On optimum fish tail propellers with two blades

ON OPTIMUM FISH TAIL PROPELLERS

WITH TWO BLADES

R. DE GRAAF

ARCL-IIEF

_{tab. v Scheepsbquwlwnd.}

Technische Hogeschool

Dellt

RI JKSTJNIVERSITEI-T TE GRONINGEN

ON OPTIMUM FISH TAIL PROPELLERS

WITH TWO BLADES

PROEFSCHR1 Fr

TER VERKRIJGING VAN HET DOCTORAAT IN DE

WISKUNDE EN NATIJURWETENSCHAPPEN

AA1 DE RIJKSUNIVERSITEIT TE GRONINGEN

OP GEZAG VAN DE RECTOR MAGNI FI CUS DR. W F. DANKBAAR IN RET OPENBAAR TE VERDEDIGEN OP

VRIJDAG 22 MEl 1970

DES NAMIDDAGS TE 4OO UUR

DOOR

RUURD DE GRAAF

GEBOREN TE NIJ BEETS

PROMOTOR: PROF. DR. J.A.. SPAREÑIERG

Oan myn âlden

Aan Jetty

VOORWOORD

Als wetenschappelijk medewerker verbonden aan de

af-deling toegepaste wiskunde en technische mechanica van de

Rijksuniversiteit te Grom rigen heb ik de gelegenheid

ge-kregen dit proefschrift samente stellen. Zeer veel dank ben ik verschuldigd aan Prof.Dr. J.A. Sparenberg voor zijn optimale begeleiding. Het werken onde zijn stuwende

lei-ding is voor mij bijzonder leerzaam geweest.

Voor de verbeteriUgen, die Dr. Ir. H.W. Hoogstraten

nog heeft kunnen aanbrengen in het manuscript, wil 1k hem

gaarne dank zeggen.

Hét noodzakelijke rekenwerk is verricht met de TR4 -computer van de Rijksuniversiteit. De staf van het

reken-centrum ben ik erkentelijk voor de snelle wijze, waarop de tijdroven4e rekenprogramma's zijn verwerkt.

Tenslotte dank ik al degeneri, die anderszins een bij-drage hebben geleverd aan het tot stand konien van dit proef-schrift. Van hen

wil

ik speciaalnoemeninejuffrouw W.Bakker, die met voortvarendheid het typewerk heeft verzorgd, en

vervolgens mijn vrouw en mijn schoonvadér, de heer J. van

der Laan, van wie ik veel huip mocht ondervinden bij het

cor ri ger en

Contents. page 9 12 1-r 29 12 l9 Introduction.

Statement of the Problem.

The Thrust and the Kinetic Energy.

. The Variational Problem.

Numerical Evaluation of Integrals, of Section .

A Reciprocity Theorem Concerning Two Vórtice,. Another Description of the Wake of the Optimum

Propeller.

The. Numerical Solution of the Mixed Bouftdary-ValUe Problems.

The Solution of the Mixed Boundary-Value Problems with the kid of an Electrolytic Tank.

The Efficiency and the Quality of the Optimum 'Propellers.

il. The Norma.1 Velocities at the Segments A.B. Induced by the Vorticity.

The Normal Velocities at the Segments A.B. Induced by the Source Distribution.

The Angles of Incidence of the Profiles. 1. The System of.. Equations for the Angles of

Incidence.

Decompösiti,on of the Angle of Incidence in Relevant Parts.

Numerical Evaluation Integrals öf Section

iTo Numerical Evaluation Section 14.

The Pivotal Pont at Numerical Results for

Incidence.

References. Summary.

of the Principal Value 1'.

of Other Integrals of

the Profile.

the Optimal Angles of

52 72 81 8 97 102 106 112 119 122 129 137 1 11 158 160

1. Introd.uctìon.

Nonstationary ship propellers,belonging to the class of vertical blade propellers, are considered. These

propellers have blades, pèrpendicular to the bottom of

the ship. The blades mové along certain periodical

stir-faces and vary their angles of incidence periodically in order to deliver a mean force, called thrust, in a desired direction. Examples of vertical blade propellers are the cycloidal propellers. The blades of such propel-1ers describe cycloidal surfaces

[8,

12, i1, 11Af).

We direct our attention to vertical blade propellers with two blades, moving side by side along more or less arbitrarily periodical surfaces. In figure 1.1 a cross section, parallel to the surface of the fluid, has been

draim. The blades will be identical and. move symmetrically

with respect to a middle plane. Besides, the surfaces, along which the blades move, will have two types of

planes of' symmetry A-A and B-B per period (figure 1.1).

If the distance between the two blades is infinite,

we obtain the case of' two one-bladed. propellers without

interaction. Then, to a certain extent, each of these blades resembles the tail of a fish.

In the following we assume the fluid, in which the propeller acts, to be incompressible and inviscid. The blades will be infinitely long in the spanwise direction. Their thickness will be small with respect to the chord length and they may be slightly cambered. They will have no twist and the chord length will be the same at every height. Therefore we have a two-dimensional problem. In order to be able to use a linear theory, we suppose the

chord length of the. blades to be small with respect to the smallest radius of curvature of' the two-dimensional

orbits, along which the cross sections of' the blades move.

numbers between square brackets refer to the

biblio-graphy at the end.

B

Figure 1.1. Several positions of the profiles of the two blades, moving along their paths.

Finally the orbits will not be self-intersecting. This limitation can be removed at the cost of some complications

in the formii1tion.

It is our aim to. prescribe the oscillatory motion of

the blades in such a way that the effiiency of the

propel-1er is -as high as possible. For that purpose we express

the mean-thrust of the propeller and the kinetic energy, left behind by the .nroe1ler, into a bound vorticity function. If then we want to minimize the kinetic energy,

urder the constiaints 'that. the mean.thrust has a

pre-scribed value and the mean force in the transverse direc-tion is zero, we have to solve a variadirec-tional problem for

this vorticity function.

.Beside this we will ap1y another method 113, viz.

aeeràlization

f the thebries of Munk Crol.,

Prandtl

Eiil

and Betz

Eli,

tobbaintie.aboVe-TfleniÖned

vorticity function, belonging to the optimal notion of the p±opeller. With this:method the -vorticity function

can be' calculated fron mixed .boundary-a1Ue .p±'.oblems or

measured with the aid of an -ele.ctrol31tic-taflk.

FÏ'om the optimal vorticity function- easily tollows

the efficiency of'the öptimum ropeller. For different orbits wè examine the interaction f the blades by varying the mutual distance. It will turn out that this.

inter-actión is favourable. We compare the efficiency with the efficiency of the ideal.roDeller, the actuator disc. It is known Ei61 that the actuator disc. has the highest

possible efficiency for. a given wOrking area, velocity of advance and mean thrust; . . .

Finally optimal angles of incidence of the blades

are calculated as a function of position. These angles of incidence consist of several parts, which can be deter-mined separately. One part delivers a mean thrp.st zero, in case the profiles would have no camber and ño thickness.

Next,there is aeffective part, proportional to the mean

thrust and there are parts, giving the influence of the

camber and thickness.

To calculate angles of inciaence of a blade we need pivotal points at the profile. These points differ to a certain extent from the pivotal points, given by Muithopp in

E91,

since we make use of another

represen-tation for the pressure distribution along the profile. In particular we find that the--three-quarter-chOrd point has disappeared, when only one pivòtal point is used.

2. Statement of the Problem.

As assumed in the introduction we have a

two-dimen-sional problem. Hence, it is allowed in the following to consider a cross section, in which the profiles _move. In this plane we choose a Cartesian coordinate system (x,y), which is at rest with respect to the fluid _at infinity. The ship, to which the propeller is attached,

win have a constant velocity ¡J along the x-axis in _the positive x-direction. In figure 2.1 the orbits of the reference points or turning points T1 and. T2 of the

profiles are denoted. The profiles can perform small oscillations around these points. For convenience _{T1 and}

T2 are taken outside the profiles, _{as drawn in figure 2.1.}

Figure 2.1. The orbits with the profiles and the

13 The orbits are represented by

y=y.(x),

j = i or 2. (2.1)

Because of the periodicity and symmetry the following relations are valid for

y.(x+h) ₌

y.(-x) = y.(x) (2.2)

y1(x) = -y2(x)

We assume that the orbits are sufficiently smooth, in other words y.(x) will be differentiable as many times as we need

(sectio 14 and 5).

