A study on screw propellers

With the Compliments of the Author.

A Study on S&ew Propellers

By

Rynsuke YAMAZAKI

Reprinted from the Memoirs of the Faculty of Engineering, Kyushu University, Vol. XIX, No. i

FEIKUOKA JAPAN

196

0-Lab. y.

Schecpskuv4k

Technische

FogescìooL

By

Ryusuke YAMAZAKI

Assistant Professor of Naval Architecture

(Received Feb. 5, 1959)

Contents

Page

Introduction 2

Chapter I. Fundamental Theory 3

Introduction 3

Fundamental equations 4

Transformation of fundamental equations 6

Special cases 8

Lift-slope correction factors 11

Approximate equations 13

An attempt on a propeller design method 17

Conclusion 18

Appendix I Hydrodynamical fundamentals of screw

propellers 19

Appendix II Zero lift lines of airfoil sections of

pro-peller blades 24

Appendix III Formulas of approximate calculation 25

Appendix 1V Problems of minimum energy loss 28

Appendix V Approximate equations due to the strip

method 30

Appendix VI Approximate integration 32

Chapter II Individual Theory 34

Introduction 34

Contour of blades 35

Area of blades 37

Radial distribution of pitch 39

Number of blades 41

Pitch ratio 43

Boss radius and others 44

Conclusion 47

Chapter III Comparison with Experimental Results 47

Introduction 47

Examples of calculations on the practical screw

propellers 48

An example of screw design 49

Conclusion 50

Conclusion 51

Notation 53

References 54

Introduction

The theories of screw propellers in open

water may be classified broadly into the three,

namely, the momentum theory, the blade

element theory and the vortex theory. Among these theories, the vortex theory [the

circula-tion theory in another name] proposed by

Lanchester and Betz-Prandtl° is most

frequently adopted at present. This theory has developed in two directions. One is the strip theory or "the combined momentum-blade element theory" which was completed by Burrill having been advocated successive-ly by Helmhold,3 Schoenherr" and Lock,<

and the other is the lifting line theory based on the wing theory brought to completion by

Kawada and Kondo7 through the consistent advocation of Goldstein, Moriya and others. In comparison with the former which is a sort of approximate method including various em-pirical and intuitive factors, the latter deduces an almost accurate solution from the ideal

hydrodynamical point of view, restricting the empirical and intuitive factors to the min-imum possible extent. The calculated results by the lifting line theory agree well with the experimental values when applied to the

air-screw having narrow blades, hut for the marine propeller having wide blades, this

theory gives clearly the large values both in thrust and torque compared with the experi-mental results as long as the effect of blade width is not accounted for. This fact is con-sidered to he a serious weak point when the

lifting line theory [accordingly the strip theory] is used to calculate the performance of the propeller. As an attempt to evaluate

this effect of the blade width quantitatively,

coefficients to correct the two-dimensional

characteristics [about the lift slope and the angle of attack] of the blade sections of the

propeller have been introduced. In the pre-war period, applying the idea of the strip theory, the correction factors due to the mutual inter-ference or the cascade airfoils [so-called gap effect or cascade effect] were adopted as these correction factors. As the values of these

correction factors, there are those of Schoen-herr,° BurriIl and van Lammeren," which

were analyzed on the basis of the

experi-mental results done by Gutsche°° and others. And this idea is forcefully maintained even

now by some scholars being modified so

as to agree with the experimental results as much as possible.

On the other hand,

in 1944, i.e., the year before the termination of the 2nd World War, Ginzel and Ludwieg°2

in Germany proposed a sort of lifting

surface theory, according to which, it was

pointed out that when the propeller blades have a certain width [chord length], the flow

near the blades becomes a kind of curved

flow induced by the bound and the trail-ing vortices, and that, speaktrail-ing in the reverse way, the blade section takes a sort of added curvature in the straight flow, and that the magnitude of this curvature changes with the blade width and the radial distributions of the hound vortices. And they concluded that the values of these curvature correction factors consist,

for the most part, of the terms of

the so-called self-interference due to the bound vortex of the blade in consideration and that

the magnitudes of the terms due to the

mutual interference by the other blades are very small, for example the order of magni-tude of 5-40% for the three bladed

propel-lers. The flow around the cascade airfoils is,

of course, a sort of curved

flow, hut it is

plain that this is absolutely different in quali-ty from the means of the curved flow of

Ginzel-Lud wieg. Furthermore, the curved flow ef-fect was introduced qualitatively by Gui1loton" in 1949 independently from Ginzel-Ludwieg,

and then Hill"' gave the semiempirical and quantitative estimation for this effect. Next, Lerbs,"' who published many reports on Ginzel-Ludwieg's theory and the lifting line theory in the years later than 1950, extended the pro-peller design method for the case of shock entrance by using Ginzel-Ludwieg's curvature correction factors that had been restricted to the case of shock-free entrance together with Weissinger's correction factors arising from lifting surface effect. Eckhardt and Morgan"

showed an example of the propeller

de-sign method based on this Lerbs' theory.

On the other hand, van Lammeren and van

Manen" have proposed a design method

based on the strip theory in which the three

factors of Ginzel-Ludwieg curvature correction,

Goldstein reduction and the friction-correction are taken into consideration.

The author has so far given a general out-line on the latest development of the screw propeller theories in open water. Excepting the lifting line theory and Ginzel-Ludwieg's theory, which will be discussed later, most of theories are considered for him to be collec-tions of what are admitted as intuitive and em-pirical facts and to lack exact theoretical unity. In the theory of Ginzel-Ludwieg, assuming the blade sections as circular arcs, the chordwise dis-tributions of the bound vortex sheets at shock-free entry, whose shapes are approximated to he camber forms and therefore are nearly

circular (or elliptic), are replaced with the equivalent rectangular distributions, and the curvatures of flow at the mid-points of the chords produced by both the bound vortices and the free vortices of whole system are calculated, and then the lift slopes of the blade sections are evaluated by supposing that there are the two dimensional airfoils in the flow having that curvature. But since this

procedure deals only with the flow in the

case of shock-free entrance at each section of the propeller blades and is based on the as-sumption that the radial distributions of the

bound vortices are already known [for instance, the optimum distribution], it is unreasonable to calculate the performance characteristics of an ordinary propeller at various conditions by using their theory. The lifting line theory is also defective where no introduction of the

effect of blade width is made possible, though

it is a theory nearly complete from a hydro-dynamical point of view. The author, there-fore, will attempt to develop a theory in order to evaluate the performance characteristics of screw propellers in open water by using the lifting surface theory which is adopted in the wing theory. This theory is based on a few

hydrodynamical assumptions, where the effects of blade width and others are naturally introduc-ed. Namely, the performance characteristics of propellers can be evaluated theoretically by a method entirely different from the strip theory. The author will also try to define quantitative-ly the curved flow effect that leads the lift-ing line theory to correspond to the lifting

surface theory, and evaluate its values as

well on various cases.

Chapter I Fundamental Theory

§1. Introduction

An axi-symmetric screw propeller which is running steadily under normal conditions in a region of unlimited and undisturbed water will be taken up for a consideration in this

paper. The performance characteristics of propellers having narrow blades as airscrews will be calculated without much error by the

lifting line theory introduced by Kawada and Kondo7' in which each blade is replaced with one vortex line respectively, but when the blade width [chord length] is wide as marine

propellers, it is unjustifiable to replace each blade with one vortex line respectively.

Ac-cording to the strip theory followed so far,

the two-dimensional characteristics of the

blade profiles of the propeller have been as-sumed by cutting each blade with a cylinder co-axial with the propeller and by taking the characteristics of one airfoil of the cascade -obtained by developing their sections on a

plane. And they have been adopted in the

process of calculation of the performance charac-teristics of the propeller operating in the

three dimensional space. To cut the blades with concentric cylinders means to make an assumption that there is no mutual influence between the individual blade sections on the surfaces of different concentric cylinders, even

if the mutual interference of blade sections on the sanie cylindrical surface is taken into

account. The state of the actual flow on the cylindrical surface co-axial with the propeller, however, is complicated because of the

ex-istence of the induced velocity. Hence in

order to transform the mutual interference of the blades into the two dimensional charac-teristics of the profiles, more complicated

factors seem to be introduced physically. Ac-cordingly, in this paper, in order to discuss

the mutual interference of the blades as ac-curately as possible, it is necessary for the propeller to replace the blades with the vortex

sheets or those which correspond to them if the blades are thin.

On the other hand, consideration will be

given to the case of a single wing. When the chord length is supposed to be comparatively wider than the span length of the wing, i.e., its

aspect ratio is small, the values of the lift coefficients calculated by means of the lifting line theory in which the wing is replaced by

one vortex line are larger than the values

obtained by the lifting surface theory

sup-posing the wing as a vortex

surface. And

with the decrease of the chord length [or the in-crease of the span length], the results obtained by the lifting surface theory become to agree with those obtained by the lifting line theory. When the light load condition is assumed, i.e.,

the angle of attack is assumed to be small,

the Jilt coefficients evaluated by the lifting surface theory fairly agree with the

experi-mental results."°>° Such a qualitative

tenden-cy of the single wing as this example seems to be suggestive of its applicability to the screw propeller.

