FI).i ,'r)R.II-. 1418

THF UNSTEADY FLOW FIELD ABOUT A0 RIGHT CIRCULAR CONE IN UNSTEADY FLIGHT

X00 E. A. BRONG

CENERAl. ELECTRIC COMPANY

TECHNICAL REPORT FDL-TDR-61-148

JANUARY 1967:++ r- +

W,:AY2 2 ssijjjJ

Distribution of this document is unlimited U.j J LL•

AIR FORCE FLIGHT DYNAMICS LABORATORYRESEARCH AND TECHNOLOGY DIVISION

AIR FORCE SYSTEMS COMMANDWRIC;I IT-PATTERSON AIR FORCE BASE, 01110

NOTICES

j'6When Government drawings, specifications, or other data are used for anypurpose other than in connection with a definitely related Government procure-ment operation, the United States Government thereby incurs no responsibilitynor any obligation whatsoever; and the fact the* the Government may haveformulated, furnished, or in any way supplied the said drawings, specifications,or other data, is not to be regarded by implication or otherwise as in anymanner licensing the holder or any other person or corporation, or conveyingany rights or permission to manufacture, use, or sell any patented inventionthat may in any way be related thereto.

Copies of this report should not be returned to the Research and Tech-nology Division unless return is required by security considerations.contractual obligations, or notice on a specific document.

600 - Mwb 1967 - CO192 - -61

I

FDL-TDR-64-148

THE UNSTEADY FLOW FIELD ABOUT A

RIGHT CIRCULAR CONE IN UNSTEADY FLIGHT

-&: E. A. BRONG

GENERAL ELECTRIC COMPANY

I, Distribution of this document is unlimited

I -

FOREWORD

The it.formation released in this report has been gencratO in support of studies

on the Hypersonic Dynamic Stability Characteristics of Lifting and Non-Lilting Ite-

entry Vehicles. These studies were conducted by the General Electric Company, Re-

entry Systems Department, for the Stability and Control Section of the Flight Dynamics

Laboratory of the Air Force Reseai h and Technology Division. The program was

sponsored under Air Force Contract Number AF 311(657)-11411, Project Number 8219

and Task Number 821902. Mr. J. Jenkins of The Control Criteria Branch, RTD, is

the project engineer on the contract. The project supervisor for the General Electric

Company was Mr. L. A. Marshall.

Tnis technical report has been reviewed and is approved.

C. B. WESTBROOKChief, Control Criteria BranchFlight %"ontrol UivislonAF Fligit Dynamics Laboratory

5i

4m U

ABSTRACT

The inviscid flow field about a right circular cone in unsteady planar flight is

analyzed by a perturbation technique which is an extension of Stone's treatment ofT

the cone at small yaw. A solution is found in the form of infinite series in the time

rates of change of the pitch rate and angle of attack. The linear stability derivatives,

Cm and C as well as "higher order" stability derivatives such as Cm1 and Cq M&q

are presented for a wide range of cone angles and Mach numbers.

The stability derivatives, Cm and Cm. , as obtained from this solution areq a

compared to results obtained from second order potential theory, Newtonian impact

theory, and an unsteady flow theory due to Zartarian, Hsu, and Ashley. Both the

potential theory and the impact theory predict that Cm.•rapidly approaches zero at

high Mach numbers while the present theory indicates that Cm& approaches a value iwhich is on the order of 10 percent Lo 20 percent of C mq

Numerical results obtained from the present theory are also compared to ground-

test data on Cm + Cm.. The agreement is found to be generally good, although theq a

data in some instances indicate a pronounced Reynolds number effect.

The numerical results for the "higher order" coefficients are used to predict the

effect of reduced frequency on the parameter C + C as obtained by the forcedm m.q a

ili

Ioscillation testing techniques. It is found that the predicted effect is very small over

the range of reduced frequencies likely to be encountered.

iv* ,k

: I

TABLE OF CONTENTS

PAGE

1. Introduction ......................................... I

2. Assumptions and Restrictions .............................. 3

3. Descriptionof the Perturbation Scheme .......................... 4

4. Derivation of the Perturbation tkquations .......................... 6

4.1 Statement of the Boundary Value Problem in General Form ...... 6

4.2 Reduction of the Boundary Value Problem to Perturbation Form ... 10

5. Numerical Solution of the Problem ........................... 23

6. Sample Resuii6 and Comparisons ............................ 36

References ......................................... 72

Appendix I. - Substitution Forms for the Perturbation Values ..... 74

Appendix U. Equations for the Field at Zero Yaw ............... 77

Appendix IMl. - Expressions for the Normal Force and Moment ...... 73

ILLUSTRATIONS

FIGURE PAGE

1. Inviscid Flow Field Boundary Value Problem ..................... 7

2. The Spherical Coordinate System (R. w,..) ..................... 14

3. Perturbations in the Velocity Component, u ..................... 40

4. Perturbations in the Velocity Component, u ...................... 41

5. Perturbations in the Velocity Component, v ...................... 42

6. Perturbations in the Velocity Component, v ...................... 43

7. Perturbations in the Velocity Component, w ..................... 44

8. Perturbations in the Velocity Component, w ...................... 45

T

I

t ILLUSTRATIONS (CONT)

FIGU RE PAGE

9. Perturbations in the Pressure ............................... 46

10. Perturbations in the Pressure ............................... 47

11. Perturbations in the Lntropy ................................. 48

12. PerturLauL zis in the Entropy ............................. 49

13. Perturbetions in the Stagnttion Enthalpy ...................... 50

14. Perturbations in the btagnation Enthalpy ..................... 51

15. Static Normal Force Derivatives, CN ....................... 52

16. Dynamic Normal Force Derivatives, CN . .............. 53

17. Dynamic Normal Force Derivatives, C 5.4....................

18. Dynamic Normal Force Derivatives, C.,. 5i

19. Dynamic Nor-mal Force Derivatives, CN 56q

20. Dynamic Normal Force Derivatives CN. ................... 7

q21. Dynamic Normal Force Derivat'ives, C

q22. Dynamic Moment Derivatives. C ..and C . .................... 59

q23. Dynamic Moment Derivatives, C and C ....................... 60m m

q 624. Dynamic Moment Derivatives. C and C ................... 61

q 625. Dynamic Moment Derivatives. C and C ..................... 12

m mq

27. Dynamic Moment Derivatives. C and C .................. 64

q &26. Dynamic Moment Derivatives. C and C ................... 63

q &27. Dynamic Momcnt Derivatives. C and C..........6.

q 6

29. Comparison o( Theoretical Results ......................... 66

30. Comparison of Theoretical Results ......................... 67

vi

ILLUSTRATIONS (CONT)

FIGURE PAGE

31. Comparison of Theory and Experiment ....................... 68

32. Reynolds Number Ef'ects on C + C ..................... 69M m

q33. Comparison of Theory and Experiment, M = 8 .................. 70

34. Theoretical Evaluation of Frequency Effects. .................. 71

TABLES

TABLE PAGF

I. Perturbation Coefficients for the Shock Shape,

= 10°, M a= 10, y =1.44 .............................. 36

vii

SYMBOLS

A, B Cefficients defined in Equations (5.7) and (5. 8)

AB Body base area

C Moment Coefficients, Moment _- W/8p U2 D2 L

C Normal Force Coefficient, Normal Force -- •/8p.. U2 )D2N

M, (" 4) \ Cm

C (2U) I Cmq

CN I C( N

CN..

S~L •CN L 2U.

CqN ...

