On slender bodies in non-uniform flows

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Fait O1 - 18i ON SLENDER BODIES IN NON-UNIFORM FLOWS

by R. Coene

SUMMARY

In this paper an investigation of some _{cases of unsteady'propulsionin} non-uniform flow conditions is discussed Slender body theory is applied to sw-iming flexible and deforniable bOdies. Expressions are derived for the .local momentaneous lateral forces exerted _{on the body. Moreover the}

mean thrust and the mean rate of working against the hydrodynamicpressure

forces are evaluated. It is shown that b.y properly adapting the motions of the body to the non-uniformities

_in

the. oncoming flow, energy can be

extracted from the non-uniformities in such a way that it can be used

efficiently for propulsion. The theoretical results are compared with some experimental results for the special case of a not-so-deeply

sub-merged rigid slender delta wing which has been tOwed and oscillated through

a regular train of surface waves. The results for the. lateral forces1, the

thrust and the rate of working tend to confirm thetheoretical predictions.

The conclusions are relevant to .the theory of the swimming of cetacean

mamals and fishes in non-uniform streams and to the theory Of the

resistance of ships in 'a wavy sea.

I NTRODUCTION

This paper deals with unsteady propulsion ih,non-unif'o.rrn'flows.

Firstly, slender body theory i's applied to flexible and deformablè bodies which 'are swimming in a ñon-uhiform oncoming potential flow.

Secondly, an experiment is described in which a 'not-so-deeply submerged' rigid.slender delta wing is tOwed and oscillated through a regular train'

Of surface waves; ' .

Lighthill [1] has worked out a theory for the swimming of a slender fish through water. at rest, i'.e. _{in uniform flow conditions. Several} refine-nients and extensions'have been made since t'hat 'time. Lighthil.l's original formulation, however, was found to be.well suited a,s a starting point for the statement of the swimming problem' in _{waves [2b, 3). The boundary-value}

that of problems arising in the uniform flow case. In the evaluation of the forces exerted on the body, the _{rate of working by the body and the}

energy extracted from the waves, some _{new features arise. It is found to}

be possible to adapt the swimming _{motions to the oncoming}

waves in such a

way that the propulsion is more efficient than in _{the absence.of waves.}

The propulsive force is assumed _{to balance, in the mean, the viscous drag} and the wave drag.

The power balance for a body moving through a non-uniform velocity field

at speed U and generating a mean thrust

f

_{can be expressed as follows:}

=

+ - 'waste'

max

represents the mean rate of working by the body against _{the hydrodynamic}

pressure forces. For a swiming animal this _{term is equal to the net rate} of working of the muscles. For _{a ship oscillating in passive recoil}

in

oncoming waves this term vanishes _{or becomes negative when there is internal} damping. In the experiments [4], also _{described in this paper, the term} represents the mean rate of working by an excitator against the vertical

comonents of the hydrodynamic forces _{on a rigid delta wing model.}

is the maximum mean rate at which _{energy can be extracted from} \3t,max

the non-uniformities. _{It should be noted that this term can only be made} determinate in a useful sense for special _{cases of body geometry and properly} restricted classes of motion of the body.

The 'waste' in (1.1) _{can be identified as the mean rate of shedding of} kinetic energy in the vortex wake in _{the absence of non-uniformities. With}

non-uniformities in the flow there is a 'waste' of a more complicated nature which can only partially be _{accounted for by the kinetic energy associated}

with the vortex wake.

In the theory as well as in the experiments _{the terms 1U andW are determined} directly. Then equation (1.1) permits

_{the determination of ()max}

_{- 'waste'.}

In the towing tank experiments a rigid delta wing was oscillated by means

of two vertical struts which _{were forced to carry out vertical oscillations} at preset frequencies (e.g. equal tO the frequency _{of encounter of the} oncoming surface waves), phase angle _{differences, and amplitudes. Four}

strain gauge dynamometers were used to measure the vertical and horizontal forces exerted on the model by the _{two struts during preset heaving and}

pitching oscillations. For the vertical _{forces the first harmonics and the} time integrated part were resolved. For the horizontal forces the first

and second harmonics and the time-integrated _{part were resolved.}

The experiments are comparedwith results _{of slender body calculations and}

a somewhat better than just qualitative correspondence _{is found. More than}

500 test runs were made and especially in _{cases with little or no flow} separation agreement is satisfactory.

Calculations of the free surface effects _{[8] as well as comparisons Of} deeply submerged and not-so-deeply submerged _{test runs indicate that the} free surface effects are negligible _{even when the mean depth of submergence} is slightly less than only the small _{span of the wing.}

This justifies the assumption, _{at least for the cases described in this}

paper, that the two-dimensional surface waves can be treated as being essentially undisturbed by the presence of the oscillating wing while they

pass along.

2. THE VELOCITY POTENTIAL AND THE BOUNDARY CONDITIONS

We introduce a Cartesian co-ordinate system

Cx, y,

_{z) which performs}

a steady translation in the -x direction at velocity U with _respect

to the water at rest. The plane z = 0 is at a constant depth ci below the calm free surface of the water and z is positive vertically upwards. The free surface is disturbed by _{a regular train of two-dimensional waves}

of wavelength A, amplitude a and phase velocity U+c with respect to the

(x, y, z) system in the +x direction; _{c is positive in a head sea and} negative in a following sea (See. Fig. 1).

