AD-'776-586

HANDBOOK OF BLUNT-BODY AERODYNAMICS VOL. I

.J. A. Darling

Naval ordinance LaboratoryI Silver Spring, MDI

DEC 73

*W

REPORT DOCUMENTATION PAGE READ INSTRUCTIONSBEFORE COMPLETING FORMI. REPORT NUBER 12. GOVT ACCESSION NO. 3. RECIPIENT'S CATALOG mUERNOLT4 773-24. TITLE (and Subtitle) S. TYPE OF REPORT & PEROD COVER .D

HANDBOOK OF BLUNT-BODY AERODYNAMICSVOLUME 1 STATIC STABILITY

6. PERFORMING ORG. REPORT NUMBER

NOLTR 73-2257. AUTHOR(&) S. CONTRACT ON GRANT NUMBER(@)

John A. Darling

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT, TASKNaval Ordnance Laboratory AREA & WORK UNIT NUMBERSWhite Oak, Silver Spring, Maryland 20910 Task No. NOL-533/AF

I. CONTROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE

1 December 1973Air Force Armament Test Laboratory 13. NUMBER OFPAGESE-lin Air Force Base, Florida 3 2542 151

14. MONITORING AGENCY NAME & ADDRESS(I1 dilferent from Controlling Ofice) 15. SECURITY CLASS. (of this report)

UNCLASSIFIED

IS.. DECLASSI FICATION/DOWNGRADINGSCHEDULE

IS. DISTRIBUTIOH STATEMENT (of this Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the abstract entered In Block 20, If different hoem Report)

1. SUPPLEMENTARY NOi7S RPRODUCED SY

NATIONAL TECHNICALINFORMATION SERVICE

U.5. DEPARTMENT OF COMMERCESPRINGFIELD. VA. 22161

IS. KEY WORDS (Continue on reoveze side If neceseary and Identity by block number)

Blunt Body AerodynamicsStatic Stability Mach NumberNormal ForceCenter of Pressure

20. ABSTRACT (Continue on reverse side It neceoeaar amd Identify by block nrvbr)This report describes methods for quickly estimating the normalforce and center of pressure of blunt free-fall shapes ofk/d = 0.5 to 10 over a Mach number range of 0.4 to 2,5 and up toa= 90 degrees. Charts and equations presented in this reportare from cited reference material and original sources. Methodsbased on slender-body theory were tested on blunt shapes andmodifying factors were developed where needed.

DD , JAN73 1473 EDITION OF I NOV65 IS OBSOLETE UNCLASSIFIEDDID I FOAN 73 SIN 0102-014-6601 1 NLSSFE

SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)

NOLTR 73-225 1 December 1973

HANDBOOK OF BLUNT-BODY AERODYNAMICS

VOLUME 1 - STATIC STABILITY

This report contains the results of a study of handbook methods forpredicting normal force and center of pressure of blunt free-fallshapes. The work was sponsored by the Air Force at the request ofC. B. Butler, Air Force Armament Test Laboratory.(AFATL), EglinAir Force Base, Florida. Existing handbook methods apply to slenderbodies and break down when applied to blunt free-fall shapes.Reference material on blunt shapes and systematic blunt-body wind-tunnel test results obtained by C. B. Butler were the source ofinformation used to test and modify the existing handbook predictionmethods to apply to blunt shapes. This project was performed underTask Number NOL-533/AF.

Acknowledgments are due C. B. Butler of AFATL and S. M. Hastings ofNOL, for seeing the need for a handbook on blunt-body aerodynamics,also to K. D. Thomson of Weapons Research Establishment, Australia,and J. E. Fidler of Martin Marietta, for contributions which makepossible estimating coefficients at high angles of attack.

Illustrations were prepared by M. T. Faile of NOL, and typing andeditorial corrections by A. T.. McLearen and L. L. Lord, both of NOL.

ROBERT WILLIAMSON, IICaptain, USNCommander

LEON H. SCHINDELBy direction

Preceding page blank

NOLTR 73-225

CONTENTS

Page

INTRODUCTION ................................................

ESTIMATING STATIC-STABILITY COEFFICIENTS OF NONROLLINGBLUNT SHAPES ... ....................... 2

C N Normal Force Slope Coefficient of Blunt-NoseaCylinder at Subsonic Speeds ......................... 5

Effect of Viscous Forces on Normal Force ............... 7x Center of Pressure of Blunt-Nose Cylinders at acp

Near 0 Degree for Subsonic Speeds .................... 13C N and xcp for Blunt- and Slender-Nosed C-linders of

Z/d 6 to 15 for a = 0 to 10 Degrees, for M = 0.8to 1.2 ............................................... 13

CN and x for Blunt Cylinders /d = 1 to 11, a Near

0 Degree and a = 10 Degrees for Supersonic Speeds .... 17CN and xcp for Slender Cylindrical Bodies, a Near

0 Degree for Supersonic Speeds ....................... 17Normal Force and Pitching Moment for Cone CylindersWhere a = 0 to 90 Degrees for Subsonic and SupersonicSpeeds .......................... ................. 34

Effects of Flares and Boattails at High Angles ofAttack .................. . . . . . . . . . . . . . . 52(ACN ) - Normal Force Slope Coefficient Contribution

a BT

of a Boattail Near a = 0 Degree for Subsonic andSupersonic Speeds ...... ................... 53

(ACN ) - Normal Force Slope Contribution of a Conical-af

Flare Afterbody a Near 0 Degree for Subsonic andSupersonic Speeds .................................... 54

ESTIMATING NORMAL FORCE AND CENTER OF PRESSURE OF FINS ...... 59Subsonic Speeds ....... .... ........... ............ 59Transonic Speeds - C and x a Near 0 Degree ....... 64

Naof e cpSupersonic Speeds ................................. 68Effect of Boattail on Fins at Subsonic Speeds, a Near0 Degree ...... . . . . . . . . . . . . . . . . . . . 86Nonlinear Normal Force on Fins at High Angles of Attack. 88Normal Force of Ring Tails Subsonic and Supersonic ..... 103Effect of Nose Bluntness and Body Length on Fin Normal

Force at a = 0 Degree and 12 to 15 Degrees forSubsonic and Supersonic Speeds....................... 103

Effect of Adding Fin-Tip Caps to Fins at SubsonicSpeeds and Supersonic Speeds ......................... 111

iv

NOLTR 73-225

CONTENTS (Cont'd)

Page

CALCULATION OF CN AND C FOR COMPLETE CONFIGURATION .......... 114

REFERENCES .................... ... ............................ 122

BIBLIOGRAPHY ................................................. 124

APPENDIX A - Calculation of CN with C at M = 0.85 ofN mTypical Body-Fin Configuration with Comparisonto Experiment .......................... ........ A-I

ILLUSTRATIONS

Figure Title Page

1 Static-Stability Axis System for Blunt Free-Fall Bodies ............................... ........ 3

2 C Normal Force Slope Coefficient Near 0 DegreeNAngle of Attack for Blunt Cylindrical Bodies,M = 0.6 to 0.9 .. *.................................. 6

3 Effect of Nose Corner Radius on C for t = 0.5NCylinder at Subsonic Speeds .................... 8

4 Normal Force Coefficient at Subsonic Speeds for Flat-Faced Cylindrical Bodies, k. 1.0, 1.5 and 2.0 .... 9

5 Cross-Drag Coefficient for Circular Cylinders as aFunction of Cross-Flow Mach Number .............. 10

6 Ratio of the Drag of a Finite Cylinder to that ofan Infinite Cylinder as a Function of the FinenessRatio of the Finite Cylinder ...................... 11

7 Angle of Attack of Cylindrical Bodies where Cross-Flow Drag Begins to Affect Normal Force as aFunction of Cylinder Length for Subsonic Speeds ... 12

8 Reduction Factor (e) for Cylinder Planform AreaActed on by Cross-Flow Drag for Subsonic Speeds ... 12

9 x - Center of Pressure of Normal Force in Caliberscpfrom Nose for Blunt Cylindrical Bodies at SubsonicSpeeds, a = 0 Degree ........................... 14

10 Effect of Various Blunt-Nose Shapes on Normal ForceCenter of Pressure (xc) when a = 0 Degree for acp7-Caliber Cylindrical Body .. ................ 15

11 x - Center of Pressure for Blunt Cylindrical Bodies,cpM 1.0 (Calibers from Nose) ......... 16

12a-g CN as a Function of Mach Number for Cylindrical

aBodies, kB 6 to 15 ....................... .... 18-19

V

NOLTR 73-225

ILLUSTRATIONS (Cont'd)