C CN for xqcg =

L

N. cq

CN.. cq L

viii

c Speed of sound

D Base Diameter

e2

f Frequency of oscillation

F Function def'Lning the shock shape

H Stagnation enthaipy

h Static enthalpy

j Index referring to angle of attack variation (j = 1) or pitch rate

variation (j = 2)

L Body length

M Mach number

n Subscript relating to the order of derivative of C4or q

nB Unit outward normal to the body surface

n Unit inward normal to the shock

p Pressure

q Pitch rate

q 1/2 Pý UC2 - dynamic pressure

R, w, o Spherical coordinates

R Universal gas constant--4 --4 =-

R, W, € Unit vectors of the spherical coordinate sys* m

S Specific Entropy

t Time

T Temperature

ix

U, Vehicle speed

U, V, W Velocity components along the R, w,'0 directions

V Velocity of the fluid relative to the body

V Fluid speed normal to the shockn

cg Velocity of the center of gravity

x, y, z Cartesian coordinates

-0 -4 -0

x, y, z Unit vectors of the Cartesian System

x Location of the center of gravity (aft of the vertex)cg

k Angle of attack

v Ratio of specific heats

pc2

p

6 i, j Kronecker delta

(E Perturbation parameter

p Density

V xV= CurlV

Superscripts

() Derivative with respect to time

Subscripts

CO In the free stream

0 Steady state, zero yaw condition

s Pertaining to the downstream side of the shock

SB Pertaining to the body

x

1. INTRODUCTION

As a re-entry vehicle penetrates the atmosphere, its pitch rate (q) and angle of

attack (a) vary in an oscillatory manner and cause the flow field about the vehicle to be

in an unsteady state. For most purposes, q and the rates of change of q, and a are

small enough that the differences between the unsteady field and a quasi-steady field

may be considered negligible. An exception to this rule occurs in the evaluation of

aerodynamic forces and moments. The small contributions of unsteady effects to the

normal force and pitching moment appear as damping coefficients in the equations gov-

erning the rigid body motion of the vehicle and play an important role in determining the

loads the vehicle must withstand.

In this work, the unsteady flow field about a vehicle (specifically, a right circular

cone) is examined under the assumption that the motion of the vehicle is given. The

vehicle moti.. appears explicitly in the equations and boundary conditions for deter-

mining the flow field in a body-fixed coordinate system (Section 4. 1). In order to make

the boundary value problem tractable, perturbation parameters which are measures of

the departure of the flow field from a steady-state, axis-symmetric field are introduced

(Equation 5.14) arkJ used to effect a linearization ,'/ the problem (Section 5.2). The

first-order effects of these parameters are considered, and a formal solution in the

form of an infinite series which gives the flow field in terms of the vehicle motion, is

assumed (Equation 5.26). Thus the mathematical problem is reduced to one of solving

an infinite sequence of linear, ordinary differential equations (Equations 5.27) with

variable coefficients which are determined by the steady field at zero yaw. These

equations are solved by numerical methods (Section 6). The numerical results, when

properly combined (Appendix 1), give the unsteady flow field to the first order of the

perturbation parameters. The unsteady flow effects on the normal force and pitching

mommet coefficients are obtained by integration of the surface pressures (Appendix HI).

Results obtained by the method presented here are compared to theoretical results

from other methods of varying degrees of approximation, and are also compared to

experimental results (Section 7). From the cases examined, it does not appear that

there is an orderly pattern of agreement 'between the results obtained here and the po-

tential theory of Tobak and Wehrend (Reference 13), the shock expansion theory of

Zartailan, Hsu and Ashley (Reference 14), or Newtonian impact theory. Comparisons

with experimental data do show fair agreement.

Manuscript released by the author September 1964 for publication as an FDL TechnicalDocumentary Report.

2

2. ASSUMPTIONS AND RESTRICTIONS

The flow field is determined by the body geometry, the nature of the gas, mad the

flight conditions. The assumptions and restrictions perts. ning to each are tabulated

below:

1. Body Geometry - The body is assumed to be a right circular =one. j2. Nature of the Gas - The gas is assumed to be Inviscid, non-conducting and at

chemical and thermodynamic equilibrium. It is represented analytically as a

y* gas as described in Reference 1. This includes an ideal gas with constant

specific heats as a special case.

3. Fligbht Conditions - The vehicle trajectory is assumed to be planar with three

degrees of freedom - i.e., two-degrees-of-freedom In translation in the plane

of the trajectory and one-degree-of-freedom in rotation normal to the place of

the trajectory. It is further assumed that the speed of the vehicle is constant

and sufficiently high that the flow about the vehicle is supersonic with respect

to the vehicle. This reduces the number of degrees-of-freedom to two-amgle

of attack and pitch. Only first order effects of these on the flow field are

considered.

The variation of ambient pressure and density along the trajectory is neglected in

the analysis.

3

3. DESCRIUPION OF THE PERTURBATION SCHEME

The flow field about a body in flight is determined by the solution to the non -linear

bomndary value problem gAiven to Sction 4.1. This boumdary value problem is stated In

a coordinate system fixed In the body and consequently the motions of the body 9 andcg

(I, appear as "driving functions" In the problem. For the simple planar trajectory

considered here the motionr, of the body are given in terms of two functions of time,

a(t) and q (t), and the constant speed Um= V cgj, by Equation (4.13). The flow field

variables are functionals of the functions 0(t) and q (t) In that they depend on all the

values taken on by o(t) and q (t) In the interval from the initial time, to the current

time, t, and are ordinary functions of position as given by the three coordinates,

(R, (, (o).

In the perturbation scheme utilized in this work, a (t) and q (t) L are taken to beUm

small quantities on the order of (1 and C2, respectively, which then become the pertur-

bation parameters with the substitutions:

M~) E o(t)

U.q (t) C 2qt L-

the pressure, for example, will be a functional of the functions 0 (t) and q (t) and an

ordinary function of the parameters c, and f2- It is assumed that p can be expanded in

Taylor's series in the parameters, (I and (2, to give a series of the form

P --PO (I PI " 'P2 ' • (II.O .T .)

4

11 owl

The coefficient, p0 , is the pressure field produced by the body in steady flight at zero

angle of attack and can be found by established methods. The coefficients, p1 and P29

give the first order effects of angle .if attack and pitching rate, respectively, and are

to be determined by solution of Equatios (4.16) and (4.17). They are ftmctionals of the

functions a (t) and ; (t) and ordinary functions of the spatial coordinates. In Section 4.2

it is shown that they can be represented formally by series of the type: jIn -

Pd n dn U 'coot= Ito 3+p - + jo oscP1, n d

n =0

(and a similar series for P2 as given in Appenolx 1) where the P1 t n are functions only of

the ray angle, w. This solution holds after "starting transients" have died out. The

coefficient, pi ,0 . gives the effect of small yaw in the steady state. The coefficients,

Pl I P , 2' etc., give the effects of time varying angle of attack.

It is the coefficients, Pj,n (and corresponding quantities for the other flow field

variables), which are found as a result of this analysis. The method of finding them is

numerical and is described in Sections 4 and 5. The pressure coefficients yield cor-

respondinig forces and moment coefficients (Appendix UI) which are static and dynamic

stability derivatives.

5

I

4. DERIVATION OF THE PERTURBATION EQUATIONS

t 4.1 STATEMENT OF THE BOUNDARY VALUE PROBLEM IN GENERAL FORM

The inviscid flow field boundary value problem can be stated to an observer In a

body-fixed coordinate system (x,yz), Figure 1, by a transformation of coordinates

from an inertial system. Lamb, (Reference 3), gives the appropriate transformed

continuity and Euler Equations:

the continuity Equation:

LP +_ l pVO

at (4.1)

the Euler Equations:

+ -0 - V + V - d x +d. + 0 + ('xr# x& (4.2)

The form of the continuity equation is unaltered by the transformation. The form

of the Euler equations Is altered In that the acceleration of a fixed point in the moving

frame of reference appears as a body force (per unit mass) involving the vector % elocity of

the center of gravity, 1cg(t), and the vector rotation of the body, 0 (t)M on the right hand

side. The vector rotation causes a given fluid particle to experience a Coriolls acel-

eratlon, 2 ( x V), which is present an the left hand side. The derivatives,dt

and dd . appearing in equation (4.2) are those observed inthe body-fixed coordinates.dt dc

For example, A-Mg isdt

8

-! t

BODY

V =-V +rxfl* cg

xf

S!

Figure 1. Inviscid Flow Field Boundary Value Problem

d (x Vco d (y- V 4 d V

dt dt it' dt

The continuity and Euler equations must be complemented by the energy equption

for the ,alabtic flow:

Ls - VSO0 (4.3)

and an equation of state for the gas. here taken to be:

P = p (0, ) (4.4)

7i

The function indicated in Equation (4.4) depends on whether the gas is an ideal gas,

a gas at chemical equilibrium, or a gas in some "frozen" composition. The development

of the equations cm, however, proceed without specification of the precise form of the

eQUq*on of state.