The velocity potential for these waves is given by

z, t) = ac exp [k(z-d)) cos [k [x- (U+c) t]}

where k = is the wavenumber. It _{is easily seen that the frequency of}

encounter of the waves for an observer moving steadily _{with the (x, y, z)}

system is given by

= k(IU+cI)

(2.2)

The amplitude of the _{waves at the swiming depth is given by}

= a exp (-kd)

. _(2.3)

As in [1] a flexible slender body is considered _{to be 'stretched straight'} when, in uniform flow and without free surface effects, _{no resultant}

lateral force acts on any cross-section. The velocity potential in this case is expressed as

Ux + 0(x, y, z)

To begin with, we suppose that the cross-sections of the body perform

displacements h(x, t) in the z-direction without altering their shapes. If the stretched straight body is given by F(x, _{y, z) = 0, the moving} body is given by F(X, Y, z) _{= 0 upon introduction of the coordinates}

(x,

Y, Z, 1), where

X = x, V = y, I = t, Z = z-h(x, t)

. (2.5)

In.these coordinates the velocity potential _{satisfies the transformed} Laplace equation and can be decomposed as follows:

(x, Y, Z, 1) = UX + c0(X, Y, Z)

_{+ c1(X, Y, Z, I) +}

+(X,Z,T)+2(x,y,Z,T)

. (2.6)

w

In (2.6) the function _{is the same function as in (2.4).} 7 _{is the}

perturbation potential due to the displacements h _{of the body in the}

absence of waves. _{is the potential of the oncoming waves, which follows}

from (2.1) with(2.5):

z, t) = z, I) . _(2.7)

is the perturbation potential due _{to the waves around the constrained} body (h = 0). Free surface corrections in (2.6) _{have not been made at this}

stage.

The boundary condition at the body surface _{is obtained by putting DF/Dt,}

the derivative following a particle of water, equal to zero:

h F _{( h \,fF} h F\ F F

- 0

At the surface of a slender body 3F/3X is assumed to be small compared 3'f' 3F fa'j' + - *) - = 0 at F(X, Y, z) = 0 where

u-w*=.+

* =w_*

X wZ wZ

Thus for each X, 'P becomes the two-dimensional potential in the (Y, z)

plane which results from the movement of an infinite cylinder with where the symbol * indicates a quantity at the mean swiming depth, d. Assuming that the lateral dimensions of the body and the displacements are small with respect to the wavelength A also, one obtains for the leading

terms in the boundary condition for 'P

(2.14)

(2.15) with F/3Y and/or F/Z. All perturbation velocities and

h/T

to be small with respect to U. Moreover h/X is assumed to For the leading terms in the boundary conditions one obtains

a0

3F 0 F U are assumed be small. for (2.9)

-

at F(X, Y, Z) = 0 and + 0 at infinity. With '

= + 2 one has for

'i' F (a' ac

h h\ 3F

+ U - 0 at F(X, Y, Z) = 0 . (2.10)

+ az

From (2.1), using (2.3) one obtains, in the new coordinates, for the wave

induced perturbation velocities in the Z-direction:

w _* k [z+h(X, (2.11) (2.12) (2.13) = _exp T)]} wZ with = cL* cos {h [X- (U+c) T]} wZ and = ack exp (-kd) = a* kc ,

with the following decomposition: 1

o2

p0 constant - pU -

_{p {(y-)}

I D'Y 0 2'P

!o ('

'w

P1

=P1+.+y+.2--.'L'(X, V.

Z., T) = w*(X, T) c(Y, Z; x)

_(2.16) q satisfies av2 az2

(2.17)

and, the boundary conditiOn at infinity.

3.

THE HYDRODYNAMIC FORCES ON THE BODY

For simplicity we assume that thw swimming body hasa cohstant density p which is equal to the.density of the surrounding water.Other' cases are easily handled by adding appropriate buoyancy terms If the overall

buoyancy does not 'balance theweight the propulsion is affected by an additional induced drag which is easily calculated by standard slender body methOds and such cases are not included in the following evaluation.

The wavelength A and 'the body length £ _{are assumed to be of the same} order of magnitude; The dimensions of the body. in the 'V and Z di:rections, the displacements h and the amplitude of the waves are assumed tobe small with respect to the body length ,, say of order c2, where c is

a small parameter. The derivatives of

,

and'2 with respect to V and

Zas well as

iax and /az are taken to be or order cU. The derivatives

of

,

and _{2'with respect to X are small of' order c2U log c. Then, by}

retaining terms up to and including 'those of orders and e2tJ2 log C,

one' finds from Bernoulli's equation

2

f N' Li = -P dY x Defining A(X) by D L1 =

_-p

_5T

A(X)}

r

Dc _{,3 2} ,3c% ₂ I w w

ii

w

it

w = P l + w -)

If

w\ 3'Y 1

f\2

p3

= P

_-

_{- 2 k5V)}

-V2) I

D a a

with the abbreviation = + U

We first calculate the lift force L(X, 1) per unit length.

Cetacean marrrtals exhibit left-right symmetry and their swimming motions are

essentially parallel to their plane of symmetry. This implies

F(X, Y, z) = F(X, -Y, z) _(3.3)

and

(x, V, Z, T) = c(x, -y, z, T)

.

(3.L)

for a cetacean mammal swimming in waves without velocity perturbations in

the V-direction. From these symmetry relations it is clear that there will be no torsional moments in the body. The lift forces _{act in the plane of} symmetry. (On the other hand most fishes exhibit left-right _{symmetry but} their swimming motions are perpendicular to their plane of symmetry. Fishes

are anticipated to be more sensitive to non-uniform velocity perturbations perpendicular to their plane of symetry, or rather in the direction of their swiming motions.)

