+= sn

THE PRACTICAL CALCULATION OF THE AERODYNAMIC

CHARACTERISTICS OF SLENDER FINNED VEHICLES

by

James. S. Barrowman

A Dissertation Submitted to the Faculty of the School of Engineering

and Architecture of The Catholic University of America in Partial

Fulfillment of the Requirements for the Degree of Master of Science

in Aerospace Engineering.

March 1967

Washington, D. C.

https://ntrs.nasa.gov/search.jsp?R=20010047838 2018-09-08T11:52:00+00:00Z

This dissertationwasapprovedbyDr. ShuJ. Ying

AssistantProfessorof SpaceScienceandAppliedPhysics

as Director andby

asReader.Dr. Hsing-PingPao

ii

ACKNOWLEDGMENTS

I would like to express my sincere appreciationto Dr. ShuJ. Ying andDr.Hsing°PingPaoof the Departmentof SpaceScienceandAppliedPhysicsfor theirhelpful suggestionsconcerningthis dissertation, I wouldalso like to expressmythanksto Mr. EdwardMayoinparticular andthe staff of GoddardSpaceFlight Centerin generalfor their helpfulnessandpatiencewhilepreparingthis report.

iii

TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS .............................................. iii

TABLE OF CONTENTS .............................................. iv

LIST OF FIGURES ................................................. v

LIST OF SYMBOLS ................................................. vi

I. SYNPOSIS ..................................................... 1

2. INTRODUCTION ................................................. 1

3. NORMAL FORCE AERODYNAMICS .................................... 2

4. AXIAL FORCE AERODYNAMICS ...................................... 42

5. COMPARISON WITH EXPERIMENTAL DATA ............................. 62

6. CONCLUSIONS .................................................. 74

7. REFERENCES .................................................. 75

APPENDIX A - Exerpt of Aerodynamic Theory From Reference 1 .................. 77

iv

Figure

3-i

3-2

3-3

3-4

3-5

3-6

3-7

3-8

3-9

3-10

3-11

3-12

4-1

4-2

4-3

4-4

4-5

4-6

4-7

4-8

5-1

5-2

5-3

5-4

5-5

5-6

5-7

5-8

5-9

5-10

LIST OF FIGURES

Fin Dihedral Angle ..........................................

Fin Geometry ............................................. 7

Proportionality Triangle for Chord ............................... 8

Fin Cant Angle ............................................. 12

Forces Due to Fin Cant ....................................... 13

Roll Damping Parameters ..................................... 14

Body Component Shapes ....................................... 19

Approximate Tangent Body .................................... 22

Supersonic I_¢r> ........................................... 33

Subsonic XB(_} ............................................ 34

Supersonic ×B_,r)........................................... 38

Pitch Damping Parameters .................................... 39

Skin Friction Coefficient ...................................... 44

Skin Friction With Roughness ................................... 45

Supersonic C% Variation ..................................... 49

Cylinder Drag Variation ...................................... 51

Leading Edge Drag .......................................... 52

Two Dimensional Base Pressure ................................ 54

Nose Pressure Drag at Mach One ................................ 58

Three Dimensional Base Pressure ............................... 61

Aerobee 350 .............................................. 63

Aerobee 350 Normal Force Coefficient Derivative ..................... 64

Aerobee 350 Center of Pressure ................................. 65

Tomahawk ............................................... 67

Tomahawk Roll Forcing Moment Coefficient Derivative ................. 68

Basic Finner .............................................. 69

Basic Finner Roll Damping Moment Coefficient Derivative ............... 70

Aerobee 150A ............................................. 71

Aerobee 150A Drag Coefficient .................................. 72

Aerobee 150A Reynolds Number ................................. 73

Page

5

V

Symbol

A

AB

AB r

AB s

Ar

A T

AwB

AWB^

A WN

AWN̂

A Ws

A WsA

AR

a

B 3

CD

C DB

C%T

CD f

CDLT

CDp

CDTT

%.

LIST OF SYMBOLS

Description

Area

Base Area

Base Area of One Fin

Nose Base Area

Conical Shoulder or Boattail Base Area

Reference Area

Planform Area of One Exposed Fin Panel

Total Body Wetted Area

Wetted Area of Booster Nose Afterbody

Nose Wetted Area

Wetted Area of Sustainer Nose and the Cylindrical Afterbody Between the Noseand the Shoulder. If there is no Shoulder, the Afterbody is Taken as the Remain-

der of the Sustainer.

Conical Shoulder or Boattail Wetted Area

Wetted Area of the Shoulder or Bottail and the Portion of the Sustainer Aft of

the Shoulder or Boattail

Aspect Ratio of Exposed Fin

Ambient Speed of Sound

Third Order Busseman Irreversibility Coefficient

Drag Coefficient, D

Base Drag Coefficient

Tail Trailing Edge Drag Coefficient

Skin Friction Drag Coefficient

Tail Leading Edge Dra_ Coefficient

Pressure Drag Coefficient

Total Drag Coefficient

Tail Thickness Drag Coefficient

Wave Drag Coefficient

vi

Symbol

cf

Cf c

C I

C,P

C_

C

Cq

%

Cp

C

CMA

C r

C t

D

d

d_

dS

F

fB

f_

fN

fN^

fs^

h

K,k

KI

K2

Description

Incompressible Skin Friction Coefficient

Compressible Skin Friction Coefficient

Rolling Moment Coefficient, _/qA r L r

(PL_ PL rRoll Damping Moment Coefficient Derivative, _c_/_ \TV] _ 2-_ : o

Roll Forcing Moment Coefficient Derivative, _C_/_ ,_ _ = 0

Pitch Moment Coefficient, _/_ArL r

Pitch Damping Moment Coefficient Derivative, _cm/_ (_) _ --qLr2V= 0

Normal Force Coefficient, n/_A

Normal Force Coefficient Derivative, _CN,/_ @_ = 0

Pressure Coefficient, 5P '_

Cord Length

Mean Aerodynamic Chord

Fin Root Chord

Fin Tip Chord

Aerodynamic Drag Force

Diameter

Diameter of Nose Base

Diameter of Shoulder or Boattail Base

Force

Total Body Fineness Ratio

Diederich's Planform Correlation Parameter for Fins

Equivalent F_neness Ratio of a Truncated Cone

Nose Fineness Ratio

Fineness Ratio of the Nose and Nose Afterbody

Fineness Ratio of the Shoulder or Boattail and Shoulder Afterbody

Fin Trailing Edge Thickness

Interference Factors and General Constants

First Order Busseman Coefficient

Second Order Busseman Coefficient

vii

Symbol Description

Third Order Busseman Coefficient

Length

Reference Length

Location of Shoulder or Bottail Measured from the Nose Tip

Fin Root Leading Edge Wedge Length

Nose Length

Total Body Length

Shoulder or Bottail Length

Location of Fin Leading Edge Intersection with Body, Measured from Nose Tip

Fin Root Trailing Edge Wedge Length

Roll Moment About Longitudinal Axis

Mach Number

Cotangent rL Pitch Moment About Center of Gravity

Number of Fins (3 or 4)

Normal Aerodynamic Force

Prandtle Factor, I/-

Prandtle Factor for Swept Fins, 1//1 - M~ cos2 rc Roll Rate

Pitch Rate

Dynamic Pressure

Reynolds Number

Surface Roughness Height

Fin Leading Edge Radius

Body Radius at Tail

Exposed Fin Semispan Measured from Root Chord

Maximum Fin Thickness

Velocity

Body Volume

Downwash Velocity

Symbol

XCG

XL T

Y

YMA

VL

r T

V1

C2

-y

_L

A

Description

Center of Gravity Location Measured from Nose Tip

Distance Between Fin Root Leading Edge and the Fin Tip Leading Edge

Measured Parallel to the Root

Longitudinal Center of Pressure Location Measured from Nose Tip

Spanwise Center of Pressure Location Measured from the Longitudinal Axis

Spanwise Location of C_A Measured from the Root Chord

Angle of Attack

Supersonic, - 1; Subsonic, _'_ - M2

Midchord Line Sweep Angle

Quarter Chord Sweep Angle

Leading Edge Sweep Angle

Trailing Edge Sweep Angle

First Region Boundary Sweep Angle

Second Region Boundary Sweep Angle

Ratio of Specific Heats

Fin Leading Edge Wedge Half Angle in a Plane Parallel to the Flow

Fin Trailing Edge Wedge Half Angle in a Plane Parallel to the Flow

Incidence of Surface to Flow in a Plane Parallel to the Flow

Fin Dihedral Angle

Fin Taper Ratio, c t 'C r

Mach Angle

Ambient Kinematic Viscosity

Ambient Density

Nose Tip Half Angle

s + r tRatio

r t

ix

Subscripts

B

B(T)

cr

N

i"

S

T

T(B)

t

W

1

Body

Body in the Presence of the Tail

Critical Value

Nose

Root

Conical Shoulder or Boattail

Tail

Tail in the Presence of the Body

Tip

Wetted

Value for one fin

X

1.00 SYNPOSIS

The basic objective of this thesis is to provide a practical method of computing the aerodynam-

ic characteristics of slender finned vehicles such as sounding rockets, high speed bombs, and

guided missiles. The aerodynamic characteristics considered are the normal force coefficient

derivative, CN_; center of pressure, _; roll forcing moment coefficient derivative, C_s; roll damping

moment coefficient derivative, C_p; pitch damping moment coefficient derivative, cq ; and the drag

coefficient, CD. Equations are determined for both subsonic and supersonic flow. No attempt is

made to analyze the transonic region. The general configuration to which the relations are applic-

able is a slender axisymmetric body having three or four fins.

The fins and body are analyzed separately; and the interference between them is accounted for

when they are combined to form the total vehicle. When necessary, both the fins and the body are

subdivided. Both purely theoretical and semi-empirical methods are used in deriving the equations.

Use is made of established methods when they are applicable.

Since the high speed digital computer is a tool readily available to the contempory engineer, no

effort has been made to linearize higher order theories or simplify complex relations. In many

cases, the equations are derived with the computer in mind.

The method is applied to sounding rocket vehicles and the results are compared to respective

windtunnel and flight data. The comparison indicates that the theory predicts aerodynamic charac-

teristics within ten percent of the experimental data.

2.00 INTRODUCTION

2.10 BACKGROUND

In order to study the performance of a new or existing flight vehicle the engineer must have

the vehicles aerodynamic characteristics. Windtunnel or flight tests can prov{de the needed infor-

mation; but they are both expensive and time consuming. Lacking either time or money the engineer

turns to theoretical predictions.

In this day of rockets and high speed armaments, slender finned vehicles are of considerable

interest. A great deal of work has been done on the aerodynamics of such configurations. The list

of contributors is exhaustive. However, a.number of works stand out and are of particular interest.

