On the swimming of hinged hydrofoils

INGENIEUR-ARCHIV

XXXIV. BAND _{SECHSTES (SCHLUSS-)HEFT}

On the Swimming of Hinged Hydrofoils

By 6. H. Bolus, H. R. Kelly, and J. Siekmann

Introduction. The purpose of the present paper is to examine a few of the intermediate cases

between the completely flexible hydrofoil [1, 2, 3, 4, 5] and the completely rigid one, which is included in this report and for which the theory is well known [6]. The _{studies of one and twoS.} hinges are believed to be the most significant steps in the sequence from rigid hydrofoil (zero hinges) to flexible hydrofoil (infinite number of hinges). The theory _{developed by JVu and}

Siek-man, is readily adapted to the use of two, one or zero hinges, and can be used to predict the behavior

of each configuration. A few interesting cases were chosen for experimental investigation and comparison with theory. In the tests described herein,_{an attempt is made to produce and measure}

the forces required to produce self-locomotion in swimming hydrofoils.

In the discussion of swimming hinged hycirofoils, it is _{important that the definition of the term}

"swimming" be clearly defined. The swimming of a streamlined underwater body covers time

complete series of motions of a free body which produces self-locomotion. _{It covers both multiple}

traveling waves (anguilliform) and single traveling_{waves (carangiform) [7]. It covers both the} lateral and longitudinal motions, the latter being produced in the water tunnel by the flow, using the moving water as a frame of reference. Swimming may be applied equally well to a flexible

body, a segmented body with hinges, or a rigid, streamlined body moving in a manner to produce

self-locomotion.

Lift, drag, and power input were measured _{on three configurations of a small, symnnietrical} hydrofoil that was mechanically oscillated in a two-dimensional water flow. The tests included oscillations of different amplitudes in combined heaving and pitching _{of the rigid, single-hinged,}

and double-hinged hydrofoils, and simple oscillations (waving) of _{the hinged hydrofoils with one}

point fixed. The drag coefficients (positive and negative) are compared with the Wu-Siek,nann theory of swimming hydrofoils. Their theory can successfully predict the thrust of an oscillating

hydrofoil, whether it is rigid or hinged.

Hydrofoil. The streamlined body was chosen as NACA symmetrical airfoil No. 63009. The

known flow characteristics of this body when used as a hydrofoil under the static condition (no

oscillation) at various angles of attack are shown in Fig. 1. The coefficients {8] are divided by the factor to conform to the definition used in this paper._{. The outside dimensions are 8-inch chord,}

8-inch span, and 0.72-inch thickness at 25 _{percent of the chord from the leading edge. This shape} resembles that of a horizontal section of a fish without fins or tail flukes. The coefficients of both

lift and drag are plotted to the same scale to give a true visual representation of the angle of the resultant force on the rigid hydrofoil in respect to the relative flow during oscillation or under static conditions. When the relative flow is sinusoidal in shape caused by the combination of horizontal flow and the vertical oscillations (heaving and pitching), the resultant force swings forward and backward through small angles under the controlletl conditions of the motion of the

hydrofoil.

The Oscillating Mechanism. The mechanism for producing the oscillations is a modification

of the equipment formerly used for the experiments in the_{swimming of a waving plate [3, 9, 10.}

These experiments were performed by the U.S. Naval Ordnance Test Station (NOTS) at the Cali-fornia Institute of Technology (CIT) water tunnel facilities.

The complete mechanism ready for installation in the water tunnel is shown in Fig. 2. The location of the hinges in the hydrofoil and the _{actuators was established in the previous}

construc-tion. The force cell for sensing drag is seen at the rear end of the drag-sensing frame. Main supports for lime driving motor and the main attachments that_{hold the mechanism to the top of the tunnel}

walls are showmi.

Coeflc/e/7/$ I25 8,20 0,70 0,05 0

/

renIIn/

-Lift) force ana'

narmuFtoflcw

£

50 90 (Drag) 4/7g/e ofa//a, 0,85

_-°

2° Profile kc O ₈₀ 7O 720

Fig. I. Lift 00d Drog Cof,eieot,,.

Fig. 2. Complete Meclmn.o.,i

Type of Oscillations. The achievement of self-locomotion by the swimming body was the prime object of these experiments, and this object determined the manner in which the hydrofoil was to be oscillated. Because the frequency was most readily changed, all the other conditions were preset. The amplitude was small (0 to 1 -- inch), the geometric angle of attack was small

(0 to 12 degrees), and the phase angle between operating cranks varied over a considerable range

to give the proper angle of attack to the relative flow. No part of the hydrofoil was permitted

freedom of motion as it is often done in unsteady aerodynamics. Each hinged portion of the

hydro-foil was provided with actuators that determined its movement in coordination with the other

body l)Ortions. In this way, the camber changes were predetermined for each test.

