Unsteady hydrodynamic loads on a two dimensional hydrofoils

4 ME 1913

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-NA VAL SHIP RESEARCH AND DEVELOPMENT CENTER

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Bbliotheek van

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Onderafdeling cIerSÌ4,eepsbouwkunde Techn;sche Hogeschool,

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DATUM:

Washington D.C. 20034

UNSTEADY HYDRODYNAMIC LOADS ON A

TWO-DIMENSIONAL HYDROFOIL

by

John H. Pattison

This document haS been approved for

public release and sale; its distri-bution is unlimited.

DEPARTMENT OF HYDROEIECHANICS RESEARCH AND DEVELOPMENT REPORT

t.ab. y. ScheepsbouwkUn

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

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TABLE OF CONTENTS Page ABSTRACT i ADMINISTRATIVE INFORMATION i INTRODUCTION i HYDROFOIL ÎHEÓRY 3 TEST PARAMETERS 3 UNSTEADY LOADS 6 7

ADDEDMASS

STEADY LOADS 8 EXPERIMENT EQUIPMENT 9. TEST PROCEDURE 11 CALIBRATIONS 13

Static Load Calibrations' ...-

13

Inertia Loads 16 Spanwise Velocity-Distribution 16 Other Calibrations 17 RESULTS 17 ESTIMATION OF ERRORS 18 DISCUSSION 18 UNSTEADY LOADS .18 Free-Surface Effects - 20 Depth Effeçts 20

Comparison of Hydrofoil and Airföil Data V 32 ADDED MASS

STEADY LOADS 44

Free Surfaçe Effects 47

Depth-EffeCtS .49

OTHER REPRESENTATIONS OF DATA 49

EXPERIMENTAL LIMITATIONS ...57

SUMMARY AND, RECOMMENDATIONS . 5.

t

ACINOWLEDGMENTS V

62

APPENDIX A - TABLES OF STEADY AND UNSTEADY LOAD DATA 63 APPENDIX B - GRAPHS OF STEADY AND UNSTEADY LOAD DATA .

...

i 83

REFERENCES V V . . .

LIST OF FIGURES

Page

- Hydrofoil NotatIons and Oscillatory Motions 4

- Pitch-Heave Oscillator _io

-. Static Calibration Scheme ₁₅

- Unsteady Lift Coefficients and Phase Angles iñ Pure

Heave as Functions of Reduced Frequency 21

- Unsteady Moment Coefficients and Phase Angles in Pure

Heave as Functions of Reduced Frequency 22

- Unsteady Drag Coefficients and Phase Angles in Pure

Heave as Functions of Reduced Frequency 23

- Unsteady Lift Coefficients and Phase Angles in Pure

Pitch as Functions of Reduced Frequency ₂₃

- Unsteady Moment Coefficients and Phase Angles in Pure

Pitch as Functions of Reduced Frequency ₂₄

- Unsteady Drag Coefficients and Phase Angles in Pure

Pitch as Functions of Reduced Frequency ₂₅

Figure 10 _{Unsteady Lift Coefficients and Phase Angles in Pure}

Heave as Functions of Depth to Semichord Ratio 26

Figure 11 _{- Unsteady Moment Coefficients and Phase Angles in Pure}

Heave as Functions of Depth to Semichord atio 27

Figure 12 _{- Unsteady Drag Coefficients and Phase Angles in Pure}

Heave as Functions of Depth to Semichord RatiO ₂₈

Figure 13 _{- Unsteady Lift Coefficients and Phase Angles in Pure}

Pitch as Functions of Depth to Semichord RatIo 29

Figure 14 _{- Unsteady Moment Coefficients and Phase Angles in Pure}

Pitch as Functions of Dépth to Seinjchörd _{Ratio 30} Figure 15 _{- Unsteady Drag Coefficients and Phase Angles in Pure}

Pitch as Functions of Depth to _{Semichord Ratio 31}

Figure 16 - Unsteady Lift Coefficients and. Phase Angles in Pure Héave as Functions of Reduced Frequency, _Comparison

with Airfoil Data

33

Figure 17 - Unsteady Moment Coefficients and Phase Angles in Pure

Heave as'Fnctions of Reduced Frequency, _Comparison

with Airfoii Data

34

Figure 18 _{- Unsteady Drag Coefficients and Phase Angles in Pure}

Heave as Functions of Reduced Frequency ₃₅

Figure 19 - Unsteady Lift Çoefficiexits and Phase Angles in Pure Pitch as Functions of Reduced Frequency, _COmparison

with Airfoil Data

111 36 Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9

Page

- Unsteady Montent Coefficients and Phase Angles in Pure Pitch as Functions of Reduced Frequency, Comparison

with Airfoil Data 37

Unsteady Drag Coefficients and Phase Anglês in Pure

Pitch as Functions of Reduced Frequency 38

- Theoretical Pressure Distribution for the

Airfoil Model 40

- Theoretical Pressure Distribution for the

Hydrofoil Model 40

- Added-Mass Lift Coefficients and Phase Angles in Pure

Heave as Functions of Zero-Speed Reduced.

Frequency . -. 41

- Added-Mass Moment Coefficients and Phase Angles

in-Pure Heave as Functions of Zero-Speed Reduced

Frequency

- Added-Mass Drag Coefficients and Phase Angles in Pure Heave as Functions of Zero-Speed Reduced

Freqüency . 42

- Added-Mass Lift Coefficients ap4 Pháse. Angles in Pure

Pitch as Functions of Zero-Speed Reduced

Frequeflcy

- Added-Mass Moment Coefficients and Phase Angles in Pure

Pitch as Functions of Zero-Speed Reduced

Frequency .

. 43

- Added-Mass Drag Coefficients and Phase Angles in Pure

Pitch as Functions of Zero-Speed Reduced

Frequency . . . 43

- Unsteady Lift Coefficients and Pha.se Angles in Pure

Heave as Functions of. w2 b/g . 45

- Unsteady Moment Coefficients and Phase Angles in Pure

Heave as Functions of w b/g 45

- Unsteady Lift Coefficients and Phase Angles in Pure

Pitch as Functions of w2 b/g 46

- Unsteady Moment Coefficients and Phase Angles in Pure

Pitch as Functions of w2 b/g 46

- Steady Lift Coefficients as Fúnctions of Mean Angle

of Attak

. . 48

- Steady Moment Coefficients as Funçtions of Mean Angle

ofAttack

48

- Steady Drag Coefficients as Functions of Mean Angle of

Attack r 50 41 42 Figure 20 Figure 21 Figure 22 Figure Figure 24 Figure 25 Figure 26 Figure 27 Figure 28 Figure 29 Figure 30 Figure 31 Figure 32 Figure 33 Figure 34 Figure 35 Figure 36

Figuré 37 - Steady Lift Slopes as Functions of Froude Numbér 50

Figure 38 - Steady Moment Slopes as Funcion Of Froude

Number ₅₀

Figure 39 - Steady Lift Slopes as Functions of Depth-to-Semichord

Ratio - 51

Figure 40 - Steady Moment Slopes as Functions of

Depth-to-Semichord Ratio 51

Figure 4F - Unstedy Gai-n Factors in Heave Motionas Functions of

Reduced Frequency 53

Figure 42 - Unsteady Gain Factors in Pich Motion as Functions of

ReducedFrequency

...54

Figure 43 - Unstady Lift COefficient, Complex FOrni, According to

Klose and Acosta ₅₆

Figure 44 - Unsteady Minus Added-Mass Lift Coefficients and Phase Angles in Pure Heave as Functions of Reduced

Frequency ₅₈

Figure 45 - Unsteady Minus Added-Mass Moment Coefficients and Phase

Angles in Pure Heaye as Functions of Reduced.

Fre4uency 58

Figure 46 - Unsteady Minus Added-Mass Lift Coefficients and Phase Angles in Pure Pitch as Funçtions of Reduced

Frequency.