The part p0 is the pressure distribution associated with _{the stretched} straight position and, by definition, does not contribute to the lift

force. The first term of p1 leads to a contribution of order C3 to L(X, 1):

(3.2)

(3.4)

(3.6)

A(X)=IdY

_(3.5)

where p A(X) is the virtual mass per unit length of _{a cylinder Cx moving}

in the Z-direction and _{is defined in equation (2.16). Thus, the contribution} L1 can be expressed as

L2=-p.ydY

x

With (x, Z, T) ₌

_{0, 1) + Zz(X, 0, 1) +}

_{.., one obtaines, to}

order

L2 = p s(x)

DT

where S(X) is the area of the cross-section. _{The remaining terms in}

(3.2)

contribute to higher-order terms in the lift only. _{Combining (3.6) and} (3.8) we thus obtain for the lift, to order e3,

D _wZ

L(X, T) = -p {w* A(X)} + p s(x)

DI

Without waves, L(X, T) _{is independent of the area s(x) of the} cross-sections and (3.9) reduces to Lighthill's [1] result. It may be noted that, as Lighthi'll showed, the error in (3.9) in the case of uniform flow is only of order c5. The _{error in the terms of (3.9) due to the presence}

of the waves becomes of order c. Changing the order of the differentiations

in the second term on the right-hand side of (3.9) shows that, this term results from the vertical component of the _{pressure gradient due to the}

waves. The result (3.9) was derived in [2b] and [3]. Similar results can be found in [2a] and [5] but there a term is included which would be of

order c4 in the present context.

Now, the operator _{in (3.9) suggests the possibility of an interesting} generalization. The operator

= + U is equivalent to a differentiation with respect to time only in a coordinate system which moves steadily at

velocity U in the positive X-direction.

Thus, to leading order this system is equivalent to the coordinate system

(x1, y1, z1, t1) which is fixed to the _{water at rest in the absence of the} body and the waves. _{In this coordinate system one cannot distinguish whether} the variations of w*A and S _{are due to a variable shape of the cross-sections}

in the x1-direction or to a variation with respect to time. This implies that in (3.9), A(X) and S(x) can be replaced by A(X, T) _{and s(X, I). Thus,}

(3.7)

(3.8)

2.. 2. JP S dx

=J

L dx fpxS ---- dx = xL dx

-9

reintroducing the original inertial Cartesian coordinates

_{Cx, y,}

_{z, t) one}

obtains, written in extenso, to leading order,

L(x, t) = -p + U

f.)

{w*(x, t) A(x, t)} + + pS(x, t) +

L.\

(x,

ax)

wz

The expression (3.10) makes it possible to include deformable or rotating sections. _{In applications of (3.10) though, some care must be taken. In} fact (3.10) makes sense only in cases where one is able to determine the

'stretched straight'. position of the body at _{every instant. The perturbation} potential associated with this position, '0(x, y, z, t) has become

time-dependent. For bodies which, during deformation remain symetric with

respect to Y = 0 and Z = 0 this is a trivial problem; _{even during deformation} there will be.no lateral forces _{on the body when the centers of the}

cross-sections remain on a straight line fixed with _{respect to the water at rest}

(such as the x-axis). _{In more general cases, without symmetry with respect}

to Z = 0 and Y = 0, one is faced with the problem of the determination _of

the 'stretched straight' position at _{every instant in order to obtain the}

proper h(x, t) for the calculation of w in (3.10). It will be clear that h(x, t) being of order c2., _{errors in h(x, t) of order do not affect}

(3.10) to leading order. _{In many cases one will obtain sufficient accuracy} in the determination of the stretched straight position by requiring that the centers of the cross-sections remain _{on the x-axis. This assumption is}

equivalent to defining h(x, t) as the centerline of the deformable and

flexible body.

It should be noted that the displacements h(x, t) of _{an unconstrained body}

cannot be chosen arbitrarily. The time rate of change of the _{momentum of}

the body in the z-direction must be equal _{to the resultant of the}

lift-forces and the time rate of change of the angular momentumof the body

about the y-axis must be equal to the moment of the lift forces _{about that}

axis: .

(3.10)

1

In this section some solutions of these equations will be discussed.

We are now in a position which makes it possible _{to evaluate the thrust}

and the rate of working by the body.

The thrust is the x-component of the resultant of _{the pressure forces on}

the body:

T

=

Jf

p dy dz =

ff

p dY.(.. dX + dZ) =

=fLdX+JJpdYdz

where the lift L follows from the expression (3.10) _{and the pressure p is} given by (3.1) and (3.2). By virtue of d'Alembert's _{paradox the part p0 does}

not contribute to I.

In the special case with U + c = 0 the waves induce a steady flow field with respect to the (x, y, z, t) system moving steadily with the mean motion of the body; In this case the thrust is dominated _{by the first}

term of p2 which is linear in and, steady. One finds, simply, to leading order

I =

_{-p Jf}

U dY dZ =

2.

=p

JUdX

_(3.13)

Using S(o) = S(2.) = 0 one can also write

2.

T=-pJUS

dX

0

It should be noted that (3.11+) _{is valid only with U + c = 0. Moreover,}

one may replace the coordinate X by x, to leading order, so that one may also write

2.

32*

T=_PJUS

dx 3x 0 (3.12) (3.14) (3.15)

which is equivalent to the result (29) _{in [5] for steady-state porpoise} bow-riding. It is clear that a positive thrust of the type (3,15) _is

generated simply by positioning the body _{in a region with a favourable}

lengthwise pressure gradient in the _{non-uniform flow. Obviously this}

possibility of favourable positioning _{of a body is not restricted}

to the non-uniformities associated with 'waves'. _{Baby whales have been observed} to swim in such regions of favourable _{pressure gradients near the body} of their mother, thus leaving _{to her the burden of generating}

some extra thrust to contribute in balancing its _{frictional drag. It is also}

conceivable that terms of the positioning _type

_(3.15)

_{are exploited for}

a rest by fishes, swimming in _{a river which contains non-uniformities fixed}

with respect to the-bed of the river, when they stay put for a while with respect to the bed of a river.

