On the use of lifting surface theory for moderately and heavily loaded ship propellers

ON THE USE OF LIFTING SURFACE THEORY FOR MODERATELY AND HEAVILY LOADED

ON THE USE OF LIFTING SURFACE

THEORY FOR MODERATELY

AND HEAVILY LOADED

SHIP PROPELLERS

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE

HOGESCHOOL DELFT, OP GEZAG VAN DE RECFOR MAGNIFICUS PROF. IR. L. HUISMAN, VOOR EEN COMMISSIE AANGEWEZEN. DOOR HET COLLEGE VAN DEKANEN TE VERDEDIGEN OP

WOENSDAG 11 MEl 1977 TE 14.00 UUR DOOR

WILLEM VAN GENT

NATUURKUNDIG INGENIEUR

GEBOREN TE AMERONGEN

Dit prôefschrift is goedgekeurd

dOor de ptornotôt

Prof. Dr. Ir; L D. van Manen

CONTENTS

GENERAL INTRODUCTION. . 1

PROPELLER GEOMETRY AND HYDRODYNAMICAL BOUNDARY CONDITION... 4 1.1. Introduction

...4

1.2. Mathematical förmulation for geometry f thin blades.. 4 1.3. Hydrodynamical boundary

condition...

VELOCITY FIELD INDUCED BY PROPELLER LOAD 16

2.1. introduction 16

2.2. Fundamental equations 17

2.3. Steadyload on actuator disk 21

2.3.1. First approxïmatión of flow fIeld due to

external forces 21

2.3.2. Approximations of the efféct ôf secòndary

forces 27

2.3.3. Remarks abôut flow through disk 35

2.4. Steady load on rotating lifting lines and surfaces... 39

APPLICATION OF LIFTING SURFACE THEORY

...

44 3.1. Introduction

...44

3.2. Formulation of integral equation añd solutiòn methOd 46

3.3. Slipstream rotation an contraction 51

3.4. Flow around blade sections 52

3.5. Corrections to blade load and onset flow 54

PROPELLER ANALYSIS AND CORRELATION WITH EXPERIMENTS 56

-' 4.1. Introduction 56

4.2. Effective axial velocity at propeller blade 58

4.3. Effcts of some model components 60

4.4. Blade load distributions 62

REVIEW AND CONCLUSIONS 70

APPENDIX A : Calculation of radial velocities at a

REFERENCES 84 LIST OF SYMBOLS 86 SUMMAR 89. SANENVATTING 90 DANK WOORD 91 LEVENSBESCHRIJVING 92

GENERAL INTRODUCTION

The design of a screw propeller ànd the analysis of its perfòrmance are important subjects in the field of ship hydrodynamics. Therefore a ict of work has been carried out on f rmulating an adequate

mathematical model fòr it. The flow in which a ship's propeller operates is non-unifòrm, unsteady, turbulent and affected by

-gravity. The difficultths met in describing this flow are limited by the apparent incompressibility of the liquid,

but they are complicated by the occurrence of gas and/br vapour filled cavities. A much simpler flow is obtaiñed, when the propellèr on its d±iving shaft is disengaged from the ship and placed in a uniform, steady, axially directed flòw in

which viscous effects are confined to boundary layers and in which gravity and cavitation are disregarded.

Even for this simple flow the description of the action. of a screw propeller has not yet the appropriate level of specification to allow-for exact calculations. Hence it is nbt surprising that the simple case râther than the complicated case has engaged many investigators. For the understanding of the complicated case the knowledge of the simple câse is fuñdamental. Hence also fbr that reason the latter still deserves attention.. For the simple flow model the name "propeller in open water" is âurrent.

The description of the flow of a propeller in open water is, apart from the propeller geometry, governed by only one parameter, viz, the ratio between the translation velocity in axial direction and the rotational velocity. Of all possible values

of this parameter, those corresponding to cases in which a thrust in the advance direction is developed are of main interest for the propulsion of ships. In this thrust producing domain of the

ropelÏer operation three main types of loading are distinguished The division into resp. light, moderate -and heavy loading is based on the magnitude of the flow velocities induced by the propeller action.

-1-For lightly loaded propellers the induced velocities are assumed to be negligible, as far as they. react on the blades. This model of the flow has been 'useful for the study of optimum propellers.

For the model of moderately loaded propellers the induced velo-cities are derived from the velocity field of a number of

helicoidal vortex sheets,. each trailing behind one of the propeller blades. At a constant radius the vortex strength in the sheets and the pitch of the sheets are constant in downstream direction. The vortex strength is related to the load distribution on the blades, but the radial pitch distribution of the sheets is not determinate. An obvious estimate for this pi.tch is the hydro-dynamic pitch,i.e. the pitch of the relative streamlines in the neighbourhood of the propeller blades. This 'choice leads to a good correlation between theory and experiment. In the induction-factor method a simple, but highly accurate, computation scheme for this model of the propeller wake has been developed. In combination with a lifting-Line representation of the propeller blades this model is widely used. Its application requires, however, knowledge about the lifting properties of the propeller blade sections,suppÏemented by influencing correction factors associated with the complicated propeller geometry. These cOrrection factors have to be derived from a lifting-surface representation of the propeller blades. The lifting-surface representation can do without 'fore-knowledge about the lifting properties of the blade seçtions, which is an obvious advantage

o this representätion in comparison with the lifting-line representation. A disadvantage is the relatively long time re-quired to work out the computation schemes for lifting-surface methods.

It is assumed that in the flow around a heavily loaded propeller the foregoing picture of the wake flow is insufficient. It. has been

observed, by theoretical considerations and by experiments,that the wake vortex sheets exhibit a radiaL conttaction,that the pitch of these sheets is not constant and that in the wake a reduced pressure level exists. Moreover the. effect of the radial component

of the indùced velocity, which is neglected completely in the model of moderately loading, has tobe considered. The study of the deformation of the helicoidal vortex sheets and its effects on the induced velocities is a difficult one. The formulation of the hydrodynamical equations of motion is complicated by the

instability of the vortex sheets. The transition region in which the vortex sheets roll up is located quite close behind the propeller. An establishêd model för this type of wake flow, by definition associated with the heavily loaded propeller, is not yet available. In this study a model, -ccounting for the main effects, will be investigated.

The helicoidal geometry of the propeller blades and the consequent shape of the wake are inherent difficulties in the formulation of an exact methematical model for the flow around a propeller.

It is often tried to eliminate these difficulties by consideration of the circumferential mean values of the quantities which are period-ic functions of the angular position. An approximate model

for, the mean values is obtained by the concept-ion of a propeller

with a large number of blades. In the limiting case of an infinite number of blades thé circùmferential variations disappear. This

representation of a propeller is called "actuator disk'. It has been proved only for the lightly loaded propeller that the flow behind an actuator disk is équivalent with the circumferential. mean flow of a propeller with finite blade number.. For moderately and heavily loaded propellers the exact equivalence is unlikely, but it might be a good approximation when the equivalence is

assumed. In this study it is -postulated at first that the characteristic effects of a heavily lòaded propeller model can b derived from this approximation. Estimates for these-effects, obtained in this way, -are combined with a lifting-sur-face represen-tation of the propeller. This model is verified by correlation with experimental results of systematic propeller series.

-3-/

1. PROPELLER GEOMETRY AND HYDRODYNAMICAL BOUNDARY CONDITION.

1. 1. Iñtrôduction.

/

In this chapter a description, of the geómetiy of' thii propeller

blàdes is given. The fbrmulation incorporates rake, skew and variable pitch'. The kinematical boündary condit10 at thé prôpelle blade in a flow field is formulates inclusive the radial velocity component. It is shown that, when camber and thickness of the blade sections are considered to be small perturbations of a réferende plane, a first, order approximation leads co two equations, of which

one relates the flow field to the thickness distribution and the other relates the flow fieid 'to the camber distribution.

