Practical approach to unsteady problems of marine propellers by quasi-steady method of calculation

Practical Approach to Unsteady Problems of Marine Propellers

by Quasi-Steady Method of Calculation

Desember 1980

MITSUBISHI HEAVY INDUSTRIES. LTD.

T 05 40 4 ÔQX

Deift

by Quasi-Steady Method of Calculation

Hidetake Tanibayashi*

In search for a method of simple and quick calculation of unsteady problems of a marine propeller, quasi-steady method of calcu-lation was examined in the light of recent progress in experimental techniques and numerical calcucalcu-lations of unsteady lifting surface theory.

As a result, it was found that some modification to the conventional quasi-steady method would extend greatly the range of applicability; by incorporating the effect of breadth of a propeller blade relative to breadth of wake peak, thrust fluctuation of a single blade calculated by quasi-steady method was found to be in good agreement with experimental results

Based on this finding, a modified quasi-steady method was proposed for unsteady response of a propeller operating in a non-uniform flow. From the results of the calculation, blade stress and propeller shaft forces are considered to be estimated within the accuracy of experiments. For the problems involving cavitation, however, the applicability is limited to wake variation of lower frequency and smaller amplitude, thus this method does not suit the prediction of erosion,

The advantage of the presently proposed quasi-steady method lies in its simplicity, and therefore it is valuable especially ¡n the preliminary stages of ship design. Comparison with unsteady lifting surface theory and the problems for further development are

a/so discussed.

1. Introduction

A marine propeller generally operates in a wake of ship with temporal and spatial variations. Due to the variations

of inflow to the propeller, the lift forces at any location

of the blades will be continuously changing, giving rise to variation of propeller shaft forces (thrust, torque and bear-ing forces), blade stress and cavitation on the blades.

Temporal variations arise from unsteadiness in the wake being mostly aperiodic, such as caused by turbulence in boundary layer, non-stationary shedding of vortices, wave-induced effects and manoeuvring of the ship.

The spatial variations are usually defined by the time average of axial, tangential and radial components of wake in the plane of the propeller. Such a spatially varying wake produces cyclic changes in angle of attack of any blade element and hence periodic variations in pressure distribu-tion over the blade with the frequency of shaft revoludistribu-tions. When they are summed up to an output from a geometrical-y sgeometrical-ymmetrical propeller as a unit, the sum varies with the blade frequency - shaft revolutions times number of blades - and its multiples.

This paper chiefly deals with the unsteady periodic prob-lems due to spatial variation of wake, since they are of pri-mary importance in propeller design and consideration of hull response against excitation from a propeller. Aperiodic variations may be treated, when necesitated, by an exten-sion of the solution for the periodic problems for example by means of Fourier transform technique, if the frequency characteristics of the response are known.

Such unsteady problems of propeller were first tackled by a quasi-steady method. This method assumes that the force on a blade at any instant is the same as if it were

Nagasaki Technical Institute. Technical Headquarters

working in a uniform flow of the same velocity.

The numerical calculation of bearing forces by this

method was first presented by F. M. Lewis as far back as in 1935, evaluating the variations in angle of attack and in relative velocity at each radiushll. Later this method was improved using steady-state propeller theory121, a propeller-design procedure such as by Burrill>3> being worked

inverse-ly stripwise to obtain the radial load distribution for the

local inflow conditions. Based on the same principle but using propeller open-water characteristic curves for the radial mean wake instead of calculation of change in the lift at each radius, Brehme estimated the effect of number of

blades on the amplitude of thrust and torque

fluctua-tions>4>. Schuster and Walinski extended this approach and assumed for practical application that the mean wake for a blade could be approximated in most cases by the local wake at 0.7 R>5>. In 1961 McCarthy compared the ex-perimental results of thrust and torque fluctuations pub-lished so far>6>'>7> with those calculated by quasi-steady method, and concluded that both results were generally in

good agreement18>.

In the meantime, attempts had been made to solve the problems on more rigorous theoretical basis, viz,, by un-steady lifting surface theory which takes into account the temporally varying vortices shed from the blades due to the variation of pressure on them. Then efforts were turn-ed to obtain the numerical results from the integral equa-tion of unsteady lifting surface theory>9>12>. The

numerical computation became possible with the develop-ment of large-capacity high-speed computers, but still a

number of simplifying assumptions had to be introduced. Looking into the results from the calculation, it is noted

that

(1) effect of unsteadiness on the propeller problems is by far less, as far as the amplitudes of fluctuation are concerned, than on the two-dimensional wing as given by Sears' function(13), and

(2> the numerical results obtained by theory differ much

with the methods of calculation even in the steady

statehl4) _{- a limit to frequency of zero.}

With such a background the author came to a point that the characteristics of a propeller in non-uniform flow could be predicted based on a quasi-steady method if the steady state characteristics of the propeller were given with rea-sonable accuracy(15). Then he examined this method in

comparison with the experimental data which became

more readily available and more reliable in the recent years. In so doing, the conventional quasi-steady method such as developed by McCarthy was modified to incorporate the effect of breadth of blade relative to breadth of wake. As a result, thus extended method proved to have wide appli-cability to unsteady problems of marine propellers except for cavitation with relatively high frequency, large ampli-tude fluctuations.

The advantage of the quasi-steady method lies in its simplicity, and therefore it is useful in the preliminary stages of ship design, enabling not only rapid estimation of unsteady characteristics of a propeller but also affording possibility of optimizing a hull form (and appendages) and a propeller. Further advantage is found in the problems which are too complicated, chiefly because of non-linearity due to cavitation and separation of flow, to be treated with unsteady theory.

This paper describes first the theoretical background of the unsteady theory and the quasi-steady technique, and then in the succeeding chapters, the usefulness and the limitation of the quasi-steady method are discussed, on various problems such as response of a single blade in a varying wake field, propeller shaft forces and cavitation

pattern on the blades.

2. Theoretical background and method of calculation 2.1 Unsteady lifting surface theory of propeller

The flow field around a marine propeller operating in the spatially varying wake can be treated in a manner analogous to the wing moving recti-linearly through a spa-tially varying gust. The essential deviation of the unsteady problem from the steady problem lies in that the vortices shed from the wing as a result from the temporal change of the lift have to be considered in formulating the boundary condition or the integral equation for the lift distribution on the wing.

Let us consider first the simplest case of a two-dimen-sional wing. If the wing is placed in a uniform stream with

the velocity LI to the direction of x (cf. Fig. 2.1.1), the

velocity induced normal to the uniform stream due to the bound vortex y on the wing is expressed, as is well known,

Vo

+u

Fig. 2.1.1 _{Schematic representation of an airfoil in} gust and explanation of notation

by

1 çC/2 'y(x')

v(x) = ---j

, dx'

271 _C/2xx

where C is the chord length of the wing.

If, on the other hand, there is a series of gust normal to

the wing and travelling with the uniform stream, a free

vortex system is shed from the wing and eq. (2.1.1) is re-placed by9> C/2 t) =

dX'[fy(x't

x12y2

i

22X1I

(2.1.2)

(x1+y )

j

Under the assumption of small disturbance by the vortex system 'y and the gust Vg compared with uniform velocity U, viz, the assumption of linearized theory, the boundary condition on the aerofoil is written as

xix' + X

U

(2.1.1)

where m is the ordinate of mean line of the wing. This is an integral equation for determination of the bound vortex distribution along the chord. For a sinusoidal gust

expres-sed by

v9(x, t) = V0expjw(t---)

(2.1.5)

with amplitude y0 and frequency w, the solution of eq.

(2.1.4) is obtained13 and the bound vortex distribution is given by atm

2 /(C/2)-x

C/2

_{/(C12)+x' ax'}

dx' y(x,t)

- J (C/2) + x

ir

f-ci

V (C/2) - x' x - x'

'(C/2) -x

S(k) exp(jwt) (C12) +x

where S(k) is called Sears' function expressed by the Bessel functions of the first kind J and the modified Bessel

func-(2i.6)

òfm

v(x,t)-U

v (x,t)

(2.1.3) öx or 1 1.C/2

-

dx' t x1 - x' + X

u

122

d X1 = öfm _(2.1.4) (x12+y2 2

_y-0

U _öx V9(x, t)

V) E 2.5 -0.6 -0.4 5.0 0.6 0.4 2.0 0.2 0.4 0 0.8 LO 0.5 Re(S(k)) 2 3 Reduced frequency k

Fig. 2.1.2 Sears' function for a two-dimensional wing in sinusoidal gust

tions of the second kind K, such as K1 (jk)

S(k) = [.J0(k) jJ1(k)}

K1(/k)+K0(/k)

+ ¡J1 (k)

and k is called reduced frequency defined as Cw

k 2U

or with wave length 7'. expressed by

k is written as (cf. Fig. 2.1.1)

irC

X

The first term of the right-hand side of eq. (2.1.6) is a time-independent well known solution of the thin aerofoil theo-ry and the second term shows the unsteady response to the sinusoidally varying gust. Integration of the second term along the chord leads to the total lift L, which is expressed

in a non-dimensional coefficient as

CL

L

, 2S(k)

(2.1.9)

1I2pUC L)

Behaviour of the complex function S(k) is illustrated in

Fig. 2.1.2 with a polar diagram above and with amplitude and phase variation on a base of k below. As k tends to zero (w-O or X-*o0), S(k) tends to unity in amplitude with diminishing phase difference, thus it expresses the variation of amplitude and phase due to unsteady effects.