It should be noted, that in _{general the stretched straight position}

does not yield a solution of the equations _{of motion (3.11) in this case.}

A steady equilibrium position _{can be found as the solution of}

JL(x) dx =

JxL(x) dx =

which upon substitution of (3.10) then yields

-p

J

[u

{(u

p -

_{x wz}

)

A(x)}

+ S(x) U

L

(x)1

dx = 0 wz _J 0

-p J

[

1. I(u-.!i

-

+ S(x) u

1..

,*

(x)1

_{dx = 0} .

aX\ax

wz wz J 0

which must be satisfied by h(x). it is easily verified that a particular

solution is obtained, by requiring L(x) = 0, when the body slope satisfies

UA

* dx

wz ax

The solution

_(3.17)

_{is an attractive one because it involves the absence}

(3.16)

of bending moments. Whenever the. thrust (3.15) does not balance the

frictional drag, additional swiming motions are required around the

mean position which follows from (3.17).

It should be Observed that the thrust given by. (3.1.5) is, formally of ord'er. c3, which indicates that.it can be quite. significant because the

thrust which can be generated in uniform flow conditions is only of order

4 . ..

In the following calculations it will be found that the dynamic

inter-actions are also dominated by terms of order in the mean thrust,T,'and

the mean rate of working, W, for cases with U + c 0. In these more

complicated cases themotions of the body are correlated with the on-coming non-uniformities. For a ship the motions are 1 passive recoil:and

for a swimming animal the motions may have a correlated voluntary part

We now consider the .cases with U + c p 0, where the non-uniformities

encountered by the.body are essentially unsteady: '

p0 does not contribute to the thrust by virtue of d'Alembert's paradox Assuming oscillatory (periodic) h and it is clear that, to leading order the contribution of p1 to .the mean thrust i-s eqi'al. to zero.

in the second integral of (3.12). The terms in p2 contribute to the

second integral in (3.12) by: an amount which follows from the pressure due to the waves at z = h(x, t). Thê term

f(',/aX)2

.

(w/aZ)2}

does not depend on X to order c2. Thus, in terms of the Cartesian inertial system (x, y, z, t) the, contribution of p2 to the second integral of (3.12)

can be expressed as . . where T2 =

_pJ

'f (

0' + ij

f.)

dx . . . (3.18) = + * + wz z, :t) ₌ 4(x, 0, t) + h 0, t) +

This contribution.. (3.18) consists of' a large.term of order which Vanishes'

inthe mean and a term of order which depends on the correlation of .h and .4*. by virtue of (3.19). There remains r

and

2..

=+ p

J(h

dx + 0(c5)

wz dx

0

When the body cross-sections are deformable the cross-sectional area may be time-dependent and the cross-sectional area can be correlated with the first term on the right hand side of (3.19). When the local variations of S(x, t) are large of order c2 this term may give rise to a contribution of

order c3 in the thrust:

2.

f D

(4 *) dx + 0(c4) 0

In p3 the term 4 /Z is now replaced by t7" and using (2.15) one obtains,

w wz

to order

JfP3dYdz=Pff{w*._!t.k\2_!()2}dYdz

az 2 _av) 2

S S

The integral (3.22), is formally equivalent, for a certain value of X,

to the pressure distribution due to the steady motion of a cylinder Cx

with constant velocity w* in the Z-direction through unbounded water at rest. It should be noticed that p3 does not involve time derivatives of

the potential. Time plays the role of a parameter only. Thus, in order

to evaluate (3.22), the following argument applies. 13

-The kinetic energy of the water per unit length is pAw*2 and the momentum

in the Z-direction is pAw*. A time 5X/U later the kinetic energy has changed by an amount p(d(Aw*2)/dX) SX and the momentum by p(d(Aw*)/dX) SX. In order

to bring about this change in momentum the Z-component of the force exerted by the body on the water per unit length must do the amount of work

pw*(d(Aw*)/dX) oX. The amount of work done by the X-component of the force exerted by the body on the water per unit.length is equal to

-(dT3/dx) ox _(3.23)

Equating the amount of work done by the body to the change in kinetic

energy per unit length leads to

dT pw* .- (Aw*) OX - d _{(Aw*2) OX} dX

=Pg-(3.20) (3.21) (3.22) (3.23)

dT3

-dX dX

For the total thrust one thus obtains, upon resubstitution of the original (x, t) variables, £ £

=_pJ.(w*A)dx+pJ..SD

_dx+

wz 0 0 £ £ (hq* ) dx + p

fw*2dx

Dt wz dx 0 0

In

(3.25)

the terms

(3.15)

for a steady non-uniformity and

(3.21)

for deformable sections are not included. We now assume that at the nose

of the body one has S(o) = A(0) = 0; at the tail, x = £, we assume S(i) = 0

but A(2.,) 0.

ThenT can be expressed in the form

11Th_*')2_u2(..!f'21

+ = p A(9)

[5t

wz,

x)

= 4p A(2.)

I(q*

)2 1. wz 2 (ah'\2 1 )

I,

where the second term on the right-hand side can be recognized as the

vortex drag of a span loading of elliptic form.

The mean rate of working by the body is given by

(3.24)

(3.25)

(3.27)

+ (A + S) w*

dx + pU

fh*

dx (3.26)

It can be seen from

_(3.26)

that without waves (c* = 0) the mean thrust is determined by the movements of the tail only, as in Lighthill's (1960)

result. in that case without waves thrust cannot be generated without

shedding vorticity in the wake. By putting A(i) = 0 in (3.26) and assuming

that there are no sharp trailing edges it is clear from (3.26) that thrust can be generated without shedding vorticity. This situation is reminiscent

of the distinction of the vortex drag and the wave drag of supersonic wings. The two terms in (3.26) involving integrals can be interpreted as the

forces acting on a distributionof dipoles with vertical axes in a velocity field with a vertical gradient. It may be observed here that for a

=

-

J

L(x, t) dx

Using (3.9) for L(x, t) with, again, S(0) = A(0) = 5(2,) = 0 but A() 0,

one has

= pA() U

[-w*] -I - - I ah dS 0 0

For bodies with deformable sections the expression

(3.29)

must be

supple-mented by a term which accounts for the rate of working against the pressure forces due to the waves by a pulsating section, such as may

arise in peristaltic deformations:

9,

ID

aS _p J dx 0 15

-(It is assumed that these pulsations are such that the total volume of the

body is constant or oscillatory so that in the case without non-uniformities such pulsations do not contribute to the mean rate of working by the body.)

