only give: a rather rough approximation,

so 'another method was evaluated..

2. Suggested method for calculating the performance in off-design conditions.

The method is based on the assumption that evêry working condition for the propeller, within the general assumptions made., is a calculable deviation from the design

condition. If the propeller is designed according to the design niethod of sections B and D, the distributions of CL nd

over the radius for the design condition are known. CL and B. can then be calculated in. off-design conditions as will be

5hown later in this section. If the design values of CL and for the given propeller are not known, however, a "dummy propeller" is designed, accordingto sections B and D, for

the same design conditions as the óriginal propeller. The

measureable indications of CL and

,

namely camber and

pitch angie' will then generally be different for the .oriinal and the dummy propeller. From these differences, CL and

can be approximately calculáted for the real propeller in design condition. In this way it should be possible to calculate the off-design performance of any propeller for which the design conditions are known.

Indicating the design condition of the given

propeller by subscript _D' and the dummy propeller by suscript

DF' the parameters in the design condition of a propeller

of unknown design values can be approximated as

CLD = CL

DP

L(

DP

8iD = ₊ _iDP (CLD -

cL){Ç

Expression (91) is obtained by using the relation between maximum camber, denoted f, and lift coefficient C; and

dCL

by using the relation ACL -

_-

cz. This equation assumes

(53)

as for the given propeller. _{The method developed in section D} for calculation of the meanline correction,- is however, not very accurate for reasoñs described in section D. _{.Therefore the} maximum camber of the "dummy propeller",

Dp' is determined with

another mean line correction. _{In ref. (e), the méan line} correction due to loading is calculated with _{lifting surface} theory, and the correction factors are tabulated for series of

propellers. By using these correction factors-from _ref. _(e), together with mean line corrections due _{to blade thickness} evaluated in section D, a better estimate of ffl _{is obtained.}

Expression (92) is obtained as Bflp and also ¿a - ¿a :1. e so that ¿CL ¿CL =

--

-e

How a = _e+OEi _{where ae is the angle of incidence when only the}

bound vortices are considered, and a1 the induced incidence.

e can then be expressed as

(93)

J

Sdx

Q

Here CL/a gives the- contribution from _{one single bláde;(2n-1)(1-ß)}

gives the contribution from the chordwise bound vortiòes of all blades as described in section D, _{so that the contrib.ition of the}

chordwjse _{bound vortices of the first blade must be subtracted} as it is already included by CL/a.- _{This effect of the chordwise}

bound vortices is expressed by the difference _{in the lift slopes} a and a0, as a describes the case of both spanwise and chordwise bound vortices and a _{the case of no chordwise bound vortices.}

(54)

dOE cx -e e

The induced ängle a _{is then}

a

f

dx

where

f

dX expresses the induced angle due to blade _thickness. Then the local lift slope is given as

dCL CL

dci - cx +.a.

e .

The sectional lift slope for a single blade, a, is

given by Küchemanrt (li) as

cosA 2n

-aosjn

_{l-nn(cotlTn=cotirn0)}

where a0 is the two-dimensional lift slope calculate-:i in

section Dl,- n is a parameter charadterizing the chorJwise load distribution on a flat plate of finite aspect ratio, given by expression (5); n0 is the corresponding parameter for infinite aspect ratio, and from (55) it is seen that

+ 14T

ii7'

In the off-design conditions, the vluesof CL

and B1 are calculated in a similar way, an'd the parameters are denoted by the subscript

OD Thé off--design condition

-is considered t-o occur in one of the following ways. (a)- The

advance Ì'atio is different frôm that of the design cndition, (b) the pitch is different from that of the design case, (c)

the actual cändition may be a combination of both these cases.

a

-

48

-Finally,

_{-dx gives the -contribution of}

the spanwise bound vortices of all neighbouring blades, _{as described in section D.}

-Then the sectional lift slope is given as

dC C

(55)

If the advance ratio is different from that of the design condition, will have a new value Then the values of CL and are-given in a way similar to tiiat of expressions

i dCL

dOE

dCL

The value of used here should be calculated from expression

(96)

as before, but with the value of inserted for CL.

D

If the advance ratio has the design va-lue, but the pitch is different from that of the design case, t-he calculation of CL and will be equal to that of equations (91) and (92) for Thus when indicating parameters for pitch other than design pitch by subscript CL and

are given by -P P CL _{= CLD} dCL -= + 'D + (CL -

CL)[_

- LJ dc dOE

Now, if the performance is to be calculated for both advance ratio and pitch which differ from those of the design condition, a combination of equations (98) and (100) and of

(99)

and (101) must be used, so that

dC dC CLOD P = CL + - OD + -(.103) OD,P 'D + OD - + (CLOD P -

CL)[_

-(91)

and

(92)

+ dCL

-(98)

C1 CLD OD i

()

_iQ = 'D + OD fl) _{+ (CL} CL _dC dOE