Category Model Aircraft Aerodynamics

DESIGNING NEW AEROFOILS

A mathematical method of working out new ordinates, starting from a chosen camber line and a symmetrical thickness form, is given in the standard text, Theory of Wing Sections by Abbott and Von Doenhoff. Although not a quick procedure, the calculations required are not difficult Failing this method, a modeller may devise his own wing sections by the graphical method outlined in Fig. A 8. The camber line is plotted first then at each station a circle is drawn, the radius being taken from the ordinates for the thickness form, and the centre being on the camber line at the appropriate point Finally a smooth curve is drawn tangentially to aU the circles and the nose radius to produce the aerofoil.

WORKING OUT THE CAMBER

Many aerofoil designations contain information about camber and thickness. The NACA systems are described in Chapter 7, with further information in Abbott and Von

Doenhoff as above. The Benedek Aerofoils give the camber, in percent of chord in the last digit, thus В 10355 is cambered 5 percent, 8356, 6 percent and so on. The first figure gives the profile thickness, the central figure or figures the position of the maximum thickness point, hence Benedek В 12355 is 12% thick at 35%, and cambered 5%.

In other cases it is sometimes possible to work out the maximum camber arithmetically. This is applicable only where the aerofoil ordinates are based on a chord • line running through the leading edge and trailing edge. In these cases, by finding the thickness of the profile at a number of stations and subtracting half this figure, at each place, from the upper surface ordinate, the approximate ordinate for the camber line is found and the maximum value then easily discovered. Note that in finding the profile thickness in this way, minus signs below the chord line must be allowed for. This method of halving the thickness will not produce accurate ordinates for the aerofoil camber line, especially near the nose and trailing edge, but it will produce a correct figure for the maximum camber and its location.

The camber of other aerofoils, plotted on tangential chord lines or on other arbitrary reference lines, may be estimated by measuring from the plotted profile. The true chord line, nose to trailing edge, must first be drawn in, then the half thickness plotted as accurately as possible, and the camber measured.

More accurate estimates of camber from the ordinates may – be carried out by somewhat more complex arithmetic, but this is seldom necessary.

Fig. A8 Fig. A8

i take ordinates of desired camber line, and plot.

ii take radii ot circles from desired thickness and draw on arc as shown. Hi draw smooth curved lines tangential to circles.

INDEX OF AEROFOILS

Mean lines, low drag bodies and symmetrical profiles are grouped at the beginning of this section. The N. A. C. A. ‘6’ thickness forms are arranged in order of increasing laminar flow (min. pressure point at 30, 40, 50% chord etc), and in increasing thickness (6, 9, 10, 12, 15, 18% etc.) In the section devoted to cambered profiles, the N. A. C. A. 6 series aerofoils are arranged in order of increasing thickness (9 to 18%), then by increasing camber (shown by the third figure from the right which indicates the ‘ideal’ ci in tenths -.2, .3, .4, .6 etc), and then by increasing proportion of laminar flow (given by the second digit from the left, 30%, 40%, 50% etc.) The various letters and other additions indicate minor modifications to the profiles (e. g., A =0.5 indicates the use of A = 0.5 mean line instead of the usual A = 1).

MEAN LINES

Reflex for zero pitching moment, каі« to reqd camber

NACA

210. CL Ideal 0.3

NACA A • 0.0

NACA A-0.5

NACA A-0.9

NACA A – 1.0

Chord

Upper

Chord

Upper

Chord

Upper

Chord

Upper

Chord

Upper

Chord

Upper

Station

Surface

Station

Surface

Station Surface

Station

Surface

Station

Surface

Station Surface

XU

vu

XU

VU

XU

YU

XU

vu

XU

vu

XU

YU

.000

.000

.000

.000

.000

.000

.000

.000

.000

.000

.000

000

5.000

3.240

1050

S96

.500

480

500

.345

.500

.289

.500

.250

10.000

5.770

2.500

.928

.750

.841

.750

.485

.750

979

.750

.350

15.000

7.850

5.000

1.114

1.250

.964

1.250

.735

1.250

.577

1.250

.535

20.000

8.940

7.500

1.087

2.500

1.641

2.500

1.295

2.500

1.008

2.500

.930

25.000

9.700

10.000

1.058

5.000

2.693

5000

2.205

5 000

1.720

5.000

1.580

30.000

9S90

15000

899

7.500

3.507

7.500

2.970

7.500

2.318

7.500

2.120

35.000

9080

20.000

840

10.000

4.161

10.000

3.830

10.000

2835

10.000

2.585

40.000

9.4Э0

25.000

881

15.000

5.124

15.000

4.740

15.000

3.707

15.000

3.385

45.000

8.700

30.000

.823

20.000

5.747

20800

5920

20.000

4.410

20.000

3.980

60.000

7.780

40.000

.705

25.000

8.114

25800

6910

25.000

4.980

25.000

4.475

55.000

8.880

50.000

.588

30.000

8877

30.000

8840

30.000

5.435

30.000

4860

50.000

5.480

80.000

.470

35.000

6.273

35.000

7915

35.000

5.787

35.000

5.150

55.000

4040

70.000

.353

40.000

6.130

40.000

7.430

40.000

6.045

40.000

5.355

70.000

3.040

80.000

.235

45.000

5971

45.000

7.490

45.000

6.212

45.000

5.475

75.000

1040

90.000

.118

50.000

5.516

50.000

7950

50.000

6.290

50.000

80.000

090

•95.000

.059

56.000

5.081

55.000

8865

55.000

6.279

85.000

060

100.000

.000

60.000

4.581

60.000

8.405

60.000

6.178

60.000

90.000

-.190

85.000

4.032

65.000

5.725

85.000

5.981

85.000

5.150

95.000

-.300

70.000

4.955

70.000

5.681

70.000

100.000

.000

75.000

2836

75.000

4.130

75.000

5.265

75.000

4.475

80.000

3.285

80.000

4.714

80.000

3.980

85.000

2995

85.000

3.987

85.000

3.365

90.000

90.000

1.535

90.000

2.984

90.000

2.585

95.000

.467

95.000

.720

95.000

1.503

95.000

1.580

.000

100.000

.000

100800

.000

100.000

.000

REFLEX MEAN LINE FOR ZERO PITCHING MOMENT SCALE TO REQUIRED CAMBER

NACA 210 MEAN LINE CL IDEAL 0.3

NACA A ** 0.0 MEAN LINE

NACA A – 0.6 MEAN LINE

0.9 MEAN LINE

NACA A – 1.0 MEAN LINE

МАСА 0010 L. E. radius
1.10 Percent

МАСА 0009 L E. radius
0.89 Percent

Chord

Upper

Chord Lower

Station Surface

Station Surface

XU

vu

XL YL

.000

.000

.000 .000

.500

.749

.500 -.749

.750

.906

.750 -.906

1.350

U151

1.350 -1.151

3.500

1.583

3.500 -1.583

5.000

3.196

5.000 -3.198

7.500

3.655

7.500 -3.655

10.000

3.034

10.000 -3.034

15.000

3.591

15.000 -3.591

30.000

3.997

30.000 -3.997

30.000

4.443

30.000 -4.443

40.000

4.447

40.000 -4.447

50.000

4.056

50.000 -4.056

60.000

3.358

60.000 -3.358

70.000

3.458

70.000 -3.458

80.000

1.471

80.000 -1.471

90.000

.550

90.000 -.550

95.000

.196

95.000 -.196

100.000

.000

100.000 .000

Chord

Upper

Chord

Lower

Station Surface

Station

Surface

XU

YU

XL

YL

.000

.000

.000

.000

.050

.160

.050

-.160

.100

.340

.100

-.240

.300

.350

.300

-.350

.400

.500

.400

-.500

.500

.503

.500

-.503

.750

.609

.750

-.609

1350

.771

1.250

-.771

3.500

1.057

2.500

-1.057

5.000

1.463

5.000

-1.462

7.500

1.766

7.500

-1.766

10.000

3.010

10.000

-2.010

15.000

3.386

15.000

-2386

30.000

3.656

20.000

-2.656

30.000

3.954

30.000

-2.954

40.000

2.971

40.000

-2.971

50.000

2.723

60.000

-2.723

60.000

2.267

60.000

-2.267

70.000

1.670

70.000

-1.670

80.000

1.008

80.000

-1.008

90.000

.383

90.000

-.383

95.000

.138

95.000

-.138

100.000

.000

100.000

.000

Chord

Upper

Chord Lower

Station

Surface

Station Surface

XU

YU

X

■<

r

.000

.000

.000 .000

300

.620

300 -.620

.400

.910

.400 -.910

.600

1.130

.600′ -1.120

900

1.350

.800 -1.250

1350

1.578

1.350 -1.578

2.500

2.178

2.500 -2.178

5.000

2.962

5.000 -3.962

7.500

3.500

7.500 -3.SOO

10.000

3.902

10.000 -3.902

15.000

4.455

15.000 -4.455

20.000

4.782

30.000 -4.782

25.000

4.952

35.000 -4.953

30.000

5.002

30.000 -5.003

40.000

4337

40.000 -4337

50.000

4.412

50.000 -4.412

60.000

3303

60.000 -3303

70.000

3.053

70.000 -3.053

80.000

2.187

80.000 -2.187

90.000

1.207

90.000 -1.207

95.000

.672

95.000 -.672

100.000

.105

100.000 -.105

Chord

Upoer

Chord

Lower

Stanon!

