A theoretical contribution to the investigation of the hub influence in propeller design

Department of Naval Architecture and Marine Engineering The University of Michigan

College of Engineering November 1970 No. 100 TECHNISCHE _UN1VERSJT Laboratorium 'QOr hQepshydromec,ij 2, 26p. _{CD DeJft}

TeL °'a'7a3.F

O1Ø7818

A THEORETICAL CONTRIBUTION TO THE INVESTIGATION OF THE HUB INFLUENCE

IN PROPELLER DESIGN by

This paper is in essence the English edition of my 'doctoral thesis, which was accepted by the Fakultät für Maschinenwesen of the Technische Universität Berlin in. 1969. The content of the German edition is now partially condensed, and the Section 2.1 contains some, additional comments on the special subject of a "simplified hub representation." However, there is no need for any further reference to the German text, and this. English pre-sentation of the original results can be considered as reasonably self contained.

iii

ACKNOWLEDGEMENTS

I am indebted to the department

_of

_{Naval Architecture and} Marine Engineering, The University of Michigan, _{where my} investi-gations :fl _{the field of propeller design were supported möst}

generously during a one year period in 1969 and _1970.

During that time Professor Harry Benford suggested _{an English} edition of an earlier paper of minè under department cóver,

_whih

I am now presenting with. the sincere, gratitude to the :foilowing

individuals.:

Proféssor Horst Nowacki and Professor Som D. Sharrna

for their encouragitig suggestions and fertile discussions, Mr. Peter M. Swift

for his patient efforts toward improving the literate _{part of the} text

and Christina Seidi

for her excellent typing of the manuscript.

They have all helped me in presenting this paper on a súbject with which I am personally conçerned.

INTRODUCTION.

THE INFLUENCE OF DIFFERENT HUB REPRESENTATIONS ON RESULTS OF THE LIFTING LINE THEORY

2.1

The basic concept of the lifting line theory

2.2

The treatment of a design problem under consideration of different hub-representations

2.3 Comparison of different solutions

THE CONSIDERATION OF THE DISPLACEMENT EFFECT OF A HUB

3.1

The wakè adapted actuator disc

3.2

The determination of the hub-induced flow field at the propeller disc

3.3

Examples SUMMARY FIGURES LIST OF ABBREVIATIONS APPENDIX REFERENCES V

1.

INTRODUCTION

When the design of a moderately loaded propeller is based on the lifting line theory, the results must be corrected with respect to influences caused by real fluid flow phenomena and by secondary effects such as blade-surface, -thickness, -rake and -skew or the hub body. Today the influence of the blade surface is fairly well known and can be regarded as the main source of discrepencies

between design results of lifting line theory and operating features of such a propeller. But the other mentioned influences do not attract much attention in propeller research, although, because of the increasing application of controllable-pitch propellers, at least the displacement effect of the hub body seems worth being investigated in detail. Until now only the relatively large hub diameter is considered within some design procedures, the hub body being neglected or replaced by a semiinfinite cylinder, which does not have any displacement effect on the fluid within the first order theory. Accordingly different opinions are advocated about the boundary conditions to be satisfied by the bound circulation at the hub. Thus, before dealing with the displacement effect of a hub body.it is necessary to treat the problem of using different boundary conditions. This is examined in chapter 2 but it should be emphasized in advance, that the analytical and numerical part of such an investigation must b exact, i.e. only certain design procedures can be applied to the problem.

In order to explain that statement in more detail some impor-tant results of a special design problem are shown in Figs. 1, 2 and 3 *). All functions indicated by i were attained using the so called induction factor method as developed by L e r b s, [li and

[2] **), and functions indicated by 2 and 3 were derived using the

methodofEckhardt & Morgan, [31.

Inthelattercase

*) _{Figures appear at the end of the paper.}

use was made of the so called K-functions, which were first intro-, duced by G o 1 d s t. e i n, _{[4], and reevaluated by L e r b s,}

[2]. Further investigation was made by a c h m i n d j i - & M i 1 am, [51, using a solution of the potential problem by T a c h m i n d j i, [6], which is slightly different from

G o i d s t e i n's solution with respect to the boundary condi-tion at the hub. All functions in Figs. 1, 2 and.3 which are indicated by 2 refer to k-functions as given by L e r b s

those functions indicated by 3 refer to K-functions as given by

Tac hrn i nd j

i &

M i lam.

The.r.esults of the.induction factor method and of the method of E c k h a r d t

& Mo r g a n

can be regarded as being exact if the radial distribution of the hydrodynaxnic advance coefficient, X1, is constant. Therefore, the results are nearly equal for both methods (Fig. 1) but if Xl

varies with the dimensionless propeller radius x, the results are rather different (Figs. 2 and 3). This is because the design

methòdof Eckhardt & Morgan isnotexactinthe

latter case and therefore cannot be generally applied fOr the above mentioned purpose of chapter,.2.

The results of Figs. 1, 2 and3 indicate fürthermore that the radial distribution of the. dimensionless bound circulatior GS, _is

trong1y influenced by the. radial distribution of X1. Therefore, the bound circulation at the hub should be expected to váry at the hub acöording to the type of loading distribution - as the

fúnction obviously does. From this point of view the method. of L e r b s cannot be easily accepted (See functions indicated by i in Figs. 1, 2 an 3) but an alternative solution of K e r w i n & L e o p o 1 d, [71, seems insufficient with respect to the hydro-mechanical representation of the hub cylinder mentioned äbove.

This will be discussed in section 2.1 below. For that reason

another hüb representation will be introduced and using a strictly numerical method of solving the basic equations the results will be free from any analytical side conditions.

It can be shown then that there exist some agreement between resùltsof thesimple momentum theory and those of the lifting line theory. Therefore it becomes possible to treat the problém of the displacement effect of the hub body separately using the momentum theory as is outlined in chapter 3. As a result of the momentum theory, the effective radial distribution of the wake

fraction, w, caused by the hub body in the propeller disc will be available. Applying this to the lifting line, corrections of the X1-functions will finally indicate the displacement effect of the considered hub body. It should be mentioned at this point that the treatment of the problem, as far as it is based on the vortex

theory, restricts the procedure to a special hub form in order to provide that the logic of the theory shall not be violated.

