An eximination of some theoretical propeller design methods

MEDDELANDEN FRÄÑ

STATENS SKEPPSPROVNINGSANSTALT

(PUBLICATIONS F THE SWEDISH STATE SHIPBUILDING EXPRIMEÑTAL TANK)

Nr50 GÖTEBORG 1962

AN EXAMINATION OF

SOME THEORETICAL PROPELLER

DESIGN METHODS

BY

C..A. JOHNSSON

SCANDINAVIAN UNIVERSITY BOOKS

SCANDINAVIAN UNIVERSITY.BOOKS

AkademiförlàgetGumperts, Göteborg Munksgaar4, Copenhagen Norwegian Universities Press, Oslo

Svenská Bokförlaget f P. A. Norstedi & Söner Albert Bonnier, Stòckholrn

PRINTED IN SWEDEN BY

Synopsis

Different propeller design methods, based on vortex theory, are

described and compared.

The paper begins with an outline and discussion of the essential

features òf the different methods. The lifting line theories presented are the GOLDSTEIN ,-method, the GUILLOTON method and the in-duction factor method in the form used by LERBS and STRSCHELETZKY. Approximate lifting surface methods developed by GrNZEL, GUILLOTON

and STRSCHELETZKY are also introduced.

The methods described have been used for the design of

three-bladed prepellers of a given blade form for a loading case defined by the condition. that a prescribed non'optimum circulation distribution, shall be obtained at an advance ratio, J = 0.45 in homogeneous flow..

The designs obtained with the different methods äre compared and

discussed.

Model propellers, designed according to some of the methods, have

been manufactured and submitted to cavitation tunnel and open

water tests. The cavitation properties and effective pitch thus deter'

mined are compared with those theoretically predicted.

1.

Iñtroduction

The most important demands made of a modern ship propeller are

that

the desired relation between power consumption and number of revolutions shall be achieved at the speed presumed

the propeller shall have the estimated cavitation properties together with high efficiency.

The first demand can excellently be satisfied by using results of standard series tests for such loading cases as are covered by the

scope of available results of this kind. It is when the second demand

4

becomes urgent. Also, for the design of more exclusive types of

propellers (supercavitating and contra-rotating, for example) a more fundamental approach to the problem is desired. A general propeller

theory enables the designer to separate the influence on efficiency and cavitation properties of respectively the thichiess and form of the blade sections, the blade form, the distribution of the load over

the blade and so on.

Ïhese demands have led scientists to investigate the possibilities of

developing a general theory for ship (and aeroplane) propellers.

Because of the complexity of the problem the immediate conclusion

has been that, when applying the fundamental equations, certain

simplificatiOns of the problem are necessary if a reasonable simplicity

is to be attained in the calculations.

In this work different scientists have proceeded in different ways

with the result that a number of calculation methods exist, of varying

degrees of complexity. In the present paper some of the methods have been applied in designing ship propellers for a rather extreme

loading case. To find out if the use of more complicated and laborious

methods results in propellers superior in efficiency and cavitation

properties, some of the designs have been applied to model propellers which have been submitted to cavitation tunnel and open water tests.

The methods useçi in the present investigation are applications of

lifting line theories put forward by GOLDSTEIN, GunLoToN, LERBS and STRSCHELETZKY and lifting surface theories formulated by

Lun-WIEG and GINZEL, GUILLOTON, LEREs and STRSOIIELETZKY.

As the approaches used and lines of thought followed varied greatly

when the different theories were developed, it was found necessary

to include in the present report a short description and discussion of the essential features of the methods mentioned above.

2

List of Symbols

= vector distance

A A0 = blade area ratio

b = Half span of wing or propeller blade dD

Cfi = drag coefficient

I camber

r,

i..4, i?,

j

Ic KT

- non-dimensional circulation coefficient accord--ing to LERBS

= FoURIER coefficients of 0L

R = induction factors of the induced velocities

V1

= D

= advance ratio of propeller

= camber correction factor according to LTTDWIEG and'. GINZEL T

=

- thrust coefficient. QD4n2 Q KQ

= oD5n,2 -

torque coefficient

= length of blade section at 0.75 R n = number of revolutions

po = static pressure

p,, = Vapour pressure

P

= power

P. = propeller pitch in ideal flow

Q = torque

r, r' = radius of blade section dL CL

=

- lift coefficient ldbV2

P

Cp

=

V3 - power coefficient A D = propeller diameter dD = drag of profile

dL = lift of profile in ideal flow

dF1, dT1 = transverse and axial components of the lift of blade profile in ideal flow

R

2

R0

75R

10.5R

Vv+cÓ.75 = REYNOLDS number of

pro-peller

t = thickness of blade section

u, u, UT, UR, uy = components of the induced velocity at thepropeller VA advance velocity of propeller

V = inflow velocity to propeller blade section, including

in-duced velocities

V,, = inflow velocity to propeller blade section, excluding

in-duced velocities

r

r'

z, z'

=

and

y, y' = spanwise coordinates of a wing = angle of attack of blade section

c ±

= angle of attack for zero lift df blade section

cj

₌

ß = advance angle of blade section

= hydrodynamic pitch angle of blade section

rb = circulation of bound vortex

l'i. = circulation of free vortex

A x pitch angle correction due to lifting surface effect according to LERES

Ap = pressure drop at profile

= drag/lift ratio

= propeller efficiency = ideal propeller efficiency = GOLDSTEL function VA 2

=

- advance

coefficient nD

=xtanß

y = kinematic viscosity = density

ç'

a

- test cavitation

number based on advance

ve-172

2 A

locity

*rA

ZIP

- inception cavitation number based on advance

velocity

r = angle subtended at the axis by the projection of a blade

section in the disc plane

= velocity potential

3. Formulation of the Problem

Analogously to the case of a profile of finite length, the bound òirculation round a propeller blade varies along the blade in radial

direction as well as periferally along the different blade sections. In

accordance with the general theory of vortex motion, the bound

vortices coñtirnie in the fluid as free vortices. Because of the geometry

of the blades these traffing vortices are of helical shape. Both the

bound and trailing vortices induce velocities around the blades. When

determining the shape of the blades it is of particular interest to

calculate the magnitude and direction of the velocities induced at

differeñt points of the blade surface. This necessitates knowledge of

the pitçh and strength of the vortices which in turn requires

know-ledge of both the load distributIon and the geometry of the propeller.

Thus the problem of determining the most suitable geometry of a

propeller, intended for a certain loading case, involves a certain

amount of trial and error work.

4.

Approximations and

Simplifications Involved in the

Methods Used in the Present Comparison

When theoretical methods are used in propeller design work, the calculations are generally made in steps, each step representing a degree of simplification of the problem. In the methods used in the

L

present comparison the following common assumptions and design

stages are introduced:

The propeller is assumed to work in ideal, incompressible flow and the corrections for the manifestations of viscosity are introduced in the final stages of the calculations.

The blade sections are supposed to be infinitely thin.

The magnitude of the induced velocities infinitely far back in the wake of the propeller can be shown to be twice that of the velocities induced at the blades., See [4].

The propeller is, assumed to be "moderately loaded" i.e. the

in-fluence.of.. the radial and centrifuga] velocity components on the shape of the trailing vortices is neglected.

The trailing, vortices from a certain blade section are presumed to form helical lines having constant pitch and a radius equal to that

of the blade section itself. The pitch is assumed to be equal to -that of the relative flow at the blade in spite of the fact -that the induced velocities vary in magnitude along the wake. See point

e) above. A discüssion of the degree of approximation involved in this assumption can be found in [15] where an optimum propeller is considered. '

The blade sections are a priori assumed to be of zero length.

