The unsteady lift force on a restrained hydrofoil in regular waves

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12 P. 1973

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NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER

DATUM:

Washington, D.C. 20034

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DOCUMENTATIE _{I: A}

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THE UNSTEADY LIFT FORCE ON A RESTRAINED

HYDROFOIL IN REGULAR WAVES

J.M. Steele, Jr.

This document has been approved for public

release and sale; its distribution is

un-limited.

DEPARTMENT OF HYDROMECHANICS

RESEARCH AND DEVELOPMENT REPORT

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QtSC1GO1

The Naval Ship Research and Development Center is a U.S. Navy center for laboratory effort directed at achieving improved sea and air vehicles. lt was formed in March 1967 by merging the David Taylor Model Basin at Carderock, Maryland and the Marine Engineering Laboratory (now Naval Ship R & D Laboratory) at Annapolis, Maryland. The Mine Defense Laboratory (now Naval Ship R &. D Laboratory) Panama City, Florida became part of the Center in November 1967.

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F-F

F

F

MAJOR NSRDC ORGANIZATIONAL COMPONENTS

DEVE LOPMENT PROJECT OFFICES O4l, SII, 80. 90 DEPARTMENT OF H Y DR 0M E CH 644 ICS 500 N SR DC CARO ER OCR CO8 AN D ER TECHNICAL DIRECTOR DEPARTMENT OF AERODYNAMICS 600 DEPARTMENT OF STRUCTURAL MECHANICS 700 DEPARTMENT OF APPLIED MATHEMATICS 800 N SR DL PANAMA CITY COMETANDING OFFICER TECHNICAL DIRECTOR

H

H

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COliN TE RME ASI) R ES P730

DEPARTMENT OF INSHORE

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DEPARTMENT OF THE NAVY

NAVAL SHIP RESEARCH AND DEVELOPMENT CENTER

WASHINGTON, D. C 20034

THE UNSTEADY LIFT FORCE ON A RESTRAINED

HYDROFOIL IN REGULAR WAVES

by

J. M. Steele, Jr.

This document has been approved for public

release and sale; its distribution is

un-limited.

TABLE OF CONTENTS Page ABSTRACT

i

ADMIMSTRATIVE INFORMATION

j

INTRODUCTION

i

EXPERIMENT 2 EQUIPMENT 2 TEST PROGRAM 2 TEST RESULTS 3 THEORY 6 WAVE MOTION 6 UNSTEADY LIFT 7 QUASI-STEADY LIFT 9 DISCUSSION 10 CONCLUDING REMARKS 12

PERSONNEL AND ACKNOWLEDGMENTS 12

REFERENCES 19

LIST OF FIGURES

Page Figure 1 - Reduced Lift and Phase Angle _{as Functions of Reduced}

Frequency, Head Seas, h/2b = 1.06, F2b = 2.11 /15 Figure 2 - Reduced Lift and Phase Angle as Functions of Reduced

Frequency, Head Seas, h/2b = 2.0,

F2b = 2.09

lo Figure 3 - Reduced Lift as a Function of Reduced _Frequency,

FollowingSeas, h/2b= 1.0, F2b

= 2.22 17

Figure 4 - Reduced Lift as a Function of Reduced Frequency,

Following Seas, h/2b = 2.0, F2b = 2.80

18

LIST OF TABLES

Table 1 - Steady-State Parameters of Hydrofoil ₃

Table 2 - Experimental Results for Head Seas, h = 2. 1 inches 4

Table 3 - Experimental Results for Head Seas,

_{h = 4.0 inches}

4

Table 4 - Experimental Results for Following Seas, _{h = 2. 0 inches 5}

Table 5 - Experimental Results for Following Seas, h = 4. 0 inches 5

Table 6 - Nondimensional Data for Head Seas, h/2b = 1.06 13

Table 7 - Nondimensional Data for Head Seas, h/2b = 2. 0 13

Table 8'- Nondimensional Data for Following

_{Seas, h/2b = 1.0}

14

NOTATIO N

A Aspect ratio

a Wave height

b Semichord length

C(k) Theodorsen function of reduced frequency, C(k) = F(k) + ig(k)

CL Lift coefficient

e' Lift coefficient slope

Wave celerity

d Water depth

F2b Froude number based on foil chord

F(k) Real part of Theodorsen function

G(k) Imaginary part of Theodorsen function

g Acceleration due to gravity

h Depth of submergence of foil

k Reduced frequency of encounter based on foil semichord

L Unsteady lift on a foil

M Function of i-' and k defined in Equation [131

t Time

u Horizontal component of orbital wave velocity

V Velocity of foil

w Vertical component of orbital wave velocity

w0 Peak amplitude of w

c Angle of attack of foil

i Wave elevation as a function of time

O Phase angle by which the lift leads the vertical component of the orbital wave

velocity

X Wave length

y Reduced wave frequency 2irb/X

p Mass density of fluid

T Wave period

p Phase angle by which the lift leads the wave peak at midchord of foil

w _{Frequency of encounter defined in Equation [4]}

SUBSCRIPTS.