The Froude efficiency of propulsion is given by

TU TU

Ti =-=

W TU + waste

In uniform oncoming flow, cannot exceed unityand to generate a positive

T, the rate of working W must be positive. In the presence of waves r may exceed unity an is not necessarily positive when T is positive. In fact

this _n is less suitable than the physical efficiency r defined by

TU TU

(3.32)

-+ waste - faE w + _max

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. A flexible recoil mode

1(x,

t) is derived there by equating the local

time rate of change of the lateral momentum of the body and the local lateral force exerted on the body by the time-varying part of the

pressure of the water:

-(3.28)

(3.30)

2-p 5(x) = L(x, t) _(3.33)

at

Solutions of (3.33) are a particular solution of the equations of motion

(3.11) which can be considered as the dynamic counterpart of the stretched straight position in uniform flow conditions. In [3] it is shown that a

body oscillating in this flexible recoil mode which satisfies the following

symetry relations S(x) = 5(2, - x) and A(x) = A(2, - x) experiences no

drag in the mean; moreover it is clear that the work done by the body is

equal to zero.

-The displacements h(x, t) satisfying the equations of motion (3.11) are

then decomposed as follows

h(x, t) = f(x, t) + i(x, t) (3.3,4)

where

1(x,

t) is the flexible recoil mode and f(x, t) is a solution of

the homogeneous part of (3.11), which is obtained by putting 0,

as in uniform oncoming flow.

The swiming problem then consists of adapting the voluntary and active displacements f to the passive displacement h. On the assumption then, that the situation at the tail dominates the generation of thrust it

is shown that

For motions of the tail satisfying (3.36) the resultant cross-flow w at the tail is exactly minus the local vertical component of the orbital

-

_TtI

_{p A(2,)} or alternatively (3.35a) +

2w**]

W - ru p A(2) U

[2

-(3.35b)

Without waves one has w = w* and the right hand side of (3.35) is equal to the mean rate of shedding of kinetic energy into the wake. The terms in brackets have a minimum value of -? for

w* = _{at x = 2.} , _(3.36a)

wz

or alternatively

()

_max

where 1* is a reference thrust

= jp A(9)

From the volume which is effectively being swept by the tail over long periods of time, half of the kinetic energy present due to the waves can be extracted and made available for propulsion. The presence of the

integrals in the thrust and the rate of working W indicate that these

estimates are conservative.

It is interesting to note that (3.36b) implies that the tail should be

tangential to its path with respect to the (x1, y1, z1, t1) system which

is fixed to the water at rest.

The motions of the tail (3.36) are precisely those which are trivial in the uniform flow case. For further details we refer to [3].

The integrals appearing in the expressions for Q and i depend on the details of the body geometry as well as on the motions of the body and

their correlation with the oncoming waves. In the comparisons to be

made with the experiments the complete expressions for Q and I have been evaluated numerically for all the motions that were carried out in the

experiments [4].

4. THE EXPERIMENTS

4.1. Introductory remarks

The main purpose of the experiments is to obtain experimental results

which can be compared with slender body calculations which are based on the expressions.derived in the previous sections. A rigid slender delta wing was designed which could be used to extract energy from

surface waves and generate propulsion from this energy in essentially 17

-velocity of the surrounding particles due to the waves. The resultant component of the velocity of the water has vanished just above and just below the tail. Using (3.36) in (3.35) one finds for the maximum mean rate at which the body can extract energy from the waves:

(3.37)

uniformities passing along the body. The experimental equipment of the ship hydromechénics laboratory available in Delft was well uited for

the experiments. A towing tank where a regular trainof' surface. waves, cari be generated while at the same time the model can be towed and

oscillated at a frequency equal to the frequencyof encouter, thereby obtaining correlations depending upon the modes of oscillation and a

'phase angle. The lift and'drag forces at the first and second pivots

weremeasured by application of strain gauge dynamometers.

Li.2. The model

The model consists of waterproof plywood, 0.06 m thick, a length of 2 m

'and a span of

0.L1

m, the planform being that of a slender delta wing. Two bulbs at the bottomside were needed in order to obtain' sufficient room in the compartments for the dynamometers. At the 'top side the. two

compartments were sealed off with rubber sheets which were kept flush in

the model surface by maintaining the proper air pressure in the cOrflpartmehts. In order to avoid condensation problems a small ventilating flow of dry

air was maintained. '

The sections (x=const.) are elliptic. For x <, 0.3-5 m the long axis is vertical; for x > 0.35 rn the long axis is horizontal. The 'nose ,i's

rOunded off and the trailing edge is sharp. See Fig. 3 'The front, compartment housed'two dynamometers for lift and drag respectively and

bearings 'With a single degree of freedom, rotation. The rear' compartment

also hOused two dynamometers and bearings allowing for rotation and

length-wise translation. The. two excitator columns are 1 m apart. They can 'carry

out vertical oscillations with' an amplitude up to 0.06 mat afrequency

up to a few Hz and a phase difference which can be preset at will The

pivots and the dimensions of the compartments allow fàra maximum slope

.2. _{= 0.12 The inertial 'effects of the model do not affect the power}

ax '

' ' ' ' ' ' '

and energy balances', but they are important' for the loadings of th,e dynamometers. For more details we refer to [L1].

