AFCRL-71-0093

VORTEX INDUCED ROLLING 24OMENTS ON CRUCIFORH MISSILE3S ATHIGH ANGLE OF ATTACK

by

Steven I. Kane

Engineering LaboratoriesBoston University

110 Cummington StreetBoston, Massachusetts 02215

Contract No. F19628-68-C-0148Project o. 7659 D D CTask No. 765904 fWork Unit No. 76590401 "

JUN 8 t7

SCIENTIFIC REPORT NO. 3 CMarch 1971

This document has been approved for public releaseand sale; its distribution is unlimited.

Contract Monitor: Edward S. MansfieldAerospace Instrumentation Laboratory

Preparedfor

AIR FORCE CAMBRIDGE RESEARCH LABORATORIESAIR FORCE SYSTEMS COMMANDUNITED STATES AIR FORCE

BEDFORD, MASSACHUSETTS 01730

Roproducod byNATIONAL TECHNICALINFORMATION SERVICE

Spi.nglrold, Ia 22151

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Qualified requestors may obtain additional copies frc-:, the

Defense Documentation Center. All others should apply to

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UnclassifiedSecu=rity Cl-osification

DOCUMENT CCNTROL DATA - R&D.9ecuriz) classification of tdc. bad oal-strac, and ndevo, ,nn,..'..,n mut ie rnu, ,,-d uhrn th m:-rtall .ei,,,t ..,

I. ORIGINATING ACTIVITY trdqsrafe tltho') 2,. REPORT SECURI IY CLAS5IFICATIONBoston University UnclagsifiedEngineering Laboratories /2, GROUPBoston: Massachusetts 02215

3. REPORT T5TLE

VORTEX INDUCED ROLLING MOMENTS 9N CRUCIFORM MISSILES AT HIGH ANGLEOF ATTACK

4. DESCRIPTIVE NOT ES (T pe'z..,or: ernd znclu r : t

Scientific InterimS. AUTHOR(S, (Hir.sz n,=e, .odih mi:irJ. last ranei

Steven I. Kane

6REPORT- DATE I 7L TOT A, NO. OF PAGES 71NoOFR S

March 1971 I s61 6O.O'° REVs60a. CONTRACT OR GRANT NO. ge- OPIGINATOR'S REPORT NUMBERS)

F19628-68-C-0148b. PROJECT. TASK. ,OPK UNIT NOS.

7659-04-01 Scientific Report No. 3C. DOD ELEMENT 9b. OTHER RIPORT ND(S) kly other nunm .rs that ny be62101F asvee t.,s ,ep,,tj "

d. O0 SUBELEWENT681000 AFCRL-71-0093

10. "3ISTRIBUTION STATEMENT

1-This document has been approved for Dublic release and sale; itsdistribution is unlimited.

11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY

Air Force Cambridge ResearchTECH, OTHER Laboratories (LC)

L.G. Hanscom FieldBedford, Massachusetts 01730

13. ABSTRACT

The problem of predicting the induced roll phenomena associated withfinned missiles at high angles of attack has been approached byexamining the interaction between the body wake vortex system andthe finned surfaces of the missile. The application of the classicBlasius moment integral to this problem permits the independentevaluation of each singularity in the flow field which may contributeto the moments. This method is applicable to missiles emoloyinq finschemes of any geometry and is used herein to analyze the inducedroll of a typical cruciform missile configuration.

A computer program has been developed to generate numerical solutionsutilizing those parameters which affect the magnitude and directionof the induced roll.

FORM 1473

DD OV 65 1Unclassified

Security Classificatioll

Unclassitied--

Secrty Classi i cation| . ~~LINK A NK | IK

KEY WORDS A.InK a LINK L

ROLE WT HOLE WT ROLE %

Blasius Integral

Cruciform Fin

Finned Missiles

Rolling Moment

Vortex

Unclassified

Security Classification

AFCRL-71-0093

VORTEX INDUCED ROLLING M4OMENTS ON CRUCIFORM MISSILES ATHIGH ANGLE OF ATTACK

by

Steven I. Kane

Engineering LaboratoriesBoston University

110 Cummington StreetBoston, Massachusetts 02215

Contract No. F19628-68-C-0148Project No. 7659Task No. 765904Work Unit No. 76590401

SCIENTIFIC REPORT NO. 3March 1971

ThiE document has been approved for public releaseand sale; its distribution is unlimited.