K Unsteady theory

M Measured

ABSTRACT

The unsteady lift forces on a restrained hydrofoil in regular head and following waves of various wave lengths were measured at the steady zero lift angle, two submergence depths, and several foil speeds. A comparison of the experimental forces with linearized unsteady and quasi-steady theoretical forces revealed inadequacies in both theories.

ADMIMSTRAT1VE INFORMATION

This work was authorized under the Hydrofoil Stability Program, Bureau of

Ships (now Naval Ship Systems CommandNAVSHIPS) letter S82/8 (420) Serial

420-149 of 21 May 1958. The work was supported under NS715-102.

INTRODUCTION

The increasing use of hydrofoils on seacraft requires that the forces im-posed on such foils be predictable for design purposes. In order to predict the response or stability of hydrofoil craft in waves, the unsteady lift force on a foil caused by its passage through waves must be known. This investigation is con-cerned with these wave-induced forces, more specifically with the unsteady forces caused by regular head and following waves on a thin submerged foil.

This study is closely related to those described in References i through 6. * An experimental study of hydrofoil craft response and stability in waves was

made previously by Leehey and Steele. They compared their experimental

re-suits with predictions based on quasi-steady analysis by Weinblum2 and reported that unsteady flow effects should be included in analyses for head sea conditions.

In theoretical studies, Kemp, 3 Kaplan, and Leehey, derived the unsteady lift

force on a fixed foil passing through a sinusoidally oscillating fluid. In addition, Kaplan4' 6 measured lift forces on restrained foils for a few speeds and for long wave lengths in head seas.

The author subsequently undertook this experimental study in waves of shorter length and at slower foil speeds in both head and following seas. The un-steady lift amplitude and phase with respect to the wave peak for two submergences

were measured at the zero lift angle. The experimental results are presented

here together with computations of theoretical predictions made for purposes of

compari son.

EXPERIMENT

E QTJIPME NT

The experiments were conducted in the NSRDC miniature model basin. This basin is approximately 55 ft long and has a constant-speed test section 30 ft long,

2 ft wide, and 2 ft deep. The basin was filled with water to a depth of 1. 55 ft. It is equipped with a towing carriage which has a maximum speed of 8 fps. The foil

used had a Wright 1903 airfoil section. Its aerodynamic characteristics and

off-sets are given in Reference 7. The foil has a span length of 12. 5 in. and a chord

length of 2.0 in. The foil was supported at each end by thin struts during the

tests.

Strain-gage dynamometers provided the connection between struts and the towing carriage. A pneumatic wavemaker at one end of the basin produced

regu-lar sinusoidal waves with a maximum length of 5 ft. A wave absorber installed at the other end of the basin opposite the wavemaker minimized wave reflection.

Two capacitance-type gages were used to measure the wave height. One was fixed near the far end of the basin and the other was mounted on the towing

carriage. The carriage speed could be accurately determined by interrupted

dis-tance signals and timing marks. The lift and wave signals were amplified and recorded by means of a four-channel Sanborn recorder. This provided a continu-ous record of the lift and wave height during the test run. The dynamometers and

wave-height gages were calibrated statically. The experimental error is estimated

to be about 5 percent for the lift force and 0. 01 in. for the wave height measure-ments.

TEST PROGRAM

The foil was first towed in still water to obtain the steady lift coefficient as a function of angle of attack. The foil was submerged two chord lengths for these

tests. Since in previous tests' no change in lift with depth was noted until the submergence was approximately one chord length or less, the values obtained were assumed to be applicable also for any submergence greater than or equal t.o

one chord length. The steady lift coefficients were measured at a speed of about 5 fps. The zero lift angle was found to be -2 deg and the slope of the lift

coeffi-cient curve e' was 4.01 at that angle. The test conditions and results are

sum-marized in Table 1.

