Calculation on unsteady propeller forces by lifting surface theory

'3 S2.

\98

ARCEF

Ref.: PAPER 22/7 - SESSION 2

(2.)

lab. y.

Scheepsbouwkut

Tedinische HogschooL

SYMPOSIUM ON

"HYDRODYNAMICS OF SHIP AND OFFSHORE PROPULSION SYSTEMS"

H0VIK OUTSIDE OSLO, MARCH 20. - 25., 1977

"CALCULATION ON UNSTEADY PROPELLER FORCES

BY LIFTING SURFACE THEORY"

By

T. Ito, H. Takahashi and K. Koyoma

Ministry of Transport

Ship Research Institute, Tokyo

Deift

ùnsteady lifting surface theory is presented, for which the computer

program has been devised at the Ship Research Institute. The procedure

employs a convenient numerical technique for solving the integral equa-tion for lifting surface, which was developed theoretically by Hanaoka. This approach is applied to systematic calculations to discuss the characteristics ôf the hydrodynamic forces on propeller blades

operat-ing innonuniform flow. The nature of the unsteadiness of the flow and

the three-dimensional effect are clarified quantitatively. The effects

of the skew and width of blade on vibratory forces as well as the effect

of operational condition of the propeller are shown.

Applicationsto numerical calculations of vibratory forces are

presented for five ships : a liner ship, a single-screw container ship,

a twin-screw container ship, a tanker, and a fishing boat. The

calcu-lated results of the thrust variation on one blade show good agreements with the experimental ones.

Áa-t

.

-

4:

1. Introduction

It is very important to predict the magnitude of propeller

vibra-tory forces in the early stage of ship design. The accurate prediction

of it is still difficult. It is not easy to get accurate experimental

data on the vibratory forces, although there are some experimental

studies [1].

There are a few accurate theoretical approach for the prediction

of unsteady propeller blade forces [2], [3], [4]. There are some

prob-lems to be solved, for getting the practical accurate prediction method.

One of them is to devise the numerical teçhnique for the integral

equa-tion for lifting surface.

Theoretical evaluation of hydrodynamic forces and moments acting

on a propeller requires the blade loading distribution obtained from

lifting surface theory. To evaluate the blade loading distribution of

a propeller with a given geometry under a given operational condition in a given nonuniform inflow field, is reduced to a two-dimensional singular integral equation, which cannot be solved analytically.

In general laborious computations are necessary for solving the

integral equation numerically, although there are many procedures

devised to solve it efficiently [5]. The method employed in the

pres-ent paper is based on the theory developed by Hanaoka[6], [7].

2. Theoretical Method

2.1 Integral equation for.lifting surface . .

The propeller with itsI-identical and symmetrically spaced blades

is consideréd to operate in weak nonuniform flow. The fluid

surround-ing the propeller is assumed to be inviscid, incompressible, free of

cavitation and infinite in extent. Assuming that the blades of the

propeller arelvery thin, the mean surface of them can be replaced with

bound. vortex sheets. The disturbed velocity induced by the boss is

assumedto be neglected. The theory dealt with here is.basedon a

small perturbation approximation, sothat

the steadyandthe unsteady

phenomèna can be analyzed separately.

In general the boundary conditions should b satisfied on ail

blades.. In the present case, as it is considered that the velocity

distribution in the hull wake varies nottemporally but spatially and

the blades are rigid, the phenomenon is a symmetric oscillation [8]

The boundary condition satisfied on one

blade is sufficient for the

symmetric oscillatión phenomenon as in the casé of

steàdyphenonenon.

Concerning the surface. on which vorticés are distributed,

the

non-linear propeller theory based on the assumption of constant

hydrodynam-ic pitch [7] is employed in this paper. tn the theory the influence

of the inducéd velocities on the shape of free vortex sheets is :taken

into accoünt on a spatial and temporal average. In this case the

pitch of the hydrodynamic helicoidal surface on which the vortices are

considered to be distributed (this surface is called a datum plane"),

differs from the pitch of the

hei i coi dal surface whi

ch is the locus of

the moving blade That means the non-linear effect

is taken into

account in a simple way.

The helicoidal coordinate system , 6 , JL (see Fig 1)

is used

in connecting with the cylindrical coordinate system .,r, G in the

relation . . .

6= 9-zIfL, ,LL=r/#L

(2.1.1) lt.

The pitch 2r

Fi. of the helicoidal surfacé

6COfl.St.

is taken

as equal to the hydrodynamic pitch;

L

2tf. = 2

(V t) i

U'

+

uit

I r

(2.1.2)

V,'

where

y :

velocity of advance

angular velocity of rotation

axial component of mean induced velocity ¿iTt at a blade

u3/r

: angular velocity of tangential component of

uJ

at a

blade

a

_2

2

uJ, uY

+(r.iiìt/r)

The requirement that the fluid particle should remain in contact

with the blade is expressed by

-(

LLT +V) /

(W*+WiV)

= tari (E0Ei)

(2.1 .3)

on the blade (see Fig. 2), where

E.0 :

geometrical pitch angle of mean camber line

Ei

: hydrodynamic pitch angle ( ta'n E

'1tir

I/fi )

W':

mean inflow velocity to blade section

C

W'V2+(flr-L4Ti)

U..)4- IAn : component of induced velocity normal to the datum plane

component of induced velocity tangential to the datum plane

1jft

: component of local velocity variation normal to the datum

plane

component of local velocity variation tangential to the datum plane.

