LEV~kIvP 05AFFDL-TR-76-55Volume I

SAERODYNAMIC STABILITY TECHNOLOGY FORMANEUVERABLE MISSILES

Volume I. Configuration Aerodynamic Characteristics

MAR TIN MARIETTA CORPORA TIONOR LANDO DIVISIONP 0. BOX 583 7ORLANDO, FLI.RIDA 32805

f- MARCH 1979

TECHNICAL REPORT AFFDL-TR-76-55, Vol. IFinal Report for period February 1975 - December 1976

C.2_

Approved for public release; distri ution unlimited.E E C

JUN 22 I979

AIR FORCE FLIGHT DYNAMICS LABORATORYAIR FORCE WRIGHT AERONAUTICAL LABORATORIESAIR FORCE SYSTEMS COMMANDWRIGHT-PATTERSON AIR FORCE BASE, OHIO 45433

Reproduced FromBest Available Copy

NOTICE

When Government drawings, specifications, or other data are used for any pur-pose other than in connection with a definitely related Government procurement.operation, the United States Government thereby incurs no responsibility nor anyobligation whatsoever; and the fact that the government may have formulated..furnished, or in any way supplied the said drawings, specifications, or otherdata, is not to be regarded by implication or otherwise as in any manner licen-sing the holder or any other person or corporation, or conveying any rights orpermission to manufacture, use, or sell any patented invention that may in anyway be related thereto.

This report has been reviewed by the Information Office (O1) and is releasableto the National 7ecznJcal Information Service (NTIS). At NTIS, it will be avail-able to the general public, including foreign nations.

This technical report has been reviewed and is approved for publication.

W. H. LANE R. O. ,ANDERSON, ChiefProject Engineer Control Dynamics BranchControl Dynamics Branch Flight Control Division

FOR THE COMMANDER

R. STANLEY, Col USAFk.. Chief, Flight Control Di sion

Air Force Flight Dynamics Laboratory

"If your address has changed, if you wish to be removed from otr mailing list,or if the addressee is no longer employed by your organization please notify

AFFDL/FGC ,W-PAFB, OH 45433 to help us maintain a current mailing list".

Copies of this report should not be returned unless return is required by se-curity considerations, contractual obligations, or notice on a specific document.

AIR fORCE/56780/21 May 1979 - 55

---------------------------------------------------------------

UNCLASSIFIED

f" EPORT DOCUMENTATION PAGE READR CMINS lIN F( UNMUOR 4~t&-..~- -2. OVTACCESSION NO. 3 ET'S CAT AL GV. NUMBE R

Aerodynamic Stability Tee ology for I5 i .n7alManeuverable Missiles. i.I. Co~n- 417jw.c~hVN17

figuration Aerodynamic Ch~aracterisitics, t7R -47 27 ER7 A NOR (.I );'0TX'W RN UOR.

Gennaro F./Aiello F33615-75-C-3O52 I bMichael CIaea

9 PERFORqMING ORGANIZATION NAME AND ADDRESS 11o PROGIIAM ELEMENT. PROJECT. TASKAREA 4 RORK U NIT NUMIOERI

Martin Marietta CorporationOrlando Division, PO Box 5837Orlando, FL 32805 -

I I. CONTROLLING OFFICE NAME AND ADDRESSREOTDEU.S. Air Force Flight Dynamics Laboratr rd*7Wright-Patterson Air Force Bass I~E OF PAGESDayton, Ohio 360

16MONITORING AGENCY NAME A ADORESS~it dIjfto.~. f.. controling Offce) IS. SECURITY CLASS (0D IA,. o*po.f)

Unclassified.

IA DISTRIBUTION STATEMENT (of (hit Rovart)

Ap proved for public release; distribution unlimited

I? DisTWRieuUioN STATEMEN4T (of IA. o.I,..t orte,.d I. Mock 20. Of diII.,.ot I-~. M.p.,f)

I0 SUPPLEMENTARY NOTES

IV KEY W011110 *r.non,, ,. . #do *. It noe..wv aid Identiv~ by .inc.V. IOuf

TransonicSupersonicHigh AnglePredictions

I0 mR. A c c.,, ,. ild. It end..r Lcdowif,I~II P., bl"., k nrmn,

This study developed empirical 'methods to predict aerodynamic characteristicsof body-tail, body-ving-tall and body-strake-tail missile configurations.Methods cover the Mach number range from 0.6 to 3.0. Methods COVIer Ehe indi-vidual body and tail characteristics at angles of attack from 0 co 180 degrees.For winged bodies the methods encompass angles of attack up to about 30 degrees.All mutual interference effects are accounted for, allowing accurate predictionof force and moment coefficients.

DD ~2~I 113 ,~ P I OV '. ON'L~TI.UNCLASSIFIED

FOREWORD

This report was prepared for the U. S. Air Force Flight Dynamics

Laboratory, Wright-Patterson Air Force Base, Dayton, Ohio, under contract

number F33615-75-C-3052 as part of Project 8219. The work was performed

at the Orlando Division of Martin Marietta Aerospace in Orlando, Florida.

The reported effort began in February 1975 and ended with the submittal

of the draft of this final report in December 1976.

The principal investigators were J. E. Fidler and G. F. Aie'io. The

technical monitors for the Flight Dynamics Laboratory were Dr. Robert Nelson,

Lt William Miklos and Mr. William Lane.

The authors wish to express their gratitude to the aforementioned

contract monitors for their guidance and support and recognize a special

debt to Mr Lane for his extraordinary effort in reviewing this report and

the significant contribution towards the readability and overall quality of

the report. The authors would also like to express their gratitude to

Mr. William Baker, Arnold Engineering Development Center, for his cooperation

in providing easy access to the 180 degree, body plus tail data bank. Many

sincere thanks are due the following associates at the Martin Marietta, Orlando

Division: G. S. Logan, Jr., D. T,. Moore and R. L. Swarn.

ACCssio1 For

11'IS G&UI ',

DDC TAB'nnouc( Du D C___________ i UN 22 1979

Avail and/or Dspecial

tii

TABLE OF CONTENTS

Page

1.0 Introduction.................................. . . . . . 1

2.0 Experimental Data Sources and Models ....... . .. . 7

3.0 Aerodynamic Data Trends............. ...... . . . . . 12

4.0 Formulation of the Aerodynamic Prediction Equations . . ... 35

5.0 Aerodynamic Methods . . ............................... . .. 39

5.1 Isolated Components........ . 39

5.1.1 Body Normal Force ......... ................ ... 39

5.1.2 Body Center of Pressure ...................... 61

5.1.3 Body Axial Force .................. .......... 77

5.1.4 Fin Normal Force .... ........ ......... 91

5.1.5 Chordwise Center of Pressure ................. 122

5.2 Body-Tail Configurations .l. ....... ................... 143

5.2.1 Tail-on-Body Normal Force .......... ............ 143

5.2.2 Tail-to-Body Carry-over Normal Force ........... .161

5.2.3 Tail-to-Body Cerry-over Normal ForceCenter of Pressure.. ....... ............... .. 171

5.3 Body-3trake-Tail Configurations ......... ............. 190

5.3.1 Incremental Normal Force Due to Strakes ..... 190

5.3.2 Center of Pressure for Incremental NormalForceDue to Strakes ....... ................. 202

5.3.3 Incremental Normal Force Due to Tails. ....... .. 220

5.3.4 Center of Pressure for Incremental NormalForce Due to Tails ......... ................ ... 232

v JO PAM ;W p A

TABLE OF CONTENTS (Concluded)

Page

5.4 Body-Wing-Tail Configurations. .. ........................259

5.4.1 Incremental Normal Force Due to Wings. ............ 259

5.4.2 Effective Center of Pressure for IncrementalForce Due to Wings. .. ............................274

5.4.3 Tail Incremental Normal Force Due to WingVortex Interference .. ............................289

5.4.4 Effective Center of Pressure of the IncrementalTail Normal Force Due to Wings .. .......... .. .... 306

5.5 Thrust Vector Control Effects. .. ........................310

5 .5.1 Incremental Body Normal Force Due to, Plume

Effects. .............. .......................... 310

5.5.2 Effective Center of Pressure for Incremental Body,Normal Force Due to Plume Effects. ................ 323

5.5.3 Incremental Tail Normal Force Due to PlumeEffects. .............. ..........................334

5.5.4 Effective Center of Pressure of Incremental TailNormal Force Due to Plume Effects. ................ 351

6.0 Conclusions and Recoimmendations. ............ ..................356

7.0 References .. ......... .................. ....................358

vi,

LIST OF ILLUSTRATIONS

Figure Page

Ia Methodology Requirements for TVC Missiles ..... ......... 5

lb Methodology -4uirements for AerodynamicallyControlled Missiles. ..... ................. .......... 6

2 Schematic of Total Data Base ........... ................ 9

3a Martin Marietta Main Body Model in the NSRDC 7' X 10'Transonic Tunnel at Sixty Degrees Angle of Attack ......... .10

3b MartinMarietta Tail Models ............... 11

4 Vortices Produced by the Reattachment of Lower SurfaceBoundary Layer . . .................... 13

5a Fin Normal Force Coefficient (M-0.8, Aspect RatioEffects) ..................... ......................... 19

5b Fin Chordwise Center of Pressure (M-0.8, AspectRatio Effects)...................... .. ...... . 20

5c Fin Normal Force Coefficient (M-2.0, Aspect RatioEffects) ............. ....................... . . 21

Sd Fin Chordwise Center of Pressure (H-2.0, AspectRatio Effects)................................... . . 22

6a Fin Normal Force Coefficient (M-0.8, Taper RatioEffects) ........... .............. 23

6b Fin Chordvise Center of Pressure (M-O.8, Taper RatioEffects) ...... . . . . . . . . . . . . . . . . . 24

6c Fin Normal Force Coefficient (M-2.0, Taper RatioEffects)................. ... ....... . . . . . 25

6d Fin Chordwise Center of Pressure (K-2.0, Taper RatioEffects) . . ................. . . . 26

7a Fin Normal Force (Mach Effects). .. . . . . . . . .. . 27

7b Fin Chcrdwise Center of Pressure (Mach Effects) ......... 28

8a Variation of Induced Out-of-Plane Forces and

Moments (M-0.6) ........... . ...... ................ 29

vii

-LIST OF ILLUSTRATIONS (Cont'd)

igurePa

8b Variation of Induced Out-of-Plane Forces andMoments (M-2.0) ........... ....................... ... 30