TABLE i

Steady-State Parameters of Hydrofoil Steady-State Experimental Conditions:

Depth of submergence h/2b = 2.0

Forward velocity V =5.Ofps

Steady-State Experimental Results:

Zero lift angle

ao =-2deg

Lift slope (at c) C' _=4.01

AH of the unsteady wave force measurements were made with the foil at the zero lift angle. The runs in waves can be conveniently classified in four groups according to the direction of the wave progression and the submergence of the foil as follows:

Group I: Head Sea, one-chord submergence (h/2b = 1.06)

Group II: Head Sea, two-chord submergence (h/2b = 2. 0)

Group III: Following Sea, one-chord submergence (h/2b = 1. 0)

Group IV: _{Following Sea, two-chord submergence (h/2b = 2. 0)}

Within each of these groups, the wave length was varied from about 2 to 5 ft. The wave height was varied from about 0. 03 to 0. 08 ft in head seas and from about

0. 01 to 0. 05 ft in following seas. In addition, the foil speed was varied from about

4 to 8 fps, except in Group I where it was nearly constant at 4. 9 fps for each run. TEST RESULTS

Results of tests in waves are presented in Tables 2 through 5. The foil velocity and phase angle were computed from the recorded data by the method described in Reference 1. The phase angle was not computed for the following-sea runs since the forces were observed to be very nearly in phase with the verti-cal component of orbital wave velocity for such low frequencies of encounter. The wave height a was determined from the moving wave-probe record and the wave

period T from the fixed wave-probe record. The wave length X was then

calcu-lated by the use of the relation:

Experimental Results for Head Seas, h = 2.1 Inches

TABLE 3

Experimental Results for Head Seas, h = 4. 0 Inches Rim No. a ft x ft C, fps V fps LM lb °M deg 97 0.038 4.96 4.94 4.86 0.565 58 98 0.038 4.70 4.83 4.88 0.559 54 99 1.035 4.08 4.53 4.85 1.514 72 loo 0.038 5.22 5.04 4.86 0.516 62 101 0.043 4.45 4.71 4.85 0.608 65 102 0.070 4.43 4.70 4.88 0.997 50 103 0.055 4.41 4.70 4.86 0.773 52 104 0.036 3.78 4.37 4.83 0.520 60 105 0.038 3.74 4.35 4.90 0.530 52 106 0.036 3.42 3.17 4.91 0.488 44 107 0.045 3.02 3.93 4.91 0.598 47 108 0.040 2.41 3.51 4.88 0.452 43 109 0.033 1.85 3.08 4.90 25 110 0.034 1.85 3.08 4.89 0.301 28 111 0.038 1.84 3.07 4.88 0.333 25 112 0.033 2.16 3.33 4.84 0.335 40 113 0.040 2.18 3.34 4.86 0.419

--Noter Deftnition of sy.rbol used in column heads th Tsblu. 2-5 are as foUnwa:

a iatIiew.ebeight, . isthewavelength. cisthewavecelerity. Visthe

velocity of the foil. L. is the m,asuri anplilude of unsteady lift, and Q1ta thr neasured pbaae axie by which the lift lends the wave crest.

Run No. a ft X ft C fps V fps LM lb °Mdeg 81 0.070 4.73 4.84 4.93 0.890 50 82 0.075 5.18 5.03 4.54 0.877 50 83 0.071 5.03 4.97 4.68 0.837 44 84 0.058 4.99 4.95 4.68 0.675 56 85 0.062 5.07 4.99 4.65 0.709 47 86 0.077 5.07 4.99 4.63 0.892 58 87 0.079 5.04 4.98 4.89 0.975 52 88 0.043 4.83 4.89 4.90 0.562 66 89 0.048 4.78 4.86 4.92 0.600 53 90 0.056 4.76 4.85 4.92 0.698 60 91 0.051 4.74 4.84 4.96 ---- 68 92 0.057 4.ò 4.82 5.02 0.736 50 93 0.061 4.57 4.77 5.02 0.762 68 94 0.064 4.85 4.90 4.bd 0.816 58 95 0.068 4.84 4.89 4.b9 0.867 54 96 . 0.071 4.90 4.91 4.86 0.880 60

