l-ABORATOR'UM VOOR AERO-EN HYDRODY"'~MICADER T. H. COMMENTS ON THE AERODYNAMICS OF LOW
-ASPECT RATIO WING-BOD-f.'nfil
ëo
~""",,H
\ç;-.-r~4'lcf"l
IN STEADY SUPERSONIC FLOW
78
DECEMBER 195 2
BY
Bibliotheek TU Delft
Faculteitder Luchtvaart- en Ruimtevaarttechniek Kluyverweg 1
'2629HS Delft
..
..
COMMENTS ON THE AERODYNAMICS OF LOW ASPECT RATIO WING-BODY-TAIL COMBINATIONS
IN STEADY SUPERSONIC FLOW
BY
G. V. BULL
Head, Äerodynàmics Section,
Canadian Armament Research and De ve loprnent Establishment, (C.A.R.D.E,,)
Defence Research Board
A Seminar giVeh at th e Institute of Aerophysics 11 December 1952
ACKNOWLEDGEMENT
The Defence Research Board has kindly permitted the author to present this seminar; much of the work included here is presented against the background of projects on which the author has been employed at the Canadian Armament Research and Development
Estab Is hrnent, .
The aid of Professor B. Etkin and Mr. L. R. FoweL of the University of Toronto and Mr. G. H. Tidy of C~A"R. D. Eo in
the general proof-reading of the manuscript is gratefully acknowledged; while stncer-e appreciation is tendered to F /L J. P. Dixîe, R. CoA. F.., for drawing Figure 1 and to Mr. B. Cheers of C.A.R"D.E. for drawing the remaining figures.
SUMMARY
During the course of this s em.inar- an attempt is made to illus t r a t e the nature of some of the problems encountered in the de sign of missiles to operate in the supersonic Mach number r-ange; Some basic theoretical :ne th o ds applicable to missile configurations, are re -viewed and applied to configurations empJ.oying rectangular plan form panels arranged in cr-ucifo rm , Assuming no interference between body and pane1, the pressure distribution for a thin, flat, rectangula r panel is used to compute the chordwise element of lift, the panel lif t cqeffic:i.ent s lope, the spanwise-c e n t e r of pressure and panel aileron power. The results obtained by Beskin for a body of revo ution in supersonic flow are reviewed and the cruciform wing-body problem considered by dis -cussing the effect of the body on the panels, the effect of the panels on the body, and the effect of one cruciform panel on the othe r, The wing-tail int erf e r e n c e problem is discussed and some results presented. A briefdis c u s sio n of viscous effects, skin temperatures and elastic defor-· mations is in clu d e d .t o wa r ds the end of the seminar .
NOTATION T ABLE OF CONTENTS • • • • • • • • • • • • • 0 0 0 . 0 0 · . 0 · • • e . · . 8 • • • • • • • • • • • • • INTRODV CTION • • • • 0 0 • • • • • 0 • • • • •• • • • • • • • • • • 0 • • • • • • • • • 1 I. Il. 111.' IV.
V
.
VI. VII. VIII. T'HEOR Y .••0 • 0 • '"• • • • •Q 0 • • ~ • • 0 0 • • • • • • • • • • • • Cl • • • 1.1 General . .• . . . •. . '! . '1. 2 Th e Lineariz ed Pote n tiàl Equa tion .•• .. •• •• • THE GENERAL THIN WING PROBLEM, ••..•..• •• •,
I
PARTICULAR RESULTSFOR THIN RECTi\NGULAR. ,
.., WÎNGS •• • . 0 fit• • Cl• • • • • •,• • • • •• • • • " • • • • • • • •'• •• • • ~ •
'-3. l Solu t.ion'of the Fundamental Equation ••. • • •0:
3. 2 Lift Dist ribution Ac ross a Thin Wing Panel ••
,3, 3 Chordwisè Elemen t of Lift o • • • • • • • • • • • • • • • •
3.4 Panel Lift Coeffi cie n t Slope •• • . •• • •~ •.•0 • • 0
3,5 Panel Aileron Power. 0 • •• •• • • 0 • • • • • • • • • • • • 3.6 Spanwise Centr-e of Pressure of a Panel •• •0 •
BODIES OF REVOLUT ION IN SUPERSONIC FLOW •• THE WING-BODY PROBLEM •• ••••" •••••• ••, ••••• 5. l Inte ra c t i on Potent tal , 0 • • • •" • • • • • • • • • 0 • • • • •
5.2 The Effect of the Body on the Panels •••.•••• 5.3 The Effect of th e Pane l on thë Body •••• .. • • • 5. 4 Comp1etnentary Pa ne1Int e r f e r e n c e ••• • ••• 0 0 THE WING - B O DY - TAIL PROBLEM ••0 0 0 • • • O ' 0 " 0 0
THE EF FECT OF THE BOUNDARY LAYER ..•. 0 "
CONCLUSIONS AND DIS C USSION •.• •0 0 • • • • • • • ~ • • •
3 3 3 8 13 13 14 14 16 17 18 19 20 20 2 1 22 24 3 1 35 37 REFERENCES •••.. •••.• 0 • • • • • • • • • 0 • • • • • • • • • • • • • • • • • 0 . FIG UR ES 1 to 11 . 0 • • 0 • 0 • 0 c 0 0 • • • 0 . . .0 • • • • 0 • 0 Cl 0 • • • • • 0 • 0 • • •
NOTATION
In general, symbols are defined in the text when in t r- o du c e d,
M
V ·
u. v,W
Mach number-,
Potential function (throughout the major portion of this work
it is the ac eleration potential).
Free stream velocity (except where it denotes a volume of integration) •
Free stream perturbation velocity components,
Axes Systern s
-(1) During the first part of this work the -axe s system origin
does not particularIy affect the work and hence the ord
in-aryco-ordinates xyz have been us e d,
(2) During the part of the work dealing with the finite wing
pr-ob lern the axis system origin is important and two reference axes systems have been employed :
(a) Tip axi.s system (origin at the tip leading edge) and co-cor-dina.te s with respect to.this system have been denoted by the primed quantities,.x'
s'
z'.(b) Body axis system (origin at the intersection of the wing leading edge and body center lin e) and co-ord
in-ates with respect to this system have been denoted by the quantities xyz.
r d b s Body radius Body diameter
Wing span (tip to tip).
Wing semi-span
Deflection of toe wing panel measured from body center lin e,
q
r
Dynamic pr-es sur e , Free stream density,
S Reference area (usually the body cr o s s - s e c tion area AB)'
c Wing chord
n c/r
Lift coefficient = Lift/ q AB
Local or section lift coefficient = Section Lift/ qc Rolling moment coefficient = Moment/qcAB
Downwash coefficient. Sidewash coefficient •
The induced rolling moment coefficient due to carry-over of the pressure fields on to the complementary panels, assuming that one planar pair of panels are undeflected whilst the other planar pair are asymmetrïcally deflected
+6
and-6
from the body center Iine ,IJ
·r
M~-i
.,
.$.Ir ~~~
(
~CI)
db W....'"-...
-
'
(1)
INTRODUCTION
As a suitable s olution to the problems-of performance, stability and contr-o l, missile configur-ations have been adopted which
employ Iow aspect ratio wirigs and tails arranged in cruciform .on r-ather.Iar-ge diameter, bodies , "T'ypical, of .s u c h ,c on fi g u ratio n s is that shown in Figure 1.. Becauae the analyses of the system require close _.integration of the contr ol, guidance and ae r odynamic characteristics,
'.accurate pr-edicticnof the configuration aer-odynamdc properties is a prerequisite to the missile de'sign problem.Consequently many of the tnterïér-encèmd Inter coupltng effects , neglected or approximated at low velocities, must be treate d in detail for the high speed case.
