Unsteady lifting surface theory for ship screw. Part I: Text

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r

P.J. Verbrugh

Unsteady Lifting Surfacé Theory fòr Ship Screw.

Report no. 68-036-AH April 1968 Part I : Text

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UNSTEADY. LIFTING SURFACE THEORY for SHIP SCREWS by P.3. Verbrugh. Introduction.

DuMng the last years many reports on ship screws operating in a real wake have been published. The objective of this

continuingseries of theoretical investigations is to approximate

the physical reality ever closerby.studying more realistic mathematical models.

Early studies in this field were undertaken by Tim.man and: Van de Vooren [i] on unsteady rotating wings. Ritger and Breslin

[2] developed a strip theory, based upon the ùnsteady twodimensiona: airfoil theory of Von Karman and Sears [3], corrected for three-dimensional effects. Compared with experient, however, this theory was found to be inadequate for an unsteady propeller.

A better approximation for the unsteady propeller problem was given by Shiori and Tsakonas [k]. Their starting point was a threedimensional unsteady theory, formulated, by Hanaoka [5], a theory that leads to a quite complicated integral equation. Shiori and Tsakonas simplify this integrai equation according to the well-known Weissinger - _{method. They could not prove} that the Weissinger model was fully satisfactory for the unsteady case.

More recently Tsakonas and Jacobs. [6] have solved the surface integral equation for a mathematical model n which the chordwise loading is taken. to be the lat-plate. distribution, with the.

boundary conditions satisfied by a weighted average over the chord

NEDERLANDSCH SCHEEPSSOUWKUNDIG BLZ.

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NEDERLANDSCH SCHEEPSBOUWKUNDIG

PROEFSTATION WAGENINGEN NO. 68-036-AH

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In none of these theories, however, one has the ful]. freedom in choice of the chordwise pressure distribution; one is restricted to distributions of the type

a. CO_° - o 2

wheref' is the chordwise angular coordinate ((p = O corresponds to the leading edge).

To overcoijie this objection one is logically led to the conclusion to apply a more rigorous three-dimensional unsteady lifting surface theory. This is what Laschka [7] did in his dissertation. It is true indeed that he applies his theory to vibrating wings, but his work proves it does no longer hold to say that the kernel involved is too complicated for numerical computations. Application of Nuithopp's [8] collocation method leads to a rather elegant solution.

The present report develops a consistent lifting surface theory fora ship's screw in a non-uniform flow. The method is based upon the integral equation derived by Sparenberg

[9]

and [10], by means of anacceleration potential. This theory leads to two equivalent integral equations, one following from a

vortex model, the other appearing when representing the liftig surface by a doublet layer. The first integral equation leads to Cauchy-singularities, the latter to Hadamard-singularities. The last representation is chosen here for further calculation. It is true indeed, that the Hadamard.-singularity needs some analytical work before we can start the numerical computations, but the other integral equation needs also a lot of preparatory work, moreover from a calculatory point of view this representation is less compact.

After giving a short review of the derivation of the integral equation the author continues making the kernel accessible for numerical computation by separating singular and non-sIngular parts in a similar way as Laschka does in his doctor's thesis. Finally a method is given for spanwise and chordwise integration.

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method. The pressure distributiön along the chord is represented by a series öl .Tschebyscheff-pOlyflOmialS, the integration of

which is carried out by numerical methods.

The author is very much indebted to Prof1 dr. ir. A.I. v.d.' Vooren, whose valuable suggestions and guidancewere of great. help in accòmplishing the work presented here. He also wishes to express his thanks to the members of the panel, Prof. dr. ir. J.D. van Nanen, Prof. dr. ir. J.A. .Sparenberg, Prof.. dr. R.

Ti.inman and Prof. dr. ir.'L. van Wijngaarden, underwhose super-vision the work has beenperformed.

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PROEFSTATION WAGENINGEN NO.

68-036-All

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¿I.

Formulation of the problem

The wake behind a ship, in which the screw operates, can

not be treated as a homogeneous flow. Although this flow is assumed to betixne-independ.ent, the rotating blades, which cut through

this disturbance, experience a time-dependent flow, provided the inhoniogeneous character of the flow is not axial symmetric. The

r

problem, therefore, must be attacked by non-steady lifting surface theory. For simplicity we assume no interaction

be-tween propeller and aftership. The coordinates that fit best

Fig. no. I in the theory are cylindrical

coordinates X ,

r ,

Lp. If we take the disturbance velocities to be

induced by the ship's hull,

inducedby the screw and

the homogeneous flow velocity,

the three orthogonal directions:

U

¡vo+vvt

J

W0

+

The non-linear model is, from a mathematical point of view extremely difficult to treat, so.we assume the disturbance

velocities small compared with the undisturbed flow veiocityU.

¡1_jo

,f/o

LJ,V

u,o

we have for axial radial tangential:

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1.

Hydrodynanlic .equatiotis

The starting point of our thèory is the set of hydrodynamic equatiois

(us) (u

)ì

(LJ)

_('4}

(L))*(1J)

rp

I

U)

_T

(

-

(.\L ) (u)(' ) (V)('J)

d

4(V)

=

t

(4) (u)(Ç)+

(Ve)

+(ì)ip(&JÇ»

The last assumption in the foregoing section simplifies these equations considerably. The linearized set of equations, taking into account the time_independency oU andu0.,'0 W0, is

-'

(2) ç

+

U

If we state f rther that thé pressure in the wake is a composition of the pressure due to 11e ship hull,,, which is continuous over the propeller plane:and the pressure due to the disturbance inducéd by the screw,

p ,

which is definitly nOt

continuous at the propeller plane,we can, according to the superposition principle, decompose. the equations into

(3)

_u

Uc

-

j_

o

t-(4),

-ti-

-NEDERLANDSCH SCHEEPSBOUWKUNDIG. BLZ.

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In the following we shall pay our attention to eq. (4),

:

combination with the

continuity

-equation

an incompressible, irrotational fluid. we can define a velocity

potential. The pressure also satisfies the Laplacia.nApo,

which follows from eq. (4). (by differentiating the equations with respect tox,) andLp, dividing the last equation throughr and. by making use of eq. (5)), so that we can défine an acceleration

functionJ-3-P.

These two

functions

andy are relatêd by .

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a solution of which is given by

(7).

UJ00

U

assuming all disturbances zero at infinity.

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2. Boundary conditions.

Consider a rotating helicoidal surface

J

(w., r. ii',

)

_ o.,-+

where i.represents the angular velocity. This surface does not

disturb an incoming flowU, o,ojfo . In a non-uniform flow,

however, it induces disturbance velocities u1,1 anthJ. Furthermore the screw blade P = O does not ly at the helical surface H = O, but differs slightly from it

fLc.,r,Lr

tj

+Ab+E

The function

Fbçi-'iS

a measure for the deviation of the scroz-blade from the helical surface. This deviation induces

disturbancesu,v andw . The sum of those disturbances is, as

2.2.

2.

we defined earlieru 1+LJ2,VV1+V2 ,

and WW,i-W

To satisfy the boundáry conditions the velocity near the blade has to be tangeñtiai to the surface

Substituting the expression for F' and neglecting second and higher order terms, we obtain

(io)

c.-pUE+$1_c

If we takeS to be zero we arrive at the boundary condition for a cambered profile in a homogeneous flow

(ii)

u

ForE= O eq. (iO) yields the boundary condition for a flat plate in a non-homogeneous flow

(12)

_6u04-LJ,

U

p-NEDERLANDSCH SCHEEPSBOUWKUND!G BLZ.