Let us consider next a screw propeller operating in a

spatially varying wake field. A cylindrical coordinate sys-tem (x, r, O) ¡s taken, as shown in Fig. 2.1,3, with the origin O at the center of propeller, x-axis in the direction of

pro-k

Fig. 2.1.3 Coordinate system

Q

V Fig. 2,1.4 Velocity diagram for a blade element

peller shaft being positive afiward, O starting from top in

the right-hand direction toward negative x, and r in the

radial direction from the x-axis. The y-axis ¡s taken hori-zontally normal to the propeller shaft positive on starboard side, and the z-axis taken upward normal to x- and y-axes. The propeller is placed in a stream in the direction of positive x, and rotates with angular velocity 2=27rn, where n is the number of revolutions per unit time, in the direc-tion of positive O. The undisturbed velocity of the stream is V and the disturbance of the flow present in the vicinity of the propeller is dictated by the velocity components v,

t'0 and Vr.

A helical free vortex sheet is shed from the bound vor-tex sheet representing each blade. The strength of the

bound vortex sheet conforms to the condition of null

normal velocity against the blades. Whereas the shape of the free vortex sheet follows this normal condition of the

velocity and the condition of continuity of flow, it is

approximately assumed to be composed of helical elements

of constant radius and pitch - without contraction and

deformation due to variation of wake and interference of the vortex sheets themselves - to avoid excessive complica-tion. The velocity component in radial direction is neglect-ed for the same reason.

Hence, referring to the velocity diagram in Fig. 2.1.4,

1.0 0.8 - 0.6 0.4 0.2 250 200 150 un

t

100 50 (2.1.7) (2.1.8a) (2.1.8b)

2iri-1\

a i

tr

N )

av R1

+

r'2dr

/h2+r2

m Os

- vn

(2.1.10) with

(ír+ v0)h - (V+ v)r

R = h2 (Ö O' + 2r)2+ r2+r'2 - -, _{. 1I2} - 2rr cos(O + T + 2iri - l/N)

The left-hand side shows the velocity induced by the vortex system on the blades and their downstream, while the right-hand side shows the contributions from the inflow velocity, the first term showing effect of chordwise varia-tion of slope of the blade mean line and the second term showing effect of difference between the direction of in-flow and the pitch surface. The definition of each symbol is summarized in Nomenclature, and therefore brief expla-nation is given in the followings.

r is the radius of the blade section concerned

2irh is the mean pitch defined appropriately for the geometry of the blades and the free vortex sheets

Ç2r+v0 is the tangential velocity component of inflow to the blade (with effect of tangential wake y6)

V+v is the axial velocity component of inflow (with

effect of axial wake v)

y shows the strength of bound vortex on the

propel-ler blades

o is the angular coordinate fixed to the propeller

blade measured from its generating line

t

is the time when the boundary condition on the propeller blade is considered

T ¡s the time when the bound vortex elements shed from the propeller blades

0/0v denotes a derivative normal to the pitch surface a/as denotes a derivative in the direction of chord

length

shows mean line of the blade section, and denotes the values on the vortex sheets

The load distribution y on the propeller blades can be obtained from this equation. In solving this, all the terms depending on time are expressed by Fourier series, and equating each coefficient of the same order, a set of

simul-taneous equations for the load distribution y is derived

which are independent of time. Firstly y is expressed by

y(r,O, t)= ym(,;Theims:2t (2.1.11)

m=0

Next the spatially varying inflow velocity is expressed

by a temporally varying function with reference to a co-ordinate fixed to the propeller. Namely with

:2.1.12) the inflow velocity components are expressed in terms of Fourier series in such a way as

V+y,

E vxmeimO.eimt

(2.1.12a)

m0

ÇZr+v0= vameimO.eimt (2.1.12b)

m=0

The velocity component normal to the blades - the second term of the right hand side of eq. (2.1.10) - can also be

expressed as

v(r,O)=

y0 me1mO1mt

(2.1.13) m=O where m =

for m0

(2.1.16) for rn>1

the boundary condition on the propeller blades is described by equating the velocity components normal to the mean pitch surface, viz.,

471 ¡=1 Rb

r(r'l

i N a

_\/h2+r2

R OL{r') dr'dÔ'

5:

(r',

Ö', V9 mt /h2 + r2

Then eq. (2.1.10) is reduced to the following expression by

Fourier series

R ÖL(r'l

mOS

Ym(r', O'). Kfr,

4r',

')eimtdr'dÖ'

Rb 0T(r')

=h2+r2

òfm Os Vfl m (ñe JmÖ. eJmt (2.1.14) m =0 where N 1 (h2+r'2) 471 /= 1

_f°e_im(r

27r1_1/N)a(i)d

_(2.1.15)

Next, equating each Fourier coefficient of the same

order we obtain the following equations independent of time t, viz., R OL(r')

f

f

y,, (r', Ô') Km (r, Ô, r', f9') dr'dÒ' Rb OTlr') h2 + r2 Ofm Vn (r) Os m(rle/m

The coefficients of these equations are generally plex numbers, therefore the solution is expressed as

com-plex, showing both the amplitude and the phase of the

response of a propeller to the spatial variation of inflow velocity.

With the load distribution y determined, the forces and the moments acting on the propeller blades can be calcu-lated. First, thrust and torque of a single blade expressed by T and Q respectively, are

R _0L(rl

T(t)pf

_f

y(r,û,t)(2r+v00+w60)

Rb OT(r)

v and y0 are axial and tangential components of the wake m = 0, 1, 2,

at the position of propeller, and w and w0 are velocity Neglecting the viscous effect, side forces and bending

components induced by the bound and free vortex sheets. moments defined in a manner shown in Fig. 2.1.3 are given v,,0 denotes the circumferential average of the axial com by

ponent of wake, while the fluctuating terms v,, are neg- _N R _{0L (rl}

lected because of smallness of magnitude. The same applies

F(t) = p E

f

_j'

[y(r, O, t)(V+v0

to the other components of the wake and the induced Rb OTIr)

velocities.

i

_{f- 2iri-1}

TD and à0 are the contribution due to drag on the

+wx.o)j

2ir11 . cosO+2t+

N

blade, which are regarded as constant with time since the

drag coefficients of blade sections do not vary much with

.h2 + r2 dr dO

(2.1.28)

angle of incidence. _N R _ÖLlrl

If the thrust and torque of a single blade are expressed

F(t) = p EJ

f

Ly(r, O, t)(V+v0

by Fourier series, such as Rb

0lr)

R _6L1')

QmNP1f f

YmN(ÇO)(V+Vx,o Rb 01-lr)

+wxo)rh2+.dr.dÔ

(2.1.27)

2iri i

2iri-1 .sin(Ô+t+

N )

./h2+r2 .drdÒ

(2.1.29) N Rp BL(rl

I

_f

y(r,O,t)(r+v00

¡=1 J OT(r) Rb

O+t+

+ wo. ₎₁

2iri i

cos 2iri 1\

N

/

.Vh2+r2dr dO

(2.1.30) N R OL(rl (t) = p

'Rb

[yfr. O, r) (W + y0 '=1 _OTIr)

2d-1

+woo)1 2ir/-1

sin(Ò+t+

f2N

.Vh2+r2drdO

(2.1.31)

where

F is side force in y direction,

F is side force in z direction,

M1, is bending moment about y axis and M ¡s bending moment about z axis

+ 2irTÏ shows that t in F I is subject to cyclic

f,2N change with differing blades

Fig. 2.1.3 shall be referred to for the sign convention. When these quantities are expressed by Fourier series in

such a way as F1,(t) =

FymNeim1t

m0

(2.1.32)

F(t) =

F mNejm! (2.1.33) m=O M1,(t)= MymNeimA1t (2.1.34) m=0 R

6lr)

6(t)=

f

_$

fr, Ô, t) (V+ V + wx 0)r Rb _0T( .\/h2+r2 _{. rdi} dO + (2.1.18)

i(t) =

Tmeimt

(2.1.19) m=0

Ô1mt

(2.1.20) m=0

and if the propeller is symmetrical with its axis - having

identical blades with equal angular interval -, the total

thrust and torque of the propeller are written as

T(t) =

i1

(+ 21r1_1) (2.1.21a) N °° . 2iri-1 = _TmeJmt.e1m N /=1 m=0 N _{- 2iri-1} Q(t)=

i1

Q(t+

2N

)

N 00 _2iri-1 = _Qm01mt.6Jmì &ZN ¡=1 m=0

Then by virture of the formula

N 2iri-1

ei'°'

_cN

-

Nform'mN

'=1 _{= O for}

_m''mN

where m = 0, 1, 2,

the total thrust and torque of the propeller are shown to be dependent solely on the components of blade frequency

and its multiples, viz.,

T(t) = Tç

0JmNt, TmN

= NTmN (2.1.24) m =0

Qmie1', QmNNÓmN

(2.1.25)

m=0

The Fourier coefficients TmN and °mN are written,

with some simplification applied to eqs. (2.1.17) and

(2.1.18), as follows.