1.2. Mathematical formulation for geometry of thin blades

For the description of the geometry of the propeller blades we use a reference plane built up of helIcal lines and given by the mathematical function F

F = q>

-

a(xzR) + wt = (1-1)

in which a and _ZR can be functions of, r

x, r, q>) ïs a set of cylindrical _{coordinates'; the poitive} x -direction is oppôsite to the axial propeller motion. Looking into thé positive x-direction the positive q> -direction is-clockwise and q>=O ata radial line directed vertically upard.

wt _{is the angular rotation with reference to the position at}

time t=O. For positive angular velocityu (right-handed propeller) the direction of 'the ròtation is opposite to the positive q> -directiön.

-5-DIRECTION OF PROPELLER

TRANSLATION

DIRECTION OF PROPELLER

ROTATION

Fig. 1. Coordinate system for description of propeller geometry.

This notation is illustrated in Fig. 1. The meanin of a and _ZR becomes clear, when the intersection of the reference plane F=O with an axial cylinder of radius r is.corisidered. The line of

inteEsection is a helical line with a pitch angle defined by

tan

=

=-- (F/q)/(3F/ax) = 1/ar. (1-2) The pitch is simply related to a according to

p 2irr tane = 2î/a. (l-3)

This relation shows that when a = constant,a constant P is recovered.

The rake ZR describeS the shape of the so-called generatorline,

which is the intersection of the reférence plane F0 with the

(1-4)

ZR is the distance between the generatorline and the plane x=O,

which is callèd thé propeller plané. A sÙrvey of the recommended nomenclature for the description of ship propellers is given by Cuiriming /1/.

For a propeller with z blades an equal number of reference

planes ¿an be defined by substituting in eq. (1-1)

k for which aré related by

k

4-k2/z

, k=b(l)Z-1 (1-5)

For the description of the propeller blades they are also inter-sected with the cylinder of radius r. Whenthis cylinder is

unrolled the picture given in Fig. 2, which is taken from Ref. /1/, is obtained. The helical reference line becomes a straight line; its pitch can be chosen equal to the pitch of the iose-tai1 line of the blade section. In this way the blade section is situated. between two points on the helical line viz, the leading edge

(x , ) and the trailing edge (xt

'

). This coordinates are still functions of the radius r and describe the blade contour.

We will consider the propeller blades as thin lifting surfaces. Consequently the blade section shapes reduce o cambered méan lines..

The following derivation, however, also holds for blade sections with thickness if it is applied to the upper side or to the

lower side. This feature will be used later on to f-md a

cor-rec-tion for thickness effects.

The camber is measured in the direction normal to the heiicàl line. The mean line is described by adding to évery point of the hélical line, between the leading and trailing edge,ä point accor-ding to

TOTAL RAKE Zt BLADE SECTION AT RADIUS r BLADE REFERENCE

.

r9

BLADE ROOT SECTION

PROPELLER PLANE PLANE OF ROTATION RAKE ZR SKEW-INDUCED RAKE

-GENERATOR

- PROPLER

-

- LINE (PROPELLER PLANE)

VIEW OF UNROLLED CYLINDRICAL SECTIONS AT BLADE ROOT AND AT ANY RADIUS r OF A

RIGHT HANDED PROPELLER (LOOKING DOWN) SHOWING RECOMMENDED LOCATiON OF PROPELLER PLANE

REFENCE POINT OF BLADE ROOT SECTION

PROPELLER HUB

-7k-FORWARD

STARBÒARD-'.... PLANE CONTAIÑING SHAFT AXIS AND PROPELLER REFERENCE LINE

PROJECTED BLADE OUTLI E TRAILING EDGE PROPELLER HUB

r

REFERENCE POINT OF BLADE ROOT SECTION AND PROPELLER

REFERENCE O (PITCH ANGLE OF

LINE SECTION AT RADIUS r)

HELICAL LINE

INTERSECTION 9F GENERATOR LINE AND PLANE AT RADIUS r

PROPELLER REFERENCE

LINE AND GENERATOR

L INE BLADE ,REFENCE LINE (LOCUS OF BLADE SEcTION REFERENCE POINTS) HOE POINT OF ROOT SECTION SHAFT AXIS LEADING EDGE FORWARD - STARBOARD-DOWN DOWN

Fig. 2. Diagrams showing recommended reference lines and terminology.

INTERSECTION OF BLADE REFERENCE LINE (LOCUS OF

BLADE SECTION REFERENCE X 8s SKEW ANGLE

where ( x,r, ) are the coordinatesôf the helical line and

p 0) are the coord-inates of the mean line.

The camber f is a function of the position ketween the leading

edge and trailing edge, which is given by the parnetêr s

s

= (x-z,)

sine + r cose . (i-7,)

s lies within the range f-rom -c/2 to +c/2 , whére c is

the chord. _ZT and are the coordinates of the midchord point on the helical line. They are related to éach othet via eq.(1-1):

- a (zT_zR) wt =

ZT is called the total rake. When a skew angle is introduced,

ea(zTzR), we can write

ZT = ZR + 05/a =

ZR + POS

tn

(l--C)

>t)T = Os -

wt-The mathematical formulation of the mean line sections of the lifting surfaces is obtaine1d by means of eliminating the

coordinates x, r and q> from -eq. (l-1, 6. and 7) . The result is

p? ₌

_{O + wt -}

_{(_zR)/(ptan) - f/(psin) = 0 (1-9)}

where

f

=

f

(s,p)

.

(1-IO)

and

s = (zT)

+ p(8cT) cos

A useful relation, which has an obvious geometrical interpretation, can be derived from eqs. (1-8, 9 and 11) and is given here for refe-rence

ZT s sine - f cose' (1-12)

The set of eguation (1-8, 9, 10 and 11)constitute the complete

formülation for the lifting surfaces, provided that the functions for rake zR(p) , skew angle

05(p)

, pitch angle c(p),

chord c(p) and camber f(s,p) are specified. As stated beforethis set of equations can also be used for blades with finite thickness. In that case the function f(s,p) is a double-valued function of s, thus specifying the upper and lower surface of the blade.

At the leading edge and the trailing edge f is not necessarily

zero.. If _t f, however, the lenth of the projection of the

nose-tail line on the helicallirie differs from the chord. In practice this difference is negligible.

1.3 Hydrodynarnical. boundary condition.

When a propeller is rotating in a moving fÏuid, the velocities of the blades and the fluid have been matched. The blades have to move continuously with the fluid and can be considered as a material surface whih properties do not change in time. This means in mathematical terms that the value of the' function describing the blades, does not change. The material derivative of this function should be zero. Let the vector with

components U, V and W in the (X, r, ) set of cylindrical

coordinates describe the velocity field, then the boundary condition is

+ (i.grad)F = O (1-13)

-9-or + .

q-;

+ w..Ç o where: 1

af3s

(A)

3t

psinc' 3s 3t

3F_

i i 3f 3s T - p:an psin 3s 3E R p dD f {tan4 +-

} +

2 2 2 dp

2.2

p tan. cos p sin

d4 dzR

Csin + pcos

-}+

r

3p

as

ap

i i 3f as

- p

psiri'

3sp36

These derivatives càn be evaivatèd fürther by üsing thè equations (1-3, 8; 9,11 and 12). Substitutïon of these expressions into eq. (1-13)

and multiplication of this equation with psin finally

results in

{VN + GN VR } - f V,.+ GT

3f

(1-14)

where: VN, VT and VR are newly defined velocity components according

to

= (wp +W) sine - U cos', (i-15)

VT = (wp +W) COSAA + U

Sjfl'

(1-16)

(l-17)

3

Bp

while GN añd GT are expressions,whïch depend only on the geometry of the propeller blades

s dz 8 +-cosi

-GN

{R

S

2ir - cos +

sin2-H

(1-18) dz_R 8 de

sdP

.s p s 2' = - {- + ---}_dp si.n

- {=

- cos 2iî dp dp cose p + {2sin -

£!