Surface

Station Surface

XU

YU

XL

YL

.000

.000

000

000

.200

490

.700

– 490

.400

.870

400

– 870

.600

1 010

.600

■1 0Ю

.800

1.170

BOO

-1 170

1 250

1.420

1.750

-1 470

3.500

1.961

2500

-1 961

s. ooo

3666

5.000 -7666

7.500

3.150

7 500 -3 150

10.000

3512

10.000 – 3517

15.000

4.009

15.000 -4.009

20.000

4.303

30.000 -4 303

25.000

4.456

25.000 -4.456

30.000

4.501

30.000 -4.501

40.000

4.352

40000 -4 357

50.000

3.971

50.000 -3.971

60.000

3.423

60.000 -3.473

70.000

2.748

70.000 – 2.748

80.000

1.967

80.000 -1.967

90.000

1.086

90.000 -1.086

95.000

.605

95.000 -.605

100.000

.095

100.000 -.095

NACA 63 009

XL YL.000 .000 M0 – IMS.7» -1.837

1.350 -3.537

3.500 -3.577

5.0 -5.0S5

7.600 -5.183

10.0 -7880

15.0 -8.441

30.0 -9410

30.0 -10413

40.0 -10.398

50.0 -9JOS

80.0 -7441

70.0 -5390

80800 -3.064

90.0 -1.113

95.0 -.392

100.000 .000

NACA 64 0Ю LE RADIUS 0.720 PERCENT

NACA 64 000 LE RA0IUS 0.576 PERCENT

NACA 65 A 006

NACA 65 006

NACA 64 2 016 L. E. rebut 1.590 Percent

NACA 64 1012

Chord

Upper

Chord Lower

Chord

Upper

Chord

Lower

Chord

Upper

Chord Lower

Chord

Upper

Chord Lower

Station

Surface

Station Surface

Station Surface

Station

Surface

Station

Surface

Station Surface

Station

Surface

Station Surface

XU

YU

XL YL

XU

YU

XL

YL

XU

YU

XL YL

XU

YU

XL YL

800

.000

.000 .000

800

.000

.000

.000

800

800

800 .000

.000

.000

800 800

.500

.615

800 -815

800

.476

.500

-.476

800

1808

800 -1808

800

878

800 -878

.750

.746

.750 -.746

.750

874

.750

-.574

.750

1.456

.750 -1/456

750

1.179

.750 -1.179

1.260

851

1850 -851

I860

.717

1850

-.717

I860

1842

1850 -1842

I860

1.490

I860 -1.490

2800

1.303

2800 -1803

2800

.956

2800

-856

2800

2828

2800 -2828

2800

2.035

2800 -2835

5.000

1.749

5.000 -1.749

5.000

1.310

5.000

-1810

5800

3804

5800 -3.604

6.000

2810

5800 -2810

7800

2.120

7.500 -2.120

7800

1869

7800

-1.589

7800

4840

7800 -4840

7800

3894

7800 -3894

10.000

2.432

10.000 -2.432

10800

1824

10.000

-1824

10.000

4842

10.000 -4842

10800

3871

10800 -3871

15.000

2826

16800 -2826

15.000

2.197

16800

-2.197

15800

5.786

15800 -6.785

15800

4820

15.000 -4820

20800

3.301

20.000 -3.301

20.000

2.482

20.000

-2.482

20800

6.480

20.000 -6.480

20800

6.173

20800 -5.173

30800

3.791

30.000 -3.791

30800

2852

30.000

-2852

30800

7819

30800 -7819

30.000

5844

30800 -5844

40.000

3895

40800 -3.995

40.000

2888

40800

-2888

40800

7.473

40.000 -7.473

40800

5881

40800 -5881

50.000

3895

60.000 -3895

50800

2800

50.000

.-2800

50800

6810

50800 -6810

50.000

5.480

50800 -5.480

60800

3.456

60800 -3.456

60800

2818

60.000

-2818

60800

6820

60800 -8820

60800

4848

60.000 -4.548

70.000

2.763

70.000 -2.763

70800

1835

70.000

-1835

70.000

4.113

70800 -4.113

70800

3850

70.000 -3860

80.000

1898

80.000 -1898

80800

1833

80800

-1833

80800

2.472

80.000 -2.472

80800

2.029

80800 -2.009

90.000

860

90.000 -860

90800

810

90800

-810

90.000

850

90800 -.950

90800

.786

90800 -.785

95.000

.489

96800 -.489

95.000

.195

95800

-.195

95800

846

95800 -845

95800

286

95.000 -886

100.000

818

100800 -.018

100.000

800

100.000

.000

2

100.000

93

800

100.000 800

>

100.000

800

100800 800

NACA 65 3 018 L. E. radtu*

XL YL.000 .000 600 – 765 .750 *628 1650 -1.163 2600 -1.673 5.000 -2.182 7600 -2.650

10.0 -3.040

15.0 -3658

20.0 -4.127

30.0 -4.742

40.0 -4.995

50.0 -4663

60.0 -4604 70600 -3.432

80.0 -2.332

90.0 -1.188

95.0 – 604 100.000 -.021

Wortmenn FX LIIM42/K 25
For 25 percent flep

CLARK Y

Gottingen 548
(1930 to 1950 period)

Gottingen 804 (Eppler EA 8 (-11
-12 06) L. E. radius.5 camber 0.67
at SO

Chord

Upper

Chord

Lamar

Chard

Upper

Chord Lower

Station Surface

Station

Surface

Station

Surface

Station Surface

XU

YU

XL

YL

XU

YU

XL YL

.000

OOO

.000

.000

.000

OOO

OOO OOO

.000

050

.200

*.430

OOO

1000

000 – ooo

200

1050

.400

-000

.400

1060

OOO -.770

MO

2000

.600

‘040

OOO

.1800

OOO ‘050

Я00

2.270

OOO

‘060

OOO

0060

OOO ‘1.120

1250

1730

1050

-1030

1050

2.440

1060 -1.430

2.S00

3000

2000

‘1040

2000

3.390

2000 -1060

SJOOO

6060

5000

-1.990

5.000

4.730

6.000 -2.490

7900

6.570

7000

‘2050

7000

5.780

7000 ‘2.740

10900

7080

10.000

‘1090

10.000

8090

10000 -2060

15.000

9.180

15.000

‘1070

15000

7080

15000 -2080

20.000

10040

20000

-1060

20000

8000

20.000 ‘2.740

»j000

11.140

25000

-.780

25000

9010

25000 ‘2000

30.000

11050

30000

-080

30000

9.780

30000 -2060

40900

IIOOO

40000

OOO

40000

9000

40.000 -1000

50000

11.180

50000

050

80000

9.190

50000 ‘1.400

60Л00

9050

60000

.780

80000

9.140

60.000 -1.000

70.000

8030

70000

050

70.000

6080

70000 -050

80000

6030

80000

.730

80000

4080

80000 – 090

90.000

Э. ЗЭ0

90.000

.390

90.000

2.710

90000 *230

95.000

1.790

95.000

.180

95000

1.470

95000 -130

100000

.120

100000

-.120

100.000

.130

100.000 -.180

Chord

Upper

Chord Lower

Chord

Upper

Chord

Lower

Station

Surface

Station Surface

Station

Surface

Station

Surface

XU

YU

XL YL

XU

YU

XL

YL

OOO

OOO

OOO.000

.000

OOO

.000

.000

OOO

020

OOO ‘.610

1008

1094

1.402

-1.448

.400

1.280

OOO ‘OOO

2.297

2.411

2.703

-1027

OOO

1010

OOO ‘1.100

4.742

3.420

5058

-2.462

OOO

1.730

OOO ‘1091

7317

4.168

7.783

-2009

1.260

2.150

1050 -1060

9.710

4.786

10090

-3.016

2.600

2080

2000 ‘2070

14.722

5065

15078

-3.227

5000

4.130

5.000 -3.010

19.781

6076

20039

-3076

7000

4080

7000 ‘3.480

24014

6.668

25.166

-3.230

10000

6030

10.000 -3.750

29076

6075

30.125

-3.125

15000

6010

15000 -4.100

40000

6037

40.000

-2037

20.000

7060

20.000 ‘4030

50.049

6.356

49051

-2.468

25.000

7070

25.000 -4.220

60.005

5.580

59015

-2.024

30000

7080

30.000 -4.120

79.W2

4.551

89098

‘1.551

40.000

7000

40000 ‘3000

80087

3.296

79003

-1 074

60000

7040

50000 -3040

90087

1016

89033

-.594

60000

6080

80000 ‘2.780

95.041

090

94059

-.352

70.000

5.180

70.000 ‘2.140

100000

.105

IOOOOO

-.105

80000

3.750

80.000 -1.600

90000

2.080

90000 -020

95000

1.140

96.000 -.480

100.000

.130

100.000 -.130

NACA 4415 LE RAOIUS 2.49 PERCENT

NACA 64409

NACA 63*209

NACA 23012 L. E. radius 1.68 percent

NACA 4415 L. E. radius 2.48 percent

Chord

Upper

Chord Lesser

Chord

Upper

Chord

Lesser

Chord

Upper

Chord Lessor

Cherd

Upper

Chord Lesser

Station

Surface

Station Surface

Station Surface

Station

Surface

Station

Surface

Station Surface

Station

Surface

Station Surface

XU

VU

XL YL

XU

YU

XL

YL

XU

YU

XL YL

XU

YU

XL YL

OOO

.000

OOO.000

OOO

OOO

.000

OOO

.000

.000

.000 900

900

.000

.000 .000

077

029

023 ‘029

.437

.798

.583

‘098

OOO

1.300

900 -.488

.000

940

200 -.550

013

1.021

087 -.741

080

073

020

-033

.400

1680

.400 -.730

.200

1950

.400 -.920

1095

1031

1.405 -003

1.170

1056

1030

-1041

OOO

1920

900 -.910

900

2950

.600 ‘1.180

2022

1095

2078 -1.151

2.408

1.768

2092

-1093

OOO

2.200

900 -1930

900

2950

900 -1.400

4003

2.732

5.197 ‘1.468

4097

20Ю

5.103

-1078

1050

2970

1950 -1930

1.250

3.070

1.250 -1.790

7.297

3.383

7.703 -1087

7094

3077

7008

-2.229

2000

3910

2900 -1.710

2.500

4.170

2.500 -2.480

9.798

3026

10002 -1057

9094

3038

10.106

‘2006

5000

4910

5900 -2960

5.000

5.740

6.000 -3.270

14010

4.795

15.190 -2.104

14001

4083

16099

-2.917

7000

5800

7900 -2910

7900

6.910

7.500 -3.710

19030

5.458

20.170 -2072

19012

4.792

20088

‘3000

10000

6930

10900 -2920

10.000

7940

10.000 -3.980

24054

5067

25.146 -2077

24025

5.169

25075

‘3079

16000

7.190

15900 -3900

16.000

9.270

15.000 -4.180

29082

6.315

30.118 -2.427

29040

5414

30.080

-3.470

20000

7900

20.000 ‘3.970

20.000

10.250

20.000 -4.150

34.912

6038

35.088 ‘2.418

34056

6030

36.044

-3.470

25.000

7900

25.000 -4260

25.000

10920

25900 -3.980

39.942

8.632

40.058 -2048

38071

5.518

40029

-3078

30.000

7950

30900 ‘4.480

30.000

11.250

30.000 -3.7S0

44.972

6.564

45.028 -2.174

44088

5091

«014

-3001

40.000

7.140

40900 -4.480

40.000

11.250

40.000 -3.250

50.000

6042

50.000 -1030

50.000

5.159

60000

‘2053

50.000

8.410

50900 -4.170

50.000

10.530

50.000 -2.720

60.045

6094

59056 ‘1.310

60.022

4.429

55978

-2.287

80.000

6.470

60.000 -3970

60900

9.300

60900 -2.140

70089

4004

60031 -016

70.033

3.430

88087

-1.486

70.000

4980

70900 -3900

70.000

7.630

70.000 -1950

80088

3.154

79031 -.030

80.032

2087

79068

‘075

80.000

3980

80900 -2.160

80.000

5.550

80.000 -1.030

90.043

1044

89079 .424

90019

1067

88.981

‘.033

90.000

1980

90.000 -1930

90.000

3.000

90.000 -.570

95.021

058

94.979 .406

95008

.612

94.991

.120

95.000

920

95.000 -.700

95.000

1970

95.000 -.360

100.000

.000

100.000 .000

IOOOOO

.000

IOOOOO

.000

100.000

.130

100.000 -.130

100.000

.160

100.000 -.160

Chord

Upper

Chord

Lower

Chord

Upper

Chord Lower

Chord

Tppar

Chord

14МЄГ

Chord

Upper

Chord

Lower

Station

Surface

Station

Surface

Station Surface

Station Surface

Station Surface Station

urface

Station

Surface

Station

Surface

XU

YU

XL

YL

XU

YU

XL YL

XU

YU

XL

YL

XU

YU

XL

YL

.000

.000

.000

.000

.000

.000

.000 .000

.000

.000

.000

.000

.000

400

400

.000

450

402

450

-478

499

473

401 -.723

.424

466

470

-.744

.423

468

477

-756

М2

1.112

418

-.796

.638

1.068

462 -458

4S5

1444

.035

-488

464

1.058

436

-.900

1.0S9

1.451

1.441

-409

1.123

1.379

1.377 -1.067

1.153

1442

1447

-1.100

1.151

1487

1449

-1.125

2476

2495

2.724

-1451

2.353

1461

2447 -1.403

2457

1455

2413

-1.473

2454

1444

2416

-1422

4.740

3.034

5451

-1492

4437

2.759

5.163 -1447

4474

2465

6.126

-1463

4405

2.769

5.131

-2.047

7.230

3465

7.770

-1419

7432

3.436

7468 -2.164

7469

3455

7.931

-2416

7464

3.400

7453

-2.428

9.737

4490

10463

-1496

9432

3.970

10.168 -2.420

9459

3.792

10.132

-2400

•463

3417

10.137

-2.725

14.748

5466

15452

-2444

14442

4419

15.158 -2409

14474

4482

20.116

-3430

14469

4.729

15.131

-3.157

10.770

6.126

20430

-2.406

19459

6.464

20.141 -3475

19485

5400

10.116

-3440

19482

5428

20.118

-3.469

24,800

6.705

25400

-2.499

24479

6446

25.121 -3462

24400

5456

25.100

-3464

24498

5.764

25.102

-3462

29434

7.13»

