A slender delta wing oscillating in surface waves. An example in unsteady propulsion

(1)

DEPARTMENT OF

,HIP HYDROMECHANICS

AEROSPACE ENGINEERING

LABORATORY

REPORT LR -257

_Q

REPORT 456

A SLENDER DELTA WING

OSCILLATING IN SURFACE WAVES

An example in unsteady propulsion

(\'V

by

Coene

DELFT - THE NETHERLANDS

December 1977

(2)

DEPARTMENT OF

SHIP HYDROMECHANICS

AEROSPACE ENGINEERING

LABORATORY

REPORT LA -257

REPORT 456

A SLENDER DELTA WING

OSCILLATING IN SURFACE WAVES

An example in unsteady propulsion

by

R. Coene

DELFT - THE NETHERLANDS

December 1977

(3)

In this report an experimental investigation of some cases of unsteady propulsion in non-uniform flow conditions is discussed. A not-so-deeply submerged rigid slender delta wing-like body is towed horizontally through a regular train of surface waves. At the sanie time the body is forced to carry out heaving and pitching oscillations at the frequency of encounter. The results are compared with theoretical predictions based on slender body potential flow assumptions. The experiments tend to confirm the theoretical prediction that energy can be extracted from the waves in such a way that it can be used efficiently for propulsion. The results are relevant to the theory of swimming of cetacea and f ishes in waves and to the theory of the resistance of ships in a wavy sea. The report contains some additional theoretical considerations and several suggestions for future work.

(4)

CONTENTS,. page Summary I Contents 2 List of symbols 3 I. Introduction 5 Theory 8 The experiments 24 Results 32 Concluding remarks 50 Acknowledgement 52 References 53 Appendices

A.I. The distributions of A(x) and S(x) for the model. 54 Numerical evaluation of the slender body results. 56 The correction for the inertia of' the model. 74 The extraction modes with the tail tangential.to its path

with respect to the tank. 76

The waves generated in the towing tank. 78

Data reduction. ' 80

(5)

LIST OF SYMBOLS.

a wave amplitude (at the free surface)

amplitude of the motions of the water particles due to the waves at the mean swimming depth d

A area related to virtual mass effects of the body cross-sections b span of the model

d the mean swimming depth

e1, e2 amplitude of oscillation of the model at pivot points I and 2 respectively

max the maximum mean rate at which the body can extract energy

from the waves

h(x,t) vertical displacements of the body cross-sections

L length of the model L steady uf t on the model L(x,t) the lift per unit length

S surface area of the body cross sections T period time excitator

e

T period time wave generator T the man thrust

the thrust generated by the waves without oscillation of theniodel

U the forward velocity

w resultant crossfiow in uniform oncoming flow

resultant crossf low at the mean swimming depth in waves

W the mean rate of working against the z-component of the hydrodynamic forces

(6)

X

y Cartesian coordinates moving steadily with the model

z t _time angle of incidence C perturbation parameter Froude efficiency physical efficiency A wavelength

11 wavelength of the path of the tail

V factor proportional to the amplitude of the motions of the tail

p density of the water

velocity potential of the waves

vertical component of the orbital velocity at the mean swimming depth

phase angle by which the displacements of the trailing edge lead those of the oncoming waves

phase angle by which the displacement of the first column leads the displacement of the second one

circular frequency

(7)

I. INTRODUCTION.

This report deals with an experimental investigation of a case of unsteady propulsion. A rigid slender delta wing-like body is towed horizontally and forced to carry out vertical oscillations through a regular train of surface waves. Strain gauge dynamometers are used to measure the horizontal and vertical components of the forces exerted on the model.

Fig. 1.1

When certain heaving and pitching oscillations are carried out at the frequency of encounter of the oncoming surface waves it is possible to extract energy from the waves in such a way that a given thrust can be generated at an efficiency which is higher than in the absence of waves.

in [2] and [3] a slender body theory was derived for the swiing of

unconstrained flexible slender bodies in waves. The case of a rigid slender body carrying out forced oscillations as in the experiment can be handled by the theory and a comparison of the experimental results with the theoretical predictions will be made in this report.

(8)

i outline of the problems we have in mind can be obtained by considering the following power balance:

+ = TU + waste

3t max

Here represents the net mean rate of working by the body against the lateral components of the hydrodynamic forces in the theory but in the present experiments this term represents the mean rate of working by the excitator against the vertical components of the hydrodynamic forces on the model. For a ship oscillating in passive recoil in oncoming waves this term vanishes.

(E!) _{is the maximum mean rate at which energy can be extracted from} max

the waves. It should be noted that if the actual extraction rate is smaller than this maximum rate, the difference of these two rates is accounted for as a waste.

T is the mean thrust in the direction of mean motion at speed U.. For an unconstrained swimming body at constant U, the thrust T should balance, in the mean, the drag forces on the body. In the towing tank experiments described in this report this requirement has been ignored.

The "waste" in (1.1) can be identified as the mean rate of shedding of kinetic energy into the wake in the absence of waves and apparently this is also correct in the more general case with waves.

The four terms in (L. I) are calcúlated and discussed in detail in [3]. The experimental determination of the first three terms in (1.1) for some special cases is described in this report.

It is found in [3] that the mean rate of wasting energy is minimized in motions of the body which are characterized by the fact that the tail is tangential to its path with respect to a reference system fixed to

the water at rest far from the surface. One may say that the best motions for the tail are precisely those iñ which the tail would be

inoperatIve in the absence of waves. The same result can also be stated as follows: The resultant crossflow at thé téil should be exactly minus the local momentaneous vertical component of the orbital velocity of the. surrounding water particles due to the waves. The vertical component

(9)

of the velocity of the water has vanished just above and just below the tail.

in chapter 2 the slender body theory will be explained in as far as it is relevant to the present experiments. Moreover., some additional remarks will be made in a more general theoretical context. In chápter 3 the model and the experimental arrangement will be

described. Moreover, the selection and presettings of the test runs will be explained.

In chapter 4 the main results will be presented and it will be seen that in several cases a somewhat better than just qualitative correspondence between theory and experiment is obtained.

(10)

2. THEORY,.

In [2] and [3] expressions are derived for the mean thrust and for the mean rate of working of flexible slender bodies swimming in waves. The bodies are assumed to have a constant forward velocity U normal to the crests of a regular train of two-dimensional waves and move at a nearly constant depth, d. The bodies exhibit left-right. Symmetry and the oscillatory vertical displacements h(x,t) of the sections which are

assumed to be uÌideformable are parallel to the plane of symmetry. See Figs. 2.1.. and 2.2.

For the leading term of the lift force per ünit length, omitting the buoyancy part,, one obtains:

L(xt)

-P(}

U )(w'À)

+ p S + U )

Here w is the resultant crossf low due to the displacements of the body sections and the vertical velocity 4? due to the waves at the mean swimming depth:

x h h x

w ,,

pA is the virtual mass per unit length for motions in the z-direction and S is the cross-sectional area. For the mean rate of working against

the z-component of the hydrodynamic forces one can write:

-W-í -. L(x,t)dx

(2.3) o

where L(x,.t) is given by (2.1). The bar indicates that the time averaged value is to be taken.

