Effects of various linkage ratios on the free-stream hydrodynamic characteristics of an all-movable flapped rudder

j

THE DAVID 'W.

TAYLOR MODEL

BASIN

EFFECTS OF VARIOUS LINKAGE RATtOS ON THE FREE-STREAM

HYDRODYNAMIC CHARACTERISTICS OF AN ALL-MOVABLE FLAPPED RUDDER

September 1955 ilL 01*

ia-r

WASHINGTON 7, D.C. by C.R. Olson Report 991

--.

-;;.---I

EFFECTS

F VAF.ICIJS LINKPE RATIOS ON ThE

FREE-STREAM HDRODZNAMC CHARACTERISTICS OF AN ALLMOVABLE FLAPPED RUDDER

by

C. R. Olson

i

Report. 991

TiBLE 0F CoNTENTs

Page

ABSTRACT i

INTRODUCTION i

MODELS AND TEST APPARATUS 2

TEST FROOEJRE 2

DISC'JSSION OF RESULTS 6

Lift Ctaracteristis 6

Drag Crracteristics ₇

Torque and Effective Center of Pressure Cracteristics ₇

CONCLUSIONS 8

RERECES

9

NOTATION

L

Lift tcre

D Drag force

Moment about leading edge of

p Mass density

S Projected area of rudder

U Velocity

Mean geometric chord of rudder a Rudder angle in degrees

Flap angle relative to rudder-chord plane, measured perpendicular to flap hinge line

in degrees

:p-1inkage ratIo

CL Li±t coefficient

CD Drag coefficient

_p/2

E Moment coefficient about leading edge of mean

/

M

geometric char-d

by

I

C R0 OLSON

ABSTRACT

The effects of various linkage ratios on the lift, drag, and torque characteristics of an all-movable-flapped rudder,

having either a 20-per centor 0per;ct:flap,have been

determined from free-stream tests, at low Reynolds number, for both ahead and astern conditions.

The results indicate that for the ahead conditton the highest lift, coefficient, is obtained with a 3-per nt flap

using a l.5 flap-linkage ratio. An increase in either the f

lap-chord or the flap-linkage ratio reduces the lift-drag ratio of the rudder and

shifts t.he center of

pressure rearward.

INRODUCT ION

The Bureau of Ships has been concerned recently with the design of all-movabie-flapDed rudders which have been proposed-in an effort to improve the turnproposed-ing characteristics of surface

shIps and submarines. At. the present time there is very little

information, especìlly of a systematic nature, for low aspect ratio rudders of this type. The results of investigations

con-dtei by

ve _{ct-er agenies on low aepecD ratio flapped}

wings ara helpfui in nakírg estimates of the effectiveness of

fIaped rudder desigs. Such data, which apply only to ahead cdt.L-ys, can be f:und in References I through

6.

T prese't. mnvestlga.ticn was made to determIne the optimum

comt'ìatc:. of flap-linkage and flap-area ratio for a proposed

suharI.r.e rudder, This was cteermined by conducting free-stream for a range of flap-linkage and flap-area ratios using a reflection-plane model of the rudder.

'Refererces are listed on page 9 of this report,

2

MODELS AND T.EST APPARATUS

The lower rudder of 4ode1 _24538, representing a preliminary design cf the SSN SMALL), was used in these tests. The basic ruder, shown in Ffgure 1, is an all-movable (spade) type with a linkage-Dperaed full-span flap. It has a NACA 0018 section,

an effective asp_ect ratio twith reflection plane) of 2.40, a

taper ratio ofO82 and a balance area of 40 per óent.

1though the original rudder was equipped with a 20-per

cent flap; provision was also made for a 240-per: cent flap. This

invDlved aesimple modification to the original rudder since both the f ia _{trai1ig edge and hinge line were unswept. The} flap-1iikage mecanismprovides for settings at fixed flap angles of 0, 5, 10, 20, and 30 degrees. With the flap locked at zero degrees the resulting configuration was equivalent to an

all-movable rudder.

The rudder was mounted with the root chord adjacent to a

flat. plate (reflection plane) which in turn was rigidly attached

to the towing carriage, as shown in Figure 2, The rudder stock

wa attached to a 3-component. strain-gage balance which measured t.he lift, drag and torque acting on the rudder. A small clearance, about 1,/8 tnch, was maintaIned between the rudder and the reflec-. tion plane0

TEST PROCEDURE

The tests were conducted at towing carriage speeds of 4 krcts for the ahead dIrection and 3 knots for the astern

dire-ticn.