In the present chapter, the author will

evaluate the fundamental equations to calcu-late the performance characteristics of screw propellers applying the lifting surface theory and define further a coefficient that renders

the lifting surface theory correspond to the lifting line theory. This coefficient is one of the curvature correction factors.

§2. Fundamental equations

Assuming that a screw propeller of axial

dço

jo

Va2(+e)2+ r"2+ r'2-2r"r' cos(++ 2kJN)

]=,

e=0

symmetry with N for number of blades, r, for propeller radius and rB for radius of boss is advancing alone at a given constant linear velocity V, while revolving at a given con-stant angular velocity Q through open water

of density p. The cylindrical coordinates whose axis coincides with the axis of rotation will he taken for consideration in this paper. The positive directions of x and O are the

opposite with those of V and O respectively. r', r", (i', O' and etc. represent the values at

points on the blade surfaces. The geometrical

expression of the thin blade of the propeller is given as follows:

x=x0(r',O')...(2.1)

where O, (r')<O'O3(r'), rBr<r(T.

And the strength of the vortex sheet expres-sing the blade in the radial direction is denot-ed by T(r', O'). Replacing the blades with

helical plates of constant pitch 2ira and con-sidering them as bound vortex sheets, vortices on such a section of the blade surface as constant and O"=variable, will be assumed to leave from the point (r", O') on this section all together. The axial and circumferential

components of the induced velocìty at a

point (x', r', O') on the blade surface are de-noted by w,', w9' respectively. Then, using the clasical potential theory in this case, we have [cf. Appendix I]

(2.2) Va(çr + c,J + r"2 + r'2 2r" r' cos (ço + £g+ 2irk/N) j._{=0, =0,}

riJo

'ro 1(1-(r")

[1

6

r" O

h O

= _dr" r(r", 8")do" r,, Oi(r")

[r'

4ir N-1 r co > dço 1

ô (r" O

a O \ jO /h2(W+,)2+ r" + r'2 2r"r' cos ( ++ 2k/N) r' 6Z9

- r"

VX (2.3)

4 Ryusuke YAMAZART (Vol. XIX.

47r rr0 dr"\ r(r", O")dO"

_{r" a0)}

Oi(rT')

[

1 61r"O

h 6\

xç

dço

_1

(r"ô

a 6

a Oea, \a Oe

r"

N-1 ro"-e' dço

Here the bound and the trailing vortex surfaces are considered to have approximately constant pitch without contraction. Though

the boss radius is considered to be finite, the condition that the velocity component normal to the boss is zero is neglected here for

simpli-city.

Boundary condition will be evaluated next. Denoting the radial component of the induced velocity as Wr', there will hold on the blade surface / , ôx5(r', O') (Qr +lV) , ,

(Vi-w,.)

- ao ± 0 , (2.4) Or C° O2V) 'AYo

j

(ir" r(r", O")SB(r", O" ; r', 0' ; a)dO"= ôx0(r' o')

y

r(r")S7(r", r' ; h)dr",

r _{Yj 0fr")} r'OO' r _YB

where

CO2fr')

r(r')=

r(r',O')dO',

(ìj(r')

the term due to the vortex sheets on the blades

i fr' O

a

O \fr" ô

a O \

SB(r"o";r',o';a)=-4(--0--Ï0_)(\_-0_»0)

dço

where 0x0(r', O')/Or' is the value representing the magnitude of the rake of the propeller, hut assuming wy', w,' and w' as the values of the same order, the 3rd term of the left side

of (2.4) is usually negligible being small as

the second order. This signifies that the rake

has little effect upon the characteristics in

open water. This is confirmed also by the ex-priments. (2.4) will be therefore as follows:

(Qr' + w9/)OX0' '?' - (V+ w,.') =0---(2.4') r 00

Substituting (2.2) and (2.3) into (2.4') and

neglecting small quantity of higher order, the following basic equation is obtained:

V a2( + )2 i- r"2 + r'4 2r"r' cos (

_{+ 0+k7N)}

_=0,

the term due to the trailing vartices

1fr' ô

h

O \Ir" O

h 0

Sr(r , r ;h)

,,,

-

a,A7 ai,.

o,

XV'1 dçi'

Jo Vh2(ç+s)2l r"2+ r'2 2r"r' C (-l'so--l-2irk/N) ],=0,E0=O.

Here hold obviously

(2.5)

(2.7)

SB(r", 0" ; r', O' ; a) = SE(r', 0" ; r', 0' ; a) = - SE(r", 0'; r', O" ; a), S,-(r", r' ; h) = S7(r', r" ; h). Since (2.5) is linear with respect to r(r', 0'), using a function a(r') which

near Ox,(r', 0')/aO' independent of O', the ist and 2nd terms of the right side divided into the two parts as follows:

Qr' 0x0(r',O')

-

V=!(OX0 0')

-

_{a(r'))+ (Qa(r') V),}

and (2.5) will be replaced with two integral equations. That is,

Çro Ç02(r") Çro

dr"\ r,(r", 0")SB(r", 0"; r', 0'; a)dO"=.Qa(r')- V- \ F,(r")S(r", r' ; h)dr",

r J0i(r") ' Jr,,

(2.6)

takes the value of (2.5) will he

(2.5')

X

-i

il

('r

__,_ (Lv" rfr, O")Sß(r", O" ; r', O' a)dO"

TB 01(r") =Q(

ax)(r', O')

_-

_a(r'))

80' V)(r")SO.(r", r' ; h)dr", (2.5") Ç02(r')

where r,(r')

_{= \} r0(r', O')dO', JOi(r') (02(r') ['1(r') = r(r', O')dO', JOi(r') r(r',ô')=rfl(r',O')-r(r', O')

Considering that the induced velocities w,' and w0' are so small as negligible compared

with V and Qr' for the light load condition

and ultimately the total thrust T, the total

torque Q and the

efficiency will be

ex-pressed as

T=To+TD, Q=Q,+QD, -,,=(VT)/QQ).

Evaluating the value of r(r', O') by solving

the integral equation (2.5), T, Q and ' will

be calculated by (2.8), (2.9) and (2.10). The

next problem is to solve (2.5), for which it is necessary to transform SB and ST of (2.6) and (2.7) into such forms as being calculable numeri-cally. Furthermore, when Reynolds number is sufficiently large and the blade has a moderate

angle of attack, it is necessary that the flow of fluid at the trailing edge of the blade is of finite velocity in continuation with its sur-roundings without being rolled up. In order to satisfy this condition, it is sufficient to put

fi

y) = - F(e')S(e", C' ; ('r0 f 02(r') ôx,(r', 0') TD= -

pN\ dr'\

c{i +(

_r'80' )2}(2r'+ w,')(V+ w,')r'dO', Jr,, JO,(r') ro ('f)o(r')

PN\ dr'\

cali

(ôxo(r'. O)\2)

.,r,, j01(r') r't90'

)

j(.Qr'+ w9')2r'2d0',

(2.10)

analogous to the single wing, and putting

a(r')a=const., (2.5')

is chiefly related to the

lift slope of the blade section while (2.5") to the angle between the zero lift

line of the

blade profile and the line corresponding to

the pitch ratio 7ra(r')/r).

Now considering the fluid as nonviscous and incompressible, the thrust T, and the torque Q, of the propeller will be as follows:

('ro (02(r') T1=pN\ dr' \ r(r', O')(S2r' + w,')dO', JrB JOj(r') (fo (02(r') (2.8) Q0=pN\ dr'\ r(r', O')(V+w,,')r'dO'. J,',, JO,(r')

Denoting by C,, the local drag coefficient of the blade section, the thrust T,, and the torque Q,, by drag will be

(2.9)

r(r', 0') to tend to zero toward the trailing edge.

§3. Transformation of fundamental

equations

In the first place, non-dimensional sign will be defined as follows:

E' = r'/r0, =

rt/r,,

E,, = rB/rB,

v,= V/(Qr,), v=a/ro, vh/r,.

And rewriting ['(r') and r(r', O') as ['(C') and

r(E', O'), and O(r'), O (r') and x,(r', O') as O(e'), O(C') and x,(E', O'), and also S,, and S,- as

S,,(r", O" ; r', O' ; a)

= (1/r0)S,,fC", 0" ; ¿', O' ; (3.2) Sr(r", r' ; h) = (1/r,)S(E", C' ; y),

respectively, and putting

i(e')=

r(E',O')dO', ...(33)

(3.4)

6 Ryusuke YAMAZAKI (Vol. XIX.

Cl û2(')

AN (C', O'; y) de" r(C", O")Sß(C", O" ; C', O' ;

CB

J.