30.166

-2437

29.902

6494

30498 -3478

29417

5.994

30.083

-3.688

29.916

6.060

30.064

-3.764

34471

7.414

35.129

-2.518

34427

6413

35473 -3.423

34435

6.192

35405

-3.744

34436

6419

36465

-3.771

39410

7452

40.090

-2.436

39452

6401

40.048 -3489

35455

6474

40.045

-3.718

39456

9447

40.045

-3489

44450

7422

46.050

-2.266

44477

6436

45.023 -3452

44475

6406

46.026

-3.680

44.976

9.151

45425

-3423

40009

7444

50.011

-2424

50.000

6434

50.000 -3.030

45475

6414

50405

-3464

49.994

5443

50.005

-3483

60.067

6424

69443

-1.418

60.039

5427

69461 -2.415

50425

6423

59472

-2.719

90.029

5445

50472

-2.641

70.108

5.490

69492

-.760

70.063

4.584

69437 -1468

70454

4410

•9445

-1.944

70.052

4427

09448

-1461

90.151

3.967

79449

-429

80470

3496

79.930 -.908

50.076

3407

79424

-1.167

90.074

2.974

79426

-1.104

90.104

2438

90496

-.076

90466

1436

89444 -.296

80.062

1.551

85445

-471

90.090

1419

•9460

-439

95.053

1428

94447

-.048

96.038

1414

94482 -485

96427

.795

94474

-405

95.029

.769

94.974

-479

100.000

421

100.000

-.021

100.000

.021

100.000 -.021

100.000

421

100.000

-421

100.000

.021

100.000

-.021

МАСА 63-1-212

г

2

2

о

e

1

МАСА 94-A410 ІАЦ>4МОО)

