On the efficiency of optimum finite amplitude sculling propulsion

by W. Potze and J.A. Sparenberg.

Summary.

Using a linear theory the highest possible efficiency is considered for sculling propulsion by means of one wing or by two winos. The span of the wings is allowed to be finite and their interaction in case of two wings is taken into account. For a number of motions of the wings the quality number is calculated.

1. Introduction.

The type of propulsion we consider is by one wing or by two wings, mounted vertically behind a ship. The wings are moving sideways back and forth while their angles of incidence are adjusted such that a thrust is created. For practical purposes two wings are appropriate because then inconvenient moments acting at the ship can be avoided. The fluid in which the propeller is working is assumed to be inviscid and incompressible.

We will try to obtain insight in the maximum values

of the efficiency which can be obtained by this kind of propulsion. It is expected that its efficiency will be larger than the

efficiency obtainable by a conventional screw propeller because "the amount of water" that can be influenced by the wings is

larger than "the amount of water" influenced in general by a screw propeller. Another advantage of the wings is that their velocity with respect to the water is constant over their span, while the velocity of the profiles of a screw propeller blade

increases towards the tip. Of course a disadvantage is the complexity of the mechanism needed to move the wings.

This hydrodynamical efficiency investigation is part

of a more comprehensive one proposed by E.V. Kon. M±j. "De Scheide", about the feasibility of unsteady propulsion. That investigation is also directed towards the construction of the required machinery.

2 8 _AUG,

1983 Lab.

y.

Scheepsbouwkunde

ARCHIEF

Technische Hogaschool

Deift

ON THE EFFICIENCY OF OPTIMTThI FINITE AMPLITUDE SCULLING PROPULSION.

-2-We calculate the highest efficiency that can be obtained for three types of motions of the wings. There are however some more or less serious approximations and assumptions we will make. First, our theory is linearized, this means that it does not

hold for propellers which are heavily loaded. Second, we assume the wings to work in an undisturbed fluid which fills the whole three dimensional space. Hence we neglect the influence of the

free water surface and of the shiphull both as a boundary of the region in which the propeller is working. Also we neglect the

influence of the ship's hull as a cause of an inhomogeneous inflow. It is not difficult to adapt the theory to an inflow with a small inhomogeneity, when however a large amount of incoming vorticity is admitted more serious difficulties occur (for instance Cl]). Third, we calculate in the following the free vorticity which has to be shed by the propeller when it is an optimum one. The question is, to which extent can this free vort.icity be created by rigid wings. This has to be a subject of further research.

2. Formulation of the problem.

A Cartesian coordinate system (x,y,z) is embedded in an inviscid and incompressible fluid. The undisturbed fluid is at rest with respect to this coordinate system. First we consider one wing w moving with a mean velocity U in the positive x direc-tion. The wing is assumed to remain in a close neighbourhood of a reference surface H which is described by

where g(x) is a sufficiently smooth function with period b. H: y = g(x) = g(x+b) g(-x) , z

3

h

2

Fig. 2.1. Reference surface H and wing W.

Hence in the case of rigid wings a necessary condition for the theory to be valid is that the chord length of the wings is

sufficiently small with respect to the smallest radius of curva-ture of g(x). Cf course the theory is valid for wings with

arbitrary chordlength of which the profiles are flexible and can follow by this closely the surface H. The reach A of the motion is defined by

X

-4-A = max

g() -

g() I

. O

',

. (2.2)

The wing W has to deliver a. thrust with a mean value , with

respect to time, in the positive x direction. This thrust has to be generated at the cost of as little energy as possible, then the wing is called optimum.

Carrying out its task the optimum wing leaves behind free vorticity which, because our theory is linearized, lies

at H and which can be characterized as follows [2J. The reference surface H is placed as an impeiuteable and rigid surface in a

homogeneous flow with velocity À in the positive x direction. Then we need vorticity on H in order to make it a stream surface. This vorticity, when ,\ is chosen correctly, is the

free vorticity left behind by the optimum wing. The way in which we calculate ,\ will be described later on. Hence in order to

find we have to solve an obvious boundary value problem for the potential function

(x,y,z) = (x + (x,y,z)) , (2.3)

where

)(x,y,z)

is the disturbance potential caused by the surface E In terms of (f2 (x,y,z) the boundary value problem reads

)XZ

j

±

) £f(X,L)=O

(2.4)

(2.5)

-1L,_

_(2.6)

where is the unit vector in the x direction and

ff

is the unit normal at H, pointing from the - side (HT) towards the + side (Ht) of H (figure 2.1).