(2.5) will be written as follows: In solving (3.5), it is necessary to evaluate

first such forms that enable to calculate

A(E', O' ;ii)= 2r112( xo(c',O')_,) SR(E", O"; E'. O' ; y) andST(E", '; y) numerically.

ra o ,,

Calculating S7( , E ; y) from (2.i) and (3.2),

(3.5) the result will be7

S,.(E",E';v1)= _2irv

N (f'+ vi)(Eí'+±)

i'1 E E

m'N'K,,(mN"/1)I,,N(mNE'/v) cos mN i for E">E', m'N21,,N(mNE"/vl)K,,A-(mNE'/v) cos rnN,i ] for E"<E',

c=O, where K,,(mNE"/v1), i,,N(mNE Iv) and etc. are

Bessel functions of imaginary argument.

De-veloping them asymptotically, we have as

follows

I !

,E vi

-

vi' (pepf/)3/2 5FNIIN

4rv e' E"

(s" -

s/FN)2

where P' = i/i + (E'/v1)', P" = 3/i + (E"/J',

(3.6)

s'= C'/V e"

,,E"/v

e"

dS'N

Npsv

i+p'

i+p"

' dE' e'

Then S(E", O" ; E', O' ; y) will be as follows

when calculated from (2.6) and (3.2)

_.,, ,, , I

.S,k 0 ;Ç,O ,v)

±YM,(E" E'O"()'v)

(37)

4n- kl)

where Mk(E", E'; O"O' ; y)

0"--

\

0(3v'{E"oE'sin(ço+2irk!N)}{E'çE" sin (c+kLN)} 3/v'ç' + E"+ E"- 2E"E' cos (ço + 2irk/N)"

E"E' + y' cos(ç + 27rk/N)

- Vv'' + E" + E"- 2E"E' cos ( + 2k/N) . )

- OH-v' sin (O" 0' ± 2rrk/N) y' sin(2irk/N)

= Vv'(o"o')'--E

'+E'2--2Ç"e' cos (O"O' + 2irk/N)' _{Ve"±E"-2e"E'cos(2vk/N)'}

+ 3(E" + y') (e" f y2) ço sin (ço + 2irk/N)

4

V + E"2 + e" 2E"E' cos ( + 2rk/N)'

2' I

= y O O')cos(O"O'+ 2irk/N) +Ç" sin(O"6' + 2rrk/N) E" sin(2rk/NJ - O')' + E'' + E" - 2E"E' cos(O - O' + 2k/N)' + _{V '+ E" 2E"E' cos(2k/N)'}

..,, ,. a

tO

{e" E' cos(çc' + 2irk/N) } + y' {çoE" Sifl(ço + 2irk/N) + E" cos(ço + 2th/N)

- '

dço o 1/v'c+?"- E"2E"E' cos (ç' + 2rk/N)'

= a [{p'(O"O')cos (O" O' ± 2rk/N) +E"sin (O"O'+2rk/N) } {E"E'cos (O"O'+ 2irk/N)} OE"{ {v'(O"- 0')' + E"sin'(O" - O' + 2irk/N)}

/(io)2+ E"2+

E" - 2E"e' cos (o" O' + 2'th/N)

E"E' cos(2irk/N)

sin (2rk/N) 1/E" + E" - 2E"E' cos (2rk/N)

Ç

-- ' _'

+ E {e E'cos(ço 2irk/N)} +v'fçoE"sin(+2irk/N)

I EIcos(co+27rk/N)_El}d1 (3.8) - ì 'i" + E" + E" 2E"E' cos (o + 2th/N)

provided that the 2nd term in the bracket of the above 4th expression is taken as zero for such k as sin(2irk/N)=0. And it is obvious

that there hold the following expressions: 14-(", C' ; O" - O' ; y) = A4k(C, C" ; O" - O' ;

= MN(E", e'; O'O";

Now when the value of (O"O') is not very large, neglecting the terms of higher order by expanding sine and cosine in power series, the expression of (3.8) will be as follows:

Mk(C",C'; O"O'; y)

»+ (O"O') +Ì' (O" O')9+È(O"O')' i/d+2c(6"O') H- b(O"O')

where

d =e"9+E'2-2C"C' cos (2irk/N), c =C"C'sin (2irk/N), b cos (27th/N), H=[d/(c2bd)2j[2cd cos (2,rk/N) (c2+bdd2) sin (27tk/N)J, G = (3c/d)H, F = (3/2d) sin (2rk/N) + [3(c + bd)/2&]H, + 4 7t li C" f1Lj 3 ç y 2(E" + e')2 '

where ' represents that for N= even

num-ber, the value at k=N/2 is

omitted and the

term after the broken line is added, and that

for N= odd

number, the term after the

broken line is omitted. Next step is to

evaluate r(C", O") by solving (3.4) and (3.5),

but its approximate solution will be treated in §6, the relation y0, y and t being unknown yet.

§4. Special cases*)

(i)

Case of very short radiai length of the blade compared with chord length

E = (l/d) cos (2irk/N) - (c/2d2) sin (2irk/N)

+ [c(3bd - c2)/2d3]H,

H y2 sin (2irk/N) + (e"2 + y2) (e'9 + 2)H,

= e"e' + v cos (2irk/N) + (e"2 + y2) (e'9 ± 2)G, - (v9/2) sin (2irk/N) + (C"2+ y2) (e'2± v2)F,

E

-

(v/6) cos (22tk/N) + (Ç"2 + v) (Ç'2 ± v2)E.

Here Mk(Ç", e', O"O'; y) varies as if the points

satisfying the equation c2bd=O on (Ç",Ç')-plane were singular points, but S(Ç", O" ; Ç', O'; y) has finite values in general excepting the points of Ç"=Ç', because the two terms of the right side of (3.9) cancel mutually and Mk(Ç", C'; O"O'; ) has finite values. Further, there hold the following relations:

IV-1

Ñ//3=O, [Hj50=0, [i ,=-[F']N_k,

k0

{Ö]k=[]N_k, [F]k= - [F]_, [Ejk= [E]N_k, where it is provided that [Ï1]k, for example, represents the value of H corresponding to

blade number k of MA(C",C'; (i"-O' ;ii).

Assuming that O"O' is small and calculat-ing from the 4th expression of (3.8) neglect-ing the term independent of e", (3.7) will be

as follows

Sß(C', O" ; e', O'; y)

i

OÍ/.t?+C"C') (o'o')2+(C"Ç')3

(o" - O') (e" - e') / (v2+ C"') (O" O)2 ±(" e')2

(

{'cos(2irk/N) -1e'} + (y3Ç'2 {Ç" cos (2irk/N) Ç'}+vC' sin2(27rk/N)l0l)

L

+ Ç'2 2Ç"Ç' cos (27th/N)3 C'2 sin (2ith/N) 1/e"2 + e'2 2Ç"E' cos(2irk/N) _J

(3.10)

Consideration will be given to such a case

as

e'C"<C' of

the blade whose aspect ratio is nearly zero, i.e., 8 is nearly 1. By (3.7) and the 2nd expression of (3.8), we have

as follows :**)

'O Strictly speaking, both (i) and (ii) should be considered about sector bladed propellers whose 02

and O are constant analogous to the single wing theory.

*) The condition to reduce the value of Vv2p2+ e"2+ Ç'2 - 2Ç"Ç'cos(ça +23rk/JV)

to zero is no other than the case of kO, çr=O. That is, except the blade in consideration, none gives

much effect to the value of SB(C", O"; Ç', O' ; y)

-coo

SB(E", (J"; ', O'; = - (E"3+ y2) (e'2-1- ?)

-o

+ (/2 + 2) (e'2 p2)1 Sifl pdço +

3 /?o2 E"2+ Ç'a 2Ç"Ç' cos

and, assuming the value of whose absolute value is appropriately small as

or O"O'<ço<O, the 2nd and higher terms of the above expression take certain finite values, but the ist term is expected to take nearly an infinite value. As the absolute value of ÇVo is appropriately small, we have

1 42o WdW o V(v + E"E')W° + (e"e' O Wo3

3(E"E')°i/(v2+ E"ç')ço0 + (e" E')° '

and, as C'E' is so small as negligible compared with ', we get approximately 1v'E1°+v3 £!

Sß(E", O" ; E', O' u)

')2 IWoI

Since the sign of ço, is related only to that of O"O', we obtain as follows: i 1/e'°-i-2

o"O'

SB(C", O" ; e', O';

(e" eT I o"

o' i

Further, the following expression is deduced from (3.6) Ç°° W sin çodço o

EE'

2E"E'cosç

s '

i 1/e'+

' ,uj)

-("?)

Rewriting (3.3), (3.4) and (3.5) by using (4.1) and (4.2), we have

V C'2+v ç2(Eh/)

o"o'

do"

-- dE' r(E", O")

4irE' _B I o" o' (e" E')2

(4.1)

(4.2)

(4.3)

2(âx0(E', O') _+VT+VLÇ'

f(e')

d

--\

ro'

°'°) 47rE' ₎

(e"e')2

C

Therefore, assuming here u'--v as lightly loaded condition, we have i

'

dE" Ç O")l'l

'd0"

(ax0(e',o')_

"

''

_iO:(E) "

IO"O'I )

,/EI-p2 \,

r,O'

0

In the left side of this equation there holds

O for o'<o1(e"),

,, , ÇO'

\ r(",

o")(i__

)cio"= 2 r(C", o")do" for _{01(e")<o'<02(e"),}

J(11(e")

2r(E") for 02(e")<O'.