This corresponded to Reyoids

numbers based on t mean

ge:'metric chord of 070

feet, cf approximately 0.4 x 10 and .

x 106, respectively.

These ICW

velocitIes were used to avoid ds'cr-ton or cL&mage ro the wooden rudder and its fIttings.

Tsts wr-e conduc.e:. for both 20-per cent and 40-per cent

f lat, For each fixed flap angle setting, the rudder angle was

varied a _{5-degree Increments up to 35 degees whIle the towing} c.rr1age wac )oenatìng a test speed. The flap-linkage ratios

reuiting

rom these combir.ations are given in Table 1. The

o

o

L

0.40 FLAP 0.20 MEAN 7, 57"

Section

_-

_{NACA 0018}

Taper Ratlo_..._.___

0.82

Geometric Aspect Ratio...j.2Q

Effective Aspect Ratio__2.40

GAP 4. 2.0" 8.44w GEOMETRIC CHORD

o

I-a: a:

Total Area ¡n Sq. ln... 85.0

0.20 Flap Area in

Sq In. ...._l 7.0

0.40 Flop AreqinSq.ln..34.O

Balance Area in Sq. ln._....34.0

F ai ri ng

Pieces

W.L.

Boundary

P la te Dy n a mom et e r /

Test Rudder

Figure 2 Sketch of Rudder Towing Apparatus

Support Bracket

to Towing Carriage

Gap:O.012 Span

Wave

Suppressor

W. L.

TABLE i

TABLE 2

NOMINAL TAB LINKAGE RATIO

Rudder Angle in negrees

5 10 ₁₅ 20 ₂₅ O O O O O 1/2 1/2 1/2 1/2 1/2 1 1 1 1 1

11/2

11/2

11/2

11/2

2 2 2

21/2

21/2

3 3

TPIB LINKAGE RATIOS TESTED

Rudder Angle in degrees

Tab Angie ₅ 10 15 20 25 3 o O 0 O _o O O _O S

i

1/2

1/3

1/5

i/6

_1/7

10

2 i _2/3 1/2 _2/5

'/3

_2/7

20

4 2

4/3

i

4/5

2/3

4/7

30

6 ₃ 2 1 1/2

_6/5

i

_6/7

6

DISCUSSION OF RESULTS

The effects of variation in flap-linkage ratio on the

lift, drag, torque' and effective center of pressure coefficients

for the 20-per cent and LIO_per cent flap rudders are presented in Figures ₃ through 8 and in the Appendix. These results are presented as a function of the nominal linkage ratio and were

obtained from cross-plots of the basic data. Data were obtained for rudder angles up to ₃₅ degrees. The data for rudder angles

above 25 degrees are questionable because of early stalling due to the low test Reynolds numbers and have been deleted. However,

the lift coefficient versus rudder angle curve, below stall, does not appear to vary in the Reynolds number range between

0,6 x i6

and

3,5 x 10° (5),

Therefore, it is reasonable to assume that the lift coefficient curve for rudder angles of less than 25 degrees is valid at full-scale Reynolds numbers.

It should be noted that data given in this report are for free stream and are not directly applicable to a rudder operating on a ship since the effects of hull interference and boundary layer on the hydrodynamic characteristics of the rudder are not

included

LIFT CiARACTERISTICS

Figure 3(a) shows that, for the ahead condition, the lift coefficient increases with an increase in flap-linkage ratio, The amount of this increase is greater for the 140-per cent flap than for the 20-per cent flap at all rudder angles below 25 degree

The opt±rnum flap-area ratio for the present rudder was found to be:

approximately 30 per cent as shovm in Figure 4.