9a Out-of-Plane Forces and Moments Due to VortexAsymmetry (AR - 0.5, A - 1.0, d/s - 0.5) .... .......... .. 31

9b Out'of-Plane Yorces and Moments Due to VortexAsymetry (AR - 0.5, A 0. dis - 0.4) .............. .. 32

loa Comparison of Tail Normal Forces ......... ............. 33

10b Comparison of Rolling Moments ........................ 34

11 Compdrison of Experimental ond Predicted Results(C) Mach 0.6 ..................... ............. ... 48

B12 Comparison of Experimental and Predicted Results

(CB ), Mach - 1.15 ................................ .. 48

13 Comparison of Experimental and Predicted Results(CN ), Mach - 1.30 ................................. 49

B.14 Comparison of Experim~ental and Predicted Results

(CN ), Mach - 2.0 ................... ............ . .49

15 Coefficients for Calculation of CM (A1 ) ..... ........ 50NB

16 Coefficients for Calculation of CNB (A2 ) ........ 50

17a Curves for Transonic C14 (1N/d - 1.5) ...... ............ 51

a17b Curves for Transonic CM (tN/d = 2.5).................. 52

a17c Curves for Sransonic CN (zN/d - 3.5) ................ 52

al8b Curves for Supersonic CN (LN/d - 2.5)........... .... 53al8b Curves for Supersonic CM (L,/d - 3.0) ....................

a

18c Curves for Supersonic CN (1N/d - 3.5). ..... .......... . . 54

18c Curves for Supersonic CN (IN/d - 4.0)... .. ......... o54

19 Correlation Factor for End Effects .... ............. .. 55

20 Variation of n withMach Number ..... ............... 55

21 Curves for Determining Transonic Values of n ... ....... .. 56

viii

LIST OF ILLUSTRATIONS (Cont'd)

Figure Page

22a Basic Values of Cd .................... 57c

22b Crossflow Drag Coefficient (Subcritical Crossflow, MH90.4). 57

23 Comparison of Experimental and Predicted Results(CN ), Mach - 0.6 ........ ................ . 57NB

24 Comparison of Experimental and Predicted Results(CN ), Mach -1.15 ................. . .. ......... ... 58

B25 Comparison of Experimental and Predicted Results

(C "), Machb- 1.30 ..... .......... .............. 59

26 Comparison of Experimental and Predicted Results(CBN ) Mach - 2.0 .... ... .................. 59

27 Comparison of Experimental and Predicted Results(C ), Mach - 2.86 ............. ................... ... 60NB

28 Comparison of Experimental and Predicted Results(C). Mach - 0.85, 1.20, and 2.25 ..... ............. ... 60

B

29a Transonic Tangent Ogive-Cylinder Zero Angle ofAttack Centers of Pressure (IN/d - 3.5) .......... 70

29b Transonic Tangent Ogive-Cylinder Zero Angle ofAttack Centers of Pressure (t /d - 2.5) ......... . 70

29c Transonic Tangent Ogive-Cylinder Zero Angle ofAttack Centers of Pressure (i N/d - 1.5) . . . . . . 0

30a Supersonic Tangent Ogive Cylinder Zero Angle ofAttack Centers of Pressure (fN/d - 4.0) ...... .......... 71

30b Supersonic Tangent Ogive - Cylinder Zero Angle ofAttack Centers of Pressuce (t Id - 3.5) ...... .......... 71

30c Supersonic !-ngent Ogive - Cylinder Zero Angle ofAttack Centers of Pressure ( /d 2.5) ........... 71

'31 Increment'in Center of Pressure Between Angles ofAttack of 0 and 20 degrees ......... ...... 72

32 Polynomial Coefficients , Low Angle of AttAck ........ ...

33 Polynomial Coefficients , High Angle Of. Attack ......... ... 73

ix

'! 4

LIST OF ILLUSTRATIONS (Cont'd)

Figure page

34 Comparisons Between Predictions and Experimental

Data XCP , Hach - 2.86 .. . . . . . . . . . .... ... 74

35 Comparisons Between Z,.dictions and ExperimentalDate Xp, Mach - 2.25 . ..... . ... ............... .... 74

36 Comparisons Between Predictions and ExperimentalData XCp, Mach - 0.85 ... ........ *. . . . . . . . .. . . 75

37 Comparisons Between Predictions and ExperimentalDataX p., Mach 0.80 .... .................. ....... 75

38 Comparisons Between Predictions and ExperimentalData XCp , Mach - 3.0 .... ... .................. . .. 76

-- Bd

39 Variatioi with Mach Number of 180-Degree Axial ForceCoefficient ........ ............. ........... ......... 84

40A Comparison Between Predicted and Experimental CA(a-f) (Transonic) ...... ... .................. B ..... .... 85

41a Curves for Determining CA (tN/d - 1.5) ....... . ... 87

41b Curves for Determining CA (IN/d - 2.5) ....... . ...... 87

lb

41c Curves for Determining CAb (IN/d = 3.5).... . ... ........ 88

42 Scaling Factor for C.A. ..... .................. .88

43 Variation of CA with Mach Number ..... ............. ... 89

44 Basic Curves of f(M, a) Calculated from Power Series 89

x

LIST OF ILLUSTRATIONS (Cont'd)

FigurePage

45 Comparison Between Predicted and Experimental DataC (Supersonic) ............. . ..... ............ ... 90

46 Power Series Pareters for Equation (24) ...... ....... 104

47 Lift Curve Slope for Taper Ratios 0-1.0 . .... ........ ... 105(a-d)

48 Variation of CN (w/2) with Mach 'Number .... ......... .. 107

49 a, Angle of Attack Above Which ACN Must be Applied(Subsonic only) .......... ... .......... ........ .. 108

50 Dimensionless CN Increment Above a.' ................. 109

51 ACI, Maximum Increment of Normal Force Above a'

(Subsonic Only) . . .. .......... .................... .. 110

52 Comparison of PredicLed and Experimental CN , Mach -0.8 110

53 Comparison of Predicted and Experimental CMT, Mach - 0.98 111T

54 Comparison of Predicted and Experimental.C ,NMach -i.02 111T

55 Variation of Fin Normal Force at a - 90" withMach'No ........... ............. ........... ......... 112

56a Variation of Normal Force Coefficient, CN (30), withMach No., a - 3,0" (A - 0) ...... . ...... T....... ... 113

56b Variation of Normal Force Coefficient, CN (30), withMach No., .a - 30' (A - .5). o ...... .. ...... 113

56c Variation of Normal Force Coefficient, CN (30), withMach No., a 30" (A - 1.0) ........ To ..... . .. 113

57 Variati.on of C (30) with Mach Number ..... ........... 114NC&

58 Power Series Parameters for Equation (26) ......... 115

59 Comparison of Predicted and Experimental CNTfrom 30 to9 0 degrees .... . .... ....... 116

60 Curves for Modifying CN Method, (X - 0, AR - 1.0,Subsonic) ............ ........... ........... ......... 116

xi

/f

LIST OF ILLUSTRATIONS (Cont'd)

Figure Page

61 An Example Using ACN. . ...... ........... 116N

62 Comparison of Method and Test ,C (A = 0, AR - 0.5). . . 117N T

63 Comparison of Method and testCN X0.5,AR=0.5;AO0,AR-l.O0 118T (C

64 Comparison of Test to Methods to 1807, M = 0.6 (C) 19T

65 Comparison of Test and Method, M = 2.0 (CN .... ..... 120

66 Comparison of Test and Method, M - 2.5 (C ) ......... . 120

67 Comparison of Test and Method, M - 3.0 (CN )(k=1.0,AR=1.0) 121T

68 Comparison of Test and Method, M - 3.0 (C N)(I- 0,AR-l.0) 121

69 Chordwise Center of Pressure Variation to 180Degrees ................ ......................... 136

70 Chordwise Center of Pressure Variation withTaper Ratio at Alpha of 90 Degrees .............. .... 136

XCp71a Basic Curves for - at Reference Mach Number 0.98

(0-180 Degrees, R AR-0.5). . 137

71b. Basic Curves for .-- at Reference Mach Number 0.98(0-180 degrees, CR AR .1.0) ............ 137

71c Basic Curves for C at Reference Mach Number 0.98(0-180 Degrees, R AR - 2.0).. . . . . . . . . 1..

72a Basic Curves for CP at Reference Angle of AttackC,175-180 Degrees -R (M - 0.6 to 3.0, AR - 0.5) .... 138

x CP72b Basic Curves for - at Reference Angle of Attack

175-180 Degrees CR (M - 0.6 to 3.0, AR - 1.0) .... I3R72c ~ stcurve forxcp72c BasicDCurves for - at Reference Angle of Attack

1-180 Degrees C R (M - 0.6 to 3.0, AR - 2.0) .... 138

73 Power Series Constants Versus Angle of'Attack . .... t39

74. Mach Number Correction Factor for a, 90 Degrees , . 40

75 Variation of Al(XCP/CR) with Mach Number at

Alpha of 160 Degrees .................. . . 110

76 Comparison of Predicted and Experimental Center ofPressure Location, X M 1.15 ............... 141

T

R

dii

LIST OF ILLUSTRATIONS (Cont'd)

Figure Page

77 Comparison of Predicted and Experimental C.P.Location, XCP N - 0.80 ........ ........ .......... 141

CR

78 Comparison of Predicted and Experimental C.P.Location, X M - 1.3 ........ ................... 142

CR

79 KT(B) Ratio at Zero Angle of Attrack .. ....... 150

80 General Coefficients for Calculation of Rr(B) (A 0 ) . . . 11

81 General Coefficients for Calculation of '(B) (A1 ) . 152

82 General Coefficients for Calculation of RT(B) (A2) " " . 153

83 Interference Factor at Angle of Attackiof 90 Degrees . . 154

84 Comparison of Experimental and Predicted Results,CN ) ,H - 0.6 ........... ................... ... 155

85 Compariso4 of Experimental and' Predicted Results,

TH)' M - 3.0 ........ ..................... .... 156

86 Comparison of Experimental and Predicted Results,CNT , M 2.0 ...... ..... ..... .............. ... 157T (B)

87 Comparison of Experimental and Predicted Results,C (B), M -3.0 ..... .......................... 158

88 Comparison of Experimental and Predicted Results,M M , 1.15 .... ......................... 159

NT (B)'