The equation (4.3) is to be satisfied at every point of the blade surface.

Next, 01(e') and 02(e') will be approximately assumed as c-onstants independent of C'. Then (4.3) ought to hold even when taking 02(f") as O'.

Since 1'i and EB<E'<l, it

will be assumed that the right side of (4.3) is approximately independent of '. Considering thus, (4.3) will be written as follows:

IO Ryusuke YAAzAic (Vol. XIX.

i

'

f(e") _de" .Qr02Ç' (ôxo(C'O')

_)1

₍₄₄₎

= e' + 2

rO'

_]'=O(E"),C'=(1 ± Ca)12.

Denoting the right side of (4.4) as a, and putting approximately as

F) we obtain from (4.4) or f 0=2a, therefore, r(e")=2a,1/ (1-- e"7. Putting next

iE

the direction rendering a to zero is that of the zero lift line. That is, we have

âXO(Ç, O')l (4.6)

°

rßO JO' = 02(E'), ¿' = (1 + EB)/2,

Both sides of this

Then putting

r(E")=r'ij/( 1_Ea)2 (e" 1±EB)2 =r,,

("LB) (1

-(4.5)

*) Strictly speaking, it exists on the assumption of 82(fl>O">Oj(E').

and the tangent direction drawn at the trail-ing edge O'=02(e') at e'=(1+E)/2 ultimately becomes the zero lift line. This is different

from the direction of the zero lift line of the

two-dimensional profile.

(ii) Case of very long radial length of the blade compared with chord length

A case where the aspect ratio

is almost

infinite, i.e., where O"O' is nearly zero, will

be taken up for consideration. Neglecting

small terms of higher order about (O"o'),

we get from (3.10)

i

E"e'

i

SB(C",o" E', O';

_{e"Iee'I o"O'}

Hence, substituting (4.7) and (3.4) into (3.5), the following equation is obtained:

ri f02(E") i ô E"

-4Je"!E,'E'I0

r(E", dû" O - O

= sr (

ôx0(E', o')

-

J, Ç _{F(e")S.1 (e", e' ;} rôO

equation will he multiplied by integral operator

2 , ,

(e'))(e))51

/02(E')'O'

(4.7) (4.8) .(4.9) 02(E') 2 O(E) /o'-01(Ç')

- o(e')) , '&o' V 02(e')O' dO'=-r,tan ag, (4.10) where ag is the angle formed by the zero lift line

of the blade section and the plane of

rotation [cf. Appendix II], defining for simplicity

tan (lg'C)(E')f E', (4.10')

and taking consideration of the relations

02(fl 2 _dO'=i,

-

_0i(E') Ok') - o' 2 O2(E') 1

'o'o(e')

dO'=

2

OO

V' ;(e')è7

02(e') -we have

ci

i

' p')

a" "E'

a

"Ç'

de" = Qr02(®(E') V5)

fi

-

r(E")ST(E", E'; v1)dE".

2'(02(E')-01(E')) E

Hence

î(E")S(e", E';

S1

irE' (02 (E') - T(e')) = Si r,2 (( (E') - y,) _Ei

This is the fundamental equation of the

lift-ing line theory, by which the performance

characteristics of the screw propeller can be evaluated. [cf. Appendix III]

It is known from (i) and (ii), similarly as in the case of the single wing, that the zero lift line of the blade section varies with the aspect ratio of the blade. As it is justifiable, however, to adopt the idea of (ii) for the single wing when the aspect ratio is not very small,*) the procedure based on (ii) is applied in this paper which deals with the usual type

propeller.

i5. Lift-slope correction factors

In the case of the two-dimensional airfoil, i.e., when considering the section lift coeficient, the lift slope is almost 2 ir in radian in the theory and in the experiments of N.A.C.A.

Wing Sections)** and others. [When the

airfoil is thick, it gets larger theoretically.] Contrasting with this case, when the wing of finite span is in the three-dimensional flow, it receives the same effect as in a sort of curved

flow due to both the bound vortex that

is

replaced the wing itself with and the trailing

*) According to Scholz,(20) the zero lift angle a6

of the single wing whose contour is rectangular and whose skeleton is a circular arc shape (the tan-gential angle at the trailing edge is denoted by ¡3) varies with the aspect ratio .1 as in Fig. I

(4.11)

vortex produced from it. Usually the wing lies in a curved flow where its camber tends to decrease, that is, it has the "curvature effect ". And the decreased camber of the

wing due to the curved flow is proportional to its lift. Therefore, the lift slope of the three-dimensional wing becomes 2irk in radian [generally k-11] taking curvature effect into

consideration. The correction factor of the

lift slope of the wing, i.e., the curvature correction factor k1 should be introduced when the idea of the two-dimensional airfoil is ap-plied extending to the characteristics of the three-dimensional wing. And the value of k,

at each section of the wing is expected to be dif-ferent. Then it is assumed the effect of the chord

length upon the characteristics of the three-dimensional wing is expressed with only the correction of the lift slope, and not with the variation of the zero lift line being related only to the skeleton of the wing. In other words. the zero lift angle will be considered to be approximately related only to the shape of the wing section.

The idea of the above mentioned "curva-ture correction factor" about the single wing

will be applied to the blade section of the

propeller. [cf. 4] Now it is plain from

(4.11) that the lift slope of the blade section

is equal to 2ir

or k,"l when

the blade width is very narrow. When the chord is broad, however, k(E') ought to be introduced, which denotes the value of the curvature correction factor k,

at the

blade section E'.

Then (4.11) may be rewritten as follows: = sir,(((E') - e,)

-

Ç _{r(e')S(E", E' ; v)dE"...(5.1)}

This is the generalized basic equation of the propeller due to the lifting line theory. The value of k,(E') of this equation will be

evaluated quantitatively in the next place.

1.0 0.1'

4 s&t

04 02 o ₂

41t. ?

_io Fig. i. Zero-lift angle of rectangular wing **) Reynolds number is the order of 3-9x106,

12 Ryusuke YAMAZAKJ _{(Vol. XIX.}

Si

At.(E'; y) = Qr02((E') - i F(E")ST (e",e' ; where S 02(E') -2

2,(ç',O';v)j/

dû' 02(e') û' S

i502(E')

= dE" r(E", û")dO" 2

/i'-01(e')

Cu f1(C') e'(o2(e') - O (e')), (C')

V (e') o' SR(E, O";e',O' ;

and the value

of ®(e') at the appropriate

radius ' will be taken approximately as the

value of y. Since the right side of (5.2) is

equal to that of (5.1), putting the left sides as equal, we obtain

k1(e ) -

'-

,,

/ ,, ['(E') , , . . )

r ,v

(5.1) is the equation generally based on the

lifting line theory, and k(E') in (5.1) may be

said to be the correction factor due

to the

lifting surface theory and can be evaluated

from (5.4). As plain from this, k(E') is

re-lated to the number of blades, the radius of the boss, the radial distribution of the chord length, the pitch distribution and others, and takes further the various values for the

variation of y, respectively as ['(E') changes with y. Hence the value of k(E') is not to be

determined uniquely. Here the fluid in the pre-sent consideration is perfect, and the decrease of k(E') due to the viscosity of the fluid and

r(E"o") 0 0

v '+

S1

d'

502W) ,,

AN(Ç', 0'; p)-

-4irÇ'

(e" ¿)2

o"o'

dû

CB OW) Ç 02(C") -1 S' dE" F

r'\_o

n""

I 47rE' CB(e"

Therefore, we obtain from (5.4)

41/(E'EB)(lE')

7r1/E'2+v'(02(E') 01(e'))

Assuming here that the contour of the blade is elliptic and the maximum chord length is

denoted by r)c, we can put by primary geometry as follows:

(5.6)

(5.2)

(5.3)

the roughness of the blade surface is

neglect-ed. According to the experimental results

about the airfoils conducted by N.A.C.A.,<21 as mentioned in the beginning of this section, for the variation of the viscosity of the fluid and the roughness of the blade surface [artificially made], the lift coefficient hardly changes, but

the drag coefficient increases or decreases

considerably and the position of stalling also changes tolerably.

Now, when the skeleton of the blade

sec-tion is straight, that is, _{ax,(E', O')/O'=a(E'),} the value of A(E', (i';11) is independent of O' since the comparison between (3.5) and (5.2) proves that their right sides are equal and

independent of O'. That is, there holds

v)=A.(E', O'; ). (5.5)

For instance, it is obvious that k1(C')1 in case of §4 (ii), and k1(E')

of §4 (i)

will be evaluated next. Using (3.4) and (4.5), the

value of A(E', O', y) will be as follows:

(03(e') - 01(ç'))VEf2 + y2

=(2c(,/(1EB))V(E'Eß) (1E').