We will now make use of a linearization procedure. We

replace the profiles by defoririable line segments A.B.,

covered with time-dependent bound vorticity,moving exactly along the orbits (figure 2.1). Besides these segments are covered with sources and sinks, representing the thickness of the profiles (section 12). However, we notice that for minimizing the kinetic energy, left behind in the wake,' these source distributions play no part. The segments have

a fixed length 1, viz, the chord length of 'the profiles

(section 13), and deform during the motion in accordance with the varying curvature of the orbits. Hence we must

assume that the profiles move in the direct neighbourhood of the orbits. From this it follows that the length 1 has to be small with respect to the smallest radius of

curvature of the orbits. We remark that a consistent linear theory for profiles with a finite chord length 1 can only be set up, when these profiles are flexible (section 13).

By a point x of the orbit y = y.(x) we mean the point

(x,y.(x)) in the (x,y)-plane. A point P of segment A.B will

have two coordinates, viz. x and r (figure 2.2). The

x-coordi-nate.is the x-coordinate of the corresponding, point of the orbit, just 'at that moment coinciding with P, and r is the

along the orbits. The distance r is positive in the translation direction of the segment A.B.. The distance

from T. to the trailing edge is denoted by a and therefore the distance from T. to the leading edge B.

is a+l.

igure 2.2. The coordinates x and r of a point at

Next we introduce a length parameter s, representing the length from the point O of the orbit under consideration

(i.e. the point (O,y.(0)))to P:

s = s(x) = El

{'Ç')}2i3

def . (2.3)

Hence

r = r(xT,x)

f()di,

(2.)

XT

where XT denotes the x-coordinate of T. Let the position

o T travels

älong the orbit:

V{xT(t)} st(xT) iç(t) f(xT) x(t). (2.5)

It is assumed that this will be the veocity of all points

of the inextensible senent A.B..

The bound vorticity per unit of length, in the direction of the tangent to the orbit, at a point of A.B. with coordinates x and r is represented by r.(x,r). This vorticity is giien a positive value, if i causes a counterclockwise circulatory flow. Now it is easy th calculate [lii., 14A1 the free vorticity y.(x,r) left behind at the point (x,r) of the segment AB by the variations

of the bound vorticity, which has passed the dint x of

the orbit. Because our theory is linear, we may assume that the free vorticity will remain at the place, where

it is created. We find: a+l

y.(x,r) = - {f(x)}1

f

(2.6)

where, as well as in the following, an accent above a function means that the function has been differentiated with respect to the first variable. When the whole

segment AB has passed the point x of the orbit, the

strength of the free vorticity remains constant. This free vorticity behind the profile has the value

a+l

=

_{f(x)}_1f

r(x,r)dr. (2.7)

15-It may be stressed that .(x) represents a vorticity distri-bution per unit of length' in the direction _{of the tangent}

to the orbit y = y.(x).

We now introduce a vorticity fwiction

g (x), representing

the integral over all bound vorticity

of

_{A .B ., which passes} the fixed point x of the orbit:

a a

a+l

g.(x) = r.(x,r)dr. (2.8)

This formula can be easily understood in terms _of

conser-vation of vorticity.

In the next section we will express the mean thrust,

the mean force in the y-direction and the kinetic energy,

left behind in the wake, in terms of the above-introduced vorticitv firnctions g(x). After that we treat two methods

for optimizing these functions. a

According to (2.7) and (2.8) we obtain the following

relation between y.(x) and g.(x):

3. The Thrust and the Kinetic Energy.

In order to calculate the thrust we need the unit normals ÏT. at the points x of the orbits y = y.(x)

(figure 2.1). With the aid of the function f(x), defined

by (2.3), we find:

n. = (n .(x), n .(x)) = {f(x)} (-y!(x),1).

xJ yJ

Introduce the thrust K(t) as the sum of the

x-components of the forces, per unit of length in the

spanwise direction (i.e. in the direction perpendicular to the (x,y)-rlane), acting on the bound vorticities of both segments A.B., at time t. Then it holds for the

thrust power ET per period and per unit of length of

the blades:

E U I K(t)dt,

T

i

X

o

where U represents the constant velocity of the ship in

the positive x-direction and T = h/U the period of time.

In the following we assume that the mean value

K ₌

of the thrust is prescribed.

We introduce a new function for the bound vorticity

distribution at A.B.:

33

de f

G(xTr)

= (3.1) (3.2) (3.4) 17

where XT, x and r are connected according to (2.1).

By application of the Kutta-Joukowski theorem

_[61

_we obtain with the use of the formulas (2.5) _{and (3.1):}

2 K = X

T.

j=1 aa+l

{

G.(xT,r)n .(x) f(xT)dxT}dr, (3.5)

where p is th density of the fluid. We remark that this

eJcpression for K _{does not depend on the periodical}

velocity, with which the segments A.B. move along their

orbité. :

Considering in (2.1) x as a function of XT and r and keeiing r constant, differentiation with respect _to XT gives:

f(xT) x'(xT,r) _{- f(x)}

Hence, replacement of the variable XT in (3.5) by x

gives according to (3,1f) and (3.6):

a+l

2 p

K =

J

x(h,r) X

T.

j1

a x(O,r)

Because of the periodicity of Fj(xr) n(x) and

f(x), we finally obtain using (2.8) and (3.1):

h g.(x) y(x) dx 3 ₃ (3.6) r.(x,r) n .(x) f(x)dx}dr. X3 (3.1) (3.8) 2

K

=-X

T.

j1

o

The mean value K of the sum of the y-components of

the forces is obtained analogously:

-2 K .2 _E y

t.=1

J o h

Next we want to express the kinetic energy E, left behind per period and per unit of length in the spanwise

direction, into the vorticity function g.(x). For that

purpose we suppose the propeller to be at x = +. The

strip O < X < h, _ < y < is divided into 3 regions, viz. H1, H2 and. H3 (figure 3.1), by the orbits of the profiles.

(3.9)

Figure 3.1. Determination of the kinetic energy, left

behind in the fluid.

After having introduced a velocity potential _{(x,y) we}

obtain for E by virtue of the second theorem of Green: E = P 2 + 2 } dx dy 3

=-p

i= i

2

(x,y)

= Re Z 2iri j=1 i

where 1. represents the contour of H. _{and the}

deri-i _i

3m.

vative of _{a-t a point of 1. in the direction of the unit} normal pointing into the H. under _{consideration.}

In order to express Einto terms of g.(x) we need some properties of the Potential _{(x,y). It holds for}

apart from an arbitrary additive _constant:

.!_ _dl.

ein. .

(3.10)

z.(x.))

log sin .(z- f(x.)dx.,

(3.11)

with z = x+iy a fixed, but arbitrary, _{point and}

z. = z.(x.) = x. + iy.(x.) a variable point _{at the orbit}

J J J J -1 J.

y.(x), in the complex _{(x,y)-plane, while Re means " the} real part of". This potential _{can be derived from the} potential of a row of equally spaced vortices of equal

strength _E6, 121. From

(3.11)

_{we can easily deduce for}

every y, with the exception of the points at _{the orbits,}

+ h,y) =

x,y).

(3.12)

For, if ll > n br. y < 6, where n and 6 are respectively the maximum aid the minimum of y1(x) (figure 2.1), it holds that

(x + h,y) - (x,y)

= +

o

x1) f(x1)dx1

2(x2) f(x2)d.x2, (3.13)

where the signs have to be combined as follows. If y > n, then - and -. If y <' -n, then + and. +.If lyl <6, then

respectively tand -. Application of (2.9) gives (3.12).

Now we immediately find (3.2) for 6

< Il

n.-For example the potential difference between and S in figure 3.1 caribe found by cònsidring thé pötential

*

*

*.