TABLE 4

Experimental Results for Following Seas, h = 2.0 Inches

TABLE 5

Experimental Results for Following Seas, h = 4. 0 Inches

5 Run No. a ft X ft C fps V fps LM Ib 152 0.040 3.03 3.93 5.15 0.881 153 0.029 3.01 3.92 5.12 0.618 154 0.041 2.52 3.59 5.14 0.897 155 0.028 2.60 3.65 5.13 0.619 156 0.025 2.06 3.24 5.14 0.519 157 0.039 2.06 3.25 5.15 0.752 158 0.035 3.56 4.25 5.17 0.686 159 0.048 3.56 4.25 5.15 0.964 160 0.039 4.03 4.51 5.15 0.674 161 0.041 4.09 4.54 6.46 1.107 162 0.032 4.05 4.52 6.65 0.875 163 0.030 4.43 4.70 6.68 0.827 164 0.035 4.66 4.81 6.56 0.952 165 0.031 4.98 4.95 6.63 0.857 166 0.025 5.00 4.96 6.63 0.710 167 0.011 4.99 4.95 6.61 0.354 Ran No. a ft X ft fps V fps LM Ib 168 0.020 4.944 4.93 6.51 0.548 169 0.024 4.06 4.52 6.51 0.591 170 0.030 4.14 4.56 6.49 0.776 171 0.025 3.60 4.27 6.49 0.615 172 0.034 3.15 4.01 6.49 0.814 173 0.028 2.72 3.73 6.50 0.622 174 0.033 2.83 3.80 6.48 0.778 175 0.031 2.44 3.53 6.49 0.643 176 0.028 2.14 3.31 6.44 0.470 177 0.018 2.06 3.24 6.51 0.332 178 0.029 2.04 3.23 6.49 0.465 179 0.027 2.04 3.23 4.52 0.358 180 0.029 2.04 3.23 3.99 0.316 181 0.031 2.05 3.24 7.96 0.553 182 0.020 5.08 4.99 7.97 0.632 183 0.020 4.94 4.93 7.96 0.651

2ird

tanh

27r X

where d is the average water depth. The wave celerity c was obtained from

2.

tanh 27rd1 2

Cw== [2

X j

THEORY

WAVE MOTION

In a coordinate system moving with the foil, the surface wave profile above the midchord of the foil can be described by the expression

= a sinwt (3)

where is the height of the wave surface above the undisturbed water. surface,

a is the wave amplitude, t is the elapsed time, and w is the circular frequency

of encounter

w =-- (V ± c)

(4)

The plus sign is used for head seas and the minus sign for the following seas. The vertical component of the orbital velocity of the fluid particles at depth h in an in-finitely deep fluid is

2ir h

27rac

-w = -w0 cos -wt

_-

e cos wt (5)

For the values of A/d used in the tests, the waves would be classified as deep water waves for the shorter wave lengths and as Stokian waves for the longer

wave lengths. 8 However, the effect of the finite water depth on the magnitude of

the vertical component of orbital wave velocity was evaluated by use of the theore-tical expressions for waves in water of finite depth. Calculations were made for both foil submergence depths. The error in the vertical component was less than 3 percent for the longest wave length.

UNSTEADY LIFT

Expressions for the unsteady lift on a fixed foil in surface waves were derived

by Kaplan.4 He assumed two-dimensional potential flow in an infinitely deep fluid

in which the wave height was assumed to be small in comparison to the wave length. The horizontal and vertical components (u and w, respectively) of the orbital wave

velocity were assumed to be small compared to the foil speed. Kaplan retained some of the higher order terms in the orbital wave velocity components and in the

foil attack angle . He showed the first-order term to be proportional to w

and two higher terms to be proportional to w and to uw. Since the latter terms

are much smaller than the first order term, they were neglected in the present analysis. * Thus the lift on a restrained foil in waves is equivalent to the lift on a

foil in an infinite fluid (no free surface) which is oscillating with a vertical velocity w0 . This is the assumption made by Kemp3 and Leehey5 in a completely linear

theory. Their results are identical to Kaplan's first order term. Kaplan notation

is used in the present study but only first order terms are considered in the theore-tical development.

The linearized, two-dimensional, unsteady lift force due to sinusoidalwaves of small slope acting on a restrained hydrofoil submerged to _{a depth h has the form}

LK= Re c'p b Vw e1Wt _[J0(v)

_{- i}

_{J(v)] C(k)}

_{+ iJ1(v)}

₍₆₎

where Re denotes the real part of the expression,

c' is the slope of the lift versus angle of attack curve (c' = 2ir in two-dimensional flow),

b is the semi-chord,

z' is the dimensionless wave frequency,

k is the reduced frequency of encounter,

J0 (u) and J1 (u) are Bessel functions of the first kind, and

C(k) is the Theodorsen function which has real and imaginary parts F(k) and G(k) , respectively.