59

Figure 47 - Unsteady Minus Added-Mass Moment Coefficients and Phase Angles ir Pure Heave as Functions of Reducéd

Frequency ₅₉

Figure 48 - Steady Lift Slopes as Functions of Froude Number, Error

Limits on Data ₆₀

Figure 49 - unsteady Lift Coefficients in Pure. Heave as Funçtions of Red.ice4 Frequency, Error Limits on pata 60

Figure Bl - Unsteidy Lift Coefficients and Phase Angles on an NACA 16-209 Hydrofoil in Pure Heave Motion as Functions of

Reduced Frequency 84

Figure B2 - Unsteady Moment Coefficients and Phase Angles on an NACA 16-209 Hydrofoil in Pure Heave Motion as Functions of

Réduced Fxequency ₈₇

Figure B3 -lJnsteady Drag Coefficients and Phase Angles on an NACA 16-209 Hydrofoil in Pure Heave Motion as Functions of Reduced Frequençy

. 90

Figure M - Unsteady Lift Coefficients and Phase Angles on an NACA

16-209 HydrOfoil in Pure Pitch Motion as unctions of

Reducéd Frequency . 92

Page

Figure

B5 -

Unsteady Moment Coefficients and Phase Angles on an

NACA

l6-2Q9

Hy4rofoil in Pure Pitch Motion as

Functions of Reduced Frequency 96

Figure

B6 -

Unsteady Drag Coefficients and Phase Angles ön an NACA

16-209

Hydrofoil in Pure Pitch Motion as Functions of

Redüced Frequency 100

Figure

B7 -

Unsteady Lift Coéfficients and Phase Angles on an NACA

16-209

Hydrofoil in Pure Heave Mötion as Functions of

Depth-to-Semichard Ratio 102

Figure

B8 -

Unsteady Moment Coefficients and Phase Angles on an

NACA

16-209

Hydrofoil in Pure Heave Motion as Functions

of Depth-to-Semichord Ratio 105

Figure

B9 -

Unsteady Drag, Coefficients and Phase Angles-on an NACA

16-209

Hydrofoil in Pure Heave Motion as Functions of

Depth-to-Semichord Ratio

108

Figure

BiO -

Unsteady Lift Coefficients and Phase Angles on an NACA

16-209

Hydrofoil in Pure Pitch Motion as Functions of

Depth-to-Semichord Ratio . 110

Figure

Bil -

Unsteady Moment Coefficients and Phasé Angles on an

NACA

16-209

Hydrofoil in Purè Pitch Motion as Functions

of Depth-to-Semichord Ratio 113

Figure

B12 -

Unsteady Drag Coefficients and Phase Angles on an NACA

16-209

Hydrofoil in Pure Pitch Motion as Functions of

Depth-to-Semichord Ratio

116

\Figure

B13 -

Steady Lift Coefficients as FunctiOns of Steady Angle

of Attack 119

Figure

B14 -

Steady Moment Coefficients as Functions of Steady Angle

of Attack

-123

Figure

BiS -

Steady Drag Coefficients as Functions of Steady Angle

of

Attack

127

Figure

B16 -

Steady Lift Slopes as Functions of Depth-to-Semichord

Ratio 130

Figure

B17 -

Steady Moment Slopes as Functions of Depth-to-Seinichord

Ratio...131

Figure

B18 -

Steady Lift Slopes as Functions of Froude Number 131 Figure

B19 -

Steady Moment Slopes as Functions of Froude

LIST OF TABLES

Page Table 1 - Unsteady Hydrodynamic Load Coefficients ₄

Table 2 - Unsteady Test Conditions ₁₂

Table _{3 - Reduced Frequencies at Unsteady Test} Conditions

12

Table 4 - Steady Test Conditions ₁₂

Table 5 - Uncertainties in Experimental Results

19

Table _{Al - Experimental Unsteady Load Coefficients and Phase} Angles in Pure Heave Motion on an NACA 16-209

Hydrofoil ₆₄

Table _{A2 - Experimental Unsteady Load Coefficients and Phase} Angles in Pure Pitch Motion on an NACA 16-209 Hydrofoil

66

Table _{A3 - Experimental Added-Mass Load Coefficients and Phase} Angles in Pure Heave Motion on an NACA 16-209

Hydrofoil

68

Table _{A4 - Experimental Added-Mass Load Coefficients and Phase} Angles in Pure Pitch Motion on an NACA 16-209

Hydrofoil

69

Table _{A5 - Experimental Unsteady Load Coefficients and Phase}

Angles in Pure Heave Motion on an NACA 16-209 Hydrofoil, Translated to a New Axis for Comparison with Airfoil

Data

70

Table _{A6 - Experimental Unsteady Load Coefficients and Phase}

Angles in Pure Pitch Mtion _{on an NACA 16-209}

Hydrofoil,.Translated to a New Axis for Comparison

with Airfoil Data ₇₀

Table _{A7 - Experimental Unsteady Load Coefficients and Phase} Angles in Pure Heave Motion on an NACA 0012

Airfoil

- 71

Table _{A8 - Experimental Unsteady Load Coefficients and Phase} Angles in Pure Pitch Mòtion on an NACA 0012

Airfoil

71

Table _{A9 - Experimental Steady Load Coefficients}

72

Table AlO - Experimental Steady Lift and Moment Slopes and Intercepts

74

Table All _{- Theoretical Unsteady Load Coefficients and Phase} Angles

75

Table Al2 - Theoretical Added-Mass Load Coefficients and Phase Angles

80

Table A13 - Theoretical Steady Load Coefficients

Table A14 - Theoretical Steady Lift and Moment Slopes and

Intercepts

Page

81

NOTATION

a Location of pivot axis, positive downstream from midchord in semispan

length

b Semichord length of hydrofoil = 12 in.

C Amplitude of load coefficient

c Complex unsteady load coefficient = Ce

D Amplitude of drag load in lb/in, span

d Complex unsteady drag load, lb/in, span = De' or De'8

e Exponential

F Froude number based on semichord = U/J/

Fc Froude number based on chord =

FH Froude number based on depth of submergence = uij/'iÎ.

f Frequency of oscillatory motion in Hertz

G Unsteady gain factor

g Acceleration due to gravity, 386 in./sec2

H Mean depth of submergence at pivot axis in inches

h Complex heave displacèment in inches, =

het

h0 Amplitude of heave motion in inches

YT

Unit vector normal to free stream

k Reduced frequency = wb/U

L Amplitude of lift load in lb/in, span

iq iO

2. Complex unsteady lift load, lb/in, span = Le or Le

M Amplitude of pitching moment in in.-lb/in. span

iq iO

m Complex unsteady pitching moment in in.-lb/in. span = Me or Me

Unit vector normal to fôil surface

p Local pressure in pounds per square. inch

p,, Ambient pressure at the free surface 14.7 psia

q Interaction coefficient from calibrations, load/voltage Vector from pivot axis to element on surface of foil, in.

S Amplitude of added mass load coefficient

s Complex added mass coefficient = SeiO

s Zero-speed reduced frequency = w2b/g t Time in seconds

u Local horizonta-1 velocity relative to U in inches per second

V Measured voltages for load ceUs in volts

ci Complex pitch displacement, rad

Amplitude of pitch displacement in radians Mean angle of attack in radians

O Phase angle by whiçh added mass load leads displacement in radians

ir 3.14159

p Mass density Of fluid = O9348 x lO lb-sec/in.4 for water

Phase angle by which unsteady

load

leads displacement, radians

w Angular frequency of oscillatory motion in radians per second 2irf

SUBSCRIPTS AND SUPERSCRIPTS A Added mass (zero velocity)

D Drag

i Imaginary part of complex number

L

Lift

M Moment about pivot axis

P

Pitch

motion about pivot axis

p Pressure

r Real part of complex number

S Steady flow

(w = O)

T Heave motion

U Unsteady motiOn

ci Kiose-Acosta coefficient (Reference 29)

*

References are listed on page 133 ABSTRACT

Unsteady hydrodynamic lift, moment, and drag on a

two-dimensional NACA 16-209 hydrofoil are determined experi-mentally as functions of depth of submergence, forward speed, mean angle of attack., and frequency of oscillation in pure heave and pure pitch motions. Various theoretical and experimental results are compared with, the experimental

steady and unsteady load coefficients on the hydrofoil model.

The comparisons include both finite and zero forward speed

effects. Added-mass coefficients are obtained from the zero forward speed case. Theoretical and experimental results do

not agree because of limitations in both the theories and.

experiments.

ADMINISTRATIVE INFORMATION

This work was authorized under the Hydrofoil Accelerated Research

Program by Bureau of Ships letters S-FOil 02 01 Seri'al 420-5S of 3 Octobei 1962 and Serial 341B-l25 of 13 August 1963 and supported under Project SS600-000. The continuing effort was funded under S4606, Task 1703.