(1-19)

The meaning of the velocity components VN and VT is illustrated

in Fig. 3, which gives a vector diagram of the velocities in

the unrolled cylinder plane-of radius p. The effect of the ve'locity component VR is neglected in this picture. It can be seén that VN and VT are the components normal to the helical line resp. älong the helical line;

Fig. 3. Diagram of velocity components in unrolled cylinderplane.

1-For the interpretation of eq. (1-14) as a boundary condltiòn at the propeller blade, it has to be realized that it has to be.

fulfilled at both sides of this surfacé. The velocity components consist of two parts viz.. the velocity components of the flow

field withoüt propeller and the velocity' components indúced by the action of the propeller.

Whènthe propeller blades are approximated by lifting surfaces without thickness the induöéd velocities can have discontnuities

across these. surfaces and across vortex sheets in the flow field.

The corrèct aproximaton will be derived from eq. (1-14) by

star-ting from the case of finite thickness. This is done by substitustar-ting

for.f

where f =f(s, p) is the

camber and

is the thickness.

The geometrical expressions GN and GT become

GN = + g _{± ½} (1-21) = + g ± ½ (1-22) where

f= f

C

½

f

t s dz e +-cos cose (i-23) fc, 2 g = sin -Nc p 2 =

sin

--p ap ' dz e a ₊

-}

sincl -dp

2Tdp

dB S .P dp cos4'

(1-20)

(l-25) s -2 Cos p (1-26)

af

ap C (1-24 )

dPCOSC

{.2

sin-¡i- }- cost

dP cosc

{2 s1nc_a _.- }- cos

In this way GN and GT are split into terms which depend resp. on blade contour,, camber and thickness.

Application of e. (1-14) to the upper resp. lower surface of

theblade and addition or subtraction of thesè equations leads to

(v±Ç)+(g+g) (V±V) +g t(VV)_ Uv±v)+(g+g) (V±Ç)

+ - + - + -

+-+½g(VV)} --

- =0

In the description of the geometry the camber and the thickness are small parameters. A fitst order approximation

of the above equations is

(V+gV) -

, (1-29)

tVR) - (V+gV) 0

(1-27)

(1-28)

(1-30)

where V = ½(V+V) and AV=V+_V for the three components and i.t is assumed that VT« V, and _{VR «VR.}

This approximation is not valid at the blade edges when the section shape is rounded there.

It is consistent with the first order approximation to treat eqs.

-13-(1-29 and 30) as boundary conditions to be f ifilled at the elical referérice plane instead of at the blade surface; the velocities

VN? VR and VT are average velocities at the réference plane, the velocity jumps VN arid VR are discontinuities across this plank.

Eqs. (1-29 and 30) are two conditïons in which the camber effect and the thickness effect are separated. Only the velocity

components occur in both equations. For eq. (l-30) it is easy to

find a satisfying hydrodynamical model. (V+V)

are resp. the velocity and the velocity jump in the direction normal to the reference plane. Eq. (1-30) specifies the velocity jump in relation to the thickness distribution and the average velocities VT and VR according to:

= (1-31)

It is weflknown that the flow field fulfilling this condition is given by a source distribution over the reference plane

with a strength equal to the velocity jump. So he right-hand side of the above equation is the source distribution. In general, however, such a distribution also has a continuous normal

velocity component.. This component is one of the velocities,

induced by the propeller action, which has to be taken into acóount in eq. (1-29), and can be considered as a thickness correction.

In eq. (1-29) only continuous velocity components occur in

combination with camber effects. To satisfy this equation, together with a flow condition at the trailing edge, a vortex distribution is required. This contribution

is related to the flcM field in which the propeller is placed.

The velocity components in eq. (,1-29) have to be split in velocity compòriènts of the flow field without propellér, velocity components

induced by the thickness of the blades and velocity components induced by the lifting action of the blades. When these components

tane = Q af.

where tan =

C

In the foregoing derIvations the radial velocity component is taken into accont ifrespective of its magnitude. The coefficients

and g of the radial velocity can be of order one for propeilers with rake and/or variable pitch. It ha already been noted by

Brockett/2/ that also in the case of no rake and constant pitch, however, the latter coefficient is not zero. In /2/ a strict derikriation of the boundary condition is given in terms of the velocity

potential. Here we have used the velocity components, which are

requIred when we develop calcuation sòlìemés.

-15-{(vÑo+vNt+VNÏ) +g +VRt+Vi)

(1-32)

2. VELOCITY FIELD INDUCED BY PROPELLER LOAD. V

2.1. Introduction V

In this chapter we will derive the relations which exist between the three vector components of the velocity field and the lifting forces on the propeller blades. Two methods are

in use.

The first method is based on the knowledge that there is a direct relation between the lifting forces and the production of vorticity. This vorticity is distributed over a narrow region downstream of the lifting surfaces, whidh is called the wake. From a specified vorticity distribution the asso-ciated velocity field òan be calculated by means of a vector relation, which is a generalisation of the law of Biot-Savart for a line vortex.

In the second thod integration of the differential equations for the fluid motion is pursued. To handlé the non-linearity of these equations the non-linear terms are partly combined with:.

the pressure and patly considered as induced forces. The

combination with the pressure is called the àcceleration potential. The solution can be obtained by means of successive approximations. The principal scheme for this method was already given by Von Kârmáh and Burgers /3/. To investigaté how this method can be used for the propeller induced velocities first an idealized ôase will be treated in this chapter. This case is an actuator disk for which the associated velocity field is steady. This model is a representation of a propeller with infïnitely many narrow blades. To allow for the effect of a finite number of blades, the results of the steady actùator disk are modified. The load is no longer constant in circumferential direction, but

is concentrated in narrow regions on the disk. The positions o these regions rotate with constañt angular velocity. In this way a lifting line representation of the prcpeller blades is obtained. The wake of thèse lines are trailing heliäòidal sheets. In a lifting surface representation of the propeller blades the chord-wise distribution of the load is accounted for, but the wake is V

2.2. Fundamental equations

We start with the equation of- motiôn for a frictionless'

fluid under influence of external forces:

+ (U.grad) U

-

grad p + (2-1)

where p = pressüre,

= force vector,

fluid specific mass,

t = time,

U = velocity vector.

Let the velocity of the fluid be composed of a steady and constant component U in axial direction and an additional component

U =

(uy, Ur u), so the total field is descrïbed by the

vector

U =

(U+u, Un u). For an incompressible fluid, this

field satisfies:

div U = div U = O. ' (2-2)

Using also the vector relation

(U.grad)

U =

grad (½u2) - U x rot

U,

-the equation of motion can be trañsformed into:

+ U =

-f-

grad (p+½u2) +

f-

+

(Ux

rot ii).

It is convenient to define the following new variables:

q = p + ½Pu2 (2-3)

= (U x rot u) , . (2-4)

-(2-5)

with which the equation of motiOn becomes

X

w (2-6)

In this equation we see on the left-hari side the linear acceleration terms, while on the right-hand side twO terms representing forces 'occur:

grad q _{is a rotation free force and.} is a generalized force composed of the éxternal force and

the secondary or induced fórcé

When we take the divergence of eq. (2-6) and apply also eq. (2-2) we obtain a relation between q and

div grad q = V2q = div . (2-7)

A solution of this Poisson equation is

f

dG k

f

dG

(L

grad (2-8)

The integration extends throughout the region G where

the force vectot is non-zéro. The equality of both integrals can be derived from the vector rélation:

f dG (Lgrad U = f dS

(L) -

fdG

div A

G S G

when S is the bourfding surface of the region G and is the local normal to this surface, pointing oútwards. Substitütion of

K =

i, with 1=o on S, and = proves eq. (2-8).

