REPORT1252 THEORYOF WING-BODYINTEWERENCE AT SUPERSONIC QUASI-CYLINDRICAL

REPORT1252
THEORYOF WING-BODYINTEWERENCEAT SUPERSONIC
QUASI-CYLINDRICAL
COMPARISONWITH EXPERIMENT
1
By JACKN. NIINAZWIN
SUMMARY
wing-bodycombinations.The trend towardusinglarge
A, theoretical
methodis presentedfor cahu.luting
i%eflow bodiesand smallwingsat supersonicspeeds,especiallyfor
jield abouta wing-bodycombinathnemployingbodiesdeviat- missiles,is the primereasonfor the increaaedimportance
ing only slightlyin shapefrom a &&r ylinder. If .L18 of wing-bodyinterferenceat thesespeeds.
Muchsigni60ant
workhasalreadybesndonein thefield.
combinatwn
pos8e8sesa horizmt.a.!
pf.aneof symmetry,no
In
reference
1,
Spreiter
has shownthatwhena wing-body
restricttis arerequiredon wingplanform in theapplication
combination
i
sslender
i
n
thesenseof hispapersimpleexpresof the methodto tlw zero angle-o
fut.tackcondition. If the
sionsfor
the
lift
and
momentcoefficients
can be derived.
combinti’on& lifting,themethodrequirwW thewi~ LmdThese
resultswere
obtainedby
reducinga
three-dimening edgesbe MLpersonic.2%enth8*t
ofti$m&ld
sionalproblemfor
thewaveequation
to
a
two-dimensional
thatcan be cakula.ted
dependson thewingaqwctratioand
whether
ornotthetrailingedgeeare supersonic. Twometlwok problemfor Laplace’sequation. Anotherapproachis that
of calculating
the$OWjield, the W-functionmethodand the of simplifyingthe differentialequationby using conical
boundaries.Followingthis approach,Browne,Friedman,
mu.ltipole
method,
are prmenti. The nwti& m pramli
andHodesin reference2 obtaineda solutionfor theprekure
areaccurate
totheorderof quasi-cylindrical
theory.
field
of a wing-bodycombinationcomposedof a flat triThemetltod
is appliedto ti caku-?ution
of thepre+wurefild
angular
wingand a coneboth witha commonapex. The
actingbelweena circu.?ur
cylindricalbodyanda rectangular
use
of
all-conical
boundariesreducestheproblemto oneof
wing. Thesecalw?atti arefor combinatti for whichthe
conical
f
lowforwbioh
powerfulmethodsof solutionareavaile$ectivempectratioof thewingpanelsjoinedtogether
h greater
able.
than2 andfor whichh e~ectivechord-radim
ratwis 4 orle8s.
Severalinvestigators
havepresentedmethodsfor deterTwocamxare calcululd,t.luwe in whichthebodyremains
at mroangleoj a.t&ckwhilethewingimidenceix variedand thepr~e field,includingtheeffectof interference,
thecasein whichthewingremain8at zeroangleof incidence acting on wing-body combinationsemployingcircular
slender. In reference3,
whilethe bodyangleof attackh varied. It wasfound that fuselagesandwingsnot necessarily
methodof obtainingthe
four I’owriercomponent8
of theinterference
jield arerequired Rrrari hasgivenan approximate
“interferenceof the wingon the streamlined
body, assumto e8tabli8h
thepressurefield, butthatonlyone component
ti
nece.wary
b establtihthe8panloading. A detui.?eo?
di.scnuwioning that the inducedfield generatedby the wing is that
oj thephy8icalnaJure
of th8in&rjerence
pre88ure~ h given. whichwouldexistaroundthewingif it wereplacedin the
.4n experirn
ent wa8performedespeoia.?-ly
for Lb purpose uniformstreamalone.” Similarly,the interferenceof the
oj checkingthe calcu?utwe
examples. The iwz&.ga&nww body on the wing has been determined.The resultsof
Ferrarithusrep~esenta first approximation,
and while a
performed
atMachnumbers
of1.48and$.00M arectie
secondapproximation
usingthe
methodis
possible
in priuwingandbodycombination.Bothtlwoariable
wing+wi.d+mce
and angle-o
f~ack UMeSwere cavered.It w found tha$ ciple it appearsthat too much labor would be involved.
solujor 8@h&ntiysmd ang.h,about2?0orkxs,i!lM
prewntmethod Morikawain reference4 hasobtainedan approximate
problemand has also
predictithepreeeuredi.stribti withinabout&10 percent tion by solvinga boundary--value
obtaineda closedsolutionby approximating
the three-difor bothCUB. ImportantnunlineareJectswerefoundfor
anglesof affackand incidenceof 4° to 6°, and important mensionalmodelby a planarmodel. Bolton%hawin referuiscowe$ectswere w.swd?y
jound whereLzminarbowndary ence5 has obtaineda solutionby satisfyingboundaryconditionsatafinitenumberofpointsratherthanoverasurface.
layersencountered
8hockwavei.
bother methodfor estimatingthe effectof ‘interference
INTRODUCTION
on the aerodywnic propertiesof wing-bodycombinations
In recentyears the problemsof supersonicwing-body whichare not necxwwilyalenderis given“inreference6.
interference
haveoccupiedtheattentionof manyworkersin In thisreferencothe method-isappliedto detarminin
g the
t-aerodynamics.
,Thelargeamountof effortexpendedon the dragof symmetrical
wing-bodycombinations;
it is alsoapsubjectis a resultof theimportanteffectsthatinterference plicableto the oalculatiogs
of theliftingpressures
actingon
can have on the overall aerod~amic characteristicsof
combinations~ploy@g wingswith supersonicedges. In
1SnperacdM
NAOATN 2677
‘“Wing-Body
Interfmence
atSnpcxsmto
SPASWItb‘anAppUmtion
to CbmbinatIom
WithReotm@arW~,by
J&.N: Nkdssn
md WILIInm
O.
Betwean
‘ThwryandExperbnont
forInterfaxmw
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~~d B~jwean
Wlqga@ [email protected],S_;~y
mm O.Pitts,Jnok
Pit@1062,
fmdNAOATNW28Worn-n
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.
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N.Nldsen,
andManrka
P. O1onMddo,
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“.-’.
-<, . .
1299
.,
1300
FORAFJRONAU’ITCS
REPORT125%NATTONAL
ADVISORY
COM3DTTEE
reference7, an essentiallynew methodof solvinga wide
classof wing-bodyinterference
problemshasbeenpresented.
Th~methodis basedon decomposingthe interferemw
of a
wing-bodycombination
intoanumberof Fouriercomponents
and solvingthe problemfor each componentin a manner
simih to thatusedby von KiirmtinandMoorein reference
8 for bodiesof revolution.