Putting the aspect ratio of the one blade as

i, 1/r, times of the area

of the blade will

become equal to (1EB)2/A=2rc,(1EB)/4, and so (5.6) will be

= 2(1

-Now when the chord length, accordingly 02(e')o(e') is not very large, the following equa-tion will be obtained by multiplying the integral operator (4.9) to both sides of (3.5) and by taking (3.4), (4.10) and (4.10') into consideration:

This relation may be also said

to be the

mean value of (5.6).*)

As plain from (5.6), the value of k[(Ç') sometimes increases or decreases from the boss toward the tip according to the form of the

blade outline. This tendency is expected to be

valid even if extended to the case in which the aspect ratio takes a certain finite value.

If the approximate values of k(E') about

various propellers have been evaluated in

sorne way previously, the results calculated by the lifting line theory are nearly identical to those obtained by the lifting surface theory.

§6. Approximate equations

Evaluating r(E', O') by solving (3.4) and (3.5), calculating w,,', w0' by (2.1) and (2.2) and substituting them in (2.8), (2.9) and (2.10), the performance characteristics of the propeller

will be obtained. It is, however, difficult to calculate with the above procedure, and Sc), replacing

the blade with some number of

vortex lines, the strength of the vortex lines will be evaluated so as to satisfy (3.5) on the same number of lines as the vortex lines.

The simplest of these procedures is to replace each blade with one vortex line when its chord

is not very large. Scholz,° for example, replacing a single wing of an elliptical plate with one vortex line of the span length, has

obtained favorable approximation of k which is

the mean value of k(&'), i.e.

the value at

the center of the wing. This leads to a

suggestion that it might be permissible to

replace the propeller blade whose pitch is

almost constant with one vortex line.

In the neighborhood of both the ends, i.e.,

4=0 and oo as stated in 4, it is valid about

k1(E') to replace each blade with one vortex

line. And therefore, this idea will be adopt.

*) Since the following is valid from (5.6): lrki(e')VE'2 (O2(e') - O(')) ==4V(E' - eB)(l e'), defining k1 which is the mean value of k1(e') as

rk _/E'2(02(E')

-=4

k1(e')Ve'+ (O2(E')

-the following relation independent to -the Contour of the propeller blade is obtained:

k== 4/2, where A = AT(1 4n)2/r(B.A.R.).

ed when A of the blade is finite. First of all, the skeleton of the blade section of the propeller will be replaced with a straight line of finite length whose pitch ratio is ir((E'). Then

assuming the strength of one vortex line in radial direction as ['(E') and its

posi-tion as (E", O"O0"(e)"), we can satisfy (3.5) at the position of (E', O'O0'(E')) corresponding

to it.

Now it is equivalent to using only (2.5') instead of (2.5) to assume that the skeleton of the blade section at E'=constant can he replaced with a straight line. Then it is

plain to be able to replace a(r') of (2.5') with r0E(E') of (4.10) and (4.10'). Though there is taken the value of ') concerning with the zero lift line of the two-dimensional profile,

the zero lift line somewhat approaches to the tangential direction of the skeleton of the blade section at the trailing edge when the blade are operating in the three dimensional flow as mentioned in the footnote of §4(u). When the aspect ratio is not very large,

however, there seems to be hardly any error even if the zero line of this two-dimensional

profile is adopted instead

of that

in three dimensional flow.

Adopting Weissinger's method, we can put as follows [cf. Appendix III]:

= (3/4)o1(e') + (1/4)02(E"), o0'(E')= (1/4)o(e') + (3/4)02(E').

And letting the chord length of the blade be c(r') and putting

c(r')/r9==c1(E'),

the following expression about the blade sec-tion at E' is obtained:

c1(E')=(O2(E') 01(E'))1/Ç'2+ E)(E')2

(6.2)

In this case, of course, there holds the

ex-pression (5.5).

Calculating now (3.4) by using (6.1), the re-sulting expression can be written as follows:

A(E'; y)

Sit

i dt"

-

r(t")1,'(t", t' ; y)

_{(t_t)2}

', (6.3) ir

1

(6.1)

where

= (1 + e)/2

(1-e' = (1 + EB2)/2 (1 EB2)t'/2, / 1 ..2

Ft" t' "

(1 ' , ,

', r

4(1CB2)E'E" i PN(C; y,) = -- r(tt)F,T(tF, t';y1) dt _a' 7nJ-1

(t t)

r J'T4v(p"p')312s"s"' IC"2$'2 \2

J' (t , t ,y,)

-

_{4 ..If.1.J2 \, nfit} - N)

'ikIÇß)Ç Ç

S S

F,(t', t'; v)= o"(e") 0,'(e') I

Fr(

O 1_i/fElB+vi:i i ÇB ['(t") ¡'(e").

Then there holds as follows from (5.2)

-

T (t")[F,(t", t'; y) + F,T(t", t': y)]

'2

= r,2((C') -

(E

7rJ1

(

t)

Inducing this equation to the form of simulaneous equations by using the method of J

pendix VI, ['(t") will he evaluated, and then ,k(C': ii) and ,IN(C'; ) of (6.3) will he calculat Then k(C') will be evaluated from (5.4).

On the other hand, zv,' and w0' will be expressed approximately as follows at the posit] of the bound vortex line:

= e'2,,.1(e' ; v,)/r,(C'2 ±y12),

w0'= e'v1p(C'; v1)/r,(C'2+ y12). «

T,= pNQrÇ ['(e') ' (i ± zve'

)d'

Qo=pNVrF(C')E'(1+

' )e'

r'

TD = -- pNVQr, CD(E'2 + 2)(02(C') oi (e'))( i

_S2r')\

wo'

\(i +

wa')_V de,

i

Q1= pNQ2ro5 _{CD(e12+v2)(02(C')_Ol(e'))(l+} w,'

Ca Qr,C' J

-i/e'2 + ? o," (e' ) o' (e') 1/e'2 + ?

Furthermore, (2.8) and (2.9) will be also written approximately as follows:

Defining for the convenience of calculation D=2r,, ¡'(e') Qr0 (vy0) -F. À,..(C' ; y) --(e' ; y,) AN( ,y)

r(-vj'

_j (C'; y,)

_2r,(vv0)'

.8) (6.9) (6 .4)

i1'fk(E", e'; 00"(ç") - o0(e)

14 Ryusuke Y.&zoAzAKI (Vol. Xl X.

.5) (6 } N-1 'p-ed. on

c_T cQ

V

Tp274

_0pn2D5'

_i5'

s=1 " :

slip ratio, : revolution per unit time,

y 2ir

then, (6.3) and (6.6) will he written as follows:

y) =

-1t(t")

_(t",t';

(t";')2'

-- dt" i.',) = -__-, 1r(t/)F,,T(t/, t';y,)

(t"t')2 '

I dt" (e') v,,

1'

_{[F0B(t, t'; y) + F0T(1", t' p,)]} (t" _j)2 -

vv,

where r(t") represents the same function with t'(e") satisfying the relations of (6.4). By (6.7), (6.8) and (6.9) we obtain

JIPI

-B'(

/&'(C ;

C =

CQ = Nv2s _t'(e')

'(i _(i

e'2 _íiN(E'

C71, _NvCD(C'2+v2)(O2(C')_Ol(C'))(1_s

C'°+v12 /iN(E' ; vi))

((i

Ç'°±v12

CQD NCj,(E2+v2)(O2(CF)

O(E')(ls

_{C:'2 /1N(C' ; y,) )2C'd'}

C7 = C7,) + D2,, CQ= CQ,, CQ,,, ' ' (v,,/2) (C/C0), J= irv,,= 7rv(1 - s).

Using the expressions (6.10), the performance characteristics of the propeller will be

calcu-lated.

Next p and y ought to be determined

ap-proximately. The following relation is

as-sumed as an average in this paper:

(6.11) This y has no relation

to the pitch

of no thrust line strictly. And assuming the pitch

ratio of the free vortex as

íi)(C') [cf.

Ap-pendix I], the following equation is made valid:

S1(')

_V±w'1

' .Qr' + we']!)'=

Therefore we get approximately from (6.7) and (6.9)

+ ji.(e' ; y,) (y - vo)E'2!(C'2 + y,2)

1j,r(';

y,) (vu,,) (e'2H- v,,v,)/('2+ vi2).

(6.12)

J

(6.3')

(6.6')

(6.10)

Then y will be approximately defined as follows:

(6.13) Putting the nominal pitch

tion as p,,(') and letting he a [provided that (Ag by the chord line and the profile in radian], there is

tiOn:

)(e') =[p9(e'/ + C'agJ/I

ratio of each sec-the zero lift angle is an angle formed zero lift line of the the following

rela-i (p,,(e')/re')aej.

(6.14) The above expression can be deduced from the expression (4.10') and the expressions defined as follows:

= E' tan ago, ag0 + (1g = (Aa.

In the lifting surface theory, since the

free vortex created from an arbitrary point

on the

blade surface is obviously carried

16

this part. According to Flam's collection(2D of the photographs of cavitation of scew pro-pellers operating in open water, however, the numerical values of measured pitches of trail-ing helical cavitating traces a little detached from the blades are obtained as shown in Fig.