Equatons (4.1) through (4.4) are a complete set of flow equations for the determina-

tion of the p"msure, p, the density, p, the entropy, S, and the three components of the

fluid velocity (measured In the moving frame of reference), V, for prescribed motions

of the body, Vcg and 0. In order to solve them, inltlal conditions at some instant of time

and boundary conditions at the body and shock must be given.

As Initial conditions It Is assumed that -U t = 0, the fle!d is the steady-state, axi-

symmetric field produced by a umiform forward translation at speed U,, of the body

along its axis of symmetry.

At the body surface, the flow must be tangent to the body surface ano the boundary

condition is:

V =0 (4.5)

On the shock surface:

Fsa (x, 0.t)--O (4.6)

iwhich must be foiwd as a part of the solution) the shock equations give the flow varlables

as tmctions of the relattve velocity of the shock and the free stream flow, and khe in-

stantaneous unit inward normal to the shozk.

SII

& -

FIn. 8+ (4.7)

The sign in equationi (4.7) is to be chosen so that u• n. is negative.

The shock equations can be reduced to the following set of three algebraic

equationi . I

P V = V

5 V~ s Pp• n V2 ÷p (4.8)

V +s=pV2 + V2

p )p - h (pmo) + -

2 2

These equations express, respectively, the constrvatin of mass, momentum

normal to the shock, and energy in a form which utilizes the result that the velocity

component tangent to the shock is unchanged in crossing the shock. In these equations

Vn- is the component normal to the shock, of the relative velocity betwedm the shock

and the flow on the upstream side of the shock, and Vns is the corresponding compo-

nent on the lIown stream side of the shock.

The flow velocity on tler upstream side of the shock is given by:

¢ - c g '.•

and the component of shock velocity along its normal is given by (reference 3, page 7):

S

aLFI It

Therefore, V n, is gven by:

-F

i Fsn. + (4.10)

V

FVcg •ns +(r x ) n. + -

EquaLons (4.8) are presumed to be solve'I in tle form*:

PS = PS (Vn , P9 , PQ)

S =Ps (VN-, P-9, P) (4.11)

Vns - Vn r- &^ Vn = 16Vn (Vn- - P-, ID)

The first two of these equations give the pressure and density downstream of the

shock explicitly In terms of the function giving the sbock, F., and the motions of the

body, Vcg and f)by use of equation (4.10). The third Equation gives the three velocity

components of the flow on the downstream side of the sbock by further use of the equation:

VS = -Vcg+ (rx f)+ 1 Vn ns (4.12)

4.2 REDUCTION OF THE BOUNDARY VALUE PROBLEM TO PERTURBATION FORM

The equations and conditions of Section 4.1 define the boundary value problem for

determination of the inviscid ilow field about a body whose motions, Vcg and C), are

*There are numerous procedures in the literature for obtaining such a solution numer-

ically. Reference 4 presents the procedure used in the subsequent numerical work.

10

known. In this Section, the boundary value problem is reduced to a perturbation form

applicable to right circular cones which move at constant speed, fVcg I = coust. = U,

in such a manner that the z -axis is always parallel to a fixed reference direction.**

For the type of motion just described, the vector velocity and rotation can be

written in terms of two functions, ot (t) and q (t):

Vg =U(-X cos C+ ysina)

(4.13)

-q z

qL

To put the problem in perturbation form it is assumed that a << 1 and L << 1,U C

and the parameters ci and C2 which measure the magnitudes of a and are intro-U,0

duced by

(14

and C 2and -. ) (4.14)U00

In Equation (4.14), i (t) and i (t) are assumed to be of the order of one in the time

interval of interest, and el and C2 (which are constants) are assumed to be much less

thaD one. The flow field variables can be regarded as functionals of the functions ot (t)

q (t) (in the sense defined in Reference 3) and ordinary functions of the perturbation

parameters c and c2.

**Although this analysis does not employ stability axes the force and moment derivatives

obtained are conventional.

11

I

In the perturbation scheme, it is assumed that each flow field variable can be ex-

panded in Taylor's Series in the variables e and c 2" For example, it is assumed that

the pressure can be written in the form:

P = P0 (x,y,z) + (1 P1 (x,y,z,t) + (2 P2 (x,yz,t) + .... H.O.T.. (4.15)

These expanded forms are then substituted into the equations defining the boundary

value problem and the resulting set of perturbed equations is separated into a number

of sets of equations for the perturbations by the usual process of equating forms in like

powers of cl and C2. (Only the three lowest order terms are considered.)

The boundary value problem so defined for the lowest order coefficients, popo,So

and Vo, is, of course, the problem of determining the steady-state, axisymmetric field

produced by the body as it translates with constant speed, U , parallel to its axis of

symmetry. Under the assumptions listed in Section 2, this field will also be conical

and can be solved by the methods of Taylor and Maccoll (Reference 5), or Romig (Ref-

erence 6). In the subsequent work it is assumed that this field is known*. The equations

defining the steady state field are given in Appendix U1.

Substitution of Equation (4.15) and like expressions for the density, entropy, and

fluid velocity into Equations (4.1) through (4.4) to obtain the two sets of equations gov-

erning the coefficients of cl and C2 yields.

*The numerical procedure used to obtain it is given in Reference 7.

12

t8

Pi~

+ vi, (Po V j+ p V 0)0;a 0Sat oj - jo " •PJ •P -

'4 - - vp i p 0.

S+ (V v + (v . ) V0 + - - P = F (4.16)0Po Po 2 = 1,2

i+ V 7S + .V S =o0at 0

2p =c +e 2 S

j 0 j 0+

d&.where: -U - y j 1

dt,.o =(4.17)

2 (Vo0xz) q **+ -- qy + rrxz3) - - , J=2.

L L L dt

The boundary condition at the body surface, Equation (4.5), separates into the two

sets of conditions:

Vj nB= 0, (j =1,2) (4.18)

For the conical geometry, the equation giving the shock, Equation (4.6), is most

conveniently expressed in terms of the spherical coordinates, (R, w, 4), I-Igure 2, in

a form giving the ray angle, &,, as an explicit function of the radial coordinate, R,

meridional angle, €p, and time, t:

Fs s cw(R. 0, t) - 0 = (4.19)

13

!-

y

IR

Le J BODY, w = B

x

z

Figure 2. The Spherical Coordinate System (R, w, ()

The perturbation form for the shock is then (to the three lowest order terms):

,,= Ws = %so + E 1 &s81 (R, 0, t) + (2 Ws2 (R, 0, t) (4.20)

where cso is the constant shock angle from the steady-state, axisymmetric, conical

flow solutions. The unit Inward normal to the shock (Equation 4.7) becomes:

aa-ss0R sinWso aC (4.21)

i-1

and the flow velocity on the upstream side of the perturbed shock is:

V =R wU + "I '3,1 + ' 2 ~ u.2 v c 1v-1 £2 v. 2 ~ +

" w W + 2 ww2 (4.22)

14

Iwhere:

U =U cooWcco 0O so

uI U csinW cosop •.l sinu w0 ccso si so

u - U x sin cos - U - .sinnu (4.23)C2 L cg s0 .s0(4o

v U sin wOD 0 0 so

v =-U (dcosw co CosCP+ U) COS )S O so so w so

V,2 = qU_ (x cgCosWSsO- RI coF p - U w2Cos s

L

w =U sin

w - ( cus -) x sin 4*= 2 L so og

The perturbation form for the normal component of relative velocity between the

upstream flow and the shock is obtained from Equation (4.10) ab:

2

V V - + V(.4JIM: W 0 E J nj(4. 24)

1=1

wheret

V u R 2W sJ + RW •j-v j =1,2.nj ®o - - j-

a R •t

Substitution of Equation (4.24) into Equations (4.11) and (4.12) gives the pertur-

bation forms of the pressure, density, and velocity components on the dowstream side

15 I

of the perturbed shock. These resulting expressions must be equated to expressions

for the corresponding variables in the field evaluated at the perturbed shock to obtain

the proper boumdary conditions for the perturbation equations. The resulting condi-

tions are: (J - 1, 2 in each equation)

os _p (R. w so 0 T 0 -- 2 k W - V n

WO Vn=W V" vO

s •R son 0

PS

so V =-

nsj

so VVn v

V n --W V-W (4 .25)

i i i ( 7ý no soo a OUs) S

)( awsj --0

+ti V (n.2V5) , pa l p -c R e Ri a

nlw

V .Vn~j

110 v M (4.25)

In Equations (4.25), the partial deri% atives with respect to w. arise as a result of

evaluating the variables in the field at the perturbed shock. The partial derivatives with

respect to Vni .1 re to be obtained by differentiation of the expressions indicated In

Equation (4.11); expressions for them are given in Reference 9.