I?hinney,reference9, hascomparedthemethodsof references3, 6, and7 by applyingeachto the calculationof the
pressurefieldactingon a circularcylinderintersected
by an
obliqueshockwave. In reference10thetheoryof reference
7 hasbeenappliedto the computationof the pressuredistributionsactingon a rectanguhw
wingandbody combinationwiththebody at zeroangleof attackandthewingat
incidence. In reference11 Bailey and Phinneyhave applied the methodof reference7 to the calculationof the
pressures
on thebody of a rectangyilaz
wingandbody combinationat angleof attaokbut withthewingat zeroangle
of attack. b reference12 the same authorshave comparedtheircalculationswith some experimental
measurementsmadeat a Machnumberof 1.9. In reference13the
experimental
pressuredistributions
actingon a rectangular
wingandbody combinationat Machnumber1.48and2.00
me extensively
comparedwiththeoreticalcalculations
based
on themethodof reference7.
In part I of the presentreportthe theoryof wing-body
interferencefor combinationsemployingquasi-cylindrical
bodiesis presented,includingrecentdevelopments
not previouslyreportedin references7, 10, or 13. The theoryis
applicableto combinationsat zero angleof attack with
horizontalplanesof symmetryor combinations
at angleof
attackif the wing leadingedgesare supersonic. In part
II the theoryis appliedto the calculationof the pressures
andspanloadingsfor a rect.angukwingandbody combinationfor the caseof thebody at zeroangleof attackand
variablewingincidenceandfor thecaseof thewingat zero
wingincidenceandvariablebody angleof attack. The calculationsforthesecondcasearemorecompletethanhitherto.
In partIII extensivecomparisonis madebetweenthe calculationsof partII andtheresultof experiments
at Mach
numbersof 1.48and2.00especiallydesignedto check the
cnlcuhtions.
SYMBOLS
:
c
C*
CA)
G
Cg
C&(s)
D,.(s)I
f2n(x)
{Jp
bodyradius,in.
aspectratioof wingformedby joiningexposed
hal-wingstogether
chordof rectangular
wing,in.
effectivechord-radius
ratio,~
m
strengthof multipoleof order2n at point x of
bodyaxis
chordat wing-bodyjuncture,in.
chordat wingtip,in.
1,=(8)
K,.(8)}
kw
K.
L
L-1
L
Lrvo
;
ill,,(z)
n
P
Po
PI
PT
P
modi.iied
Besselfunctionsof thefirstandsecond
kinds,respectively
& ~B=o
b
&va
. o
—$
z~=
L
lift of combinationback to wing tmilingedge,
lb; Laplacetransformoperator
inverseLaplacetransformoperator
lift on exposedhalf-wings
joinedtogether,lb
lift on exposedhalf-wingsin combinationwith
body,lb
indexidentifyingsetsof multipolesolutions
free-stream
Machnumber
characteristic
functionsfor obtainingmultipole
strengths
numberof Fouriercomponent
staticpressure,lb/sqin.
staticpressurein freestream,lb/sqin,
staticpres-sure
at anyparticularoriiicoof wingbody combinationwhen a~=iW=O,lb/sq in.
staticpressureat wind-tunnelwallori.flco,lb/sq
in.
pressurecceflicient,~;
—$! for theoretical
calculations
interferencepressure coefficient due to nth
Fouriercomponent
free-stream
dynamicpressure,lb/sqin.
!10
dynamicpressurebased on conditionat wall
~T
orificeof windtunnel,lb/sqin.
r,ejx
cylindricalcoordinates:y=r cos 0, Z=T sin O
(Seefig. 1.)
R
Reynoldsnumberbasedon wing-chordlength
realpart
R.P.
semispanof wind-bodycombination,
in,; Lnplaco
8
transformof x coordinate
U,o,w
axial,lateralandverticalperturbation
velocities,
respectivey in.lsec
free-strewn
v‘d ocity,in./sec
v
W,n(x,r) characteristicfunctionsfor calculatingpressure
coetlicient
Carte9ianccordinatxwx, axial coordinate;~,
S,y,z
lateralcoordinate;z, verticalcoordinate,in,
(Seefig. 1.)
ffB
bodyangleof attack,radiansexceptwherootherwisedesignated
up-wash
angleof body-aloneflow,radians
%$
wingangleof attack,radians
~w
P
m
DA
effectiveaspectratio
Dirac delta function;
6(Z)
P1.
arbitraryfunctionsof ~
0
velocityamplitudefunctionof nth Fouriercomponent,in.lsec
wing-incidence
angle,radiansexceptwhereotherwisedesignated,
positivefor txailingedgedown
A
$
A
a(z)=o,
2#o;
J‘“8(X)*=1
polarangle(Seefig. 1.) ‘“
dummyvariableof integration
sweepangleof -wingleadingedge
.-.
“.
—--—-
-.
.-
—
‘--
‘- ‘--
- ‘--
interference
perturbation
velocitypotentiaJ
nth Fourier componentperturbationvelocity
potential
combination
perturbation
velocitypotential
wing-alone
perturbation
velocitypotential
wing-alone
perturbation
velocitypotentialdueto
theexposedrighthalfof thewing
wing-alone
perturbation
velocitypotentialdueto
theexposedlefthalfof thewing
wing-alone
perturbation
velocitypotentialdueto
theportionof winginsidetheregionoccupied
by body
Laplacetransformof q
suR90RlPm
lowersurfaceof combination
uppersurfaceof combination
I. GENERAL
INTERFERENCE
THEORY
PHYSICAL
PRINCIPLEX3
Prior to a mathematical
formulationof the wing-body
interference
problem,it is wellto defineinterference
andto
explainhowit arises. Withastationary
wingora stationary
body in a uniformparallelflow, thereare associatedthe
wing-rdone
andbody-aloneflowfields Thewing-alone
flow
field does not, in general,produceflow tangentialto the
positionto be occupiedby the body surface. As a result
an interference
flowfieldmustariseto canceltheflowfield
inducednormalto the body by thewing. For thisreason,
the sumof the body-alonepluswing-aloneflow fieldswill
not be theflowfieldfor thebody andtig together. The
ditl’erence
betweenthe flow fieldof the body andwingtogetherand the sumof the body-aloneandwing-aloneflow
fieldsis defied to be theinterference
flowfield.
The effectsof wing-bodyinterference
on theflowfieldof
rLwing-bodycombinationare illustratedby considering
separately
theeffectsof eachcomponenton theothers. For
the purposesof thisdiscussionfigure1 showsa wing-body
combinationdividedinto the part in front of the leading
edgeof thewing-bodyjuncture,henceforthcalledthenose,
thewingedpartandthepartbehindthewingtrailingedge,
henceforthcalledthe afterbody. If the combinationpossessesa horizontalplaneof symmetryandtheangleof attack
is zero, no restrictionson wing plan form are necesmry.