The aforementioned trends may be interpreted in terms of the effects of the lift developed by a rudder on the tactical d1ameer of a given vessel. The increase in lift of a rudder does not decrease the tactical diameter in direct proportion, For example, the lift coefficient of a rudder having a 20-per

cent flan and a linkage ratio of 1,0 is ₅ per cent higher than a comparable all-movable rudder. However, the tactical diameter

of the Eutmarine model using such a flapped rudder is only 12

per cen. less than that with the all-movable rudder (7). _Assuming that this ratio of percentage increase in rudder lift to percentagt

decrease n tacticaldiameteris typical, i,e,, approximately 4:l, t is not necessary to obtain precise values of maximum lift coef-ficients for this purpose.

lt may be noted that a lift increase equal to that of a flapped rudder could be obtained by increasing the area of an all-movable rudder by about 50 per cent. The drag for this

enlarged rudder would be roughly 20-per-cent less than for the

various flap ar-ra-gements. These results lead to the speculation

that flapred rudders may be cf more advantage than plain all-rncva'ie rudders manly when space limitations prevent adjustment

of rudder area. This deduction is based on the larger rudder

angles only since the drag penalty ot the flap rudder does not apply at the small angles.

For the astern condition, the results shcwn in Figures (b.)

ad

5 ird1cate a dec;rease in lIft coefficient over most of the

rudder angle range with an increase in either flap-linkage or

flap-area ratio, However, there ìs less loss in lift for the

20-per cent than for the 40-per cent flap rudder.

DRI CHAPJCThRiSTIOS

The drag coefficIent cf the flapped rudder in the ahead condition increases at all rudder angles with an increase In

flap-Ïincage ratIo, as shiwri in Figure 6(a). This increase is

greater for the 140-per cent than for the 20-per cent flapped rudder, in general, the increase in drag with an increase In

f lap-iiniage ratio or flap area, is greater than the

correspond-ing increase in

lIft.

This corresponds to a reduction in the

11f t-drag ratics.

For the astern condItion, Figure 6(b), the drag increases with an increase in flap-linkage ratio but not as sharply as for the ahead condition, The drag values for the 140-per cent

flapped rudder break down at increasingly smaller angles as the

f l-l1nkage

ratto

increases.

This

breakdown prevents any

ccnsist.ent cornoarison between the 20-and 14-0-per cent flapped

rudders but In any case

che ditferences are small,

:Q7E AN

E ETT1VE CjNER 3F ESSTJRE AFiCTERISTICS

Por- the ahead corait1cn, the torque coefficient increases

with an inr-ease in either flap-linkage or flap-area ratio, as

.scow

ifl rigure

7a).

owever, the converse is true for the

astern condition s show in Figure 7 b), These trends are

sinIiar to those exr1.Ibited by tne lift

coefficient.

The effective center of pressures for both the ahead and

astern ccndìtlon.s which are shown in Figures 8ta) and 8(b),

were derIved from the torque and lift coefficient -data. For the

ahead condition, there is a large rearward shift In effective-center of pressure wIth an increase in flap-linkage ratio but

-only a small change with an increase in flap-area ratio.

For astern motion, however, the effective center of pressure

tends to move f orward rapidly with an increase in linkage ratio,

and for sorne cases is located beyond the leading edge of the mean

geometric chord. For simplificat±on of the effective center of pressure calculations for the astern condition a nominal chord for the rudder was used, This nominal chord is defined as the

straight l±ne connecting the forward edge of the rudder with the extreme edge of the flap in its deflected position.

CONCLUSIONS

A free-stream investigation made to determine the optimum combination of flap-linkage and flap-area ratio for an all-movable flapped rudder indicates the following conclusions:

1. The highest lift coefficient is obtained with a 30-per

cent flap and a flap-linkage ratio of 1.5. The lift coefficient for this configuration is about 50 per cent higher than an all-movable rudder of equal area,

2 An increase in either flap-linkage or flap-area ratio results in a reduction of the rudder lift-drag ratio, Consequently, flapped rudders are recommended mainly when space limitations prevent adjustment of rudder area

to produce the required lift.

3, increases in both flap-area and flap-linkage ratio tend

to shift the center of pressure of the rudder rearward. 4. For the astern condition, an increase in either flap-area

or flap-linkage ratio reduces the lift of the rudder.