In e. (2-8) R is the distance between the point (x, r, ),where q is calculated,and the variable point (,p,e) of the integration

i U w X f dx' q {x' - 19 (2,-15) R2 = (x-)2 + r2 + p2 - 2rpcos (e-e). (2-9)

For R the notation R(x;) w±ll be uséd,. whére x resp. indicate the positions of the two points invOlved in the expression

Inthe regions where the fOrce field i is zero, we see from

eq.. (2-7) thàt q satisfies the Laplace equation V2q = 0. In this case the acceleration terms in eq. (26) are derivèd orn

grad q and thereforé q is called he acceleration potential.

To find à _{olution for the system of eqs. (2-6) and (2-7) it} is convenient to split the elocity field in a rotation free component and a component .

Writing

grad . (.2-10)

and

=+

. (2-lI)

and substitûting these relations into eq. (.2-6) leads to an

equation for which a solution has been suggested iñ /3/., by separate solution of

at ax

p

(2-12)

and

}+U_.q

₍₂₁₃₎

In steady cases, when and are zero, añd can be

derived f-roth iñtegrai expressions

It is assumèd that O for x'-,.--.

To find the velocity field i = + grad ,when the external force field is given,suçcessive .approximations can be used.. the first approximation the secondary forces are neglected. We then have k1 =. ,according to eq. (2-5), while the

integra-tions in eq. (2-8) and eqs. (2-14, 15) can be performed, resulting in the first approximations

,

q1 and.1.

For the second approximation eq. (2-4) yields the secondary

forces g2 = (

xrot

) and with k2 = + g the

inté-grations can be repeated. An example of a steady case is treated in section

2.3h

where it is observed that, besides a usefül first approximation, also general relations can be derIved.

In unsteady cases, integral expressions similar toeq. (2-14) and (2-15) can be used.

t =

- f

_dt' i {x-U(t-t'),t'} i {x',t

.L}

i t

= -

_--

_f

dt' q {x-U(t-t'),t'}

f

dx

q (x,t

In section 2.4 an uns.eady case will be considered.

(2-16)

2.3. Steady load on actuator disk.

In the áonCeptiOfl of a steady actuator disk the external force field

is confined to the region inside a circular cylinder with outer radius r and with a small length . Let the axis\of the cylinder

be in c-direction. The axial, radial and tangential components of the force vector can vary in radial direction, but are constant in circumferential direction. The distribution of the

force components in axial direction is such that in the limiting

case, where the length L' becomes zero, finite values for the

integrated force components remain. At first it is left out of consideration in whiôh way the forces are generated. Later on the relation to infinitely many lifting blades is introduced.

2.3.1. First approximation of flow field due to external forces.

For the first approximation the generalisedforce vector is equal to

i1 =

+ f 6I

+ (2-18)

The force components are written as products of the factors -and , which depend on the radial position only, and the factor 61,giving the axial distribution. This distribution is such that always

Jfó1dÇ = f

JcS1dF

= f,

(2-19)

-A/2

-L/

even in the limit

From eq. (2-14) we obtain as a first approximation of the vector field w:

for p > r and for p

r,

<-L/2 0, (2-20)

-21-for the region inside the disk far p<r,>L/2 1

=_JUE

f

6d

w -1/2 '

pU

w (2-21) (2-22)

The region downstream of the disk is called the wake.. It ca?I be concluded that only inside the disk and the wake the velocity

field is affedted by the extérnal fbrce field f. In the limit A-*O this velocity field is discontinuôus at the disk. As

div i=ü, thé irrôtatïònal fièld must have a discontinuity o opposite sign.

Heréafter we will refer to the components of separately ànd use the notation:

= w1 I+w1I+w01I8

The velocity potential from which is derived will be considered for each vector component of the force ? separately. The symbols q and are given the corresponding indices. According toeq.

(2-8) we obtain for q:

div(f 011)

=_f

dG G 4TR(x;) D div(f O

I

q1=_JdG

pip

p 4iîR(x;) div(fOiI0)

= o,

= -J

dG q01 4TR(x;) - GD (2-23) (2-24) (2-25)

/

=

dS

I:

R(x;)

o

where SDis the circular disk surfäce.

The corresponding v1ocity potential field is obtained from eq.

(2-15)

using

_3

i 3 i

R(x;)

3x

R(x;)

This result can be interpreted as the potential of a sink dis-tribution of strength f ¡p U over the disk surface S . The axial

D

velocity component at the disk is:

I3

3 1 + =

_4ïU

_J

dS

R(0)

-x=0 x-*O w D

for x-O resp. +O

1(x,r)

_J

4TTPU s - 23 D (2-27) wheré is thevolume of the disk..

The thIrd expression is zero, because the force field does nôt vary in circumferentiàl direction. Consequently there is no corresponding velòöity potential according to eq. (2-15)

ei

= (2-26)

For a point oùtside the disk(R O) , êq. (2-23) can be further

evaluated:

1

d6d

O Ô

-A/2

-*

(u )-.

where the iñtegral over C is an integration along a circle with radius p at tlie axial position = constant. For further use we recall tkat /R3 = grad (1/R)

Acóording to a variant of Stokes' theorem the following relation

holds:

Jdciix.Ä}= ¡dS{(.ixV)xA}.

Substitution of this relation with

A =

grad and use of a rule for vector triple products

(xV)xV =

_{V{.V}-CV2 Ì'}

yields (*) = 4rPwU

J

J

dp(_)JdSgrad(

R(x;)

_(2

R(x;)1

(2-29) where we have taken fOr S a circUlar disk surface with radius p.

i XR r df o

__&)jdC_3

-=

¡d

dp (;_

dp _{4rR (x;)}

The jump in this velocity component is equal to the discontinuity found for thé axiàl componént of and consequently the complete field

()

w1i+ grad

(2-28)

is continuous at the disk.

It is usefül to shbw that this field can also be described by

a vorticity distributi.on in the wäke. The strength of the vorticity

- - i

in the wake is: rot(u)1

rot (w1i) = --

- i.

The associated velocity field can be derived from a general vector relatiOn

Patial integration over p is possible and when it is assumed-that

f = O for pr0 the result is:

=

Jd

f

dS

f

[rad(

R;d

(V2

R(x;F)]

o _SD

The first part of this integral can be evaluated by performing first the integration over . Further the operation grad

R(x)

applied at the coordinates(F,p,8) is the opposite of this operation

applied at the

ord.thates (x,r,).

Consequently the first part of the integral becomes:

4iîpU grad'

[

f

dS

= grad (2-30)

2 i The second part of the integral has an integrand i-n which V

_R(x)

is only non-zero when the points (F,p,8) and (x,r,) coincide. Therefore the value of the. integral reduces to

41TPtJ

J d

JdS f[V2.

R()1

pu

J dG

f[V2

R(x;E)

o _SD . G

= -h-

f iñside the wake

ì= (w1 i).

= O outside the wake

1-Eqs. (2-30) and (2-31) prove the equality of the vector fields and (), given in eq.. (2-29) resp. eq. (2-28).

Now we hav to proceed with the evaluation of eq. (2-24) to find the velocity potential field associated with the radial force in the disk. For a point outside the disk we have:

r q1(x,.r,) = -

-J d(-8)

¡äp (pf) Jd

R(x;).