However,if thewingis twistedor camberedor if the nose
is at angleof attack,thenthe wingleadingedgesmustbe
supersonic
for thefollowingdiscussion
to apply.
,/
/
/
/
..
,/ Nosewove
/“
“h% line,
/
/
{ A/’
,/’
y
/
I ,.A
1., . .
\
\
\
\\
\
/
,/
~Fonvord boundoryof
1 regionof influenceof
) oppositewing ponel
1.\
//
‘. A
I-x
1
z.-fj+-
‘-t’
e=lr
port
FIGUREI.—timpmentsof typicalwing-body
combination.
JNTDRFERENCE
ATSUPERSONIC
SPEEDS
1301
Effectof nose on wing.—Considarnow the flow as it
progresses
pastthebody. At thebodynosetheflowis that
arounda bodyof revolution,andit canbe treatedby misting
methodssuchas thoseof references8 and 14. TVhenthe
bodyis at angleof attacka,, thereis anupwashfieldin the
horizontalplaneof symmetryof the body. If the body is
sticiently slender,the flowfieldin a planeat rightangles
to the body axis correspondsto that arounda circular
cylinderin a uniformstreamof velocity,V sin @?. This
givesan upwashfieldin the horizontalplaneof symmetry
of thebody of
%=aB(l+a2/&)
(1)
The effectof thisupvmshon the wingcan be obtainedby
consideringthe wingto be at angleof attackand twistedaccordingto equation(1) and by applyingthe formulasof
fieldso obtained
supersonic
wingtheory. Thewingpressure
is exact,tithin thelimitations
of thetheory,for thatsection
of the wingoutbomdof the Machlineemanating
fromthe
leadingedge of the wing-bodyjuncture. If the wing is
locatedcloseto the body noseso thatthereis a chordwise
variationin theupwashfielddueto thebody,thenthewing
is effectivelycambered,and the solutionis mom difiicult.
However,for mostwing-bodycombinations
it is possibleto
disregardtheeffectof thenose,andto assumethatthewing
is attachedto a circularcylinderthat extendsupstream
indefinitely.
Mutualeffeots betweenbody and wing,-The mutual
interference
betweenthebody andwingon thewingedpart
of a combinationcausesan interference
fieldacting.on the
body andon thewinginboardof the Machline emanating
from the leadiagedge of the wing-bodyjuncture. The
wing-aloneflow field does not, in general,produceflow
tangentialto thepositionto be occupiedby the body surface. An interference
flowfieldmustarisethatcancelsthe
veloci~ inducedby thewing-aloneflowfieldnormalto the
body whilenot changingthewingshape. Alternately,the
originof theinterference
fieldcanbe explained
in thefollovring manner. The wing and body can be thoughtof as
sourcesof pressuredisturbances
that radiatein all directionsin downstream
Mach cones. The wing disturbances
whichradiatetowardthe body are,in part,reflectedback
by thebody ontothewingandin parttransmitted
ontothe
body givingrise to interferencepressures.Likewise,the
disturbances
originating
on thebodypas-sontothewingand
affectthepressuresthere. It is apparentthatthedeterminationof theinterference
pressurefieldon thebody andon
thewinginboardof theMachlineof thejunctureis thecrux
of thewing-bodyinterference
problem.
Mutualeffeotsbetweenwingpanels,-To detx+mine
the
regionof influenceof onewingpanelon another,it is necessaryto tracathepathof a pulsehornonewingpanelacross
thebody onto the other. The pathtracedacrossthebody
by thepulseoriginating
attheleadingedgeof thewing-body
junctureis theforwardboundaryof theregionof influenceof
onewingpanelon thebody. (See@. 1.) It is clearlythe
helixintersecting
allparallelelementsof the cylinderat the
Mach angle. The boundarycrossesthe top of the body a
distanceof ~M_
downstrewandreachestheopposite
wing-bodyjuncturea distancemz~=l
downstream.A
1302
\/
P
FORAERONAUTICS
REPORT125&NA~ONAIJADVISORY
CO~
pulseoriginating
at a pointon onewingpanelandtraveling
to a pointon theotherpanelcantravelaroundthebody on
its surfaceto theoppositejunctureandthenalongthewing
to a givenpoint,or it canleavethebodytangentially
before
reachingtheoppositewingjuncturein a straightpathto the
point. The secondmeansof transnu
“ttingthe impulseis
shorterin distancethan the first and is the one which
determinca
theforwardboundaryof theregionof idluence
of onewingpanelon theother. Applyingthisconsideration
to thepulseoriginating
at theleadingedgeof onewing-body
juncture,it is easyto showthattheforwardboundaryof the
regionof influenceof onewingpanelon the oppositewing
panelis givenby the equation
(2)
This boundaryis also shownin figure1, and it becomes
pmallelto theMachlineatdistances
farfromthebody.
EfFeotson the afterbody.-~ far as the interference
effectof thebody on thewingis concerned,it is confinedto
thewingedpartof thecombination,
buttheeifectof thewing
on thebodyisfeltalsoon theafterbody. Fora symmetrical
configuration
atzeroangleof attackthereisno dowmvash
in
thehorizontal
planeof symmetryandtheafterbodypresents
noparticular
problem. However,behindaliftingwingthere
is a dowmvash
field. If the dowmvaihwerelmowneverywhereinthewingwake,thenthewakecouldbe considered
as
an extensionof thewingwithtwistandcamber. Thewing
wake and afterbodycould then be incorporatedwith the
wingedpart of the combinationand treatedin the same
manner. However,the actual dowmvashpatternin the
wingwakedependson theinterference
effectof thebody on
thewing. It is thusapparentthatthesolutionof theafterbodyproblemrequiresthattheinterference
problemfor the
wingedpart of the combinationbe solvedfirst. Only the
wingedpartof the combination
is analyzedin detailin this
report.
Regions of applicabilityof the theory,-The present
interference
theorycanbe appliedto all or part of a -ivingbody combinationdependingon the configuration
and the
lift. If thecombinationis not liftingandpossessesa horizontalplaneof symmetry,then the interferencepressure
fieldcanbe determined
for the entirecombination.For a
liftingcombination
withsubsonicleadingedgesthe up-wash
field in front of the wing makesthe presentmethodinapplicable.
For a liftingcombinationwith supersonicleadingedges
severalgeometricfactorsconsiderably
influencethediflicuhy
of calculatingthe interference
fitildor indeedthe extentto
whichit canbe calculated. Theei7ectof oneof thesefactors,
the sweepof the tmilingedge,is illustratedin figure2 (a).