Note that in order that a potential function exists in the whole space with the exception of H, it is necessary and

sufficient that the total circulation around H is zero. This makes (f a "single valued function" outside H and causes the kinetic energy belonging toq to be finite.

We also consider the probably more important case of two wings W1 and W2 moving with a mean velocity U in the positive x direction along two different surfaces H1 and H2 (figure 2.2). The mean value, with respect to time, of the thrust of both wings together is also prescribed to be ¶T. We assume H1 and H2 to be placed symmetrically with respect to the (x,z) plane. They are described by

H: y

= (-l)g(x) (-l)g(x + b) = (-l)g(-x)

(2 .7)

Izi

--

j=l,2

We distinguish between two different cases, the reference surfaces H1 and H2 intersect each other or they do not intersect, as drawn

-5

(b)

Fig. 2.2. Two wings ; (a) intersecting reference surfaces,

-6

in figures 2.2 (a) and 2.2 (b) respectively. In case (a) the wings Wi and W2 cannot move side by side but have to pass one

after the other the lines of intersection of H1 and H2. In

case (b) there is no objection that W1 and W2 move side by side, although then a gap has to exist in order to avoid that the profiles come too close to each other. It will turn out that suchs a gap is from the point of view of efficiency unfavourable. The disturbance potential Ç2 has to satisfy in this case (2.4),

(2.6) and

LP

Li-

_e

.72 C.'-3 C'nK) ,

(X))

(ì,z).

_(2.8)

Here an analogous remark can be made with respect to the potential ÇIl being a single valued function and to the

kinetic energy per period b being finite, as has been done in the case of one wing. In this case the total circulation around H1 and around H2 separately has to be zero.

-7-3. The lost kinetic energy.

Suppose the disturbance potential (f

is known by

solving one of the boundary value problems mentioned in the

preceding section, then we need to deteiiuine the unknown

coefficient >

-

This scalar follows from the condition that the

mean value of the thrust is

.

We first consider the propeller

with one wing W.

The period of time '

of the motion of W is given by

The potential À ¿f is the potential of the flow far behind the

optimum propeller induced by the shed tree vorticity at H.

Hence the condition from which

A

follows reads

where

is the density of the fluid andf(b) is the momentum

of the fluid f ar behind w over a period of length b in the

x direction induced byte. So we have to calculate '()

Th)

_(3.3)

Because the disturbance velocity field is a periodic

function of x with period b and because (( is single valued and

for (y2 + z2) -

we have

_{(f -}

O, it is not difficult to argue

that also

_If

is periodic. Then we can rewrite (3.3) as

-

-(')

=JJELJ

(e

.

H)

where H(b) is one period of H and[(fJ

is the jump of the

potential over H

(3.1)

(3.2)

(3.4)

and dUis an element of area of H(b). Using (3.4) the coefficient À follows from (3.2).

Next we calculate the mean value of the kinetic energy lost per unit of time by the propeller

2

E=BJJ

JfLVd

T

_(3.6)

Z - -

() - _____

2

o-

pLJ

By this we find for the efficiency

NL

-i,

tL+E

- L

2rLLJ(4)

J

We now consider instead of the propeller an actuator disk with the same rectangular working area Ah as the working area of the propeller. The disk has constant load f in the negative x direction such that its total thrust is , hence

/=

/5

(3.8)

and its velocity in the x direction is U. The loss of kinetic energy per unit of time of this disk equals

ci-c

- zpALL

The quality number q of the propeller is then defined by

-

_<i

(3.10)

The inequality in (3.10) follows from the fact that the propeller and the disk are assumed to work in an undisturbed

fluid (no wake of a ship) and the kinetic energy loss of the actuator disk is the smallest possible loss compared with any propeller with the same mean velocity of advance U, the same

mean thrust and the same working area Ah. Using q the efficiency can be written as

-8-(3.7)

-9-=

['

₊

_{zpLL1 1A}

(3.11)

Next consider two wings W and W2. In this case we define the reach A of the wing motions by

A =2maxtg(x)I , O x b . (3.12)

The momentum(b) becomes

2..

-

-4

(_Z

1f rq

(e

X

1='

(3.13)

It is seen that we have to replace A and

'J

in the foregoing formulas by (3.12) and (3.13) respectively in order to obtain the corresponding quantities for two wings.

