On computer aided propeller design

TECHNISCHE UNIVERSITEJT oum

&heepshydromechanlca zchIef

Mokeiweg 2,2828 CD Deift

T 015-. _{Facois mia}

ON COMPUTER-AIDED PROPELLER DESIGN

Carsten Östergaard

Department of Naval Architecture and Marine Engineering

The University of Midhigan April, 1970

No. 088

'

PREFACE

The following comments on computer-aided propeller design

were prepared as notes for a leçture given at the

Depart-ment of Naval Architecture and Marine Engineering of The

University of Michigan. These notes may not be understood

as a program description, although they are based on a

com-puter program developed at the Lehrstuhl für

Schiffshydrody-namik of the Technische Universität Berlin. All

recornxnenda-tions given here may therefore be taken as well proven, but the problems treated in this context should be regarded only

as the more striking examples of a whole series of related

questions.

May 1970 Carsten Östergaard

1. ON THE GENERAL DESIGN PROBLEM

A design procedure generally depends on the kind of the design

problem that is to be treated. Propeller design is no exception

and a computer progam for propeller design must be limited

therefore to a special range of design problems.

Considering e.g. the design of a tug boat propeller, at

least two design conditions are of interest, namely the behind

condition at maximum ship speed and the towing condition which

is necessarily defined by a lower speed of advance. Thus any

treatment of this problem requires the propeller characteristics

over a certain range of ship speed and results of systematic

test series will determine the design procedure. This can be

done by the aid of a computer, of course, but the related

problems are nothing more or less than interpolation problems

and therefore not of specific interest from the viewpoint of

computer-aided propeller design. However, when the design

prob-lem is based on only one condition, e.g. maximum ship speed,

it can be treated by means of propeller theory, the advantage

of which being the consideration of such important parameters

as radial distribution of loading, wake, stress, etc.

Theref ore such a procedure is open to optimization in any

direction, i.e. according to any performance index, including

manufacturing costs, etc. But, although the theoretical

approach in various details of the propeller design could

possibly provide results of very good accuracy, it is basically

the amount of necessary computing time that leads to an

-2-approximative description of the design conditions, e.g. the

consideration of a steady fluid flow. Those basic assumptions

have a series of consequences which must be kept in mind when

results of a theoretical design procedure are to be interpreted i

A propeller designer is therefore accustomed to correcting the

final results by experience or even by experiment. Because

this final step cannot be done without having a good under-standing of the foregoing treatment of the design problem it might be useful to have at least a general outlineof it and in the following sections some different aspects of the

pro-peller design problem are described for this reason. In.

addi-tion it is assumed that the numerical part can be accomplished.

with the aid of a computer. This leads to a somehow unusual

handling of the design procedure and therefore seems worth

being covered as a special subject.

2.

ON A BASIC DESIGN PROBLEM

For given values of VS (ship speed), T (propeller thrust),

n (rate of revo].ution) and D (propeller diameter) the delivered

power at the propeller shaft, D, is to be determined. It

should be mentiond that D is not only a fünction of these

given values but any other influence may be regarded as some kind of side condition that will be explained when occurring

in the following context.

Following now the first of the attached flow charts (Fig.

solùtion of equation (5)

yes

(step 1) (step 2) (step 3)

H

given parameters V, T, n, D determine no (step 7) yes

()

L(step

(step 5 i (step 4) 6) yes (step 3a) Fig. i yes = n + i, t

cT5

= T

p _s 2'ïD2

4

-4-Ste. 1: Definition of a thrust coefficient

t should be mentioned that this coefficient. cannot tell

any-thingabout the specific loading of the propeller disk if the speed of advance of the propeller, V, is different from

V.

In order to treat the hydrodynamics of the design procedure separately for real and ideal fluid flow a somewhat higher thrust coefficient must be assumed for the latter mentioned

condition;

cThSi _k cThS

and a first guess for k1, e.g. k1 = 1.04, will be verifjed änd possibly corrected after having treated the real fluid flow

problem (step 7).