89 Comparison of Experinental and Predicted Results,CN , M - 0.8 ..... ....... . ................ 160T()

90 Transonic IB(T)' Schematic ..................... ... 166

91a Curves for Estimation of Transonic I (all A) ...... 167a

xiii

LIST OF ILLUSTRATIONS (Cont'd)

Figure page

91b Curves for Estimation of Transonic Ib(all A and d) . . ......... . . . ...... 167

91c Curves for Estimation of Transonic I(all A and M) . . . .. . . . . . . . . . . . . . . . . .. 167

92 Comparison between Predicted and ExperimentalIT . .. .. .. .. .. .. .. .. .. .. ..... 168

93 Supersonic I(T) Schematc ....... 168SCue for E, S

946 Curves for Estimation of Supersonic 12 ......... 169

94b Curves for Estimation of Supersonic 12 ............... 169

94c Curves for Estimation of Supersonic 13 .. .. .. . .. 169

95 Comparison Between Predicted and Experimental I(T) . 170

96 Curves for Determining X C with AlterbodietSp

for Supersonic Speeds ....... 183

97 Curves for Determining XCP for No Afterbodies at

Supersonic Speeds ........ ........... ......... 184

98 Curves for Determining XC. for SubsonicB- (T)

CRSpeeds (Zero Leading Edge Sweep) .... ............ . . 185

99 Curves for Determining X ( for Subsonic- 1(T)CR.

Speeds (Zero Trailing Edge Sweep) . . . . . . . . . . . 186

100 Coefficients Required for Evaluation of

XCP-PB(T M . . 187

101 Comparison Between Predicted and Experimental DataCNOT ....... ........... ............................. 188

xiv

LIST OF ILLUSTRATIONS (Cont'd)

Figuree

102 Comparison Between Predicted and ExperimentalData, X ........................ 189

ST

103 ACN BS General Curve Form .... ................. 196

104 ACN' Peak Factor K ........... ..... 197

105 Coefficients for Calculating ACN ............... ..... 198BS

106 Comparison of Test Data and Method, ACNBS ........ 200(a-d)107 General Curve Form, XCPA,.S ............... . 211

108 Straka Parameters . ..... ........... . ....... 212

109 Polynomial Coefficients for Calculating XC S . .. . . . 2i3

110 J and K Values for Calculating XP .. . . . . . . 216X ABS

111 Comparison of Test Data and Method, XCp /d ....... .... 217ABS

112 Comparison of Test Data and Methol, XCp BS/d ........ 219

113 Coefficients for Calculation of ACNBST (A) ..... ... 227

114 Coefficients for Calculation of AC N (A2 ) ...... 228

BST

115 Coefficients for Calculation of ACNs (A3 ) ......229

BST

116 KT(B) and %B(T) Ratios (Slender Body Theory),.. .... 230

117 Comparisons of Predicted Results with ExperimentalData, AC, BST . . . . . . . 231

xv

LIST OF ILLUSTRATIONS (Cont'd)

Figure Page

118 KT(B) and KB(T) Ratios (Slender Body Theory) . . . ... 245

119 Tail Alone Center of Pressure at Subsonic Speeds . . . . 246

120 Tail Alone Center of Pressure at SuFersonicSpeeds............ ............. ......... . . . . . 247

121 Curves for Determtning CPB(T) !or Subsonic Speeds

CR

(Zero Trailing Edge Sweep) ....... . ........ 248

122 Curves for Determining for No Afterbody

CR

at Supersonic Speeds ......... . . . . . . . . . . . 249

123 Curves for Determining for Subsonic Speeds

CR

(Zero Leading Edge Sweep) .. .... . .... .. 250

124 Curves for Determing XCPBMT with Afterbodies atSupersonic Speeds I CR. ...... ..... . . 251

125 Coefficients for Calculation of XCP, (A,) . . . . 252

CR

126 Coefficients for Calculation of XCp. (A32) .. ......... 253

CR127 Coefficients for Calculation of X cp1 (A 3) .. .. . .. 2.54

C R

128 Comparison Between Predicted and Experimental

Results, XCP s/ M4-0.6 ........... ......... 355EP ST

129 Comparison Between Predicted and Experimental

Results, XCP BST/d, 4-0.85 ............ ... 256

xvi

LIST OF ILLUSTRATIONS (Cont'd)

Figure Pg

130 Comparison Between Predicted and ExperimentalResults, Xp /d, M-1.2 ...... .................... 257

CPBST

131 Comparison Between Predicted and ExperimentalResults, X CPBST/d, M- 2.2 ...... .................. 258

132 Comparisons of Existing Method Predictions vithExperimental Data, ACN.BW ........ ................ .. 266

133 K Rt) Iato at Zero Angle of Attack .... ............. 267,

134 Comparison Between Predicted and ExperimentalResults, AC BW, Configuration 2, H-1.1 ......... . . .. 268

135 Configurations (Body + Wing) ....... .............. .. 269

136 Comparisons Between Experimental and PredictedResults, ACNBW, Configurations I and 3, M-1.- ...... .... 270

137 Comparisons Between Experimental and PredictedResults, ACN , ?3.08 .. . .. .............. 271

138 Comparisons ,Between Experimental and PredictedResults, ACNN , M-"9. .... ............... . . . . . 272

BU

139 Comparison Between Experimental and PredictedResults, ACN . M- 0.85.. ..... ............. . . . . 273

140 KI (B) and KB(W) Ratios (Slender Body Theory) .... ....... 279

141 Wing Alone Center of Pressure at Subsonic Speeds. . . . . 280

142 Wing Alone Center of Pressure at Supersonic Speeds. . .. 281

143 Curves for Determining XCP I/CR at Subsonic Speeds. . . 282

CB (W)R

144 Curves for Determining XCP (W)/CR with Afterbody). ... 283

at Supersonic Speedi

xvii

LIST OF ILLUSTRATIONS (Cont'd)

Figure Pae

145 Configurations (Body + Wing) . .............. 284

146 Comparison Between Predictions and ExperimentalData, X CPw/d, H-.0.85 . . . .............. 285

"ABW

147 Comparidon Between Predictions and ExperimentalData, Xc Ip/d. M-.1................... ...... 286

ABW

148 Comparison Between Predictions and ExperimentalData, XCP, /d, M-21.9 . . . . . . . . . . . . . . 287

BW

149 Comparison Between Predictions and ExperimentalData, XC Id,.M-2.86 ...... . ......... * . . . . . . 288

150 Transonic Wind Tunnel Test Configurations . . . . . . .. 299

151 Wing Vortex Location ........... .................... 300

152 Wing Vortex Induced Tail Angle of Attack ..... ......... 301

153 Comparison Between Predicted and ExperimentalResults, AC I TWV .Mi. ........ . .............. 302

154 Comparison Between Predicted and ExperimentalResults, C M-0.*7. ................... 303

NEW

155 Comparison Between Predicted and ExperimentalResults, CN. .1,W0.85 ................. 304

156 Comparison Between Predicted and ExperimentalResults, CBW, N 42.36 ..... ... ................. .. 305

BWT

157 Comparison Between Predicted and ExperimentalResults, XCP /d, M.0.85 ... ..................... 308

BWT

158 Comparison Between Predicted and ExperimentalResults, XCP BwT/d, M..2.36 ...... ............... ... 309

xviff

LIST OF ILLUSTRATIONS (Concluded)

Figure Page

159 General Curve Form, AC P . ............. 317

I P

160 Power Series A for Calculating AC , .. . ... 318

161 Amplification Factors for Calculating ACM ....... .319'P162 Comparisons Between Predictions and Experimental

(a-e) Data, ACN ....................... 320NBP

163 Comparison of Body Alone XCp /d (Jet-On and Jet-Off) 328(&-.)

164 Comparison Between Predictions and Experimental(a-.) Data,p B/d.................. ............. 331

165 General Curve Forms, [AC j3..........43

166 Amplification Factors for Calculating AC.NTp ........ 344

167a Pover Series A for Calculating ACM , M(

LIST OF SYMBOLS

_-0 AI, A2 , A(a) General coefficients

AR Aspect ratio, (2b) 2 /S (2 panels)

AR Strake aspect ratio

a, I Body radius "- inches

a, of al, a... Polynominal expansion coefficients

- o B0 , B2 , B(a), General coefficients

b Exposed semispan 1 inches

c 1i c 2 , ... General coefficients

CA Axial force coefficient

CA Axial force coefficient, omitting base effects

C Basic value of CACAlb A1

CAbase Axial force coefficient due to base effects

C A C A 1+ CA~ 1 CAbase

Cd Drag coefficient

C Pitching moment coefficient

CN Normal force coefficient, based on Sref

C NB CNBP Body alone C., jet-off and jet-on, respectively

Body + strakes normal force coefficientBES

CN , C N Total CN on the body in the presence ofB(ST) B(ST)P strakes and tails, jet-off and jet-on.

Cu Normal force coefficient at a - 90"CNx

xx

LIST OF SYMBOLS (CONT'D)

CTotal CN of four strakee in presence of body,

totall ttal Jet-off and Jet-on

CNT Single tail panel alone normal fo:ce coefficient

S.. .TTotal CN of four tails in presence of body, jet-offT (B)total T(B)Ptotal and jet-on

- ( -CT Single tail panel normal force coefficient in the

T(B) T(B)P presence of a body, jet-off and jet-on

CR Root- chord A inches

CR Strake root chord length ev inches

Cs Base stagnation pressure coefficient.

d Body cross-sectional diameter 1v inches

dnoz Nozzle exit diameter % inchra

K5 5 , K160 Amplification factor at - 55%, ll0, andK10' 160160 - f(M)

K.7o Value of AC /MR at a - 70" (-f(M)3

K-..5 Value of ACNBp /MR at a - 1453 [-f(M)]

F(HACH) Mach number cornection used in conjunction with

SM Iq

LIST OF SYMBOLS (CONT'D)

AC @a *(S .8 AR

JNs - 0".K(W)+((B()) (- AR

ref

2 Scale factor for XCPABS at a-600

a Amplification factor, peak value of ACNBS ata - 57 and 135".