Figure Title Page

13a-g CN as a Function of Mach Number for Cylindrical' a2Bodies, ZB= 6 to 15 to.................. o......... 20-21

14a-g (X cp) as a Function of Mach Number fora=00

Cylindrical Bodies, 1B = 6 to 15 ................. 22-2315a-g D, as a Function of Mach Number for Cylindrical

Bodies, kB 6 to 15 ...... 0.................. 24-2516 CN ~ Blunt Cylindrical Bodies, M 1.2 to 2.5 ..... 26

17 x - Calibers from Nose for Blunt Cylindrical

Bodies, k < 10, M = 1.2 to 2.6 ................... 2718a-b Center of Pressure Xcp ) at a = 10 Degrees for

Cylindrical Bodies at Transonic Speeds ........... 28-2919 Normal Force Coefficient Gradient for Tangent

Ogive-Cylinder Configurations .................... 3020 Center of Pressure for Tangent Ogive-Cylinder

Configurations ................................... 3121 Normal Force Coefficient Gradient for Cone-

Cylinder Configurations .......................... 3222 Center of Pressure for Cone-Cylinder Configurations. 33

23a-b Variation of the Function A with X and M,0 X S 2.4 .... *...........o......o..... ?........... 35-36

24a-b Variation of the Function B with X and M,

0 f X 2.4 ......................... ........... 37-3825a-b Variation of the Function C with X and M

0 :5 X :5 2.4 ............. *..... s.... ................ 39-40

26 Geometry for Cone Cylinder at Incidence ............ 4127 Extent of Upstream Influence of Edse (E) ........... 4228 Variation of Vortex Strength Parameter F with

Cross-Flow Mach Number and Incidence ............. 4329 Reduction Factor H to Allow for Effect of Base

Influence on Viscous Normal Force and PitchingMoment . . . . . . . . .,. . . . ....... 0......... 44

30 Reduction Factor J to Allow for Effect of Nose onViscous Normal Force and Pitching Moment ......... 45

31 Effect of Incidence on K CD (turbulent separation)/c

CD (laminarseparation)......................... 464 c

32 Drag of Circular Cylinders of Finite Span MountedNor'nal to Free Stream (the Factor N) ............. 47

33 Basic Curves for Determiiing Distribution ofCross-Flow Drag Coefficient ...................... 48

34 Cross-Flow Drag Correction Factor G due to AxialPressure Gradient ............................. 49

35 Illustration Showing Composition of Cross-FlowDrag Coefficient for Three Types of Bodies ....... 50

vi

NOLTR 73-225

ILLUSTRATIONS (Cont'd)

Figure Title Page

36 KBT and Center of Pressure for Boattail at

Subsonic Speeds .................................. 5537 Normal Force Coefficient Gradient for a Boattail 5638 Center of Pressure for a Boattail .................. 5739 Incremental Normal Force Coefficient Gradient for

a Flare .......................................... 5840 Subsonic Fin Normal Force Coefficient Gradient ..... 6041 Fin Panel Geometry for Use With Figure 40 .......... 6142 Fin-Body Interference Factors - Subsonic ........... 6343 Normal Force Coefficient Gradient for Rectangular

Fins, Transonic Speeds ............. ...... 6544 Center of Pressure for Rectangular Fins, Transonic

Speeds ........................................... 6545a Transonic Force-Break Mach Number for Zero Sweep ... 6645b Transonic Sweep Correction for Force-Break Mach

Number ........................................... 6645c Correction to Lift-Curve Slope at Force-Break

Mach Number ....... ........ ................ 6745d Chavt for Determining Lift-Curve Slope at M. ....... 6745e Chart for Determining Lift-Curve Slope at Ma ....... 67

46 Rectangular Wing Planforms ......................... 6947 Normal Force Coefficient Gradient and Center of

Pressure for Rectangular Fins ..................... 7048a-c Fin Normal Force Coefficient Gradient at Supersonic

Mach Numbers ................................. .... 72-7449a-c Fin Center of Pressure . ............. ..... 75-7750 Fin Normal Force Coefficient Gradient Correction

Factor for Sonic Leading-Edge Region ............. 7851 Lift Factors-Influence of Fin on Body, Supersonic

Speeds ..................................... ...... 7952a-f Lift of Fin (Body) for at/m = 0, 0.2, 0.4, 0.6, 0.8

and 1.0 .......................................... 80-82

53 Normal Force Coefficient Gradient of Multiple Finsat Supersonic Speeds ............................. 85

54 Subcaliber Fin Effectiveness Factor Variation withNose Bluntness ....................... ...... 87

55 Variation of f(ff/2) with Mach Number, 0 a ao 30 9155 a' - Angle of Attack Above Which ACN Must be

Applied (Subsonic Mach Number Only) .............. 9257 Dimensionless CN Increment Above a'................ 93

58 ACNM - Maximum Increment of Normal Force Above a'

(Subsonic Mach Number Only) ...................... 9459 Variation of Fin Normal Force With Mach Number

(a = 90 Degrees) ... .................. 9560a-c Variation of Normal Force With Mach Number

(a 30 Degrees) ................................. 96-97

-p vii

NOLTR 73-225

ILLUSTRATIONS (Cont'd)

Figure Title Page

61 Variation of Normal Force Curve Slope withMach Number (a = 30 Degrees) .................... 98

62 Chordwise Center of Pressure at 90 Degrees VersusTaper Ratio ....................................... 99

63 Basic Curves for Xcp/C at Reference Mach Number

0.98 (C 90 Degrees) .i....................... 10064 Variation Factor of x /C Versus Mach Number ....... 101

65 Variation of Sign Factor N with Mach Number(Pelta Fins Only) ............................ 102

66a-f Inc;:'emental Normal Force Coefficient Gradient forRing Tail Mounted on a Cylindrical Afterbody ...... 104-106

67a-b '.;fect of Nose Bluntness on Fin Normal Force atSubsonic Speeds for a = 0 to 6 Degrees, anda-= 12 to 15 Degrees .............................. 107-108

68a-b Effect of Nose Bluntness on Fin Normal Force forLow-Aspect Ratio Fins at Supersonic Speeds ........ 109-110

69 Fin-Cap Effectiveness Factor (Kc) for SubsonicSpeeds .................................... 9........ 112

70 Fin-Cap Effectiveness Factor (Kc) for SupersonicSpeeds ....... ................................... .. 113

71 Sample Blunt Body Showing Equations for CalculatingNormal Force and Pitching Moment Coefficients ..... 119

A-1 Typical Free-Fall Shape ............................. A-11A-2 Development of Fin Normal Force Coefficient ......... A-12A-3 Normal Force and Pitching Moment Coefficients for

a TypicalBlunt Free-Fall Shape, M = 0.85 .......... A-13

TABLES

Table Title Page

1 Comprehensive Review of Calculation Procedure forFinding CN and Xcp of Blunt Shapes ................ 116-118

2 Analysis of Contents of Items in Bibliography ....... 120-121A-1 Results of Handbook Calculation of CN and C for a

Typical Free-Fall Shape, M 0.85 ................. A-8

viii

NOLTR 73-225

SYMBOLS

A - function used in Equation (9) and shown on Figure 23

4/7 (CDc/F)dx (Ref. (10))

0

Ap - area of body as seen in crossflow when a = 900 (dxl)

AR - aspect ratio of fin panel (be2/Sfe)

B - function used in Equations (9) and (11) and shown onI xFigure 24 4/7 (CDc/F)zdx (Ref. (10))

0

be - span of lifting surface X

C - function shown on Figure 23a, C = 4/ (CD /F)x2dx0

c or cr - root chord of fin panel (Fig. 41)

CA - axial force coefficient

CDc - drag coefficient of an infinite cylinder in

crossflow (Fig. 5)

Cm - pitching moment coefficient [N(kcpj

A~v - viscous contribution to total pitching moment

coefficient (Eq. (10))

CN - normal force coefficient

CNc - fin normal force at a = 900 (Fig. 59)

CNa - slope of normal force coefficient per radian near

a= 00

4. ix

NOLTR 73-225

SYMBOLS (Cont'd)

ICNv - viscous contribution to total normal force coefficient

(Eq. (8))

CNafb - fin normal force slope referred to the body cross

section area '

CNafe - fin normal force slope coefficient per radian

referred to the exposed fin area CSfe)

CN- fin normal force slope coefficient per radian

referred to the wetted area (Sfe + cd)

8CN - dimensionless CN increment above a' (Fig. 57)

ACNm - maximum increment of normal force above a' (Fig. 58)

d - body diameter also referred to as caliber, used as

reference length for all body and fin dimensions

(inches)

