On the theory of screw propellers in non-uniform flows

Lab.

'i.

Sche.p5boiwknJ

Technische Hogeschool

Dellt

With the Compliments of the Author

On the Theory of Screw Propellers in

Non-Uniform Flows

By

Ryusuke YAMAZAKI

Reprinted from the Memoirs of the Faculty of Engineering, Kyushu University, Vol. XXV, No. 2

FUKUOKA JAPAN

By

Ryusuke YAMAZAKI

Assistant Professor of Naval Architecture

(Received October 4, 1965)

ABSTRACT

On the basis of the unsteady lifting surface theory the author attempts to develop the general expressions for the performance characteristics of a screw

propeller working unsteadily in a non-uniform flow, and then, as an actual example,

presents the method to calculate the performance characteristics of the propeller advancing with constant speeds and rotating with constant angular velocities in a

non-uniform flow. Applying this method, further, he calculates numerically the unsteady loads on the blade sections and the total forces and moments acting on

the propeller in given wakes and oblique flows, and compares them with the ones

obtained by using the quasi-steady method.

TABLE OF CONTENTS

§ i Introduction

The method to evaluate hydrodynamically

the performance characteristics of a marine screw propeller based on the vortex theory

has been studied by many researchers, partic-ularly by H. Lerbs and others., .since A. Betz. The author, also, developed the lifting surface theory of a screw propeller working steadily

in a uniform flow, and further derived

mathematically the approximate formula,

that is, the 14- and 3/4-chord lines theorem,

which is applied to calculate the performance

characteristics of actual propellers, and the

theoretical results coincide quantitatively with

those of open test.° However, actual screw propellers are usually operated in wakes

behind moving ships.

In other

words, a

Page

§1 Introduction 107

§2 The Unsteady Lifting Surface Theory of Screw Propellers 108

§3 _{The Theory of Screw Propellers in Non-Uniform Flows 115}

§4 The Quasi-Steady Theory of Screw Propellers in Non-Uniform Flows 128 §5 Numerical Example (1)For a Given Propeller in a Given Wake 129

§6 _{Numerical Example (2)---On the Effect of Blade Area Ratio} 131 §7 _{Numerical Example (3)For a Given Propeller in an Oblique Flow 132}

§8 Conclusion 134

References 135

Nomenclature 136

Appendix I The Unsteady Wing Theory in Two-Dimensional Flows 13S

Appendix II Process of Numerical Calculation 144

propeller is working unsteadily in a

non-uniform flow which is induced by a ship hull.

Therefore, the effect of the fluid on the propeller in the non-uniform flow differs from

that in a uniform flow. For example, during

one rotation of the propeller, the unsteady pressure acting

on the

one blade surface

causes several appearance and disappearance

of cavitation on the blade, occurrence of

erosion on the bade, and bend-vibration of the blade. The fluctuations of force and

moment acting on the propeller against time are generated mainly by the pressure differ-ences between both sides of the blade varying

with time, and are transmitted to the ship

hull through the bearing and cause the

vibra-tion of the ship, while the average values of thrust and torque with respect to time relate to propulsive coefficients of the ship, i.e.,

effective wake fraction, relative rotative effi-ciency etc., which are obtained by comparing them with those in open water. Furthermore,

the average value and the osillatory parts of

pressure with respect to time induced at a point on the hull surface by the movement of the propeller blade differ from those in a

uniform flow, and the so-called "thrust

deduc-tion" depends mainly on the average value

of the sum total of pressure all over the hull surface with respect to time.

Thus, it is very significant to clarify the

performance characteristics of a screw pro-peller working unsteadily in a non-uniform

flow. These characteristics have been studied

recently by S. D. Ritger and I. P. Breslin, T. Kumai, S. Shuster, J. Krohn, H. Schwanecke,

T. Hanaoka, J. Shioiri and S. Tsakonas, S.

Tsakonas and W. R. Jacobs, M. D. Greenberg,

N. A. Brown and others. In the present

paper, by extending the vortex theory, that

is, the steady lifting surface theory which

was developed in the previous paper,° the

author attempts to deduce the theoretical method evaluating the performance character-istics of the screw propeller working unstead-ily in the non-uniform flow, neglecting the

mechanical conditions of engine, shaft and ship hull. Applying this method to the

propeller advancing with a constant velocity and rotating with a constant angular velocity

in the non-uniform flow, he develops the

formulas to evaluate its performance

charac-teristics, that is, the instantaneous relative

inflow velocities and lift coefficients of the blade sections and the instantaneous and

time-average forces and moments acting on the propeller as a whole. Finally, he calculates

two numerical examples: one is for given

propellers in given ship wakes, and the other is for given propellers in oblique flows.

§2 The Unsteady Lifting Surface Theory of Screw Propellers

In this section, the general theory to evaluate the hydrodynamic performance characteristics of a screw propeller working unsteadily in a non-uniform flow is developed on the basis of

the unsteady lifting surface theory. The fluid surrounding the propeller is assumed to be inviscid, incompressible, free of cavitation and infinite in extent. Assuming that the blades of the propeller are very thin, the mean surfaces of-them can be replaced with bound vortex

sheets, and then the free vortices shed from these bound vortices at an arbitrary past time

can be considered to recede freely from the blades retaining their strength. Furthermore, the

disturbed velocity induced by the boss may be neglected to be very small,° and so the

boundary condition on its surface is not satisfied.

The propeller consists of a set of identical, symmetrically spaced blades attached to the boss, having number of blades N, radius

r0 and boss radius r. We define a rectangular

coordinate system O-xyz which is fixed in space, and transform it further to a cylindrical

coordinate system O-xrO by the expressions

x=x,

y=rcosû,

z=rsinû.

(2.1)

The x-axis is chosen so as to coincide with the axis of rotation of the propeller, and the

x-axis in the negative direction of O. The coordinates of a point P fixed in space are

denoted by (x, r, O) [vide Fig. 1]. The position of the propeller at time t can be represented

z'

X "

/

/

FREJ VO8TEX

/

FIXED TO TI-/E P/?OPELL EP

BOUND VÛA'TEX (t'=C2N5TANT)

O

/Px.a

I \

/

Fig. 1. Coordinate systems of the propeller

by the space coordinates (x(t), 0, O(t)) of the representative point fixed to the propeller where

x(t) and O(t) are functions of time t. The position of a point P1 fixed on the mean surface

of the k-th blade moving with the propeller can be represented by the parameters (u, y). Here we may not loose any generality by setting umr. Then by using r and y, the space

coordinates (x1,, r, Os)) of the point Pfl,, at the time t can be expressed as

x=x,(r, y) - x(t), r,=r, O,=Oh(r, y) - O(t) + 2r(k- 1)/N, (2.2) where the subscript b refers to quantities related to the point P10, on the mean surface of the blade, and xb(r, y) and Oh(r, y) are functions of r and y which represent the relative coordi-nates of the mean surface of the blade fixed to the propeller. And we can define for (2.2)

vL(r)vvT(r), rßrro; k= 1, 2,

, N,

where the subscripts L and T refer to the leading and trailing edges of the blade.

The axial, radial and tangential components of velocity of the non-uniform flow without

the propeller at the point P(x, r, O) at the time t are denoted by v, v and y0, respectively.

Hence pa,, v and y9 are functions of x, r, O and t, and must satisfy the condition of continuity: Ôv

F ayr + r F ÔV0 = _(2.3)

fr

r

rao

If the components of this flow velocity parallel to the x-, y- and z-axes are denoted respective-iy by y,,, y5 and v, we can define from (2.1) as follows:

v,,,=v,,, v=v,. cos O-v9 sin O, VZ=V,. sin O-Fv0 cos O

Then the bound vortex is assumed to he arranged along the line of v=constant on the

mean surface (2.2) of the blade, and its strength contained in the elemental width dv at the

point P,, at the time

t is represented by the function rh.(r, y, t)dv, which corresponds to the

pressure difference between both sides of the blade, Le., the load on the blade. The bound

vortex located at the point P, at the past time z- remains behind at the later time t as the

where

(T2 _(vrCr)

dr' dv' Yk(r' y' r) {( r7'OOI r1' r,.'a01' 0r1''\ O

4 k=1

' ' Or Or' 0r) Ox7'

(Or7' Ox1' Or1' Ox1' \ O

+(Ox1' r/087' 0x7 r7'001' O ) i

Or Or' ' Or ) rJOO/ Or Or' Or' Or

)

Or/

_f R1 dr , (2.6)

R1 i/(x1' x)2 + r7'2 ± r' - 2r1' r cos (Of' - O),

and x1', r1' and Of' represent respectively the functions x1, r1 and 0 as referred to r', y', r and t in the expressions (2.4). The expression (2.6) can be surely assumed to be convergent from the physical stand-point. Denoting the axial, radial and tangential components of velocity induced by the propeller at the point P(x, r, 0) by wa,, w,. and w9, respectively, and

using the velocity potential , we obtain

w,.= , w,= O2P_ (2.7)

Ox Or rOO

The components w, w,. and w0, of course, can be proved to satisfy the condition of continuity by using (2.6). Substituting (2.6) into (2.7), the integrals expressing w, w, and w0 are con-vergent in all over the space except the domain of the bound and free vortices, and are usually divergent at points Pfk, which contains P as a special case, on the bound and free

vortex sheets. Thus in the similar manner to the lifting surface theory in steady state,' in order to obtain the finite parts of induced velocity at the point P1,, or P, we must integrate them associated with the point at an infinitesimal normal distance o apart from the vortex sheet containing the point Pfk or P,, and take the limiting values of them when o tends to zero. That is, the Cauchy's principal values must be adopted after integrating them by parts.