2. _{Accordingly, we may put v=rO.9 y as the}

ist approximation. 10 Po 0.5 û Fig. 2.

If a screw propeller is given, i.e., N, Eß,

(lgj, c(') and others are already known,

(E'),v, O,(E'), 02(E'), O)"(E"), o'(E'), F01(E", E' ;

and others are evaluated from (6.2), (6.1), (6.14), (6.11) and (6.5) etc., _{and the} values

of 7, F,T(E, E' ; y), T(E') and others for given or s are obtained by successive approxima-tion from (6.5), (6.6'), (6.3'), (6.12) and (6.13). Then from (6.3') and (6.10), C, CQ and , for

each J will be evaluated.

On the other hand, in order to calculate

(6.10), the values of CD have to be known. Solving (6.6'), TÇE') is obtained and the lift coefficient of each section

2(v 0)t(e')

c(/

can be calculated. The effective angle of attack a, for this section will be expressed by

u, = CL/27r.

If the shape of the profile is already known,

the profile drag coefficient CD corresponding

to CL or a, will be obtained theoretically or

experimentally [cf. Fig. 3]. Substituting this C in CD of (6.10), C1., CQ and r will be evaluated.

(6.15)

Ryusuke YAMAZAKI

CD

CL

Fig. 3.

With respect to a given propeller, C is near-iy proportional to (z.vo)1'(E') from (6.10), and by using the idea of the equivalent section of Appendix V, we can get the following

ex-prrssion from (6.15)

since E'>v where E'

represents the equivalent section

CTO=[E'WcI(E') -CL]j'_0. (6.16)

It is, therefore, justifiable

to take CD with

C as the base instead

of CL in

Fig. 3.

Furthermore, C7. may be also used instead of

C,.1) because C7.007..

Evaluating the basic equation for the lift-ing line theory by uslift-ing (5.1), (6.9) and (6.5), we have as follows:

2rk(E')Ç'(Oa(') -01(e'))

1

S1 i'(E')FT(t" t'; y,)

_®(E')vU(6.17)

_{il - J/))}

This equation is corresponding to the basic equation (6.6') in the lifting surface theory. It is one of the objects of this paper to evaluate the values of k,(E'). If k,(E') is

known, we can obtain the same results as

the lifting surface theory by solving the

in-tegral equation (6.17) instead of solving (6.6'), and furthermore, instead of solving (6.17), there is no great inconsistency in using the

strip theory, in which there holds an

approximate method containing the idea of

the equivalent section [cf. Appendix V]. (Vol. XIX.

/

In the next place, as the method to determine the quality of the characteristics of propellers, one of C1., CQ and J has been hitherto taken

as the base and

of various propellers are

compared, hut in actual cases the base ought to be taken in accordance with the data given

in the beginning, even though the former method is more convenient for the comparison

of data.

The author uses B and B

that were adopted by D.W. Taylor,°4 which will be defined as follows in this paper when

taking kg, m and sec.

as the gravitational

unit and denoting the number of revolution per second as n [where 2irn=[d]:

B = n (siQ)°/ V2

} (6.18)

B=nT°1/V2=1/pC/J2.

When the ship speed is given, V and T of the propeller are approximately determined, and n is determined from the engine, and so B

will be evaluated. Otherwise if V, n and QQ [power of the enginel are given, B will be

evaluated. Accordingly, in order to compare the characteristics of some number of

propel-lers, it will be sufficient to take ,

and J

with / B or -/ B

as a

base. It is clear that the larger on a certain value of i B

or B is, the better the performance

characteristics are.

§7. An attempt on a propeller

design method

The screw propeller is operating in the uniform flow. Q, V and T are given, and it is assumed that the propeller diameter D=2r0 is known by sorne way. Namely, r,, i, and C7. are evaluated. Then a design method

of the screw propeller whose loss of energy

is minimum will he next taken into considera-tion. The radial vortex distribution that satisfies this "condition of minimum energy loss" tentatively calculated applying the

lift-ing surface theory, agrees with the result of the lifting line theory. Namely, there holds the theorem of the rigid vortex sheet formulat-ed by Betz as it was. [cf. Appendix IVJ

Letting w be the rearward displacement velocity of the helical trailing vortex surface at infinity, and putting

V+w/2 V .Qr, Vo'9r,

. s1=1 --,

y1 we have =Sdr,(1v1). (7.2) Putting further

r(e')_

E(E1) r0w/2 - Sr2(y_v)

defining the following:

AN(E'; _{S2ro2(vi_vu)'}

(I

iiE ;)

J. (E'; V1)_Qrt(yi_Vo) (7.1) (7.3) (7.4)

and substituting (3.4), (7.3) and (7.4) in (IV. 5) of Appendix IV, we have

;N(; y1) 1. _(7.5)

Therefore, there will be obtained the follow-ing equation from (3.4) (5.2) and (7.4):

(E')=(vo)L(E'; v)+v,

where

'N (E'; = t' ; V)

(7.6) Rewriting (7.5) again, we have

-

i(t1FoT(t/l, t';

(t't') =

1. ..(7.5')

Evaluating (i (E'), the pitch ratio p0(e') of each section will be

p0(e')

1+E(Ç')agi/E'

- (niE') (E (E')2 + E'2)ag i. (7.7)

Determining properly N and and

assum-ing the values of y1, .ET(E, ¿' ; p1) will be

calculated from (6.5), and therefore ¡'(E') will be evaluated from (7.5'). Assuming further 1i0 or s, C. and CQ will be calculated. That is, from (7.1), (7.3), (7.4), (7.5) and (6.10), the following expressions will be obtained:

18 Ryusuke YAMAZAKI 2 1

-CT)=NVISI

r(e')e'(l_si

)5 4 _ie CQo=M C70, ir2 ri 2

-

Niil C(e"+ v13)(O(E') - o(e'))

(i

s1

8 + ) de',

2 \2

CQD= NÇ1 C(E'2+ v2)(82(E')

-

0(e'))(l - s Vj

₎

F7 C = CT7 + C,, CQ= CQQ -F- CQD, = (v,/2) (Cr/CQ), Jirvo, O(Ç') - Ot(e')

3/e'2 +

Assuming now the shape of the contour of the blade and giving the chord length c(C'), 02(C')O(C') will be obtained from the last expression of (78).

Hence, giving c(C') in the beginning, tak-ing some number of , assuming further C,, in accordance with the shapes of the profiles, calculating J, C7., C,, and r1 corresponding to

each ii by (7.8) and drawing the figure in which C7., CQ and are taken on the hase of

J for the parameter y, the value of ii for

given C7. and i',, will he determined from this figure, and then I' e') and i corresponding to

this will he evaluated, and finally the power

required for the propeller will be obtained.

In this case, since the contour of the blade

is known, F17'(E", '; y) will he calculated from (6.5), and such i', consequently

(')

as

satisfying (7.6) will he evaluated by successive approximation. Furthermore, when the shape of the blade section is known, a will he de-termined and the

pitch ratio p(e') of each

section will be evaluated from (7.7).

In-cidentally, substituting ('; Vt) = I instead of ii) in (6.12), 1(E) will he constant in-dependent of C',

i.e. ()7(C')1.

In the above example of the design methQd,

it is needless to say that

n, V and T are

given from the beginning of the design and D=2r is obtained by some way. It will be, therefore, discussed next how to evaluate the optimum diameter D. Now since n, V and T are known from the beginning, assuming some number of propeller diameters D, then ii, and C,. will be evaluated by the following expressions:

V -. T

VO=1rfl.ö

'-r''

_pn2D4'

and y1 and corresponding to them will be

determined.

The relation of D with

can

be, therefore, drawn in a curve, and so,

evaluating such I) as r becomes maximum, this D will prove itself the optimum D

re-quired. In this case, the values of C,, ought

to be chosen in accordance with the shapes of the blade sections to he adopted. By the

way, it should be added that the design

method dealt here is limited to the case in

which the shapes of the profiles and the chord length of the blades are given.

8. Conclusion

In this chapter, the author has stated on

the method to evaluate theoretically the per-formance characteristics of the screw propeller

in open water by using the

lifting surface

theory. He has also shown an expression to evaluate the lift slope correction factor [a

sort of curvature correction factor] which is deduced to correct the lifting line theory by corresponding the basic equation of the lifting

line theory with that of the lifting surface

theory. Once this lift slope correction factor

k(e') is obtained in some way, the same re-sult can be got by using the lifting line theory as

that calculated by means of the

lifting surface theory. And the result obtain-ed by the lifting line theory almost agrees with that by the strip theory. Furthermore, in the strip theory, if the hydrodynamical characteristics of the profile of the equivalent section are evaluated, the performance charac-teristics of the propeller can be also calculated without difficulty [cf. Appendix V]. Accordingly, if the lift slope correction factor at the equiva-(Vol. XIX.

lent section has been estimated, the

characteris-tics of the propeller will be calculated in a

simple way.