Under the assumptions

I Eo E11 ( j

and' ¿tJ,

ifs

(W,

boundary

con-dition (2.1.3) becomes

4

-.-(

UJyat+lJvuL) /

WN=

O

(2.1.7)

W&L,t(L

= tant (E0-E)

(2.1.4)

on the blade area projected on the datum plane, where' £ denotes advance

angle,

lz'nE

=V/fLr

(2.1 .5)

Equation (2.1.4) can be divided in two parts:

tAJrt lAY nc + (J)

iJn =

JflS+1JnL4 (2.1.8)

bJns

and lins are independent on time, whereas ¿jJ,

and unu

dependent on time relating with the blade. 1Jis the component

vary-ing not circumferentially but radially.

The right side of equation (2.1.6) is given by the geometrical and

operational conditions of the propeller.

The local velocity variation in hull wake about the propeller shaft at any radius can be thought of as a sum of harmonic functions of

bladeposition angle

9

lía

= 2

I C(r) 42/n.

9 +

(r) c& p6 L

lJ

= Z.

Ie(r)42/n6

-f.

fcr

co

'6},

J

where

9

is measured from the vertical -axis in the right-hand sense

for the axes defined in Fig. 1, p is order of harmonic,

VZ

is the

axial velocity or ,-component, iJ is the tangential component in

-direction, and 00

P (fl+ uìt/r

)t - Lpt/2

=

_e

P-o since

8 =

2.1.9) (2.1 .11) 00

ìp(+Lt/r)t-Lpt/Z I

l/t = Z.

er)+frcrJe

p-o

C r) 4, cr)

p(r)

cr) -

=

- tttAìifli

(2.1.10)

9 ( .Çj + ¿.Ù / r) t

is to be put in place of

û

in (2.1.9)

in order to express lJ and

I7

in connection with the point

which moves along a helicoidal path with constant velocities

V+ i

and

_ft

+ LlÌt

/ r

. Then 1./a and lit at the blade

( 6=

o)

are given by real parts of

j

coordinates system is transformed to _fr7, co-ordinates system with the relation

= (,t2-j.L)I,ü, ,ü=Çuo-/4)/2,

,t.1=y10+,Ui)/2

(-Z)/2,?=(72+c,)/2

(2.l.i2)

where and ,Uj, are

,U

coordinates corresponding to the

propeller radius r0 and the boss radius _fl, respectively, and

and U7 are T coordinates corresponding to the leading

edge and the trailing edge respectively.

We consider the expression of induced velocity in fluid motion

oscillating with circular frequency

p

. If

eJt

denotes upwash on the blade, and 'iuu _{denotes circula-.}

tion density of bound vortex, the following relation has been known

[7].

p

= 9/ (fl-i-

z.U/r)

R*=

_{/X2/L2+L?_2,a,a1cq,}

,

6

The kernel function K C il;

p, p')

presents the

compli-cation arising from the high-order singularity. For infinitesimal

value of I,LL LL'I the limiting value of is

K( ;,)

()=

(2.1.13)

-I. -I

jf,

'r'

wherethe integral means the surface integral extended over a blade.

And the kernel function

K (

if;

_,,

p')

is given by

1+ 2 1-%

i2pmc/1

-ip(LÌ-1Y')

e

S

e

K

A,ì..t)

=

_3c

2:

+

c&. çO

(1uX -

_{ß'4in Q)(}

u'X Lt AL#fl (P)]

R"'

_R

(2.1.14)

_t

2

17

[Ia+y*

J

i-r

ç

4_2

l4(i+f)

(2.1.17) where

Yz= (,LL JL')2/

(I+,U2)

(2.1.18)

Details of the derivation of the formula (2.1.17) are given in

refer-ences [7] and

_tig].

These formulae (2.1.13)(2.l.l8) correspond to the th harmonic

of

= L

uS (2.1.19)

P.O

and then correspond to the p th harmonic in the wake velocity (2.1.11).

If we put p= O

the above-mentioned formulae become those for

the steady case corresponding to equation (2.1.6).

Since the left side of equation (2.1.13) is given by the p th

term in equation (2.1.11) and boundary condition (2.1.6) or (2.1.7),

the equation (2.1.13) can be regarded as the th integral equation

for lifting surface, in which complex amplitude of circulation density

' is unknown and

Ee

,

E

, and tT, are known as qeometry

and operational condition of the propeller, and wake distributions.

The boundary-value problem is expressed in terms of the singular

inte-gral equation relating a prescribed upwash distribution to an unknown circulation density for propeller blades.