*Ogilrie points out that a nonlinear term in ua5 becomes important for moderately large steady angles of attack»0

In Equation (6), V = 2'rb/A cw)k=--= 2irb

(i±

C(k) = F(k) + i G(k) J12(k) + Y12(k) +

- i

[1

Y0(k) + J1(k) J0(k)] J02(k) J1(k) + Y02(k) + Y12(k)

where Y0 (k) and Y1 (k) are Bessel functions of the second kind.

With the use of Equations [7] and [8] , w0 in Equation [5] becomes

- y h/b

wo= e

The absolute value in the equation makes it valid for both head and following seas. Then the unsteady lift becomes

LK= Re c' p V2 a y e-ph/b + iwt M(k,v)eiO

(11)

where M(k,v)

e0

= [Jo(v)

- i

J()] C(k)

+ iJ1(y)

(12)

or M(k,v) [F(k) J0(v) G(k) (13) 2 + [G(k) J0(v)+

(!- F(k))

Ji()]

and (14)

'k

G(k)J0(z.') +

-- F

(k)) J1(v) O = arctan F(k) J0(v) + G(k) J1(p)

4

=

In deepwater where the length of the waves is less than twice the water depth, the wave celerity is

vh/b 7r

M(k,v)e

cos

(ça--- )

9

(16)

Equation [18] shows that the Froude number is not an independent dimensionless

parameter but is a function of k and y Therefore there is no advantage in

substituting the Froude number into the expression for the lift coefficient since M

already a function of both k and t,

QUASI-STEADY LIFT

When the wave lengths in a seaway become very 1on, becomes small and

the frequency of encounter with the waves becomes small. Then M (k, u) ap-proaches unity and Equations [16] and [15] become

CLQ b W -uh/b

-=

_a e V (20) cw k

i

JXg/27r ₍₁₇₎ V V _V

From this equation the chord Froude number may be written in the form

V

i

F2b = _k

i

t, (18)

In these equations e is the phase angle by which the unsteady lift leads the

verti-cal component of the orbital wave velocity. The phase angle ça by which the lift leads the wave crest

ça = -f + e (15)

is more easily measured.

The unsteady lift coefficient is the real part of Equation [11] divided by

In quasi-steady flow the lift coefficient is no longer a function of k (the frequency

of encounter) and the loading is in phase with the vertical component of the orbital

b

velocity of the waves. If Equations [17] and [181 are used, CLQ - becomes

the following function of y and F2b

C _!!

C' _{e_V1V'JD}

LQ a

_F2bJ2

(21)

DISCUSSION

The experimental data have been analyzed in termsof both unsteady and quasi-steady theory. There was some question in the preliminary analysis as to

the correct value of the lift slope parameter c' . In steady flow, the lift slope

of a finite span hydrofoil with aspect ratio A has the theoretical value

2ir

- i + 2/A

(22)

Since the experimental foil had an aspect ratio of 6. 25, the theoretical value of c'

was 4. 76. This compares reasonably well with the steady-state value of 4. 01 mea-sured in the calm water tests. The influence of the end plates and proximity of the

free surface could be responsible for the observed deviation. Therefore c' was

given the experimental value 4.01 in the analysis.

In the unsteady analysis, no fixed value of c' could be found which would follow the trends of the data. This can be seen by the solid curve in Figure

la.

The experimental data indicate either

that e'

is a function of the wave frequency

y or that there is an additional dependence on y which is not included in Equation [161 . In an attempt to fit a curve through the experimental points, it was

as-sumed that c' varies linearly with

C' = c0 + c1p

Values of c0 and e1 were

determined by fitting Equation [16] to the data by the

method of least squares. This semi-empirical relation is shown as the dashed curve in Figure la.

The dimensional data in Tables 2 through 5 have been reduced to nondimen-sional form in terms of lift coefficient, Froude number, and dimensionless fre-quency parameters in Tables 6 through 9. Although Froude number is not an in-dependent parameter in the unsteady analysis, this was the only parameter which was held constant in the experiments. Therefore the analysis has been made for constant Froude number which is a function of p and k

Figure 1 shows the head sea data for F2b = 2. 11 and h/2b = 1. 06 plotted

as functions of the wave frequency y . The points are drawn with different

sym-bols which are indicative of the value of k/v . Theoretical unsteady curves are

drawn for constant e' and for varying c' on the same graph. The variation in k/v is marked off in appropriate intervals on these curves. It is seen that by forcing the theoretical curve to pass through the points, the values of k/p are pre-ducted with reasonable accuracy.

