AD-785 247

A LIFTING-SURFACE PROGRAM FOR TRAPE-ZOIDAL CONTROL SURFACES WITH FLAPS

Justin E. Kerwin, et al

Massachusetts Institute of Technology

III I I I __ I RI i I I I I

Prepared for:

Office of Naval Research

August 1974

DISTRIBUTED BY:

National Technical Information ServiceU. S.°OEPARTMENT OF COMMERCE5285 Port Royal Road, Springfield Va. 22151

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4. TITLE (1id Subtitle) 5. TYPEOf REPORT & PERIOD COVERED

A LIFTING-SURFACE PROGRAtM FOR TRAPEZOIDAL Final ReportCONTROL SURFACES WITH FIAPS 9. PERFORMINO ORO. REPORT NUMBER

7. AUTHORsf) I. CONTRACT OR GRANT NUMBER(S)

Justin E. Kerwin N00014-67-A-0204-0067Bohdan W. Oppenheim

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT. PROJECT. TASKAREA & WORK UNIT NUVBERS

Dept. of Ocean EngideeringM.I.T.Cn=hriAye_ MA 02-119

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August, 1974Office of Naval Research 13. HU -.F PAGES

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1. SUPPLEMENTAI'tNOTES

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Control Surfaces Rp,dt -d byNATIONAL TECHNICAL

Lifting Surface Theory INFORMATION SERVICERudders V ;e Ppartmn-nt of Commerce

Ship Maneuvering Spin,,fid VA .2 1

20. ABSTRACT (Continue on reverse alde It neceeeat' ernd Identify by block number)

A numerical lifting surface procedure is developed specificallyfor flapped control surfaces with trapezoidal planforms. Theprocedure uses a discrete vortex approximation with spanwisevortex lines located at constant percentages of the chord. Useof the procedure in obtaining an optimum flapped rudder aesignis demonstrated. A listing and user's description of the computerprogram is included.IJ

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MASSACHUSETTS INSTITUI!E OF TECHNOLOGYDepartment of Ocean Engineering

Cabridge, MA 02139

Report No. 74-15

A LIFTING-SURFACE PROGRAM FORTRAPEZOIDAL CONTROL SURFACES WITH FLAPS

by

Jttin E. Kerwinand lob.P

Bohdan W. Oppenheim

August 19740

Reproduction in whole or in part is permitted

for any purpose of the United States Government.

N

This work was supported by the Office of Naval Research,Contract N00014-67-A-0204-0067, NR 062-467, MIT OSP 80464.

10)

.L".t.r. N-

I

CONTENTS

AbstractNomenclature

1. Introduction -

2. Discrete Vortex Arrangement - 43. Continuous and Discrete Circulation Distribution 104. Computation of Induced Velocities- - ----------- 18

5. Computation of Forces -------- --------- 246. Test of Program Accuracy - ------------- 287. Design Application ----- ------------ 338. References -------- ------ --------- 37

Appendices

1. Instruction for Preparing Computer Program Input Data - 402. Sample Output ------- ------ ----- --- 423. Computer Program Listing and Particulars ------ -- 464. Positioning of Control Points Within Their Own Elements 59

Figures

1. Coordinate System and Notation for Planform Geometry --- 22. Illustration of a Typical Horseshoe Vortex Element - - - 6

3. Complete Lattice Arrangement- -------------- 7

4. Chordwise and Spanwise Mode Shapes ----------- 125. Spanwise Lift Distribution Obtained by Present Program

Superimposed on Fig. 13 of [51 ------------ 30

6. Spanwise Distribution of Lift Due to Angle of Attackand Flap Deflection for Rudder Designs of Table 6 - 36

7. The Element Used for Estimation of the Error Causedby Control Point Position Deviations --------- 60

Tables

1. Relative Vortex Strength for Eight Chordwise Intervals - 152. Relative Vortex Strength for Fifty Chordwise Intervals - 16

3. Variations of Parameters With Extreme Precision Numbers- 284. Indentification of Symbols in Fig. 5 - --------- 315. Effect of Sweep and Taper on Flapped Rudder Characteristics 34

[ II

1 LVt

-. -: -- --.- ,

ABSTRACT

A numerical lifting surface procedure is develcped specifically for

flapped control surfaces with trapezoidal planforms. The procedure uses

a discrete vortex approximation with spanwise vortex lines located at

constant percentages of the chord. Use ef the procedure in obtaining an

optimum flapped rudder design is demonstrated. A listing and userts

description of the computer program is included.

Iii

F.

-- *-

NOMUTURE

(In the order of appearance in the text)

a effective aspect ratio (2A)2A

s semi-span

A rudder area

f flap area ratio A/A

Af flap area

A taper ratio CT/CR

CT tip chord

cR root chord

A sweep angle of 1/4 chord

x chordwise coordinate axis

z spanwise coordinate axis

XL(z) leading edge chordwise coordinate

XT(z) trailing edge cbordwise coordinate

i spanwaae panel index of the lattice

j chordwise panel index of the lattice

m spanwise index of control points

n chordwise index of control points

XFR flap chord at the rudder root

XFT flap chord at the rudder tip

xSR skeg chord at the rudder root

xST skeg chord at the rudder tip

IV spanwise precision number

IR chordwise precision number

I number of chordwise panels on rudder

J number of spanwise panels on rudder

NF number of spanwise panels on flap

NS number of spanwise panels on skeg

G(x,z) nondimensional circulation distribution

a rudder angle of attack

j, 6 flap deflection relative to skeg/IV

(cont.)

Y (xs 3 dime sional circulation distribution

U velocity at infinity

ckY, mode amlitudes

k spanwise mode index

I chordwise mode index

K number of spanwise modes

L number of chordwdse modes

fk(z) kth spanwise mode

p£(s) ith chordwise mode

z. transformed spanwise coordinate

transformea chordwise coordinate on rudder

c(z) rudder local chord

t: transformed chordwise coordinate on flap

integral of kth chordwise mode

r(z) spanwise circulation distribution

E,4 coordinates of a general point on a vortex

v ( s ) velocity induced by spanwise vortex

v(t) velocity induced by trailing vortex

vmpnpij velicity induced by the (i,j)th element at the (mn)thcontrol points

vm,n,k,l velocity induced at the (mn)th control point by the(kl)th mode of unit amplitude

Vmn velocity at the (m,n)th control point

p mass density

CM(Z) local lift coefficient per unit angle of attack

CL6(Z) local lift coefficient per unit flap deflection angle

CIa overall lift coefficient per unit angle of attack

CL6 overall lift coefficient per unit flap deflection angle

CDi induced drag coefficient

fl lifting surface efficiency

xH M ) first moment of the Xth chordwise mode function

XH(Z) local disLance of the center of pressure as a fraction

c(z) of the local chord from the hinge line

NUM~ECLATURE(Cont.)