Munk, reference 20, developed the technique of slender body theory and applied it to low speed

airships. Tsien, reference 29, later modified Munks technique to apply it to pointed projectiles in

supersonic flow. Jones, references 7, 8, 9, extended Munk's work to finite wings and also provided

excellent application of conical flow theory to wing lift and drag estimation. Nielsen, Pitts, Ferrari,

and Morikawa, references 22, 3, 18 and 17, have thoroughly analyzed wing-body interference effects.

Even though they provide valuable understanding, many of the contributions have limited appli-

cability to practical calculations. Their inherent assumptions and approximations reduce the ac-

curacy and reliability of their results. Those that provide accuracy are usually so complex or

abstractthey cannotdirectly providethedesiredaerodynamiccharacteristics. This thesisis anattemptto form concrete,usefulexpressionsby applyingthebestabstract theoreticaltreatmentsaswell as reliable empirical data.

2.20 STATEMENTOF PROBLEM

Determinea set of expressionsfor thepractical calculationof the aerodynamiccharacteristicsof slender,axisymmetricfinnedvehiclesat bothsubsonicandsupersonicspeeds.Thecharacteris-tics of interestare the normalforcecoefficientderivative, CN_; centerof pressure,X; roll forcingmomentcoefficientderivative,C_; roll damping moment coefficient derivative, C; ; pitch damping

_p

moment coefficient derivative, Cmq; and drag coefficient, CD.

2.30 GENERAL METHOD OF SOLUTION

1. Divide vehicle into body and tail.

2. Analyze the body and tail separately. Subdivide either when necessary.

3. Analyze wing-body interference.

4. Recombine to form total vehicle solution.

2.40 GENERAL ASSUMPTIONS

1. The angle-of-attack is very near zero.

2. The flow is steady and irrotational.

3. The vehicle is a rigid body.

4. The nose tip is a sharp point.

3.0 NORMAL FORCE AERODYNAMICS

There are a number of aerodynamic coefficient derivatives that are directly dependent on the

aerodynamic forces acting normal to the longitudinal axis of the vehicle. The normal forces acting

on the fins are generated by four main sources; angle-of-attack; fin cant angle, roll rate, and pitch

rate. These excitations produce the normal force coefficient derivative, CN_, at a center of pressure,

& V; a roll forcing moment coefficient derivative, C4_; a roll damping moment coefficient derivative,

Cy ; and a pitch damping moment coefficient derivative, C q respectively. The only normal force,p

acting on the body is due to the angle of attac_k. This is the normal force at a center of pressure.

3.10 TAIL NORMAL FORCE AERODYNAMICS

3.11 TAIL NORMAL FORCE COEFFICIENT DERIVATIVE

In subsonic flow, the semi-empirical method of Diederich from reference 2 gives the normal

force coefficient derivative of a finite thin plate as;

Af

42 +Fp +--

where;

%_0 = Normal Force Coefficient Derivative of a two dimensional airfoil.

FD = Diederich's Planform Correlation Parameter.

According to Diederich;

ARF D : 1

2---#%=o cos Fc

By the thin airfoil theory of potential flow (reference 26),

C__a 0 = 27

Correcting this for compressible flow according to Gothert's rule (reference 26)

(3-1)

(3-2)

(3-3)

27 (3-4)CN ° - 13

Thus;

J_ (3-5)FD - cos Fc

Substituting equations (3-4) and (3-5) into (3-1) and simplifying,

) 27 _ (Af "Ar] (3-6)C-Na --

This is the value for one fin.

The value of (_)_ for one fin in supersonic flow is computed using the method of reference 1,

as given in Appendix A. It is essentially a strip theory which analyses two dimensional strips using

Busemann's Third order expansion of the compressible flow equations.

2 K3 3 B3V_ (3-7)Cpi : KI _i + K2 "_i + "Oi +

where;

Kz = 2/z

(T + I) M 4 - 4,_ 2K 2 =

4Z"

K3 = (W + i) Ms + (2_ - 7W- S) M_ + 10('y+ i) M4 - 12M 2 + 86/3 7

1B3 = (5,+ I) M4 [(S - 37) M4 + 4('>'- 3) Ms + 8] 4-'8

The strip values are then summed in a spanwise direction with a correction applied to those strips

that are intersected by the fin tip mach cone. The most distinct advantage of the method in reference

1 is that it includes a thickness distribution in the analysis.

To apply the C.N_ of one fin to a multi-finned vehicle the effect of fin dihedral must be con-

sidered. Only three and four finned vehicles are analyzed, since these are the most commonly

used. Both are analysed assuming that one fin is in a plane parallel to the flow. Reference 13 de-

rives the effect of dihedral on the CN= of tWO fin panels at very small angles of attack.

Figure 3-I defines the dihedral angle, A.

A three (3) fin configuration has two effective panels at a 30 ° dihedral angle. Thus the tail

normal force derivative is,

(3-9)

=)T 4 (4 fins) (3-10)

From equations 3-9 and 3-10 we can generalize for three and four fin vehicles.

N

Since the dihedral angle of the two effective panels of a four (4) fin configuration is zero, the tail

normal force derivative is;

_y

Figure 3-1-Fin Dihedral Angle (As Viewed From Rear)

Substituting 3-6 into 3-11, for subsonic flow;

(c__) T Nv PR (Af/A) (3-12)

V/4 f 'f2 + + \cos Fc]

3.12 TAIL CENTER OF PRESSURE LOCATION

From the thin airfoil theory of subsonic potential flow, reference 26, the center of pressure of

a two dimensional airfoil is located at 1/4 the length of its chord from its leading edge. Thus, on

a three dimensional fin, the center of pressure is located along the quarter chord line. By defini-

tion, the center of pressure is also located along the mean aerodynamic chord of the fin. There-

fore, by the above argument, the subsonic fin center of pressure is located at the intersection of

the quarter chord line and the mean aerodynamic chord. In this treatment, the subsonic center of

pressure is considered to remain constant with mach number. It remains to determine the length

and position of the mean aerodynamic chord.

The mean aerodynamic chord is defined as;

f0

1 c 2 dy (3-13)CMA -- _f

Figure 3-2 shows the fin coordinate system. The generalized chord is a function of the spanwise

coordinate. To find this function, a proportionality relation is set up. See figure 3-3.

c,_ c = ct (3-14)L_ L-y L* -s

It is assumed that the fin edges are all straight and form a trapizoid. From the first two terms of

equation 3-14;

c = c - Y---c, (3-15)• L*

From the first and last terms of equation 3-14;

CtL* : c rL* -c S

Simplifying, and defining;

C

_. =_L (3-16)C t

iMEAN

AERODYNAMIC

CHORD

lIMA

\\

\

Figure 3-2-Fin Geometry

Ict

,QUARTER-CHORD

LINE

)-CHORD LINE

c t

c r

-,_-Y _-_ L*-Y

L*

Figure 3-3-Proportionality Triangle for Chord

we find;

Lo = s (3-17)

Substituting3-17 into 3-15,

c = cr (1 + ky) (3-18)

where,

I )_-Ik - ----

L* s(3-19)

Using 3-18 in 3-13;

c2 fo'rc_^ =_ (1 + ky) 2 dy(3-20)

Integrating

C 2r

c_^ - Af 3k--- [(1 +ks) 3 - 1]

= r--L- 1 +ks +3 sCMA A f

(3-21)

Substituting3:19 into 3-21 and simplifying;

1 C2 S

• [}2 +_+1]CM^ = 3 Af

(3-22)

But;

1A,=_(% +c t) s (3-23)

thus;

2 c2

[\_ +_+1]CMA 3 C r + C t

or,

CMA ='3" r _ Ct Cr + Ct

(3-24)

To find the spanwise location of %^, equate 3-24 and 3-18 and solve for y.

3- r+Ct =c r 1+ yC r + C t S

or

I'_ 2cr ct 1 sc

Y_^ (% +c,) 3(% +c t) " c -c,(3-25)

After much algebra;

s ICr +2Ct 1 (3-26)

The spanwise subsonic center of pressure coordinate is then;

YT ---: r t +_- + C t J

(3-27)

As can be seen from figure 3-1, the longitudinal coordinate is;

1= 17 + lu^ + _-cM^

(3-28)

But;

YMA

J'MA = YMA tan r L = -- x t .s

(3-29)

Combining equations 3-24, 3-26, 3-28, and 3-29;

x t c + 2c t + Cr Cr + CtT_r : 17 +Y ; c--_-j 5 +c (3-30)

The fin center of pressure in supersonic flow is found by the method of reference 1 in appendix A.

10

3.13 TAIL ROLL FORCINGMOMENTCOEFFICIENTDERIVATIVE

A forcearises ona fin at zeroangleof attackif thefin is cantedat ananglewith the longitu-dinal axis. Seefigure 3-4. Thecantangleis aneffectiveangle-of-attack. As canbeseeninfigure 3-5, thetotal force resulting from cantingall the fins at the sameangleis zeroonthreeandfour fin configurations.However,a roll forcing momentaboutthe longitudinalaxis is produced.Sincethecantangleactsas aneffectiveangleof attacktheroll forcing momentis subsonicflowmaybewritten in terms of (C_)_;

a I

The values of (C_), and VT are found using equations 3-6 and 3-27 respectively. The rolling moment

coefficient is defined as,

(3-32)C_ : _A L

thus; using equation 3-26;

N YT CN_ 8

' (3-33)Cj_: Lr

The roll forcing moment coefficient derivative is defined as;

c _C4 : (3-34)

thus; the subsonic roll forcing coefficient derivative is,

7 r

In supersonic flow, C_:s is computed using the method of reference 1 as given in appendix A.

This method determines the roll forcing moment of each strip and sums them in a spanwise direction.

3.14 TAIL ROLL DAMPING MOMENT COEFFICIENT DERIVATIVE

When the vehicle has a roll velocity, the superposition of the local tangential velocity of the fin

panel on the free stream velocity produces a local angle-of-attack which is a function of spanwise

position. The local angle-of-attack produces a local force distribution along the fin. For three

and four fin configurations the total resulting force is zero, as shown previously. The moment pro-

duced about the longitudinal axis opposes the roll rate (see figure 3-6) and is a roll damping moment.

The local rolling moment at _ is,

11

V

J

Figure 3-4-Fin Cant Angle

12

(A)

F"

-'-F

(e)

F

T/,

F"

F

VIEWS LOOKINGFORWARD

Figure 3-5-Forces Due to Fin Cant

13

TOP VIEW

• I

X

,.y

F(()_a(()

S+r t

VIEW FROM REAR

F

_v

LOCAL

ANGLE -OF-ATTACK

Figure 3-6-Roll Damping Parameters

14

The local angle of attack at 4 is,

dt : (F(_) (3-36)

(3-37)

But;

v(_) : - P _7

Thus;

(3-38)

For v >>P,_ the local angle-of-attack is near zero, thus;

_,(,_) - - -- (3-39)V

In subsonic compressible flow the local normal force on an airfoil strip is found using equation 3-4.

v(:) - %,0 h ,(=) c(_) ,1-_ (3-40)

Where CN_° is defined in equation 3-4. Knowing that,

y - _ - ,- (3-41)

equation 3-18 gives

c(_') - c r 1 ---I - (: _ rt )• S

or

c(4) : ' . s (3-42)s 1----_ -_ -_ r

15

Combining equations 3-36, 3-39, 3-40, and 3-42;

Simplifying;

d- 1 = - CNa0 _ pcr (1 - 4)(L • ÷ rt _ _) _2 d_

VS(3-43)

where L ° is defined by equation 3-17. Applying equation 3-32 provides the local roll moment

coefficient.