Different 'lifting line theories" have been developed for use in this stage of the calculations.

Using' the results of the calculations according to lifting line theory

aS a 'starting point, the induced velocities at different points of

the blade surface are cakulated by the aid of "lifting surface

theories".

The main differences between the methods considered in the present

comparison are the basic ideas behind the lifting line and lifting, surface theories and the number of points of the blade surface for

which the induced velocities can be calculated by the different lifting surface theories.

5. Common Basic Equations and Mathematical Aids Used

in the Different Methods

The relation between lift dL and circulation Tb for a blade section of a propeller, working in ideal flow, can, according to the K.UTTA JOIJKOWSKI theorem, be expressed

Fig. 1. Velocities and forces at a propeller blade section in ideal flow.

dL=rV

(1)

where V Is the èntrnce velocity including the induced velocities

u and UA, as shown in Fig. 1.

Given the notation of Fig. 1, the dimensionless thrust and power

coefficients CT and C in viscous flow for the whole propelIer can be written - T CT 2

= 4z f GL (

-2VA ₄

'h

i GP =

-

f

(i

+

;Á)

(1

+

tai

dx 4 1h

with 0L = - dimensionlesscoefficient of the bound circulation D VA

of one blade section according to LEEBS [14-- 15].

(3)

T

tanß)dx

(2)

'o

The determination of the circulation distribution r(r) and the

induced velocities UT and UA being a potential problem, the solution can be reached by finding a velocity potential which satisfies certain

boundary conditions.') As the flow is supposed to be jrrotational there is the additional condition (Li'rcE's equation)

A2P=O

. (4)

An alternative approach is to üse the law of BIOT-SAVAT. From this lawthe induced velocity d at a pöint P, emanating from a vortex element ds of a vortex line of circulation r, can be expressed

-r dsxa

4y: a3 (5)

where is the vector distance between the vortex element and the

point considered

A typical application of Eq. (5) is the case of a wing of finite length.

If the wing is replaced by a hftmg line of varying circulation E(ij)

dr

straight vortex lines of circulation -a--- ¿y extending to infinity are

shed downstream. Then the inducéd velocity at a point P of the

lifting Eñe can be asôertained, by. usirg Eq. (5), to

dr

uy,. =

J

4t(yy')

(6)

b

being directed as shown in Fig 2..

The main problem when the induced velocities are calculatedirorn

Eq (5) or (6) or from similar equations is that the equations tend

to infinity when the distance yy' tends to zero. This difficulty is

generally overcome in the following manner which refers to Eq. (6)

but which is also typical of a propeller blade.

The new variables and rp' are introduced which are defined

y = b cos 9:)

y' = b cos

1,) Also the acceleration potential can be used as starting point, an approach which

was proposed by PRANDTL [1] and which was recently used by SPAENBERG [2] when formulating the fundamental equatioñs: of a lifting surface theory for propeUers.

Fig. 2. Velocities induced at a wing by the trailing vortices.

and the distribution of the bound circulation I'(p) is expanded in a

FouillER series e

f(q:)= 2Asinnq'

(7) n=1 which gives e VA

4b

f

:

4

(8) o

For the solution of Eq. (8) GLAUERT [3]. has deduced the following expression

f

dç

-o

cos9cos'

Siflq'

which has finite values also when q = p'.

In the case of a propeller the trailing vortices are of helical shape. For this case, the following expressions (see [15]) can be obtained for

Ii

with

C

zS

A

Pf

z

Fig. l. The determination of the induced velocities from a helical vortex.

determining the magnitude of the tangential and axial components

UT and UA of the induced velocities at a point A0, emanating from a helical vortex, starting at G:

du =--tanßrf

--[r'rcos(9+?p)--- (rO tan ß+zs) cot ¡9 sin (O+)] dO (10)

r,

i

duA=-,--rf j[rr'cos(é1-{-tp)]dO

(11)

a = [r2+r'2-2rr' cos (O±)+(rO tan ßj+zs)2]h12

For lifting line calculations the induced velOcities at a line through

the starting point are dèsirable, corresponding to p = z5 = O in the

equations above. (Ç coincident with S in Fig. 3).

6

Lifting Line Theories Used in the Present Comparison

Here follows a short description of the means by which, in the

diffèrent liftiñg line theories chosen for the present. investigation, a relation is achieved between the circulation distribution I'(r) and the distribution of the induced velocitycomponents uA(r) and uT(r) over .the blade. This relation established, the hydrodynamic pitch angle in non-viscous flow can then be obtained, referring to Fig. 1, as

UA

tan ß.. = _X (12)

- VA

Further the dimensi6nless thrust- and power-coefficients can be

calculated from Equations (2) and (3)..

Goldstein's - Theory for Propellers with Minimum Kinetic Losses The special case of a propellér with maximum induced efficiency

or minimum kinetic losses and hub diameter zero was first treated

by BETz [4] who formulated the following theorem:

"For a propeller with the circulation distributed along the blade in such a way that for a given thiust, the energy loss is a minimum, the flow far behind the propeller is the same as if the vortex sheet,

formed by the trailing vortices, was a solid membrane, moving back-wards in the direction of its axis with constant velocity."

Accordingly, fär such a propeller, the velocity u in Fig. i is inde

pehdent of radius

Starting from BETZ' theorem the following "condition of normality" can be established:

"If the voftcès, shed from a propeller, form a vortex sheet of true helical shape,. i.e. if the velocity u is independent of radius, the

in-duced, velocity u at each blade section is perpendicular to the vortex sheet.."

For a propeller with minimum kinetic losses (an optimum propeller) the induced efficiency m for a blade section is independent of radius ánd can be expressed:

VA dT VA

rour

tan ß

- dP

- rw

VA ± UA - tan ß, - VA ±u (13)

with the notation Of Fig. 1. From this figure we obtain for the axial

an4 tangential components of the induced velocities:

U U = -w:-- CO52 ß "A "A UI U

=

cos ß sin ß Y4 _'A

The problem of fiñding the relation between induced velocities and circulation distribution for. optimum propellers with a finite number of blades was first studied by GOLDSTErN [5] who defined, the boundary

conditions for the velocity potential to be

u cos

= 7;_cos

rôEI

at the blades (see Fig 1) and

grad 0 = O when r = co

z, r and O being cylindrical coordinates.

As an additional condition LAPLACE'S equation O should be satisfied.

A ondlition to be fulfilled by the potential was that it should be

continous everywhere except at position angles corresponding to the blades.

The solution obta.ine4 was strictly applicable only to the region far

behind the propeller. For a moderately loaçled propeller however it could be valid also at thé blades themselves. If the discontinuity of the potential at a bladesection is denoted /40/, the circulation round

this sectiOn is obtained as

(14)

In GOLDSTEIN'S paper [5] a solution of this potential problem for a

two-'bladed propeller was presented and values of the circulation

distribution given for different combinations of i/Aç and 1/2-values where

i _,ì

2 _VA

Equatiens applicable to any number of blades were also given.

A more complete evaluation of the GOLDSTEIN potential was later given by Ï(BAMER [6] who related his results to the corresponding results for propellers with an infinite number of bla4es. For the latter

type of propeller the circilation round a blade section can be

ex-pressed s

f =

(15)

according to STOKES' theorem.

For a propeller having a finite number of blades the magnitude of the velocity component U varies with the angular coordinate 'nd

Eq.. (15) can be modified to

zi' = 2rirx(2uT) . (16)

or

zF= (1')z_'

where can be regarded as a "mean value factor".