INTRODUCTI ON

When the U.S. Navy began a program for acquiring oceangoing

hydro-foil craft, very little was known about the nature of the forces, vi-brations, and stability of fully submerged hydrofoils operating at high

speed in a seaway. To study these problems, the 'Bureau of Ships (now the

Naval Ship Systems Command-NAVSHIPS) initiated the Hydrofoil Accelerated

Research Program (HARP). As a part of this overall program, the David

Taylor Model Basin (now the Naval Ship Research and Development Center)

undertook a study to determine the unsteady loads on two-dimensional

hydro-foils oriented horizontally beneath a free surface.

Much theoretical and experimental work has been done to study

*.

unsteady loads on airfoils; see Refereñces 1 through 15. Of the two ex-perimental studies referred to here,3'6 the measurements of Halfian appear' to be more inclusive. However, these rêsults ate not applicable to the case of hydrofoils for the following reascns:

Water is a much denser fluid than air, which implies that added-mass effects are much more important to oscillating hydrofoils than to

oscillating airfoils.

Cavitation can occur on hydrofoils, producing a departure from the normal flow pattern about an airfoil.

Hydrofoils operate near a free, surface. As a result, there is a free-surfàce êffect which depends on Froude number based on a hydiofoil dimension and depth of submergence.

Hydrofoils operate in a higher reduced frequency range than do airfoils.

The theoretical studies of von Karinan and Sears,2 Woods,4

i . . 8-11,13,14 15 . 12

Theodorsen, Giesing, Smith, and then and Wirtz may be ex-tended to include the effects of the free surface, added mass, and

cavi-16,17

tation in a manner similar to the hydrofoil:studies of Kaplan, Crimi

and Statler,18 Grace and Statler,121 Widnall,22 and Wldnall and Landahl.23 It is only since the initiation of HARP thàtork on the unsteady

loads on hydrofoils has started. Unsteady loads have been measured on 24-26

three-dimensional, cantilevered hydrofoils by Rans leben, on a strut-pod-foil configuration by O'Neill,27 and on twö.dimensional hydrofoils in

28 29

heave motion by Klose and by Klose and Acosta. To determine unsteady waké effects, flow visualization studies were done by Magnuson.3°

The unsteady loads and phase relationships to unsteady motion are related to the stability of oscillating hydrofoils and, as such, are im-portant parameters in the determination of hydroelastic effects, such as

flutter. These relationships are discussed by Jewell,31 Abramsofl and

32 33 . . . 34

Langner, Chu, and Cieslowski and Pattison.

Several survey reports and analytical studies on unsteady loads and hydroelasticity have been written, including those by Isay,35 Yegorov,

36 . 37

and Sokolov, and Abramson, Chu, and Irick.

Steady loads are also included in this investigation as a zero frequency case of the unsteady loads. Many theoretical and experimental investigations have been made of hydrofoils in steady flows. In 1965, the author summarized several of these studies in a report which concentrated

- 38 . . .

.9.

on flat plate analyses. A later analysis by Giesing and Smith included profile effects on the loads on a hydrofoil beneath a free .surface.

This paper presents unsteady load data obtained in the NSRDC high-speed towing basin. A hydrofoil with a 2-ft chord and 6-ft. span was mounted

between large end plates and towed at speeds up to 30 knots. An oscillator

was used to force the hydrofoil in pure heave and pure pitch sinusoidal

motion. Measurements were made of these motions and of the forces and moments they caused. In addition, comparisons are made between hydrofoil and airfoil unsteady load data.

Details of the experiment aré described in Reference 34. The

results presented there were incOrrect, however, because interactions

be-tween the various components of the dynainometer were not fully accounted for in the original calibrations óf the hydrofoil model. The interactions were found to be smal,l when pure forces wereapplied in the original

cali-brations. However, in later calibrations when combinations of lift,

moment, and drag were applied, the interactions were found to be large and

nonlinear. The data have now been corrected and all the data obtained to date are presented in this report.

HYDROFOIL THEORY

TEST PARAIvIETERS

In the experiments described below, the hydrofoil model was forced

to oscillate in pure heave or pure pitch. These motions are defined by

the following two equations:

h =h

et

(1)

iwt

e (2)

o

In response to these motions, unsteady lifts, moments, and drags act on the hydrofoil. The positive directions of the motions and loads

are shown in Figure 1. The unsteady loads are also defined ïn coefficient

form in Table 1. A phase angle or O is associated with each of the unsteady loads. It is the angle by which the unsteady load leads the motion. For example, the lift coefficient in pure heave is the complex

LEADING EDGE PIVOT AXIS MID C HO RD

-

ob PITCH FREE SURFACE

Figure 1 - Hydrofoil Notations and Oscillatory Motjons

TABLE 1

- Unsteady Hydrodynainic Load Coefficients

H E AV E

T RAILING

EDGE

Unsteady Loads, U > O Addèd-Mass Loads, u = O

Pure I-leave Pure Pitch Pure Heave Pure Pitch

Lift _CLI Lut CLP Lup SLT LAT SLP -AP - 2 2

ipU k h0 - ipU bk2 2 - upb w2 2h0 npb

32

Moment C M ur CMP M - SMT M SMp M AP

ipUbk h0 upU b k2 2 1/2uïpb w h02 1/2npb w2

Drag _CDT Duî -CDP 0UP 5DT 0AT --SDP -DAP - 2 2

LT = CLT

eLT

(3)

At zero speed, the added-mass lift coefficient is the complex quantity

SLT =SLT eLT

(4)

where C or S are absolute values of the coefficients. In this

investi-gation, the utisteady load coefficients and phase angles were determined as

fùnctions of frequency of oscillation; epth, mean angle of attack, and

speed.

i

For a flat plate in an infinite Uuid, Theodorsen found that the

lift and moment coefficients depended on a reduced frequency defined by

k = wb/U (5)

In the case of finite depth, it is convenient to study the unsteady loads in terms of this reduced frequency along with additional effects due to

depth H, finite mean angle of attack and finite Froüde number

For the case of zero speed, however, the reduced frequency k becomes

in-finite and is no longer useful. Instead, a zero-speed reduced frequency defined by

- 2

s

= w

b/g

is found to be a convenient parameter to show the dependence of apparênt

mass loads on angular frequencies of oscillation w.. At finite forward

speed, the zero-speed reduced frequency is related to reduced frequency k and Froude number F by the expression

For the case of zero frequency, the steady loads _{are obtained in} coefficient form as shown below:

Steady Load Coefficients

L Lift: _CLS -PU b M Moment: C MS 2pU2 b2 D5 Drag: _CDS - pU2 b

Since steady lift and moment coefficients are foind to be nearly linear

with angle of attack, it is convenient to study t1e steady lift and moment

slopes (dCLs/daS and dCMS/doS) as functions of the Froude number or the

depth to seniichord ratio H/b.

Theoretical predictions of the unsteady, added-mass, and steady

loads on hydrofoils use airfoil theory as a basis. Classical airfbil

theory should be valid fOr the case of fully wetted flow past a hydrofoil

which is at deep depths. This theory includes work on unsteady loads by

Theodorsen.1 To extend the Theodorsen work, a nUmber of investigators

have attempted to introduce free-surface effects. _{A few analyses for a}

thin hydrofoil or a flat plate have been programmed satisfactorily for a

high-speed computer. For this investigation, the following programs were used.:

A program developed by Widnall22'23 for the unsteady loads on a

flat plate beneath a free surface.

The Douglas-Neumann program developed by Giesing and Smith9 for

steady loads on a hydrofoil near any boundary.

UNSTEADY LOADS

Langan and Coder7 used the Theodorsen theory to compute. the

un-steady lift and moment coefficients for pure heave and pure pitch motion

for infinite depth. _{In parallel investigations, Crimi, Grace, and}

18-21 - 22,23 .

high-speed digital computers to find .the unsteady lifts and moments. How-ever, only the Widnall program has been run successfully on the IBM 7090

computer at NSRDC for the actual experimental conditions. As a result, the Widnall theory is compared with the Theodorsen theory and the

experi-- 8-11,13,15

mental results. Results from the studies by Giesing could not

be evaluated because the computer programs were not available for use.

ADDED MASS

In unsteady airfoil theory, von Karman and Sears2 suggest that the

unsteady loads can be separated into the following three contributions: The quasi-steady contribution due to the circulation about the

foil alone.

The wake contribution due to vortices shed into the wake to satisfy the Kutta condition at the trailing edge of the foil. The added-mass contribution due to all noncirculatory terms.