16

To complete specification of the boundary value problem for the perturbatitn vari-

ables, it is necessary to give the initial conditions at t = 0. Corresponding to the

assumption stated in Section 4.1, this condition is that the perturbation variables vanish

at t =0.

Equations (4.16) and (4.17) together with the boundary conditions at the body sur-

face, Equation (4.18), the boundary conditions at the shock, Equation (4.25), and the

initial conditions define two separate, linear boundary value problems for the perturba-

tion variables. In these problems, the "motions" of the body, & (t) and '(t), appear in

inhomogeneous terms and play the role of "driving functions." Because of the linearity

of the problems, the solutions for arbitrarily prescribed functions, F and 4, can be

divided into parts which can be called the particular solution and the complementary

solution. The particular solutions are defined as solutions of the inhomogeneous prob-

lems which reduce to the trivial solution (all perturbation variables equal zero) when

and ý are identically zero; the complementary solutions are defined as solutions of the

homogeneous problems (obtained by setting & and ý equal to zero) which cause the com-

plete solution to satisfy the initial conditions.

On physical grounds, it is known that the complementary solutions must '"ie out"

in time, and their decay tim,- e i •,,,mMn..I .MUttple of the time it takes the body to

move through its own length, LAJ, . Subsequent to this decay time, the flo,, field %.bout

the body is given by the particular solutions alone. It is, therefore, only the particular

solutions which are of interest here.

17

£ P

The problm of determining the particular solutions of the boundary value prob-

lems for arbitrarily prescribed functions, - (t) and i'(t), can be approached in several

ways - e.g. , by use of the Laplace transform. The approach takan here is one which

reduces each problem to a problem of solving an infinite sequence of sets of ordinary

differential equations having the ray angle, w, as an independent variable. The substi-

tutions which accomplish this reduction are tabulated in Appendix I. The form of the

pressure perturbation, p,, is typical:

Pl(R p, t1= Pn (M) coosC(d tn (4.26*)

nI=O

In writing these series, it is assumed that of (t) and if(t) are analytic functions. Also,

it is recognized that there is no a-priori guarantee that these series are convergent for

any given functions, N (t) and i (t). The factor I/Un has bee introduced so that the

perturbation coefficients, PJ-n, have the units of pressure.

The two infinite sequences of sets of ordinary differential equations for the varia-

bles Pj, n(4) etc.. which arise from the substitutions listed in Appendix I can be

written in the common form. (j 1, 2), (n - 0, 1,2,3....).

-d ' jn + Vopj, sinw +(n+ 2+ 6 J.2) OUj +uopi, sin w

"". + n ,,j 'in (4. 27a)

*The "zeroth" derivative, is equal to i, by convention.

18

I,

m• •

Vo d n vo Vj,n + /_Ejn (4.27b)

dv dvo 1 dp jn I dpovo +%uovj,n +Vo uj,n + vj,n - + - _f-0 dw po du, p0

2 d-'

(n+ 2 ) u vjn 4.27c)

dwj nVo cd w + uo w j, n Vo w , a cot W o s +,(n + J2 ) % wj,n .n (

p~sim~.(4. 27d1

V0 + 6J,2) % SI,n n (4.27s)

Pj,n C Co 2 Pj,n + eo2 Sj,n (4, 27f)

where:

*jn -(1-6o n)U. 6 n~Sin•:

Aj,fl 0-(- o,n) Uf j.n-I in

11;,n 1 - (I - 6o,n) Ut Uj,nI - 61,n 6J.1 U 2 sinu:+ 6o, 2 Um (2 vO + U.5sinw)

n = -11 -6o, n)Um Vj,n-1-61,nUM2 1(6j,1COW +ý,.2) +

6o.n 6 j,2Um (-2 uo+UCcoSw)

•J,n =-1-o,n) U-, wJ,n1+ 61,0nU' ,, 1,+ ý,2 COOW) +

6o," 6-,2 12 U. (uo cos W- v slnw)- U- '

,.n= - (I - 6o,n) U- s,n-l

The condtiou of irrotaUonality of the field at zero yaw (Apxdl1x E) has been used

to simplify equation (4. 27b).

19

-I

46

The boundary conditions for these equations can also be written in a common

form. At the body, w - wB, the condition of tangency of the flow becomes:

viI no 0 atw•( • B (4.28)

At the shock w a wso, the perturbation variables, PJ n etc., are related to the

quantities, wa which describes the perturbation in the shock shape by:

p'n=~nl 0 1+o

-. so 2 )

+ k . ;ýP2•'Jni

" MIIV a, =- V0o (4.29a)

an [ u* j +(n+I+6+, 2 ) U. cos 5 0 nPj~n -80 VnI V nl -v jo

~P2+'j,n ýV n"-1 Vn1 " -vo (4. 29b1

PJ, n Co 2

. n 02 (4.29c)

20

lj, n =,2 n - o, P., + a'j 6 -6 o, U6,u -in it;, (4.29d)

V - (uo+ +)-

jjn,•. a = 80sov ni n

- (n+l + 5j,2) U,- Coos so 8V n---l V V0Wj,n

"- o, n(U" [6j, Cos ° + 6 j, 2] ,n + k Vn

nI Vnl -Vo)

(4. 29e)

w.Ln - vn 0).o n + 6°'n U-(6j. I + 6j, 2 cos wso) (4.29f)

where:

Xj,n 60o, n U-(6j,I Cos Wso +j,2)+(1-0o,n)UM1 ,n-1

Equations (4.27a) through (4.27f) are ordinary differential equations with variable

coefficients which are determined by the solution lor the axially symmetric conical

field, u., v., p. and io. For j A I and n a 0, the Equations are equivalent to tnose de-

rive-d by Stone (Reference 9) for the problem of cores at small vaw. For eithlet value

of ; and any value of n (except n 4 0), the equat~ions and boundary conditions are corn-

piltc prvided that 1he solution for the same value of ) and a value of n 'hhich i.. Une

less. has been previcxsl. solved. For n - 0, the tquations atnd bourtiarv contiAt.un.

are complete wiu-out a knowledge of the solution for any other value of; awW n. For f21!

4-t

any values of 3 and n, the problem Is a two-point boundary value problem in which the

boundary conditions contain an unknown constant w3j,n.

The equations have a singular behavior at the body surface where vo = 0 since vo

multiplies the derivatives of the perturbation variables in equations (4.27b) through

(4. 27e). The method of treating this singularity is given in Section 5.

22

F5. NUMERICAL SOLUTION OF THE PROBLEM

The numerical method of solution of the problem is based on manipulated forms

of Equations (4.27a) through 4.27f). The objective of these manipulations is to reduce

as much as possible of the Dumerical work to performing quadratures and also to pro-

vide a convenient method of handling the singularity at the body surface.

The manipulated forms of the equations are obtained as follows:

Dividing through equation (4.27e) by v and solving it by use of an integrating0

factor gives

SOS

S ( , n )L In + T fn J~n dLu (5.1)

where

m +2,

and

o 0 0

has been introduced by use of the equation

vo d oU =-- -

0 0 dW

from Appendix H.