CNa° Pc r (1 -k)

dC,_ -: - A L VS (L" ÷ r t - f') 52 d_(3-44)

Integrating over the exposed fin,

where;

Performing the integration;

C£ : k /'[ S÷rtJ:

;t

[(L • ÷ rt ) _ f] z2 d_

pc r (1 - 4) CNa 0

k = -

ArL_ VS

(3-45)

(3-46)

ks 2

CL : 12(1->0 [(1 + 34) s 2 + 4(1 + 24) sr t + 6(1 _ k) r_] (3-47)

Reintroducing k and simplifying;

pc r S CNa 0

C_ : 12 _ A¢ L r V[(1 + 34) s 2 + 4(1 + 2k) sr t +6(I +_ r2] (3-48)

The roll damping moment coefficient derivative is defined as;

\2V/IPLr oI-re °

16

(3-49)

Then,from 3-48,

crSCsa°[(1 +3k)s2 +4(1 +2>0 sr t +6(1 ÷k) r_]

6L2r Ar

(3-50)

However, CN_° must be corrected for three dimensional effects using the Diederich correlation

parameter. From the results of section 3.11, equations 3-1, 3-5, and 3-6, the subsonic roll damping

moment coefficient derivative for N fins is,

Nc Sr

C; ×---

p 6L 2r

l

(3-51)

In order to determine the roll damping moment coefficient derivative in supersonic flow, the

angle-of-attack distribution of equation 3-37 is introduced into the equations of reference 1 in

appendix A. The resulting rolling moment is divided by PL 2v to form the roll damping derivative.

To insure that C_ is truely a linear derivative, the value of %.x must be near zero. To provide_p

this condition we assume

PL(3 52)--:k

2V

where k is any defined constant; and,

k<<l

From equations 3-37 and 3-41:

max t Rll -I -

Using equation 3-52,

-_ tan -I

For the slender, finned vehicles under consideration;

2(s _ r)--:,4

L

Thus

- tan -_ (4k)mRx

(3-53)

(3-54)

17

To satisfy thecriterion,

equation3-54dictatesthat,

:: tan

k <_10-a

Using the equality, the angle- of-attack variation becomes;

.002 (y . r t)_(y) - (3-55

L

This variation is introduced into the equations of reference 1 and a value of the damping C is de-

termined. The corresponding C_ _ is found by,

For N fins;

C; C_

C/ = IO00NC_p

(3-56

3.20 BODY NORMAL FORCE COEFFICIENT DERIVATIVE

3.21 BODY NORMAL FORCE DERIVATIVE

The axisymmetric body of a slender vehicle is generally a combination of a nose shape, circular

cylinders and conical frustrums. The nose shape is usually conical or tangent ogival. See figure

3-7. It is required that there be no discontinuities at the interfaces of these components. This

allows integration over the entire length of the body. The effect of compressibility in subsonic flow

will not be considered in this analysis. From a static stability point of viewi ,this is a conservative

assumption since the normal force coefficient derivative increases near roach one.

For a slender axially symmetric body in subsonic flow, references 15 and 4 derive the steady

state running normal load as,

:' [A(x) w(x)]n(x) = _,Y__x

(3-57)

Where A(×) is the local crossectional area of the body. The rigid body downwash is a function of the

angle-of-attack is constant in ×,

18

(a) NOSE

CONE

OGIVE

(b) CYLINDRICAL SECTIONS

(

(c) CONICAL FRUSTRUMS

SHOULDER

Figure 3-7-Body Component Shapes

BOATTAIL

19

For angles-of-attack near zero

Then equation 3-57 becomes,

w:Va

n(x) = _V2 a dA(x) (3-59)dx

From the definition of normal force coefficient derivative,

Cn(x ) _ n(x) _ n(x)qA 1

_V_A

Applying equation 3-60 to 3-59,

(3-60)

Integrating over the length of the body,

2_ dA(x)Cn(x) : A dx (3-61)

Thus,

._ °dA dx 2 _ fo%=a_ d-_ __-- '°da

By definition of normal force derivative,

(3-62)

2_2

Cn : _- [A(Io) - A(O)]- (3-63)

equation 3-63 provides,

c_ : (3-64)

2

cN : _ [AO0) - AC0)](3-65)

Immediately equation 3-65 indicates that C_ is independent of the body shape as long as the independ-ent of the body shape as long as the integration of equation 3-62 is valid over the body. The

2O

bodycomponentsbeinganalysedare slenderand haveno discontinuities in their crossectionalareaor its derivatives. Thus,equation3-65 maybeappliedto them. Sincethe nosetip is assumedto bea sharppoint,

A(0). 0

Thus,thebodynormal forcecoefficientderivativein subsonicflow is;

Cu _) = 2 ABN:B _ (3-66)

Reference 27 presents a very useful method for determining the normal force coefficient

derivative of pointed bodies of revolution in supersonic flow. The second order shock expansion

theory approximates the body by a set of tangent conical frustrums. See figure 3o8. The frustrums

are each analyzed using an exponential series expansion for the pressure distribution

2 3s

(S0s +Sl ___ +S 2 s_+...)

P = - Po + C e s (3-67)

where Po, C, and S° are constants for a particular frustrum. By applying the boundary conditions

at the leading edge of the frustrum and at infinity while retaining only the first series term the

pressure distribution becomes;

where;

P = Pe - ('Pc -P2 ) e-_ (3-68)

_ = <p -_os _

-- surface coordinate

Pc = pressure on a cone having the same slope as the frustrum

= angle between surface and longitudinal axis

( )2 = indicates the value is taken at the front edge of the current frustrum.

21

TANGENT

BODY

Figure 3-8-Approximate Tangent Body

22

The notation of reference 8 is used and defined to simplify correlation between this presentation

and that reference. The value of (_P';:,s) 2 is determined by an approximate solution of the general-

ized characteristic equation in axially symmetric flow,

_P - k _ 1 _P -,k (_ sin_,sin_) (3-69)_s T_s cos _ _C 1 cos/a r

where

2)'p

sin 2_

c, = First Characteristic Coordinate

= Mach Angle

r = local radius

, - ratio of specific heats

The solution is;

3P B2 _')l - sin (3-70)

where;

B -

M

"Tp.M 2

2(M 2 - 1")

, ('21--)M_ 2{_-__21 J

)

( )] indicates the value is taken at the trailing edge of the preceeding frustrum.

( )3 indicates the value is taken at the trailing edge of the current frustrum.

The value of (_P/.s)_ is just (:_P ':_s) 3 of the previous frustrum.

- P3 _P

By applying equations 3-68, 3-70, and 3-71 to each element in succession, the pressure distribution

over the tangent body is determined. The local nondimensional normal load is given by;

23

A = n(x) 2 if" d(P/P_)- -- cos ¢d¢ (3-72)CL_ d TM_ JO da

where

4_ = Circumferential Angle.

d = Local diameter.

The derivative of the pressure distribution is evaluated by differentiating equation 3-68 with respect

to angle of attack; and then applying physical restraints to simplify the result.

d(P/P_) d(Pc/P _) X2 d(P 1 /P_)-- : (1 - e -v) + e -v

da da X1 da

Using 3-73 in 3-72 and simplifying,

(3-73)

X1(3-74)

Where "tcx" indicates the value of an equivalent cone. The running load distribution is determined

by successive application of equation 3-74 to the tangent body elements. The value of :xl, for the

initial cone is just,

Where "v" and "tcv" are the values at the initial equivalent cone vertex. The normal force coefficient

derivative is determined by the longitudinal integration of the running load distribution.

(CNT)B =_BB A r dx(3-76)

where;

AB = Body Base Area.

1 = Body Length.

The initial cone and equivalent cone characteristics are obtained from references 24 and 25; and are

presented in tables 3-1 thru 3-3.

24

000o.,.dr/l

,.....-i

iI

..Q

00p..cxl

t_0012.-.--i

0...-i

00

O0

0___

_O

0[_--

¢Q0

L_

I__t_

_

_1__0

"_O

__

_""_

__0

'_•--_

¢Q_

O_

_--_¢_

__'-

O_

__

_1_L

__

[_"

¢Q

o,g

_Z_

_"xl_

_-_

O_

_.1___-

_p-

__

_1_C

_1'_

_

__

__

_._

__

_-_o

__

__

o

_-o

__

,-__

_o_

_,_

_:o

_-,-_

LO

••

25

0

_'__

_'_

_'_

_1_

__

__

_L,_

D--

"C'4

Lt'_

_DO

0L""

O_

0_'-

_1"_'_

C

,,__o

_,_

_,_

co_

_0_

o_.

__

0c_•

_

©c_

00

0o

__

o_

,__0

_;o

o_

_,_

,__

o_._

__

_.o

_-_-

o.-,

_-_

__;

_.o

_$

__°

__

_o_

.-,o

0

,o

o•

o26

00co¢xl

00t_0o

_•

._o0

_o._

,__

_o_

__

_.,_

o._

0

©

0u-_

_¢o

co0

0_l

0co

co

o__

__

__

o_

_

•-_

0CO

t_"

__

u')

tt_

__

_

•_o

_o0

o_-

_o

__

_-_-

__-

_

0IcO

A

-_.

.__.

__

_-_

_o_

__

__

,--40

00

00

00

00

00

C_l

cO

0

_¢_1

_1_

_O

__1

Ob

_

NN

_o

oo

oo

oo

oo

oo

oo

L_

27

3.22 BODY CENTER OF PRESSURE LOCATION

As in the subsonic body normal force coefficient derivative analysis, it is required that there

be no discontinuities between the body component shapes in order to allow integration over the en-

tire body.

By definition, the pitching moment of the local aerodynamic force about the front of the body

(x = 0) is;

5('x_ : xn(x) (3-77)

The pitch moment coefficient is defined as;

thus,

Cm : --5 (3-78)A L

Since,

c _ xn(x) (3-79)_ Ar L

equation (3-79) becomes

n (3-80)% -_A

x %(x)

Cm(x) - L (3-81)

The value of the subsonic local normal force coefficient is given in equation 3-61.

equation 3-81.