1, NACA TN 1517 - "1ind-Tunne1 Investigation of an NACA 0009 Airfoil with 0.25 and 0050-Ajrfoil-Chord Plain Flaps Tested

Independently and in Combination' by M. Leroy Spearman dtd March 19L1.8

NACA TN 2080 - "Wind-Tunnel Investigation at Low Speed óf an Unswept, Untapered Semispan Wing of Aspect Ratio 3.13 Equipped with Various 25-Percent-Chord Plain F1aps",ì?y Harold S. Johnson and John R. Hagerman, dtd April 1950. NACA TN

2288 -

"Estimation of Low-Speed Lift and

Hinge-Moment Parameters for Full-Span Tr'ailing-Edge Flaps on

Lifting Surfaces with and without Sweepback",. Jules B. Dods, Jr., dtd April 1952

14 _{NACA TN 3)497 - "Summary of Results of a Wing-Tunnel}

Investi-gation of Nine Related HorizOntal Tails", by Jules B. Dods, Jr.,

and Bruce E. Tinling, dtd July 1955

5 NACA Report 938 - "Summary of Section Data on Trailing-Edge

High-Lift Devices", by Jones F. Cahill, dtd 19)49

6 Cornell Aeronautical Laboratory, Inc., Report No. AF-7143-A-2

of January 1953, "Aerodynamic Characteristics of Low-Aspect-Ratio Wings with Various Flaps at Subsonic Speeds", by

H. N. Stone

7. TMB CONFIDENTIAL Report C-703 of April 1955 - "Surfaced Turning,

Maneuvering, and Rudder Torque Tests on Model 4-538 Representing the SSN (SMALL) 3S578", by C. R. Olson and F. D. Bradley

¡.40 ¡.20 I.00 0.80 0.60

0.0o

o LOO

0.80

0.5

060

N

' L

0.20

0.00 o

lo

20% FLAP

40% FLAP

1.0 _'.5

Wa

25

a

(deg) 15

a) Ahead Condition

ao

'5

AE 2.5

b.) Astern Condition

a

(deg)

Figure 3 Lift CoefficienT

_Curves

3.0

5

1.50 ¡.40

[30

.20

L'o

0.8C C L 0.70

0.60

0.50

0.40

0.30

0.20

0.10

(tcb linkage ratic)

25

I.0

(.eg)

nu

0.5

2.0

[.5

_'=5

0.0

o

¡0

20

30

40

FLAP AREA IN PERCENT

Figure 4 - Effect of

_{Flap Area on the Lift}

_Coefficients

For the Ahead Condition

LOO

0.80

0.70

0.60

C L

0.50

0.40

0.30

0.80

0.70

0.60

CL

0.50

0.40

0.30

0.40

C

L030

0.20

o IO

20

30

FLAP AREA IN PERCENT

12

¿

40

Figure 5

Effect of Flap Area on the Lift Coefficients

for the Astern Condition

(tab Iinkge ratio)

L

_____

LO

î.-0.5

--Iìi.

-____

e g)

I.51IPJ!

0.5 0.4 0.3 0D

02

0.I 0.0 O

---40% FLAP

a

(deg)

A25

a) Ahead Condtion

Wa

15

Figure 6 - Drag Coeflicient

_Curves

b.) Astern

a

(deg) I

Condition

I

-

---0.5 LO 1.5 _{2.0 2.5. 3.0}

0.6 0.5 0.4 0LE 0.3 0.2 0. o. o 0.4 0.3 ' L E 0.2 0.1 o

o

0.5 0.5 L0 1.0 14 1.5

8/a

1.5

8/a

ao

2.0

Figure 7 Torque Coefficienl

_Curves

2.5 3.0 2.5

ï - i

I

T

_i

--20%

40%

FLAP

FLAP

_{a) Ahead Condition}

a

(deg) 25

b.) Astern Condition

a

(deg) 25 I

I

A 15 5

°LE

0.4 0.3 0.2

0j

0.3 0.2

°LE

0.0 o

0.l

0.0

o

-ci

- . -S.4 ... WOE

Figure 8 - Center of

_{Pressure Curve,}

2.5

-í

i I

20%

40%

J

1

FLAP

FLAP

a

(deg)

-

5--a

(deg) f5

t

a.) Ahead Condition

Il-,

I I t

b.) Astern Condition

25

a

'5 A 0.5 1.0 '.5 2.0 2.5

30

0.5 I.0

i6

APPENDIX

The faired testdata are given as a function of rudder angle in the following figures:

Figure _T Lift Coefficient Curves Figure 10 - Drag Coefficient Curves Figure 11 - Moment Coefficient Curves Figure 12 - Center of Pressure Curves