*

-*

*

alongP

-Q R S . The potential differences overt' Q

and S R have to be equalbyr.easoflSOf.ProdiC.ity. Further the following relation ±or the potential jump across the orbit y = y(x) is valid:

.(x)-

.(x) g.(x) : (3.1)

where .(x) are the limiting values of the potential

at the +side of the orbit (figure 3.1). This can be shown as follows. Denote the x-coordinate of the trailing

edge A of th segment by ' and the leading edge B. by Consider the moment when A. is at the point x (of

the orbit y y. (x)) undei c6nsideratiori. IÍ thé sum of

Figure 3..2. The potential. differences across the orbitso

the free and bound vorticity at a point (,i) of the segment is given by G. tOt(XT,r), then integration along the contour,

denoted n figure 3.2, from the - sideto the + side gives:

a+l

Becaus e

a

a

as will be proved bélow, we see that

a-I-1

tOt(xT,).

(3.15)

r

+ +(xA) -

.(XA)

(3.18)

If we replace the variable by according to (2.4) and use (2.9) and (3.17), we immediately find (3.14).

Let us now prove formula (316.).. By definition we have:

G. t(xT,) = + y.(i,). (3.19)

J, o J J

If an elementary bound vortex of A.B. pases a point x,

it generally gives rise to the creation of free vorticity.

The sum of this created free vorticity and the variation of the bound vorticity remains zero at every moment. Hence, using (2.3), it is obvious that (3.16) must be valid.

A more analytical derivation of (3.16) can be given as follows. From (3.19) we find:

23

+(xA) - c.(XA) = (3.17)

Hence formula (3.14) is valid for a point x, if

coin-cides with it. However, if the segment AB. has passed

the point x, the potential difference remains the sanie. For at every moment the variation of the bound vorticity is compensated by the creation of free vorticity.

Assuming the validity of (3.16) we can also verify (3.14) in an analytical way for the point x ofhe orbit (figure 3.2), if the segment has passed. Integration along a contour from the - side to the + side gives

Because for an

arbitrary

point (ic,) of A.B.

JJ

we have: a+l a+l

GjtOt(xT)d

=

r(c,)= r.(xA,1)

+ a a+l x(x ,r*) =

.(x) +

dr a a

r (,r*)dr.

a+l a+l _x(XT,r) =

f

r.(xA,)a +

f

a

f r,)d

(3.20) (3.21) (3.22)

if we replace the integrationvariable by r*. By changing the order of integration of the last integral we get

(figure 3.3): a+1 Ft(,r*)d

JB

*

f.

r!(,r*)dr* XA r a+l (3.23) a+l a+l

X

XA

a+l dx

a a+L

Figure 3.3. The change of the order of integration.

Next we replace the variable by i with the use of (2.):

a+l J

r,r*)dr*=

f

f(Y-r,r*)dr*.

r(xT,x) a r(xT,.x) (3. 21)

At last, by using (2.6), we find

r)

' _{r= r(xT, )}

From (3.20), (3.22) and (3.25) follows (3.16). Now we return to the discussion of the kinetic energy E, given in (3.10). With the aid of (3.12) and

(3.1k) we find: 2

E=-p

g.(x) f(x) dx _(3.26)

j1

a-I-1 =

-

f

y(i,i)dí. a _(3.25)

where _{is the derivative of the potential}

unit

noiin1 . (formula (3.1)) at a point o y = y.(x). Because (x,y) assumes. a constant

y = + the parts of i. at y = + yield no to the contour integral in (3.10). We derive

for an arbitrary point (x,y)

h

2 1 /

(.)_.j

J-I0

2rr . 2ff

Cslnh -

(y-y. ()),-sin

_{-- (x.-Y}

bf()d.

1-cosh

4- (y-y.())

-cos - (x-)

Hence, we find with the aid of (3.1):

along the the orbit value at contribution from (3.11) (3.27)

(-) L.(x)

f()

L.41(x)

f(x)

where the following index convention is introduced:

if j = 1, then j + i = 2,

if j=2,thenj+1=i.

The first integral in (3.28) is a Cauchy-principal value integral and the interval l is defined as follows:

if 0 5 x .5 d, _{then 1} d.f { -d. _.5 .

5 h-d},

if d < x < h-d, then

df

I 0 5 < h} , _(3.30) if h-d 5 x 5 h, then lx

df

{. h

.5Ç.5

h+d}.

By d is meant a small positive value. The shift 'oÍ he interval of integration in the neighbourhood of x O

and x = h is carried out in order also to get a principal value integral at x = 0 and at x = h. The functions

L. .(x,) and L. .1(x,E) are defined in the following

3,3

3,J_

way:

27r 27

'.(x) sinh -r-- 1y.(x)-y.()} + sin - (x-a)

L. .(x,) 3 3

h

2h [cosh

fr.(x_v.() } -

cos (x-e)] h "j

--"3

(3.28)

(3.29)

2(

y! (x) sinh (x)_Y+ C

) }

+ sin

x-2TIr 2 [cosh

T

(x)1

C

) } - cos r(x

(3.31) 27 3(x) def _{.(x). =} an. fl3 3

Îith the use of (2.9) we find, for the kinetic energy E

after substitution of (3.28) into (3.26): h 2 E = p Z

f

.(x){f

_{g() L. .(x,)d}

J 3 3__10 i X (3.32) h +

f g!()

L+1(x,)d}dx.

For the sake of completeness we still have to discuss the case that the orbits touch each other, i.e. 6 = 0. In

order to define in that case a potential in the inner regions as well, we let approach 6 to 0. The limit of the

potential for 6 + O will be called, the potential for

6 = 0. It is obvious that the relations (3.12) and (3.1I) remain valid, if 6 = 0. Then again the kinetic energy E

can be expressed by (3.32). It may be noted that the

second. integrai over in (3.32) is a principal value integral with respect to those values of x, for which

k. The Variational Problem.

It is our aim to minimize the kinetic energy E under the constraint that K has a prescribed value. It will

X

turn out that the transverse force K can be taken equal y

to zero without energy losses. Before discussing this variational problem we introduce dimensionless quantities.

If barred symbols are dimensionless, it holds:

t

tT,

X = X

h, y(x) =

K =i pTh,

EX=

r(x,r) =G,)('h,

g.(x) = ¡)

L. .(x,)

1. .(,j)h'.

3,3

3,3

In future, unless stated explicitly we use dimen-sionless quantities and we omit the bars. Hence, making

use of (li..i) we find for the thrust and the kinetic

energy instead of (3.8), (3.9) and (3.32):

2 1 2 1 = -

j1

g.(x) y(x)dx, K =

f

g(x)dx, 29

2 j=1

() L; .(x,)d

3

3,3

2E cosh 27r

-

cos 27F (IL.3) + _I

g!1()

dx

where

y!(x) sinh 2IT

{y.(x)-y.()} +

sin 2(x-)

L. .(x,) =

3 3

,y!(x) sinh 27F {y.(x)+y.()} +

sin 2ir(x-)

=. .

- ..J

-2[cosh 2ir {y.(x) +

y.()} -

cos 2Tr(x-))

The

meanirg

of the index

j+1 has been given by (3.29).

Because

of the. syetrical Dosition of the two orbits

(formula (12.2)) we have

replaced y.1() by -y..() in

th&1at

expresion. In (1.)

the iterval of -integration

i is now defined as follows:

X

-if O < x d, then l

df

-d. 1-d} ,

if d <x 1-d, then. .l

1±' 1-d. x 1, then del' d < 1+d} o

First we remark that the following equalities

are

valid:

L

(x,) = L

(x,),

,1

2,2

L

(x,)

L

(x,).

1,2

2,1

(.6)

These relations can immediately be deduced from the meaning of the functions L. .(x,) and L.. For

3,3

3,3

from (3.28) and (2.3) follows that L.

.(x,T

I

f(x)

3,3

represents the normal velocity at the point X of th& orbit y = y.(x) as a result of cOncentrated unit vortices

r(1)

at the points + k (k = O, ± 1, + 2, ...) of the orbit under consideratjon. Likewise L. . 1(x,)

I

f(x)

repre-3,3

sents the normal velocity at the paint x of some Orbit

as a result of concentrated unit vortices r(1) at the

points + k (k

_{= 9}

_{± 1,+}

2,...) of the other orbit.