Contract Monitor: Edward S. MansfieldAerospace Instrumentation Laboratory

Preparedfor

AIR FORCE CAIBRIDGE RESEARCH LABORATORIESAIR FORCE SYSTEMS COMMANDUNITED STATES AIR FORCE

BEDFORD, MASSACHUSETTS 01730

iii

ABSTRACT

The problem of predicting the induced roll phe-

nomena associated with finned missiles at high angles of

attack has been approached by examining the interacticn be-

tween the body wake vortex system and the finned suirfaces

of the missile. The application of the classic Blasius

Moment Integral to this problem permits the independent

evaluation of each singularity in the flow field which may

contribute to the moments. This method is applicable to

missiles employing fin schemes of any geometry and is used

herein to analyze the induced roll of a typical cruciform

missile configuration.

A computer program has been developed to generate

numerical solutions utilizing those parameters which affect

the magnitude and direction of the induced roll.

*

iv

TABLE OF CONTENTS

List of Symbols ........................................ v

Section 1: Introduction ............................... 1

Section II: General Formulation of the Problem ........ 3

Section III: Contour Integral Evaluation .............. 10

Section IV: Numerical Analysis and Digital Computer

Results .................................. 29

Section V: Conclusions ................................ 45

References ........................................ 48

Appendix A ............................................. 49

Appendix B ........................................... . 51

V

LIST OF SYMBOLS

a (c 4 _R 4 ) 1/4

b Fin semi-span (see Fig. 1)

c 2 = radius of circle in theV plane (see Fig. 3)

M,Rolling moment coefficient (l/ 2 b20% 1/2 L

Cr Vortex strength coefficient r= 2T- )

zdZ/dt

) See Appendix A

f-i See Equ. (32)

h(r) See Eau. (27)

Im Blasius integral around pole of order m at(see Equ. (8))

I Blasius integral around branch cut betweenI'II and "SII

I VII,VI Blasius integral around branch cut betweenl VIIand : VI

LUnit pure imaginary number

vi

List of Symbols (continued)

J "27R12 (see Equ. (18))

K (see Ecu. (19))

L Chord length of fin

M Cross flow Mach number

MI Rolling moment/unit length of fin0

Mo Rolling moment

m Order of pole

q Dynamic pressure V!

R Radius of cylindrical body (see Figure 1)

r Polar coordinate (see Figs. 5 and 6)

U Cross flow veiocity=V.Sljno<

u Velocity in x direction or real component of com-plex velocity

V0 Free stream velocity

v Velocity in y direction or imaginary component ofcomplex velocity

7V Complex velocity potential

vii

List of Symbols (continued)

w Complex velocity=u- iv

x Fin axis in tail body plane (see Figs. 1 and 2)

y Axis perpendicular to x axis in tail body plane(see Figs. 1 and 2)

Z Complex plane (see Fig. 2)

CK Angle of attack of missile

1Residue (see Equ. (7))

.r Vortex strength

Arbitrary circulation around body

Radius of contour about tranch point (seeFigs. 5 and 6)

Complex plane of the circle (see Fig. 3)

Position of a pole in the i plane

Position of stagnation point on the circle in theto" ' plane (see Appendix B)

,K, !_-Branch points in the plane

r() See Equ. (26)

(2.

viii

List of Synbols (continued)

Polar coordinate (see Figs. 5 and 6)

01 Argument of (see Appendix B)

See Equ. (42)

See Equ. (45)

Fluid density

See Equ. (41)

Roll angle (see Fig. 1)

Stream function (imaginary part of W)

1

SECTION I. Introduction

Rockets and missiles employing fins of cruciform

arrangement have been found to develop anomalous aerodynam-

ic rolling moments which are functions of, among other

things, the angle of attack, the bank orientation of the

wings and Mach number. These rolling moments tend to up-

set the stability by rolling the missile, and unless they

are counteracted by roll-control measures, the actual flight

path control accuracy is apt to deteriora-c at a crucial

time. It is generally accepted that the absolute magnitude

of "the induced rolling moment is of primary concern.

Induced roll phenomena, whic. could be significant

for some configurations at low angles of attack, become of

increased importance as the angle-of-attack range is extend-

ed for the purpose of increased maneuverability. The source

of high angle of attack problems has been found to be aa-

sociated with bOdy-wing or body-tail interference.

Without the mitigation of these induced rolling

moments, high-angle-of-attack maneuvers in some cases may

be precluded or in other cases result in increased control

system complexity or increased size of the wing surfaces.