2.2 Numerical method for the integral equation

The integral equation for lifting surface (2.1.13) is treated

numerically with the aid of high-speed computers. There are many

nu-merical methods for solving the integral equation [5]. The method

adopted here is the one presented by Hanaoka [6], which is a kind of mode function method.

circulation density is the same as that in two-dimensional flow given

by the Birnbaum series. Hence the circulation density is given by

fl-I (N)

-LA VÂN() ;

W0

N.O where f-a) (0)

rl

L

_k

= 2?r,iV'

_{AN(')K (o,7)d'}

f

I _(M)

_(M-l)

KCHN)

_27c)l

:,

_{K (o,vH-ÂWj<}

io,p}cL', (i1

o)

=

zLr0

()

',v=(1) uJ(,V

ii<

(çjZ)=(_i)hK(IJ;,LL,ltlI)

()

(2.2.3)

(2.2.1)

Cs)

Spanwise distribution

A

is determined later.

The basis of the Hanaoka's method is to convert the two-dimensional integral equation for lifting surface into simultaneous one-dimensional integral equations, in expanding the two-variable integral equation in

power series of one-variable by Taylor's theorem and equating the

co-efficient functions of successive powers to zero. Onsubstitutinn

equation (2.2.1) for integral equation (2.1.13), expanding the integral

equation in power series of , and manipulating the coefficients of

the series, the following simultaneous one-dimensional integral equa-ti.ons (line integral equations along the propeller radius) are obtained:

(M) _{j (N)} (MN)

-

t:iz)

₌

A o) K

d' ,(M=o,1,2,»,n-1),

(2.2.2)

8

N,n.

(M)

Ac'p

(M)

"

j

=-(M) . CM-f)

uY.

(o,))

-4- A.WLAY0

,

(M

o)

1<

c.l2)

C

(0)

02)-LA); (4,)

u.)0 C,')

and

2

_-ipE

E

..i

ip(9

c.1

Sa. S]

___ e

z e

3

R"

iip-im/L

K (0,)p

₌

_{_Iowo}

co

re)

-

-ip2mi

r c,

K (o,Y)+íwK '°,

_iTti,+,u2

m=oe

_a

Ss.]

E

ipE-zmW/á,

(a)

f')

_-

_I 2 Ii

-ìp2m/I r

.

Ze

K co.v

+tLL)

_K

_°'1

_ïE(-)J,

M_O

SC,+S3Cz+SaC

]x.i

g'

9..E-2mtVI,

.13) C1)

K (0VIWK Co,) =

¡I

4,U' co

bp2mc/E

[

t 14U2

3

C4C1+2C3C2-2 SeS4+,«'S.SaSea

+ 2

S4S3C21 254S2C3+1S353C4

Io,.]

_{?E_2m,d/2}

5.2iitP

Co=CPP

Ci

,44,U'+CSdQ

52 = ,L1X-p'ALI?LP,

C2 =

1L4

-C3

)4',4Lc94'P

5dÇ=

_{X+pjz'2Lin'P,}

C4-

_{I+1L4,LL'C6'9,}

E

C,, s2,

53

50 S4

C2, C.5

co, C4

(MN) CMN) CNN)

K

4.

₁

.Lt

1(oN)

jI4,LL

tu,Nbo)

2

CM-l)

(MN)

_.

À

(o)

,

(M4o),

2J

2 (ON)

_Ai

_rr4-,u

LN(0)

'

-

_{ji +'}

__

₁₁₄₍₁

+p

4

- A,

(0) +

j

2

(0)]

(2.2.4) (M)

(M-l)

The function

K (o,+LWK

(o,)

for any _M

except for t4O

does not include the integral calculation, and the function for any p

can be evaluated by simple manipulation of the value for

O

The characteristic.s of this function are considerably advantageous for numerical calculation.

CMP)

The modified kernel function

K

in equation (2.2.2) contains

singularities at

_,U

J4' , the forms of which are of importance to

the numerical calculation of the integral equation. After lengthy

manipulation, the singular behavior of

K

(MN)

is described by the

following expression [7], [9]

MN

-

z 2

= ____

_{ri+' L 4(f+) 4}

r 4-1v

(M-I)

N

AN

(o)]

(Mo)

J

(2.2.5)

CNN)

1< contains a high-order singularity with Hadamard finite

con-tribution and a logarithmic singularity, where

lo

(MN)

I

(M-i)

(P4)

AN (V

(d/d)MX)

I,0 (0)

To(W)+LTo°((t)}

I Cl)

1,2(0)L1clTo

(W)+ T0 «InI

()

T,

v=

a/d.w) T0(w)

1Lw'

T0

=

J(W)H0(Ca))

:-

_Ç01-:,

d'

(MN)

R

((p,

where

=

Jr/(iniI)

s (2)

,i(°

r{Toci.u+To (w)},

(2)

i,(o)=

7r1T0

+T4w)},

Substitution of equations (2.2.7) and (2.2.9) in equation (2.2.2)

yields

O'%

U=

(2.2.6)

and

Jo

(W)

and 1-fe

(W)

are the Bessel function of the

first kind and the Struve function, respectively.

Next, by the collocation method, the simultaneous one-dimensional integral equations (2.2.2) are reduced to a set of algebraic equations. Integrands in equation (2.2.2) are rewritten as follows [10]

A'K"/2

=

i«

' tn i

-n'i

(2.2.7) where

= C43 (

'Z'=

c&;S (p'.