D1 viscous component of Equation (6) for finding xcp

of a body

dBT - diameter at small end of boattail (inches)

dc - diameter of cylinder at large end of boattail

F(MCH)(Eq. (11)) or small end of flare (Eq. (17)) (inches)

df - diameter of the flare (Eq. (17)) (inches)

E -base influence factor (Fig. 27)

e -reduction factor for cylinder planform area acted

on by crossflow drag for subsonic speeds (Fig. 8)

F -vortex strength parameter (Fig. 28)

F(MACH) - variation factor of ANcp/cr (Fig. 64)

Fz - normal force (pounds)

Fx - axial force (pounds)

x

NOLTR 73-225

SYMBOLS (Cont'd)

f(4) - fin crossflow drag coefficient at a' = () (Fig. 55)

f(30) - fin normal force at a = 300 (Fig. 60a, b, c)

f'(0) - fin normal force slope at a 00

(Fig. 40, 44, 47j 48a - c)

f'(30) - fin normal force slope at a 300 (Fig. 61)

G - crossflow drag correction factor due to axial

pressure gradient (Fig. 34)

H -reduction factor to allow for effect of base

influence on viscous normal force and pitching

moment (Fig. 29)

J - reduction factor to allow for effect of nose on

viscous normal force and pitching moment (Fig. 30)

K -effect of incidence on ratio of

CD (turbulent separation)(Fig. 31)

CD (laminar separation)

c

K1,2 - Munk's apparent mass factors (Fig. 2)

KB - boattail factor (Fig. 36)

Kb(f), Kf(b) - fin-body interference factors (Fig. 42)

Kc - fin-cap effectiveness factor (Fig. 69)

Kf(n) - nose bluntness fin effectiveness factor

(Fig. 67a, b, 68a, b)

Kf(sc) - subcaliber fin effectiveness factor (Fig. 54)

- total body length (N + kB) (calibers)

kB - body length (calibers)

ZN - nose length (calibers)

xi

NOLTR 73-225

SYMBOLS (Cont'd)

'BT - length of boattail (calibers)

Lf - length of flare (calibers)

M - Mach number

Mc - crossflow Mach number (M sin a)

My - pitching moment (inch-pounds)

MFB - force break Mach number (Fig. 45a)

N or n - ratio of drag of finite cylinder to drag of infinite

cylinder (Fig. 32 same as Fig. 6)

N - sign factor for delta fins (Fig. 65)

q - dynamic pressure at M (psia)

rc - radius of corner of blunt nose (inches)

S - Strouhal number (S = 2 Eqs. (9), (11))

SB or Sref - cross-sectional area of body (square inches)

bt- projected area of boattail (square inches)

Seff - effective fin area (Fig. 53)

Sf - projected area of flare (square inches)

Sfe - exposed area of fin panels in horizontal plane

(Fig. 41) (square inches)

Sw - wetted area of fins (Sfe + cd) (square inches)

t - fin thickness (inches)

Xcp - location of center of pressure of normal force

(calibers)

L, XN -nose and body lengths as expressed on Figure 26

XeP - center of pressure of fin panel as percent of rootcr

chord (Fig. 62, 63)

xii

NOLTR 73-225

SYMBOLS (Cont'd)

y - vertical distance between two horizontal planes as

on Figures 69, 70

a- angle of attack (degrees or radians)

at' - see Figure 56

- Mach number parameter where M

NOLTR 73-225 V

INTRODUCTION

The design of blunt free-fall shapes requires a source of

aerodynamic performance characteristics for use in predicting the

free-fall trajectory. The blunt shape may be described as a body

symmetrical in both planes with a characteristic blunt-nose shape

and a low-aspect ratio tail configuration. In calculating the

aerodynamic characteristics of the shape, the order of procedure is

to find: (1) drag, (2) static stability and (3) dynamic stability,

magnus, and roll damping. The drag is dependent primarily on the

nose shape and usually may be quickly fixed because of fuzing

requirements.

The procedure used by handbooks such as the "USAF Stability

and Control Handbook" sometimes referred to as DATCOM (Ref. (1))

and the AMCP 706-280 "Design of Aerodynamically Stabilized Free

Rockets" (Ref. (2)) is to calculate the drag and lift of the com-

ponent parts of a body and then add them up, with appropriate

influence factors, to arrive at total aerodynamic coefficients. The

two handbooks mentioned are comprehensive and are good guides to this

procedure. For use in blunt body design, some of the charts in the

handbook are not applicable since both handbooks start with the

premise that every aerodynamic shape is first of all a slender

streamlined body.

The purpose of this report was originally to collect available

blunt-body aerodynamic data from test reports on shapes already

* 1

NOLTR 73-225

designed. This collection of data was to form the foundation for a

handbook for blunt-body design.

After collecting the available data, the author attempted to

use the AMCP handbook to calculate the normal force and center of

pressure for various blunt shapes and compare the calculated values

with the experimental values taken from the collected reference data.

This started the basis for putting together a modified version of

the Static Stability section of the AMCP handbook.

ESTIMATING STATIC-STABILITY COEFFICIENTS OF NONROLLING BLUNT SHAPES

A body in flight experiences two primary static forces: lift

and drag. The total of the pressures distributed over the body sur-

face are resolved into these two static force components. Lift and

drag are aircraft forces related to the ground as a reference plane.

In the free-fall shape, the pressures over the body are referred to

the body axis system which coincides with the ballistic path during

fall. The forces are given in dimensionless coefficient form to

ease scaling of the forces to body size and operating altitude.

Instead of lift and drag, we have normal force (Fz) and axial force

(Fx ) which in coefficient form are CN Fz/qSref and CA = Fx/qSref.

For a body in flight to be statically stable, the center of pressure

of the normal force (xcp) must be behind the body center of gravity.

Generally on bodies, it is necessary to add a tail configuration to

move the center of pressure back of the center of gravity.

The procedure in finding the normal force coefficient and the

center of pressure of a blunt shape is to start by estimating the

normal force and center of pressure for the nose-body combination

alone and then select a tail configuration and calculate normal force

2

NOLTR 73-225

zw

z4CENTER OF GRAVITY X

YYwF Y w

cp apz

ww

FIG. I STATIC STABILITY AXIS SYSTEM FOR BLUNT FREE FALL BODIES

3

NOLTR 73-225

and center of pressure for the tail. The two component forces are

added together and the overturning moments they cause, acting about

the center of gravity, determine the location of the resultant center

of pressure. The resultant normal force acting at the resultant

center of pressure causes a resultant moment about the center of

gravity which is referred to as the pitching moment (My). Generally

the aerodynamic coefficients are functions of the Mach number

(speed) and angle of attack to the wind (a). Figure 1 shows the sta-

tic force axis system with the normal force caused by the body being

at angle of attack, a.

Following are methods with accompanying figures for use in

estimating the normal force coefficients (CN) and the center of

pressure (xcp) for blunt-shaped bodies and typical stabilizing

configurations. The handbook first tells how to find CN and Xcp

near a = 00. Up to a = 4 to 8 degrees, the values are linear with

angle of attack. The nonlinearity begins when the body crossflow

drag force comes into play. So to construct a curve of CN and

Cm or (Xcp) versus a up to high angles of attack, the curves must

be built up in sections. Another point that will be noted is that

there are separate curves and charts for subsonic and supersonic

speeds. The handbook starts with the nose-body at subsonic speeds,

goes through supersonic speeds and then does the same with the

stabilizing tails. On the figures and illustrations, nose and body

lengths are given as IN and kB with the total length as L. When

shown this way, it is understood that all lengths have been divided

by the body diameter, d, such that the dimensions are said to be in

calibers.

4

NOLTR 73-225

CNa NORMAL FORCE SLOPE COEFFICIENT OF BLUNT-NOSE CYLINDER AT SUBSONIC

SPEEDS

The slender-body theory provides a basis for obtaining CN, at

subsonic speeds at low angles of attack where viscous crossflow

forces are small. The theory states that:

dCN (K2 - K1 ) (ds( ) ~ sin 2a (Ref. (2)) (1)dx Sref \dx

which when integrated from x = 0 to x gives

SBCN 2(K2 - KI ) (2)Sref

a=0

(K2 - 1 is the apparent mass factor defined by Munk in

Reference (3). The theory does not apply very well at low-fineness

ratios (9/d < 4.0) because the nose bluntness begins to overpower

the body effects. IFor fineness ratios less than 1.0, normal force is very

sensitive to nose radius for flat-face noses. As the nose shape

approaches a hemisphere, the sensitivity to nose shape disappears.

CNa for blunt-nosed bodies may be obtained from Figure 2 which

includes experimental data from References (4), (5), and (6) and a

plot of CM using the slender-body theory, as presented in

Reference (2).