X. U(Z) local distance of the center of pressure as a frac~tionC(Z) of the local chord from the leading edge

'XH resultant chordwise position of the center of pressurec relative to the flap hinge line as a fraction of the

mean chord

cmean chord - (XpR + XT+ xR+ xST)/2

r-DV viscous drag coefficient

a R transformed chordvise coordinate evaluated at the posi-

tion of the flap hinge

I i*

1. INTRODUCTION

This study arose from the need to develop a rational basis for the

selection of optimum geometric characteristics for rudders with relatively

small flaps. In this case, very minor changes in sweep or taper can effect

a major change in the spanwise distribution of flap chord. This, in turn,

could be expected to influence the spanwise and chordwise distribution of

lift when the flap is deflected. To optimize rudder performance one would

like to have a distribution of lift whicb results in the maximum lift/drag

ratio, highest possible stall angle, and minimum control moment for both

skeg and flap. Consequently, it is necessary to have the means for estimating

spanwise and chordwise distributioi of lift for a given geometry.

Ship rudder effective aspect ratios typically fall in the region where

neither high aspect ratio norlow aspect ratio zheories are valid. One must,

therefore, resort to numerical lifting-surface theory to ortain meaningful

results.

It seemed most expedient for this application to write a specialized

computer program designed to accommodate only trapezoidal planforms of the

form shown in Fig. 1. The flap hinge is required to be at right angles to

the root section, and the tip chord is required to be parallel to the flow.

No restriction is placed on aspect ratio, swfeep or taper, provided that the

flap hinge emerges from the tip, rather than the leading or trailing edge.

A trapezoidal planform makes a discrete vortex lifting-surface model "1

relatively simple. Unlike the original work of Faulkner [1], or current

schemes for propellers (21, the present work employs spanwLse vortex lines

located at constant percentages of the flap and skeg chord, rather than at A

right angles to the oncoming flow. This eliminates the problem of vortex

fIc(z)

[ CT

r SR 0 FRx

TRAILING EDGE1/4 C HORD

LEADING EDGE SE

ST FTI

z

Fig. 1 Coordinate System and Notation for Planform Geometry

2

elements ruuning out of the leading and trailing edges, which is inevitably

a source of inaccura,-y. This would be particularly objectionable in the

tip region of a small flap.

As a result, the spanwise vortex lines are not purely bound vortices,

since they include a component of vorticity parallel to the oncoming flow.

However, as long as the accompanying system of trailing vortices is arranged Iin such a way that continuity of vorticity is preserved, this is an equally

valid, discrete representation of a continuous vortex system.

It was decided to use a strictly linear lifting-surface theory, since

an examination of results for two-dimensional airfoils with flaps [3] indi-

cates that flap effectiveness is reduced due to viscous effects before any i

non-linear augment of lift becomes apparent. Consequently, the vortex system

is located on a plane, even when the flap is deflected, and the influence

of thickness on loading vanishes. One may then solve the problem of a rudder

with zero flap deflection and unit angle of attack, and a separate problem

of a rudder with zero angle of attack and unit flap deflection. The solution

to any combination of angle of attack and flap deflection is then simply a

linear combination of the preceding two results.

3 ,

3 .~ I

A:2

2. DISCRETE VORTEX ARANGEMENT

The computer program is designed to accomwodate any quadrilateral

flapped control surface with a hinge axis normal to the flow, and with a

tip parallel to the flow, as indicated in Fig. 1. The planform geometry is

uniquely specified by the four nondimensional quantities tabulated below:

Symbol Definition

a Effective aspect ratio (2)2

f Flap area ratio AF/A

A Taper ratio cT/C R

A Sweep angle of 1/4-chord

From these, we may obtain the coordinates of the four corners of the control

surface. The coordinate system, as shown in Fig. 1, is located with the

x-axis situated at the root section and the z-axis coincident with the flap

hinge axis. The semispan, , is taken to be unity, so that all length

dimensions are nondimensionalized at the outset with zespect to this qr tntity.

The x-coordinates of the four corner points then become:

3(l- ) + 4f(l + X) 1XFR = 2a(A + 1) 2 a

4

XSR xFR - TXTT ___

3(X -) + 4f(l + X) + 1 tan AxCFT 2 tn

2a(X + 1)

XsT xFT -a + 1) (2.1)

4I

-t -

If we do not wish to have the flap binge emerge from either the leading or

trailing edge, it is necessary that the choice of input quantities be such

that XR and xFT > 0 and S and. xST < 0. If these conditions are not met,

an error message is printed.

The two trapezoidal regions representing the skeg and flap may now be

subdivided into a lattice of spanwise and trailing discrete vortex lines.

An individual segment of a spanwise vortex, together with the two trailing

vortices originating at the endb of the segment, forms a horseshoe vortex of

constant strength, as illustrated in Fig. 2.

The complete lattice arrangement is shown in Fig. 3. The fineness of

the grid is controlled by specifying a spanwise precision number, IV = 0,

1, or 2, and a chordwise precision number, IH = 0, 1, or 2. A zeroIII precision number denotes the coarsest possible grid spacing, which is the one

illustrated in Fig. 3.

The semispan is divided into 51 + 7 chordwise panels of equal width.V

The panel nearest to the tip is further subdivided into 14 equal intervals,

and this pattern is dupJicated on the image side of the planform. The total

b number of chordwise strips over the span, I, for each precision number is as

follows:

Precision Number Number of Chordwise Panels

II" Iv I

~0 40 ,'

2 60

5

--

-It 11I!~ i'1

!A

XL( Fg Icontrol pointle

i ,j 1 - m i

f,,

- iFig. 2 Illustration of a Typical Horseshoe Vortex Element A

6

gw Alm

-I VI..UI' IL'1 4

I

'1 i

-I

'1

hii'I

I:,~ j~. Ir

Fig. 3 Complete Lattice Arrangement Corresponding to ZeroPrecision Number '1

4 7

I - - - k~IML~0~ZS ~ -

The total chord is divided into J spanwise panels in arcrdance with

the specified horizontal precision number I.,

J = 50 + 10I . (2.2)

In general, it is not possible to have the chordwise spacing the same on

both the skeg and the flap. However, the two spacings will be very nearly

equal if the number of panels on the flap, NF, is th integer closest to

the flap area ratio times the total number of intervals over the chord,

NF > fJ . (2.3)

The nimbez of panels on the skeg, NS, is then:

NS - J - NF . (2.4)

The determination of NF and NS is internal to the computer program, based

on the specified input value of f.