Step 2: Determination of some hydrodynamic parameters for

the. ideal fluid flow

According to cThS1 a radial. distribution of he.

lift-chord-product Ct' c and the relatéd geömetry àf the free vortiçes behind the propeller can be derived, making the usual assumptioñ

that the pitch of the "free" vortices does not change in axial

dIrection so that the desired value of the hydrodynamic angle

of advance, determines the whole vortex system This leads

5

blades which must be replaced by straight "lifting lines" of

bound circulation, rb, at this second step of the design

pro-cedure. Neglecting radial velocity components for moderately

loaded propellers, the resultant velocity vector, V, at any

given radius of the lifting line may be replaced by its

com-ponents in the following way:

Fig. 2

Assuming shock-free entrance as the desired operating condition

for the blade section no additional angle of. incidence, must be

considered and the lift acting on the blade section is given by

the law of Kutta-Joukowski

dL = pvrdr

so that L dL 2r1 c = (Vr1)2 _dr -rw

vSu

_w)

and using a relation easily derived from Fig.

rw -. uT

vSu

- w).+ uA

.1

cos sin

the lift-chord-product is then given by

C-

2lTDs.inGS

(1)

(1-w)+--V

or

C =

Dividing now equation (1) by equation (2) the so called

hydro-dynamic advance coefficient is given by

(1-w) +-.

i i V

X

=xtan

-with X and x = . And the additional velocity components

A,T may be expressed by

i 2 2nD coS T rw U

Vsvs

T X U C. V dG:(X°) dx0 (2) (4)

X.,

(3)

i 2

i

i X

i

'V

A

-tan

----=

V A

with AX x tan = AS(l - w) (see Fig. 2).

Neglecting mathematical difficulties for the moment, equation

(5) defines the relation

A: A1 (x) = A1 (Y (x)), Y (x) = GS (x)

(5a)

or B:

G5x =G(Y(x)),

Y(x) A1(x)

So for any assumptionpf'(x) a solution of theproblem exists and the remaining prob1emis only the definition of a function

7

H

(x being the dimensionless hub radius).

Here, the so called induction factors .A,T are geometrical

functions of the shape of the free vortex lineswhich are

helices of dimensionless radius

x0 and pitch angleand

also functions of the radius x at the lifting line where uA1

are to be determined. For further information see ref-vs

erence [1] or [2}.* In this concept the functions A,T are

considered as known functions whenever the parameters x,

and Z (the number of blades) are given.

Combining equatiöns (3) and (4), it turns out that

+ tan

dGS(x0)

dx0 =

*A list of references is àttached at the end.

o

X,

cT _{= 4Z}

f

cThSi = 4Z

f

X As V 8

ThSi.

Y(x) in such a way that the desired value of C is also

given by

G5dx

UAxS

+- --Gdx

(These equations follow from the defjnition of

cT,.

applying

1

the law àf Kutta-Joukowski in order to find T1 =

dT', see

Fig. 2). With the aid of a computer a first assumption for the,

distribution of Y(x) is rather arbitrary and is usually made under the consideration of such side-conditions as cavitation

behaviour, weight, moment of inertia and so on. (The relation

between Y(x) and these side-conditions will be understood in

connection with step 4 and step 6 below.) Now running the

solution of equation (5) iteratively changing' the function Y(x),

the desired value of cT can be.obtained. But there remains

the question of how to change Y (x), in such a way that the just

mentioned side-conditions are satisfied to the same extent at the first and final steps of this iteration procedure. The

following recommendations are made in order to get an invariant

kind of loading distribution but do. not claim to be of general

advantage in this direction. So at the th iteration one might

vs _{vs X1}

.ThSi

_{- cTh} _i(0..7)Í1

nl ]

XX(O7)

cA_1(0.;) -(8)