The restriction to undeformable sections can be removed. This is relevant im applications to the swimming of bodies carrying out peristaltic motions. Such peristalic effects also arise when a surface

ship oscillates in a wavy.sea and it may be interesting to investigate the validity of extensions of the formalisms introduced in [2] and [3]

to surface ships,.

(2.1)

(11)

z

X = constant

hix, t)

y

(12)

For periodic motions one takes the average over n periods where n is a natural number. In the experiment n was chosen in such a way that an acceptable signal to noise ratio could be expected. Assuming SA=O for

x=O and SO but AO for x=i, one obtains:

lah M1

i _aw

W pA(i)U

_{(y w ji + P}

f (A+S) d dx +

o

+ p U

/

- dx

dS

at wz dx

On the other hand one has for the leading terms of the mean thrust:

pA(i) - -

u2()2

Il

xi +

3 dS

+ p f

(A+S) w dx+pUf h i. dx . (2.5)

The terms in square brackets were obtained earlier by Lighthill [i ] for the case of uniform flow.

The two terms in (2.5) involving integrals can be interpreted as the forces acting on dipoles with vertical axes in a velocity field with a vertical gradient. The last term in (2.5) has been omitted in [2]

[3] and[4] but it is consistent with the usual slender body assumptions

to retain it.

The Froude efficiency of propulsion is given by: TU

(2.6) W

In uniform flow, i.e. without waves,, the propulsion is uniquely determined by the situation at the tail only and the vorticity left behind in the wake is uniquely related to the rate at which kinetic energy is being lost. In uniform oncoming flow, r cannot exceed unity and to generate a positive , the rate of working must be positive. In the presence of vaves T may exceed unity and is not necessarily

positive when is positive. In fact this r is a wild function and it is possible to obtain an efficiency r which behaves better. Using the

(13)

TU

i (k.)

_max

TU TU+waste

In[5] this number is referred to as the "physical efficiency". in the non-uniform case the integrals in the expressions for and T are not related to the kinetic energy of the vorticity in the wake. In this respect there is a certain similarity with the far field considerations of the flow field around a supersonic wing where one can distinguish a vortex drag and a wave drag.

For an unconstrained swimming body the displacements h(x,t) cannot be chosen arbitrarily because the equations of motion of the body as a whole must be satisfied. In[3] this problem is discussed in detail. In the experiments described in the following sections the model is rigid and the motions are forced by two excitator columns carrying out vertical oscillations with the same frequency but with amplitudes and

phase angles which may be different,

h(x,t)(I .82-x)e1sin(wt+i12)+(x-O.82)e2 sin ut (2.8)

e1 sin (wt+(12)

e2 sin wt

Fig. 2.3

(14)

Fig. 2.4.

Here e1 and e2 are the amplitudes of the oscillations of the first and second columns respectively; w is the circular frequency and is the phase angle by which the displacement of the first column leads the displacement of the second one.

The expressions for i and T indicate that the thrust and the efficiency of propulsion will be mainly determined by the situation at the tail, or

the trailing edge, when A increases to a maximum at x. For the

essentially planar delta wing used in the experiment A(x) is.proportio-nal to z2 and,excepting the apex, much larger than S(x).

(15)

X

0.05 2.05

Fig.2.6 The A and S distribution for the model (see also appendix J).

So the propulsion will be dominated by the flow conditions near the trailing edge but this effect will be less pronounced than can be expected for a slender body with the A and S distribution of a cetacean mammal.

Another aspect that may be noticed is the following:

A flexible, freely swimming, body can produce favourable propulsion with its tail while at the same time the crossflow at anterior body sections is kept small. In contrast,with a rigid model which is forced into motions which lead to a favourable situation for propulsion at the trailing edge, i.e. the proper correlation of , U and at x=R., relatively large cross flows and flow separation may occur at anterior sections. It is clear that the amount of separation depends on the phase angle of the periodic motions of the model with respect to the oncoming waves.

For a proper understanding of the comparisons to be made between the slender body calculations and the experimental results it is helpful to consider first three special cases of motion of the rigid model in uniform flow.

(a) The model is towed at a constant angle of incidence without carrying out oscillations. Then (2.4) yields =0 and (2.5) leads to a drag force:

(16)

= -p A(9)(U 3h)2 (2f9) At the other hand oné obtains from (2.1) and upon' integration for the

total lift':

L-pU2A(2.')

(2.10)

The hydrodynamic center is at x= With a weak .Jôukowski condition at the trailing edge the lift distribution is the same as for a flat tri:angular plate at incidence with the same planform which induces a conical field.

A more realistic strong Joukowski condition would predict a slightly reduced effectiveness of the par,t of the wing near the trailing edge, where the pressure jump would then vanish with the square root of the distance from the trailing edge. Application of the strong Joukowski condition would lead to a lift slightly smaller than (2.10) and a hydrodynainic center at x < P,. Satisfying this strong condition however would not be consistent with the slender body assumption in the usual sense and therefore no such corrections were applied.

Comparison of (2.9) and (2.10), and replacing by for the angle of incidénce, one finds:

This is the well known result for the vortex drag of a sleiider wing with elliptic spanwise lift-distribution. The resultant fórce on such a wing is 'obtained by inclining the lift force backward over one half of the angle of attack. At sharp leading edges the velocity field and the pressure distribution are characterized by square root singularities and' finite suction forces with a thrust component in the -x direction to the amount of L act at these edges. 'In order to realize such forces physically the leading edges must not be sharp, they must be rounded off. Then the potential flow solution without separation leads

to:

T aL , (2.12)

suction

-T'=D

=aL

(2.11)

(17)

and compari8on with (2.11) shows: D = aL - T _{(see Fig.2.7)} vortex suction D_vortex

aL

x--L)

L (1 + O(a2))

Fig.2.7 Steady wing at incidence (case(a)).

O vortex

Fig.2.8 Heaving wing (case(b)),

(18)

The model carries out heaving motions given by:

h(t) = e sin wt . (2.14)

From (2.4) one then finds: - p A(e) U

()2

, (2.15) and from (2.5): = p A(i) (eu»2 (2.16)

It should be noticed that for oscillations at a frequency ù, the

horizontal components of the pressure forces have a productive component with frequency 2. In this heaving case the efficiency of propulsion is:

TU

(2.17)

w

One half of the work done by the excitator against the lift forces is wasted as kinetic energy in the wake. The other half is spent usefully and yields propulsion. Comparison of Figs. 2.7 and 2.8 indicates that in both cases (a) and (b) the momentaneous lift is tilted by one half the momentaneous angle, ia,, and that in the heaving motion this leads to a thrust force which is generated at an efficiency n=..

Again the hydrodynamic center is at x= and the considerations related to the Joukowski-condition at the trailing edge in the previous case also apply in this case.

The class of oscillations which are characterized by the fact that no vorticity is being shed into the wake. In this uniform flow case the mean thrust and the mean rate of working should vanish. From (2.4) and (2.5) it can be inferred that this can only be accomplished by motions satisfying;

at ax (2.18)

In this case there is no crossf low at the tail. The lift and the

(19)

body approximation, as can be seen from (2.1). In geometrical terms (2.18) implies that the tail of the model (xL) should be tangential to its path with respect to a coordinate system fixed to the water at rest. It will be explained below that these motions which are trivial for propulsion in the case of uniform flow are "optimal" in the case of non-uniform flow for the extraction of energy from non-uniformïtes. Moreover, the amplitude of the motions of the tail satisfying (2.I8) and the phase angle with respect to the waves can be chosen to yield a given thrust for a given

4'.