.4 .3 1.2 LO 0.9 08 0L 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 CL L O 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 O lO 15

a

20 25 CL 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

Figure 9

_{- Lift Coefficient Curves}

0 5 IO 15

a

- ...--... 20 25

iii

Ahead 40% Conditions FLAP

iiiiIJF

A

i 30AYA

ii

2.5 2.0

iiVWA,Amaii

iwimraiiia

IINI1I1II

Ahead Condition FLAP

'

20% 1.5

A

iiiii

_{ii UVA}

2

:4a

'.i

Ä4

wr,i,maiii

riauiii

iii

iiiii i

Astern 40% Condition FLAP

'a

All

_0.5

iiVAiUi

mai

lau

mii

Astern 20% Condition FLAP

ihm

amai

I.0

iiA

I.5

r

0 5 lO 15 20 25 _{o lO 15 20 25}

a

_a

.4

.3 .2 1.0 0.9 0.8 CL _0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

CD C D 0.5 0.4 0.3 0.2 0.I 0.0 o 0.5 0.4 0.3 0.2 0.I 0.0

:0

5 IO 15

a

5 lO 15 ci 20 20 25 25 18 CD 0.5 0.4 0.1 0.0 o 0.5 0.4 0.3 0.2 0.! 0.0 o

Figure 0 - Drag Coefficient Cyrves

5 IO 15

a

20. 25 Ahead Condition FLAPI

iIii

40%

u..

ji

2SJ.d

maI10

ÍO

aw 6.5

00111

Ahead 20% Condition FLAP

I

-

,iiIAVA

304

JA

.

420

__Ø

11V

%5

ll

III

44

i.114l

II-I

I IIVArAlU

i lIar4rÁr4alI

0.0 Astern 20% Condition FLAP

Ill R

I-óJaa

I.

_,-R.l.

AÀRRRR

Astern Condition 40 % F L AP 5 IO 15 20 25 a 0.3 Cb 0.2

LE 0.6 0.5 0.4 0.3 0.2 0.1 0. o 0.3 co Li. 0.2 0.1 0.0 -0.1 -0.2 o 5 IO 15 a 20 lO 15 a 20 25 25 CQ co LE. 0.6 0.5 0.4 0.3 0.2 0.1 LE. 0.0 o 0.3 0.2 0.1 0. -0.1 -0.2 o 5 IO 15

a

Figure ¡

- Moment Coefficient Curves

20 25 5 IO 15

a

20 25 i Ahead 40%

1-i

Condition FLAP i 2.5 3.0

14

2.0

arÁ!1 Ali

U

I T I Ahead Condition 20% FLAP

- ...

UI 30

UAV'

Ui

uil

0.01111

U

P

Ui

0.5

UAU

,,,

Ua

Astern

r40FLAP

Condition

1,09

4I.O

-.--__ 2.5

-3.0

t

___/ .- Astern 20 % Condition FLAP -1.0

.-»

L15 :o. 2.5 3.0

I

1 I

0.0-0 5 lO 15 20 25

a

a

20

CP LE. 0.3 0.2 0.I

Figure 2 - Center of Pressure Curves

[

Ahead Condition -40% FLAP I T 3.0 2.0

-l.5

-I

0.5

P..

Astern 40%

III

Condition FLAP 0.5 4 0.3 0.2 -0.0 0.3-I O. I

i

0.2 1L.E. 1

-

LO 0.0 l.5 -0.! Astern Condition FLAP 20% I _0.0 -0.2 -0.I 0 5 lO 15 20 25 0.5 Ahead 20% Condition FLAP Î 0.6 0.5 0.4

0.4

--t-

-3.0

'a.

-2.5-2.O

0.3

_j.o ô 5

io

15 20 25

a

CP LE. 0.2 o.' 25 20 o 5 lO 15 a

I

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t

David W. Taylor Model Basin.

Rapt. 991.

EF'FECTS OF VARIOUS LINKAGE RATIOS ON TIlE FREE.

STREAM JI'DI1ODYNAMIC CHARACTERISTICS OF AN AI,L- MOVABLE FLAPPED RUDDER, by C.R. Olson.

September 1955

iii, 21 p. md. figs., tables, refs.