In air, this contribution is often negligible. However, in water the added-mass effects, which are discussed below, cannot be ignored.

In the course of oscillating the hydrofoil, the surrounding fluid

must be accelerated. This has the effect of adding mass and moment of

inertia to the foil. Grace and Statler2° suggest that the magnitude of the added mass and the added moment of inertia is likely to be affected by the proximity of other surfaces and by waves radiated on the free surface

of the fluid. Furthermore, there is some question about including wake

effects in the added-mass contribution at finite forward speed. In the

airfoil case, von Karman and Sears argue that symmetry eliminates the

noncirculatory wake effects. However, it is quite likely that the

presence of the free surface may upset this symmetry; hence, wake effects

should be included.

Although the added-mass effects of a hydrofoil cannot be separated

from other forces and moments at finite forward speed during experimental studies, these effects can be readily obtained at zero forward speed. In

this case, there are no circulatory effects and the inertial forces and

moments on the foil can be readily measured in air and eliminated. The remaining terms in the equations of motion are the mass and added-moment terms.

Bispiinghoff, Ashley, and Halfinan39 use the Theodorsen theory to find the apparent-mass lift and momeflt on a flat plate In an infinite

fluid. For an experimental pivot axis, a = 0.167, the expressions for

mass loads yield the following limiting values for the apparent-mass coefficients, defined in Table 1, and phase angles:

SLT = 1 SLP = 0.167

OLT_It

S = 0.333 S = 0.306

For infinite depths, then, these coefficients are independent of frequency. In the finite depth case described by Grace and Statler,2°

wave-making effects cause the apparent-mass coefficients to depend on frequency as well as depth, especially near a critical zero-speed reduced frequency

of = 1.

In all the analyses described above, the loads and motios are

determined about various axes. For direct comparison with experimental

results, complex algebra is used to refer all the unsteady loads and

motions to the same axis. In most cases, this is the experimental pivot

a.xis a = -0.167. However, the axis a = -0.250 was chosen for comparing

unsteady hydrofoil and airfoil loads.

STEADY LOADS

Giesing and Smith9 modified the Douglas Neumann program to compute pressures in potential flow about an arbitrary profile hydrofoil in the

presence of a free surface and any other boundary. This provided a gQOd

simulation for tests in a basin of finite depth. Actually, on the basis öf this theory., very little bottorn effect would be expected for a basin as deep as the .NSRDC high-speed basin.

The modified Douglas program was run on the IBM 7090 digital computer at NSRDC to .ôbtain the pressure distributions at all of the steady-flow test conditions, which are discussed below. The steady lift and moment coefficients were then obtained by integrating functions of the

CLS = -. - C n j dS p 9 (9) C =

J

C ( x dS (10)

where C is.the pressure coefficient represented by

p - p0, +

pgy

P

_1/2pU

2

In the above equations,

g is acceleration due to gravity,

is a unit vector normal to the direction of motion, is a unit vector normal to the surface of the foil,

p is the local pressure on the surface element dS,

P0, is the ambient pressure at the free surface,

is a vector from the pivot axis to the surface element dS,

y is the location of the surface element dS with respect to the

undisturbed free surface (negative downward from the free sur-face), and

p is the density of water.

EXPERIMENT

EQUIPMENT

The theories that have been discussed apply to an infinite, thin hydrofoil oriented horizontally beneath a smooth free surface. The test equipment was designed to approximate this condition as nearly as possible. The hydrofoil model (2-ft chord and 6-ft span) was mounted between two

large, faired struts to simulate two-dimensional flow. The struts enclose an oscillator mechanism which can drive the model in prescribed sinusoidal motions in pitch or heave and a mechanism which can vary the mean angle of

I;::::::::JçT____

HYDRAULIC POWER SUPPLY

-.

BASE.

TYPICAL

WATERLINE ST R UT HYDROFOIL MODEL

control and hydraulic power for the oscillator mechanism. Depth was varied

by mounting the system on the towing carriage in several vertical locations

and by varying the water level in the high-speed basin.

To simulate a thin hydrofoil, an NACA 16-209 cross section was

chosen for the hydrofoil model. To eliminate end effects, a floating

section at the center span was instrumented to measure accelerations, lift, drag, and moment about an axis 2 in. ahead of the midchord (ab = -2 in., hence a = -0.167).

A digital data acquisition system (DIDAS) was used on the carriage to convert measurements to digital values and record them on magnetic tape

for later analysis on an IBM 7090 digital computer. The experimental equipment is described in more detail in References 34, 40, and 41.

TEST PROCEDURE

The pitch-heave oscillator and foil were mounted in the ay of

Carriage 5 in the high-speed towing basin. For unsteady tests, the foi.1

was set at an average angle of O or 5 deg and towed over a range of five

depths and fourforward speeds. At each test condition, the foil was

oscillated first in heave and then in pitch over a range of frequencies. The test conditions are summarized in Tables 2 and 3. A complete set of

conditions was tested at O deg, but fewer frequencies and speeds were used at 5 deg. Measurements made at each of the test conditions included

amplitude of motion in pitch and heave, unsteady lift, moment, and drag.

The hydrofoil model was towed without being oscillated to obtain steady loads at six depths, seven forward speeds, and five mean angles of

attack. These conditions are summarized in Table 4; not all combinations of H, U, and were tested. Measurements made at each of the test con-ditions included steady lift, moment, and drag. Data for both the steady

and the unsteady tests were reçorded on DIDAS for later analysis on the IBM 7090 digital computer.

*

Hydrofoils with similar cross sections are used on U.S. Navy hydrofoil

TABLE 2

Unsteady Test Conditions

(b = 12 in., ab

-2 in.,

ci.

= O, and 5 deg)

TABLE 3

Reduced' Frequencies. at Unsteady

Test Conditions

TABLE 4

Steady Test Conditions

Freq., f Hertz Reduced Frequency k U = O knots 10 20 30 'F 0 2.98 5.'95 8.93 2.5 1.23 7.67 0.372 0.931 0.186 0.466 0.124 0.310 5 30.7 1.86 0.931 0.620 8 78.6 2.98 1.49 0.993 14 241.0 5.21 2.60 1.74 15 276.0 5.38 2.79 1.86 20. 49.1.0 7.44 372 2.46 Foil Depth H H -Speed knots Oscillation U Freq Hz Amplitude Heave Ptch 6.88 0.573 0 1* '0.2 10 ' 2.5 .0.2 0.5 20 5 0.1 0.5 8 0.05 0.25 14 0.03 0.25 20 0.03 0.25 9.75 0.812 0 10 i 5 0.2 0.1 1 .0.5 20 14 0.03 0.25 20 0.02 0.25 12.88 1.013 0 1* 0.2 1 23.88 1.990 10 2.5 0.2 0.5 47.25 3.937 20 5 OJ 0.5 30 8 0.05 0.25 14** 0.03 0.25 20 0.03 0.25 *

Runs with l-Hz frequency were restricted to

o and 10 knots in nios,t cases.

**

Frequency was 15 Hz for 23.88-in, depth

(heave and pitch) and 47.25-in, depth (heave only).

Foil Depth H ifl H ' Speed U knots Froude tlumber F Attack Ang'l.e deg 6.88 0,573 5 1.49 -2 9.75 0.813 7 2.09 O 10.75 0.896 9 2.68 2.5 12.88. 1.073 10 2.98 5 23.88 1.990 15 4.47 7.5 47.25 3,937 20' ' 5.96 30 8.94

CAL I BRAT IONS

The hydrofoil model was statically calibrated by _{applying known}

loads and measuring corresponding voltage outputs. _{In addition, the model}

was oscillated in air to determine loads resulting from the inertia _{of the}

measuring system, so that these loads could be eliminated _{from measurements}

of unsteady load in water. _{Velocity surveys were done to determine the}

flow velocity over the model as _{a function of towing carriage speed, the} angle of attack indicator was calibrated,, and the accelerometers in the hydrofoil model were calibrated.

Static Load Calibrations

For the static calibrations, known lift, _{moment, and drag loads}

were applied to obtain loads as funçtions of measured _{lift, moment, and}

drag voltages. _{The loads were applied one at a time before the tests.}

The voltage output from each load-measuring _{element was nearly linear with}

load applied. _{Moreover, this voltage was much larger when the}

correspond-ing load was applied than when the other _{loads were applied. As a result,}

it was assumed that the loads could be found from the voltages by using the following linear matrix equation

D

cMM

In the first matrix on the,right side _{of Equation (12), the primary}

multi-pliers are the diagonal elements and the _{interaction coefficients}

are the off-diagonal elements. The matrix elements _{were obtained by linearizing} the calibration results. _{Unsteady loads reported in Reference}

34 were determined in this manner.