On the other hand, the drag measured in motions satisfying (2.I8) in the uniform flow case gives some indication of the extent to which one is justified in treating the viscous drag as a force depending où the forward speed U only.

The simplest possible case where the non-uniformity of the onëoming flow gives a contribution to the thrust (2.5) arises when the constrained model is towed horizontally through a regular train of surface waves without carrying out any oscillations, i.e. the case with:

h(x,t) = O and O . (2.19)

From (2.4) and (2.5) one obtaines =O and for the mean thrust:

= pA(R,) (2.20)

For a sharp trailing edge with span b one has A(i) = b. It is clear

that in (2.20.) the details of the geometry of the body do not appear and that only the vertical component of the velocity field induced by the waves at the swimming depth does occur. From the volume which is effectively being swept by the trailing edge over long periods of time only one half of the kinetic energy present due to the surface waves can be extracted and made available for propulsion. It is inherent to the slender-body approximation that the other half of the kinetic energy cannot be extracted, because the lateral perturbation velocities

(20)

due to the body are O(e) while the streamwise perturbations are only

2

-O(e ). The thrust T as given by (2.20) does not depend on U but it must be remembered that in order to avoid violation of the usual slender body assumptions it remains necessary to require U. »

i is clear f ron the expressions for the mean rate of working (2.4) and the mean thrust (:2.5), involving time averages of products of terms related to the motions: of the body on the one hand and the oncoming waves: on the other, that interesting interaction effects can arise only when the motions of the body are properly correlated with the oncoming waves. In the case of a freely swimming body h(x,t) will always contain an involuntary responsive. part which is correlated with the oncoming non-uniformities. In [2]. and [3] it has been shown that it is worthwhile to correlate the voluntary swimming motions as well. In the experiments described in this report the frequency of excitation of the model could be preset and taken equal to the calculated

frequency of encounter of the oncoming waves, permitting an evaluation of the dependency of the interaction terms on the phase angle between, say .the displacemént of the trailing edge and the displacement of the water surface above the trailing edge..

A power balance can be expressed asfollows:

+ (.!) = U + waste . (2.21)

tmax

F.or an unconstrained swimming body represents the mean rate of

working against the z-component of the hydrodynamic forces exer.ted on the body. In the present context of the experiments this term represents. the rate of working of the excitator.

On the. assumption that the propulsion is dominatéd by the terms in (2.4) and (2.5) with square brackets, involving the situation at the trailing edge only, it is possible to obtain some qualitative insight. it follows frOm (2.4) and (.2.5) that the "tail terms" lead to the following, estimates for the respective contributions in (2.21):

1M2

.K

Pd 4pA(L)U 1[w +2v

(21)

2 -(.!) _d _{pA(L) U} _{= TTM U}

3t max wz

and for the waste one obtains:

3(TM2

2 waste 3 _{pA(R.)(w +}

) pA(L) w

Wz x=, x=2,

It should be observed that in this "waste", the difference between the mean rate at which energy could be extracted from the waves and the mean rate at which it is actually being extracted has been included. Without waves one has w=wTM and the right hand sides of (2.22) are equal

to the waste (2.24) and can be interpreted as the mean rate of shedding of kinetic energy into the wake.

From (2.22a) it is clear that it is favourable to correlate w and wz negatively at x=2. In fact the right hand sides of (2.22) have a minimum value: r

2'

I I TM -c

at x,

pA(2)U for w = which can also be expressed as:

(2.23)

(2.24)

(2.25a)

wOatx

. (2.25b)

TM TM.

It may be observed that (2.25) implies î I, where n is given by (2.7).

From (2.25a) it is clear that the resultant crossf low wTM _{at the}

trailing edge is exactly minus the local vertical component of the orbital velocity of the surrounding water particles due to the waves. The vertical component of the velocity of the water has vanished just above and just below the tail. Moreover, from (2.25b) it follows that the tail is tangential to its path with respect to a coordinate system or alternatively, using (2.2):

J pA(2) u

[w2_:

(2.22b)

The maximum mean rate at which the body can extract energy from the waves is estimated by:

(22)

fixed to the water at rest (the towing tank). With perfect correlation !'the wavelength" p of this path is, in case of a regular train of on-coming waves with wavelength À and phase velocity e,

p = ÀU/IU+cI

. (2.26)

In _[31 it is shown that for perfectly correlated motions satisfying (2.25) it is possible to write:

ç

d \)..0

_(2.27)

where V is proportional to the amplitude of the motions of the tail. Whenever is insufficient to sustain a speed U the body or the excitator, must do work and one may say that the reference thrust can be amplified. For further details we refer to [3].

Obviously y cannot become too large and in the case of our rigid model the requirement that the flow should not separate from the model implies a rather severe restriction on the amplification factor (I+v).

The integrals appearing in (2.4) and (2.5) depend on the details of the body geometry as well as on the motion of the body and its correlation with the oncoming waves. In the comparisons to be made with the

experiments the complete exprèssions (2.4) and (2.5) have been evaluated numerically for all the motions that were carried out in the experiments. See appendix 2.

Some additional remarks..

It should be noticed that the simple geometric and kinematic criteria which follow from (2.25) do not determine the motions of the tail uniquely. In this respect there is a certain analogy with results obtained by Sparenberg

_[sJ

for cases of unsteady propulsion in steady non-uniform flow conditions where the maximum possible amount of energy that can be extracted from a non-uniformity can be obtained from

considerations involving suitable layers of vorticity only. The steady non-uniformity can be replaced by a layer of vorticity in a reference strip containing the wake of the wing system and yielding the same non-uniform velocity component perpendicular to the reference strip. The

(23)

free vorticity being shed from the trailing edge should then annihilate (to first order) the vorticity representing the non-uniformity in order to extract the maximum amount of energy from the non-uniformity through which the wing is moving. One may say that the non-uniform component of the velocity normal to the reference strip must be relaxed for maximum extraction.

In general the lines of constant vorticity in the wake of a wing carrying out oscillatory motions with oscillating valües for the bound circulation will exhibit a cellular structure. In the slender body case the streamwise length of these cells is large with respect to the lateralj, spanwise, dimensions and the velocity component normal to the wake induced by the vorticity in the wake is dominated by the streamwise (trailing) component. it appears that the slender body results obtained in [3] for unsteady non-uniformities are a natural counterpart of the results obtained in [1j for wings moving through steady non-uniformities. The assumption of slenderness in the streamwise direction makes it possible to remove the restriction to steady non-uniformities which is necessary in[5J. The requirement which the free vor.ticity in the wake must satisfy along the wake with steady non-uniformities is replaced by a requiremett on the local and momentaneous resultant crossflow at the trailing edge for maximum energy extraction.