Reduction of Data and Test Schedule. The results of the tests were in the form of oscillo-graph records. From these records, the forces were tabulated for each test condition, along with frequency of oscillation and actual water speed. Tile tabulated results were then converted to

dimensionless parameters of force coefficients and reduced frequency for comparison with timeorv.

The test schedule for the experiments included rigid, single hinged and double hinged

livdro-foils. Time tests of single-hinged and double-hinged configurations were made principmmlly to examine

the situations where the hydrofoil camber was changed continually to conform more nearly to time

curvature of the relative flow path. It was thought that this might minimize tite drag. A few

tests of "waving" hydrofoils were made also.

Propulsion Theory. Several theories have been published to describe tile swimming of fish-like bodies [1, 2, 3, 4, 5, 11, 12]. The special study of hinged bodies has also heeti treated by Ro,m-thron and Fejer [13] in which side forces were carefully balanced for the test of a freely swittlImmilIg

body. Since the 1)resent exhierinlemits concerned a constrained body, it was decided to use time 0

I

(4;c;/ 7j

1

r

340 G.H. Buwlus, H.R. Kelly and J. Siekrnann: On the Swimming of Hinged Hydrofoils !nf,.e,,iersr..trcI,, V

2° f°

Sin

XXXIV.Beod 1965 _{G.H. Bowlus, H.R. Kelly and J. Siekmann: On the Swimming of Hinged Hydrofoils 341.}

Wu-Siekrnann theory for comparison. This theory was written with a thin flexible plate in mind, but is completely general and can be applied to a body with a finite number of hinge points.

The double-hinged hydrofoil is represented in Fig. 3 as a thin-hinged plate with six degrees

25*

Fin. 3.Doublc_Uiogcd,Thrn-Ph.LuHydrofoil witi,Six Degre of Frcodou.

where h1, the (dimensionless) displacement of the hinge, is measured in units of the half chord,

while 6, 6, 62 represent the angular positions of the hydrofoil segments. The circular frequency is

denoted by , the time by t. K is a nondimensional amplitude; q, q1, and q. are nondimensional

parameters; and e, and s2 are three phase angles. The imaginary unit is (lenoted by i =

/-

1.

The displacement at any location, x, is

h(x, t)

= h1 ± 6 (l - x) + H (x - 11) (6 -

6) (l _{x) + H (x - 12) (62 -}

_{ô) (1 -}

x) , (5)

with H (x - 1) as tile ileaviside step function:

Ii(x -1) = 0,

_{x - 1<0;}

H(x -1) = 1,

x -

0.

Letting x = - cos 0, 1 = - cos

_Pi'

h = h1

₊

61

_±

+ (62 6) where 12 = - cos 1p2 a 6

cos 0 + (ó 6)

f('p)

+ 2

f0()

0 1 Fourier

+

cos IL

series expansion gives

2f0(1) cos n

(6)

=

f(cos 0

cos ) cos n 0 dO, n 0. (7)

0

The latter becomes, for the two special cases of interest,

1.

fp) = -

(sin ip

_{- p cos}

and ₍₈₎

f(uy) = _{( sin} cos

It follows froni Eq. 6 that

=

6

(61-6) ii(x

(62 6) iI(x

---(6 6)

+ 2 sin 'Pi

no] ₍₉₎

J_(62_

₆₎

₂ SIH flip2

+

cos

Substitution of Eqs. (1) to (4) in (6) and (9) and setting

h[Bo+2'B0cosnfJ]eiPt*

(10)

=

[c0

+ 2' C,

cos ii

oJ e'

(11)

of freedom. Referring to the figure, we let

= .KeP' (1)

6 = q K e(/iL+) (2)

61 = q1 K e(#t±Ei)* ₍₃₎

we can calculate the variousBand C needed as

B0

1 + q 1 e" + (q1 e'' - q ee)f0(ip1) + (q2 e1

_{- q1 ei)f0(2)}

B1

q e + (q1 e' - q e) f1(v1) + (q2 e

- q1 e) f1(2)

K2

= (q1 e'

q e )J,(ip1)

+ (q2 e1' - q1 e)f,,(2)

= -

I

_{q e16 + (q1 e1' - q e) + (q2}e162

-

_{q1 eu')}

2}

= - --

_{ei - q}

i6) SiflflIpj _{+ (q2 ele, -} 16) sin

Now, as in Ref. [2], we set

A,, = C,, + iw B,, (17)

with w = j/U as the reduced frequency, which for a chord of 2b is given byfi b/U, cf. [9]. U denotes the freeestream velocity.