The orbits y = y1(x) and y =

y2(x)

are syunetrical

with respect to the xxiTherefoè it i-s plausible -that a symmetry relation exists between the optimal Vorticity functions g1(x) and

g2(x)

Indeed, we will

showthat

g(x)

def g1(x) +

g2(x) E

('t.T)

with c as some constant, if the kineticienergy is minimal and the mean thrust has a prescribed value.

We find for tie kinetic energy E0, belonging to the

optimal vorticity, functions g1(x) and

g2(x)

-g1(x). + according to (1..3) and '(1.6):

E0

s.

o g1(x)[ g1(x) i X

i

_X i

Now, suppose the relation

_()4.T)

is not valid. Then instead of g1(x) and g2(x) we respectively consider the vorticity

functions g1(x) - g5(x) and. -g1(x) + g(x) and determine the kinetic energy E, belonging to these:

-

g'()} L

(x,)d]dx

5 1,1 ¿/ {g;() -

g()} L

_(x,)d]d.x 1,1 (4.9) -

g()} L12(x,)d]dx

( ) -

g'()} L1 1(x,)d] dx

g() - g()} L1 1(x,)d] dx

- g()} L12(x,)d]dx

g) L12(x,)d]dx

(1.8)

K = -2 X

It is obvious rom the symmetrical position of the orbits and from (1.2) that remains unchanged. When e take

g1(x) = g2(x) =g5(x) in (1.3) and use ()-.6) we find that

the difference o

g1) - g()} L

(x,)d] dx.

1 2

{

gs(x){I g'() L

(x,)d}dx

5 1,2

(h.io)

has to be a positive expression, if g5(x) is not

identical-ly constant. However, this is contrary to the supposition, that E0 represents thernininial kineticenergyi.ii.er the

constraint that K has a prescribed value.

X

-It follows from (1.î) that in case of optimal vor-ticity functions g1(x) and g2(x) we can write for the

thrust (formula ()-i..2)) and kinetic energy (formula (1.3)):

1(x) y(x) dx ,

by takin _{into accöunt respectively thé setrical}

position of the orbits and the relations (4.6)

As assumed in the introduction the orbits _{y y (}

idil be even ft±ictiöns. Then _{we are able to prove that}

the optimal functions g(x) have to be odd functions,

apart from some additive constants We split g (x) into an even and an odd function

g < ) =

+ ej j(_ t

oi oj

(4.i)

where the upper and lower signs belong respectively to

the upper and lower subscripts _{Substitution into (4 12)} gives:

L

(4.1?)

()

g'1()}

L12(x,)dJ

No we make use of the following yetry properties of the functions L11(x,) and L12(x,):

+ g0

L1 1(x,) = - L (1-x,1-),

, 1,1

L (x,) = - L (i-x,i-).

1,2 1,2

Figure 4.1. The syietry property of L11(x,).

(li..15)

As has been remarked, L1 1(x,) /f(x) and L1 2(x,) If(x)

represent the normal velocities at the point x of an orbit

as a result of concentrated unit vortices r_(i) at the

points + k (k = O, + 1, + 2,...) respectively of the

orbit under consideration and of the other orbit. There-fore the first relation of ()4.15) can be deduced from

the synnetry and periodicity of the orbits (formulas (2.2) and (2.3) and figure 14.1). Because, besides this, the orbits are synnnetrical with respect to the x-axis, the second relation of (14.15) has also to be valid.

Because of (I.15) thè expression ()..i1). can be reduced to:

fg

o -- x &1 ()' IJ1,

(x,)d

o.

()

L1 dx

Comparing his with (.12) we see that

(1.i6)

cQnsists Of two positive integrals'. From (1.ii) it foflows that gj(x)

does not contribute to _K Hence, considering p4.7), (I ₁₃₎ and (1.i6), for a minimal value of E we have to take:

(.i)

with c. as soe onstants. In other: wOrds the optimal

vorticity funtions g.(x) are odd, if we disregard some constants. Paying attention to the finite dimenaions of

real pröpeller blades we have to ta.ke into account the occurrence o tip Orticity. Using this fact, it is dis-cussed in section 7 that thesé êonstants have to vanish

inthe.case o.f Îni-nimal kinetic energy losses. There-fOre

we choose them equal to zero, although this choice is arbitrary within our two-dimensional theory. We remark that then the mean force K of

R.2)

_{is zero automatically.}

However, it is obvious that

the condition K = O is

_y

already satisfied, if we

shouLd take C1 = -c2.

We will solve the variational

problem with the use

of Ritz's method. Supposing

that the fupction g1(x) in

the optimal case can be aiproximated sufficiently accurate

by a finite Fourier sine

series of N terms, we replace

g1(x) by

(N.)

def

(N) ..

g

(x)

=

a.

sin 2irjx.

.L.i8)

j=1

SubstltutiOfl of

(14.18)

into the right hand side of

(14e

ii)

yields:

-2'

af

sin 2irpx y(*)dx

(14. 19)

2irpx y' (x)dx

def

L e

1

p=i,

Hence the condition on the thrust

becomes K

N

= K

-,

X X

Substitution of (14.18) into the right hand.

side of..(iL.12.).

gives

N N

(N) df

_{2ir E}

p=l q1

i

q a

a

p

q O

sin 27rpx

{

fcos

2q L1 1(x,)d

-

f cos 2q

L12(x,)d}

df

(14.20)

37 E

a

s.in

p=1

N N

(N) (N)

27r E

qa

a

p1q1

p q

With the multiplier rule of Lagrange we now calculate the unknown coefficients aÇ. Because the condition K = O

J y

on the transverse force K has already been satisfied,

we only need the condition on the thrust K. If

F = 2rr E E q a a

(P)

- x { 1 E a(N)

L+ K }

p q _D

X'

p1 q1

p1

-V4.21)

where A is an unknown constant parameter, then the fol-lowing relations have to be.satisfied for a minimum of

E(N)

under the constraint that the value K is prescribed:

= O (j = 1,2,...,N),

ff

= O. (1. 22)

J

This yields a system of N+1 linear equations for A and (N) the. N coefficients a. J N (N) ( ) E a p + A = O (j = p=1 p lt (Ii. 23) N

(N) (p)=_,

Ea

L p=1

Then

= -

p b

p

It may be noted that all the coefficients aÇ1\T) depend

on K linearly, as follows from system (li.23). Thus the

optimal vorticity function g1(x) is Droportional to K.

This fact will be used in section 10.

Now we disciss a special case. Suppose the orbit y y1(x) is given by

co

and

the mean thrust is represented by

N (N) K 2ir

Z pa

b.

X p p y1(x) = b + Z .b1 cos 2rr(2m-1)x.

m1

(I .26) (IL .27) (Ii. 28)

This means that y1(x).-b0 is antisyimetrical with respect

to x = (2k1)/IL, where k = 0, + 1,

-i- 2,...

. If the dis-tance between the orbit and the x-axis (figure 2.1) becomes infinite, it can be proved that then the even

39

q ₌ p (.21)

which will be proved in section 7.

We suppose that we may develop y = y1(x) in a Fourier cosine series:

y1(x) = b cos 2irmx.

m0

coefficients aPT) of the optimal vorticity function

vanish. It is clear from ()4.)4) that

L11(x,)

= -

L11(-x,-),

if the orbit y = y1(x) is represented by (1.28). Of course this can also be deduced from the meaning of the function L1 1(x,)/f(x). We notice that L1 1(x,) is

a periodical function of x with period i and likewise

a Deriodical f'unction of with period i and that L1 becomes independent of , if 5 - . Then we

see that defined by (.2O), is equal to zero, if

p+q

is odd. Further we remark that defined by (1.l9),

vanishes, if p is even. Thus, solving the linear system

of equations (1.23), it follows that the coefficients

of the optimal vorticity function are zero, if p is even.

Finally, let us consider two orbits y = y1(x) with the same odd terms and opposite even terms:

y1(x) ₌ b0 +

(b1 cos 2ir(2m-1)x+b

cós 14xnx). m= i

(4.3o)

This means that the two functions y1(x) - b0 are opposite to each other, if one orbit is shifted over an odd number of half periods, hence over a distance x = (2k+1)/2,

where k = O, + 1,

+ 2,...