In the slender body approach described in {3J one is led firstly to a criterion which assures the extraction of the maximum possible amount of energy from the waves and only secondly to a more precise specificat-ion of the motspecificat-ions of the body and tail for efficient propulsspecificat-ion. Such a separation of criteria does not become apparent in the two-dimensional linearized approach by Wu et al [2cl to the problem of flat plates carrying out heaving and pitching oscillations in waves., They discuss a class of global optimum propulsion solutions. In two-dimensional thin wing potential theory (not necessarily steady or linearized) an important

role is played by the three quarter chord point and it may be worthwhile to investigate whether a similarly important róle can be assigned to this point in the two-dimensional problems indicated in [2c]. There appears to be an analogy between the kinematic condition at the trailing

(24)

edge in the slender body case and this condition at the three quarter chordpoint in the two-dimensional flow case. If in uniform flow

conditions the wing remains tangential to the paths of these respective points there is no energy transf er in the mean for periodic motions. In the slender body case this class of motions yields maximum extraction of energy from non-uniformities and it would be interesting to find out to what orders of approximation such simple criteria can be applied to the two-dimensional case. Without entering into further detail we note that the result obtained in the slender body cäse [3] with a single wave is readily generalized to a spectrum of waves. For maximum extraction the tail should still remain tangential to its path with respect to the water at rest. This would be an important feature for an observational investigation of the motions of the tail flukes of Cetacea swiuming in waves when they are in some sort of a hurry or when they must cover a large distance.

There j:5 another aspect of the propulsion problems discussed in this report which shoûid be mentioned It appears that surface ships in waves generally experience a resistance increase due to the waves. An example of that phenomenon is discussed by Gerritsma and Beukelman [6]. The present slendér body results sggest. that it must be possible to design submarines which benefit from a spectrum of surface waves and one can speculate even further aboût the possibilities for surface ships. Obviously the problem for surfäce ships is much more difficult than the problem related to submerged ships where the free surface effects remain relatively unimportant and can be treated s corrections. if one feels it is necessary to do so. In view of the towing tank experiments described in the next chapter the free surface effects for the steady towing case and for some more general ocillations with and without waves were evaluated numerically [7] for certain distributions of pulsating singularities of which the strength was derived from the

theory [3]. It turns out that there are complicated interference effects which, interstingly enough, do not always lead to an increase of

-resistance. The main conclusion as far as the experiment was concerned was that the free surface effects, although definitely present, do not

(25)

severely affect the results obtained. In fact the viscous effects turn out to be relatively more important as a source of uncertainty.

(26)

3. THE EXPERIMENTS.

3.1. Introductory remarks.

The main purpose of the experiments is to obtain experimental results which can be compared with slender body calculations carried out in relation to the swimming problem of cetacean mammals and fish in non-uniform flow. Obviously such results are also relevant to the propulsion

and resistance properties of ships in waves.

Rather than reproducing the geometrical features of a constrained porpoise at rest, a rigid slender delta wing vas designed which could be used to extract energy from surface waves and generate propulsion from this energy in essentially the same way as a freely swimming flexible body can attempt to do by properly correlating the voluntary part of its motions to the non-uniformities passing along the body [2,3].

Fig. 3.1. A regular train of oncoming waves in the towing tank of the ship hydromechanics laboratory..

(27)

The experimental equipment of the ship hydromechanics laboratory avail-able in Deif t was well suited for the experiments. A towing tank where a regular train of surface waves can be generated while at the sanie time the model can be towed and oscillated at a frequency equal to the

frequency of encounter, thereby obtaining correlations depending upon the modes of oscillation and a phase angle. For comparison approximately uncorrelated motions could also be carried out. The 11f t and drag

forces at the first and second pivots were measured by application of strain gauge dynamometers. For the vertical components of the forces, including the inertial forces due to the model, the first harmonics

(in phase and quadrature terms) were extracted. For the horizontal components the second harmonics and the time integrated terms were resolved.

3.2. The model.

The model consists of waterproof plywood, 0.06 ni. thick, a length of 2 m. and a span of 0.41 ni., the planform being that of a slender delta wing. The two bulbs at the bottomside were needed in order to obtain sufficient room for the dynamometers and in order to avoid time-consuming key-hole surgery. At the top side the two compartments were sealed off with rubber sheets which were kept flush in the mode1 surface by maintaining the proper air pressure in the compartments. In order to avoid condensation problems a small ventilating flow of dry air was maintained.

The sections (x=const.) are elliptic (the sections of the bulbs are circular). For x < 0.35 m. the long axis is vertical; for x > 0.35 ni. the long axis is horizontal. The nose is rounded off and the trailing edge is sharp. See Fig. 3.2.

The front compartment housed two dynamometers for lift and drag respectively and bearings with a single degree of freedom (rotation).

(28)

k

s

N!uIuup

-f&'iLo/

a/4i2ID

A

I

ra

dt

;.orsneQcc

d y.

1AwlvDw.n.

_{( 144)}

.Z7Dop,ft7gdgD.

Fig. 3.2

Lt

3scpj,,ide-LE

.VeoP3,,.Q//, Gr

3. vi4/wh4ai7 ¿,'X dgm.,,hpa/.

Vo., c.ipd,4',.#v' già /..ua io'.//a,, ,#,diW,

fl,a4P Cd'4dJ,a/fl gJiIIaií!p Oft ISF4OQh ioIi?Êkx,

"m d..,a,,d .,4pon

, c.bd,,d.n, moa' wppdv, Md./ W,,Q«#,

/

Ñh'. 1g. imrÁi.ding.s saIaid

1.?d? qIMtl.n

,,t

ci. 7-Z

t

-rn

Oo.a.'g

_m,iiihck,

A_... RDmp. .Do/fji model V LIEGTUIGOOU WO UN 0 1 T IC H N DELIT

îH-go. oo.o4

MM.I 01.; Th_ ,...b.M,t. .1

(29)

The rear compartment also housed two dynamometers and bearings allowing for two degrees of freedom (rotation and translation; during oscillation of the parallel columns of the excitator the distance between the pivots

is variable when the oscillations of the columns are not identical). The excitator columns are at a distance of I m. from each other. They can carry out vertical oscillations with an amplitude up to 0.06 m. at a frequency up to a few Hz and a phase difference which can be preset at will. The pivots and the dimensions of the compartments allow for

ah

a maximum slope -

0.12.

ax

r

Fig. 3.3..

The loading on the four strain gauge dynamometers should never exceed 400 Newton and in view of the oscillatory nature of the effects to be studied it was important to have a model with a mean density approxi-mately equal to the density of the water so that the weight of the model is balanced by the buyoancy forces.

(30)

dimensions of the model. Another consideration which is worth mention-ing is that for a planar model (A » S) the relation between the hydro-dynamic forces on the model and the corrections to be made for inertial forces of the oscillating model is more favourable than for a non-planar (A S) model. Moreover, a non planar body would cause larger and more complicated free surface effects giving a smaller signal/noise ratio.

The center of gravity of the model is located at x-1.354 n. The mass is 1.63 kg. The radius of inertia for pitch oscillations is O.48 n. The

inertial effects of the model do not affect the power and energy balances, but they should be accounted for in the loadings of the dynamometers. In order to single out the hydrodynamic part of the forces, the measurements must be corrected for the intertia of the model. See. appendix 3.