Scale factor for XCPABS at a-600

Value of XCPB/dREF at a - 120'

KT(B) Ratio of normal force on tail in the presence ofbody to tail aloae normal force

%B(T) Ratio of normal force on body due to tails totail alone normal force

KB(W) Ratio of normal force on body due to wings towing alone normal force

%(B) Ratio of normal force on the ving in the presence ofbody co wing alone normal force

L Mean value of ACN B, 80O < < 120O

Length ou inches

1/d Fineness ratio

L Missile total length inches

'A Length of missile cylindrical asction ' Inches

SHMissile nose length I inches

Le Distance between wing trailing edge-and tailleading edge at a lateral distance Y

M Freestream Mach number

MR Jet momentum ratio - q /q.

M.S. Missile station (inches from the nose)

N Normal force - lbf

xxrii

LIST OF SYMBOLS (CONT'D)

p Tail seemspan, measured from body centerline inches

qj Je;t dynamic pressure at nozzle exit - lbs/sq. ft.

q., q Freestream dynamic pressure u' lbs/sq. ft.

R Tangent ogive nose rad',s or value of CP5at m - 60 %. inche- d

R aReynolds number

RT Tail area ratio - ST/Sref

RT(B) lIterference factor (CN T(B)IT)

RW(B) Interference factor (C,),/%)

r Body radius ", inchesrv Radial distance measured from vortex core u inches

S Area N sq. ft.

Sp Body planform area P' sq. ft.

Sref' or SB Reference area - wd. 1. sq. ft.4

SR , or S Strake single span exposed area %, sq. ft..RS a

SSB Aree of two strakes + planform area of bodybetween strakes

T Tail single panel exposed area %' sq. ft.

Sw' -Wing single panel exposed area % sq. ft.s Total tail span including body

T Value of XCpS at a - 120"

d

V Vortex tangential velocity at a distance r

X Axial distance, %, inches

X Ditanci to center of planform area inches

X A Location of forward strake segment centroid relativeto LE

xxiii

LIST OF SYMBOLS (CONT'D)

xB Location of aft strake segment centroid relativeto LE

xS Location of net strake centroid relative to LU.

XCP Center of pressure of carry-over loading on bodymeasured from tail root chord leading edge

XCP Center of pressure of the tail measured from thebody nose

X Chordwise center of pressure of tail in theCPT(B) presence of a body neasured from tail root chord

leading edge

o Cp Strake CP location at a - 0*

0

X X Body alone center of pressure station, jet-offand JeL--on , relative to the nose

A Body + strakes center of pressure, relative toCBS the nose

XCPS , CPs Center of pressure of a body-strake-tail combination,BST BSTP jet-off arnd jet-on

X , CP9X CPp Effective center of pressure (M.S.) of total carryoverC P CN die to strakes + tails, jet-off and jet-on

xX CP Effective center of presaure (4.S.) of strake carry-I(S) I(S)P over on body CN, Jet-uff and jet-on

x CP XCP Effective center of pressure (M.S.) of tail carryover1(T) 1(T)P on body CN, jet-off and jet-on

X Cp Center of pressure of AC Ns, relative to 'strakeSLE NBS

X CP ABS Center of pressure of ACN BS, relative to the nose,

XC Center of pressure of AC1 s as a percentage -fCP ABf N STFR root chord measured from the wing root chord

leading edge

XCp C-nter of pressure of AC measurcd in diameters-T ABW from the nose NBW

xxiv

LIST OF SYMBOLS (CONTUD)

X Effective center of pressure of .TCIP ATWV

xCP /CitChordvise center of pressure (nondimensionalizedby panel root chord, C )

xCf Center of pressure at a i i degrees

Q-1

XrpjTail center of pressure at , - 160" for

a - 160 basic Mach - 0.98M' -0.98

XCP Tail center of pressure at a 160*" a,- 16corrected for Mach nu.ber

C R,- 160S~XCaPI Ao.XC

H M-0.98Effective center of pressure of the incremental

XCZS(T) force on a body strske-configuration due to theaddition of a tail

C .10.90 Initial slope of tail chordwise center of20 pressure at a - 160"

XLE Strake leading edge station from nosetip

a Angle of attack

a, Angle at which linear variation of X begins

CPS

4-17 or 4=-

wCBP Incremental CN on body alone due to Jet -

CN P - NB

ACNB-S Incremental normal force coefficient due toBS strakes

ACN Slope of AC vs a curve - 3AC /ga

xxv

LIST OF SYMBOLS (CONT'D)

AC N Increment in aormal force due to the addition of,IR/ wings to a body

ACNS Incremental CN on strakes due to Jet =CN NSP SNs

ACNT Incremental CN on tails due to jet = C N - CNT

ACN Increment in normal force due to the tails of aT(BS) body-strake-tail configuration

AC Total incremental C N on body + tails configurationTr due to jet effects on tails - (CN +IB(Tp) -

(CNT + IB(T))

ACN Incremental normal force coefficient producedWV on a tail due to wing vortex interference

AlB(ST)P Incremental interference C on body due to jeteffects on strakes and taiLs - IB(ST)P I(ST)

Al B(T)p Incremental interference CN on b'ody due to jeteffects on tails .IB(ST)P - IB(T)

Y Spanwise distance'between wing root and locationof trailing vortex

AX CP~ Change in CP location of strake + tail interferenceCN duv to the jet - XCP - XCP

IP I

AXCP Change in CP location of strake-on-body inter-I(S)P ference in C N due to the Jet - X CP - XcCP

,(S)P I(T)

AXCp Change in CP location of strake + tail inter-IP(s) ference CN due to jet effects on XCpS

AX Cha-ige in CP location of the strake + tail irter-I (T) ference CN due to jet effects on XcpT

AXCi' Change in strake CP location due to Jet effects =S P XCP S XCpT

xxvi

LIST OF SYMBOLS (CONCL'D)

"AX Change in tail CP location due to jet effects -C x. xcPCPTP CPT

^X jDifference betveen tail chordwise centers of" CR Ia

SUBSCRIPTS (CONT'D)

B(W) Body in the presence of a wing

BWT Body plus win& plus tail

c Crossflow

D.P. Double panel

a Exposed

I 1n(T)

I General indicator

L.E. Leading Edge

N Nose

n Nonlinear

p Planform area

POT Potential

ref Reference

S Strake

SF Skin friction

7 S.P. Single panel

T ,Tail

T(B) Tail in presence of body

T.E. Trailing edge

V Vortex

W Wing, or wave drag

W(B) Wing in the presence of a body

a Denotes differentiation with respect to a

ABW ACNBW

v/2 a - 900

t a-180

xxviii

//

SUBSCRIPTS (CONCL'D)0z

0 C = 00.,

16 a=16'

20 20*

160 Gi-1600

xxix

SUOIARY

This repozt doecribes the construction and use of methods for pre-

dicting the pitch plane aerodynamic characteristics of a class of missile

configurations. The configurations include body alone, body-tail,

body-strake-tail and body-wing-tail configurations at high angles of

attack. An assessment is also provided of the effects of a rocket exhaust

plume on the pitch plane characteristics for a range of thrustqr conditions.

The methods, semi-empirical in nature, were developed through corre-

lation of test data obtained during several independent test programs.

These data, when taken together, form a rather extensive data bank in

which configuration geometries and flow conditions are systematically

varied. Except for the methods pertaining to winged missile configura-

tions, which are limited to 30 degrees angle of attack, all methods are

applicable to angles of attack between 0 and 180 degrees. In several

instances lack of test data imposed Hach number limitations; however, in

"the majority of cases the methods apply to K:tch numbers between 0.6 and

3.0.

Methods are provided to predict the characteristics of isolated

components and interference effects produced when various components are

combined. The methods pertain to bodies of circular cross-section. When

tails are added, they are mounted in cruciform (plus attitude) with the

tail trailing edges in line with the base of the body and undeflected.

Forward lifting surfaces (strakes or wings) can also be a~ded.

The methods enable the user to estimate the normal force and center

of pressure of a variety of configurations by calculating the character-

istics of individual missile components and their mutual interactions

xxx

produced when in combination.* Where possible, predictions have been

compared against data which were not used in the development of models.

In general, these comparisons have demonstrated good agreement.

xxxi

1.0 INTRODUCTICN

"A recurring problem in missile engineering is the lack of accurate methods

for predicting configuration aerodynamic charaeteristics, for all Mach numbers,

at high angles of attack. The situation is aggravated by the long term trend

toward increased missile maneuverability and angle of attack. Historically,

maximum angle requirements have increased steadily. The greatest increase

has occurred relatively recently to meet advanced air-launched system

- -~ maneuverability requirements. These now dictate angles of attack to 90 and

even 180 degrees.

The missiles which fly at these very high angles are usually of the slewing

.type, i.e., their angle of attack is generated by thrust vector control (TVC)

(for example, AIR SLEW and AGILE). Aerodynamically they tend to be somewhat

simpler than missiles which achieve high maneuverability through use of

aerodynamic surface deflection because of the large control forces available

from the deflected TVC nozzle. Non-TVC missiles usually can deploy wings and

canards as well as tails, and their maximum angles of attack are limited to

about 40 degrees. Air slew missiles usually deploy tails, but any forward

lifting surfaces are generally small (e.g., strakes). Basic aerodynamic

prediction methods are required for both types of vehicles.

The aerodynamic performance of TVC type vehicles is further compli-

cated by plume interference; therefore a method is required for calcu-

lating this effect in addition to methods for estimating the basic aero-

dynamics.

It has been well-established (References 1, 2, 3, and 4) that the best

means of constructing methods for estimating basic aerodynamic character-

istics at high angles of attack is through correlation of experimental

data generated by testing over systematically-varied ranges of the relevant

geometric and aerodynaalc parameters (Reference 1). This report describes

the generation of methods using that technique. The methods deal with

the aerodynamics of aerodynamically controlled missiles and TVC missiles

with and without plume effects. A summary of the d-ta u.sed in the develop-

ment of the methods is presented in Reference 5.

The objective of this work was to evaluate existing methods, to

improve upon these existing methods if possible, and, where necessary, to

develop new methods to predict the pitch-plane aerodynamic characteristics

for aerodynamically controlled and TVC missiles. The m%,thods addressed

were applicable to the configurations, angle of attack and Mach number

ranges indicated in Table I.

Table I

Scope of Methodology Requirements

"Control MechanismV Aerodynamically Control TVC Jet

CONFIGURATION a - 0 - 300 ai -O0 - 180" InterferenceM - 0.6- 3.0 M - 0.6 3.0 Effects Included

Body Alone / / /

Body-Wing-Tail(Canard)

Body-Tail / / I.