Chord

Upper

Chord

Lower

Chord

Upper

Chord Lower

Chord

Upper

Chord

Lower

Chord

Upper

Chord

Lower

Station Surface

Station

Surface

Station

Surface

Station Surface

Station

Surface

Station

Surface

Station Surface

Station

Surface

XU

YU

XL

YL

XU

YU

XL YL

XU

YU

XL

YL

XU

YU

XL

YL

.000

.000

.000

.000

400

400

400 400

.000

.000

400

.000

400

400

.000

.000

.417

1.032

.583

-432

.435

419

485 -.712

415

477

.795

-427

414

479

.786

-426

JS57

1460

443

-1.120

478

499

422 -459

494

1461

1418

-499

481

1j550

1419

-.666

1.145

1422

1.355

-1.408

1.169

1473

1431 -1469

2472

2.470

2428

-.799

2.084

2.476

2439

-.767

2478

2494

2422

-1412

2.408

1.757

2492 -1485

4420

3405

5.480

-466

4408

3.719

5.494

-432

4453

3438

5.137

-2406

4498

2.491

5.102 -1469

7403

4450

7497

-453

8484

4.703

8419

-411

7458

3463

7442

-3.115

7494

3489

7406 -2421

9403

5.457

10.457

-.434

9.479

6441

10421

-.771

9459

4454

10.141

-3420

9494

3456

10.106 -2421

14430

6414

16.470

-.756

14.500

•402

16400

-456

14488

5.470

15.132

-4.124

14499

4438

16.101 -2492

10479

7433

20.422

-496

16443

7468

20.457

-.526

19482

6.137

20.118

-4445

19409

4438

20491 -3446

24439

6420

25451

-.595

24401

8.795

25499

-483

34400

6.606

25.100

-4416

24421

6497

26479 -3407

29.707

0402

30 493

-.454

29468

9.420

30432

-432

29420

6401

30490

-4457

29436

5.732

30484 -3.788

34.790

9.596

35420

-.328

34.742

9457

35459

-.065

34441

7430

35450

-4470

34451

6454

35.049 -3494

39455

9413

40.145

-.174

38420

10.107

40.190

.123

39482

6491

40438

-4449

39.966

8467

40432 -3425

44430

9422

45470

.034

44.900

10.150

45.100

.354

44482

9.799

45.018

-4409

44494

6488

45418 -3488

50400

9449

60400

400

49477

10405

80.023

437

90.000

6.473

50400

-4467

50.000

5416

50400 -3.709

00.117

9439

50453

401

•0.114

9425

89496

1.187

80429

5.491

59471

-3449

80427

6417

59473 -3476

70.150

7.496

50411

1453

70415

7450

69.795

1410

70443

4.182

69457

-2438

70443

4.128

89457 -2.194

•0405

6475

79.792

1.495

•0400

5419

79.700

1457

80442

2498

79.958

-1.109

90444

2.793

79458 -1.191

50.166

3475

•9435

1477

90404

3.004

89.799

420

90425

1.224

99.975

-.190

90428

1427

89472 -493

•6.112

1451

94488

493

•5.104

1412

•4499

.450

95.012

468

94498

496

95414

422

94496 410

100400

.009

100.000

400

100.000

421

100400

-.021

100400

400

100.000

.000

100.000

400

100400 400

МАСА 64-А-410

МАСА 64^910

МАСА 64^.210

МАСА вЗА-210

Chord Lower Station Surface

XL VL.000 300

MO -.350 .400 -.410 300 -.560 1.250 -500 2.500 -1500

5.0 -2300

10.0 -3300

20.0 -4.000 30300 -5.000

40.0 -4.900

50.0 -4.500

60.0 -4300 70300 -3.500

80.0 -2300 90300 -1.500

100.000 .000

Sigurd Isaacson 53507 L. E. radius 0.5 percent

Sigurd Isaacson 64009 L. E. radius 0.3 percent

Sigurd Isaacson 73508 L. E. radius 0.4 percent

Sigurd Isaacson 53009 L. E. radius 0.8 percent

Benedek 894OSB LE. radio*
l. OParcant

Benedek В669вС L. E. radius

Benedek 995668 LE. radios

Benedek 984S6F

Benedek 8635568 L. E. radius

0Л Percent

0.7 Percent

0.7 Percent

Chord

Ifooer

Chord

Lower

Chord

Upper

Chord

Lower

Chord

Upper

Chord

.ewer

Chord Upper Chord Lower

Station Surface

Station

Surface

Station

Surface

Station

Surface

Station Surface Station Surface

Station Surface Station Surface

XU

YU

XL

YL

XU

YU

XL

YL

XU

YU

XL

YL

XU

YU

XL

YL

.000

1.000

.000

1.000

300

1.000

.000

1300

.000

.760

.000

.750

.000

.700

.000

.700

.200

1.450

.200

.590

300

(300

300

300

.200

1310

.200

.320

.200

1.200

.200

320

.400

1.700

.400

.440

.400

1.770

300

360

.400

1300

.400

.160

.400

1.400

.400

.080

jBOO

1.900

.900

.480

.600

1.960

300

300

300

1330

300

.090

300

1300

.700

.000

JBOO

2.100

300

.330

300

2.150

300

.280

300

2.100

300

.030

300

1.800

.800

310

1.250

2.500

1.250

350

1350

2300

1350

300

1.250

2.600

1.250

.000

1.250

2.180

1.250

.030

2.800

3.400

2300

.000

2300

3300

2300

.000

2.500

3.900

2.500

.200

2.500

3.140

2.500

.150

5.000

4.600

5.000

.200

5.000

4.280

5300

360

5.000

4.950

6300

.500

6.000

4.550

6.000

.420

7.800

5.400

7.500

.450

7.500

5300

7300

.400

7300

6.000

7.500

300

7300

5350

7.500

.780

10.000

9.160

10.000

.750

10.000

5.750

10300

.700

10300

6.900

10.000

1.100

10.000

6.530

10.000

1.120

15.000

7.250

15.000

1.300

15,000

6300

15300

1.200

15300

8300

15.000

1.500

16.000

7.780

16000

1.850

30.000

8.000

20.000

1300

20.000

7.700

20.000

1.750

20.000

8.700

20.000

2.200

20.000

8.550

20.000

2.450

25.000

9.550

25.000

2.350

25.000

8300

25300

2.250

25.000

8.950

25.000

2300

25.000

9.000

25.000

2.920

30.000

8.980

30.000

2.700

30.000

8.750

30300

2300

30300

9.000

30300

3.250

30.000

9.150

30.000

3350

40.000

9.400

40300

3360

40300

9.150

40300

3300

40300

8.900

40.000

4.000

40.000

8.990

40.000

3370

80.000

9.300

50.000

3300

50.000

9.100

50300

3.750

50.000

8300

50300

4.500

50.000

8.230

50.000

3350

00.000

8.750

60.000

4.000

60.000

8350

60300

3360

60.000

7.500

50.000

4.500

60.000

7.100

80.000

3.500

70.000

7.950

70.000

3300

70300

7300

76.000

3300

70.000

6.400

70.000

4.050

70.000

3750

70.000

3.000

80.000

5.900

80.000

2350

80.000

6.000

80.000

3.000

80.000

5.060

80300

3.300

80.000

4.080

80.000

2.220

90.000

3.550

90.000

1.500

90.000

3.700

90300

1.750

90.000

3.700

90.000

2.000

90.000

2.230

90.000

1.190

96.000

2.000

95.000

300

98300

2.100

95300

350

95.000

2.600

95300

1.100

100.000

320 100.000

.000

100.000

.400

100.000

.000

100.000

.450

100.000

.000

100.000

.500

100.000

.000

Benedek 993569 L. E. radius

Benedek 883068 L. E. radius

Benedek 974560

Benedek 87406F

0 A Percent

0.9 Percent

Chord

Upper

Chord

Lower

Chord Upper

Chord

Lower

Chord

Upper

Chord

Lower

Chord

Upper

Chord

Lower

Station

Surface

Station

Surface

Station Surface Station

Surface

Station

Surface

Station

Surface

Station

Surface

Station

Surface

XU

YU

XL

YL

XU

YU

XL

YL

XU

YU

XL

YL

XU

YU

XL

YL

.000

1.110

300

1.110

.000

1.180

.000

1.180

.000

350

.000

350

.000

300

.000

.900

.000

1.400

.200

.MO

.200

1.740

.200

300

.200

1350

.200

350

.200

1.500

.200

.420

.200

1.810

.400

Ж

.400

2.050

.400

.420

.400

1320

.400

350

.400

1.900

.400

.300

.400

2.070

300

.390

.900

2.310

.600

320

.600

1330

300

.ISO

.600

2.200

300

.220

300

2.530

300

.300

300

2350

.800

.270

300

2.080

.800

.030

300

2.490

300

.170

1.250

3.000

1.250

.170′

1.250

3.020

1.260

.170

1350

2.500

1.250

.000

1350

2.950

1.250

.100

2.500

4.150

2300

.030

2300

4.110

2.500

.000

2.500

3.480

2.500

.200

2.500

3.950

2.500

.100

5.000

5330

5.000

.050

5.000

5330

5.000

.070

5.000

4.900

5.000

.450

5.000

5.800

5.000

.450

7.500

7.090

7.600

.250

7.500

7.130

7.500

380

7.500

5.950

7.500

.700

7300

8.800

7.500

300

10.000

8.000

10.000

.500

ta. ooo

8.180

10.000

.960

10.000

6.700

10.000

350

10.000

7.400

10.000

1.000

18.000

9.150

15300

1.190

15300

9.500

16.000

1.470

15.000

8.000

15.000

1.450

16.000

8.550

16000

1.500

20.000

9370

20.000

1370

20.000

10.220

20.000

2.130

20.000

8.700

20.000

1350

20.000

8.200

20.000

1.950

25.000

10380

25300

2350

25.000

10310

25.000

2.560

25.000

9.000

25.000

2.600

25300

9.580

25.000

2.400

30.000

10.370

30.000

2.700

30.000

10300

30.000

2.830

30.000

9350

30.000

3.000

30.000

9.850

30.000

2300

40.000

9.910

40.000

3.050

40.000

9.900

40.000

3.000

40.000

8300

40.000

4.000

40.000

9.300

40.000

3.400

60.000

8380

50.000

2.980

«0.000

8330

50.000

2.900

50.000

8.000

50.000

4.500

50.000

9.800

50.000

3300

90.000

7.500

90.000

2.670

60.000

7.470

60.000

2.820

60.000

6300

60.000

4.000

60.000

7.700

60.000

3.750

70.000

5.900

70.000

2.220

70.000

5350

70.000

2.170

70.000

5.500

70.000

3.000

70.000

8350

70.000

3.400

80.000

4.200

80.000

1320

80.000

4.150

80.000

1.530

60.000

3.950

80.000

2.000

80.000

5.400

80.000

2350

90.000

2320

90.000

390

90.000

2.330

90.000

.830

90.000

2350

90.000

1.000

90.000

3.950

90.000

1.600

100.000

.330

100.000

.000

100.000

.350

100.000

.000

95.000

1.400

95.000

.500

95.000

2.900

95.000

.900

100.000

.500

100300

.000

100.000

.500

100.000

.000

Benedek 885578 I E. rediu* 0.6 Percent

Banadak 883088 LE. radiu* 0.7 Percent

.000 .400

.000 .790

.200 1.000 .400 1.220 800 1800 1.290 1.970

2800 3.000 9.000 4.730 7.900 8.170

10.0 7830

19.0 9.*20

20.0 10890 29800 10.470

30.0 10830 40800 10830

90.0 9.470

90.0 8.200

70.0 8.800 80800 4870

90.0 2.900 100.000 .100

Pfenning Laminar 11 LE. radiu* 08 Percent

AIRFOIL

64

6.45*

2*00

e. oo

AIRFOIL 65

e

•88«

6.00

N

A

Y

N

A

T

V-DISTR. FOR

the above

ALPHA REL

. ZERO-LIFT LINE

0

00000

0.0000Q

0*862

0*854

0

1.00006

0

00000

J7»19

0.S7T

l

99679

0*00076

0*891

0*888

1

0.99660

0

00070

0.699

0.895

2

98761

0*00311

0*947

0*948

2

0.96762

0

00260

0.969

0.951

Э

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0*00686

1*006

1*013

3

0.97319

0

00639

1.005

1.012

4

95409

0*01145

1*042

1.05S

4

0.95367

0

01060

1.035

1.066

5

93022

0*01659

1*057

1.077

5

0.92963

0

01563

1.062

1.061

6

90185

0*02223

1*073

1*099

6

0.90067

0

02079

1.0S1

1.077

7

86942

0*02829

1*090

1*123

7

0.06763

0

02075

1.063

1.09S

8

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0*03455

1*105

1*146

8

0.03062

0

03321

1.076

1.116

9

79408

0*04081

1*119

1*167

9

0.79010

0

06006

1.092

1.139

10

75200

0*04691

1*132

1*188

10

0,76727

9

06713

1.110

1,166

11

70756

0*05267

1*143

1*208

11

0.70230

0

05621

1.130

1.195

12

66122

0*05793

1*153

1*228

12

0.65560

0

06100

1.1S2

1.227

13

61344

0*06257

1*161

1*246

13

0.00056

0

06739

1.170

1.262

14

56469

0*06646

1*168

1*264

14

0.56091

0

07286

1.201

1.300

15

51543

0*06950

1*173

1*281

15

0.51352

9

07697

1.229

1.362

16

46615

0*07162

1*177

1*298

16

0.60626

0

07915

1.222

1.366

17

41731

0*07275

1 • 1179

1.315

17

0.61697

0

07976

1.215

1.356

18

36938

0*07288

1*180

1*331

18

0.37226

9

07905

1.206

1.361

19

32280

0.07197

1*178

1*348

19

0.32656

9

07716

1.197

1.369

ao

27803

0*07004

1*175

1*365

20

0.26233

9

07622

1.165

1.377

U

23547

0*06710

1*168

1*382

21

0.26007

9

07026

1.171

1.385

22

19553

0*06322

1*159

1.401

22

0.20017

9

06530

1.154

1.395

23

15858

0*05844

1*145

1*420

23

0.16305

9

05966

1.133

1.605

24

12494

0*05285

1*124

1*441

24

0.12905

9

05326

1.100

1.617

25

09490

0*04653

1*095

1*464

25

0.09651

9

06627

1.070

1.630

26

06872

0*03960

1*052

1*487

26

0.07170

9

03680

1.022

1.666

27

04658

0*03221

0*986

1*511

27

0.06660

9

03119

0.956

1.661

28

0?863

0*02453

0*861

1*531

28

0.03022

9

02363

0.652

1.660

29

Of 496

0*01680

0.70S

1.543

29

0.01569

9

01563

0.686

1.502

30

0<) 560

0*00931

0*382

1*525

30

0.00600

9

00605

0.361

1.519

31

06056

0.002S8

0*286

1*429

31

0.00061

9

00261

0.306

1.519

32

00075

•0*00255

1*503

0.752

32

0.00079

•9

00236

1.506

0.752

33

06675

•0*00664

1*390

0*199

33

0.00693

•9

00629

1.390

0.199

34

01818

-0*01038

1*318

0.S29

34

0.01636

•9

00993

1.316

0.5Э0

35

03477

-0*01348

1*260

0*683

35

O. OSSOO

-9

01295

1.260

0.663

36

05640

-0*01582

1*215

0*766

36

0.05066

-9

01522

1.216

0.766

37

08293

-0*01735

1*178

0*814

37

0.06317

-9

01670

1.178

0.616

38

11415

•0*01810

1*146

0*843

38

0.11639

•9

01761

1.166

0,663

39

14979

•0*01809

1*118

0*861

39

0.15006

•9

01730

1.116

0.661

60

18954

-0*01740

1*093

0*872

40

0.16979

-9

01006

1.096

0.673

41

23300

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1*071

0!* 879

41

0.23325

•9

01533

1.071

0,679

42

27973

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1*051

0*882

42

0.27990

•9

01352

1.05І

0.663

4*%

32924

-0*01212

1*032

0*884

43

0.32950

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01131

1.033

0.666

44

38098

-0*00965

1*01S

0*883

44

0.36126

•9

00866

1.0І6

0.6*6

45

43439

-0*00701

0*999

0*882

45

0.6Э66Т

•9

00021

0.999

0,6*2

46

48884

-0*00434

0*984

0*879

46

0.66912

•9

00356

0.966

o. tao

47

54370

-0*00173

0*970

0*876

47

0.56399

-9

00095

0.970

0.876

48

59831

0*00069

0*956

0*873

48

0.59661

9

00165

0.956

0.673

49

65199

0*00284

0*943

0*869

49

0.05230

9

00357

0.963

0.669

50

70409

0*00461

0..9Э0