. _{Now it holds that the}

coef-ficients of the optimal vorticity functions for

= _{are equal to each other, if j is odd, and opposite}

to each other, if j is even. For it an be shown that

L

(x,)

= L

(x,+),

1,1 1,1

(.29)

where the superscripts + and - rèspectively correspond to

the orbit with even terms +b cos 1iîrnx and. the orbit

with even terme -b cos 1rmx. Thus we see that the values

of L p,q for the two orbits are equál to each other, if p-4-q is even, and oposite to each other, if p+q is odd.

Next, it immediately follows that the values of for

the two orbits are equal to each other, if is odd and

opposite to each other, if p is even. Hence, considering the linear system of equation.s (.23) itis clear that

the coefficients a for the two orbits are equal or

opposite to each other.

5. Numerical Ealuation of Integrals of Section 4.

In this section we first examine the integrals

defined by (4.20.). may write

p,q)

= _{i (5.1)}

and

2 sin 2TrPx{] cos 27rq L1

2(x3)d}dx.

(5.3)

As follows from the meaning of L1 1(x,)/f(x) (section 4)

the function L1. 1(x,) in

(P)

s singular, if x = Therefore we replace x and by more suitable variables. Because of the periodicity we may change the area of integration as denoted in figure 5.1. Introducing the

variables

u =

x+),

y = (x-), (5.4)

we cari write for

L1):

1

L'

= 2 sin 2p(u+v) cos 2q(u-v)

where

=

:si1

2lrDx{

os 2çL11

,)d}d

(5.2)

O

With the aid of (1.15) we find:

L'

=

J{

f

sin 2p(u+v) cos 2q(u-v)

Q C

1(u+v,u-v)d } du

+ 2

f

{f

sin 2p(u+v) cos2q(u-v)

L11(u+v,u-v)dV du

where c has a small positive value.

x =1

Figure 5.1. Change of the area of integration.

(5.6)

]n order to calculate the prindpa1 value

intgrai

in (5.6) we consider L.1 1(u+v,u-v) for

small

talues

_of y. The f1lowing. expansions äre üséd:

dy (u)

d2y

(u) d3y (u)

dy

(u)

du

du2

du3

1u

sin 2rr S I7TC a2Tr 3 3 - -'-,'--

I,

cosh ?r{y1(u+c)_y1(u_c)}

= 1+8 [

'3Ji?

{dY1u)

du du3 du dy1 (u) du

I

'-y1 (u) du o(c'), ? [d3y1(u) 5 C2 + (5.7)

We suppose that y = y() is differentiable sufficient times. Substitution of (5.7) yields:

i L (u+c,u-c) = + 1,1 where dy1(u) d?-y1(u)

c(u)

--du

du2'

c0(u) + c1(u)c +

d0(u) + d2(u)c2

,

[dy1(u).

&y1(u) d211(u) d3y1(u,) e

(u)' =

2 du du

du2

du3

J _{?y '(u)':}

':h3

i

+1671

1 -du2 L

du

d(u) =

17F

rdyi(u)

d3y1(u)'

1jdy1(

I

- +1Lff

Ldu'

du3

+ 0Cc3),

(5.8)

Hence for the principal' value integral in :(,5.6).hoids according to (2.2) and (5.8):

c(u)

{(dYl(u)

=

f{

sin 2p(u+v) eQS

2(_d}

du

¡[

I

.si 2p(u+v) eQs 2q(u-v)

(5.9)

r

c0(u) + c1(ù)v-i- c2(u)v2'

1

..

? dv

L

d(u) + d.2(u)v2

The first integrai in the right händ side of (5.9) can e. òhanged i.±to -- 2j-1

f

dv=

(.io) -C. where O _{Ô, if p q and = 1, if p q.}

By using . power series epazsion -for L1 1(u+C,u_c) the

second integral i-n the right hand side of (5.9) öan

expressed into one-dimensional integrals, whereas in their turn these integrals can be determined analytically, if y1(x) is given by a Fourier cosine séries.

In order to accelerate the numerical integration of the first integral in the right hand side of

_{(5 6)}

and to

perform the integration with a greater accuracy we subtract

the singuLar part from the.integrand:

2f{

sin 2itp(u+) cos. 2r(u_v)L.1,1(u+vu_v)dv} du

i

i

i

f{ f sin 2p(u+v) cos 2q(u-v) L

(u+v,U_v)dv}

du

1,1

o

=L

y sin 2rrp(u+v) cas

2ïtq(u-v) L

1,1

(u-4-v,u-v)

-

sin 2ipu cos

2iTqu)dv}dU

The integral of the singular part

equals zero.

We choose e in such a way that the expansion (5.8)

for L1 1(u+v,U-v) is

sufficiently accurate for O

y < e.

We may not take

e arbitrarily small, for the integral in

the right hand side of (5.11) has also to be calculated

suf-ficiently accurately.

As follows from (5.7) both terms

in

the numerator of L1

1(u-i-v,u-v) deviate from i of the order

y2, if y is small. Hence, we expect extra loss of

signifi-cant figures during the numerical evaluation of

L11(u+v,u-v) for small y. If we want L1 1(u+v,u-v) correct

to

significant figures, while the calculation is carried

out in m* significant figures, then the

following inequality

has to hold for e:

n*_m*+ 1

10

_(5.12)

8ir2

Next we consider the integral L'

in (5.3). Because

of (.15) we write for

(5.11)

L'

= 2

2

sin 2lTpx {

cos 2q

L12(x,)d}dx.

We inédiatêly dèduce from the meaning of L12(x,)/f(x)

(section 1) that L1 2(,) behaves singular at such points

(x,.) o tiè arè. of integration, where x = and

y1(x) =y2(x) =

O, in other words, if the minimäl distance. 2(5 between: the two orbits y = y1(x) and y =

y2(x)

(figure 2.1) becomes zero. In this limiting case it is

easier to make use of the theory, discussed in section 7, in order to determine the otimal vorticity function g1(x).

Here e only state for which válues of

_{the inteals}

cari b _{calculated sufficièfitly accurately without}

neediig an expansion of L1 2(x,) in thé nèighbourhood of the points, where theorbits nearly touch each other. Suppose L1 2(x,) is panted correct to n _significant

figures .ar. the calculation is carried out in m* significant figures Then from the expansion of

cosh 2iry1,(x) + y1()} in the vicinity of those points (x,), where x = and y1(x) , easily tollöws that

>

--8ir 2

6.

A Reciprocity Theorem Concerning Two Vortices. In the following section we need a property of

L11(x,F) and L12(x,), which can be derived from a

general theorem about two vortices.

Consider in a plane two curves e1 and c2 with

res-pectively the unit vortices r1 at P1 and r2 at P2

(figure 6.1). These vortices are again positive, if they

cause a counterclockwise irculatory flow. Let the points of c1 and e2 be represented by the distances from some

fixed points Q and Q2 at c1 and c2, measured along the

curves. Suppose P1 and P2 are given by s1 and

2

respec-tively. The normal velocity at the point P2 of the curve

Figure 6.1. Two unit vortices at curves e1 and e2 with

induced normal velocities.

c2, induced by the vortex r1 at the point P1, will be

denoted by K2(s2,s1). Analogously the normal velocity at

P1, induced by r, will be denoted by K1(s1,s2). We agree

Hence, in other words, the variation of K1, as a result of a shift of r2 along C2, per unit of length, is equal or

opposite to the variation of K2, as a result of a shift of

r1 along e1, per unit of length.

The proof of (6.1) is carried out as follows. We denote the angles between the tangents to the curves at r1

and F2 and the line, connecting F1 and. F2, by c and.

(figure

6.2)

respectively. After a sign convention these

Figure

6.2.

Shift of a vortex along the curve.

(6.1)

that the normal velocities K1 and K2 _{are positive, if they} form counterclockwise _{an angle of 90 degrees with vectors}

and

2 respectively, which are tangent to the curves

and point in the direction of increasing values _{of s and} s2.