Turning now to the solution of equation (5), in order to treat

problem A of the relations (5a), the left hand integral of (5)

can be evaluated by making a first guess for the unknown function

e.g. X1(x) = ÀX(x). The right hand side of equatïon (5) supplies then a new function X1(x) which can be used as a new

guess in a second evaluation, of the left hand integral. This

again leads to an iteration procedure easily accomplished with

the aid of a computer. But in addition these may occur a

numerical problem related to the evaluation of the left hand integral itsèlf; if, for example, the concept of Lerbs is used

for this purpose,. [3],.the bound circulation is represented by

a Fourier-sine-series and the induction. factors are expanded

into a Fourier-cosine-series. In addition, this procedure use

G(x)

=

G_(x)

or X(x) = C

X(x)

(7)

and also X1(x) = C(À1(x) - XX(x))

The value of can be obtained in the way given by Lerbs in

[31 and C can be determined at x = 0.7 using.a formula given

10

-requires the assumed functionY(x) at some very special radii which are given by

H H

l+x

1-x

x(c)

= 2 2 COS c with O,

(a), r, and N being the considered numbers of terms

in the Fourier-series. There is no problem if Y(x) is given as!

a continuous function (see e.g. [3]). But in order to have a

completely free choice of the input function Y(x), a finite number of related input data (x Y) is desirable and the re-quired values Y(x()) will be obtained by interpolation.

There-fore, a.solution of equation

of the interpolation method.!

Turning back to problem A of the relations (5a) it is worth dG(x)

noting that. the preparation of the input data _dx at x

usually gives some trouble and requires furthermore a good deal of experience, because the solution X1(x) is grossly

dS

changed by a slight variation in

Gx

Therefore, problem B in (5a) has more practical advantages than the prcblem A and

most of the advanced propeller programa are based on this case.!

Equation (5) is treated then as an integral equation of the first kind the kernel function of which being.

K(x,x0)

(A

+ tan iiT),(x - x°)

Existing solutions depend of cóurse on the boundary conditions

being introduced änd a discússion of this problem can be found

in [5]. As to different methods being developed in Order to

solve the integral equation (5) see e.g. [li, [5] and [6]. In

any case, A or B, a relation between X1(x) and GS(x) can be

found which satisfies the side condition (6). Using equation

(4) the desired value of cL c is given by equation (1) or

i . ... i

(2), being defined in (3) as a function of X

It should be mphasized that it is still impossible to make

use of a lifting surface theorythe blade outline is not given

yetand as all following steps will necessarily depend on the

results of the lifting line theory it is obvious' that this basic

concept is of utmost importance in propeller design. In addition,

all second order influences which are related to the blade sur-face, e.g. skew-back and section thickness distribution must be

skipped for the moment. But some other second order influences

which are not covered yet, e.g. the influences of rake and hub,

should be treated at this early step of the design procedure.

Unfortunately, the state of the art in propeller theory is not as advanced as might be desired to enable the problem of rake

to be handled. As far as the hub is concerned, some useful

information may be found in [5] and [6].

Anyway,. these corrections do not influence the design

pro-cedure itself which may be continued as follows:

Step 3: ' Determination of the radial thickness distribution

For the moment a first, guess for the maximum thickness of any

considered blade section is necessary. This can be done by

12

-published also in [4]. The blade outline can be determined then

as descrIbed in step 4 below.

Step 3a: Variation of th thickness distribution

If the stress analysissee step 6leads to values of maximum

stresses that are different from the greatest allowable stress

values given by the propeller material at any considered

pro-peller

radius the following relation leads

to smaller differences after a subsequent strçss analysis:

s

n-1

_tn_1

s

being the maximum stress as derived in step 6 in a

fore-going (n 1)t11 iteration and e being 3 or 4 or even higher, de-pending on the desired accurac.y. It is practical to change e

while the iteration goes on.

If the difference between 5n1 and s is sufficiently small

at all considered radii, the design procedure is to be continued

with step 7. Otherwise step 4 and following steps 5 and 6 must

be repeated But this procedure is not practicable near the tip

of the propeller blade where the attained maximum stress tends

to very small values or even

to zero at the tip itself.