2. _{THE INFLUENcE OF DIFFERENT HUB}

REPRESENTATIONS QN THE RESULTS OF THE LIFTING LINE THEORY

2.1. _{The .basjc cOncept of the lifting line theory..}

In order to describe the action of a propeller on its envirar-ment, the environment must be idealized _{as well as the propèller}

itself. _{To begin with, the following conditions are assumed to}

be. valid for the environment:

(Cl) _{The fluid is incompressible and inviscid.}

The free-stream velocity is axisymmètri.c and stèady. In lifting line theory each propeller blade is _{repläced by a} bound vortex for loading effects, the circulation _{of which is}

distributed in spànwise direction _{over a straight line. Therefore} the idealization of the propeller blade _{may be expressed by the} following condition:

Section-length and -thickness as well as rake and. skew

of a propeller blade are not, considered.

The 'vortex theory demands that free vortices are shed from the lifting line wherever the djstribution of bound circulatiòn _varies with the span of the propeller blade and in _{a propeller-bound}

cordinate system all free vortices form a general helical _surface behind the propeller. _{These free vortices cause additional} velo-city components at the lifting line which must be knowfi in order to obtain the strength of the bound circulation for the desired design conditions of the propeller. However, there are serious, analytical and numerical difficulties of determining these addi-tional velocity components so that the foliöwing conditions are

introduced mainly to overcome this problem: The propeller is moderately loaded.

The free vortices have constant pitch and constant diameter in the downstream direction.

The last condition, (C5), is usually implied, when the theory is

characterized as being of the first order. This also means that additional velocity components induced by the free vortices do

-5-not effect the shape of the helical vortex sheet. It should be noted that this statement is true Only at the lifting line itself but is false far behind the propeller. Therefore condition (C4) is introduced to assure the validity of condition (C5) as a first order approximation to the exact solution.

Now dealing with the hub problem, a semiinfinite hub cylinder in practice approximately represented by the long shaft of a test device for a free moving model propeller - can be regarded as compatible with the above mentioned conditions for a first order lifting line theory. However this would not hold for a finite

hub body downstream, because condition (CS) is violated just behind the propeller disc and the introduced set of free vortex sheets can no longer be thought of as a f.irst order approximation to the exact solution. Therefore, any hub body considered within a treatment of the design problem by means of the vortex theory in the context of this paper will consist of a semiinfinite hub cylinder behind the propeller. Thus the treatment of a finite hub body in front of the propeller disc will be the main concern of the following investigation and results can be applied without restrictions only to the above ientioned propeller test device.

As mentioned in chapter 1 the problems of the boundary condi-tion which has to be satisfied by the bound circulacondi-tion at the hub as well as the hub représentation itself are to be treated at the very beginning. In the solutions of M c C o r m i c k,

[8], K .e r w i n & L e o p o i d, [7], and T a C h m i n d j i,

[6], for example, the following boundary condition was used:

(Bi) The bound circulation at the hub has zero-slope. This boundary condition is to be introduced whenever the hub is

considered as a rigid cylinder within the flow, because the strength of the free vortex, lines behind the propeller is defined by the

slope of the bound circulation at any given radius. As there is no free vortex line in the boundary of a rigid body the slope of the bound circulation must be assumed to be zero at this radius. The strength of the bound circulation at the hub will be deter-mined by the desired radial loading distribution over the

is given by a constant pitch distribution is based _{o the} cònsid-eration of a hub body of f mité length behind the propeller,

which is not taken into account within this investigation because of the reasons outlined a.bove). _{Referring for example to. I s a y,}

[9], the principles of hydromechanics demand .an extension o.f the bound circulation at the hub into the hub in a way that all vortex lines are closed curves or traverse the fluid, beginning and

ending on its boundaries at infinity. _{Therefore, another system} of free vortex lines might be set up within the hub cylinder. Thi

system must not induce a.ny additional velocity components öutside the hub because it is to be. regarded as a hydromechanical represen tation of the considered hub cylinder. (This hub cylinder does not have any displacement effect on the flow according to the applied first order theory because it is compatible with the outer vortex system to the first order by definition.)

K e r w i n & L e o p o i d, [7], introduced a hub rère-sentation which cannot be accepted according to the. above stated restrictions - as already mentioned by S t r o m T e j s e n,

[7], when discussing the approach of K e r w i n & L e o p o 1 However, it will be seen in Section. 2.3 below, that the results do have some practical advantages in propeller design. This is because of the pressure drop between adjacent propeller blades which is not described by the vortex theory but .is. approximately

taken care of by this hub representation. As this pressure drop is of no interest when dealing with the hub problem the following attempt of representing the hub cylinder according to the above outlined restrictions cannot be expected to provide practical results for pröpeller design purposes although it will provide some interesting information about the characteristics of the f irst order vortex theory of the propeller and finally lead, to a solution of the hub problem.

To.begin with, the lifting line is extended into the hub cylinder, the bound circulation simply being constant and equal

-7-to its value at the hub. This vortex has to be extended in the downstream direction on the centerline as a semiinfinite vortex

so that the whole system of free vortices will be closed at mf in-ity behind the propeller. However the. semiinfinite vortex will induce tangential velocities outside the hub cylinder which are to be canceled by superimposing another vortex - or vortex system - within the hub. As the following investigation deals only with the flow characteristics at the lifting line the super-position of an infinite vortex on the centerline of the propeller will have the desired effect on the flow at the lifting line if the strength of its circulation is opposite to, but half as mudh as the circulation of the semiinfinite vortex described above Far

ahead of and behind the propeller an additional rotation of the whole fluid can be easily seen to be of the already accepted second order if the hub diameter is not taken smaller than practical and condition (C4) is observed. Therefore, the introduced "simplified hub image" can be regarded as compatible with a first order vortex

theory.

If the hub body is completely neglected another boundary con-dition for the bound circulation .at the hub radius can be formu-lated, namely:

(B2) The bound circulation at the hub has zero-strength.

This boundary condition leads again to reasonably satisfactory design results because of the already mentioned pressure drop between adjacent blades, in this case dealt with approximately by neglecting the hub body. Advocated for this reason by L e r b s,

[1], the boundary condition (B2) is accepted by many scientists as a basis for any further investigations, e.g. by

S t s c h r e i e t z k i [li], M o r g a n and W r e n c h,

[12], T a c h m i n d j i, [6], and by many others. Obviously

it cànnot be used for the purpose of this paper but in order to obtain comparative results which can be accepted to be exact within the stated limits Qf the vortex theory and different assumptions about the hub representations the method of L e r b s. as well

saine example which was chosen. to study the effect of the "simpli-f'ied hub image" on the vortex theory. .

