Controllable pitch propeller in transient flow: An application of the asymptotic matching technique

-HELSINKI UNIVERSITY OF TECHNOLOGY'.

"SHIP'MyDROOYNAMICS'i'LABOitATORY-,

OTANIEMI TINLAND. REPOkicNO:f3"P

EONTROLLABLE.;,TJ.iCH:PROPELLER:IN TRANiIEN:r7kL,OW:

APPLICATION OF THE ASYMPTOTIC

MAtCHIN&JECHNiQUE:-,..

..mukkgq_pyL0)41-0-ABSTRACT

The equations for determining the performance and the spindle torque of a controllable-pitch propeller in transient flow are derived using the asymptotic matching technique. The inverse aspect ratio is taken as the perturbation para-meter. Velocities calculated from acceleration potential connect the outer and inner regions. The pitch must be changed in such a way that the boundary conditions can be linearized. There are three solution domains depending on the rate of change of the incoming flow at the lifting-line.

At present no numerical results are available. Steady state is obtained as the long-time limit of impulsive accele-ration to constant speed. By making appropriate assumptions, simple analytical formulae result, from which the effect of various parameters on performance can be intuitively

CONTENTS ABSTRACT .CONTENTS: NOTATION., 1. INTRODUCTION .2. COORDINATE SYSTEMS ..,130UNDARY CONDITION FORMULATION CF-GOVERNING-EQUATIONS 4.1 Cuter Limit

41 Inner Limit

..ANGULAR VELOCITY OF SECTIONS OF ORDER,,A-3

- 5.1 Outer Limit Induced. Velocity

5.3.='Inner Limit ':

5.4 Steady State

5.5 Steady 'State, Disturbances in the Main 'Flow ANGULAR:VELOCITY OF SECTIONS OF ORDER

.1 .:General SoiUtion for

Angular

Velocity,Of-Order A-ImpulsiveAdceleration

6.3 Formulation of the Solution for Angular Velocity

-2 -,of Order A ' 3 5 6 13 15 , 21 23 23. 26 -32. -46 .' . -50 51 52

7.. _{ANGULAR VELOCITY OF SECTIONS COMBINATION OF'}

DIFFERENT

_Ttzuoius

53

:.7.1 Physical and Mathematical. Background

7.2,-:Steady' State following the Acceleration Phase 1.3 Acceleration following Steady Phase

-5.§' ACKNOWLEDGEMENTS.

REFERENCES

-1

'APPENDIXA Components 'of -Grad

APPENDIX B Equations Defining a1(I13,10' .

APPENDIX C: Inner_Solution for Case (c) for t t

, r

LIST OF FIGURES'',

Fig. 1:: Coordinate System Fig. 2. Blade CoOrdinate System. Fig. 3. Section Coordinates

-a acceleration

Br:. = 1- iK(;);

b' integer

b. ' coefficient of Fourier cosine series :define

by Eq. (4.9.) chord length' unit',base vector esubscript forcesubsOripte.denote,thedirection_{_} ,

-;kUnctions,describing the lielidoidal:SUrfaces.

=

+cOMPlek accleratiOn Potential

_ B'

= F8/V(t)

definedby%Eq.- (v.20)

1

mathematical abbreviation defined by''.:Eq..15'.1ij

0

subscript mathematical abbreviation defined

by -Eq.:'(-5A9):

'

mathematical expression 'defined by Eq. ,(B,. 1 8 i ).

.. .mathematical expression :defined by Eq 8 iv)

Tr(T.,) mathematical 'expression 'defined- by Eq.. (.7.1,),, transverse displectment'r,of the mean chord line

F'Fb

mathematicalabbreviation Subscript Struve. function = 171 integer divided by aspect ratio of z blade, .(R-Rh)2

the blade area

mathematical expression in the inner solution. w(t)/U(t), 21T divided by ,the-.pitch;of heli coidal-surface

7

modified Bessel function of second kind Kn

integer lift

Lp localized pressure jump

9. = c/2

lift per unit span moment per unit span

fl

integer

unit normal vector pressure

variable for Laplace transformation P. static level pressure

Ap _{pressure jump}

torque

QSH hydrodynamic spindle torque

ie absolute velocity

ie velocity due to ihe relative motion between the inertial and body systems

velocity in the body system propeller radius

Rf distance defined by Eq. (2.8)

Rh hub radius

RA distance defined by Eq. (5.10)

distance defined by Eq. (5.4) Rsubscript

radial coordinate in (x,r,e) set

4

position vector of a field point (in Ch. 2) rb distance defined by Eq. (5.16)

rf = defined by Eq. (5.3)

'

,t2,1/2

wm wp

advance velocity of propeller

perturbation velocity defined by 14. (4.14) vos 0 - wpsin .0, perturbation velocity body frame

axial disturbance velocity

in

the main flow axial perturbation velocity

perturbation velocity in body frame in

ar direction

integration variable defined by.Eq. perturbation velocity in body frame in

abn direction

= 1J(t)sin S(t) + r(t)cos O(t) normal velocity defined by Eq. (3.5) induced velocity

radial disturbance velocity in the main flow radial perturbation velocity

- iui, complex representation Of absolute velocity

-tangential disturbance velocity in the main flow tangential perturbation velocity

axial coordinate in'.(x,i,13) set; also the longitudinal axis which concides with the shaft, directed forward .

radius vector

of

afield point (defined by Eq.

(4.6))

Bessel function of second kind

in dS element of a surface

T

thrust time velocity vector 'subscript ui Um up ur ^ U8 Un V V V vm X Yn , component (5.26ii) 2 component

eb

: Dirac delta functicai

-'defined by Eq..,(5.18iv) _{(t > .9)} scr. t _variable

- " _{rd1 cobrdinate in--th}

(E"; v ) ' &est',

angular .coordinate in

_(x,r,84)::

_set

= 2.1rb/Z (b-= t. f:14(`-i)'dY ' ..D 111 transverse axis, 0-2 number of blades defined by Eq.

third Cartesian- axis, ,sdirected 'downward

a

J4,6-4,71; Aefined by Eq 1 i.ii )

geometric'.

angle of attack

.= 1-an-1(U/mr),

pitch.

of incoming flow

lifting line _ = 43(t =

0)1

gamma.funct.icar-= 0.57721----Eurer's constant variable defined

py,ER.

9

axial irrtegration variable

irected-.starboard

defined- by -Eta: ( 5 . 2 0 )

-defined by Eq.,

(5

:;21 )

angular coordinate in the ( )" -set

axial coordinate in the integration variable fluid density

body coordinate system

radial blade fixed coordinate, bn _{of which is at radihs. z}

,dIbn radial inertial coOrdinate,thi"origin.of:Which

_is

a-tradius zr1 defined by Eq. (6.:5) defined by Eq. (4.2). r

r.

defined by Eq. (4.9) = defined by Eq. (4.20) _ . = T;T + X , defined. integration. variable _

integration Variable in Api6andiX1) integration variable, in-Appendixj3:

variable for definingTOurier'CoeffiCients acceleration. potential'

modified acceleration potenfial defined by

tu

subscript modified acceleration Potentialan'annervariables, subscript denotes the parameter defined by Eq. (,4.12)

Imodified-aeZeldratiOn,potential%defined by

imaginary PoATorien-tof comple]Cacceldration.

potential -:

-scalar potential:defined by Eq.

(4.15)-= rotational Velodity.cfpropsller,

A

an radians per Unit: :right

hind rotation

-.

-angular velocity of the coordinate tystem, defined by,..Eq. (2,15ii)-H

SUPEiLSCRIPTS

simspita7s

a.symptotically equal to.(iii some giveii limit) apn-roximately equal to

(in

any useful sense)

one .writes /12 D. 24/ 'f(i) = 11(s()) 0 -.if -lim{f(i)/5())<=.

one writes /12 p . 2/ f ( )I= o.(.5

a))

as

.-E-6.0 if = O.

-point an a helicoinal surface inertial fratie

disturbance in the base floy../.

1.sertUrhat_ion velocity

,transition from one

domain

to another (Ch. 7)

-in

Ch. 3 the point where the boundary condition

is satisfied

-- ,

in eChs. 4 .

.and 5 field

inner vaidable.

inixtmucTioi

-Singular perturbation technique -simplifies the task of

-computing the Propeller performance Instead of solving an integral equation only quadratures need be evaluated. In same Special cases most ofthese can be integrated

in

closed.

form. Consequently it is possible to consider the influence of some of the parameters without performing any numerical computations.