Using results from work, by LöscH [7] for propellers with an infinite number of blades and his own extended evaluation of the GOLDSTFIN potential, KRAMER calculated fl-values for different numbers of blades

which were given in tables atid diagrams in [6]. Re Jso established the diagrams showing the relatión between induced efficiency j, advance ratio .2 and thrust (power) coefficient C (C) which have

now become classical.

In his evaluation of the GOLD STE [N potential KRAMER used rat.her approximate interpolation methods. More accurate calculations were

later made at the D a y i d T a y 1 o r M o d e 1 B a s i n [8]. These

calculations were recently extended to cover the case of a propeller with finite hub diameter [9-10].

GuiUoton's Lifting Line Theory

When the circulation distribution differs from that corresponding: to minimum kinetic losses, the vortices shed from the blades do not

form a true heliôál surface. In such cases the velocity u in Fig. i is

not independent of radius and the relation (14) 'between the compo-nents of the induced velocity s no longer valid. The induced velocity

components have to be calculated separately by the use of other

lifting line theories. A tbeòry f òr such cases characterized by simplicity in application is that of GuIrJoToN [11-13]. Diagrams necessary for

ma1ing the calculations according to this method are, however,

available only for three-bladed propellers and cover only a rather

limited range of pitch ratios.

When calculating the induced velocities, GUILLOTON uses the law of BIOT-SAVART in the form shown in Equations (10) and (11). The

values of the velocity components dud and dur are obtained by

direct integration along the free vortices and summation for three

blades.

The difficulty of infinite u-valuds at r' = r is overcome in the

:following way,

Fróm the calculated, values of u and UT the induced velocity

component perpendicular to V0 (notice not V) in Fig. 1 is related to

the corresponding induced velocity from a straight vortex having

'the same direction as the tangent to the helical vortex at the starting

point at the blade by

Rdr/

i

=

fi

I

NO

j

\y'_,y

(17)

where H remains finite when y'-. The discontinuity is thus taken

account of by the'first térm in Eq. (17) and overcome by using Equa-tions (8) and (9).

Strictly speaking there is still a discontinuity in the function H

but this is shown in [11], however, to be of no practical importance.

By the aid of the expressions in Equations (8) and (9) the first

part of'Eq. (17) can be transformed

with b

-i

dT

=

--

r

Y' Y

-i ¡ 2x sin 2p'

3i sin 3p'

== --- A1 -- +A2 -i;-.

±A3

b sin q'

Rr,,

9

2

(20)

The solution of Eq, (20) and the analogous expression for du can

According to GUILLOTON the second term is transformed

i B

VO2

=

dr H = -i-- (k1A1+k2A2+k3A3)

(19)

The coefficients k1 and k2 were given in [12, 13] for three sections

along the blade (Sectións 1, 3 and 5 in Fig. 6) as functions of Yh/l? and pitch ratio while k3 was judged as negligible. The values were calculated on the assumption of constant pitch ratio over the blade

for the vortices shed from the different sections. The diagrams cover

3-bladed propellers with pitch ratios within the range O.62<P/D<

<1.25.

If the pitch ratio varies over the blade, it is recommended in [12] to use the P/D-vaIue at the tip when reading off the diagrams, as

the vortices near the tip are of predominant influence when calculating the induced velocities.

In [12] and [13] a scheme is given for calculating the induced

velocities and circulatiòn distribution for the different blade sections,

which makes the performance of the calculations easy and rapid.

Le'rbs' Lifting Line Theory

A more complete lifting line theory for the case of a moderately

loaded, non-optimum propeller than that ofGUILLOTONwas developed by LEBES [14, 15]. In addition, the diagrams, necessary for performing

the calculations were established not only for three-bladed but also

for fbifr- and five-bladed propellers.

The basic equations for the method were deduced by Lnns starting both from LALACE'S equation and from the law of BIOT-SAVART.

In this short description the latter starting-point will be used.

Like GIJILLOTON, LEEBS deduces from Equations (10)(11) the expressions for the different components of the induced velocity.

Instead of attacking these expressions with numerical methods as

GuILL0T0N

[11-13] and

STRSCIIELETZKY [17] LERBS solves them in

an analytical manner.

The case = Z5 =0 Eq. (11) can thus through direct calculation

be expressed in terms of the modified BESSEL functions I (first' kind)

and K (second kind) and their derivatives [14-15].

To overcome the difficulty that the resulting equations tend to

infinity when r'-+r LERBS uses the concept induction factor originally introduced by KAWADA.

The induction factor corresponding to the axial induced velocity

component is defined

- du

-i.e. as the axial velocity induced by the helical vortex divided by the

induced velocity from a straight vortex placed in r parallel to the propeller axis and reaching to z = +oo. (See 'Eq. (6) and Fig. 3).

The induced velocities from these two vortices become infinite in the

same order when r'-r and the induction factors remain finite. For the induction factors expressions dependent only on the number of

blades, the ratio nr', and the pitch of the vortex can now be obtained. By replacing the BESSEL functions and their derivatives appearing

in the final expressions by asymptotic expansions according to

Ni-CHOLSON, LEnEs was able to calculate the induction factors i4 and jT for z = 3, 4 and 5. The results were given in diagrams in [14] and

[15]. Later WRENCH [16] made more accurate calculations using

LEKMER'S asymptotic formula. The results of these calculations were given in tables.

The induction factors once determined, the induced velocities can be calculated from the following expiession

UA

i 1dGL dx

(22)

and a corresponding expression for UT.

The only difference between Eq. (22) and the corresponding

ex-pression for the induced velocity at a wing of finite span (see Eq. (6))

is the presence of the induction factor A in the former equation. Again Eq. (22) -+00 when x'-*x. This difficulty is overcome by

LERBS in the same way as shown above for Eq. (6) i.e. the dimension-less circulation distribution GL(X') is represented by a FOURIER series

0L

2'Gmsin(mç')

(23)

m=1

with

₌

+ (l±) (l) ces q'.

(24)

Further the valués of the induction factors i(x/x', ß) and A(X/X', _ß,)

read off in the diagrams are resolved into an even FouniR series

with respect to q:

i =2I(p')cos(nTp)

Eq. (22) is then transformed into

'

mGhp')

VA

lXh

m-i

The coefficient h (and the corresponding coefficient for the

tan-gential component h,) is made up of two terms of the same type as the expression in Eq (8). As was done in that case, LERBS uses the

transformation according to GLAUERT

shown in Eq. (9) and gets

the result

E

smm

l.(ç")=

. I

5inq 92 n=O (ip')cosnq2'+

+cos rnço'

I

(97') sin np' (27)

nm+i J

and an analogous expression for h'(p).

This expression has finite values except at the end points of the

blade, p' = O and p' = 1800. For these points

LERBS deduces

ex-pressions by using l'Hosrxmx's rule. See [14] and [15]. Strscheletrky's Lifting Line Theory

The concept induction factor is also introduced inSTRSCHELETZKY'S

lifting line theory [17], but, in contrast to LERBS, the values of the

induction factors A,j and R(the radial component is considered in

addition) are obtained by direct numerical integration of expressions of the type shown in Equations (10) and (11). The results, only covering three-bladed propellers, are given in diagrams in [18]. Values read off in these diagrams show only small deviations from the corresponding values in [14] and [15].

In addition to the induction factors

_{T, A} and _R obtained by

numerical integration from zero to infinity of equations deduced from the law ofBIOT-SAVART,,SRSCHELETZKY gives diagrams of induction

20

factors ¿1 T, i and 1 R corresponding to small parts ¿1 s of the free

helical lines. These induction factors have been obtained by stepwise

integration of the equations mentioned above. An element As of a

helical vortex corresponds to an angle A 9. See Fig. 3. The induction factors jT, and 4R are given on the basis of the hydrodynamic

pitch angle ß and the ratio x/x' for the values of J O shown in Table 1., making together sixty diagrams.