Velocities are computed at 80 control points distributed over the span

and chord. Their spanwise placement is permanent, as indicated by small

circles in Fig. 3. Their chordwise placement is arbitrary, and is specified

as input data by the user. Spanwise vortex lines are placed in the middle

of each panel. Each control point is located midway between adjacent span-

wise and trailing vortex elements, which is essential in order for the dis-

crete system to converge to the Canchy-Principal Value of the corresponding

continuous singular integral. Thus the control points are placed on the

panel boundaries.

To avoid errors due to edge effects, control points should be located

at least 2 spaces away from the leading and trailing edges, and 4 spaces

away from the tip. The choice of 8 spanwise and 10 chordwise stations

follows from our experience with similar computing schemes for propellers [2].

Since the flap, when deflected, acts to some extent as an independent lifting

8

%I

J4

surface: it. is important to include as many co)ntrol points as possible 3ver

the flap chord, while avoiding the immediate vicinity of the hinge line.

This requires an unusually large number of panels over the chord, and a

flexible system of specifying control point locations.

This lattice arrangement, which concentrates at least half of the

elements over the outer seventh of the span, is the result of considerable

numerical experimentation. Lifting surfaces with trapezoidal planforms may,

und.r certain circumferences, have a spanwise load distribution which falls

to zero very abruptly at the tip. To obtain accurate results it is nececiary

to locate a sufficient number of control points very close to "he tip.

However, to avoid edge effects, as we have noted before, it is essential to

have several grid spaces between the outboard control point and the tip. One

must, therefore, have an extremely fine grid in this region. Since a

relatively coarse spacing provides ample accuracy in the midspan region of

the lifting surface, it would be an extreme waste of computer time to extendI

the fire grii over the entire planform.

'9

3. CONTINUOUS AND DISCRETE CIRCULATION DISTRIBUTION

One r-juld, in principle, solve for the circulation of each discrete

spanwise vortex element necessary to satisfy the flow tangency boundary

condition at each control point. However, this would require that the

number of control points be equal to the number of vortex elements, and

would result in the need to solve large systems of simultaneous eqtutions.

For zero precision number, for example, there would nave to be 800 control

points rather than 80, and there would be 800 simultaneous equations to

solve.

We will, therefore, employ a modal approach, which greatly reduces the

number of unknowns. The continuous distribution of circulation over the

span and chord is assumed to be given by a series of known forms with

unknown coefficients, following the classical work of Glauert (4]. The non-

dimensional circulation distribution due to the angle of attack, G('), and

flap deflection, G( / is assumed to be given by the following series:

.,a)¢ .a) ¢ , = K L-1 °" (3.1)G4z (x,z) Y 4 fk(Z) ptss)

k=l k=l

(6) (x,z) K L (3.2)

S(x,z 4 fk(z) pt(s),k- k= 1=

In the above equations c Vare unknown mode amplitudes, fk(z) are the

spanwise modes, and p,(s) are the chordwise modes. The spanwise modcs are

given by the following expression:

f sin[(2k - l)] , (3.3)

10

where z is the transformed spanwise coordinate

z 1 (-Z) • (3.4)

The chordwise modes are

2 1 + cos sPi 0 )- ' - (3.5)

7rc(z) sin s

4 sin[((,-1)s'P~S) rc z) Z 2, 3, ... , L - 1 (3.6)

if

= 2 1 + cost (3.7)

L ) T(Z) sin t

where s = cos-[l ( (3.8)

-,2x (39)t = cos-[I XT( ) (3.9)

The last chordwise mode, corresponding to £ = L, contains a square-root

singularity at the leading edge of the flap and is, therefore, omitted in

(3.1). The first six spanwise and chordwise modes are plotted in Fig. 4. One

can readily see from Fig. 4 that control points must be located very near the

tip in order to resolve the fifth and sixth spanwise modes.

The chordwise modes may be integrated to obtain the total circulation

around any element of chord length:

S) (')dx 1( + sin (3.10)

Sf 1% sin 2 (3.11)P2(s) p2 (sjdx -I(s 2

rzp fI! Iib

1/2 ~

All it 'I N\\ I:t, P4 f4I

P5 f5 1 i 'I Iv

1/41 6P 6 f6

S'0 S--jS.ILEADING H'IGE TRAILING V V VEDGE LIN- EDGE ROOT TIP

IChordwise Modle if Sponwise Modes

Tx

Fig. 4 Chordwise and Spanwise Mode Shapes

- (3.12)

P(a) =f p (t)dx -(+sin Q (3.13)

The total circulation around any spanvise station due to each mode isi

obtaioed by subsz 4 tt':ting the leadinsg and trailing edges as limits in the

preceding integrals. It, therefore, follows that

xT(Z)

f p (s)dx I

11XT(Z)

f p/.)dx I .=2 ..

XL(Z)

xT(z) V

f = td 1 .(3.14)

0

We can now combine (3.14) with the spanwise modes to obtain an expression for

the spanwise circulation distrlibution

XT (Z) K(o(ax)r"~ (Z) = f y (xz)dx 4U ~ c~+c. f(z). (3.15)

XLjz) k=l

13

... .... ..

X(z) K

(Z) f Y (X, ) dx-4 0 (ckl + k2 +CiL )fk(z)(Z) k-i

(3.16)

We can now obtain the circulation strengths of each discrete spanwise

element corresponding to each mode of unit amplitude. The circulation must

be, of course, constant over the span of the element. We chose to make this

value correspond to that of the continuous distribution at the midspan of the

element. Consequently, the spanvise mode value is obtained by substituting

the value of z co-.responding to the midspan of the element in question into

(3.3). The circulation is then obtained by multiplying this value from (3.5)

by the integral of the particular chordwise mode over one chordwise interval

from (3.10) - (3.13). The sum of the strengths of the spanwise vortices over

any chordwise panel will, therefore, be equal to the total circulation around

the midspan of the panel in the continuous case.

The circulatiol. of the twL trailing vortices shed from the ends of each

spanwise vortex segient must then have the same magnitude with an appropriate

algebraic sign. Finally, since corresponding elements on the image portion

of the lifting surface have the same strPni;n, this computation needs to be

made only over the semispan.

An alternative procedure for determining individual vortex strengths

is a method originated by Faulkner [1). The relative vortex strengths

corresponding to any chordwise mode are obtained by requiring that the

downwash at each panel boundary be exact in two-dimensional flow. The

results are given in Table 1 for 8 chordwise panels, and in Table 2 for

50 chordwise panels. The only significant differences occur in the first

14

a'. a -C '

U% N~ur- a1! No 0 -. -I aQU lN @0. 0.0 Ill @0I

O000VCV-0c -w-1C0O0-

ai 10 ga

H~~~ ~~ v4COC~. c- N NCo~

.0 4

) 1*1 1 - 1

~* o0 e0 4@ 0000 00 0

0'. n 00) 4 .I:j4 II N.~s C'Jf II ~ .4N

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PC -C C A

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:..I' ; t WW.