R Oh(r)

TmN =pN

f

f

YmN('i. O)(Zr+ y00

Rb 8T(rl + woo)*V/h2+r2.dr.dÔ (2.1.26) (2.1.21 b) (2. 1.22a) (2.1.22 b) (2.1.23)

R OL)r)

FzmN JPN$J

r) _YmN+1fr,0>10Ì

(V+v0+w0)

.h2

+ -dr- d R _OL(r)

MymN =PNf

Rb {YmN_ 1(r,)eiÔ Tir) YmN+1VO)e-18} (2r+ y0 + w0 0)r - h2 +- dr- dÒ 1 R _OL(r) MzmN

jPNf

_f

_{{7mN_1VÒ)e1°} Rb er(r) YmN+1fr,O)0_10) (r+v00+w00)r - h2 +r2- dr- dÒ

Non-dimensional expression of them follows the conven-tional definition as Kr = T/pn2D4 (2.L40) K0 = Q/pn2D5 _{(2.1A 1)} KF =F/pn2D4 (2.1.42) KFZ = F/pn2D4 (2.1.43) KM = M/pn2D5 (2.1.44) = M/pn2D5 (2.1.45)

and these are expressed as a function of advance coefficient based on the undisturbed inflow velocity

J V/riD (2.1.46)

or the one based on mean velocity of inflow

J0= (V+v0)/nD

(2.1.47)

Bending moment Mh with respect to pitch surface at the blade root due to hydrodynamic loading is given by

R L(r) Mh(t)

n'Cb fÒTIT) 7(r,ù,t) (V+

v0

+wxo)2+(r+ ve0+we0)2}2

.cos(ß+çb)(r_Rb)Jh2+r2

drdÒ

(2.1.48) where ß is hydrodynamic pitch angle at any radius and 'Pb is the blade angle at the root.

Cavitation pattern on the propeller blades cannot

direct-y be predicted from the above theordirect-y, unless boundardirect-y condition on the surface of the cavity conforming to the condition of constant pressure is taken into account. This means further complication of the theory and treatment of the problem, and is at the time of writing beyond reach of the purely theoretical approach.

2.2 Quasi-steady method of calculation

In the previous section, a procedure was described how to evaluate the unsteady forces and moments etc. based on

unsteady lifting surface theory. All the calculations are

based on the load distribution y on the propeller blades, but the solution of the integral eq. (2.1.10) for y distribu-tion had not been obtained until the recent development of high-speed computers has enabled the numerical calcula-tions of the type of eq. >2.1.16).

Formerly therefore an approximate approach was made to the unsteady problems by quasi-steady methods of calcu-lation. Quasi-steady method is based on the assumption that a blade section in a spatially varying wake will develop a lift which would be equal to steady lift if this

instanta-neous wake were uniformly encountered by the other

blades. In this case the distribution of the shed vortices in the slipstream of a propeller is considered to be independ-ent of time.

According to this assumption, the load distribution y

in eq. (2.1.10) no longer varies with r and the number of blades, thus

y(r,O,t-T-

) y(r,O,t) (2.2.1)

- 2iri-1

-and r is included only in R1, showing the location of the

vortex element shed at a past time (t-r). Therefore eq.

(2.1.10) results in the one for the steady lifting surface

theory, and the following equation is derived from eqs.

(2.1.10) and (2.1.12).

i

N _a R OL(r)

i1

a

f f

\/h2

+ r'2- dr'dÔ' . y(r', Ò', t) b OT{r')

xf

k/72+r2Ç2dr

h2+r2 a

-v(r,Ò+r)

(2.2.2)

lt is to be noted that time t appears in the above equa-tion solely to show the blade posiequa-tion angle by 2t, and does not affect the integral of the left hand side. In other words, eq. (2.2.2) corresponds to the problem of a propel-ler operating in a circumferentially equal but radially vary-ing wake distribution. Solvvary-ing the equation for a series of

t, with which the radial distribution of v, ¡s varying, we

obtain load distribution y and in turn thrust, torque etc.

in a quasi-steady sense.

With such simplification, however, the solution of eq. (2.2.2) requires still considerable amount of computation. Therefore further approximation had been made either by inverse application of steady-state propeller design theo-ry(31O17>, or by making use of open-water

character-M(t)=

MzmNejmt

(2.1.35)

,n=0

The Fourier coefficients are expressed, with the same de-gree of approximation as applied to thrust and torque, as follows. R, OL(r) FymN 2

pN$

$

YmN_lfr,0)e Rb er(r) +ymN+1(rO)010}

(V+v,0+w0)

.Jh2 +r2 .dr.dÒ _(2.1.36) (2.1.37) (2,1.38) (2.1.39)

e0(deg)

20 40 60 72

w-

--istics of a propeller14 (5)18)

Each method has its advantages and disadvantages, but the method based on experimentally obtained open-water characteristics is considered to be more practical and favourable to further extension and examination with experiments, owing to the simplicity of calculation and

the recently increasing availability and reliability of experi-mental data.

As a representative method of calculation along this line, McCarthy's short method11 is briefly described in the

following, on whose basis the present investigations were carried out.

First it is assumed that, at each angular position of a propeller blade in a wake, the average inflow conditions from root to tip may be determined by the volume meant

longitudinal wake (1+v/v) and tangential wake (v0/V),

viz. (0)

fRvx(r,0)

dr V _V(Rp2Rb2)/2

1v0(r,0)r.

dr v0(0) Rb V _V(RP2Rb2)/2

Having determined them, the local mean advance ratio f and the local effective number of revolutions n can be calculated by

V+v(0)

J(0)

-/ vo(0) = n. (1 + \

2nr)

Next, by use of the open-water characteristic curves of the propeller, it is possible to enter at a local J and obtain the

instantaneous thrust and torque for each blade. If the

angular position of the generating line at the instance t is denoted by

00= (2.2.7>

the thrust and torque of a single blade coming at that posi-tion are calculated by

o

0'O

* McCarthy examined other types of weighted mean, such as

thrust mean wakes for different loading distributions and local wake at 0.7R, but did not find significant difference in the

re-Fig. 2.2.1 Examples of McCarthy's calculation and comparison with measured results (32000 DWT single screw tanker model)181

T(00) KT{J*(0o)}

p{n*(0)}2D4

N KQ{J*(0o)) p{n(Oo)}2D5 _(2.2.9) 0(0e) -N

Then the total thrust and torque of a propeller are

N .,

_{2iri i}

T(00)= T(00+ N ) ¡=1 N /

_2iri-1'

o(oo)= 0Ç00+ N ) i= 1

and the average total thrust and torque T0 and 00 are

T0=

î20

T(00)d00 or 2îr/N

2/Ni

T(00 )d00 (2.2.12) 00₌ 0(00)d00 or 2rr/N

= 2iT/Ni

0(00)d00 (2.2.13)

The fluctuations of thrust and torque can now be deter-mined by

AT TT

1 N - * = = T - 1 (2 2 14) T0 T0

N11K7-00)

AQ

QQ0_

N

r

KQ (fl21

_{- 1 (2.2.15)}

-PI AI

KQ0\J J

0 0 1=1

McCarthy examined his method of calculation by com-paring with experimental results for a single-screw tanker obtained at NSMB16 and HSVA171 using then most modern instrumentation. Sorne of the results from the comparison are shown in Fig. 2.2.1. Agreement betwecn the calcula-tions and experiments is generally good, but it should be kept in mind that the above comparison was made for a very limited number of çases for the fluctuations of thrust and torque depending on small differences between large quantities. This is due to the nature of total thrust and

torque fluctuations of a propeller, and for verification of the method, the experiments and the calculations need to

8(deg) 20 40

/

60 (2.2.8) 7 - Measureo - - - Calculated 6-bladed propeller with moderate skew 5 bladed propeller

with moderate skew 5-bladed propeller without skew (2.2.3) (2.2.4) (2.2.5) (2.2.6) (2.2.10) (2.2.11)

be compared in terms of the large quantities themselves. Thrust and torque of a single blade and blade stress belong to this category, which, however, were not available at the time of ref. (8).