Such alternatives usually result in overall performance

degradations.

The possible major sources of anomalous induced

rolling moments for typical rocket configurations were

discussed and evaluated in Ref. 1. The magnitude of some

of these rolling moments have been found to be negligible

compared to the induced rolling moments observed from ex-

perimental investigations. Strong evidence is available

(Ref. 2) to support the argument that the interaction of

2

the wing or tail surfaces with the vortex system generated

by the body is largely responsible for the induced rolling

moment phenomenon.

The vortices formed in the wake of a body affect

the attached lifting surfaces by altering the local flow

characteristics--i.e., angle of attack, Mach number, static

pressure, etc.--acrosS the span. The variations in the

magnitude and spanwise distribution of the panel normal

forces, which are produced by these changes in local flow

characteristics, are, therefore, functions of both the

strength of the vortices and the geometry of the wing-

vortex system. Consequently, comprehension and prediction

of the effects of wing-vortex interactions must depend

critically on the description of the vortex system itself.

Fortunately, experimental data are available (Ref. 31 which

describe the wake vortex system generated by the body of

the wing-body configuration considered in the present stvdy.

These vortex data eliminate the additional complexity of

relying on theoretical estimates of the vortex system and

are used herein as a basis for analysis.

It should be noted that the Blasius moment inte-gral has been recently applied (see Ref. 1) in evaluating

the induced roll of planar missile configurations.

3

SECTION II. General Formulation of the Problem

In order to simplify the problem we will assume

(as was done in Ref. 1 for the planar model) that the flow

is steady; incompressible, and inviscid. The assumption of

negligible missile roll rate will allow for the steady flow

condition.

A typical body-tail configuration utilizing four

equally-spaced rectangular fins will serve as our model for

analysis; (as. will be shown later the fin can be of any de-

sired shape such as triangular, trapezoidal, etc.). The

model i the cross-flow plane (Fig. 1) shows the cross flow

velocity,U, the missile roll attitude, , and pertinent

model dimensions.

+

Figure 1. The Cross-Flow Plane

We may now represent the complex potential for

this flow in the Z plane (Fig. 2) by"W(Z )where the two

body vortices are shown relative to the fins and the

4

cross-flow velocity U. Because of the missile roll the

vortices are not necessarily symmetric or of equal strengths.

Figure 2. The Z Plane

For steady state rolling moments about the origin

the Blasius moment integral is qiven as

where the contour is along the outside of the body surface.

The analysis can be considerably simplified if

the configuration in the Z plane is conformally mapped into

a circle. The mapping procedure for the cruciform shape

can be found in Ref. 4. Thus the plane (Fig. 3) repre-

sents the mapping of the missile contour into a circle of

radius c. (Note that the magnitude and direction of the

5

cross-flow velocity and the magnitude of the two body

vortices are unaltered by the mapping.*)

,C'c

-i5r

Figure 3. The plane

The relationships between corresponding points in both

planes are

ZX (2a)

and

z~ +~ ±(2b)

where 12 b n )i.

See Appendix A of Reference 1.

6

The choice of signs in (2a) and (2b) is based on the fel-

lowing rules:

Cl) The "outside" plus and minus signs hold for

the right half and the left half of the Z

or planes, resprectively.

(2) The "inside" plus and minus signs are chosen

accordingly to insure that the exterior of

the body maps to the exterior of the circle.

The reader is referred to Ref. 1 (p. 15) for the

derivation of the complex potential W ) which was ob-tained in the following form:

-Cr +<,? e L4>~

27r

- LUjLz2rr

where the velocity potential takes into account the pos-

sible presence of a Kutta condition by including an arbi-

trary votex of strength Y with center at the origin.

As shown in Ref. 1 (p. 16) the Blasius integral

may also be expressed as

li4 _ze:(4)

where Z

C d

7

and wC&)

The use of the expression for Z as a function of (Equ. (2b))

and its first derivative[it<Jz -+ @C4> -f Nff F ' C4/ ____- ____

gives, after algebraic expansion,

The reader is referred to Appendix A for a list of deriva-

tives of

The expression for W 2() ;an be found in Ref. 1

(p. 17) where the singularities are shown to be poles,

located at

C z C 2

(The poles. a-t and are inside the circle.) Since

z ' = C, they need notand 'Fare. exterior to the circle =,thyneno

be accounted for as contributions to the integral.