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PROEFSTATION WAGENINGEN

NO. 68-036-AH

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We can treat these equations seperately, according to the superposition principle (linearized model) and we will òciIpy ourselves first with the case, for which éq. (12) serves as a boundary condition: a rotating helicoidal surface in a non-homogeneous flow.

We now

try

to find a mathematicál representation for this

surface. The nbrmal direction of the surface H O is given by

(13)

(-1

I C being

_/j»4\2.

V

+rJlLrb(f1.

after substitution of C and calculation of the partial derive.

tives we get .

o

_'

U

Ç'TP+

r'

hence the disturbance velocity in the normal direction is

tor

-)

₊

. U

Vu

.

y

u +

which is zero according to eq. (12).

The normal acceleration. is . . .

1-

'42

iff 1UàLva-A.

Ì1L1.1).

.

c)L] U.s

rSu0.)+ U

+43 [.ru+iJ

From eq. (15) it follows that this is zero. Hence the normàl acceleration is independent of the side of the surface. From the hydrodynamic equations, the definition of the acceleration

.potentialand

the

result deduced from eq. (16) we can conclude

thatis continuous over the surface.

Knowing

that anybody in

aioV'can

be replaced by a distribution of sources (sinks) and doublets, we choose in favour of the doublets, being the only distribution with a continuous normal derivative.. This means that

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we can represent the screwbiade (helical surfaòe) by a doublet-layer of pressure dipoles.

NEDERLANDSCH SCHEEPSBOUWKUNDIG BLZ.

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3. The velocity field.

The pressure field induced in a point with coordinates

(x,'-,) by a pressure pole of strength,A.tat a point (,çe) is

________

I

'-.'c

e9))jk

From this expression we can derive the field of a pressure

doublet by differentiation in the direction normal to the helical

surface. This means that we have to calculate the inner product

of eq. (1k) and the partial derivatives of

CO% L9p)jL

with respect tof andO.

If. we, in the result,

replace9by4w., since the doublet

lies on the helical surface, we obtain

bc.,r,

_(o

having the diménsion of a pressure pole times a length.

From eq. (6) we have for the velocity potential

I 2..

'j

_J

(-af)t

_{Lt P}

--OQ

where the integration variableX has been replaced by X Ci

andis the new integration variable.

To obtain the velocity potential of the doublet layer we have

to integrate over the screwbiade, the range of integration being

r).x(Í)and r;fr0

where)andL(3,)determine the leading and. trailing edges,

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11

r andr0 the inner and. outer radii of the blade.

Keeping in mind, that the integration over the projection of the propeller blade in the plane, in stead of over the actual propeller blade, gives rise to the factor

øI

d

we get

I, I

The veloc±ty components can bed.erived from the velocity potential by differentiation with respect tox,r andL. We are not

interested in the radial component of the disturbance velocity, so we give only the expressions for the axial and. tangential components

I, 2

and.

3

and. form the expression for j.i-t./

Eq. (24) is valid. for one screwbiade. A n-bladed. screw we can represent by a set of helical surfaces 2.1C,k.e,),2.

We find the expression

forXu-/ for an arbitrary

screwbiade by replacing in eq.

(2k)(fbyt2.1

The contribution of all the blades together is obviously a summation over the number of

blades,('), since we deal with a linearized theory

I, 5

The boundary condition, eq. (12) states that in the immediate neighbourhood of the propeller the velocities ind.aced by the propeller must cancel the distuibance induced by the ship's hull. However, to obtain the propeller induced. velocities in

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a point lying at one of the helical surfaces, we meet with difficulties in the form of singularities in the integrand of expression (2), due to the fact, that the distance

R.=(f(.-

..ar1cosL9..4)3

between (u,r,iq) and. at least one of the points of the doubletlayer becomes zero. To overcome this difficulty we take the point (x,P,t) a small quantity,E, outside the helical

surface, thus lying at the surface?=o.c.f, and carry out the

limiting proces in such a way, that the integral does not loose its sense. This yields the integral equation

(26) II,.

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4. The singular behavior of the integral equation.

In order to investigate the correct we first take into account the fact that is periodic with a period

T,

so tbat'OT=.

into a Fourier series

o,

IvnA3b

(27)

,(.ç'1)=.

/"rr('f)

1

Inserting this in (26) we get

limit of equation (26), the doublet strength 2X. We therefore expand.

S3 : the remaining part

Wemay exclude fron our considerations the region S3,because

the integral remains finite everywhere within this region. An analogous reasoning shows that we only need take into account a small part from the interval(-o,-)for integration oven ,

namely that part, which includes the pointCo, say we take

where

g,Pe

o<.. The choice for seems unnecessarily complicated, but it originates from a transformation, which we will apply further on. That part of the integral on the righthand side 6f eq.

(28)

which then

needs further examination is

(28)

II, 2

We now focus our attention to the term for k.0 at the right hand side of (28), which gives rise to a singularity in the

neighbourhood of t.o( 2t.)and-4r. To establish the behavior of this singularity, we divide the integration region into three parts

NEDERLANDSCFI SCHEEPSBOUWKUNDIG BLZ.

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(30) i

(29). II, 3

Investigation of eq. (29) shows that not all ternis in the integrand of the triple integral contribute to its singular behavior. In view of the integration interval of the inner

integral we can evaluate for sinailt the integrand. in a powerseries tot , keeping in mind. thatis small too. Only

those terms in the expansion will yield a non-zero contribution

to eq. (29) for?_oandE_o, which become infinite for

where.-I. This means that we can exclude from further considerations terms inthe integrand of order

withÇ-!. This is obvious for the terms containing positive powers of since these terms tend to zero forr-*o. It is also trivial for the case C, since these terms are

indépendent 0t and remain finite fort-O, while in the

limiting case j-0the integration interval of the outer

integral tends to zero. It needs a closer examination to show that it is even truè for the case p.-I. Integration of the terms in the integrand, containingt overt, under the assumption

tbatjr, yields an expression, from which we can, by

elementary calculations, extract a term lnI_rl . Only this part in the expression may cause trouble forJ But

since

11m.

We can also confirm the last part of the statement, that only

terms containingtwith4-%in the integrand of the inner

integral eventually give rise to the singular behavior of expression (29).

The substitution

=t_6

in the inner integral of eq. (29) leaves us an integral with a symmetric integration interval

and an integrand that contains odd. and even terms. So we can effectuate a further reduction of terms in the integrand,

since integration of an odd function over a symmetric interval yields zero result.

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The remaining part, eq. (30), consists of integrals of the type

4-OC

oLTT

(p,)c)

It can be shown that the integral (31) is convergent

under

the conditions

lo )c#i

2°.Ag and

_F>0

.50

_and

The terms in (30) which satisfy these conditiOns do not contribute to the result of the triple integral, eq. (29), in the case what can be shown by an analogous reasoning as above. We integrate the remaining terms in eq. (30) overg

and evaluate the results for small-r_1andE, which leads to

II, ₅

Expanding,Mß,jin

a Taylor series in the neighbourhood

ofpr. and substitutingr_f), we meet integrals of the type

j' _c1ts

(p,>o)

Again we can show that for

l

2°

and

_>0

the integral (33) equals zero whenÇ-+o. The remaining part

of (32)

can

be integrated, resulting in

(311-) III, i

We thus can replace (29) by

z 2.' 4 i

yr (c.o -

r

(35) Um

L. L

,P:k

;vn4

NEDERLANDSCH SCHEEPSBOUWKUNDIG BI.Z.