No mention is made by McCarthy on side forces and bending moments, but once the instantaneous thrust and torque of a single blade have been determined, they are calculated by assuming the radial position of application

of thrust and torque. Namely, if this is denoted by r0,

-í

2iri-1\

2iri i \

N cos(Oo+ N ) (2.2.16) N

F(O0) =:;

i=1 F(O0) = r0

N ä(o0+ 2lri_1)

_2irii\

sin(Oo + N ) (2.2.17) ¡=1 r0 N

M(O0)=

T(00+ 1= 1

2iri i \

N

)0

cos(Oo + N _{/ 2iri-1} M(O0)= ¡=1 N )ro sin(Oo+

2.3 Numerical results from unsteady lifting surface theory

With the recent development of the large capacity high-speed computers, attempts have been made to solve the integral eq. (2.1.10) based on unsteady lifting surface theory.

Most of the methods proposed so far assume a Birnbaum series for load distribution along the chord at each radius, viz., Yin (r, Ô) = g0 (r) -

- +g1(r)

- O 21r/_i) (2.2.18)

27ri_i)

(2.2.19) a = V '3 a G, I I 0.2 0.4 06 0.8 1.0 r/Rp

Fig. 2.3.1 Numerical example of Yamazaki's calculation (1) amplitude of bound vortex and wake distributions

ing inflow velocity are generally integer multiples of propel-ler revolutions, while the chord length varies along radius. Therefore the reduced frequency is not constant, but varies along radius. For convenience of comparison, the reduced frequency around a representative radius 0.7R shall be used

in this paper as a common measure. According to Fig. 2.1.4,

U2n./h2+r22rrnr

and w = 27mm therefore, k is approximated by

k=mR)

₍₂₃₂₎

2(r/R) and at 0.7R m(C/R)0.7 k07= (2.3.3)

Yamazaki presented the results of calculation on a pro-peller of a cargo ship n a non-uniform flow and further on a series of propellers with variation of expanded blade area

in the same wake distribution9. Fig. 2.3.1 shows the

former results, illustrating the radial bound vortex

distribu-tion G=irg0

QR and wake distribution. In Fig. 2.3.2, the ratios of the bound vortex and the wake fraction are given, in which no significant variation in response ampli-tude can be seen with the order of harmonics, while the phase angles increase with the order. Fig. 2.3.3 has been extracted from the latter results showing the effect of

ex-panded area ratio in terms of total bound vortex [' at a

blade section,

\/(ûÔT)(OLê)

+ (2.3.1)

whereas the radial distribution is left to be determined by the integral eq. ( 2.1 .16).

Mathematical complexity of the problem, however, has imposed a series of concessions with regard to propeller

geometry, helicoidal wake and a number of chordwise

modes.

Yamazaki took the first term of the Birnbaum series

analogous to the case of a 2-dimensional wing in a gust, and applying an integral operator /(öL-Ö)I(Ö-ÔT)to both sides of the equation, he obtained the radial distribution of the loading on the blades°. Tsakonas et al. took into account a finite number of chordwise modes of load under the basic assumption of approximating the blade helicoidal wake in a staircase fashion, and solved them as a set of simultaneous equations of complex variables(10). Koyama expanded eq. (2.1.10) into Taylor series around the mid-chord, and equating each coefficient, obtained the radial distribution of the load functions11.

In presenting the unsteady characteristics of a two-dimensional wing, the reduced frequency k defined by eq. (2i.8) is used. In the case of propeller, frequencies of

vary- 0.06-0.04 0.02-0 C) o 001 o 00.8 0.7 -'n C) 01- 0-'n 'n 0.1

10 w 'O 0.05 o no i' 100-E' no E' 0.15

0-N

Nm=4

N

OL(r) F(r,O)

_{= f}

y(r,Ò,O) - \/h2 +r2 dÒ (2.3.4) OT(r)

lt is noted that amplitudes of F scarcely changes with ex-panded area, while phase angles advance steadily with in-creasing expanded area ratio.

Breslin et al. conducted calculation for 3-bladed propel-lers operating behind a 3-cycle wake producing screen. (Such a configuration was planned with a view to providing comparison data of experimental and theoretical thrust!

Fig. 2.3.4 Numerical example of Breslin's calculation 3-bladed propeller in 3-cycle wake

bi

t

50 150- 100- 50-0.015 - 0.010 0.005

N

Ae/Ad rn=2 m =0 No. of cycles of wake variation in 360 deg k 0,74=1.86 = 1.24 = 0.62 =0

Fig. 2.3.5 Numerical example of Koyama's calculation (1) variation of bound vortex with cycles of wake

torque fluctuations which are cited later in Chap. 5.). The

calculated thrust and torque fluctuations are shown in

Fig. 2.3.4(10). lt can be seen that the amplitudes do not decrease until the reduced frequency reaches as large as 1.8, while the phase difference between the thrust or

tor-que variations and the wake increases approximately

linearly with the reduced frequency.

Koyama made a calculation for an idealized wake, radially constant and circumferentially varying with

m-periods during one revolution of a blade111. The radial

0.0 15 K53= Qilpfl'D' -150 bE 0.010 -100 0.005 -50 o-o

-0

0.08 K3= TJ5n20' -150 I I na 0.06 -loo 0.04-50 0.02-0.3 0.6 0.9 1.2 0.9 1.8 27 3.6 0.05-I I I I 0.2 0.4 0.6 0.8 1.0 r Rp

Fig. 2.3.2 Numerical example of Yamazaki's calculation (2) variation of response with order of harmonics

90 180 270 360

Oo(deg)

-180 -90 0 90 180

o (deg)

Fig. 2.3.3 Numerical example of Yamazaki's calculation (3) effect of expanded area ratio

0.2 0.4 0,6 r/Rp 0.8 «I 1.0-o E'

0.5-150 bu Q

t

100-E o 5 0-0.06 H .i 0.04-E o 002-m No. of cycles of wake variation in 360 deg ko74 1.2

Fig. 2.3.6 Numerical example of Koyama's calculation (2) variation of bound vortex with cycles of wake

distributions of bound vortex F and the first term of the

Birnbaum series g0g0,, are shown in Figs. 2.3.5 and

2.3.6, respectively, where the effect of unsteadiness is

shown to increase in the inner radii and to decrease with nearing the outer radii.

Summarizing the above examples of the numerical

results of the lifting surface calculations, we may conclude that the amplitude response is affected little by the

un-steadiness of inflow velocity, whereas the phase difference increases monotonously with the reduced frequency. In order to illustrate this conclusion, the amplitude and the phase responses were plotted to the base of reduced fre-quency; the amplitudes in terms of ratio to the

quasi-steady value according io Breslin(12) and the phases assum-ing no lag at the steady condition. The quasi-steady ampli-tudes were derived differently from each source; Yama-zaki's values had been tabulated in ref. (9) itself, Breslin's

according to his publication'2 and Koyama's taken for

m0 Figs. 2.3.5 and 2.3.6. The plotted results are shown in _{Figs. 2.3.7 (amplitude) and 2.3.8 (phase). Response to} a two-dimensional wing is also entered for reference. From these figures, characteristics of unsteady response of pro-pellers can be clearly seen: the response amplitudes are very near to unity at small k values, ¡n contrast with the considerably rapid reduction of amplitude with increasing reduced frequency of a two-dimensional wing. Phase angles, on the other hand, increase with reduced frequency with a tendency very similar to a two-dimensional wing.

Therefore it is inferred that applicability of quasi-steady method will be much wider for propeller problems com-pared with two-dimensional wings, if due attention is paid for the phase angles of the response functions. The present investigations started with such a review on the available results of unsteady lifting surface calculations with a view

1.0 0.8 -o ° 0.4 0 o 's

0.2-Fig. 2.3.7 Results from unsteady propeller theory (1) ratio of amplitude between unsteady and quasi-steady theory 250 on Q -o 200 -o o on 50 150

/

/

/

,

/

,

100 o OA \ O Yaniazaki O Fig.2.3.2 Breslin - Fig.2.3.4 Koyarna A Fig2.3.5 Koyama A Fig.2.3 6

Two -dim. - - - Fig.2.12

A Yamazaki O Fig.2.3.2 Breslin Fig.2.3.4 Koyarna Fig2.3.5 Koyama A Fig.2.3.6 Two-dim. --- Fig.2.1.2 2 3 Reduced freti, k0.7

Fig. 2.3.8 Results from unsteady propeller theory (2) phase difference between unsteady and quasi.steady theory

to exploring the range of applicability of a quasi-steady method. In the next chapter principles of the quasi-steady method proposed in this paper will be explained.