As the

maximum thickness of,a section

cannot be smaller than a certain

quantity of the propeller diameter

(e.g. t 0.0025 D)

be-cause of manufacturing

limitations, equation (9) cannot be

(9)

must be determined by a f airing process there.

Step. 4: Derivation of the blade outline

It is necessary now to make a choice of the kind of blade

sec-tions... The use of NACA 16, a = 0.8 is recommended here. The

so called cavitation number, given by

q =

p0 +

pg (h -q max '100 13 -V

-p

with q = + (wr)2)

defines that pressure drop at radius r which leads to cavitation.

(The term Po is the static pressure at the water surface, the

vapor pressure and h is the, distance between propeller shaft and

water surface.) So, as the greatest allowable pressure drop max

must not exceed the value of this cavitation number-if cavitation is to be avoidedand as the lnumber-ift at any section has a prescribed value, given by cL . c from step 2, the length

of the section cannot be smaller than a certain chord length

min max

c . And because the pressure drop tp must be derived under

two-dimensional fluid flow assumptions, a safety margin, S, is introduced so as to reduce the allowable pressure drop;

The safety margin as defined in this formula usually has an

average value of S, 20, but this value must be discussed

14

-maintained for a propeller working in a circumferentially vary-ing wake (e.g. behind struts)see [7]and t:he computer program must be open to a free choice of this safety margin at any

con-sidered radius. The blade outline., given by as a function

of cL, 0V and S, is therefore easily tò be influenced by the

choice of Y(x)see equations (5a) and (1) or (2) in step

2-and also be the choice of the safety margin S which on the

contraryif the blade outline is givendescribes the cavitatiön

characteristics of the propeller. (This case can occur when,

a ducted propeller with Kaplan-type blades is to be designed.)

As to the details of the procedure itself complete

inora-tion can be found in [4] or [7]. It should be mentioned here

'that this procedure supplies also the camber f of the raeanline

which is to produce the required lift predominately.

Step 5: Corrççpn of the blade outline (optioafl

If the bound circulation tends to very small values na.r the hub,

the smallest 'possible chord length c, as derived in step 4,

would tend to very small values toosee equation (1) or (2),

But then the maximum thickness of the section, which is related

to a given value, of maximum allowable stress by an iteration

pro-cedure already described as step 3a with equation (9), would possibly grow to such an extent, that the section cannot be

re-garded as slender in the sense of the theory of wing sections,

namely 0.25, and the derivations of step 4 would become

15

-for the hub section should be prescribed by the user of a

computer program. If the value of . given by step 4 at the

hub exceeds that prescribed value, the blade outline must be

corrected in such a way that all results of step 4 are ignored

whenever the given chord length, is smaller than a chord

length extending fore arid aft to two straight lines passing

through the nose and tail points of. the section at the hub and

drawn tangent to the blade outline.

Step 6: CalculatiQp of stresses

Up to this step, the loading distribution over the propeller

blade is given (CL c from step 2 as a function of Y(x)) and

i

its geometric shape is known too: from step 2 (no additional

angle of incidence is required for the shock-free entrance

con-dition), the section shape chosen in step 4, t from step 3 or

3a, f and c also from step 4. If there were any valid method

for a stress analysis it should become possible now to determine

the maximum stress value within any blade section under

considera-tion. However, stress analysis remains, the weakest point in a

propeller design proc.e.dure, and that is because of various

rea-sons, computing. time being only one of them., So the simple

beam theory is still to be used in a design procedure.. Referring

e.g. to [4], the stresses can be determined only at nose, tail and at a back point of the section., Assuming that these points are

the critical ones, the maximum stress within the contour of the

16

-at one of these three points.

As seen in step 3a, it is mainly the radial distribution of stress that determines the blade thickness distribution and therefore the related values öf propeller weight and moment of inertia are strongly influenced by the loading condition which is given by the choice of Y(x)see step 2.