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a similar way we

can

show.that the divergent part

0±'

is given by

Lin_

-1-ev i-.r.

'I,,

-(f,r)"°

which is a

mere extension of the upper integration

limit

(35),.

We now define .

(38).

as the Hadamard

principal

value of the, integral on the righth,and.

side of (28). . , "

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NO. 68-036-AH

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5. Separating regular and singular parts.

In order to make the integral equation

(39)

III, '4.

accessible to numerical calculations, we have to extract the singular part from the kernel, to treat this part

separately. We have already seen that the singular behavior of the .kernelKLz,r,,j))iS restricted to the term fork.Q. Noreover the term

(ko)

3 $

(orC Oi.1)

(oft

-r3fr)

O.t),

jrrA.t

_dC

{ri-AyÂ_3

t3'I

remains bounded forT0and=r. To show

this we only need consider the integral over the interval(O(,4-O), because if

the integrand becomes infinite, it will occur within that

part of the interval, which containstC. Evaluation of the

integran.d in a powerseries totyields

('4i) 11 (r..y(ti. ¡rnO c:'c2':4 Lrr)2'(

O (te')

_dt

JJ_

J

Taking the limit forj'Dr before performing the integration, we can see easily, that the integral is bounded. The same holds when integrating first and taking-the limitr.r

afterwards. However, the latter sequence reveals the fact, that the integral is discontinuous, when integrating over

the pointtO forrP. . This is caused by the first term in the integrand

__

______

r

(4.2)

_cz-rr

r

(i +

J

-T

-

L _J

The magnitude of the discontinuity is

('4.3)

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Having shown that eq. (40) contaiñs no singularity, we can pay full attèntion to the term

ri

(114) e.

dt

i)

[t-2+f-.

cctJ

Here again we need only to consider the

integrand in

the

neighbourhood 0fto Evaluation in a powerseries totleads

to O

rr

tir)_4

).

J

L ii- L.

j'

)t3 .

Li cr)i- t

-The part

certainly is singular. To subtract this singularity.underthe integration sigñ ofthe kernel we extend the integration

interval of eq. (46) to(-o-), which is allowed, since

the integral exists for the lower limit-co.

The remaining terms in eq.

(45)

can be written slightly different

(11.7)

IV,i

(4-8)

IV,. 2

Ít

can

be shown, simply by integration, that the terms

L -

i.

-

(r'

r

cit

- -. . ,---

.

110.

t2dt

(20)

are finite for1

r.

However, the integrals appear to be discontinuous, when

integrated over the pointtzOforf.r. The magnitude of

the

discontinuity can

be calculated to be

I

t2.\2

.tu_Sor+I+O1r)rn

(I+or)i2.

The singular part of (47) and (48) is now reduced to

J

-

1.

Extension of the integration interval can not simply be done, since the integral in eq. (53) does not converge for the lower

limit-. To remedy this we multiply the integrand

by

a factor that does .not change the character of the singularity for

Co

, but effectuates the convergence of the integral for

the lower limit-oo.A factor, that satisfies all this is

z

_/

\

P3 .

LT-j'J

,

wherer>r0

, so that

r7

_t+2rp

f

all possible values oft-

arlar .

The last

demand

is just for mathematical reasons: it defines the integral.

So we can write for (53)

I..

Y)rP135

(t)&p4.

We are now able to subtract the singular part of the kernel,

kL-)nosf)

, under the integral sign, leaving a regular kernel,

cxr4).

Integrating the eq. (4k) and

(54)

analytically

over Twe obtain for the singular part of the kernel,ks.(.r.r),

Obviously the term

(21)

(58)

_-

+(i+2p.)rn3p

rj

(r)

{r

is non-singular, so we add. (58) to

Kn.&rif)

Finally we can write for the integra], equation

V'I

V,2

V,3

The singular part 01' the integral equation needs. some more investigation. In the next section we will try to express the singularities in a more explicit way.

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(62)

6. Separating chordwise andspanwise singularities.

To simplify the calculations we introdue a set of

new

variables

lLr)

X

eL)

y

r..ç'

Furthermore we suppose that,

_{can bé expanded in a Biriibamn}

series

(63)

'i')c,

c()41jx'

where(C)are

linear cörubinations 6f Tschebyscheff polynomials

of the first kind. with the argumenti_.X.

(6L)

This enables, us to separate the variables

and

_Ç'.

Wé consider the first term in eq. (61)

Lrr)

_L)

'ce.)

and substitute (62) and. (63) into (65)

v,z.

The integrand has a logarithmic singularity in the

second

derivative toYroroana>c'x, asis already stated by

Nuithopp

[8]. It seems advisable to treat the logarithmic

_singularitr

NEDERLANDSCH SCHEEPSROUWKUNDIG. BLZ.

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separately, so we will investigae therecise nature of ït.

Tó do so we evaluate h (X')and

°' x-x).

a powerseries to

(X._)C) .- However the

oordinate)C-X

is not a convenient variable to use for this expansion. In the first.placeZrnay

exceed unity forY# o, which means that the point under consideration does not ly at the screwblade any longer.

NoreoverXdepends ony

We therefore prefer the variable

X(y=o')=xsee

eq. (62)), sinceX,is independent 0fr

while the relationO.XI

d.oeshold.

g

Thus evaluating j ande. in the neighbourhood

of)c.and substituting the results into (66), we get

VI, i

The part

. VI,.2

in (67) yields, when integrated, a term of order In view of the spanwise integration it is advisable to isolate this

logarithmic singularity, thus obtaining

VI,3

Returning to our original coordinates we can also write.

2.. i..Q

Q

L. C

tr _..f

rp5

_p

where

(72) . . VI,.5

The áccent in

{('Y4 states differentiation with respect

(71) VI,

and

ÑEDERLANDSCH SCHEEPSBOUWKUNDIG BLZ.

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to the argument

IL

The next term to be. considered is

-

-(73)

_(I+c.rf)a.J

1

--

(?)-

j

_'J

This integral gives rise to a logarithmic singularity

for alivalues

We can isolate this singularity in the following way, keeping

in' mind (63)

(7Li.) ' ' VII, i

The second integral can be calculated to be

'

VÏI, 2

Isolating the logarithmic singularity

'

vii, 3;

we can write (77) where (78') (80)

9

Cr,.) ¶r'

,rp)i-E_

pDO

vp

Pt:,o

;n10--

_'Lr)

p(f)

(1r)

NEDERLANDSCH SCIIEEPSBOUWKLJNDIG BLZ.

PROEFSTATION WAGENINGEN NO.. 68-036-AH 23

and

(79)

VII,

(25)

To make

VIII,

a regular term, we subtract

VIII, thus obtaining

'p

(8k) £

Crnp(f)

where and.

This terni contains for.x-Oa1so a logarithmic singularity

£nJrrL

namely

(81)

VIII, 2

(2)

L. to

The integrai equation finally takes the

forni

IX, 2

where. _t'x1r.? is given by

(71) and

Lw..,v',r) by

VIP J _OIP 4

tx,t-,rÇ

h

) IP _C

j

P , .

L-)

J Furthermore is

(9)

Î

_-'

r"

R L, R s NEDERLANDSCH SCHEEPSBOUWKUNDIG BLZ.