• • ' . • . • ,:: , ' I . • . ", '~ ' '.. • " •
. Most'I11issiÏe configu;roations,.whether
:
lau~ched
frorn the.grburid
or
.f'r-om a cárrie'raircraft, 'are requiredto·perform thr-ough:,;the s~bsonic~n'd'fransonicspee(frangesase'weu' asatrsuper-sonic Mach
" number-s, The field of mfs sîle ae r-odyriamic s thussernbr-ace s the whole
, ,..,field offluid'mottón, compr-essfbIe and Incompr-eseib le,s teady and non-e teady, Whilst yielding many useful res ul.ts in practtce, the cus -tomaryas sumptîons ofHnear-ity and ideal gas flow, made to r educe the
, equatdons ofmotion toa mathematically tractable form, represent only
,"lim iti ng cas es forthe low aspect ratio configuration. At.very low sub-sonic speeds, where the Hui d may be consideredla s incompr-es s ible , the
Iinear-Ize d, Invi.s cîd theory yields results valid both quantîtatdvely and qua lit:ativeIy , At high subsonic velocities nori-linear effects become
.i mpor t a n t and quantitative agreement no longer exis ts between the ex-perimental r-esults and Iänear-ized, inviscid theory peedictton, Flax and Láwr ence, reference I, have presented an excellent treatinent of the,case of low aspect ratio configurations in high-subs-erric flow, where
non-Iinea r and viscous effects are predominant,and the problem here is e ssent.ia.Lly that associated'wi.t h shock wave -boundar-y layer interaction phenomena; Fora treatmentof this pr-oble m, the reader is referred to the work of Liepmann arid Roshko., reference 2. -Thr oughout the super-soriic,Mach number range,, lin e a r i z e d'inviscid theory yields results which generally are s uff'ic ierrtly accurate for practical-design computa -.tions, .pr- o vi ding the inddences enc ounte r-ed arenoLlarge •. The
assump-t.iona.of zero viscosity break down, of course, when.b ou n dar'y layer effects become large; such as near a wing-body juneture or near the
trailing edge of a wing where conside rab le thickening of-t he boundary layer due to interaction between the boundary la y e r and the trailing edge
,s h oc k wave occur-s , The zero-lift drag characteristics are largèly de-pendent on the nature of the boundary layer, and failure to pr-e dict accurately boundary layer transition points can result in fairly large
(2 )
errors in the drag estimates. In addition foifailure at high inc idence s,
the assumptions of linearity break down in the case of the wing-body com-bination where the linear addition of the isolated fields of the wing and the
body produce a resultant field which does not satisfy the boundary
condi-tions of the physical sys te m,
In this serniriar-, we shall confine ourselves to the supersonic
regime or-, more specifical y. to that range of Mach number where the
effective panel aspect ratio is greater than unity(that Is , to that range of
Machnumber where the tip Ie a ding edge Mach cone is contained inside the
cone generated by revolving the panel diagonal aroundan axis through the
tip leading edge point a....nd parallel to the free s tr-eam velocity vector).
F'urther-, we shall consider only configurations employing rectangular
plan-form pane ls , although the basic methods are not necessarily
res-tricted to this type of configuration. Although some of the most
impor-tant aspects rela ting to the automatic control of guided missiles concerns
the dynamic characteristics of the ve hîc le, we shall lim it our scope to
those aspects relating to steady state fli.ght, For a treatment of the
non-steady case. the reader is referred to the work of Garrick and Rubi.now , reference 3.
(3)
I. THEORY
1.1 General
A body or wing, immersed in a fluid s tr-eam, sets up abou t
itself a disturbed flow field. In the case of a free stream velocity some
-what less than the velocity of sound, the fie:ld of disturbance about the body can be considered as extending to infinity in all directions. In the case of a supersonic free stream velocity, the field of disturbance is confined to a finite region in general, the region bounded by the envelope
of the after Mach cones , If a body is immersed partially or totally In the field of disturbance about another body its aerodynamic character
-istics will be modified. If the perturbated ve locity of the disturbance field is very sma l l, as in the case of very low s ubs onic, free-stream velocities, the modification to the aerodynamic cbaracteristics will be
correspondingly smalle On the other hand in the-case of supersoni c free
stream velocities in which a body is totally immersed in the fie l d of
dis-turbance associated with another body (as is gene rally the case for
al-most all low aspect ratio missile configurations) considerable alter a tion
oI'.the aerodynamic chara teristics may be f'ound,
The ma in in t e r fe r e n c e fields associated with a missile con -.figuration are shown in Figure 1. The body disturbance fie d is conf i n e d
to the region in t e r i o r to th~ con ic a l shock wave origînating at the body .nos e , Unde r the conditions depicted, both the wing and tail He comple t ely
inside the body dîsturbance field. (At higher Mach numbers the vert ex
angle of the conical shock de c r-eas es, and eventually t he nos e cone shoc k will lie in s i de the wing leadîng edge tip, 50 that onlypar-t of the win gs wï 1 be affected by the body disturbance field . ) The tai l, in addition to lying
within the body disturbance field, also lies within the disturbance field of
the wings , The body, besides being im m e r s e d in its own field. is partially
.im m e r s e d in the dîsturbance fields of both the wings and tail,
1. 2 The Linearized Potential Equation
In this report, as is customary throughout most compr e s s i b l e
flow work, it is assumed that the linearized form of the potential equation
is valid; 50 that disturbances to the original s tr eam, produced by the imrnersed bodie s, are sufficiently small that only terms in v o lving their
first powers need be retaîned. Some dîscussion of the lim ita tion s intro
-duced by this assumption will be made later. The potential e quat i o n , in
(4)
(1)
where
rp
may be either the ve locity potentia1 defined by the re1ations :~= )
(2)
or the acce1eration potential defined by the re1ations
(3)
or any one of the perturbation ve loc ity c omponents , Following the method of Prandtl (reference 4). we define
rp
as the acce1eration potentia l, -a n d by in t e gr a ti o n of equations 3 it can be seen :~=..k-
(/
V
.o:
~ ~
17<;
rI(-t,
10-)
J-I-1~
p
-=:..à-
r
..1. (/)
(~
q
"I)
d"tj.':'0
-00Y
(5;
There are numerous mathematical methods ava.i lab le for fin ding so ution s
to the linearized potential equat.ion, By the affine transf6rmation :
'.t hi s. equation may be put into an analogous form to the two-dimensional
wave equation :
(5)
Some of the most usefu generalized mathematica l approaches are thos e
presented by Heaslet, Lomax and Jones in references 5 and 6, and Robinson in reference 7. These methods employ Green's theorem to relate the volume integration to the surface in t e g r a tion :
where
D'7t
is the directional derivative given by-Yl..L.' Y"l,a... Y1..
3 • being the direction cosines of the in wa r d vector norma l to
the.s urface, By defining the co-normal vector-
J!
in the form :~=
-n.t
)1.'2-
=
Yt4;)3 ::: Yt3
(6)
i. e .
DY\,
is now the directional derivative along the co - n o rmal, IfrIJ
and (T' are two continuous and single valued funetions whichsat-is f y the potential equation, then :
~(rr)
=
0and Green's theorem reduces to the fo r m :
ff
[i
0",~
-rI
o;
( Y]J.
s ;
0s
whic h , upon using the relations :
can be written :
[[CfJt;-rr1/]JS
=0S
(6)
If
sr
is a known particular solution to th e wave equation, and the boundary co n ditions on the surface S are known, then the integral equation (6), ca n be solved for the potential funetionrl .