In the present paper a linearized theory is developed for controllable-pitch propeller in transia'nt-fiim. The propeller blades have large aspect ratio. The fluid ix_ inViscid and incompressible The rectilinear and

AngplaY

velocities are not constant. Also the propeller

. .

is not allowed to ,get into its slipstream: The blade shape and motions are -specified. The time-dependent thrust,

-torque, and hydrodynamic spindle torque are obtained -A propeller in steady flow is a special case of the above approach /6 p. 754/.'

The present way of attacking the Problem is an extension Of-earlier work of James

/14/,'

/5/, /6/, van Gent /13/, and Sparenberg /11/. All these calculations are based on the concept of Acceleration potential. In this paper the geometric notation and general formulation of the problem have been adopted as far as possible from References /13/ and /11/. The flow conditions around a blade of-a cycloidal propeller resemble those of a controllable-pitch propeller

/51- Also a wing in curved flight forms another analogy

with siviiinv,

physical features /4-/. The above viewpoints must be modified to suit the controllable-pitch propeller

case and then appropriately connected together. This can be _done by applying the matching technique of James /6/.

-

14-'The:IliethOd of solution as taken froMlitef::76,-p,._ 753/

is as follows. Two simplified problems, in the

ouferah-ci

inner -region, relayed to the full :problem are solved In the outer region,,theblade is represented by a line-distribution of pressure singularities, Whoge:-Strengths, ,vary with...time:end in the radial lOcatiAnt.lhe :loweSt7.

Order singularities, occuring on

the lbeidecie;

,

preSSUrediPoles-.:wifh axes: parallelthe."biadenormar. 'These.

Are tOilowecl7by

pressure quadrapbres-andhighefer

..Multipoles.. In the inner region, the problem bedbMes.a.

local

:investigation of the

two-dimen-sional airfoilor blade Section.' TheUndteady'indnced.. velocity on section, owing to the:Nidirtexwake, .introduces .aiecond7order correction to

iheaceieration-"-potential in the inner- region Ille:_:inauce&:VelOCity is -obtainectfrom analysis

in

the otite-regiOn. IrC.the4resent,

case'dOnStraintiwith7reSpect to the:nuMber.0-154d6C4tid'11;_{_} . -baseflOW;wil/prob-Nienecessary due the mora.CoMpkicafed:'

(compared geomdfri-6)donfigUration.'

2. COORDINATE SYSTEMS

The following assumptions concerning the propeller are made /13 p. 243/. The thickness of the blades and the presence of the hub are not taken into account. The geom-etry and the position of each lifting-surface are approxi-mated by the projection of the blade contour on a helicoidal surface with constant pitch and without rake. However, the pitch changes with time. The propeller blade is symmetric and of large aspect ratio /6 p. 754/. The development can be extended to include minor chordwise and radial asymmetries.

The inertial frame

(mi,yrzl)

is fixed with respect to the undisturbed fluid (subscript I denotes the vari-ables in inertial coordinate system). At time t= 0 a 2-bladed, right-handed propeller is at rest in a quiescent fluid and subsequently achieves the forward speed U(t) in the negative x1-direction, and the angular velocity w(t) at a later tine t. The axial and angular position of the blade are given by (Figs. 1 and 3)

(2.1) Z (t) = - U(y)dy

e (t) m(y)dy

0

The Euler equations can be integrated in a coordinate system advancing but not rotating with the propeller In this frame two sets of cylindrical coordinates are -used;

(xir,O) refer to a field point, and (E,n,v) to a point on the helicoidal surface (Fig. 1) /13 p. 244/.

The coordinates

(tir ,o-,o )

(Fig. 2) are rotating and n

advancing with the propeller blade. The inner expansion of the acceleration potential will be developed in this

16

to the radial vector through the tip of the blade. /3p.

16/. Positive _0r values point in the downward direction (i.e. the scalar product of the unit vector in the x-direc-tion. and the unit vector in or-direction is positive). The angle $ is the pitch angle of the incoming flow at

the lifting-,line. .Coordinates (mr,ai) are called "body coordinate system" in Ref. /5 p. 6/. To within the order of A-2 sections, the length of.which is' of. the order of

-1

A defined on a cylindrical surface or on z plane at the same radius are in agreement /3 p. 76/.

The transverse displacements (i.e. pitch and changes of pitch) of blade sections are described by /6 p. 754/

(2.2)

ai = h(ar,on,t), jar&(a)/A

t > 0 R11-< a < R

1(an) = e(0.n)

where c is the chord length

A is the aspect ratio of a blade Rh is the hub radius

R is the propeller radius section under consideration

h is the transverse displacement of the mean chord line.

The symmetric blade outline is-tiven-for -ai = 0, ar-= =, ±i(o)/.A < an < R). The function. h(ar,an,t) is constrained by admitting

only

displacements that are within the scope of a linearized theory /6 p. 755/.

For defining the angular velocity of sections,

two

sets of coordinate frames,a_{( Ir'aIi'aIbn)} and _(Gzocii'abn) are introduced (Fig. 3). The origin of the radial coordinate is

21.

(2.3) + b a(t)E, + W(t)t = 0, b=

-.a(t) = w(t)/U(t)

a(t)ci .= 1/tan

- The .boundary z condition is

(2.4) F

- a()+ os(t)t =0 .

unit .normar Vector to this surface

is--= b =

atradiUs a. The subscript

I denotes inertial. frame.

-Equidistant surfaces ,are given- by the !--expressions

/13,

p. 244/

satisfied on the

In terms of the sYgteM; .the :Cartesian

-and the-

cyrilididcal

polar co...)rdinates

are:

(2.7 )

0.60s

s

-(2.6)

ocos B

+ C7 sin B

sin(13 +e ) - -a.sin + cos Sicos(6 +6

n w b r w b

--=

1

In the following, the point (a ) ; is situated .orf

one of the helicoidal this fact.

-The distance, between a 'field point and a.presSure doublet' . -on the helicoidal surface is given by

+ Ob + tan IC-a-sin +

-112 .cos(8 -7- - e tan.7,1 7.a -sin0-4%-a c.cos 13)/,a-1)}

. r

- n

,

-When writing down:the:hydrodynamic equation grad

as applied to a point on the helicoidal -Surface is needed; . ,

3R-1 aR

-'R

f f

,+1

f'

I;grad.Rf. 7 ex at. + ex, .+

-1.

MT

.18

surfaces. The, subsCript, --denOte's

1 r

aRI

oa: 3R '

f

3an Da 3R-1.-3a. '3R71

a.

r

_-3v.

f

f

.311i aai

Tr)

i

The: main components of this expresSion, are given n Appen--2

,dix A, -where 'some -tdrmi of order A have been neglected, _however. The unit' base yectors- are -denoted by -esubscript.

2

s{

(x'- la .co + arsin 13] ) + +, 0 +[-a. +cr co

2r /an I-o.sin_3. + arcos 8)2'

The unit vectOrs in a .and'

a.

-,clirectibris are

613-tallied 'by differentiating /3 'p. 13/

[(x-E)2 + r2 + n2 - 2rn

cOs(.87y.)31

(2.8)

_

and

they amount ,to-. T

Dr,

-(2.11 ) = - sin B e + (-cos B cos (0 +0 ).);y

+ (-cos -+0 zr

w-4- ' ar. -,

-

-I.-:cos f, + (sin B cos(.8 + 0 ))e

Y _.

+ .(sin B 's,i.n(6 +6 ))e -..

7

e = e

an - r

=(i)

+ cos(00;141)ei .

4 4

The surface element- is given by

- -.(2.1 2 ) .dS = .=-da n e al dS +do da

+an+ L-;in

1.1/7a7-12 2.2 in 13 11 +.a a

coste-(ere )]

where is the -unit normat to the eai = aci:

-

=-the

last eXpression terms of second 'Order

haiie

'been discarded .-

.,The'relatiOnshipS"--betweeri the

coordiriltdiin

e-Ii

sin B sin 8 - a cos sin 0. +:0 cos

+ a' cos 0 - a

w n '

cos 13 e .

+ sin a

e

al ar

= -sin 13 cos

8Z .

+ cos 13 cos e e

al u!

-arc

+ sinew eabn --'e = sin Ibn 8 sin 0 ;oi - 13 in 0 e w or + cos ew eabn

From the time derivates of- the unit base vectoi" 20 -..

de,

_or 4. -8 ecri + {-'cos 8 0 e , at w a ri . d;

-8 e

+ sin e ti:. +e. , dt or w abn ..

ttie angina',

velocity of the origin

4.

-(2.15ii) .13 eObn + sin 0 ear + cos 8

, w

WG

w

2 n 2 [U- U U ( w a )

IS":oi5tained (see Eq. (10.2.4) of Ref. 418/)'.- Then

,

--..the angular velocity of the section is; seen, to .be, :.