0.6

3. Proposed quasi-steady method of calculation and verification of the method

3.1 Extension of quasi-steady method of calculation Referring to the investigations on the available methods for calculating the unsteady problems of a propeller, the

author came to consider that the applicability of

quasi-steady method would be fairly wide and that it would be

worth while to improve it further in view of the results

/

/

/

,

/

/

,

,

f

,

0.2 0.4 0.6 0.8 10 2 3 4 5 r/Rp Reduced freg. k)

from unsteady lifting surface theory. Firstly, therefore, it was intended to take into account the effect of blade chord length relative to breadth of wake, and secondly to consider radial distribution of wake. This was done by specifying the

types of chordwise and radial distribution of thrust and torque in uniform flow. Then the elemental thrust and torque at any position on the blade are calculated as

described in 2.2 corresponding to local advance ratio and local effective number of revolutions, for which the open-water characteristics were entered.

Namely, if the open-water characteristics of thrust and torque are given as a function of advance ratio J in such a

thrust and torque of a single blade with its generating line located at 0=80 are written as

R OLlr)

t(00)=f

_I

_N Rb fc(Ö)1Crfr)01 dÓ R OL(rl * Ó(00)= P

f

f

KQ(J ) Rb OTlr)

whereJ is the local advance ratio defined by

J*_

VVX

n *D

with the effective number of revolutions given by

r ( 0

n =njl+

(3.1.7a(

\

2rnr/

v and y0 are given ordinarily as a spatial variation depend-ing on r and 8. In calculatdepend-ing the response of a sdepend-ingle blade at a specified position angle, they are turned into a function of blade position angle putting 00=r in eq. (2.1.12), viz.

0=0+00

(3.1.8)

Then KT and K0 are read out of the open-water charac-teristic curves at

V+

_vx(r,00)

n with

n=n 1+

-

v0(r,+00)

27rnr

1(0) is a function showing the type of chordwise load

distribution, such as = 1/{ÖL(r)

-

rfr)}

1c3 -1T{ÖL (r)_T(r)}

0L (r)Ô

2 where

fc i corresponds to a lifting line approximation with

°L (') 1-OT(r)

Ici

6{0-0Mr),

OM(r)-2

'r3 r 12

Comparing eqs. (3.1.4) and (3.1.5) with eqs. (2.1.17) and (2.1.18), it is noted that the following approximations have been made.

y(r, ,t)(&lr+ V0 + w0) /h2+r2 KT(J*) = iv fc«))fr)T)fl*2D4 (3.1.19) With T and thrust and N T(00)= obtained by eqs. (3.1.4) torque fluctuations are given

and (3.1.5), the by (3.1.17)

2iri-1'

i-(o0

¡=1 N

/

Q(00)= N _2iri-1 ä(o0+ (3.1.18) ¡=1

N)

way as T/(pn2D4) = KT(J) (3.1.1) Q/(pn2D5) = K0 (J) (3.1.2) J= V/(nD) (3.1.3) where

r1 corresponds to assumption of the load distribution

concentrated at a representative radius 0.7R r2 is the load distribution increasing linearly from hub

to tip

r3 corresponds to an approximation of load

distribu-tion derived from circuladistribu-tion theory by a simple polynomial

Which type of and r among the above is most suited for practical application will be examined and discussed in the next section in view of comparison with experimental

results.

Eqs. (3.1.4) and (3.1.5) take a form of Kn2 and

K0n

*2 being multiplied by f, and _r,and integrated over the blade area. In this sense, f and 'r are regarded as weight functions for loading, and further, if a linear relation be-tween Kr or K0 and J is assumed, they are regarded as a weight function for averaging the wake in the McCarthy's method. With this in mind, the coefficients of and _r

are so determined for convenience that

f(Ò)dÔ=1 (3.1.15)

R

f

f1(r)dr= i (3.1.16)

Rb

McCarthy's method corresponds to the special case of them in which and r2 are taken in considering the non-uniform wake.

total

respect to the midchord, with O being Dirac's

delta function

corresponds to chordwise constant distribution

rC3 corresponds to the load distribution on a

two-dimensional wing in a sinusoidal gust as given by eq. (2.1.6)

1rrfr) is a function showing the type of radial load

dis-tribution such as

ri

=O(r_0.7R)

(3.1.12) 'r2=2(r_Rb)/(RPRb)2 (3.1.13) (rRb)2(R r) (3.1.14) (3.1.4) (3.1.5) (3.1.6a) (3.1.6 b) (3.1.7b)

y(r,,t)(V+v +wx)rh2 +

Ko(J*)

-

N fc(Ö)frO)T12D5

with this relationship substituted into eqs. (2.1.28) through (2.1.31), side forces and bending moments are given by N R

F(00)=p

1R

10L ¡=1 07-Ir) D5 -cos(8+00) 00+27r/ 1/N 1'c(Ô)'rfrfrnT dO (3.1.21) N R B&(r)_K J*)

F(O0)=p

_f

f-

Q - Rb

0(r)-D5 . -sin (0 +00

r

_00+2iri-1/N fc(Ô»r(r)dr dÓ (3.1.22) N R OL(r)

M(00)=p

_f

_f

¡=1 Rb OT(r) *2D4 cos(O+00) - 00+21ri 1/N 'c(Ö)fr(r)rdr.dÖ (3.1.23) N R 0L11 r (J*) M (0e,) = P

f

_f

I T ¡=1 Rb

°r)

L N D4 sin(Ò+00)1 J00+2lTi 1/N fc(Ò)fr(r)r. dr. dÖ (3.1.24) where 00+2iri-1/N shows that O in [ j is subject to

cy-clic change with differing blades.

Thus as is evident from eqs. (3.1.19) and (3.1.20), the load distribution y as a solution of the integral equation has been substituted by Kr and K0. They can be estimated, even though no data are available for a geometrically similar model, from the published data such as NSMB's Bseries(18) and SRI's AU-series propeller charts(19). This means not only remarkable reduction of computing time (ratio more than lOto i by UNIVAC 1100 computer), but also omis-sion of the laborious work of writing blade offset as input data. Such features of the quasi-steady method will be ap-preciated especially in the preliminary design stages, during which similar sorts of investigations are repeated to com-promise the design requirements conflicting with each other.

Blade bending moment at the root due to hydrodynamic

loading can be written, with reference to eqs. (2.1.48),

(3.1.19) and (3.1.20),as R _OLlr) -Mb(OQ)

_{=p f}

[{KTJ*}2 {KQ(J*)}2 1/2

Rb

0(r)

(rID) Ko(J*) N (3.1.20) KT(J*) N .n*2D4f(Ö)ffr)co5(ß_Øb)(r...Rb)dr.d (3.1.25)

This formula is useful when the bending moment has to be estimated from thrust and torque coefficients of a pro-peller, but if the measurement of blade bending moment M has been made in uniform flow (which is nowadays a relatively easy work and accumulation of data, if required, will be possible without too much labour), the variation in a non-uniform flow may be estimated in a manner similar to thrust and torque, such as

R eL(r)

In cavitating condition, elemental thrust and torque are a

function of not only the local advance ratio, but also of

local cavitation number. With this additional variable, even the quassteady method becomes considerably comp) i-cated. Since, however, in principle the local operating con-dition of a blade element can be determined ri a quasi-steady sense, it would be worth studying the applicability of the method. This will be discussed in Chapter 6.

3.2 Verification of proposed method of calculation by experiments using one-blade dynamometer As mentioned in the foregoing chapters, the

quasi-steady method has been applied so far chiefly to the problems related to variation of propeller shaft forces.

Magnitude of the forces and moments handled ¡n such problems, however, is of the order of a few percent of the

total thrust and torque of a propeller. Therefore

com-parison of calculation and experiments made so far did not suffice to verify the method of calculation. Results from measurements did not have sufficient accuracy either, since even nowadays the measurement of fluctuating propeller shaft forces is one of the most difficult experiments be-cause it requires to pick up small quantity of forces and moments with relatively high frequencies which are integer multiples of blade frequency.