2.2 The treatment of the design problem by consideration of different hub representations.

It can be shown analytically that the system of free vortices behind the propeller does not induce radial velocity components of the first order on the lifting line, [.1], which is in perfectiy

good agreement with condition (C5), Section 2.1. Therefore, only tangential and axial components of the induced velocity are to be determined. For practical reasons they are non,-dimensionalized by the speed of advance of the ship.:

T,A

o T,A

r

.T,A

o S o

U (x,x ) U 1

j i (x,x )

dG(x)

_dx0

(2.2.1)

V V 2

(x-x°) dx0

This équation. follows from the definition of the induction factors: T,A

iTIA(x,xo) _{= i}

' = 2

GS(x0)/(x_x0)

(Detailed information can be found in [1] and [13].) Fôr the purpose of this paper it should be noticed that the induction factors are geometrical functions of the already defined shape of the free vortex lines - which are helices of dimensionless radius x0 and pitch angle and also functions of the radius x

T,A

at the lifting line,, where U ¡V are: to be determined. The functions T,A can be considered as known functions whenever the parameters x, x0, and Z (the number of blades) are given:

T,A (x,,x°, _{3, Z)}

It follows from the vector diagram of the velocity components at radius x at the lifting, line (Fig. 4) that:

tan

+ uAiv x/AS _¿"/v

(2.2.2')

Combining equations (2.2.4) and (2.2.1) a fundamental relation bewteen the bound circulation and the hydrodynamic pitch angle can be derived:

4 f

(iA _{+ tan}

BuT)

o dGSxo) dx0 xli

(x-x)

dx i :1. tan B (2.2 .5)

This equation. descrïbes the situation as given by the hubless design case according to L e r b s and others and also as given by the simplified hub representation as described in section 2.1.

If the radial distribution of the bound circulation GS (x°)

is given, equation (2.2.5) defines the related radial distribution of the hydrodynamic advance coefficient X1(x). An iteratjon

procedure to solving the problem in practice is outlined by

L e r b s in [141. On the other hand for a given À1-distribution, equation (2.2.5) represents an integral-equation of the first kind, the kernel-function of which having a logarithmic singularity.

Therefore a solution exists if the unknown function Y(x0) =

So

o . ..

dG(x )/dx remains finite within the limits of the integral but is strongly influenced by the assumed boundary conditions.

Introducing t'he boundary condition (B2), (Section 2.1), the method of L e r b s will be applied. The function GS(xD) is

then represented by a F o u r i e r - sine series and the induction factor by a F o u r i e r - cosine series. Thus solving the inte-gral in equation (2.2.5) analytically, a system of N linear equa-tions is derived in order to provide N F o u r i e r - coeffi-dents for the bound circulation which can be determined at N positions x on the lifting line.

When using the boundary condition (Bi) in connection with the simplified hub representation the method of K e r w i n & L e o p o i d, [7], could be applied. However their representa-tion of the bound circularepresenta-tion by a power series provides no

numerical advantages and théreforè a striçtly numericaIso1utin of equation.(2.2.5) will dò it as well or evén better - at least

òne uncertainty within the solution, namely the questionablé quality of a power series representation, cn b avoided.. As in

themethod..of K e r w i n& Le op o i d., the lifting 1in

between hub and tip is divided into (M-l)/2 intervals of equal length x, (Fig. 5), on which the bound circulation isassumed tó be constant. The field points are chosen to be situated on the

lifting line at = (x/2) (n-l) + H, n=2, (2), M-1, and the freé vortices are shed from the lifting line at x.= (x/2) (m-l) +

m=3, (2), M, their strength being equal to (Fig. 5). Mäking M a sufficiently large number the integrai of equation (2.2.5) can be replaced by a sum and the integral equation becomes a system

of linear equations . . 1 = . tan - = . (2.2.6 .T,A .T,A i with i = i (X ,x , , Z). K e r w i n & L e o p o 1 d, [7], who obtained a similar system of linear equations to be solved for the trnknown coëfficiénts of the above mentibnèd. power series, solved the equations systematically changing M. Referring to their results a number M = 65, i.e. 32 field points, can be considered to provide a solution of acceptable. numerical accuracy, which consists then of the M unknown values AGJ. To obtain the bound circulation the following equation is to be solved

i

GS(x) =

_-

_f

(dGS(x9)/dxP) dx0 (2.2.7)

Thus, beginning with GS = O, the desired function values at x are given by m3, (2) i .T. i .+tan

.1

n,m n n,m (x -s s s s G (x) G

= G2

-rn (2.2.8)

with = O,

(Bi), G1

¿G/2 and n = M-3,(-2), 3. i 2 i=3, (2) M

V

(iA _{+ tan}

iT

1 _LGS rim n mn (X -x ) m m=3,(2) n m = _tan n

Now introducing the hub representation of K e r w i n & L e o p o 1 d the outlined numerical scheme remains basically unchanged. In addition there is a system of f ree vortices within

o

H2 o

the hub, each vortex starting from a point = (x ) /x at the extêndèd lifting line. The strength of the free circulation at

is equal but opposite to that at x0 and the hydrodynamic advance

io.

io

coefficient X ( ) is the same as X (x ). At the lifting line

outside the hub there are additional velocity components now given

by

uT1A(x,x0)+ UTlA(x,0)and

from equations (2.2.4) and (2.2.1) the following equation can be derived:

H X

r

. 2.o1 S o + tan j. G (

H2

(x-°)

d°

(x)

4

(iA1x° _{+ tan}

i.T,x°)

1 dGS(x0) _{dx0 (2.2.9)}

(x-x) dx

X

=tan

--xs

(The induction factörs have additional superscripts referring to the free vortex system to which they belong.) Applying the above outlined numerical scheme the following system of linear equations will replace the integral equation.

M

(i + tan. .

T,A T1A _Cx

ni n

2.3 Comparison of different solutions.

In order to compare the. solutions of equation (2.2.5) with: (2.2.9) on a common basis, the thrust coefficient shall have thè same value in every example and any Originally assumed function A1(x) will be iteratively changed until the thrust coeffidïèht is

alsogivenbythelawof Kutta-JoÍ'.kowoki:

1 = 4z J (x/X - UT/v) _dx . . k H . X

= 4Z

J(/V

+ /V) (x/X1) S _dx H X

i

_i

While AN_l(x) is changed into a function _{AN(x) at the N-th} itera-tion, the solutions are intended to maintaIn the _{relatioflship}

S S

GN(x)/GNl(x) = constant. It can be shown numerically for solu-tions of equation (2.2.5) with the boundary _{condition (Bi) and the} simplified hub reprèsentation that an iteration scheme given by

(A - XX)/(X1 - AX) _{= constant has the desired effect on the} resulting distribution of bound circulation. _{For the first}

itera-k

tion, N = 1, three different X -functions _{are taken under}

considera-tion:

X : constant between hub and tip,

: maximum value at the hub, slightly decreasing toward the tip,

A : maximum value close to the hub, slightly decreasing toward hub and tip.