1

3. BOUNDARY CONDITION

The kinematic boundary condition is stated in the body system as in Ref. /5 p.12/. The linearization admits displacements such that

- 1 Dh 8h ah

(3:1) TaTT TT 0(1),(1) --- = O(i), W 0(1)

Dan aar

where

V(t) U(t)sin (B) + w? cos ? .

denotes the radius at which the boundary condition is to be satisfied. In this Chapter the supercript

(0)

refers to quantities at this radius.

The velocity vector of a fluid particle with respect to the moving (body) coordinate system is /8 p. 52/

(3.2) ir =

qa qe

where

+ +

qa = upeIx +-vp_{e1r +}

we

ple

ue_z + ve

pr

+

wee

0 Q

)IXIe

w ar w al

qe = -U(t co(t)r e

4

(sin e e + cos 18 e e ,

+8 eabn) x (a e a.e . +

O--

_ 8 )

r Or

1 0.1

bn abn

qe is the transformation velocity vector:, and ia is the perturbation velocity vector in inertial frame;

_{-(`8_ 8.}

₎

r' b

is- the radius vector of the fluid particle. However, the .radial coordinate is delt with by varying ?, that is 131.)=0. Within the linearized theory

cc

is set as 'zero.

i

On the index blade: b is zero, hence eb = 0.. On the blade

surface:

' grad:Q.17 y e ÷ (cos B e -

sin

8' C''' '°a

_S[er where a = r . ri-'W sin 13 =

ch-8i)

Dx e

The kinematic boundary condition is that

the

normal ... '

- .

velocity of the 'blades (relative to the ,body

-frame)

mist be

eqtidf- to the

normal

--:velocity of the;!fluid-;,.7k.e:

.,6h

-

0

'(3. ) + -grad(h

7 a.) =

_i

0

Where

lcar4.f.i,, Neglecting terms of second order this leads to

-(. . 5) t + ._wancos 1:3 / ah ; Dh u 6.0s, w 1 = "v 57-rf.: ' v (8 - T.) , 8. = ±-0, :8 1 , .i.(8 )/A-, ' r i n;... i , t

4.. FORMULATION OF THE GOVERflIN6 EQUATIONS

4.1. Outer .Limit

The basic hydrodynamic assumptions are similar to those in Ref. /5 p. 3/ . The Reynolds number, based

on

the blade speed and the chord length, is assumed to be large. The ordinary boundary layer is nt,glected and the free vortex Sheet is taken to represent the effects of viscosity- in the wake. Then the problem can be solved by deriving the equations for the acceleration potential.

The present solution is not valid in the tip nor in the root region. Also the propeller is not allowed to cross its own- wake /4 p. 2/. No perturbations 'may exist within

-1

the distance 0(A1) of the index blade /7 p. 35/, that

is

2178

(4.1) n sin

e >>

.

in stopping. or deceleration manoeuvres the theory also demands that no perturbation::, (trailing vortex sheet) exist

:

in the imMediate'vidinity of al:dad-6. 7

is introduced. For positive-definite U(t), there is a, one to-one correspondence between t and tu",^ so that t = t (t() exists /15-. 755/. In the following U > 0 for t > 0

is assumed.

- The lin'eari'zed Euler equotion,

in

adVanEifig b-ut.- not

p. 45/ where where tb uP P P ,v ,w '7 ,,(A x)

= 7

9 denotes

Is

the fluid density.

p

- p(x,r,e,t(7 .)))

The modified, aCceleration potential 17) measures the verde

-tioh of the 'pressure p from the static level The perturbation, field is 'divergence and curl -,free in the fluid. bounded by the blades, the shed Vortices, and the encom-passing surface at infinity.; By taking the divergence of

-

-z: (4.3), the .Laplace equation

- ,

is .obtained.

'The velociti is derived by :integrating the

hydrodYnairii6-ai-equations _along a streamline /13 p. 245/, and, assUnii.ng zero disturbance velocities far

in

front of the lftgsufaces

-The.- veloci zy comparientS aMOunt to

)

(x,r,13,7)=

I

4) (A,r,0,7(X;x))dA. p

-x-T

- U (x,r, ,T )1 (X,r,O,T(XiX)MX:. ,

. .XtU

TU

!W-(X,r'

I

-

(11' r 0 7-''('AxfldX P 0 ' ' '

...

The boundary value problem defined by

(i)

,

'tends to zero

inity

where

;

is the raciii..S- vec-tor,,of.;44f ield. point

(ii) the :kre-SSUre is .continuous eyeryWhere exCept

across the blade,

_'''

. .

(iii) the 'perturbation velocity across the ropeller

blaile'Zis §PeCified by Liq..,(3.5-),.

(1v) the pressUreiS' bounded aiOngthe,,Sharp'trailing

edge,

'(v)

Eq.:(4,4)

can ..bc solved by diStrIbuting.presdurc

- .

over the

tiladei, projected Onto the plahes a-. =0

76

p., 756/.

-.

The pressure, field at

..

...

,..,

_.,, . _,,, .

x = (x,r,(3),

_:.

--- ( ;4 . 7 ) -: -- 1 .

'."(1):'14.

Tr p, .

"

rad ()ApdS

-+1-:

E(0,)/A,v--.2=1

Linpu. ,

'Cda da

/17.27}

Rh- -R (a -')/A-

71, a R

a

DR- 3 .

r

'AT:T7 1-.

_.-al

-ad . ati7-

aa

' f .?rf

i .

are , . I (C - ,r " n'' J - r r !.ar'l 1?, s ;.;

,

'V g(a

an

)"-=

bb(a 't-.

. o ^ ^ b ( ,t( a) )co:,

pi

(0

--1 3

I

-' ' = Ah(a -1: ^ -1 Rh. -'3'n

where the Folirier coefficienta are

'

.;b)

the preSsure Jump on the

,blacte.--.at;:a2tpoint-(o

,p; 415) after the propeller

hastraverS'ad.

forward di-Stance

to.

Inner7Eimi

+ kt

a

I r

,8

a - -3ar' -;

-The details of the flow close to abl,a-de are investigated

by

stretching the coordinates normal to the radial direction'

/6

15; 7 5 9 / The inner variables

, .

=A81

, 8 :

;

= Ai

,

.

, r r 1 1 . .

-are introduced Maknifying time

t,

simplifies te-task of writing down equatiOns.

The pitch disPIabeinent function: h(8 --t)and the asS6Ciated nor:nal :velocity expressed in the

inn:6r

variables .

21T _^

b.(8n8

))=

v

Og

(ar

T"8 )COS UdU

i

n"

/A

T =AT = A

f

(U(ti)sin (3)(t1)+ w(t1)8hcos B(t.1))dti

0 0 ₀

= A t IA

f

_V(t1)dt1

.

0

In Chapter 5.5 an additional parameter, the angular position of the blade, is assigned to Vg.

On the basis of earlier research /6 p. 760/ the normal

o

velocity u.(a

r,a.;a ,T

,A) cn the blade is tentatively

0

assumed to be of the form

, ,^

-.o

o (4.10)

U.t0

_{0.-C1 ,T}

tU

0.'0

T 1 r'

1'

n B 11 r'

1'

n'

B 1

,^

0 +

ui2(.0r,0i;0n,Ta) +

larl

<L,

Rh < 8,

< R, = 0±, A 0 % U.

_{11 r" n' B}

(0

_Vg(cirTB)

G.

6;

0.8

)

= -hisAi)I(8n'B'A)

12

r" n' 0

= VI(8n,i0A-1,A) where

VI is the induced velocity.

In the inner region the-acceleration potential Gu is

given by

(4.11) Ou(;;Tu) = Ou(ar,ai;8n,.7u,A) = 08(ar,aiayis,A)

depends parametrically on 3 and TB or Tu and is analogously expanded in an asymptotic series in the inverse aspect ratio as /6 p. 760/, /9 p. 704/

28 -A ^

.0

^ ^ (4.12) _{4,8(0,00i,an,I139.)} ar'ai;an'T0' 4. A 4)2(0r'0i;3n'.78) 1 where 0(01(ar,ai;c9n,%)) - A .

Next the equations that d2fine the ok:s must be derived. In non-inertial body frame the equation of the motion is

/5 p. 20/, /8 P. 53, Eq. (2.7.10)/

(4.13) + v(-4.a qe + / qaa) = - Vp .

q

7

at

a

In the above equation the velocities (see Eqs. (3.2))

A

(4.14) qa = urear + uieai

n

ueabn

4

qe = -Ue - r w cos eweIr +

K)

_{sin eweIbn}

.

(sin 8

0.