Moreover, as one of the conclusions in this chapter, it has been made clear in §2 that the variation of the rake gives hardly any effect to the characteristics of the propeller.

Appendix I Hydrodynamical F'undamentals of Screw Propellers

It will be assumed that the screw propeller [the number of blades N, the radius r0, the

boss radius r5] is operating steadily in open water at a constant velocity of advance V and a constant angular velocity of rotation Q. The cylindrical coordinates (x, r, O) will be taken into consideration so as that xaxis coincides with the axis of rotation. The propeller is ad-vancing with V in the negative direction along

xaxis and rotating with Q in the negative

direction around x--axis. t denotes the time, and a position of a point on the propeller

blades is marked with a prime. Moreover,

the propeller will be assumed as axial

sym-metry, and for simplicity no consideration will

he taken to the boundary condition on the

boss. [cf. Fig. 4]

Fig. 4.

The thickness of the propeller blade will he assumed to be thin. The mean surface of the propeller blade is replaced with a bound vortex sheet [i.e., the lifting surface].

Ad-opting the parameters of u and y,

a point

on the blade (x', r', O') will be expressed as

follows

*) The vortex sheet is known to be a singluar surface. The vortex line whose strength is T ex-tends round a closed curve, and is replaced by a sheet vortex on an arbitrary surface S' bounded by this curve, n' denotes the normal to the element dS' of the surface and its positive direction is

defin-ed to satisfy the right-handdefin-ed screw convention when

the direction of the vortex line takes the right handed screw convention. Then the velocity po-tential q due to the vortex line will be as

fol-lows :(20)

fÇ _{f Is'}

4irjJ,, ân'\R'

provided that R is the distance from the point P' on the surface element dS' to any passive point P.

tx'=x0(u, o) - Vt, (p,) r'=r0(u, y),

O'=8((u, y) -QE,

where x11(u, y), r1(u, y) and 00(u, y) are all the functions of u and y, which express the

posi-tion of the blade at t=O. A posiposi-tion of a

trailing vortex (x.,, r,, O,) leaving from a point

on the bound vortex sheet (x', r', O') is got

as functions of not only u and u

but a

parameter ço, and so we can put as follows,

it being relatively steady for a point (x', r', O'):

y, ço), (s) r,=r'+r,0(u, y, ço),

0, O' + O,0(u, y, ço) 4- 2rk/N,

where, it is considered that (x', r', O') satisfies

(p') and k=O, 1, 2, N-1, and x,5(u, y,

y, ç) and O,0(u, y, are all such func-tions of u, y and ço as tending to zero at ço=O. And the position of the trailing vortex at the infinite rear will be expressed as

çy=. Choosing y so as that the line of the

bound vortex may agree with the curve v=

constant [the two qroups of curves as u=

const. and v= const. do not necessarily cut one another orthogonally], and expressing the strength of the bound vortex of the elemental width dv as r(u, v)du, such a velocity

po-tential as follows will be obtained apply ing primary differential geometry :*)

v)Q(x, r,O; u, v)du dv, (1.1)

20

N-I ¡r

Q(x,r,0;u,v)=

1V\

('r5dO.0r_r55O3 0r5\ O

4ir / R 0go Ou Ou OgoI Ox5

o JO

+ r5.00, 0x r500,\ O +(0 Ôx Or, Ox,\ O

i1

d

\ 0go Ou Ou Oço

I

Or, \ 0go Ou Ou 0ço

I

r,O0,J R3]

'°

R8=i/(x, -x)2+r,2+r22r,rcos (0,-0),

and the suffix (s) of [ j means that the values of x5, r and O, in [ ] satisfy the

expression (s), and the surface integral about

du dv is taken over the whole surface of the

one blade.

Letting the axial, radial and tangential components of the induced velocity at a point (x, r, O) be w,,, w and w0 respectively, there

Ryusuke YÀMAZAKt hold Ox aq, , w0= Or rOO L(Qr' + wo') (

6x0(u, y) 0r0(u, y) Ox11 (u, y) 0r0(u, v)\

Ou 0v 0v Ou

)

± (V+ w')(0r0(u, y) r0(u, v)000(u, y) 0r0(u, y) r0(u, v)003(u, v)\

Ou 0v 0v Ou

)

0x0(u,v) r0(u,v)000(u,v) 0x0(u,v)V7

Ou 0v 0v Ou

)]y)

where [ ]() signifies that the values of x',

r' and O' in [ ] satisfy the expression (p').

And the propeller is operating steadily and so the relative position of the free vortex for the propeller ought to be always constant

in-dependent of the time, while the free vortax is moving with the induced velocity, of its position. Let us now consider the axial com-ponent of the position of the free vortex. [w,j(,)

denotes the axial component of the induced velocity of the free vortex at x=x,=x,(u, y,

go, t)

at the time

t,

and the free vortex is

moving with this velocity. Here the position of the free vortex after the infinitestimal time

At moves to

x=x,(u, y, go, t)_+[w,,](,)4t,

while the position of the same vortex at the

time t+4t is expressed by adopting go

inde-pendent of go as follows: x==x,(u, y, ço, t+4t).

Since these two helical vortices ought to agree, there holds

x,(u,y, go, t) ± [w,](,) 4t=x5(u, y,go1,t+ 4t). As the infinitestimal movement is considered in this place, we can put

ço,go=Aço,

(Vol. Xl

X-(1.2)

Furthermore marking dashes to w,, and others, they express the values on the blade surface. Then satisfying the boundary condition that the velocity normal to the blade surface is zero, the following equation will be obtained:

(1.3)

and neglecting the higher order of 4t and

4go, the following equation is made valid:

0x3+ Ox3 go =[wj(,). Ot Ogo4t Similarly OrS+OrsAço_r 0go O0 00, 4go Ei i = wo I Ot 0go At _L r jc)

Substituting the values of (s) in these equa-tions and transforming them, we have

V+[w,,](,)= go, r i ,0',4go ( LWrJ(,) 0go At

2+r.9i

_Lr J(o 0go At

where [w,,](.,) and others denote the values of

zv and others satisfying the expression (s)

respectively, and Ago/4t is considered as a con-stant appropriately taken so

as to

let the above three equations hold simultaneously, for

the two free

vortices at the time

t-FAt are congruent by shifting the variation go

ap-propriately according to At.

and the surface integrals are defined as the

same as (1.5). Therefore the total thrust T, the total torque Q and the propeller efficiency y will be

T=T,+T,,, Q=Q,+Q,,,

=(VT)/(sìQ)...(1.7) Further, in this general theory, the condition that the velocity component in the direction normal to the boss is zero is neglected for simplicity.

Putting now as u-r', rB.rrO and v,(r')

vv2(r'), instead

of (s) and (p'), mak-from (1.1) Çro Çv(Y') q= _\ dr' \ r(r', v)Q(x, r, O; r', v)dv, (1.1') Jr,, Jvi(r') where Ar-i Q(x,r,O;r',v)=1-:: _ 47r Ox Oçc' rOO kO J'-'

-

r'

Ox,,(r',y, o) 0O,(r', y) (Oxo(r', y)

+ Ox,0(r', y,ço)\\ 0 1.

' i

d

\

ôço Or'

\

Or' Or'

ing use of the simple expressions as follows: 'x=x'+x,,(r, v,ço), (s)

r,=r',

0,

= O'+ ç' + 2irk/N; 'x'=x,(r',

y) Vt,

(P1') r' = r', O' = O,(r',y)

-and considering that this theory retains

uni-versal validity even at t=O, the following

expressions are obtained

R'=i/(x,,(r', y, ço)x,(r', v)x)2+ r,2+r-2r,r cos(ço+ O,(r', v)O+ 2n*/N) from (1.3) and (1.2)

Evaluating r(u, y), w, w and w9 by solving

(I.1)(I.4), (s) and (p'), the thrust T, and the

torque Q, in perfect fluid will be calculated from the following expressions T,=pNr(u, v){(Qro(u, v)±w9')

0u,

y) ,r,(u,

Q=PNjr(u,

y) {(v+ w;) Or,(u,y) Ox,(u, V)}r( v)du dv,

(1.5)

where the integrals about dudv represent those of the whole surface of the one blade of the

propeller. Since no allowance has been given for the effect of viscosity in the above expres-sions (1.5), letting separately the thrust and the torque due to viscosity be T,, and Q,, re-spectively and adopting the local drag coefficient C,,, we have

T,,= - pN CDV (V+ vt)2+ ( lTv)I1 w91?(V+ w')F(u, v)du dv,

(1.6)

Q,,=--pN C,,1/(T'+ w')+(Sr(u, y) -Fv'$'(Qr,(u, y) + w,')r,(u, v)F(u, v)du dv, where

F(u, y)

_-

[(0xo(z y) Or,(u, y) ôxo(u,y) Or,(u, v)\2

au

oi

0v

+(0ro(u

y) r,(u,v)000(u,y) Or,(u,y) r0(u, v)0O,(u, v)V'

Ou 0v 0v Ou

)