In.o r de r to i lustrate the regions in v o l v e d in the preceding Inte gr-ations, et us consider a soure e of dis turbance 0 in a supersonic free stream (Figure 2). This source sets up disturbances at all points in terio r to the after con e through 0 and in turn makes them sourees of disturbances for all points ly i n g within their aftercones. Thus it is seen that th e potential at any in t e rio r point P will be due to all sour es (o r 'p oi n t s ) ly ing in the volume contained by the after-cone through the point
o
an d the fo r eco n e through th e point its elf', Thus for the conditionsde-picted in Figure 2, the vo urne of integration is the volume enclosed by
the con e s O-ABCD ani p_ABCD, and the surfa e of in t e g r a tion is the -surface bounding V.
The general problem of lin e a riz e d aerodynamics thus re -duces to solving the in te g r al equation (6) fo r th e potential funetion by fin din g a particular solu tion to th e wave equation tr , valid everywhere within the volume V (to satisfy equation (5 )) and which together with its firs t deri va tive, is known on the enc los ing surface S. The potential at
•
(7)
any point then can be determined in terms of the boundary potential and potentlal gradient along the co-normal directional derivative at the boundary.
,
.
(8)
H. THE GENERAL THIN WING P!lOB L E M
The integral form of the potentîal equation is of particular use in thesolution of general wing problems, that is in the derivation
of general wing characteristics and the flow field about wings , This -amenabil.ity sterns fr om the interjectîon of the wing as an additional
boundary, which exhibits symmetrie properties and on which the poten-tial function and its directional derivative are known, In order to clarify this point, consider the conditions depicted in Figure 3, where P is a point lying on the top side of a thin flat plate wing, In this case, since condit ions underneath the wing cannot affect the potential at the point P, the surface of in t e g r a tion is made up of Sl (the surface of the
forecone through P lying above the plane of the wing), S2 (the surface of the forecone through P) and S3 (the surface of the top side of the wing enclosed by the trace of the Ior-econej , Thus the integral equation (6) may be written :
(7) •
A s imila r integral equation ma y be written for the underside of the wing but the potential at the point P due to this part of the field, desig
-nated by
rp-
must be zero. We may write then :!f
[?'-~
-
<r--tt-]
J
S
s: 0s;
«s;
.,.S;
(8)
Before proceeding further, it is necessary to consider the nature of the
0' functîon. In general, if
o:
is a solution to the wave equation,then it wiIl possess singular points in s i de the volume V which must be excluded from the region of in t e g r a tio n, (XYZ itself wiU be such a point). By the-usua l process of surrounding the singular points with a surface and taking the lim it of the integral as this surface.s h r inks to zero, the
.pot e nt i a l at the point can be expressed in terms of the potential and the potential gradient
-i!
on the boundary. Determination of the physical conditions on the surface of the fore Mach cone .Is , generaUy,not pos s ib le , This suggests then, that 0- be chosen such that no
knowledge of conditions on the Mach forecone is required, i.e., that
both o: and ~cr' vanish everywhere on the surface Sj , This
(9)
cr=
(9)
In this.cas e the singularities are contained along the axis of the forecone, i.e. ,along the line
éJ-
~ ):a.~()E.-
~.1.)2.;::0, this line running from thepoint X YZ to Xi, YZ, its point of intersection with the envelope of the aftercones thr-ough the leading e dge , By eonstructing a cylindrical sur-face of radius
E
about this Iine, and taking the limit of the ïntegral over the surface of this cyltnder asf-Po
,
these singular points can be excluded fr-om the region of integration. Heaslet and Lomax (reïe renc e 5)have shown that .:. .
where
K
is the cylindrical surface. Using this r es ul t, .to g eth e r with the fact- that the integratton over the surface of the forecone varrishe s,equation (7) may be written : '
. ·X
~7fL
rKx.-
YlE)
d
XL
=
l:
(rP!: - rr}/)
d,S
and differentiated to yield :
. .
(10)
The integrations of equations (10) and (11) involve in te g r a t i on s over the surfaces of the envelope of the afte r cones , Although no general proof exists, it has been found for specific cases (in particular, for thin flat plates satisfying the assumptions of this work) 'tha t the integrations over the envelopes of the leading edge Mach cones are symmetric, i. e.
Hence by adding equations (10) and (11) an expression for the potenttal : function.can be derived which in vólve s in te g r-a tion only over the surface of the wi~g, S3 :
If we ass ume the wing to be a thin fla t plate in the g = 0 plane, then
'""1.:::Y\il,=0 · .and the normal and co-nor-mal vectors become co-linear,
and using the re lation : .
valid for thin flat plate wings, equation (12) may be.written :
(13)
Differ,entiating equation (9) along the normal (-::;;
11:
~-:f
in thiscase) and tr-ansför-ming back to the original spatial co-ordinates x, y, z , the potential function becomes (equation 17 of reference 3). .
- -- -
-(11)
where x, y.,ZJ are the co-ordinates of a point on S3. I , ,
If it is desired·to maintain an analogy with the subsonic cas e,
then the choic e of o: will be
1
whichwill be r ecognized as a source function, The problem of obtainin g
a s olution to the în te g r-al..e q ua tt on paralleling that obtainedfor- Volterra' s
choice of (]'" (which is the integral of the source function), is complicated
by-the.fa ct that inthis ca s e o: is infin ite everywhere on'the forecone as
weIl as at' t~e point XYZ., ItLs necessary therefore to r esor-t to Hadama r d 's
method of finite part integration of an in fin ite integral (reference 8), equa
.-tions analogous toequations (10 ) and (11) being obtained by excluding the
-p oint XYZ from the region of in te g r a tion. Since equation (14) provides the
necessary equipment for the future developmen t of this semtnar , th e finite par t method wiU not be discussed îur-ther, other than to quote Hadamard's result that for sources confined to the Z
=
0 plane :'(15 )
wher-e
*
denote s the finite part of the in t e g r a l e Ward (reference 9) has inte g r a te d thîs expression and obtain e d :. : ' .
(16)
It is not meant to convey the impression, by not developing fu11y the th e o r y for the case when 0" represents a sour e functton, that the finite part method is less suitable than the Volterra approach. Indee d
the extension of the source concept (even if the compressible source doe s
r-epr-esent a somewhat physically nebulous quantity) to supersonic flow
permits computation of the induced flow fields due to specified distrib u
-tion s of sourees and doubtets..: :an èlin particular is of use In determini ng
the in duc e d flow fields behind wings of finite span. It may be said the n ,
(12)
those for which Volterra's solution is meaningful, With regard to down-wash fields. equations (15) and (16) win be referred to later. For a complete treatment of the basic theory of the finite part m ethod, as weU as some applications of it to particular pr-oble rns, the reader is referred to the work of Robins on (reference 7) and Heaslet and Lomax (reference 6).
(17) (13)
lIl. PARTICULAR RESULTS FOR THIN·RECTANGULAR WINGS.
3.1 Solution of the Fundamental ·Equation.
Equation ( 4) may be considered as the fundamental equation of thin wing theory. If the potential jump across the wing inside the trace of the Mach forecone on the wing (i. e , region S3) is not constant. that is if :
-then the integrations of equation (14) become of.the elliptic type and it is not-po s s ib le to obtain a closed analytical expression for the potential function. However, if the potential jump may be considered constant over the region S3 then the integrations may be carried out. This has been done in reference 5 for the case of a thin, flat, uniformly loa de d,
re c t a ngula r wirig, and it has been shown that if the potential jump is represented by Co' for all points lying behind the eading edge wedge, ahead of the trailing edge wedge, and bounded laterally by the inside of the leading e dge tip Mach cones
cP=
±~
s:
An d that, under the same as sumptions, for all points lying within the
Ma c h cone from the leading edge tip, outside of the Mach cone from the
tra ilin g edge tip, and forward.of the trailing edge.w e dge
(18)
Th e reference axi.s system for equation (18) is shown in Figure 4. along
with the tip reference axis system.