The remaining singularities of the integrand all

lie on the circle and are due to 4(V) (Equ. (5)). These are

poles at = _ cand + c and branch points where the

8

square root radical vanishes

or

Rearranging, we have

0

and

Thus branch points are located at

where

That is, eight branch points appear at

z LV t - -K7

V .(- % z ±UVC-J-1R2 )6Z__ ' CZRZ

V I(6

C Cs---

W_

__ __

5~Jv~Y7~-wuv )

9

These are all on the circle since in each case

[ ()-

10

SECTION iII. Contour Intearal Evaluation

The procedure for evaluation of the poles is

shown in Ref. 1 (p. 20). Thus, if a complex function has

a pole of order m at = 'O ,then its residue there is

S(rn-I

Referring to Eau. (4), a contour around a pole of order m

of the Blasius integral located at = Tocan be expressed as

where 1 can be defined as

~) w 2 (~)(9)in Equ. (7).

For the simple poles at l =c the integral is

Similarly, for the pole at ' =-c the integral is

The total contribution due to the poles at c and -c is thus

1I

+I -C JZC. 7( (10)

Since e W(C)='eW-C) =O , it follows that

4 iW' 2 ('C)=k W2-(C) and therefore

Similarly, the total contribution due to the poles

at. ic. and -Zc is

Since Im w(1c) = It w(-Lc)= 0, it follows thatRe 1w2 (Ic) = Re iw2 (-jc) = 0 and therefore

_ .. (13)

The contribution to the total moment due to the

simple pole at 0 is

Since (O)=O(Equ. (5)),

(14)

The contribution of the second order pole at

0 is. of the identical form as Equ. (21) in Ref. 1 where

4(0)=o and (0)=-l (see Appendix A). Thus, after differen-

tation, elimination of imaginary numbers, and collection of

terms we have, as in Equ. (22) in Ref. 1,

12

J. -- f(15)

The contributions to the total moment due to the

third and fourth order poles at! =0 are

B O(16)

since fv(01=0 (See Appendix A), and

1O: O (17)

since i ! (0)=0 (see Appendix A).

For the simple poles at 1 = and -= the

contribution to the total moment is of the identical form

as Equs. (251 and (26) of Ref. 1, respectively.

Finally, the remaining second order poles at

-= and =- are similarly evaluated as Equ. (27) in

Ref. 1. Expressions for - in Equs. (25) and (26) and for

in Equ. (27) are contained in Appendix A.

The remaining singularities to be evaluated are

the eight branch points --I'-II' -..., VI) which lie

on the circle. We may conveniently choose branch cuts

joining these branch points as straight line segments as

shown in Fig. 4.

13

4!

Figure 4. Branch points and branch lines of

Let us define the coordinates J and K as

j- =\F c2 + 2 (18)

c3~ 2(19)

We can enclose each branch line, e.g., - to etc.,

with the contour ABCDEFA as shown in Figure 5.

g r A L n C to

Figure 5. Branch Line Contour

14

Referring to equations (4) and (5), we wish to

evaluate the integral

MCDEFA [ (± F4+C4J

or

I'1r5 ~ 52~d ~(20)

48DCEFA

The evaluation of the integrals around ABC and

DEF was shown to be equal to zero, (see Appendix C in Ref. 1).

Using the analysis contained in Ref. 1 for the

evaluation of the integrals along the lines FA and DC, we

find that

Tic r ', on FA (1, = (21)

- l - rr on CD

On FA,

'Fand dJ=cF r

15

and

(22)

Substitution of (21) and (22) into (20) gives

2K

JFA - J r c-y .1C -V a r r,

Dropping the subscript on r and replacing rI by 2K-rii,

we have

16

f(23)

(z c-)(2r-r) ....- ( Ic- (r--a)) t-- )=6+c")]

Similarly, by extending the analysis along CD we find that0

-,~ ~zr- r')~W 2 -Fir -~ r_21(2K

(24)

S- r(2=-r)--

where the only change in the integral is indicated by

Equ. (21). Addition of (23) and (24) yields, after algebraic

reduction

2K

whr) + (q+C L2

FA CC>

where

17

and

h (r) --+ 4(2-r) r (2J-r)(2K+2 T-r)(KJ'+ +(J-K)-r) .