Every computational method needs to be verified by comparison with reliable experimental results, however rigorously the method may have been developed. In other words, agreement with a more sophisticated theory does not necessarily mean that the method has been validated. In this sense the existing experimental data ori the pro-peller shaft forces have not been reliable enough to be used as a tool for evaluation of computational methods. In the meantime, however, a number of casualties which occurred on propeller blades, and the resulting needs for detailed investigations into fluctuating blade stress afforded good opportunities for more fundamental approach to

fluctuat-Mh(OO) with where

=pf f

Rb OT(r) 1 KMt(J) N KMt(J)n*2DSfc(Ö)fr(r)dr.dÖ 1/2 (3.1.26) (3.1.27) (3.1.2 8)

rRb

(rID)2 -D KMt(J) = M(J)/pn2D5

ing loads on propeller blades operating in a non-uniform wake. On a number of occasions blade stress was

meas-ured(221) and further,

attempts have been made to measure the thrust and/or torque of a single blade of multi-bladed propeller by a special dynamometer developed for this particular purpose(22)25). With such measurements, it has become possible to evaluate computational methods in more detail since the blade stress as well as loads on a single blade varies with frequency of propeller revolutions, and its multiples including frequencies lower than the blade frequency.

With such a background it was ¡ntended first to examine the effect of breadth of wake peak in relation to breadth of a blade. For this purpose the one blade-dynamometer as shown in Fig. 3.2.1 was developed by the author(26). Unlike the similar attempts having been made so far, the propeller blade is supported by a dynamometer of a paral-lelogram type by which the natural frequency can be in-creased as high as 300 Hz because of very small displace-ment of the blade.

With this dynamometer, effect of varying breadth of wake on the thrust of one blade was examined

experi-mentally in cavitation tunnel. Only the axial wake compo-nent was changed locally, since it was considered that absence of tangential and radial wake component

appear-ing in the flow field at the stern of a model ship would

give a more direct comparison between theory and experi-ment regarding the effect of breadth of wake.

Fig. 3.2.2 shows the wake distributions used. Both of them have the wake peak with nearly the same height, but the breadth was different. Radial wake distribution at the wake peak was nearly uniform, thus enabling comparison of different wake breadth as clear as possible.

The model propeller tested in these wakes had the

principal dimensions as shown in Fig. 3.2.1 and the project-ed blade outline is entered in Fig. 3.2.2 to give an idea

about the breadth of the blades and wakes. Fig. 3.2.3 shows further the circumferential distribution of the wake at the representative radii together with the breadth of the blade

- Wake A of Fig. 3.2.2 Wake B of Fig. 3.2.2 60-40-20 0 20 40 e9o(deg) 60

40-20 0

20 40 Go (deg) -40-20 0 20 40 d (deg)

Particulars of the model propeller

Fig. 3.2.1 One-blade dynamometer for a propeller

Q z

Wake A

Q

Fig. 3.2.2 Wake distributions used for examination of effect of wake breadth

sections.

Fig. 3.2.4 shows the typical results from measurement of the thrust of a single blade as a function of angular posi-tion O of generating line of the blade. From this figure, effect of both wake distributions can be interpreted that

amplitude of thrust in wider wake is larger than

those in narrower wake, and

the phase difference of the thrust relative to wake

Q Q z Wake B Strain gauges - In wake A of Fig. 3.2.2 In wake B of Fig. 3.2.2 Top

90

0 90 Go(deg)

Fig. 3.2.3 Comparison of two wake distributions Fig. 3.2.4 Variation of thrust of one blade in with different breadth two wakes with different breadth

13

Diameter 268mm

Pitch ratio(const.) 1.000

Expanded area ratio 0.652

Boss ratio 0.191 Thickness-Chord length ratio at 0.7R 0.0507 Number of blades 5 0 08 0.06 Q 0.04 0,02 o

008 0.06 o 0.04 0.02 o 0 Wake A Top Top

Fig. 3.2.5 Comparison of measured and calculated thrust variations in wakes - effect of chordwise weight functions

Full load condition Trial load condition

Fig. 3.2.6 Wake distribution of a container ship

peak is larger for narrower wake than for wider wake. Then the calculation of thrust variation of a single blade was made according to the proposed method described in 3.1. The type of radial distribution of thrust was assumed to be

r3 4

12(rRb)2 (Rp_r)

(3.2.2)

V'pflb)

in common through the calculations concerned since the radial wake distribution was nearly uniform as mentioned above. The effect of circumferential variation was taken into account by the three types of function described in 3.1.

The results from the calculation are shown in Fig. 3.2.5,

in which ¡t is clear that the use of function f03 yields a

best fit to the experimental results in terms of both ampli-tude and phase, while yields overestimation of ampli-tude and underestimation of phase difference. A simple chordwise average f02 mitigates the spatial wake distribu-tion, but does not cause the phase shift between wake and thrust variation.

Therefore it may be concluded that the conventional quasi-steady method paying attention to wake variation at

one representative spot of the blade will overestimate

z

0.08 0.06

o Experiment

0,7R fri

Volume mean fra

Weighted mean fr3 o

0.08 O _Experiment

0.7R fri

- 0.06

o _{o °} Volume meanWeighted mean

fr2 fr3 0.04-z.:L:-o \ 0.02-o 90 180 270 3(0

Blade position angle Oo(deg)

Fig. 3.2.8 Comparison of measured and calculated thrust variations in model wake (trial load) effect of radial weight functions

amplitude, and that f is the best to incorporate the effect of circumferential variation of the wake.

Now that the effect of breadth of blade in relation to

the breadth of wake has been made clear, further

investiga-tions were made ori the effect of radial distributons of

thrust and torque. For this purpose, a propeller with the one-blade dynamometer was run with a model of container ship, since it was considered to be more convenient for examination of effect of radial distribution of wake because of more rapid change of wake in radial direction than for full ships. The wake distributions are ahown in Fig. 3.2.6. For the calculation, three types of radial distribution of thrust and torque were examined, which were explained in 3.1. The chordwise load distribution was assumed to be of the type r3 as concluded from the foregoing examination.

Comparison of the measured thrust of a single blade and calculated thrust is given in Figs. 3.2.7 and 3.2.8.

Agree-ment between the measureAgree-ment and the calcijation is

generally good with comparatively large deviation when 1r2 was used. Choice of either

ri or

is difficult to be

decided. If time for calculation is so limited as to take a single representative radius,

_ri

would be a choice. For

establishing a practice, however, it would be safer to take into account the wake fractions from hub to tip, and not to depend on the data of a single representative radius. Therefore we had better use the distribution more realistic, viz., f03, which is more alike the distribution obtained by theory and experiments.

0.02

-o 90 180 270 360

Blade position angle Oo(deg)

Fig. 3.2.7 Comparison of measured and calculated thrust variations in model wake (full load)effect of radial weight functions

90

o Oo(deg)

90

o Oo( deg) 90 90 Wake B 0 _Experiment Midchord fci -Simple mean fc2 Weighted mean fc3

4. Blade stress

Measurement of blade stress of a propeller working in a non-uniform flow has provided valuable information for strength problem of propeller blades which should be con-sidered on the basis of corrosion fatigue characteristics of materials. Since the blade stress varies with the frequency of revolutions of the propeller, - not as high as blade fre-quency - measurement can be made with reasonable ac-curacy without special device for pick-up and data

process-ing system. Thus investigation into the blade stress, especially at the root of a blade, has served a good example for examining the applicability of quasi-steady method of calculation.

Stress c at the blade root is the sum of the one due to the hydrodynamic bending moment Mh and another due to centrifugal force, namely

Mb M F

Ot=Uh+Uc, 0h=

7c= + (4.1) b b b where 0h oc Zb F Ab

According to 3.1, in which radial and chordwise distribu-tion of thrust and torque are assumed by f,.. and the bending moment induced at the root section of the blade by hydrodynamic loading is given by

R OL(r)

Mh(Oo)=Pf

_f

KMt(J)n*2.DS

is the stress due to bending moment Mb caused by hydrodynamic loading

is the stress due to centrifugal force, being sum of the contributions from bending moment M

caused by rake of the blade and tension F is section modulus at the blade root is centrifugal force

is bending moment due to the centrifugal force is cross sectional area at the blade root

1 Rb OT(r)

Therefore the blade stress at the root can be calculated by eq. (4.1).

lt is to be noted that radial distribution of the integrand in eq. (4.2) includes the terms of fr(r).r and fr('r)/r as well as fr(r) considered so far. However, as mentioned in the previous chapter, the type of radial distributions of thrust and torque is not so much influential on the quasi-steady value, and therefore KMrf(r) may be further approximated by a type of function f(r).