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Lt)

fp&u1

&'Crf)

and respectively given by ('72), (78) and. (85)o Finally

IX,3

. (.zryi) and rp ,,,Í)respective1y given by

(79), (86) an(60). As long asJ+re.(9O)is equivalent

ith

Ix,

where

x,1

;which follows from eq. (77) and (84.). Eq. (91) is froni a

numerical viewpoint, easiér to handle, In the casef- however,

we have to use eq. (90) in which we sübstitute bfor. . For

n,p-) and Lz,r,rwe get

J

X,2

x, 3

and.

(95) XI,

In the next section we shall occupy ourselves with the spanwise

integ-ration.

NEDERLANDSCH SCHEEPSBOUWKUNDIG .BLZ.

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7. Spanwise integration.

To perform the spanwise integration we make use of the Nuithopp-procedure. We treat the three essentially d±fferént parts of (87) separately.'

cJ(îTh f0

ay

r0

j

F (y)

Jr...yi oL1ç

ro

'

Py)QLy

where

Fi(Li+otrtL. CrnÇ)2

q::) 2O J

r2.cT)LL

CLy)

L-?srj')

We introduce a new coordinate

r0+fl _-r

_cosG

J

2.

which makes the integration liinitsoandt.. As a consequence

ø'=.Jr

0ie

-J .

.p.. ro1.-_Pb.r;,s

z t.

+r,..-r;r0-P

_sLet)

16 .*

-

_{!!ì ÇOOS.(r+ ¿)}

- co b

Neglecting terms of order this last expression yields. for ¿

9 )

NÈDERLANDSCH SCHEEPSBOUWKUNDIG BLZ.

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1ff 1 F,(e).e

- ' L 1 J

J

J (cot Br C e,.. O

(loo)

r0-r1

fe

_{e de}

(ioi)10fl5

F

C9c

(99)

Substitüting this in (96),. (97) and (98)we get

We start our calculations with expression (99.). We püt

(102)

gL)

being an interpolation function, determined by the

condition that

.r

Lgg.i,,...

where _{denotes the integration stations. The derivation}

of1G

,..is given in appendix A. We give here the result only

(103)

q

ce

L£Ir.X.9Irl)%

Inserting (102)

and (103)

_{in(99) we obtain}

(104)

. XI, 2

Pcr4r is

, so that the integrals In (104) exist. in

the sense oP' Cauchy. Using Clauert's formula

r

bI3II

d

(29)

1G I Ç G>%G

de =

G-cos

o we obtain

n.

Ç

_{.sir A G} ₍ .

A e .sir 9 d O

i

j

e,..- e

? e

At

r

P1+1

forL.r we have in a similar way

er-ec

hI

stA$r.[f

₊₅

d.9.=

a'

)

h.pf L...

_{coz OosB J}

_3tr%O

1=

Furthermore

is .

f.

-i'

_e) A

en

i

'

____

= -

_-

_£I

9r

.

t51v &r

U,_

This r suit inserted in (1011.) cancels the term

.FiLe

so that (10k) leads to the simple expression

XI, ₃

where

2rL

- .

L

X

r

h*-

_)=i

_slriÖ,.

This sum is calculated in appendix B and yields

o

eri-e

der)_cr(er

rfP_r.(er'1OL4E)

2i.LGr)

2.

NEDERLAN DSCH SCHEEPSBOUWICUNDÍG BLZ.

(30)

NO.

_68-036-AH

NEDERLANOSCH SCHEEPSSOUWKUNDIG PROEFSTATION WAGENINGEN

(107) ,

(I)

(108)

'r

L.').

2iL.;r

_{e t}

flf)

a

-e e

rsr

e

Q Jo

we exclude th.e integrel

forÀ.i

. to treat this one in a different

Yo For )we find

sn ), B

sn

O

Ocos.Od9

-OS9rOOSJ

fTt

_{T(10_I.i c;i(ìt)ocn 0}

dO

=

Ic.os e-.cc

I,

L ?

T ECoS IÀ.#t)e._ coS. )O

ZT tcoA

_{9_cos(A_t)}

el

cOS_CoS e

o

(Lr

odd)

dO=

BLZ. 2 ?

Tt r

j'(+a)s.

sin A. eT i

in1

sP-L)

L °«- £-%

j

(ion)

p

_r,

_r

evén)

We apply the saine procedure to (100). Substituting

!°=Ê

0e).0)

in (loo) give.s

XI,

In calcuiaing ..the integrals

_

r.

r'i-/i-.I

(31)

BLZ. NEDÉRLANDSCH SCHEEPSBOUWKUNDIG

PROEFSTATION WAGENNGEN NO.

cos(A.#i

I

_t)L+t

For

we find

-I

Ç.zre2licosOr

_o

the second term of which can be calculated as before, yielding

The first term goes as follows

2.

CeS (1_i) B!L

J.ncose.._cosOJdBj1n IISIfl8PL46

e.

c9=.

kjT0

.cL-..

)co2_cDOc& -

1-f: coeç.

ç, 11

JcB+iJ

e.-e

ide

o-e

P

i+f

1n(snD)d9#J

1ns;nD)d9=

h

= 2.

1r

14-2 J2

L(.in e) d94fi

2.

ln(SIn B)d e+j

_ìnlsne1cìG

2L(s.n

_{d9+zLt1n(sine)a9#Ij}

(32)

Finally we öbtain

(112)

-_rìj

£

where

(113) XI, 5

This sum can not be simplified anymore.

The integrals

eJerc

r+

o

in (iii) are all zero except for

. t2) oLi)

(il 4) = A'

_-r*I

The integral (loi) leads with

F3te'L

XII, I

Combining the thus found quadrature formulae for the three terms in eq. (87) we get for the integral equation

(117) XII, 2

The chordwise integration is the only thing left to take care off. This will be d.one in the next section.

NEDERLANDSCH SC-HEEPSBOUWKUÑDIG BLZ.

(33)

NEDERLANDSCH SCHEEPSBOUWKUN DIG

PROÉFSTATION WAGENINGEN

BLZ.

32

8. Chordwise integration

It is sometimes more convenient to handle problems, when they are formulated in dimensionless coordinatesánd parameters. So before going on integrating eq. (117) over the chord of the screwbiade, we return for a moment to eq. (28), in order to make the integral equation dimensionless. This can be done by dividing the geometric coordinates by some reference léngth,

say,-0: thus = , etc. Furthermore we replace by

tot

u

the so called reduced frequency, based on the screw radius. Also

- 4. _U0

and .

= ) 1.00.

-rn O

u

Keeping in mind thatz1r etc. stand for.,r etc. we can just as weil omit this suffix,getting for the dimens.ionless integral equation

XIII, I

As a consequence we can write eq. (117) as adimensioniess equation XIII, 2 where XII], 3 XIII, LI. and XIV, 1

(34)

where we takeri. Furthermore is

_______

and dimensionless, because this is true for}.J(1f) .

We now continue our line of attack, whici i the chordwise

integration of the integrals

iptf)

and'

It is possible tO do this partly by. means of analytica methods, but for reasons o.f generality we prefer a pure nume±icai integration method designed by'Roinberg-Stiêfel [ii], which has the advantag.e that the integral can be calculated within an' accuracy, prescribed beforehand. Of course we have to write the integrands of the several integrals in such a way that the numerical method can be applied to them. This will be done. here.

a) First we consider the function

t(-rf)

The function

k

-R.fonta ins' for

a square root singularity.

(12.3)

h t'x')T

tI_2..)C)4-.tp-i L1.é)

',tV'7( (i-x).

where

(124)

X'-

-L1

.

('x

*Lr))

The functions1L,_d2DÒare Tschebyschef f polynomials of the f irst kind of orderX with argument ¿i-ZX'). To eliminate this square root singularty we can introduce a new variable

X' -k

L_cop)

from which follows

XLIOS?