O O 25 -(2-31)

or the 1ixiiting case t-O the result is

q01(x,r,4)=

dS

and

:1

_{J dx-J d}

-

I

dG

-

R(xO)

4irp c

R(x;.)

D w,

where G

is the volume of the wake. If we define a function

F

w p1

x

ld

by:

_d

(F )

= f

dp - pl

-p

with

F01

= 0 for 0

we obtain:

1/2F 1(x,r,)

_'4pt

pl Gw

where C

(2-32) -(2-33-)

To this expression the. Green's second theorem cän be applied with

the final result

=

4p'U

f dS

F01 +c

(2.-34)

D

= F

(r)

inside the wake,

pU

Pl

C _û

outside the wake.

The complete velocity fIeld associated with a radIal force in the

disk -is, according to the definitions in eqs.

(2-10, 11)

= grad

_pl

+ w

I.

(u)1

_{- 4U}

grad f dS F1

R(X)J

grad (2-35)

SD

The integral represents the potential of a doublet distribution

over the dik SD. It has to be noted that the field ()

,

is irrotational.

The vector field (ii) can alsn be described by an equivalent

distributiön of ring voi'tices with strength f /pÜI0, over the disk. Using the general vector relation

r

-p

=fdp

8:3

and using ägàin eqs. (2-28) and (2-29.) the equality (*)1 = can be proved in a similar way as has been done for

2.3.2. Approximations o.f the effeât of secondary forces.

Now we have derived expressiöris fOr the complete first approximation of the velocity field associated with a general steady force

distribution inside an axisymmetric actuator disk We recapitulate that this field consists in three parts

= i + grad

p1 = w i + grad 1=grad *il (2-36)

= w01 1

.

in Fig. 4 the streamlines of these velocity fields are illustrated. At the diskthe axial velocity is contihuous and equal to

UDWl+

].+1)=½ w1+

where f +

w1 =

= w1.

-27

e

(U)1 .w1 T .GRAD

w91 7g

The radial and tangential velocities have discontinuities at the

disk

w1

f/(pU),

w1 = f6/(pU).

The mean values are

uPD =

W1 +

_1p1

U8D = eR)wT3) ½w0 =

½w1.

On the base of this velocity field we can derive approximations for the secondary forces and their influence on the flow.

At the disk the secondary forces are surface foráes instead of volume forces, similar to the external forces, see eq. (2-19).

An estimate for theroation of the velocity at the disk is

+ +

(rot

=

.JL

'o

A suitable definition for the secondary forces at the disk, accor

Ing to eq. (2-4), is

= lin _{J PWUDX(r0t u)D}d} o

-/2

(2-37)

r

+ + 2 - +

= Pw{UpD w1+½(w01) }

The secondary forces in the wake can be evaluated starting from

the definition in eq.. (2.-4)

* *

*

= p (uxrotu) =

p(v + w )

x

rot w (2-38)

-29-It is useful to consider two separate parts of g2, viz.

= rot

_* -*

= p(w x rot w ).

It will be observed that the force is constant in downstreàm direction, but the force disappears far downstream as the velocity componeñt goes to zero. Therefore the latter force can be related with the contraction of the wàke in the region behind the disk. To find thè effects of on the flow through

the disk, we have to investigate the associated velocity potential.

'rom eqs. (2-8, 15) it fòllows

div

v2 = 4pU

f

dx'

G'

dG

R(x';) (2-43)

where Gw is the volurhe of the wake.

div = div (*x rot

= 'w1°

) . (rot (rôt rot

-*12-*

-=

y .1V

(w )- grad (div w = +

_}V{w

_}. i pl (2-4 4)

With eqs. (2-43,44) the velocity potential _v2 is related to the firstapproximation of the flow field.

Evaluation of the expression for eq. (2-42), reveals that it is a radially directed force

ii th

= grad(1+1)

(2-39)

h1.L (Pw01)jI

This equation can also be written as

g

I

çl.ivg2

w2 =

P;jtJ!

dx' dG R(x';)

div =

h

_V2FW

The vóluine integral in eq. (2-47) can be transformed into surface integrals by app] cation of Green's second théorem.

Without going into

the

details of: the manipulations involved., we givè the result

= w2

4*p U

_w

-Fw(p)

/

ds

_R(x;0)

+ SD -31-(2-45) (2-47) (2-48) (2-49) wheré r Fw(P) =

P{½w1+½w1-

.wdP}

(2-46)

The function _Fw depends on p only.

Ït will bè proved that the effect of the force

2 on the flow

can be described bya.secondary fôrce at the disk, in addition to the force in eq. (2-37).

The proof. can be given in two ways.

One way is based on the investigation of the associated velocity potential:

where

_{C =_XFw(r)/(pU)}

inside the wake,

C =0

_{outside the wake.}

It has beenasumed that Fw and dFw/dp are zero at p = r0.

Comparison of the twoterms on the right-hand side o eq.(2-49)

with eq. (2-27) resp. (2-22) reveals that

w2 is the potential of a velocity field completely similar to () associated with the

the extetnal fôrce f on the disk.

Consequently _Fw Tcan be considèred as an axial secondary force at the disk.

The second way to prove this relationship is based on the considera-tion that radially dIrectedforces can be associated with vorticity rings. In sèctiòn 2.3.1. it has been shown that a radial force

f cbncentrated at the disk induces a vortex ring with strength

f/(PtJ)ie. It has also been shown that a vorticity distribution. in the wake with strength _(df/dP)/(PwU)T6 is induced by an axial

force fI concentrated at the disk. ReplacIng f by Fw then, in

vièw of eq. (2-45), 'i can be concluded that the raia], force _2w th the wake is equivalent with the axial force -FIat the disk. saving obtained an approximation for the secondary forces at the disk, we can cônsider the second approximation for the generalised

force according to eq.(2-5). At the disk we have

k2 = f + _{- Fwi}

In view of eq. Ç2-14) the components of this force are directly connected with the discontinuities ih the velocity at the disk:

a PwU W

f+pwtupDw+l_½(l)2+J(tl)2dPj?

= p

Uw+ _{= f -p u} w w p2 p w D pl

k20 = PUw2 = fO_puDwl.

(It has to be noted that w1= w1 and W[: w61). A review of the

steps which are made to obtain this result reveals that similar relations will apply to higherorder approximations. We write

4n+1

-,

P½(w)2

₊ -32-+ PwU Wpn+l + pU Wen+l = = f + - PW uflwfl + - w UDWOfl? (:2-50)

= k/(PU) = (f+g)/{p (U+½w) },

w

= k 1w

_{= fp/{Pw(U.+uD)}}

p p

w

_{= k8/(pTJ)}

=fe/w+D1

It has been assumed tacitly that at each approximation step the

total velocities at the disk, UD and u, are also improved. It

has to be realized, however, that these velocities also depend on the velocity potential

' for which an approximation is given in

eq. (2-43). Successive approximations of Vv seem not very easy to handle. Neglection of means that the effécts of the contractión of the wake are disregarded. An estimate for these effects can be made by investigating some general features of flow in the wake.

We start with the general formulation of the secondary forces in the wake, according to eqs. (2-4, 10, 11).

= x rot i)

= p(+)x rot

(2-55)

= P{(v-1-w) (--

-

+ w--}i

w +

_wp

aw aw (___a

(Pwe)_(v+w)

_{a ap p} aw a

+p{-(v+w) -ii _(v+w)

In the wake the generalized force i is equal to and, accordinc

to eq. (2-12), we have for a steady wake -33-(2-52) (2-53) (2-54) where =

Jo

*(wn) 2dP] (2-51) p

When we assume that for a h.gh value of n the successive approximations are convergent, the following relations can be derived

= (2-56)

From the eqs.(2-55, 56) three relations can be derived, -iz.