0*864

50

0.70660

9

00531

0.930

0.666

51

75392

0*00595

0*918

0*859

51

0.75626

9

00661

0.916

0.660

52

80082

0*00678

0*906

0*8S5

52

0.60115

9

00760

0.906

0.655

53

84416

0*00709

0*894

0*849

S3

0.06650

9

00705

0.696

0.650

54

88331

0*00686

0*883

0*844

54

0.06305

9

00730

0.663

0,666

55

91770

0*00612

0*871

0*839

55

0.91603

9

00656

0.672

0.639

56

94675

0*00491

0*861

0*833

56

0.96709

9

00522

0.661

0.636

57

96990

0*00333

0*857

0*83$

57

0.97019

9

00350

0.660

0.637

58

98661

0*00169

0*861

0*843

58

0.98660

9

00175

0.670

0.652

59

99665

0*00046

0*865

0*853

59

0.99071

9

00066

0.660

0.667

60

00000

0*00000

0*862

0*854

60

1.00000

9

00000

0.679

0.871

ALPHAO «

4«SS

DEGREES

CMO a-0*1222

ALPNAt * 6.56 oegrees

CNO >«0.1206

ETA «

1*071

ETA «

1.072

WARNING * SUBROUTINE SMOOTH HAS SLOPES WARNING – SUBROUTINE SMOOTH HAS SLOPES

•0.3B0 *N0-0*482 BETWEEN POINTS 32 ANO 3 «0.406 ANO-0.691 BETWEEN POINTS 32 AND

EPPLER 205 EPPLER 207 EPPLER 209

PROF1L E 205 10.484b PROFIL E 207 12.04% PROFIL E209 13.78%

N

X

Y

X

Y

X

Y

0

100.000

0.000

100.000

0.000

100.000

0.000

1

99.655

.039

99.647

.045

99.639

.052

2

98.649

.174

98.625

.202

98.600

.232

3

97.049

.427

97.011

.489

96.969

.557

4

94.916

.778

94.870

.881

94.821

.992

5

92.285

1.196

92.238

1.337

92.187

1.488

6

89.175

1.668

89.128

1.841

89.077

2.027

7

85.624

2.199

85.576

2.400

85.523

2.615

8

81.684

2.786

81.633

3.011

81.577

3.253

9

77.412

3.419

77.357

3.666

77.295

3.930

10

72.866

4.088

72.806

4.352

72.738

4.634

11

68.108

4.777

68.043

5.055

67.968

5.352

12

63.204

5.470

63.132

5.759

63.049

6.065

13

58.218

6.147

58.139

6.441

58.047

6.753

14

53.217

6.782

53.129

7.079

53.028

7.393

15

48.265

7.342

48.169

7.638

48.058

7.949

16

43.410

7.785

43.306

8.075

43.185

8.380

17

38.680

8.081

38.567

8.362

38.436

8.657

18

34.101

8.214

33.981

8.483

33.841

8.764

19

29.699

8.177

29.573

8.430

29.424

8.694

20

25.496

7.970

25.363

8.205

25.208

8.448

21

21.508

7.606

21.371

7.819

21.211

8.037

22

17.764

7.111

17.626

7.300

17.461

7.490

23

14.302

6.507

14.162

6.669

13.997

6.830

24

11.157

5.811

11.018

5.944

10.854

6.073

25

8.360

5.040

8.225

5.143

8.065

5.237

26

5.937

4.211

5.808

4.282

5.656

4.341

27

3.909

3.344

3.791

3.383

3.651

3.406

28

2.292

2.461

2.189

2.468

2.066

2.454

29

1.097

1.589

1.015

1.565

.916

1.517

30

.331

.766

.279

.714

.216

.635

31

.002

.055

.000

-.015

.007

-.106

32

.233

-.506

.304

-.626

.398

-.775

33

1.065

-.988

1.212

-1.204

1.379

-1.467

34

2.419

-1.420

2.628

-1.750

2.852

-2.140

35

4.291

-1.776

4.543

-2.234

4.804

-2.765

36

6.669

-2.053

6.943

-2.649

7.219

-3.329

37

9.534

-2.252

9.807

-2.991

10.073

-3.821

38

12.864

-2.378

13.109

-3.257

13.340

-4.233

39

16.627

-2.436

16.817

-3.448

16.986

-4.561

40

20.783

-2.435

20.893

-565

20.974

-4.799

41

25.290

-2.384

25.292

-3.611

25.258

-4.943

42

30.097

-2.292

29.966

-3.586

29.793

-4.985

43

35.149

-2.168

34.861

-3.487

34.524

-4.909

44

40.388

-2.021

39.937

-3.302

39.430

-4.683

45

45.751

-1.859

45.165

-3.044

44.516

-4.322

46

51.174

-1.689

50.495

-2.744

49.749

-3.884

47

56.591

-1.516

55.860

-2.425

55.057

-3.409

48

61.938

-1.345

61.189

-2.102

60.368

-2.923

49

67.149

-1.180

66.414

-1.787

65.609

-2.446

50

72.160

-1.023

71.467

-1.488

70.708

-1.993

51

76.911

-.876

76.283

-1.212

75.594

-1.576

52

81.343

-.740

80.796

-.963

80.197

-1.205

53

85.400

-.614

84.948

-.744

84.451

-.884

54

89.034

-.380

91.942

-.390

91.665

-.400

55

92.195

-.380

91.942

-.390

91.665

-.400

56

94.860

-.252

94.699

-.239

94.522

-.222

57

97.017

-.125

96.930

-.108

96.834

-.089

58

98.635

-.036

98.598

-.026

98.558

-.014

59

99.651

-.003

99.643

-.000

99.633

.003

60

100.000

.000

100.000

.000

100.000

-.000

CM =

-.0460 fi «

2.37°

CM = -.0499 fi

= 2.33е

CM * -.0547 fi

= 2.28°

XU

YU

.000

900

.026

.190

.466

.915

1.344

1.740

2.652

2906

4.363

3.487

6.525

4.352

9.061

5.161

11.957

5.957

15.218

5.663

19780

7.284

22920

7905

26.595

6.213

30.967

8.487

39403

8.603

39.979

8951

44973

8.332

49.458

7.954

54.305

7.438

59.186

8908

64.052

8.112

56939

6.381

73.484

4.842

77.923

3.914

82.096

3.214

85.945

2.558

89.414

1957

92.452

1.415

95.023

932

97.106

.522

98.674

.220

99.661

.051

100.000

.000

XL YL.000 000 .129 -.375 919 -.938 2.044 -1.252 3.791 -1.598 6.049 -1841 8901 -2.010 12.026 -2.098 15.697 -2.112 19.778 -2.061 24.227 -1.955 28.998 -1.807 34.035 -1.628 39.280 -1.430 44.672 -1.244 50.145 -1.019 55.630 – 824 61.059 -.645 66.384 -.486 71.479 -950 76.339 -.239 80982 -.153 85.050 -.091 88.788 -.048 92.048 -.018 94.794 .010

97.003 .032

98.640 .034

99.655 .014

100.000 .000

Wortmann FX 38*153 Camber 2.1 fercant

AIRFOIL

0008 10

.80X

AIRFOIL

12

9.27X

AIRFOIL 14

8.47X

N

X

Y

N

К

Y

N

X

Y

0

100.000

0.

0

0

100.

.000

0.0

0

100.000

0.0

1

99.664

0.

082

1

99.

.665

0.052

1

99.667

0.045

2

98.714

0.

338

2

98.

.701

0.219

2

98.707

0.195

3

97.253

0.

740

3

97,

.190

0.497

3

97.194

0.446

4

95.321

1.

211

4

95

.174

0 .837

4

95.169

0.760

5

92.908

1.

712

5

92

.660

1.213

5

92.645

1.112

6

90.028

2.

250

6

89

.671

1.629

6

89.647

1.506

7

86.722

2.

819

7

86.

.251

2.080

7

86.218

1.937

8

83.033

3.

407

8

82

.447

2.557

8

82.405

2.394

9

79.007

4.

001

9

78

.304

3.047

9

78.255

2.867

10

74.689

4.

586

10

73

.875

3.538

10

73.817

3.344

11

70.131

5.

147

11

69

.209

4.018

11

69.145

3.811

12

65.381

5.

672

12

64

.361

4.472

12

64.292

4.256

13

60.492

6.

147

13

59

. 384

4.889

13

59.310

4.666

14

55.516

6.

558

14

54

.331

5.256

14

54.254

5.029

15

50.504

6.

895

15

49

.257

5.562

15

49.178

5.334

16

45.508

7.

146

16

44

.213

5.795

16

44.133

5.570

17

40.577

7.

303

17

39

.251

5.948

17

39.172

5.727

18

35.760

7.

359

18

34

.419

6.013

18

34.343

5.800

19

31.103

7.

309

19

29

. 765

5.984

19

29.692

5.780

20

26.649

7.

149

20

25

.331

5.857

20

25.262

5.666.

21

22.440

6.

878

21

21

.158

5.631

21

21.095

5.456i

22

18.512

6.

498

22

17

.282

5.308

22

17.226

5.151

23

14.898

6.

013

23

13

.>36

4.892

23

13.689

4.755

24

11.627

5.

431

24

10

.551

4.391

24

10.513

4.275

25

8.724

4.

763

25

7

.750

3.815

25

7.725

3.721

26

6.211

4.

023

26

5

.357

3.176

26

5.344

3.104

27

4.105

3.

230

27

3

.389

2.492

27

3.388

2.439

28

2.421

2.

405

28

1

.859

1.779

28

1.867

1.744

29

1.169

1.

577

29

0

.772

1.067

29

0.786

1.048

30

0.360

0.

779

30

0

.140

0.393

30

0.150

0.390

31

0.004

0.

079

31

0.

.026

-0.161

31

0.022

-0.139

32

0.219

-0.

521

32

0.

.486

-0.652

32

0.465

-0.595

33

1.024

-1.

097

33

1

.483

-1.155

33

1.447

-1.055

34

2.328

-1.

645

34

2

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34

2.938

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35

4.124

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137

35

4

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35

4.931

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36

6.398

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559

36

7

.462

-2.382

36

7.412

-2.116

37

9.136

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903

37

10

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37

10.363

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38

12.318

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170

38

13

.781

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38

13.756

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39

15.914

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363

39

17

.558

-3.074

39

17.559

-2.629

40

19.884

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484

40

21

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-3.190

40

21.730

-2.690

41

24.185

-3.

538

41

26,

.153

-3.252

41

26.227

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0.0000

0.0000

20

1.0000

0.0000

0.0000

HQ-1,0/10

HQ-1,5/8

X(i)

Yo

Yu

_

X(i>

Yo

Yu

0.0000

0 . UUOO

1

0.0000

U. 0001)

0 . OOUO

2

.0050

. 0092

– .0070

2

.0050

. 0082

– .0048

3

.0125

.0141

-.0114

3

.0125

.0128

– .0082

4

.0250

. 0214

-.0166

4

.0250

.0189

-.0115

5

. 0500

. 0307

-.0228

5

.0500

. 0273

-.0156

6

. 1000

.0421

-.0301

6

.1000

. 037V

-.0199

7

. 1500

. 0499

-.0351

7

.1500

. 0451

– . 0229

8

.2000

. 0540

-.0375

8

.2000

.0489

– . 0243

V

.2500

. 0569

-.0393

9

.2500

. 0517

-.Ubjj

10

.3000

. 0586

-.0400

10

.3000

.0534

-.0255

11

.3500

. 0596

-.0404

11

.350 0

. 0544

-.0256

12

.4000

.0589

-.0393

12

.4000

.0534

-.0246

13

.5000

.0554

-.0354

13

.5000

.0513

-.0213

14

.6000

. 0478

-.0284

14

.6000

. 0450

– . 0159

15

.7000

.0368

-.0192

15

.7000

.0356

-.0092

16

.8000

.0243

-.0104

16

.8000

.0243

-.0035

17

.8500

.0177

-.0066

17

.8500

.0181

-.0014

18

.9000

.0117

-.0037

18

.9000

.0121

-.0002

19

.9500

.0054

-.0013

19

.9500

.0057

-.0004

20

1.0000

0.0000

0.0000

20

1.0000

0.0000

0.0000

U5/» І7І —

HQ-1,5/9 HQ-1,5/10

X (i) Yo Yu X(l) Yo Yu

HQ-2^/8

SD7032

SD7037

1

1.00000

0.00000

17

0.44745

0.07211

33

0.00127

-.00393

49

0.60914

-.00549

2

0.99672

0.00042

18

0.39862

0.07410

34

0.00806

-.00839

50

0.66197

-.00349

3

0.98707

0.00180

19

0.35101

0.07504

35

0.02038

-.01227

51

0.71305

-.00168

4

0.97146

0.00436

20

0.30508

0.07488

36

0.03800

-.01541

52

0.76178

-.00014

5

0.95041

0.00811

21

0.26125

0.07358

37

0.06074

-.01777

53

0.80752

0.00104

6

0.92450

0.01295

22

0.21989

0.07113

38

0.08844

-.01934

54

0.84964

0.00182

7

0.89425

0.01865

23

0.18137

0.06754

39

0.12084

-.02017

55

0.88756

0.00220

8

0.86015

0.02490

24

0.14601

0.06286

40

0.15765

-.02032

56

0.92071

0.00218

9

0.82261

0.03141

25

0.11410

0.05715

41

0.19850

-.01987

57

0.94859

0.00185

10

0.78201

0.03788

26

0.08586

0.05049

42

0.24296

-.01891

58

0.97077

0.00132

11

0.73865

0.04413

27

0.06146

0.04300

43

0.29055

-.01754

59

0.98690

0.00071

12

0.69294

0.05011

28

0.04102

0.03486

44

0.34071

-.01586

60

0.99671

0.00021

13

0.64539

0.05572

29

0.02462

0.02632

45

0.39288

-.01396

61

1.00001

0.00000

14

0.59655

0.06085

ЗО

0.01232

0.01770

46

0.44643

-.01190

15

0.54693

0.06538

31

0.00418

0.00936

47

0.50074

-.00976

16

0.49706

0.06917

32

0.00021

0.00185

48

0.55519

-.00760

SD7090

1

1.00000

0.00000

17

0.43649

0.06457

33

0.00345 -.00734

49

0.61442

-.01948

2

0.99655

0.00050

18

0.38699

0.06674

34

0.01238 -.01318

50

0.66605

-.01658

3

0.98664

0.00219

19

0.33890

0.06795

35

0.02624 -.01834

51

0.71605

-.01363

4

0.97113

0.00512

20

0.29269

0.06814

36

0.04514 -.02262

52

0.76381

-.01083

5

0.95062

0.00882

21

0.24878

0.06724

37

0.06903 -.02605

53

0.80869

-.00829

6

0.92522

0.01284

22

0.20759

0.06522

38

0.09771 -.02873

54

0.85007

-.00609

7

0.89500

0.01718

23

0.16945

0.06206

39

0.13087 -.03067

55

0.88735

-.00426

8

0.86036

0.02188

24

0.13467

0.05780

40

0.16817 -.03188

56

0.92001

-.00279

9

0.82176

0.02691

25

0.10352

0.05249

41

0.20926 -.03240

57

0.94757

-.00160

10

0.77972

0.03218

26

0.07624

0.04621

42

0.25371 -.03229

58

0.96974

-.00066

11

0.73479

0.03760

27

0.05297

0.03907

43

0.30106 -.03161

59

0.98622

-.00009

12

0.68754

0.04304

28

0.03384

0.0312S

44

0.35077 -.03046

60

0.99650

0.00004

13

0.63860

0.04834

29

0.01891

0.02298

45

0.40227 -.02889

61

1.00001

0.00000

14

0.58850

0.05329

ЗО

0.00827

0.01458

46

0.45499 -.02696

15

0.53777

0.05773

31

0.00196

0.00637

47

0.50832 -.02472

16

0.48692

0.06153

32

0.00005

-.00097

48

0.56166 -.02221

SD8000

[1] N. A.C. A., National Advisory Committee for Aeronautics, U. S.A., now replaced by N. A.S. A.

[2] For explanation of the first, second and third digits of these aerofoils, see Chapter 9.

[3] The author is grateful to Andy Lennon for drawing his attention to this work and for subsequent discussions of it.

Appendix З

Ordinates for nearly 200 different aerofoils are given in this appendix, together with four low drag bodies (Young bodies) for fuselages, fairings, wheel spats etc. Not all profiles will be satisfactory for models, some have been included only to amplify comments in the text The modeller should choose his aerofoils with discretion, bearing in mind the general principles discussed in the relevant chapters.