A subsonictrailingedgegivesrise to multipleMach wave
reflectionswhich greatlycomplicatethe determination
of
theinterference
fieldovertherearpartofthewing. &Iother
importanteilectlimitingthe applicabilityof the theoryis
illustratedin figure2 (b). This figureindicatesthat the
interference
fieldbehindtheincidentwavecaninfluencethe
tipupwashfieldwhich,in turn,influencesthe presmrefield
behindthereflectedwavefromthetipin a complicated
way.
‘1
-1
\A/
\
/’
,,
i//
I // Mul!lple
a
X
A
Mach wave
reflections
A
(a)
‘\\
\\
\
>
/
(b)
\
\
\
\
‘\A,
;X:
\\
c
E-
/
/
>
/
>< B
/
/
/’D
4
\ \
\\
\<
“\
d)
(a) Subsonic
trailing
edge.
(u) Simplecase.
(b) JIffeot of wing-bodyintmferencoontipupwaeh
field.
(d) Tractable
case.
FIGURE2.—Classes of interference
problems
forliftingwingondbody
wmbinations.
Avoidanceof thiscomplication
requiresthattheincidentwave
intersectthe trailingedgeratherthan the wing tip. To
assurethiscondition,the aspectratiomustbe greaterthnn
a certainminimumvaluein accordancewith the following
inequality:
One of the simp16casesof wing-bodyinterference
for a
liftingwingandbody combinationis shownin figure2 (c).
Heretheleadingandtrailingedgesarebothsupemonic,and
the root-chordMach wave intersectsthe trailingedge.
Alsothewing-tipMachwaveintersects
thebodydownstream
of thewing-bodyjunctureso thatno wing-tipeffectsoccur
on the wing interferencepressurefield. This condition
imposesthe aspec~ratioinequality:
Under the circumstancesof this figure, the interference
problemproceedsas if the combinationhad a horizontal
plane of symmetry. ‘Any body upwashfield in front of
1303
INTERFERENCE
ATSUPERSONIC
SPEEDS
QUASI-CWMNDRICAL
TBEORYOFWING130DY
thewingcanbe treatedas equivalentto a changein thickness distribution.The rectangularwing of aspectratio
gnmterthan twois an exampleof the simplecase, and it
willbe treatedas anillustrativeexamplein thispaper.
An exampleof a tractablealthoughfairly complicated
camto whichthepresenttheorycanbe directedis shownin
figure2 (d). In regionA the pressurefieldis determined
as a purewing-aloneproblemwithanybody upwashbeing
treatedas equivalentto a changein thiclmcssdistribution.
In regionB theproblemis stilla wing-aloneproblemwhich
is complicatedby upwaahoutboardof the tip. In region
C thereare body interferenceeffectsbut no tip effects.
In regionD both effectsprevail. In regionE the tip has
influencedthe flow at the body surfaceand produceda
secondaryeffecton the interference
pressurefield.
MATHEMATICAL
FORMULAnON
OFPROBLEM
Throughouttheanal@s, thebodyradiusis takenasunity
and W is takenas 2 so that 19=1. Any formulacan be
generalized
to anybodyradiusby dividingalllengthsymbols
by a, andto anyMachnumberby dividingall streamwise
hmgthsby p, by multiplying
allpressureandlift coefficients
by /?,andleavingallpotentials,lift forces,andspanloading
unaltered.It is necessaryto specifythewingalonebefore
any detailedinterferencecalculationcan be carriedout.
However,in the theoreticalsolutionof the problemthe
wing-alonedefinitionis arbitrary. The flow field about
the combinationdoes not dependon the definitionof the
wingalone.
General.decompositionof boundary-valueproblem.—
Tlmgeneralcaseof a combinationat angleof attackwith
the wing at incidenceas shownin figure3 is considered.
The mdhematicaldetailsof the decompositionof this
ccdguration into tractableconjurations is carriedout in
detailin AppendixA followingthe suggestions
in reference
15. A simplifieddiscussionof the decompositionis now
prcsonted, The completecombinationcan be decomposed
intothreecomponentcodigurationsasshownin figure4 (a)
in whichthe wing boundaryconditionsare to be applied
in the z=O plane and the body conditionson the ~= 1
v
cylinder. Component(1) is simplythe body alone,which
createsan upwashfield a=in that regionto be occupied
by thewingin accordancewithequation(l). Components
(2) and (3) are combinations
withwingsof the sameplan
form;butwhilecomponent(2) hasa wingat angleof attack
i~, component(3) has a wingwith angleof attack—au.
The significanceof this particularmethodof decomposing
the generalwing-bodyproblemis that component(l), the
body alone, can be solvedby knownmethodsand components(2) and (3) with bodiesat zero angleof attack
canbe solvedby themethodsof this report. In thewingincidencecasewherea~=O,onlyconfiguration(2) remains.
This configurationcan be decomposedinto a wing-alone
problemanda distorted-bodyproblemas show-nin figure
4 (b). We confineour attentionto this wing-incidence
casefor thetimebeing.
+“
+3
\
(
k
-1-
—
(1)
(a)
aB=O
-c
k
h
(2)
“o
+
+6 ‘o
.
-1
au
‘au
{
(3
~:v-1
l!
II
r
iw=o
;W
—-
>
il
0
I
Y
Zu
(b)
(a)
Deoompoaition
of general
wing-body
combination.
(b)Decomposition
forwing-inaidence
case.
FIGURH4.—Deaomposition
of wing-bodycombinations
into simpler
combinations.
l?nxmn3.-Genera1combination
undercombinedeffectsof angleof
attaokandwinginciden’m.
Considernow a combinationwiththebody at zeroangle
of attackandlet ~ be its potential. (Seefig. 4(b).) This
potentialcan be consideredthe sum of a wing-alonepotential~ andof aninterference
potentialq.
Pc=#v+P
(3)
1304
E FORAERONAUTICS
REPORT125%NATIONAL
ADVISORY
COMMITI!E
Sincethebodyis aninfinitecircularcylinderat zero.angleof
attack,it producesno flow field. If the body werequasicylindricdwith smalldistmtions,a potentialdue to the
body couldbe includedin equation(3). If thebody hasa
horizontalplaneof symmetry,the inclusionof a potential
due to body distortionwill not changethe interference
potential.
The essentialproblemis to determinep. l?irst,selecta
convenientway of extendingthewingthroughthebody to
formthewingalone,therebyspecifying~. The -wing-alone
am
flowfieldin generilproducesvelocities~ normalto the
with
ap2.
