right circular cone in unsteady flightmarshall. tnis technical report has been reviewed and is...
TRANSCRIPT
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
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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
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