Because of the large scatter in _{the results, of Reference 34,}

a

more careful examination was made of the _{calibration method. This time,}

combinations of pure loads were applied, resulting in interactions _which

were large and highly nonlinear. _{As a result, complete calibration}

tables

of lift, moment, and drag

as functions of combinations of lift, moment, VL

and drag voltages were computed and are discussed below. The tables were used to correct all the data for interaction effects. Correct loads were found from measured voltages by interpolation within the tables.

A special sequence of measurement, interpolation, and averaging was necessary to obtain the calibration tables needed. The steps are summarized as follows:

Lift, moment, and drag voltages wère measured and recorded for several cOmbinations of applied lift, moment, and drag load,

as shown in Figure 3 where the darkened squares indicate data points.

The reminng unfilled squares shown in Figure 3 were filled

in by linear interpolation and averaging. In most cases, the linear interpolations were made in more than one direction and then the results wêre average4. In all cases, however, original data were retained (darkened points n Figure 3) and not

averaged oüt.

Curves of constant lift, moment, and drag voltages were determined b' level. Thus, The values of intersections The tables of interpolation

of least sensitivity for each type of load, as shown below.

Load

Li ft

Moment

Drag

interpolation and plotted for each applied drag constant voltage surfaces were defined.

the loads were determined graphically at the of the constant voltage surfaces.

loads as functions of voltages were filled by based on least-squares fitting in the directions

Direction of Least Sensitivity

Least-squares fitting was necessary to smooth the slightly jagged function

caused by the graphical determinations. Adjustments were made for zero offset. Also, because of a diagonal bias, the tables could not be

con-veniently fitte4 into a cube of voltages but were divided into four

quadrants corresponding to positive and negative lift and moment. The Drag voltage and moment voltage

Drag voltage

s

-ss --sss

5-sss---f , / 5-sss---f 5-sss---f

- -

i. M

______________

_, 7 7 7 J

wm

Figure 3 - Static Calibration Scheme

15

DARKENED SQUARES

resulting errors in lift and iiioient caljbrations were estimated to be less than 5 percent. Since the drag dynaniómeter was particularly ensitiVe to lift and moment, the drag calibration was less accurate.

Inertia Loads

The hydrofoil model was oscillated in air over the complete range of frequencies in pitch an heave. Unsteady lift, moment, and drag voltages and motions were measured. The inertia loads were then determined from

the voltages by interpplaton within the tables describbd above. In order to analyze utisteady lads data, Fouriet analys$ was

applied to the raw data to obtain mean voltages and unsteady voltage co-éfficients. Then the steady and total unsteady loads were obtained from sets of three voltages by interpolation within the static calibration tables. The appropriate inertia loads were then subtracted from the total unsteady loa4s to obtain the unsteady hydrodynamic loads. The

inertia-cancelling system (described in Reference 39) Was iot intrneite4 at the

time of the tests.

Spanwise Velocity Distribution

Three attempts were made to measure the spanwisé velocity distri-bution over the hydrofoil by replacing the hydrofoil with a pitot tube

rake. It was suspected that the flow might be grossly affected by the blockage and wake of the large supporting struts. The results Of these

investigations were inconsistent for a variety of reasons.

In the first two survys, the pressure changes from the rake were

mechanically transmitted through tubing to measuring equipment ön the carriage. This equipment consisted of a seven-tube manometer board in one case and pressure traïisducers in the other. In the third survey, pressure gages were mounted within the rake itself. The speed range varied frOm 3 to 30 knots.

The three sets of data showed erratic velocity variations up to

15 percent of the carriage ve1oçty a midspan and larger vaÌiations to

the side. In addition, the variations at the test sectioti beçame smaller as the speed increased. This erratiç behavior may have resulted from

varying flow conditions as the speed increased. On the average, the

carriage velocity appeared to be the best value for the flow velocity at

the test section.

Other Calibrations

The angle of attack indicator and the frequency counter in the pitch-heave oscillator and the accelerometers in the hydrofoil model were

calibrated prior to tests. Each of these measurements was accurate to

within 1 percent.

RESULTS

The experimental measurements were reduced to steady (see page 6) and unsteady (see Table 1) load coefficients and phase angles between the

loads and motions.. These data are tabulated and illustrated in Appendixes

A and B for all the conditions tested. The graphs and tables also include comparable experimental data obtained by other investigators and theo-retical results obtained by computer programs based on the Theodorsen

analysis and its extensions.

Comparable experimental unsteady load results were obtained from

the Halfman wïnd-tunnel tests on an oscillating airfoil.3 In order. to

compae his results with the hydrodynainic tests, the hydrofoil results

were algebraically shifted to the measuring axis f the airfoil (a = -0.250)

and the airfoil results were reduced to coefficients defined in Table 1. Comparable theoretical results for steady and unsteady loads carne from four sources:

Results for an oscillating flat plate in an infinite fluid7

based on work by Theodorsen.'

Results for an oscillating flat plate operating at finite

Froude number beneath a free surface from work by Widnall.22'23 Her results

were limited to reduced frequencies of five or less by the numerical scheme

used in her computer programs.

Results for an oscillating flat plate beneath a free surface

operating at zero forward speed (added-mass loads) from work by _{Grace and}

Statler.2° Only results for two published depths were used herein since

their programs have never been successfully _{run on the IBM 7090 digital}

(4) Results for a steady NACA 16-209 hydrofoil beneath a free sur-face from work by Giesing and Smith.9

ESTIMATION OF ERRORS

A detailed error analysis of the experimental system w3s not

per-formed. However, all inputs, including the load dynamometers, the

accelerometers, and' the angle of attack indicator were carefully calibrated using the digital data acquisItion system.

Also,

changes in sensitivity were accounted for in the data processing programs.

Errors from the load calibrations and velocity uncertainties were

much greater than errors from any other source. These errors resulted in

the uncertainties shown in Table 5 for the load coefficients, Froude num-bers, and reduced frequencies at low and high speeds.

DISCUSSION

For the discussIon which follows, certain data were selected fröm

the complete results shown in Appendixes A and B to illustrate the effects

of self-generated waves, depth of submergence, and mean angle of attack on

the unsteady, added-mass, and steady loads on a two-dimensional hydrofoil.

Accordingly, it is çonvenient to present the discussion in separate

sections. In the discussion, data are coiipared with theoretical

pre-dictions and other experimental data. Finally, the effects of experimental limitations are discussed.

UNSTEADY LOADS

This discussion Is subdi.vided into the effects of self-generated waves and the effects of depth of submergence on the unsteady loads. Angle

of attack effects were noted to be very sma11. Some discrepancies were found between theoretical and experimental results. In an effort to ex-plain t1 discrepancies, another subsection compares the tesults with unsteady airfoil data.

TABLE 5

Un-certainties in Experimental Results

19 Type of Result Percent Uncertainty u Low Speed 10 knots High. Speed u > 15 knots

Steady lift and moment

slope

Unsteady lift and moment

coeffi ci ent

Semichord Froude number

Reduced frequency 28 8 11 12 14 8 4 5

Free-Surface Effects

The effects of the self-generated waves on the free surface can be

evaluated by examining the variation in unsteady loads and moment with Fròude number, defined in quation (6). The effects of Froudê numberat

single depth to semichord ratio H/b = 1.073 are illustrated in Figures 4 through 9. All these graphs are plotted as functions of the reduced fre-quency k, defined in Equation (5). Figures 4, 5, and 6 show the variation of lift, moment, and drag coefficients in pure heave motion for various

Frouçie numbers. igures 7, 8, and 9 show similar data for pure pitch motion. The lift and moment data are compared with the theoretical

pre-2223

dictions based on work by Widnall.

Böth theory aÑd experiment shOw little effect frÒm the presence of

the -free surface except for a. small but interesting effeçt in pitch results.

Each set of experimental coefficients in Figures 7, 8, and 9 appeared to

level off toward high reduced frequencies. The leveling off occurred

earlier at the- higher Froude -numbers.