The boundary condition must be satisfied. In other words, the resultant relative velocity normal to the mean surface of the blade is zero on its surface. Hence, at an arbitrary time

t, the following equation can be deduced at an arbitrary point P=P,,k(r, y) on the k-th blade: O0,,(r, y) (dx9(t) Ox(r, y) ( d0(t) } +[vj(hk)+[w0](k) }

Ir

+[v9](,,)+[w9](,) 6v

ldT

rôz, dt 1

-where [ ] expresses the quantity at the point P,0, which is obtained by replacing x, r and

O in [ I

with x,, r

and 0,, of (2.2). Then substituting (2.7) into (2.8), we get for the k-th blade as follows:

r 5 000(r, y) O Oxh(r, y) 0 (Oxh(r, y) OO(r, y) Oxb(r, y) O0,Xr, y) \ O

Lt 0v Ox rOy rOO

\

Or 0v 0v Or JOn PJ

Ç Ox(r, y) O0,(r, y) Oxb(r, y) O0(r, y) 1

- - - r i [v,.](,,,) + Ew,.](,) - O , (2.8)

Or 0v 0v Or )

fixed to this free vortex at the time t is represented by the space coordinates (x1, r1, ô,), that is,

x1=x12(r, y, r, t), r1=r10(r, y, r, t), 01=O1(r, t', r, t) , (2.4)

where x1(r, y, r, t) etc. are functions of r, y, r and t, and the subscript f represents

quanti-ties related

to the point P» on the free vortex.

From the definitions (2.2) and (2.4) there results:

: x0=[x1),.,, rh=[rf],.=t, ûh=[Ofj,._, (2.5)

Since the fluid is considered to be inviscid and incompressible, the system of the bound and free vortices can be represented by superposition of closed elemental vortices, and the

ax(r,v)

(

d09(t) ± [voJ?)) OOb(r, y)

₍

dx(t)

+ [vio))

ray dt av dt

f

y) aôbfr, y) ôxb(r, y) &O7Xr, y)'\

+

} [v,] . (2.9)

av ôv

ar

However, the integral of the left hand side of the equation obtained by substituting _{(2.6) into} (2.9)

is divergent at the point P,

_{on the mean surface of the blade, too, and so we must} take the finite part of this diverging integral according to the same procedure _{as (2.7) at the} point PR. Furthermore, when the blade holds a moderate angle of attack, the_{extended Kutta's}

condition is assumed to be satisfied at the trailing edge of the blade. _{In other words, the}

pressure difference between both sides, _{namely, the load function rk(r, y, t) on the blade}

vanishes at this edge in general. Thus the integral equation obtained by substituting _(2.6)

into (2.9) is the fundamental to calculate the strength of the bound vortex.

Let us next furnish the condition concerning the position of the free vortex. While the

free vortex drifts in accordance with the outer flow, it also varies as a function of time t.

Consequently, the free vortex at the point R at the time t satisfies the following equation:

aXf

[a]

+ [V,]fk),

-

[-'-]

+ rf

[]

+_[ve](fk) , (2.10)

at ax fk) at (fo-.) at r60

where L expresses the quantity of L I at the point Pft on the free vortex. _{Now we}

shall consider the space coordinates of the two points at the time t: the _{one point} Pfo-, at the

time _{t on the free vortex which shed from the bound vortex at the point P,o-,}

_{at the past}

time r and the other point

at the time t on the bound vortex fixed to the blade. And

we assume rt-it where it is an infinitesimal time interval, and denote the differences of

the x-,

r- and 0-coordinates between these two points P.'.o- and P,»1.

by 4x-, 4r and iOf,

respectively. Then using (2.2), (2.4) and (2.5), for instance, we get 4x1 = [x1]0_ - x = [x1]1_ - [xf]r=t =

-L aaxrf Jr= 4t

in the x-direction. On the other hand we obtain

r ôXf a[x1J

-

_{r ôx1}

L ar Jr=t at

L at ]=.

Hence, using (2.10), (2.5) and (2.2), we can obtain

(L

UJ

-

[]r)it

=

([w]o-+[vi+ ')1t= (4j'

+[vi(k)+[wio)k))it.

In the same manner, we obtain

iif= ([Vr](bk) +[Wr](hk))it , J0.=

(d0(t)

+

l

[vo1(o-) + - [w9J(o-)) it.

Substituting these iX1, ir1 and 40. into (2.8), the following equation is obtained:

aob(r, y) _{ôxixr, y)}

-

(r, r

ix ix

)

ir1.=0 . (2.11)

The equation (2.11) represents the tangential plane to the mean surface of the k-th blade.

and so the

free vortex shed from the bound vortex is found to follow initially along the

mean surface of the blade.

The above-mentioned is the outline of the theory of the screw propeller working unsteadily in a non-uniform flow. Now, let us compare the number of independent unknown functions with the number of independent equations representing the conditions satisfied when we give

1 4ir D (D') I dr' dv' r(r',y', t-ç'!çr0(r')) DB VL(r') O a ôî(r', ça) ô r- -. r'ôça aU R. YAMAZAIU

x,,(r, y), Oh(r, y), x(t), O(t), vD,, y,. and y9. The number of unknown functions X1, r1, O, rk(r, y, t) and ç is 4N±1. The equations (2.6), (2.9) and (2.10) for the unknown functions are

independent mutually and the number of them is 4N+1. Therefore the 4N+l unknown functions can be obtained by solving the simultaneous equations (2.6), (2.9) and (2.10). In

other words, the equations (2.6), (2.9) and (2.10) are necessary and sufficient as conditions

to evaluate the hydrodynamic performance characteristics of the screw propeller.

However, when a given propeller is working unsteadily in a given non-uniform flow, it is usually very complicated to solve the equations from (2.2) to (2.10) simultaneously, for, even

from the physical view point, the "strength" and the "position" of the bound and free

vortices are fluctuating with time and location when the propeller is under such an unsteady

state. In the next place, we shall treat the case of an unsteady state near to a steady state. Then, the working condition of the propeller does not deviate so far from the steady state,

and so can be determined by superposing linearly a steady state and small oscillations with respect to both the movements of the propeller and the non-uniform flow. Hence, the inflow

velocity to the propeller is divided into two parts: one is a steady part corresponding to the

moderate load and the other is an oscillating part which is assumed to be small. Considering the conclusion that in the steady state the performance characteristics is little effected by

changing a little the pitch of helical free vortices,0 we may safely neglect the fluctuation of

the geometrical positions of the helical vortices with respect to time in the unsteady state,

too. And as to the geometrical position of the free vortex shed from a point on the bound vortex, we may also neglect the radial contraction and the transverse drifting, for these

effects on the performance characteristics of the propeller are considered to be very small in

the steady state» Thus we assume that the path of the free vortex shed from a point on the bound vortex is located approximately on a circular cylinder whose axis coincides with the x-axis. Therefore, supposing the relation çD=ç(r)(t--) where p1(r) is a function of r and

independent of t, the position of the free vortex (2.4) is approximated as

xf='xb(r, v)-x(t)+i(r, ço), r1r, Of=Oh(r, v)-O±çD+21r(k-1)/N , (2.12)

where (r, cp) is a function of r arid ç, and further t(r, 0)=0 holds from (2.5). The quantity ço1(r) and the function _{(r, p)} can be approximately determined from the average positions of

the free vortices with respect to time. Then, substituting (2.12) into (2dO) and averaging these equations with respect to time, we get

a(r, ç') dx(t)

+

[i']

+ [,]u-A 0='

[1

+[,.]-dt - ax (ft) &r J(fk)

ao(t)

r

₊

r0

at + Lri5Oi&k) L r

where the quantity symbol with a bar on top indicates the average value with respect to t

at the average position of an arbitrary point on the free vortex. The right hand sides of (2.13) are functions of r, y and . The second equation of (2.13) is, of course, not true

because of neglecting the contraction of slip stream, and the first and third equations can

not hold exactly.

If the values of î,

, j,, and th,, at a point Pfk can be evaluated

accord-ing to some procedure, by averagaccord-ing (2.13) with respect to y and ç', we can obtain çc0(r) from

the third equation, accordingly ç from ç'=ç'0(r)(t-z-), and further ai(r,ço)/açc, accordingly (r,ç'),

can be obtained from the first equation.