(k~-F4-LT-K)-r)'&--I J- L (J- K) -r)(K+4 L(Th-K)-r). (7

The same procedure is usei for the evaluation of

the integral around the branch cut connection g with

and produces

2K

T~~r0(74 I?-T-~~~r V2 ? (28)Lh r +( + C') J;

where

.(r' r - (-,)- J (29)

and

(30)

(K- J- (J +K) -r (k/-;(L- ¢) -r) (k/ 4- J (J-K) -r

It should be noted that the only difference between the

integrals of (25) and (28) is in the sign of Li.

18

We may combine (25) and (28) to obtain

2K

0

where

_(32)

and

H +C47 (33)

It was shown in Ref. 1 (p. 30) that Equ. (31) can be

modified to

d r .(34)

0

W2 is given in Ref. 1 (p. 17), H by (32), by (26), and

hby (27).The procedure used to obtain (34) is now used to

evaluate the contribution to the rolling moment due to the

four remaining branch points V VIII- Referring to

Figures 4 and 6 we can enclose the branch cut connecting

andI VII with the Contour A'B'C'D'E'F'A'.

19

CI

Figure 6. Branch Line Contour

The integral we wish to evaluate is

Now, we may use the polar coordinates

where both GV and V are allowed to vary from - toVI VII2

(TI

20

On FIA', i andS(9 =--VI 2 vii"

On CID' VI 2T and -VII 77vi -2Thus,

o on FA'

;27T on C'D I

Therefore,

~ - ru WJr e

e on F'A'

"-', . (37)

_ r..~ Fxon CD'

From (36b), along F'A' and CID'

+ r1 e

and

Also, on F'A' we note that

N -Z( r LtI) ,

where r=rviI and similarly

21

~-~= ~2JT+Ltr

WL-L+- (-k,&Lj-)= j KI<+ YT+/<)

- T-- L-k+L(r-(- -)=-- K/+K Lr+ (--<

Also

Substitution of (37) into (35) gives, on FIA'

2<

FO (38)

(J +K+ L+K)---(2J-+ ) ( - c+)]

Similarly, on C'D', where r varies between 2K and 0

.O

± (... [ ='- (39)(g=+ir± dr)21< (39)

22

Adding (38) to (39) we obtain

I~4(:+ (2e[2(r-CY (40)

Since pKr+Lj (see Equ. (26))

thenTM(1

and therefore,

-r5 =-iq.Also,

p(r) + J(kr 64(#r & K)cKr-(rKj-F +K LT+K (42)

[ (cr-K Lr j&T-K 3 K+zri(WT-K) Z2T-2!1K#r 2 +Lr)

We may also show that

(see Egu. (27)).

23

Thus, 2 -2+h =-P 43)

The same procedure is used for the branch cut

connecting g V with VIrII Replacing T by -tand p bywe thus obtain

L Ref~j(2)+C2i(+T-C- C3-2+(-4+c9J (44)0

If we let

2mx; +-ESPc~)~ (45)

then

-PPJL- (46)

Therefore

2K

~=T i'eL zf f [PW~r) +iPw2'(-T)]cr (47)0

or, alternatively K

4f~ (48)0

We have proved that

(see Equ. (41))

and (see Equ. (43)).

24

Substitution into Equ. (45) gives

(49)

5h-H

and therefore,

I+± & ~Hw(-l)w-~]r (50)

Adding this result to Equ.(34)will give the total moment

contributed by the four branch lines

2K

_~ £~Lffui ~w&i~~wki](51)0

In summary, the total rolling moment is the sum of the

contour integrals around the singularities

MO= I I ~ + -c +I. , +-To

4- +/IC + (52)

.4-

25

The equation for each integral in (52) is conveniently

tabulated below.

Table 1

Integral Equ. Number

IC + I-C -= o(i

(13)

J7 0(14)

110 (15)

Le (16)

-T O (17)

VeK (Ref.l, Equ. (25))

(Ref.1, Equ. (26))

Ic / t I cy/ (Ref.1, Equ. (27))

(51)

26

The final result presented by Equ. (52) is pro-

grammed (see Section IV) to obtain numerical solutions with

the use of a high speed digital computer. It is apparent,

however, that results may be obtained for limiting cases.