Table i

The diagrams can for instance be used for calculating the

contrac-tion of the slip-stream and the variacontrac-tion of the pitch of the free

vortex lines with the distance from the blades. One can thereby

con-sider the influence of these factors on the induced velocities at the

blades, an influence neglected when using the methods hitherto

discussed.

Having arrived at an expression of the type shown in Eq. (8)

STBSCHELETZKY proceeds in a way somewhat different from that of GLAUERT and LEEBS. The evaluation of the integral in Eq. (8) is

performed by dividing the blade in radial direction into three parts

and integrating

'e

r

f=f±f+f

0 .0

ç'e

'+e

The first and third integral are evaluated by using SrMrsoN's rule.

For the small region 2e around the secfion considered, where the

expression passes through infinity, the expression

r.

I

' is deduced in [17]. Length of element Position of element /16

O-2r/16 /16/8

t/Sr/4 t/4-3t/8 3z/8/2

/4 2t/2-3/4 ... 7,t/4-2 2r-2.52z . 4.5-5.Ojr 31 52t-631, 631-731 3 731-1031 2e 1 ôi âGL I-.. 2Gii I (28)

cospcos

511192 _{L592 92}

+i

As the vortices are shed downstream their strengths are added. The total strength of the free vortex system at a point z1 is then

z=;

=

-

dx (30)

Analogously, för a propeller blade the circulation distribution can be

replaCed by a bound vortex system in radial direction [rb(r)]8, to which a system of free vortices [I',(0)],. is added. See Fig. 4 b. The free vortices, shed downstream, are of helical shape. For this case

expressions analogous to Equations (29) and (30) are applicable.

7.. Main Characteristics of the Lifting Surface Theories Used

for the Calculations

Lifting Surface Theory in General

When formulating a lifting line theory for a whig or a propeller the

problem is reduced to the calculation of the velocities induced by a system of free vortices at their common starting line. In the case of

a wing the vortices are straight lines while in the case of a propeller

they are of helical shape. The result of the calculations is for the

wing a velocity field that varies only in spanwise direction while for the propeller only velocity variationsin radial direction are considered.

For the blade widths generally used for ship propellers, however,

knowledge of the variations of the induced velocities along the different

blade sections is desirable. To furnish such information different

lifting surface theories have been developed..

For a wing of finite span in ideal flow the lift coefficient and thereby the circulation vary both in chordwise and spanwise direction. As is

shown in Fig. 4 a the continùous circulation distribution can be

replaced by a system of straight vortices [F b(Y)]x of varying intensity,

bound to the blades and directed parallel to the y-axis (spanwise

direction). To fulfil the condition of irrotational flow this system has to be completed by a system of free vortices [I',(x)], shed downstream

as shown in Fig. 4 a. The strength of the free vortex, shed from a

certain point z1, Yi, can be deduced from the expression IOFb

I

+

or i

'1=0

(29)

22

r

Fig. 4 a. Fig. 4 b.

Fig. 4. The vortex systems of a wing and a propeller blade.

Vortex systems of the type shown in Figures 4 a and 4 b induce velocities at the blades which vary both in spanwise (radial) and

ehordwise direction.

The slope of a streamline relative to the inflow velocity V is at a

point z1,

tan c =

uz1 (31)

where t2 =. induced velocity perpendicular to V at the point x1.

Thus, if Un varies along the section, the profile must be given a camber and/or an angle öf attack in addition to that cakulated from 2-dimen-sional profile data to give the desired lift. To make possible the

deter-mination of this "induced" camber or angle of attack the magnitude

of the induced velocity component u2 must be calculated with suffi-cient accuracy for as many points as possible along the blade sections.

When developing a method for solving a problem of the type

outlined, above, the degree of thoroughness desired in the conclusions must be weighed against the amount of calculation work necessary.

This consideration can lead to dissimilar results, as is illustrated by the brief descriptions of the diffèrent lifting surface theories, used

which for a óircular arc meanline can also be related to t/ by

12

= f(2R,f)

23 The Ludwieg and Ginzel Theory

An approximate lifting surface theory which is extensively used

for ship propeller calculations is that of LUDWIEG and GrNZEL [19-23]. The resulting diagrams generally referred to are strictly applicable however only to

three-bladed propellers

the combinations of blade form and circulation distribution used in the calculations underlying the diagrams (only symmetrical blade forms, i.e. blade forms having no skew back, are considered) blade sections having circular arc meanhines, working with shock

free entry

The condition of shock free entry means, for a blade section with

circular arc meanline, that the whole lift is realized by giving the

section a camber le/I' resulting in a flow that is symmetriòal relative to the midpoint of the chord. Assuming the hydrodynarnic pitch angle

ß. to be correctly determined by lifting hiné theory, the boundary condition in Eq. (31) can be fulfilled at the midpoint of the blade

section by giving the profiles an additional camber find. According to LUDWIEG and GIIcZEL the total geometrical camber is expressed

Igeom = fe//+ lind (32)

As the boundary condition is. fulfified only in one point of the section it is tacitly assumed that the form of the mean line is un-changed, i.e. that the geometrical mean line will also be of circular

arc form.

The values of / and k are determined by putting the expression in Eq. (31) into the ordinary expression for the radius of curvature

for a function which gives

i

du

i 11 ds i du

24

which for small values of l/R can be approximated

t

1l

l'8 R

Thus lind i du i

8V da

du

The derivative -a--- is obtained by differentiating an expression of

u established by using the law of BIOT-SAVART. To facilitate the

calculations of u, the bound circulation distribution in chordwise

direction, which for circular arc mean lines is elliptic, is replaced by

an equivalent rectangii1ar distribution extending over an angle

r' = 0.707 r where r is the angle corresponding to the projected chord.

See Fig. 4 b. Accordingly in this case the radial vortices, bound to

the propeller, will have the circulation

r.)

rbr_{= x'(r)}

Then for the strength of the free vortices (see Eq. (29))

d (r

dx\r'

except for the leading and trailing edges of the blade. Remembering

that vortex lines can only begin and end at infinity we get for the

leading edge according to Fig. 4 b

r'

151' 1

d 'r'

r

1__L1

ô9j.

dx+(---)dx±--

dx r r dx

dx=0

r'

[61', 1 61',

_{d f r '}

r

(1

60 - - dx\r'/

r'

dx

By using the law of BIoT-SAvnT, expressions for the induced velocity components due to the bound and free vortices according to Equations

(34) and (35) can be established for different points of the blade

(33)

H

BLADE AREA RATIO - 100%

1.0 Os (14 0.2 BE12' OPTIFIUII CIRCULATION DI CIRCULATION OISTR SIRS. A,,(I-,) O - IO --0.2 0.4 0.6 0.6 I 0 0.4 0.6 0.8

BLADE SECTION X r/R BLADE SECTION 'x nR

Fig. 5. Camber corrections for two different combinations of blade form and

circu-lation distribution according to Gxl.2ZEL [23].

surface. By differentiating the expressions deducéd with regard to the

chord and specializing on the mid-point of the chord the induced

curvature can be obtained from Eq. (33). From Eq. (32) the geometrical camber and thereby the curvature correction factor k can be obtained if the effective camber feil is known from lifting line calculations.

du

In [19] and [23] expressions of are given in which the influence of ail the blades of a three-bladed propeller is considered. In [19-22]

these expressions are used for calculating the k-values for 3-bladed

propellers having a prescribed blade form 'and BETZ' optimum

circu-lation distribution. Values for one of these eases are given in the

diagram in Fig. 5.