2ax dA(x)

C (x) = Ar L_ dx

Integrating over the length of the body,

Using it in

(3-82)

C _ mm 10X 1

28 (dA(x_

A L \--_xdx (3-83)

28

This canbe integratedby parts;

(dA(x)_u : x dv : \-_--x /

du = dx-----------v = A(x)

dx

giving;

A L oA(Io)- A(x) dx(3-84)

But, by definition, the integral term is the body volume, therefore;

C = 23 rtoA(10) _ VB]m Ar Lr L

By definition, the pitch moment coefficient derivative is;

O_ [a:O

(3-85)

(3-86)

Then; from equation 3-85,

(Cm:)s:A-_L [10A(I0)-VBI(3-87)

The center of pressure is determined by,

Using equations 3-65 and 3-87 in 3-88 and simplifying yields,

(3-88)

V B

Re : I o - __%

(3-89)

29

The volume formulae for cones, conical frustrums and circular cylinders are well known and

do not merit repetition in this paper. However, the volume formula for a tangent ogive is not

readily available and is given herein for the readers convenience. The volume of a tangent ogive

nose shape is determined in reference 12 as,

(')dN A(L) - - 1 (3-90)

where;

L N

fN :--

In supersonic flow, the method of reference 27 determines the pitch moment coefficient derivative as,

C =---2-_ A rx dx

m A B d(3-91)

Where \(x) is found using equation 3-74. The center of pressure location of the body in supersonic

flow is then determined by the use of equations 3-76 and 3-91 in equation 3-88,

\%/ (3-92)

3.30 TAIL-BODY INTERFERENCE EFFECTS

The effect of interference flow between the tail and the body on the normal force aeTodynamics

of a slender, finned vehicle can be expressed in terms of interference factors (reference 23).

(C_,,)T< B} : (%')r Kr(s) (3-93)

(C_)T(B): (C,_,) T kT(B) (3-95)

The interference factors are determined by applying slender body theory to the flow over the total

vehicle.

3O

3.31 TAIL IN THE PRESENCE OF THE BODY

Because the fins under consideration are of low aspect ratio,the value of KT(B}from references 7 and 17.

is available

(3-96)

Reference 23 determines that the longitudinal tailcenter-of-pressure is independent of the inter-

ference flow.

Xr_B> : XT (3-97)

3.32 BODY IN THE PRESENCE OF THE TAIL

The only portion of the body affected by interference flow is the portion between and behind the

fins. Subsonically, slender body theory, reference 3, determines;

(3-98)

where KT(_) is given by equation 3-96. In supersonic flow, the interference effect on the body can

be strongly affected by the fin tip mach cone. The fin tip mach cone intersects the body when,

(1)fl _ (1 + _,) _ + 1 > 4 (3-99)

Until this condition is reached equation 3-98 is still valid in supersonic flow. Once the fin tip

mach cone impinges on the body, the function becomes quite complex and is more easily presented

in the form of curves. Figure 3-9 shows the variation of the corrected interference factor. Using

the result of this curve, K_(r_ ;

31

Cu)r_(r) : K'B(T):(1 • ') (,_ - 1) (3-100)

By the extension of the fin lifting line into the body, reference 23 derives the subsonic variation

of the center-of-pressure location of the tail interference flow over the body.

f - r 2 cosh 1 s - s , _ r /

--1

_ c • - r,l tanF I/4 (3-101)

XB(T) = IT + "4- rt s s "7

rs2)f_'--_"-_ , r t 2

Where, s > r and ;i_ _>4. In addition, slender body theory is applied subsonically to find XB(T) at

_i_= 0. Slender body v_(r) values are obtained at intermediate .;_values only for _ = 1. For other

values of _. and sweep the Y--BeT) curves are interpolated between the lifting line values and the

slender body value at 3_= 0. The shape of the interpolated curves is assumed similar to the _ = 1

curve. Figure 3-10 shows the full subsonic '_(r) variation. In supersonic flow, the center-of-

pressure of the interference flow over the body is found by applying equation 3-92;

(Cm:JB(T )-- z

_(B(T) -_B(T )

--d+l v(3-102)

Reference 23 gives the body interference pitch moment derivative as derived from slender body

theory;

x- 8 (m_)3/2 dy x cos -I

y

This is for fins with supersonic leading edges, m_ > 1. For subsonic leading edges, 02 < 1 and,

foX?(C % = _ 4,2m dy dx (3-104)ma (T) A r _,'-_ ._32m 2 - 1 v m)r_ - y

32

I

q_

#_rn T

6 QO

.,L5 ,X

2 (/94)(_+X) (_+i) >44 RADIAN ON ORDINATE SCALE

0.8X

,_. 0.6

( /©

0,4 0,8 1.2 1.6 2.0 2.4 2,8 3.2

2 Brr

(CrlT

Figure3-O-SupersonicKB(T)

3.6 4.0

33

(33

IX

,.Err23u3u)uJr'r"(3_I.J_0

LIJI--ZI.iJ£b

0.3

0.2

O.l

0

0.3

0.2

O.I

0

0.3

i

..... EXTR APOLATION

_6"f

f/

/ I'

_°

T =O

04\ o.2,,,,,

0.6 i

.I

- ---..--. --. -wF

0.4\x.

O.2\"r=O

0.6 '2

0.2

0.1

J

Q) NOLEAOING-EDGE SWEEP, X=O.

_k_T=0,0.2,04, 0.6 --

2 3 4 5 6 7 8EFFECTIVE ASPECT RATIO, _.,4R

(b) NO LEADING-EDGE SWEEP, X =_--.(c)NO LEADING-EDGE SWEEP, X=I

Figure 3-10-Subsonic XB(T)

34

0.4

0.3

0.2

0.1

0(d)

0.2 r- 0.6

h

A k\0.4

0

0.4

(e)0

,"f 16...,

,f

r =0.6

'_'----- \ \ 0.4

_, \02\0

0.3

0.2

0.1

\r =0,0.2,0.4,0.6

0 2 3 4 5 6 7 8EFFECTIVE ASPECT RATIO, /_AR

I(d)NO MIDCHORD SWEEP, X=O. (e)NO MIDCHORDSWEEP, k = -_-- (f)NO MIDCHORD SWEEP, k=l.

Figure 3-10 (Continued)-Subsonic XB(T)

35

0.6

0.5

0.4

0.3

0.2(g)

.ram.

r =0.6

_"0.4

\0.2

0

0.4

m

J,.o0.3

IX

J

_0.2

W

U_

o 0.1n-

WI---

Z

W

o 0h_

r

"S_y'--r =o6//I /

0.2

/0

/0.4

0.5

0.2

0.1

0

/r :0, 0.2,0.4,0.6

2 3 4 5 6 7 8EFFECTIVE ASPECT RATIO,/_.,_

I(glNOTRAILING-EDGE SWEEP, k=O. (h) NO TRAILING-EDGE SWEEP, X=--_. (ilNOTRAILINGEDGE SWEEP, k=l.

Figure 3-10 (Continued)-Subsonic XB(T)

36

Wherethe coordinate system has been given in figure 3-2. _(T_ is determined by using equation

3-100 along with either 3-103 or 3-104 in equation 3-102. The result is shown in figure 3-11.

3.33 CANTED TAIL IN THE PRESENCE OF THE BODY

The effect of fin cant at zero angle-of-attack is determined in a manner similar to the effect

at an angle-of-attack alone. Reference 23 determines the corresponding interference factor,

[-

1 J_2 (T _ 1) 2 ,,,_-2 . 1,)2 _2 _ 1 2:'(_ 1)

kT(B) _72 ?2 -r2 (T - 1) 2 T(_ - 1)

2

, (72 + 1)2 (sin-, T2 - I I 4(T + 1) sin-, ('r2 - 1_

8 in (T2+ I_ (3-105)* CT - I_---5 _--5_/J

No center of pressure value need be considered for the canted fin in the presence of the body

since only the roll moment derivative is considered.

3.40 TOTAL NORMAL FORCE AERODYNAMICS

The total normal force coefficient derivative is the sum of all the components.

CN_ CN_: + = (B) ; B(T)

The overall center of pressure is the solution of the moment sum of the components.

XB (CN=_ +XT(B)(CN_(B ) +XB(r)(CN_(T)

(3-106)

(3-107)

The value of total V is zero. The total cz value is determined from equation 3-95.

3.41 PITCH DAMPING MOMENT COEFFICIENT DERIVATIVE

When the vehicle has an angular pitch velocity, the superposition of the local pitch velocity at

the fins on the free stream velocity produces a local angle of attack which is constant across the

fins. See figure 3-12. The moment produced by the induced force is;

37

0.7

rn

LJ

_c

GoGo,,, 05CC

EL

I,0

rri,i

0.4Z

0.3

I

m# =0.2

(b)

0 4

|

).8 1.2 1.6 2.0 2.4

EFFECTIVE BODY-DIAMETER,ROOT-CHORDRATIO C--;-

2.8

Figure 3-11 -Supersoni c XB (T)

38

Lx CG

•

XT(B)

OVERALL PARAMETERS

F

1V

LOCALANGLE -OF-ATTACK

Figure 3-12-:Pitch Damping Parameters

39

in FC.x] (3-108)

where;

But,

and,

For V >>v,

L_× = XTCB) - XCG

' '(v)a = l:an-

v

v

(3-109)

(3-110)

the local pitch velocity is,

v : q(Ax_ (3-111)

Combining equation 3-108, 3-109, 3-110, and 3-111 and simplifying;

T(B)

(3-112)

Knowing the definition of pitch moment coefficient, equation 3-78,

c°:1%.,'_-/(v) (3-113)

The pitch damping moment coefficient derivative is defined as,

k2V/lqcr:02V

Thus, using equation 3-113 in 3-114.

(3-114)

4O

(3-115)

Equation3-123is equallyapplicableto threeandfour fin configurationssincethetail normal forcederivativeis correctedfor dihedraleffects.

3.42 ROLL DAMPINGIN THE PRESENCEOF THE BODY

Anytail bodyinterferencefactor maybeexpressedastheratio,

s÷r

fv %(_) c(f') d_, K : (3-116)

s+r

_ %(_) c(f_ df

(See figure 6) where;

:_,(O -- the angle-of-attack variation over the fin in the presence of the body

• 0(f) = the angle-of-attack variation over the fin alone.

Reference 23 gives the value of %(0 for a tail-body combination at an angle of attack, _, as;

r_ (3-117)a i (O : a + f2 %

where % is the angle of attack at the body surface. For the case of the tail-body combination at an

angle of attack,

% = _ : %(0

However, for a roiling vehicle at a zero angle of attack, equation 3-39 gives;

and

Then, equations 3-39, 3-117, and 3-118 give,

Pf%(0 : -_-

Pr%- V

=i (O = - V +

(3-118)

(3-119)

41

Therefore,theroll dampinginterferencefactor, kR_B), becomes

kR(B) :

; -_ f + c(,f)d{

_+r

jr PSimplifying;

kR(B) : 1 +

+rs

f_ c (5)d#q-

s÷r

(3-120)

(3-121)

Integrating equation 3-121 using the chord variation given by equation 3-42 yields,

kR(B) = 1 +

v - L 1 - L111

T - 1

(z + 1) (_ -X) (1 -s) (r a - 1)2 3(_ - i)

(3-122)

The roll damping moment coefficient derivative can then be expressed as,

Where kR(B) is given by equation 3-122.