Thus, for example, if we consider that no external vortices

are assumed to be present in the flow fields (I=rO),

then Equation (15) equals zero as well as Equations (25),

(26), and (27), of Ref. 1. Therefore the total sectional

rolling moment in (52) is reduced to

0

(53)

Using the expression for W/A: in Ref. 1 we may note that

and

2___ C2

therefore,

= U I - =- e)+ - , J5cif

27

By the same procedure,

After subtracting (55)from (54) and substituting the resultin (53) we obtain

Alo~~~~~ +9e*ZF* r ros)d

0C

or, 2

Moo ()k e8f (Sino -cosoPf f )dr (56)

0

The expression for H (obtained from Equ. (49)) is now sub-stituted in (56) to obtain the final result:

MO.-,;= '-q++,,4T+)' (57

(See Appendix B for a derivation of Y6(O) )

Equation (57) may now be investigated in the fol-lowing for two dimensional flow, in the absence of externalvortices, past certain configurations which are orientedat a roll angle 0 with respect to the cross flow velocityvector U.

A simple cross configuration is obtained if wereduce the body radius R to zero. If R=O, then it isobvious that (57) equals zero and that this configuration

28

experiences zero net rolling moment at any roll angle ,

free stream flow density f and velocity U * The other

limiting case to consider is that of a finless cylinder

where R=b (see Fig. 1). Because a=b (by the conformal trans-

formation of the Z plane to the plane) it .s obvious that,

since K=O (see Equ. (19)), Equ. (57) equals zero since the

upper limit of integration vanishes. Thus the rolling moment,

M0 , for a finless cylinder is zero for this case, as ex-

pected.

See Reference 5.

29

SECTION IV. Numerical Analysis and Digital Computer Results

The sectional rolling moment Mo has been found(see Equ. (52)) to be a function of the following para-

meters:

As is shown in Ref. I (Equ. (43)), M may be expressed in

dimensionless form where

MO 4r). (58)

Equation (58) is used to obtain numerical data with the aid

of a digital computer. For presentation of the computed

data, however, the following changes were made in the above

parameters:

(1) The reciprocal form b/R, is plotted with b

used as the independent variable.

(2) Vortex positions i and - are presented

as IZII/R and 1'22 1/P, , respectively, which

are obtained by mapping the positions in the

V plane to the corresponding positions in

the Z plane.

(3) The free vortex circulation, ?, is dependent

on the presence of a Kutta condition and is

therefore not specified independently.

The two dimensional (rectangular) fin and vortex

data used for the wind tunnel model in Ref. 3 provided the

inputs to the computer program to generate numerical data.

For a two dimensional fin, the total moment in dimensionless

30

form is defined as

CMj = - -LbOC407 *2-f -Tz2 LL

where Mo= the total moment, L = the chord length.

The wind tunnel model geometry and flow conditions used for

the parametric study are as follows:

b= 2.8 in.

L= 2.5 in.

R= .75 in.

U= 430 in./sec

f= 1.45 X 10-6 slugs/in.3

0X= 20 degrees

Z= 285 in. 2/sec.

1.312 in. (Separated by an included angle of36 degrees)

(It was reported by C. Wong (Ref. 3) that these vortex

positions are essentially symmetric with respect to the

1body.)To illustrate the vortex-fin interaction at various

roll angles of interest, Fig. 7 shows the relative positions

of the vortices with respect to the fins. The numerical

results obtained from the computer study are shown in Figures

8-13.

(To check the validity of the above results this

writer performed a hand calculation of the individual terms

of Equ. (52). Except for Equ. (51), which proved difficult

and time consuming to hand calculate, the results agreed to

within five percent.)

I

31

VORTEX POITIONS IN THE Z-PLtA1E

-, CVORTEX I"-- VOR-TE,

#:,zg") I8.I')

F7

32

cM vs ¢

b/R -3.73

.08FI I /RIj'-f/R= 1.75.08-

.06 C r C -11.

.04--

C-c

•7---

.12.

0E

-.08-

-10

-AZ. FIGURE 8

33

Cm vs.1-,il/R

b/R = 3.7312/R =I. 75C, CI- -"14I

.10

.08 .

.0:

0 z 3 4- G 7 8 9 10

-. 06

- 0z17.9

--. z -= 71.9'

FIGURE 9

34

cvs VS- e z/R

b/Ft= 15-73

Cr C Fn. 41

101.08] 0

0=1.17.9.9

-. 08- AT 0 =: 7.9

~.~01

i .1FIGURE 10

35

CM VS. b/R

£Ili

25 (Ii

w

.40

72.1

.. 0 3. 4.0 7.0

772.

FIGURE I I

36

cm V5. r

b/R B3. 73

Cr .14

P.3

37

cm vs. cr

b/, 3.731 ,I P Ijj/R =1.75

.8 1,4-I

.4

, 45 4 9o*

2.

o .I .2. .3 .4.2- 2 . 1 0

---

j6: 17.5

Fr2.