~=f2z(x)
cos 2n0
(9)
atr=l
Thenthecombination
givingtheinterference
potentialp can
be decomposed
in a seriesof combinations,
eachgivingoneof
the~. valuw. The decomposition
is illustratedin figuro6.
m
*=A&,f$
l-l
surfacethatwill enclosethe circularcylinderas illustrated
in figure4(b) for theregionabovethewing. In figure4(b)
and subsequentfigures,all bodiesare shownas cylinders
parallelto the z axis. Whilethe bodiesof the component
configurations
in somecasesareslightlydistortedcylinders,
they are nevertheless
shownas true cylindexs. This procedureis compatiblewith the fact that the boundaryconn.o
/7=1 nx~3-.
ditionsareto be appliedon a tie cylinder. The valueof
Fmmm6.—Decompoeition
of interference
combination
intoseriesof
a$m.
— varieswith8 andwithz. Thismeansthata body conFouriercomponent
interference
combinations,
br
formingto thewing-alone
flowfieldis distortedin a compli- Forn=O
catedfashion. NTOW
sincethe body mustbe circular,there
!!$’=fo(z)
mustariseaninterference
potentialp thatidenticallycancels
~w at thebody surface,therebystraightening
it.
andthereis no variationof thenormalvelocity,pressure,
or
b
potentialwith0. Thusthefirstinterference
combination
is
(4) a bodyof revolution. Thepressurefieldactingon thobody
of sucha combinationcanbe determined
by themethodof
Therearetwootherconditionsto be fuliilledby q. It must reference16. This n=O interferencecombinationhas the
not distorttheshapeof thewingwhenaddedto * to pro- verysimplesignificance
thatitsflownormalto the~= 1 cylducew Thuswhen0=0,
avo ap
inder,~ subtractedfrom~ reducesthe flow acrossthe
lap o
——.
(5) bodyto zerowhenaveragedfrom0= Oto o=r atanystroamT ae
wiselocation. For n=l,
or iW=Ofor theinterference
combinationasshownin figure
g=f,(z) Cos20
4(b). The last conditionis that the interference
potential
mustbe zeroaheadof thewingedpartof thecombination.
andthenormalvelocity,pressure,andpotentialwillvaryas
(6)
cos 20.
To summarize
briefly,it hasbeenshownthatthegoneml
Equations(4), (5), and (6) are the essentialboundaryconinterference
problem
of a body andwingat ditlerentangles
ditionson p.
of attackcanbe brokendownintowing-bodyproblemswith
Thenormalvelocity~ to be inducedat thebody surface bodiesat zero angleof attackas shownin figure4 (a).
by theinterference
potentialcan be analyzedat any given Combinationswith the body at zero angleof attackam
combinations
intowingsaloneplusinterference
streamtie positionas a Fouriercosineseries. The ampli- decomposed
asinfigure
4
(b).
Theinterference
c
ombinations
me finally
tudesof thevariousFouriercosineterms,f~m
(z),varywithz,
decomposed
i
ntotheirFourier
components
a
sin
figure6,
thestreamwise
distance. Thus,
A generalmethodfor determiningthe characteristics
of
anyFouriercomponentwillnowbe given. It willbe shown
(7)
COs
atT=l
that good accuracycan be obtainedfor the interference
n-0
potentialwithfewFouriercomponents.
Onlyevenmultiplesof 0axeconsidered
becauseof thevertical
SOLUTION
BYMZTHODOF W’FUNCTIONS
plane of symmetry. Considerthat the interferencepotentialis decomposedinto a seriesof potentialssuch that
Theproblemto be solvedis thatof a supersonic
wingand
eachcancelsone Fourkr componentof the velocityat the body combinationsubject to the conditionsalreadymenbodysurface;thatis,
tioned,but with the wing and body possiblyat different
anglesof attack. Thisproblemis reducedto a body-alone
(8)
P=& $92n
problem
andtwowing-bodyproblemswiththebody at zero
n-0
-4@=*+++
$0=0,
X<o
*=Afs.(4 %8=–93
1305
AT BUPERSONIO
SPEEDS
QUASI-C~RICAL THEORY
OFWG-BODY INTERFDRENC!E
angleof rtttackM shownin figure4 (a). The body-alone
problemcanbesolvedby existingmethodssuchasreferences
8 and14. Theprocedure
neceasagtosolveeitherwing-body
problemasgivenin reference7 is nowsummarized
together
withrecentimprovements.
Thepotentials~, pr, andp mustallfulfdltheequationof
linearizedcompressible
flow
(M’–l)fo=-pw-p:z=o
(lo)
If werestrictourselves
for thetimebeingto thecaseM= @
andtransformequation(10)to polarcoordinates,
wehave
$%r+:
w++
W-%r=o
independentof the boundary conditions. The inverse
transformof theproductof thetwo transformscanthenbe
determinedby the convolutionintegral. The part of the
transformindependentof the boundaryconditioncan be
thoughtof as defininga set of characteristic
functionsor
influencecoefficients.A tabulationof thesefunctionsallows
a numericalsolutionof the problemfor all boundaryconditions.
Themannerof splittingequation(17)intotwo transforms
dependsalsoon the existenceof the inversetransformsof
thepartsintowhichit is split. Letuswriteequation(17)as
(11)
(18)
withthecoordinatesystemof figqre1. In solvingtheprob- Withtheaidof thefollowingrelationships
lemwe chrmgefromthephysicalspace,z, r, O,to thetrans-,v-lq=j,m(z-r+l)
formedspaces
Z-1[F2J~)e
9)9)r o by meansof theLaplacetransformation
L[9(X)]=
J‘e-%@fa=@(s)
L-I(s@)=p=
(12)
0
%++
@ee–s@=o
(20)
andthedefinitionof thecharacteristic
functions
Withtheboundaryconditiongivenby equation(6) thatPis
zero2for x<0, equation(11)cm be transformed
to
%++
(19)
(13)
Expanding@ in a cosineseriesof multiplesof O,we can
aatisf
y the boundaryconditionsgivenby equation(5), and
sincothereis a verticalplaneof symmetry,we can confine
ourselvesto evenmultiplesof 0. Withthisrestriction,generalsolutionsto equation(13)canbe written
1
L-1 ~ti-u ‘~.(m) ~J_ =W,n(z~)
K,/(s) J
‘
[
(21)
we obtain
P==&
cos2n6 ~,
n=o
LJ fln(~–r+l)W~~(z–gjr)d&f’’(z;r+l) 1
(22)
cb=n~oCos2ne[c,n(s)K2a(w)+D2z(8)I,n(w)] (14)
Withthe aidof the Wj,(z,r) functions,the valueof q., and
hencethepressureor potentialanywhere,canbe calculated
where lz.(sr) and F&(m) are modifiedBesselfunctions. horn equation(22) by numericalintegrationfor as many
Tho constantsC~%(s)
andDzs(s)arearbitraryfunctionsof s. harmonicsasdesired.Thisresultwaspreviouslygiven(refs.