For blunt-face cylinders where k/d < 2.0, CN can be negative

at angles of attack from 2 to 20 degrees.

The cause is explained in Reference (7) as a separation bubble

along the top surface of the cylinder starting at the corner of the

NOLTR 73-225 i

4.0 M=0.8 'M=0.9

M=0.8

3.0 M=0.6

M=0.9, 1.0M=0.8

M=0.6

CNa&= 0 2.0 =0.

PER RADIAN M06

b2(K 2-K1 ) SLENDER BODY THEORY

NOSE SHAPE

a 0OFLAT1.0 b r/d0o.1OTOO.25

CHEMISPHERESSECANT OGIVE r /d=1 .25

SECANT OGIVE r /d=4.25

FIG. 2 CN NORMAL FORCE SLOPE COEFFICIENT NEAR D= DEGREES ANGLE OF ATTACK FORBLUNT CYLINDRICAL BODIES, M=0.6 TO 0.9 (REF 4,5, AND 6)

6

NOLTR 73-225

flat face and body juncture which gives a negative normal Force.

As the cylinder becomes longer, the flow reattaches and past the

point of reattachment the flow is normal and gives a positive normal

force. Figure 3 shows the effect of nose shape on CN versus ,- for

z/d = 0.5 cylindrical body. Figure 4 shows Cq versus a for flat-

faced cylindrical bodies of L/d = 1.0, 1.5, and 2.0.

EFFECT OF VISCOUS FORCES ON NORMAL FORCE

At low angles of attack, generally only the nose shape affects

the normal force on a cylindrical body. For blunt bodies where

rc/d < 0.5, the length of the body also affects normal force. As

the angle of attack increases, the body is exposed to the wind; and

a crossflow drag component is added to the normal force.C Ap sin 2a (a)

ACNm = lCDc Sref

This normal force component is the Allen-Perkins Viscous

Crossflow component (Ref. (8)). Values of CDc and n may be found

on Figures 5 and 6.

In testing Equation (3) with the existing blunt-body data, two

things were found: (l) that the onset of viscous crossflow was

affected by the body length and (2) that the crossflow component so

predicted was too high. Figure 7 shows the angle of attack (a),

where the crossflow component becomes part of the normal force

plotted as a function of body length. Figure 8 shows a reduction

factor (e) which reduces the effective cylindrical planform area Ap

of the cylindrical body.

7

NOLTR 73-225

0.40.

0.2-

0.4-0.4

0.2

C N jVc20 20

-0.2-0.2 o-0-0.O 0.05M=0.4

-0.4- M05 ~ 01 -0.4

0.2 0.2~~00.50

FI.3EFCCFNS ONRRDU NC O 2~. YIDRASUSNCSEDNRF4

8

NOLTR 73-225

00

NC

00

III--

0

00

00

I~uo U..

LUou

a-

LL

co i0

Z9

NOLTR 73-225

2.4

GOWEN-PERKINS

20- ALLEN-PERKINS2.0 - M"% M=M SIN a

BELOW TRANSITION I \ C

u I, 1.6-z

0 1.2

0.8-

0

0.4

ABOVE TRANSITION0 I I I I I I I0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2

CROSS MACH NUMBER, Mc

FIG. 5 CROSS-DRAG COEFFICIENT FOR CIRCULAR CYLINDER AS A FUNCTION OFCROSS FLOW MACH NUMBER (REF 8)

10

NOLTR 73-2251

0~0.9

0.8

zo 0.7-

00

0 0.5I0 4 8 12 16 20 24 28 32 36 40 44

EFFETIVELENGTH-TO-DIAMETER RTO 2~

FIG. 6 RATIO OF THE DRAG OF A FINITE CYLINDER TO THAT OF ANI* I INFINITE CYLINDER AS A FUNCTION OF THE FINENESS RATIO

OF THE FINITE CYLINDER. (REF 8)

NOLTR 73-225

16-

14

12 B

10

8

6

4

2

II I II

2 4 6 8 10 12

FIG. 7 ANGLE OF ATTACK OF CYLINDRICAL BODIES WHERE CROSS-FLOW DRAG BEGINS TO AFFECTNORMAL rORCE AS A FUNCTION OF CYLINDER LENGTH FOR SUBSONIC SPEEDS (REF. 6)

1.0

cAp= exlB xd0.8

0.6 - rc/d =0.50.4 C 2

0.2 -aA..rC/_... r /do

0 I I2 4 6 8 10 12

FIG. 8 REDUCTION FACTOR (e) FOR CYLINDER PLANFORM AREA ACTED ON BYCROSS-FLOW DRAG FOR SUBSONIC SPEEDS (REF 6)

12

NOLTR 73-225

The complete expression for normal force of a blunt body for a

between end of linearity and 150 is:

CN= CNa (a) + nCDce(Ap/Sref)sin2a (4)

Xcp CENTER OF PRESSURE OF BLUNT NOSE CYLINDERS AT a NEAR 0 DEGREE

FOR SUBSONIC SPEEDS

At low angles of attack, the center of pressure of the normal

force (Xcp) for a flat-faced cylinder has an apparent shift in

location from ahead of the flat face for Z/d < 1.0 to aft of the

flat face between L/d = 1.0 to 3.0. This shift in location is

caused by the separation bubble over the upper surface of the cylin-

der. Figure 9 shows Xcp for blunt-faced cylinders up to k/d = 11.0

for subsonic speeds.

Generally increasing nose bluntness increases the static

stability of a body. Figure 10 shows the effect of nose corner

radius (rc/d) on the location of Xcp for a 7-caliber cylindrical

body. Summarizing for Figures 9 and 10, static stability decreases

with increasing rc/d, increaeq with the addition of an oversized

ring, and is not much affected by concaving the flat face.

Figure 11 shows Xcp for blunt 5odies M = 1.0. Here again, stability

is improved by bluntness of the nose.

CN AND xqp FOR BLUNT AND SLENDER-NCSED CYLINDERS OF L/d 6 TO 15

FOR a = 0 TO 10 DEGREES, FOR M = 0.8 TO 1.2

A method, developed by H. Barth (Ref. (9)), gives

charts for calculating CN and x from a = 0 to 10 degreesN cp

angle of attack. The charts are good over a range of M = 0.8

to 1.2 and body length /d 6 to 15. Nose bluntness varies

13

NOLTR 73-225

o 0

LI~u 10L L .

Li 0

I z

U)

-U

ui

0

____ 0 ju1 1A LA

I ILn0(A L LL'00

L L I oJLii

ce

UL

( AU..

0 0

UL

14z

NOLTR 73-225

NOSE r/d NOSE r,/d MOIIA I LOCATIONSHAPE rCd SHAPE C MOICAON4 FROM TIP

13 0 Ii 0 ].1 -DRING 0'10 0.125 0 1.1-D RING -0.58D

Ir 0.250 I3 0 1.0-D PLATE 40.25D33.50 0 0.8-D CAVITY -0.2D(WITH 12-0.05D BLEEDS)

0-0125 1.1-D RING -0. 58D

-2

0 -3z0

-4 4-4Lu

(L -5

-6

-7

0.2 0.4 0.6 0.8 1.0 1.2MACH N',UMBER

FIG. 10 EFFECT OF VARIOUS BLUNT NOSE SHAPES ON NORMAL FORCE CENTER OFPRESSURE (Xcp) WHEN a =0 DEGREES FOR A 7-CALIBER CYLINDRICAL BODY

(REF 6)

15

[ NOLTR 73-225

0 2 4 6 8 10(NOSE) 0 UI '

-2 o

-4 1

-5

NOTE: SEE SYMBO. KEYON FIG. 10

FIG. 11 X p-CENTER OF PRESSURE FOR BLUNT CYLINDRICAL BODIES,M=1 .0 (CALIBERS FROM NOSE) (REF 6)

16

NOLTR 73-225

from flat to a 2.5-caliber ogive and cone. By this method which is

based on experimental data:

a2CN = CNa (0a) + CNa 2(a ) 2(5)

Xcp (Xcp)a=0O + Dla (6)

Figures 12, 13, 14, and 15, a through g, are carpet plots of CN,

CNa2 Xcp , and D1 as a function of M and B/d. The ordinates of

the plots do not give the desired values directly. As an example

for Figure 12a, the ordinate gives CN + 0.4 ZB/d = A so that

CNa = A - 0.4 kB/d.