FIGURJE 13

38

CM mvs. 0~

,Oe b/R=2.oo

(Z,1/R =1 F-2I/ R =1.753 C

e- /R=3.00

.1I 6/R=372.0b/R 5 00

b/R=:Zoo

0 107 io O 40r 50 0~

b/?= 3.700

FIGURE 14

39

Cr .141

UoCp.=.40 f' Cr,=.40

0 'o I0 30 40 5- C GO 80 90

C = .40 ..

FIGURE IS

40

CM VS. Z

b/PR 5.-I iS*I/ R =.1 .75

.04-

U00 10 2.0 30 40 5 0 70 80 90

H "".0.. j~I~ j75I/=1/~.75

-. 04"

.10.

FIGLURE fG

4.1

CM Vs. Cr,

0= 71.90

b/R = 3.75i Z21 /R= 1.75

Cp .1.41

I0

o

0 .2_G .3 .4

I ,)/R: 3.35

--0

--4

FlexURE !7

4i

42

CM mvs. I 7I /R

0 =71J9 0

b/R~ = 71 B2.1/R = 1.75-

c .14i

0.11

.2..2-

.3=30

FIGURE 18

43

CM Vs. c

0 7Z.Ia ib/R= 3.73

1z2 j/R = 1.75C - .141

IiI/R= 1. 75

.3. 1 1/%.~G

.4

I.3

- ' I

FIGURE 19

44

CM VS. I1, I/R

7=7Z. 1"b/R - 373

I ~z/R~1.75I Cz .141IG C = .4-0 -o

.5

.4-

3c.- CpZ = .o - / '

2 C .141

5 7

FIU 2 ,I/R

F IGURE 20

45

SECTION V. Conclusions

Induced rolling moments at high angles of attack

are due principally to the interaction of the vortex pair

generated by the body with the attached fins. It is evi-

dent that these rolling moments are strong nonlinear functions

of fin span and vortex strengths and positions.

One important assumption made for this report is

that the flow is two dimensional. If this restriction is

removed, the problem is further complicated in that wing

tip vortices as well as the body vortices must be accounted

for. This is necessary as the body vortices are undoubted-

ly influenced by the wing tip vortex system as they pass

over the fins. The resulting interaction between the vor-

tices of both systems depends on their relative positions

in the flow, relative strengths and directions of rotation.

For example, two vortices of equal rotational directions

will induce velocities on one another causing them to move

in opposite directions, while two vortices of opposite ro-

tations will move towards each other. The above inter-

actions, combined with vortex-fin interaction and missile

roll motions disturb the flow symmetry which would other-

wise prevail if these effects did not take place.

The vortex data used for this report was obtained

by experiment with the use of the subsonic wind tunnel at

Boston University's College of Engineering. The preliminary

data (see Section IV) indicates that the body vortices are

symmetric with respect to the body, even in the vicinity

of the model fins. This observation was made after reviewing

photographs of the missile model and vortex geometry ;rn the

wind tunnel.

46

The variation of roll moment with roll amgle is

shown graphically in Figure 8. it is observed that, as

the roll angle approaches 18 degrees, vortex 2 approachesthe horizontal fin causing a rapid asymptotic increase in

roll moment and stability. In the physical sense, however,

the vortex may not be capable of moving so close to the fin

as to cause the roll moment to approach infinite values.

The interaction between the vortex and its image results in

the vortex (and image) moving outboard and parallel to the

fin. (The reader is referred to Reference 6 for a theoreti-

cal treatment of this interaction.) Also, the sign of the

rolling moment in the vicinity of a fin depends on which

side of the fin the vortex is located. In Figure 8, for

example, the sign convention of the roll moment indicates

that the vortex attracts the fin as it approaches it.

Figure 9 shows the sensitivity of the roll moment

to perturbations made in vortex position (1ZI/R ). This

sensitivity is most pronounced when vortex 1 lies in close

proximity to the vertical fin. In the vicinity of the ver-

tical fin tip, the roll moment becomes highly stable when

vortex 1 is to the left of the fin and highly unstable for

positions to the right of the fin. (Refer to Figure 7.)

The influence is less pronounced at roll angles where the

vortex is located further away from the fin. A similar

situation exists in Figure 10 where perturbations in posi-

tion are made for vortex 2. Here, the roll moment shows

greatest sensitivity when vortex 2 is in close proximity

to the horizontal fin. Included in this figure is the

effect of simultaneously increasing the perturbations of

both vortices at = 17.9-.