(C_p)T(B) = (C,c,p) kR(B) (3-123)

4.00 AXIAL FORCE AERODYNAMICS

The only coefficient dependent on the aerodynamic forces acting parallel to the longitudinal

axis of the vehicle is the drag coefficient. Drag forces arise from two basic mechanisms, skin

friction and pressure distribution. Since both mechanisms, and their interactions, are extremely

42

complex,mostpractical theoreticaldrag computationsmustbe temperedwith statistical andempirical data. Themajority of theequationsderivedin this sectionare either theoretical gen-eralizationsof statistical-empirical dataor direct curve fits of experimentalresults.

4.10 SKINFRICTIONDRAG

The skin friction coefficient is defined as the skin friction drag of a plane surface divided by

the dynamic pressure and the wetted area of the surface.

_friction

Cf _A. (4-1)

4.11 INCOMPRESSIBLE FLOW SKIN FRICTION'COEFFICIENT

The skin friction coefficient for incompressible flow can be expressed in terms of the reynolds

number, see figure 4-1. For laminar flow, their relation has been established both experimentally

and by theoretical boundary layer considerations, reference 6.

1.328 (4-2)Cf : _/-'_

For reynolds numbers greater than about 5 × l0 s the boundary layer flow becomes transitional and

then turbulent. No good theoretical solutions have been found for turbulent flow. Figure 4-1 shows

the experimental data for turbulent skin friction. Schoenherr applied the theoretical boundary layer

work of von Karman to the experimental data and established the turbulent flow relationship;

tog (R of) : 024_____22 (4-3)

Hama formed a simpler formula which agrees with equation 4-3 within ±2%.

Cf : 1 (4-4)(3.46 log R - 5.6) 2

The form of the transitional curve may be approximated by subtracting an increment from equation

4.4. The increment has the form

k (4-5)ACr =

Based on experimental data Prandtl indicated that k = 1700. This fact, as well as other possible

transitional curves are shown on figure 4-1. Combining equations 4-4 and 4-5,

43

II

I[

I

0

o

0

oo

.//

o

"G,:.,z

LI-

.EI"7,

,i

44

_o0

O_o

ood

o

uii

rr_,

o

ii

f

_oo

6od

45

Table 4-1

Approximate Surface Roughness Heights ofPhysical Surfaces

Type of Surface Approximatemicrons R s in mils

Surfaces like that of a "mirror" .

Surface of average glass

Finished and polished surfaces

Aircraft-type sheet-metal surfaces

Optimum paint-sprayed surfaces

Planed wooden boards

Paint in aircraft-mass production

Steel plating - bare

Smooth cement surface

Surface with asphalt-type coating

Dip- galvanized metal surface

Incorrectly sprayed aircraft paint

Natural surface of cast iron

Raw wooden boards

Average concrete surface

0

0.1

0.5

2

5

15

20

50

50

100

150

200

250

500

1000

0

0.004

0.02

0.1

0.2

0.6

1

2

2

4

6

8

10

20

40

46

Cf: 1 1700(3.46 log R - 5.6)2 -_ (4-6)

forR > 5 × 105 .

The above considerations are for surface roughness sizes that are submerged in the laminar

sublayer. For appreciable rough surfaces the turbulent skin friction coefficient is shown in figure

4-2. The "smooth" curve is followed until the critical reynolds number corresponding to the rough-

ness is reached.

IR \-i 039(R•)_ = Sl (__) (4-7)

\'_r/

For reynolds numbers higher than the critical value, the skin friction coefficient can be considered

independent of reynolds number, reference 6.

Cf = 0.032 (4-8)

The flow is considered to be turbulent throughout the reynolds number range since the roughness

trips transition even at low reynolds numbers. Table 4-1 is a list from reference 6 of the approxi-

mate average roughness heights for physical surfaces.

4.12 COMPRESSIBLE SUBSONIC VARIATION

In compressible subsonic flow reference 6 indicates that the laminar skin friction coefficient

remains unchanged while the "smooth" turbulent value varies as,

c, = c, (i - .o9 M2) (4-9)c

and the change for turbulent flow with roughness is,

Cfc = Cf (1 - .12 M2) (4-10)

4.13 SUPERSONIC VARIATION

Theoretical solutions in reference 16 indicate that the supersonic variation of the Laminar skin

friction coefficient is;

Cf (4-11)Cf =

¢ (1 + .045 m2) I/4

This has been confirmed by experimental data, reference 6. Experimental data, reference 6, for

the variation of "smooth" turbulent Cfc in supersonic flow indicates a strong dependence on the wall

47

heattransfer as well as machnumber,figure 4-3. Thewall temperaturealternativesshownarethetwopossibleextremes. Theactualconditionwouldprobablybeclosestto therecoverytempera-ture. Thecurvefit of figure 4-3 is,

Cf

Cf = (4-12)c (1 + k M_)-ss

where k = .15 for no heat transfer and k = .0512 for cooled wall.

According to reference 10, the turbulent skin friction with roughness drops off according to the

relation,

Cf (4-13)Cf =

c I + .18 Ms ..

However, this equation must be used with care. It is apparent the value of c% calculated for the

flow with roughness should never be less than the corresponding smooth value.

4.14 DRAG COEFFICIENT

Once the skin friction coefficient is determined, the skin friction drag coefficient, as based on

reference area, is found by,

C% = cfc (_) (4-14)

For the fins, the wetted area of each fin is twice it's planform area.

drag coefficient is,

CD :2NCf (AT_ (4-15)ft c \%1

The body skin friction coefficient must be corrected for the fact that the body is not a flat plate,

reference 6.

A

CDfn = + Cfc Ar (4-16)

Thus, the tail skin friction

4.20 PRESSURE DRAG

4.21 TAIL PRESSURE DRAG

The pressure drag on a fin in subsonic flow arises from three basic contributions:

radius leading edge; thick trailingedge; and the overall fin thickness.

a finite

48

{,3

I,I

1.0

0.8

0.6

0.4

0.2

0

.i

,_-II

Twall=Tomb

=Trec. )-

I0 II

Figure 4-3-Supersonic Cf¢ Variation

49

A finite radius leading edge can be considered a circular cylinder in cross flow that effectively

has no base drag. Figure 4-4 from reference 6 shows the variation of the total drag coefficient of

an infinite circular cylinder perpendicular to the flow and the variation of its base drag. The dif-

ference between these two curves is the drag coefficient of the leading edge. This difference is

plotted in figure 4-5.

Subsonically up to M = .9 this curve is fitted by,

AC D -- (1 - M2) --417 - 1 (M < .9) (4-17)

Between .9 < M < 1, the curve is essentially linear.

ACD : 1 - 1.5 (M - .9) (.9 < M _<1) (4-18)

Supersonically, the curve may be approximated by a high order polynomial, reference 6.

ACD 1.214 .502 .1095 .0231: ---+-- +-- (M> 1) (4-19)M 2 M4 M6

From cross flow theory) reference 6) the drag coefficient of a cylinder inclined to the flow is,

CD_" = COS 3 F

c1

The frontal area of a typical leading edge is;

ALE = 2 LLE r L

the length can be expressed in terms of the span,

thus;

S

LLE cos F L

ALE 2 S r L

A r A r cos F L

(4-20)

(4-21)

50

From equations 4-20 and 4-21, the total leading edge drag of N fins becomes,

CDLv = 2N (_--_-rrL)cos 2 FL (ACD)(4-22)

where ACo is obtained using equations 4-17 thru 4-19.

The trailing edge drag is two dimensional base drag and can be expressed in terms of the cor-

rected skin friction coefficient, reference 5, for incompressible flow.

0.135 (4-23)

CDBT = _fB

o

I I 1 l 1 [ ] 1 1i

2.5 ' " -

2.O -- - ÷

i'_ _.' /TOTAL".._t_,a + "_.% /DRAG (38. K) -

--,_;_ ; " ",..,_;_ _ ,i._..-- _-...-; --.=..--_=j -.=,e-..t_, _e_A "_r %

I.O

0.5

0

0.1

i % /I.43/M z

- BASED._G"'_,\/

I _ I [ J I -r_-_-7=-_-O. 2 0.3 0.4 0.6 0.8 I 2 :3 4 6

MACH NUMBER

I

8 I0

CYLINDERS BETWEEN WALLS

ARC, TUNNEL TESTS

o AVA, WIRES i TUNNEL

• NACA TUNNEL---

+ NACA,FLIGHT TEST

,, BRITISH, STANTON

A DVL_SUBSONIC-----

• JUNKERS_TUNNEL

• NACA SUBSONIC---

BASE DRAG

/_ SUBSONIC CYLINDER

. SUBSONIC CYLINDER

- TRANSONIC CYLINDER

_- SUBSONIC SUPERSONIC

SPHERE (CORRECTED)

,k NAVORD, SPHERE

Figure 4-4-Cylinder Drag Variation

51

JQcJ

JLt_

IOQ

_co

mf_

--cJ

OO

JLr_

c_

ta_

W_Jf_

L_

LL

52

where;

C%=2Ce(c) (4-24)

For subsonic compressible flow, this may be corrected using the prandtl factor for swept wings,

reference 6,

1Pf = (4-25)

1 -M 2 cos 2 Fc

However, in order to maintain a finite value at

the prandtl factor is corrected to,

M2 cos 2 Fc : 1

1P; - (4-26)

K - M2 cos 2 F;

where K is a constant to be determined from the value of CDBT at M = 1. Also, for compressible flow,equation 4-24 becomes,

CfB:2Cf ( C ) (4-27)

After correcting to the reference area, the subsonic trailing edge drag coefficient for N fins is,

.135 N ('a'Bft

\At/

,/,<_ M2 ro(4-28)

Reference 11 gives a curve for the two dimensional base pressure coefficient at supersonic speeds

figure 4-6. This is essentially the base drag coefficient refered to the base area. An analytic ex-

pression which remains well within the cross hatching is,

1 - .52M -t't9

M2

But, this must be corrected for possible boattailing of the fin, reference 6.

1 - .52 M -_ .19 (4-29)

+ 18 Cf M 2

53

-0.80

-0.72

-0.64

-0.56

ZLul

-o.481.1_i1I.,iJ00

-040UJOd

u_cO

"' --0.32(]C

(3_

L_O9

m --0.24

--0.16

-0.08

0

, |

NOT E :REFERENCE NUMBERSARE THOSE OF REFERENCE II

2 5

FREE-STREAM MACH NUMBER, Moo

Figure 4-6-Two Dimensional Base Pressure

4

54

For N fins, the supersonic trailing edge drag coefficient, as corrected to the reference area is then,

In order to determine the constant I<, equations 4-28 and 4-30 are set equal at _1 = 1.

'135N QA_) N(1 - 52)(Au_/\A/

- _c , 18C ti

where the superscript * indicates sonic values. Solving for K,

[t< cos 2 223 . 4.02 C_ \l_7]j (4-31)

]+Cr C_

This is used in equation 4-28 to evaluate the subsonic compressible trailing edge drag.