23

Id

Multiplying equation (4.27b) by u, equation (4.27c) by v and equation (4.27e) by

T, 0 adding the results and combining terms gives

dH1

v -" ,n + m u H, u If. + v + T •. (5.2)0 d 0 J,n o J,n o 3,n o J,n

where H Jis a quantity related to perturbations in the stagnation enthalpy

H. Pjn + u u. + v v. + T S. (5.3)J,n PO o J,n o J,n o 0.,n

The condition of irrotationality,

du0 d

has been used in deriving equation (5.2). Also a quantity,

dT p. p p dPo

S. o Pjn dpo j,n o

J,n d 2dj dw

which appears in the derivation is shown to be zero by use of equation (4.271), the

relation,

? Do 1 ?0o

3W c 2 dwo0

the identity,

•T 2oP/s 2 c 2

and the fact that

S = const.0

24

The form of equation (5.2) is the same as equation (4.27e) and the solution is

noHn + Tmond

so

Solving eq. (5.4) for p. , substituting into equation (4.27b) and re-arranging thej,n'

result gives

du~- mH, +mT SJ - (m + 1) v - m H.n + 0 Sn (5.5)j,n d(U j,n v

Substituting for p.,n from equation (5.4) and vj,n from equation (5.5) (in terms

of J j,n) in equation (4.27d) and re-arranging again gives a result of the same form as

equations (4.27e) and (5.2). The solution is

W , n J, sn lm+ 0

jn j,n (m + 1) sinxv + sin

08

so (5.6)

m+l 0_ ( S__ VoJ,

sinin w +oH( o SJ n (m+) Lwsi n x J, n m o + l1

x o080

so 0

The preceding results can be used to eliminate v and W fromj, no P),n Wj,n

etuation (4.27a) and thus reduce it to an equation for u. . Equation (5.5) gives v.duloll )o

in terms of J and -d and equation (5.4) subsequently gives p.,n in terms of H.Jn

d jLL u jn and S . Substituting this result into equation (4.27f) then gives Pj,n inj, n' - t )on

25I

terms of the same quantities. Using this result toge'her with the expression for v ,n

from equation (5.5) and w in terms of W and u from equation (5.6) in equation~ J,n J,n

(4.27a) gives

d Ujn 1 dA n +Bu (m+1) r (5.7)

dw2 A dw dw j,L 2

where

1 Uo VO

A = -2- 0 sin w exp -(2m+ 3) f 2 dw (5.8)0 c CO - V2

-1 + (m+1) (m+2) 0 ) (5.9)B=1 -v2-• si (e)re)c2 2

0wSsind u ovsinw

- o0o1 (mt d )

P 1 - 1sin w)

II2r - + r (5. 10)dv 2

w ith: P i uV2 v si w

TI "l= (m+I) 1-2 n 2 J,,!c c

oT +e 2

vosin wu 0-0 Sj~

co

26

U

A

T- 0 W - (m+2)sinw uo'2 J,n o j, n 2 HJnc

(5.12)

Puv T +e+ 0 0 0 11,n 0 0 0

c2 (m+l) 0 c 2 , n0 /

Equations (5.1), (5.4), (5.5) and (5.6) give Sn, n J and W as linearJ~' j~n ,n J,n

functions of the shock perturbation parameter, w Jn by substitution from the boundary

conditions, Equations 14.29). That is, they give the variables In the forms:

- S (1) S (2) WSJ,n J, n +J,n J,n

H = H(1 ) + H(2) wj,n J, jn j,n

j 1 (1) + j (2) (5.13)Jj,n J, n J, n Wiln

and:

( W(1) + W(2)J,n j,n Jn Wj,n

where the superscripted variables are kno.wn functions of w (or are known constants in

some instances) which can be determined numerically by quadratures. With the super-

scripted variables in Equation (5.13) known, the quantities, 1 and I-2 . which determine

the righthand side of Equation (5.7) are also known in the sense that the superscripted

quantities in the expressions:

r "(1) + n U1 (2) WJn

(5.14)

F2 2 2 Wj, n27

6

are known functions of w. Thus, Equation (5.7) is a second-order ordinary differential

equation of the form:

d uJ'n I dA djun + Bu T= rW(.5

d dw) d J, 1 2 j'n

where r"1 and 7- 2 are known functions of w, and w j,n is an unknown constant. The

solution of this Equation involves two additional constants of integration, K and K2 ,

and can be written in the form

u u1 + U(2) W +g K +z K (5.16)Jnn Jn nJn J, n 1 J,n 2

where:

z is a solution of the homogeneous equation,

d2 zjn 1 dA dZ J, n 0

A dK d u + j,n

Subject to the conditions,

dzzj, n (WB) = 0, d '.; = 0- Bo

(These boundary conditions are chosen so that zl,0 = v );

g J,n is a linearly independent solution of the homogeneous equation (Reference 10,

pg 33).

C -t, (5.17)J n nn Az j,n

so

2 I8

and u. u(2) + is a particular solution of equation (5.15), (Reference 10,J,n J,n •J,n

page 30)

u(1) + u(2) w Agn (TI+ r dwjn ujn °jn = fzj A (j1n j3 n 2 ;

Az (,+w idw

"-j, n f J, n + w, 2 i

The particular solution is separated into the equations

u(k) -A g dF-g Az dw fk=1,2)j,n j,n f jn kjn k

Wso so

Putting

(m +1) r1(k) r (k)-el+

and A from equation (5.8) into the expression for u , k and Integrating by parts to i

d r0 k)

eliminate 1_ gives •

sin w

d 0(kk)

wid fro eqatio (5.) ito te exresion or u andinteratig (5.ars8t

d r (k+) j-geliminate,' gie

U J.•n - (m+ 1) gi, n (ziIn + (re+) z ~n g -J~n

29

I

i ,l~ I i • | i -- -

I where: d

(I (k) (( e 0 dw (k 1,2)

(JJ30

U V

F~ -(2+3) 2 2V -cO O

The perturbation variables, v. , can also be expressed in terms of the three con-

stants, a~,K! and K2 by use of Equations (5.5), (5.13), (5.16) and (5.18).

v-(2m + 3) (2o I dg~2 d1 2'_c

_. ._. + d (5.19)J. n J,n Jn J. (n(+) dFu

where

d IIg j, 'gLo )j.n J, (5.19a)j, n J, n d A (in1)

so

I (k) dzJ,n -(k) dgJ, n

TiT z di.

30

a,I

The equations, as written, contain several limiting forms of the type 0. and

- was the body surface is approached. With one exception, J(k) these limiting

forms have finite values and the final results for the perturbation variable are all

finite at the body surface. The following limiting forms are encountered.

1. Equations (5.1), (5.4), and (5.6) contain terms of the type

W

Z / f M da wheren m o(form =0, f 0 in all cases) andm*I 0 °f v *tm

W 0 0so

f (Ws) is non-singular at the body surface. As the body surface is approached,

U.)

* -- 0 and _W d -. However, use ot L'Hopitals rule and0

su

f (w B)the equations of Appendix 1I gives lir I m u (. These terms are

handled numerically by use of t qtdrature formula having the same singularity

as the integral.

2. Equation (5.5) requires division by zero at the body surface. For m = 0,

the numerator of the ft , I, Is identically zevo so the limit is zero; for

other values of m, the limit is not finite. However. this infinity does not

appear in the subsequent numerical work as explained in items 4 and 5.

31

I -

IP

3. Equation (5.17) for g approaches the form 0.-,at the body surface due to

the boundary conditions imposed orn z J,n The limit is again finite and is:

I

Sn #" B)o-2 (B)A B)

The first derivative of g.jn (which is needed in Equation (5. 18) approaches

the form . - aat the body surface and the finite raiue of this limit is ob-

tained as follows:

differentiating gJ,n in equation (5.16) gives

dgj dz, f _.~n I 1J,n d w

dw Az. d w AJ,n Az "so

As an identity we have

I _ d uIAz-. - A-z. -dA~' ;j,n _IL Az. 4

I dA d7 dn j~ n

or, using the equation for )

A - d __)._

Jj, n In•5n(L.'12

I9

Substituting this result into the expression for the derivative of g,'In

gives

dg )onI

(F- Az dz (U.,2d gJn d w f A ( Jn)

Cso

which is an equivalent expression whi-h does not approach a limiting

form at the body surface.

4. In the computation of u(k) , Equation (5.18), integrals of the form:j,n

I _ f Jj, nd

so

where ce L- vr'e of singularities, are required. Although J is infinite atI'n

the body surface, these integrals are finite. By substituting for.A fromjln

equation (4.27e) in equation (5.2) and solvtng the result for 4 , equation

(5.5) can be written:

d H. d S.J. n - T

di 0 di J.nJ. n u

0

dH,Using tMs expression for ,! and insegrating by parts to -'emove

dS j.n d u:

and gn gKies the following cxpression for the Integralsdc3

12 TB - 11j, n• I)., D T I,

so

rlld T• /u n ' d

+ S -- (.2 Hj' T-u 2 df Sj,n Kc Hu o n u°

so

There is no problem with singularities with this form for the integrals.

f•dJ

(k)e so (k)e J.

5. At the body surface, the term 1 Jn apperring inA m+1

Fquatien (5.19) assumes the limiting form - (except when m = 0). How-

ever, when rh- expression for 1 arising from Equation f5.11) is sub-

stituted algebraically into this expression, the infinities cicel without the

application of L'Hopitals rule to give a finite result.