The results of a recalculation of k-values for the optimum case,

covering additional blade fbrms and numbers of blades, including a

somewhat different approach to and deduction of the analytical ex-pression, were recently published by Cox [24], who used 'a digital

computer for carrying out the rather laborious calculations.

The use of the k-values for the optimum case for design work, is

complicated by the fact that these k-values tend to zero when x

tends to unity, giving infinite camber at the blade tip. Later on ZEL, however, introduced the circulation distribution

Tb'

Ax(1x2)

(36)

by which the calculations resulted in k-values giving finite camber at the blade tipJ See Fig. &

Because òf the great amount of work involved when calculating'

new k-factors,LUD WLEGand GrNZEL'S original values are often included

26

as camber correction factors even in propeller design methods intended for general use, although they are strictly applicable only to thé cases specified above. See for instance [15] and [25-27]. In the procedures

described, in these references the k-factors are reproduced in ways

differing from the original one and minor modifications of the values made.

The dingram in Figure 5 suggests that neglecting the influence of

the circulation distribution on the k-factors leads to a rather crude approximation while the results in [24] indicate that the influence of the number of blades, the blade shape and the hub diameter is

perhaps of less importance.

Lerbs' Approxinwte Method of Correcting the Pitch for Lifting

Surface Effect

The lifting surface theory of LUDWIEG and Gixz1L is incomplete in so far as the hyd±odynamic pitch angles are calculated by the use of lifting line theory.

An approximate pitch correction in order to achieve a correct

virtual pitch has been evaluated by LERBS [15], [28], starting from an approach used by WEISSrNGR [29] for wings of finite span.

According to WEISSINGER the bound vortex system for a wing is replaced by a straight vortex in spanwise direction through the one-quarter point of the respective sections. Further the boundary

condi-tion that the flow shall be tangential to the profile is fulfilled at the three-quarter point only. The problem is then to find the velocity, normal to the inflow velocity in the threequarter point, induced by

the straight bound vortex with spanwisé variable éirculation and the

straight free vortices starting from the one-quarter point.

'!transferred to the propeller case the method gives the angle o' of zero lift for the bladé sections relative to the inflow velocity V0. See Fig. 1. As the angles x0 and c are known from profile data and lifting line calöulations, the additional pitch angle 4 o due to lifting surface effect can be expressed

= x'(oc+o)

(

When calculating û' LERBS [28] makes some assumptions and simpli-fications.

The calculations of the influence of the free vortices are made only

for the section z 0.7 for which the deduced velocity in the three-quarter point is expressed in relation to the one induced at the

one-quarter point itself, i.e. the one which is obtained by lifting line theory.

Thus, by this method,, no provision is made for correcting the in-. accuracies in the lifting line theory. The ratio UA o.75!UA_0.25is calculated assuming an infinite, number of blades in which case the axial velocity

component can be deduced from the velocity potential for an

axis-symmetrical distribution of sinks over the disk. The tangential velocity component is supposed to be constant and equal to UT_O25 behind the one-quarter point.

The velocities induced by the bound vortex lines are calculated by the law of BIOT-SAVART, assuming the bl'ade sections to be straight lines instead of circular arcs. Rake and skew back are not considered hut the influence of all the blades is calculated.

The resulting equations for calculating the pitch angle corrections

due to influence of the bound and free vortices are given in [25],

[27] and [28].

A more thorough lifting surface theory for the propeller case in

which WEISSINGER'S approach is used as starting point was recently published by ALEF [30].

The Guilloton Theory

A simple lifting surface theory that makes it possible to cónsider

the influence of the blade form and the radial and chordwise

circula-tion distribucircula-tion on the induced curvature and the hydrodynamic

pitch angle of the blade sections is that of GUmL0TON [31].

When using GuuioToN's method, the induced velocities are first

obtained by lifting line calculations. To calculate the additional

induced velocities due to lifting surface 'effect GU1LLOTOic uses the vortex system reproduced in Fig. 6.

The bound vortex surface is replaced by 5 radial vortex lines having

a dividing angle of 20° to which the circulation is concentrated in

proportion to the chordwise load distribution.

To six points along each bound vortex free helical vortices are

added the circulation of which are approximated to

jl = '0.25' r,2 = r6 b r025,. r,3 = r1 b r065,

r,4

_{= "l.4b"ib' r,5 = r185rl4b, r,6}

_r185

The induced velocity component, perpendicular to the inflow

28

Fig. 6. The free vortices of a propeller blade according to GtJILL0TON [31].

free vortex system, relative to the corresponding component calculated by lifting line theory, was determined by GTJ1LLOTON for all the 30 points in Fig. 6 by the use of the law of BI0T-SAvÁI1T. The results are

given in tables in [3Ï] as coefficients r and F for the pitch ratios

0.416, 0.834, 1.042 and 1.25. In these coefficients the influence from

the bound and free vortices from the two other blades of a

three-bladed propeller is also included.

A drawback when using the tables mentioned above for calculating camber corrections for propeller blades is that the dividing angle as

well as the number of vortex lines of the vortex system in Fig. 6 is fixed. For narrowbladed propellers the number of vortices covring

the blades is insufficient near the tip, while for broadbladed propellers

the same is true near the hub.

The SIrscheletzky Theory

Diagrams for calculating the induced velocities from the free and

bound vortices at a great number of points of the blade surface have been published by STRSCHELETZKY [17]. The equations on which the diagrams are based can be found in [18].

The method of calculating the influence of thefree vortices

by STRSCHELETZKY, described in Section 6. The calculations are

per-formed by the use of the induction factors z-1A and 1R

men-tioned in Section 6. By using thi type of induction factor, the

in-fluence can be calculated of a small part ds of a free, helical vortex, on the induced velocity components at the starting point or at a line

through it in radial direction (point S and line OA0 in Fig, 3).

Accordingly the resulting induction factor can be written

i(40) = I z' i (A0)

using the values of ¿li corresponding to the different steps 0n-1

-of Table 1.

By lifting surface calculations, however, the induced velocity

components at points A in Fig. 3 are aimed at. As the angles E) and

co are taken perpendicular to the z-axis, the same values of ¿li are

obtained for the points A and A0, if STESCHELETZKY'S diagrams are

used. The correct induction factor for the point A may be related to

the one for the point A0 by an expression of the type ¡a0

¿lt(zU9, A) = zIa(zl E), A0)

. f1jJ +F2(--

(38) a11

In STRSCHELETZKY'S book [17] equations of this kind for accurate determination of the induction factors at different points of the blade surface are given, the use of which, however, involves a considerable

amount of calculation work. To simplify the calculations to some

extent it is recommended in [17] and [18] tO use as an approximation ¿U(A) ¿U(A0)

and for the resulting induction factor

i(A) I ¿U(A0)' (39)

w=0

In [17] it is emphasized that this approximation is applicable to cases

- when r'

r in the first place hut in [18] it is recommended for use in all the calculations.

Eq. (39) presumes constant intensity of the trailing vortex. At a

point E) of the chord of the blade section z however, the total strength of the free vortices is obtained by adding the strengths of the vortices

30

shed from the points ahead. Abaft the trailing edge of the blade this

summation gives

-

dGL

dx

in non-dimensional form.

If the total strength of the free vortices at a point is expressed

-

dGL

r,(6,

r) = k(9, r) (40)

the induced velocity component at the point A, emanating from the blade section r can be calculated by putting into Eq. (22) the

induc-tion factor

i(r)

= f

k(9,r)Ai(A0)de (41) 8= Wi

or, if the real intensity variation is replaced by a stepwise variation. in line with the values given in Table 1,

i(r) =

kmAI(Ao) W' Wn+1 fkdE with km wn

As regards the distribution along the blade section of the coeffi-cients k, i.e. how the free vortex system, corresponding to different

distributions of the bound vortices, shall be expressed, no proposals are made in STRSCHELETZKY'S book. Only a special case istreated as

an illustration of the use of the diagrams.