On a finite rectangular wing the loading is not uniform in sid e
the tip Mach cones and consequently the expression for the potential
func tion as given byequation (18) is not valid for points the tr-ace of
wh os e forecones lie within the tip Mach cone tr-ace, Heaslet and Lomax
ha v e constructed the loading on the rectangular wing by superposing a series of uniform loads; each uniform load Incr-em ent is applied over a trapezoidal surface corresponding to the wing with its tips r-ake d, The
(14)
amount of each load increment is determined by the in c r e m e n t in rake angle from the preceding load in accordance with coni ca l flow theory. By solving the ensuing inte gr a l equation by use of the boundary conditions , the pressure coefficient inside the tip Mach cone is expressed in the form :
(19)
theco-ordinates in this case being with respect to the tip reference axis sys tern , ando<, being the panel incidence , .
3.2 Lift Distribution Aéross a Thin Wing Panel.
In Figure 5 a rectangular panel in presence of a body is shown, By putting equation (17) into the standard form for an infinite wing, we may write :
(20)
Equations (19) and (20) then determine uniquely the pressure distribution over the body-panel combination providing no in t e r f e r e n c e effects occur ,
This distribution is shown in Figure 5.
3.3 Chordwise Element of Lift (Section Lift Coefficient).
When wing panels such as that shown in Figure 5 are im m e r s e d in the induced flow fields behind lifting s urfa ce s, both the incidence and local lif t coefficient slope vary spanwise across the panel. In order to ob-.tain the tota1panel lift coefficient it is necessary then, to integrate across
the span the product of the local lift coefficient s lope and local Incrdence , By def'inition, the lift coefficient for any surface S is
(15)
Consider a chordwise str ip of constant width
A't'
and 1ength-e
shown in Figure 5(b).. The lift force per e ementa1 area
Al
J~AL
-::.rr,p-~IA-)
"r
J~
so that the tota1 force L acting on the strip
.c:
Aa
is:as is :
where the subscripts 1 and 2 refer to the pressure differentia1 over the pàr-ticula r area of integration. From this re1ation we can now fo rm the lift coeff.icie nt for the e1emental strip . ~Ár based on its own area, depending on whether
d'
bA,; - ~ or4>
q>
...Q,- ~ • .For the first condition, that is efficient becomes :
the section lift co
-(21) (22)
,
~L
=
:/~At[Ar[Z~f~1J,,'
+
AJ'~"-cf;t-t'~)./"1
'
. .=
1:- [
[/31'
7
J~'
-r,[e
~ ~4~~,/(.J{;-'J?(]
~,' ( - .=~/oL~Pi
:-+J~
~J
-,(.e-r
(3~
r»
J
••>(16)
Transforming back to the body axis system by putting
1'=..4-.,
and diff e r e ntia tin g to obtain lift coefficient slopes, equations (21) and (22) may be written :(23)
and :
(24)
for A.
> (
> -4 - ~Equations (23) and (24) define the s e ction lift coefficient at any spanwise pos ition Y» on the panel.
3.4 Pane1 Lift C oefficient S lope
Using equations (23) and (24) the lift coefficient for a thin rectang:ular panel such as that shown in Figure 5 may be obtained by span-wise inte g r atio n . Using Figure 6 we ma y write :
Lift of pane 1,
L
=Lj.
+
L:z-
.
A-%
~~
r::
f
ë
-C-L~)/,L
+
raI
(~·,,-t
+
.a.- - {4-~
=i
0<-<-f~-~J-J)7
+,o<-.e-fA;
Çgr~Jf)7
r ~_~(17)
Il sin g the body cross section area as the reference area, we may write the panel lift coeffi c ient and differentiate to obtain the slope
L
P'O....e!and
whieh, wh en the int e gra tion s are ca r rie d out becomes
(25)
Pr o viding the thin wing conce p t is va Iid, equ a tion (25) presents the panel lift coeffici e n t in a for m that is very usefu l for optimiza tion studies, [I,eg for dete rmina tion of chor d length fo r any given span to yield a maximum lift coe ïfi cte n t) ,
30 5 Pane l Ailer on Power
The formulae of the preceding sections may be used to compute the rolling moment du e to pane1deflection providing no iIit e rfe ren ce effects oc cur , Under this assumption we shall caIl the Induce d-m o m e n t due to panel deflection the panel aile ron power, which is determined by the spanw ise in t e g r a tio n of th e element of rolling rnom erit due toa clnrdw ise element of lift:
(18)
and inte g r a ting yields :
which ma y be differentiated to yield fîna ly :
Eq uation (26) is th e rolling moment coeffîcient slope of one wing panel.
3.6 Spanwis e Cent re of Pressure of a Panel
Know i ng the ift co e ffî cient slope and aïleron powe r sIope of
a panel, equa ti o n s (25) and (26), the spanwis e panel centre of pressure
can be de ri ved by using the relation :
d'e
.P.
=Cl.
~ .
(19)
IV~ BODIES OF REVOLUTION IN SUPERSONIC FLOW
A cylindrical body pitching in a supersonic free stream set s
up about itself a circulatory flow field. The flow conditions inside th e body disturbance field can be determined by replacing the body by a family of sourees distributed along its axis, the distribution strength being determined by the boundary condition (s ee reference 10). Ferri
(reference ·lO- A ) has also calculated the flow field about bodies by dire-ct application of the method of characteristics.
Using linearized theory it has been shown that for a body with
a conical head and cylindrical afterbody, the afterbody does not affect the lift of the configuration, the result be ing obtained that
C
N==-
t1.-. ' . . ~8
per radian based on body cross section, and the centre of pressure is 66..70/0 of the cone height aft of the nos e, Experimentally, the linearized
theory result does not agree with measured values, the normal force co
-efficient being found about 500/0 higher than theory predicts and the ce ntre of pressure located considerably further aft (the exper-irnenta l centre of
pressure position corresponds more closely to the 1000/0 cone height point
(20)
v.
THE WING-BODY PROBLEM 5. lInteraction PotentialIn the preceding sections we have been concerned with the problerns-of determining the aerodynamic characteristics of isolated wings and bodies and the flow fields about thern , It is now necessary to consider that problem which is fundamental to missile design, that is, the problem of the wing-body combination where due to wing-body interference effects, the aerodynamic characteristics of the combination are considerably different from the characteristics to be expected by linearly superposing the characteristics of the isolated cornpon érits,
H, in considering a wing-body combination, the potential
fie lds of the body and wings are linearly s upe rpose d, the resulting potential distribution will not satisfy the boundary conditions of the physical sys tern , In order to overcome this difficulty a correction potential ( the interaction potentialof referencë 11 and the interference potentialof reference 12) has been derived which is a solution to equa-tion (1) and which when superposed linearly onto the combined body and wing fields yie lds a resultant fie ld which satisfies the physical boundary
con di t ions , The mathematical cornplexity of the derivation of this cor-rection potential makes the methods of references II and 12 extremely laborious when applied to practical configurations. Further since boundary ayer effects are neglected, the method yields onlyapproxi-mate ans we r-s , For practical design work it has been found that sufficient accuracy is obtained from linearly combining the fields of the wings and bodies, and the amount of labour involved is reduced considerably. In the remainder of this work we sha11 derive the
characteristics by linearly combining f'Ie l ds, leaving to the reader the consulting of references 11 and 12 for more exact treatments.
Considering a pair of rectangular wings arranged in cruci-form on a body as shown in Figure 1, the wing body interference
problemmay be divided into three parts:
C
)
the effect of the body on the panels, (2) the effect of the panels on the body,(21)
5.2 The Effect of the Body on The Panels
A pitching body sets up about itself an upwash or downwash field, depending on whether it pitches positively or negatively. For a panel-body combination pitching positively, Beskin (reference 14) under the assumption that the flow field immediately upstream of the wing is cylindrical, has derived the re lation :
(28-i).