Now the following theorem holds:

K1(s1,s2) K2(s2,s1)

angles are determined uniquely; in figure

6.2

and c are supposed to be positive. If the vortex T1 is shifted

a little along c1 from s to s1 + s1, then the distance

r between r1 and r2 is changed into r1 + Lr1. Further this shift will change

.

into

+ 2

With the aid of

figure

6.2

we now find:

dOE2 sin . dr1

-

=

, - = .1.

COS .OE

r1 ds1

so that

aK2(s2,s1)

1(cos

OE) cos(OE1 ₊ 2ir s1 r1

However, if r1 remains at P1 and r2 is shifted a little

along C2 from 2

to s2 + M2, where the- distancé r1 between

r1 and r2 is changed. into r1 + r2 and the angle c into + then we find: dOE1 S1fl

2

dr1 r1 so that K1(s1,$) 2

Comparison of (6.3) with (6.5) gives (6.1).

From the above it is clear that only the directions of v2 are essential and not the curves.

ds2 = -c.os ci. 2ir r (6.2) (

6.3)

(6.1)

(6.5)

2ir r 51

7. Another Description of the Wake of the Optimum

Propeller.

In section we discussed Ritz's method for solving a variational problem for the optimal vorticity function g1(x). Because it is possible to adapt the theories of

Munk ['io] , Prandtl [ii] and Betz [i] for optimum wings and

optimum screw propellers to optimum vertical blade propel-lers [13], this vorticity function can also be obtained in

another way.

In accordance with the theory of Betz we can inline-diately find the linear part E of the variation of the kinetic energy E, belonging to the variations 1G.(xT,r)

of the bound vorticities G(xT,r) defined by (3.1v). For

that purpose we have todetermine the induced resistance, which bound vorticities ¿G.(xT,r) experience by moving

along the orbits y = (x) an entire number of periods

"infinitely" far behind the propeller. It is clear that

this variation E of the energy has the magnitude

2 a+l

E=

{

f

G(xTr) Wn(X)

V(xT)dr}dt, (7.1) where w .(x) nJ by (2.5). We of XT and r,

is given by (3.28) and V(xT), with _{XT = xT(t),} remind that x can be considered as a function

as is obvious from (2.)). Besides we remind

that all quantities are dimensionless here (see (4.'i)).

The vorticity G.(xT,r) is positive, if it causes a counter-clockwis circulatory Ílow, and w(x) is positive in the direction of the normal ii'. _{(figure 2.1). If we replace the}

variable t by the, variable xyith' the aid of (2.5), (3,1.) and (3.6), we find:

2

j=1 _a

Beqase df (2.8) and. the periodicity of Lr.(x,r) and.

w .(x)this caribe written as:

nJ

i

- E

J

g.(x) w.(x) f

Ç)

.

j=1

Howevèr (7.3) can also be d.ed.uóed in an analytical

way. According to (3.26) and,(3.28) it holds for»E,

neglecting sècond. order quantities in the variations:

2 E. i= i where 1,1 def '2 = E = 2 E 1,2 a+l i+r

f

o

r(x,r) w(x)

f(x)x}

Ir. g.(x) nj f(x)dx (7.2) (7.3)

By .makïng use of (2.9), (3?8) an.d (1î.6) '4ïe, see that the

foflosdng relation has tO be shown:

tE

+E

E

+E.

1,1

.1,2

2,1

2,2

g.(x)

g()

2,1

def

j1

/{

g() g;() L1 1(x,)d} dx,

def -= E

2,2

L

(x,.)

l'i 2 u

{

f g.(x)

L12

First we need some properties of L

(x,) and.

L12(x,),v±z.

1,1

i

2(x-)

+ N(x,), L

1,2

(x,)

2ir(x-,)

(x)

(7.6)

where r(X,) _{= O, if y1(c) > O and}

= 1, if y1(x) o, wlúle

i(c) and. N(x,)

_{are continuous functions in the}

area of integration, and

L11(x,)

L11(,x)

L12(x,)

DL12(,x)

- X ,

(x

)

(7.7)

The relations (7.6)

can

be verified easily by starting from the meaning of L1 1(x,)/f(x) and L 2(x,)/f(x) and

1,

using the fact that s'.(x) = f(x), as follows from (2.3).

We recall that L1 1(x,)/f(x) and L1 2(x,)/f(x) represent

the normaÏ velocities at the point x of an orbit as.a result

of concentrated unit vortices at the points + k

(k = O, + 1, + 2,...) Íespeótively of the orbit under

We remark that the first relation of (.6) is in accordance with (5.8). As for the relations (7.7) their validity can be shown bymeans of straightforward differentiation.

However', it is also possible to verify (7.7) by starting

agair from the meaning of L1 1(x,)/f(x) and L12(x,)/f(x)

and making use of the theorem, derived in the preceding section.

Suppose the normal velocity at the poin.t x of the

orbit y y1 (x), caused by a unit vortex at the point of the orbit, is denoted by K(x,). We recall that the normal direction is given by (3.1). Analogously let K(,x) be the normal velocity at the point , caused by a unit vortex

at the point x. Now it follOws from (6.1) and (2.3) that

f()

I f(x).

(7.8)

If we replace by +k (k = 0, + 1, +2,...) and take into

account the periodicity.of the orbit , we'may write:

f(x)

K(+k)

= K(+k,x) = K(,x-k) ,

(î.9)

By definition it holds: N L (x,)/f(x) = K(x,) + 11m (K(x,-i-k) + K(x,-k)) 1,1 - N-o k1

(7.10)

(cf. the definition of the velocity potential of a row of equally spaced vortices of equal strength in [6], which has been used in section 3). It is easy to check that this

series behaves as a hyperharrnonic series with exponent 2.

Hence we have to consider the summations of the right hand side and the left hand side of (7.9) over all k, where terms with opposite values of k have to be taken together.

Next we interchange the sumtions and differéntiations.

It can be verifiea easiÏ that this interchange iS allowed.

Now we in»nediately find the first relation of (7.7).

Thé second relation of (7.7) can be. proved analogously.

We notice that this second relation is also valid for

x = if y1(x) > O.

We ar now able to prove (7.5).. The quantity E can be written as =

1[/ g.(x){f

g() LÏ

1-d +

f

()

L11(X,)d}dX

x+c. g'.() L

(x,)d

1,1 + +

f'

.(x){

7E

g() L11()d

1-d d 1 +d g'.() L (x,)d , i x+E: g'.() L J i_,1 dx 1,1

Integration by parts and substitutián of the first relation of (7.6) yields, making use of the periodicity of g.(x) and

= um

g.(x-c)

C-L J g.(x) 11ml J L TF 1-d XE: g

L11(x,)

X £

2 /E

=1im

E

-1,1 £-90 j=l ire

) L(x,)d

1-X+C 1-d + N*(x,x_c)} _{- g.(-d) L (x,-d)} J

1

g ()

L (x,)

-g()

dJ.

X+c

This is a Hadamardprincipa1 value. Hence we find, treating the other Integrals of (7.11) in the same way:

1-d X+ C L11(x, ) Ag.(x) g.(ç)dx DL1 1(x,)

-ig.(x)

g() -

dxd,

g)

L11()}

DL (x,.) d (7.13) 57 + g.(1-d) L1(x,1_d -

g.(x-i-c)

_{C -}

i _{+ N(x,x+s)}} (7.12)

where 01 is represented by figure 7.la. An analogous ex-pressionis valid for E2

. After changing the integration

variables c and , we obtain:

E lim 2,1

If

0R 2 ij=1 pp

!pJOIHhIIIIppp

iiiiHhlO

F1111 1-d L1 1(,x)

-g(.) .g(ç)

_'*

.dd.xt-

-where is reprèsentedby figure 7.lb. Because of (7.7)

the integra.nds in both two-dimensional integrals of (7.13)

and (7.1l)are equal. Only the rei6ns oiÍitegration differ. Hoéver the ihtegrands have the same value for

(x,) and (1+x,1+). Thi means

= E21.

In an analogous way we can show that =

l-d i

g() 4g.()d

(a) (b)

Figure 7.1. The regions of inteSration of the

two-dimen-sional integrals.

point x0 of the orbit y = y.1(x) with

if' the orbits y = y1(x) and y = y2(x) then we

must

consider the integrals

0 < <1}asCauchy-principaÏ

value integrals for x

X.