(tJ+u)

aw 3w _3We

U(

U +W = O, ₍₂₅₈₎

3(pw0) ,3(pw8)

(U+.u) + u = O (z-59)

The first and the third êxpression reveal that along a streamline w and (pw0) have constant values.

This feature of the flow in the wake can be used - in the

intêr-pretation of an eqúation which is derived in the general moentuin: theory for an actuatôr disk. Suffice ït to say that it is obtained by application of Bernoulli's relation to the flow on both sides of the disk. For the details is réferred to Glauert/4/. In our

notation this relation is (inclusivé the effect òf radial forces)

{f(p)+½pu2 (p)

_{+ pw}

u}0=

(2-60)

p{Uu(r)+½ u(r)+½u(r)}

-p0-p}.

This relation holds along a streamline of which the radius at the disk (x=O) is p and for downstream (x=) is r.

We can substitute:

u(r)

=

W(r)}

W(p),

{rue(r) rw0(r)} 2

!

we(r)dr.

The third relation follows directly from the equation of motion (2-57)

(2-1), when applied far downstream where the óuter radiu of

thé wake is rw. The result of substitution in eq. (2-60) is:

1 -w - - +

(f+g)

P(U+½w)

where g _PwIupDwp

{1_(_)}.J

(w)2{()3

}dj.

p

When comparing eqs.. (2-51, 52) with eqs. (2-61, 62), wè see-that we have Obtained a slightlydifferent éxpression for w. Instead of g, which was derived from successive approximations, we now

have g , which has. been derived from generai considerations. The difference between the two expressions is connected with the radiai contraction Of the streamlines through the disk. Eq. (2-62) can be considered as the correöt expression, incorporating the effect of contraction of the wake on w. Its evalùation, however, requires the knowledge aboüt the radial contractiOn of the streamlines.

The considerations in this section lead to the conclusion that the velocity vector can be derived from a generalised force at the disk. For the axial and tangential components of the courses along

a streamline have been found. For the complete flow through the disk.

thè magnitude of the irrotational velocity vector has to be known. Its relation to the secondary forces in the wake is less simple. In the next section some remarks about the effects of thése secondary forces will be made.

2.3.3. Remarks about flow through disk.

F'or the total axial velocity at the disk we write

= ½w

[h

v]=O

(2-63)

We recall that in this notation, it is incorporated that the dis-continuity w at the disk always corresponds to a continuous velo-city, ½ w. For the other term we write

-35-(2-61)

(2-64)

where is thevelocity potential associated with the secondary forces in the wake which can not be included ïn the generalized force at the disk.

The velocity potential connected with hè secondary forces in the wakè has to be found from a solution of the equations of motion for the flow in the wake, eq.. (2-57,58,59). This solution will also yield thé radial velocity UD? which appears in eq.

(2-62). A relation between the radial contraction of the stream-line's and the axial f lôw through the disk is given by the condition of Qontinuity of the flow inside an annular streamtube:

(U,4-w)rdr = (U+üD)PdP (2-65)

For the solütion of the equations of motion for the flow in the wake, we refer to the work of Wu /5/ and Greenberg /6/ They

use a formulation in terms of. the vortïcity and the stream function, wh±ch is more convenient. -Their results indicate that y is rather small. For the présent purpose we write

= ½w + VÇD

VD= £

½w (2-66)

and assume that c is has a small value, which is known, either from theory or from experiment. Its actual valué will depend on the radial position, on the load and on the load distribution.

Another conclusion, whidh can be drawn from /6/,concerns the ràdil velocity component at the disk. It appears that, except for the region close to the outer radius

of the disk, 'the effect of the contraction of the wake on the radial velocity component is small.. Consequently a good stimâte

of this velocity component UPD can be obtained from, the first approximation of the flow field, treated in section 2.3.1. If there 'is no radial force at the disk, the velocity vector field

(u), see eq. (2-36), has to be investigated. This is done in Appendix A, where special attention is given to singular behaviour

encountered in the evaluation of the integral expressions. A computational scheme for the distribution of': the radial velocity component UD results. An example is presented. These results supplement the calculations of Hough and Ordway /7/, who

also calculated the radial velocities from such a model, but not at the disk itself.

Hitherto the treatment Of the actuator disk has been rather general,. From the requirement that the disk has to represent a propeller with infinitely many blades, rotating with angular velocity -w, an additional equation can be derived. To the

flow relative to the. propeller blades Bernoulli's relation can be applied. See Glauert /4/. The result can be written as

(2-67)

-37-Combination of this equation with eqs. (2-53, 54) yields

(U+uD)f + u.pDfp.+ (½w-wp) f= O

or in vector notation

(UI _{- WPie+ UD)} = O. (2-68)

This result shows that the resulting velocity vector at the disk relative t the blades is alaays normal to the force vectOr.

For further reference we rewrite the equations for w and w eqs. (2-52,54).(The effect of a radial force òomponent is

disregarded and therefore is irrelevant.)

= + Wg (2-69)

= Wf9 -

W99 (2-70) with añd = (2-71) Wf0

e1w1m'

(2-72)

Wg=

_{I (Pwm)=F½(W)} 2

2}+f°

(w)2{ 3 Pj/U (2-73) Wg0=

g/(pU) =I(UD/(D)]fe/wUmD'Urn(274)

+ Um = U+½w (2-75)

The components _WfE and Wf6 can be conidered aC thè components. of a vector:

= ?I (PU).

(2-76) This composition o the velocities-can be interpeted tentatively. FOur contributions can. be recognized.

The vector Wf is parallel to and simply relàted to the external fOrce vector

L

he relation (2-76) is similar to eq. (2-22), whiòh is derivéd for the first approximation of w. In the nominator

the velocity U is replaced by the velocity _Um We conclude that

for Wf a quasi-linear acuator disk model holds.

The vector (w i) Is an axial velocity component induced by the

g

secondary force (gi). This force is related to the pressure

for the axial momentum of the flow through a propeller shows that this pressure decrease means a loss of effective thrust of the propeller. Obviously the expression for Wg is the proper des-cription of this connection.

The vector (Wg818) is proportional to the effeòt of the secondary forcés in the wake on the disk, riz. proportional to vD. It has been observed that, for ari actuator disk representation of a propeller, VD is small. If such a representation is extended, to include e.g.. the velocities induced by a propeller duct, vFD and Wg6 may become of importance;

Just like for

,

the expressions for (WgI) and (WgI6) correspond

to a quasi-linear model of an actuator disk with U as refèrence m

velocity in the nominator. Consequently the velocitïes at the disk are the halves of these vectôrs.

In Fig. 5 these velocities at the disk are combined in a vector

diagram. The velocity component vD , for which we' have

an approximation in eq. (2-66), describes the deviation from the quasi-linear model. In the diagram of Fig.5, the condition of

normality, eq. (2-68), for the resulting velocity vector and the external force vector is satisfied.

2.4. steady load on rotating lifting lines and surfaces.

Various distinötions have to be made between the representation of the propeller as a finite number of thin blades and the re-presentation .of the propeller as a thin, circular actuator disk. Formally the former representation can be obtained by a modification of the latter. Let the actuator disk rotate about the axis arid allow its load to be not constant in circumferential direction. The associated flow field becomes unsteady in the original reference frame, but is steady in a reference frame that rotates with the disk. When the regions on the disk where the forces are non-zero

-39-Fig. 5. Velocity and fôrce veçtors at actuator disk.

shrink to a finite nuniber of identical arid equidistant lines, the

lifting line representation is obtained. Lifting lInes are,however,

idealisations of material lifting surfaces, which stretch in

streamwise direction. The flow field of sùch ah actuator disk

with a rotating load distribution can also be split in a

irrotati-onal and remaining part, as defined in the foregoing sèctions..