For any of these functions, the following hub representations and boundary conditions, Section 2.1, will be studied:

i : (B2), method of L e r b s, no hub body,

2 : (Bi), method of the author, simplified hub

represen-tation,

3 : (Bl), method of the authôr, hub representation by

Kerwin& Leopold.

(The retho.s are described in Section 2.2 and thé design data the first example, Figs. 1, 2 and 3, remain unchanged.)

Figs. 6 to 11 represent sorne propeller characteristics which were obtained. It can be seen that the hub representations under cdn_

sideration do cause remarkably different results. From a purely practical viewpoint the functions 1 and 3 are' most attractive as explained in Section 2.1. On the other hand some striking facts

indicate that the functions 2 represent the characteristics of the first order vortex theory in a most natural way - confirming that the simplified hub representation was indeed the appropriate approach. To state these facts a brief comment on the so called "optimum condition" seems necessary.

Defining the efficiency of a propeller in ideal fluid flow by

i

n

i:í

VAdT

i.t follows from Fig. 4 that

1

J

(VA/y) (x/XS -

uT/v

.GS.dx H n 1. çbR

f

xdF H X -13-1 (VA/v)(x/AS -

uT/v).GS(xu/xdx

(2.3.1) (2.3.2)

If the A1-distribution is constant between hub and tip it follows for the free running condition of the propeller that

Xi

n = A /A (2.3 .3)

Referring to B e t z, [151, this equation defines the optimum efficiency, so that the results given by Figs. 6 and 7 represent optimum conditions. Referring to the simplified hub representation,

functions 2, the optimum condition _{can be seen to be:}

= constant, uT = o _(2.3.4)

This will be proven in Section 3.1 to be the optimum conditïón also given by the momentum theory of the propeller, which is base on the same linearization principle as givén by condition _(C5), namely the neglecting of radial velocity _{components. Thus having} introduced the simplified húb representation it _{turns out, that.} the propeller flow, given by the momentum theory at the propeller.

disc is qualitatively similar to the propeller flow _{given by the} vortex theory at the lifting line when the optimum _{case is}

consid-ered.

With reference to the assumption that no hub body is consid-ered, functions 1, the well known normal condition óan be seen tó be valid:

tan _(2.3.5)

This was derived analytically by L e r b s in [21 from a more general expression for the optimum. case, namely

(2x/Z) (tan uA/v - uT/v) S H S

G (x ) - G (1)

Now introducing the condition (2.3.4) into equation _{(2.3.6) it} follows that

GS(xH) ₌ (X1/z) (2uA/v)

. (2.3.7)

and replacing the variables in equation (2.3.7) by their values as obtained from the numerical procedure, equation (2.3.7) can be

"identically" satisfied within the numerical accuracy of the applied procedure:

S H

G Cx ) = (0.303/5) (2) (0.244) 0.0296 (2 . .3.8)

-15-Therefore any results can be interpreted as being due solely to the related hub representations, because there is no specific influence caused by the applied methods themselves.

Turning finally to the functions 3 in Figs. 6 and 7 it can be seen, that these functions are very close to functions. i although the hub body is represented in an entirely different way. Since the approach of L e r b s as well as that of K e r w i n & L e o p o 1 d causes the bound circulation to drop near the hub,

the influence of the pressure drop between adjacent blades is approximately taken care of by both "hub representations.t' (Any

exact treatment of that problem has to consider, parameters like blade number, radial loading distribution and hub radius etc.)

The. following investigation of the displacement effect of the hub body will be based on the results of the lifting line theory

as given by functions 2 in Figs. 6and 7., which are not "practical" when compared with function's i and 3. Therefore it should be

empha-sized that these results provide hub corrections rather than final propeller design. parameters.

3. THE CONSIDERATION OF THE DISPLACEMENT EFFECT OF A HUB

In general the procedure of considering the influence of a rigid body in the nearfield of the propeller on propeller char-acteristics at the design point includes in addition to codition

(c2,) the assumption that any axial variation of the flow field with the propeller present does not violate cOndition (C5) more than is already accepted wihin a first order theory (Section 2.1). If this limit is observed it becomes possible to consider the

effect of a rigid böd.y only on the flow field at the lifting line. This is usually done by introducing the sb called wake fraction w, which relates the actual inflow velocity at the lifting line to

the speed of advance of the considered body.

The following sections deal with the determination of te radial distribution of the wake fraction. This is caused by the hub bàdy which consists of a semi-infinite cylinder bèhind the propeller and of a finite continuation into the flow field ahead of the propeller. (Such a hub bodycan be regarded to be compat-ible with the limits of the theory restated above a±id also out-lined in detail in Section 2.1.). It was shown in Chapter 2 that thelow pattern at the lifting line can be simulated by means of the momentum theory if the optimum condition of B e t z an a

"simplified hub representation" are used withIn a solution of the iòrtex theory (Figs. 6 and 7). $ince the flow f ièld around a H propellér can be descrjbed much better and more easily by the

momentum theory than by the vortex theory, the following method of determining the desired radial distribution of the wake fraction is based on the sink disc representation of the propeller. The radial distribution of the sink density on the. propeller disc is then determined by the additional velocity component at the

lifting line (multiplied by 2), which is in turn dependent on the waké fraction. Thèrefore, starting with the results of Section

2.3, Fig. 7, an iteration procedure becomes necessary in order, finally,.to obtain the same flow pattern at the lifting line and at the sink disc.

-17-Furthermore, it has not yet been clarified in which way the radial distribution of X1 has to be related to the radial distri-bution of = XS(lw) if the condition (2.3.4) shall hold also in the case of w being a non-constant function of x. The follow-ing Section 3.1 deals with that problem on the basis of the momen-tum theory, the results of which are then numerically verified by the vortex theory as will be seen in Section 3.3.