The numerical solution of the problem proceeds in the following order (the axisym-

metric conical flow solution is assumed to have been pre-computed).

1. Compute 10 and S (2) from the formulae which arise from substituting1,0 1,0

the boundary conditions at the shock, Equations (4.29a), (4.29b) and (4.29c)

into Equation (5.1).

2. Compute H (1) and H(2) from the formulae which arise from substituiijkg

1C0 1,0

the boundary conditions at the shock, Equations (4.29a) through (4.29e) into

Equation (5.4).34

4(1) (2)3. Compute WIo and W o from the formulae which arise from substituting the

boundary conditions at the shock, Equations (4. 29d) and (4. 29f), into Equa-

tion (5.6).

d Zl, 04. Compute z o and as described following Equation (5.16), using

finite difference methods when m ý 0.

d g,

5. Compute gl, 0 and d w from Equation (5.17) and its derivative.

6. Compute 41, 42 iI• 1, and (2) and the remaining factors present in Equa-

tion (5.18).

7. Compute v(1) and v (2) from Equation (5. 19a)." 1,0 1,0

8. Compute the constants, W 1, 0, K1 and K2 in Equations (5. 16) and (5. 19) from

the boundary conditions, Equations (4.28), (4.29d), and (4.29e).

9. Compute ul, 0 from Equation (5. 16), v 1 , 0 from Equation (5.19), $1,0, H1 , 0 ,

and W1 , 0 from Equations (5. 13), Wl, 0 from Equation (5. 6), pl, 0 from Equa-

tion (5. 3) and p1 , 0 from Equation (4.27f).

10. Repeat the preceding sequence for n = 1, 2, 3... Nmax and for j = 2,

n = 0, 1,2... Nmax to generate as many perturbation coefficients as

desired.

35 I

6. SAMPLE RESULTS AND COMPARISONS

Figures 3 through 14* show distributions of the perturbation coefficients defined in

the preceding Sections, across the shock layer of a 10-degree cone at Mach 10 for ideal

gas conditions. These results show a tendency for the magnitude of tI.e perLurbation

coefficients to decrease with increasing n. This tendency continues through n = 10 (the

highest value for which results have been obtained) and supports the use of series of

the type given in Equation (4.26) in obtaining a solution to the unsteady flow problem.

The perturbation coefficients are fairly well behaved. Near the body, some - such

as those in entropy, Figures 11 and 12 - have derivatives which vary like (W - I.B)

due to the v which multiplies the derivatives in Equations (4. 27b) through (4. 27e).0

This behavior does not cause any difficulty in the numerical work, however.

The perturbation coefficients of the shock angle corresponding to the results shown

in Figures 3 through 14 are given in the table below.

TABLE I

PERTURBATION COEFFICIENTS FOR THE SHOCK SHAPE

W B 100, M , 1.0, ) = 1.4

W = 0.215 radso

1, = -0.0969 2,0 .176

1, = 0.0742 W 2,1 0.0112

1,2 0.0319 2,2 0.0104

, U - - 0.00737 .2,3 -0.00250

*All numerical results shown in these and other figures are for an ideal gas with) 1.4.

3 6

If the body were pivoted at 50 percent of its length and oscillated such that

a, 0 sin 2 Ift

q =2 i f E cos 2 ii ft

with V fL < <1, the shock would be given approximately byU_

220. ~O215 - 0. 0969 1 + 3. 65 x - 0. 92 -L + 0. 125)(127rfL)

s L 2 L L L.

sin 2rrf t+ 2.62 (x-0.191L))

The amplitude of oscillation in the shock is essentially the quasi-steady valiqe as-

sociated with the cone at small yaw under steady-state conditions and the time lag,

2.62(x-0. 191L) is quite small, in general.

U_

Figures 15 through 28 give static and dynamic force and moment derivatives

resulting from integration of surface pressures. The static normal force derivatives

given in figure 15 are identical to those given in References 11 and 12. The normal

force derivative CNq , applies for rotation about the cone vertex. For other locationsqo

of the center of gravity it is given by:

x

CN ý CN L2 Nq qo a

Figure- -19 and 30 show comparisons of the numerical results for unsteady normal

!o,.-•-,- C•.vat,;es \ith results obtained by several other methods. These are: 1) the

second-order potential theory due to Tobak and Wehrend (Reference 13); 2) the unsteady

flow theory due to Zartarian, Hsu and Ashley (Reference 14); and, 3) the Newtonian

impact theory. In order to obtain the form of results shown in Figures 29 and 30 from

the results of Zartarian, et. al., it was necessary to substitute expansions of the form

a (t -X/U) = (t) - cx +

in Equation (38) of Reference 14.

Figure 29 shows a fair agreement between the present results and potential theory

for a 10-degree cone at low Mach numbers. The agreement for a 20-degree cone

(Figure 30) is not so good, however. At higher Mach numbers, the impact theory

does not predict CNqo and CN& as well as might be expected based on results for

CN . The agreement with the theory of Reference 14 is good over limited Mach No.

ranges for both the 10-degree and 20-degree cones but the two sets of results tend to

diverge at both high and low Mach numbers.

Figure 31 shows a comparison of numerical results with experimental data for a

10-degree cone at Mach 10 over a range of center of gravity locations. The data also

cover a range of Reynolds numbers. The agreement between theory and experiment is

quite good except for the highest Reynolds number. Figure 32 shows the experimentally

determined variation of the dynamic stability parameter, Cmq + Cm& , with Reynolds

number and gives some indication that tie disagreement just noted may be due to

boundary layer transition.

38

I'

Figure 33 shows a comparison of theoretical results and experimental data for .1

20-degree cone at Mach 8. The data are for varying angle of attack but show very little I

influence of angle of attack. The agreement between theory and experiment at zero

angle of attack is seen to be quite good.

t

Figure 34 shows the theoretically predicted effect of frequency of oscillation on the

dynamic stability parameter, Cmq + Cm&, for a 10-degree cone in a forced-oscillation

wind tunnel experiment. The values of the constants, K1 , K2 , etc., indicated on the

Figure 34 are related to the unsteady force coefficients, CN. CN4 , etc. The result

is that there is no detectable effect of frequency over the range of frequencies en-

countered in wind tunnel testing. This is not in agreement with experimental observa-

tions which show a fairly large effect of frequency (Reference 15). The effects of

frequency appear Lo be a strong function of Reynolds number.

3i

39 1

a

1.0uU

CONE SEMI-VERTEX ANGLE, wB 100

IDEAL GAS, Y= 1.4

M -10

0.6

U1 .3

0.x 100

U U1 ,11u at-u x 10

0.2

u• X1

SUl1,2

-0.2 U1,0

U,

-0.4

-0.6

10.0 10.5 11.0 11.5 12.0 12.5

w - DEGREES

Figure 3. Perturbations in the Velocity Component, u

40

1.0 u

CONE SEMI-VERTEX ANGLE, wB = 1000. 8 IDEAL GAS, Y := 1. 4

M =10

0.6

0.4

UM

U 2.3x10u

0

U .2x 10U

-0.2 /

-0.4

-0.610.0 10.5 11.0 11.5 12.0 12.5

w DEGREES

Figure 4. Perturbations in the Velocity Component, u

41

tI

I ,

0.6

CONE SEMI-VERTEX ANGLE, w = 100IDEAL GAS, y =1.4

M= 1 0

0.4 -_._ _10

U. Ix 10

0.2

-0.

v 1,3 I 1 0

-- 0.2

U 0 ,,

-0.

-0.62

-1.010.0 10.5 11.0 11.5 12.0 12.5

SJ. " DEGREES

• • Figure 5. Pertuarbations in the Velocity Component, v

42

-0.

0.6

CONE SEMI-VERTEX ANGLE, w = 100

IDEAL GAS, Y 1.4M =10

0.4

vI

i0.2

-0.2

-0.4

-0.6 -2

UU-0.8

-1.010.0 10.5 11.0 11.5 12.0 12.5

w-- DEGREES

Figure 6. Perturbations in the Velocity Component, v

43

1.0

0.8 ,

0 .6 ,_,_

CONE SEMI-VERTEX ANGLE, w B = 100IDEAL GAS, y 1. 4M. = 10

0.4

0.2Wi

W 1 .2 x 1 0

-0.2 ..