(2.2.8)

(MN)

And

(P,P )

can be expressed as

(MN)

2

IR

'm+i

-I

(2.2.9)

(2.2.10)

(,4

---A

7I

(N) (Mid) I

()T

N0

2

2d1CYfliI

_*111

, ('l')

'

C .M=

o,

i,

2,', n-i)..

(2.2.11)

a

T0 (W)

J0

1W)

i-Ia (W)

J,(f7

After applying the formulae

d)r=

CbS2'P

2,t4

-)(Z,i)Z

'?

-

4M«P

to equation (2.2.11), the following simultaneous algebraic equations are obtained.

j4N

r1.

c=

_{NO 31}

where

c

N (N)

AA (c.dP)

MN

_AN

_{( t1= o,}

I, 2, , )1- i) (2.2.13) ¡J.

J

) «II,2,3," ,

vit I

-

(I)T1

A1fl

(MN) fj

(1431

z

2

n4-I

K

4+,u

j'.ea6ø. ,4N (14W)

r

7fl,4.

-

L.)

4,(p

7(1411).I

f_cS2-.ß,,4

AM(fL

g

+

It3fcece6

and denotes the summation with respect to J except for

J=x.

Numerical calculations in the following séctions are performed on

the condition that

fl-4 and

n.

7.

2.3 Resulting forces and moments

The blade lift density (pressure difference between the upper and lower sides of the lifting surface) can be obtained from the solution of the integral equation for lifting surface, because the relation

(2.2.12) 2. 2 14) 12 I MP4 r A

,1J

between the blade lift density

ff(,Y)

and the circulation

den-sity of the bound vortex is given by

=

(2.3.1)

dL

=

dr

vvi

where is the mass density of the fluid [7]. Therefore themagnitude

of the force acting on an elementary strip of unit width of blade is given by

a'-dr

C,

7*(yl)d

(2.3.2)

where

c'

is the length of half chord, and the direction of the

force is normal to the datum plane.

The elementary forces and moments of the various components can be

easily determined by considering the resolution of the force

dL./dt

and the position angle of the blade

9'

which is measured in the di-

9'

rection of propeller rotation (

9' = -9

, see Fig. 3). Fig.3

The characteristics of summing the effects of all blades is

re-vealed as follows (for example, reference [4]). The thrust -

F

(see

F

Fig.3) and torque

¡4

aredetermined by the loadings associated with

¡"k

wake harmonics at blade frequency or integer multi ples of that

frequen-cy, i.e., at

=

, whereas the transverse forces

F

,

F,

f F

and bending moments fr1 ,

fr1,

are determined by the loadings fr1

,1z

at frequencies adjacent to blade frequency or integer multiples of that

3. Characteristics of Unsteady Blade Forces

.3.1 Effect of elocity distribution

Calculation has been. made of the vibratory forces generated on the

propeller. Li operating. in a series of simple velocity distributions.

The propeller LT has the same geometry as that of the propeler used

for the comparative calculation on unsteady propeller blade forces

proposed by PrQpeller Committee of the 14t.h I.T.T.C. [2]. The

cha,rac-teristics and outline of the blade are in Table.1 and Figs. 4 and li, Tablel

respectively. The series of thé velocity distributiOns used in the Fig.4

calculation are the axial wake varyingsinusoidaily with respect to the

position angle of the blade as shown in Fig. 5. Amplitude and phase Fiq.5

of the variation are independent on the radius. The ratio of the

am-plitude of lJ to the advance velocity

V

of the propeller is taken

as 0.3. The velocity distri.butions are expressed by each term of

equa-tion (2.1.9) on conditiOn that O , _0.3

V

The operational condition isthat the mean advance coefficient

j

,vV/(flr.)is

0 646 and the hydrodynamic pitch is

27r_-.0.9.39

X

C 2 To Y.

The results .of thé calculation are shown in

Figs. 6 and 7. . .

. t I

Fig 6 exhibits the complex amplitude of the non-dimensional dr- Fig 6

culati on around the blade section

Gr e

.

,. where

c*=rP*/20,croV

/dr=

rr0 pWV.

¡4

The calculation reveals the characteristics that the. amplitude decreases

and the phase advances with increase in

b

, and that the tendency is

remarkable for blade section far from the tip, which means that the efféct of unsteadiness Of the flow becomes larger with increase in

re-duced frequency W proportional to and

c*(

see equation (2 2 3) or

(3.1.2) ). . : . . .

Fig 7 exhibits the vector diagram showing the real and imaginary Fig.7

parts of

Sp(,V)

and. as functions of the reduced frequency

W

EfrC.P_')

is the complex amplitude of the non-dimensional lift

(cv)

on the element of the blade as shown by . .

J.