This method of maintaining finite values of the prandtl factor at mach one is used throughout

the remaining derivations.

From reference 6, the drag due to thickness of two dimensional airfoil in compressible sub-

sonic flow can be expressed as,

For a three dimensional fin, based on the reference area this becomes,

(4-32)

CDTT +-:c+ (/, _M_co,_i(4-33)

55

Correctingit for a fin sweepandintroducingthe correctionconstantin theprandtl factor,

(4-34)

The value of the thickness drag of finitefins at mach one has been determined and presented in

reference 6. A curve fitof the result is, "

CDTT. = 1.1S \cr] 1.61 + _f - ¢('_ - 1.43) 2 + .578 -. (4-35)

where

Solving 4-34 for K,

CDTT A t t

K=_os_r c + - ___t5 (4-36)

Equation 4-35 is used in 4-36 to determine K. The result is used in the following equation to de-

termine the subsonic tailthickness drag. For N fins,equation 4-34 is;

30 cos 2 7¢

4N t cos r" + (4-37)

CDTT = Cf c (_-K _M2 cos2

At supersonic speeds the effect of a finite radius leading edge is found by using equation 4-19 in

equation 4-22. The trailing edge drag is calculated using equation 4-30. The thickness or wave

drag, C%r, is computed by the method of reference 1 in appendix A.

The overall drag on the tail is the sum of the different contributions. In subsonic flow

(4-38)(CDT)T=CDfT+CDLT+CDBT+CDTT

56

At supersonic speeds,

CDT)T = CDf T ÷ CDLT ÷ CDBT + CDwT (4-39)

Equation 4-38 uses the results from equations 4-15, 4-22, 4-28, and 4-37 respectively. Equation

4-39 uses equations 4-15, 4-22, 4-30, and results of appendix A respectively.

4.22 BODY PRESSURE DRAG

The body pressure drag may be broken down into two contributions, for body and base. Accord-

ing to reference 6, the forbody drag in subsonic compressible flow may be written,

C D

p

6 AWB C%: (4-40)

Again applying the correction constant to the prandtl factor,

C D

p

6 AWB C c= (4-41)

f3 A (K- M2) "6

From experimental data, reference 19, the forbody drag values of both ogives and cones are known,

figure 4-7. In the range of values that are of interest, 1 <_ fB _ 10, these curves may be fitted by

the following equations.

i:.pl 8s (4-42)D cone (fN + "T) 1"29

(C'p)D : .88 (4-43)ogive (fN + 1")2'22

Solving equation 4-41 for K;

K=I _-6 AwB C_] s/3 (4-44)

57

0d_

[I

II

II

I

0d

dd

6d

o

I'o00

Z

0o_EaD_i

I.L

z

58

Then

6.82 AwB C_ (fN + "731"29] s/3K =1+ - ! (4-45)

6.82 Awe C_ (fN + 1)2"22ts/3__ . (4-46)

The subsonic compressible forbody pressure drag is found by using equations 4-45 or 4-46 in equa-

Aws^ fs^tion 4-41. For the nose AwN̂ and fN^ should be used. For a conical shoulder and are

proper.

The supersonic value is determined by the second-order shock-expansion method, reference

27, as explained in section 3.21 of this thesis. The axial component of the pressure distribution

is integrated around each element and then over the length of the body.

4 P sin ,, d: d× (4-47)CDp : )_]2

where

P comes from equation 3-69

_, = the local angle between the surface and the longitudinal axis

_ = Circumferential Angle.

The Base drag of a body of revolution in incompressible flow is presented in reference 5 as,

(4-48)

where,

CfCfB =

Correcting this to the reference area and for subsonic compressibility.

CDB :(4-49)

59

Introducing the correction constant into the prandtl factor term.

0.29

(K - M23

From reference 11, for no fins near the body base, or for fins of zero thickness, the body base

pressure coefficient at mach one is

At machone with h /c r = 0.1,

See figure 4-8.

P_ : 0.185

P_ _ 0.30

The results of reference 11 indicate that PB is roughly a linear function of fin trailing edge

thickness at a constant mach number. Therefore,

Solving equation 4-50 for K,

c;B=E 185+115

K=I +

(4-50)

(4-51)

(4-52)

Using equation 4-51 in 4-52 and simplifying

1K : 1 + (4-53)

The subsonic compressible body base drag is computed by using equation 4-53 in 4-50.

References 11 and 6 both present curves of the base drag in supersonic flow. Figure 4-8 is

from reference 11. The free flight test curve is most representative of the variation. It is an ex-

ponential over an initial range and an inverse square over the remainder of the mach numbers. To

find the exponential, the free flight curve in figure 4-8 may be fitted in terms of the value at mach

one.

60

013(3.

LL

L.)

Wrr

(..f)qJrrCL

LLI(.'3

(13

-036

- 032

--0.28

-0.24

-0.20

--0.16

-0.12

-0.08

0

ESTIMATE B,

\ _..-"W,THF,NS

ESTIMATE A / _'_¢

WITH FINS %_%_" D_

PRESENT ESTIMATE, NO FINS, Mo_'M,:,_

----- FREE-FLIGHT TESTS

OF FINNED MODEL

REF 12, R :10 TO 12OxlO 6

WIND-TUNNEL DATA,WITH FINS:

0 REF. 26

0 REE 26

0 LANGLEY 9"SST

FLAGGED SYMBOLS, NO FINS

- 0.04

0

1.0

INOTE: REFERENCE NUMBERS ARE THOSE OF REFERENCE II.

1 I I I I1.2 1.4 1.6 1,8 2.0 2,2 2.4 2.6

FREE-STREAM MACH NUMBER, M

Figure 4-8-Three Dimensional Base Pressure

2.8 3.0 3.2 3.4

61

CBB :: CDB I0.88 + .12e "J'58(M-11 ] (I _M_Mr,.) (4-54)

where M is the point where the two variations cross.

variation as,

Also, reference 5 gives the inverse square

(M > Mcr3 (4-55)CDB M2

Ifitis assumed the ratio of the value of CoB at .M=M_rto C" remains constant for all curves, (inspec-

tion of the curves in reference 6 indicates this is a good assumption) then the value of the critical

roach number can be found,

M 0.892 (4-56)z __

The value of C* is given by equation 4-51. In either subsonic or supersonic flow,DB

COt) + C (4-57)B = CDP DBB

For subsonic values, equation 4-41 and 4-50 are used. In supersonic flow equations 4-47 and 4-54

or 55 are used.

4.23 TOTAL DRAG

The drag of the totalvehicle is the sum of the tailand body contributions.

:(%1,+

No interference drag effects are considered.

(4-58)

5.00 COMPARISON WITH EXPERIMENTAL DATA

In an attempt to obtain as broad a range as possible without being voluminous, each characteris-

tic is compaired to the data from a different configuration. In some cases, the choice of configuration

was dictated by the availabilityof the appropriate data.

5.10 NORMAL FORCE COEFFICIENT DERIVATIVE AND CENTER OF PRESSURE

The windtunnel data of the Aerobee 350 sounding rocket, figure 5-1, are given in reference 14.

Figures 5-2 and 5-3 show the results of the present method as well as data points from

62

I

"Tu'3+_

u-

63

C Na[

33_

52-

30

26

22--

20--

14-

12 --

IO--

f

33.4

X

A = 381 in 2r

×

X

I I I I2 3 4 5

Mm

Figure 5-2-Aerobee 350 Normal Force Derivative

THEORETICAL

WIND TUNNEL

x

X

i i I6 7 8

64

410 --

Xcp

4OO

390

38O

370

360

350

340

330 --

320 --I"

0

x

A r = 381 in 2

L r = 22 in.

THEORETICAL

x WIND TUNNEL

I I I I2 3 4 5

Moo

Figure 5-3-Aerobee 350 Center of Pressure

I i I6 7 8

65

reference 14. The maximum derivation of the theoretical CN, values is 7.83_ and the maximum

deviation of the theoretical center of pressure values is 1.18%. The degree of accuracy of the

windtunnel tests is indicated as about 6%.

5.20 ROLL FORCING MOMENT COEFFICIENT DERIVATIVE

The only roll forcing moment coefficient derivative data over a broad range of mach numbers that are

available to the author are those of the single stage tomahawk sounding rocket, figure 5- 4. The data for this

vehicle are unpublished at this time, but have been provided by the test coordinator, reference 28.

Figure 5-5 shows the comparison between the theoretical result and windtunnel data. No data are

available in the subsonic region. The average deviation of the theoretical C. is 7.54%. The

accuracy of the windtunnel data is about 5%. It is apparent, both from the windtunnel data and the

subsonic theory that the theoretical value at M = 1.5 is no good.

5°30 ROLL DAMPING .DERIVATIVE

Reference 21 presents a full set of data for the basic finner test rocket, figure 5-6. It is the

only config-uration for which roll damping moment coefficient derivative data are available to the

authors. Unfortunately, the range of the data is relatively small. Figure 5-7 shows the comparison

between the theoretical C: is 5.68%. Reference 21 does not indicate the degree of accuracy of its.p

data. No subsonic data can be found for Ca •-p

5.40 DRAG COEFFICIENT

Aerobee 150A, figure 5-8, drag coefficient data are available from reference 29. The theoretical

drag coefficient results are compared to the windtunnel data in fig-ure 5-9. In order to duplicate

the test conditions, the test reynolds numbers were used to set up a Re vs. M curve, figure 5-10.

This curve was used in generating the theoretical data. The surface of the windtunnel model is

highly finished. From table 4-1, the appropriate surface roughness is about .01 mils. The equiva-

lent full scale surface roughness is .1 mils. The maximum deviation of the theoretical drag co-

efficient is 8.52%. Again no subsonic data are available.

5.50 PITCH DAMPING DERIVATIVE

There are no C experimental data available. The data taken in windtunnel experiments are amq

combination of the pitch damping derivative and a time log derivative due to the rate of change of

angle-of-attack. There is no way to separate the two effects. Therefore, no comparison of the

pitch damping derivative can be attempted.

66

_gooZLdt'--XhiC

:3£0

m-

wF-

xt.l,J

E3

Fi,

I_

i

--(

o_

_10'_

a0

dd

--.d

%

<_lU

-I

I--0z

o3c)n"LL

J.J

I

£0

oI-'-I

7,&

6'7

50 m

cz 8

45

40

55

25-

20-

15-

I0-

5 I0 I

30

A r -- 64 7 in 2

Lr= 9 in.

THEORETICAL

x WIND TUNNEL

x

x x

x

I I I I I I I2 .5 4 5 6 7 8

M

Figure S-S-Tomahawk Roll Forcing Derivative

68

oJt

r

°1llo

iI

c_0

I

(._Zw-_c/)

c_zL

L_"

Ow

o_c_Q

OW

!

uZ.uI_o

69

-45 --

czp

-40

-55

-30

-25

-20

-15

THEORETICAL

X WIND TUNNEL

-I0 I I0 0.5 1.0

A r = 786 in 2

L r = I0 in.