The details of the stress analysis may be looked up in [4]. It should be noted only that the obtained values are generally

smaller than thöse derived by the use of other methods and a safety margin should be recommended when the allowable values

of stress are prepared as a part of the data input. It is alsö

useful to reduce these values to a certain extent at the outer blade sections..

Step 7: Thrust verification in real fluid flow

Fig. 2 shows the basic assumption that the real fluid causes

ari additional force, D (drag), acting in the direction of the

resultant velocity vector V. Reactions on the lift, dL1, or

on the circulationif there are anyare ignored. Similar to the lift coefficient c*, a so called drag coefficient can

be defined and its value must be determined by experiment. For

the chosen NACA 16, a = 0.8 blade section an average value of

a formula cT _{= 4Z} 17 -T s (1 - tan 1)G dx v, (10)

can be derived from Fig. 2 (compare with equation (6)). If

this thrust coefficient is equal to the desired onesee step

1the calculations are now brought up to a level where an

investigation of those second order effects which are related to

blade surface, becomes possible. But it must be emphasized

once more that there is no propeller design based on lifting

surface theory alone. And that is because the contribution of

this theory to the propeller design consists of nothing else

but corections of camber and angle of incidenceaccomplished

mainly in order to adapt the blade sections to the propeller

induced flow curvature, so that the two dimensional section

characteristics remain unchanged when the section is used as

part of the three dimensional propeller blade. Furthermore,

the bound circulation as derived in step 2see equation

(5)-is d(5)-istributed in a certain manner over the chord of a section

in lifting surface theory, but the already determined radial

distribution of the bound circulation over the lifting line

remains untouched. Therefore, any result of the lifting

sur-face theory is tobe discussed together with the foregoing:

treatment of the lifting line theory and that is basically the

with

ThS

(This formula follows from Fig. 2 in a similar way as C

using

Di

=

rwdC1, dci given by the law of Kutta-Joukowski.):

Otherwise anew thrust coefficient for the ideal fluid flow.

must be assumed, e.g.

cThS i

ThSi

ThS

n-2

Cn

-c

_ThSi

and an

th

iteration through the steps 2. to 7 will be necessary.

3.

ON A BASIC OPTIMIZATION PROBLEM

By the aid of ä high speed computer

it becomes possible to

accoinplisha series of comparative runs of the

basic design

1

=4zL(;:k_Ï.

j. (i+

18

-As already mentioned in the final note of step 2, these

second order effects havé no significant influence on the

de-sign procedure itself so that the equality of the attained and

the desired thrust coefficient mean the end of the basic design

problem covered in this second Section and the delivered power

is then given by

Çp uP2

E

i

n n = n ± .ln

f

n-variation determine n

t

yes

L.

J = .804 .5 11 full optimi - zation determine yes

.j

D-variation Given parameters VS, T, n or ,. determine D or n e D

-t

D = D ± .1D Fig. 3

20

-problem just descrí.bed in Section 2 (y, v-problem according to

Fig. 1). So iteratively changing n or D in every run it is

easy to find the smallest value of the delivered power The

related values öf n or D are then called optimal. The

treat-ment öf this very special optimization problem is outlined on

the second of the attached flow charts (Fig. 3, , yr-problem).

Beginning with a first guess for n or Daccording to any syste-matic series dataat least three runs are necessary tò find the

smallest delivered pöwer by the use of a vertical quadratic

parabola. This procedure is called full optimization" in this

context, while thé first assumption for D or n gives only an approximation but leads to adequate results for certain steps

in a design. problem covered in the following Sectjon, called

there "partial optimization."

In connection with this optimization problem the striking advantage of the outlined procedure may be seen in the fact that the actual radial wake dist:ribution behind the shipat least the nominal wakecan be considered. On the other hand the handling of wake propeller design by theory in the

de-scribed waysee Section 2seems to be questionable according

to several assumptions, one of them being the definition of

constant pitch of the "free" vortex sheet, disregarding the

in-fluence of the axial decrease of the potential wake behind the

ship. This should be kept in mind when the propeller efficienôy

B ₌

T.VS

D p

Using this formula any thrust deduction must bé taken under consideration when the magnitude of T is defined as one of the

input parameters and the average wake may be derived from the

relation; T -- - (1 - E tan 1)GSdx = xS _V (1 (1 -T - _{(1 - e tan} 3.)GS _dx, X V 21

-(compare with equation (10)).