NEDERLANÖSCH SCHEEPSBOUWKUN DIG BLZ.

(35)

Substitution of (125) in (120) leads to a regular expreèsion

XIV, 2

with

=

_(f

4.f- COS ('+i)

P

The derivative of with respect to its argument, which we need also in our further calculations, is given by

2- '

p

lt

_-

_{.scr)L?c.LI.cos9o)}

For5pthe integrand shows a discontinuity

2. for

ìk-+ -rk -

_L

_j

_(-FO'

o

for

Z

L)

Furthermore is (r..f £riIrjio.so we replace eq. (127)

forp-

by

,bcr)

=

2J' {copIf'#c.os

1p

J'

In the.foregoing forinulaelis a function of

b) Next we consider

The expressions (128), (129)

and

(130) being known, the function ) r-p can be calculated

J

XIV, 3

(36)

c) Finally we cOnsider the function

We havetwo distinct cases, viz.y.*.-and3r. The exPression,

foro.r,r is quite d.ifferent fron the one for.

for

s +

-J We start with the casej'r. Ilaking use of the expressions

(124),

(125) and: (126) we get

XV. i

given by. (128),

(130)'

(131).

The calculation of the eq. (122), which

ocòurs within the integrand äf sp Lxrj) , will be taken care off in the next section. At this point we

assumeQxi-being known as a numeriòai quantity. The calculation 91' eq.

(133)

is now straight forward.

This is not the case whenÎ_. We now have to integrate

the formulae

XV, 2

where

LL,r,,fis

now given by XVI, I

Applying (124), (125) and (126) to (134), the last equation is transformed to

XVI, 2

the square root singularity being eliminated. is given, by

(131), withp' substituted for

The integrand in the expression for L'çiis discontinuous

at

LXXO

ç

.i.0)

, the limiting valtes being

(37)

nC)

L

The

functionL'X,rr)contains

a logarithmic singularity at.x..

To remOve this singularity we proceed as follows. The integrals can be written (in the original coordinates) as

xvii,

in which the first integral is non-singular. Rewriting the second and third integral in the following way,

XVII, 2

we can calculate the second and fourth integral to be

-

L

)oLa.[_.Er

LfLr9

(1ko)

E f-L-

kLP:)xJ

while the integrands of the first and third integrals tend to zero Thus we -get (again in the new coordinates)

(iki) XVII, 3

For the choice of the pivotal points over the chord we follow Muithopp's suggestion [8] (1/42)

.;X

-).jt')1

22+?

J

;m.Er-A.

QJ*sr

LJ .Çbr.ip4

T:

.1. I NEDERLANDSCH SCHEEPSBOUWKÚNDIG BLZ.

(38)

PROEFSTATION WAGENINGEN NO.

68-036-AH

BLZ.

Integration orQL1i-j).

The integraiQ&x1r) gives rise to two difficulties0

First it is an improper integral, so that.it needs some manipula-tions to make this integra], accessible for numerical computamanipula-tions.

SecoÌidlcx,4,Y is the inner óne of adoubie integral..Since

the alcor compiler of the TR4-computer does not allow for. recursive calls and moreover it is not necessary the best ].ine of' attack to compute the double integral by aid of simple

double integration, we have to find another way out. It is easy to verify, that the integrand

which in the following will be denoted tends to

zero fort-co, which allows us to cut off the range of

integration below some fixed but arbitrarily choosen value

and.to approximate the integra].for_4t._-C2_ by analytical

means.

(1k3) Lr,ÍioLt

])Q

_rp

_{Lr,1i) o}

The first integral on the right-hand side consists of' two essentially different parts

(144)

Q1rirtct

Lr')+tatt

where (145)

flLt)=L[

ko

*

iLo+.zic)

and

sfr,Lr42.4 ct -r

tt+Ay'2- 2.v3)co(at 1.zT)

(39)

(111.6)

The part B(t) is added to the part A(t) to keep the latter

finite fort=O .

But

sincé the range of integration (_oo,_C) does not containt=o we can treat both parts separately. B(t) can easily be integrated overt.

-£2. (i 11.7) (1k8) For Imaginary part

1+ jrpD.t

r

l4-Pf)k

1r.

!Î

L

_Ii.,-j

NO. _68-036-AH 2

.f-.ot.

r'2

Ci)(c.i)rt.

t

¡i.r.j'

I

dt

irr).J

fr+ (r")2.

_ocL

r

VøC ..L ? I

Lr'T

/+cr')

ft

+

-

L

i+&2ry J

(lk7a)

0c

i +o.

J Ltti){c2..

(ii-.st)

í--L

j

the part A(t) we find an asymptotic expansion. The

2.

Lr)Z

I101 BLZ. 38

(--

3L)

L 1-' 2.

'r

.tLL

1-c--i-'-j"

i-.1 2. 4. i+c integrand can be evaluated in a powerseries to tof negative powers.

Separating real and. imaginary parts we obtain for the real part

rL1

(1k9)

p-coLcc+

1-ItI

Jrt

z

y

Ç,cctry

Lrf)3cos

rj'cs cttiic1

L

_{16ß2. z)}

+__

3tn(tt-r

riJ

Ir ¡ t3

(40)

PROEFSTATION WAGENINGEN NO. 68-036-MI

BLZ.

39

IL[-J ko

₀₀

). ₂

- CoS

Lot

i. ₊ o (r i ir Lait 2.11'Ç

It')3

Jt1:

#cLr+y

i

..].rrn

t+2Tldt

By partial integration we

can

throw back the first

_two

terms of the integrand at the third one. This leads to the following expressions.

Real part for

w-I

oes rn(c-

k)c'

!

_{cc Lrn)}

(i5i){

î lLrrtH) Lrni-i)J -'-

_-

'I

_co.Lrn

_{i) 'c}

k 9.Lrvl_I) I '1

Forrni

andrn the formula is slightly different. For _yp

E. J

(152)

.

[-&vyos (oM..

2.1c)+. L

-

+

iS. tq For -

{a.ry3ir2..(os2.

(153)

rfl.Lj,

etc.

Lin

rn(O.Q..4.J.._

.&n Lmt

cxJ

J_I

mfc2J.

4 _2LrVH-1)4. W 2L'n..4)a,. 2. 2.

-

-r )

_w

_e

f Jt

k

w

-(c_

213ffc,La

_2.Tc:.L1

W

2AL

N _J

(41)

(154)

For

(155)

I _c.o

(o.S2.-

____

vi

LcJ.

ic

2(rv,_I).

i

w

f

_{Lr?Zfr 1rnt7.c-}

-

zit3+[_j

)

f2

2.(.fl)-')

t arn4-i)"o!

sCvr*)

J

L N

Lrni

. I

jJ-

-Ñi

Z1

_.ìt)_J_

o(o.cL_.Z1t)i

k0

L '.p- f4

_k

JCr2f)3J... [éT .sir

f c....)_

-L N

(.L

2

a)sn

Tc\Zj...

eci. L Lg q

"J...Q1

IL

For

P7).9.

rL

(ÖL .Q

-

2.

_L cas

.Q.. - $_ N

(,.

N J

Iko

(156)

k

_sf.._c)J..P 7sìr(r' ZI&(.L +.L4LpM

_2c.