X

i I i I1.5

M

2.0 2.5 5.0

Figure 5-7-Basic Finner Roll Damping Derivative

7O

0

z0(D

w

i

(;9

0n-OC]

D0r_32or)

WOr)0

!o

_-

+

0w

--

0T

mt,-c_(M

±

tt<¢_I-w

tozu_wt--o90

0

-tr0if-,

_._jI

if)ZU

_

rri11Zo')

9iitl

U-

71

C D

0.6

0.5

0.4

0

A r = 225 in 2

I I I I2 :5 4 5

M

Figure S-9-Aerobee 150A Drag Coefficient

THEORETICAL

WIND TUNNEL

X

I I I6 7 8

72

i

0

X

n.-

9

8

7

4

00

x

X-VALUE USED IN

WIND TUNNEL TESTS

• (REFERENCE 29)

x

x

x

\

THE SHAPE OF THE CURVE

IS FAIRED BY COMPARISON

WITH NOMINAL FLIGHT

REYNOLDS NUMBERVARIATION.

X

I I I I

3 4 5 6

M

X

7 8

Figure 5-10-Aerobee 150A Reynolds Number

73

6.00 CONCLUSIONS

On the basis of the comparisons in section 5, the method presented in this dissertation pre-

sented in this dissertation predicts the values of CN_ , x, C_, ¢t , and c__ to well within ten (10)P

percent of experimental results between mach 2 and 8. Although no comparison can be made for

Cmq , the results of section 5-1, coupled with the dependence of Cmq on (_)T<B_ and X_<B), tend to

indicate that the theoretical C is also accurate to about 10%.mq

The lack of subsonic experimental data prevents any concrete conclusions to be drawn about the

accuracy of the present method at subsonic spbeds. However, the figures in section 5 show that the

subsonic variations of all the compared characteristics are quite reasonably valued with respect to

the supersonic values.

In general, then, the method presented in this dissertation can be used to confidently predict

the aerodynamic characteristics oI slender finned vehicles.

74

7.00 REFERENCES

1. Barrowman, J. S., FIN a Computer Program for Calculating the Aerodynamic Characteristics

of Fins at Supersonic Speeds. NASA-GSFC X-721-66-162, 1966.

2. Diederich, F. W., A Plan Form Parameter for Correlating Certain Aerodynamic Characteris-

tics o[ Swept Wings. NACATN-2335, 1951.

3. Ferrari, C., Interference Between Wing and Body at Supersonic Speeds-Theory and Numerical

Application. J. A. S., p. 317, June 1948.

4. Goldstein, S., Lectures on Fluid Mechanics. Interscience Publishers. New York, 1960.

5. Horner, S. F., Base Drag and Thick Trailing Edges. J. A. S., p. 622, October, 1950.

6. Horner, S. F., Fluid-Dynamic Drag. Published by the Author. Midland Park, N. J., 1958.

7. Jones, R. T., Properties of Low-Aspect-Ratio Pointed Wings at Speeds Below and Above the

Speed of Sound. NACA Rep. 835, 1946.

8. Jones, R. T., Theory of Wing-Body Drag at Supersonic Speeds. NACAReport 1284, 1956.

9. Jones, R. T., and Cohen, D., High Speed Wing Theory, Princeton Univ. Press, Princeton, 1960.

10. Liepmann, H. W., Goddard, F. E., Note o,1 the Mach Number Effects Upon the Skin Friction of

Rough Surfaces. J.A.S., p. 784, October 1957.

11. Love, E. S., Base Pressure at Supersonic Speeds on Two-Dimensional Airfoils and on Bodies

of Revolution With and Without Fins Having Turbulent Boundary Layers. NACA TN-3819, 1957.

12. Mayo, E. E., Cone Cylinder and Ogive Cylinder Geometric and Mass Characteristics. Memo

to Code 721.2 Files, NASA-GSFC, 1965.

13. Mayo, E. E., Equations for Determining the Effects of Fin Dihedral on Vehicle Stability, Memo

to Code 721.2 Files, NASA-GSFC, 1967.

14. McNerney, J. D., Aerobee 350 Windtunnel Test Analysis, Space General Corporation, E1Monte,

California, 1963.

15. Miles, J. W., Unsteady Supersonic Flow. Section 12.4 A.R.D.C., Baltimore, 1955.

16. Moore, L. L., Yound, G. B. W., Jansen E., Compressible Boundary Layer-Laminar Solution,

J. A. S., p. 229, April, 1952.

17. Morikawa, G., Supersonic Wing-Body Lift. J. A. S., Vol. 18, No. 4, April, 1951, pp. 217-228.

18. Morikawa, G. K., The Wing-Body Problem for Linearized Supersonic Flow. PhD Thesis,

California Inst. of Tech., 1949.

75

19. Morris, D. N., A Summaryof the SupersonicPressureDragof Bodiesof Revolution. J. A. S.,p. 622,July 1961.

20. Murk, M. M., TheAerodynamicForcesonAirship Hulls. NASAReport184,1924.

21. Nicolaides,J. D., Missile Flight andAstrodyuamics,BuWepsTN-100A,Washington,D. C., 1959.

22. Nielsen,J. N., andPitts, W.C., Wing-BodyInterferenceat SupersonicSpeedswith anApplica-tion to Combinationswith RectangularWings. NACATN-2677,1952.

23. Pitts, W. C., Nielsen,J. N., andKaattari; G. E., Lift andCenterof Pressureof Wing-Body-Tail Combinationsat Subsonic,Transonic,andSupersonicSpeeds.NACAReport1307,1959.

24. Sims,J. L., Tablesfor SupersonicFlowAroundRight Circular Conesat SmallAngles-of-Attack.NASASP-3007,1964.

25. Sims,J. L., Tablesfor SupersonicFlowAroundRight Circular Conesat ZeroAngle-of-Attack,NASASP-3004,1964.

26. Shapiro,A. H., TheDynamicsandThermodynamicsof CompressibleFluid Flow,Vol. II,RonaldPress,NewYork, 1954.

27. Syverston,C. A., andDennis,D. H., A Second-OrderShock-ExpansionMethodApplicabletoBodiesof RevolutionNearZero Lift, NACAReport1328,1957.

28. Thomas,C. G., UnpublishedWindtunnelDataona .25ScaleTomahawkSoundingRocketTakenat LRC UnitaryandARDCTunnelB, October1966.

29. Thomas,E. S.,WindtunnelTestsof the Aerobee150A. Aerojet-GeneralCorporation,Azusa,California, April 1960.

30. Tsien, H. S.,SupersonicFlowOveran InclinedBodyof Revolution,JAS,p. 480,Vol. 5, 1938.

76

X-712-66-162

APPENDIXA

FIN

A COMPUTERPROGRAMFORCALCULATINGTHE AERODYNAMICCHARACTERISTICSOF FINSAT SUPERSONICSPEEDS

(Exerptof TheoreticalTreatment)

JamesS.BarrowmanSpacecraftIntegrationandSoundingRocketDivision

April 1966

Goddard Space Flight Center

Greenbelt, Maryland

77

LIST OF SYMBOLS

CDW

C L

CL_

C t

C

Ca

C

CP i

C r

d1

d w

F a

F l

K

L

1L

1L r

1 r

1T r

i w

MP

Reference area

Length of airfoil strip chord

Wave drag coefficient : "'_LqA

Lift coefficient FlqA

dC_/a_ (_ = O)

Rolling moment coefficient -Mr

q AL

ac,/_I _ ( 3 = o)

Pitching moment coefficient - Mpq AL

Local pressure coefficient

Length of root chord

Component of ii parallel to the freestream

Portion of (t behind the fin-tip Mach conet

Drag force (parallel to freestream)

Lift force (normal to freestream)

Iv.terference factor

Reference length

Length of airfoil-strip leading-edge region chord

Length of root airfoil leading-edge region chord ,

Length of center region and of center-region chord

Length of airfoil-strip trailing-edge region chord

Length of root airfoil trailing-edge region chord

Distance between the leading edge and the intersection of the Mach cone with the airfoil

strip

Length of local region surface

Freestream Mach number

Pitching moment about the reference axis

78

M r

n

Pt

P_

ct_

I"t

s

x L

XP i

V

Y

_'_y

j)

FL

rT

Pt' P2

S

_L

_T

SUBSCRIPTS

B

B(T)

i

J

LIST OF SYMBOLS (Continued)

Rolling moment about the root chord

Number of strips

Component of 1 normal to the freestream

Local static pressure

Ambient static pressure

Freestream dynamic pressure

dw/d _ ratio

Semispan length

Distance from the reference axis to the airfoil-strip leading edge in a streamwise direction

Center-of-pressure coordinate measured from the reference axis

Distance from the reference axis to the forward region boundary of a local region in astreamwise direction

Center-of-pressure coordinate measured from the root chord

Distance from the root chord in a spanwise direction

Width of airfoil strip

Angle of attack (degree)

. _12 - 1

Leading-edge sweep angle

Trailing-edge sweep angle

Ragion boundaries sweep angles

Ratio of specific heats for air = 1.4

Fin cant angle

Leading-edge wedge half-angle in the streamwise direction

Trailing-edge wedge half-angle in the streamwise direction

Inclination of a local surface to the freestream

Mach cone semivertex angle

Body alone

Body in the presence of the tail

Local region number

Airfoil strip number

79

SUBSCRIPTS

o Value at _ = 0

T Tail alone

LIST OF SYMBOLS (Continued)

Tail in the presence of the body

80

FIN

A COMPUTERPROGRAMFORCALCULATINGTHE AERODYNAMICCHARACTERISTICSOF FINSAT SUPERSONICSPEEDS

BACKGROUND

Calculating the aerodynamic characteristics of the fins of a sounding rocket is one of the basic

steps in calculating the overall aerodynamics of a sounding rocket. Because sounding rockets fly

at supersonic speeds for the greater portion oi_ their flight, the regime at these speeds is of particu-

lar interest.

Methods previously used for calculating supersonic fin aerodynamics (references 1, 2, 3, and 4)

are lacking in accuracy and applicability because of the many assumptions and approximations in-

herent in using them to reach a closed form or nearly closed form solution.

By using a high-speed computer to numerically solve basic theoretical equations, one may ob-

tainanswers rapidly and as accurately as desired. The only restrictions to accuracy are the

assumptions inherent in deriving the basic equations. The theory found to be most amenable to pro-

gramming in Busemann's Second Order Supersonic Airfoil Theory as described in reference 1.

PROGRAM CAPABILITIES AND LIMITATIONS

Given a supersonic fin with chord plane symmetry, at a given Mach number, FIN computes as

functions of angle of attack ( ) the following:

• Lift coefficient, CL

• Wave drag coefficient, CD_

• Pitching moment coefficient, cm

• Center-of-pressure coordinate measured from the reference axis,

• Center-of-pressure coordinate measured from the root chord,

Since , as defined in the program is equivalent to the fin cant angle (:) at _ = 0, the rolling

moment coefficient C, is considereda function of >.