(~9)
where
3:
is the distance along the span measured îr-omthe body centre Iine , andr
is the radius of the body. Applying strip theory to a rectangular panel, the ave rage incidence over the wing panel can be shown to be :(28)
Thus the ave rage Incidence across the wing panel is increased by the
.factor
(1
.
+
cJ;b)
.
and hence for a panel in the presence ofa:
body pitching in the same direction as the wing, the Uft coeffieient as given byequation (25) must be modified and becomes :(
J
CL \
_
~
~o(..~
-,<3A
e Pbne.1 in r~es.e.nc.e 0/ fOSiti'fe.'r fitc./'dn( lx.JyEquation.(29), perhaps requires some clarification. It represents the panel lift coefficient of a pitching wing-body combination. where 0<.. is the angle of pitch. If a deflection of the wing'p a n e l
Ó
is super-posed on the pitch, then the lift coefficient of the panel is given by the relatîon :d
Cl.
do<..
=
(Equation 29)0<.. 0<.+
del.
dó
+ (Equation 25) Ó(22)
5.3 Effect of the Panel on the Body
The pressure field developed on the wings is carried over on to the region of the body contained between the root leading and trailing edge Mach helices. These regions are LlIus trate d in Figure 7. A method of computing the magnitude of this ca r r y - o ve r ,,-b a s e d on the application of'-equati.on (14), becomes apparent if the assumption is made that the flow fields may be linearly s upe r pos e d, If two-dimensional flow is assumed
everywhere over the panel root cho r d, then equation (14) can be integrated
to give a general expression fo r the poten tia l function, valid everywhere
inside the region contained between the root le a din g edge and trailing edge Mach cones, Under the assumption that the potential remains unaltered by the body surface, the resulting expression would then define the
poten-tial distr ibution over that portion of the body lying inside this r egion,
Using the linearized form of the pressure coefficient :
c
r
=
and expressing U in terms of the acceleration potential function (equa-• tion 4) :
(30)
the pr e s s u r e coefficient on the body could be dete nmin e d, This method has not been worked out, and hence no mdtcation of accuracy can be g ive n, Generally spe a king, theoretic al methods negle c ti ng both boundary layer and
tipeffe c t s give only fair agreement with experimental observations, and the
gr eat am o un t of labour invol v e d in thei r applica tio n justifies recourse to mor e "in the la r ge " analyses. Typica l of this more approximate type of analy s i s is that presented by Morikawa in reference 13. In this case the
as s um ptio n is made that on the shaded portion of the body (see Figure 7)
the press u r e ca r r i e d over is constant (a t any gi ven Mach number) and equal
to the asymptotic pr es sur-e at a la r g e distance downstream from the trail
-ing edge, In this case ~ CL. (th e increment in lift slope due to
0(
a....
wth e carry - o v e r from the wi ng onto the body, based on the exposed area of a
•
(23)
(The above integration is carried out by projecting the shaded area of the body, SB' onto the x-y plane; Sw is the wetted area of two wing panels).By putting in the value of the pressure coefficient far down-stream of a rectangular wing which has a lift slope based on Its wetted area:
Morikawa derives the relations for the carry-over of lift onto the body :
and: for
-t:
d
. ~1.
l)..-tJî ]
.
1-
n
(32) where for :-
1.
.1..-+
~
at-,.
the aspect ratio of a single wing panel.
being
It should be observed that equations (31) and (32) are ap-plicable only under the fundamental assumptions of this work as d
is-(24)
cussed in the introduction, particular note being made of the fact that the effective panel aspect ratio must be greater than unity,
50 4 Complementary Pane 1 Interference
Generally the nature of the in t e r fe r e n c e effects between two panels at right angles is extremely complex j it is a function of the amoun t of pitch and yaw of the combination as weIl.as the amount of de-flectio n of the panels, body and wing geometry, and Mach number. In the ca s e of arge deflections of a wing panel in the presence of a body, a gap appears at the wing-body juncture and con s i de r a b l e alteration in the-flow fields of he body and wing may oc cur ,
The nature of the complementary panel interference effects
is suc h -as to alter both the total lift and lift distribution (both spanwise and ch or dwi s e ), of the panels. Few experimental results des cr-ibtng these effec ts are available arid, because of the complex nature of the problern, the theoretical investigations are rather l.im ite d, Regarding the o r etic a l wor k, it is apparent that equation (14) or equation (16) would be suita ble for this type of in v e s ti ga tion providing.s u c h- e ff ects as root
-gap flo w could be taken in t o account. Martin (reference 15) has applied the finite part method outlined by Robinson, to non-planar systems and derive d.the surface pressure distribution and damping in roU
charac-teristics of a fa m ily of rectangular fdns, At C. A. R..D.·E., the author
an d M.G..H. Tidy have applied equation (14 ) to the problem of computing the indu c e d rol in g moment (due to carry-over of the pressure field onto the un d efle c t e d panels) of a pair of rectangular wings arranged in cruci-form on a cylindr i c a l body where one plana r pair of wing panels are dîffere ntially deflected to produce a rolling moment. The nature of this
indu c ed rolling moment, which tends to reduce the aileron power of the wings, is iUus t r a t e d in Figure 8.
In deri ving an expression fo r the in duc e d roll.moment co-efficie n t of the configurat ion depicted in Figure 8 two approaches were use d, In the firs t, it wa s assumed that the body may be replaced by two flat plates equal in semi-span to th e body ra dius and arranged in crucifor m at zero inc ide nce to the fre e stream. The pre.ssure field from the deflected panel was then assumed to act on the area of the
un-de flec ted panel normally enclosed by the body as weU as on the enclosed
area of the exposed panel. The contribution to the induced moment of the added flat plate section lying in the plane of the differentiaUy deflec-ted panels was.assumed to be zero, this plate being considered to act on ly as a reflection plane preventing any alteration of the pressure
dis-(2 5)
tribution over the inb oa r d portion of the deflected panel due to circulatory flow ar ound the root. In this approach the n, the limit of in t e g r a tion with respect to
3'
(see equation 37- i) is zero. and a rathe r. simple expres-sion for the in d u c e d rolling moment may be der-ived.. In the second ap-pr-oac hernp loy e dj the area of the undeflected pane nor-ma.Ily ene losed by the body is exc luded from the re gion of in t e g r a tion •.and a somewhat more complex expression for the in d uc e d moment coefficient r-esults ,Because of the na ture of the flow (M ~ 1). it·i s possible to employa reflection plane technique, and hence the.flow field need be con-sidered only in ·on e quadrant. This will be true providing that the diagonal joining the root leading edge poi nt of the deflected panelto the tip trailing
edge point of the undefleeted paneI lie s ahead of the..root leading edge Mach cone.. If this condition is not satis fied, then the reflection plane technique must be abandoned and a lift ca ncellatio n method ernployed, The assumed wave field in one quadr a n t is sho wn in Figure 9. In addition to those dis-cussed in the introduc tion , the followin g assumptions are inherent in this work : (i) (ii) (Hi) (iv) (v)
ro ot - ga p -flow effe cts are negligible;·
the bo dy' is at zero incidence;
wing deflect.ion angle s are sufficiently smal1 that the
wing panel s ma y be considered to tie inthe'~ = O plane;
the wing panels are thin , flat plates; .
th e configu r a tion has zero rate of roU •._
The significanee of this last assumption , aside fr om the fact thatlt re-presents acondition readily duplica ted in the supersonic wind tunnel,
will appear later. Using the lin e arize d re lation for the pressure co-efficient as given byequa ti o n (30). the in du c e d rolling moment co-efficient (based on body cros s sec tion area and wing chord), may be written as :
S
=-f:-cfC1r
JS
=-~
8~1qJ(~OJ)jJS
(33)'Ç;r
~r
-where, as shown in Figu re 9. ~g, is the appropriate area en-closed, beneath the trace of the röo t le a din g edge Mach cone from the deflected panel. on the vertical pan el.