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*" 4.f. ft N 2 *C P *........ *4****. *

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&j -C4-V

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.3 ft s' t4.It.44JQ f Z

P3 . . . . . oo cC CC 0C 0C .o ~ ~ ~ o c t cC * C O C.c

144

C. 0

14 v P. i-e eW.1 _ , Ie Zz ~ ~ ~ f C-, ItL. 41t~ . .pIcP. P V 'c 0t t 0a v

oc % . ...

ZI C-r NN C.NN 4 fn I V0t0g.a r tC %IUCw I

:7 w~ rMW ,W. r C pVatfr Tr rt!C -C'oz or* $4

- - ~P d t P pa.00C 0 .~0 0 C.~ . a *.- -. -; .( .o .. .~ % - % v low 0

%n N C P a- c 4 * * r Z i4 * * " * f- -7 o- 4 **** 4 r *, *

161

ev- ,,--,- r 10 1,

chordwise mode near the leading and trailing edges, and both methods result

in essentially identical final values of lift distribution. However, the

Faulkner method requires a" greater amount of corn atation, since the relative

panel widths on. the skeg and flap are functions of spanise position. For

zero precision nuberk, the Faulkner method would require the solution of

120 different sets of simultaneous equations, each with 50 unknowns. With-

out the complication of the flap, the Faulkner procedure for obtaining

relative vortex strengths would be much simpler, since the relative vortex

strengths could be pre-computed.

17

^ 1( ._

4. COMPUTATION OF INDUCED VELOCITIES

The lattice arrangement as described in Section 2 results in a set of

discrete, skewed horseshoe vortex elements, as shown in Fig. 2. Consider

a particular horseshoe element consisting of a spanwise vortex of strength

r extending from (xl,zl) to (xz,z2), joined by semi-infinite trailing vortices

of strength F starting at (xZ,z2) and -r starting at (xl,zl). For each

such element located in the interval 0 < z < 1, there will be a torresponding

image element of the same strength extending from (xa,-z2) to (xl,-zI). The

K trailing vortex shed at (x2,-z2) will have a strength -r, while the vortex

shed from (Xi,-zl) will have a strength of +r.

Let us first consider the spanwise vortex segment. Defining ( ,i) as the

coordinates of a general point on the vortex, and (x,z) as the coordinates of the A

control point, the velocity may be written in accordance with the law of Biot-

Savart as follows: x21Z247 if (x,z) - r (x-)d - (z-0d(

4 r [(x-_) 2 + (z_ )21 3 / 2 (4.1)

XIZ2

Along the vortex, we have:

t d =const , (4.2)d z2-zI

so that (4.1) may be expressed in terms of the variable alone and readily

integrated to give the result:Z2

47v s (x,z) 2a +r 2daC2+b +c zi

18

L44~44

ii

where

a - l+t 2

e W x.-xl+tz1

b - -2(et+z)

c - e2+Z2

d -tz .(4.3)

Equation (4.3) is not suitable for numerical computation if d becomes

small. As is evident from Fig. 2, d is the horizontal distance between

the control point and the spanwise vortex line or its extension. For the

vortex arrangement shown In Fig. 3, d can never be less than approximately

one half the chordwise grid spacing for vortex elements occupying the semi-

span interval 0 < z < 1. The distance d will only become small enough to

cause problems if the aspect ratio is extremely high. For this situation,

the limiting form of (4.1) valid for d<<l is:

47T v (xz) 2€ d - 2 2d 1(4.4)r LT (2az +b)2 (2az2+b)2 (4.4)

provided z1 < z < z2 . In that case, the velocity approaches that of an

infinite vortex as given by the first term in (4.4). The second and third

terms represent small corrections to the result for an infinite vortex. It

has been found by numerical experimentation that round-off error is minimized

if (4.4) is used for Idl < 0.002.

If z is not in the interval z1 < z < z2 , the first term of (4.4)

disappears, and the velocity tends to zero as d + 0. If (4.3) is used in

this case, catastrophic round-off error can occur. This is because the

integral becomes very large, but independent of ?. Hence, the integral

19

from z, to z 2 becomes the small difference between the two large numbers.

This situation does not occur in practice for elements over the semi-span.

However, due to the inclination of the spanwise vortex lines, it is evident

from Fig. 3. that a control point could easily be aligned with the extension

of an image vortex segment located in the interval -1 < z < 0. This can

result in seemingly random errors which come and go with minor changes in

grid spacing or planform. This problem is eliminated entirely if the velocity

is set to zero for jdl < 0.002 provided z is outside the interval

z, < z < z2 . Inclusion of the small correction similar to the second term

of (4.4) makes no difference, and is therefore an unnecessary complication.

The velocity induced by the two trailing vortices is:CO

7rv(t) xZ = (z-zi) + (zz)2]3T-

r 2

Xl

O0oE

- (Z-Z2) [(x_)2 + (z3z)2]3/2

X 2

[,Ix-xD +___________ +

Z-Z1 (_X) (Z-zi)2 +11 2 )2C +i

(4.5)

Since the trailing vortices are all parallel, the round-off error situation

is much simpler than for the spanwise vortices. Difficulties could only be

encountered for aspect ratios far below those of practical interest. Equation

(4.5) may therefore be used for all elements.

The total velocity induced by a horseshoe element is the sum of either

(L.3) or (4'.4) and (4.5). The contribution of corresponding image elements

20

A. -

may then be obtained by repeating this computation following the substitution:

X2 X2

: ~xl + I ) '

*@ which results in the correct algebraic sign for each of the individual ele--

ments. We shall use the notation v to denote the velocity inducedam,nrj

at the control point located at (z , ) by the complete unit strength

horseshoe element located between (z and (zi+ 1,x together

with its image.

The FORTRAN function HSVEL listed in the appendix performs this calcu-

lation.

The strength of the (i j)'th vortex element corresponding to the ktth

spanwise and k'th chordwise modes of unit amplitude is '.

ru 1XIf (Z P s(4.7)

Il'I

where zi is the transformed coordinate of the midspan of the element-i (zi + zi~)'

zi cos (z) , (4.8)

and Pk(si,j) is the integral of the chord load function obtained by substituting

the x-coordinates of the leading and trailing edges of the midspan of the -

(i,j)'th element as limits of integration in (3.14).

The velocity induced at the (m,n)'th control point by the (k,k)'th mode of

unit amplitude is, therefore, obtained by summing the product of the mode

strengths from (4.7) with the velocities for unit circulation obtained from

" (4.3) - (4.5) over all I x J horseshoe elements:

2104:

, pX(sij)Vmsnpipi (4.9)i-,i j-l

Since the contribution of the image is included in vm , the sumation in

(4.9) is only over the elements in the semi-span interval 0 < z S 1.