In order to compute the average thrust and power, we need only use B0, B1, _0. C, A0, A1.

When separated into real and imaginary parts,

B,,

B + B:,

_{C,, = C:, + i C,}

_{A = A:, + i A}

₍₁₈₎ we get

K -

+ q 11 cos e + (q1 cos e - q cos )f0() + (q2 cos

-

COS 1)fO('2); (19)

B'

= 'J e + (q1 sin

- q

E) fo(tP) -F- (q2 sin 2 - q1 sin e) f0(v2) (20) and so forth.

We may now use the results of Ref. 2 directly. For the average power E dissipated into the

wake we obtain

U3bw

=

[(C

-

Cl) w (B -

RI')]2 + [(C' - C)

+w(B-

B)]2. [

+

2)]}, (21) where denotes the density, w the finite width of the plate, and _{(w) and (i(w) the real aid}

ima-ginary parts of the Theodorsen-f unction _{(w) [14]. Further, the average power input P to the} hy-drofoil is given by

U b w = (0 (w) {(C - C)

(B +

BI) - (C - Cl) (B + B) + w ftB)2 + (B)2- (B)2 - (Bl')2]}

+ (0 (w) [(C' - Cl') (B' +

_{BI') + (l - Cl) (B + BI)}

_{2w (B' B - B, BI')]}

- w [(C' - CI') B - (C - Cl) Bfl

(2 [(B

_{- B) BI + (B' - BI') Bfl .}

₍₂₂₎

*

The average thrust (drag) T may be calculated from tile results of Eq. (21) and Eq. (22) yieieltiig

* *

T P E

23

U2bwr U3bw

'r U3bw

Then the efficiency factor can be expressed as

*

= I

E (Z4)

P Thus, the thrust (drag) coefficient c used is

*

T

CT = _{r U2 b w} (25)

and so forth. It should be pointed out that the terms "thrust" and "drag" referto the streamwise component of the resultant iky(lrotlynamnic force, where thrust is directed Opposite to the (liteeli011 of flow, while drag shows in flow direction.

XXXIV. Band 1965 _{G. H. Bowlus, H. R. Kelly and 3. Siekmann: On the Swimming of Hinged Hydrofoils 343} fl li,e

ue'''

oe1Ct_v, _i&Lt4) off/ow % 31,0

Fig. 4. Double-Hinged, Thin-Plate Hydrofoil with Four Pairs of Actuators.

Fig. 5. Rigid Hydrofoil, Drag Coefficient

(cT>O: drag; CT<O: thrust).

I

x

0-S Amp/liude un un Pbu.se 300 00 I f/us,

7. Discussion of Results. It will be shown in this section that, while considerable scatter exists

in the data, well established trends may he found, and good agreement with propulsion theory is evident. The torque data were least accurate; and since they only indicated qualitatively the power input to the whole mechanism and not just the hydrofoil, they are omitted. Lift coefficient data are also omitted because the scatter made them almost meaningless. Representative drag

coefficient data for each configuration are presented here for comparison with the Wu-Siekrnann

theory. A more extensive presentation of theoretical curves is then made to show the effect of

changing the phase angles on the propulsion force and efficiency.

a) Rigid hydrofoil. We shall first examine the results of the rigid hydrofoil tests. The runs

with zero angle of attack are shown in Fig. 5. This test was designed to have both the geometrical

angle of attack and the angle of relative flow oscillate between + 10 degrees, withan amplitude of motion of about 1 inch. Tile angle of attack of the hydrofoil then remains atzero, producing

no lift, no thrust, and a minimum of drag. The wave length 2 of this motion is 3 ft, and from the

0

}frhe

-408 -/212 N.N

\

In order to apply the theory that is outlined above to the-actual experiments, we note that the double-hinged hydrofoil is driven by four pairs of actuators attached at x-values of s1, _{2' s3,}

and S4. as shown in Fig. 4. The amplitudes of these stations are a1, a2, a3, anda4, and the phase

angles of the motion are

=

, ?7, and l4, respectively. A simple metho4l of adapting the

theory is to let the dimensionless amplitudes be complex:

K=keiy,

(26)

with k as a dimensionless amplitude and y a phase angle. Since the phase is arl)itrarily set to

zero, we have

a1euilt = h1 + (ii- s1). (27)

The factor ehJt will cancel, and, separating real and imaginary parts in the equation

a1 = k (cos I' + i sin y) [1 + q (1 - s1) cos e ± i q (l - s1) sin ] , (28)

we find

a1 = k [cos y + q (l - s1) cos e cos .' - q (l - s1) sine sin (29)

and

0 = k [sin y + q (l - s) cps e sin y + q (l - s1) sine cos . (30)

Tile displacements at tile other three actuators lead to the equations:

a2 ei(Pt_ui) = h1 -f- 5 (l - s2) (31)

a3 e' (#')s) h1 -!-- (1 - $3) (32)

and

a4 e(fl_'is) = h1 + l (12 - 1) + (12 - s) (33)

These equations, when treated as above (Eqs. (28), etc.), lead to similar, but somewhat lengthy

formulas and will not be reproduced here.

D,'og eoet/Ycientc,.

344 G.H. Bowlus, H.R. Kelly and J. Siekmann: On the Swimming of Hinged Hydrofoils Iuaernur.Archis relations v A = U, where v denotes the frequency, and w = 2 r v b/U, we find w = 2 r b/A. With the half chord, b, being 4 inches, then w = 0.7 satisfies the design condition. We shall call this the design frequency, under which the hydrofoil maintains a zero angle of attack at all times in its motion. Of course, at lower or higher frequencies, the angle of attack will not remain zero, and there will be an oscillating lift and a net negative or positive average thrust. It can be seen in Fig. 5 that the Wu-Siekrnann theory indicates zero thrust at about w = 0.7 and the data extra-polate to about the same value.

One fact is quite apparent iii Fig. 5, and is confirmed in the data for the other rigid hydrofoil tests: the data points follow the theoretical curve very closely, and tend to lie below the curve (toward greater thrust), if anything. In order to avoid bias in the comparison, a least-squares fit has been made to the data, and it is shown by a dashed line. The least-squares fit assumes a

qua-dratic function passing through the origin.

The comparison is surprising when one considers that the theory does not take skin-friction drag into account. Skin-friction drag was not measured directly in these experiments, and

its estimation is somewhat controversial. The drag coefficient for this airfoil shape is found to be 0.0028 for zero angle of attack (Fig. 1), but the NACA data are for higher Reynolds-number. The

Reynolds-number of the present tests, varying from 70,000 to 200,000, may be used to estimate the skin friction. Assuming a laminar boundary layer, the drag coefficient will be about 0,002 to 0,003, while a fully turbulent boundary layer will give about 0,004 to 0,005. An effect of this

magnitude is not evident in the data for the rigid hydrofoil. There are three possible explanations.

One is that the thrust produced is greater than predicted by perfect fluid theory. The others are

that the motion of the hydrofoil somehow changes the boundary layer and reduces the skin friction

to a small value, or that the zero of the drag balance mayhave shifted.

In addition to the drag at zero angle of attack, there is the induced drag that is due to lift that must be considered. In these experiments, the geometrical angle of attack was chosen to have a maximum value of about 10 degrees. The angle of relative flow at the upper limit of the

frequency range that is used in the experiments, would have a maximum value of about 30 degrees.

This would result in an angle of attack of about 20 degrees, which should cause an induced drag

that is due to lift of about 6 times the drag at zero angle of attack, according to extrapolated NACA

data. This effect is obviously not present in the experimental results. The reason for this may be surmised from the flow visualization tests. When the hydrofoil was stationary at an angle of attack with respect to the water flow, the dye pattern showed an unmistakable separation effect

on the hydrofoil and a broad wake. Immediately on starting the oscillating motion, this separation and broad wake disappeared, the flow followed the body contour and was shed in a narrow wake

at all the frequencies that were used. This flow pattern would tend to confirm the idea that the

induced drag that is due to the angle of attack would not be present in oscillating motion. In fact,

it tends to cast some doubt on the quantitative value of any steady-state coefficients as applied to dynamic behavior. This deserves more detailed study than it has had.

b) Single-hinged hydrofoil. Tests of the single-hinged hydrofoil were made in three groups: one group used a constant crank amplitude of one-half inch, the second group used a constant amplitude of one inch, and the third group used a linear variation of crank amplitude from zero at the front actuator to about one inch at the rear actuator.