For the performance of the calculation of the induced velocities

from the bound vortices diagrams are given in [183. When

performing the calculations the bound vortex system of each blade is replaced by step-like vortex lines, each step containing straighl lines of constant strength in radial direction.

The coefficients necessary for determining the induced velocities UT, UA and aR at a point A(YA, 0A' z') emanating from a vortex step of radius

Xa + X

2

can be read off in diagrams given in [18]. Here Za = outer dimensionless radius of the vortex step

= inner dimensionless radius of the vortex step

YA = 4- = dimensionless axial distance between the point A and

the vortex step

= the angle corresponding to the projected chord between the

point A and the vortex step.

The radii Za, x and z' considered are given in Table 2.

Table 2

The values of the coefficients given in the diagrams also include

the influence of the corresponding vortex step of the two other blades of a three-bladed propeller.

The induced velocities being calculated for a sufficient number of

points A of each section the induced curvature due to the bound vortex system can be determined and added to that due to the free

vortices.

The coefficients given in the diagrams were determined by STRSCHE.-LETZKY by the use of the law of BIOT-SAVABT without introducing

any important simplifications or approximations. This means that the resulting equations were very !engthy and that an appreciable

number of coefficients had to be included in the general expressions for the induced, velocities finally deduced.

The final equation for calculating the dimensionless tangential induced velocity component u at a point A is expressed in [18] as

r r

-

_{21' yAcos9Ads} m Ï

Ut = 2 f(ft)m1'mdm+fCf)pI'pdp_ _j, °

_{+ .'}

f (tg),pI'mdm (42)

m=1O O A) YA _p+1O

'with the blade divided into m vortex steps, the step p corresponding to the section z'.

Za 0.25 0.35 0.45 0.55 0.65 0.75 0.83 0.88 0.92 0.96 1.00

x 0.20 0.25 0.35 0.45 0.5Ç 0.65 0.75 0.83 0.88 0.92 0.96

32

The vortex strength I', the length of the chord , the chord element

d and the axial distance YA are all non-dimensional quantities. See

Analogous expressions are given for the axial and radial components

u1 and .

The coefficients f, Ir, f, / and f for determining the influence

of each of the sections Xm of Table 2 at points A(YA, 91) of each blade

section x' of the table are given in diagrams in [18], making 159

diagrams in all.

Only diagrams applicable to three-bladed propellers are given in It is evident from thé short summary given above that the use of STRSCRELTZXY'S diagrams for propeller calculations requires a

considerable amount of labour, especially as the calculations involve some trial and error work.

8.

Propeller Calculations

Starting Points for the Propeller Calculations

In order to be sure of marked differences in the appearance and

characteristics of the propeller designs resulting from the application of the different design methods, calculations were first performed for a rather extreme loading case (Loading Case 1), characterized by low

advance ratio and a non-optimum circulation distribution. Data for

this case are as follows:

Dimensionless circulation distribution

0L = 0.2073 (X-Xh) (1x2) (43)

Hub radius Zh = 0.167

Design advance number J = 0.45

Preliminary design thrust coefficient K = 0.12

Number of blades z = 3 Blade form, see Fig. 7.

The circulation distribution chosen is of the same type as that of

Eq. (36) the only difference being zero circulation at the hub for the distribution of Eq. (43).

As Eq. (43) was used as starting point for all the methods the

Dimensons n mm

Fig. 7. General appearance of the model propellers in the present investigation (L o a ing Case 1).

quality of the methods in this respect was therefore judged by com

paring the values of KT for the design value of J, obtained at the open water tests, with the values calculated according to Eq. (2).

In addition to the complete calculations according to different

methods performed for the above-mentioned loading case, liftiñg line calculations were performed for the following case (Loading Ca8e 2):

Dimensionless circulation distribution

= 0.09295

(XXh) (1x2)

(44)

Hub radius Xh = 0.167

Design advance number J = 1.02

Preliminary design thrust coefficient KT = 0.12

Number of blades z 3.

The Results of thé Lijiing Line Calculations The hydrodynamic pitch ratios in ideal flow.

P1/D = ixtanß

3

34 0.8 Q o:-s 0.6 o J

j

LU o 0.4 z Q I-a2 = '-J I- o-O

DENOTES CURVE FROM CALCULATIONS ACE. TO òf -METHOD

o o o o o n GUILLOTON'S METHOD

n n n o n ii LERBS' INDUCTION FACTOR METHOD

n n n STRSCHELETZ)(Y'S INDUCTION FACTOR M.

0.1 02 0.3 0.4 05 0.6 0.7 0.8 1.0

BLADE SECTION x=r/R

Fig. 8. Pitch ratio curves in ideal flow, calculated according to different lifting lilie

theories. Loading Case 1.

for the different blade sections x, obtained by calculations according

to the different lifting line theories are shown in the diagrams in

Figures 8 (Loading Case 1) and 9 (Loading Case 2).

Results from the following lifting line calculations are included in the diagrams:

GOLDSTIN'8 x-method with x-values taken from [15]. (The in-fluence of the hub not considered).

GuILL0T0N's method.

LEEBS' induction factor method with induction factors taken from [18].

SThSCR1TTZKY'S induction factor method with induction factors

taken from [18]. Calculations of the influence of the contraction

of the slipstream on the induced velocities for the performance of

which additional diagrams are given in [18] were however not

performed.

From the diagrams in Figures 8 and 9 it can be seen that the pitch

curves resulting from the different theories vary, considerably in

shape, the most extreme one obtained with STRSCHELETZKY'S induc-tion factor method while the calculainduc-tions according to the x-method

e-O .4 L2 0.2 00 :A110p rl. ND. ACT.r1.

1/

/

DENOTES CURVE FR011 n i n n n ii CALCULATIONS n i n .4CC. u n LERBS' TO èC-METWOO n n GUILLDTON'S n n 51RSCHELETZIÇY'3 INDUCTION METI

-- --.

n n n 0.1 0.2 0.3 04 05 0.6 07 0.8 l.a BLADE SECTION x=r/R

Fig. 9. Pitch ratio curves in ideal flow, calculated according to different lifting line

theories. Loading Case 2.

rences between the two different inductión factor methods should especially be noticed. In these calculations the same values of the

induction factors were used while different numerical methods were

applied for calculating the induced velocities (see Section 6). It is also interesting to notice the rather small difference between the

pitch distributions resulting from the 'calculations by LERBS'

induc-tion factor method and the far simpler method of GUILLOTON. A

common feature of these two methods is however that in both cases GLAUERT'S expression Eq. (9), is used for overcoming the

difficul-ty, of infinite values of the induced, velocities at the section x' = x. As regards the accuracy of the calculations it must be mentioned

that in Loading Case 1, owing to the low design advance ratio and the

extreme unkading of the blade tips, very low values of the angle_ß were obtained for the outer blade sections. For such low values of

ß1 the accuracy of the values of the induction factors read off in the

36

to ordinary pitch values. When using GuliLo'roic's method the low

values of ß made some extrapolation necessary when reading off the values of 1c1 and k2 in the diagrams in [13].