CNa AND Xcp FOR BLUFF CYLINDERS 1/d = 1 TO 11, a NEAR 0 DEGREE.

a=0AND a = 10 DEGREES FOR SUPERSONIC SPEEDS

Figures 16 and 17 give CN and Xcp for short bluff cylindersca=O

over the Mach number range of 1.2 to 2.5. These figures are based

on the AEDC data (Ref. (6)). As a companion to these,Figures 18a

and b give the center of pressure of CN when a 100. This accounts

for the nonlinearity above a 4 to 8 degrees. Here CN is found

with Equation (4) and:Cma = CNa = 10 p=0 (7)=i1 0o C io Xcpa=.10o

CNa AND Xcp FOR SLENDER CYLINDRICAL BODIES, a NEAR 0 DEGREE FOR

SUPERSONIC SPEEDS

Carpet plots from Reference (2) are reproduced on Figures 19

through 22. These plots based on experimental data are for ogivesand cone-nosed cylinders where n/d 3 to 7 and M 1.4 to 7.0.

17

NOLTR 73-225[NB ~N: O 5d TdC =2

12 N CN+N (a)12 a212 -CN CN ()+ C ()a Na 20a211r 1 15 10-10-

ft 9 c118 ~ 8

+ 8- 10 +.

z 7 7 I U

S I 6

6 1061

I0.8 91.0:m:5 .8 0. 0.8 0.9 .01.2

2 10 C CNa = CN1

15 8. 1lB IG ~

0. 0. 1.0 M2aIN

MaN 0.5 0. IN.12

FIG 1 C N S F N T O F C NU MBER FO CYL NDR CA BO IE oB = O 5 d (R F 9

10- 1- N C~e C a 0a

NLR73-225

ZN- 1.5dTL 13 -

12 C N C N (c) + C N (a 2

22

ZN- 2 .5dT101

N N (a) + CN (c 2a

92

081

6

z 6 t

66

5 5 F0.8 0.9 1.0 1.1Ma1 .2 0.8 0.9 10 IIa.

4 -ZB=5 Ma0. 1 4r ZB5 Ma01FIG;. 12e

FIG. 12f

B d

10- CN= CN(a) +CNa (l) 2 T2 N =2 .5d

9 1

7-+

11

5 6

0.9 1.0 l Ma 1.2 FG ~4-Ft B= 5 Ma 0.1

FIG. 12 CNa AS A FUNCTION OF MACH NUMBER FOR CYLINDRICAL BODIES ZB= 6 TO 15 (REF 9)

19

IN

180- 80-

160- 70-

140156

120- cB 50--1

N100 -10 + c 0z t

80- 30

60F 0.8 0.9Ma6 20[

85 Ma 0. 14oL 10FIG. 13a .- .4 -0~. FI.13B 5 Ma = 0.1

F'B IN -*'N

N 1.5d

120- 60-

100-5

80 15-10 ZB + 40-

10 30-1

40- 01.0 6B 20

0 0.9 L1a 2 0.8 0.9M 0 1.01 .0520L lc = aC5 10 IB= Ma. 1

FIG. 3c Ma 0. 1FIG. 13d

FIG. 13 CN AS A FUNCTION OF MACH NUMBER FOR CYLINDRICAL BODIES Z 6 TO 15 (REF 9)

20

NOLTR 73-225

'N 1

A 1. 5dN* 90-

80- N

70- Kr- -dN 2.5d

th 60- 60-

+ C450-1 50-1

z40 UF 4040

B

60 I

10 40I

S 30 -4 301 .zz20 6 20 - 660.8 0.9 1.0 1 a1208 09 io 1 ~,.10 lB =5 Mar 1B 0.1a .FI .l eBM .1FIG. 131'

21

[ NOLTR 73-225 'IN

-~. d(X ) Z N 0.5dx

16- (Xp =p a=0D+a

a=0

15 10

+ 1

10 - 8 -

8 101X 6- 4

0. a0.8 0.9 ma.60. 0. .0 1 . 1ma. 6

4r 1.0 B =5 Ma=0. 1 2 1B =5 Ma0. 12

FIG. 14a FIG. 14b

IN

INNA I18- D + aIcL a= 00

16-

X +0.5+~ Dia*

14 8 c X pa 0N1 -2

4 40.8 0. 1.0 1.0M 1 5

01.0

FIG 0.91a a .

FIG 14c (X8, =5 M .0 B= M.=0

FIG.14 (cp) OPAS FUNCTION OF MACH NUMBER FOR CYLINDRICAL BODIES, t B= 6 TO 15 (REF 9)

22

7 NOLTR 73-225ZN

ZN

(Xcp) tN 1.5d d:

14 Xc (X p a= o+D a 1 6-X = (Xc)_ + D a (X cp) Z 2

~co 15 .0 1

+o 10- 001

00I01 8- B 'Z BB 8-Y

10 a.

x 6 0. 6 .4. 0.8 0gMa 1.01.05 4 0. _09 1. 1Ma 12

ZB 5 Ma. 1 ZB =5 Ma=O. I

FIG. M~e FIG. 14f

(Xc) ZN =2.5d

16 X (X) + Di

010- 15+

0000 8

-1

x 6[

0.Z8 0n.9 1.0 ljMal 1.

FIG. 149

FIG. 14 (Xp 00 O AS A FUNCTION OF MACH NUMBER FOR CYLINDRICAL BODIES, ZB= 6 TO 15 (REF 9)

23

NOLTIR 73-225

T' IN =0B5

d160- ~ =X Da N

120- =c(X p) oo +D140-

100- 120 2

1580- ioo

co 1

1B 60 105810

4006 6

0.8 0.9 Ma 1.020- jB 5 a . 40 0.

B a=010.8 0.9 1.0 1.l1M01.2FIG. I~a ' 20 =B5 Mao. I

__IN 1 .dB

/N0IN T15 d D

IN0.dd 160- (X)p =0

120- X=Xc C~ 0 0 140-

100- 120-2

80 -100-

60 - + - 80

4 - 60 -01

FI .8 . . 9 l.,cl .

2Ma. 400.8 0.9 1.0 1. 1Mal1.2=520IB= 5 Ma=0. 1

FIG. 15 D1 AS A FUNCTION OF MACH NUMBER FOR CYLINDRICAL BODIES~j B=6 TO 15 (REF 9)

24

NOLTR 73-225

ZNZN

d~.S 7 1 B~

tZ=2.5d tf120 x X ~120

100 i~ i~ -0 20 =(X a;.~OO 4-Dj a

8015 1015

ZB 80-Z

10 '010

6 0[

0.8 0.9Mal.0l.05 400.8 0.9 1.0 1.1IMal.2

20L ZB=5 MaO02.1B5 Ma0

FIG. 15e20Z= MO.

FIG. 15f

ZN z I~ d

Z =2.5dN

80-

'0

20FIG.15 Ma.

FIG. 15 D1 AS A FUNCTION OF MACH NUMBER FOR CYLINDRICAL BODIES, ZB=6 TO 15 (REF 9)

25

NOLTR 73-225

SYMBOL NOSE SHAPE

O3 FLAT

35~~ '- r/d=0. 2503. '-3 C(3 HEMISPHERE

S rc/d=1.25 T04.25 OGIVE

3.0 U FLAT -RECTANGULAR CROSS SECTION

2.0 /1.513

2.26

C T3

NOLTR 73-225

1.0

2 4 6 8 10 12 14 16 180 0 i r

x 1.0- C1 C3Xcp 0 a

-2.0_

NOSE SHAPE n

O FLAT-3.0 a HEMISPHERE

r /d=0.TO 0.2

r /d=1.25 TO 4.25 OGIVE

U FLATRECTCROSS SECTION

FIG. 17 X CALIBERS FROM NOSE FOR BLUNT CYLINDRICAL BODIES Z

NOLTR 73-225

24 6 8 100

FLAT NOSE rc/d=0

-1.0

-2.*0

xcp

-3.0 -- M=O0.6

-- M=1.0-M1.2

-4.0

-5.0

2 4 6 8 10

HEMISPHERE NOSE

-1.0- r C /d=0.5

Xp -2.0

-3.0

FIG. 18a CENTER OF PRESSURE (Xcp) AT cc=10 DEGREES FOR CYLINDRICAL

BODIES AT TRANSONIC SPEEDS (REF 6)

28

- NOLTR 73-225

2 4 6 8 10

SECANT OGIVE NOSE rld=4.25

Xp -2.0-- M=0.6

M=0.8

-3.0 -M=1.2

X cp -2.0

FLAT WITH ROUNDED-3.0[ CORNER r,,d=0.25

FIG. l8b CENTER OF PRESSURE (Xcp) AT a=10 DEGREES FOR CYLINDRICAL

BODIES AT TRANSONIC SPEEDS (REF 6)

J]

29

NOLTR 73-225

d --

('. -

It UU-/-0

-- L

_ -, z- 0

---- L

L~~~ju Iv -m~~~-jL~ - -- iJ~IU C)

/ -- ,

CD z~ 0

It it

- -

U, Q' I

30 0

NOLTR 73-225

C14U-

-4-4 / -z0

- z

0 -u

- LUJ

- -- 0-L

- 0 .1ItLCDz

0 /

\ ---- - -- 31

NOLTR 73-225

U.

i~'h zC,' 0

ki z_0 0

LU

-> z

-w 0

- d -

N LL

cm /

C--j

C, W! 0I

z

C..' I I.