ThefunctionsIzm(sr)
canlogicallybe eliminated
atthispoint 7 and10)for the.=1 caseonlyas
sincefrom their asymptoticforms they can be shownto
representwavestravelingupstream. The function(72.(s)
(23)
q==~ cos2n0 :j2m(t)w2.(z–t)dt–j2m(z)
canbo evaluatedby meansof theremainingboundaryconn-o
[s
ditiongivenby equation(7). If welet
andthe W2S(Z)
functionsweretabulatedfor numericalcalonly. The genF,#(s)=L~,Jz)]
(15) culationof the body pressuredistributions
eralizationof the WI.(Z)functionto WZ,(Z,~) functionsby
then
meansof equation(21)is a naturalextensionthatpermits
(16) the simplecalculationof thepressureanywherein theflow
field. Somemathematical
propertiesof W2.(Z,~) functions
Wo thenhnveastheoperational
solutionto ourproblem
and methodsfor theirevaluationby automaticcomputing
machineryhave been studiedby Dr. W. Mersmanof the
NACA. A r&wrn6of hisresultsis reportedin reference18.
(17)
Propertiesof the W2n(x,r) functions.-Two important
propertiesof the Wz.(z,~) functionsthatmakethemuseful
The solutioncan be splitinto the productof two trans- for numericalworkarethattheypossessno singularities
in
forms,onedependenton theparticularboundaryconditions the fieldandtheirmagnitudes
neverbecomelarge. These
m representedby the ~zn($) functionsand anotherpart advantages
arein distinctcontrastto severaldisadvantages
of
the
multipolemethod
subsequentlyto be described.
fmquontly
atntdfnderiving
eqnatfon
(fS),thatA.O forz-W, b Mt
~ThemndItfon
requlmd
mproven
fnmfemnm
(1?).Thisb bramoral
wftbthofnhrltfve
physkdideathat
Curves
of
theW&(z,~)functionarepresented
in chart1 for
onrvewhlobforengfnedng
anyatcnIn$, @ theorfglnmuhoreplactdby a mnthmons
n=O,
1,
2,
and3
for
usein
numerical
computations.
pnrpmw
mmbnvoanolkctdlffarwnt
fromtlmtofthe*P onlyina lhnft2d
lKSIregfon.
1
—
1306
FORAERONAUTICS
REPORT125*NATIONAL ADVISORYCOMMJTT13E
A simplephysicalpictureof theWSE(Z,
r) functionscanbe
obtainedtim equation(22). Writethe interference
pressurecoefficient
3dueto anyharmonicas
2 Cos2nd jJz-r+l)
P,n=– + ~=
~
()
[
$–
a(+-ak+:+:)+”(
’28
w.(X,7-)=
w,m(o,7’)–
–~
(16n’+3)-@’’;-l)
(29)
8+ [
It shouldbe noted that tbe valueof W*.(0,r) is known
precisely.
I
SOLUTION
BYMETHODOFMUL’1’fPOLES
Let the veloci~ amplitudefunctionbe a deltafunctionat
theoriginasshownin figure6. Then
2 cos 2n0 @=-r+l)_W,x(~-~+l,r)
(25)
P2n= ~
1
[
$
‘u
,#Jnit arm
Multipoletypes.-In references7 and 10 a multipole
methodwasusedto determine
thepressure
fieldoffthebody.
The singularities
arisingin this method,togetherwith the
lossof accuracyforthehigberharmonics
duetolargenumbers,
led to the developmentof the W,x(z,r) methodjust described. Since the multipolemethodhas applicationto
certainproblemsand sinceits connectionto the WZa(x,r)
methodis of interest,it willbe givenhere. In the W2~(z,?’)
methodthe pressurefield-isdetermined
by usingboundary
conditionson the body surfaceandcontinuingthepressure
fieldoutwardfromthebody. It is intuitivelyobviousthat
any quasi-cylindrical
flow canbe generatedby distributing
sourcesand multiples along the body axis in vmioble
strength.If thestrengthof theaxialmultipoledistribution
canbe relatedto the body shape(velocityamplitudefunctions),thenthe entireflowfieldcan be calculatedoutmwcl
fromthetis.
Considerequation(14)which,withDSn(s)equalto zero,is
@=&2.(8) Cos2neK’&r)
(30)
Thisequationcanbe interpretedto meanthatthepotentialisbuiltupfroma distribution
of multipolcscorresponding
to the inversetxansformof cos 2ndK~n(sr)
alongthe x axis
in Shengthq.(z). However,therearemanypowiblesets
of multipoleacorresponding
to x integralsor derivativeof
the set just mentioned. These are generatedsimplyby
rewritingequation(30)as
(31)
.
Theiirsttermrepresents
thetial strengthfunction,andthe
secondtermrepresents
thefundamental
multipolesolutions.
component
interference
comb-tion with“dolta For eachvalueof theindexmthereis obtaineda distinctset
FIGUEII6.—Fourier
funotion”protuberonco
atz=O.
of multipolesclutions. For selectedvaluesof m themultiIt is seenthatphysicallytheW2,(Z,r) functionrepresents pokah&e thefollowingforms:
the pressurefield due to a deltafunctionin the velocity m=()
amplitudefunction. The fit termrepresentsan iniinite
cos2n6cosh(2ncosh-lz/r);~>r
pulsepropagatedalongtheMachconewithapexat z= —1
Cos27LeL-yK2=(&r)]
=
~+
andattenuating
inverselyasf. The W2x(Z—r+
1, r) term
representsthe overexpansion
behind tie bumpwherethe
X<T
=0;
presmrewouldbe zeroif theflow-weretwo-dimensional.
m=l
Formulasfor theW,s(z,r) functionsfor smallz andlarge
=CosZti sinh(2ncosh-lx/r);~>r
x c-anbe obtainedhornLaplacetransformtheory. In fact COS2m3L-~K,,(w)
—8
2n
theseresultsare
(32)
(26)
=0;
x<r
m=2n
(2n)I
K,&)
—- 2 2“
cos2nOL-1—@
(4n)10r
eqosuorl
(23)C8nontbeomtkd
S!rhoewntlonfcup
mme aB3mdmttobe115@wfh
groM&beoh8ng@lfrl
gofngfrom
uk3*blth8bodYoftie conflm-atfonsscu==ed~
(??-~-’l’;
Z>r
formoftbeBemonJllequatfonbretaImd
[1
[1
foralmplfolty,
thelharbed
rofemnm
IL Howmr,
of theqwadratio
terlmto the
throusbout
tbatiecdfcd@daumM. TheCOrltz-fbutfon
bodyp~
ccamclent
h UllWmenuy
dbalmedforW mwofthem*tfon atfmgle
ofattaok.