3.1 The wake adapted actuator disc.

D

If the radial distribution of the sink density over an

actuator disc is such that the energy loss has its smallest possible value, the actuator disc is called "wake adapted" in the context of this paper. In order to derive a general expression of for the wake adapted condition the principle of variation will be applied to the problem - following a logicj which was already used in [12] in order to treat a slightly different optimization task.. Since the energy loss - as well as the energy gain - is a matter of definition in propulsion theory, two different cases are to be investigated simultaneously:

Case 1: in this case the power to be gained from the propeller is defined by

vA _dT

X

and the power to be delivered to the propeller, namely

=

J(vA

+ D/2)dT 1 ) dT H H X X (3.1.1) (3.1.2)

is expected to have its smallest possible value. Now introducing the thrust of a ring element of an actuator disc (see also Appendix)

it follows from (3.1.1) _{and (3.1.2) that} =

:J

ED( ED/2)2 D dxD i V4 + e1/2) 2ffxD CIXD _R

Dj

R2

The gained power, (3.1.4), has _{a prescribed value and therefore}

must not change if the function ED is _{varied by a comparatively}

small

function cD:

= PT(CD + scD) - pT(ED)

(3.1.6)

If the delivered _{power, (3.1.5), has a}

_{minimum value}

for a certain radial distribution of D,

then the Small variation of CD _into

+ _{will not. cause any additional power input. Thus:}

= PT(CD ₊ cSED)

-

PT(CD) _o

(3.1.7)

Çombining equation _{(3.1.4) with (3.1.6) änd equation (3.1.5) with} (3.1.7) the following expression will be obtained:

J+

SED)

(V4 + (CD +

_ôcD)/2)

-

_{CD(V4 + ED/2)]xD dxD} = O (3.1.8) [(ED + CD)

(V4 + (6D +

D)/2)2 - _{+ ED/2)2]XD dx} = xH . (3.1.)

As is small by definition, all terms with (cD)2may _{be neglected.}

Hence:

J

6EDÇJA(\JA CD) D dxD _{= O}

-19-+ ED/2) +

3ED/2)

D dxD _{= (3.1.11)}

The integrands of these two integrals contain the same functions in different combinations but the integrals have the same value. This can be true if - and only if - the integrals are

propor-tional:

(vA) 2 _vA

2D

(D)2 (D/2) 2

VAVA

+ ED) = constant (3.1.12)

Because of condition (C4) it can be assumed that ED A and therefore

(VA ₊

cD)/vl

_{= constant (3.1:13)}

or

ED = _{2c1vA (3.1.14)}

(C1 is a constant which depends on the required thrust). Equation (3.1.14) can also be interpreted as follows:

uA=cvA

_(3.1.15)

Under this condition the efficiency of the actuator disc

+ D dxD

(3.1.16)

has its maximum value. If vA = constant, and tjA= constant accord-ing to equation (3.1.15), then equation (3.1.16) is the well known F r o u d e -efficiency. Turning now to the lifting line

repre-T

sentation of the propeller, s. Fig. 4 with U = O, it follows from

H X 11=

J

uAvAu2 D

D x dx,

equation (3.1.15) that the optimum condition _{can be expressed by}

= KJXx (3.1.17)

It will be shown numerically in Section 3.3 that the condition

uT _{= O holds for vA being an arbitrary function of the propeller}

radius if the condition (3.1.17) is observed.

Case 2: it can be derived from the momentum theorem that _the force acting on the system "hub plus propeller" is given _by

i

= p

+ CD!2) 2TTxD dxD _{R2 (3.1.18)}

(No radial contraction of the jet behind the propeller is öonsidered,

(C4)). In this case the power to be delivered to the whole _system

i i

=

J

(V +

CD/2) _dT

= J (V + UA) dT (3.1.19)

is expected to have its smallest possible vá1ue. _{In order to} ôbtain the related function CD(x) the principle of variation will be applied to the problem in the same manner as described in Case i

D.

D D

above. Thus, a change of c into E + c5c will lead to the

follow-ing expressions: i

J[(CD

± cD) (V + (CD + 6CD)/2) - CD(V + CD/2)J D dxD = 0 (3.1.20)

J[(D

D) _{(V +}

(D

_{+ eD)/2)2 -}

_Dv

+ CD/22] D dxD _{= o} o (3.1.2i)

Again this can be true if and only if - the integrands are proportional and hence:

-21-or

= 2C2V (3.1.23)

(C2 depends on the required value of ). Equation (3.1.23) can also be interpreted as follows:

uA=c2v

(3.1.24)

Turning again to the lifting line representation of the propeller it follows that the optimum condition can now be expressed by

= K2 + AX (3.1. 25)

In this case a relation V/ was optimized, which turns out to be the F r o u d e -efficiency for constant.

3.2 The determination of the hub induced flow field at the propeller

disc.

In order to investigate the displacement effect of a hub, the following boundary condition must be satisfied:

(B3) Related to a coordinate system moving with a rigid body there

is no velocity component normal to the surface of the body. When distributing co-axial sink ring over the surface of the hub, condition (B3) determines uniquely the only possible sink strength

for each sink ring, as will be outlined in detail below. For this purpose it will be made use of a generalized velocity vector

which is related to the actual velocity vector as follows: at

p _p

_PQ.

a field point (xe, zj) the velocity vector V9, is induced by K sink rings of radius

4

with center points on the z-axis at z. If these

sink rings represent surface ring elements of small width on which the sink density =

c(4,

Z)

can be assumed to be constant

then

PQ

_'

Q -*PQ

L.... k. 9k k

k=l

defines the generalized velocity vector ,, which can be obtained as described in the Appendix. (The subscripts and k are intro-duced for counting purposes in a numerical scheme. Thus equation (3.2.1) gives the 9,th velocity vector that is induced by K. vortex rings being situated at a kth position, k = 1, (1), K).

Now using the superscripts H for hub and D for' propeller disc, and counting sink rings on the hub by n,v = 1, (1), N and on' the

propeller disc by p = 1, (1), M, the condition (B3), introduced

H H

above,, applied at a point (xv, z) on the hub contour, can be expressed by In detail: -tHD

'

D-*HD D H D y L1 C V tx' ; z < z 1_L Vii (3.2.3) M

r'

DHD

D - D D H

D

= C v x + c(x, z'); > z ij=l and - +HD

IIH)O

e

(V+V

+v

-1 N H HH AZH1 H H ; AzH/cosc (3.2.4) n=l n=V+l

Equation (3.22) may be solved, for the unknown values of by an iteration method as developed by Am t s.b e .r g, [116], and his

students, or by directly solving the system of linear equations, basically following the concept of ' H e s s & S m i t h, [17].

In either case it should be observed that M is chosen to be the greatest integer number which is smaller or equal to (D/2)/AzH so

that AzH

When the N values of. are determined, some other interesting functions can be derived. First of all:

=

VA(x,

_{z') = y}

¡ZDH

(3.2.5)

with -23-N -tDH

'

H DH H H V11 C\, V1 tz /CO5.)