The quasi-steady analysis in which c' = 4.01 gives values of the CL b/a which are several times as large as the measured values. Therefore this curve

is not recorded on Figure la. Neither the unsteady theory nor the quasi-steady theory predicts the experimental phase relationships. This poor agreement is shown in Figure lb.

Figure 2 shows the head sea data for h/2b = 2. 0 and F2b = 2. 09. These

data were confined to such a narrow band of reduced wave frequencies that itwas not possible to fit a curve through the points. Since the Froude number was nearly the same as that used in Figure 1, coefficients were computed using the

_{same e'}

in the unsteady theory. The agreement is fairly good but does not prove that e'

is a function of F2b only. For this particular submergence, the quasi-steady

theory comes into better agreement with the experimental results; see Figure 2a. Again, the phase relationships are poorly predicted.

Figures 3 and 4 show the variation of CL b/a as a function of z-' in the

following seas. The phase angles are not plotted because q was approximately

ir /2, the quasi-steady value. Both the unsteady and the quais-steady analysis

are presented in these figures.

r

CONCLUDING REMARKS

The data presented here represent only a limited range in the pertinent non-dimensional parameters. However, they are sufficient to show that quasi-steady theory cannot be applied to predictthe loads or phase angles. Also, the unsteady lifting line theory is grossly inadequate. A valid analysis should account for the

variation in lift slope with wave frequency and also give better prediction of phase

differences between the unsteady loads and the wave motions.

The lifting line theory indicates that there are only three independent

dimen-sionless parameters: k, t' , and h/2b . (The Froude number F2b is a function

of k and

uì .) Any future test program should be designed to vary only one of these independent variables at a time.

PERSONNEL AND ACKNOWLEDGMENTS

The investigation described in this report was undertaken by the late John M.

Steele, Jr. He designed and conducted the experiment, reduced the data, and set up the theoretical computations. Subsequent to Mr. Steele's untimely death, the

report was completed through the joint efforts of Dr. Avis Borden, Mr. John

Pattison, and Dr. David Jewell, members of the staff of the Department of Hydro-mechanics.

TABLE 6

Non1dmensiona1 Data for Head Seas, h/2b = 1. 06

TABLE 7

Nondimensional Data for Head Seas, h/2b= 2.0

13 F_2b -X k P b CLM _0M 2.10 29.8 2.02 0.106 0.309 1.01 2.11 28.1 1.99 0.111 0.305 0.94 2.09 24.5 1.93 0.128 0.314 1.26 2.10 31.3 2.04 0.100 0.286 1.08 2.10 26.7 1.97 0.118 0.297 1.13 2.11 26.6 1.96 0.118 0.295 0.87 2.10 26.5 1.97 0.119 0.293 0.91 2.11 22.7 1.90 0.139 0.299 1.05 2.12 22.4 1.89 0.140 0.289 0.91 2.12 20.5 1.85 0.153 0.276 0.77 2.12 18.1 1.80 0.173 0.261 0.82 2.11 14.5 1.72 0.217 0.237 0.75 2.12 11.1 1.63 0.283 0.175 0.44 2.11 11.1 1.63 0.283 0.184 0.49 2.11 11.0 1.63 0.285 0.183 0.44 2.09 13.0 1.69 0.242 0.218 0.70 2.10 13.1 1.69 0.240 0.219 F2b 2b -h--p cLMa _M 2.13 28.4 1.98 0.111 0.258 0.87 1.96 31. 1 2.11 0.101 0.280 0.87 2.02 30.2 2.06 0.104 0.268 0.77 2.02 29.9 2.06 0. 105 0.263 0.98 2.01 30.4 2.07 0.103 0.261 0.82 2.00 30.4 2.08 0.103 0.268 1.01 2.11 30.2 2.02 0.104 0.255 0.91 2.12 29.0 2.00 0.108 0.268 1.15 2.12 28.7 1.99 0.110 0.259 0.92 2.13 28.6 1.99 0.110 0.254 1.05 2.14 28.4 1.98 0.110 --- 1.19 2.17 28.1 1.96 0.112 0.257 0.87 2.17 27.4 1.95 0.115 0.244 1.19 2.11 29.1 2.00 0.108 0.265 1.01 2.11 29.0 2.00 0.108 0.264 0.94 2.10 29.4 2.01 0.107 0.259 1.05