-0.4

-'.g

-0,x to-0.2

-0.4N

-1.0 ___ _

10.0 10.5 11.0 11.5 12.0 12.5

)DEGREES

Fitgure 7. Perturbation*b in the Velocity Component, w

44

I

1.0

0.8 .w 2 ,0U.

0.6

CONE SEMI-VERTEX ANGLE, wB 100

IDEAL GAS, y = 1.4M = 10

0.4

wU

0.2

-202Su---:x 10

SW~~2,1i.0 U x t. o-

-0.2 w2'x 100U.

-0.410.0 10.5 11.0 11.5 12.0 12.5

W DEGREES

Figure 8. Perturbations in the Velocity Component, w

45

l.0

CONE SEMI-VERTEX ANGLE, 10°t

IDEAL GAS, Y 1.4

0.8 --" 10 Po-. m 10 -0.Pl

_____O__

0.6

0.4 Pl,2

P q.

0.2

0

x 10-0,2

10.0 10.5 11.0 11.5 12.0 12.5

DEGREES

Figure 9. Perturbations in the Pressure

46

1.0 I

CONE SEMI-VERTEX ANGLE, W. 10 0

IDEAL GAS, Y 1.4M 10o~s o

0.8

!_0x 10

0.6 p 29

0.4

p P2,2

0.2

-0.2

P, x 100q.

-0 .4 ..

10.0 10.5 11.0 11.5 12.0 12.5

-DEGREES

Figure 10. FerWrbation, In be Pressure

4.,

8

CONE SEMI-VERTEX ANGLE, B 100

IDEAL GAS, Y 1.4M= 10 S/RI,0

S/R0 x 10

- S/R,

00

00 S/R 1,3-2

S-2

-6

iL 10.0 10.5 11.0 11.5 12.0 12.5

WU - DEGREES

Figure 11. Perturbations in the Entropy

48

b

60/

CONE SEMI-VERTEX ANGLE, UU B IrIDEAL GAS, Y 1.4MaJ= 10

5 ~S/R 2,0

4 -- /-S/R 0 x 10"

SR

2

0

-210.0 10.5 '1.0 11.5 12.0 12.5

SDEGREES

Figure 12. Perturbations in the Entrop

49

• l ra • m mi • w m mm -

6

H x 10

CONE SEMI-VERTEX ANGLE, u= 100 U co24 IDEAL GAS, y = 1.4

M =10

2

H-2 _" "_Ix 100 -,

U.2 2

-4 ______

-6

10.0 10.5 11.0 11.5 12.0 12.5DEGREES

Figure 13. Perturbations in the Stagnation Enthalpy

50

6

H..x I0U 2

5

CONE SEMI-VERTEX ANGLE, w = 10°0

4 WDEAL GAS, y= 1.4M =10

00

3

H2 0 x 10

2 _U2

-- I

H 2___ __ _23 x 10

2 U7

0H'2.2x 100

U 2

H2 1 x100

-3 - -U

-4

10.0 10.5 11.0 11.5 12.0 12.5

) •DEGREES

Figure 14. Perturbations in the Stagnation Enthalpy

51

r 0 to 0 0 O

0 Ir4 C-4 m

C4

OD

0

= EI 0

z

0 L

4 -4 04

52_

° -4

104

z

1a

C.) 0

AO..

/ 00or C_ _ _000 _ _ _ _0 -0_ _

- x

44

-4

zC.)

00o0 (Dig 0

NVI 11

-44

z

W44

100

Noo

_ 5_

00

z

000

CC.)

NVI z b N.0

- q

_________ ________00

0-4

ac cla a

___b __: __

_ _ _ _ _ _ _ _5

Ill~I__O

II ii/ a K

j.,j

C

-__to-__.P 4

40,0584

I

0.4

Cm& w B =5 DEGREES

M0 1

-C7.8 -0.4 -. .3 -

50 . .. .

-1.2 MC O

--1. ,6 ____20

-2.0

1

!I

-2.40 0.2- 0.4 0.6 0.8 1.0

x-2L

Figure 22. Dynamic Moment Derivatives, C and C

59n

0.4w = 7 DEGREES

Cm&

0

-2.0 1

-2.4 "

0 0.2 0.4 0.6 0.8 1.0

X

N Figure 23. Dynamic Moment Derivatives, C and C

Cm q

•. 60

L

*10

0.4

Cm6 0 ERE

0 a

Cnq M

]/'q

-1.2 1.5

-1.6 ___20__

-2.4__ _ _ _ __ _ _ _ __ _ _ _ J0 0.2 0.4 0.6 0.8 1.0

x

L

Figure 24. Dynamic Moment Derivatives, C mand C

61

- - JI

0.4,

C -- 15 DEGREES

-0.8

-1.2•2. ....

-1.6

-2.02

-2.4 -

-2.80 0.2 0.4 0.6 0.8 1.0

xegL

Fip"r 25. Dynamic Moment Deriv*Uiws, C and C Mq

62

0.4

Cma &ERE

0

-0.8

-1.2

-1.6

-2.0

-2.4

-2.80 0.2 0.4 0.6 0.8 1.0

x

L

Figure 26. Dynamic Moment Derivatives. C In %W C Iq &

63

w8 = 30 DEGREES

Cm6L -M = - -- -

-0.4 m

-0.8

Cmq

-1.2

-1.6

-2.0;_ 1 -- - - -- -/, "20

-2.4 -

-2.8

-3.20 0.2 0.4 0.6 0.8 1.0

x

L

Figure 27. Dynamic Moment Derivatives, C and Cmq

64

Cm'

-2

-4 2

_________________ _________________ _________________ ________5________ _________________ ________________ ________________

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4x

L

Figure 28. Dynamic Moment Derivatives, C and Cm Meq

65

f16

10-DEGREE CONE

14 FLOW FIELD ANALYSIS

CTN 100 TOBAK AND WEHREND. REFERENCE 13Nq% (Me= 5. 8)

ZARTARIAN, HSU, AND ASHLEY,I REFERENCE 14

12 ---

8 N M WTONIAN

6 -

CN i

2 TAN 100

4

0

0 4 a 12 1s 20 24

MACH NUMBER

Figure 29. Comparison of Theoretical Results

66

I

720DEGREE CONE

FLOW FIELD ANALYSISTOBAK AND WEHREND, REFERENCE 13

. = 2.9)6 - - ZARTARIAN, HSU, AND ASHLEY,

REFERENCE- 14

CN 5

2 TAN 200

3

2

CN&

2 TAN 200 -

% % / NEWITONIA0-

-

0 4 8 12 16 20 24

MACH NUMBER

Figure 30. Comparlosm of Thmoretical Results

67

I

REFERENCE 15

Re/FT2. 22 x 106

A t,2 x 106

0 0.3 x 106

--- UN UNSTEADY FLOW FIELD- -.-.- NEWTONIAN

x_9L

0 0.50 0.55 0.60 0,65

-0.2

+ -0.4

-0.6

-0.8 -

Figure 31. Comparison of Theory and Experiment

68

_____

6z

0

00

0 b0

0

IIl_ _ _ _ -vim

o r$4

0 (0J

69

-4

0.1I

I--- DATA TAKEN IN A.E.D.C. TUNNEL B, AEDC-TDR-62-11

A MODIFIED POTENTIAL THEORY DUE TO TOBAK AND

0. - WEHREND, N.A.C.A. TN 3788 (M - 2.9) REFERENCE 13

C mq = -0. 349

U RESULT FROM FLOW FIELD ANALYIS Cm= -0. 073

-0.1 - Cmq+Cm - -0.422 -

X ZARTARIAN,. HSU AND ASHLEY, REFERENCE 14

-0.2 -20-DEGREE CONE, = 0.667

L*ti NEWTONIAN

+ -0.3POTENTIAL THEORY(M = 2.9)

-- F 7-0.4 C-0

FLWFIELD

-. FROM REFERENCE 14 EST, MAX UNCERTAINTY-0.5 IN DATA 0.10

-0.6 I t i i I A0 2 4 6 8 10 12 14 16

ANGLE OF ATTACK, a DEGREES

Figure 33. Comparison of Theory and Experiment, M 8

70

1o 08

Cinq+Cmm"l1 klU2 +k 2 •4 ''"t (Cmq+Cm) 0

10-DEGREE CONE M " 10

S• ~x3

1.00•+ e g Io

S0.96 - x 2

0.92 ... ....