3.3. The experimental arrangement.

The test arrangement is given schematically in Fig. 3.4.

The mean depth of the model could be selected by adjusting the level of the excitator. The frequency of excitation, the two amplitudes of

the columns and the phase difference could be preset. Due to the arrangement of the tank and the wave generator and also in view of the pilot character of the experiment the test runs were restricted to

cases with head waves and to cases without waves for comparison. The conductive wave probe took its signal from the waves in a vertical plane through the mean position of the trailing edge of the model at a

spanwise location where the perturbations of the oncoming waves due to the columns and the model looked negligible. The wave signal was visualized on a Uy recorder together with signals representing the displacement of the second column and the loads on the dynamometers. The 13V recordings were mainly used to determine the amplitude of the waves and the phase angle by which the displacements of the second

(31)

I. strain meters

phase

settin9N\4

scotch yokes

air

outlet

Motor

'7

R1 D1 ₀₂ R2

I.

strain gauge dynamo-meters

wave probe

Fig. 3.4: The experimental lay-out

quodrature comp.

print outE 2 resolvers w (R1.R2)

2resolvers 2w(D1,D2)

pressurisation

ventilation

oscilloscope

¡r

e1sin(t.41

(32)

Moreover, the signals from the wave probe and the displacements of the second column were combined on an oscilloscope to

yield

Lissajous curves for a visual check on the quality of, correlation of the two signals during the test runs.

The four signais from the dynamometers were handled as follows: For the test runs without oscillàtions of the model th four time-integrated outputs for the lift and drag forces at the two pivot points were printed out.

In the test runs with the model oscillating the two first harmonics were resolved and printed out for the lift forces. For the drag forces the two second harmonics were resolved and printed out as well as the

time integrated parts.

It may be observed, that in the absence of friction and obstruction in the linear and pivotal bearings at the second pivot point, one .has D2

-L2,, suggesting that the direct measurement of D,2, is redundant. in fact however, the comparison of D2 and L2 provided us with a useful check on the proper functioning of the mechanisms,.

For every test run the momentaneous towing speed was printed out several times, permitting an accurate estimate of the mean value for the towing speed U, which in general was very nearly constant.

The dry air for pressürization and ventilation was. supplied by high pressure air bottles. The- pressure. in the model was maintained at a value slightly higher than. the surrounding hydrostatic pressure by, adjusting 'three reduction valves.. During the nights the model was pressurized by the pneumatic power system available in the laboratory.

3.4. The test runs.

Three types of motion of the model, all with a constant forward speed U, were carried' out.

(i) Steady translations with . O and several

'(ii.) Heaving motions with h(x,t)e -sin

t.(e1e2e,

Motions for which the tail remains tangential to its path with respect to the tank; '( + U 0.

(33)

For comparison these motions were carried out in three types of oncoming flow:

Uniform flow.. At a mean depth d=I .08 m. the presence of tiSe free surface and the bottom and sidewalls of the tank are assumed to be negligible.

Almost uniform flow. The model is towed at a mean depth d=0.37 rn. without oncoming surf ace waves.

Non-uniform flow. At a mean depth d0.37 n. the model is towed in regular trains of head waves.

For the types of flow (c), with head waves,, correlated runs were made. The correlated runs were made by presetting the upper limit of the

frequency of the excitator equal to the cal'culated frequency of encounter of the oncoming waves at the preselected U and A. By delaying the moment at which the preset excitator frequency was attained and at which the actual measurements started, a rough choice of the relevant phase angle

could be made by freezing the desired Lissajous curve on the oscillos-cope. Moreover the resulting Lissajous curves provided essential informat-ion on the quality of the correlatinformat-ion, or rather the constancy of 4i. A

subjective criterion for the constancy of the Lissajous curves during the test runs waS used as a criterion for the constancy of . When the deformation was seen to become too large the run was stopped at the completion of the last acceptable period. In this way the variation of

o o i during a test run with correlation could be kept smaller than 20 -30 for most test runs. When the UV recording indicated a larger variation the run was discarded. Two wavelengths were selected, A4 and 6.25 and

in most cases with waves the towing speeds U3 and 4 were taken.

For the uncorrelated runs the frequency of the excitator was taken equal to 90% and 110% of the calculated frequency of encounter and the

measurement was extended over some 10 or 20 periods, the Lissajous curves at the end of the run becoming roughly the same as at the beginning. In view of the absence of the phase angle dependency of W and T in the

(34)

4. RESULTS.

In Fig. 4.1. the drag is plotted against the speed for = 0.18. For the deeply submerged case with = 0.54 no accurate results for the drag are available and therefore it is not possible to isolate the free

surface effect on the drag. In this deeply submerged case the horizontal displacement of the first pivot point under the drag loading caused an unexpectedly large deformation of the second rubber sealing, leading

to unreliable results for the drag. The vertical (lift)-components are unaffected by this displacement. In the not-so-deeply submerged cases with = 0.18 this problem was circumvented by installing some supports assuring sufficient constancy o.f the horizontal distance of the two excitator columns o

In Fig. 4.2. the lift force is plotted against the square of the speed for different angles of attack. flue to the asymmetry of the model the results are not symmetric with respect to the line 0. The bulb at the rear leads to a theoretical nullif t situation with = -0.025,

which is the slope of the line of centrods of the sections at the

trailing edge with respect to the x-axis of the model.

1 2

Fig. 4.1. The total drag force on the model with - O for 0.18 o (N) :30 + +

++

20 1+

++

io ₊

+,

+

(35)

1 2 3 4 Fig. 4.2. The lift force on the model for = 0.18 and 0.54

At the small angles of attack the slender body theory predicts too large a lift. For the larger angles of attack the non-linear effects related

to the vorticity shed at and near the leading edges and the loss of lift at the trailing edge combine in such a way that the correspondence of theory and experiment is better. Comparison of the experimental results for the two depths indicates that the free surface effects in these cases

are less important than the viscous effects in explaining the difference between theory and experiment.

(36)

A similar conclusion follows for the, oscillatory, heaving, motions. In Figs. 4.3 and 4.4 the amplitudes and phase angles of the vertical components of the forces in the front and rear dynamometers respectively are plotted against the speed. It should be noticed that in this case the inertial effects of the model are included. The experimental results for the amplitude at the front are slightly larger than the calculated values, while those at the rear are. smaller than the calculated values. This is due to the fact that in the slender body theory only a weak Joukowski condition is being satisfied. In reality the pressure loading vanishes at the trailing edge and a strong Joukowski condition is being satisfied.

This effect explains the fact that at the rear the loading is smaller and at the front slightly larger than predicted by the slender body calculation. The discrepancy in the phase angle at the lower speeds is due to flow separation. This discrepancy tends to be smaller at the higher frequencies with-the same values for the amplitude of the cross flow, ew and at a certain speed. This is probably due to the fact that in the higher frequency cases the boundary layer effects associated with the crossf low get less time to develop and lead to less flow separation. Another detail that may be observed is that at the lower speeds the Joukowski condition effectively satisfied at the trailing edge tends to 'be weaker than at the higher speeds. In the limiting case with U-O one can expect suction peaks even at the trailing edge.