+ cos a eeai + Be

.+

.abn)

( 4-

8.Z

+bnabn)8

lar or

1 oi

bn = 0

=[-U cos 8 + w r sin B +

i

r ea

- 4 NU sin 13 (A). cos 13 Ei.f;]

or

4--eobrit-wcollsint38.-cos

_0r]

will be substituted. Neglecting quadratic perturbation

00

00

ternisandalsoos B a Ju

) as being of

i r n

second-order when expressed in inner variables, one obtains

(4.15) a

+ a

[3. + v(t)] -

a.8

l(p

-p)

at r r p

Vip

(4.16) (-FE t..k;Cf )

,

Introducing

complex,

Variables

(4.17) -, A 6,-0 0

!°n

= ,

-and ,defining complex .acceleration potential 1

94

5 -(4.18) = .:41 f'" 7T1-7

'.Equat

i 4.16 beconies

-

a-(4.19)fcz,T.

az

In the continua-tiOn",,the 'notations, ..

%(4-20)

are used to shorten tht,for4lae.

-'

;

Mow the inner Sr.3olefit'Cur. be st-.4ted

(4.25)

P-277 la la-r h(Ilr'cri'an'T) o ^ a :az where: k. = 1,2 30 ; ,T = 0

-or f is an analytic function of a ia.

for all time and has a discontinuity across the blade,

'a

a (4,24-)

fk(aroi;OnT)0as

r =

(a-_,r + (IL IL):, .(2,q)-3:1c(q): a-7 k

az

<It,

Rh. <

<

= 0-±

From the last equation it can be seen that there exist .three different types of soiu,,:ion according to the*St::

. ._{_} .

;(4.26) r=. 0.(11A), this is denoted as Case' (a-),

x-(i).= 0(1/A2), this is denoted

as,CaSe.(15).,-K(T) =

.0(1/A3), this is denoted as Cse'(d),.

-

--In Case (a) only the first-order term can be computed.: -Formally two-term inner and outer solutions can be written

but the linear connection between, TU7 and TB has -e-fidnvanishing: constant coefficient. In Case (c) Equations

(4..25)'become

where k r

In Case (b) fl (or 01) is determined by

on

equation of the form (4.27) and

f2 is: determined by an equation of the form (.4.2:6).

In practical ship configurations the flow 'defined by the inequality (see Eq. (2.1,6))

(4.28)

0(b -

U !) <<: {both, ()(c1)) and

₀₍₆₎₁

does not appear. ..That is, components-of k(r) cannot be of higher order than K(T). This is due to the resistance of the

ship.-The solution to Case (c) has-been given by X4111 /15,/, and

James, /,6/, _And to Case (a) by James ./4/., /5". The' main

task in the present work is to derive the inner :expansion of the outer solution.. The inner solutions for Cases :(a) and (c) can be taken directly from the references 'cited-When all this has been done the solution to Case .(b) can obtained.by_appropriate minpr. changes in notation and nomenclature. Also the problem of how to combine different solution regions (defined by Eq.- 4..26)) will be. discussed.

, -; R i(b .)-/A

f

-- . E _{AP(ar,o_n--y'b):} -(5 ' (xiTU) 4171:-7gi -)/A- r b=1 -..f (x,rie,a ,a,;b)_r n .3f r Bo z-1 .

f

:darda -

I

Ap(.a 0--"r 'Cx R - n b=0 1 where (x r e a a b) = fluci +a 1 5 r, n, R3- U x. f 2,Tb.

_irrc

B.1 \ ,, 1---' Ci + --.(172 cos2B) .r. .sin(e - e - -- -can a os ' to Z .,, L an lign _an 1 2 fa + (arcos R Z-1 -, -1 . (x,t ) og

f

da 1 (f (x,r, -LI LirriDU , -- Rh n b=0 afo(x,r,e,0,a ,b) ;-B.a.;; 32

. ANGULAR VELOCITY OF SECTIONS OF ORDER

Outer limit

The limiting form of the modified aCceleratidn,pbtent in the outer region is obtained frdm'Eq. (4-.7) by

-.lett.ing the 'Chord c(or)/A tend to zero /5. p:' 757/._ the limiting process Rf is (xpandedin '.a'''Taylcir series

_ .

(5.2) where L(an)/A =

J

n)/A

Man)/A

m(cylv,A,b)

-f

-Jt(an)/A 0(1(071;Tu,A,b)) A-2

0(M(0

n'

_''TU AO))) = A-3

1(0 ;TU ,A,b)

n Ap(or,an;Tu,b)dar

arAp(ar ,an ;TU ,b)dar

AP(ar,on;Tu,b) = AP(ap,on;Tu,0)(1+ 0(A-1)) .

The lift 1(an;Tu,A,b) and moment m(an;Tu,A,b) depend parametrically on the aspect ratio A. Their .orders of magnitude can be stated on the basis of earlier research

/14 p. 40, pp. 44...45/, /9 p. 701/. The parameter b accounts for the case when the base flow is not homogenous

(or contains perturbations). In the present paper the perturbations in the base flow are zero except in Chapter

(5.5) where they are of order A-1 compared with the base flow.

Rf is to be expressed in the body coordinates

o . x = a.cos B + arsin 2 o2 o 0 2 r = a + [-o.sin B + orcos B] -1 o ° o

B=Bw+tan[(-a.sin

+rcos B)/a ]

and in the stretched body coordinates

(5.3)r = r cos y

_{= Aar}

-2 -2

= a + a..

or thatching purposes the aaYMptoti'c limit of the outer

o2

sOldtion . as (a. +

is /6 758/.

.r - ,

-?-.The.conirlbutionS.t.(5,-theodified acceleration

from the index blade (b ..=-0)

'and

.froMiother'bladesill be

"discussed separately j'orthe index tlade,.-Rds expanded using the Fourier transformation of R /6 T.,756/: The

:reSulting6tieseel

functions arc developed in ascending !eries,

-/1 p. 375/, /14p 36/, Applying this method tb the eZpOgsions (b 0)--wa (x,r,(3,00 ,0 ;.+sin 8 u n cospi

-1 9

-3sir1

.84 fla

U 34 is cg + ceri, Co S 1)1

o'clpk;

+

-8

in A, linner variableg' Bf +sin ,Rfwan

'1

U o o 0 B - cos B oa 0 . .0.

icoS 134Or

)}{sin Salicos (3.+ = (02 .o2 o2 2 + n n,

,

-2mt '

raisin.B.+

ardos e -0 ) sin 17z1).112 an 13) sin (3)+cos

(i»1'

A 1 0 o = 0 -tinher 'variables OrCOS

_

-(

1 b = ( --) + (a + 0 (1.0-0§-1 ,T.,b1) 112 . 1 27r :28O (1 - o,-%s Ial 5(1.1 )1/2' a

/

,

.. or for the .

index

'blade

o2

0".

an)2 1 /2. R = (a. +o2. and ob:gerv'ing . (5.5) 2(Ccif2 92. 4. (8 :G., ) -3/ 2 - 2 i n n =

f

_K-1 + o k}cosilq..0o - hlk 1

1.:

'0 /62 o2 a-. car 37r (52 , o2 o - 2 -5/2

(a.

a + (a -a---). ) -2 a r n r,... 7-'.. K

(582

itk casild a 'Mk , o 1. + . one Obtainth

tl(ao

3)s4n y

2iTpU -Ds 7

-Provided' A- 1 ' O'(-A- 1n7 Ai) 5 ::7 ) cos(Lrb)1 >> 1 A

fb 1, -

7.

the magnitude of' f and are

-c,

7

:8)Y .6(f 36 OC/28 a .11- co (2.ffb);)

n n

. .

cfo

= {0(1) + - 0(1) aor This implies (5.9 0 OD T I U U (1)0(1(-8-A

b= 1, ..,(Z-1)

0(m)0{0(1)+

7,01'-z. ="

that is; no additional terms,

blades ,, appear in 'Eq;.'(5,..6);:"

,

-5...2 Induced velocity.

7.The induced flow on the blade connects the ifiner.

'outer limits of the full problem /6 p: 758/.- The pertur-bation velocity u. defined by Eqs. (3.5.).and -(4.5)

Will

be applied as a correction to the specified :normal velocity

',of the

blade in

the inner region. The velocity. components.'

nyA

-='

5.'10

).'p

1 )1c!. dx ,da ".E=0;' ta);( iT" 06X,) ,b)

." 4npU Rj T1 -2.(cr)/b U h T1 a TT f4,(),,r,es0r,on,b)1 :Tehei,e WOn + 3 I 3 -UR ().-cr )2 /2 3 (1 + 5 2 R an R5 cos2 (3) Ok-o sin B)-. due 1-A

to the influence, of Other

-215-f, r sin 8- - -z (A r 9' ar a ,b)

_-'

where 3ff. sin0 -wa / 2

an+ araos2

( 6) 2,Tb. 1ar.'1rAi -a-cos 5 ...-+, 'Fo, 131?)i-} 1 1

{7 A,

f = 47pU

X-u

CT (-11.-.COSB r.coS gre evaluated at

8.