For convenience we

use

i

interval i instead of the interval { j O < E < 1 nd consider for all points x of the interval {x 0 <

x< i}

the integrals over l ,as Cauchy-principal value integrals.

Then, analogous to (7.13) and (7.114,) we find tha

2..

E =lim. E 1,2 .

-If

tg.(x) g1()

L1

_2'

dxd

(7.15)

Oi If there exists a y1(x0) = 0, hence touch each other,

over the interval

2 E

=lini

E

-2,2 c-'O

j=l

-ff

g() g.1(x)

02 x)

i

(x) ô(x)dx

g)

ddx}.

(î.i6)

This completes the proof of (7.5).

With the use of (7.5) we can easily. prove the sretry

property (14.214) for the integrals

För thatpupose

we take in and AE21 for g1(x) and g1(x) respectively the functions sin 2px and sin 2îrqx and. choose g2(x) and

g (x) identically zero. Then we immediately find that

q(P)

_{= , where}

is given by (.2). In the sae manner we obtain that qL2P =

where _{is given by (5.), if we take}

_{iti ¿E1}

2

¿E2 _{2 for. ¿g1(x) and g2(x) again respectively the functions} sin, 2rrpx and sin 2irqx and óhoose ¿g2(x) _{and g1(x)}

iden-tically zero.

In order to obtain the optimal _{vorticity functions}

g.(x.) first we have d.edued. the _{expression (7.3) for the}

linear par _{of thvariationof the kiietiä}

belonging io the variations j:(X) of g.(x) down the cbrrespoiding _{ariat-ion .K of the}

It follows frorn (h.2) that

2 z X J=1 ¿g.(x) y!.(x)d . . (7.17) energy E, Nèxt we write mean thrust K X

Now we amt only variations ¿g(x), which _{leave K}

un-changed. Îf g.(x,) represent the optimal vorticity

func-tion, the fwictional ¿E in _{has to vanish for all}

variations g.(x), for which thé functional

AK in

(7.17)

vanishes. Hence1 the liñear functionals ¿E and ¿K , definea

X

on the product space of the linear spaces,. ons±sting ¿f

the continuous functions Ag1(x) _{and ¿g2(x), satisfy the}

rlation

-, u ¿K(g1,Ag2)

_{= 0}

_(7.18)

for arbitréry variatiöns ¿g.(x), where u is an unknown constänt. rus _{e find with theaid of (7.3), (.17) and}

(3.1) the following necessary condition:

In other words, the free vorticitie8 (z) behind our

ap-timwn vertical blade propeller, with a prescribed mewi thrust, are equal to the bowzd vorticities along the aur faces, described by the blades, when they are considered

impermeable ccvid rigid wid are placed in a homogeneous

flow

with a velocity u in the positive x-direction. This velo-city u is determined by the thrust K. It is easy tQ verify

that for a positive value of the velocity u has to be

positive. Therefore in

(718)

we have a priori written -u

instead of u.

Nbw the validity of (i.7) also follows directly from

(7.19) and (2.9).

Furthèr we can also deduce from (7.19) that the opti-mal vorticity functions g.(x), apart from some additive

coÈ.stants, are odd,if the fupctions y = y(x),

represen-ting the orbits, are even. To this end we split y.(x) into

even and odd functions of x: -y2(x) y1(x)

'e + y0(x), y.(x)

±

(T.2Q)

o o

Suppose the segments A.B. are at x = 00, Conside±tô

sym-metrical points P.1 and P2 at the orbit y =

_y1(xY

(figure 7.2). The floral coniponents and

7n2 of the velocity u,

respectively at Pl and P2 , are oppbité

in

sinazd have

the same absolute values. These -normal components have to

be compensated by the normal components ïfl1, and

n2' which are induced by the vorticity at both orbits. Now consider the elèentary vortices Ye(Q)dS at two symmetrical points

Qi and Q2 of y = y1(x) and the elementary vortices 1e(R)ds

at two symmetrical points R1 and R2 of y=y2(x), where s re-presents a-engtIi parameter along the orbits (formula (2.3)).

These vortices yield contributions to w and r , which also nl n2

are opposite in sign and also have the same- absolute values.

However, the contributions of the elementary vortices y(Q)ds and -y0(Q)ds, respectively at Qi and Q2, and

Figure

7.2.

_{The rigid örbits in a homogeneous flow.}

the elementary vortices y0(Eds and _{..y ()ds, respectively} at R2 and R1, are not opposite in sign0 Now, integration over the whole vortex layers y(x), stretching from x =

-to x = + at y _{1(x).and y =y2(x), may give no} contri-bution to the northl òornonents * at P1 and. at P.

i±ice this holds for arbitrary points P _{and P2 at the} orbit y y1(x), we see that

o, _(.21)

if we assue uniqueness Of g(x). Application of (2.9)

gives that g'.(x) äre evén functions. Consequently g.(x) are odd fuñctions, apart from some additive constants.

We now choose the additive constants for the optimal

Figure T.3..The surface,described. by one propeller blade

of finite length, with its free vorticity.

have a finite length. This we will nbw

veri.

Consider one of the propeller blades and suppose the length of it is finite A reference line of this blade in the spanwise direction will describe a period:ical,

cylindri-cal, surface S in the 3-dimensional (x,y,z)-space with x, y and z Cartesian coordinates (figure 7.3). It was assumed

(see (2.2) with h = i.) that this surface possesses planes of

symmetry, perpendicular to the x-axis, namely x O and. k

x = ±- (k = 1,2,...), hence two planes per period. These

planes cut S along lines 6f symmetry, e.g. 10, 11 and 12 in

figure 7.3.

In [13] it is proved that the optimal free vorticity at S can be foimd by making rigid the surface S and by placing it in a homogeneous flow. This is analogous to the

above-men-tioned statement for the two-dimensional case. Let the free. vorticity at S be given by components y(x,z) and 'y'5(x,z),

where y(x,z) is the, component in the z-direction and y5(x,z)

is the component in the direction of the velocity of the blade,

hence' perpendicular to the z-direction.

(

(

NOW _{the optimal free vorticity has two properties of} symmetry. C:onsider at S four points

_{Q1, Q2, Q3}

_and

which lie symmetrically with respect to a line of symmetry

The poiits _{Ql and Q2 have the sä.tne distance to the} upper edge as _Q3 and Q to the lower edge. of S (figure

First it is to be shown that

Figure 7.1.The symmetry properties of the optimal.ree

vorticity.

= (YZ(QL.)

;(Q)),

(Q.2Y)

(?(Q), y(Q)).

(1.22)

For that purpose we rotäte the surface S with _{hé free} vor-ticity through an angle of lßO° around the hue through P, perpendicular to S, where P lies at l _{as far from the upper} edge as from the lower edge of S. After that we reverse the direction of the field of the free vorticity. _{Then again}

the frozen surface is a stream surface _{in the}

above-men-tioned homogeneous flow (cf. _{[13]). Néxt it is to be shown} that

(y (Q1),

₁₎₎

-(t.23)

This can be done by splitting both components y(x,z) and.

y5(x,z) in even and odd functions with respect to x = 0 or

k

x = (k = 1,2,...), analogous to (7.20). After that consider the normal components of the velocities at points

P1 and P2, where P1 and. P2 lie symmetrically with respect to induced by the even and. odd terms of the free

vorti-city. Then we can find (7.23) in an analogous way as

formula (7.21) was proved.

With the use of the symmetry 'elations (7.22) and

(7.23) we can verify that at l the optimal free v-orticity

has only a component y(x,z). For, suppose the free

vorti-city lines would cut

-' over a finite part, under a finite

angle. Then this happens from both sides under the same angle. This gives rise to concentrated free vorticity

Y(x,z) at as follows from the fact that the vector field of vorticity is a field without divergence. Such a concentrated free vorticity will have infinite kinetic energy. Hence no free vorticity can leave the lines

We obtain a picture of the optimal free vorticity as drawn

in figure 7.3 (cf. [13]).