(26)
The acceleration potentia1 function
cP
(f.O!)
is given by equation (14) wherer
=l!::
0 and S3 is the region enclosed on the de-flected panel by the trace of the forecone through the point (x, 0, z) asshown in Figure 9. The equation of the Mach forecone through (x, 0, z)
is :
and the equation of the inte r s e c ti on of this cone with the z
=
0 p1ane is -.-
-Et---I-J.)
z -
(3
(1J.'-
+
't:l.) -
0 (34) .The region of In t e gra tion S3' is then the region bounded by the root and 1eading panel edges and thehyperbo1a defined byequation (34). Hence in making the in t e g r a tion s of equation (14), the t.VJ. in t e g r a tion is made
over the int e r va l: d'
r
to
and the int e g r atio n with respect to -t-.L is made over the in t e r va l :
The in t e g r a 1 expression for the potential furiction on the y = 0 plane may then be written in definite integral form :
~-(31'f~+1~'
cfJt!-0'j)
-=
rfJ-
rfJ-~fî,('i-7.~)J-t.L
o
~1f'r)'X
,
o
(2 7)
Under the assumptions of line a ri z e d theory :
and for effective panel aspec t ratios greater than unity,
where
J
is the panel defle c tion (see Figure 8). Thus:and the potentia l fun ctio n on the y = 0 plane may be written :
(35)
Fina lly equa tion (35) may be s ubs titut e d in t o equation (33) and the
rolling mome n t coefficie nt written as :
s
=
~:8~~
f!ft-
~~;:~~'yw~7JK?
(36)"Xl
The in t e g r ati on s of equation (3 6) can be performed, once the area
-r
is kriown,(28)
First Approach :
As previous ly discussed ~ in this case is the region
en-c losed by the traen-ce of the root leading
e~e
Mach cone on the undeflectedpanel and that portion of the undeflected panel enclosed by the body. The
equation of the root Mach cone is :
and the equation of the trace on the vertical fin is
Thus the integration with respect to
't'
is made over the interval fromo
to t'i
;~'J..
-1" where 'n =~r
and the~tegration
with respect t0?C is made over the interval fr-om
13)
r"+'t~
to )"tr
.
Putting inthese limits the induced rolling moment coefficient (e quat.ion 36) may be
written in the definite integral form :
(37-i)
which , when integrated, yields :
(3:7 )
Equation.( 3 7 ) represents the contribution of one quadrant only and must be
multiplied by four to give the final result for wings in cruciform as
depic-ted in Figure 8. Thus by differentiating equation (37) and m ultiplying by 4.
the total roll moment coefficient derivative due to complementary panel
carry-over (designated by the subscript VI,...,
w
)
may be written :(29)
Second Approach
In the second approach to solving this problem, the body is
excluded from the region of Integr-ation, 50 that the expression for the
induce d rolling moment coefficient is :
which when integrated (substituting in the expression for
tf-f'-
pre-viously derived), yields :
+(~
-
ffi-)~-'.fl-
-I-
~~-'..2::-
.
.
\~
6Y1-
YY1.:I._(3~'
...
Ix
~-.2.(J~"
1~ -J)~-I (~"_p{!
..,
.
n
- f
~~'
/Y1
~-:L/t.a.
a'
(39)Under the restrictions of this derivation (
/J
Á2
p>l ).
the values of the induced rolling moment coefficient as given by equations (38) and(39) agree rather closely in magnitude, although their percentage
(30)
The above work deals with aconfiguration with zero rate of
r-ol l, and is of particular value in determining the roll charaeteristics of
a r-oll stabilized configuration. The results should hold for conf'Igur-ations
with very.a rn a ll roU rates. In the case of configurations with rather large
rolling r-ate s , computation of the induced rolling momentdue to
comple-mentary panel carry-over becomes more difficult because of the rolling of
the undeflected panels into the root Mach cone and .th e spanwisevariation
in local lift coefficient along the deflected panel due to the roll velocity •
The fir s t effect results in a reduetion of the area ene losed by the trace of
th e root leading edge Mach cone of the deflected panel on the undeflected
panel and may be calculated r-eadily, The secondeffeet, the spanwise
variation in local lift coefficient, presents a serious difficulty, because
the potentia 1 jump within the trace of the forecone on the.deflected pane 1,
~-
tI-
',
becomes a function of "t and '( so that:the integrations ofequat ion (14) become of the elliptic type, and hence a closed, analy.tic
form of solution is not obtainable. Several methods of c.ir-cumventing the
diffi culty are being in ve s tig a t e d presently at C. A. R. D. E., but as yet no
(31)
Vlo THE WING-BODY-TAIL PROBLEM
In addition to the wing-body problem already discussed, the
general wing-body-tail problem reduces to consideration of the wing-tail
in t e r f e r ence pr oble m whi ch involv e s determining the induced flow field
at the tail due to the wings, includin g any alteration in the velocity
dis-tribution cause d by the body. Because of th e la r ge spanwise velocity
gradie nt s usuaIly en countered ac ross a tail surface lyin g downstream
of finite Ufting wings , the total lift coe ffici e n t of the tailmust be
com-puted by spanwise int e g rarion of the pr o du ct of local lift coefficient and
ocal incide nce , The local lift co e ffi cie n t of a r éc t a n gu lar-.tail panel with
no flow in du c e d on it by the body, is defined byequations (
2 3) and (24).
In the cas e of a pitching body, the variati o n in lo ca l-in ci de n c e across the
tail span due to body upwash mu s t be superposed onto the induced inc
i
-dence due to th e wing wash fie lds,
In gener-al, detèrmina tion of the downwashand sidewash
Helds behind lo w aspe c t ratio wings is considerably co m pli c a t e d by the
la r g e amo unt of distortion as sociated with the trailing vortex patte rn;
Additional co m p lic ation is int r o duc e d by th e cruciform arrangement of
miss ile wings as weIl as the rather larg e body diameter -to-wing span
ratios usually ernploy e d , The standar d co n c e p t s of line vortex theory
have been ex tend e d from the sub s o nic ca s e to cover compressible flow
conditions by Mirel s and Haef eli (re f e r e n c e 16). Spreiter and Sac ks
(r efe r enc e 17) have considere d the general problem of the rolling up of
th e tr-afldng vort ex sh e et, as weU as le a p -fr o gging of the vo r ti c e s at
high bank angles, Lagerstrom an d Graham (r e f e r en c e 18 ) unde r the
assumpti on of no r-ol.Iing up of th e vortic e s, have applied Bus e mannts
con ical flow method (refe r ence 19) to the problem of deter.rnining in
-du ced flo w fields behi nd semi- in fi n ite wings at supersonic speeds. This me thod consists of expre s si ng th e downwash and sidewash
asso-ciated with th e plana r se mi- in finite qu a d r a n t in a clos e d analyti ca l
form and the n obtaining the induced field due to a semi- iniinite
rec-ta n gula r wing by su p er posing ont o the field alon g the Iine of th e tra
il-ing edge , an infinity of ne gatively lifti ng conic al semi- i niinite wings
in such a fashion as to dec r ease to zero the perturbation velocity
(lifting pr essur e) in the plane of th e wing outside of the wing plan f'or-m ,
The summation of these wi ngs cannot , in general, be expressed in a
closed analyt ica l form except fo r the ca s e of calculations.made in the
Trefft z pla n e , that is, the plane infinite ly far downstream from the
wings, In this cas e Lagers t rom and Graham have obtained the
ex-pressions for dow nwash and side was h (referred to the axis system
-
-
~1r'
(32)~+j/+(~t
(4fr+(4J!-t
(40) (41) (42 )($
is the ang1e of attack of the wing).For reasonab1e tail span-to-body diameter ratios, experiments have
shown that the Trefftz plane calculations are reasonab1y accurate for
tails located more than two wing chords downstream from the wing tr ai
l-ing edge.