Substituting (2.12) into (2.6), we obtain

f

or

+ r'

(ôx(r' y')

50(r', y')

0(r',

c' ₊ O(r', ço) \ a ¡ i

Or' ar' oç Or' ) ai-1'f R11

where

dço , (2.14)

R11 _{ì/((r', çc)±xb(r', v')-x-x(t))2 ± r1'2+ r2 -2r1'r cos (ç+ Ob(r', v')-O-02,(t)+ 2r(k- 1)/N)}

Further, for convenience, denoting as

dx(i) dO»(t)

-dt ' dt '

and using (2.14) and (2.2), the boundary condition (2.8) or (2.9) ori the k-th blade is rewritten

as follows: (i + [V(hk) + [Wihk)) (ôxh(r y)_\ / ôOb(r, y) ) (rÖ5 + [V9] + [WØ](k)) ray

/

av (Oxb(r, y) 0x5(r, y) 005(r, y) / 005(r, V)

-

a 0v Or

/

0v

)

([v,]Qk)+[w,.](k))=O y dr' 5 dv' 5

rk(r' y' t-/(r'))

r

a

_-

₍

0x(r, y) / OOb(r, y) ê 1 N ° rß _{'L '} ' Ox rOy

/

0v 4irr k-'=I

(Oxb(r,v) Oxh(r, y) OOh(r, y) / OOh(r, y) ' O

O _0(r',ço)

O

0v Or

/

0v

)ôr

Ox r'Oç 00

+

Or

O0,(r', y') O(r', y')

+ O(r', (p) \ O I

i

I

Or' Oçt' Or' ) Or1'i?

r1' = r'

x=xs(r,v)-xp(t) (Oxb(r,

y) / _{O0,,(r, y)) O 0s(r,v) - O(t)+2x(k - 1)/N}

- rOy 6v

+

(8x(r y)

6x,,(r, y) O0,,(r, y)

/

O0(r, V))

[]

_(2.15)'

a,- 5v av av

The number of these equations is N.

Furthermore let us consider the force and moment acting on the propeller. If we can

assume that vorticity of the flow velocity represented by v, Vr and r, is zero or negligibly small, the pressure difference between the face and the back of the blade, i.e. the load, is expressed by the product of the strength of the bound vortex and the velocity component

normal to it and the density of fluid p. Denoting, for convenience, as

+ V,, + iv, V,. = V,.+ W,., l7= rÓ + y, -F w, , _(2.16)

we can express the force acting on the blade element dvdr as follows:

the tangential and axial components d2F,, and d2F,.,. of the force acting on the radial

com-ponent of the bound vortex are respectively expressed by

d2F,.o=p[V,.](,,k) rk(r, y, t) dvdr, d2F,.= pEVei(h5 YkO, y, t) dvdr,

the axial and radial components d2F,., and d2F,, of the force

acting on the tangential

component of the bound vortex are respectively expressed by

d2F,P[V,.])

rk(r, ,t) rOOb(r, y) dvdr, _{d2F0r P[VJ(u,) rk(r, V, t)rOO;} y) _dvdr,

the radial and tangential components d2Fr and d2F,,, of the

force acting on the axial

component of the bound vortex are respectively expressed by

d2F,,,P[ V01 (bk) Tk(r, , t) Ox(r, y) dvdr, d2F,,9= P[Vr](b) rk(r, y, t)Oxh(r, y) dvdr.

Or Or

Hence the axial, radial and tangential components d20, d2F0 and d2F0 of fluid force acting on the blade element dvdr are obtained as follows:

d2F0 d2F,.,, + dF0,,, d2F,.0 d2F0,. + d2F,,, d2F00 = d2F,,0 ± d2Fr9, that is, d1F0= prk(r, y, t) (Lvol(bk)- rOO(r, y) [VTJ(bk)) dvdr, Or ( Oxh(r, y) d1Froprk(r, Or

d2FOO=prk(r, y, t) (Ev]Cb- Oxh(r,v)

Or

rOOh(r, y) [VJ(hk)) _dvdr,

Or

[Vrl(k)) dvdr.

On the other hand, in case of an actual propeller, we must take into account the effect of

frictional drag caused by viscosity which is neglected in the above derivations. The direction

of the local frictional drag on the blade surface is assumed to agree with that of the stream

line flowing on it, and further approximately the value of V,. is negligibly small compared

with V,, and V0. Then defining the local drag coefficient

by CD, the axial, radial and

tangential components d1FD, d1F,.D and d1FOD of frictional drag acting on the blade element dvdr are respectively obtained as follows:

d1F,.D'Cn p

{i + (0x1

/

ôOi(rv))l} V8V,,r ôOb(r,v) dvdr

d1F,D=CD . -F (Oxh(rv) / 6O,Xr,

y) )1}

v9v,. r E30b(r, u) dvdr (2.18)

d2FOD=CD . + / OO(r, y) )2} ÌOVO r

ôO,,(r, u) _dvdr

Moreover the suction force acting on the leading edge of the blade is of almost same order of magnitude as the frictional drag, and so we take account of the extended CD includ-ing this suction force [vide Appendix III. The axial, radial and tangential components d2F, d2F,. and d2P'0 of total force acting on the blade element dvdr are obtained from (2.17) and

(2.18) as follows:

d1F = d2F, + d2F,,D, d2F,.=d1F,.0 + d1FrD, d1F0 = d1F01 + d2FOD

i

(2.17)

(2.19) Now the new rectangular coordinate system 01-xyz, whose axes are parallel to those of the 0-xyz system and whose origin O is moving with the propeller along the x-axis, is defined as

x1=x+x(t), y1=y, z=z,

(2.20)

and then the angular coordinate O is measured from the y1-axis. Thus the position of the blade element represented by (2.2) is expressed by the new coordinates as follows:

x1=x+x(t)=x(r, y), y1=rcos O, z1=rsin O, where Oh=O(r, v)O(t)+2r(k-1)/N. (2.21) Then the components of the force parallel to the x1-, y- and z1-axes denoted respectively

by d2F,,, d1F and d1F, and the components of the moment about the x-, y1- and z1-axes

denoted respectively by d1M,,, d2M, and d2M., acting on the blade element dvdr, are obtained

as follows

d1F,,=d1F, d2,=d1F,. cos Ohd1F9 sin O, d1F=d1F,. sin Oh+d1Fo cos O;

d2M rd1F0, d2M=d2F r sin O d1F,x1,(r, y), d2M= d2. r cos O± d2Fx,(r, y).j (2.22)

Hence the components of the total force parallel to the x1-, y1- and z1-axes denoted

respective-ly by F;,, F;, and F;, and the components of the total moment about the x1-, y1- and

z-axes denoted respectively by M, M and M,, acting on the propeller can be obtained by

summing up and integrating the elemental force and moment (2.22) all over the blades of the

propeller. For instance, we get

,V

Çr0 _Ç"r

M N Ço Ç'r(')

F,= dr F dv , M,= dr d M, dv . _(2.23)

'

J"L'' drdv Jr irLO) drdv

And denoting the thrust and torque of the propeller by T and Q, we get

T= F;, Q=M,, .

(2.24)

Finally, we consider the geometrical position of the bound vortex, which can be determined exactly when the functions xb(r, y) and Ob(r, y) in (2.2) are given. However, it is allowed to

consider in order to simplify the procedure of calculation that the bound vortex can he

approximately placed on the helical surface which nearly coincides with the mean surface of the blade.

We treat the blade section cut by the circular cylinder of r=constant.

The

0-coordinates of the leading and trailing edges of this section are respectively denoted by OL(r) and 02.(r),

and the pitch of

the helical surface is assumed to be 2ira(r) which varies

with the radius r. Then there are two forms of the expressions. The first form is as follows:

Oh(r, v)-0M(r)+O(r)v

where (2.25)

OM(r) = (Oz(r) + Or(r))/2, 0(r) = (OT(r) - OL(r))/2, - 1v i .

And the helical surface of the bound vortex is expressed by

x(r, v)=rx0(r)±a(r) t(r) y , _(2.26)

where the function x0(r) represents the x-coordinate of the middle point of chord v=O, i.e.,

O(r, O)0M(r) of the blade on the helical surface. Hence the expressions (2.25) and (2.26)

show that the line representing the bound vortex agrees with one of the "equi-chord lines"

expressed by v=constant on the helical surface with radially varying pitch 2ira(r). Next, the

second form is

based on v00, where

0 is the angular coordinate fixed to the propeller.

From (2.2) we get

0(r,v)00 and

0L(r)0O0T(r) , (2.27)

and from (2.26)

x,,(r, 00)x0(r)±a(r) (0lOM(r)) . _(2.28)

The expressions (2.27) and (2.28) exhibit that the line representing the bound vortex coincides

with the intersecting line between the helical surface and the plane containing the propeller axis. The second form was adopted in the previous paper.'

Further denoting the chord length of the blade section at the radius

_{r by c(r), we can}

obtain

c(r)_2O(r)1/riia(r)1 (2.29)

from (2.25) and (2.28).

§3 _{The Theory of Screw Propellers in Non-Uniform Flows}

In this section, the hydrodynamical theory of a screw propeller advancing with constant velocity ,= V and rotatIng with constant angular velocity Ó=Q in a wake of a ship, i.e. a

116

non-uniform flow, is developed on the basis of the unsteady lifting surface theory presented in the previous section. In this case, the velocity induced by the propeller has an influence upon the stream near the body, i.e. the ship hull and the rudder, and so the wake field

induced by the body

is varied to satisfy

the boundary condition on the body surface. Accordingly, the performance characteristics of the propeller obtained in this way are different from those in case without mutual interference between the propeller and the body. However,

these second effects of fluid velocity is so small as be neglected. For simplicity, we adopt (2.25) and (2.26), i.e., the first form in the last part of the previous section. Setting x(t)= Vt

and O(t)=.Qt, we obtain

xb=xl(r, y) Vt, r0=r, O=OM(r)+O(r) vQt+27r(k-1)/N,

where 1<v<1

(3.1)

from (2.2) and (2.25). Assuming that the bound and free vortices are located approximately on the helical surfaces with constant pitch 22rh, there result a(r)=h and î(r, ç)=hç in (2.12).