Hence also in the case of figure 2.2 (b) we take as the working area Ah although there is a gap between H1 and H2. However this gap is an unavoidable drawback of a propeller of which the wings move side by side, so it seems more realistic to reckon this gap also to the working region because it could have been used by a slightly different propeller. In this way we estimate the penalty on letting the wings move side by side.

4. Small values of h.

In case that the span h of the wing or wings is small with respect to the period b, the stroke A and the radii of curvature of g(x) we can apply slender body theory in order to calculate the potential(f.

Again we consider first the case of one wing moving along the reference surface H. The potential

j2 is then determined

by (2.4), (2.5) and (2.6). In agreement with slender body theory we consider each vertical cross section of H to be placed separ-ately in a parallel flow normal to it, with velocity

(X)

Then it is well known that, when the total circulation around the cross section is zero, the vorticity becomes

10 --i - _-,

r i

U 2

-

t

L 5. (4.1)

is reckoned positive when it has, with a right hand screw, a positive component in the positive x direction. Here we assumed in order to avoid complications, that g(x) is a one valued function of x. The vorticity is an approximation of the component of the vorticity at H parallel to the plane z = 0.

An approximation of the vorticity component on H in the z direction can be obtained from (4.1) by the demand that a vorticity field is free of divergence. Because in (3.4)

+

we only need[LfL we do not calculate . From (4.1) we find

1 -p

-,

P - -p

2(.7î)J

_dzce

(4.2)

t

-Hence (b) becomes

/

z "2.. -p ₁

i)zJ

J )(

-

_-

_{I ( ,fl)}

I+(X)2JX.

_(4.3) - .', J r

Then q and follow from (3.10) and (3,11).

In case we have two wings W1 and W2 of small span h, we can simply replace(b) (4.3) by

2

because by the smallness cf h there is upto the accuracy of the theory, rio interaction between the vorticity at J- and at I-i2.

Possibly the part of the theo of this section as given for one wing, can be used for estimating the highest possible efficiency of the large amplitude swimming of eel

12

-5. Finite values of h and h=

When the span of the wing or wings is not small, we will use a vortex lattice method

f3],

[4] in order to obtain an

approximation of the basic quantity

CfJ,

which occurs in the formula for(b) which in its turn detetiuines the efficiency and the quality number q. We first indicate the method for one wing, hence for one reference surface H.

From the symmetry properties of g(x) (2.1) and from the symmetry of H with respect to the (x,y) plane it follows that only the vorticity at the part

Oz4h of H need

to be considered as unknown. Along the line of intersection of H :ith the (x,y) plane we introduce a length parameter s, with

s=O at x=O and 5=5b at xb. On half of this interval we choose N+l points _5n (n=O,l,,N). The x coordinate of so equals zero

and the x coordinate of 5N equa1sb. On the z axis for OzLh, we choose M+l points Zm (m=O,l,,M), with z0=O and zM=h. The points in the s direction as well as the points in the z

direction need not to be equally spaced. Their position for a number of cases will be mentioned later on.

Next we draw on H the straight lines parallel to the z axis, through the points 5n and the curved lines y=g(x), zz These lines intersect at (N+l) (M+l) points an,m. Each point

anm is connected by straight line segments with its neighbouring points in the z direction and in the s direction. In this way a

lattice is constructed which is in the neighbourhood of H. The rectangles of this lattice, which have one side along the upper

edge cf I-I (z= h) are slightly changed. The uppersides of these rectangles are brought downwards over a distance of one quarter of their hight in the z direction, hence over a. distance _(zM-zM1)

Around each of the N.M rectangles of this lattice we put a concentrated vortex of still unknown strength 7

(n0,l,",N-l; mO,l,-,M-l). These are the N.M unknowns of the problem. This lattice is now extended in an obvious way over the whole of H. Note that by using the closed rectangular vortices of strength 1'm we automatically satisfy two conditions. First, the divergence of the vortex lattice is zero. Second, the

- 13

-total circulation around H is zero, as was demanded previously.

n,m+1 _°n+1,m+1

m

anm

mm

0r+1,m

Fig. 5.1. Closed rectangular vortex r

When the whole of H is covered by the lattice of concentrated vortices, we calculate by the law of Eiot and Savart the induced velocities at the midpoints of the vortex rectangles (figure 5.1). Then the component of these velocities in the direction of n is equated to the prescribed value -(.) at H. In this way we obtain N.M linear equations for the N.M unknown vortex strengths . Although the vortex

lattice at H stretches from x=-

towards x=+

, we take into

account only a finite number P of periods to the right and to the left of the considered part of H.