P4

citz

t4k

_w

%Z.k

₎

N

j2i1

s..;

The remainder terms are given by

wI

_k

_I

L. Ç.c*.r'

_a--o---

I

co1rrs+s) £5L__

21C11

rr _W _ZLrrltI)Cu.

t4

J

(42)

NEDERLANDSCI4 SCHEEPSBOUWXUNDIG

PWEFSTATION WAGENINGEN NO.

68-O6-AH

-n

-co

cor

.{+('*)} +f)21 Co5 (2at4.2x)

1t21

(i»r)

Co4(2t) '

!(

)

_'j')

CoS(3

+

t3

+...

-t;.

t 2) t a) W4

t5

b

_.k

i

i.

f4)2

_+(,,'+ _. CoS f(tm_i)&r.. .ti4

t5

.:tp

M4)O_2EJ

t;

pJ4t+32)t).c0s (t..+2x)}

-t;

t,

ïß+

_pJ

so(m t).

_{Iw+lI)At 4 t4}

(nit

-

í

t5

Sm (Piii-I)Ot + &xj + touI_i1 T

.

t,,s;i (,r.i')x4nI + Sin [(ni.t

-...]

a

ti5

+ BLZ. ¿Il +

-1

+ 41: +

Sin (tar+..x)

t2A. ¡41 ¡1'

4tp{sin(3ut42x)_sin (o.t+

t5

-'tp1

(tax)

_Sn(3ct

.i.1j)

t5.

1i(a+pt)

sin(r__p

(4t#!*

t)

¿-r

(43)

NEDERLANOSCH SCHEEPSROUWKUNDtG PROEFSTATION WAGENINGEN

BLZ.

It can be shown that these remainder terms are of

orderÇ'(-'-Consider the expression

po0

J

..r r)

O

Jt r

rrAt

-.

r

rrt

The last term on.the righthand side isof

orderO), as can

be easily seen from

t'

Zfr

t

O6)

The first term needs a closer examination

s;f1 )

r'

r I wa ç

(k2,)

Thus

t

.ts

(kt

SinT ct *

-

p

wt

_d-c

Stn t

_ct

_)

t

t5

2,t

t Li-

(mo)

ÇLr

ssit

dt +

)

ki

+

Qf..L

I. ,2 k

s) 1-ir

(eìtt

Mat

t +

)k Ç

_dt.

Jr+ kx)

(fr.z

-(k.'

TI

(kir

NO.

_68-036-AH

₄₂ -w a

(44)

1c I

Ar5

according to

Thus the concluion is that (14.4) can be computed sufficiently

accurate fortoo ,

without consuming too much running time. To overcome the second difficulty, integration of a double integral, we tabulate the inner integral and, when calculating the outer integral, find the wanted value of the inner integral by interpolation within the tabel. To tabulate the second

integral on the righthand side of (14-3) to the argunient, the integration variable of the outer integral, we notice that

only occurs in the upper limit of the integral, so that tabulation means calculation of the integral for different values of the

upper limit. Instead of tabulating towe take the full upper

iimit,x4

as argument, which means a slight gain in computation time., since we do not have to tabulate the integral for each value of the parameterX. again.

From eq. (131) and the integration limits of eq. (133) it

follows thatvarieswithin the intervai(Ly)xj))

of length

the chordlength of the scewblade; the same holds fork, eq. (124.) and (126). This means that the value ofvaries

within the interval(_Ç1+». To prevent unnecessary computations

we need only tabulate the variable part of the integral for different values ; for the parts that is independent of...

one computation suffices

(-1 58)

ot.I..)JQ (ry

dt

The integrandJQv)in the first integral at the righthand side

(45)

behaves for a wide range oftquasi-periodic, i.e. with a fixed periodand a descending ainpi-itude. As long as the arguments of the trigonometric functions are not too great the best way to deal with an integral like this is to interateJO.(..fì) over each period separately, to keep terms of the saine order of magnitude together

(159)

J

JQ C

cit

J'

i

L frj.t) dt

The second term at the

righthand

side of (158) needs tabulation to the argument%4. We divide the integration interval, in a number of parts, enought to garantee sufficient accuracy during interpolation

and

compute the integral over every interval, including those from (1114)

and. (159),

we find a table of

aequidistant points for the integral in (143) for different values

of the argunient. To make inteipolation at the endpoints of the

tabulation

interval_Qi.y)

i-eLj')) as accurate as possible, we compute one value ofQL,,p,y'ion each side outside the interval(_&f),.Qy)) The choice of-Q.is determined by the choice of eq and by the mutual distance of the tabulated points '

.(160y.!L.

The calculation of the integrand

of(4leads

for -o

andP-to very inaccurate results if we maintain the form of the

integrand as given in eq. (111-5) and (111.6). This is caused by the

fact that the terms in eq. (111.5) and (111-6), if separately treated,

tend. to infinity fort-+0. In a computer subtraction of two almost equal, but very big numbers given rise to a considerable loss of

accuracy. To overcome this difficulty we evaluate (145) forko and small values oft

(t.tr,

wheret _r is determined, after some experimental calculations)in a powerseries tot . This gives us

the possibility to remove the. ''infinite'' terms by analytical means. We have to continue

(46)

NEDERLANDSCH SCHEEP$BOUWKUNDIG

BLZ.

the expansion to rather high powers oft, to get

forTt9ra

reasonable accuracy. For we compute the integral in thc usual way. LI I

I2O .-'Z.r

2. '3,/

_j

d7

c.t

n+i4*(i1.o:2)rr)

_i

_______

_j.0'Lp

)j+

4..

(161 a

(161b

f '+....L

(rn#

'

_____

C L

c2-r)

J

o

(e_

. I

Q (i.i.c-...LI+Oar )rt1/

+ IprtA. V"L 'z,2- 2'. 3

(IAtr)t2.

4.0'.

¡

\

2p%'Z.

(i... (D+IOtfl2P.L1+C

)rfl )_

&r-! (+

io

LjO

_iJJ

ìrr,4t

cit...Sjp,Oct)

XTCI

[t'1i-

i-.

0C14j .,x)3]31i

t.c

r

tr,Lc1.2m)27

im(or'ic)

ttt1c.csLcAt1.*

21d33J

e.

+

2.2\ 2.\ I+C r'

(47)

PROEFSTATION WAGENINGEN No. 68-036-AH

BLZ.

46

The part (161b) represents those terms in the integrandQ(rçt'

which do not contain ''infinit&' terms fort-O.,so this

part needs no evaluation. The function sign

(r)

equals zero. for t.O .Since expression (161a) is

special arrangements for that. To complete the integration with some remarks concerning the

not zerO, forto, we need

ofpL'xr,

we end. this section interpolation, which is

necessary to calculate the valueof the 'in.ner.integral

each time we compute the integrand of&çr).

An interpolation formula that sufficés is the threepoint.. Lagrange interpolation formula [12]. 'Since we can take the. coordinate, for which we .av to calculate the tabulated

function arbitrarily in the left or. right interval of the i.terpo1ation formula, we calculate. the

for both cases and take the arithmetic mean as the final result. At the ends of the. interpolation interval we are sometimes in a position .that we have only one way to calculate a value of .

QLn'

since we have only One way to apply the threepoint .

(48)

BLZ.

10. The solution of the integral equation.

We now return to eq.

(119),

in which we can compute the and. the coefficients to solve this equation. The functionsL1012 ,r19) and

WLr,LM')

on the lefthand. side of eq. (119) are, as defined before, the axial and tangential disturbance velocities, due

to the ship's hull at the place of the screwdisc. The inhornogeneou

velocity field, in which the ships propeller operates (the

wake) is from the viewpoint of a screwblade a periodic function. Obviously the way to proceed is to develop the disturbance

velocities in a Fourierseries

00

{rt#o

X7)_f,

O

Here use is made of the fact that there is a relation between

X.,i- ,& ande. , the point lying at the helical surface. The integra]

equation now takes a somewhat less complex form

ni't

.i.