Body- Strake 1 /Body-Strake-Tail /

2

Prediction of the aerodynamic characteristics for the configurations indicated

in Table I requires methods for predicting the aerodynanmcs of individual

/ components and mutual interference effects. Figures Ia and lb show the

extent of existing capabilities prior to this contract with respect to total

methodology requirements.

Although it Is not shown in Figures la and b, a certain level of

capabilities existed in each of the areas indicated. In general, the accuracy

of these methods is poor at angles greater than a few degrees; therefore,

these methods were not indicated. Under the present work, methodology was

developed to fill In the gaps indicated in the overall requirements of

Figures la and b. The methods developed are of an engineering type and

include charts, graphs and formulations which facilitate ease of use by hand.

By and large the methods are empirical and therefore are limited to the

range of test conditions and geometric parameters tested. The specific

conditions tested are discussed in Section 2.0 and the Mach number range of

interest, namely 0.6 to 3.0 is adequately covered. However. as Is usually the

case, the flight combinations of Mach and Reynolds numbers were not achieved

in the wind tunnel test programs. Therefore the resulting methods do not

contain all the effects of Reynolds ni-mber variation that might be desired.

Until better matching of flight conditions is achieved in wind tunnel tests,

the user of such methods must exercise care and Judgement with regard to

this point.

/3

//

/z

Finally it is noted that methodology was developed to predict Induced

yaw forces and moments and induced rolling moments,* and was provided as

pert of, this program. Reference 39 describe* the development of the methods

-. and the computerized version of the methods.

The general layout of the report is as follows: First, a general

description of the equipment and models used in data generation is given in

Section 2.0. Then a limited amount of data analysis is presented in Section

3.0. folloing this, Section 4.0 describes the forimulation of the aerody-

namic prediction equations and the terms for which methods are constructed.

The methods themselves are described in Section 5.0. Where applicable each'

description includes background discussions, treatment of data, approach of

construction, use of methods, and where possible, checks of method accuracy

against data not used in the construction.

4

ExistingMethodology

Methodology

Requirements

Body 18

Strak-Take l Angle ofBody -90 sl o

_ Body , --0.

" Tall1 1.5 .-. 3.0

0.6 1.3 OMach)

(Mach)

Figure la. Methodology Requirements for TVC ?fisalles

/

Exis ting

Methodology

Methodology

Requirements

dy Angle ofWing (Canard) Attack

Tail ar)Wody (deg)

70. 1. 1.3

(Mach)

Figure lb. Methodology' Requirements for Aerodynamically Controlled Xisuil..

6

2.0 EWUERIENTAL DATA SOURCES AND MODELS

The majority of data available for correlation (see Figure 2) were

generated using either U.S. Air Force or Martin Marietta, Orlando Division,

supplied models. Reference li, which is based on 485 hours of testing in

tunnels 4T and A at AEDC, is the primary source of data. The TVC data are

taken from a 312 hour test program in tunnels 16T and 16S at AEDC. Typical

missile compotents were tested separately and in combination. A Martin

Marietta supplied reflection plane and fins were tested to provide isolated

fin data to 180 degrees angle of attack. Isolated body and non-rolled body

tail data were generated using both Air Force and Martin Marietta models.

The Martin Marietta main body model is shown in Figure 3a with the selection

of tails which can be mated to the body shown in Figure 3b. The Air Force

and Martin Marietta models are both 10 cilibers in length with tangent ogive

noses but the Air Force nose is 2,.5 calibers compared to 3.0 calibers for

the Martin Marietta nose. The Air Force and Martin Marietta model diameters

are 1.25 and 3.75 inches, respectively. Tails of identical planform

geometry, arranged in cruciform and undeflected, were tested on each body.

Tail taper ratios, asptct ratios and diameter to span ratios were varied

between 0 - 1.0, 0.5 - 2.0 and 0.3 - 0.5, respectively. Angles of

attack varied from 0 to 180 degrees. The maximum angle of attack attained

by the Martin MarieLta sting mounted model was limited to 60 degrees. Through

a combination oi stings and struts, the Air Force model was tested to 180

degrees. The Martin 4arietta model was equipped with four 3-component tail

balances compared to a single tail balance for the Air Force model. These

7

f/

balances measured t'ail normal force, hinge moment and rLot bending moment.

Six-component main balance data were' available from each model.

Body-wing-tail configurations were tested to 30 degrees angle of attack

at a non-rolled altitude using the Martin Marietta model. Data consisted

of 6-component main balance atid 3-component fin balance outputs. This

model can accomodate sets of half wings mounted in cruciform at several

different axial stations between the shoul'ar and after body section containing

the tail balances. The wings are not attached to recording balances. Wings

tested were of constant aspect ratio 2.0 and taper ratio 0.0 with diameter to

span ratio vatying between 0.35 and 0.5.

A more complete description of the sources of test data, test conditions

and model configurations is contained in the Data Report (Reference 5) submitted

as part of this study contract (CDRL Item No. AO05).

//

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oc

92

0

0a

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/~a 41 /14 / 441 1

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Figure 3a. Martin Marietta Main Body Model in the NSRDC 7'xlO'Transonic Tunnel at Sixty Degrees Angle of Attack

10

Figure 3b., Martin Marietta Tail Models

"3.0 AERODYNAMIC DATA TRENDS

Before proceeding to the various methods, a qualitative analysis of

some of the test data will be presented. The discussions are intended to

illuminate the basic phenomena underlying model aerodynamic behavior and

provide the user with more than simply a recipe for calculating the

v.rious force and moment quantities. Many of the basic ideas used were

presented in References 2 and 3. They will be sumarized here for the

sake of convenience. The discussions here will be limited to isolated

fins and bodies and body plus tail configurations.

"3.1 Fin Aerodynamics

Most of the disaussions in this section are based upon those of

Reference 2. No attempt will be made to reproduce all of the previous

material. The reader is referred to the original document for a detailed

treatment.

The discussions center on the effects of fin geometry (planform

taper and aspect ratios) and Mach number on the aerodynamic characteristics.

Fin flow patterns are discussed briefly along with the associated stall

r '" characteristics. The implications for fin normal force coefficient and

chordwise center of pressure location are outlined. Discussions begin

with a consideration of delta fins.

ii1'

i 12

st ANGL2 OF A17ACK-DO.

LOCUS OF URATTATCUIINT

Figure 4. Vortices Produced by the Reattachment ofLower Surface Boundary Layer

At high angles of attack the flow aroui:d delta fins is char-

acterized by the presence of large upperesurface vortices fed with vor-

ticity from the boundary layers which separate at the leading edges (See

Figure 4). Stall on such wings is brought about by vortex "bursting".

This is accompanied by a breakdown of the well-ordered vorttx flow and a

sudden pressure increase at and downstream of the "burst" point. Upstream

the pressure in the vortex remains low and produces a suction which in-

creases the normal force. As angle of attack is increased the "burst"

point moves upstream towards the trailing edge. When it crosses the edge,

stall begins and is characterized by a loss of normal force and a forward,

movement of the center of pressure. As aspect ratio increases, the stalling

angle of attack decreases. These effects ore shown in Figures Sa and 5b

at transonic speeds. The figures also show the following:

13

.1) The normal force curve slopes, C N, at a - 00 and 1800

are numerically equal - this result is predicted by

Slender Body Theory.

it) At a - 900, the centers of pressure and of area very

nearly coincide. This is inituitively obvious.

iii) At a - 1800 the centers of presiture of these delta fins

lie right at the "leading" edge. This bears out the

Slender Body Theory result that all of the loading on a

fin occurs over the region where the fin span is changing

(increasing). The predicted effect of retreating side

edges (i.e., -to push the center of pressure upstream) is

not evident. A similar result is found for non-delta

fins also.

Still confining the discussions to delta fins, Figures 5c and d

show their behavior at supersonic speeds. It will be seen that no stal-

ling is visible at thisMach number. During the reflection plane tests

from which these data were obtained, it was found that near a - 900 at

supersonic Mach numbers, the fins behaved like forward facing steps, re-

sulting in low values of C.. Accordingly, the CN value at a - 900 was

obtained from Reference 6 and the data faired through that point as shown.

Also worthy of note is the center of pressure behavior, particularly near

a - 1800.

When the fin planform is not triangular, the upper surface vortices

referred to earlier are modified or joined by yet other rotating flows.

14

(

For rectangular fins, the large suction-producing vortices now spring

from the side edges, while a laminar separation bubble can exist at

the leading edge. When stall occurs on such a fin, it is frequently a

result of laminar bubble lengthening, spreading low-velocity, high

pressure flow over the upper surface. The result is a loss of normal

force and a rearward shift of center of pressure. A clipped delta fin

'displays behavior somewhere between that of a delta and a rectangular

fin. This behavior is shown in Figures 6a and b at transonic speeds.,1 00

Note the centers of pressure for the rectangle at a - 0 and 180". They/lie right at the "leading" edge as predicted by Slender Body Theory. At

a - 1800, all three fins show this predicted behavior. As before, the

supersonic data show no visible stalling and have been faired through

CN /2 from Reference 6, Figures 6c and d.

The effect of increasing Mach number on a delta fin is to move the

vortex "burst" point downstream. Thus a fin which is stalled at one

Mach number may be unstalled by simply increasiug Mach. This behavior .s

* shown in Figures 7a and b for an AR - 2.0 delta fin. The stalling

behavior at M - 0.8 is entirely removed at M - 1.3 and higher.

3.2 Body Aerodynamics

As in the case of fins, the aerody-.amic characteristics of

bodies at high angles of attack are largely influenced by viscous, stpa-

rated flows. The discussions below deal with these, especially in the

case where the body wake takes the form of an asymmetric vortex pattern.

This phenomenon has recently become of considerable interest for high

15

incidence missiles (Reference 7).

When a slender missile body is placed at angle of attack in a uni-

form flow,' the boundary layer generally separates on either side of the

body and forms a lee-side wake. Separation usually begins near the rear

when the missile reaches about 6 degrees angle of attack. The wake takes

the form of a pair of symmetrically-disposed, counter-rotating vortices

fed by vorticity shed from the separating Loundary layer. As angle of

attack increases, the axial extents, sizes and strengths of vortices

increase also.