Denoting the rake angle of the blade by r, we have x(r)=h OM(r)+rc, and so the following

expression is obtained from (2.26):

xh(r, v)=h(OM(r)+(r) v)+re . (3.1)'

Since the rake angle c effects little on the performance characteristics, we assume e=O in

the greater parts of this section, and so obtain

xb(r, v) :h(O(r) + "J(r) y) (3.1)"

from (3.1)'. Hence from (2.12) we get

xf=h(ç2+ OM(r)+ (r) y) Vt, rf=r, O,-ip+ OM(r)+ (r) vQt+2m(k 1)/N . (3.2)

Now let us examine whether the expressions (3.2) which represent the position of the free vortex will satisfy the first and the third equations of (2.13), or not. Considering that the performance characteristics of the propeller are mainly effected by the vortex system near its blades, instead of [Jcfk) + [zJ-- and [veJfk) -I- [zi'9]0-,, we may approximately adopt [,,lChk) -1- [zJ

and [vo(h)+[üol(,,,,) at the representative point on the blade or the average values of [i3,ihk)

+[alflk) and [i39](hk)H- [iOi(k all over the blade, which are denoted by Vu, and .Qru9,

respective-ly. Thus we get

ç0(r) = 2(14-u0), h = C V/Q)(1 ± u)/(1 -- u0) (3.3)

from (2.13). In this section, however, the case of light load condition, that is,

(3.4) is adopted in order to simplify the calculation.

The velocity of such a non-uniform flow as a wake of the ship is the function of the

position only, and further can be assumed to vary little even if the x1coordinate shifts to

some extent. Thus the components vr, Vr and r0 of flow velocity are assumed to be functions of r and O, and so are expressed as follows:

v,= v,(r, O), v= v,(r, O), vo= vo(r, 8) . (3.5)

Hence their values on the kth blade are obtained from (3.1) and (3.1)" as follows: [vr1cbk)=v(r, OM(r)+(r) vQt±2r(k-1)/N)

[Vr)(?O,) = vr(r, OM(r) + J(r) y - Qt + 2ir(k 1)/N)

[vo](bk) = v0(r, CM(r) + O(r) y - Qt H- 27r(k - 1)/N)

Defining the harmonic number as integer n, we expand the components of velocity V+[vJ(hk),

where

{V+v.(r,O)} e° do

v(r, O)e1Th9 _do

(37)

22V

-{Qr±ve(r, o)} e° dO 2ir

-Considering that the propeller is working in the above-mentioned non-uniform flow and consists of a set of identical, symmetrically spaced blades, we find that the quantity rk(r,y,t) is equal to r1(r, y, t-2'r(k-1)/(NQ)), _{i.e., the strength of vortex on the first blade before time} 27r(k-1)/(N.Q). _{For convenience, denoting r(r, y, t) by r(r, y, t), we obtain}

r(r, y, t)=r(r, y, (i1t+2ir(k-1)/N)/Q)

. _(3.8)

Substituting (3.2), (3.4) and (3.8) into (2.14), we obtain the following expression:

-5° dr' 5 dv' [ 5

rO', y', (j'Qt+2ir(k'1)/N)/Q)

4ir c -I k'=t

h

-

_d11

(3.9)

\ h 0v,. r' 0E0 _-/0)'

where L ] represents the value of [ i when 4!', r,. and v1 _{satisfy the following relations:}

o/'=OM(r')OM(r)+(J(r') v'(r) y,

v.= 0MO) + (](r) v(x+ Vt)/h, vo= 0MO) + (l(r) y (O + Qt).

Then from (2.7) and (3.9) we can calculate the following induced velocity:

W,= , ¿09= . _(3.11)

Ox Or rOO

Here, since the point P,,, on the mean surface of the k-th blade _{can be represented by}

x=h(OM(r)+ O(r) y) Vt, O=OM(r) + O(r) y Qt+2r(k 1)N,

we get v=0, 19=-22r(k-1)/N at the point PIk. _{Hence the following components of induced}

velocity at the point P,0, can he obtained:

[wXJ(ik) 5°dr' 51 dv' [ ₅ flr', y', - (çc' 4!I ô)/Q) 4ir _'j -1

k't

q,0

Oír'

O h

ô\1

1 X hOv,

'7î ôr, -

_{) R} dço1

j00

EWri(hk = 5° dr' 5' dv' [-t k'=i rO', V', (çp ç!' + 0.)/S2) O h O i X Or \ h 0v, r' Oc0) Rf2 dc1 [W9](bk.) 1 dr'

5 dv' [ TO', u', (çuISbl+òk')/Q) 1V rß 1 0. i O

(r'

O h

O \

1 X rOe9

h O

r' 0e) --

dc0 V+ [y ,l(hk) = v(r) e"°M' +0(r)v-2i+ (k- 1)/N), [VrJOk) =

v(r)

.Qr+ [V0k) = v0(r) e OM)+ ).Qi+2,«k 1)/N), (3.12) } (3.10) f (3.6)

In a way siniilar to V+[v,j(,k) and Qr+[v91(bk), the function r(r, u, t) contains the terms of oscillating functions whose periods are 2ir/n with respect to W, and so can be expanded as

y_t

_(3.14)

r(r, y, t) = r(r, u)

Ea

(r'O

h 6\1

AOk'k(r, r, h ; ) = _{LrOe0 \ h Oc,}

-

_{r ôe0j}

-

_{R1 £=O, e9=-2rQc--1)/I}

and the expressions of (3.13), of course, hold in (3.16). Then substituting (2.25) and (3.1)" into

(2.15), we obtain the boundary condition on the k-th blade as follows: Oxb(r, y) V+ Lv1(k + [W,J(k) - (Sir + [VIl(h) +LW91(hk)) = O , (3.17)

r(r)6v

hence, h êx,Ár, y) (Sir +Lvo](k)) - (V+ [v,,]cbk))

-

Lwo1 = r

r(r)0v

Substituting (3.15) and (3.16) into this equation, we get

1

dr' Ç r,1(r', y') dv' Ak.I(r', r, h; h AOkl(r', r, h ;

n 4 .J - t

r

= Ox,,(r, y)

n

{(r)

Ou

ven(r)_vn(r)} eL (9M(?)+O(r)V+')

This equation must be satisfied independent of k and t, and so for each of n we get

dr'

r(r', v')[r B(r', r, h ; ) h B9(r', r, h ;

)] dv' 47rr rB 1 ( Ox(r, y)

- i r (r) 6v

v9,,(r) - v(r)} ei" (0M(r)+(r)) where 1

dr'

1

r,(r', y') dv'k'1 Ç Aen'k(r', r, h ; ) e

lo' d1

n

Substituting (3.14) into (3.12), we get

i o = dr' r,(r', dv' [w](bk) y')

A(r',

r, h; ) e r, h; çv1) e1 Ok' dçoj D8k' dp1 _(3.15) ° dr'

r(r',

[Wrj(1)k) = v')dv' 4ir

i

k'=l where r, h; r, h ) = ) =

ra Ir'

O h E

6fr'

;

O

h' 0\11

- -;

) R JCO, £9=-2nC)N

h 0'11

-

(3.16) L

r' ac;}

Jo, C-2(k-1)/N

l's

R. YAMAZAKI where

= O(r') - Ok(r) + (r')v' - (r)v, ô = - Sit + 2ir(k' - 1)/N

(3.13) Rf1/h2(pl+ Cn)2+ r'2+r-2rr' cos (ço + e9+27r(k'l)/N)

(3.17)'

n B,,(r', r, h; = AkI(r', r, h; i)

d1

Brn(T',r, h; ') = S Ak'I(r', r, h;

) e_2'_/

, (3.19) B9,(r',r, h: ')

5Ao.(r'.

r, h;

) ei+'0/

d1 b=OM(r')O(r)+((r') v'O(r) y.

The equation (3.18) is the fundamental to calculate the value of r(r, y). In order to solve the integral equation (3.18), at first, the function T(r, y) must be expanded into series of proper functions w(r) [m=0, 1, 2,...) as follows:

r(r, y) ₌

1v

(A w0(r) + w1(r) + ..) + /f_ v (A w0(r) + _{w1(r)+ ...)}

1+v

+ v1/1_v (A w0(r)-I-Ar w(r)+ -)+ (3.20)

where etc. are unknown constants. Substituting (3.20) into (3.18), and then satisfying

this equation at all the points P,5(r, y) on the mean surface of the blade, we can obtain the simultaneous equations with respect to the unknown constants etc. The simul. taneous equations obtained here are practically solved on the assumption of containing finite

number of unknown constants, and the two methods are considered as the procedures to

introduce the simultaneous equations. The first is the method that the simultaneous equations are derived by satisfying (3.18) directly at the proper points (r, y) chosen on the mean surface

of the blade, whose number is equal to that of unknown constants. The second is the

method that the simultaneous equations are derived by multiplying both sides of the equations (3.18) by the weight functions of y, which correspond to the finite number of functions of y

included in (3.20), then by integrating definitely with respect

_{to y between 1 and 1, and}

finally by satisfying these equations at a certain number of the blade sections r. In this paper as well as the previous paper,° the simplest form of the second method is adopted.