We can now write the final expression for the velocity induced at the

(m,n)'th control point in terms of the unknown mode amplitudes ck,X:

K LVm,n =l P• C mnkY (4.10)

k=l 2=,--

Equation (4.10)represents a set of simultaneous equations for the mode

amplitude coefficients, once the values of V are prescribed by the boundarym~n

conditions of the problem. The mode amplitude coefficients corresponding to(a)

zero flap deflection and unit angle of attack, ci,,' are obtained by solving

(4.10)with V +1 for all values of (m,n). Similarly, the mode amplitudem,n

coefficients for zero angle of attack and unit flap de)lection are obtained

by setting V = 0 for all values of (m,r: corresponding to control points on

the skeg, and V = +1 for all values of (m,n) corresponding to control pointsmn

on the flap.

If the number of modes is equal to the number of control points, the

boundary condition may be satisfied exactly at all the control points. If

the number of modes is less than this, (4.10)may be solved by least squares

to provide the closest possible fit at all the control points. The latter

approach is preferred, since the higher modes amplitudes are generally of

22

questionable accuracy. For this application we solve (4.10) for K.L unknown

mode amplitude coefficients by least squares through 80 control points.

23A

5. COMPUTATION OF FORCES

Lift and induced drag may be obtained most directly from the spanwise

circulation distribution given in (3.15) and (3.16), employing the well-

known results of classical lifting-line theory (4]. The local lift coef-

ficient per unit angle of attack is:(a) K

C (Z) P(z) . 8 - r ' (ai) + cc) (5.1)La IpU2C() C(Z) '(z){ck1 k7

noting that the coefficients c U and the circulation r have been obtained

from (4.10)for an angle of attack of unity. Similarly, the lift coefficient

due to unit flap deflction is:

C() 8 K (6) c(6) (6)CL6(z) L(Z) f {Ck + c k2 + cu (5.2)

Overall lift coefficients are obtained by multiplying (5.1) and (5.2)

by the local chord, c(z), integrating over the span, and dividing by the area

of the lifting surface. Since only the first spanwise mode contributes, we

obtain the result:

C - a(c + c12)

CLS rra(c1 ) + c12 + iL (5.3)

where a is the effective aspect ratio.

The induced drag coefficient, in accordance with lifting line theoLy,

may be written as follows in terms of the present notation:

24

K.~ (a) (C)

CDia 7ra.( :(ki (a) (a)1

k- 2 C1 2 +C 1 2

C 2 K(6 () (6r6 'Ckl 4Ck2 +i

(~ 1 + IX(2k-1) %.(') (6) (6) (5.4)k- c +C 4-C

a 6The efficiencies n ' of the lifting-surface are defined as the re-

ciprocals of the quantities in square brackets in (5.4), aad are equal to

one if the spanwise circulation coefficients are zero for k>i.

To obtain the chordwise position of the center of pressure at any span-

wise location, we must obtain the first moments of the chordwise mode func- itions, Pt. 'T

MI f ()dx = Yz) +4j

X

M fJx -P 16) dx xL z+ c (z)

XL

x

M Jx.p,(O') dx 0foZ=

XL

x)

i TJX 'IVt X 2z)

[1 0

25

The results for M,, M, and "L can be readily indentified as the positions

of the center of pressure of these modes, remembering that the origin of the

coordinate system is at the flap hiuge, rather than at the leading edge. The

third chordwise mode contributes a pure moment, while the reiaining modes

contribute neither force nor moment.

The local distance of the center of pressure resulting from the joint

effect of all the modes is expressed as a fraction of the local chord from

the hinge line as follows:

Xi (Z) 1 k-l 9-1

c(z) c(z) K f-1 (a)Sfk~z) I isk-l 9£-l

(5.6)K L

xia (z) 1 k-i 1- k,

C (Z) c (Z) K L()fk( z ) P

k=lk 1 1 k ,

Distance between the local center of pressure and the leading edge as a

function of local chord can be easily found from (5.6) as:

(a)x)(z) a)

C(Z) = 1 [XT(Z) [(Z)(5.7)

(6)(Zx() 1 [x (Z) -( ) 1

c(z) 1 cN) T

Finally, the resultant chordwise position of the center of pressure

reiative to the flap hinge line can be obtained by integration over the

semi-span,

26

Xli 1 ~ J@c (Z) C (z) c(z)dzc c CM (58

-f xa Wt) CL6,(z) c(z~dzC cC Lot 0III

This integration is performed numerically, using ten equally spaced stations

and an integration formula which has been developed for functions with a

square-root singularity in slope at the tip [2].

The preceding results may be combined to generate lift, drag and moment

characteristics for any set of combinations of angle of attack and flap de-

flection. By adding an empirical viscous drag term of the form:

=V 0.0085 + 0.0166 C2 (5.9)

a realistic approximation to the characterists of a flapped control surface

can be made, provided, of course, that stall does not occur. The constants

appearing in (5.9) were obtained from experimental airfoil data with "standard

roughness" given in [3]. A sample tabulation of this type is given in

Appendix 3.

11I

27"_ _i i

6. TESTS OF PROGRAM ACCURACY

[

The effect of the spanwise and chordwise g8-id spacing on computed

lift-slope for a typical control surface is given in Table 3. Variations

with precision numbers are quite small, and it may be concluded that zero

precision lumbers are satisfactory for plauform shapes similar to the case

examined.

Table 3

Variations of Parameters with Extreme Precision Numbers

CDi6 CS

0 0 0.114 0.116 3.138 1.776 0.996 0.984

0 2 0.114 0.116 3.132 1.877 0.996 0.980

2 0 0.114 0.116 3.101 1.756 0.995 0.980

a 2.8 A 150 f 0.2 0.6

Convergence of the solution with increasing numbers of elements is a

necessary but obviously not sufficient test of program accuracy. Fortunately,

the results for zero flap deflection may be compared with corresponding

solutions obtained by other current numerical lifting-surface theory tech-

niques. A recent publication by Langan and Wang [5] is particularly helpful

in this regard. This reference compares the spanwise distributions of lift

for a tapered wing of aspect ratio 5 obtained by fifteen different lifting-

surface computer programs. Fig. 5 shows the results of the present program

28

plotted on a reproduction of Langan and Wang's Fig. 13 which

are in excellent agreement with the average results of the fifteen programs

compared in [5]. The overall lift coefficient obtained by the present

program for this example is 4.106 which agrees exactly wi, the results given

in [5] obtained by the Tulinius progr=! Induced drag was also found to

be in good agreement.