Recently Leslie (reference 20) has obtained a general
ex-pression for the downwash behind wings by in te g r a tin g by parts Ward's result (equation 16) and differentiating to obtain :
(43)
(33)
J«.,.~)
=
r-t:-"ti)~-'i)
(1('--
(J)~
'P..
[~-""i)':-fd'-lJ [~-,l
+
r~
and the integration is made over the part of the wing lying inside the forecone through x"y, z, Equation (43) proves .extremely 'usef'ul, By assuming conical flow Leslie shows that it can be in t e g r-a t e d to yield the formula of Lagerstr-omand Graham,'whiLst a similar formula to that of Mirels and Haefeli can be obtained by a.s sumdng-the wing to be r-eplacedby a lifting line a long the line
-x.
J.=
0 •The effect of the body on the wing dowriwash and sidewash fields -at.t h e tail is twofold in nature and becomes of particular impor -tance when large body-diameter to span ratios are employed. First, the-presence of the body causes the vortex spacing to be -in c r e a s e d, experimental results indicating a vortex spacing ratio (i. e. the ratio of {he distance between the tip trailing vortices and the tip- to- tip wing span) inthe region . 94 to 1 for body diameter to wing-span ratios in
the range .1 to . 3. The second effect is the blanketiing by the body of part of thé tail panel from the effect of the tip vort ex on the opposite side of the body. A method of accounting for this blanketting effect, based on the assumption of conical flow and assuming tha.t any portion of the tail panel optically shielded is totally shielded from any ln duced flow -effe ct, has been derived at C.A.R. D. E., and when applied to specific configurations has yielded good agreement w-ith experimental r es alts, For convenience of illustration a configuration with its tail panel out-of line with respect to the wing panels, will be considered (Figure U). The origin of the conical flow system is assumed to lie onthe panel tip chord located chordwise aft of the leading edge at the centre of pressure. Further, it is assumed that the tail panel lies we Il inside the after Mach cone or-Iginating at O. It is apparent then, that there will be a family of rays emanating from 0 bounded by the rays OAB and OCD (Figure 11) which will in t e r s e et the tail panel, the ir trace on the tail pane 1 being given by the line BD. In this method it is assumed that any ray lying below the family of tangent
(34)
rays wil. be com ple t e ly reflected and con s e q u e ntly th a t por-tion of the tail enclosed between the roo t chord and the Ii rie BD wiU have no induced field due to the wing panel on the opposite side of th e body. (In practice, re-fraction and ref ec tion of the rays at th e boundary-layer would be expected,
but this effect would be s m a.Il }, For any geometri cal arrangement th e line BD may readily be f'ound,
In com putin g the resu.tant lift of th e tail panel due to the wing panel on the opposite side of the body th e blanke Ued portion of th e panel
is exc luded com pletely an d the produ ct of th e section lift co e fficie n t and
local incide n c e inte grat e d fr om BD to th e panel tip . For con fi gu r ati on s where thee- blanketted area is a large percentage of the panel area, it was found necessary to apply this correction. The theoretical basis of this
method is very much lacking in rigour, but the agreement with experiment
(perhaps for t uit ou s) was good in the cases in ve s tig a t e d . It is envisaged
(35 )
VII.' THE EFFECT OF THE BOUNDARY LAYER
For missile configurations such as those considered he re, the vis cous problem is more that of the shock-wave boundary--layer inter-action type than of the boundary-layer a lone , Studies of the
characteris-tics of compressible boundary layers have been made by Van Driest (ref-erence 21) and Lees (ref(ref-erence 22). The shock-wave boundary layer interaction problem has been s tudie d by Liepmáhn, Barry et al in refer-ence R, The effects of surface r oughnes s and skin temperature gradients enc-ountered on most practical missiles introduces appreciable..errors into the theoretical predictions of transition point and boundary layer thickness.
Generally in missile wer-k, the ultimate -effect of the boundary layer must be obtained from experiment. For the class of configuration discussed her-e, Ithas been found that the effect of the boundary layer on the-ae r odynam ic characteristics (except drag) is small in the case of
moderate pitch angles, but ca n be appreciable whenr-ectangula r panels are defleCted in the presence of cylindrical bodies (due lar-gely to the effect of the-boundary layer onthe flow around the root gap). The effect of the boundary. layer on the zero lift characteristics is. inalmost all cases, fairlylarge.{'A,ttempts at estimating the friction drag of this type of con-figuration by u sing the predicted transition point and compressible bound-àry layer theory have yielded results which are not too satisfactory. A semi-empirical approach. adequate for most initial design estimates, has evo lve d, It consists in applying the incompressible turbulent data, in-creased by a factor of 1.5, to the supersonic case. A typical formula which may be used to estimate the drag of a configuration at supersonic Mach numbers is Prandtl's subsonic formula (see reference 23) increased by a factor 1.5 :
--
(44)based on the wetted area As, and Re is the Reynolds number of the com-ponent. the characteristic length being taken parallel to the free stream. It has been found that reasonable estimates of configuration drag can be obtained by applying equation (44) and.linearly superposing the component drags.
(36)
The boundary layer affects the transfer of energy to the sur-face of the body, and hence the skin temperature .. Recently (F e b . 1952) an excellent report discussing the factors affecting skin temper-atur es has been published (reference 24); in this report afather complete set of solutions to the skin temperature prob ems are pr e s ented, The authors present an empirical law for temperature rise which because of its sim-plicity and surprising practical accuracy is quoted here:
where
AT
is the ultimate rise in body temperature in degrees centi-grade and V is the velocity in miles per hour , For a more complete treatment of this problem, the above referenced work is recommended.(37)
VIII. CONCLUSIONS AND DISCUSSION
-Dur i ng the cou r se of the seminar we have attempted to illu s-trate the nature of some of th e proble m s encountered in the design of missiles to operate in th e sup ers onic Mach number r-ange, We have re-viewe d some basic theoretical methods applicable to missile configura-tions and by app ly i ng them ha ve de rive d expressions for specific charac-teristics in a form somewhat differe n t than usually encountered. These characteris tic s expressed in the form gi ven he r-e, have proven particu-lady useful in determi n i ng the deta iled aerodynamic characteristics of low.aspect ratio wing-body-tail combina tions and have been found to be reasonably acc ura te. Because of tim e considerations the application of these fundamen t a l metho ds has been re str i c t e d to the case of configura-tions which have rec tangu lar wing panels. However, they may equally we l.lbe applied to config ura tion s employing delta and other type wing panels.
In thi s wo rk th e components of th e configuration we re con-sidered tobe per fe c tly rigid bodies . In practice th e deformation of the components due to loadi ng may in t r o d u c e cppreciable alteration in th e aerodynam i c characteristics. Hen c e in computing the performance, stab ility an d contr ol characteristics of miss ile configurations it is
essen tia l to include the interwea ving between the aerodynamics and elastic properties of th e config u r ation components. Generally then, the computa tio ns of th e res u ltant ch a r act e ri s ti c s of a configuration in vo l ve the simultane ous solution of a set of equations, one group of which represen t s th e ela s tic charaet e ristic s of the components, the other--group repres enting the aerodynami c properties .