=Coa
Zti
=0;
x<r
INTERFERENCE
AT SUPERSONIC
SPEEDS
QUASI~DRICAL THEOILY
OFWING-BODY
For positivevaluesof m it is clearthat at the Mach cone
(z=T) no nonin@grableeingularitica
occur. For negative
valueaof m, derivativesof the m= O multiples are encounteredand the singulmitiesoccur on the Mach cone
ratherthanon the rmis. Sincethesesingukwitiea
occurin
theflowfield,theyarenot wellsuitedto numericalmethods
of analysis. Anothersetof simplemultipleswithsinguhwitics on the Mach cone are thoseof reference19 givenby
g“ cos 2n0
~“
Multipolestrengths,-Thefirst step in determmm
““g the
interfmencepotentialby the methodof multiples is to
determine
themultipolestrengthhornthevelocityamplitude
functions. For the m=O set of multiplestherelationship
betweenthesetwo quantitiesis alreadygivenby equation
(16)
(33)
1307
Faltungintegralof equation(34)we obtain
%(x-l)=
J0
*A=(W2A-W
(36)
Tbe functionM2=(z)has a square-rootsingularityat the
originso thatc.2Jz—
1)willbe iiniteif$n(z) is finite. However,fz.(z) mayhavea singularity
whichin confluencewith
the square-rootsingulwi~of M2~(z)producesa singularity
in C.2n(z-1)
.
Increasing
the”indm
mbyunityhastheeffectofintegrating
thesetof multipleswithreapedto z andof differentiating
the axialstrengthfugctionsby z. Whilethisdecreases
the
orderof the singularities
of the multipolesolutions,it incrwes the orderof the singuhwities
of the axialstrength
functions. Thehighcs.t
indexm thatdoesnotleadto singularitiesthusdependson how manynonsingular
derivatives
G,(z—1)possesses,which,in turn, dependsonthesmoothness
If thepotentialat a pointP asshownin figure7 is desired, of $%(z). In the calculationsfor the wing-incidencecase
(ref. 7) f~~(z)has a square-rootsingularityat $=1 so that
the
c2,’(z-1) has a logarithmiceingulariw. Since cx@-1)
y,z
correspondsto m= O,it waspossibleto usemnltipolesolub.
/ M~hlines
tionsof the m= 1 classandstillobtainintegrablesingular/ - ‘.,,
/--””
‘.y/
P(XA
z)
itiesin thetial strengthfunctions.
P
/
/
/
Properties of the M2.(z,r) functions,-The Ik.&(z,r)
/
zr
/’
/
o
functions
havesimplephysicalsigniflwce. Letthevelocity
/
,/
/
amplitude
functionbe thatcorresponding
to a deltafunction
Y
/
1/
(
x
as
shownin
figure6.
Thenby
equation(36)
x-r
-1
‘?-
u
)
$?
Fmum7.—Geneml
pointat whiohpotential
is to bedetermined.
multiples mustbe distributedfrom —1 to x—ralongthe
body ti.
Since Laplacetransformsmust be zero for
z< O, the axialdistributionmustbe shifteda distanceat
leastunityto therightby introducing
e-’ intothetransform
(34)
Equation(34)definesa Fahnngintegralinvolvinga newset
of characteristic
functionsgivenby
“*@)=L-’Lk’@l
Thus the Mz.(z) functionrepresentsthe distributionalong
theaxisof multipolestrengthfor them=O setof mnltipolw
necessaryto makethe velocityamplitudefunctiona delta
function. Corrwpondingly,
it is the distributionnecessary
to produceapressure
f%ddcorrespondhgto
theWZ~(z—r+l,r)
function. Equations(25) and (30) yield the relationship
betweenthe W,n(zjr)andL&(z) functions.
(35)
Thesecharacteristic
functionshavebeenstudiedinreference
7, are tabulatedin table I, and are plottedin figure8.
J?orming
nowthe
(37)
Seriesfor theA&(z) functionsfor smallandlargevalues
of theargument
havebeenobtainedby thestandard
methods
of Laplacetransformtheoryin reference7
Lo
(
~ $
J&(z) = ——
[
.5Ma(x) C
(16n~+3)
~---25632n3+33+33
~fl+ . . .
96
1
(38)
The square-rootsingularityof MSJZ)at the originis noteworthy. For the asymptoticresultonly a singletermhas
beencalculated
-51.0
M.(z)-—:
II
II
5
LO
1.5
2.0
(39)
J
30
3.5 40
25
x
Fmmm8.-&aphiaalrepresentation
of M*=(z)functions.
o
x>r—1
(40)
1308
REPORT 125%NATIONAL ADVISORY COMMI’ITEE FOR AERONAUTICS
Spanloadingand pressurecoefficient,-l?romequation gukwitiesaretractableusingthe methodsof analysis,they
arenot adaptedto thenumericaI
methodsusedherein, The
(31)thepotentiilcanbe written
method utilizhg the W2.(z,r)function moids difficulties
@=>O cos2nd[e-’C,.(s)] [K2%(8r)d]
(41) withlargenumbersandwithsingukwitiea.
IL APPLICATION
OFTIWiORY
TO COMBINATION
OF
so that
CIRCULAR
BODYANDRECTANGULAR
WING
g-l-l
. Q.(x—l—f)cosh 2n COSh-l
—
In thispart of the reportcalculationsare cnrrieclout to
P=g coa2n0
r ) @ (42)
r-l
n=o
determinethepressurefieIdactingon a wing-bodycombinaJtion employinga rectangdaxwingwithno thickness Tho
Fromthisresultthe potentialcanreadilybe obtainedand calculations
arefirstmadefor thebodyatzeroangleof attack
henm the spanloading. The pressurecoefficientfollows with the wingat incidence-thewing-incidence
case. Tho
directlyfrom equation(42) using the linearizedform of
calculationsare thenmadefor the body at angleof attack
l%rnoulli’aequation
withthewingatzeroangleof inciden~the angle-of-attack
attentionis focusedon theupper
(43) case. For thecalculations
halfof thecombination
sincetheexpsrimentrd
measurement
weremadefor theupperhalf.