When the body angle of attack reaches about 25 degrees, the symmet-

rical nature of the wake disappears. The two vortices are joined by a

third, beginning again at the body rear, and the wake becomes asymmetric. As

angle of attack is increased further, more vortices Join the flow until the

: - wake contains several which have been shed from the body. A section

taken through the wake shows it'to resemble the von Karman vortex street,

well known in the literature on two-dimensional flows.

The asymmetric nature of the wake produces an asymmetric distribution

"of pressure forces along the body. This results in out-of-plane forces

and moments being induced, whether the body has lifting surfaces deployed or

not. These forces and moments can be significantly large, requiring special

means to be found to counteract or remove their effects (Reference 8).

Figure 8a shows the force and moment coefficients induced #. a body at

Mt's 0.6. The effect of increasing Mach number to supersonic values is

u~tidly to reduce those effects to negligible prcportlions. This may

be seen in Figure 8b for M - 2.0. Later discussions will illustrate the

16

" " " 1'/ "./ i ",

"additional effects of adding lifting surfaces to such a body. The steady,

asymmetric wake persists up to angles of about 50 to 60 degrees. At higher

angles the wake becomes unsteady and vortices are shed asymmetrically.

3.3 Body Tail Configuration Aerodynamice

The addition of tails to a body generally increases the out-of-

plane forces and moments induced by &symmetric vortex effects as well as

produring rolling moments. Several examples will be given of these

important effects. Figures 9a and b show out-of-plane quantities at M - 0.6/

for two typical sets of cruciform tails fixed to the 10:1 caliber body

("plus"' attitude). It is of interest to note the correspondence between the

peaks of force and moment. The angle of attack has generally been limited

to 90 degrees because:/

i) By 90 degrees the wake flow is unsteady and the out-of-plane

quantities fluctuate rapidly.

ii) Above 90 degrees, the presence of the strut support might cause

alterations in the wake pattern and its effects.

By the time Mach number has reached 2.0, no induced effects are visible

(not shown here).

Another illustration of the asymmetric wake effect is contained in

Figures lOs and b. Previous testing on a MO( model with four instrumented

tails yielded the forces and moments on the individual tails. Complete

configuration rolling .wment was obtained from separate (main balance)

instrumentation. Figure 10a shows the tail forces for a "cross" configuration

( 45") at angles of attack to 60 degrees. If the moments of these tail

17

-/K

"forces about the missile axis are summed and the result -%spared with the main

"balance reading, the comparison of Figure lOb is obtained., Clearly, the induced

roll Is generated by the unequal tail forces, which thenselvee are induced

by the asymetric wake.

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Figure l0a. Comparison Of Tall Normal Forces

33

-/o

0 Main Balance j. I0 t (Fin Force x Dist To Model Centerline)

S-0.86 -4 5 -,Degrees____

2

10 230 40 50 60

U

10 ._ 31 0 4-s-6

-2

-6

-114

ANGLE 01 ATTACK-DEG.

Figure lob. Comparison Of Rolling Moments

34

4.0 FOIMULATION OF THE AEROODNTAMIC PEDICTION FUATIONS

Because of the nature of the information available, the following

formulations of body-tail. body-etrake-tail and body-wing-tail configuratlon

"pitch-plano serodynamtc characteristics are necessary. These formulations

will vary dependirg on whether the configurations are to be aerodynamically

or thrust vector controlled (TVC).

Aerodygmically Controlled

"ody-Tail

C Br CN B + 2 CRT (B),ST_ + B(T) (1)SB

NT B1 1 d d d

Body-Strake-Tail

"CH CH + 'C" + &C.s (3)BST B as BST

,Cs xC BX+ + AC~ BST (4)"1RST BS S ST -

II&

N ST + B(T) + n(5)

C man +NWB+ 2P ANTS C B 2()j C NTWRr()L X' BX p *Xcp~iw +T+. ...__! m -CF +T(B)5 Tj +iI+

d d SB d

B3(T) M.x ( + ACWv xCP TV (6)

d d

35

Thrust Vector Controlled

Body-Tail,

CNB -aCNB + ACNBP + 2 CNT RT(B)ST + ACNTp + 'B(T) (7)

""B

"ITz~ B d, cB + "D x + 2 Nr: T(B)T s T c B~) +S (8)

"ACNTp XCPTp + 'B(T) XCPI(T)/Id d.

Body-Strake-Tail

CN C S + AC1 + ACN + ACN &CNBST B IPS + S 1ST l (Bo)

xx1C S PBT+ BHP+ NCT (0

d d

Hence, the following quantities are required in order to conduct

"aerodynamic analyses on body-tall, body-wing-tail, or bedy-strake-tail

configurations which are either aerodynamically or thrust vector controlled.-I

* The section of this report In which each quantt.cy is developed is listed

as follows.

36

. . . *.-

/

Quantity Section Page

CNB 5.1.1 39

xCp 5.1.2 61

T 5. 1. 4 91

XCPT 5.1.5. 122

Use in either"RT(B) 5.2.1 143Aerodynamically Controlled

I B(T) 5.2.2 161 or TVC modes

Xcp 5.2.3 171

5.3.1 190

XcpBS 5.3.2 202.

'N BST 5.3.3 220

xCPABST 5.3.4 232

ACNBw 5.4.1 259

XC~P Bw 5.4.2 274

".5.4.3 289

XCPTWV 5.4.A 306

37

Quantity Section Pais

ACN 5. . 1 310

BP

xcpP 5.5.2 323

TVC Mode OnlyAC,33

XcP'P 5.5.4 351

As indicated above, certain of the quantities are applicable to the

equations for aerodynamic control ar well as the equations for TVC. Others

are used only in the TVC model. Limits of applicability for each method are

indicated in the appropriate sections.

38

/'

5.0 AERODYNANIC METHODS

5.1 Isolated Components

5.1.1 Body Normal Force

Summary

A method is presented for predicting body normal force coefficients,

CB , for angles of attack between 0 and 180 degrees and Mach numbcrsB

from 0.6 up to 3.0. Comparisons between predicted results and experi-

mental datashow good agreement. This'method represents an improvement

over existing methods in that it accurately predicts CN both transonicallyB

and supersonically.

Background

The aerodynamic force directed normal to a body in its pitch plane

can be separated into potential and viscous flow contributions. Using

slender body theory, Munk found the potential flow contribution to be

equal to sin 2o, where 2 is the slope of the normal force coefficient

curve at a w 0 degrees. In later work by Ward (Reference 9), it wan shown

that this force is actually directed midway between the normal to the

stream and the normal to the body axis. Taking this into account, poten-

tial contributions to body normal force can be expressed as:

C oT sin 2a cosa (11)

At very low angles of attack, this potential term dominates body rormal

force. However, for angles of attack greater than 6 degrees, viscous

effects are introduced and rapidly become the dominating factor. Existing

theories do not adequately predict viscous effects. Empirical procedures

39

have been developed based on the early work by Allen and Perkins10 and

Kelly U which introduced the concept that the viscous crossflow around

inclined bodies of revolution is analogous to the flow around a circular

cylinder normal to the flow. In accordance with standard'notation, these

empirical procedures relate the viscous normal force contribution to Cdc

the crossflow drag coefficient defined by analogy with two-dimensional

flow. Thus

S 2C * n d -P sin2 a (12)

C d c Sref

Experimental data have shown Cd to be a function of both Reynolds andC

crossflow Mach numbers. Values of i have been determined empirically

from two-dimensional and finite length cylinder data.

Combining the theoretical potential and empirical viscous contri-

bution results in the following expression for body total normal force

coefficient:

Sp 2CM - sin 2a cos + Cd n--sin a (13)

12

This iS the same expression used by Jorgensen to predict transonic and

supersonic values of CN for angles of attack between 0 and 180 degrees.

The procedure outlined by Jorgensen in Reference 12 was found to be

inaccurate at transonic Mach numbers when predicted results were compared

r^-. the data of Reference 13. These comparisons are presented in

figures 11 through 14. Accuracy is only fair when all Mach numbers and

angles of attack are considered, but does improve with increasing Mach

number.

Two avenues are available to improve accuracy. First, develop a new

method to improve transonic capabilities. The second, and perhaps most

40

desirable approach, would be to develop a single procedure which would be

accurate both transonically and supersonically.

Method Development

A power series approach is used to develop a method which predicts

the combination of potential and viscous effects on body total CN'

Boundary conditions were sought which would adequately define the character-

istics of CN between angles of attack of 0 and 180 degrees. Values ofaCN

C and - at a - 0, w/2 and w were taken as boundary conditions.

Experimental data indicated that values of CN at a - 0 and a - r are

zero. Also from experimental data, it was observed that -- - 0 at a - w/28CN

and W. The remaining boundary conditions, i.e., CN at a - '/2 and - at a - 0.

were retained as free variables.

Applying these boundary conditions to the expression:

'2 3 4 5CNB oa +aa+ a2a + a3a + a4a + asa

yielded

2 3 4 a

w itNw/2

which can be rewritten as

C - +A2CNCN A1 CN + 2 N Sref Sba (14)B a W/;2an

where

6a 2 I3a3 12a4 4ct5A1 , + 13a w +

16a 22 32a3 16a4A2 .'2 W2 X 3 4

41

...

Values of A and A2 are plotted'as functions of angle of attack in

Figures 15 and 16. Values of C and CN still require definition.x/2

Transonic values of CN presented in Figure 17 as a function of

Mach number, nose length and afterbody length were taken from References

14 and 15. Supersonic values of CN presented in Figure 18 were taken0

from Reference 16 as a function of Mach number, nose length and afterbody

length. The data, of Figures 17 and 18 represent improvements over

existing correlations. Linear literpolation is required for values'of

C between Mach 1.2 and ].5.N

Values of CNw/2 can be calculated with Equation 13 recognizing

that the "potential" teim goes to zero and utilizing the published data

for values of n (Reference 17) and Cdc(Reference 12). The available

valueb of n (shown as no 0 n Figure 19) are derived from subsonic test

data and are typically assumed to apply up to croseflow Mach number (M c)

equal 1.0. Above Mach one n is normally assumed to be 1.0. Rather than

continue to use such a discontinuous rapresentationa procedure is

employed here which produces an estimate of the variation of n with M

through the transonic regime. The transonicvariation of n is developed

as follows:

The potential component of normal force Is still defined as in

Equation 11 with the change that CN replaces the 2. The intent is toN

make use of the test data (Reference 13) as a source for CN rather than

rely on the theoretical value of 2. Then the viscous contribution to

the normal force is defined as as follows:"CvIS

42

CN -CN si (CN ax) cos (a/2)

and CN are both obtained from the test data. Then. . D C S I S 2

- P ' s i a. "CDC _. si2a

ref

The quantity n CDC was calculated utilizing this expression at crossflow

Mach numbers ranging from 0.2 to 2.0. Values of Cd. were taken from

Reference 10 at the corresponding Mach numbers to permit solving for n.