But thIs is only a suggestion that seems to be compatible

with the treatment of the basic design problem in Section 2.

4. ON THE DETERMINATION OF THE DESIGN POINT FOR A GIVEN VALUE

OF THE DELIVERED POWER USING THE RESULTS OF A SELF-PROPULSION

MODEL TEST

In this case, it is assumed that the required propeller thrust

isa known function of ship speed, namely T = T(VS). It could

be found for example by means of a model self-propulsion test

using a stock propeller. As shown in Section 3 there is no

difficulty in treatjng an optimization procedurè according to n or D being a function of the maximum efficiency. So only in

22

-order to simplify the following explanations it is assumed for

the moment that n and D are given.

As shOwn on the third of the attached flow charts (Fig. 4,

cx, w-problem), at a first step (i 1) a velocity Sm is chosen

as the arithmetic average of the lower and upper limit of the

span over which the thrust values are given:

Sm (VSU

V)

₊

By interpolation the related thrust T(VSm) can be found and

the basic design procedure of the y, v-problem (as describèd

in Section 2) can be applied.

This procedure provides a value P which must be compared with the given iälúe It is then possible to define a

smaller range of values v <

VS <.V'where

_{the design}

point-given by V (ED) and T(VS) _{--should be found.}

Repeating this procedure once more (i = 2) this-range can be expected to be small enough for the following final procedure

(i = 3)

For the three given conditions T(VSi, Sm, three values

of the deliveredpower P(V,

.vSm,..V) are dtermined by three different runs of the:y, v-problem and by interpolation a value

VS(PD) _{can be found. Obtaining the related thrust} T(V5) in the

already described way by interpolation, a final run (i= 4)

leads to all desiredpropeller data, and usually verifies the

equality of the given and the finally attained value P. But if thiá equality cannot be verified to the desired extent an

23 -design f or vsml I

Îtt

'i .Ie I I vS.Q = vSTfl j _I

v

=Sm

_j

L__j L

D <"att

-i

Fig. 4 given parameters D _T(VS) n (and, or) D determine V, T

(Dorn)

design -

_S,m,n

j

forV

f

I. design.fOrVS

T = T(V

If the problem is such that one of the parameters n or D are to be optimized (in the sense as described in Section 3) the basic steps of the procedure (i = i and 2) are accomplished under the assumption-of "partial optimization" (seeSection 3)

and Only the steps 3 and 4 are accomplished by making. use öf

a full optimization procedure which is outlined as the ,

rr-problem in Section 3.

However, if the automation of a design procedure is

de-veloped to that. extent there are usually two related problems

occurring:

The first problem leads to the question, whether the amount

of computiflg time and cost is still acceptable or not? There

is, of course, no general answer, but in this specal case it

might be worth knowing .that the execution time of the above

described

,

w-problemincluding optimization of n or Dis

of about 70 to 80 minutéS on an IBM 360/40-processor using the disc operatioñ system. DOS!... .

The second problem leads to the question, whether the process

24

-additional run of the basic design pröbleinsee Section 2-canbe accomplished using

v

1

25

-of decision making by men might be more efficient than the automated process when very complex procedures are to be run

on a computer? Up to now, propeller design seems to remain the

domain of a relatively small number of specialists, although

computer programs for this purpose are available to nearly

everybody in Naval Architeçture. It can be assumed therefore

that a prôpeller design program is usually run by well trained staff so that the interaction between man and computer offers

some real advantages, especially in the interesting case of

the final Section 5 below.