.a.. ¿ .L

1-+.!I

(2s-

2LÇ

We have to solve the integral equation for each harmonic

in the wake separately. The unknown coefficientsC.(r? are a

measure for the pressure distribution along the screwblade (they are the coefficient of the terms in the Birnbaumseries, which describes the chordwise pressure distribution.

Substituting the known values of for

certainand'in the lefthand side of eq. (163) and calculating

the known functions in the righthand side of eq. (163) for

cor-responding values òf.andr, we arrive at a set of linear

algebraic equations. The coordinates ( ,- ) represent a nunber of points along the screwblade,h in span- and?+I in chordwise

(49)

In other words we satisfy the integral equation in the pivotal

pòinto reduce the integral &quation to a set of linear algebraic.

equations, which can be solved applying one. of the wèll-known

methods. Tie coordinates of the pivotal points are given by

(IL.2) as far as the chordwise coordinates are concerned; for

the spanwise coordinates we take the stationpoints of the spanwise

integration

. . . . .

(16k)

-?

.9e

where

h+i a

1=

...3-7

We. have already seen that eq. (163) delivers for each wake

harmonic

tWÒ

a set of equations. Thus we have to calculate the

coefficient matrix of this syste

for éachrvi separately, the.

rank ishtL'P*')

,

the number of pivotal points. To gain some

rumling time, we do not compute the matrix elements row- or

colinnnwise, but in partmatrices of rankL'), the number of

pivotal points ovei the chord.

The matrix elements are complex numbers as long asrflO.

We can easIly convert a set of complex equations into a set

of real equations. It can b.e shown by elementary means that

(165)

N

.riJl. ).jx+)

(o+;6)

is aequivalent with

(166)

tri,., M'10

f/c.

t%f1.

_fir'

The raflk o.f the. real set of equations is twice the rank of th ....

complex one.

The known lefthand. side of (163) consists for eachn of the

amplitude of one of the harmonic components, which charácterize

the velocity field of the wake. Since there are no or at least

very few measurements of the. xïistiirbance velocitie.s as. a function

of )., we assume the variation in axial direction small, so tiiat.

in our computáttons, the lefthands.ide of (163) is a Thnction. of

the span

. ... . .

NDERLANDSCH SÇHEEPSBOUWKUNDG BLZ.

(50)

only. The measurements of the

disturbance velocities over the span do of course not

coincide with the coordinates of the pivotal points over the span. To

calculate the.amplitudesfor

these coordinates we make use of the

already mentIoned interpolatjo

formula..

The Solution of _{the set of equations finally is}

peíforthéd

by applying the method of Crout, _{a modification of the}

well-. known Gauss _algorithme.

NEDERLAND$cH SCHEEPSBOUWKUNDIG BLZ.

PROEFSTATION . WAGENINGEN NO.

(51)

11. Extension of the theory to.. steady state calculations.

The casemo represents the homogeneous part in the velocity

field of the wake. Since we choose the rotating helical surface in such a way, that it does not generate a lift force in a

homogeneous flow, the solution for the case

r=O

is consequently the trivial one, the zero vector.

A more significant solution we obtain, when we start from

the boundary condition (ii), whereU stands for an angle of incidence and eventually the camber of' a profile. This means that we replace in casern.=othe lefthand side of eq. (163) by

the known quantityEU-4. In the following we will derive the

meaning of this quantity more precisely.

Since we apply a linearized theory, the boundary conditions are satisfied on the projection of the screwblade to the helical surface, in stead of at the screwblade itself. If' we denote the angle between the helical surface and the screwblade

by5'

, the angle of incidence,., the angle between the helical surface and. the discplane byo((see fig.

3,

pag 51) and the pitch angle bye',

we can express,5in terms of pitch,I-') and reduced frequency,..

(16?)

aò_or

where

(168)

A simple calculation yields

-The chordlengthLr), which occurs in our formulae, is measured in axial direction, so that the relation between&Àand the

chord-lenght measured along the screwblade,k2L-) , is

Li-)

k 21r) *t-os)-e

p-,r. kiL-) csJ

(Z,,2.)

NEDERLANDSCH SCI4EEPSBOUWKUNDtG BLZ.

(52)

To find the abscissae of the leading and trailing edge,.resp. we calculate the position of the generator

Lrr.ir

Lpc.L)

the rake angle being the angle between the vertical an the generator of the. screwbiade. So we get for the leading, edge the abscis

cosft

1+

T.rsir

and for the trailing edge of course

(172) .

L= ,Lr)

+

Chord length

p-o

From the camber of the screwblade we can derive the local

angle of attackjj. We need tó know this angle to calculate

the ,lefthand side of the integral equation. We state'.tha

(13)r-

J3

Li+o'r''),

This formula can be derived as follows. The équation of the helical surface is

(174)

that of the. screwbiade

(175),

heIicI Si-sr'

Fig.

'r

From the fig. 3 one can see that the

angle between the helicalsurface and the

1' _{plane, or between screwbiade and}

YZ-plane is formally given by Li

i

ci»cIP'=CohS.

NEDERLANDSCH SCHEEPSBOUWKUÑDIG BLZ.

(53)

Applying this to eq.

₍₁₇₄₎

and

(175)

we

cancaiculate)in

every point of the screwblade. by elementairy means

The value ofJfoIlows from. the camber of the .screwblade profile

(sée fig..

3,

page

51)

.

. ...

= cambereffect

(q.e.d.).

Fig. 4

st

(176)

If'

we denote the plane in which the

chord of the screwbiade is lying,

by,

than is the angle between this

planeand the helical suiface

determined as), the angle

of attack

The local angle of attack,), is

expressed in terms used in Fig. 4,

defined by .

In a practical case the camber of a screwbiade is often said to be a parabola. the shape of which is characterized hyits height at the midchord.. The equation of the parabOla is. the

-

1°

.The derivative of this equation to the arguments' is

L_?o

OLs

f

____

2 . Pç. L. Loc+

furthermore

35 IOC?

+Oc

Loci 5' L Ç)

cr-r+EP_.

So

In our linearized theory this expression simplifies to

NEDERLANDSCH SCHEEPSBOUWKUHDG BLZ.

(54)

(179)

X.

Eìcpressingthe coordinates in fractions of the chordlengthL=5-3 and. at the same time translatin the. coordinate system in such a way,that. the leading edge of the profile maps at the poInt

(o,o) we arrive at . .

-

(x-v)

whereT

and % Z!L.

L s

_L

For the X-coordinate we can take the same numerical values as are calculatéd. for the chordwise coordinates of the pivotal points., eq. (142)b Substituting eq. (169) and

(177)

in eq.

(176),

we get . .

XI+cr)

Consequently the lefthand. side of the integral equation finally

takes the form. . .

NEDERLANDSCH SCHEEPSBOLJWKUNDIG BLZ.

(55)

12. Formulae for some hydrodynamic quantities.

Apart from the pressure coefficients from the Birnbaumseries

C

J the values of which follow directly from thé solution rnpYj

of the set of linear equations, we can compute sorne hydrodynamic. quantities as lift force (tbrust),moment and. application point. The lift force per section is

Ç.

. .