A comparison was also made of the lift coefficient of a rectangular

planform of unit aspect ratio. Watkins, Woolston, and Cunningham in a

1959 report [6], cite the following values obtained from several sources,

to which we have added the results from the present program:

Source C La

Jones (low aspect ratio limit) 1.571

Lawrence 1.400 IHsu 1.497

Watkins-Woolston-Cunningham 1.455

Present Program 1.508

As a further check in the low aspect ratio range, J. Dulmovits of

the Grumman Aircraft Engineering Corpcration k-tndly offered to run his

program for a tapered planform of effective aspect ratio 2.8. He obtained

a lift slope of 3.140, which agrees almost exactly with a value of 3.138

obtained by the present program.

A test of the portion of the program dealing with the flap was made

by running a rectangular planform of aspect ratio 60, to provide an essen-

tially two-dimensional result. The exact solution in this case, as given

in [7], is C [('-s) + sin( r-s)]L6 [(fsH H

C La IT (6.1)

29

\L% , - ., i,. ....... . r u -" 4,,O, ...... .. __.."__ . .., ,*" " ' " ; ' " " .....

i'I

0.967 0.970

0.92 Ii i I I0.91 o 0

0.90 0 0 00 ap (o_~o 0 A

0.88- o

0.86- o-

0/ 0 00;

08 - ISO

CL

0.82 PRESENT PROGRAM NOQ80 -

0.780. 77-

I I I I . , I I IQI0 0.1 0.2 0.3 0.4 0.5 0.6 07 0.800.7539 Y

Fig. 5 Spanwise Lift Distribution Obtained by Present ProgramSuperimposed on Fig. 13 of (5]

30

" .... ..." "- = 't ":'l ' .k,, = , £,:' ;- .; 4= , - - ..- . . . .. . ,-,

SYMBOL PROGRAM

o TULINUSo3 DULMOVITS

0 MARGASON- LAMARA GI ESING

ts RUBBERTrl LOPEZ-SHEN (EVD)

0 HAVI LAND0 JORDAN

o LAMAR (MULTHOPP)

0OINLO BANDLER (ERA)

Q ROWEo CUN NINGHAM

0 JACOBS-TSAKONAS

LOPEZ (KUCHEMANN)

Table 4 Ideiatificatiou of Symbols in Fig.5

31

where s is the transformed chordwise coordinate evaluated at the position

of the flap hinge. For a 50% flap the presrit program gives C 6/Gn .807

compared with a value of .818 obtained from (6.1). For a 20% flap the com-

puted value is .526 vs. ,550 obtained from (6.1). We should expect slightly

lower results for an aspect ratio of 60 so that this result seems reasonable.

The calculation for an aspect ratio of 60 also provided an additional

check on C in the limit of high aspect ratio. The value obtained was 5.915

which compares closely with a value of 6.080 which one would obtain from

the lifting-line theory.

It may also be of some interest to compare the value of CL given

by the present program with that given by Equation (23a) of Chapter VIII of

Principles of Naval Architecture (8]. The latter is a simple empirical

equation:

(0.9) (270aC~st "a(6.2)

(cosA /-=---+ 4) + 1.8cos A

0

which, for an aspect ratio a = 2.8 and a sweep angle A = 15 gives a

value of 2.997, which is about five percent below the computed value of

3.138. Since (6.2) was developed to provide correlation with expermental

results, the lower value of lift slope is not at all unreasonable.

32

7. DESIGN APPLICATION

The present program was initially developed to provide a rational

basih for selecting optimum planforms for an experimental series of flapped

rudders [9]. The first two rudders in this series were to have an effective

aspect ratio of 2.8 and flap area ratios of 20% and 10%, respectively. These

were planned as an extersion of an earlier series of experiments on flapped

rudders published in 1972 [10].

With such small flaps, minor changes in sweep angle and taper ratio

have a large effect on the spanwise distribution of flap chord. The objective

is to determine the sweep angle and taper ratio in such a way as to optimise

the following parameters:

a) Maximize lift slopes Cla and C L6

b) Minimize induced drag;

c) Provide nearly uniform distribution of CL over span.

This requirement is based on a decision to use a constant

thickness/chord ratio of 15% over the span. Uniform

lift coefficient is then optimum both for delay of cav-

itation inception and delay of stall.

d) Provide sufficient flap tip chord to permit installation

of a hinge.

Calculations were made for five combinations of taper ratio and sweep

for the 20% flap rudder and the principal results are given in Table 5.

Plots of spanwise distribution of lift-slopes are shown in Fig. 6.

33

Table 5

Effect of Sweep and Taper on Flapped Rudder Characteristics

effective aspect ratio = 2.8

NO. f A A C~a CL6 ia CS

1 0.2 110 0.9 3.101 1.670 0.115 0.1212 110 0.6 3.146 1.802 0.114 0.115

3 15° 0.6 3.138 1.775 0.114 0.1164 " 18 0.6 3.129 1.738 0.114 0.117

5 19.570 0.5 3.136 1.774 0.114 0.115

6 0.1 150 0.6 3.138 1.355 0.114 0.119

It is clear from Table 5 that overall lift and drag characteristics

are very insensitive to sweep and taper within the fairly limited range

permitted due to the small flap. This is, of course, a characteristic

of low aspect ratio lifting surfaces, so that this result is not surprising.

If one looks closely, one can see the trend of decreasing lift slope and

increasing drag with increasing sweep angle.

The effect of taper ratio on spanwise distribution of lift is much

more pronounced, as is evident from Fig. 6. Here it is clear that a taper

ratio of 0.9 unloads the tip too much, while a taper ratio of 0.5 does the

reverse, and that a value of 0.6 seems about right. A change *.j sweep angle

from 11 to 18 degrees has essentially no effect on CW, and a small

effect on CL.

Rudder No. 2 appears to have the best spanwise distribution of lift as

well as maximum lift and minimum drag. However, the tip chord of the flap

is extremely small, which would cause difficulties in the model and possibly

34

in the full-scale hinge design. An increase in sweep angle from Ui to 15

degrees overcomes this problem with very little compromise in performance.

Rudder No. 3 was therefore selected for the test program. This has the

additional practical advantage of a nearly vertical trailing edge and a

constant flap chord.

The constraint on flap chord for the 10% flap area ratio rudder is

even more severe. Rudder No. 6 shown in Table 5 and Fig. 6 has the same

planform as No. 3, but with a flap area of 10%. Again, we find that CLS

is a little too low at z = 0. Increasing this by a decrease in sweep angle

is even more impractical in this case due to the small flap chord. We

conclude, therefore, that this planform seems to be nearly optimum for both

the 20% and 10% flap rudders.