4.3. The experimental arrangement

The test arrangement is given schematically in Fig. 4. The conductive wave probe took its signal from the waves in a vertical plane through

19

-the mean position of -the trailing edge of -the mdel at a spanwise

location where the perturbation of the oncoming waves due to the columns and the model looked negligb.le. The wave signal _{was visualized on a UV} recorder tôgetherwith' signals representing the displacement of the second colUmn and the lOads on the dynamometers.

The UV recordings. were mainly used for the determination.of the

amplitude of the waves and the phase angle by which the displacements

of the second column lead those of the water surface.

The four signals from the dynamometers were handled as follows:

(i) _{For the test runs without oscillations Of. the model the 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 outfor the lift forces. For the drag

forces the two second harmonics were resolved and printed out as wel.l as the time-integrated parts.

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.

4.4. the test runs

Three types of motion of the model, all with constant forward speed U,

were carried out:

V V

Steady transl.at ions with = 0 and several

at ax.

Heaving motions h(x,t) = e sin wt (e1 = e2 = e,

12 = 0).

Motions for which the tail remains tangential to its path with

respect to the tank; w

=

(.-+

u'.') E 0 as indicated by equations (3.36).

The general. form for the oscillations of the model is:

h(x, t) = (1,82 - x) e1 sin (wt

+ 12 + (x - 0,82) e2 sin wt

For comparison these motions were carried out in three' types of on-coming flow:.

V

Uniform flow. At a mean depth d = 1.08 m the presence of the free surface (and the bottom and sidewalls of the ta.nk) _{are assumed to} be negligible. , ,

Almost uniform flow. The model is towed at a mean depth d = 0.37 m withOut oncoming surface waves.

(c) Non-uniform flow. At a mean depth d = 0.37 m the model is towed in regular trains of head waves.

For the types of flow (c), withhead waves, correlated runswere made. A rough choice of the relevant phase angle could. be made by freezing

the desired LissajOus curve on the oscilloscope when presetting the upper limit of the frequency of the excitator equal to the calculated frequency

.of encounter of the waves at the preselected U and A. The variation of p during a test run with correlation could be kept smaller than 200_300 for most test runs. When the UV recorder. indicated a larger variation

the run was.discarded. Two wavelengths were selected, A =4rnand A =6.25 m

and in most cases with waves the towing speeds U = 3 rn/sec and U = Li rn/sec

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 the beginning. In view of the absence of the phase angle dependency of W and T in the

uncorrelated runs just a few of such runs for comparison purposes were

sufficient. .

RESULTS

In Fig. 5 the drag is plotted against the square of the speed in the not

so-deeply submerged case with = 0.18. For the deeply submerged case with

= 0.54 no accurate results for the frictional drag are. available. In this

deeply submerged case the horizontal displacement of the first pivot point under the drag loading caused an unexpectedly large deformation of the rear

rubber sealing, leading to unreliable results for the drag. In the not-sO-deeply submerged cases, with = 0.18 this problem could be easily

circum-vented by installing some supports assuring sufficient constancy of t!he

horizontal distance of the two excitatOr columns and negligible deformation

ofthesealings.

.

For reference the steady lift force is plotted against the square of the speed for different angles of attack in Fig. 6. At the small angles of

attack the slender.body theory. predicts too large a lift. For the larger

angles of attack the non-lineareffects related to the vorticityshed 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

21

-better at the larger angles of incidence.

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.

A similar conclusion follows for the, oscillatory, heaving motions

In Figs. '7 and 8 the amplitudes and phase, angles of the vertical components

of the forces in the front and rear dynamometers respectively are plotted

against the speed. The experimental, results for the amplitudes at the front

are slightly larger than the calculatedvalues, 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 Joulcowski' condition, is being satisfied. In fact the pressure loading vanishes at the trailing edge and a strong

Joukowski condition is being realised. This explains the fact that at the

rear the loading is smaller and at the front slightly larger than predicted

by slender body calculations. 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 amplitudes for the crOssflOw, ew

and at a certain speed 'This is probably due to the fact that in the higher frequency cases the boundary layer effects associ'ated with the cross flow 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

'effecively satisfied at the trailing edge tends to be weaker than at the

higher speeds. In the limiting case with U = 0 one can expect suction peaks

even at the trailing edge. The difference between the results at the two

different depths

f

= 0.18 and O.51 is small. The free ,surface effect

results in a slight reduction of the amplitudes of the forces and a slightly

increased phase lag, bu't it does not significantly alter the general picture of the results.

In Fig. 9 the. corresponding values for 'the net rate of working of the 'excitator have been.plotted against the speed.

In the cases with small crossflows (a = 0.03,'w = 2tr) it appears that' there is no sign i,ficant flow separation for 'U > 1. For the cases with larger ,crossflows this is true for say U > 2. In these regions of attached floW the slope of curve's 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 trái 1 ing edge. One may say that 'here the calculation slightly over-estimates the effectiveness of the wing as far' as, the transfer of energy

In Figs. .10 and 11 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 amplitudes 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 horizon-tal force on the model is predicted to be equal to the thrust produced:

A()

()2

= pA(9) U

_(5.1.)

with A(i) = b2

In these cases one half of the wOrk done by the éxcitator against the lift

forces is predicted to be wasted as kinetic energy i.n the wake. With frequency

w for the oscillations the horizontal components of the pressure forces have

a productive component with frequenèy 2w.

In Fig. 10 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 subtraction. Especially from

the high frequency case., with w = Lrir 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 closetO 8rr. 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 sub-merged cases.

In the not-so-deeply submerged cases the columns were supported horizontally

and no inadmissible deformations were detected. The results with1 = 0.18 are shown in Fig. 11. The second harmonics yield larger values for the thrust

thanthe integrated terms obtained by subtraction.. This indicates that

there is a small increase of the viscous dragdue to the oscillations.