The curve faired through the values of n which result from this exercise

is shown in Figure 20. The subsonic value is seen to apply up to about

- C.8 with the upward ,trend continuing to about M. a 1.4. A

polynomial expression was then derived as follows to represent the

variation of n with Me,

- 0.0 at M -0.8 atid 1.4c

n - n at Mc -0.8

n - 1.0 at M - 1.4c

Applying these boundary conditions to the following expansion:

a3Hcn a+ aM + a," a -~Melc -- 3

yielded

n no (-9.0741 + 31.1111 Hc -30.5556 me2 + 9.2593 a 3)

+ (10.0741 - 31.1111 Mc + 30.5556 M 2 - 9.2593 Me 3

which can be rewritten as:

SBo n + B I(16)

43

wh're

"B - -9.0741 + 31.1111 HM - 30.5556 1 2 + 9.2593 M j3

1 -10.0741 - 31.1111 H + 30.5556 H 2 92593 M

1 C CC

Equation 16 is applicable to croseflow Mach numbers between 0.8 and 1.4

Values of n0 are contained in Figure 19. Values of 1% and B1 are pre-

sented in Figure 21.

Values of Cd from Reference 13 modified on the basis of the

results of Reference 3 are presented in Figure 22. These data cover a

wide range of croseflow Mach numbers and come from a number of different

sources.

Using the above information and Equation 12, it is now possible to

calculate the value of C required for the calculation of Cbetween

N /2bewn

a - 0 and 180 degrees.

Methr._d Evaluation

Check cases were made using the same configuration and conditions

represented in Figures 11 through 14. Figures 23 through 26 show com-

parisons between these predictions, experimental data, and predictions

using Jorgensen's procedure (Reference 12). These comparisons indicate

improved accuracy at high angles of attack in the transonic Mach regime

and equally good accuracy at all angleo of attack in the supersonic

regime.

Use of Method

The method for predicting isolated body normal force in applied in

the following way.

44

i ~

1 Depending upon the Mach number, use either Figure 17 or 18 to

determine C as a function of nose and afterbody length.

2 Calculate the value of C /2 using Equation 12.

a Use Figure 22 to determine C

bDepending upon the Mach number, determine the value of n.

. For M. < 0.8, use Figure 19 to determine n as a function

of LId.

* For 0.8 < H 1.4, use Equation 16 and iigure 19.

* For HM 1.4, n - 1.0.

3 Using Equation 14, the results of steps 1 and 2,and Figures 15

and 16, calculate the values of C from 0 to 180 degrees.N B

Numerical Example

Calculate CNB between 0 and 180 degrees at H M 2.86 for a body with

the following characteristics:

"-i - 3.0 (tangent ogive)

L - 6.0d

S;Re 10.2ref

1 Usins Fiure 18b, % 3.05/rad

2 Use the followinp equation to calculate Cw/2

SN C n zNw/2 d c Sref

45

* - I. ~ ~ ' .. ....~* P M

a From Figure 22, Cd - 1.34S~C

b For M - 2.86, n = 1.0

c Therefore CNI2 - 13.67

3 Using the following equation and Figures 15 and 16, calculate

CKB iACNO + A2 CNw2 SrefinS.ase

"A, A22;

0 1.0 0.0 0.05 0.074 0.01 0.36

10 0.123 0.045 0.98915 0.153 0.095 1.7620 0.167 0.155 2.6330 0.162 0.305 4.6640 0.13 0.475 6.8950 0.09 0.645 9.0960 0.051 0.79 10.9570 0.023 0.905 12.4480 0.005 0.975 13,3485 0.001 0.99 13.5490 0.0 1.0 13.6795 0.001 0.99 13.54

100 0.004 0.975 13.34110 0.015 0.905 12.42120 0.026 0.79 10.88130 0.034 0.645 8.92140 0.037 0.475 6.61150 0.033 0.305 4.27160 0.022 0.155 2.19165 0.014 0.095 1.34170 0.007 0.045 0.636175 0.002 0.01 0.143180 0.0 0.0 0.0

Data Comparisons

The results of the numerical example are compared with experimental

data,(Reference 18) in Figure 27. Because these data were not involved In the

development of the method, this comparison represents an independent

check of the method. Agreement is quite good throughout the angle

46

of attack range transonically. Figure 28 represents further Independentchecks of predicted results against experimental data from Reference 19.Comparisons between predicted results and experimental data have shownthe method of this section to be more accurate than the Jorgensen methodIn the majority of cases. However, the Jorgensen method has proven moreaccurate in the 0 to 40 degree angle of attack range transonically.Therefore, it is recommended that the Jorgensen method be used in thisregion and the method of this sectlon in all others.

47

dI Z ii4

RN4.134 x 101

12

Cs ____ (M. 12)

4

/ 12

0 i0 40 60 so 100 120 140 160 ISOMI3I Of ATTACK, DIM233

Figure 11. Comparison of Experimental and Predicted Results (CNB) K Mach - 0.6

20

010 1 -- 3.010

0 -0

44

g2

021 -0 20 40 r0 s 0 2 4 6 S

/"O Of ATAC -, w"asts

/T

0 A

200S S O R 4.149 ' 05

, A

" II_d

/ 0 lf 0' _II A

12

Cl.

10T

JORGKENSEN (IV?. 12)

0

o 20 40 40 80 100 120 140 160 ISO

AWGLS OF ATTACK, DIgIcZS

Figure 13. Comparison of Experimental and Predicted Results (CN) ,Mach- 1.30B

20

o ........ 0 40. . 80 4 N 137700 80S

14 1

' .... : .....' .. .. ......... .,.o' -

d "0

* ( 1

0/

0 20 40o 60 80 100 120 140 10 ISOANlCLEI OF ATT'iACK. DE:IUiIS

Figsure 14. Comparison of Experimental and Predicted Results (CNB Mach -2.0

49

0.1

0.06

0.02

00 20 40, 60 so 200 120 140 .160 te

AIIOLE OP AtTTA". OuagSFigure 15. Coefficients for Calculation of c%

.0.

0.7

0.4

0.3

0.2

00 20 40 4 0 so to 120 240 160 Ito

AMLSL OP ATACXt, DINRtx3

Figure 16. Coefficients for Calculation of C

waS d

10

9

7

0.8 09 1. 1.2 (TANIGENT OGIVE MOSES)

1 .4

Figure 17a. Curves for Transonic I. N/d -15

51N

............

10

9 _ _5

7

6

0.6 0.9 1.0 1.1 1.2 N

Figure 17b. Curves for Transonic CN. (LN/d 2.5)a

10

9 1

8

1.1 6 (TANGENT OGrVZ NOSES)

Figure 17c. Curves for Transonic C N NId -3.5)

52

).3 _ _ _

3.21

3.0 _ 3.0

).10

102.

-2.4

Figure 18a. Curves for Supersonic CN (I /d' 2.5)

3.' -- - - -- - - - - -.-. 3.0_

I .~~~~~~~3.1 0 -- - - - - - - . . -- - -2.9 ----- -10------- - - --

2.62.5-

2.8

2.6 --- --- O

Figure 18b. Curves for Supersonic C N (I N/d -3.0)

al

53

3.3

3.2

3.1 L"*%

. 43.03.C

2.9

2.82.7

Figure.18c. Curves for Supersonic CN (9../d - 3.5)2. -A- e

2.3

Figurel8d. Curves for Supersonic C N (IN/d 3 .5)

-7-d.4.0

3.2

3.1

3.0 iI ./

C% 2.9

%a

54.

Figure 13d. Curves for Supersonic C N (t N/d -4.0)

a

54

0.8

0. 6

0 10 20 30 40L/d

Figure 19. Correlation Factor for End Effects

1.0

00

o11

0 0.I Iv.. .I I I."0 0.4 0.6 1.2 1.6 2.0 2.4

caossvtov MACH mMMU, K

Figure 20. Variation of n With Mach Number

55

'I

0.4 -

S,,+,! ~~. . . . . . . . . . . .. .. ......... .. .... [. ..

0.4 -.

0.3 - *-

0.2

o i

0.9

0.S . . . .. . ... ... .. .. ... -

0.3 6

0.4 . - -- ,

0.1 . . . . . - . ... . . .. . . . . . . ..+.-

+0 o - _ a . + --..

"0.8 0.9 1.0 1.1 1.2 1.3 1SCOSSPILV RACK IJMilMf, MC

Figure 21. Curves for Determining Transonic Values of tj

56

2 01.2.3.

Fiue2aCaicVledfCC'

1 .2 - 0 ---

01.0 2. 3.0--- --

1.4

10410 10 6 10 1

CROSSFLOIE REYOLDS MUN3R

Figure 22b. Crosaf low Drag Coefficient

(Subcritical Crosaf low, M' 0.4)

57

20

I M.~~0.6 ___

IN 4.134 x10

14 ~~.-----..-- 3.0 - 01 _ _

0 2 0 40 60 S0 100 120 140 160 IgoANGLE OF ATTMK. MIW219S

Figure 23. Comparison of Experimental and Predicted Results (C%) *Mach -0.6

14 4

1110 PI4 P 0 80 10 10 14 6 8

AEL 0? AflAIN (. 12)IE

Figure~~~~~~~~~O 24 oprio31Epeietl n rdctdRslt93),Mah-11

4 P58

M1 - 1.3W--

0 0

a JI

------r- - - - - -- *--I --IS

0 so 60 aA.