The profile drawings given here have been plotted by a computer. In many cases, the results are accurate enough for the drawing to be enlarged or reduced photographically to produce a perfect template outline for modelling. In other cases, where the ordinates give a coarser outline, the mechanical plotter produces an angular drawing with segments of straight lines and other small irregularities. In these cases the modeller should smooth the outline with a ‘zip’ or ‘French’ curve before cutting the template. The ordinates for the Wortmann M 2 aerofoil in particular produce an irregular, corrugated leading edge. This should be smoothed in practice. The corrugations are a result of the approximate methods of calculations used by Wortmann for this very early, low speed aerofoil.

Hand plotting is laborious but can produce accurate results if carefully done. The method is described in most elementary books on aeromodelling. The more advanced aerofoils given in this appendix have been produced by computer and the old, standard plotting points are not used. Instead there may be different chord stations for upper and lower surfaces.

FINDING THE AERODYNAMIC CENTRE OF A WING

For trimming a model it is often necessary to know the position of the wing aerodynamic centre. The model’s centre of gravity should normally be at this point or very slightly behind it, as discussed in Chapter 12. If the c. g. is too far aft, parasite drag will increase since a larger tail will be required for stability and induced drag also will rise, due to load being transferred from wing to tail. If too far forward, a down load on the tailplane will be required for balance, which also leads to increased tail drag although the model’s stability will be high.

If die wing is not swept back or forward, with respect to its quarter-chord line, the aerodynamic centre for all practical purposes may be taken as one quarter of the chord aft of the root leading edge, and for wings of normal plan, with only a very slight sweep, this point will as a rule be close enough, for trimming, to the true a. c.

With wings of complex form, or swept, locating the a. c. is both more important and more difficult, important because for preliminary trial flights the centre of gravity must be as close as possible to the right place before launching. For a straight-tape red wing, the graphical construction shown in the upper part of Figure A7 may be used. Where t is the tip chord and r the root chord, by drawing on an accurate plan of the wing half panel, extend the root chord by t, and the tip chord by r as shown, and join the extensions with a diagonal line. Where this line cuts the line joining mid-points of root and tip chords is the geometric centre of area of the wing panel. The chord through this point should be drawn parallel to the aircraft centre line. The wing aerodynamic centre lies a quarter of the way along this chord line, measured from the leading edge. This point should be projected onto the centre line to give the required balance point

With more complex wing planforms, the following procedure may be used. Although fairly laborious, the exercise is worthwhile whenever a wing of unusual form is designed.

1. Calculate the mean chord, c = S/b, where S is the wing area and b the span.

2. Consider the wing on one side of the centre line and divide into a convenient number of panels (With a curved outline divide the wing into panels with nearly straight lines to arrive at a close approximation.) (See lower part of Fig. A7.)

3. Find the area, dS, of each panel.

4. Find the centre of area of each panel, using die construction method of Fig. A7.

5. Find the centre of area of the wing by taking moments about the centre line:

– _ (dSi x yi) + (dS2 x У2) + (dS3 x уз) etc. y S/2

<In this formula DSi 2 3 4» etc. refer to the areas of the successive panels of the wing marked out in step 2, and yi 2 3, etc. refer to the distances of each panel’s centre of area from the centre line. The whole formula gives у which is the distance from the centre line of the centre of area of the half wing.)

6. Locate the quarter-chord points for the chord through the centre of area of each panel.

7. Find the chordwise distances, xi X2 хз x4 etc. of these quarter-chord points measured from some convenient transverse axis such as the line through the leading edge at the root of the wing.

8. Find the fore and aft location of the mean quarter-chord point, x by taking moments about a transverse axis (i. e. the line through the leading edge at the root, as before).

Подпись: x =(dSi x xi) + (dS2 x X2) + (dS3 x хз) etc.
S/2

9. Locate the mean chord c (calculated in step 1) in a plane through the centre of area (calculated in step 5) with its quarter chord point marked the distance x aft of the reference line.

Подпись: Here, the meanings of the symbols are:

CALCULATION OF THE NEUTRAL POINT AND STATIC MARGIN A formula which gives satisfactory results for the position of the neutral point is

hn — position of the neutral point as a decimal fraction of the wing standard mean chord.

ho = position of the aerodynamic centre of the wing with allowance for fuselage effects, on the standard mean chord. (Fuselage may be ignored for rough estimates, in which case use ho = 0.25)

щ = stabiliser efficiency. This must be estimated: — 0.9 for a‘T’ tail, — 0.6 for a normal tail, — 0.4 or 0.3 if tail is near wing wake or on a fat fuselage in disturbed flow. A canard foreplane may be assumed.95 to 1.0 efficient

Vs = stabiliser volume coefficient calculated from the formula:

у _ U x S(stabiliser) с x S(wing)

See Chapter 12, 12.16.

ai = slope of the lift curve of the stabiliser. [From wind tunnel charts with aspect ratio correction.]

a = slope of the lift curve of the wing [Wind tunnel, corrected for A.]

de _ change of downwash angle at the stabiliser with change of wing angle of attack. The

da average is Vt to W, i. e. downwash at tail changes about half to one third as much as the wing angle of attack, in disturbance.

Canard foreplanes are in the upwash ahead of the main wing, which also must be allowed for.

A worked example:

‘BANTAM’ sport power model.

Dimensions:

Span = b = 1.25m Wing area = Sw = 0.29m2 Aspect ratio = A = 5.39 Mean chord = c = 0.232m

Stabiliser span = bs = 0.5m Stab, area = Ss = 0.07m2 Stab. Asp. ratio = As = 3.57 Stab mean ch. = cs = 0.14m Length of tail arm = ls = 0.557m

ho = 0.25 [Fuselage ignored]

tjs = 0.65 [‘Normal’ tail]

„ ls x Ss.557 x.07 0.039

Vs =J——— 1=—————- =———- = 0.58

c x Sw 232 x.29 0.067

Slope of wing lift curve: Wing is a 13.7% thick profile similar to the thickened Clark Y or Gottingen 796. Assume a lift curve slope of a~ = 0.11 (i. e., at ‘infinite’ aspect ratio, about 2Tr/Radian which is — 0.11 Cl per degree e°)

Slope of tail lift curve: Tail is a thick flate plate section with rounded leading edge and knife trailing edge. (9.5 mm thick balsa) Assume lift curve slope similar to flat plate, a~ = .095, which is a conservative (pessimistic) figure.

These must be corrected for wing and tail aspect ratios, using the formula below:

Подпись: a

^(corrected) —

[This means that if the wing Cl changes by 1.0, the stabiliser Cl would change by 1.0 x 0.8 for the same angle of attack change. But, because of downwash, the change of angle at the tail will be less than the wing which is allowed for in estimating de/da below]

For estimation of de/da, various elaborate methods are available (see D. Stinton, book listed below). For a rough calculation, use the formula:

de

— = 35a/A for monoplane (55a/A for biplane)

In this case:

Then the neutral point position is

h„ = 0.25 + (0.65 ж 0.58 x 0.8 ж [1 – 0.519]) = 0.25 + 0.145 = 0.395

That is, the neutral point lies at about 0.4 or 40% of the wing mean chord aft of the wing’s aerodynamic centre

The recommended position of the balance point on the plan is 33% of s. m.c., i. e. 0.33 x mean chord. The static margin is found by subtraction-

sm = hn – 0.33 = 0.07

This is a stable position but perhaps would prove ‘hot’ especially since propeller and fuselage destabilising influences have been ignored. By moving the c. g. forward 0.5 cm the s. m. would be increased to about 0.1, and a full 1.0 cm movement would give s. m. = 0.12.


Note that for a canard layout, the foreplane is destabilising, i. e. it brings the neutral point

н

 

N 60 R

 

L

 

(kxtingan 795

 

5 0 <х* 5 10 15 -0-2 cm

Подпись: ЇЙ J?

т

 

о

 

О cd

 

 

Подпись: гй-п

*5

 

І

 

Подпись: 272

273

-0,2

276

а,.,15 20 0 0.02 0.04 0.06 Q08 0.10 0.12

ПЛІ

-5

0

5

10

280

283

THE ANGLE OF INCIDENCE

The problem is to set the fuselage at such an angle to the flight path of the model that it creates the least possible drag. For a racer this will depend on die angle of attack of the wing when it is flying at maximum speed. For a soaring sailplane the angle of attack when flying at maximum Cl1 VCd is what counts, though the performance gain caused by

1

2

3

4

5

6

7

8

Cl

V*

Vm/sec

Assumed

Induced

Profile

Total

(4) X V2

Parasite Cd

Cdi

Cd

cL

0.4

120.12

10.96

0.010

.00679

.0120

.02879

1.2

0.6

80.08

8.95

0.010

.01528

.0098

.03508

.8

0.8

60.06

7.75

0.010

.02716

.0101

.04730

.6

1.0

48.05

6.93

0.010

.04244

.0107

.06314

.48

1.2

40.04

6.33

0.010

.06115

.0113

.08245

.40

1.4

34.32

5.86

0.010

.08318

.0125

.10568

.34

1.5

32.03

5.66

0.010

.09549

.0137

.11919

.32

1.6

30.03

5.48

0.010

.10864

.0149

.13354

.30

1.675

28.69

5.36

0.010

.11925

.0162

.14545

.29

FINAL DRAG BUDGET, Kg. FORCE

PARASITE INDUCED PROFILE

TOTAL

9

10

11

12

13

14

15

(5) X V2 (6) X V* (7) X

V* (8) X fcpS (9) X WpS (10) X HpS

(11) X ttpS

0.82

1.44

3.46

0.735

0.502

0.882

2.119

1.22

0.79

2.83

0.490

0.747

0.484

1.721

1.63

0.61

2.84

0.368

0.998

0.374

1.740

2.04

0.51

3.03

0.294

1.250

0.314

1.858

2.45

0.45

3.30

0.245

1.501

0.276

2.022

2.86

0.43

3.63

0.208

1.752

0.263

2.223

3.06

0.44

3.82

0.196

1.874

0.270

2.340

3.26

0.45

4.01

0.184

1.997

0.276

2.457

3.42

0.47

4.17

0.178

2.095

0.288

2.561

saving parasite drag at this low speed will be very small. For a ‘penetration’ sailplane, correct fuselage alignment at high speed is important, less so at low speed.

For a racer, ensure that the wing camber is such that the wing profile minimum drag is at the average operational Cl for speeds at which the model will fly (see Chapter 6). From wind tunnel test results if available, or if not, by assuming a lift curve slope of 0.11 ci per degree (from zero lift angle of attack) find the angle of attack of the wing profile at the operating c[. This angle is for a wing of‘infinite’ aspect ratio. It must be corrected for downwash effects as shown below.

For a penetrating sailplane the section angle of attack chosen will depend on the extent of the aerofoil’s low drag range or*bucket’. Flight at a lower angle of attack than this will bring a marked deterioration (steepening) of the glide due to increased profile drag. Wind tunnel results at the Re appropriate to flight at this Cl and airspeed allow this to be estimated, or, if NACA 6 series aerofoils are used, the extent of the low drag range can be judged roughly from the third digit – e. g. 643618 gives a low drag range of 0.3 c| above and below the ideal cl of 0.6, hence the low drag bucket ends at cl 0.3. From this it is possible to estimate the angle of attack (infinite a. r.).

For a soarer, the operating Cl must be found by the methods given in this appendix and the angle of attack found from the wind tunnel results.

Knowing the angle of attack of the wing at infinite aspect ratio, the correction to the angle for the real model wing, affected bv downwash. is found bv the formula:

Angle of attack increment: (18.25 x Cl) (Assuming a nearly elliptical lift distribution)

A

Thus, for the model considered earlier the Cl for minimum sink with a. r. 7.5 was found to be 1.55. The induced angle of attack for this a. r. is:

18.25 x 1.55/7.5 = 3.77 degrees

From the tunnel tests (unfortunately not at the correct Re, so this is useful only as an example of method) the aerofoil yields C) 1.55 at 8.0 degrees. The rigging angle of wing to fuselage should be 8.0 + 3.77 = 11.77 degrees.