The assumption of attached flow conditions has tacitly been made through out this report. Howe ve r, in many cases (moderate in-cidences at low Mac h numb ers) se pa r ation occurs and the aerodynamic charaeter istics are altere d by the res ultin g mixed subsonic and super-sonic flowfields , An ex p e r im e n ta l studJl of the effect on the aerodynami c cha r a.ctèristics of transition from an atta che d to a datached bow shock wave at the le adin g edge of a finite span liftin g we dgejls presented in reference 25 . For the is olate d we d g e it is concluded that no radical changes occur in the charaeteristic s at separation,.rather the change is a gradual one , The th e ory develop e d in the referenced lite r a tu r e of reference 25 can be applie d to determ ine th e charaeteristics of an
isolated body, but for a comp ex wing-body-tail combination it is doubt -ful if any satis fa c tory results can be obtain e d from theoretical predic-tions because of the inh erently larg e boundary layer effects•
REFERENCES
1. Flax, A. H., and Lawrence, H. R.
"'T-he Aerodynamics of Low-Aspect-Ratio Wings and Wing-Body Combinations" • Paper presented at the third Ang1o-American Aeronautical conference held in Brighton 4-7 September 1951 and reprinted by the Roya1 Aeronautica1 Society.0 (Availab1e in the
U. S. from the Institute of Aeronautica1 Sciences).
2. Liepma nn, H. W •• Rdshko, A., and Dhawan, S.
"On Reflection of Shock Waves from Boundary.L a ye r- s " , N.oA.C. A. TN 2334, 1951. and
Barr-y, °F . W. , Shapiro, A.H., andNewmann, E.:p'.
"The Interaction of Shock Waves with Boundary Layers on a Flat Surface" • Journalof Aeronautica1 Sciences, Vo1~18, p.229. April 1951.
3. Gar-rLc k, I.E., and Rubinow, °S . I .
"Theoretical Study of Air Forces on an Oscillating or Steady
Thin Wing in a SupersonicMain Stream" • N. A. C. A. Rept. No. 872, 1947. 0
4. Pr-andt I, L.
"Theorie Der Flugzeugtragf1uge1 In Zusammendruckbaren Medium". Luftfahrtforchung Bd, 13, October 1936. 0
5. Heaslet, Max A •• Lornax, Har-var-d, and Jones, Arthur L. "Volterra's Solution of the Wave Equation as-Applied to Three-Dimensiona1 Airfoil Prob1ems". N.A.C.A. Rept. No.889. 1947.
6. Heaslet, Max, and Lornax, Harvard.
"The Use of Source-Sink and Doublet Distributions Extended to the Solution of Boundary- Va1ue Problems in.Su p e r s o ni c Flowll
•
N. A. C. A . .Rept. No. 900. 1948.
7. Robinson, A.
IIOn Source and Vortex Distributions in Linear-i aed.Theor-y of Steady Supersonic Flow' . Quarterly Journalof Me charric s &
Applied Mathematics,
vor
.t,
Part 4. December 1948. 8. Hadamard, J."Lectures on Cauchy's Prob1em in Linear Partial Differential Equat ions ", Ya1e University Press. 1928.
9. War d, G~No
"6alcu1ation of Downwash Behind Supersonic Wings". Aeronautî ca1Quarter 1y, voi. I, page 35. May 1949.
10. F'e rr-i, A.
"Ele m e nts of Aerodynamics of Supersonic F10ws ". MacMi 11an
co.
.
N. Y. 194 9"WA. Ferri, IA .
"The Method of Characteris tics fo r the Determination of Super-soni c F low over Bodies of Revolution at Smal! Ang1es of Attack";
NoA.C.A. Rept. 1044. 1951.
11. Browne , S. lI. , Friedman, L., and Hodes , 1.
"A Wing-Body Prob1em in Supersonic Conica l Flow".
Journal of Aeronautica1 Sciences, Vol. 15 , No.8. August 1948. 12. Ferrar i, C.
·"In t e r f e r e n c e Between Wing and Body at Supersonic Speeds-Theory and Numerical Application" • Journalof Aeronautica1 Sciences , Vo1,, 14, No.6, page 317. January.1948.
13. Morikawa, George.
"Supersonic Wing-Body-Lift". Journalof Aeronautica1 Scie nc e s , Vo1,,18, page 217. April 1951.
14. Beskin, L.
"De t erm ina tion of Upwash around a Body of Revo1ution at Su p e r
-sonic Ve1ocities". APL/JHU Rep t. CM-25 1, 27 May 1946.
15~ Mar tin, Jo hn C.
"A Vect o r St udy of Linearize d Supersonic F-low Applications to
Non-P1anar Prob 1ems :. N. A. C. A. TN 2641. June 1952.
160 Mi re1s, Har o 1d, and Haefe i, Rudo 1ph C.
"The Ca1cu1ation of Supersonic Downwash Using Line Vortex
'Theor'y ". Jou rnal of Aeronauti c a l Scierice s , Vo1.17, No. I, page 13. January 1950. (Se e a1s o NACA TN 1925),;:..
17. Spre ite r , John R. and Sacks, A1vi n H.
"The Ro ling U p of the Traili ng Vortex Sheet and its Effect on the Downwash Behi nd Wi ngs" . Journal of 'A e r-onautica l Sc ience s, Vo1.18, No.1. Jan uary 1951. and Sacks , Al vin H.
"Behaviour of Vortex System Behind Cruciform Wings-Motions of Fully Ro11ed -Up Vortices". iN.A.C.A. TN 2605. Jan. 1952.
18. Lagerstrom, P.,A•• and Graham, M. E.
"Sidewash and Downwash Induced by Three .... Dimensional Lifting
Wings in Supersonic Flow".
Douglas Aircraft
.
ce
..
Rept. No. SM-13007, Apr-i l! 1947~;,
19. Busemann, Adolf,
"Infinitesimal Conical Supersonic Flow".
N~ A. C.'A. TM No.
noo.
,
Mar-eh 1947. (Translàted from ane ar-l.ie r German paper). .
20. Leslie, D. C. M.
"Supersonic Theory of Downwash Fields". ,. Quarterly Journal of Mechanics and Applied Mathematics, Vol.V~, Part 3.
September 19 5 2 . 21. Van Driest, E. R.
"Investigatton'of Laminar Boundary Layer in Compre ssible Fluids using the Crocco Method" •
.N .'.A. C. A. TN No. 2597. January 1952. and "Turbulent Boundary Layer in Compressible Fluids".
Journalof Aeronautical Sc ie nce s, Vol.18, No.3. March 1951. 22. Lees, Lester, and Li.n, C. C.
·" I nv e s tiga ti on of the Stability of the Laminar-Boundar-y Layer in a Compressible Fluid" . N. A. C. A. TN No.. lU5. Sep.1946. 23. Durand, W. (Editor).
"Aerodynamic Theory". Vo1. 3, Dw G. page- 153ot
24. Da vi.es , F.Vo, and Monaghan, R.J.
"The Determination of Skin Temperatures Attained in High Speed F light". R.A. Eo Report No. Aero 2454.
25. .Htlton, John H.
"Flow Characteristics Over a Lifting Wedge of Finite Aspect Ratio with Attached and Detached Shock Waves ata Mach Number of .40".
v
FIG. 2.
r
p
v
SOURCE DISTRIBUTION DUE TO O,AFFECTING P
FIG. 3.
z Tip z' Axis System y y' / b
x'
FIGURE 4REFERENCE AXIS SYSTEMS EQUATION (18 ) ••• (x,y, z) TIP SYSTEM ••••• (x'y' z' )
. r
h
I
v
I Z X / \ \ (b)/
6y x=O- j
r-y! x-x',
Pi/I x=c . , !~
_ 801_ - SM'\.•-j"Vy"
CL ;,
/
'\,lT~
x'
i 'Leadlng Edge
Tip Mach Cone
Trailing Edge
Tip Mach Cone
y' I I (aJ I ~p _ 40<
q-
r;
Y'Reglon of
2-dimensional
flow !""'C- -+..._- ...- - -....-- . S xI
",'
--FIG. 5.Lift = L 1. , -• FIG URE 6
REGIONS OF APP LIC AT ION OF EQUATIONS
•
Le ad ing Ed ge
Root
Mach
Helix
REGIONS OF PRESSURE FIEL D CARRy -'OVER ---WINGS