C=!I . 4 I I

N IV U I No

32c.II N

NOLTR 73-225

I 7.0 'Pol

6.04 10

l..921 .>I

0. olLU

-l D

-*~. uL z. -- Az

040

rl:L

-a

U., / LU

53~

NOLTR 73-225

NORMAL FORCE AND PITCHING MOMENT FOR CONE CYLINDERS WHERE0 TO 90 DEGREES FOR SUBSONIC AND SUPERSONIC SPEEDS

K. D. Thomson formulated a rapid method for calculating CN and

Cm for cone cylinders for the angle-of-attack range of 0 to 60 degrees.

The method is good for bodies 1/d > 3 where cone half angle does not

exceed 750 and the crossflow Mach number (Mc) does not exceed 0.8

(Ref. (10)). By Thomson's method:

CN NAC1 + ACNv (8)

where: ACNI - Cr2 CNa sin 2a (inviscid contribution)

CN is found on the appropriate figure

in this report and ACNv is the viscous flow contribution. IACN S Xv j XN L-E XL

:-cot a [B] 0 + [A]x + H[A] E (9)KF sin a cos a S N L-E

For the pitching moment referred to the nose tip:

Cm = ACml + ACmv (10)

1where: ACml = [ CNa xcp sin 2a (the inviscid contribution)

Xcp is taken from the appropriate figure

(ACm )2 X X -E Xv N L L

cot a [C] + [B] + H[B] (11)2 S 0 X LX-EKF cosa N

L

34

NOLTR 73-225

0.61.8

0.6- OA

1.07

0.6

------ 7

4- >Lt. rF -

0.4A

0...4060.2.

0.6.

NOLTR 73-225

5.0 A A 4(D- - 2.4

0.6

4.5

.0

33.I

A_ _ _-0

NOLTR 73-225

1.4 .

I 1 /1-.-0.7

* II 9 , 9 . . 9;0.8

1.0 - 0.599 * 9

I I I i t ! I OA

0.8 j.4..

9 * ..9 h

f i l l : ; ! I t : 1 : 9 ! 1 t

i t : I t : ;9** ' ! i I f

9~~~ ~ V 9 1 9 .1I;

0.4 9 9 . .9

0.2

0 0.2 0.4 0.6 0.8 1.0x

FIG. 24a VARIATION OF THE FUNCTION B WITH X AND Mc0

NOLTR 73-225

6.5 -- --

FOR X 2.4 - .

6. .c D 0.752. I- (D(X2- .76)5. -L..... 2.4

- -- 0.6,0.8

5.5 -E 1 - -

4.5 0.--

--- - -- - -

---- - --- - -

B- ..--. . ..4 ......

3.30

-. 1 7 7. .

NOLTR 73-225

1.2 *'

1.0.

I J 1..

I III ; - 0.5

.Im OA .I

C 0.6 if1 1I ii . il

Ic 0

A I.

0.4,

0. 0.2 0. .608 .

00.204X 01. (REF 10)

39

NOLTR 73-225

7.0 0.7 0.75 0.8 0.6

4.0

.5 ~ f I L .

1.0 ~ 4:~=~.4=~..

1.0 ... 2 4 R F 0

3.00

Aj

NOLTR 73-225

Nose MainbodyBaseNoseme ain odyinfluence

IN e

a DF _ _

M IONXNH_

_ _ _

XL -E E

XL

NOTE: IN, and e areXN I S tan a; XL S Stan a; E: e Stan a in calibers

FIG. 26 GEOMETRY FOR CONE-CYLINDER AT iNCIDENCE (REF 10)

41

NOLTR 73-225

~ I NOTE:, ON SHORT BODIES IT IS POSSIBLEFOR E TO BE GREATER THANXL, IN SUCH CASES TAKE (XL-E):O

Reference diameter D

E sta C7 rl -Ito shedvortex

0.2

E 0.1

00 0.2 0.4 0.6 0.8

FIG. 27 EXTENT OF UPSTREAM INFLUENCE OF BASE (E) (REF 10)

42

NOLTR 73-225

1.

Quasi-steady regime

F

0.4 Steady reigime

0.2 ___

0 20 40 60 80 90a (degrees)

FIG. 28 VARIATION OF VORTEX STRENGTH PARAMETER FjWITH CROSS-FLOW MACH NUMBER AND INCIDENCE (REF 10)

[4SC ~ .43

NOLTR 73-225

1.0

100,.

0.4

0.2

0 I I I I j. j, I0 0.5 1.0 1.5 2.0 2.5

XL

FIG. 29 REDUCTION FACTOR H TO ALLOW FOR EFFECT OF BASE INFLUENCEON VISCOUS NORMAL FORCE AND PITCHING MOMENT (REF 10)

*4

.. _.._.._.._.._

NOLTR 73-225

ON 450

0.8/

Di , /.306&

0.6 200

150

0.4

0.2 ___50

0 10 20 30 40 50 60a (degrees)

FIG. 30 REDUCTION FACTOR J TO ALLOW FOR EFFECT OF NOSE ONVISCOUS NORMAL FORCE AND PITCHING MOMENT (REF 10)

45

- NOLTR 73-225

x Perkins and Jorgensen(ref.6) Mc = 0.340 Bursnall and Loftin(ref. 16) MC = 0

Jones, Cincotta and M =09l Walker(ref.15)C

1.0

0.8

0.6 .TranscriticalK

0.4

..........

0.2 PostciticaI

00204608

a (degrees)

FIG. 31 EFFECT OF INCIDENCE ON K -C (TURBULENT SEPARATION) (REF 10)CDC (LAMINAR SEPARATION)

46

NOLTR 73-225

RD 8 . 8x1 04

M =0

0.8

0.6

(CD)L/DN-

(CD)L/D= o. 0.4

0.2

00 5 10 15 20 25

I

FIG. 32 DRAG OF CIRCULAR CYLINDERS OF FINITE SPAN MOUNTED NORMAL TOFREE STREAM (THE FACTOR N) (REF 10)

47

'\a-

NOLTR 73-225

0.4

U-

--- I.

0

LI-

U. ** - - - u(

.1

I C-

00

~ 0U-1

U AU0 0uTHCI'll

00>.

04

- a %S.A5.IWt n...aL.

NOLTR 73-225

1.0

0.8-

0.6

G /

0.450.2

0 0

0 20 40 60 80 90

, ANGLE OF ATTACK (DEGREES)

FIG. 34 CROSS-FLOW DRAG CORRECTION FACTOR G DUE TO AXIAL PRESSURE

GRADIENT (REF 10)

49

_ I

NOLTR 73-225

[.1-NXL-E E----

CDC

XL

CD C

XL

CDC H~

XL

FIG. 35 ILLUSTRATION SHOWING COMPOSITION OF CROSS-FLOWDRAG COEFFICIENT FOR THREE TYPES OF BODIES (REF 13)

50

NOLTR 73-225

theseto te ofXL-EIn these equations, to find the value of terms such as [A] ,find the value of A at XL-E and at X and take the difference.

Figures 23 to 31 give A, B, C, E, F, H, J, and K. in both

Equations (9) and (11), S = 0.2 and is the Strouhal number. On

Figures 23, 24, and 25 for finding A, B, and C as a function of XN

XL and XL-E as defined on Figure 26 where:

XN kN S tan a

XL Z S tan a

E e S tan a (found directly on Figure 27).

The following table gives the limits of Equations (9) and (11)

and shows how to determine the values of K and F.

RANGE CHARACTER OF WAKE K F

00 to 400 Steady, Laminar Separation 1 Fig. 28

Turbulent Separation Fig. 31 1

300 to 600 Quasi-Steady Laminar 1 1

Turbulent Fig. 31 Fig. 28

For the range of a 600 to 900 the body is in the unsteady wake

regime and:

2 ___ACN sin a[CD (12)NV D ref

where: CD KNCD , find CD on Figure 3S

c CX-2.4 x=24

and N on Figure 32.