We now discuss briefly the case of two wings. Then we cover both reference surfaces H1 and H2 with a grid of the type we discussed before, while the grid on H2 is the reflected one of H1 with respect to the (x,z) plane. Note that the number

of unknownl, in the case of two wings is the same as in the

case of one wing when the fineness of the vortex lattices in both cases is the same. This is caused by the symmetry of H1 and H2 with respect to the (x,z) plane, by which only the vorticity on for instance H1 has to be considered as unknown.

In both cases, one wing or two wings, the problem

becomes two dimensional for h . Then we need only a grid

the problem becomes easier to handle from the numerical point

of view.

For infoiivation we mention the values of N, M and P and the distribution of the points s and z1 which, by carrying out

some tests, turned out to give satisfactory results for a number of specific cases. We take

(X)

O-5 Lc

t1Z) (5.1)

for the case of one wing and also for the case of two wings with intersecting reference surfaces H1 and H2 (figure 2.2 (a)). For two wings with non intersecting H1 and H2 (figure 2.2 (b)) we take

In the latter case the maximum slope of the reference surfaces is the same as the maximum slope of g(x) given by (5.1), however the length period of the motion is halved. To denote these three cases we introduce a pararneter , the value =1 denotes one wing;

=2 denotes two wings with intersecting reference surfaces H1 and H2 and =3 two wings with non intersecting H1 and H2.

Further we take in some cases the points _5n or Z to

be equidistant and in other cases as given in figure 5.2. We

¿s

¿s

¿s

1<

¿s

¿Z

¿z

- 14

-SN2 SN1 SN

¿z

¿zz

T

2

TT

Z1 _ZM...4

_{ZM3 ZM..2 ZM1 ZM}

Fig 5.2 Distributions of points in s direction and in z direction.

introduce a pararneter(3 withj3 1 to denote the points 5r to be spaced equidistantly andp =2 for the points to be spaced as in figure 5.2. A parameter _{is introduced analogously for the points} z, when ' =1 the z are equidistant, when 2 the z are spaced as in figure 5.2.

Then it turned out that the schemes of table 5.1 for - 15

-Table 5-l. Schemes for calculating the quality number q.

calculating the quality number q were satisfactory. A criterion was that q became stable rather soon with respect to a refinement of the vortex lattice, hence for an increase of the numbers N and M.

We also compared two ways for the calculation of the momentum(b) (3.4) from the found discrete vortex system. First, we determinedCL(J directly by means of the concentrated vortices hence the integration was carried out over a step function.

Second, the discrete vortex system was replaced by a continuously distributed vortex system of the shape

C(){

₄

_'*

L c ()(

_L

-z)5

(5.3)

where K is a suitable chosen number, anyhow smaller than M. The coefficients cj (kO,l,,K) were calculated by means of least

squares. The differences of these two methods however were neglicible, hence the first method, "hich is the more simple one, has been used

in the following.

h=

hl

h0.l

1 2 3 1 2 3 1 2 3 1 1 1 1 1 1 1 1 2 30 30 30 15 15 12 12 12 12 2 2 2 1 1 1 18 18 12 12 12 12 64 128 120 1 8 8 I 2 2 2

16

-6. Numerical results.

We now give a number of values of q which is t.he only unknown quantity needed to calculate the efficiency") (3.11)

for a certain propeller. In the case of one wing (l) or in the case of two wings with intersecting reference surfaces (2) we take

In the case of two wings with non intersecting reference surfaces (=3) we take

px)= k

fL

(e).

(6.1)

(6.2)

In all three cases the reach A of the wing motions equals 2a. We consider three values of a, a=0.l2, a=0.25 and a0.38. For the half width S of the gap between the two reference surfaces

s- c-

ç-(6.2) (case 3), we take five values, =0, '.-' =O.0l,c =0.02,

=0.04 and=0.O6. The span of the wings ranges from h0 towards h3, while at the right hand sides of the figures the values of

q for hare given.

1.00

0.80

a: Ui

0.60

z

>-0./.0

°

_0.20

a=0.38 0.25 0.12

0.00

1 _I i i

/

0.00

0.50

1.00

1.50

2.00

2.50

3.00

SPAN

Fig. 6.1. Quality number q,

i wing, casel.