For penetration the low drag range of the aerofoil ends at approx, ci = 0.5, which develops at -2.3 degrees.

The induced angle of attack will be 18.25 X 0.5/7.5 = 1.22.

The rigging angle should thus be – 2.3 + = -1.08.

(The aerofoil is highly cambered and not very suitable for a fast flying model sailplane. Its performance is very good at high ci, for which it was designed. The figures above illustrate the method only.)

A pylon racer with a symmetrical aerofoil, flying at a Cl of 0.05 with the same aspect ratio would require an angel of incidence as worked out below:

18.25 x 0.05 : 7.5 = 0.122.

The symmetrical profile would reach cj 0.05 at 0.11/0.05 = 0.45 degrees. The rigging angle should thus be 0.122 + 0.45 = 0.57 degrees.

ALLOWANCE FOR PARASITE DRAG

After correcting wind tunnel results for aspect ratio effects, to arrive at an estimate of drag for the whole model, parasite drag must be taken into account This may be estimated from tunnel tests on fuselages, undercarriages, etc. with an additional quantity for interference drag. As a rule such detailed estimates are not necessary for modelling. To illustrate the effect of parasite drag on performance, however, the above example is taken further. It is assumed (arbitrarily but not unreasonably) that the model with a. r. 7.5 of the previous example has a parasite drag coefficient of 0.01, this remaining constant throughout the flight attitudes considered. This assumption breaks down if the fuselage is not aligned with the average flow direction, but is sufficiently accurate for purposes of illustration.

Fig. A4 Aspect ratio and parasite drag effects

Note: parasite drag has a considerable effect on the trim for best performance, shifting the peaks of the curves to the higher Cl (higher angle of attack) side.

The calculation proceeds as before: the parasitic increment of drag is added to the total of profile and induced drag, to give the total drag coefficient of the whole model. This new total is then used to work out the model power factor and L/D.

1

2

3

4

5

6

Cl

Cdi x Cap

Add parasite V’ci3

у/ ci3/Cd

Ci/Cl

a. r. 7.5

Cd = .01

Total

.4

.01879

.02879

0.253

8.788

13.89

.6

.02508

.03508

0.465

13.255

ІГШ

Max. L/D

.8

.03726

.04726

0.716

15.150

16.93

1.0

.05314

.06314

1.000

15.837

15.84

1.2

.07245

.08245

1.315

115.9491

14.55

Max. Power Factor

1.4

.09568

.10568

1.657

15.679

13.25

1.5

.10919

.11919

1.837

15.412

12.58

1.6

.12354

13354

2.024

15.156

11.98

1.675

.13545

.14545

2.168

14.905

11.52

The power factor and L/D worked out in columns 5 and 6 are graphed and from this graph the desired Cl for trimming is found. The values for a. r. 15 are also plotted for comparison (Fig. A4).

Assuming the model weight and wing area are known, the sinking speed can be computed and plotted as shown to give the ‘polar1 curve of the model. This gives the minimum sinking speed by inspection. The sinking speed of a model glider (from page

232) is given by j-^—————- £-

Sinking speed =y—

Assuming a model wing loading of W/S = 3 kg/sq. m, the first term becomes 6.932. Thus sinking speed at each Cl and V value is as shown for the a. r. 7.5 model:

1

2

3

4

Cl

Vm/sec

1/CL15/Cd

Sinking Speed, M/Sec

.4

10.96

0.11379

.7895

.6

8.9

0.07544

.5229

.8

7.75

0.06601

.4576

1.0

6.93

0.06314

.4377

1.2

6.33

0.06269

.4346* ‘MINIMUM SINK

1.4

5.86

0.06378

.4421

1.5

5.66

0.06488

.4498

1.6

5.48

0.06598

.4574

1.675

5.36

0.06709

.4651

A more complete performance estimate may be attempted if full wind tunnel test results are available for the aerofoil to be used. Interesting studies are possible to illustrate the effect of increased wing loading and aspect ratio. If the wing loading is increased, the glider flies faster, which raises the Re and may improve the drag characteristics of the aerofoil. Given complete tunnel results this improvement may be estimated, and the profile drag figures in the calculation table will show the difference. Whether the improvement in profile drag is enough to offset the increased sinking speed due to the higher wing loading will depend entirely on the aerofoil and how much it is affected by the ‘ increased Re. Similarly, with a model of given wing area and wing loading, increasing the aspect ratio decreases induced drag but reduces Re, and so increases profile drag. Given adequate wind tunnel tests results, it is possible to compute the effects on model performance of such changes, and so determine the best aspect ratio.

Some examples have been published by the author and are listed among the references at the end of Chapter 10.

As an illustration of the results possible, the following tables and polar curve are included, but a full explanation of die method would occupy too much space.

Velocity, M/sec: 8.30 Mean Reynolds number 113043 Root angle of attack 15.721

Cl

Chord

Re number

Profile Cd

1.09

0.210

119345

0.02062

1.09

0.210

119345

0.02062

1.08

0.210

119345

0.02055

1.05

0.210

119345

0.01950

1.02

0.203

115634

0.03117

1.00

0.192

109108

0.03355

0.94

0.182

103534

0.03430

0.84

0.174

99050

0.03280

0.65

0.169

95766

0.03212

0.37

0.165

93763

0.02542

Подпись: Mean Lift Coefficient =1.00 Profile drag coefficient 0.0256 Induced drag coefficient 0.02256 Efficiency 0.961 К factor = 1.041 V =8.30 L/D = 20.7 Sink = 0.400 Result of calculation of glider performance at speed of 8.3m/sec. The mean Re is 113043 but the wing is tapered so Re varies across the span. Note the Profile drag and section CL coefficient vary as the chord and vortex-induced downwash change toward the tips, and die Re falls.

THE DRAG BUDGET

Figure 4.9 was produced by applying the methods outlined above. The contribution of each type of drag to the model Cd at each Cl may easily be found in the foregoing tables;

245

□ WING A = 20 ★ WING3A=8

 

POLARS

 

■ WING 2 A* 14

 

M/S “ 4.3kQ/m*

 

Fig. A6 Polar curves of three sailplane wings of different aspect ratios but all of the same aerofoil section.

[The low A models are ballasted to bring them to the same M/S as the A = 20 model.] Note the marked superiority of the [ballasted] A = 8 model at high speed. At low speed there is little to choose between A = 14 [ballasted] and A = 20. The low Re of the high A wing is responsible.

these are entered in the table below.

To find the dragforce contributed at each flying speed, first the velocity of flight at each Cl is found by applying the standard formula:

= / W x 9.81 / 3 x 9.81 = /48.048

л/liSx 1.225 Cl w й x 1.225 x 1 x Cl * Cl

For the purpose of the example calculation, W/S (wing loading) was assumed 3 kg./sq. metre (approx. 9.8 ozs./sq. ft), and a model weight of 3kg. assumed (6.6151bs.).

Next, each Cd is multiplied by VipSV2 to produce drag force in kg. The results are of course only approximate since no proper allowance has been made for varying Re. The assumed model would achieve Re approx. 280,000 to match the wind tunnel results if its VL (velocity multiplied by chord) was about 4. This corresponds to a model with chord 0.3 metres flying at 13.3 metres per second (11.8 ins. at 29.8 m. p.h.). As the figures below show, the model would reach this speed at Cl between 0.2 and 0.4. The aspect ratio of 7.5 gives a span of 2.74 metres, wing area 1 sq. metre.

The drag budget is worked out in the table overleaf, and plotted as in Figure 4.9.

ASPECT RATIO CORRECTIONS

The section power factor, and 1/d ratio, as calculated above is directly taken from wind tunnel results at infinite aspect ratio. To discover the power factor for the wing on the model, the wind tunnel figures must be corrected. In the example worked below, based on

241

the Stuttgart figures for the FX63-137 aerofoil at Re 280,000, correction for two aspect ratios has been applied to show the great effect of a. r. on soaring ability. This Re is of course high for a model, but the method is the same at any Re. Induced drag is found from the standard formula:

Ctv = Cl2

1 3.142 x Aspect ratio X

At the two example aspect ratios, 7.5 and 15, the Cji, the vortex-induced drag coefficient of the wing thus is found. The plan form correction, k, is assumed to be 1 for purposes of this calculation.

Подпись: Coi (a.r. 7.5) =Подпись: CDi (a.r. 15) =CL2 = CL2

3.142 x 7.5 23.562

CL2 = Cl2

3.142 x 15 47.124

In the table, this allows the drag increment for each Cl and A. R. to be worked out in the first four columns, this is added to the profile drag coefficient read from the tunnel tests,

ASPECT RATIO EFFECTS CALCULATION

1

2

3

4

5

6

7

Cl

Cl2

Cdi(7.5)

C<u(15)

PROFILE

Cd

Cdi + Cdp

Cdi + Cdp

from tests

Cl x Cl

Ci2/23.562

Ci2/47.124

From tests

a. r. = 7.7

a. r. = 15

0.4

0.16

0.00679

0.00395

0.012

0.01879

0.01595

0.6

0.36

0.01528

0.00764

0.0098

0.02508

0.01744

0.8

0.64

0.02716

0.01358

0.0101

0.03726

0.02368

1.0

1.00

0.04244

0.02122

0.0107

0.05314

0.03192

1.2

1.44

0.06115

0.03056

0.0113

0.07245

0.04186

1.4

1.96

0.08318

0.04159

0.0125

0.09568

0.05409

1.5

2.25

0.09549

0.04775

0.0137

0.10919

0.06145

1.6

2.56

0.10864

0.05432

0.0149

0.12354

0.06922

1.675

2.81

0.11925

0.05963

0.0162

0.13545

0.0783

8

9

10

11

12

Cl/Cd

Cl/Cd

VCl3

/CL3/Cdi + Cdp JCtVCdi + Cdp

a. r. 7.5

a. r. 15

a. r. 7.5

a. r. 15

21.3

25.078

0.253

13.46

15.86

123.92

34.401

0.465

18.54

26.66

Compare Max. L/D

21.47

33.78

0.716

І19.22І

30.24

18.82

31.33

1.000

18.82

31.33

Compare Max.

16.56

28.66

1.315

18.15

[зЩ

Power factors

14.63

25.88

1.657

17.32

30.63

13.31

24.41

1.837

16.82

29.21

12.95

23.11

2.024

16.38

29.89

12.36

22.09

2.168

16.01

28.59

and the calculation of the L/D ratio and power factor for the wing proceeds exactly as in the table on p.163.

N. B. For an accurate result the 5th column should be modified to allow for different Re effects on profile drag, for the two different wing chords.

THE POWER FACTOR AND L/D RATIO

For a soaring glider or duration model when gliding the total power factor, Cl1 5/Cp should be as high as possible. The most suitable wing profile is that with a high section power factor, Cl’ VQj. This may be calculated from the wind tunnel results.

The table opposite indicates the method. A pocket calculator should be used to speed the work. The section 1/d ratio is also worked out in the tables. This gives a general idea of the profile’s efficiency, while the minimum drag coefficient indicates the potential of the aerofoil for high speed flight A low «у at low q is essential for a speed model wing.