A.:

35

0

soo

pIrt 3 1

- U)

Zi

_ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ (5

CjI

to0 toL 0r~i C~i C N-

Fig. 6 Spanwise Distribution of Lift due to Angle of Attack

and Flap Deflection for Rudder Designs of Table 6 S43

REFERENCES

[i] Falkner, V. M., "The Calculation of Aerodyanmic Loading on Surfaces

of Any Shape", Aerodynanic Research Counsel, TR R&M, 1973

[2] Kerwin, J. E., "Computer Techniques for Propeller Blade Section

Design", Second Lips Propeller Symposium, Drunnen, Holland,

May, 1973.

[3] Abbott, I. H. and Von Doenhoff, A. E., Theory of Wing Sections

Including a Summary of Airfoil Data, Dover Publications, Inc.,

New York, 1959.

[4] Glauert, H., "The Elements of Airfoil and Airscrew Theory", Cambridge

University Press, London, 1926.

[5] Langan, T. J. and Wang, H. T., "Evaluation of Lifting Surface Programs

for Computing the Pressure Distribution on Planar Foils in Steady

Motion", NSRDC 4021, May, 1973.

[6] Watkins, C. E., Woolston, D. S., and Cunningham, H. J., :A Systematic

Viernel Function Procedure for Determining Aerodynamic Forces on

Oscillating or Steady Finite Wings at Subsonic Speeds", NASA TR R-48,

1959.

[7] Durand, W. F., (Editor in '2hief), Aerodynamic Theory, Dover Publications,

Inc., New York, 1963.

[8] Comstock, J. P. (Editor), Principles of Naval Architecture, The Society

of Naval Architects and Marine Engineers, New York, 1967.

[9] Kerwin, J. E., Lewis, S. D., and Oppenheim, B. W., "Experiments on

Rudders with Small Flaps in Free Stream and Behind a Propeller",

M.I.T. Report 74-16, July, 1974.

[10] Kerwin, J. E., Mandel, P., and Lewis, S. D., "An Experimental Study

of a Series of Flapped Rudders", Journal of Ship Research, December,

1972.

37

Y..~,t... e.*,..,,.~~.. - ~ ~ .... ,,~bfr..aa{jam...ta~ntt4.Al

iri

Precdingpag blak 3

APPENDIX 1

Instructions for Preparing Computer Program Input Data

Integer variables must be right-uijusted in their fields - Real variables

must contain a decimal somewhere within their field.

A. FIRST CARD

SYMBOL MODE FIELD LIMITATIONS DESCRIPTION

KDM Integer 4 1-6 Number of spanwise modes-KRecommended value is 6

LT Integer 8 3 Number of chordwise modes-LRecomended value is 6

IHF Integer 12 0,1 or 2 Chordwise precision index

Recommended value is 0

IV Integer 16 0,1 or 2 Spanwise precision indexRecommended value is 0

ASR Real 17-24 * Geometric aspect ratio, a

AF Real 25-32 O.I<AF<0.9 Flap area ratio, f

T Real 33-40 * Taper ratio, X

PP Real 41-48 * Sweep angle in degrees, A

KOPT Integer 56 0 or I I If KOPT=l, the subrottine. OPTION will be called and

I 'will perform the calcula-tions appearing on the thirdpage of the output

B. SECOND CARD

This card contains the indices of the ten chordwise panels which contain

control points, NCP(N),N=I,10. These are integers which are read in ten

*Combinations of these quantities must be so chosen that flap chords are

positive for all z.

40

A

fields of 8 columns. The recommended values for zero precision number

are 3, 8, 13, 18, 23, 28, 33, 38, 42, and 48. However, the intention is

that these may be altered for a more advantageous placement relative to

the flap hinge if so desired.

C. MULTIPLE RUNS

Upon completion of the calculation, the program returns to the read

statement for t i> f'rst card. A blank card (or more specifically, a zero

value for KDM) terminates the run.

41

.m41

:|j

APPENDIX 2

Sample Output_

The correspondence between the output and the text symbols is as follows:

I text output text output

z A

.i:CL(z) CLA Ci~~(z) c C(z) C(z)A

CL(z) CLA (6)Z)

CLA (Z) aW ( ) (z)z 0-

CL6 CLDR ALPHA

CDia/C CDIA 6 DELTA

cI 1 / C CID S/t DELTA/ALPHA

C CL a + CL6 6 CLTPEFA L Le S'

71 EFD C+ V CDti '16 'D i + DV c

V m downwash ° a2 + C C 62)/CL CMVmn velocities (C La CM 1+ CL L6L

c C-ALPHA a6CKLN • C a + O. )/CL XHL/C

CL C-DELTA c Tc -L

c(z)x(6) (Z)

XD/LC

(a)x( L (z)c)ALE/LC

(6)X.L (z) XDLE/LC

42

0~ -g~ c ;Eoa

Maa * 0M~ C, aO

40 00 4? n 000000-0 30a 0 30 0~

* U Coc 0000000

a 00-0 a00?oC

.

C aC.NC'

OILV~ 4mf r4 a.I~I ., 3.0000 *0000C.

U. - - - -3 - ---- ---- -- 0 '30 M ,a of00i I A 00 COCOC

. CI9 %-W " o c

x C j -: w DI- C 4 Cw om4 - 6 r

oy Ma I. - C cI3 0U-t

41 N3

P- CCC *0- 0 .00 00"w* C, P. No n-N 00t NV t .- r 0 %7 20 0 )0 v

*- 0,C 4333 . 0000 000000

-t.: In -. " N. Z. :zI tu 0 .a- C. 01 ;C

U~~~~~~n~L U*.* ** .W U 4 U @ U UN ffcjM 21 'A.3 U M'(~.

4A I- Ix %n. ** . r ..6L C. -Z m -OUO 't0

4A f. *c 0- Sli N 3-

CIS 3- D UC- Q., KI- 0C 0 %00 a

*4 .4 MJ . 2

s 30 3 .U. J 4.GO'Is43 0 .Z~ O'O*

::& ILU 0 N ~ U N C ' V 0 'C S O C. U C '4 I 0 M

4W w 3 4A ~ n.44.4 -. N U~. -rN s >;r~. u .N~ t W4) C ONOnU-OV%

4U, ... M. U30'.C~0U00

-. Z~ 0 JSW C~ . .. . V .43

@4-O.-0 04Qa0 ~ e

CC000 0- CA C

--- o---O W 9 Wok % 0 0 00

*4tt .'a'.0 1 -4 a -c:

.#~~~~ * d %' 4 0-aI.. r-. a. ke 0 .C p p%*4A.'0...u. "n -c c ~,.n c c

Z C, wo nc co& . 2 P

A 6- C

g 0* -000 . 00 0 4e0 e0 0 C C : r - C CI-II

-0 A v

0 Ea%*9piC wV'. 0I

z* N- V...... r 0 a . c

a x VO-C- c-,--f o OC, o 7

~i C

- Ca I-U CwrO %WWC

8- - N .S 4 , 4CO' C' W 2

C - uj .0 '

&L i* A 0 C P%. aW4a

P4. C, *, *4 .* m. *

04I -q 0'00 0 0

owl

* V1142

U.. W ZZ - .0U.. . Co o - -aC - ~ - U . P ' N . A E J '

ab z e O a O o 00 ------

- CIUX a .... * * ............