151

\51

NOLTR 73-225 jFor the pitching moment:

ACmv sin 2a[CD Sref ] Xcp (xcp at high angles of

(13)

attack is the centroid of the body)

EFFECTS OF FLARES AND BOATTAILS AT HIGH ANGLES OF ATTACK

Thomson also describes how to calculate the normal force and

pitching moment of bodies having flares and boattails. The method

is not rapid as the one for cone cylinders aescribed in the previous

section but requires integrating the viscous crossflow drag over the

body so that:

ACv 4)XL

- si~ S 0 CD T- dx (47r S 0ref)

XL

ACm - cos2 CD Dcd )Xdx (15)0 S2 ref)

where: CD is obtained from Figure 33, S 0.2 andc

XL = 9 S tan a, 9 = total length of body.

The body is treated as four sections:

(a) nose, where CDc for XN is multiplied by G from Figure 34

to account for the effect of pressure gradient on the nose.

(b) body, where CDc is taken directly from Figure 33.

(e) upstream influence of base, determine E from Figure 27 and

H from Figure 29. Find CD for E from Figure 33 and

multiply by H.

(f) flare or boattail. For a flare multiply CDc by G using

the flair angle as 0 on Figure 34.

52

I I-

NOLTR 73-225

For the boattail, take CDc at x = 2.4 from Figure 32 as the

crossflow drag coefficient for the boattail. Figure 35 shows what the

developed crossflow drag coefficient would look like. Keep in mind

that such a curve of CD versus L has to be corrected to the localc

body diameter when integrating for the total crossflow drag. Also

when finding the turbulent viscous forces, multiply by K from

Figure 31 except for the case of the boattail where only the region

ahead of the boattail is multiplied by K.

(ACNm)BT - NORMAL FORCE SLOPE COEFFICIENT CONTRIBUTION OF A BOATTAIL

NEAR a = 0.DEGREE FOR SUBSONIC AND SUPERSONIC SFEEDS

A boattail causes a negative normal force which subtracts from

the normal force of the cylindrical body nose.

S(aCN)BT = -KBT - (16)

K = 2 where dBT/dc > 0.8 (Ref. (2))

where: Sbt : cross-sectional area of boattail at

its smallest diameter

dBT = diameter of boattail at its smallest

diameter

d c diameter of boattail where it joins body

SB = reference area of nose-cylinder CNa

The center of pressure of the boattail is assumed to be located at0.6 of the boattail length measured from the body boattail juncture.

The above expression was derived for boattail angles of 100 or less

and ratios of dBT/dc > 0.8.

53

NOLTR 73-225

Where the boattail is replaced by a cylindrical boom of

diameter ratio to body of 0.3 or less at attachment point, the

CN of the boom may be ignored.

* For boattails where dBT/dc is less than 0.8, use Figure 36 to

find the factor KBT and the center of pressure of the boattail. At

supersonic speeds, Figures 37 and 38 give CNBT and Xcp/kBT for

boattails on infinite cylindrical bodies. This means that local

flow conditions upstream of the boattail are equal to free-stream

conditions. These charts which were taken from the AMC Handbook

(Ref. (2)) were derived from linearized theory calculations and

slender-body theory predictions.

(ACN) - NORMAL FORCE SLOPE CONTRIBUTION OF A CONICAL-FLARE

AFTERBODY a NEAR 0 DEGREE FOR SUBSONIC AND SUPERSONIC SPEEDS

The contribution of a conical-flare afterbody to the normal

force based on slender-body prediction is:

(ACN)f = 2 - (Ref. (2)) (17)

SB (f]

and the center of pressure

is:

-[(18)rf()2]

(an overall value of 60% may be used)

5[1t L4

NOLTR 73-225

16

12

8I

xV4

0.4 0.8 d /dcbt

KBT FACTOR USED IN EQUATION (16) FOR C NtBOATTAIL

a ~ ~ ~ - 0.6x- --

0.4

eZI

u 0- 0.x

040.8 d /dcILOCATION OF CENTER OF PRESSURE ON BOATTAIL

FIG. 36 K AND CENTER OF PRESSURE FOR BOATTAILBT

AT SUBSONIC SPEEDS

55

NOLTR 73-225

. .v- .1L ....d ldbt

NOTE: M > 1

Sre f =4

2.0-

1.8'

=. 1.4-

: ,,, 1.2"

0 1.0

a L 0.8

. 0.6.

0.4'

0.2-

00 1.0 2.0 3.0 4.0

l bt/dcr2-I

FIG. 37. NORMAL FORCE COEFFICIENT GRADIENTFOR A BOATTAIL ( REF. 2)

56

NOLTR 73-225

d 4.

; ' LL

0Ci c

I.D LLCJi

U, U-

LL. CD U,0

LL Y

57IC

.4, ~>

NOLTR 73-225

------------ ~ T

2.0'.0-

1.6CH'~i10MAHNO .

SIC

w

0 .2 . 6 . 0

* Idc/di 24.6. .

'C/d

2.0. 0

0 .6

Wcdtdcd

FIG. 39 INRMNA1OMA-OC .OFIIETGAIN O AFAE RF2

CA58

NOLTR 73-225

The important geometric parameter is the ratio of forebody

cylinder diameter to base diameter. Flare angle and Mach number do

not influence the flare normal force within the limitations of

slender-body assumptions. The effect of upstream flow conditions

ahead of the flare are not accounted for. Since separation is likely

to occur at the body-flare junction, these predictions are not apt to

be realistic.

Figure 39 gives ACNaf over a Mach number range of 0 to 2.5

(Ref. ANC Handbook) and should be used in preference to Equation (8).

ESTIMATING NORMAL FORCE AND CENTER OFPRESSURE OF FINS

The basic fin configuration is four fin panels attached to the

body in the horizontal and vertical planes. In estimating the normal

force contribution of the four panels, only the two panels in the

horizontal plane are considered. All of the methods outlined here

for estimating CNafe and Xcp are for isolated panels not attached to

a body. Body-fin and fin-body influence factors are used to correct

the isolated panel normal force to a panel force attached to a body.

SUBSONIC SPEEDS

a. CNafe AND Xcp, a NEAR 0 DEGREE

The normal force of the fins is first calculated as a flat

plate detached from the body and positioned in the horizontal plane.

The normal force coefficient slope based on lifting line theory is:

-_ _ _ _ __ _ (Ref. (2))

AR 2 + (AR)2(02 + tan --) + 4 (19)

59

L NOLTR 73-225

NOTE: AR= ASPECT RATIO = e2fe

1.6 - b = EXPOSED FIN SPAN

Sfe = EXPOSED FIN AREA

z. A c/2= FIN SWEEP ANGLE AT MID-CHORD5 1.2 - \ :In :o ==

N OLTR 73-225

--

ALE

b etan A ctan A LE 4 [.(1- X)1

S e b________ e LE (I + I

2 2A

ARf Sfe =2b(1eX/C( X

FIG. 41 FIN PANEL GEOMETRY FOR USE WITH FIG. 40 (REF 2)

61

NOLTR 73-225

Figure 40 is a plot of CN /AR and is good up to M = 0.6. Thea

center of pressure of the fin is assumed to be 25 percent of the

mean aerodynamic chord aft of the leading edge.

Figure 41 illustrates the fin panel and shows how to use

Figure 40. It should be noted that there is no significant effect

of fin roll o.,ientation on the normal force or xcp of fins as long

as there are an even number of fin panels,

b. FIN-BODY INTERFERENCE (SUBSONIC)

When fins are attached to a body of revolution the

interference between the fin and body causes an increase in the

effectiveness of the fins and brings about a lift over the body por-

tion enclosed between the fins. Figure 42 (Ref. (11)) is a plot of

the multiplying factors Kf(b), effect of fin on body, and Kb(f),

effect of body on fin. In the figure the span (b)

referred to in the ratio d/b is now the full-fin span as projected

through the body. Using CN(e) from Figure 40 for the fins alone,

we have:

cS C-- Kf(b) I Kb(f ] (Ref. (11)) (20)Ca fw Ca (fe ) Sf w

where: Sfw =(fe) + (projected fin area passing

through body)

Sfe exposed fin area.

62

NOLTR 73-225

2.0

1.8

1.6 K Kf(b) oo

1.4

S1.20

U-

LUj

zu 1 . 0 1LLJ K f

0.8

0.4

i ~~0.2_ _ __ _ _

00 0.2 0.4 0.6 0.8 1.0

d/b

FIG. 42 FIN-BODY INTERFERENCE FACTORS -SUBSONIC (REF 2)

63

NOLTR 73-225

To use CNafw with the body normal force components the

coefficient must be referenced to the body area

CN Cf (Sfw) (21)

afB afuSB

c. FIN-FIN INTERFERENCE

The fin normal force calculations have been for a four-fin

configuration where two fins have been assumed to act in the hori-

zontal plane. For six-and eight-fin configurations, multiply the

four-fin values by 1,37 and 1.62,respectively (Ref. (2)).