Hence the desired wake fraction can be found to be

w = w(xD, zl ₌ L - VA/V

11 11 11

The tangential velocity component on the hub is given by

=

_z)

yX . H Z H

sinct + V cosa V

+X(HH

HD)

v

= y

+

+Z(HH

+ HD)

and

then the pressure distribution can be derived if the discon-tinuity

2

H2

tp = T/{'rrR [1 - (x ) 1}

is taken into account when passing through the sink disc in direc-tion of the z-axis:

H o

TH2

pv P V _{H D}

- ;z <z

p/2V2

V V (vTH) 2 i 'J p H D H =

--

_{I Z}

>Z ,X

_<i

p/2V2 y y

Finally, the axial force on the hub can be determined by

= _{p 2irx} H _{tana R} v=i (3.2.6) (3.2.7) (3.2.8) (3.2.9) (3.2.10) (3.2.11) (3.2.12) (3.2.13) (3.2.14)

3.3 Examples.

The following examples have two fundamental objectives:

l. For hub form 3, Fig. 12, the displacement effect of the

hub body, given by the radial wake distribution as derived. by the sink disc representation of the propeller, shall be taken into consideration, when calculating the

distribution by means of the lifting line representation of the propeller.

2. For different hub forms, hub contours and hub radii, Fig. 12, the behavior of some characteristic parameters shall be investigated, by means of the sink disc represen-tation of the propeller.

In either case, the design parameters of Fig. 1 remain unchanged.. Therefore any conclusion is to be interpreted with respect to that special propeller loading at a given advance coefficient.

In order to take care of objective 1, the design results Of Figs. 6 and 7 for a hub radius H = 0.2 with a itsimplified hub

representation," Section 2.1, define initially the radial sink

distribution over the actuator disc according to equations (3.1.17), case 1, or (3.1.25), case 2 of Section 3.1. This will also be

H H

accomplished for hub radii x = 0.3 and x = 0.4.. Then at the ontour (a) of hub fòrm 3, Fig. 12, the boundary condition (B3), Section 3.2, will be satisfied and the potential wake distribution över the radius can be derived as outlined in Section 3.2.

D ,. .

Since the sink distribution e (X), which is given by the. addi-tional axial velocity components tjA, _{depends on the wake}

distribu-tion in case 1, see equadistribu-tions (3.1.14) and (3.2.7), CD(x) will be

derived by an iteration procedure e(x) = 2C1V1, V = V.

However, this is done only once, Fig. 19, because. the difference in the results, of cass i and 2. turns out to be very small in the

considered hub problem. (This might be quite different if a ship hull is taken,into consideration).

Finally the wake of the hub body isobtained on the basis of a sink disc representation of the propeller. Now the propeller

-25-blades, represented by lifting lines, aré thought to run in a flow field determined by this wake distributIon. _{(It should be} re-em-phasized that this very common procedure neglects any axial varia-tion of the potential wake, and is therefore restricted to the treatment of hub form 3.) As a result of the vortex theory, a new distribution of additional axial velocity components tjA _{can be}

found. Hence, new values of = ED(UA) _{can be determined and}

making use of the sink disc propeller representation _{once more, a} slightly different wake distribution may be derived at this step of an iteration procedure, which makes use of both propeller

representations and cornes to an end when the wake remains unchanged at subsequent steps.

At each step the results of the lifting line theory

_{- based}

here on the "simplified hub representation" as introduced in Section 2.1 - verify numerically the relations (3.1.17) and (3.1.25). _In all considered examples the additional tangential velocity _components do not exceed the following limitations:

case

i

: IüTivI 0.006 òase 2 :

luT/vi

0.0003

This characteristic of the vortèx theory is the essential basis _for the above outlined iteration procedure of determining "hub correc-tions" on the distribution of the hydrodynamic advance _{coefficients.} Results of the earlier used example, Fig. 1 et ai., _{are shown in} Fig. 13. _{The corrections can be seen to be of the order of "lifting}

surface corrections" as far as blade sections close _{to the hub are} concerned and therefore cainot be ignored when running _{or evaluating} propeller tests which are accomplished by the aid of _{a test device} that consists of a long shaft with the propeller ahead of the

bear-ings.

Turning now to the objective 2, the above mentioned character-is.tic parameters are thrust, axial force on the hub, efficiency and

radial distribution of sink strength, tangential velocities and also pressure distribution over the hub contoir in the axial

direc-tion. _{These parameters shall be observed when varying hub form, hub}

over the propeller disc as was obtained for the hub form 3 at the. last step of the above outlined iteration procedure _{on the basis} of different propeller representations. _{When changing the hub}

radius, = 0.2, (0.1), 0.4, the relation xH/(L/R - 1) is kept constant, i.e. the length L of the hub body varies with its radius In the case of the semi-infinite cylindrical part of hub form 2 and 3, Fig. 12, this cylinder is represented by sink rings onlyto a distance R in front or behind the propeller disc.

The results are given in Figs. 14 to 20. At first it can be seen from Figs. 17, 18, 19 and 20 that the sink density on the hub has very small valués at distance ±R from the sink disc so that the above introduced simplification of making the semi-infinite hub cylinder a quasi-finite one seems to be valid. Fig. 14 repre-sents the zero-loading case, or the case of the body moving in an undisturbed flow in the direätion of its axis, showing already the basic influence of the hub contour. When comparing these results to those of Figs. 15 and 16, one influence of the propeller 1ow immediately becomes obvious, namely the occurence of an axial force acting on the hub body. This is. because of the non-Symmetrical behavior of the characteristic function and also because of the pressure. discontinuity at the propeller disc. Comparing now Figs..

17 and 18 with respect to the pressure distribution, it turns out that in the case of the hub form 3, Fig. 18, the pressure gradient is negative nearly all over the hub but in the case of hub form 2, Fig. 17, the pressure gradient rises to high positive values just behind the propeller disc. Therefore flow separation at the hub will probably occur in real fluid flow and a "viscous hub wake" behind the propeller must be expected. Thus, any theoretical

treatment of,the hub form 2 by means of conditiòn (Ci), Section 2.1, seems very questionable, but an investigation of hub form 3 by

means of a potential theory can be expected to hold under real fluid flow conditions as well. Finally comparing Figs. 18, 19 and 20, the irfluence of the hub radius can be seen to be of relatively Small influence on the considered characteristic functions. An

interest-ing fact may be seen in the influence of. the hub wake on the thrust values which have a maximum value at a certain hub radiué when the

-27-sink distribution remains unchanged. This is easily understood when comparing the increasing loss of thrust caused by growing hub radii with the gain of thrust that is caused by higher speed of

advance of the propeller disc, when the displacement effect of the hub is taken into consideration.