THE BEST ANGLE OF CLIMB

It is assumed here that a duration power model has enough power available to achieve any desired angle of climb. The problem is to know which angle of climb, at maximum power, will give the best rate of ascent

A COMPARISON OF TWO AEROFOILS AT Re 100.000

1

2

3

4

5

6

GOTTINGEN 801

Re 100000 (KRAEMER

C1

cd

‘Vcj

q3

AT

V7/cd

TEST)

l

FROM

FROM

Column (1)

(Cold))3

JCo (4)

Col (5)

TEST

TEST

Column (2)

Col (2)

0.4

0.0289

13.84

0.064

0.253

8.754

0.5

0.0241

20.75

0.125

0.3535

14.668

0.6

10.02181

27.75

0.216

0.4648

21.32

Min Cd at Cj0.6

0.7

0.0220

31.82

0.343

0.5857

26.53

0.8

0.0240

33.33

0.512

0.7156

29.82

0.9

0.0260

34.62

0.729

0.854

32.85

1.0

0.0300

33.33

1.000

1.000

33.33

1.1

0.0317

34.70

1.331

1.154

36.40

1.2

0.034

1 35.291

1.728

1.315

38.68

Max 1/d at q 1.2

1.3

0.0378

34.39

2.197

1.482

ПЙШ]

Max power factor, c j 1.3

1.4

0.0518

27.03

2.744

1.656

31.20

0.4

0.0222

18.02

0.064

0.253

11.396

107,000 G. Muessman Test

0.5

ІЙ.0221І

22.62

0.125

0.3535

15.995

Min. C^atcj 0.5

0.6

0.0222

27.03

0.216

0.4648

20.936

0.7

0.0223

31.39

0.343

0.5857

26.26

0.8

0.0224

35.71

0.512

0.7156

31.95

0.9

0.0235

138.301

0.129

0.854

36.34

Max l/d at cj 0.9

1.0

0.0265

37.74

1.000

1.000

Г37Л7І

Max

1.135

0.0400

28.37

1.462

1.209

30.23

Starting from level flight trim, the power is increased step by step. In level flight, as already seen:

Lift (Level flight) = Lo = W = *pV*SCL This may be re-arranged to give an equation for speed:

V2 (level flight) = V02 = W/V4pSCL

In the climb Lift (Climb) = Lc = W Cos в Also, if Vc = speed along the inclined flight path then

Lc = 14pVc2SCl =W Cos в

Re-arranging this in turn to obtain equation for Vc2, Vc2 = W Cos 0/WpSCL – From the foregoing:

Vc2 = (WCos flj ^ / W = WCos в ISpSCL

V02 WpSCL/ : [^pSCl/ WpSCL W

which cancels down to:

-rr-r = Cos в and so-гг— = y/CosB

*0* ’’O

(This is on the assumption that Cl remains unchanged, i. e. the model is not retrimmed.) In the small diagram Figure A3, Vc, the flight speed along the inclined path, is

ч.

 

Fig. АЗ The best angle of climb for a high-powered model

represented by a line at angle 9 to the horizontal. The length of this line is proportional to Vc = V0 x VCos 9

The rate of climb on this diagram is proportional to the length of the line marked C. From basic trigonometry,

-^-= Sin 9 or C = Vc Sin 9 *c

And therefore:

C = V0 x v/Cos 9 x Sin 9

For a particular model and trim condition, V0 is constant The factor /Cos 9 x Sin 9 may easily be worked out with the aid of standard tables of Sine and Cosine, for any value of climb angle, 9. The result may be plotted against 9, as has been done in Figure 4.6. The maximum rate of climb is then found to occur when the graphed curved reaches its maximum close to 55 degrees. The result is approximate. Departures of four or five degrees either way make little difference. The practical trimming procedure is thus to aim at achieving the desired climb angle by adjustments of trim, wing camber, flaps etc. then to ensure that the engine propeller combination yields maximum thrust at that angle.

WHAT SHOULD THE CAMBER BE?

A mean camber line, such as those presented in Figs. 7.2 and 7.3, is designed to operate at one ideal design value of cj. Once the camber is determined, any symmetrical thickness form or envelope may be fitted to it to give an aerofoil which will operate most efficiently at the design ci.

For a racing model, it is first necessary to decide the speed at maximum power, straight and level. This may be measured from a real model in flight, or estimated for a new design, from previous experience. Then the model Cl may be worked out as described above. Knowing the Cl, the type of mean line required should be chosen from those in the tables (or from any similar source). Most of the NAC A mean lines are worked out for a design ci of unity. This allows the designer to arrive at the camber for his aerofoil by simple multiplication of the camber line ordinates in the NACA tables.

A worked example follows: the figures used are not intended to be representative of any modern racing model.

Model weight 1 kg. wing area 0.2 sq. metres, designed speed 20 metres per second.

„ , , „ lx 9.81 _ 9.81 _ . .

Model Cl и x 1 225 x 202 x 0.2 49.00 0-2

Hence the ordinates for the NACA a = 1 mean line may be multiplied throughout by

0. 200 to give the desired camber line.

For example, at 50% chord, the maximum camber point on this mean line, the camber should be:

.200 x 5.515 = 1.103, i. e. a 1.1% camber approx, for flight at this speed with this model

If the model is in a steep turn, the required lift force, and hence the effective weight, increase perhaps to three or four times die above. The formula then must be modified:

Cl (in steep turn) = 4 x.200 = .8, requiring a camber of.8 x 5.515 = 4.120%.

Подпись: .032 = CLA flight speed of 50 metres/sec yields: 9.81

306.25

The required camber is then.032 x 5.515 = 0.177% and in the steep turn 0.708%. Cambers of less than 1% are thus required for pylon racing models with speeds over 180 k. p.h. or 100 m. p.h. Note also the width of the low drag range or ‘bucket’ on modem laminar flow symmetrical aerofoils (see Chapter 9).

THE POWER FACTOR DERIVATION

Lift in level flight may be taken as equal to weight Then by re-arranging the lift formula:

Подпись: V =W

V4pSCL

For a glider the rate of sink is given by V x Sin a where a is the glide angle. As the ratio of drag to lift is also (very nearly) equal to Sin a the above expressions may be combined:

Подпись: V Sin оSinking speed =

W CD

fcpSCL x cL

 

Fig. A2 Chart showing variation of Reynolds number with air temperature and pressure [altitude].

The formulae in the right hand margin give the equations for Re extremes of winter and summer at near sea level. The shaded area indicates variations with seasons and altitudes up to 1000m (3281ft).

This is simplified as shown:

Sinking speed = / W 1 Cp _ / W Cp

V WpS X C1У1 CL V ^pS CL3/2

From tliis it is seen that two factors affect the sinking speed, one of these contains the wing loading, W/S, the other is the factor Ср/Сь3/2- To decrease the sinking speed, the wing loading W/S may be decreased, but this factor appears within a square root, so the effect of a large decrease of wing loading is relatively small. To obtain a larger improvement in sinking speed, Cp/CL15 must be reduced, or, what amounts to the same thing, Cl1 5/Cp must be increased. For steeper angles of glide, more than 10 degrees, the wing loading factor remains unchanged but the other factor is slightly modified to:

Cp (CL2 + Cp2)3/4

For a power model the lift formula is accurate for level flight, so the minimum power to sustain flight is arrived at thus: Power = force x distance in unit time

= Drag x Speed = DV

Drag = D = W-^ = (Formula for V is given above)


L CL

Power = W X/ ^ X S’R/7

‘V fcpS CL3/2

As with gliding, the wing loading, W/S, and the power factor must both be adjusted to achieve flight at minimum power. In addition, the weight alone plays a major part and the formula shows that a heavy model necessarily needs greater power for sustained flight

WORKING OUT THE LIFT COEFFICIENT

To calculate the lift coefficient of a model in a given trim condition it is necessary to know the speed at which it is flying, its wing area, and flying weight The speed is easy to determine if the model can be flown several times over a measured distance and timed with a stopwatch. Allow for any wind. For free flight models such timed flights can be done by making a series of straight glides from a high point in calm air. Radio controlled models are, of course, easier to time accurately.

The standard lift formula may be re-arranged to give Cl in terms of model weight If it scales 1 kg. this indicates a mass of 1 kg. It then needs a lift force of 1 x 9.81 Newtons to support it The formula then may be applied as shown:

Cl (whole model) =

A numerical example: Suppose the model mass is 1 kg., area 0.2 m2, speed 12 metres per second. Assume air at standard mass density of 1.22S kg./m3.

Cl = 9.81 (Й x 1.225 x 12x 12 x 0.2)

= 9.81 – 17.64 = 0.556

The same example worked in Imperial Units:

1 Newton force equals 2.205 lbs. force, 12m/sec equals 39.37 ft/sec., 0.2 sq. metres equals 2.1492 squ. ft Assume standard density of.002378 slugs/cu. ft

CL = 2.205 – (Vi x.002378 x39.37 x39.37 X2.1492)

= 2.205 + 3.9608 = 0.556

This establishes that the coefficient of lift is the same whatever units are employed in the calculation, providing a coherent system of units is adopted.

The value of performing such a calculation in practical modelling is that it enables a modeller to improve his choice of aerofoil, particularly its camber. It may also indicate possible improvements in wing rigging angles, tailplane size, and fuselage design.

Suppose an F3B sailplane of span 3 metres and wing area 0.75 sq. m weighs, with ballast, 4 kg. It is hoped to complete the 60 metre (4x150m) speed task in 17 secs. This represents V = 600/17 = 35.3m/sec. on average, which includes the turns. (The actual speed on the straight will be greater.)

The lift coefficient, on average, is found from the formula:

Cl (mean) % x j 225 x 35 32 x 0 75

39.24 л

(More in the turns, less on the straight)

Suppose now the same model, unballasted, weighs 1.5 kg. and when trimmed for minimum sink flies at about 5.5m/sec. on a timed glide. The Cl then becomes

Подпись: CL (min sink) =9.81 x 1.5

V4 x 1.225 x 5.52 x.75 -1-4,715 =

13.896 106

Fig. A1 Chart showing variation of air mass density with air temperature and pressure.

[Drawn by Lnenicka]

An aerofoil for such a model would require a low drag ‘bucket’ extending from Cl < -05 (nearly zero) to at least 1.06. This might be attained by using flaps.

Since the lift coefficient in the speed task is so low, very little camber is required here. A symmetrical profile would be satisfactory except in the turns. Much more camber is needed for soaring. Cl for the distance task can also be calculated.

Parasitic drag is very important at high speed, less so at soaring or distance task speeds. The fuselage should therefore be set, relative to the wing, at the angle for Cl.07 with corrections as explained below.

SYSTEMS OF UNITS

In measuring mass and forces in engineering, various systems of units are used. Which system is adopted is a matter of convention and convenience, but the Systeme Internationale or S. I. system is adopted officially in many parts of the world. In this system the unit of mass is the kilogramme, the unit of acceleration is the metre-per- second-per-second, or m/s2. A force, as the second law of motion indicates, is measured in terms of the acceleration of a mass, or Force = Mass x Acceleration. In S. I. units, the unit of force becomes the Newton, one Newton being the force required to accelerate one kilogramme of mass at one m/s2. The mass of a model, on or near the planet Earth, is constantly acted on by the acceleration due to gravity, which has for practical purposes the value of 9.81 m/s2. Hence a model of 1 kg. mass exerts a downward force or weight of 1 x 9.81 Newtons. Metric kitchen scales in common use do not usually read in Newtons, but so long as they are used on Earth, they may be taken as reading kilogrammes directly as units of mass. In aerodynamic figuring, however, the forces must be expressed as Newtons to maintain consistency. Many modellers are accustomed to other systems, such as the British Imperial system, or some variety of it In this, the unit of mass is the slug, of acceleration the foot-per-second-per-second, of force, the pound-force. Scales reading in pounds measure the force exerted by one slug of mass under the influence of Earth’s gravity. The acceleration due to gravity is 32.2 ft/s2.

There are other systems of units. Which is used is a matter of individual preference, but whichever is employed it must be consistent, so that one unit of force always equals one unit of mass multiplied by one unit of acceleration. If this rule is not observed confusion results. A fuller explanation of the rival systems of units with conversion scales may be found in Metrication for the Modeller (M. A.P. Technical Publication, 1972).