2CfC*W*,f

toe

jt

2c*

(3.1.2)

and

_..SV)

is Sears' two-dimensional response function for the

sinusoidal gust [li], where iJ is the amplitude of the normal gust

velocity across the blade iJ and is the wave length of the

sinusoidal gust. Fig. 7 shows that the three-dimensional effect is

extremely large and the degree of the effect decreases with increase in

(L)

3.2 Effect of operational condition

Calculation is performed for three operational conditions : advance

coefficient

J

0.404, 0.646, 0.888. The geometry of the propeller

and the wake distributions are the same as those in the case of Section

'3.1.

The result of calculation exhibited in Table 2 reveals that varia- Tabie2

L)'t

tion of complex amplitude of non-dimensional circulation

G4-/e

with advance coefficient J is very small. It means that there exists

little variation in in spite of changing of hydrodynamic

pitch. Therefore, according to the equation (3.1.1), it can be said

that the variation of lift is proportional

toW*on

condition that the

geometry and the advance speed of the propeller and the wake variation are kept constant.

3.3 Effect of geometry of blade

It is considered that number of blades, skew back, and blade width are important geometric factors for vibratory forces.

Number of blades

According to the description in Section 2.3, it is clear that the

effect of number of blades is very important from vibration point of

view. An example is illustrated in Section4.3.

Skew

Calculation is performed in the case of three propellers SO, Si,

S2 with the different niounts of skew (see Fig. 4). Propeller Si is

the same as the propeller Li used in Section 3.1. Other propellers

16

skew, whereas the amount of skew of S2 is twice qreater thri that of Si. The wake distributions and operational condition are the same as those

in the case of Section 3.1. .

The complex amplitude of non-dimensional circulation around blade

section at

nr0

=0.584 is exhibited in. Fig. 8, which shows two Fig.8

characteristics on the effect of skew. Theone effect is the variation..

of phase with increase in skèw. The shift of.position of blade section

due to skew can explain the phase variation. As the amount of skew

becomes greater, the difference in phase between radial sections

in-creasés, so that the vibratory force acting on a blade decreases. The

. .

L)'t

.

other. effect is that the amplitude of

G! e

decreases as the

amount of skew becomes greater. . .

To compare the magnitudes of the. two .effects, the vibtatory thrust

coefficient for eàch blade

T/ prL2v4

of the propellers

operat-ing in the wake harmonic 4, is calculated in some cases. The

results are exhibited in Table 3, in which the value in b) is the co

efficient obtained from the amplitude of

dL/d.r

for SO and the

phase for S2, and the valye in c) is the coefficient. obtained from the

amplitude of

d.L /dr

for S2 and the phase for SO. The results

reveal thät the magnitude of the coefficient in the case of S2 is 68 percent of that for SO by thé two effects due to skew, and that the

effect of decrease in amplitude of

_{- ¡e}

is as noticeable as

that of the variation of phase C) Blade width

Calculation is performed in the case of three propellers Wl, W2,

W3 with the different blade width (see Fig 4) All propellers have

no skew. The propeller W2 is the same as the propeller SO. Wi and W3

have, the same gebmetry as W2 except for the biadé width. The blade

width is 0.5 time for Wi and 1.5 times fo W3 as wide as that of W2.

The wake distribütions and operatioñal condition are thé same as in the

case of Section.3.l. The results of the ciculation are shown in Figs.

9 and lO. . .

The complex amplitude of non-dimensional circulation around the

blade section at

nr0

0.878 is exhibited in Fig. 9, which shows

that at larger the amplitude increases at first and then décreases

and thé phase advances continually as the blade width increases. This

tendency can, beregarded as that on the vibratory force

dL/dr

corresponding to - (see equation (3 1 1)) The explanation of this

4T.

D

Tablé3

tendency is that the amplitude increases at first due to the

quasi-steady forces increasing with blade width 2C. and that the amplitude

decreases because of the effects of unsteadiness which is shown by the

function

(W) (

see equation (3.1.2) and Fig. 10 ).

Fig. 10 exhibits the functions and

(w)

, the corn- Fig.l0

parison of which reveals the characteristics of the three-dimensional

effects. The amplitude of

_.Sp(w)

decreases and the phase of it

advances as p or

_c?

increases. This tendency agrees qualitatively

with that of

ß cw)

in the case of two-dimensional flow. However,

quantitatively, the amplitude of is not close to that of

And further it is not the function of ) alone. For the

same W ,

amplitude of is smaller in the case of larger C,'

or

r ,

which is remarkable in the case of small

ct)

..

This

characteris-tiçs can be explained reasonably as the three-dimensional effects. The

slope of the

W- Sp(W)

curve with constant is steeper than

that with C constant. The function

Sew)

seems to be a function

of W alone in the case of large W, although it is not close to

4 Application toShips

4.1.Propellers and wakes

of calculations on the thrust, torques bearing forces . (1ml

moments arepresented for the six cases; Li : Series 60, 0.60e.

single-scréw model with alteration in vicinity of stern tubé bossing, whose wake distribution and the geometry of the propeller have been employed

forthe comparative calculation proposed by Propéller Committee of

the 14th LT.T.C. [2]; Cl : a single-screw container ship [12]; C2

a twincrèw container ship; Ti : asingle-screw tanker; Fl and F2 : a

single-screw fishing boat fitted alternately with 3- and 4- bladed

pro'-peller. The chacteristics of, the propellers and the conditions for

calculation are summarized in Table 1, and projected outlinés öf the

blades. are shown in Fig. 11. '

Calculàtions are performed using the model wakes obtained from

5-hole pitot tube or Prandtl type pitot tube The circumferential

dis-tributiôns of wake's at,propeller disk are 'shown in Figs. 12 and '13.