Meanwhile let us consider about the two-dimensional airfoil in the gust [Ref. Appendix I). It is assumed that the forward velocity of the airfoil is constant and the velocity distribution

of flow perpendicular to the forward direction is _{sinusoidal and, of cource, the other}

com-ponents of movement of the airfoil and velocity

of the flow are

neglected. Then the

instantaneous chordwise distribution of bound vortex located on the two-dimensional flat plate airfoil in the above-mentioned flow is the same shape as that of the flat plate airfoil advancing steadily with a constant small angle of attack in a still fluid. Assuming that such an

instantaneous shape of chordwise bound vortex distribution on the two-dimensional airfoil in

the sinusoidal gust can be applied to the case of the propeller working steadily in _a non-uniform flow though this case must be considered as a kind of unsteady three-dimensional problems, the bound vortex distribution r(r, y) on the blade for the harmonic number n is

expressed approximately by the following simplest relation of (3.20):

r,(r, v)-(1/ir) T'n(r)V(lv)/(l+v) ,

where 1<v< i

(3.21)

and P(r) represents then n harmonic component of the circulation around the blade at the

section r. Further the circulation f'k(r, t) around the k-th blade can be defined as follows:

lk(r, t) 5Tk(r y, t)dv

and so from (3.8), (3.14), (3.21) and (3.13) we get

Substituting (3.21) into (3.18), and multiplying both sides of this equation by the weight operator i we obtain where

= s+2ir(k 1)/N,

ç!', =O,,f(C')OM(E)+ 6(C')v'(C)v, B,,,(E', E, y; ç!',) =

, Ç

Ak',(C', E, y; çi) e',1''»'

dip

B(Ç', E, y; q,,)=

Ar'1(', E, y; ip) e,+2'I>/

dip B9,(C', E, y; q,,)=

:,A(E', E, y; ip)

2'-I)/N)

dip

Aç'j(E', E, y; ip)

-+ ¿'-2E'Ecos(ip-+2ir(k' 1)/N)' 3v'ip {E'ip- E sin (ip + 2ir(k' - 1)/N) }

i

_{Ç° r,(r') M(r',}

_{r; h) dr' = v(r)} ir where

v(r)=

Ç'

/i+v

faxb(r,v) -i V

1v

r (r)

v

v9,(r)v(r) } e(8+t dv

M,,(r', r; h) _Ç

1/i+v

dv Ç

/1v' {rB(r'

r, h; q',)hB9(r', r, h;4b)} dv'. ir

r 1v

-IY 1+v'

For convenience, next, defining the following non-dimensional quantities: s= Qt, = nr,, f = rß/r0, oì = h/r,, , =V/(Qr,), G(e) =

vn(E) = v,,,(r)/(Qr,), v(e) = v(r)/(Qr,), v(E) = vgn(r)/(.Qr,), v(E) =

OM()=OM(r), (C)=(r), Oh(E, v)=Oh(r, y), x(C, v)=xb(r, v)/r,, G(e,$)=rk(r,t)/(Qr,'),

and setting

Ox(r, y) h y

r'(r)Ov

r

E,

cos (ip+ 2ir(k'l)/N)'

we obtain from (3.15), (3.16), Ezo1o/(Qro)= _n n (3.13), e" (3.19) and (3.21)

d'

-G,(C') dC' G,,(E') dC'

i f'

as follows: (3.26) B,(E', C, y; q,,) dv', 4ir e" B(E', C, ;q,,)dv',

e" ---

_4ir

/1v'

B(C',

, y;çb,)dv', 120 (3.22) (3.23) (3.24) (3.25)

where y sin (çø+2r(k'-1)/N) Ak'5(Ç',E, v;ç)=

---/Ø +

e'2 + e2 2E'Ecos(çø + 27r(k' - 1)/N) Z

3V {-' cos (ç+2r(k'-1)/N {'ç

sin (c'+2r(k' 1)/Np +C'2+ç2-2e'E ¿s(çD +2rr(k'- 1)/N) y cos (ç-I-2ar(k'-1)/N) Aok'l(E',C, y i/vç +'+ e 2C'Ecos(tp+27r(k'-1)/N)

+ 3vE' sin (ç + 2r(k' 1)/N) E'ço C sin (ç + 2ir(k' 1)/N} + '1 + S 2C'Ç cos (w + 2r(k' 1)/N)

E LfiO()(D'-V) (i v'v \

[B,,,,(C', E y; =

(E'E)8/2 + 2

e v'

[B9,', E, y;

= (e'E)21/

e1"

(i-Further, using (3.25) and (3.27), we rewrite (3.23) and (3.24) as follows:

4E

G(E)M(C',C;v) dE=vn*(E),

where

v(E)=-1-- Ç

i

/i+v

ÇOx(E,v)

V

1v l()av

v()v(E)}

dv,

M,,(E', e; y) =- "

/1+v dvÇ ,/Ii { B(e',

_{, y; b5)v B9(C', C, y; i')} dv'.}

j. (3.29)

Now using (3.6), (3.26) and (3.25), the axial, radial and tangential _{components of the inflow} velocity expressed as (2.16) at the point P,,, on the blade is obtained as follows:

[,j(bk)/(.Qro)= V _{[Vr1c)/(Qrs) Vr,, e"8, [VoJ(,,,.)/(S2rO)= V e1'}

, (3.30)

v= v(E)

V,=v(Ç) é(OiI» +

V=v(E)

+ 47r + _{B,(C', E, p; 4b0) dv',} 47r J

L

_G,,(E')dE'-L i 1+v'

G(e') dE'- Ç

/1v'

Brn(E', ¿, v;çl'0) dv',

2V _iV 1+v'

ÇG(E') dC'1

/i-

B9(e', , y; ç) dv'.

(3.28)

Substituting (3.30) and (3.31) into (3.17), and setting &xb*(f, v)/(E) _{v=w approximately from} (3.1)", we get

= (v/C) [Vo1(h) , namely, V= (v/C)

e" V

(3.32)

By solving the equation _{(3.28), the unknown function G,/E) can be obtained, and then the} non-dimensional circulation G(E, s) around the k-th blade can be evaluated by using the

following formula:

Gk(E,

s)=G,,(E) e3k

_(3.33)

obtained from (3.21)' and (3.25).

Assuming that the fluid is _{inviscid and incompressible, the force of fluid acting on the} }

(3.31) (3.27)

1 4r

U,(e;m,l;f(v))--Ur(E;m,l;f(v))--

/1._v

Uo(e;rn,l;f(v))--

v/i__v

bound vortex in the elemental region dvdE of the propeller blades is represented by (2.17),

and so, from (3.1)', (3.1)", (3.14), (3.21), (3.25), (3.30) and (3.31), we can rewrite (2.17) as follows: d2F p9 r04 -- em+0 _{G ,(E) j/}

iv

.{ vi,,, Ob(E, y) } dvdE ir 1+v

- v

¿Ob(E, dvdE Ob(E, y) _v dvdE '-"j where

Ob(E, y)= O,(E)-I- (E)v, O= Or(E) + E(E)v 4- ô. (3.35) Further the force caused by frictional drag acting on the blade element dvdE is expressed by

(2.18), and so from (3.1), (3.1)', (3.25), (3.30) and (3.31), we get

d21D=p9°rO4 -

(i+

-

)

e""k V

Vr,, dvdE

2 m,m

d°FDp9°rD4 'CD p1(E) (i+

4)

''k V Vm dvdE

(3.36)

2 E rn,,,

d°FD=p9°rO4

-CD](E) (i

_{-) e} e°"'°k V2 dvdE.

The force and moment acting on the propeller are obtained by using (3.34), (3.36), (2.19),

(2.22), (3.25) and (2.23). In this case the effect of the rake e on the length of lever xb*(E, y)

can not be ignored, and, for simplicity, we neglect the effect of the components of the bound vortex in the tangential and axial directions. Therefore, defining the new functions

where i, ni and n are integers and f(v) is a certain function of y, and denoting the components of force and moment of ideal fluid acting on the propeller by F,0, F0, F,0, M,0, M and M in

a way similar to (2.23), we obtain the following expressions from (3.34), (2.22), (3.25) and (2.23):

4

- -

f'

e'° G,(e) U9(E rn 0 1) d

p9 r0 1fl,n c=i

V 1

Ç [

e'"'»6k G,,(E)_{U(E; rn,}

1; 1)i

e°""'3kG,,(E) U,(E;rn, -1; 1)1

p9 ro 2z B

-

4 =

i

f1 [

e«"''° G,,() U (E in 1 1)+

e'"''

G 0(E) U ( in -1. 1)1 d

p9 r0 2 m,n J

O -

e«"'°G (E)U(Enz 01)Ede

p9 r0 m,fl Je

-

ri

G(E) U9(E; m, 1; 1)

f

e"'k G,,(E) U9(E; in, 1; i)}

p9° r1° m,n ißL2i t k1

(3.34)

d2Fro p9° r04 e m+n)&k G(e)-/ 1v J±_

v

°' V 1-i-v (E 7V mm dF90 p 9 r04 _

ém+k

G(E)1/ 1v {

1+v itm,m Vr.,,, e0M0) f(v) dv v°,, (OM()+e(e)v) f(v) dv (3.37)