C,Lr?,bC)QLX

The moment per section, as usual related to the. one quarter

chordpoint,is .

I

.Yflr)=ft

1. C

."

.X1f&.%

J

Applying eq. (125) and (128) we get after some. calculations

Lir)11'') C0

1r)

mL}

CLr

The lift force as calculated in (180) is perpendicular to the flow (IA,o,o). The contribution of the lift force to the thrust (force in axial direction) is therefore

,LLrL

£L-).C,0L,-)

c.rILr)C,.,0Lr)

For the moment,fl)Lr), a simulár transformation holds

.Lr)OrZ

ic1()

'1

To find the total thrust of the screwpropeller, we sum the

contribution of each blade, which results only then in a non-zero force, when the number of blades corresponds with the wake

NÉDERLANDSCH. SCHEEPSBOUWKUNDIG BLZ.

(56)

PROEFSÏATION WAGENINGEN NO 68-036-AH..

.01-z.

55

harmonic

mXPJ

, )o12,...and integrate (1.82) over the span.

Ñ .

(18k)

k10

.

The point of application of the mean lift force per blade section is found from the fact that the moment related to this point will be zero.

C0JL1xll%_%:OOLfr)3::O.

For the abscis of the point of application follows

(185).

C,,Dfr)

To find the total torque we need the cOmponent Of the

lift force, which lies iri. the discplane. This is given in eq. (180) Multiplying this sectional liftforce by the radius.we get

after an anaiogous,.reso

fling

as before .

4Q'

r4gr c

Lr otr

k0o

From eq. (181) and (186) we deduce the simple. relation

TL,

To calculate the bending inoinent,m, we note, that we get only a contribution, when the number of blades corresponds with the wake harmonic plus or minus one. .The components of the bending moment about the vertical and horizontal axis are

(1 88a) IT

1

'i

rr±

ti1c)

Çot

(57)

NEDÈRLANDSCH SCHEEPSBOUWKUN DIG

PROEFSTATION WAGENINGEN NO.68-036-AH

BLZT1

56.

13. Resultl.

.

Numerical computations are made for a hypothetical case, the results of: whiöh, from a mathemat±cal point of view., were very satisfactory.

.

The screwpropelier isa two-bladed one, with elliptically shaped. blades. The hubrad.ius,v is .2.of the screwradius,r ,. .

which is 3.000 mm. In the developed. position the small BJCiS of '

the ellips (the greatest chord.length) is .2 of. the screwradius . too.The parameter of the helical surface,a-is calculated from the homogeneous flow velocity,1t, the angular velocity,c..O and the screwradius, respectively 7.5398 m/sec, 12.5662 rad/sec., and 3000mm. The rakeangle is taken to be zero.

The wake we composed 'of, apart from the homogeneous flow velocity,, the second harmonic of the Fourierseries only. 'The amplitude of this harmonic is choosen to depend linearly on

the radiùs-, being twenty percent of the undisturbed flow velocity at the tip and zéro at the root of the screwbiade.

To find an average lift in the undisturbed flow we take

the angle of attack,1to be 3 degrees, independent of ihe radius. The camber of the blade we takezero, i.e. the flät plate shape.

The resulting curves for the lift force per section as. function of the radius show for an increasing number óf pivotal points over the radius a good convergence and a reasonable

smoothness 'for more then'one pivotal point over the chord, (see fig. 5., age 57),

In. this diagram we plotted, the modulus

of the liftforce per blade section as a function of the radius i.n case

in=oandm=2.

. ., .

The, three dimensional effects tend to shift the point of. application slightly to the front, which is in agreement. with th& results of Truckenbrodt [iL], The running time per wake harmonic for a twobladed screwpropeller is about -O min. for 12 pivotal. points, all arranged along the span. For 2 resp. 3 pivotal points in chordwise direction (and 12 over the span) the running time

(58)

o .20 .10 o 0.2 .30

=0

.20 L ½Pu2t_(r)

t.

L Y2PU2L(r) m=2 O 0.2 QL 0.6 118 r LO r_o PAGE .5?

NETHERLANDS SHIP MODEL BASIN

WAGENINGEN

NO 68-036AH

08

- 1.0

(59)

ØLZ.

58

References.

Timman, R. and Vooren, A.I. van der:''Flutter of a hiicopter rotor rotating in its wake''. J.. Aero. Sci,

1957.

Ritger, P.D. and Breslin, J.P.: ''A theöry for the quasi-steady and unsteady thrust and torque of a propeller

in a ship wake''. S. I. T., D. L. Report 686, 1958..

Karman, von.Th. and Sears, W.R.:''Airfoi]. theory for non-uniform motion''o J.Aero. Sci., 1938.

Shirori, J. and Tsakonas, S.:''Three-dimeflSioflal approach to the

gust problem fora screw propeller''. S.I.-T., D,L

Report 940, 1963.

Hanaoka, T.: ''Thtroduction to the non-uniform hydrodynamics c6ncerning a screwpropeller''. J.Zos.Japan No.

109 and: ' 'On the integral equation concerning an

oscillating screw propeller by lifting line theory''. J.Zos.Japan No. 110, 1962.

Tsakonas, S. an.dJacobs, W.R.:''Unsteady lifting surface theory for a marine propeller of low pitch angle with chordwise loading distribution''..S.I.T., D.L. Report 944, 1964 and: ''Unsteady propeller liftin surface theory with finite number of chordwise modes.'' S.I.T., D.L. Report 1133, 1966.

Lasbka, B.: ''zur Theorie der harmonisch schwingenden tragende. FThche bei Unterschallanstrmung''. Dissertation T.H. Hünchen, 1962.

Nuithopp, H.: ''!'letbôds for calculating the lift distribution of

- wings (subsonic lifting surface theory). ARC, R.

(60)

Sparenberg, J.A.: ' 'Application of lifting surface theory. tó

ship screws''. Proc. K.N.A.W., Ser. B., 1959..

Sparenberg, J.A.

'Note on the ship screw in an inhomogeneous

field of.fiow'.' N.S.N.B. Nemoranduin, 1962.

Stiefel, E:''Einführung in die Numerische Nathematik''.

Teubner Verlag, Stuttgart 1961.

Kopal, Z.: ''Numerical Analysis'' Chapman & Hall, London,. 1955.

Triconi, F.G.: ' 'Vorlesungen iiber Orthogonaireihen'' Springer,

Berlin, 1955.

1. Truckenbrodt, E.:''Tragflchentheorie bei Irilcompressibler

Strömung'' Jahrbuch der Wissenschaftlichen

Gesellschaft fir Luftfahrt, 1953.

NEDERLANDSCH SCHEEPSBOUWKUNDIG BI-z.

(61)

Appendix A.

Derivation of the function 9)

71e try to Lind a polynomial which satisfies the. orthogonality relation on the interval ( o,'t'. ?'e know that the trigonometric functions A. 9 form a orthogonal and complete system on (oit) for X an integer

(A1

fit SIh

1. 8 s'n

9 ae

= o (). dp.integers

o

Putting %= cs 9 we have

S\n

where

()

is a polynomial of degree

A-

.. The integral (A1)

becomes (A2)

L'

2' TA-t ('u)

sn2)LB

d=

Normalizing expressIon lt t

dx.=o

we know that

A recurrence relation for (Ç _'x) _is _{asily derived0}

Analogous with

Sfl (.+!)

4 Sn

8 z s X 9 9 we have ) A- = A BLZ. NEDERLANOSCH SCHEEPSBOUWXUNDIG fPROEFSTATION 68-036-All 60