4.2 Thrust variation on one' blade . ' '

Two examples of the thrust variations acting, on one blade. are shown

in Figs. 13 and l& . '

Fig. 13exhibits the two kinds of calculations for the case of

'Li for the combined axial and tangential wake and for the axial wake.

As fór vibratory forces and moments acting on propeller shaft, they are not so affected, by the tangential component except.for the mean values of bearing forçes and mòments, because the tangential wake is composed of weak harmonics exceptfor the first féw.harmoni'cs and because the effect of' tangential wake on blade loading is small at the elements ,of a propeller b'lade which produce the largest part of the thrust.

Fig'. 14 shows the comparison of results between calculation and, s

experiment [12] for the case of Cl. From Fig. 14. it is seen that good

agreement is obtained, althOugh tangential wake is not taken into

ac-count in càlculat'iOn. The good .agreements"betwêen calculations and

experiments have been obtained on,the other models C2. Fi. Moreover,

to make sure of .the accuracy of the calculation, correlations with model data have been made in the steady cases (propeller operating in

úniform flow),and the agreement bétween theory andexperiment. was good

[13].

4.3 Vibratory shaft fbrces and moments ' . . , . '

The results óf the calculation on vibratory thrust

.-F

. , torque

18

Fig.11

Fig. 14

Fig.l2

M

bearing forces F , F1., and bearing iiiornents

M

,

M for the

six cases, are shown in Figs. 15 and 16.

The vibratory forces and moments for Fi and F2 are very large due

to the large wake harmonics at blade frequency ( see Fig. 17 ). The

bearing forces and moments for Fi (3-bladed propeller) are in excess of

those for F2 (4-bladed propeller). Thrust and torque variations for Fi

are nearly equal to ones for F2. The reason is that the amplitudes of

wake harmonics for = 3, 4, and 5 are about 1/2, 1/2, and 1/4 of

that for

p = 2

as seen in Fig. 17. Special attention must be paid

to the following fact. If the 4-bladed propeller is adopted instead of

the 3-biaded propeller for a fishing boat, reduction of beari.ng forces and moments is achieved without increase of thrust and torque varia-tions.

The thrust and torque variations of Cl are. larger than those of

Li. The reason is considered to be the fact that wake component for

4 of Cl is larger than that of Li on the whole radial range as

shown in Fig.l7. It is very dangerous tÓ ju1ge the amount of forces

only from the magnitudes of the harmonics on a certain radius, because the wake harmonics vary radially in general.

Fig. 15 Fig. 16 Fig. 17

5. Conclusion

The numerical procedure for determining hydrodynamic forces on

propeller blade operating in nonuniform flow has been developed. The

procedure employs the Hanaoka's method for solving the integral equa-tion for lifting surface which is very convenient especially for the case of unsteady lifting surface.

Ir

applying the procedure to systematic calculations, the nature

of unsteadiness of the flow and the three-dimensional effect are

clari-fied. Concerning unsteady lift coefficient on blade section of the

propeller operating in nonuniform flow, the amplitude of its variation decreases and the phase advances with increase in reduced frequency,

which s the same nature as of the two-dimensional wing in the

sinu-soidal gust. The amplitude, however, is far smaller than that of the

two-dimensional wing at the same reduced frequency, which is explained

as the three-dimensional effect. The magnitude of the

three-dimen-sional effect varies with reduced frequency. Furthermore, the

magni-tude varies according to the frequency of wake harmonics, chord length

(expanded area ratio), or radius of the section, even at the same

reduced frequency. The Iecrease of the amplitude of the variation

with increase in reduced frequency is rapid for the case of varying chord length.

The relation between vibratory blade forces and geometry of the blade is clarified by the results of the systematic calculations. When the blade width becomes large, the amount of vibratory blade forces increases at first and then decreases, which is explained by the effect of quasi-steady force increasing with increase in blade width and of unsteady force decreasing with increase in reduced

fre-quency. Vibratory force on a blade can be decreased by increasing

skew, which arises from the effect of the shift of phase and the effect of the decrease of amplitude of the lift acting on the blade section.

The two effects have the same magnitude for the reductionof vibratory

force.

As for operational condition, non-dimensional circulation/e'

varies little with the advance

_{coefficient J}

.

Results of numerical calculations on vibratory thrust, torque,

bearing forces and moments, are presented for five ships. Comparisons

of the calculation results with experimental ones show good agreements. 20

Acknowledgment

The authors wish to acknowledge their indebtedness to Dr. T.Nana-oka for his valuable advices and discussions, which initiated the pres-ent study.