V e°" f(v) dv

± e'' G(Ç) U(E; in, 1; Xb*(, v)/E)+ G(Ç) U(Ç; in, 1; xb*(E,

2 s=i

M0

- ç rI

ei(m+1s G(E) U9(E; in, 1; 1)± G(e) U9(E; in, -1; i)}

p.Q r06 m,n EBL 2 lk=1 k.1

e« G.,,(e) U(Ç; in, 1;Xb*(ç,V)/)

-

G,) U;rn, -1; x(e,

2z

si

where

v)=v Oh(, v)+e=vO(E)±viJ()v+Ec

(3.38)

The

lift L per unit length acting on the section

of the first blade at the time t is evaluated as follows. From (3.32) the force L is approximately normal to the helical vortex

surface with constant pitch 27th. It is assumed that the chordwise gradient of inflow velocity

streaming on the blade section is negligibly small and the effect of the axial and tangential components of the bound vortex on the lift is very small. Then, denoting respectively the

axial and tangential components of the magnitude of the lift

U =L1(r,t) by L(r,t) and

Lo(r, t), from (3.34) and (3.7) we obtain

L,(r, t)= d2F0 dv=pS21r63 G(e, s) V91 -i rd.dv

L0(r, t)=

rv

dv=p.Q1r02 G1(E, s) V1

where

T/=j eth8k U(e; n, 0; 1), Vr5. e1' Ur(Ç; n, 0; 1), V95=>j e6k U9(E; n, 0; 1) . (3.40)

In this case V, Vrk and Vok express respectively the axial, radial and tangential components

i of the non-dimensional average inflow velocity streaming along the k-th blade section. From

(3.39)

On the other hand, the circulation around the

first

blade section at the time

t can be

expressed by 9r12 G(E, s) from (3.21), (3.25) and _{(3.33), and then, denoting the magnitude of}

chordwise average inflow velocity streaming to this section by Qr0W, there results:

L(r, t) :pQ1r01 WIGI(, s)

And so, by comparing the above two equations, we obtain

V912 (3.41)

Therefore, neglecting the extremely small quantities, the non-dimensional chordwise average inflow velocity W streaming to the k-th blade section can be expressed by

w=

Vv01+ V952 = [W0j (3.41)'

Defining the non-dimensional chord length by c1()c(r)/r1, we get

c1() = 2(e)/2 + . _(3.42)

from (3.1), (3.1)", (2.29) and (3.25). Denoting the section lift coefficient of the k-th blade by

CLk, s), the coefficient CLI(E, s) can be defined by

CLlf, s) = L(r, t)/ {(p/2)CQr0 )2(r0c())} . (3.43)

Then from (3.45) and (3.47) we obtain (3.39) we get approximately

CLI(E,s)=G1(E, s)/(1(E) T,4T /E2+.,2 ) (3.44) and further the relation

CLk(E, s)=CLI(E,ôk) (3.44)'

holds. In this section we have obtained the unknown quantity G,,(E) not by solving the

equation (3.32) exactly, but by solving the equation (3.28) which is multiplied by the weight

operator (3.22). Therefore the quantity G,,(E) obtained by solving the equation (3.28) can not exactly satisfy the following equation:

1'Ç=(v/) V , (3.45)

where V,,,. and V,,. are defined by (3.40). This equation is obtained by multiplying both sides of (3.32) by the integral operator

I

dv/1

However, considering that the equation (3.45) holds approximately, we can get

Wk/1+V/E2 V9,. (3.46)

from (3.41). We have deduced the expression (3.36) according to this idea.

The force and moment caused by frictional drag acting on the propeller can be obtained

by substituting (3.36) into (2.22) and then by integrating definitely these quantities with respect

to y and r all over the blade surfaces.

We denote these components by F,,,, FED, F,D, M,, M1/D and M. Now, for simplicity, we consider the definite integral (2.23) with respect to y

along the blade section as follows. Namely, using the section drag coefficient CD as C of

(3.36),

this new CD depends not only on the shape of the blade section but on the lift

coefficient which varies with time, and further the local variation of relative velocity on the mean surface of the blade section is assumed to be small. Then we can use U,,(E; n, 0; 1), Ur(E; n, 0; 1) and U0Cf; n, 0; 1) defined by (3.37) in place of V,,, Vr,, and Vi,, in (3.36). Thus

the chordwise integral can be simplified. Hence, neglecting the small quantities of higher

order, we can obtain the following expressions:

= Ç' _{i(E)(1 + 4'\ C» U0rn U,, EdE}

p.Q r0 _JB

\

E )

k=1 ln,n

)sin e(E) C» {e 1,.+W.w() U9rn(Ur,,+jU0,,)

F»

1Ç'(1±

N

p!d°r04 2 E' i =,1 m,n

+

e'6k°M

U9,,( U,.,, - iU0,j} EdE

F» i

Ç -_)sin ti(E) CD Uo,,,(iUr,,+ U9,,)

p!?r0 2 e .=i

F e L+nI)OM( U0,,(jUrn+ U0,,)} Ede

M

Ç (E)(1+ °

)

C» e1""

U U e°d

p!? r0 E mju

i

_{Ç (i+}

:

CD U9,,,[e' +"+' _{{i Esin (E)U,,}

p2 r0 2 EZ E k=l

((ec +vO(Ç)) sina(E) +iv(sin_o(E)e(E) cos o(E))) U9,,)}

JWW i

_{' (i}

+

4-)

C Us,,, [ei< 1)+IOM(f) {

-

sin ) U,,,

pS? r0 2 k=1 m,fl

(i(e+VOM()) sin (Ç)+i(sin ()o(E) cos ())) U9,} +e"°M

x {E sine(e)U,,,(i(e+vOM()) sin /)(ç)+(sin 7()-J() cos L))) U0}] ede

where

U,,,=U(E;n,0;l),

(J,=U(Ç;n,0;l),

U9=U0(4;n,0;1).

(3.47) Here the section drag coefficient C is a function depending on the section shape, the section

lift coefficient C1(, s) and the roughness of the blade surface, and accordingly, varies with time. Namely, CD is the known function of the section shape and CLI(E, s). For example, for the section shape of the Troost type propeller, J. E. Kerwin gave the following formula :0

CD=0.0085±O.001 (

c1,

J80

where a is the zero lift angle from the base line of airfoil.

Therefore using (3.38) and (3.47) we can calculate the components F,,, F.,, F., M, M and

M, of force and moment acting on the propeller as follows:

F,,=F,,O+FD, F5=F5Q+FD, F,=F,Q+F,D;

M,,=M0 + MD, M, M0 + MMD, M,=M,, + M.,.

Furthermore, for simplicity, we define the following non-dimensional coefficients:

KF- K 7r2F5 4p90r44 4pD2r0

- 8p2r

KM,, 1 e1"k = { _o Ne_im

_{for n=rnN}

for nm.2V

KF, KM, = 4p.Q0r04 .2W, 8pS2r00 (3.48) (3.50)

These coefficients which express the instantaneous force and moment acting on the propeller can be calculated by (3.49). Furthermore we can simplify the expressions (3.38) by using

the following formula:

(3.51)

where rn is integer.

Next we consider the thrust and torque of the propeller. Denoting the thrust and torque

except frictional drag by T0 and Q0, the thrust and torque caused by frictional drag by T and QD and the total thrust and torque by T and Q, there results:

T0=

--F.,

Q0=1W,0, TD-- F,0, Q1 lW,, T== Fa., Q=A1.

Defining the following nondimensional coefficients by using revolution per unit time n=Q,(2r) and diameter D=2r0:

C70=T0/(pnD'), CTD=TD/(Pflr0D1), C7=T/(pn2D1)

=

C =Q0/(pnD0), CQD =QD/(Pflr0D'), CQQ/(pn2D0)

and using (3.38), (3.47), (3.48), (3.40), (3.37) and (3.33), we obtain

(3.52)

(3.53)

1 N 1 2 N I

C= --

Gk(E,$) V9-dE, CQO= _\ G(E,$) VÇdE,

4

Jß

8 k=1

2 N

C7

_-

a(E) (1+ --) (CnVeV,k) EdE

EB 7c=1

CQD= Ç1 6(E) (i+ (CDVek2) E2dE

.)EB

\

E-'='

Cr=CTO± CTD, CQ=CQO+CQD

Thus we can calculate the instantaneous thrust and torque of the propeller by using (3.53). Now let us further consider the concrete, supplemental procedure according to the

above-mentioned theory. Denoting respectively the local longitudinal and tangential wake fractions

by w and W, we can express as

w= v,JV, wT=vo/V . (3.54)

If the local distributions of the wake fractions are given experimentally or theoretically, we

can obtain the values of

v, and y0. The wake fractions are usually symmetrical about the

xy-plane which corresponds to the center plane of the ship, and vary little along the x-axis.

In this case we can expand w and UT as follows:

(V+ v)/V= 1w=a0(E)+ 2a(E) cos 0+2a2(E) cos 20±

= a) e°,

(Qr+ve)/V=(E/vo)±wr ib(E)+2b1(E) sin 0±2b2(E) sin 20+

=(ib(E)) e°,

where

an(E) = a_,(E) = (1w) cos nOdO, b0(E)=iE/v2

b,1(E)= bfl(E)

.0 w sin nOdO

for nO

From (3.55), (3.6) and (3.25) we get

v(E)=v0 a,(E),

v(E)= iv2 b,) .