The Induced Camber Resulting from the Lifting Surface Calculations Lifting surface calculations were only performed for Loading Case 1,

i.e. for the advance ratio J = 0.45

The effective and geometrical cambers resulting from the

calcula-tions according to the different methods are shown for the blade sections z = 0.3, 0.5, 0.7 and 0.9 in Figure 10 In the figure, results

obtained with the following methods are included:

The method of LUDWIEO.-GINZEL with k-values according to [27].

Of the k-values available in the literature these were judged to

be the most representative for the circulation distribution chosen. The induced and geometrical curvatures were obtained from Eq.

(32), starting from circular arc mean lines for the effective camber. The method of GUILLOTON [31].

The results of calculations using STRSCHELETZKY'S diagrams [18].

As no proposal regarding the appropriate representation of the

free vortex system is given in STRSCRELETZKY'S book, GINzEL's

approximation, Eq. (35), was used for these calculations. The

second term in the equation was however modified in such a way

that the influence of skew back could be calculated. As for the

corresponding l.ifting line calculations the contraction of the slip-stream was not considered in these calculations.

As can be seen from the diagram in Fig. 10 the induced cambers

calculated according to the different lifting surface theories included

in the present comparison differ materially. Also the variations of the camber along the blade in radial direction as calculated by the

different methods are very marked.

Pitch Corrections Resulting from Lifting Surface Effect

When the lifting surface theory of LUDWIEG and GINZEL is used,

it is presumed that the corréct pitch P. can be obtained by using

lifting line theory.

By the use of GUILL0T0N's lifting surface theory the local hydro

dynamic pitch angles of a blade section are obtained by summation of one part calcuiate4 by lifting line theory and one part emanating from lifting surface effect. Results from lifting surface calculations

DENOTES

9JOfl ACCORDINO TO GIJILLOTON'S METhOD

LUDWIEG ANO GINZEL WITh k-VALUES ERO1I [27] OBTAJNEO WITH STRSCWELETZKY'S DIAORAM USINO GINZELS

APPDXIMA110N OR THE FREE VORTEX SYSTEM.

- - __t_. I I I F I r.e. 0.9 0.7 0.5 0.3 0.1 Le. BLADE SECTION AT x-0,9 o I I

--II

tre. 0.9 - - - - 0.5 0.3 0.1 I.e. - BLADE SECTION AT x - 0,3

Fig. 10. Geometrical camber for the blade sections z 0.3, 0.5, 0.7 and 0.9 calcii. lated; according to different lifting suFface theories. L o a d i n g C a s e 1.

io3f BLADE SECtION AT,, -0,7

tre. 0.9 0.7 0.5 0.3 0.1 I.e. BLADE SECTION AT,,-0,5

io3f 30 20 IO O 40 30 20 10 o

-.

-tre. 0.g 0.7 0.5 0.3 0.1 I.e.

38

0.2

T

03 04 05 08-. 07 08

BLADE SECTION x=r/R

Fig. il. Induced pitch corrections from lifting surface effect, calculated according

to different theories. Loading Case 1.

according to this method show that for symmetrical blade forms the

latter part of the m.uced velocities will be zero at the midpoint of

the blade sections. Moreover the induced camber will be symmetrical

relative to this point. For blades with skew back on the other hand

an assymetriàal induced camber is obtained. Thus, if the phóh angle is defined relative to a line connecting the leading and trailing edges, the result of the calculations for this case will be an induced camber as well as a pitch correction. The magnitude of this correction for the actual case is shown in Fig. 11.

When the lifting surface ca1culations are performed by means of

STRSCHELETZKY'Sdiagrams, the total induced pitch angles for different

points of the blade sections are obtained without using any lifting

line theory. As the method is an extension of the lifting line method of STRSCHELETZKYit may be significant to compare the pitch curves,

defined as above, obtained with these two methods. The pitch. correc-tion thus dedi.iced is shown in Fig. 11.

Finally the magnitude of LERBS' approximate pitch correction for

lifting surface effect for the present loading case is shown in Fig. ii:

As this correction is dependent on the pitch angle ß at x = 0.7 two

curves are shown, one applicable to lifting line calculations according

NOTATION LIflINOSURVACE LIflNG LINE GALC. ALC. TO

LEPOS GUILLOTON LEPBS STRSCIIELETZKY o'f -1ETh0D GU1LLOTOÑ

to the -thethod and one applicable to the inductor factor calculations

according to STRSCHELETZKY.

Complete Propeller Calculations for the Design of Model Propellers

For some combinations of lifting line and lifting surface theories

complete propeller calculatiòns were performed and model propellers manufactured according to the resulting designs. The starting points, equations, references and diagrams used in the calculations are given

in Table 3. The design values correspond to Loading Ca8e 1. The

general appearance of the propellers is visualized in Fig. 7. Table 3

Particulars of Calculations for Model Propellers

Common starting points and equations used in the calculatiOns

Design advance ratio

Dimensionless circuiation distribution Number of blades

Blade form, rake, blade section thickness

Mean line of blade sections

Blade thickness distribution of blade

sectioñs

Relation between effective camber_le/Iand lift coefficient, CL

Viscosity correction on pitch angle

Cavitation inception calculated by the use of values in

J = 0.45

0L = 0.2073(xO. 167)

z = 3

Fig. 7

Circúlar arc (NACA

NACA 66 mod. [27]

(lx)

65) le/IR -4z+168/t = 1.824 CL [15] Ref. [32] and [2]

Combinations of Lifting Line and Lifting Surface Theoriés

Propeller No. P867 P868 P870 P950 P977 Lifting line theory. GOLDST. -values from [15] Hub not cons. md. f. acc. to

Sascx-LETZKY. md. fact. [18] GUrLL0TON Same as P868

-Lifting sur-face theory: Camber corr. GrNZEL k-values from [2 i] Same as P867 -No correction -Same as P8i0 STRScHELETzKY's diagrams [18]. GJNzEL's approx. of free vortex system Lifting sur-face theory:: Pitch corr. LERB5 [28] - . No correction

Early in the investigation it was realizèd that the lifting line theory of GuuLOToN or any of the two induction factor methods in combina-tions with the lifting surface theories of GLNZEL or GuntOToN would

result in propellers giving greater thrust at the design advance ratio

than calculated. Because of this the comparison between these lifting line theories was accomplished by testing propeller designs based on lifting line calculations only i.e. desigñs in which no camber correc-tions were included.

It can be mentioned that propeller No. P867 is calculated almost

entirely according to the procedure proposed by ECKHARDT and MOR-GAN Ifl [27], the only deviation being a slight difference in the

viscos-ity correction ¿1 x.

In Fig. 8 the pitch distributions resulting from both hERBS' and

STiSCHELETZXY'S induction factor methods are represented. As the distribution resulting from .the calculations based on LERBS is very similar to that obtained with GuIrLoTöN's method, no propeller was manufactured according to LERBS' method.

LERBS' approximate pitch correction for lifting surface effèct is

applicable to propeller No. P867 as well as P868 As the latter could be seen to be definitely overpitched without correction no correction was applied to this propeller.

9. Results of Open Water Tests Performed with Model

Propellers

According to the different designs shown in Table 3 propeller models of diameter D = 0.2128 m were manufactured with which open water tests were performed at a constant number of revolutions n = 16 r/s, corresponding to a Reynolds number, R075R = 4.8 10 at the design

advance ratio J = 0.45.

The results of the open water tests are given in the thagrm in Fig. 12 and in Table 4.

The calculated values of KT and were obtained from Equations

(2) and (3) by putting the minimum drag coefficient for the blade

sectiòns, 0D = 0.008.