References

Wereldsma, R. : Comparative Tests on Vibratory Propeller Forces,

13th I.T.T.C. Report of Prop. Committee, Appendix 2a, (1972)

Schwanecke, H. : Comparative Calculation on Unsteady Propeller

Blade Forces, 14th I.T.T.C.Report of Prop. Committee, Appendix 4, (1975)

Tskonas, S., Jacobs, W.R., and Ali, M.R. : An "Exact11 Linear

Lifting-Surface Theory for a Marine Propeller in a Nonuniform Flow Field, J. of Ship Res., Vol. 17, No, 4, (1973)

Tsakonas, S., Breslin, J., and Miller, M. : Correlation and

Appli-cation of an Unsteady Flow Theory for Propeller Forces, S.N.A.M.E. Vol. 75, (1967)

[5] Langan, T.J. and Wang, H.T. : Evaluation of Lifting-Surface

Pro-grams for Computing the Pressure Distribution on Planar Foils in Steady Motion, N.S.R.D.C, Report 4021, (1973)

Hanaoka, T. : A New Method for Calculating the Hydrodynamic Load

Distributions on a Lifting Surface, Report of Ship Research In-stitute, Vol. 6, No.1, (1969)

Hanaoka,T. : Numerical Lifting-Surface Theory of a Screw

Propel-ler. in Non-Uniform Flow (Part 1 Fundamental Theory), Report of

Ship Research Institute, Vol. 6, No. 5, (1969)

Hanaoka, T, : Hydro1ynamics of an Oscillating Screw Propeller,

4th Symposium on Naval Hydrodynamics, Washington,D.C., (1962)

Koyama, K. : A Numerical Method for Propeller Lifting Surface in

Non-Uniform Flow and Its Application, J. of the Soc. of Naval Architects of Japan, Vol. 137, (1975)

Mangler, K.W. and Spencer, B.F.R. Some Remarks on Multhopp's

Lifting-SUrface Theory, A.R.C., R. and M. No. 2926, (1952)

Sears, W.R. Some Aspects of Non-Stationary Airfoil Theory and

Its Practical Application, J. of Aero. Sci., Vol. .8, No. 3, (1941)

Takahashi, H. : On Propeller Vibratory Forces of the Container

Ship--Correlation between Ship and Model, and the Effect of Flow

Control Fin on Vibratory Forces--, Papers of Ship Research Insti-tute, No. 44, (1973)

Koyama, K. A Numerical Analysis for the Lifting Surface Theory

of a Marine Propeller, J. of the Soc. of Naval Architects of

Japan, Vol. 132, (1972) .

Table i

Characteristics of the propellers

Li Cl C2 Ti Fi type of ship liner container tanker fishing boat no. of prop. no. of blades boss ratio

exp. area r. pitch ratio J

( mean ) tang. wake p ( max. ) 9 9 ii 6 10 1 4 1 4 2 5 1 5 1 3 0.167 0.307 0.200 0.184 0.313 0.474 0.605 0.730 0.650 0.515 1.025 1.209 1.248 0.692 0.851 0.646 0.819 0.984 O.297 0.481 Inc. exc. -Inc. inc. Inc.

Phase lag in rad. 1 2 3 4 5 6 8 9 0.404 -0.01 -0.19 -0.42 -0.68 -0.97 -1 .25 -1.53 -1 .80 -2.08 0.646 -0.01 -0.18 -0.41 -0.67 -0.95 -1.23 -1.50 -1.77 -2.05 0.888 -0.01 -0.17 -0.40 -0.66 -0.94 -1.21 -1 .47 -1.74 -2.01 1 2 3 4 5 6 7 8 9 0.404 0.0179 0.0174 0.0165 0.0154 0.0138 0.0124 0.0111 0.0098 0.0086 0.646 0.0182 0.0177 0.0167 0.0155 0.0139 0.0125 0.0111 0.0099 0.0087 0.888 0.0185 0.0179 0.0169 0.0156 0.0140 0.0125 0.0112. 0.0100 0.0087 Table 2

Complex amplitude of the non-dimensional circulation

Gr/et

around blade section at

r/r0

= 0.743 for the propeller Ll

/1r?D4

Table 3

Amplitude of vibratory thrust coefficient for each blade

0.0189

of propellers SO and S2 operating in the wake harmonic p=4

0. 01 57 0.0157 0.0129 (a) (b) (c) (d) effect of phase effect of amp. effect of phase and amp. SO SO S2 S2 SO S2 SO S2

List of Figures

Fig. 1 Coordinate system

Fig. 2 Flow around _a propeller blade section

Fig. 3 Forces and moments

Fig. 4 Expanded outlines of the blades

Fig. 5 Axial wake distributions

Fig. 6 Complex amplitude of the non-dimensional circulation

around blade section for the propeller Li

Fig. 7 Vector diagram showing the real and imaginary parts

of4ccv)

for the propeller Li and Sears' function

_V)

Fig. 8 Complex amplitude Of the non-dimensional, circulation

around blade section at

nr.

=0.584

for the propellers SO,

Si, and S2 .

Fig. 9 Complex amplitude of the non-dimensional circulation

around blade section' _at

_r/r.=0.878

_{for the propellers Wi,}

W2, and W3

Fig. 10 Complex amplitude of thé non-dimensional lift

Spw)

_Ofl

blade section at

nr.