The first approximation forthe wake, that is the region where

the vector

is non zero, is a ntímber of helicoidal sheets. This

can be seen from eq.

(2-16)

where we put the generalised force

k equal to the external forcé f at x'=O. Let the distribution of

on the disk be given by a niflnber of Dirac functions:

where Z = nuiitbei of blades and k0(1) Z-1.

Evaluation of eq.

(2-16)

yields

w=

(2-78)

The argument of 6 describes helicoidal sheets with a constant'

pitch

2iîU/w.

These sheets are wakes of zero thickness. Just like

we have done f br the actuator disk a description can be given of

these wakes starting with finite thickness. Studying the limiting

properties for vanishing thickne'ss reveals that the velocity

vector vanishes too, but not the rotation of this vector. Thus

the weliknown picture of the wake as a vortex sheet is obtained.

The vector product of the vortex strength vector and the velocity

vector is a measure for thè cncentrated secondary forces in the

wake. So the main part of the flow through the actuator disk is

irrotational. To distinguish this flow from the wake of an actuator

disk with constant circumferential load distribution it is called

slipstream.

The flow in the slipstream can be described by a velocity pOtential

and has to be matched with the flow induced by the secondary

forces in the wake. An obvious question can be posed conòern.ing

-41-the relation between -41-the time averaged flow in -41-the slipstream and the flow in the wake of an actuator disk with a circuxn-ferentially averaged load equal to the load of the lift-ing lines.

It has been shown by Hough and Ordway /7/ that these flows are equivalent in the limiting case of light loading. For heavy

loading such an equivalence is unlikely as shown by Greenberg /6/.

Nevertheless it can be assumed that some effects of heavy loading on the flow through the propeller disk are described sufficiently accurate by the model of an actuator disk wIth the same time-averaged load. (For convénience we use the terms propeller disk and actuator disk to distinguish between the disk wiLth load on

lifting lines or surfaces and the disk with circumferéntiall

-constant load). In the foregoing section the flow through an actuator disk is separated in locally induced velocities and

wake induced vélociti.For propeller disk flow it is assumed

that the velocity induced by the rotation of the slipstream is equal to the corresponding velocity of the actuator disk, so circumferential variations are disregarded.

For the locally induced velocities holds that they are directly related to the external forces (see eq. (2-76)). The relation is quasi-linear 'as the effective mean flvelocity Um depends on the induced velocities and cañ var' with the radius. Application of this quasi-linearity also to the propeller disk flow means that

in eq. (2-16) has to be replaced by and U by Um which results

in

= PwUm

fdx

{x ,t

XX'}

=

-h--

?6{e+wt-- x + 2iî} (2-79

wm

in

Comparison of this equation with eq. (2-78) shows that the pitch

of the helicoidal wakés is changed to 2U/, which means that

this pitch is adapted to the local flow.

-42-The total velocity in the neighbourhood of the propeller disk has to be derived from the velocity potential. According to eq.

(2-17) we have X 1

c

-43-m (2-80)

where qç follows from eq. (2-8)

qf

f

dS (.grad

) (2-81

SB

The integration area SB extends over all the lifting surfaces.

For the velocities induced by the coñtraction of the slipstrèath it can be expected that they differ considerably from the veloci-tieà induced by the contraction of the wake of the actuator

disk. In the slipstream the secondary forces, which describe the contraction, are located in the trailing vortex sheets close behind the blades. Therefòre it seems reasonable to assume that

these forces induce a velocity which is directed tangentially to the pitch line of the lifting surface. See the velocity v in

Fig. 6. This velocity is larger in magnitude than the circumfe-rentially constant values of VD and ½WgO in the actuator disk model. A consequence of this modification is that also the secon-dary forces concentrated on the lifting surface have to be

correc-ted. An estimate for these forces is based on the following

consideration of eqs. (2-22) and .(2-76) . The former gives a first approximation of the locally indùced velocity at the disk, without

the effect of secondary forces. The latter, includes the effect of secondary forces. The apparent difference. is the replacement of

the main flow velocity Uby the local axial velocity Um In Fig.6

we see thàt Um is proportional to the resultant velocity, which

is tangential to the liftingsurface. Weconclude that the

secondary forces at the lifting surface associated with the increase of this tangential velocity can be accounted for by replacing Um by Ub.

3. APPLICATION OF LIFTING SURFACE THEORY.

3.1. Introduction

In chapter 1 and 2 the basic principles are gïven, with which it is possible to find the geometry of the propeller blades and their load. The formulation of suòh a relation is the subject of this

chapter.

In chapter 1 the hydrodyhamical boundary condition is given as an equàtion, (1-32),for the flow velocity, which is locally decomposed

in radial component VR, a côinponent VT tangential to the pitch line ànd a component VN normal to the pitch line and ñòrmal to. the iadial direction. The ptch line is the intersection of the blade reference plane with a cylinder of constant radius and coincides with the nose-tail line of the blade sections. It i assutied that eq. (1-32) has to be satisfied on the blade reference plane

instead of on the blade surface.

For the hydrodynamical load a similar approximatiôn is used. It is described by a distÉibution over the blade reference plane instead of over the blade itself. The blade force normal to the reference plane is given by à pressure jump across the reference plane. The blade force tangential tò thé ±eference plane, connected with camber, friction and leading edge suction, is much smaller than the normal force and its effect will be dealt with äs correôt-ions to the model with a normal force only.

The analysis of the flow through an actuator disk, in

chapter 2, has revealed that this f low'ôan be decomposed in locally induced velocities and wake-induced velocitiès. The 'locally in-duced velocities are directly related to the external forces

(f-forces, see eqs. (2-71, 72)). The wake-Induced velocities at the disk can partly be described by sécondary forces at the disk

(g*_forces, see eqs. (2-73,74)) which account for the effects of the. rotation of the wake.' The other part of the wake-indùced

2

wake, which takes place in the region close béhind the disk. For the effects of the f-forces and the g*_forcs quasi-linear actuator

disk relations cn be used, but the effects of the contraction require a solution of the equation of motion for the flow in the wake.

The corisideratJ.ons about the flow through a propeller disk, which

are also presented in chapter 2, have to be matched with the

fòrmulation of the boundary condition. This is accomplished by the following prescriptions.

- The velocities induced by the rotation of the slipstream are derived from an actuator disk model with the saine radial load distribution as the propeller.

- The locally induced velocities are derived from a velocity potential, which is obtained by integration of the distribution of normal forces over the blade reference plane..

The velocitiés induced by the contraction of the helicoida]. wake sheets behind the blades are assumed to be tangential to the blade reference plane (see Fig.6).

The magnitude of the vjocj.v is still unknown. It will be found by correlation of the theoretical model with experimental results. Also the velocity factor c as defined in eq. (2-66) has to be fixed to complete the model.

-45-3.2. Formulation of iñtegrai ßquation and solution method.

To introduce the properties of the load-induced velocity field, as discussed in.the foregoing section, into the hydrodynamical boundary condition, we use a modificatioñ of eq.. (1-32):

tan (ß-B)

= -

(+gj).

(3-1)

To obtain this equation from eq. (1-32) we have used the following relations and assumptions:

- The velocity components induced by the thiòkness öf the bladés are expressed in a correction angle .Bi which has to be subtracted

from the caii,ber an1e

- In the flow field without propeller, there is. no radial velocity

component,hence VR=O.