The runs at constant small amplitude were apparently a failure, since not only was there a wide scatter in the data, but a large positive drag was found for all conditions. The agreement with theory is also very poor. The data appears to indicate that there was a positive-drag force present whose value was independent of the water speed. Dividing by the square of the speed would then make the coefficient of drag appear to be smaller for the greater water speed. This is

confirmed for each set of data (Fig. 6).

An examination of the recording tape reveals that there was indeed a positive constant-drag force added to the data. This occured immediately before these runs, when the recording needle

was somehow shifted toward the positive drag side. This was not noticed at the time, and because

no calibration was made after the runs, the zero shift was not discovered. This means that the data for the first group are of no quantitative value, though the measured trends are qualitatively

correct.

The runs at one-inch amplitude (Fig. 7) show much better results. On the average, a small

XXXIV. Band 1965 G.H. Bowlus, H.R. Kelly and J. Siekmann: On the Swimming of Hinged Hydrofoils 345 The runs with linear variation of amplitude were successful and agreed well with theory.

Con-trary to the case of the rigid hydrofoil, there is apparently a measurable skin-friction drag. It is

of the order of 0,009 to 0,015 in drag coefficient, higher than would be expected. There is stillno

significant evidence of induced drag that is due to lift so we must assume that this does not occur

for the oscillating hydrofoil. A least-square fit of the data was made, but (Toes not appear to work

well. Apparently, forcing the curve to the origin does not represent the actual data. Another scheme is to assume the theory to be correct and to solve a least-squares of the deviations from this theory. This turns out to be a simple average of the deviations. The results are shown in

Fig. 8. 416' al! LW! ao4' 0 402 0 aD! 0,04' Fig. 8.

Fig. 6 to 8. Single.I-Iinged Hydrofoil, Drag Coofficient (OT>O: drag; CT<O: thruat).

c) Double-hinged hydrofoil. Part of these tests were essentially a repeat of those of the third group of the last section, but with four pairs of actuators and two hinges instead ofthree pairs of actuators and one hinge. The theoretical results are identical with those for the single hinge, and the experimental data are very similar (Fig. 9). A small skin-friction drag is evident,

just as with the single-hinged hydrofoil, but there is no appreciable increase in drag with angle of 404 O'ug coef/?i*ntC a 0 0 0) 0 0 flTesf AJd1l3P111 points runs Pro a a 0 40 ,Tes/poi,n/sfroii, j dmuli','ent runs 00C 0 a) 0,4 1,2 1,8 2,0 2,4' Actuator 4' 3 I flow Amplitude un un un Phase 1000 _0° 48 1,2 1,6' 2,0 24' Actuator 1, ₃ 1

Amp/lode lie 45in Din

Phase 300 0° 0°

Ac/ua/or 4'

Amplitude 451n

Phase 80°

0 _{\.Tesl piils from}

Jd'fferen/ runs 0 44 48 1,2 1,6' 2,0 2,4 £8 3 1 0,Sin 0,Sin 40° 00 Fig. 6. Fin. 7. Dt'agcoef/7cieii/ C)'

346 G.H. Bowlus, t.l.R. KeIIy and J. Siekmann: On the Swimming of Hinged Hydrofoils Ingcrncor-Archiv attack. Taking the average deviation, as before, we find a value of about 0,008. Other runs (liffered

only in that the phase angle of the motion changed between each successive actuator, instead of all at the last actuator.

Another group of tests is represented by Fig. 10. The motion here has been labeled "swimming"

instead of "waving", because the second pair of actuators is held fixed instead of the first pair. Actually, it is not a true swimming motion, and the theoretical efficiency will later be shown to

resemble the "waving" case. Thug coet'fioiern'Cr 0,08 ao' 0 -aO4 aa. -aLt a 0,08 408 -410 0,12 Center/bieOtOSCiIlCtIO/7

Osdll/a/io,,ofhitigedhydrot'o//s with onepoiiil/7xeo

Di"ecfion of f/ow isth'mr'i:qhf toleft

Dr coefficient CT

Ao/uator 3 1

-Amp//hide fin fin

Phone 48 as 44' A SW 42 0 Di'ag coeflb'eii/ CT te,,/er/liw ofastht't,n O,iin 300 Oin 421n 100 0° 2 Athiolor/Jo. 2 is fied

Fig. 9 and 10. Double.Hinged hydrofoil, Drag Coefficient.