Because here the bound vorticity on the blade. is also

a function of the third coordinate z, we can introduce a bound vorticity ftnction g(x,z) analogous to g(x) in the

two-dimensional case (section 2). Thus at the moment when the optimal bound vorticity g(x,z) is passing

-' it is

independent of z. Now suppose that g(x,z) O at This gives rise to the creation of free vorticity at the edges

of It is possible that this free vorticity is spreading out. Consider then for instance the point Q, where cuts

the upper edge of S, and surround Q by

an

arbitrarily small

sphere. It is obvious, in connection with the symmetry,

given by (7.23), that there will be a net outflow of free

vortiáity from this sphere (figure 7.5). This is, however,

contrary tò the fact that the field of free vorticity is a field without divergence. Another possibility is the creation of concentratéd free vbrtices along the edges of

S. This woùld give infinite kinetic _{energy. Hence g(x,z)}

has to vanish at 1.

Figure. 7.5.. The vanishing of the bound vorticity-g(x,z)

ati.

s

We still remark that now the following syxetry

properties for the optimal g(x,z) _{can be deduced easily:}

g(x,z) = -g(-x,z),

g(x,z) = g(x,2z - z),

(7.25)

where z, i _{the z-coordinate of the point P (figure 7.1&.).} The first relation of

_(7.25)

_{states that the optimal bound} vörticity functions g(x) are odd.

In the preceding the proof has been given for _{a finite} propeller blade, which is notinfluencedby the bottom of the ship, under which the propeller _{is acting However, if} we may represent the bottom by a two-sided jnfinite plane, the results remain valid _{For, then by the second relation} of (7.25) the bottom can be simulated by lengthening the

3=

Figure 8.i The five-pointed star of mesh points dth

center at

it holds

(8.2)

+ c + . + .

- 1.

i+i,j i,j+1 i-1,j

i,j1

_{+ O(h) = o}

h2

Hence. for every interior mesh point (x.,y.) the -value will be the average, of the four neighbouring -values. The

deviation of the differential operator from the difference operator is of the order h2. The difference equations, which

contain 4-values at the boundaries IC[ and LM, cari be reduced

It us now consider the other boundary conditions. _Fòr the boundary mesh points along KL (with the exception of the cornerpoin-ts K and L) we again use the five-point formula. These difference equations contain -values at

exterior mésh points, lying below KL. We give for these exterior mesh points (x1,y1) difference equations by

taking int account the boundary condition. It is possible to replacethe normal, derivative by a difference

expres-sion, agaix deviating order h2:

H.

-i3O i,1 i,-1

2h

Next we treat the curved boundary M, along which the normal derivative of the potential also has to vanish. For the boundary mesh points along (with the exception of

the points at KN and LM) we again apply the five-point formula. Here it is more complicated to find a suitable difference equation, approximating the boundary condition

up to and including0(h).. In literature mostly (e.g. [lii

and [7]) a method is proposed, which gives a deviation of order h. We will deduce a difference equation, which is

more accurate. This can be done by taking in-to account

the rate of change of the normal derivative of the poten-tial along the curved boundary (compare the method of Viswanathan in [171). However, such a difference equation can only be applied, if

Iy(x)

1, as will be explained at the end of this section.

First we give a relation betwêen the first and second derivatives oÍ' the potential at a point of Ifl'T, represen

-ting y

311(x).

For a differentiable functioñ F(x,y),

defined at will hold at (x,y) (figure 8.2):

3F(x,y) def F(x,y)

n (x) -+ F(x,y)

n 1(x), (8.4)

x xl y y

where = (n1(x), n1(x)) is the unit normal at the point

X Of y = y1(x), as given by (3.1). Hence, if (Xjy) 1is at

Ìvfl, then _{répresents the normal derivative of the}

y y1 (x)

(x ,y Cx))

.fi.inctior. F(x,y) at that point. y

Figure 8.2. The derivatives of a function F(x,y) at a point (x,y) with respect to two directions,

perpen-dicular to each other.

Next we define a derivative at a point of

the region KL in an analogous way:

3F(x,y) def F(xy)

t Cx) F(x,y)

- x xl

where

= (t (x),t (x)) = {f(x)} (1,y(x)) xl yl

is the unit vector at the point x of y = y1(x) in the direction of the tangent to y = y1(x). As the normal

x,y) .

derivative

n along vanishes, we find that

(8.5)

(8.6)

L

(x,y)

nt

X

1(x) t1()

(x_i1,y) y y1(x) (x,y1(x))

n1(x) t1(x)

(8.7) + i(x) t1(x) ±

i) t1(x)} = O,

if y = y1x).

Consider the perpendicular to the bodary _through a boundarr mesh point (x1;) âlong Jv _{(figure 8.3).}

This

perpendi&alar will cut the orbit

_{y =}

y1(x) at the point (x,y1(x)) The points (x1) and (c,y(x)) rna _coincide.

The positive distance from (x1,y.) to (x,y1(x)) is _denoted

(x1y1)

(xj, y

Figure

8.3.

_{A bouida±y meh oit (x.,y.) along the orbit.}

a(x,y1(x)) a(x,y1(x)) aCx,y1(x)) -- - n (x) + - - n (x), an ax xl ay

yl

(8.8)

-so that a2q,. a .

1,3

L + k . Cx) an ax

i,j

xl ax2 +

k.

. n (x) + O(h2)} n Cx) +

{

13

i axay xl ay + k. . n Cx) 1,3 + k. . n Cx) +

O(h2)}n

(x).

1,3

xl axay

i,j

yl

a12

yl

Fron (8.7') we find, if the derivatives of at the point

(x,y1(x)) are repiaçed by derivatives at the mesh point

a2.

.

ra.

j_,J :1,3 axay

LX

X

(x) t1(x)

n'i(x) t1(x) (8.9)

a2.

1,3

n (x)

t

(x)

1,3

n Cx) t

(x}

(8.10) ax2 xl xl ay2 yl yl

{ni(x)

t1(i)

+ n11(x) t1(x) ¿(h).

We make use of (3.1), (8.6) and. (2.3) and. substitute (8.10) into (8.9):

(x,y1(x)Y [dYi(x)

j-

(dY1()2.1

an

LLdX

_{l_}

\d)J J

_x [2k.

1,3

dy1(x) d2y(x)

{

dx dx

r

(dy1(x)

3

₊

_{-1+ I}

ax _L \. dx 1 (dYl(x)\2 -/

r

(dy (x).\2

1,3

2k I L i,j\. dk

dy1(x)

(dY1(x)2_1 21

at..

dx2

)x

L J

ay

-(8.11)

[k..(x))2

_{(dYì(x))2}11}

a2:..

r

i dy1(x) 2 -1

a2.

a2.

Hence we have eliminated

a1'3

This is necessary i-n order

a(x,y1(x)) ax y

to express into -values at the five mesh points, drawn in figure 8.3. Otherwise we should also need

mesh points (x+1,y.1), (x11,y.1), (x.1,y.1) and

(x1,y.1).

}2J

Lki,j

l

-(

_d±

We next simplify formula (8.11). As there has been given

(x,y1(x)) . .

that = 0, we can express and. into

each other with the aid. of (8.8), in which an mikiown term

of order h then occurs. Making use of this, we may write

for the sum of the second and fourth terni on the right hand side of (8.11):

r

dy1(X) d.2y1(x)

(dy1(X)'\21

_I

dx2

li(\ dx)1

11+\

dy (x)

c

( i,p i i,j dx y

r

d2y1(x)

-I

2k. . (8.12) L 1,3 2

dy1(x)\.2-1

(dy'(x)\\2}_1I . + 0(h2). c

)1

dx)

It is obvious from (8.11) that we can now find a difference

operator, deviating 0(h2) from the differential operatör

Suppose that y(x)j 1. Owing to the application of the five-point formula there occur as many exteriör mesh points as boundary mesh points along MI. We now are able to express the potential at these exterior mesh

points into the potentials . ., . ., . . and

1,3 i-1,3 i,j-1 sufficiently accurately. If the substitutions 1

₃

- . 1,3 141,3 i-1,3 + oh2' ax . 2h ' '' a.. . . . - .

1,3

i,j+1 i,j-1

+ oh2

ay - 2h ' '' (8.13) 79