Agreement between theory and experiment was better for the moment

coefficient (Figures 5 and 8) than fOr the unsteady lift coefficient

(Figures 4 and 7). No theory is available for the unsteady drag (Figures 6

and 9). Several reasons for the discrepancies between theory and experi-ment are discussed in a later section.

The data presented in Figures 5 throUgh 9 showed no definite trends

with changes of angle of attack from O to S deg. The theoretical analysis

is independent of the angle but applies to a fully wetted flow with a

Kutta condition at the trailing edge. At the 5-deg angle of attack -and

higher Froude numbeis, some cavitation occurred along the leading edge of

the hydrOfoil. Cavitation appeared to affect the unsteady drag shown in Figure 7 more than any of the other unsteady loads.

Depth Effects

The effects of foil submergence on the unsteady load and moment coefficients and phase angles-of foils oscillated in heave and pitch are

illustrated in Figures 10 through 15. These graphs are plotted as functions

of the depth-to-semichord ratio H/b for several reduced frequency ratios. The lift aíid momènt data are compared with the theoretical predictiois

2.0 CLT LT 3 2 0.2

04

06

08

I.0 21 K

Figure 4 - Unsteady Lift Coefficients and Phase Angles in Pure Heave as Functions of Reduced Frequency

(H/b 1.073, a = -0.167)

20

40

60

8.0 N KEY as

Fb

00 50

0

02.98

D

G G

8.935.95

(wNALL)

THEORY,

4

F=2.98

t.,

o

iI.---.

'4 r4 -6.0 4.0 LO 0.8 0.6 .6

08

I.

20

40

_{6.0 8.0}

2

0.8

0.6

CI11

0,4

0.2

0.I

4nT

o

O

0.3 0.4

K

Figure 5 - Unsteady Moment Coefficients and Phase Angles in Pure Heave as Functions of Reduced Frequency

(H/b = 1.073, a = -0.167)

a

,

002.98

KEY F

5.95

8.93

THEORY (WIDNALL)

-G

-

o

4 6 8

0.6 0.8

LP

0.203 04

5 2 o 10.3 _{04 06 08 .0} 20 40 60 80 K

Figire 6 - Unsteady Drag Coefficients and Phase Angles in

Pure Heave as Functions of Reduced Frequency

(H/b = 1.073, á = -0.167) 0.3 3 2 04 06 08 I. 23 o

Do

%. 04 06 08 1.0 - 2.0 40 6.0 80 K

Figure 7 - Unsteady Lift Coefficients and Phase Angles in Pure Pitch as -Functions of Reduced Frequency

(H/b = 1.073, a = -0.167) KEY OES Fb 0° 5° 0 0 2.98 P 0 5.95

G8.93

o

-th0

KEY as ò° 5° Ö Ø Fb 2.98 8.93

(waLL)

THEORY

-

-

- -- = ----

=z

-

An an Q 4 DT 3 06 08 I. 'J 20.0 0.0 6.0 4.0 2.0 C LP I_0 0.6 0.4 0.2 0. 2.0 1.0 CDI 0.6 0.4

6 4 2

0.8

Cp

0.6

0.4

0.2

O. i o P

i

_{0.3 0.4}

_{0.6 0.8}

_{I 4}

k

Figure 8 - Unsteady Moment Coefficients and Phase Aug lês in

Pure Pitch as Functions of Reduced Frequency

(H/b = 1.073, a = -0.167)

o

KEY 00 o

o

0

2.98

O 0 5,95

8.93

THEORY (WIDNALL) 3

0.3

04

2 o

o

u

KEY as Fb 00 50 O

Ø 2.98

D

Z5.95

8.93

¿.3

0.4

06

08 1.0

u

II

G

O

'p

25 K

Figure 9 - Unsteady Drag Coefficients and Phase Angles in Pure Pitch as Functions of Reduced Frequency

(H/b = 1.073, a = -0.167)

60

8.0

06

08

1.0

20

4.0 6.0 8.0 I0 o. i

20

40

CDP DP

4 CIT. 2 0.8 0.6 3.5 3.0 I .5 I .0

Q

o

. O I 2 3 4

Figure 10 - Unsteady Lift Coefficients and Phase Angles in Pure Heave as Functions of Depth to Seniichord Ratio

(F = 5.95, c 0, a = -0.167)

k=

26I

o

I .49

0.93

g.

0.47

0.19

-KEY

k=0.I9

- - EXPERIMENT

k

0

0.19

p

0.47

¡

O

3.72

- --

-p

o

-

0.47

--

-o

---

0.93

.

- - - -

- --THEORY

H/b

1.49

_{- -}

__-_--

- =

(WIt1ALL)

----1

-(THE000RSEN)

_____

2.5 *LT 2.0

4 2 0.2 O.' 2.0 1.5 4IIT 1.0 0.5 0.0 EXPERI tIENT

k

THEORY

H/b

- FINITE

(WIONALL) KEY (THE000RSEN) k = 0.19

o

. e 0.47

0

$ 0.93 a a

G

9 -9 1.49 9 2.61 6 3.72

o0

o

oo9

°

o

k

=0.19

0.47 0.93 1.49 2 61

-

3e"

o

0.19

p

0.47

û

0.93

G

1.49

9

2.61 2.79

O

3.72 O I 2 3 4

H/b

Figure 11 - Unsteady Moment Coefficients and Phase Angles in Pure Heave as Functions of Depth to Semichord Ratio

(F = 595 = 0, a = -0.167)

0.8

C111 0.6

sot

3.0

2.5

2.0

I .5

o

n

D

:

oe

o

D

u

C KEY EX PERI tIENT

k

o

0.19

a

0.47

o

0.93

G

1.49

9

2.61

2.79

O

3.72

Oô

00

o

ri

O

0

o

D 3

H/b

Figure 2 - Unsteady Drag Coefficients and Phase Angles in. Pure Heave as Functions of Depth to Semichord Ratio

(F = 5.95, = O, a =-O.167)

G

_o

4

0.8

0.6

0.4

40 20 2 2.0 I .5 ,LP 1.0 0.5 0.0 o

-f

s' 4

H/b

Figure 13 - Unsteady Lift Coefficients and Phase. Angles in Pure Pitch as Functions of Depth to Sernichord Ratio

(F = 5.95, c = 0, a = -0.167) 0.19 KEY

0.47

EXPERIMENT

k

o

0.19

P

0.47

0.93

1.49

9

2.61 2.79

Ó

3.72

D

o

D

D

093

O O

--o

1.49

o

-THEORY

H/b

FINITE

-

(WIDNALL) (THE000RSEN) Z.b 3.72

8'---..._

k

3.72 I 2.61

o

000

o

o

-o

0

o

1.49

C

C

0

O47-1;

2 3 Io 8 6 C1p 4 0.8 0.6 0.4

0.2 0.0 $MP -0.5 20 0. -1.0 I o i 7 I .49 = 3.72 2.61 0.19 0.9 2 3

s.----¿I

.49 A 0.47

H/b

Figure 14 _{Unsteady Moment Coefficients and Phase Angles}

in

Pure Pitch as Functions of Depth _{to Semichord Ratio}

(F = 5.95, = 0, a = -0.167)

k =0.19

9.

., 'b KEY

jEXPERIMENT

-o

0.19

. 1

0.47

0

_0.93

G

1.49

9

2.61 "9

2.79

O

_3.72

o

Î., ' ci

O

-093

,

o'

-- , 1'

r,

THEORY H/b 1.49 -. - 2.61 FINITE (WIONALL) (.THE000RSEN) 3.72 ...: CM P 2 0.8 0.6 0.4

20 0.1 0.5 0.2 1.0 0.0

o

o

D

C KEY

o

, EXPERIMENT

k

o

0.19

P

0.67

0

0.93

2.79

O

3.72

n

Ö

O

QQ

o

o

Q C

o

000

D

o

C C

D

O 2 3 4

H/b

Figure 15 - Unsteady Drag Coefficients and Phase Angles in Pure Pitch as Functions of Depth to Semichord Ratio

(F = 5.95, = 0, a = -0.167)

lo

8 6 4 2 CUP 0.8 0.6 0.4

Both theoretical and experimental results showed that depth had

only a slight effect. However the saine discrepancies between theory and

experiment shown in Figures 4 through 9 were present here also.