0 0 0) I °}Tes I poio/s from differenh'u,i 0 0 °) Tesipoi,ils Prom, difPeren/ runs

rniu_

ii._

Theo.'yfor lest conditions (see be/ow) different 0 1

J

-,4chilors

1 Phone 0° 00 0° 0° 3 300 84° 78° 0° o Thyfordnrere oites/condh'isns iv, A j some um,ok7ude(r forOcluotOf'S 3

'

0 30° 00 o _{4'° 0°} 180 _0° 00 00 40 2,4' 2,8 Ac/uclo,' * 3

__

81

flow

Amplitude tin 0,liin 0,lSSin Oin

P/,ose 30° 0° 0° 00 0 04 a8 1,6 ₂₀ 2,4' 8,8 Ac/uo/or 4

21

F/ow fin Amplitude Phone 800 04' 05 1,2 20 44' 2,8 3 1 2 0,04' 0 0,04'

Oi'og coef'ftc/efl/ CT 0,01 0 -1401 -1403 -403

XXXIV. Band 1965 _{G. 11. Bowltis, H. H. Kelly and J. Siekmann: On the Swimming of Hinged Hydrofoils 347} 8. Theoretical Efficiency. It will be shown here that the theoretical efficiency of self-locomotion may be very high for some of the test configurations and very low for others. It has been

demon-strated that the JVu-Siekmann theory predicts the thrust very well, and because the power and efficiency could not be measured accurately, _{we shall examine the efficiency of self-locomotion}

from tile standpoint of theory alone. The choice of configurations to be compared will follow thc experimental program very closely.

A family of drag-coefficient curves for the rigid hydrofoil is shown in Fig. 11. These range from those shown in Fig. 5 to those of a heaving wing, where the chord remains horizontal (parallel to the free-stream direction). The heaving wing gives propulsion at all frequencies, while tile others

give a net negative thrust at low frequencies. Corresponding propulsive efficiency is shown in

Fig. 12 (remember that efficiency has no meaning for negative thrust). At first glance, the heaving wing appears to be su1)eriOr, having a greater thrust at any given frequency and efficiency

approach-ing unity at very low thrust. llowever, self-propulsion requires that a small drag be overcome.

if one makes the arbitrary assumption that the NACA value of about 0,003, which _represents

zero-yaw drag, is the thrust coefficient required for self-propulsion, then the dimensionless

fre-quency required for each mode is as marked in Fig. 11. The corresponding efficiencies are marked in Fig. 12. The heaving wing is found to have a theoretical efficiency of 0,7, while the others show an efficiency of 0,65 or better. Ac/so/or:4' 3 1 04' Actug/op as Amp/i/ode Phase 4 0,5 in 1,2 3 asin 2,0 I 2,4' flow

Fig. 13. Singlc.Hinged Hydrofoil, Theoretical Drag.

2,8 0 S A 0 oU) 0,6' as 44' 42 0 ?00 000 1 2

Fig. 14. Single.Hinged Hydrofoil. Theoretical Efficiency. 'rem '1./n

5in)

3

a,

The drag coefficient of a single-hinged hydrofoil, is shown as a family of curves in Fig. 13,

with phase angle as a parameter. The same criterion of self-propulsion is used here as for the rigid

hydrofoiJ. Figure 14 shows corresponding theoretical efficiencies, with efficiency being nearly optimum at 0,80 for phase angles of 40 and 80 degrees at actuators 3 and 4.

,

The theoretical curves corresponding to a waving, single-hinged hydrofoil are quite different. Figure 15 shows that large thrusts are still available at high frequency; but, as shown in Fig. 16, the efficiency of self-propulsion is quite low. It appears that for the waving motion the efficiency never rises much above 50 percent.

The theoretical behavior of the double-hinged hydrofoil is not much different _{from that of the}

single-hinged hydrofoil. Typical theoretical curves for the double-hinged hydrofoil at constant

ani1mlitude arc found iii Figs. 17 and 18. In contrast to tile "waving" case, a decrease in amplitude gives not only a smaller force, but a smaller optimum efficiency.

800 40° 00 0'0'7T0"0lT' 1000 50° 0OA ff51 cndi/iasi 1200 07 0° some unzplitode( 140° 70 00 o fir ocluolors 1, I0 A 0

-Il

A f5çf (see be/ow) 1 ieoryioraiffei'eii oJ 0 A 0 500 80° 1000 120° 140° 30° 40° 50° 80° 700 0° 00 0° 00 0° 0

348 G.H. Bowlus, H.R. Kelly and J. Siekmann: On the Swimming of Hinged Hydrofoils Ingenieor.Arehio Dxvgcoefficient c--0,02 -0,0',' -0,08 -408 -410 £402 0 -0,12 Drag coeffiieiif c 1,6 Amp/i/ude fin 451n Cm Phase o 30° 0° 00 °40° 00 00 A59° o° 00 3 20 I .44 F/ow 2,9 A 0 (1) ow 0,4 0,8 42 .3 Pise 30° 0° Theoiy/'&'r 40° 0° 000'th1fre,uf -50° 0° 0° AJ_cono'iffons"est A

Ampli/zate Tin 0,5in Oin

2 3

Fig. 16. Single-hinged Hydrofoil, Theoretical Efficiency.