If, according to the principle of the quasi-steady method treated in this paper, open-water characteristics of blade stress are known as a function of advance ratio, the blade stress in a non-uniform flow can be calculated by

0.75 0.50 025 R _eL(r)

u(0)

f

_f

Ka(J*)n*2D2./c(0)fr(ñdr.dO Rb er(r) (4.4) where K0(J) = KMt(J) . = Uh(J)/Pfl2D2

For example when the open-water characteristics are given by Fig. 4.1, the blade stress in the wake distributions shown in Fig. 4.2 is calculated. The results from the calcula-tion are presented in Fig. 4.3 compared with those meas-ured by strain gauges stuck at the root. Good agreement can be seen between the measured and the calculated

re-suits. 400- 300-(T 200- 100-0 0.4 0.5 0.6 0.7 0.8 0.9 Jv/nD

Fig. 4.1 Blade stress as a function of advance ratio in uniform flow

z

/ I

120

Sh,p wake

Fig. 4.2a Comparison of wake distributions of model and ship

180 where KMt(J) N fc(Ò)fr(r)d0'0 {KT(J)}2+

{K(J)}2

(4.2) 1/2

_{r- b}

o (r/D)2

cos(ßb)

D N K0(J)

rRb

KT(J)cosb+

r/D sinøb - _- D _(4.3) Wake distribution at 0.7R - Model wake - Model wake

Ship wakewake

Blade section! Blade section! - - - -i-' at 0.7R - - - -i-' at 0.7R i t i t 0

U!'

U!'

₃₀₀

//

/I

/I

₀ '' ₆₀

Measurement

-- - Quasi-steady calculabon Model wake

5. Propeller shaft forces

By propeller shaft forces are meant the force and the moment exerted on the ship through the propeller shaft and the bearings as a sum of the contribution from all the blades. They are further classified into thrust and torque,

and side force and bending moment according to their directions (either parallel or normal to propeller shaft).

5.1 Thrust and torque fluctuations

Suppose the thrust fluctuation of a single blade is ex-pressed by a Fourier series of the angular position 0 of the generating line such as

t(00)= me1m80 (5.1.1)

m=O

then the fluctuating thrust from the propeller as a unit

becomes, as described in 3.1,

TJeo,

TmNNtmN

(5.1.2)

m=Q

and in a similar manner the fluctuating torque is

Fig. 5.1.1 Wake distribution for comparative experiments of 13th ITTC Propeller Committee

0.1 0.1 0.2 0.2 0.2 0.1

012345678

m 0.2 0.1 m m

012345678

m O.7R

012345678

m 0.5R

012345678

Fig. 5.1.2 Fourier coefficients of velocity distribution in Fig. 5.1.1

mNmN

(5.1.3)

m =0

Thus only those components of the blade frequency and its multiples of one blade response appear in eqs. (5.1.2) and (5.1.3). Since they are dependent on the corresponding Fourier coefficients of wake VXmN, VOmN which are small

quantity, experimental verification of eqs. (5.1.2) and

(5.1.3) is a difficult problem because of difficulty in measuring thrust and torque of high frequency and in

measuring velocity distribution so accurately as to evaluate the higher harmonic components.

Fig. 5.1.1 shows an example of wake distribution pro-vided for comparative experiments of thrust and torque

0 60 120 180 240 300 360

'I (deg)

Fig. 4.3 Comparison of calculated blade stress with

measurement

The quasi-steady method will be applied further to the stress of the blade, which emerges partially above the water surface during one revolution, as observed in connection with propeller racing in waves. This problem is at present

far beyond the scope of unsteady lifting surface theory

because of strong non-linearity of the blade characteristics, and quasi-steady method is the sole approach which can provide certain information on the variation of blade stress

Difficulty arising even from quasi-steady approach, is

lack of open-water characteristics of a propeller whose blades emerge out of water always by a definite portion. If they are supplemented by calculation based on eq. (4.3) for a reduced diameter of propeller corresponding to partial submergence, the blade stress in behind condition with

partial emergence can also be estimated by the quasi-steady method. 60 120 180 240 Oo (deg) 300 360 0.2 0.1 0.2 0.1 OE9R o 2345678 o 2345 678

Q) -tu = a-.. E E 1.0 tuo = un

_o.

O, 0.5 Q) tu o F- 0.06-- 0,05-0.04- 0.6 tu 0.03- 0.5 0.02- 0.4 0.01- 0.3 0- 0.2 0_1

Fig. 5.1.3 Open-water characteristics of the propeller for ITTC comparative tests

A NSRDC NSMB MHI QuasiSteadY ca. vws 10 15 20 25

Propeller speed X model speed VQ (m/s2)

0 10 15 20 25

Propeller speed x model speed VQ (m/s2)

Fig. 5.1.4 Thrust and torque variations of ITTC comparative tests

fluctuations organized by 13th ITTC Propeller Commit-tee(27). _{Fig. 5.1.2 is the results from Fourier analysis of} the circumferential variation of wake fraction as shown ¡ri Fig. 5.1 .1. In this comparative study, 6 organizations of the world took part, and the measurement was made by their own measuring instrument on the same model and propeller circulated ¡ri _{turn. Fig. 5.1.3 shows the}

open-water characteristics of the model propellers.

The measured blade frequency components are shown

in Fig. 5.1.4 together with those calculated by the pre-sently proposed quasi-steady method using eqs. (5.1.2) and (5.1.3). The measured values reported from the partici-pated organizations are, however, scattered so widely that it is difficult to draw a definite conclusion whether or not the quasi-steady method provides with good estimation of

tu

Three-cycle wake screen Four-cycle wake screen

Fig. 5.1.5 Wake screen for three and four cycle wake

0.10 0.090.08 - 0.070.06 - 0.05- 0.04-0.03 0.020_01 -

0-\

"K

0.6 '5 0.5

\

'5 S

\

\

'S s '5

\

S 0.4 0.3 0.2 0.1 '5 '5 '5 '5

\\\

\,

'5'

\

'5 5

\

Kr \

\

5' 05 Propeller No. 5' 5' J VnD

Fig. 5.1.6 Propeller open characteristics of NSRDC propellers

the thrust and torque fluctuations.

Considering such difficulty in evaluating the adequate-ness of the method of estimating the thrust and torque fluctuations, another comparison is attempted for the

measurement in NSRDC28 carried out on the 3-bladed propellers in a 3-cycle wake. Fig. 5.1.5 shows the screen set in a cavitation tunnel to produce a 3-cycle wake

togeth-er with the screen for 4-cycle wake to be cited in the

following section. With such a scheme, blade frequency components of the measured thrust and torque could be obtained which were large enough to be compared with the calculated values. The measurements were made on the 3-bladed propellers with their expanded area ratio varying from 0.3 to 1.2. Open-water characteristics of the propellers are shown in Fig. 5.1.6. Comparison of the measured results with those calculated by eqs. (5.1.2) and (5.1.3) is shown in

(P/D)o-i 4132 1.086 0.3 3 4118 1.077 0.6 3 - - - 4132 1.037 1.2 3 Ae/Ad N 1.0 5' 17 P.1538 D =253mm P' D=1.000 Ae/14d = 0.485 N=4 0.5 1.0 JV/OD

.

-Fig. 5.1.7 Thrust and torque fluctuations of 3-bladed propeller in 3-cycle wake (KTOxO.15)

Fig. 5.1.7. Except for the amplitudes of the propeller with expanded area ratio of 0.6, agreement between the measure-ment and the calculation is good for both amplitude and phase.

5.2 Side force and bending moment

Side force and bending moment of a propeller arise from fluctuation of elemental torque and thrust, respectively. They are calculated by quasi-steady method based on eqs. (3.1.21) through (3.1.24) using the open-water charac-teristic curves of a propeller. Like the thrust and torque fluctuations mentioned in the previous section, they are presented in terms of amplitude and phase as

F(O0)= E FymNe!mfs16o m=0 F(O0) E FzmNEjmN0o m=0 M(O0) =

m0

MymNeimNeo M7(00) = E MzmNejmNeo (5.2.4) m-O

As is inferred from a simplified set of eqs. (2.1.36) through (2.1.39), they are contributed by (mN±1 )th order (blade frequency and its multiples plus/minus 1) of harmonics of elemental thrust and torque fluctuation. This means that, like thrust and torque fluctuation mentioned before, side force and bending moment are caused by wake components of small quantity.

Few experimental data of side force and bending mo-ment are available, however, which are competent to com-parison with theoretical calculation, because of difficulty in measuring small amplitude, high frequency force and moment.

Figs. 5.2.1 and 52.2 show the results from the measure-(5.2.1)

(5.2.2) (5.2.3)

oc

k1

Fig. 5.2.2 Bending moment fluctuations of 3-bladed propeller in 4-cycle wake (Kr00.15)

ment on the 3-bladed propellers mentioned above in a 4 cycle screen shown in Fig. 5.1 .5. Correlation between meas-urement and calculation is not so good as illustrated for

thrust and torque fluctuations, especially again for the

propeller with expanded area ratio of 0.6, but considering the difficulty in measurement, both results may be said to be in reasonable degree of agreement.