FIN also computes as functions of Mach number the following:

• Lift coefficient slope, CL

• Wave drag at zero angle of attack, CD%

• Pitching moment coefficient slope, Cm

• Rolling moment coefficient slope, c,

81

• Center-of-pressure coordinate measured from the rgference axis at zero angle of attack, x0

• Center-of-pressure coordinate measured from the root chord at zero angle of attack, Y0

The range of -. is from zero to _ma_' as determined by the user, in increments of 1 degree. The

program allows for a maximum of 100 Mach number points, the user specifying the range and

increment of the Mach number.

Since FIN has been designed for use with a four-finned vehicle, the output values of CL, CDw,

Cm, CL_, CDwo, and C_ are for apair of identical fins; the output values of C_ and C_ are for two

pairs of perpendicular fins. The values of _,'_, x0, and T 0 apply only to a single typical fin.

Busemann's Second Order Airfoil Theory is applied subject to the restrictions outlined in

reference 1, pages 192 to 241, with special attention to Figures 10.3 and 10.28. However, the use

of the third-order terms in this program slightly enlarges the applicability shown in this reference.

Busemann's theory is applied to small streamwise airfoil strips, and the strips are summed in a

spanwise direction. In addition, the fin-tip Mach cone is accounted for by applying a correction

factor to the necessary portion of each strip. This correction-factor technique was obtained from

references 1 and 3. The fin-tip Mach cone correction may or may not be included at the user's

discretion.

Figx_re 1 shows the types of hns to which FIN is applicable. These configurations cover all

of the shapes normally used on sounding rockets and missiles. The program input pattern for each

type is given in the section on usage. A listing of FIN in FORTRAN IV is given in Appendix A.

AERODYNAMIC THEORY

Busemann's Second Order Airfoil Theory has been applied to two-dimensional airfoil strips

with the following assumptions:

• All parts of the airfoil surface are in supersonic flow and make small angles with the flow.

This implies low angles of attack.

The leading edge is sharp. The trailing edge must be sharp for a double wedge fin or a

modified double wedge. For a single wedge and a modified single wedge, the effect of the

blunt base is neglected.

• The shock waves are all attached.

• Each region of flow over the surface acts independently of the others.

The basic local pressure coefficient equation of the Busemann theory is a third-order series

expansion:

82

(a] Modified Double Wedge (b) Double Wedge

-_--_- t--

(c)Modified Single Wedge

----------'---------1

(d) Single Wedge

Figure 1-Fin Configurations

83

where

K I

2

(2a)

K 24 _4

(2b)

K_

I)M _ (2 2 _ 7) -5)M 6 _ 10(, * I)M 4 - 12M 2 * 8

6"77

(2c)

If the flow-over the surface region inclined at v_ to the flow has been preceded by a single

compressive shock wave anywhere upstream in the flow, the value of K" is:

K* (3), , t)M* [(s - 3,)M_ + 4(_ - 3)M_ , s]48

If the flow upstream of the region in question has contained only expansive shock-free flow,

K" = 0. % is the inclination of the leading edge surface. Essentially, the first three terms of

equation 1 represent the first-, second-, and third-order pressures on the surface region respec-

tively. The K" term is the third-order irreversible pressure rise through the bow shock. The

index, i, indicates the surface for which the pressure coefficient is being calculated.

The pressure coefficient of equation 1 is applied to a typical airfoil region, including the fin-tip

Mach cone, by dividing the region into two portions at the intersection of the airfoil strip and the

Mach cone (Figure 2). A correction factor of 1/2 is applied to the local pressure coefficient for the

region within the Maeh cone. The center of pressure of each portion is taken as the midpoint of

the length of that portion. By this means, equations are obtained for the local lift (F_), drag (Fd_),

local center of pressure (_,), and, hence, the local pitching moment (Mp_):

F| : C v d 1 - (4)

Fd, = Cp n I - (5)

½d i 1 - r i + --7 +_'- - ri

E, : (6)r_

2

%, : r, _ (7)l I

84

STRIP _X IIw

\REFERENCE

AXIS

Xp i

MACH

\CONE

Iw \ :- I

_, \

Y

REGION

BOUNDARIES

AIRFOIL

STRIP

d

Figure 2-Mach Cone Correction Geometry

85

The airfoil-strip lift (Fij) , drag (Fd), and pitching moment (Mv ) totals are then computed by summingJ J

the local characteristics.

6

F l : _ F l (8)

i=1

F d • F dJ

5

V _

l=l

(9)

6

ZMpi Mp

t=l

(I0)

The strip rolling moment about the root chord is calculated from the airfoil lift and strip span-

wise position, y:

: (ii)Mrj YFIj

The total characteristics over the entire fin are then computed by summing the airfoil strip

characteristics in a spanwise direction.

(12)

(13)

Mp = _ M% (14)}:1

(15)

The forces and moments computed in equations 4 through 15 are actually divided by the ambient

dynamic pressure (q_) and strip width (Ay). Thus, the associated coefficients are:

86

2F, ":_.y

CL : A (2 fins)(16)

2 F d • E y

C_w = A (2 fins)(17)

2 M • _, yp

Ca : AL (2 fins)

4M r • _y

CI : AL (4 fins)

(18)

(19)

The center-of-pressure coordinates are calculated from the respective moments and the

appropriate lift forces.

M

: o (20)F, cos

M

_( : _ (21)F t

To obtain the linear slope coefficients near _ = 0 at a given Mach number, the values of CL,

C, and Cz are calculated at a = 1° and are taken as the slope values ct_ , c, and C_ (all per de-

gree). The slope values per radian are obtained by the proper conversion of radians to degrees.

Thus:

CL-_ 180 (22)rad. _r CL _ = 1

% , ) (23>tad. _ a = I

Cl 8 1_0rad. - --(cJi_ = ,) (24)

The zero angle-of-attack values of CDw, )(, and _ are simply their respective values at a = 0.

CONFIGURATION AND GEOMETRY CONSIDERATIONS

The user indicates the shape of the fin configuration by specifying the root chord length (Cr ),

root airfoil leading and trailing edge region chord lengths (1L, and IT,), leading and trailing edge

87

sweep angles (F u and V t ), region boundary sweep angles (T 1 and V2 ), and the fin semispan length

(S) (Figure 3). The program then computes the distance from the reference axis to the airfoil-strip

leading edge in the streamwise direction (X L), airfoil chord length (C), and the airfoil region chord

lengths (l L and I t) as functions of the spanwise coordinate of the airfoil strip.

X L : y tan[ c cos ._ (25)

C y (tan[ T tan L) " Cr (26)

1L : y(tan;' 1 - tanF'L) + 1L (27)

1T : y(ta. m- ta T )+ (28)

The width of the airfoil strip is determined from the number of strips (n) desired by the user.

SZy : -- (29)

n

The airfoil strip crossection is assumed to be symmetrical about the chord line. The most

general crossection configuration is the modified double wedge (Figure la). The other three types

(Figures lb, lc, and ld) are special cases of this basic shape. Figure 4 shows in detail the general

crossection, which has six separate flow regions. As can be seen in this figure, the surface region

lengths are determined by specifying the leading- and trailing-edge wedge half angles in the stream-

wise direction, _L and _T"

1 1 : I 4 =--

lL

vcos L (30)

1T (31)13 [6

COsr

T

12 : I s : C - l L - 1T (32)

The program computes the local-surface inclination angles (_i) as functions of _:

l _ L (33a)

_;2 : - , (33b)

(33c)T)3 : - *'T - ':

74 : L + "_ (33d)

88

S

1_ -_ /'_ 12

14

SECTION A-A TYPICAL AIRFOIL

Figure 3-Fin and Airfoil Geometry

89

L --i _

'L "4

/J

c

IM

Jj _'2

mf

J

J

Figure 4-Airfoil Angles

!

go

"_s : ' (38e)

% -' +o (38f)'T

From these, the streamwise and normal components of the region lengths are calculated.

as:

d = 1 cos _, (i I - 6)

(34)n : 1 "sin ":i _i :: 1 " 6)

The distance of the forward local-region boundaries from the reference axis are then determined

xp, ×p_ x L (35a)

xp2 :: x L _ {t, (35b)

Xp_ : x L + d z + d 2 (35c)

Xps XL _ d_ (35d)

×P6 : XL _ (t4 * ds (35e)

FIN-TIP MACH CONE CORRECTION

The Mach-cone angle (_,) is a direct function of the Mach number.

a, : Arctan (1) Arcsin (1) (36)

For an airfoil strip at a spanwise direction y, the intersection of the strip and the Mach cone is a

distance I w from the leading edge.

1 w : (S - y)tan_ L ÷ tan(90 --) (37)

But, by trigonometric identities and the use of equation 37, this may be reduced to:

1 w : (S - y) (tan, ,. + ) (38)

If I w is greater than or equal to the chord length, there is, of course, no actual intersection and no

correction is necessary. If i w is such that

C " 1w Z C- l T

91

thentheratio of theportionof Ir falling behindtheMachconeto Ir is:

c- 1W

r 3 : r 6 -- l T

AIso,

r I : r 2 : 1- 4 : r 5 0

If 1, is such that

C- lv _ 1w _ I L

then the corresponding ratio for the middle regions is

(39)

(40)

Also,

and,

If 1w is such that

C- 1T 1,

r2 : rs |M

r 3 r 5 ],

C- lT

r I r 4 0

1 1 " 0

(4t)

(42a)

(42b)

then the ratio for the leading edge region is

r I r 4

I L - 1 w

1L

(43)

Also,

r2 r 3 r s r 5 1.0(44)

These ratios are the values used in equations 4, 5, and 6 for calculating the local lift, drag, and

center of pressure. In these equations, the pressure coefficient in the portion behind the Mach cone

is taken as half the value of the C calculated by equation 1.P

92

REFERENCES

1. Hilton,W. F., HighSpeedAerodynamics.Longmans,GreenandCo. NewYork, 1951.

2. Pitts, W.C., Nielsen,J. N. andKaattari, G.E., Lift andCenterof Pressureof Wind-Body-TailCombinationsat Subsonic,Transonic,andSupersonicSpeeds.NACAReport1307,1959.

3. Shapiro,A. H.,The DynamicsandThermodynamicsof CompressibleFluid Flow. Vol. H.RonaldPress. NewYork, 1954.

4. Harmon,S.M. andJeffreys,I., TheoreticalLift andDampingin Roll of ThinWingswithArbi-trary Sweep and Taper at Supersonic Speeds Supersonic Leading and Trailing Edges. NACA

TN2114. May 1950.

5. Thomas, E. S., Windtunnel Tests of the Aerobee 150A. Aerojet General Corp., 1960.

6. McNerney, J. D., Aerobee 350 Windtunnel Test Analysis. Space General Corp. January_ 1963.

93

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