UNCLASSIFIED

The effects of various linkage ratios on the lift, drag, and

torque characteristics of an all-movable-fl appod rudder, having either a 20 percent oc 40 percent flap, have been detormined from free-stream tests, at low Iboynolds nuinbor, for both ahead and astern conditions.

The results indicate that fo.- the ahead condition the highest

lift coefficient is obtained with a 30 percent flap using

a 1.5

flap-link age ratio. An increase in either the flap-chord

oc the

flap-linkage ratio reduces the lift-drag ratio of the rudder and

ahifta

the center of pr.iaur. rearward.

Rudders (Marine)

Control surfaces Model test

Rudders (Marine)

-Hydrodynamic character- istics

Submarines

Maneuverability _I.

Olson, Clifford R.

IJ.

NS7IS-102

David W. TayIQr Model flítsin.

Rept. 091.

EFFECTS OI VARIOUS LINKAGE RATIOS ON TIlE FREE.

1. Rudders (Marine)

STREAM HYDRODYNAMIC CHARACTERISTICS OF AN

ALLControl surfaces

-MOVABLE FLAPPED RUDDER, by C.R. Olson.

September 1935.

Model test

iii, 21 p. md. figs., tables, rafa,

UNCLASSIFIED

2. Rudders (Marine) Hydrodynamic

character-The effects of various linkage ration on the lift, drag, and

istica

torque characteristics of an all-movable-flappod rudder, having

3. Submarines

-either a 20 percent

or

40 percent flap, have boon determined from

Maneuverability

free-stream tests, at low Reynolds number, for both ahead and

I.

Olson1 Clifford R.

astern conditions.

II. NS 715-102

The results indicate that for tho ahead condition the highest

lift coefficient in obtained with a 30 percent flap using a 15 flap- linkage ratio. An increase in either the flap-chord

or ute flap.

linkage ratio reduces the lift-drag ratio of the rudder and

hstt

the center of prea.ur. r.arvard.

David W. Taylor Model Basin.

Ropt. 091.

EFFECTS OF VARIOUS LINKAGE RATIOS ON TIlE FREE.

1. Rudders (Marine)

STREAM IIYDROI)YNAMIC CIIARACTER ISTICS OF AN

ALLControl surfaces

-MOVABLE FLAPPED RUDDER, by C.R. Olson.

Septeniber 1955

Model test

lii, 21 p. mcl. figs., tables, refs.

UNCLASSIFIED

2.

Rudders (Marine)

Hydroalynamic

character-The effects of various linkage ratios on the Lift, drag, and

istics

torque characteristics of

an

all-movable-flapped rudder, having

3. Submarines

-either a 20 percent oc 40 percent flap, have boon determined from

Maneuverability

Iree. stream tests, at low Reynolds number, for both ahead and

I.

Olson, Clifford R.

astern conditions.

II. NS 715- 102

The results indicato that for the ahead condition the highest

lift coefficient is obtained with a 30 percent flap using a 1.5 flap- _{linkage ratio. An increase in either the flap-chord or the (lap. linkage ratio reduces the lift-drag ratio of the rudder and huft} th. center al prs.aure re.x-wsrd. David W. Taylor Model Basin.

Rept. 991.

EFFECTS OF VARIOUS LINKAGE RATIOS ON TIlE

FREE-1.

Rudders (Marine)

-STREAM IIYDIIODYNAMIC CHARACTERISTICS OF AN

ALL-Control surfaces

MOVABLE FLAPPED RUDDER, by C.R. Olson.

September 1955

Model test

iii, 21 p. md, figs., tables, rofs.

UNCLASSIFIED

2.

Rudders (Marine)

-flydrodynarnic character.

The effects of various linkage ratios on the lift, drag, and

istics

torque characteristics of

an

all-movable-flapped rudder, having

3.

Submarines

either a 20 percent

or

40 percent flap, have boon determined from

Maneuverability

free-stream tests, at low Reynolds number, for both ahead and

I.

Olson, Clifford R.

astern conditions.

U. NS 715-102

The results indicate that for the ahead condition the highest

lift coefficient is obtained with a 30 percent flap using a 1.5 (lap- linkage ratio. An increase in either the flap-chord

or the

(Ial>-linkage ratio reduces tho lift-drag ratio of the rudder and

ahi fr.