0.878 for the propellers Wi, W2,

and W3

Fig.. il Projected outlines of

theblades

Fig. 12 Wake distributions

Fig. 13 Wake _distribution and thrust variation _{on one} blade of Li

Fig. 14 Thrust variation on one blade of Cl

Fig. 15 Vibratory shaft forces and moments

Fig. 16 Vibratory shaft forcés and moments

Fig. 17 cosine components af axial wake harmonics,

deer/v

i

Coordinate system

Fig.

2

Fg 4

ExPafld

0ut1

of the blades

SO = W2

G.L.

= u

S2

_1800

p=I

-90°

Fig.

5

Axial wake distributions

90°

0.02

10.01

0.00

y

r

I

r0

0.968

0.878

0.743

0.584

I I I - I i

0

2

4

6

8

order

of

harmonics,

p

Fig. 6

Complex

amplitude

of

tne

non-dimensional

circulation

G/e1Ut

around

blade

section

for

the

propeller

Li

D

D

w

-

e U)

o

-c

w4

W429

w=2.79

imag.

0.3- w2

I

W4955

6

O.I

II

0.4

't

II

)

--0.1

w=0

0.88

0.8

Fig. 7

Vector diagram snowing

the real and imaginary

parts of Sp(w)

for the propeller Li

and Sears' function S(w)

Sp..(w)

nr0

nh

0.968

3245

0.743

2.490

y 0.424

1.421

0.199

0.666

Sears Funct. S (w)

(AJ=Q

I.0

G) 'I-

00.0I

G)

E

o

0.02

so

SI

p

S2

0

2

₄

6

8

order

of

harmonics,

_p

Fig. 8

Complex

amplitude

of tne

non-dimensional

circulation

G/e1)t

around

blade

section

at

_nr0

₌

0.584

for

the

propellers

SO,

9-

o.

E

o

0.00

o

-c

o.

w'

W2

W3

I I I I

0

2

4

6

8

order

of

harmonics,

_p

Fig. 9

Complex

amplitude

of

the

non-dimensional

circulation

G/e10t

around

blade

section

at

_nr0

₌

0.878

for

the

propellers

Wi,

0.6

(J,

t'o.5

ci

0.4

Q)

0.2

0.1

0.0

0.0

o

C

a

2-7r-

D

W3

WI

W.2

W3

Sears'

_function

Fig. 10.

Complex

amplitude

of

the

non-dimensional

lift

Sp(w)

_on

blade

section

at :nIro ₌

0.878

for

the

propellers

Wi, W2,

and

W3

Q.) . p i i i i (1)

2

0

2

3

4

o-

reduced

frequency,

_w

il

0.8

Q6

O.4

0.2

0.0

0.2

(1)

>

ao

41

0.2

FI, F2,

Q850r0

C2

Fl,F2

I I I Fig. 12

Wake distributions

-180°

-90°

0°

900

1800

bottom

top

bottom

blade position

8'

04

0.2

0.0

0.2

0.0

'I'

7

\

thrust variation

on one blade

tang. wake. inc.

.\\'tang. wake exc.

0.08

117

-('p

-0.06

/

\'%._.

-

F F

-.

-%.,. %...____

-wake distribution at O.700ro

0.04

Fig.

13

Wake distribution and thrust variation on one blade of [1

-180°

-90°

0°.

90°

180°

bottom

top

bottom

blade position

9'

0.12

Ojo

o

c'J

cO.08

F-0.06

0.04

-calculation

experiment

Fig.

14

Thrust variation on one blade of Cl

-180°

-90°

0°

90°

180°

bottom

top

bottom

blade position

1.05

.00

0.95

0.00 r

-Q05

0.20-0.15

0.10

-.'

T: mean thrust

Q: mean torque

I_%

I

F F F % F

I

/

%

/

LI

cl

o.

00

90°

1800

0°

90°

180°

blade position,

0'

blade position, 6'

Fig.

1.5

Vibratory shaft forces and moments

I

'i

/

I F I

-F

.'

I

.

-'

LO

0.9

\

\

FI

F2

TI

T: mean thrust

Q: mean torque

/ /

%_ -____ s__s

____ 55

I I

00

90°

1800

blade position, 9'

IO''

X

0.9

0.5

IO

>

0.0

1.0

IO

N

0.0

I

/

I

I'

_5%\

\

'I'

/

/

s S

/

s' i

...-

r...-00

90°

180°

blade position, 6'

Fig.

16

I

/

I

o04

-p

e_e_e e O

-p2

LI

CI,

Fl,F2

E-0.I

-o

0.2

-C

__

CC

---a

o

-0.i

a,

0)00

o-U)

a,

o-t.'

c

'J.

!01

0.0

-D.

4-E

o

C-,

0.0

I I I

GO

0.5

1.0

0.0

axis

r/r

tip

axis

Fig.

17

cäsine components of axial wake harmonics, dp(r)/V

p=5

a

a

a a

- ___ - a e e__ea

- a

__ _eee

-I I I

0.5

Lo

nro

tip