TRANSONIC SPEEDS - CN AND Xcp a NEAR 0 DEGREE- c~~~afe DcaERDGE

For rectangular fins of very low aspect ratio linearized

slender-wing theory predicts that CN, = (n/2)AR. Utilizing transonic-

similarity laws, McDevitt (Ref. AMC Handbook) obtained from experi-

mental data the correlation shown on Figures 43 and 44. Note that

here the wing thickness ratio t/c is an important factor.

For M = 0.6 to 1.4 to calculate CNafe for either rectangular

or swept fins, use the method from the U.S. Air Force Stability and

Control Handbook (Ref. (1)). This method calls for plotting a

curve of CN from M 0.6 to 1.4. For thin wings or low-aspect

ratio wings, the curve shows an increase in slope from M 0.6 to a

high value near the critical Mach number (MFB) and then through Ma

and Mb to the value at ,. 1.4. The charts cf Figures 45a to e are

used as follows:

(1) Find CN(f) for M 0.6 from Figure 40

(2) Calculate MFB from Figure 45a for zero wing sweep

(3) Fo. ,,ding with sweep obtain a corrected M(FB)A from

Figure 45b.

64

NOLTR 73-2252.5-

NOTE: Sre f FIN PLANFORM AREA

-2.0

1.50

-- 3

-4 / 1.25

(-'40

1.5- M2_ 1 -6 5 71.0

2I

A .575

00

FIG. 43. NORMAL-FORCE COEFFICIENT GRADIENT FOR RECTANGULAR FINS,TRANSONIC SPEEDS (REF 2)

- b

0.5-

NOTE: Sref FIN PLANFORM AREA0.4- AR(t/c) 1/ 3 -"1.5

;

0.42

0.3- 1.0

.75.

0.2-

0.1- M2 -1

0

FIG. 44. CENTER OF PRESSURE FOR RECTANGULAR FINS, TRANSONIC SPEEDS'(REF 2)

65

I 1- n

NOLTR 73-225

1.0 TRANSONIC SPEEDS2

0.92

U-

~ j 0.8A~~ 6/ DG

FIG. 45 6TRANSONIC SWREPCORECTIONAFO

FORCERBRAKEMAC NUMBE (REF 1)

1.66

0. 9

NOLTR 73-225

TRANSONIC SPEEDS (CONT'D)1.2

A 6 8

LL

0I 1.0 2

z z,u u 4

0.80 4 8 12 16

THICKNESS RATIO (% CHORD)

FIG. 45c CORRECTION TO LIFT-CURVE SLOPEAT FORCE-BREAK MACH NUMBER (REF 1)

1.0A

0.8 8

60.6 4

0.4- 22

0.2-

0

-0.2 ' I0 4 8 12 16

THICKNESS RATIO (% CHORD)

FIG. 45d CHART FOR DETERMININGLIFT-CURVE SLOPE AT M (REF 1)

a

0'21

-OI I I0 4 8 12 16

THICKNESS RATI 0 (% CHORD)

FIG. 45 e CHART FOR DETERMININGLIFT-CURVE SLOPE AT M6 (REF 1)

67

NOLTR 73-225

(4) Find CN using Figure 40 for MFBa (fe)

(5) Find actual [CNa(fe)]FB/CNa(fe) from Figure 45c using

(tic) ratio

(6) The value of Ma MFB + 0.07 (22)

(7) [CNa ( ] = (1 - a/c)[CNa(fe)]FB where a/c is found onafe)a af)FFigure 45d (23)

(8) The value of Mb MFB + 0.14 (24)(9) [CN 1f)](1 - b/c)[CN fe IF where b/c is found on

(9)5Figure 45e(25)

(10) Find CN for M 1.4 use the methods in the sectionNa(fe)

on C of Rectangular Fins at Supersonic SpeedsNaSUPERSONIC SPEEDS

a.CNafe AND Xcp RECTANGULAR FINS, a NEAR 0 DEGREE

On rectanaii1ar fifns at supprsono speeds there is a loss

of lift on Lhe tips which according to Evvard's theory (Ref. (12)) is

half of the two-dimensional value. For a finite fin panel

C 1 when AR > (26)Na(fe) a 26AR w

When the span is short enough that the Mach cones from the tips

intersect, then the expression for normal force slope becomes,

2 [2A1 +A 2_+ A3] 1C= I -2when AR < - (27)Na(fe) 6 AT

(See Fig. 46 note A4 drops out in the

solution)where: BAR >0.5 use the chart on Figure 47

68

L

NOLTR 73-225

b I

/ / /9cA 2j

/ AR -.118i

FIG. 46 RECTANGULAR WING PLANFORMS (REF 12)

69

NOLTR 73-225

c CHORD b SPAN

AR L' 1. FOR RECTANGULAR FIN)

Si C LEADING EDGET T

.4 '4

-P xio

C-)N

00.4 0.8 1.21.20

FIG. 47. NORMAL FORCE COEFFICIENT GRADIENT ANDCENTER OF PRESSURE FOR RECTANGULAR FINS.(REF 2)

70

NOLTR 73-225

The above expressions are based on linear theory and are good

up to a__ +0 .

b. CNafe AND Xcp SWEPT FINS, a NEAR 0 DEGREE'

The charts shown on Figures 48 and 49 (a, b, c) were taken

from the AMCP Handbook (Ref. (2)) and are part of a more complete

coverage of wing planforms made by E. Lapin. Note that X is the tip-

to-root-chord ratio.

c. FIN-THICKNESS EFFECTS

When Mach lines lie on or near the fin leading edge, there

is a loss in normal force from the theoretical value. Figure 50

(Ref. (2)) gives a correction factor for this loss. To use the

figure:

(1) find AyL equal to the diffzrence between upper surface

ordinates at the 6 and 15 percent chord stations (in percentof chord).

(2) AyL = Ay/cos Ate

(3) for a double-wedge leading edge AyL = 5.85 tan 6L where

6 is the wedge semi-angle

d. FIN-BODY INTERFERENCE

Mach number and the fin plane geometry are important to

supersonic fin-body interference when considering the effect of the

fin normal force carryover to the body, Kb(f). Figure 51 (Ref. (2))

gives Kb(f) for the case where the cylindrical body extends past the

fin trailing edge and for the case where the body is flush with the

fin trailing edge.

NOLTR 73-225

6

4 - 4

a (f,

TAN Ale

6

C-,-,t,

C N72I-e af)7

NOLTR 73-225

5 5

IIV

33 3__

TANAl 1

0 0"

0 .2 .4 .6 .8 1.0 .8 6 .4 .2 013 TAN Ale

TA N Ale

7 -77

NOLTR 73-225

7 7

(E))

-62~~~

ARTN_5

ATANANfA

IIFI.~ I NR OCECEFIINT GRDETA2SPROI

MACH~~ NUBES RE 2

(F 77

NOLTR 73-2251

(A)1.2- -I I-

E 1.0 AR-- TAN Ale.8 -ARTAN Ae-- -

..

0_ANAl /3 I _ANAl_

(3 TAN -Ale TAN Ale (

(B)1.4

'k- AR TAN .'fe

1.0. -ARTAN A~e --

.6m

_ UNSEe T.E

C_ L

NOLTR 73-225

.4

(.2 TAN ATeNTNlAe

~AR TAN Ale

.4

H S SO C :S2PE 3SONI C_______

10 TAN Ale 1 ( 0 le 1 TAN Ale 03 TAN Ale TAN Ae

IIX I6

NOLTR 73-225

1.4NI UEROI IL I I

AR TAN Al TN

1.6'0~ 1-10.8

-C-

- rA) e

SUBSONIC - S)ES~I _____

0TAN Ale 1 P 0 P 1TAN Ale 0p TAN A 6 TAN Ale

FIG.49 cFINleNEOFPSUR(EF2

77)

NOLTR 73-225K

- - -- - AY1 *.51(deg)

.9 - OE:M,>- - 0 =585TA

SOICLEDIGEDE EG. (RF 2)0;0

78 IOE >1*yI5.5TNs

if NOLIR 73-225

Cree

I I CreN~

F,

NOLTR 73-225

E

00

.54

r im wU-u

ca

U- U

00

- 0-

-D A

i-o -_ _ _ m.

-

4 U-00

L LL.

00 0L

LL

806

- - I

:1 NOLTR 73-225

C4

-

U0

FF. 11. -om c

00

0 U-

0'-

-

U..

0.

U

81

NOLTR 73-225

0 U

U- L

LL

-W~

C ___

NDCD

ELLC

a~

0