M0. aF tc. ;C f 1C 0ff no ,P %,&S C, -v F& -0 1edNf Yf %- kotw9 vw4 io.A.- tef %- r4f

COc ;9oC *0C; cC ,C ;4 .

3- 1 1 11.

011 f tF .f ew 0 P3o ~ 04 Z m- 4 r r0a

a *V% w -f ry 00rEty

. It r. . . . . . ..... **. * ***

0

4 wN0w Ap- N o N -CMP.- 'o -0 op N a. .. %I -91 0 e ocr - f a -~ a

ftC 4 ft 4 -*- * * * C *4 -rt m2P z-It t IOO CO O 0 0 *. tr ev*" 4- w. 40 *1 g- it *y A* 4** *4* (P w N

o 0 ~~00000000000000000000000

0U,

a9 u U)~~~e~.. I

*~~~~~ l -N W.. 0 M r4 * * N100e~.~Nna.~i.NPP

a 4 C

If %

0 x4

4A

*. c * ...

Ua 0e0aeclCc ccc cc OC.L..000 00000a UUNNOI000

u . .0 0

inca a., a-v~fr~E.5a411 LLZ a- ::4~P P %-'

% NIPNNNIINN. 3ACu. C * . ... . . .

at ooacoooo45

APPENDlIX 3

~ Computer Program Listing and Particulars

Timing Information: 0 precision with 36 modes;

1.081 minutes execution time.

IBM1 System 370/165

NMovry Requirement lOOK

46

-0U

ww

0. I

at 3 - -a,

00

ty M or I-.

W. -u J- r I- Mc

c3w0 !- 0 t

-Q 0. 1.i. ., C2 9* C. I

a : C6 C. W, I L a -U~~~ 3 . 0 7

IL 0..- u cf .0. t4n. a U q. 0 . a a I I*C c C&

4 X C 01 z ,,.0.tW e 4- -;w Ond0 3- a a Z.. U*4 v t.

T' 'I S C Co.0 0. 0. 4, 0 2 .

t.6N ft ft InI..

cf -n64 a -0 0 m 3'. 2:C C Nciii LL C. .1 ~ u a (. %t K 0UWL 4" C0 Z,7~ U. x .xt x 0 4 t U44L. cC

UCVu cL Qw OC. 0 3 --0 C.f014L J. j 0.Pq w I%. - -9 N .~~-U.9 0 Z-. . . cf C -*4 wZ - 0. 2 N 0 0I

N 00..vNv Irv%-* *Cf- M& 04 £ r- vJ . . * C'1 0oc0 0 NL ~ &C IN ~ I. l N f4 ' 00 .

91 0 3C00 co 4 1 *~0 61. - ~ '... IU 60 00..LU oU O 00-I.. coo .'- t .JJ. S2 Z 4

* .t 0. O f' .- 80 *b I CC~4f 66 3 1.6.016 4d7

e031~ ~ ~ ~ ~~~~~~~~~~I A-o OI34- '- . CCuf ''.4*.V It

[II

NN0Z

00

0

%L

*i z

x I u A*0 - a w. WA-G C, c 20.a

14a0 1 a -cz qt o :

- : 0 00 3I

* U

*x 0 -** 1, 0 -. .

%m 0 V9 a a-

kn wo 4-t~~ z f2cw c

-00o O00 000 concoU 0000 OC .

* a a =&N Va k"

ILI0e00

2- I

.3 4.

- OC Z 7 4v

4q 7 4

o 0 W9 .0 D

-j 10 1-4

-. - -

a OC 0 00co 0000 00 0 Oc OIoO OC a OOC OO OLLt

P. As 49

416

0

.9 itl jq

co i *

3e .4 c ay*~- Uf

2f c 4490 4c -W

I I&IC4 MC rCc ;a m - -2 7a

x 3too - Mt t,. . N7, 3 t -fta 4-Ir Uo Z ...WU

C7 OP 0 4.-L:4

.0J C.L C-

- 43 CV 22,C. C -

O-f~ft V- nVo

.4ft *MY I. 10 9.. CO O CXCS 00 00 *Naoocoo 0 J0**2 0 0 00 0U.4IJ1

50* M 4~~ *.J

to C

of4CI

49. PA f U -%

OI 04 -i Iss-C.

.W 3-c . it PS * 3:M. .0u C a. 0 ' C £K

do.: 7 A 3C U. *

P4 M&J 6l me IL Pw -U r

S- LL, z - * K aU

6- C. zx8 XI I U

- ~ ~ I - -j ~~U4 at. %- 4 U. KU' .

cc I.% - 4A . Or c a, N' 2e 3 C O-

a o8KU LL U. 4A 4t KLL . 3IL .K -. 5- = . .5 w- I *I -7

KN*U - - 0 . 4 W J,.Z

up- Y T Ju 0 j zs . ' -u a - ~-. 4 8' - C . U - 53 C5 C* U.W *V,

... t~0 I-- .8~ n . ' C 3 0 N C u. U'U.3.7VC~~ ~ ~ W53 N U 5 -c r 7 - Is e 5

U4 K C, a I t81 K . .f

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APPENDIX 4

Positionina of Control Points Within Their Own Elements

Control points should be located at a point within the element where the

effects of the velocities induced by all four boundaries of the element cancel.

This is essential in order for the discrete system to converge to the Cauchy-

Principal Value of the corresponding continuous singular integral. For a rec-

tangular element this will be, of course, at the centroid of the element.

With the present vortex scheme the elements may be both swept and tapered,

so that a question arises as to whether the placement of the control point is

critical. A detailed calculation was therefore made for an extreme element

with geometric characteristics as shown in Fig. 7.

The velocity induced by the four line vortex elements was computed at

finely spaced intervals throughout the interior of the element. In this case,

the null point, 0, was found to be displaced from the mid-chord/mid-span point,

H, by the amount shown in Fig. 7. This deviation, as well as the velocity at

M is extremely small.

In order to obtain an estimate of the error introduced by locating control

points at M rather than at 0 in the complete lifting surface program a test

run was made with all control points displaced by two percent of their panel

chords. This resulted in a change of 0.92% in predicted lift slope. Since

this displacement was far in excess of the value shown in Fig. 7, it can be

concluded that the error introduced by locating control points at the mid-chord

/mid-span position of an element is negligible.

59

I250co

NULL POINT C(location exogger.&id)

0.00405

Fig. 7 The Element Used for Estimation of the Error Caused

by Control Point Position Deviations

60