- The velocity components induced by the blade load are written as

= VNg + VNf

V.=V +V

_{Ri Rg} Rf

v..=V

+v

Ti Tg Tf where VNf - = , VTf = neglected, V . -½ (w cos B +w

smB

Ng g

.p

ge p

Vg =

+½(WgSifl

Bp_wgocosBp)+v*D, with

pitch angle; the notation B is introducéd to avoid confusion with the velocity potential c

3JN and /R are differentiations in the directions of VN and

VR. resp.VR are related to the radial distribution of w resp.

wg according to the relations between radial and axial velocity components given in Appendix A.

On the left-hand side of eq. (3-1) the, terms are colleò.téd which

aré either known from the flow withoijt propeller, viz. VN0 and

V , or estimated from the actuator disk model, viz V , V

To Ng Tg

arid On the right-hand side of eq. (3-1) the terms

are collected which are derived from the velocity potential of the lifting surfaces.

The evaluation of the expression on the right-hand side is laborjo. The integral/expressions from which cari be derived, see eqs.

(2-80, 81) involve integration of the load distribution over the lifting surfaces which represent the propeller. Further the integrand has singular points

to make calculations possible.

for which eq. (3-1) is used is the analysis of the load

distribution on a propeller of given geometry. To obtain a solution of. eq. (3-1) fòr that case, it is necessary to approximate the integral by a linear combination of a number of load function parameters with coefficients depending only on the geometry.

The formulation of the velocity otential on the base òf an external load distribution on the blades of a ship propeller has beeñ studied by Sparenberg /8/, Hanaóka /9/ Pien and Strom-Tejsen /10/ and Tsakonas et al. /11/. Most of these studies are not restricted to the case of steady blade load but include also

unsteady blade loads which are periodic with the sh'aft rotation rate or a multiple of it as basic frequency. The case of steady load corresponds to zero frequency.

Calculation procedures for the' evaluation of the right-hand side

ôf eq. (3-l) have restrictions concerning the blade _{geometry, i.e.}

the pitch is constant and mostly there is no rake. To keep the computation time short it is appropriate to evaluate the expression

-47-and requires a détailed analysis Finally' thé more difficult case

as a whole and not in separate terms. Fôr this study one of the two calculation procedures developed at the Netherlands Ship Model Basin is uséd. The first one, which applies to proel1ers with constant pitch and'without rake, is'prepared by Verbrugh /12/ and applied by Van Gent /l3/ Recently the second one, which applies to propellers with constant pitch and constant rake angle, has bécome available and has been prepared by Vis /14/.

For constànt pitch and no räke = O and the right-hand side of

eq. (3-1) reducs to

aN

Tò allow for the application of the correspondin calculation procedure also in cases of small radial pitch variations and small rake angles, we do not neglect the term

but transfer it to the left-hand side of eq. (3-1), and approximate

it by This term can be oinbined with the term

Vg'

as

VRi = g(V+Vf)

and vRi is derived from the actuator disk. Hence the equation

which will 'be investigated is'

-+ tari

= (3-2)

For the details òf the calculation procedures we refer to the

fôrement4.oned literature. Some main point are reviewed here.

The distribution of. the normal' forces over the blade reference

piane is represented by a, truncated series of elementary chordwise

f t ions + g 'p = .E C (p) H (s)

p=O

p (3-3)

is a chordwise parameter defined by

cos q) = 1-2

(S+c/2}

s c

thus q)=O at the leading edge and =T at the trailing edge.

C() are the weight factors of the functions Hr(s) _{and they are}

functions of the radial position. It has to be notéd that the func-tions H(s) are zero at the training edge, thus satisfying _{the Kutta} condition for the flow. At the leading edge all the f-unctions

_H(s)

hae a singulir behaviour. In the section 3.4 some properties of

these functions will be further investigated.

By eq (3-3) a separation of variablês is introduced, which can be further exploited by using an interpolation formùla for radial distributions and functins. The fïnal result of the manipulations involved, see /13/ , is that the calculation procedure for Dq)/N

can be formally written as

P N

,s.) = - = sin S Z Z C (p )M (r ,s.)

J N

a01 p

n pn j where

ru and s. specify the radial and chordwise position of the considered point on he reference plane,

C (p

Pn

) is the weight factor of the function H at the radial -p

position p

-M (r 5.) are the coefficients of complicated structure, but

pri y _j

depending only on geometrical properties of the reference plane,

is the pitch angle at the radial position r of the helicoidal plane at which the load is situated. It should be equal to the pitch angle of the reference

plane, but the distinct notation is appropriate for approximations iñ cases of pitch deviations. H (s) = H (q) ) = -p _p s 7T sin q) -49-p=O(1)P. _(3-4) (3_5) (3-6) where , cos(pq)) +cos(p+1)q)

he radial positions r of the considered point oñ the reference plane and of the interpolation stations havé the same distri-bution derived from

r+r.

r-r.

(3-7)

where r and r. are the outer radius resp.

hub

radius of the blades.'

o i.

For r G = \nh/(N+1) e 1(1) N!

for

n : O nu/(N+1) n = 1(1) Ñ.

The chordwise positions s are distributed according to eq. (3-5) with

= 2(j+l)1T/(2P+3) , j=O(1)P (3-8)

I-n this way the calculation procedure 'indicated in eq. (3-6) gives

for a number of N(P+'l) statiòns on a blade the right--hand side

of eq. (3-2). When the terms on the left-hand side are known in

the same points,a set of linear equations for the unknowns C(p) is' obtained. This is the formulation the analysis problem of a propeller with given geometry. The sâme set of equations can be used for the main part of the design problem. The coefficient

matrix of the set of equations, which is gïvenby Mpn (r,s.),

depends on the geometry of the blade reference plane and on the blade contour. The details of the blade geometry are not involved i-n the coefficients. Hence, when a starting form for the reference plane and blade contour is available from initial design consider-ations, the- final design for given Cp(Pn) follows from the -set of equatiDr1s. The deviations from the reference plane which describe the details of the final blade shape are comprised in the left-hand side of eq. (3-2). Therefore àñ iterative solution of this equation is necessary to òbtain an accurate answer. This applies also to the solution of the analysis problem, in wh-ich the correction terms on the left-hand side of eq. ('3--2) have to be found by iteration.

In this formü-lation of the- application of lifting-surface theory

rnatrix are calculated only once for the problem. The iterations involvé repeated calculation of the vector èlements and matrix-vector multiplication. The iterations require much less computation time than the calculation of the matrix elements. About 4 itera-tion steps suffice to obtaiñ a soluitera-tion. This takes about 1/30 of the time required to calculate the matrix elements.

3.3. Slipstream rotation and còntraction

In section 3.1 It is stated that the velocities induced by the rotation of the slipstream are derived from an actuator disk model with the same radial load distribution as the propeller. In this

section this equality is further specified. The relation between the actuátor disk and the propeller is derived from eq. (2-67) . For

the present case (w0), this equation can be rewritten as

F

f : =

Zç--2lTp

(3 -.1 0)

where F is the chordwise integral of the axial component of the load distribution on a cylindrical blade section of radius p. For the description of the actuator disk, it Is assumed that the value of F is known from the lifting surface calculations.

By combination of eqs. (2-65) and (2-66) an expression for the ra-dial contraction of the streamlines through the disk is obtained:

d 2 _(3-11)

dp U+w

As stated in chapter 2 the coefficient c is assumed to be known.

The eqs. (3-11) and (2-66, 69 to 75) deter-miné the velocity

compo-nentsw+, _{w and w} , by which the effect of slipstream, rotation

g gO

w =

wp_wp)2_2

(3-9)

For a propeller with a. finite number of blades, we replace f by

the circumfrential density of the axial compônents of the blade section forces according to