From the results given in Fig. 12 and Table 4 it can be seen that for three of the propellers the agreement between calculated and measured thrust coefficient at the design advance ratio is good. It should be noticed that of these designs the one based on lifting line

0 70 60 50 20 0

IO _T 100 K NOTATION PROPELLZR LITINO LINE :UCTING SURCACE

Ç - NO ThEORY THEORY 6 5 40' 4 30 3 - 2 O 2 - ---rn---P 867 P670 P 868 poso p977 -R - METHOD GUILLOTON STR.1N0,CACT n

-GINZEL+LERBS PITON CORS: -GINZEL

-STOSCHELCTZKY 2o

-:-.

,.-'

Ns _Ii-t \\\ -

k

\

A

.0 I

VADESIGN POINT o' 02 0.3 0.4 q.5 0.6 07 - On

Fig. 12. Results from open water tests with model propellers.

caiculatiöns according to the k-method (P867) includes a camber

correction according to GINZEL as well as a pitch correction for lifting

surface effect calculated according to LERBS while the other two

designs (P870 and P950) are based merely on lifting line calculations.

The design based on induction factor calculations together with

GINZEL'S camber correction (P868) gives as result a propeller having

a thrust at the design advance ratio that is 10 % greater than that

calculated according to Eq. (2). With the propeller designed according to STRSCRELETZKY'S lifting surface theory (P977) a still greater value of the thrust is obtained.

Of the three propellers showing good agreement between calculated and easured Kr-values propellers Nos. P870 and P950 show

42

Table 4

Results of Open Water Tests

able discrepancies in the corresponding comparison as regards effici-ency while propeller No. P867 is more succesful in this respect. The

discrepancy observed for the two former propellers awaits further explanation as it can hardly be explained by wrong estimate of the drag coefficient 0D because this would require CD-values of about 0.003.

An interesting result is the high peak efficiency of the propeller

P870 (GUUL0T0N lifting line theory) and the increase in peak efficiency when for the induction factor design the GLNZEL camber correction is omitted (compare propellers Nos. P868 and. P950).

10. Cavitation Tests Performed with Model Propellers in

Homogeneous Flow Carrying out of the Tests

With the five model propellers designed according to the principles shown in Table 3 cavitation tests were performed in uniform flow in

the SSPA cavitation tunnel [33], using a test section 0.5 mx 0.5 m, the water velocity being 3.5 rn/sec. during the tests.

Propeller No. P867 P868 P870 P950 P977

,e-method lad. f. GUILL. md. 1. L. surf. Notation GtNZEL- (Srs.)- No corr. (STR.) STR. diagr.

LERBS GINZEL No corr.

K1' cale. at design advance

ratio J = 0.45 0.120 0.121 0.120 0.121 0.121 K measured at design advance ratio J = 0.45 0.123 0.134 0.119 0.120 0.150 KT as % +2.5 61.1 59.0 +10.7 54.9 58.1

0.8

55.9 60.2

0.8

54.9 60.5 +24.0 54.2 55.4 me . 1)

cale, at design advance ratio %

meas, at design advance ratio %

2 2 o o 5 0.40 a4 050 V J= IO IO 8 8

4b4

2 2

DENOTES INCEPTION OC BUBBLE CAVITATION SWEET

TIP VORTEX

Fig. 13. Incipient cavitation curves

for model propellers.

0 55 PROPELLER NO. P 670 -0.8 R -0.65 R -O.9p PROPÇLLEP NO. P867 R PROPELLER NO. p 950 - 0.75R PROPELLER NO LJCTING UNE THEORY LIFTING SLIRCACE THEORY P 867 4.1 -METHOD &INZEL.LERBS' P 870 GUILLOTON PITCIl CORP.

P 865 STR.IND.CACT. GINZL P950

-P 977

-

STRSCI4ELETZKY 0.40 045 o. so 0 5s 0 40 0.45 0 50 0 55 040 045 050 0.5 V Io IO 8 6 o o

44

IO

O

05

NOTE. NO TIP VORTEX OBSERVED ON

PROPELLERS ND P 950 ANO NO 977

/

DENOTES INCEPTION 0F BUBBLE CAVIT.

/

SWEET

n /

TIP VORTEX

- THEORETICAL. CURVE 0F INCIPIENT CAVITATION

LOWEST CAVITATION NUMBER REACHED IN THE TESTS

ROUGH THEORETICAL ESTIMATE OF TIP VORTEX INCEPTION

Fig. 14. Comparison between the cavitation properties of the model propellers at the design advance ratio J = 0,45.

The water velocity in the tunnel was measured by means of a pitot tube placed between the propeller and the tunnel wall. The measured

values of speed and static pressure were corrected according to

equations given by LERBS f34] and good agreement was obtained between propeller characteristics from the tunnel at atmospheric

pressure an4 the corresponding results from open water tests.

PROPELLER NO UcTING LINE THEORY LIniNG SURVACE THEORY P867 P 867 .e-METWDO GINJZEL.LERBS'

P 870 GUILLOTON PITCI4 CORP.

P 868 STR.IND.ACT. GINZfl.

P950 ii ii

-P 977. -

-

STRSC$4ELETZKY

LO

06 0.7 08 0.9

Fig. 15. Cavitation picture of propeller No. P867. Loading case: J = 0,45, a = 5.

The air content ratio of the water was kept low during the tests.

The tests included observations of the inception of different kinds

of cavitation at different advance ratios. In addition photographs

were taken of the propellers at interesting loading conditions.

Results of the Tests

The results of the observations on the different propellers are given

in the diagrams, Figures 13 and 14.

In Figures 13 ae incipient cavitation curves a for different kinds of cavitation are plotted with the advance ratio J as basis.

In Fig. 14 the cavitation properties of the propellers are compared at the design advance ratio J = 0.45. The diagram shows the extension

of different kinds of cavitation over the blades.

From the diagram in Fig. 14 it can be seen that on propeller No. P867 (-method) inception of sheet cavitation and tip vortex

cavita-tion took place at rather high cavitacavita-tion numbers compared with the other propellers on which very little or no sheet cavitation was noticed.

Very little bubble cavitation was observed on P867.

For the other propellers the inception curves of bubble cavitation were rather similar, propeller No. P950 (induction factor+no lifting

46

surface corr.) being a little better and No. P977 (STRSCHELETZKY'S

lifting surface) a little worse than the others.

No tip vortex was observed at the design advance ratio on propellers

Nos. P950 and P977. Because the strength of the propellers was in-sufficient, the water velocity could not be increased further and the lowest cavitation number attained at this advance ratio was o = 2.5. During the tests observations of hub vortex inception were also made. As the start of this type of cavitation was very diffuse for all

the propeller models the observations gave very inconsistent results and have not been considered in this report.

From the diagram in Fig. 14 it is evident that propeller P950 is

better than the other propellers included in the present investigation

from the point of view of cavitation.

In Fig. 15 a photograph is reproduced, showing theextent of sheet

cavitation over the blades of propeller No. P867 at the loading case

J = 0.45, o = 5. On the other propellers the predominant type of

cavitation was bubble cavitation which was difficult to illustrate by photographs.

Possibilities of Predicting Inception of Different Kinds of Cavitation If the laws governing the inception of different kinds of propeller cavitation can be formulated in analytical expressions, a comparison between such theoretical values and the corresponding experimental curves of cavitation inception will give information on how the desired

load distribution over the blades has been realized in the different

designs.

In the diagram in Fig. 14 a theoretical curve is given, representing the incipient cavitation number

(45)

for different blade sections

where ¿Ip = the maximum pressure drop in ideal flow at the profile considered, calculated by the use of profile data given in [27] and

[32].

The curve representing Eq. (45) in Fig. 14 can be looked upon as

an approximate curve of bubble cavitation inception. From the

diagram can be seen that in the experiments the inception of bubble