This effect is slightly malle in the higher.frequenc.y cases. The

23

-smaller at the higher speeds. The results from the second harmonics

correspond satisfactorily'with the calculation for Ii = 3 and V 4, and

they are quite consistent with the results found for the mean rate of working W given in Fig. 9. The corresponding experimental propulsive

efficiencies, for these cases are very close to the, predicted , when

their. evaluation is based on the second harmOnics. The integrated terms yield efficiencies which are too small.

It should be noticed, that in cases with waves and in cases which involve

pitching motions of the mOdel, the thrust producing hydrodynamic forces are not in phase along the model and no simple 'direct.' comparison of these

second harmonics and the integrated terms is possible.

In Fig. 12 the mean thrUst 1* (3.38) 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. The model

is ted steadily and

no work is done by the excitator. 1* is being generated at. a Froude efficiency

which is infinite and at a physical efficiency r* equal to unity. The

correspondence between' the' experimental results and the' calculated values is satisfactory. The dynamic angles of attack remain small, with 'U = 2, 3, 4

and a.. 0.08, and presumably the flow remains attached. Typical values for,

1* _{in oncoming waves with wavelengths A}

= 4 and :6.2.5 m, and an amplitude of

0.075 m, are ,about 0.8 N

.at.4

= 0.18. . .

In Fig. 13 the mean rate of working by the excitator, W, and the product TU

have been plotted for heaving motions at the frequency of encounter as a funct'ion of the. phase angle i. The theoretical curves were obtained from the

complete expressions (3.26) and (3.29) fOr waves with A = 6.25 m and an

amplitude a = 0.075 m. In the test runs the mean value of' the. waveamplitude

was somewhat smaller than a = 0.075 m and a correction by making the

re-presentation dimensionless with respect to 42 would improve the correspondence between theory and experiment. On the other hand it should be remembered that the interaction terms are linear in the amplitude of the Orbital velocity

at the mean swimming depth' for given oscillations of the model. Ther,efore a dimensionless, representation 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 e'= 0.03 m. In the larger amplitude cases with e =0.06. rn the experimental results for W are close to the calculated values. The experimental results for. 1u 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. For further details we refer

to[iI].

In Fig. 14 the mean rate of working W and the product TU have been plotted

against the phase angle p for the extraction modes.where the tail is

tangential to its path with respect to the tank as in (3.36) while the

model oscillates at the calculated frequency of encOunter. The theoretical

curves were obtained from the complete expressions (3.26) and (3.29) for

waves with A = 4 and A = 6.25 at a nominal ampl itude a = 0.075. It may be observed that the mean value of W, with respect to. , is predicted to 'be

zero, while the mean value of iU is predicted to be T*U where T* is given

by.(3.38). . .

In the cases with high speed, U = 4, and long waves, A

= 6.25,the

correspondence between theory and experiment turned out to be better than

in the cases with lower speed and shorte.r. waves. This t.rend appears to be

quite consistent with the earlier results for cases without waves where flow separation was found rtO 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 section 3 these extraction modes only approximate motions

with optimum propulsion to the extent 'in which the integrals appearing in the expressions for and ci are relatively small. In [4] the values for

and c are calculated with and without the integrals.

As expected the integrals lead to a phase shift with respect to '. = 180°,

which is predicted as "optimum" when the' tail terms only are considered.

As could.be expected they also have an effect on the extreme values for

lii and ci.

6. CONCLUDING REMARKS

It may be not iced that the simple geometric and kinematic criteria wh'ich follow from (3.36) do not determine the motions of the tail uniquely.

In this respect there is a certain analogy with results obtained by Spareriberg [5] for case of unsteady propulsion in steady non-uniform flow conditions where the maximum possible amount of energy that can be extracted from a

25

-non-uniformity can be obtained from considerations involving suitable layers

of vortici'ty.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 freevorticity being shed from the trailing edge should

then annihilate (to firstorder.) the vortici'ty 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. It appears that

the slender body results discussed in this paper fOr unsteady non-uniformities are a natural counterpart of the results obtained' in [5] for wings moving

through steady non-uniformities. The asSumption of slenderness in the stream-wise direction makes it possible to remove the restriction to steady hon-uniformities which is necessary in [5]. The requirement which the free

vorticity in the wake must satisfy along the wake with steady non-uniformities

is replaced by a requirement on the local and mOmentaneous resultant crossf low at the trailing edge for maximum energy extraction., It is easily shown that this.. remains true for two waves and therefore also for a spectrum of waves. There is another aspect of the propulsion problems discussed in this, paper which should be mentioned. It. appears tha.t surface ships in waves generally

experience a resistance increase due to the waves [6]. The present slender body results suggest the possibility to design submarines which benefit from a spectrum of surface waves and one can speculate even further about

the possibilities for surface ships 'or semi-submersibles'. In view of the towing tank experiments described in this paper 'the free surface effects 'for the steady towing case and for some more general oscillations with and.

without waves were evaluated numerically [7] for certain distributions of pulsating singularities of which the strength was derived from slender

body results [3]. It turns out that there are complicated interferenceeffects

which, interestingly enough, do not 'always' lead to an increase of the

resistance. As far as the experiment was concerned it was concluded that

the free surface effects do not significantly affect the results obtained.

REFERENCES

1. Li'ghthill, M.J. Note on the swimming of slender fish.

Newman, J.N, Wu,

Coene, R.

Wu, T.Y, Chwang, A.T.

3. Coene, 1. Coene, R. Newman, J.N. .Sparenberg, J.A. Gerritsma, J. Beukelman, W.

2. _{Swimming and Flying inNature, Vo1 2.}

Proceedings of. the Symposium on Swimming

and Flying in Nature, held at the California

Institute Of Technology, Pasadena, California1 1974. Plflum Press, New York (1975).