Comparison of HydrofOil änd AirfOil Data

In an effOrt to expkain the discrepancies between theory and

ex-periment demonstrated in thp above subsections, compariso1s are made with

unsteady airfoil data taken from Halfman.3 Because the loads and motions

were taken about. the axis a = -0.2S0 in the airfoil studies, the hydrofoil

motions and loads were transformed to the same axis. In the following

equations, subscript 1 refers to measurement about the original hydrofoil axis a1 = -0.167, and subscript 2 refers to measurements made about the

new axis a2 = -0.250. The transfOrmations for heave motions are

h2 = h1

CLT = CLT

- a1) CLT _{+ CMT}

CDT =

The transformations fOr pitth motons are:

= - (a2 - a

cLP

-

(a - a1) CLT1

_{+ CLP}

(a2 - a1)2 cLT1 + (a2 - a1) _cLP

cDP - a1) CDT1 + CDP (20)

Figures 16 through 21 show the load coefficients and phase angles about a = -0.250 as functiops of reduced frequency fOr the most deeply

6 CLI 40 20 IO 8 4 2 0.8 0.6 3

\

KEY EXPERIMENT MODEL

_ReXIO6

AIRFOIL HYDROFOIL

0

0.7 0.8 1.0 2.8

bfF2.98

THEORY

F8.93

CURVE

H/b

INFINITE (THE3-DORSEN) 3.94 (WIDNALL)

o

-O

Do

D u

.

F8.93

--

s

*

F2.98

02

04

06 Ö8

I 2 4

68

K

Figure 16 - Unsteady Lift Coefficients and Phase Angles in Pure Heave as Functions of Reduced _{Frequency, Comparison}

with Airfoil Data

(a = -0.250, 5=0)

I LT

MT 0.2

0004

KEY EXPERIMENT MODEL

_ReXIO6

AIRFOIL HYDROFOIL

0

0.7 0.8 1.0 2.8 5.6 8.4 F 2.98

F8.931

THEORY CURVE

_H/b

INFINITE (THEO-DO R SE N

3.94

(IJIDNALL

-o

o

F8.93

.,

F= 2.98

00

Ö.I

02

04

06081

4 K

Figure 17 - Unsteady Moment Coefficients and Phase Angles in Pure Heave as Functions of Reduced Frequency, Comparison

with Airfoil Data

(a = -0.250, a= 0)

Io 8 6 4 2 CMI 0.8 0.6 0.4

DT

2

K

Figure 18 - Unsteady Drag Coefficients and Phase Angles in Pure Heave as Functions of Reduced Frequency

(a = -0.250, a = 0) 35

o

KEY SYMBOL Re X IO O D

0

8.4

o

D

.

oo,

D D

02

0.4

06081

2 4 6 3

o

o

D

u

D 2

o

o

D CDT 0.8 0.6 0.4

LP 200 20 Io 8 3

\

t'

Fr8.93 2.98 o 2 0.04 0.1 Ò2 0.4

06 08

I K

Figure 19 - Unsteady Lift Coefficients and Phase Angles in

Pure Pitch as Funtions of Reduced Frequency, Comparison

With Airfoil Data = -0.250, = 0) KEY EXPERIMENT MODEL Rex 106 - AIRFOIL HYDROFOIL o / --THEÓRY cRv H/b

/

//

-

-6 4 2 0.8 0.6 0.4 D

o

I"

\

g> D 'V1

o

o o» D."-

o'

800 600 400 00 CLP 80 60 40

4MP 400 200 CMP4O 20 Io F8.93 -F2.98 K

Figure 20 - Unsteady Moment Coefficients _{and Phase Angles}

in Pure Pitch as Functions of Reduced _Frequency,

Com-parison with Airfoil Data

(a = -0.250, = 0) 37 KEY EXPERIMENT MODEL

RXI06

AIRFOIL 0.7 0.8 HYDROFOIL o 1.0 0 2.85.6 8.4 THEORY CURVE _H/b INFINITE (THEO-DO RSE N 3.94 1WIDPhU.) 4 2 100 80 60 0.8 0.6 0.4 0.2 8 6 4

4,Dp

2

0.

2

K

Figure 21 - Unsteady Drag Coefficients and Phase Angles in Pure Pitch as Functions of Reduced Frequency

(a= -0.250, ct5 = 0)

o

KEY SYMBOL Re X 106 D

G

0

D

G

2.8

5.6

8.4

D

0G

D

o

D

o

o

o

J

G0

Q

o

D.

H

D r%A .2 4 6 6 4 0.8 C DP 0.6 0. 0. 0.0

depth from studies of Theodorsen1 and finite depth from work done by

Widnall22'23 (for two Froude numbers) are included. No theoretical or

experimental airfoil data are available for drag. As a result, Figures 18

and 21 show no comparisons, but are included as a check on the axis

con-version. For the lift and moment, however, the remaining figures show that the airfoil data were in better agreement with theory than with the

hydrofoil data, particularly the lift coefficients. There are four

possible explanations for this difference in agreement:

i. Possibly the large struts which house the oscillator mechanism caused enough blockage and free-surface distortion that the flow over the

test section of the hydrofoil model was not truly two-dimensional.

Block-age would also affect the flow velocity.

In the more viscous water medium, the hydrofoil is likely to

have a thicker boundary layer than the airfoil. Thus, the boundary layer,

which is not taken into account in the potential theories of Theodorsen and Widnall, is likely to have a greater effect on unsteady loads even though the Reynolds number for the hydrofoil is significantly higher.

For instance, a significant boundary layer could change the Kutta

con-dition at the trailing edge of the hydrofoil in unsteady motion.

The radiated surface wave may contain significant nonlinear effects which are not accounted for in the Widñall theory.

Differences in profiles of the airfoil and hydrofoil models

could affect the results. The airfoil profile was symmetric, but the

hydrofoil had camber. Thus the two models would have different chordwise

velocity and pressure distributions. Figures 22 and 23 show the

theo-rectical pressure distributions for the two models. Data for thése curves

were obtained from tables and graphs of References 42 and 43.

ADDED MASS

The added-mass data, in Figures 24 through 29 were obtained at zero

forward speed. _{The data were very sensitive to changes in water depth but}

relatively insensitive to changes in angle of attack. The large

dis-crepancies between the experimental results and theoretical predictions

cast doubt on the assumption that added mass is independent of a finite

3

2

'U)

Figure 22 - Theretical Pressure Distribution for

the Airfoil Model

s 5

4

()2

2

0

02

04

0.6 X/C

Figure 23 - Theoretical Pressure Distribution for the Hydrofoil Model

(UPPER SURFACE) NACA (LOWER SURFACE

0012

L;.S(ÚPPER

SURFACE) I

r0(L0WER SURFÀÇ)____....

I- - -

-

----NACA 16-209

o

0.2

04

06

08

Io

0.8

1.0

1.2 0 MT 0.5

02

o 0 LT 3.0 H/b = i 2.5

H/b=2

2

1*-KEY a H/b os 50

O 0 0.57

D 0.81

Q 0 1.07

O

1.99

o

THEORY

- FINITE

(RAE-STATLER) - INYINITE THEODORSEN) KEY a

_S0

H/b THEORY

H/b!1

STATLER) --INFINITE (THEODORSEN)

___________

o

-.

-e

O D Q

-

-=-li/b = i

/c__

mi

0.01 0.1 IO IO lOO 1000 -J s

Figure 24 - Added-Mass Lift Coefficients and Phase Angles in Pure Heave as Functions of Zero-Speed Reduced Frequency

0.01 01

I0

IO loo 1000

S

Figure 25 - Added-Mass Moment Coefficients and Phase Angles in Pure Heave as Functions of Zero-Speed Reduced Frequency

1.0

0.9

S_LT 0.8

0.7

0.6 0.5 0.4 SMT 0.3

0.6 o B 1.0 IO 100 .1000

Figure 26 - Added-Mass Drag

Coefficients

and Phase Angles in Pure Heave as

Functions

of Zero-Speed Reduced Frequency

0.3 ci

H/b2

o KEY s

Ö1O7

- 'O

-o

---'Q KEY

- a

H/b O 50

0 0 0.57

-

000.81

o 1.07 THEORY 6 D

-

FINITE (GRAcE- STATLER) - INFINITE (TIODORSEN) ' 0.01 0.1 l'O IO 10,0 1000 s

Figure 27 - Added-Mass Lift Coefficients

and Phase Angles in Pure Pitch as Functions

of Zero-Speed Reduced Frequency

0.4 SDT .0.2 4.0 3.0 2.0 0L P 1.0 _0.0 0.2 SLP 0.1 3

0DT 2

SMP 0.4 I.0 _{0.8 0.6 0.2} 0DP 0.5 o

_-2

Q e D

û

KEY H/b 0.81 1.99 FINITE STATLER) INFINITE THEORY (THE1xRsEN) I' OS 50

OO.57

D D

001.07

_{Q 0}

0

03.94

0 o

-

(GRACE (GRACE

-o

4

--?-I-g-e

KEY of2 u/b O 50

O 0 0.57

Do.8i.