(3.57)

The expressions (3.57) agree with the results obtained by (3.7), (3.54) and (3.56). Defining ,

', G) and i'(E) by

E'= i/(1± ÇB2)/2+ (1 Eß2)'/2, E= /(1 ±Eß2)/2± (1 Eß/2, G(e)=G,(), v(E) = v(), (3.58)

we can rewrite (3.28) and (3.29) as follows:

i

Ç

G(') F(',

;y)

= v(V),

7t _J-1 where (E'2E2)2 f1

/1±v

_dv

/1v'

4(1EB2)E'

L'

V

1v

i

V i+v' x {Bn(E' E, y; b2) -B0(E', E, y;

v () = v()

(

_\

(E)

z'(E) - va(E)'\'

/ + y e010»

dv,

E

1v

(3.55)

(3.56)

(3.59)

i

I'

/1+v

x(E,S

dv.

ir -t

T iv

(i(Ç)av

When the particulars of a propeller are given, the values of N, ,

O() and )(E) are

known, and further, if the values of V, Q, w and 10T are given, we obtain 21 and then v(E)

and v() for

y, . On the other hand, though it is valid to use the approximate formula

(3.3), that is,

as the 21-value, we usually, for simplicity, adopt 21 obtained from the pitch of base line at the

representative section i=T/(1+B')/2 of the blade or y obtained from

21

+ ®(,)')(,' + v(E,)') (2

- )(E) v0())

()(,) + v0(,)),

2

at the same section. In any case, settling the value of y, we can calculate F,(2', '; y) in

(3.60) and then obtain G,) by solving the equation (3.57). Thus the non-dimensional

circula-tion G, s)

around the k-th blade is obtained from (3.33). Now from (3.31), (3.37) and (3.40) the axial, radial and tangential components V,k, V, and V0,. of the non-dimensional average

inflow velocity are expressed as follows:

e"2& { v(E)

I'

/

1v

dv - G,() F,,,', i; vi _{I }}

n ir -i 1+v ir -1 ( 72)

=

_{e'- { vn(E) 11}

}/1_ V _{dv G,(i2) i; y)}

_{_}}

(3.63) ir _-'

_i+v

_ir -1 (V ,7) , V,,.- =

e"ic { v,()

/

1v

e1 dv -

G,) F,,',

_{; vi} n ir -i 1+v ir -i ( ) (3.61) (3.62)

Using the values of G,() obtained previously, we can calculate V,. and V,,. from (3.64) and

(3.63), consequently, W,. from (3.41) and (3.41)', CL,.(, s) from (3.44) and (3.44)', CD from (3.49) and C,- and CQ from (3.52). _{Similarly we can obtain the components of force and moment}

except C,- and C0 by using (3.38), (3.47) and (3.48).

Finally we examine the pressure on the surface of the propeller blade. In this section the load

acting on the blade have been mainly treated, and so we have considered the angle of attack and the shape of camber line of the blade section. However, we must take account of the effect

of chordwise distribution of blade thickness on the pressure near the blade, too. _{The type of} blade section which includes both the shape of camber line and the thickness distribution is assumed to be given previously, and it is sufficient in this case to consider the first blade

only. _{We may consider quasi-steadily that the chordwise distribution of the pressure on the} blade surface at the time t agrees approximately with that of the steady two-dimensional

airfoil at the same load condition though the propeller blade belongs to the unsteady

three-dimensional wing. In the steady two-dimensional airfoil theory, denoting the pressure reduc.

tion on the

airfoil surface from the atmospheric pressure p, by A], and the inflow velocity

by U, the non-dimensional pressure coefficient C is defined as where

F,.,',

; 21) =

-F(',7);v) =

-F0,,(,/, ,7; V) =

-, y;41',) dv', ,_{y; 41',)}dv', , y; 9",) dv'. (3.64)

(e'2E2)2

iÇ'

_dv

/i'

_B,n(',

4(1-8)Ç'

2 )_ 1+v -i

i+v'

(2E2)2 i _Çl dv 1' /i - V' _Brn(',

v/'v

(1_ß2)I _{1+v -i 1+v'} (F2_e2)2 _i Ç'

/1v

_dv /i_ V' _B0(e',

128

- Ci,, 4p/(p U°/2). (3.65)

For a given type of two-dimensional airfoil and a given lift coefficient CL, the value of C can be obtained theoretically or experimentally.

In other words, C is a known function

of U and CL when the type of airfoil is given. Applying the steady two-dimensional airfoil

theory to the unsteady propeller theory, we may set

U=Qr0W1, CL=CLI(C, s). (3.66)

Further denoting the vapour pressure at the temperature of water by Pe and the depth of blade section from the free surface of water by d, the condition of occurence of cavitation

on the blade surface is expressed as follows:

4ppgd=>p0P.

(3.67)

Since W1 and CLI(E, s) on a certain blade section can be obtained from (3.41) and (3.44), by

using C, (3.65) and (3.67) we can decide whether the cavitation will occur or not. And further

it may be

sufficient that, in order to decide the occurence of cavitation on a blade

surface, we examine the maximum value of C on the surface of a blade section. Denoting

the thickness-chord length ratio by tQ the maximum value of C2, of the steady

two-dimen-sional airfoil is expressed by f(CL, t0), where the form of the function f(C1,, t) of CL and t

depends on the type of airfoil. Besides, the form of f(CL, t) for the face is different from that for the back. Thus denoting t0 of the propeller blade by t0(E), the condition of occurence of cavitation at the section is rewritten as follows:

1 (1,0+ 2gd \ f(CLI(C, s), t0(C)) 2l472 nODI) where (3.68

=(P0 e)/(pnr1D2/2) cavitation number.

The form of f(CL, t0) is, for example, given experimentally by F. Gutsche.°

§4 The Quasi-Steady Theory of Screw Propellers in Non-Uniform Flows

It is assumed in the above-mentioned lifting surface theory that the strength of the free

vortices fluctuates with time in their positions fixed relatively to the propeller. In this section we deal with the quasi-steady theory, in which the free vortices are considered to be following along helical surfaces from all the blades to infinite rear, retaining the constant strength

which is equal to the strength of the bound vortex at a section C of the indicated blade

at the time t.

Namely, in the fundamental equation obtained by substituting (3.9) and (3.11)

into (3.17), we may replace the strength of bound vortex Tk(r, y,t-ç/Q) with the function

r(r, y, t) independent of k and ç.

Then we can solve this equation and obtain G(Ç)

corresponding to r(r, y, t) which is denoted by GQfl(C). Then by comparing the above funda-mental equation with (3.59), we get

-

F0(2', ; y)

- v() ,

(4J)

Jr .-1

where F0(', '; y) and v,(ij) are defined by (3.60) and (3.62), respectively. In this case the

non-dimensional circulation GQk(E, s) around the k-th blade is obtained from

Furthermore in the former usual quasi-steady theory the normal component of inflow

velocity streaming to the blade section at the time t

is assumed to be constant along the

chord. Then instead of (3.6) we get

V+[v,JQ)k)= v,,,(r) eMk), .Qr+[vG](bk)= v,Xr) eiM+ki

n n

Accordingly, in (4.1) we may use _{instead of v(v) as shown in}

(3.61). Therefore denoting GQ,() in this case by GQ'fl(E), we get

-

I

_G,,(')

F0G7', ; y)

_-

v(e)v(e))

(4.3) Using the solution G0.,) obtained by solving (4.3), the circulation G9(Ç, s) around the k-th blade is obtained as follows:

GQS(, s) = GQ.k() e' _(4.4)

n

Following P. D. Ritger and J. P. Breslin, T. Krohn7 etc., we can obtain GQ'k() modified by applying the unsteady two-dimensional thin airfoil theory. Denoting this new circulation of the k-th blade by GQ»k(, s), there results:

GQ'k(t, s) = GQ.',,() e'

n where

= GQ.(E) F(IC)

F(!C) = i / [i/C {K1(i,) ± K(i,C)} j: the Sears function,

,= n (): the reduced frequency.

The reduced frequency IC and the Sears function F(IC) _{are defined fully in Appendix I.}

Furthermore according to the above-mentioned quasi-steady theories, in (3.63) _{and (3.64) we}

can use F0(r/, r; y) and F00(', ; y) instead of F,.,,(', ; ) and F0(V, ; y) for various n. Then, for example, denoting Va,. and V0 for GQ.',) by _{and VQ'Ok, respectively, we get}

= :i e1

_{{ v()}

1

VQ.'ok = e0ik

v()

1

5

_{Numerical Example (1)For a Given}

Propeller in a Given Wake

When the particulars of a propeller, the

advancing velocity V, rotating angular velocity

Q, the local longitudinal wake fraction w and the local _{tangential wake fraction} w7 are given, let us calculate the non-dimensional

circulation G1(, s) around the propeller blade, the components V,1, V11 and TV1 of inflow

r'

_d'

)

G',,(')

; y) ;,

CI _dz"

G»,') F00(',

; y)

(2'v)

- I (4.7)

sing (4.5) and (4.7) we can calculate W, from (3.41), CLI(, s) from (3.44), C _{and CQ from} (3.48) and (3.53), KFZ, ICE,. etc. from (3.38), (3.47), (3.48), (3.49) and (3.50). These new quantities 14/k, CLk(c, s), CT, CQ, KF,, K, etc. are denoted by W0., CQ.'Lk(, s), _{CQ'T, CQ.'Q, KQ'Fn,} etc., respectively. _{Similarly we use the subscript Q and Q' in the quantities V., V, Wk etc.}

obtained for CQ,,() and C0(El, respectively.

velocity on the blade, the section lift coefficient CLS(Ç, s), the thrust coefficient C. and the tor-que coefficient CQ for the advance coefficient J:

J_ V/(nD)=irv, (5.1)

according to the process of Appendix II for

m=7. Further w and Wr are assumed to be

independent of J.