Fig. 15. Single-Hinged Hydrofoil, Theoretical Drag. w

0

9. Flow Visualization. Figures 19 to 22 show typical flow visualization pictures that illustrate

important features of the test program. In some cases the dye is hard to photograph and the wake is indistinct. In each case, dye is extruded from holes in both the top and bottom of the hydrofoil

near the leading edge, for example see Fig. 23.

Figure. 19 is a typical zero-thrust case. Actually, the bottom picture corresponds to an angle of attack = 1,40 in the top curve in Fig. 15. Here the theory predicts just enough thrust for self-propulsion. A more complex wake pattern with vortices is shown in the bottom plettire of

Fig. 21. 401 309 410 o Theorj'foi' test A (sbdow)conditions different

_-U---'

_u_U

kI

°i Theory for

conditions be/ow) different ° test J (see 0,4 48 Actuator Aoipffluo'e Phase a o A 1,2 4 fin 59° 89° 89° 800 3 Tin 300 300 80° +6° 2,0 .44 2 1 fin fin 100_U 100 0 100 0° 0° .48 0 1 2

Fig. 18. Double-Hinged Hydrofoil, Theoretical Efficiency.

Fig. 17. Duuble.Hiugcd Hydrofoil. Theoretical Drag.

0,8 Ac/ua'/o,'s 4 80° 90° 80° Phass 3 2 1

30° 10° 0° o Thery for diiPe,v/

30° 10° 0° 0 1 /95/ conti/lons 14'IIk 300 10° 0°-° rs,eanpIhljda(iin) 45° 15°.0°racfu9/o,,s3,4

I

80° 0 44 08 1,2 Athig/or 4

Fig. 19.

lop: IUgid hydrofoil or:.11tiog

0

woter flow U = 1.5 ft/we. ;'

1 eps:

botto,n

Single-hioged hydrofoil oriIIotiog in sntrr flow U = 3 ft/err, r =

Fig. 20.

top Single-hinged he drofoil n fi 'e:l position.

l on log fI ow seJn rntion U = 3 ft/see, r

0 epe;

bottom: Single_hinged l

drofo,I

;,l:lishee smooth flow by oscillation U

3 ft/err, r

I

g. 21.

lg. 22.

toll: 1)o l:l.-l:iegt:l l,:lrofoil is fsed position, si:owigf1ow seporotion U

1.5f1/sec, p

Oepo;

Isp: lUgid l:s Irofoil ooeillati.g in

icr flow U

1.5 ft/see, r

1.5 cpa:

hot tori,: l)o:hle.l::nged hydrofoil, >l:o%sing details at downatrea in srJbaraliori U= 1.5ftjscc, ':

1.5 rps.

lull to,11

Doial:leIiigud liydrof,,iI

still stir, g in outer flow U

1.5 ft/see,

2 er

Fig. 19 to 22. Flow liSi101iOOtlOfl.

r

XXXIV. Uud 19U5 G.H. Bowlus, iI.R. Kelly and J. Siekmann: On the Swimming of Hinged Hydrofoils 351

Noteworthy is the effect shown in Figs. 20 and 21. In each case, one picture depicts a stationary hydrofoil, while the other depicts the same hydrofoil in precisely the same configuration alit1

posi-tion, but in motion. It appears in 1)0th figures thai. the motion eliminates the effect of separation at the hinge and eliminates the broad wake. This may be because the relative flow at any point does nut remain parallel. In any case, it seems worthy of further study.

Trah/klg edge

1/8n side p/ale Pflernbocpd'o&lic

7/i5-in-dunih/age pills

Fig. 23.

Conclusions. It is found that the Wu-Siekmann theory can successfully predict the thrust of an oscillating hydrofoil, whether it is rigid or hinged. The best operating efficiency was found when all actuators operated at the same amplitude with a small phase difference. Efficiency of self-propulsion can be optimized for a given amplitude of motion by the choice of phase angles. In general, the efficiency of self-propulsion is quite low if a point near the leading edge is held

fixed and the rest of the hydrofoil is oscillated.