= C. (5.11) 4p..(ar,Z -1. an I j)n IN)111, _z-1

f

.da

tca

)/A r b-Ap(a,,ar;Tu(A-,x),b) +(a- cost3)2 n r - 2

sinI3D--, a -+a cos an n - - cosB a- 1 t -_ /02 +a2cos.25 [-at,cos6]?)13--_, a 1+

-078tan

-+A-x,b) Cd

6(a )1(a

T +

-x,A,b)

r n U

viriaties

'

)

1 '

. setting' x -to' zero one obtains

' R - , 0.1,4 ' .., - 1

,

r

,

.f3) .;.;.1 (ai -; , 1, ) j cos(3,j.

rc.F77-3- - ( on

;',:.( 4.-apli R (1 , .U:' 1 -12 ,_,.. 3 J 1

11/2/

-..

dr da

+ 1 n LiTr.pp U..; a -;''.-,:,

L: R-F.T.T=0

- .-U -1

ft

+Ir. +n,7i-.:Ci-':',-cOs(.1-122)

.---::'

B .7--

'7.

0.,,..'l

p --ttr..li

, r" LiTrpU 4.s in - Rh ":1(r-on ). ;.;,0 27Th . 1./2' ., i R ,1,,. :Z:,1,.; ,

cos()r- ....

,,..

(

)

d-r da ---1--- f si.:/i 6'1.1"r '-2 / 2 - 1 n 1-rirpU , '- - . .,,,,.,,,...,. , .T11:!-(3.n).-',', ' ' 'Rh . r2.4.6?.2i4ZCioit21--E--" .. n --. n -

Z '

U:/_::a..,, .,..1.. 1/2 ,....-''

j-

.. 1(9 TU T-1

'A'b)I

', .

T .de.

...1

tr,,to -2ra"do(...)].

-2... .''-' )--n r-,- n .

--...r,

,...,.

.-..,. + _0..rr- a . in 6,/2z....i../STin2L',1./2Tr-ip

jf

.1"-..!a. ul':1:,....i,.: :_l.(..-60 '/-. ill-.;A%, h.)

,.. P :,R n . , /..." 1!... 2'.' - ;IT ., ,..1 2....----,2rb.,2 k.-2 i2, 2 4 -2.:' .,--; .: Lr,;.+a --zra ccioS J

T '+,,F.+0 7.2ro co's7.P.!

-. . 2 ',21Tb `ni - '

r +a

2.11:1 COS\.)1 '415 aDS,9!.7,-, J.1 Inne'r

Cr the

',basis of

velocity

3. -17 ' . B

esearCh

Z, 6

760/

,

expanded

as

-

o ^

-(5-14)

u.(a ,a *a

_n'

T ,A)

u.

(a-

,a

.a

t

A)

r

n' B'

1 ^ o

+ -A- ui2karocri;crn,TB,A),- j<Z, Rh <8n<R; iii=0±, A ,

S',/ Ca8

g n3'Ts_B) 7'1/ ((c'i -r^ ,A)

n'

.1^.1 63

; -3 ';

A) ,--.'C'

il

r' i' n'

B' g r n

u. (a

0

a.-8

.f,,

,A)

= -lim ,A

c/ (S

;

A)

r,

l' n'

B

(S

q.

,A) = V (S

q.

A-1,A)

p

n'

Observing (see Eq. (.4.,26))

(5.15) A2 ^.

U(T = 0) ^

.1

7u =

.7T7.757 Ts + sin' .13(t = 0)'7'

(t )T.

2 . a A

the induced velocity in Eq. (5.14 ) becomes

R

(5.16) VI(Sn,i-u,A cc)s' cos fit ° - 3.(o ;(U-? )A-1 ,A,O)

LurpU Rh 0 ) ^2 _g r T1 1/2 , I sin B 2 r) ' 31 A`d.l. da +'sin3 ' 1 n 14TroU J o 2 1 L(.i2+A-ca -t:, n n

(a

T1-a )n . .. TU _a

T,2

1(a ;(-7

-1i

)A-1,A-,0)[ -1

di.dan

0

D(r-i )

n U . 2.1.A2,`s

)t

u _'1 n n Z-1 TU ? R

f

cos (3

f

a i(a cu; )A-1,A,b) Rh 4Trp U b=1 0 D(iu-ti) n ^2. 1-2rtbl . r Ti- 1/2

0 R

Z-1

cds -,-,-,

2 ^ sinB Adi. da i-1

,.

Q Lci:24.A2.r2)31 1 n 47r-OU ) -111' b11 --I - 2 1 b P 11

r

b

. where 1(c7 *'(T n U 1 40 q:

zing

R. Z-1 " .0 - - -" LtT rpU I sinB8 a

I

-sin Ril.' sin 8,,- sin 0(t = 0) R2 4. 28 a COs[2n b

n n

z

where C = T Z = or + lai 8 T : r _ `Li A "' (a -I; , b= (0 ) ,1 (z-1 ) .

'Setting B = 1r/2 and 2 = 1, Eq. (25) of Ref. /6/ is

recovered.

; ,.

rain )BL+E

c), = k

n'

g (;o-

T)

1 [2,111/

-z+R,

(E -,z)

r

(5.181)g a ;8

=

-[-1;+

r n 3T aar -/ 1 .

'

(5i.

A-(8 ,T

f

(2. -E )

1--:1--)

).u- (a ,0;a=;:r)da.d

.--: IT_L

\a;

az n r .1 +` --a

(8

T) 2

n'

- '

-Now the solution to the inner problem, determined by

Equations - (14..23 ) ... (4.27), Can be taken directly - from Ref . /6 p. 76.1/ or Ref. /15 p. 3'4C4..347/.: .

E 2) T 2 r Z /L) r(a. )112

uik(E,0;cyn'T1)dT1dr,

3

,

(C) e+ieD 1 1 (5.181v)

H(T)

7.71 r exp(cT)[K0()+ K1()1 dc , e > 0 . e-ico

Similarly the inner acceleration potential is

Substituting Eqs. (5.3) into Eq. (5.19) the two-term outer expansion of the two-term inner expansion

1 o ^ 1 o " sin Y

(5.22) 00(;,T0)

=Id A [A1(an,T) +

T A2(an,T)) r f 1 o " 1

,0

_^Ni

iLl_21

4. __[,, (a

,T) +

_{A p2.cn,T),} A2 "1 n 2 rf is obtained.

Expressing the two-term inner expansion of two-term outer acceleration potential (Eq. (5.6)) in outer variables

1 (5.23) 0 T ) =

1(8

A 0) !-3-11-1 U ' U 2.11.pU -fl5

U"

rf sin 211 - m(a ;T A,O) Ti U5 2 J rf (5.19) (5.20)

(1),(a

Krin

,a.; ,T where o

r 2

Ak(an,T,

k an, ri+2.7111/2

f

!

_{ti_uk}

-1-1.1+p_k(gn' ^ ^ n,T)dE -1 n

a Ci

&k ' $ (5.21)

pk(an,T)

°

^ 7.11T-

f

(1-Z/Z)

1+E/1

1/2 gk(E,an,T)dE .

1-Ei

/

,

matching can be performed.. -FOr the PhYsical

the 1,esults T must be rep:1-aded by AT

-uble 1

of

1:ef. `/6 p.,-762/ presents the resuitt Of the

sl:eriebyLstep

'application of the asymptotic matching Principle /12/.

-

The,

main results are

o ^ o ^

-(5.24i) 1(arl,-rti,A,O) = 2npUA (a ,t)A

21

-o ^

m(a A,O) = -2nUu (a t)A'

n''ll'

1

n'

.

based

on

'Tro.c1TI) and 1T1(2T0)5 -and

-'t&.2irfi) A ,70 ) m(8

_{n' u'}

A,O)

based on..2TOOTI) and 2TI(2T0). 2T0(1TI) .denotec two-term outer ,expansion of one-term inner expansion. .