Another data for comparison is presented for the meas-urement made on a container ship model with a conven-tional propeller and a highly skewed propeller291. Highly skewed propellers have recently been increasingly employed chiefly because they produce less propeller shaft fcrces and moments, they are less sensitive to cavitation inception and they transmit tess pressure fluctuations. Of these character-istics, the first (propeller shaft forces and moments) was of

se

character-istics, the first (propeller shaft forces and moments) was of

F,, 00. o F3 KF,, =pn'D' 0.01° 300 'C , 0.005

----

_s 200 ci, _s Ae.'Ad 0.3 0.6 0.9 100 0.9 1.8 2.7 O Exp. amp. 0.015 _{Exp. phase} ,

/

150 Cal. amp.

,

/

be

-0.010 Cal. phase

,

/

o ₁₀₀

-o

g 0.005 50

-c

K3= QaJpn2D5 o

-0

-: 0.08 Kr:T,Jpn2D"

_/

150 0.06 o

/

, 100 0.04 50 0.02 o 'C 300 E o.00s 200 1.2 0 o 100 3.6 o 300 0.010 o 200 'C E 0.005 100 M,, O Exp. amp. 00. Exp. phase - Cal. amp. ---Cal. phase Q Exp. amp. Exp. phase O - Cal, amp. - 300 ---Cal. phase_ to.oio o-200 0.005 _{- -}

-100 1.2 AelAd 0.6 0.9 a-0.3

0.9 18 2.7 3.6 Fig. 5.2.1 Side force fluctuations of 3-bladed propeller kx.,&

Conventional Propeller HighIly Skewed Propeller

Fig. 5.2.3 Models of a conventional propeller and a highly skewed propeller

Fig. 5.2.4 Open-water characteristics of conventional propeller and highly skewed propeller

primary concern when examination of highly skewed pro-pellers was resumed for practical application30. Namely, in designing highly skew propellers, amount of skew and

radial skew distribution are determined in principle to mis-match the wake harmonic components so as to minimize propeller shaft forces.

Fig. 5.2.3 shows photographs of a conventional type propeller and a five-bladed highly skewed propeller with 100% skew. The conventional type propeller is identical in

geometry with the propeller fitted with the blade

dyna-mo meter mentioned in 3.2.

Open-water characteristics of them are shown in Fig. 5.2.4. The wake distribution has already been shown in

Fig. 3.2.6. Results of the calculation of propeller shaft

forces are compared with those of measurement in Figs. 5.2.5 and 5.2.6 as a function of J=V/nD, where V is the speed of advance of the model. Results from the calculation by Koyama's lifting surface theoryhhl) are also entered with fairly good agreement with the experimental results. Devia-tion of the results of quasi-steady calculaDevia-tion is larger, but both the experiments and the quasi-steady calculations show clearly that the vibratory forces and moments are

much smaller for the highly skewed propellers. Bearing in mind further that those bearing forces are very small

quan-tity, it may be considered that the present quasi-steady

K,ev= MrIpfl2D5

_.

s

o---

_-

o A s . o o 0.8 1.0 1.2 J-= V/nD

Fig. 5.2.5 Propeller shaft forces of conventional propeller and highly skewed propeller (full load)

o

Exp. CaL(QS) Cal,(L)S)

.z;

o o o 0- 0-19

\

- Conventional propeller

0.06- Highly skewed propeller

0.05-0.04 K7 's 0.03- 0,4 _5 '5 '5 0.02- 0.3 '5 '5 '5 0.01 0.2 '5 0- 0.1 's_5 o 0.5 1.0 J=V'nD

0.008 Exp. Cal.(QS) Cal.(US)

Conventional

A prop.

. O Highly skewed prop.

K,..5= Fx/pnpD1 0.006- 0.006 K,5=F»/n2D4 0.004- A 0.004-0.002

.

s

-

0.002 s s A K755= Fx./pfl2D4 O A Conventional prop.

----

Highly skewed prop.

0008-Ky,x= F,s/pn2D4 0.006-s _s

.

o 0.004

t

s s o _0.002 0.002- 0.002 0.001- ,-" o

-- -

- 0.001 °

_{_A-'"}

0.8 1.0 1.2 1.4 J= V/nD J- V/nD

Fig. 5.2.6 Propeller shaft forces of conventional propeller and highly skewed propeller (trial load)

method is useful also for design and evaluation of highly skewed propellers.

6. Cavitation

Cavitation on propeller blades takes place when the

pres-0.8 1.0 1.2 1.4 0.004-KMVS= M05/pn2 0.004 0.003- 0.003 0.0 10-0.008 0.006 0.004 0.002 0- 0.004-0.003 0.002 0.001

sure on the blade has fallen to the vapour pressure of water, as a result of increased velocity of flow on the blade. In early days the problems were confined to high-speed ships such as destroyers and torpedo boats, but as speeds and powers have increased, the erosion and vibration aspects of cavitation have become more and more important, particu-larly in high-powered, single-screw ships. In such ships, there is a large wake variation over the propeller disk, which encourages unsteady cavitation appearing at a high wake zone and disappearing after passing it. At the moment of disappearance, or collapse of the cavity, extraordinary large pressures are radiated from the cavity which cause erosion on the propeller blade and exciting forces on hull surface.

Therefore the researches on propeller cavitation which had been initiated in uniform flow have been replaced by those in non-uniform flow similar to ship's wake. In this chapter, however, an attempt is made to explore how far the quasi-steady approach is applicable to the problems

involved with cavitation; the author thinks this worth

while, since complete description of unsteady cavitation is still far beyond reach of theory and on the other hand

individual testing of a propeller in differing wake distribu-tions requires a large amount of time and expense.

6.1 Propeller cavitation in uniform flow

The principle of the presently proposed quasi-steady approach lies in that a phenomenon met with in non-uniform flow will be predicted by the data in the non-uniform

flow at the same local flow conditions. In the previous

chapters the local flow condition was dictated by the local advance ratio, but with the presence of cavitation the local

cavitation number has to be taken into account.

This section is rendered to describe the propeller

charac-teristics in uniform flow - cavitation pattern, thrust and torque variation with operating conditions - to afford a

basis of prediction of those in non-uniform flow.

Types of propeller cavitation n uniform flow can be

divided into the following three kinds. Bubble cavitation (Fig. 6.1.la)

This type of cavitation appears on the back of the blades at considerably small positive angles of attack, when very small bubbles grow steadily with decreasing

pressure towards the point of maximum thickness.

Those bubbles then move aft and collapse as they enter a region of higher pressure.

Normally the bubble cavitation does not appear on the face side, since the face side of the blade sections especially in the outer radii is flat or hollow in general and therefore no gradual decrease of pressure does not occur along the chord.

Sheet cavitation (Fig. 6.1.1 b,c)

This type of cavitation appears either on back or face side of the blades at considerably large positive or nega-tive angles of attack. In such a condition the pressure along the blade drops at an extremely rapid rate behind

(a) Bubble cavitation on back side

(b) Sheet cavitation on back side

c Sheet cavitation on face side

(d) Tip vortex and hub vortex cavitation

Fig. 6.1.1 Typical cavitation patterns in uniform flow

the leading edge because of the generally small radius thereof, thus there is not enough time to form bubbles. The liquid then leaves the surface of the blades to form a cavity, until the pressure downstream is large enough to close it. The liquid-cavity interface appears generally smooth, and it is called the sheet cavitation after its

appearance of silvery sheet at angles of vision reflecting the lighting.

At an excessively large positive angle of attack, a sheet cavity extends as far as downstream of the Trailing edge, and the blade sections are entirely covered with a cavity. This phenomenon is called supercavitation, and is often applied to high-speed propellers (supercavitating propellers) taking advantage of eliminating frictional

drag in the area covered by cavity and of avoiding ero-sion due to collapse of cavity on the blades.

(3( Tip vortex and hub vortex cavitation (Fig. 6.1.ld) The tip vortices and the hub vortex are always pre-sent, except when the propeller is delivering no thrust, and with increasing loading they increase in intensity. The pressure at the center is reduced until it reaches vapour pressure, when the core of the vortices becomes visible as a cavity taking on the appearance of twisted ribbon of semi-infinite length.

At moderately large angles of attack, tip voilex cavi-tation often appears at the same time with sheet cavita-tion on the back side in outer radii.

With extensive cavity on the blades, thrust and torque of the propeller are generally smaller than in non-cavitating condition, but they do not change appreciably as far as the cavitation is limited to tip vortex cavitation and sheet cavi-tation in outer radii. An example of thrust and torque

varia-tions is shown in Fig. 6.1.2 for a model propeller of a