Appendix. For the sake 1)1 convenience the following list furnishes a comparison between the coefficients employed by Pu [1] and Siekrnann [2, 3].

Definitions:

Siek,nann transformation Ilararneter x = - cos 0,

displaceiiicirt function g(x, t) = (B0+ 2 ', B co n

et

- slopc of ihisplacenietit function t) (c0 -F 2 ' C,, cos ii &It

JVu - transformation parameter x = cos 0,

- displacement functiomi h(x, t)

=

(__ o cos a

o) eit

- slope of displacement function

=

+

cos nO eJ0I

"Downwash'':

Siekmann - w(m, t) = U (i + 2

_L'

A cos n ei

Wu

- v(x, + 0, t) = -

u(20

cos no) ejwi.

Oyeex#/br

lows" su,'/'ace

Leadfrig

edge

Dye exit for toper surface Dye eu/apace Truap,00Nai in n T

//

b/ut

.-U 81U -e -Japnaionil4z Trunnioa//o.3

-- 351lifl.---'---TrullnionMa2451

-Relation between coefficients:

Siek,nann A = C,, + io B,,,

a0

=(w)(A0 A1) +A1,

a,, =

jA,11

-

_nAnj-Ani

Wu 2,,

= - (y,,

_+JoIJ,,),

a =t.(a)(20+A1)

a,,

=

2,, + -- (2,,_'

-Correspondence between symbols:

Hence

a0of Siekmanu's corresponds to -a0of Wu's a,, of Siekmann's corresponds to - n a, of Wu's.

Acknowledgements: The authors are obliged to Professor Dr. T. Y. Wu for valuable discussions. They also gratefully acknowledge the assistance rendered by Mr. Sui-Kwong Pao and Mr. Su-Ling Cheng.

References

T. Yao-Tsu Wu, J. Fluid Mech., 10 (1961) P. 321. J. Siekmann, Ing.-Arch. 31(1962) S. 214. J. Siekmann, Ing.-Arch. 32 (1963) S. 40.

Sui-Kwong Pao and J. Siekmann, Proc. Roy. Soc., London. Ser. A 280 (1964) p. 398. J. P. Uldrick and J. Siekmann, J. Fluid Mech. 20 (1964) p. 1.

National Advisory Committee for Aeronautics. "Propulsion of a Flapping and Oscillating Airfoil", by

I. E. Garrich, in the 22nd Annual Report of the National Advisory Committee for Aeronautics, 1936.

Washington, GPO, 1937 (NACA Report 567). C. M. Breder Jr., Zoologica 4 (1926) p. 159.

L. H. Abbott and A. E. von DoerzhoJf, Theory of Wing Sections, Including a Summary of Airfoil Data. New York: Dover Publications, Inc., 1959.

H. R. Kelly, Fish Propulsion Hydrodynamics, in: Developments in Mechanics, Vol. 1. Proc. of the Seventh Midwestern Mechanics Conference held at Michigan State University, 6-8 Sept. 1961, New York: Plenum Press, 1961, P. 442.

II. R. Kelly, A. W. Rentz and j. Sieh,nunn, J. Fluid Mccli. 19 (1964) P. 30.

M. J. Lighth ill, J. Fluid Mccli. 9 (1960) p. 305.

E. H. Smith and D. E. Stone, Proc. Roy. Soc. London. Ser. A, 261 (1961) p. 316.

R. L. Bonthron and A. A. Fejer, A Hydrodynamic Study of Fish Locomotion. Proc. of the Fourth_U.S. National Congress of Applied Mechanics held at the University of California, Berkeley, California, 18-21 June 1962, New York, 2 (1962) p. 1249.

National Advisory Committee for Aeronautics. "General Theory of Aerodynamic Stability and the Mecha-nism of Flutter" by T. Theodorsen. Washington, GPO, 1935 (NACA Report 496).

(Eingegangen am 15. Mãrz 1965.) Anschrif ten der Verfasser:

Mr. Glenn H. Bowl us, 58-E Calle Cadiz, Laguna Hills, California, USA. Mr. Howard B. Kelly, U. S. Naval Ordnance Test Station, China Lake, Calif. USA. Prof. Dr.-Ing. J. Siekrnann, Department of Engineering Science and Mechanics, University of Florida,

Gainesville, Fla. USA.

= 0, 1, 2,

a

= i

Co.

352 G.H. Bowlus, H.R. Kelly and J. Siekmann: On the Swimming of Hinged Hydrofoils IngeuieurArehiv

Siekrnann Wu

3 (complex unit) (0