-_,-F011owing -Ref. .6 p..762/ V (a -t ) is defined as Al. = 27rpU(--A-. P1 = - 21r pU(--A-(3. 25/1 ) j. coa, 2 tbosB du aa Rh- 0 .(37 i'0,- 2

+4,t52

, B 20 13 11. ,s3n -sin 13-.7./2 n.

j

Te, Lax (a 7.811 )." 1/2 . ry. 13 u, 18 + a), )

sin B

n

.(5.20)-where 13 = sin u

g^

n!t)

=--{a1 (a T , n '1.1 2 . r 7thl 'T0.4, . /, - . .... _1, ---2- " 0A 1.a - ) .. 11/2 ,.: . 1" IT1? 8-U

-

-r .1 du do

.2-2' .1

'8 n .. Ri.,:-.,.. b=1 _rb 0-:., a(-raru.d_L. ,-.2 ., - b ) - ' - j13-4. --i-75n8s

.-Z-1ls 7{2 bi 0 8 2i - 'B 3-n Z I - B

ax (a-

u 1/2 2 n nb=.1 - ,y -13. 5 ^2 -/ 21 2

r-1 + 2

;2

_ sin (.12. rb)2) B [b

(8

(;».-o . aT

27)

l'i..(,(1,Tv=7,) + 4T [b

(8

,i(i))-b

(8 ,t(T)))).

1

-n

a

Exparicling, V

FOurier

series (Eq. (4.9)) the-2.f&l.lowing

_ormulae are

obtained':

/6 p. 3631, /11,..,:§: 347/:

(5.28)a1(8n,c)

H((q-c-' vg.,.)[00(,;1).

T]cmyr

2, 2 -T o

529)2 (

= .1-1( ) )V (a

44

Now all the necessary formulae for evaluating the propeller performance have been derived. The general expressions for propeller thrust, torque, and hydrodynamic spindle-torque are Rh aibc o o o o o (5.31) T

I

Z-1

f

( + F_ar sin (3)d b an F b=0 Z-0 1 R n o o o o n

Q=

F

f

_{d(Farbcos - Faibsin 13)ddn} =. Rh A QSH = m(8n;Tb,A,b)ah Rh where 0 n

Faib is the sectional lift (= 1(d -TU' A,b))

Tax;b is the sectional force in ar-direction.

In line with the linearization _Farb is neglected /6 p. 13/, /15 p. 52/.

Next the practical application of the above approach will be considered, and as examples, short- and long-time limits-of such propeller motions that allow simple mathe-matical treatment, will be investigated. In the rest of this chapter some derivations needed in the examples are performed. The equations (5..26)(5.30) are evaluated in the Laplace transformation plane /6 p. 753/. The transformation is taken with respect-to time 'parameter T (=T/A) and the above formulae become

CO

(5.32)1(gn,

p;A)=

f

exp(-pT)A/(Sh,T A)cite

0

I '''

r

,-1,-- -,

I) -13- 'TX1(8n- ,T)dT = A-1 Xi (aon,pA-1)

... .7A la (8 pA1)-1; (8 pA-1) -1. pA-1 ti; (8 pA-1)-i; (8 pA-1)3),

(5.35)

n(R.pA-1

(.5.36)

3

'

A'1)

i; (8

PA-Tv P 1

n'

-. n A ) = i;

(8

pA

)7-H(LpA

>Lb p

-1

K1 (tpA (2.pA

) +11 (LpA._ )1

-(5.37) -_ipi(8'n;p;A) =

".'

:A -. r. fQ. _{Z-;1 '-1} 1 00s131 {3:1 (ci, jaA - Rh b=0

-sanD L

--_--s in E1,Rr . Z-11r r27r1) 1

-L Z 2

?i(aPA

R b=3

r

sinB

) 1

-

(

rb ,2,

,

2h -

p 1 ,Ln

tp

y0( p

sin0

'

-

)

_(p

_{, ,}1 slr.13

sipB

da

n b inW R 2[2.6b1-

_{._,k71.,,}

₊

-r

+G₂

-

s

_{f sin8a a ;I:Sir/}

y

,r2L, 2

2. I

2 D

_{n n}

1 b= ' b -;- b ' '

is a BeSei.

oi the second

5essel funetion of ill:: Second

and

a Struve ,undtion.

.r

b.

1

W (p

_2,-

_-77c-

)+

7 3

_d

sinB

sine, 7 -rr

CG244"/

-+

0(4 )

'

I..; -.

-..

The long-time limiting

e'cliressions. are :Obtained by

taking :theiimits. of-D..is.. (15% 3 2)-... .

( 5 . 39 )

as

.7- ir7' 3.'1 and,

, ., , .' , - -_ ,_

- inNierting.

The following limitfig

'....forths_ are need-ed :_in_.

-Chapter

5.4 /1'.14).. 258, 360, 375,

067:7:----,

, . .7;:.1_ -..-:-: cG14:"° . - 1 - 4G14. 7.(74. 1.11:2.-Y):G1 - -2: ' G 1 urn. - 1 '1 2 2 1 + 4-)2-,4 + 4 . 2 G .

where

,

.71(10.5/721

EUler!S ConStaht).

' '

--

TheeefunctiOns'ire,.,defined.wih- 4..bi4nCh cut along the

As the branch -Cut ls,

-approaChed,.:fhim;the::bottom. and-00:_top,,,. the difference-in

the funCtiOns are

( :4 0 ID_ [1-1_

) - H( H)] m

(41an''

2 ) - ^+ )1

lac - -

),

1 - G - . G1

e2cii(G2-)J'`

G2. -5.4

-:Steady. State

;._ ;(5,40i) -

_lini

4--H) ,m1:i7

(y-ln

2)4'44 -4H]in,.tf.]+.,0c.cAl.T1 41:1)

-.." 1. 1 . - - 12 1 ' 2

iiin

--...

-Steady- state is obtainedas the long-time limiting

forth

'

-

t

-

-for constant (with -respect to .ifille)angle:Cf-,-inCidence--d-.

The SitUation is-formally gqui valent jto a

Steii=function-change-in-tWangie. of att4ok at a contap=forwdra

and rate

of

rotation

/6

of the reletraht

formulap-can.'be taken ' from Ref.

/61:

The porrzerO Four lee

ie transformation of which, is

(5.4111)(8

''bA-1 2v = 0 n cgG: ". .C5.4 7) ( 5. 32)...(537) beccric.

-( 5 .42.). .8n,p;A) = ',2Vuo -5 ,1_1(2.pA

(5.43) - ,p;A) V9cz. H apA 2 1 4. (5;1-44) ,1D,J;AS =',"" V.E 1 = --

-'

,)1

°

(5.4 5)

(;A),-.=7[H(Lp -T).

j cos_13(-EVV(oil)ao(anl)

- Z71 11r n(L(o. )pik, !CG da P

n ..-

b_ 2' c R

+ sin 8 -IC .sin:13(.-t.( an )V(an')aG(itril))

H(k(a )pA-1-) d

h=0

in Br. sin138 a..(-12.(d' )1/Ca )a,.(a ))

_ n n n , .h, 2:711 2. 1:1( Z.( c .)P1-17 ) 0. sin2

1

-4-7.; (.2 G2b-G1 bi ) ' n b= r. (-2-.:2[H(dtpA

). +

ipA

(--5. ;- 12-1--1(LpA-1

ni

PA).

-rformiing. the inverse Laplace trantformatio7

and

.aspuming

-cA the above 'formulae-transform to )3.

(n'i-.3 AT ) u2. 32 , a. + -0L(A7 1 G 13

-5.49):

-5ett-ing

rB=

in these e.quations the ,sted.df,i lift and

.

moment per ,unit blade .i.e.nith-.:-,::,leeleriV.".are

'.(5.51)

,o

1.12.03n0%.y = 7 st.X2on,ATI3

cinottijo?

-icos°5 0( (2.A71 it

(0

-1

2,

+ sin V+ sin 5

.VCcin)ce ,(cr:. )

.r2nbi ;0 0

Z,1

Man) da,72s3.h. 5 L Bandn.

b=. ° rb "

Ph-I

1 '

sin[

22rrbi

727: dg i.1-113V(a--)ct-(cr

6=1 rb.

,n Ts3:

c.6sj2frbid6..(1+ 00

2--A--) 1- 2sinfta

dinBa-V( )aG( n

-1(a )% _{2F 2-ir151}

b=1" -4 J .112...

-T1 Vb. - ' .

(di = -27fpUZ 1 {V(8 aL;(8

4 sin 8 1.1(a )ce

Z-1 .

..I(

);- I _{cosi-7 do_ T.2_cro}12-rrbi,

_{sin8 f a sinfrga-)a (a}

li(o-rr

n n . .

b=0 rb

_Ph.

14 sin2{27_4121dm, 11

.,4)=1 rb

5.2)

7

1(3

n, A n

(5. 54ii)

'55 Steady state, disturban6esin the main flow

The main

flow

contains: a-disturbance velocity vector

. 4

(5.53)

qmr (

,vm,wm

Each

of.

these coMpohents Is of order A'1. ThiS.approach might be used,

.for

ekampIej, to deal with thidknes6 effects' /1:3 p. 243/. Another possible application is a propeller

in Inclined flow.