O.6/.8 O.5/.5

1.00

0.80

LU

D.

z

>-0./.0

0

_0.20

1.00

-

17

-0.00

0.00

0.50

1.00

1.50

2.00

2.50

3.00

SPAN

Fig. 6.2. Quality number q, 2 wings, case« =2.

a0.38

0.25 0.12

0.00

I I I I

'-0.00

0.50

1.00

1.50

2.00

2.50

3.00

SPAN

co

Fig. 6.3. Quality number q,

2 wings, case=3, a0.12.

0.00

_0.670

0.01 0./.09 0.02 0.3 14 0.0/.

0.188

0.06

0.103

0.80

w

0.60

z

>-0.40

0.20

0.86 1 0.823

0.755

co

1 .00

0.80

w

aD

0.60

I- 0./+0

-J c3

0.20

0.00 1.00 0,80

0.60

0.40

o 0.20

0.00

0.01 0.024

*0.0/.

0.06

0.00

I I I I 1

//

0.00 0.50

1.00

1.50

2.00

2.50

3.00

00 SPAN

0.830

0.699

0.653

0.585

0.52 8

Fig. 6.5. Quality number q, 2 wings, caseo(3, a0.38.

0.00

_0.77/.

0.01 _{0.60 1}

0.02

_0.537

0.04

_0.4/./.

0.06

_0.368

0.00

0.50

1.00

1.50

2.00

2.50

3.00

OD SPAN

Fig. 6.4. Quality number q, 2 wings, case 3, a=O.25.

1.00-0.80

w

D.60

z

>-0.0

0 0.20

0.00 1 1 i _{I I} i

/

0.00

0.50

1.00

1.50

2.00

2.50

3.00

SAN

5ig. 6.6. Comparison of quality number q, a0.38;

casec<2

, case3,=C,

It follows from the results that the quality number q increases in all cases with h and with a. The quality number

decreases strongly in the caseo( 3 of two wings with the increase

of hence with the increase of the width of the gap in between the two reference surfaces. This means that in the case of

figure 2.2 (b) the optimum efficiency (3.11) will be sensitive for the space which is needed in between the two profiles when they are at their closest position.

When we keep , U and A constant it follows from the

graphs in connection with (3.11) that in casesl and =2 for

increasing values of h with h O.b the efficiency will strongly increase. For values of h O.5b the efficiency will increase more

smoothly with h. In case3, because there the length

period is halved, we have to replace the inequalities by h b and h b . When the wings behind the ship pierce the free surface

it is not clear what we have to take for the value of the span.

To a certain extent the free surface acts as a boundary of the fluid and possibly the "effective" span is longer than the underwater part of the wing, by which a larger value of q will be more appropriate. / I 0.861

0.830

00

20

-The slender body theory of section 4 yields the slopes of the graphs at the origin of the figures 6.1 - 6.6, hence for sufficiently small values of h. The results of that theory are given in tables 6.1 and 6.2.

In table 6.1 the second row is a factor 2 times the first row. This is because slender body theory does not perceive the interaction between the two reference surfaces H1 and H2 in the cases =2, hence the results for one wing are simply doubled. In table 6.2,which also refers to slender body theory, the

quality number q changes for one value of a. This is because when S changes and the total reach A of the two wings together

is kept the same (6.2), the geometry of H1 and H2 changes.

Table 6.1. Quality nurner q, slender body theory, h0. 04.

Table 6.2. Quality number q, slender body theory, 3,

h0. 04. 0.12 0.25 0.38 1 0.782h l.l75h l.339h 2 l.564h 2.350h _2.678h O 0.01 0.02 0.04 0.06 0.12 l.564h l.346h 1.137h 0.75Th 0.442h 0.25 2.35lh 2.220h 2.090h 1.832h l.578h 0.38 2.677h 2.590h 2.503h 2.328h 2.154h

21

-References.

1 Weissinger J., Linearisierte Profiltheorie bei

ungleichfoiitiiger Anstromung, I, Unendlich dinne Profile (Wirbel und Wirbelbelegungen), Acta Mechanica 10, 1970.

2 Sparenberg J.A., Sorne ideas about the optimization of

unsteady propulsion, 0.N.R. Symposium, London, 1976.

Belotserkovskii S.M., The theory of thin wings in subsonic flow, Plenum Press, 1967.

4 Kerwin J.E., The solution of propeller lifting surface

problems by vortex lattice methods, M.I.T., Naval Architecture Department Report, 1961.