Hydromechanical aspects of fish swimming.

pp 615-634.

The swimming ofslender 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.

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

Delft University of TechnOlogy, Dec. 1977.. Dept. of Aerospace Engineering, Rep. LR-257. Ship Hydromechanics Laboratory, Rep. 456.

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 llth Symposium .on Naval Hydrodynamics, London 1976, pp. VIII 17-29.

Analysis of the resizstance increase in waves

of a fast cargo ship. Netherlands ship

research centre INO, Report No. 169 S (215/SH)

(1972).

8. Hounjet, M.H.L. . :De "golfweerstand" van eendeltavleugel. Ingenieursvers lag 'sept. .1975.

z

27

-Fig. I.

Fig. 2.

.xPSP/fh .i

23 - .ZborjnedeP-.1DAPandda 4

)DDMfla?-Z

3.

sr

MN, au /..paa

ntn

4'dsmku g. YDm V1N41P,d a.4.n C.D,dp,an m.j .,dv, aw.ndfuw41). Md./ wuV in

all. 4undung.n vp4agI.n.

P,l/a

k'A.,/.nd.y dw.n.n.

*__..__ _.j__ N.h. I. VLIIGUIGIOUWKUNDI

4 sham meters phase sething' scotch yokes . our outlQt

29

-Môtór R1

D.

₀₂ R2

4 strain gouge dynómo-meters

b(x,L)(I.82x)e1sjn(wt+4,12)+(xO.82)e

sin

Fig 4

The experimental toyout

hums integroted 1 1 inph3se camp. quadrature c.omp. untegrators 2resowers w (R1.R21 1

2resolvers 2wID,.j

oscilloscope

Ipriiit outi

pressurisat Ion ventilation

- 200

-100

-100

- 200

1 1

= 0.18

= 0.54

- theory

2 3 2

Fig. 5.

Fia. 6.

a

3 4 4 U 4' + +

20

+

*1+

+ 10 ₊ 1+ +

Experiment Theory

R,N)

_4R1 dii

0

0.54 +

y

0.18 e = 0.03

w: 2Tt

- 31 -Heaving No waves e=O.03 U): 4 Tt

Fig. 7. The ampi itudes and phase angles of the vertical loadings

on the front dynamometers.

44-+

e = 0.03

LL) 2Tt

R2(N)

_'R1

_d/2

Experiment

_.

_0.5k + Y 0.18 T neory e

0.03

W: 4Tt

0

2 e

0.06

W

2 it

Fig.

8.

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

0

-20

0 10 1 2 3 1 ₂

U'

:3 4

*

.

.

+

a

-too

e

= 0.03

(2) =2Tt

3.

-Fig. 9.

The mean rate of working during heaving.

Heaving No waves

V(Nm,sec.)da

Expenrnent 0 0.54 V 0.18 Theory

e = 0.03

2): 4TL

T

0.4

Heaving

No waves

d

= 0.54

e 0.03 U)

2Tt

0

0

0

0

0

0

0

0

T 0

0

T 0

0

6

0

0

o

0

C 2 8

-

0

T neory

Exper irnent

o

second harmonic

o

integrated drag

reduction

0

0

0

800

resonance

e: 0.03

=

41

0

0

0

e

0.06

w= 2Tt

0 1 2

3-

U40

1 2 3

Fig. 10. The thrust while heaving in the deeply submerged case.

0

0

0

0

2

00

0

₀

0

0

a 2

0

0

₀ 0

0

₀

3t

,-U4

0 1 2 3

U4 0

1 2 6

e: 0.04

w= 3 it

0 4 4 0

H a vi n g

No waves

= 0.18

e = 0.03

(A)

.2 ii

35

-T 0 1 2 3

U40

6 e

0.03

= 4TL

0

e =0.06

Fig. 11. The thrust while heaving in the not-so-deeply submerged case.

Theory

Experiment

0

second harmonic

Integrated drag

red u c t ion e w

:0.04

: 31t

3

.0 40

3'-U4

0.06

0.07

0.08

0.09

a

Fig. _{12. The mean reference thrust T* generated by waves when the model}

15 40 20 10 10

I u

(Nm/s!

40° 80°

37

-40° 80° 120°

160° 200° 240° 280° 320° 3600

4)

w

TV, Vy

I I

.

. I 1200

_{160° 200° 21.00 280° 3200 3600}

40° 80° 120° 1600

_{200° 240° 2eC° 320°4)360°}

Theory

a=0.075

Experiment

U V +

w

40° 800

_{120° 160° 20}

_{240 ,° 32}

36 4) TU

w

U :3

e O.03

e = 0.03

U:3

e: 0.06

U =4

e =0.06

Hg.

_13. W and TU in heaving with waves.' _{is the phase angle by which} the _{wave diaplacements lead the displacements of}

the trailing -edge.

Hecvirq

Wth waves

= We

):

6.25

20 10

TU

-10 20 10 -10 Toil tongentiol to tts path ,

W_O

With waves, W

_We

+ 800 _120° 1600 _200° + 0° 80° 1200

1600 20e

80° 120°

160° 200°

Theory

0:0.075.1 = 0.18

Experiment TU

Y

W

+

2;0° 3200 36

TU

2800 320° 360° 4'

TU

X=4

U:3

= 0.06

0.0195

q,2 = 41 0

X6.25 U=3

e1 =0.06

e2= 0.0246

4'tt=

4

U =4

e1 :0.06

ez :0.021

42°

A = 5.25 U : 4 e1= 0. 06 ea

0.027

c2: 1,3°

Fig. 114. W and TU in extraction modes with waves. is the phase angle

by which the wave displacements lead the displacements of