5. ON AN INVERSE DESIGN PROBLEM

The bâsic idea is thedetermination of certain character isti òs

of a given propeller on the basis of the above outlined theory

for a single design point. Such a problem occurs for example

when the radial distribution of maximum stress is of, interest in connection with an investigation of broken propeller blades

or when the reason for cavitation caused material dexnclition

must be examined theoretically. Carrying out this example it

must be recalled that the interesting steps of the design

pro-ceduresteps 3a to 6 in Section 2are outlined under certain constraints, given, for example, by the choice of a special

section shape.

But

the structure of the whole program allows any other treatment of these steps, simply accomplished by

changing the related subroutineè. Accordingly, the program

26

-geometry.

The real problem occurs in connection with the

definition of the radial loading distribution that is given

by a relation between X' and

XX_..sèe

_{for example Section 2,}

equations (7),

(5) and W. Therefore, looking at X

the

strong influence of the radial wake distribution can be seen.

But if there is any relying on the assumed radial wake

distri-bution it is possible to determine one, and only one function

for each of the radial distributions of the maximum stress,

s(x), and of the safety margin against cavitation, S(x), for

the given propellersee again steps 6 and 4.

Except for

these functions the design point must be' defined in the usual

'way.

The redornmended procedure for this purpose is based on the

possibility of running the program step by step from a

ter-minal with partial data output and input.

So making a gless

H

for the desired functions s(x) and S(x) an interruption might

be caused just before step 7 is about to be accomplished.

Then changing s (x) with respect to the desired radial

thick-ness distribution, t(x), and S(x) with respect to the desired

blade outline, c(x), the execution cai be continued with step

3a.

Repeating this cOrrective change of the input data until

the attained blade outline is equal to the given blade outline

thè execution should then be continued with step 7 which is

usually followed by step 2 after a fi±st run of the program

through the steps 2 to 6.

The second runith a modified

27

-time equality of both of the attained functions t(x) and c(x)

with the respective given functions must be reached, because

the following thrust verification of step 7 can be expected to hold after this second run which is based on a nearly

un-changed blade outline.

The whole procedure usually does not take more than eight

to ten iterations if the decisions are made by an experienced

operator, so that the amount of execution time of this inverse

design problem is of the same order as the basic design

prob-lem (Section 2) that takes about 2 to 3 minutes on the IBM

360/40-processor mentioned in Section 4. But the handling of

the data at a terminal takes a large amount of time if the

out-put is to be drawn on diagrams and the inout-put to be read from

diagrams again. Thereforeand for many other similar practical

reasonsthe following final note on computer-aided propeller

design may be worth thinking about.

If the state of the art in propeller theory is to be used extensively for propeller design purposes an additional use of advanced computer graphics is needed in order to make some

, 28

-REFERENCES

[1,] Lerbs, H.. W.: .: Moderatèiy Loaded Propellers with a Finite

Nuiriber of Blades and an Arbitrary Distribution of

Çirculà-tion. Transactions of The Society of Naval Architects. and

Marine Engineers Voi. 60, 1952.

12.] Wrench, F. W,, Jr.: The Calculation of Propeller Induction Factors Applied Mathematics Laboratories, Techn Rep 13,

1954 .

.:

[3] Lerbs, H.. W.: Einige Gesichtspunkte beim Entwurf von

Propeilern groer Leistung. Jahrbuch der

Schiffbautech-nischen Gesellschaft., 60.. Band, 1966.

':

[4]' Eckhardt, Mo .K. and Morgan, W, B.: A. Propeller.Design

Method. CtiQnS. of The Society of Naval Architects

and Marine Enginéers, Vol.63, 1955.

[5]. Östergaard, C.: gzur theoretischen Behandlung des

Nabeneinflusses beim Propellerentwurf Dissertation,

Berlin 1969, D 83.

[61 Kérwin, J. E. and Leopold,' R.: A Design Theory for

Sub-cavitating Propellers.. Transactions of., The Söciety of

Naval Architects and Marine Engineers, Vol. 72, 1964.

[71 Kruppa, C.: High Speed Qpi1er:.. Hydrodynamics and,

Design. University of Michigan, Department of 'Naval