The variation of roll moment with fin span is

shown in Figure 11. It is apparent that large moments exist

47

whenever a body vortex is located close to a fin tip. In-

creasing the span reduces the roll moment to an asymptotic

value which indicates that the unbalanced loads inducing

the roll are located near the inboard portions cf the fins.

It follows that for any particular condition of body vor-

tex-fin geometry, a particular value of span will result

in zero-induced rolling moment.

An almost linear variation of roll moment with

perturbations in vortex strength is indicated in Figures

12 and 13. Again, the greatest variation in roll moment

occurs for a vortex position close to the fin.

Results of Figures 8-13 are cross-plotted in

Figures 14-20. These figures are felt to be self explana-

tory based on previous discussions and are therefore not

described in detail.

Before analytical work is continued by including

unsteady, viscous and compressible effects, it is strongly

suggested that measured wind tunnel data be obtained to

support the results of this report. Further, since Refer-

ence 3 does not consider stagnation point 2.ocations on the

body, it is suggested that actual locations of the stagna-

tion point be determined from experiment and compared with

the theoretical locations assumed in Appendix B.

48

REFERENCES

1. Udelson, Daniel G., "Vortex Induced Rolling Moments onFinned Missiles at High Angle of Attack," Boston Uni--versity Engineering Laboratory Report No. i for AFCRL(68-0569), November, 1968.

2. Mello, J.F., "Investigation of Normal Force Distribu-tions and Wake Vortex Characteristics of Bodies ofRevolution at Supersonic Speecs," Journal of the Aero/Space Sciences, Vol. 26, No. 3, pp. 155-168, March,1969

3. Wong, C.W., "An Investigation of Vortex Path and Strengthof Finned Missiles at High Angle of Attack and NumericalCalculation of Induced Rolling Moment," BUEL Report No. 2for AFCRL (71-00?92), March, 197.

4. Bryson, A.E., Jr., "'Stability Derivatives for a SlenderMissile with Application to a Wing-Body-Vertical-TailConfiguration,' Journal of the Aeronautical Sciences,Vol. 20, No. 5, pp. 302-303, May, 1953.

5. Sacks, A.H., "Aerodynamic Interference of Slender Wing-Tail Combinations," NACA Tech. Note 3725, p. 31, January,1957.

6. Milne-Thompson, "Theoretical Hydrodynamics," 4th Ed.,MacMillan Co., pp. 358-359, 1960.

1

/

49

APPEUDIX A

Derivatives of

The derivatives of CS ) are listed here for reference.

From Equ. (5)

where

V(iV+C4) * ~(S+cL 4)

Hence

where

s (V( +c)i +( +tc.)] z(+c)¢ I] . ,

Also

2+ ?r4:"( )=~~~~ 1U090) _,, , 3 -~

56

and

It is easy to show that

c 0

Therefore

5].

APPENDIX B

The purpose of this appendix is to find an analytical

e-xpression for the free vortex, ?, as well as the location

of the stagnation point to satisfy the Kutta condition for

the flow described by Equ. (3). The limited case will be

considered where r 1= 2=0.

To find Y, the following procedure is used: The

stagnation point is assumed to "move" with the cross flow

vector, U, as it is permitted to rotate about the body.

Thus, in the Z plane the stagnation point, Zo, is located

on the body contour, as follows,

+b ,,=o

- b €c,=-7/2

To obtain the corresponding point in the plane, we

make use of the conformal mapping (see Equ. (2a))

~Lx(+'Y) + Z O 2 (60)

Substitution of (59) into (60) gives, for O<0<7T/2

Tte complex velocity is obviously zero at a stagnation

point. Therefore, if we first differentiate Equ. (3) to

obtain W/-5) Cthe complex velocity) and equate this result

to zero, we have

F>' 52

2T)

77

After algebraic reduction we obtain the genera ex-

ass

I+ +

obviously, for the simple limited case where

F. 0 we have

To~ ~ I2r -4sn+L~.)coSP]Texpress as a real quantity, we can further define

ubs in theLeo

Substitution in the above expression gives

4~5o> rnc (Cos sin-sin 6,cos)

53

It is easy to show, by substitution, that "' is zero wher-

ever flow symmetry exists about the configuration; that is

where =0, 7T/4 and 7T/2.

Ii