As an example, we adopt a propeller of the

single screw fast

"dry cargo"

ship with

block coefficient 0.65 and a non-uniform flow

(4.5)

j

corresponding to the nominal wake fractions of this ship given by J. D. van Manen." For

convenience, this propeller is entitled "Pro-peller M." The particulars of Propeller M is

shown in Table 1. In this table, ((C) obtained

from (3.61) can be calculated by using

Appen-dix II of the previous paper" and the formula

of the zero

lift angle in the Glauert's thin

wing theory. Furthermore we assume the pitch ratio as 7r11=0.9546, i.e., iì=0.3039. Now

the local wake fraction is considered to be

in the longitudinal direction only and the local tangential wake fraction WT is neglected.

Namely, from (3.56) it is assumed that b() =iE/i'

and b(C)=0 for nO, and the value

of a(E)=a_,,() is shown in Table 2.

Neglec-ting a,,()=a_,,(C) for n9, the circumferential distribution of 1-w is shown in Fig. 2. As

shown in Fig. 2 the distribution of 1-w used in the evaluation of this section is different

from the given wake in the neighborhood of

0=0. Furthermore the sign i shown in the figures and tables of the following sections represents the designation number of blade

section as ¿=[i=1,2, .-, 7, and the relation

between i and E is defined in Table 1.

At first we treat the two cases: one

is

open test, i.e, 1-w==1 and the other is the test in the circumferentially averaged wake, i.e., 1-w=a0(C). The expressions for the both

cases are independent of time and so

are sufficiently treated as the case for n=0 only. Thus we can calculate C. and CQ by using

the expressions in §3. Denoting CT and CQ

for open water by CTQP., and CQ0,,, and CT

and CQ for the circumferentially averaged

wake by CTaO and CQ, respectively, their

calculated values are shown in Fig. 4 as

ordinate with abscissa J=irv1. As shown in

Fig. 4, we obtain CTO,=0.189 for J0.679,

and this result is a little different from CT=

0.193 obtained by J. D. van Manen. Now the J-value satisfying CTO=0.193 is 0.669.

Next let us calculate the hydrodynamical

unsteady quantities varying with time when

Propeller M is in the non-uniform flow shown

in Fig. 2. Using the procedure of Appendix

II, the integral equation (3.59) can be solved

and the solution G() are obtained as the

sum of real part G(E) and imaginary part

G1(E). For example, the solutions for J=0.8748

are shown in Table 3. Thus using the

ex-pressions of §3, we can obtain the values of

G(ç, s), V.,,k, V., Wk, CL(C, s), CT and CQ and then their average values with respect to s

in the range from O to Zr. For example, the

values G1(E, s), Wi, CLI(C, s), CT and CQ versus s for J=0.8748 are shown in Figs. from 4 to

7. Denoting the average values of CT and CQ with respect to time by Crna and

respectively, and the differences between the maximum and minimum values of fluctuating

CT and CQ with time by 4CT and 4CQ, respec-tively, the values of Cr,,a,, and CQ,ea,, versus J

are shown in Fig. 3, and the values of 4CT, 4CQ, IiCT/Cr,,ean and 4CQ/CQ,,,a versus J are

shown in Fig. 8. Similarly the quantities

ob-tained by using the quasi-steady theory are

shown in Figs. from 3 to 8 in addition. Now the circumferential average values of

1-w are equal to aa), and further let us

define the volume average value, i.e.,

1-(nomi-nal mean wake fraction) as

i C CI a=- (1-w)dO CdC Zr Jtß Jo JEB Ii 1 = _{a(C) ¿ d}

/ -

_{(1- Es-)} . (5.2) EB

Then calculating (5.2) by using Table 2, we

obtain ã0=0.7571. On the other hand,

com-paring J=0.8748 obtained so as to be C7,,,,,,, =0.193 with J=0.669 obtained

so as to be

Co=O.193 as shown in

Fig. 3, under the

condition of thrust identity the effective wake fraction can w. be calculated as follows:

1- w,=0.6690/0.8748=0.7647.

This 1-w, is a little different from a0=0.7571

obtained from the nominal wake fraction. Further the values of CQ,,n and CQ»,a,, for

these J are 0.3102 and 0.3048, respectively, and

so the relative rotative efficiency eR is as follows:

eR=0.3102/0.3048=1.018

By the similar procedure, the values of 1-w,

and eu for the various values of CT,,,,0, are

obtained as show in Fig. 9. It is plain from

Fig. 9 that the values of 1-w, hold almost

constant with varying CT,,C,, and the values of e rapidly increase with decreasing

These tendencies coincide with the experi-mental resuits» Further, for reference, the

values of 1 w and eR obtained from C'-01».,,,

CQOfl, Grao and C in Fig. 3 are shown in

Fig. 9 in addition, and then eR in this case

is nearly equal 1.

Figs. from 5

to 8 show the quantities G(Ç, s), W, CL(Ç, s), C- and CQ varying with time s. It is found from these figures that all the periods of G0(Ç, s), W0. and CL,(Ç, s)

are 27r with respect to s, and C. and CQ are

periodic functions with the period 2r/N=ir/2. These results are theoretically obtained from

(3.53) and (3.51), too. The variation of Wr

with time s is almost constant.

Furthermore let us examine the occurence

of cavitation near the blade tip at the time

s. In this example given by van Manen, we

have i3O=2.79 and dv-O. From

(3.68) the condition of cavitation occurence on

the first blade is as follows:

f(CL(Ç, s), t0(Ç))0.283/Wj° , (5.3)

where W1, CL(Ç, s) and t0(E) are obtained from

Figs. 5 and 6 and Table 1.

Further as the

function f(C1,, t0), we adopt the K. Schoenherr's figure° for ogival sections obtained from the F. Gutsche's experimental results, since the

type of the blade sections near the tip,

i.e., at Ç, Ç and Ç are almost ogival.

Examining (5.3) for the sections Ç, and Ç,

the domain of the occurence of back cavita-tion are shown in Fig. 10. Let us compare Fig. 10 with Fig. 2. In response to the

domains where the values of 1w used in

the calculation of this section almost coincide

with, or are

larger than, those obtained

ex-perimentally in the cavitation tunnel, the locations obtained theoretically where the back

cavitation occurs almost agree with, or are narrower than, those obtained experimentally,

respectively. The face cavitation is

consid-ered not to occur at

all experimentally in

the cavitation tunnel, and the locations obtained

theoretically where the face cavitation occurs

hold the negligibly small domains

correspond-ing to

the neighborhood of the maximum

crests of 1w for Ç<0.92, which are greater than the experimentally measured values of 1w as shown in Fig. 2. The similar results

for back and face

cavitations are obtained

for the case using CQ'(Ç, s) instead of CLI(Ç, s).

Besides, as shown in Figs. from 3 to 7,

comparing the quantities obtained by the

unsteady theory with the results obtained by the quasi-steady theory without correction of the Sears function, both are almost equal each

other in magnitude. However, the phases

of G(Ç, s) with respect to s are advancing before those of GQ',(Ç, s) which coincide with those of wake fractions at the middle

point of chord

as shown in

Fig. 11, and

further these results are almost independent of J in the range from 0.5 to 1.0. Both the

phases of G(Ç, s) and CLk(Ç, s) fully coincide

because Wk is dependent little

on time as

shown in Fig. 5. Further the phases of C and C0 are advancing before those of CQT

and C.,, too.

The differences between the

maximum and minimum values of fluctuations of the magnitudes of GQ'k(, s) and CQ»Lk(Ç, s),

which are obtained by the quasi-steady theory

corrected by the Sears function, are far

smaller than those obtained by the

above-mentioned two theories, and the phases of

G0(Ç, s) and CQ»L(Ç, s) are between those of

the other two theories, respectively.

According to the results by S. Tsakonas

etc.,'° the similar results are obtained.

Finally the average thrust and torque of

Propeller M obtained experimentally are

com-pared with the results obtained by the above-mentioned theories as shown in Fig. 3.

Ac-cording to S. Nishiyama, both the theoretical and experimental values of the average thrust

and torque coefficients of Propeller M working

in the non-uniform flow represented by Fig. 2

for J--0.897 almost coincide each other as shown in Fig. 3. Namely, we obtain

experi-mentally = 0.1846 _and CO3,ea,o = 0.0293

for J = 0.897 and theoretically C7,,a,, = 0.184 and CO3an = 0.0295 for J 0.897.

§6

Numerical Example (2)On the

Effect of Blade Area Ratio

We shall calculate the non-dimensional circu-lations of the propellers with different blade

width working in a given non-uniform fow

in the same process as the previous example.