ANGL9 OF ATTACK. DWftUS

Figure 25. Comparison of Experimental and Predicted Results (C ) sMch 1.30B

20 -- ---

12 ------ ~ .I -

AMCLI OF ATTACK, DO3REA

Figure 26. Comparison of Experimental and Predicted Results (C,,) Mach 2.0

'9

20 1 ]M -2.86- 3. 0 1--

6 .0

d0 d

0

0 Q )(PERIENTAL (REF. 18) ____

-- PREDCTE

0 2 0 60 so 100 120 140 10 1;0

AiCLE Of ATTACK, DEGREES

Figure 27. Comparison of Experimental and Predicted Results CN), Mach -2.86B

~A~A d-0 '_ A

0, 0B 10 'O 14 16.0 -18

20--

512Body Center of Pressure

Summary

A method is presented for predicting isolated body center of pressure,

X p for angles of attack between 0 and 180 degrees and Mach numbersB

from 0.6 up to 3.0. Comparisons between predicted results and experimental

data show good agreement.

Background

New highly maneuverable missiles will encounter extreme angles of

attack. In some cases angles of attack may approach 180 degrees in

either the transonic or supersonic Mach regimes.

Effective evaluation of proposed configurations will require methods

for predicting aerodynamic characteristics at extreme angles of attack

over a wide range of Mach numbers. Current predictive techniques are

limited to angles of attack less than 30 degrees. New methods are required

to fill the void between existing and, required capabilities. This section

deals specifically with a method for predicting body center of pressure,

X CP .* The method presented .is applicable to Mach numbers between 0.6

and 3.0 and angles of attack between 0 and 180 degrees.

Method Development

The method for predicting X C was developed using an empirical

approach. The initial step involved a survey of available data (References

13, 18, and 19). The data displayed characteristics which were unique

to specific Mach number and angle of attack ranges. For Mach numbers of

1.0 or greater, X CP displayed a rapid rearward movement between angles

of attack of 0 and 20 degrees, followed by a nearly linear progression

of X between 20 and 160 degrees and passes through the centroid of the

planform area at 90 degree. Finally, between 160 and-180 degrees', another

61

-/: . .' .. . . . . . .

.- /---

rapid rearward movement of XCP was observed. Experimental data showed

that the XCP left the body between 170 and 180 degrees. As the body

approaches 180 degrees, a couple is produced as the positive potential

normal f~ree on the eorward facing portion of the body becomes equal to

the negative potential force on the trailing nose portion of the body. This

couple subjects the body to a moment and to a zero net normal force. Under

these circumstances, calculated values of X~, tend to become infinitely

large.

For Mach numbers less than 1.0, XCp displayed the same characteristics

between 0 and 20 degrees and 160 and 180 degrees. However, the location

of X tended to remain essentlally constant between 20 and 50 degrees,CP

followed by a rearward movement which is linear between 50 and 160 degrees

and passes through the centroid of the planform area at 90 degrees.

A power series approach was used to develop the method between 0

and 20 degrees. In the usual way boundary conditions were sought. The

center of pressure at a - 0 degrees was taken as the first boundary

condition. Curves presenting X!g as a function of L., tAI atsd h in0d o d d

the transonic Mach regime are presented in Reference 3. For the sake

of completeness these are presented again here in Figure 29. Similar

data in the supersonic Mach regime (1.5 k M < 4.5) were found in

Reference 16 and ore presented in Figure 30. For a second boundary

condition it can be shown that for symmetrical bodies 3XCP/d 0I a0.0.

A third boundary condition was defined by the center of pressure at 20

degrees. This was defined ar the center of pressure at zero degrees

Splus an increment. Using data frce References 3. 13, aed 20, the

percentage of body length by which X U. shifted between 0 and 20 degrees was

determined as a function of Mach number (see Figure 31). As a final boundary

condition, 3Xp/ at 20 degrees was assumed to equal the slope of the

linear variation between 20 and 90 degrees angle of attack. Experimental

data indicated that the renter of pressure at 90 degrees could he approximated

as the centroid of Lhe planform arca. At 90 degrees, when the flow is

separated along the entire length of the body, the normnal force will be

due' entirely to crossflow-drag (Reference 3). Assuming a constant 'dc alongC

the body, the centers Of pressure and of planform area should then coincide.

Collecting boundary conditions and applying them to the following

polynomial expansion

Xp2 3'd -- xao +810 + 2 + a3

d0 1 2 3

yielded F_2 3 2 223"" ~~7 al 21ai 2 3c, -8

"0 + 0 2800 Xo + 2800 28000 X20

2800'0 - 2. -0 L] X./2+ 80C0 2800] /

which can be rewritten as

X XA 0 ao + A1 X2 0 + A2 Xv/ 2 (17)

Wheree e A 1 + 7 a3 21a2

o 28,000 2800

2 3A 23a -8a

A1 2800 28000

3S- 2& ,. in radians

:. 8000l 2800

Values of A0 , A and A2 are plotted as a function of angle of attack in1 2.

" Figure 32.

Equation 17 was developed based on the characteristics of XCp at Hach

numbers of 1.0 or greater. Applying Equation 17 for Hach numbers less than

1.0 will produce good results even though 3XCP at a=20 degrees will be3a

in error.

As indicated earlier, the variation in X between 20 and 160 degrees

*:. is dependent upon Mach number. For Mach num'ers less than 1.0, the

location of X remains constant between 20 and 50 degrees and then moves

linearly toward the rear to the value of X at 160 degrees, passing throughCF

the centroid of the planform area at 90 degrees. For Mach numbers of 1.0

or greater, Xlp varies linearly between the locations at 20 and 160

degrees, passing thvough the centroid of the planform area at 90 degrees.

Using this information, the following equations were derived for determining

the slope of the linear variation and the value of X at 160 degrees.

aLX - Xw/ 2a- I = , g (18)a'- 90

70 + + (19)X160 a= Xtw/2

where ac, the angle marking the bound of the low angle region, Is 20 degrees

for Mach numbers of 1.0 or greater and 0 degrees for Mach numbers I.as than

1.0.

A pover series approach was used to develop the method between 160 and

180 degrees and in the usual wey boundary conditions were sought. The center

of pressure at 160 degrees wa tak as the first bovdary it ion

Ni

This can be calculated using Equation 19. A second boundary, j~(written x O)

at 160 degrees was assumed to equal the slore of the linear variation between

c'and 160 degrees. This value can be calculated using Equation 18. Also,

as a third boundary condition it can be shown t Ihat at 180 degrees 1C- 0.

Doi'As a final boundary condition, the center of pressure at 180 degrees was

assumed equal to the body length, rather than trying to define it as some

) Ipoint off the body as indicated earli.er. This assumption will1 introduce

no significant errors since the resulting forces and moments are sQall.

Collect ing these boundary conditions and applying them to the

following polynomial expansion

~CP X ao +al + a2 + a3 0d -

yielding

[51840000 +900000 ai -5200 a + 10 a X1j

+ L ~4000 /

+ -86000+ 86400 ai - 51 a 2 + a4000 160

which can be rewritten as

X so X0 +li1t1d +B2 X1 6 0 (20)160where + - 20c+0

23-51 0 O-64000O 000a+-52.00ci.

Talme. ~ ~ ~ "600, 8640 a -, 51d a. ar +hvia ucino ane ofat3ki240

values of 2*rd1,ar hf safucino nleo takI

Use of Method

The method for predicting isolated body center of pressure is applied

as follows:

jDepending upon the Mach regime, use either Figures 29 or 30 to

determine xas a function of I /d and Lit d. Linearly interpolate

0, N

for vola~es of Xo between Mach 1.2 and 1.5.

SUsing Figure 3 determine the rearward shift in center of pressure

between 0 and 20 degrees for the appropriate LId and M. Add this

value to the result of Step 1 toadeterziine X2 0.

3 Calculate the distance from the nose to the centroid of the planform

area using

S SPN + SPASpN1 + S PA

and where SPN and SPA are the planform areas of the nose and cylindrical

sections respectively in the case of a tangent-ogive cylinder body

PN V/- 2in + R + R sin -2(R-r) IN

SN P 2 (R 2 -IN 2 ) 7R +-1N2RLN + I R 2 sin-(R-r)N

R d t2

d

Ps A* d

and

XA S A + z A) (1 * d) Note that v/2 4

66

SUsing Equation (17), the results of steps 1, 2, and 3, and Figure 32,

calculate the centers of pressure between 0 and 20 degrees.

5 Calculate the slope of x at 160 degrees using Equation (18).

6 Calculate the value of X at 160 degrees using Equation (19)

I Using Equation (20),the results of Steps 5 and 6, and Figure 53,

calculate the centers of pressure between 160 and 180 degrees.

8 Depending upon the Hach number rL.n-e of interest, determine

the variation of X between 'O and L6,J degrees.

a. For M > 1.0, extend a straight line from X2 0 to X160'

b. For M ( 1.0, maintain a constant value of X from 20 to

50 degrees and then extend a straight line between

the values of x at 50 degrees and 160 degrees.

Numerical Example

Calculpte X between 0 and 180 degrees at H - 2.86 for a body with the

following characteristics:

tN/d. 3.0 tangent - ogive

A /d - 6.0

t/d - 9.0

d - 1.5 inches

1 Interpolating between the values ofFigure 30b and 30c, Xo was

calculated to be 1.93 calibers aft of the nose.

2 Using Figure 31, AX/t/d - 0.285 at H - 2.86. Therefore, for L/d - 9#

AX = 2.565.

X20 a X0 + AX

X20 a 4.495

67

3 For the configuration of Interest

xv/2 4.96

4 Use the following equation ard Figure 32to calculate the centers

"If pressure between 0 and 20 degrees.

x A O Xo + A, X20 + A2 X7r/2

a Ao Al A2 x

0 1.0 0 0.0 1.93

5 0.85 0.17 -0.0125 2.343

10 0.5 0.53 -0.036 3.169

:5 0.15 0.88 -0.04 4.047

10 0.0 1.0 0.0 4.495

5 Using the following equation, calculate the slope of the linear

variation between Iq and 160 degrees.

I ~Xa, .Xw/ 2

160 a'- 90

6 0.0066 O/des

6 Using the following equation, calculate the value of X at 160 degrees.

X 16 0 ' 7 0 x a / + x I /L '- 90 J

X160 0 5.425

Using the following equation and Figure 3),calculate the centers of

pressure between 160 and 180 degrees.

X oX 60` + B1 t/d + B2 X,,0

68

Bo BB2X

160 0.0 0 1.0 5.425

165 2.81 0.154 0.846 5.994

.170 2.5 0.5 0.5 7.229

175 0.91 0.846 0.194 8.455

180 0.0 1.0 0.0 9.0

8 Graphically determine values of X between 20 and 160 degrees by

connecting X0and X160 with a straight