4.

SUMMARY

Propeller design on a theoretical basis usually does not con-sider the displacement effect of the hub body. However, a theoret ical investigation öf this subject shows clearly that the radial distribution of the hub induced effective wake demands corrections of the hydrödynamic advance coefficients close to the hub, which ar of the order of lifting surface corrections. Based on a special hub form, a procedure for obtaining these "hub correctiOns" is outlined and the results of an example are presented. The special hub form in question allows the bound circulation of the lifting

line to be finite at the hub. Introducing a "simplified hub

representation" for the down stream part of the hüb body, the vorte theory is shown to provide the same flow pattern a.t the lifting line as is given by the momentum theory at the sink disc for an "optimum propeller." Therefore, both propeller representations are iteratively used for the purpose of determining the potential wake of the hub. Basic relationships, derived for the wake adapted

kactuator disc1 are app1ie to the lifting line representation of the propeller and the validity of this procedure is numerically confirme.

Investigations based only on the sink disc representation of the propeller allow the treatment of arbitrary hub forms. In this knanner, the influence of hub form, hub contour and hüb radius on

thrust, "thrust deduction.," ef.ficiency and radial distribution of wake is studied. However, the behaviour of the pressure gradient over the hub boundary in the axial direction indicates that the treatment of "practical" hub forms by means of a potential theory would be questionable.

FIGURES

-29-Design results for constant pitch distribution, different procedures.

Design results for variable pitch distribution, different procedures.

Velocity components at the lifting line, schematically.

Counting principle for free vortices and field points.

Design results for constant pitch distribution, different procedures.

Design results for variable pitch distribution, different procedures.

Definition of geometrical hub parameters.

Radial distribution of the hydrodynamic advance coefficient, with and without hub corrections. Hub form i in parallel flow.

Hub form 1 in propeller flow, 0.2.

H

Hub form 2 in propeller flow, x = 0.2.

Hub form 3 in propeller flow, = 0.2.

Hub form 3 in propeller flow, 0.2, vari-ation of the radial sink distribution over the propeller disc.

Hub form 3 in propeller flow, = 0.4.

Fig. 1 Fig. 2,3 Fig. 4 Fig. 5 Fig. 6,7 Fig. 8,9 Fig. 10,11 Fig. 12 Fig. 13 Fig. 14 Fig. 15,16 Fig. 17 Fig. 18 Fig. 19 Fig. 20

FIG. i I

08

DESIGN

Z=5

Xx=O.2i4

I

H

0.7

DATA:

0.6

2

05

PROCEDURE: DESIGN 1: 2: 3. LERBS MORGAN ECKHARDT-(GOLDSTEIN) (TACHMINDJI) J

0L

0.01

T1

0.02

01

0.03

02

03

0.04

0.05

DESIGN DATA AND PROCEDURE

SEE FIG. i

-31-FIG. 2

0.1

0.2

0.3

0.91

0.02

0.03

0.O

_0.05

0.1

0.2

0.3

0.01

0.02

0.03

0.04

0.05

0.8

0.7.

0.6

0.5

0.4

0.3

DESIGN DATA AN D PROCEDURE

SEE FIG1 i

FIG, 3

m

M

M-2

M-4

M-6

7

5

3

i

1

t

m

k I .

°IN

N-2

oN-6

FIG.

X

DESIGN DATA

0.7

_{SEE FIG. i} DESIGN PROCEDURE LERBS AUTHOR -35-AS 2 (IMAGE SYSTEM OF 1(ERWIN a LEOPOLD FIG. 6

2

A

0.1

0.2

0.3

0.01

0.02

0.03

0.04

0.05

Cs

0.9

0.3

0.2

I

0.8

FIGI

0.1

0.2

D - DESIGN CONDITIONS

AS FIG. 6

0.3

uT1Aiy

-37.-FIGI 8

0.9

los

f

AXNI

0.7

U.

6

DESIGN DATA SEE FIGI DESIGN

0.5

i

-6

/

0.4

PROCEDURE SEE FIG.

001

o.p2

01

0.03

02

0.4

0.05

0.8

0.?

0.6

0.5

0.4

0.3

0.2

UI/v

DESIGN CONDITIONS AS FIG 8

FIG, Y

0.1

0.2

0.3

UT,A/V

DESIGF4 DATA SEE FIG. i DESIGN PROCEDURE SEE FIG1 6

-39-0.1

0.2

0.01

0.02

0.03

0.04

FIG. 10

0.3

0.05

FIG. 11

T,A

_/v

N

08

L°7

2

0.6

UTiV,

05

2

V

31.

Ji

L

0h

0.3

r

-'-. __dl_ DESIGN

6

01

02

0

CONTOU R

(b)

CONTOUR

(a)

POSITION OF PROPELLER DISC

L-R

L-R_

FLOW DIRECTION CONTOUR

(a)

-41-

FIG. 12

0.9

0.8

_DESIGN DATA

0.7

SEE FIG, i

DESIGN PROCEDURE:

0.6

AUTHOR HUB FORM 3

0.5.

0.4

0.3

0.2

-FIG, 13

NO HUB

0.20

0.25

0.30

0.35

1.0

0.5

-0.5

-1.0

1.0

0.5

-0.5

-1.0

1.5-

1.0-

0.5-

-0.5-

-1.0--1.5

-FIG. 15

.1

108430kp

3630kp

CThSI= 1.22

,

0.811

1.5-

1.0-

Q5-

-0.5-

-1.0-

-1.5-vTH,v

-45-FIG. 16

-T =108 300kp

tT= 4 330 kp

CThSI =1.22

çì

=0.811

1.5

1.o

05--0.5-

-1.0-FIG, 17

T 104 340kp

tT=2 210 kp

H..02

ThS1=i.i7

=O.808

(pH.p°)/(q/2 v2

FIG. 18

T=103940kp

zT= 1180 k.p

xHO.2

cTi

=1.16V

1.5

1.0-

_0.5-

_-0.5-

-1.0-,

_/

/

_/

-,(pH_p0,(q,2

y

2)

N

VTH1V

CTI

=1.20

=

0.791

FIG. 20

1=106 960 kp

A= 3240kp

xH=0.4

j

LIST OF ABBREVIATIONS*

C Constant

Superscripts: ThSi, for thrust loading

T/(p/2 V2irD2/4)

Subscripts: 1, for a relationship

and VA 2, for a relationship

andy

D Propeller di.ameter F Circumferential component of lift G Non-dimensional strength of bound circulatiàn

Superscript: S, for ship speed: GS _{= r/NDv)}

K Force acting on a sink Subscripts: L Hub length P Power Superscripts: R Pröpeller radius S Sûrface coefficient: between between ED

1,2, for a relationship between X1

and XX K being a constant

D, for delivered at propeller shaft T, for definition on the thrust of

the propeller blades

Not mentioned subscripts are counting indices. Vectors are indicated by an arrow in the text.