001.07

O03.91

D

i

o

H/b=1

Q

f

-t-,-1.0 I0 loo (000 Figure 29

- Added-Mass Drag Coefficients

and Phase Angles in Pure Pitch

as Funçtions

of Zero-Speed Reduced Frequency

0.01 01

Io

Io loo (000 Figure 28

- Added-Mass Moment. Coefficients

and Phase Angles in Pure Pitch

as Functions

of Zero-Speed Reduced Frequency

0MP

o

0.3

DP

It is showfl in the section on theory that the presence of a free

surface complicates the determination of added-mass effects at finite

for-ward speed. However, the consistency of the data may be determined in the high frequency limit. Using the Theodorsen theory, Bispiinghoff and his

39 . .

colleagues predict that in an incompressible fluid, added-mass effects

will dominate the unsteadyloads. This should be true if the radiated

waves do not produce a contribution similar to the acoustic wave

con-tribution in the airfoil case. The unsteady loads at zero and finite

speeds are plotted in Figures 30 through 33 as functions of w2b/g ( for

22

.

-U = O and k F for U 0). The experimental results are compared with

three theories, summarized as follows:

The studies of 1'heodorsen1 yield the unsteady load coefficients

on a flat plate at infinit depth.

The studies of Grace and Statler2° yield the unsteady load

co-efficients at zero forward speed beneath a free surface.

22,23

The studies of c3-nal.l yield the unsteady load coefficients

at finite forward speed beneath a free surface.

Both the theoretical and e*perimental results show the expected asymptotic

behavior at high frequencies. The êffèct of depth appears as slight

changes in the asymptotes»perhaps due to radiated surface waves.

Further work is needed to determine the added-mass effects at

finite speeds and low frequencies. In this region, Grace and Statler2°

predict singular behavior hen , or w2b/g, approaches 1. In Figure 4,

for example, this anomoly is evident in the experimental results to a much

higher than theory predits.

STEADY LOADS

In the finite frequency case, steady loads are defined as the mean

values of the total loads. This report presents results only for the zero

frequency case where the steady loads are identical to the total loads.

The effects of oscillation on the mean loads, particularly drag, should be investigated in detail at some later date.

The steady load coefficients (defined on page 6) are found to be

functions of angle of attùk, Froude number, and depth of submergence.

or

Figure 30 - Unsteady Lift Coefficients and Phase Angles in Pure Heave as Functions of w2 b/g

(H/b = 1.07, a = -0.167, = 0) lo 8 6 4 SN, I 0.8 0.6 0.4 0.2 20 Io

!"!II!

!!i I!P!I!"

-_____

I.

-

111111

i;li!j:ulu

1!hiiihìtimlolo

F.8.93

III

I

KEY

I EXFERIMENTI THEORIES

-.

F SIM -- -GRE-STTLERb n I I N

SIUI

u U 0.0 o O

_I",

IIIIIUluiii

--THE000RSEN

H/b-"q

O b.

ÍIIIi1..

893 0.0

_.!!!!rií

m 1 - O - ______ M!IU KEY EXPERIMENT THEORIES H/b I -ORA-STATLER WiDNALL

UIIlI

s

-- -- -- THE000RSEN

.

InÍIÌ_

'

.

UIII1

liii

-

g

_____ s U

_I

I

iII

UI

!!I

.

---o I

--

---- - e I

--

_--

.

_{_____.iI.__i_p}

r

o

I

F-8.93 o 5.95 -

-;

298 C -w s o -

--

r) - o L 8 6 4 CLT or ₂ SL, 0.8 0.6 3.5 3.0 9LT or 2.5 LT 2.0 1.5 Io loo 1000 w2b/g

Figure 31 Unsteady Moment Coefficients and Phase Angles in Pure Heave as Functions of w2 b/g

(H/b = 1.07, a = -0.167, c* = 0) Io loo 1000

Jb/g

2CM, ₂ 1.5 1.0 MT or _0.5 eM, 0. -0.5

0.2 0.1 3.5 3.0 2.5 #LP 2.0 or 1.5 eLP 1.0 0.5 0. u 5.95

N

2.96 0.0

F8.93

J

KEY RIM EN T F SYM 0.0 0 2.99 0 5.95 0 8.93 0 o -û

_.

D o THEORIES H/be I - - - GRE-8TATLER

- WIDNALL

H/b - - - THE000RSEN û D

o

u

.

.

u

ál

b

S

i ill i OIL111

LII

uuInpu;!!ift

iiIIIIIIIIIIII

niuiiuunni

IR 8.93 11111 F - 0.0 -0.5 I Io loO

w b/g

Pure Pitch as Functions of w2 b/g (H/b = 1.07, a = -0.167,

= 0) 1000 NP or MP in 0.2 0.5

I'll

F-893

KEY EXPERIMENT F O 98 95 93

'L

O. 5.95 2. 5.

NII

'

N 2.98 P.\

N

II

____

u

-0.0

SYM o o o O o THEORIES H/b I - - -GRACE-STATLER

-WIDNALL

H/b

-- -- --THE000RSEN û 0. .

'J,

Il ol

P9 5.95 ci -0.5

-

--8.93 k I0 w2 b/g loo 1000

Figure 33 - Unsteady Moment Coefficients

and Phase

Angles in Pure Pitch as Functions of

w2 b/g (H/b = 1.07, a = -0.167, = 0) 30 20 Io a 6 4 CLp or 2 SLp 0.8 0.6 0.4 40 20 Io 8 6 2CMP 4 or 2 SMP 0.8 0.6 0.4

coefficients are nearly linear functions Of angle of attack. This is shown, for example, in Figures 34 and 35 where the lift and moment co-efficients are shown for a depth to semichord ratio H/b = 1.073. The

agreement between theory and experiment for the steady loads appears similar to the agreement for unsteady loads. Theoretical lift coefficients are much larger than experimental ones; theoretical and experimental .moment coefficients are nearly the same.

Figure 36 shows drag coefficient as a function of angle of attack. It is disturbing to note that negative values of drag appear. However, it was known that the drag dynarnometer was inaccurate and quite sensitive to changes in lift and moment. It is quite possible here to have an in-determinate zero offset in the drag "hich was aggravated by the calibration

scheme.

The linear dependence of lift and moment coefficients on angle of attack makes it convenient to consider lift and moment slopes (dCLS/ds

and dC/d.,

in radians) as functions of Froude number and depth of submergence. These are discussed later and jilustrated for a few experi-mental conditions. More detailed steady load results are shown ïn

Appendixes A and B.

Free-Surface Effects

At low speeds, a hydraulic jump appears directly over a shallowly

sUbmerged hydrofoil. As the speed is increased., the jump decreases and eventually washes downstream. There is some question as to whether the effect of chord length or depth of submergence is more important for this critical wavemaking phenomenon. Schuster and Schwanecke44 claim that the

wavemaking is critical when a Froude number which is based on depth

FH = ¡J/p'i (21)

approaches unity. However, observations by Parkin, Perry, and Wu45 indi-cate that the wavemaking is critical when a Froude number is based on chord length

F=

(22.)

0.8 0.6 o -0.2 o .12 .10 .08 .06 .04 Cp .02 -.0 -ao -LO O LO 20 30 OEs DEGREES' 40 60 6.0 7.0

Figure 35 - , Steady Moment Coefficients as Functions of

Mean Angle of Attack

(H/b = 1.07, a = -0.167) 8.0 KEY D O * F 149

l.8

2,9 2.68 - -KEY

o

0 O E 4.47

: :::

-F 1.49 2.09 ¡.78

2.g8

2.68 . . F 8.94 .. ' : (GIESING-SMITH) ThEORY 0 2.o9 2.0 -1.0 0 ¡.0 2.0 3.0 4.0 5.0 60 70 8.0 DEGREES

Figure 34 - Stead' Lift Coefficients as Functions of Mean Angle of Attack

(H/b = 1.07, a = -0.167) 0.4

CLS