+ (ro(u, v)ôfI ,(u,y) Ox,(u,y) r,(u, v)0O,,(u, t') Ox,(u,v)'\21'

f- r'(O0o(r' y) 5x0(r', y) 500(r', y) ôx,(r' v)\ O

} {r» O

Ox,0(r", y', ça) O

- Or' 0v

/

0v

Or')

Ox' ôça r"OO'

+ r"((0xn(r" y')+ Sx,o(r", y', co)) ôx,,(r", y', çc') t909(r", v')\ ô

\

ôr" r Oça Or" ) or',} _r,'dw_{r",x' = xo(r',v),O'}

R"= / (x,0(r", y', ça) +x,(r', v)x')2+ r,'+r'1-2r,'r' cos(ço+ 0(r", v')O'+2rrk/ 1sf),

from (1.4)

V+ [w,,jc,, ôx,,(r', y, ça) ôr,0(r', y, ça)

= O...(1.4')

ôçc' ' _S2+[w9/r]3> 0go

and further from (I.5)-...'(I.7)

'Yi V2(Y') T,=pN dr' r(r', v){(Qr'+ w9') ,_r'OOo(r', V))

'

Or' j.dv. YE vi(r') rr,, Cvi(r') ,0x0(r', v)ì _r'dv, Q0=pN

dr'

r(r', y) {(v+ wy') - w,

-Jr1 v,(r') 5r, r2fr')

=

-dr's CDV

(V+ w')2+ (Qr' -- w9')2( V+ w')F(r' v)dv, ri, v1(r') ro rvi(r')

- pN

dr'\

CDV(V+ w')2+ (Qr'+ w9')2(Qr'+ w9')F(r', v)r'dv, i where

)L(°)+(00

v))i ±(rO0n(r'. v)ôx,(r', y) rô0)(r,v)axfl(rl,v)\21

I (r', i'

Si'

äv -

- Or' 0v 0v Or'

) j

Here the expressions (1.4') show that the

as-sumption that the contraction of the helical surfaces of the trailing vortices is neglected

holds as it is.

Furthermore, putting as

v+0', 0(r')<O

<02(r') and rewriting (p') and (s1) at t=O, we can put x,=x' +x,0(r', 0', ça), (s2) r, = r', 0, = O' + ça + 27th/N, S Tç, Qr Ox,(r', y) //ô0n(r y)

- v=

dr,

r(r

,,, y )S(r , V'; r', v)dv',, r'Ov 5v _(r") Yj, Vi where N-1

S(r",v';r',v)=

_{VÇ [jr'}

_0x0(r',v) / 000(r'v) O 47r / Ox' 0v ,/ 0v r'OO' ri, Jv1(r') T=T,+ Ti,, Q='Q,±Q», =(VT)/(2Q), (1.3') ('.7,) N-I i'00

Q(x, r, 0; r', 0')

ir

0

-

ôx,,(r',O', ça)

o +r(ôX0fr' 0')

Ox,,(r', 0', ça \ Ô}1] dça,

4r

[(

Ox 0go

r

Or' Or' ) Or _{r, r'}

22 Ryusuke YAMAZAKI (Vol. XIX.

x'=x0(r', 0'), (p2') r' = r',

= 0',

and the following expressions will he obtained: from (1.1')

('ro (02(r')

= \ dr'\ r(r', 0')Q(x, r, 0; r', 0')dO', ..(I.8)

Jr,1 J0i(r')

where

N1

S(r",0";r',0')=-1

[{r'

0x0(r',O') O O

}

4'r Ox' 00' r'OO' Or' Or'

k=O O

5r" Ox,o(r", 0",ç) O +r"( 0x0(r", 0")+ Ox,0(,-", 0", ça)\ a d

t Ox' Oça r"OO'

\

Or" _Or" )0r0'J

R"= i/(x,(r", 0", ça) +x0(r", 0")x')°+ r,°+ r'2-2r,,'r'cos'ça+ 0"- 0'+ 2îrk/N). Now instead of ça we adopt the new variable ça, satisfying the following relation:

çaça+(0"O').

Then it is assumed that Ox,(r', 0')/OO' and Ox,,(r', 0' ço)/Oça are put approximately as a constant a when ça,

lies between 0"O' and

O, and as a constant h when ça lies between O and 00

And for simplicity, the term Ox(r', 0')/Or,' Ox,,,(r', fi', ça)/ôr' will be neglected assuming as a higher order than the order of r' and ôx,,(r', 0', ça)/Oça. Furthermore if we differentiate with infinitestimal variables r and r instead of x and (1 respectively, we can put

o a a a a

OxaOo,, orhO

ao=as;

Neglecting hgher orders, the following expression is obtained from (1.8)

Q(x, r, 0; r', 0')

r

N-1

1I(r'

aaa\V

dço,

4m[\ a Os,,

r' a5)L

o /a8(ça1+s,,)+r'2+r2-2r'rcos(çaj+9+2irk/N)

N-1

(r' O

h O \'Ç dço,

.h

r' a) íjo 1/h°(ça, +,,)2+ r'°-l- r°-2r'rcos(ça +se+2vk/N) _=o,=o On the other hand, the equation (1.9) is originally deduced from (1.3), and (1.3) is rewritten

as follows:

(Qr'+ w9') ôx,(r',O') (V+ w,,') + w,'ôx,(r', if)

o.

rOI) Or

The 3rd term of the left side of this equation is the product of the rake of the propeller and the radial component of the induced velocity, and so is considered to be higher order

as compared with w,' and others. Using this and the before-mentioned approximation, the fol-lowing expressions will he obtained from (1.2)

r y, t dr"l r(r", 0")dO"

4Jrj

J0,(r") >< [ a

r" a

a N-1 [ aa,,

(

-,V-1 _ dça,

a(r,,a

hO

+ hO

h O -

'ase) _5o h2(ça1+s,,)2+r"°+r'°-2r"r'cos(ça,+s9+2k/N) R' = i/(x,,9(r', 0', ço)+x9(r', 0')x)+ r2+ r2-2r,rcos(ço+ 8'-0+27rk/N), from (I.3' ro CO2(r) Qr' 0')_ dr"\ r(r", 0")S(r", 0"; r', 0')dO", r'OO' r _{r,, j0i(r")} (1.9) dço1 /a2(ço + s,,)°+ r' 2+ r'°-2r"r' cos(ça, + f- 2rk/N)

24 Ryusuke YAMAZAKI (Vol. XIX. ('ro Ç02(r') / i w = \ dr"l r(r", 0")dO" 4ir rl, Jüi(r')

r

N-1g'IJ'/_j' I

ô (r" ô

a ô \\

dço1 >< [ r'ôeo ka ô

r" 0o

)o

/i(

F )2 _r" 2r"r' cos(1 + + 2rrk/N) N-1 roe ô

(r" ô

h

ô\\'t

dp1 F

r'a6 \h

?' ôo)

jj 1/h°(ç1 + e,)2+ r"2+ r'2-2r"r' cos(ç01+Ee+ 2irk/N) _=o,

and further from (1.9) there will holds

ii

r'a

a a

r"a

a ô S(r", 0"; r', 0')=

_(a

_{ô, - r'}

ôso)(a

ae - r"

ôoe) dç'1 1/a2(ço1 + 2,)2+ r"2+ r'

2r's(ço1+ro+ 2rkfN)

a h

ô \(r" ô

h ô

k h a.

r' ô09)k h a r" a

J0 /h°(çc +o,32+r + r'°-2rr' cos(ço1+o± 27rk/N) ]o-c ij=O.

Considering the above-mentioned approximate solution physical y, it is assumed that the

hound vortices distribute on such helical surfaces as pitch r a, and that the trailing vortices on the cylindrical surface rr , flowing along the blade surfaces, concentrate at the points r='r", 0=0' and go out forming helicoidal surfaces of the pitch 7rJì.

Ultimately we obtain approximately from (1.7') and (1.4') Çr ('Oo(r') T0pN\ dr'l r(r', 0')(.Qr+wo')dO', Jr, J0i(r') ('ro _{f (12fr')} a '

Q=pN

dr' r(r', 0') {(V+W')_Wr' xo(r; O rji 0i(r)

T= -

N5 dr'5 °1(i

_{+ (} 2)(Qr'+ w0')(V+w')r'd0, r, Ui(r) ('ro (02(r') Ç 2 QD= -pN

dr\ c (1+(o;,O

)))(Qr'+we')°r'odO', Y Ui(r)

where, since ôx(r', 0')/ôr' is considered to be

very small as compared with 1, the term of w' will be omitted usually as a small quanti-ty of higher order.

Appendix 11. Zero Lift Lines of Airfoil Sections of Propeller Blades

Considerations will be given to the cylindri-cal surface co-axial with the propeller in which the blade profile is contained. This surface is developed in a plane. Then the skeleton

of the blade profile is as shown in Fig. 5. x--axis is assumed to agree with the direction to which the profile advances [cf. Fig. 5].

Then there will hold

y'=y cos agoX sin a,

x'=x sin ago+y cos &gO

= X +y'tan ago cos X COS aR

I

X dp