T Thrust of the propeller blades, axial component of the lift

U Additional velocity at the lifting line

Superscripts:

Ship speed

Superscripts: A, for sped of advance of a pro peller section

p p p

PQ, for velocity at (x , ,

z),

induced by a sik ring öf radiu

X0 with center point in, z0, its axis being z.

HH,DH,HD, see PQ with D for disc and H fo hub

TH, for tangential veiÖcity at the hub contour

x,z, for radial and axial Velocity component at the hub contour

Z Number of propeller blades

Magnitude of a f ied. point vector

Magnitudç of a unit vector

Superscripts: n, for inward normal of a surface element

x,z, for radial and axial direction

Induction factor

Superscripts: A,T, see U

-51-A, for axial component

hub

m _{Strength of a sink}

p Pressure

Superscripts: _{o, for undisturbed flow} H, for hub contour

q Magnitude of velocity

indticed by a sink.

r Radius

Superscript: o, for free vortex line

y _{Generalized velocity:}

V/(c's) _{Superscripts: see V}

w _{Effective wake fraction}

Non-dimensional radius: nR

Superscript: o, see r

D, for propeller disc H, for hub contour

for field point for sink point z _{Axial coordinate}

Superscripts: _{o, first considered hub point in} up stream direction

others, see x

p

Pressure diSCnt11ty in

the propeller disc

Width of a sink ring

Angle of advance Superscript: Sink density Superscripts: Efficiency Superscript:

Angle velocity öf the. propeller X Advance coefficient:

xtan

Superscripts: Reference angle Superscripts: p Density of water Non-dimensional radial coordinate: Subscript:

3-r Strength of bOund circulation

Angle between z-axis and its projection on the surface of a sink riñg around the same

axis Superscripts: H, for hub element

Q, for a ring with radius x0

i, for hydrodynamic angle of advance, Fig. 4

D, for propeller disò H, for hub surface

P,Q, for general cöórdinates, see x

F, for Froudè-efficiency

i.,

for hydrodynaadvance

coèf-ficient: xtan

ship advance coefficientj:

v/(R.)

for field point for sink point

o, for radius of a vortex line with±i z

APPENDIX

Defining cylindrical coordinates by (x, w, z), a sink is assumed at a point (xe, w0, z0) in a flow field. A force acting on the sink was derived for example in [18] to be

d (5.1.1)

In this formula, is the velocity vector of the flow field in Q, the influence of the sink itself being disregarded. The strength of the sink is given by dm0 and therefore

=:_dm0/(47ra2) /a (5.1.2)

is the sink induced velocity vector at a field. point (xv, w, zr). "a" is the absolute value of the field point vector , directed from Q to P.

If sinks are arbitrarily distributed over a surface S, a sur-face element ds is now defined to be small enough to allow the

assumption of constant sink distribution within its region. Intro-ducing the sink density E1 by the relation äm0 = EdS, formulas

(5.1.1) and (5.1.2) can be applied also to "surface sinks," provided that a >

On this basis, the thrust of a sink disc will be derived in the following way. If there is a sink ring of width dxD and radius XDR moving parallel to its axis z with a velocity vA, _{then the}

relation (5.1.1) defines a force

dT1 = - 6D 2TrR2xDdxM (5.1.3)

acting on the sink ring if is the sink density distributed

constantly over dxD

and

XDRdW. Considering this sink ring part of the sink disc representation of the propeller, the down stream side of the sink ring - with siik density CD/2 - needs the superposition of a velocity vector D to provide continuous fluid flow through

the disc, see e.g. [19]. Thus

PQ

-55-is an additional force acting on the propeller d-55-isc element. Corri-bining equation (5.1.3) and (5.1.4) the thrust on a propeller ring elemént turns out to be

dT ₌

_1l1

_IdT2I

(5.1.5)

dT =

PCDWA

+CD/2)

2D

dxDR2

(The negative sïgn òan be dropped because of the conventional definition of the direction of the propeller thrust.)

Applying now the formula (5.1.2) on a sink ring of width

dz/cos c - which becomes

if + ir/2 - aid defining the

field point for reasons of symmetry only by (x, z) with E O,

the velocity vector induced in (x, z) cán be expressed by 2iî

+PQ x

Qf la

_{Q Q}

dv =

_-r-

c

j - du dz /cos

6a

Dividing by the sink density of the considered sink ring, and

by

the width of the sink ring, a "generalized velocity vector" is

.gïven by

2'rr

VPQ)X, (vPQ)z}

= - 21..

1 -4

_a (5.1.6) (5.1.7)

In a rectangular coordinate system with its axes pointing in the

- -

-+x +z

directiöns of the unit vectors

{i, j,

k}, (ie

, kEe ), the vector is given by its components

a,

a', ak}= {x-

XCOS

-xsin w, z

- z} (5.1.8) Thus

a =

((x2

+

(x)2-2xx'cos

w ±. (z -

z)2)'2

(5.1.9)

Using equation (5.1.7) the generalized velocity components may therefore by expressed by 2 PQ 1 Ql x

- xcos

P

V j

_Xj

3 U(A) a O (5.1.10)

PQz

(y 4ir k= 1 with ak = ((x1')2 + (x)2 -2x1'x0cos W + _{(z -} z)2)1"2 _(5.1.14)

It is recommended that LIZ/COS (or xAw if

¶12). This can be done by assuming K to be the largest integer

number that is smaller than or equal to

21Tx/(L\z/cos

c) or

2rrx0/ix respectively. and

(v)Z

= 2ff a3 dw _(5.1.11)

The analytical treatment of these expressions can be performed according to [19], (Appendix). If a high speed computer is avail-able the integration can be replaced by a summation, so that

K

(v)C

--:: -U) 1)Q _(5.1.12) k=1 K LW (5.1.13)

-57-REFERENCES

L e r b s, H.W.: Moderately Loaded Prbpellers with a Finite Number of Blades and an Arbitrary Distribution of Circulation Trans. SNAME,VÒ1. 60, 1952.

L e r b s, H.W.: Ergebnisse der angewandten Theorie des Schiffspropellers, iahrb. STG, 49. Band, 1955.

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