TABLE 8

Nondimensional Data for Following Seas, h/2b = 1. 0

TABLE 9

Nondimensional Data for Following Seas, h/2b = 2.0

F2b -x k - b 2.23 18.2 0.237 0.173 0.416 2.21 18.1 0.234 0.174 0.401 2.22 15.1 0.302 0.208 0.408 2.22 15.6 0.288 0.201 0.414 2.22 12.4 0.370 0. 254 0.393 2.23 12.4 0.369 0.254 0.364 2.23 21.4 0.178 0.147 0.367 2.23 21.4 0.175 0. 147 0.373 2.23 24.2 0.124 0.130 0.319 2.79 24.5 0.297 0.128 0.324 2.87 24.3 0.320 0. 129 0.306 2.89 26.6 0.297 0.118 0.306 2.83 28.0 0.267 0.112 0.318 2.86 29.9 0.253 0.105 0.312 2.86 30.0 0.252 0.105 0.315 2.86 29.9 0.251 0.105 0.378 F9b X k

C-

b 2.81 29.6 0.243 0.106 0.314 2.81 24.4 0.306 0.129 0.284 2.80 24.8 0.297 0.-126 0.303 2.80 21.6 0.343 0.145 0.289 2. 0 18.9 0.382 0. 166 0.284 2.81 16.3 0.426 0.192 0.260 2.80 17.0 0.414 0.184 0.278 2.80 14. 6 0.456 0. 214 0. 243 2.78 12.8 0.486 0.244 0.199 2.81 12.4 0.502 0.253 0.217 2.80 12.2 0.502 0.256 0.192 3.44 12.3 0.593 0.255 0.141 3.44 30.5 0.474 0.103 0.252 3.44 29.6 0.481 0.106 0.260 1.95 12.2 0.285 0.256 0.323 1.68 12.2 0.190 0.256 0.356

b 0.25

cl

-a 0.30 0.20 0.15 0.1 ₀₃ 15 0.2 V Figure lb - Phase Angle

Figure 1 - Reduced Lift and Phase Angle _{as Functions of Reduced Frequency}

Head Seas, h/2b = 1.06, '2b = 2.11

D

KEY UNSTEADY THEORY -- -- -- -- QUASI--STEADY THEORY EXPERIMENTAL RESULTS 1.6±0.05

û

1.7 2.0 _{1.9 ).'L.}

A

1.8 D 1.9 1.8 .. 2.0 1.7 _c'4.Ol

'a

1.6

N.

5.83- 8.95 =

ONNI

F _{_1.7} k ;-= 2.0e 1.9 1.8 )

o

-o-Oç 1

o

A A

_û

90

1.6 80 1.4 70 1 .2 60 1.0 50 0.8 40 Q.6 30 0.4 0.] _{02 03} I I

II

30 25 20 _À/2b 15 14 13 12 11 10

0.30 c_L0 0.25 DEGREES RADIANS 0.20 0.09 0.10 0.11 V I I ¡ 31 30 29 28 27 A/2b 0 12 - QUASI-STEADY THEORY 009 0.10 011

012-Figure 2b - Phase Angle

Figure 2 - Reduced Lift and Phase Angle as Functions ofReduced Frequency

Head Seas,h/2b= 2.0, F2 b = 2.09 k 2.1 c=4.0l

_L

-f----2.0

+iOo8go

5.84 c'= - 8.95V V 2.0 j

o

OcbP

90

1.6 80 1.4 70 _1.2 60 1.0 50 0.8 40 0.6 EXPERIMENTAL RESULTS Figure 2a - Reduced Lift _k/V

o 2.0 ± 0.05

Ö2.1

0 13 0.14 KEY UNSTEADY THEORY 0 13 0 14

0.50 0.45 b

CL

_0.40 0.35 0.30

o

r

c=4.O1 V

o

k I L

-v

o

0.2 _0.3 0.4 17 EXPERIMENTAL RESULTS - - QUASI-STEADY THEORY c' = 3.48 + 5.44v 0.1 0.2 0.3 p J 30 25 20 15 14 13 12 1) 10 À /2 b