(37)

Experiiment

Theory

R1(N) _'FR1

d12

0

_0.54

y

018 R1 o e

0.03

w

2it

e O.O4 w

3 ii

RI 160 140 120

e:O.03

w 4 it

Heaving

No waves

2 e _0.06

W:

2m

Fig.4.3

The amplitudes and phase angles of the vertical loadings on the front dynamometers.

3 4

o 4+

o

R1

(38)

80 60 300 250 200 S o

.

+ Ra(N)

_lRi

_d/L

Experiment

s 0,54 +

_y

0.18 Theory S +

f

S +

S_SI.

+ e

0.03

w 21 S

o eV

+

S S +

U4

(I) R2

i'

-

30° 2O

-

id

300 5

250

200 200 100 160 o

180__w

i

S S S o S 4+

e z 0.03

w 4m

2

Heaving

No waves

S e

0.06 w=21t

Fig.4.4 The amplitudes and phase angles of the vertical loadings on the rear dynamometers..

S S S + S

-30

o

-20

o

I

2

___

2 3

..U4

(39)

i

Reduction of air pressure for pressurisation and ventilation.

(40)

The difference between the results at the two different depths, 0.18 and 0.54 is small. The free surface effect results in a slight reduction of the amplitudes of the hydrodynamic forces and a slightly increased phase lag,. but it dóes not alter the general picture of the results. One may say that even at the low value of 0.18 for , when the model is at a depth somewhat smaller than the span, the flow around the model is essentially the same as in the deeply submerged' case.

In Fig. 4.5 the corresponding values for the net rate of working of the excitator have been' plotted against the speed. In the cases with small crossflows (e=0'.03, 2n) it appears that there is no significant flow separation for U > 1. For the cases with larger crossflows this is trué for say U > 2. In these regions of attached flow the slope of the curves fitting the experimental results is smaller than the calculated slope. 'This, again, can be explained by the fact that the slender body calculation does not account for the vanishing pressure jump at the trailing edge. One may say that here the calculation slightly overestimates the effectiveness of the wing as far as the transfer of energy to the flow is concerned.

(41)

e 0,03 w

2Tt

3

ø-u4

O (N rn,sec.)d/.2 Experiment o 0.54 V 0.18 Theory

Fig.4.. 5 The mean rate of working.

Heaving Nowaves

e :0.03

(42)

T e

0.04 o

w=31L

_o

o

0.4 o

G 4

e

G

o

o 2 G

G G

4

o

2

o

Q

Fig.4.6 The thrust in the deeply submerged case. 0'

o

O

E

Theory

Experiment

o

second harmonic

a

integrated drag

reduction

e 0.06

(u=2Tt

o

1.6

o

0

T

o

o 6

0 o

o

e

Heavi ng

No waves

0.54 resonance,

e: 0.03

w 4Th e (A) 0.03

2m

4 2

o

E)

.---o

a

o

a a

o

12---o. e

o

3

U4O

2 3A

U4

1 2 3

øU40

2

3-

,.u4

(43)

In Figs. 4.6 and 4.7 the thrust produced in heaving is plotted against the speed. The experimental results were obtained in two essentially different ways. One set is obtained by subtracting the total mean drag of the model in heaving motions from the drag in the corresponding

cases without heaving. The other set is obtained directly as the ampli-tudes of the second harmonics of the horizontal forces on the model. This comparison makes sense because the slender body theory predicts that the thrust-producing suction forces which are concentrated near the leading edges are in phase along the heaving body which implies that the amplitude of the second harmonic of the horizontal force on the model is predicted to be equal to the thrust produced.

In Fig. 4.6 the results have been plotted for the deeply submerged case with = 0.54. In this case the second harmonics yield a larger value for the thrust than the integrated term obtained by substraction. Especially from the high frequency case, with u=4vr it is clear that the second harmonic yields values for the thrust which are too large.

Presumably one has to do with resonance in the first bending mode of the first column and the model. The measured value of the stiffness of the first column and the mass of the model lead to a frequency in this mode which is close to 8r. At the lower frequencies the errors seem to be smaller but all the results for the horizontal forces are felt to be unreliable for the deeply submerged cases.

(44)

In the not so deeply submerged cases the columns were supported horizontally and no inadissable deformations were detected. The

results with = 0.18 are shown in Fig. 4.7. The second harmonics yield larger values for the thrust than the integrated terms obtained by subtraction. This indicates that there is a small increase of the viscous drag due to the oscillations. This effect is slightly smaller

in the higher frequency cases. The discrepancy between the calculated values and the test results tends to be smaller at the higher speeds. The results from the second harmonics correspond satisfactorily with

the calculation for U=3 and 13=4,, and they are quite consistent with the results found for the mean rate of working W given in Fig,. 4.5. The corresponding experimental propulsive efficiencies for these cases are very close to the predicted , when their calculation is based on the

results for the thrust-producing second harmonics. The integrated terms yield efficiencies which are too small. No unsteady boundary layer

calculations have been carried out but it appears that unsteady viscous effects which are not of separation type lead to the increase in the viscous drag during the heaving oscillations in the high speed range.

It should be noticed that in cases with waves and in cases which involve pitching motions of the model, the thrust-producing hydrodynamic forces on the model are not in phase along the model and no simple direct

comparison of these second harmonics and the integrated terms is possible.

(45)

T

Heavin g

No waves

0.18 e 0.03

2E

e :0.04

w

3E

T

e = 0.03

W:

4E

Fig.4.7 The thrust in the not-so-deeply submerged case.

o

e : 0.06

w:2 IL

Theory

Experiment

second harmonic

D Integrated drag

r e duc t ion

(46)

(47)

in Fig. 4.8 the mean thrust TM has been plotted against the square of the amplitude of the waves. Here the values for the thrust have been obtained by subtracting the drag in the case with waves from the drag in the case without waves at the same speed. It may be noted that in these cases with waves and à model that is towed steadily without oscillations no

-X.

work is done by the excitator. The thrust T is being generated at a Froude efficiency which is infinitely large. On the other hand the physical efficiency M at which the energy is being extracted from the waves is predicted to be unity. The correspondence between the

experimen-tal results and the calculated values is satisfactory. The dynamic angles of attack remain small and presumably the flow remains attached. Typical values for in oncoming waves with wavelengths A4 and 6.25 n, and an

amplitude a = 0.075 n, are about 0.8 N at 0.18.

o

y

Experiment

Fig. 4.8. The mean thrust generated by waves when the model is towed steadily without oscillations.

(48)

In Fig. 4.9 the mean rate of working by the excitator, , and the product TU have been plotted for heaving motions at the frequency of encounter as a function of the phase angle . The theoretical curves were óbtained from the complete expressions (2.4) and (2.5) for waves with À=6.25 and an amplitude a=O.075. In the test runs the mean value of the wave amplitude was somewlat smaller than a=O.075 and a correction by making the representations dimensionless with respect to the wave amplitude!, or rather , would improve the correspondance between theory and experiment. On the other hand it should be kept in mind! that the inter-action terms are linear in the amplitude of the orbital velocity 4

at the mean swimming depth for given oscillations of the model. Therefore a representation in dimensionless form was felt to be too confusing in the present context.