-Equations, (3.5) and (4.9i) become now (Case (c) is considered in this chapter)

(5.54i) u 3h:

+ v(i)

LI' cos- B + w

sin-.,at

acr Dh

_V()ah

r , Dt Da im I/

,8

g r ' at where ^ o u.

(a T

,A) = (a r-A .im

n'

im B

That is4 the disturbance velocitY. is contained in the Fourier coefficients If the disturbance velocity is

Periodic (for example harmonic components ofthe wake field)

. the contributibhS:.of11-im:.-.t6 V1 can be calculated

- acCording to Ref. /6 76.61, '

',

..-There are no differences in Other hydrodynamic equations. From the last of the Eqs, it can be seep that

se'ctional lift -andMoment mut

he

CdmPuted separately for :each blade, .HoweVer, 7' ;needs be eVaida'Eed'only

once-" - 0,

,a u

ANGULAR VELOCITY OF

SECTiONS.-OFORDER-_ ,

-6.:1 General solution for angular velocity of order

-The treatment in this

chapter.

closely .folloWs that of .

Refs.

and /.5/.

Correspondingly:

the discussion , can be-7,.

kept very verse and only the main .results ',:Ther.'

seCtional

.lift and ,moment, taken directly f i75A :Ref.

/:-10/, 50, P.(

7

+ C G-1 G1

--'

26G2 12 ()-"G'2)_

-The inIt of G .,,is.of the same form ,as that ;of

for -rcll

t

the matching procedure can formally be extended z

,

-(6.1)

1(8

,tu;A) =

irpURE(Re(a(t

lc) -.b 1.(b -b )1-83 -- -0 2 A 3 pUL2I(Re P

(ai

(;n,K)+,,b_2) (b.,1-13..31] o -where

_{al (T63-K)=}

a (o _TfrX) . '

The FOUrier coefficients are differentiatedwith .resPeCt :to T . £quaiions (12) and (13) of Ref..1/4,/8efine (T4;ic. They are given "here as Appendix B.

The short-time limiting forms,for the sectiona-1 lift and moment are :obtained by setting p

co..' When

evaluating

( 5.32 ).. ; (5.37 ) use is made of the expansiOns,

to include second-order terms too. But as in Ref. /6 p.

765/ the starting lift and mom,mt are found to result from two-dimensional effects only (within the accuracy adopted).

6.2 Impulsive acceleration

In order to obtain simple analytical formulae, the impulsive acceleration from refA with constant K is con-sidered as an example. The fact that K is constant implies that U and w depend linearly on time and that the pitch ratio is changed during the acceleration. An example of

such a flow is given in Ref. /10 p. 341/. The non-zero

,

and

(6.4i1) b0(Sn,pA-1) = 2VaGA/p

E171'pA-1) = VK A/p .

Sectional lift and moment are calculated from

o 1 - -1 1

(6.5) a (a /PA_n )=

VA

H(XpA )[2Va +VidA

(3 ,p;A)

=I Li-f-H(2,pA-1)[2VaG A 1+ VKA _11}

1 n 2 A p TA 1

= -

71[2VaG+Vic]

f

H(TA-ti)dT., 0 o 1 2 1 - o -1

12.(an,p;A)

=Tit Aa(an,pA

)

7A 1 2

=cL

_{V[K-(2aG+K] I}

H(TA-ydri]

0

Fourier coefficients are /4 p. 14/

(6.4i) b0

('

8_n = 2VaG

_'

> 0

b1(8n

= VK , b.(8 3 n = 0 , t > > 0 0 ,

j

> 2

where- - = A and formulae (6.6)

1(8 ,O;A)

= - PM/L(20.G K) are Obtained.

-6. 3 Formulation of the solution for angular velocity

When computing the second-order corrections to lift and moment, Eqs. (6.1 ) and (6..2) are to be multiplied by A .

_

sections of order _ A

. ,

The complete set of formulae for Case (b), being "unwiedly, is

not

written down. - The general form of the

solution is discussed only. For, _1c = 1 the formulae of Chapter 5 are used. Only Eq. ( 5..15 )' must be replaced by

U(t = 0) 0(1)

(6'''' TU V(t= 0) '8, ,A

.

= 2 the formulae of Chapter 6.1 are applied

the -Fourier coefficients being determined from. _

(6.8)

:V (8 ,8

,

) _r -

v (8

REGIdNS.

cit_Z/E)

2

112

-I,

(R.

. .

PbySical.. and .mathematical DackgrOepd.

,

In tiractice-:AbembSt:..edmmor.-sit'iations4re,steqdy

fter Abeacceleration

_{. .} .

acOeleration' or deceler--

.

Ahe'',steady,period.

The-se'wo.cases will

e

..tion following

discdsse'd by -CObining ReOns'-.(a) and .(6)..

Mathematically.

this medns.that-the initial values are nbt zero;:theY'.are

giveivat the time tr'::The-deri:i.lationsof Refs

/i;-/

and.

/15/ must be recalc4ateci.-cio abblintl:fOr -this:fact:

. .

7,2".Ste'AdY state follo

.

-

:4ing_th:aecelerationlphase

e

. .

!There are no-cbanges,.iltirtbe,eqdationfor'induced

.

'velocity (Egr:- (5.1,6)):

One ohlY needS tb:KnOwtho,thrilete.

-ime

story of.1.(o;(tu7t..)A

b5W it was

-uri4.:ved

t

,The',detailsof tth

Inner -sOlutibb arejr.eSenteb'

replaced;by..

_

-.-,

o f '

,

1K ' n

alja

F 2 --",,2 )

1/9

-r

.f..41. '. .

-

U. CE'.",0';13,t

)dt cIE

'Where

s+im

-

H (T)f

= --k-

exp(T) eTC dc, c> 0

.

Tr 2711

_E-lw

_(C)+

The connection '- between

TU and-

Ts_ becomes:: ,(see : Eq.,. ($ . 15))

U(tr) ' - ' ' .-'1J(f ),.. 1

(7-..2i)

to(t) =

T (tr)- T (t ) --7---c-t-

T, (t ) - Irrr 1.1-.

r Vttr? a vtr, A

, .tc.

,.... If the' condition

(7.2ii) 0{TU ( _r) - T (t )U(tr )/V(tr '))_r

-is satisfiedETI (52:.5ii.) is still:yaiid and Dirther complit cations in computing the induced velocity are avoided.

7 . 3' AcCeleratiok following steady phase.

The-equationt,

defining the

Inner,.s6lution.are

given j...11

'Appendix - present case Eqs..:(131.1ii) and (B.liii)

_ . must be changed to f(2' i[v(q)-v(i- '5]

Ccic)-+(2r,i

, SC-L,'T)e, -B(2' r r y ...

4

B(-2..-r)e ....

A.v.14 If --

0-.: z

.-.

T, ,,..c ir DT' BC2'',7?.i. + f.(214.T')KIT')z .7t =

IA1(8

,...c;K)

(7.3ii)

i'...'

= -Z exP(i[(v(i)-v.(::-'')i},+

7 .cf i hi(t:- )-y(T'' )1)(17.0

v(T)

_"

_I

_("

To

7:.

-Z "excif1lv(T),,,,(i,;4)

1 + 1ex, i tv"Cr

As no analytiCal, solutions have beet. "found

this' case is not clevelOiSed-fUrther.

8. CONCLUSION

A controllable-pitch propeller having variable forward -speed U(t)' and rate of rotation w(t) has been

investi-gated. The approach is based on an asymptotic series in the inverse aspect ratio, three-dimensionality entering

as a correction to the two-dimensional results.

The theory has several limitations. One makes the customary assumptions regarding: incompressibility and inviscidness. The rate of change of pitch is such that the boundary condition can be linearized. Due to

linearization

leading edge suction is neglected. The number of blades must not be too large. The propeller is not allowed to cross its own wake nor its immediate vicinity, or to be more exaxt, there must exist one-to-one correspondence between

I

and t, and T and t respectively. The solution is not valid in the tip or root regions.

Three types of governing hydrodynamic equations. and. _

solutions exist depending on the rate of change of the incoming flow at the lifting-line. It is also possible to treat the case where the type of the governing equation changes at the time tr 0. However, this type of flow_

results: in cumbersome. formulae. By making appropriate assumptions the equations for lift and moment become simple enough for the effect of various- parameters on performance to be easily estimated. In short-time calculations three-dimensional effects are of order A compared with the primary effect.

.1'he' author expresses- hit rhaniCs.

PriSleS"Sc-r V.

Kosti-

--lainen, Mr. P.; Hervala, Mr. .V.-P. Pel2tola, and thc staff

,

of the Department Of;';MeCai-iiaal Engineering of Hels'inki''

1...Inivef,S1t-Y of Technology i: fbr their advice during the

f nishing stages .:."OT,:preparihg the manusc_pt

Also ,my-:

thanKs..,are due to Miss E. Heap- for ,correctihe,the:-English.'