0 0.1 0.2 0,3 0.4 0.5 0.6

REDUCED FREQUENCYW w ', 1Um0

Figure 34. Theoretical Evaluation of Frequency Effects

71 I

,-s4

REFERENCES

1. F.G. Gravalos, I.H. Edelfelt, and H.W. Emmons, "The Supersonic Flow about a

Blunt Body of Revolution for Gases at Chemical Equilibrium," GE TIS R58SD245,

June 16, 1958.

2. Vito Volterra, "Theory of Functionals and of Integral and Integro-Differential

Equations," February, 1958.

3. B. H. Lamb, "Hydrodynamics," 6th Edition, New York, 1945.

4. F.G. Gravalos, E. Brong, I.H. Edelfelt, "The Calculation of Flow Fields with

Secondary Shocks for Real Gases at Chemical Equilibrium," GE TIS 59SD419,

May, 1960.

5. G. L Taylor, and J.W. MaccoU, "The Air Pressure on a Cone Moving at High

Speeds," Proceedings of the Royal Society (A), Vol. 139, p. 278, 1933.

6. Mary F. Romig, "Conical Flow Parameters for Air in Dissociation Equilibrium,"

Convair Laboratory, May, 1960.

7. C. Johnson, "The Flow Field about a Right Circular Cone at Zero Yaw," GE TIS

62SD211, November 1962.

8. E.A. Brong, and I. H. Edelfelt, "The Flow Field about a Slightly Yawed Blunt

Body of Revolution in a Supersonic Stream," GE TIS R62SD111, March, 1962.

9. A. H. Stone, "On Supersonic Flow Past a Slightly Yawing Cone," Journal of Mathe-

* ,matics and Physics, pp. 67-81, Vol. 27, 1948.

72

9..P

10. F.B. Hildebrand, "Advanced Calculus for Engineers," Prentice-Hall, Englewood

Cliffs, N.J,, 1948.

11. J. L. Sims, "Supersonic Flow around Right Circular Cones Tables for Small

Angle of Atteck," Rept. No. DA-TR-19-60, Api;I 27, 1960.

12. Z. Kopal, ed., "Tables of Supersonic Flow around Cones," Mass. Inst. of

Tech. Report -No. 1, 1947.

S 13. M. Tobak and W. R. Wehrend, "Stability Derivatives of Cones at Supersonic

Speeds," NACA Technical Note 3788, September, 1956.

14. G. Zartarian, P. T Hsu, and H. Ashley, "Dynamic Airloads and Aeroelastic

Problems at Entry Mach Numbers," Journal of Aerospace Sciences, Vol. 28,

March 1961.

15. R.B. Hobbs, "Results of Experimental Studies on the Hypersonic Dynamic

Stability Characteristics of a 100 Cone at M f 10," GE ATDM 1:37, May 22, 1964.

73

J .""

APPENDIX I

SUBSTITUTION FORMS FOR THE PERTURBATION VARIABLES

Foria ____

Pi d-, a R co pn -0 dtn

= U , n coo os

0

p 8 (W) noo cPn=Od0 d •

5 Wl,a (a) sRne

nFO dt

1 dU dtn a

74

n ,• ) •i

C tn

n 0

(W1, are constants)ii~~~1 n ,U

11nSForj2 )o) -a

n 0

1-i R -q

i02, n(W) L• -Pl, n M• L d Cs C

= dtdn=O

v c I R

S2= 1 S2 , (w) ~.-Sl~ in(~)n=0

R g /R \ dcoo

Mv(w) 29dn 0

R L gJ(

2=1 2, 1 R)n. vidtn 0

75

IXI.

s Iw 09~l( \ms• = 2, n nL L I

n0 d t

4

; t02, uare constants)

S"76• •W •''• .......

U

APPENDIX 1I

EQUATIONS FOR THE FIELD AT ZERO YAW

The following equations determine the unperturbed field at zero yaw:

the continuity Equation,

d P o Vo sin w pts n odw + 2PoUo inw =0;

the condition of irrotationality,

d ud -vo =0;

the state Equation,

Po = P (Po So);

the integrated energy Equation,

S constant

and, the Bernoulli Equaion

h (ps) + 2 C cont.

The first of these equations can be re-written:

Vo )--"• 4 o Voinw ;• 4 + 0 )-7o L -( 07

77•

______

. iu

and is used in deriving Equations (5.1), (5.4), and (5.6). These eluations are used to

replace derivatives of the flow field quantities at zero yaw by equivalent expressions

in terms of the flow field variables themumelves.

78

S p

APPENDIX HII

EXPRESSIONS FOR THE NORMAL FORCE AND MOMENT

Integration of the perturbation description of the pressures on the surface of the cone

gives the following expressions for the normal force (negative of the force In the y direc-

tion) and the moment in the z direction about the center if gravity.

I!

- F (iD U naCNAY N1, n U)n+N2, n-- + NI-IN q AB d tn

n=0 n00

M Nz .dn (n+2) D d

- 2 - (n-3L q tAnBL co( )B dtn

N )(n+3)

S2, n (n4)

n 0

Xg 29 Nn~ 'A RI +

L tanx B (n+3) ) 2.B 2 U. ) dtn L

79

I

where:

N PI, n. 008 WBn q n+l

C1 (n+3) sin WB

NP2, n Sps W BN2, n - qGO (n+3) sinn+2 Iu B

The normal force derivatives defined in the symbols section are related to the

NJ, n as follows:

2CU__n a Cs

= N

a d tn )

() n+1 a CN (2 tanw u 18 N

80

UNCLASSIFIEDSO=uit Classification

DOCUMEIW CONTROL DATA - R&D(.modiey elaeellifcaetio of title. b6di of abstract OW nd .nd..M evnotstl*, mwsat be entered men NMe overall tsport is ciptallled

I flRIGlNATINO ACTIvýY (Coopmvafe auth) 20 RIEPORT0SECURITY C LAII104ICATI~ft

General Electric Company 26GOPUffCLASSIFIEP.0. Box 8555, Philadelphia, Pa. 19101 NbARU

3 REPORT TITLE

The Unsteady Flow. Field about a Right Circular Cone in Unsteady Flight

4 DPSCRIPTIVE NOTES (Type of report and inchialve date&)

Fin&.l Report: - Nov 1966S. AUTHOR(S (Last ntme. first name. initiIl)

Brong, E.A.

6. REPORT DATE 19770. TOTAL NO. O1F PAOES 76. No. or REps

do. COW'T ACT OR GRANT NO. On. ORIOINATOR'S RIEPORT HUMBIZA(S)AF33(;57)-l1411

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D~istribution of this doc')ment is unlimited

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AFFDL (FDCC)NA Wright-Patterson AFB, Ohio 45433

13 ABSTRACT

The inviscid flow field about a right circular cone in unsteady planar flightis analyzed by a perturbation technique which is an extension of Stone's treatmentof the cone at small yaw. A solution is found in the form of infinite series inthe time rates of change of the pitch rate and angle of attack. The linearstability derivatives, Cmq and Cm; as well as "1highier order" stability derivativessuch as C4and Cqqare presented for a wide range of cone &ngle and Mach numbers.The stability derivatives, C,.q and Cm&, as obtained from this solution arecompared to results obtained from second order potential theory, Newtonian imrpsct

theory, and an unsteady flow Lheory due to Zartarian, Hseu, and Ashlej. both thepotential theory and the impact theory predict that Cw;,rapidly approaches aero athigh Mach numbers while the present theory indicates that Cu& approachesa valuewhich is on the order of 10 precent to 20 percent of 11111. 1Numerical results obtained from the present theory are also t~ompared toground-test data .on m + Comm. The agreement is found to be generally good,although~ the data in some instances indicate a pronounced Reynolds number effect.tThe numerical results for the "higher order" coefficients are used to predictthe effect of reduced frequency on the parameter Caq + C.,, as obtained by theforced oscillation testing techniques. It is found that the predicted effectin very small over the range of reduced frequencies likely to be encountered.

D D .1 1473 UNiCLASSIFIEDSeculity Clasafficstice

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Dynamic stability derivativesSupersonicReduced frequency effect.Ursteady flow theory

148TRWuTIONS,

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