The correspondence between theory and experiment is not too bad in the small amplitude cases with eO.O3. in the larger amplitude cases with e0.06 the experimental results for are close to the calculated values. The experimental results for TU are clearly below the calculated values except in regions where a small thrust is predicted. Again, the dis-crepancies are due mainly to viscous effects.

In these heaving cases the Lissajous curves on the oscilloscope were used to obtain situations "inphase" and "in counterphase" with 00 andì 1800 respectively,, because the results were expected to be sensitive with respect to ip in these regions. it is clear from the clusters of symbols in Fig. 4.9 that this approach was not very productive in view of the repartition with respect to and in the subsequent test runs with = O, the extraction modes, a more

(49)

u(Nm/s

10 5 15

lo

Fig.4.9

600 800 _120°

_{160° 200° 2400 2800 3200 36cl'}

60° 80° 1200

_{1600 2000 260° 28C? 3200 3&)°}

Theory a=O.O75,

Experiment

1 V + w

Tu

40° 80°

120° 160° 20

240°3236

Tu

w

TU

40°

80° 120°

160° 200° 240° 260° 320° 360°

U =3

e =0.03

U =4

e 0.03

U=3

e

0.06

H ea y ir g

With waves

W We

).6.25

(50)

In Fig. 4.10 the mean rate of working by the excitator, , and the product U have been plotted for extraction modes where the tail is

tangential to its path with respect to the tank while the model oscillates at the calculated frequency of encounter,as a function of the phase

angle . The theoretical curves were obtained from the complete

expressions (2.4) and (2.5) for waves with À4 and X=6.25 at a nominal amplitude 0.075. Again, the results are not given in dimensionless form. It may be observed that the mean value of , with respect to ,

is predicted to be zero, while the mean value of U is predicted to be where is given by (2.20).

In the case with high speed, T.J=4, and long waves, A=6..25, the corres-pondence between theory and experiment turned out to be better than in the cases with lower speed and shorter waves. This trend appears to be quite consistent with the earlier results for cases without waves where flow separation was found to be the most plausible explanation for the discrepancies between theory and experiment.

The experiments clearly confirm the possibility to amplify the reference thrust at high efficiency by carrying out those oscillations which would be trivial in the absence of waves.

As indicated in chapter 2 these extraction modes only approximate motions with optimum propulsion to the extent in which the integrals appearing in (2.4) and (2.5) are negligibly small. It follows from the calculated values., with and without the integrals., in tables A-2-S to À-2-8 that these integrals lead to a phase shift with respect to ip=180°, which is predicted as "optimum" when the tail terms only are considered. As could be expected there is also Some effect on the extreme values for U and .

(51)

20 10

Ti

-lo

20 10

TU

-10 20 10

4. TU

-10

Tail tangential

to its path

,

With waves., w= We

0°

80° 120° 1600 2000. ++ + + 00 _80° 1200 1600 20

80° 120° 160° 200°

Fig.4.10

Theory

a0.O75,_ = 0.18

Experiment ¶u

y

W +

0° 320°

TU

W

280° 320° 3)°'I

TU

320° 36

X4 U=3

e,

0.06 e1 0.0195

o 41 °

X6.25 U3

e =0.06

e1= 0.0246

X:4

U 4 e,

0.06

0.021 I2

42°

X=6.25 U4

e,

0.06

e1 0,027 +

t

(52)

CONCLUDING REMARKS.

- In several cases described in this report a somewhat better than just qualitative correspondence between slender body calculations and experiments has been obtained. The experiments confirm that energy can be extracted from surface waves and be used efficiently for propulsion. - The investigation can be pursued further along four main lines:

Theoretical.

This part of the investigation should remain within the scope of

potential flow theory and include an analysis of the free surface effects of not-so-deeply submerged distributions of oscillating and pulsating singularities which can be associated with swimming slender bodies in surface waves. The results of such an analysis can be formulated as an extension of the slender body results described in chapter 2 of this report and they are felt to be an essential step towards a better under-standing of the even more complex resistance characteristics of surface ships in a wavy sea.

Engineering.

It is possible to design submarines and semi-submergibles which yield a passive recoil, in response to a given wave spectrum, which is

favourable for propulsion while the riding qualities remain acceptable.

Experimental.

The pilot experiments described in this report can be continued and improved. The main problems are the effect of viscosity on unsteady flows and flow separation which depend on the details of the geometry and the motions of the model as well as on the oncoming waves, the depth, etc. At present, though, it appears to be possible to design experiments which are closely related to results based on slender body potential flow theory, including steady and unsteady free surface effects. In view of the results described in this report the following modificat-ions may be considered:

(53)

- Ogee planforms have a more pronounced "tail" and with a slightly larger model, larger thickness and span, it may be possible to design a model without bulbs yielding a better signal/noise ratio.

- It may be worthwhile to design sma'll dynamometers which can function properly in water so that there would be no need for large rubber sealings. - With a rigid model relatively smaller amplitudes of excitation than the

maximum amplitudes used in the present experiments will yield a more favourable signal/noise ratio.

- For experiments with a flexible model it would be necessary to design an excitator with at least three columns so that a wave can be made to pass along the body.

(iv) Observational.

It would be most interesting to investigate observationally whether cetacean mammals move their tailLs in anything like the correlated

motions which were found to be efficient in the slender body approximat-ions. In principle it is sufficient to extract three signals:

I. the water pressure .just above the tail, the water pressure just below the tail, the acceleration of the tail.

Alternatively one may attempt to exploit the fact that the theory predicts,, for the simplest possible cases, that the tail should remain tangential to its path with respect to a coordinate system fixed to the water without non-uniformities. In the case of surface waves this is with respect to a coordinate system fixed to the water at rest far from the surface. In the case of non-uniformities such as those occurring in rivers this should hold with respect to a coordinate system moving steadily with the mean motion of the river.

(54)

ACKNOWLEDGEMENT.

it is a pleasure to thank Prof. Gerritsma who suggested the experiment and who kindly invited the author into his laboratory.

Thanks are due to the technical staff of the department of aerospace engineering for the design and construction of the model on short notice. The author is also indebted to the technical staff of the ship hydro-dynamics laboratory. Without their help the experiments would not have been possible. In particular I wish to thank Mr. Onnink for his

(55)

I. Lighthill, M.J. 2. REFERENCES. Newman, J.N., Vu, T.Y. Coene, R. e. Wu, T.Y., Chwang, A.T. 3. Coene, R. 4. Newman, J.N. 5. Sparenberg, J.A. 6. Gerritema, J. Beukelman, W. 7. Hounjet, M.H.L.

Note on the swimming of slender fish. J. Fluid Mech. (1960), vol. 9, pp 305-317.

Swimming and Flying in Nature, Vol. 2. Proceedings of the Symposium on Swimming and Flying in Nature, held at the California

Institute of Technology, Pasadena, California, 1974. Plenujn Press, New York (1975).

Hydromechanical aspects of fish swimming. pp 615-634.

The swimming of slender fish-like bodies. pp 673-686.

Extraction of flow energy by fish and birds in a wavy stream.

pp 687-702.

The swimming of flexible slender bodies in waves.

J. Fluid Mech. (1975) vol. 72, part 2, pp 289-303.

Swimming of slender fish in a non-uniform velocity field.