'

Itext of ttie:inanuscrIpt,M

LafaXSio ancl,,Miiisi-f I. Haiehius

'fcir, the typing, and Mr

_{. ,}

R: iurimo 'for drawing the._:figur,es.

,

.

The financialsiippert, Of

he Acaderi

of yliilan9 has

REFERENCES

1 Abramowitz M. & Stegun I.A., Handbook of Mathematical Functions. New York, Dover Publications, Inc., 1970.

2 Bradbury T.C., Theoretical Mechanics. New York, John

Wiley & Sons, 1968. 641 pp.

3 Brockett T., Propeller Perturbation Problems. October

1972. Naval Ship Research and Development Center, Ship Performance Department. Report No 3880. 122 pp.

4 James E.C., A Linearized Theory for the Unsteady Motions of a Wing in Curved Flight. 1973. Naval Ship Research and Development Center. Report No 4098. 27 pp.

5 James E.C., A Small Perturbation Theory for Cycloidal

Propellers. California 1971. California Institute of Technology. University Microfilms. 71 pp.

6 James E.C., Lifting-Line Theory for an Unsteady Wing as a Singular Perturbation Problem. Journal of Fluid Mechanics 70 (1975) 4, pp. 753...771.

7 Johansson B.C.A., Lifting-Line Theory for a Rotor in

Vertical Climb. Stockholm 1971. Flygtekniska Fors5ks-anstalten. Meddelande 118. 120 pp.

8 Kochin N.E. & Kibel I.A. & Roze N.V., Theoretical

Hydro-mechanics. London, Interscience Publischers, 1964. 577 pp.

9 Ogilvie T.F., Singular Perturbation Problems in Ship Hydrodynamics. Eight Symposium Naval Hydrodynamics. Office of Naval Research - Department of the Navy, Arlington, Va., 1970, ACR-179, pp. 663...806.

10 Rubis C.J., Acceleration and Steady-State Propulsion Dynamics of a Gas Turbine Ship with Controllable-Pitch Propeller. Trans. SNAME 80 (1972), pp. 329...360.

11 Sparenberg J.A., Application of Lifting Surface Theory to Ship Screws. Koninkl. Nederl. Akademie van

Weten-schappen-Amsterdam, Proceedings, Series B, 62 (1959) 5, pp. 286...298.

12 Van Dyke M., Perturbation Methods in Fluid Mechanics. New York, Academic Press, 1964. 229 pp.

13 van Gent W., Unsteady Lifting-Surface Theory for Ship Screws: Derivation and Numerical Treatment of Integral Equation. Journal of Ship Research 19 (1975) 4, pp. 243...253.

14 van Molten T., The Computation of Aerodynamic Loads on Helicopter Blades in Forward Flight, Using the Method of the Acceleration Potential. Delft 1975. 110 pp. 15 Wu T.Y., Hydromechanics of Swimming Propulsion. Journal

of Fluid Mechanics, Part 1, 46 (1971) 2, pp. 337...355, Part 2, 46 (1971) 3, pp. 521...544, Part 3, 46 (1971) 3, pp. 545...568.

"-(B.1i)

.7,

.

iy(.7!)

(B.1ii) ,iA' (t;or) = B

f-

[

-05

13.(t,.;) a?

-A : (B.liii) ' eicpfi(y(c)=y(q')]}:+-1-eii[Y(-7 ) . 0 77( 0 - . 17iic(T)2

f ,K(T.)d-r

, = A

₍₈

. .1 .,' 1 n? z,r) iA., (-14. 04') - oc) z' )

The substitution of Eq: (B.2) into Eq; (B:1) `reililts in

-a integr-al equ-ation for the determin-ation of% A, (i-oC) or a (toc) /13 p. 8/.

TAPPENDIX C Inner solution for Case (c) for

3F

..aw

aw

:

11 di du

-The problem to,be_solved is defined by the Equations ,(4.9i),(4.21)...(4.2P with Nm(T)) = A7"

'and.

i >:;.1:,-' which correspond to Eqs. (16)...(22) of Ref. 115/. In order

to

simplify the computation, the nomenclature of Ref

/1,51 is adopted in this appendix These symbols, used by .

-Wu, are not inrinded in the notation list. Also to faeili-tSte compariiSi6n the number of_{_}

the

original equation is,

. .. given,additionally.' Those formulae that .do not differ in principle from the original ones are not discussed bere.

. . . . ' _ .

(C.6)

The derivation is modelled on Ref. /15 pp 3'4'4

,

taF(z,u+Tr)

_aw(z,u+Tr)

_* aw(z,u+T )-r

. u

a z

= j.e-suF(z ,u)du 0 ST 0, _t

_a

_.'r le r

I

re-s-F(z,T)dT]

= re

dz dz T r, ST m

w(z,tr)4

ser

e-ST

w(z,T)dT -, _ e: W(ziT)ch]

It Is assumed that m(y) = 0 and F(y)=-0-for ' '.7.(

C't .

co d

-aki e

0

s f

e-ss7w(I,y)dY 0 -. "

F(z,y)dyi=lm44 eA r -sYw(z;y)dyl- "

X

-ST

_,

4

e vtx1'y'Tr)dx1 d

-ST

(C.7)

F(z,$)= s)w- e rwCz,Tr),

T > T

dz dz

- r

(27) x

(C.8i) 4/(x,y,$)=. -v(x,y,$)- s v64,y,$)dx1

-1 S(X1+1)

-ST

r =

f

e e.

v(x1'r)dx1

-1 s(x +1) 1

-+ s le

T(x1' - 0+I s)dx1 .

There are no changes in Equations (30) and (31) of Ref. /15/ when applied to the present case. Of course i(E,O,$), :i0(s), and

ii(E,$)

_{contain different terms.}

x (C.83,1) = +41(x,y,$)- s e ¶(x./ -STr s(x -x)₁ v(x1,y,Tr)dx1 (C.91) i(x,0±,$)=71"1(x,$)+A0(s), 'xi <1 where -v(x,0±,$)=V(x,$) on the -plate y = 0/,

lx1 <

1

(x .)= -(2_9.01 .i(x

_,

_ax

_-I'

_xi

X

-ST

+

f

e _{rv(x1,0-1,Tr)dxl, 1x1 < 1} -1 (CAidi) A0(5)-7- -s

f

v(x1'0±s)dx1+

f

e rv(x1'0±'r)dx1

, . .. ., '': -,.- . . . ,.--,-,2,,

,

.. _ '.. ;,-,,, , , . :

-

1 -

317"r"

(C.10i)

41(X, 0±,S )

=',A0(S)-. -I 1

1- x -1

*1(E's)

-71 1-E E

-1

.(C.111ii) sJ es_ (x4'1);(x,0±,$)dx.= A Cs)

-1

.s(x+1)

-ST

_r

4

-e

v

_{l' -9. r}

)cbi

-

₊

A (s)= a CsIIK0 (s),+-,K(s)/se

se

sx

1 (x4.' ST

₁

-C.101i1) 0 =. f SS

'e

rv(x 0-±

_9.

_r

)dx+.a-(s)I1C.(i)l-K-(s)1ses

_u

s

"LE-2 1 {EK

( s w '7a1, ' _ -

1.

f esx{

1

E-x _

.

,0

)dx

. ' , =

[E-H(s)(14rIvr

1

(32a),

2e

-7.11'

e

c-' ''' S 11( (s)+:K.:(s)].,==,

,-..

.1 ,

V(E s)

J---0F.-y(rot,T gi

e-57-s[1(0(s)+1C(s)]--.1 .,,

r -

_{.1 147:1- 37+} ...

'

I

_-

_-

,--V2.-1, i [E-H(s)(1+V

E)E,S),

-

icig.

...

-STr

2e 1e

s'[1(0-(s)+K1(s)]

-

lyt(x,t):=

u

(37)

(C.15)2

(3'3)

(t)=

-Oheere -1,-,1U(t)

I

,e

-Similarly -the Equations (33) and ( 35 )... (41) reMlaixi

Unchanged.

Only

a0(t)

'changed.

v(E;0-1,T=)ff:TrH-, (T-T')

_r,

. E+i=;.

(C.14)

' (i)

= _.Px1)(sT)

ds

-

Tr

s[-Kti(s)+Ki'.(s)]

/77

. .

j_ao(T)

. . .

,

v(E,0±,;Tr){1

dy}dZ

Y+11'4 17 1

.;

. 1?-( n1(T,t))1-1(V7T,')d-re.+:151(..T)

(a a ) PLANE

I 1"

I