Figure 3 - Reduced Lift as a Function of Reduced Frequency

Following Seas, h/2b = 2.22 KEY UNSTEADY THEORY

o

o

û

o

k/ 0.1 ± 0.05 0.2 0.3 0.4

0.35 0.30

CL

0.25 a 0.20 0.15 18 KEY UNSTEADY THEORY - QUASI-STEADY THEORY EXPERIMENTAL RESULTS

o

_{0 N}

0

k/0.2± 0.05

0

0.3 c' = 6.37 - 2.98

0

0.4 0.5 .2 0.3 0.4 0.5

- ___

% o

c = 4.01

o

c'= 4.01 0.1 0.2 .03 i! I I t I I ¡ I I' 30 25 20 15 14 13 12 11 A/2b

Figure 4 - Reduced Lift as a Function of Reduced Frequency Following Seas, h/2b = 2.0, P2b = 2.80

19

REFERENCES

Leehey, Patrick, and Steele, J. M., Jr., "Experimental and Theoretical

Studies of Hydrofoil Configurations in Regular Waves," David Taylor Model Basin Report 1140 (Oct 1957).

Weinblum, G. P., "Approximate Theory of Heaving and Pitching of Hydro-foils in Regular Shallow Waves," David Taylor Model Basin Report C-479

(Oct 1954).

Kemp, N. H., "On the Lift and Circulation of Airfoils in Some Unsteady-Flow Problems," Journal of the Aeronautical Sciences, Vol. 19, No. 10 (Oct

1952).

Kaplan, Paul, "A Hydrodynamic Theory for the Forces on Hydrofoils in Un-steady Motion," Dissertation presented to the faculty of Stevens Institute of

Technology (1955).

Leehey, Patrick, "The Hubert Problem for an Airfoil in Unsteady Flow," David Taylor Model Basin Report 1077 (Jan 1957).

Kaplan, Paul, "The Forces and Moments Acting on a Tandem Hydrofoil Sys-tem in Waves," Stevens Institute of Technology, Experimental Towing Tank Report 506 (Dec 1955).

Loukianoff, G. S., "Traglachen-Untersuchungen des aerodynamischen Laboratoriums der Technischen Hochschule Moskau," Zeitschrift fur Flugtechnik and Motorluftschiffahrt, 3, 153, (í412).

Rouse, H., Editor, "Engineering Hydraulics," John Wiley and Sons, Inc., New York, 1950, Chap. XI, "Wave Motion," by G. H. Keulegan.

Lamb, H., "Hydrodynamics," Sixth Edition, Dover Publications, New York,

(1945).

Ogilvie, T. Francis, "The Theoretical Prediction of the Longitudinal Motions of Hydrofoil Craft," David Taylor Model Basin Report 1138 (Nov 1958).

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ODFORM 1473 (PAGE 1)

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Ser reify c!a.ssi(ic.,t, o,, ci title, body u!. 1cr Ira, t ..,,iJ ,rldr'xi,,1 .Innot3lircn reus t be entere,! ,vhen litt- overa!! report Is cincsz!ied)

OOIGINATING ACTIVITY (Corporale anthor)

Naval Ship Research and Development Center Washington, D.C. 20034

2a. REPORT SECURITY CLASSIFICATION UNCLASSIFIED

2h. GROUP

7c1-PORT TITLE

THE UNSTEADY LIFT FORCE ON A RESTRAINED HYDROFOIL IN REGULAR WAVES

IESC RIP T IVE NO TES (Type r,! report arrd ,flC!nr,ivr da!es)

au TI-IORISI (First trame, ,,r,d,I!e initia!, !SI name) John M. Steele, Jr.

6. REPORT DATE November 1970

7a. TOTAL NO OF PAGES

27

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10

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so. ORIGINATORS REPORT NUMBERIS)

3.86

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IT. DISTRIBUTION STATEMENT

This document has been approved for public release and sale; _{its distribution is} unlimited.

rl SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITy

Naval Ship Systems Command I). ABSTRACT

The unsteady lift forces on a restrained hydrofoil in regular head and following waves of various wave lengths were measured at

the steady zero lift angle, two submergence depths, and several foil speeds. _{A comparison of the experimental forces with}

linearized unsteady and quasi-steady theoretical forces revealed inadequacies

UNCLASSIFIED Security ClassificatLon

4

KES WORDS LINK A

LINK B HNK C

ROLE WT ROLE WT ROLE WT

Hydro foi is

Hydroelasticity Unsteady Lift Wave Forces