J. Austral. Math. Soc 19, Ser. B. (1975) pp 95-111.

Some ideas about the optimization of unsteady propellers.

Proceedings 11th Symposium on Naval Hydro-dynamics, London 1976, pp VIII 17-29.

Analysis of theresistance increase in waves of a fast cargo ship.

Nethexlands ship research centre TNO, Report no. 169 S (215/SR) (1972).

De van een deltavleugel. Ingenieursverslag sept. 1975.

(56)

z APPENDIX 1.

The distributions of A(x) and S(x) for the model.

r

See Fig. 3.1. As far as the virtual mass terms for vertical motions are concerned the sections are essentially elliptic. For x < 0.35 the long axis is vertical and A for vertical motions of the section with respect to the water is given by the surface area of the inscribed

circle. For x > 0.35 the long axis is horizontal and A is given by the surface area of the circumscribed circle of the section.

At x-0.35 the cross-section is circular and A is equal to S. Neglecting the details of the geometry of the nose and the small perturbations of A due to the presence of the bulbs one obtains the simple expression:

A(x) 2 for 0.05 x 2.05

i

Fig. A.1.1. The apparent mass per unit length is pub2 for elliptic sections. S - u ab.

(57)

The expression of S(x) is more complicated because the effect of the bulbs on S(x) is not negligible. For the elliptic parts one has S t ab.

The sections of the bulbs are circular and their sectional areas are readily calculated and accounted for. Summarizing one has:

X S(x)

<x

O.009425x /1-l6(x-0.3)2 0.050 0.300 0.009425x 0.300 0.704 0.009425x + O.045(x-O.704)- 0.704 0.820 - 1.12'(x-0.704)3 0.009425x + 0.0035 0.820 .0.91.2 0.00942x - O.136(x0.912)2 + 0.91.2 1.076 + 0.0334(x-O.912)3 + 0.0035 0.009425x 1.0,76 1.604 0..009425x + 0.045(x-1.604)2 - 1.604 1.720 - 1.12(x-1.604)3 0.009425x + 0.0035 1.720 1.820 0.02075 - 0.00494x2 1.. 820 2.050 0.05 2.05

(58)

APPENDIX 2.

Numerical evaluation of the slender body resülts.

For the steady cases, without excitation of the model, and without waves one obtains for the lift force from (2.1):

L(x) -p U L (wTMÄ)

ax

with w' = .0 . This lift distribution is linear and the hydrodynamic center is at x i. For the total 'lift one obtains:

i

2h

L

= f

L(x) dx = -p U - A(i)

0

OX

For instance., with p=1000 (kg/m3), U=4(m/sec), = 0.1 and

2 X

A(i) O.132(m ), one has L = 211 (N ).

In the unsteady cases with waves the formulae become more complicated. For the vertical displacements h(x,.t) one can write:

h(x,t)=(1.82-x)e1 sin (Wt+i12) + (x-O.82)e2 sin

Wt,

(A.2.3)

e1 and e2 are the amplitudes of the oscillations of the first and

second columns respectively. The first pivot is at x=O.82 and the second at x=1.82. w is the circular frequency and

I2 is the phase

angle by which the displacement of the first column leads the dis-placement of the second one. (See Figs. 2.3 and 2.4).

For the crossflow due to the waves at the mean depth of the model we write:

= L cos {2T(i_X) + ₊ p} (A.2.4)

Here cz' is the magnitude of the orbital velocity at the mean depth d of the. model. x = i is the position of the trailing edge; A is the wavelength of the oncoming waves; We is the frequency of encounter which can be expressed as:

2v

w =-

_e _A

_IOJ+c)I

(A.2.1)

(A.2.2)

(59)

where U is the towing speed and tU+cI the phase velocity of the waves

with respect to the model.

is the phase angle by which the wave

displacements at x2.05 lead the displacements of the trailing edge.

For c*

one has:

2ird

M

2v

A

a =ace

Here a is the amplitude of the surface displacements. Moreover, the

phase velocity e and the wavelength A are coupled by:

c

1.25 /)

.

(A.2.7)

Substitution of (A.2.3) and (A.2.4) into the expressions for

and

yields results which depend on the phase angle iii.

One first evaluates the integrals:

£

_2ir(R.-x)

IAl = f

A(x) sin

o A

dx,

1A2 = f

A(x) COB

2v(L-x)

dx

O A £

_2ir(R.-x)

ISI = f

S(x) sin

dx

o A. L

_2ir(L-x)

1S2 = f

S(x) cas

dx

o A

2v(L-x)

1A3

fL

xA(x) sin

dx

1A4

fL

xA(x)

2ir(L-x)

o COB A

dx

L _.

_2ir(L-x)

1S3

f

xS(x) sin

A

dx

1S4 =

1L

xS(x)

2v(L-x)

o

dx

(A.2.6)

(A.28)

for a given wavelength and then substitutes these into the expressions

for

and

with selected values for U ,

a ,

e1, e2 and i12. For

further details of the calculation we refer to the list of the Computer

program.

(60)

A-2-1 to 8. The terms based on the situation at the tail only, which are the terms in the square brackets in (2.4) and (2.5), are printed

separately in view of the special role they play in [3] and as a check on the calôulations,. In case of perfect and negative correlation

(.l8O°) of the displacements of the tail and the watersurf ace as weil as in the case of perfect and positive correlation (p=O°) these terms take on extreme values. The integrals in (2.4) and (2.5) give changes in the phase angles and the magnitudes of the extreme values for and W which depend on the details of the geometry of the model.

It may be noted that the thrust can be decomposed as follows:

T=T

uniform where: interaction -

rh 2

U2(ah2-1 T . = pA(2.)

]xP

uniform (A.2.9)

which is Lighthill's result [i]. This term depends on the situation at the tail only and does not involve the waves .and the surface areas S(x). For we have:

which is the thrust obtained from the waves when the model is towed steadily at a constant depth below, the surface. The remaining term is the only one which depends upon the phase angle and is a simple trigonometric function of i4.

r

M

i

interaction = -pA(i) i + £ M

+pf

(A+S)w4dx+

dx - dx wzx (A.2 Il) (A.2.12)

(61)

The mean value of with respect to* is given by = + mean uniform o L dS h M

+pUf ----cdx

o

Again the interaction term is a simple trigonometric function of . The interaction terms are linear in the amplitude of excitation (h(x,t))

andin the amplitude of the waves.

In the extraction modes the tail is tangential to its path,

(i!

+ U O at x=L, and i . and vanish. A typical uniform uniform

Eig. A..2.1. The decomposition of .

For the mean rate of working one can write:

(A.2.13)

Wi

+interaction

where -W . p A(L) U uniform

rh(hUh

I

(A.2.14) and - r W. . = p A(2,) U interact ion (- j +

xi

+

p f

(A+S) dx + (A.2.15)

(62)

value of in the experiments is 0.8 N. Typical extreme values (positive and negative) of the interaction thrust are several times larger than ?. One may say that by proper correlation (the right choice for p), the thrust can be amplified. It is clear from the results for

U and that the Froude efficiency is a rather wild function covering the whole range f rom- to-i- and therefore not a suitable parameter for graphical representation. It appeared more attractive to plot TU and W and to compare these with the experimental results.