X+l
Gm(—1)
cosh 2ncosh- —
WING-INCIDENCE
CASE
)
The completepressurefieldwill now be calculated.As
~~g
r +
previouslymentioned,the wing alonecan be specifiedin
anyconvenientmannerand,for thepurposeof theexample,
the wingaIoneis takenas the rectangularwingextending
straightthroughthe body fromsideto side. Althoughtho
analysisas carriedout is for M=fi, the resultsam preThe practicabilityof usingthis resultfor calculatingthe sentedin a form applicableto a rangeof Mach numbem.
pressuresdependsin the &t place on the accuracywith Thestepsin performingthecalculationare:(1) to determhm
which& G.(z—1—f) canbe calculated.Sincethiscalcula- ~, thewing-alonepotential;(2) to determinethe velocity
amplitudefunctions,j~x(z); and (3) to determinotho
tion dependson the Ma.(x)functions,whichare tabulated potentialor pressure,as desired,anywherein thefield. No
at the presenttime only to the third decimalplace,only tip effectsareconsidereduntiltheresultsarepresentedaaa
threesignificant
figureswillusuallybe obtainedfor theaxial functionofwingaapectratio.
strengthfunctions. For higherharmonicsandlargevalues
Wing-alonepotential.-The wing-aloneflow, exclusive
of z, lossof rwcuracyis incurredthroughthe natureof the of tip effects,can be determined
from theAckerettheory.
multipolesolutionsthemselves.The followingtabulation Theflowat a spanwisestationout of theregionof influenco
illustrates
thepoint.
of thewingtipsis illustrated
in figure9. The potentialfor
theflowabovethewingis
cosh(2ncosh‘b)
(
s
P+%
~
(
\
(46)
~= Vz+Z&T?(z—
z) whenz<z
I ‘/j+
:,
;,
;,
Z)qw
Thesidewash
producedby sucha potential— iszero,rbncl
%
/
,A’-----ych
. ‘i”es
Althoughthesetof multiplesusedhereis not welladapted
to-thecalculationof pressurecoefficientfor highharmonics
andlargevaluesof z, it nevertheless
is usefulfor calculating
spanloadingssinceonly one or two harmonicsare needed
in this case. The ticulties of computingpressurecoeflicient me alleviatedin part by the fact that the pressure
disturbancedueto higherharmonicsdampoutwithina few
downstream
radii. In reference7 the pressurecoeilicients
werecomputedup to thefourthharmonic(n=3) but with
somedifficulty.Theuseof a setof multipolesolutionsother
thanthe?n=Osetdoesnotholdmuchpromisesinceincreasing the valueof m introducessingularities
into the axial
strengthfunctionsanddecreasing
thevalueof m introduces
sin”~aritiesinto the multipolesolutions. Whiletheseein-
/
/
/
\
\
\
.
\
\
\
\
\
\
\
\,
\
by wingalono.
FIGURE9.—FIow fieldproduced
.
QUASI-C!~RICAL THEORY
OFWINGBODY
RWERFERENCE
ATSUPERSONIC
SPEEDS
-’lz!k&
-la
~2”(z1_v&F2 Cos(27L-1)w_2 Cos(2n+l) Co_
2n—1
2n+l
T [
4
4n’—l whenz< 1
o
o
T
o
u
8
l?munE10.—Variation
of normalvelouity
inducedat bodysurfaceby
wingalone;unitbodyradius;wing-inoidence
case.
r
6
.4
1L
fo(x)
0=1
*
-.
1
1309
(51)
(52)
wherea=sin-lr. Thejz.(z)functionsareshownin iigure11.
The constantvalues of YJB(z)for z>l are noteworthy.
Thevaluesoffz,(z) aretabulatedin tableIt.
Interferencepressure distributions.-The interference
pressuredistributionshave been calculatedfor tbe first
four Fouriercomponents.
and are presentedin figure12.
In this@e theabscissais proportional
to distancebehind
theMachlineoriginating
attheleadingedgeof thejuncture,
asillustrated
in part(a) of thefigure.Althoughthecalculw
tionshave been carriedout for M=fi, that is, f?=l, and
for unitradius,they are generalized
to all Mach numbers
andbodyradiiby replacingz—r+1 by ~a—~+l and~z~by
-2 -
13PJS
ashasbeendonein thefigure. Fromthe figureit is
apparentthat the cuspsin the pressuredistributionsare
-40
.4
.2
.6
x
.8
12
Lo
propagateddownstream
alonglinesof constant~a—~l or
z—pi-;that is, alongthe downstreamcharacteristics.A
Frmmm
11.—Graphioal
representation
ofvelooityamplitude
functions; the pressuredistributionsmove outwardfrom the body
wing-hmidenoe
case.
along the downstremncharacteristics,
they are distorted
and decreasedin magnitude.
Z)*
Increasingthe orderof theFourierharmonicscausestwo
‘he
‘omwmh
--&Tis uniformly—iwV. The down-wash importanteffects:first,thenumberof pointsof zeropressure
cfmses a flow normalto the surfacer= 1 in amount—iWV
sin t?. Thismeansthat,for rLbody conformhgto tbe wingrdongflow,the deformation
is zero at the wing-bodyjuncturesand a maximumon the top of the body. The interferencecombinationswhen addedto the deformedbody
straighten
it outFouriercomponentby Fouriercomponent.
Fourieramplitudes
of bodynormalveloeity.-TheFourier
amplitudes
of thenormalvelocityinducedby thewingalone
hw at
potentialat thebody aredetermined
by expanding~
r= 1 in a Fouriercosineseriesof evenmultiplesof & The
normalvelocity distributionis shownin @e
10. For
x> 1 the body is totallyimmersedin the wing dowmvasb
field. With the usualequationfor obtainingthe Fourier
amplitudes
of a function,thereis obtained
fo(z)=:~:k’z
iwvsin eG%
(47)
j,.(z)=:Jiwv sine Cos2n0at?
(48)
Theintegrations
give
fo(x)=~
(1–~)
(49)
when$<1
●
f.(z)==2viWwhenx>1
(50)
is incr-ed and, second,the pressurecoefficientdamps
morerapidly. As a resultof the fit effect,the contributionsof the higherharmonicsto the combinationspan
loadingare proportionally
less than theircontributionsto
the pressurecoefficient;while, as a resultof the second
effect,themoreremotea pointis fromtheleadingedgeof
the wing-bodyjuncture,the fewerthe numberof Fourier
componentsthat mustbe includedto obtainits pressure
coefficient
accurately.Allinterference
pressuredistributions
ja~a”+l=l.
This beexhibitdiscontinuities
in slopeat —
havioris a consequenceof the fact that thebody becomes
totallyimmersedin thewing-aloneflowfieldfor thiscondition. Whenthepressuredistributions
of thevariousFourier
componentsare addedtogetherto obtainthe interference
pressuredistributions,
the discontinuities
in slopetend to
cancelso thatthepressuredistribution
for the combination
willbe smooth.
A detailedexaminationof the interferencepressuredistributionfor the firstFouriercomponentillustratesseveral
pointsof interest. Theimportanceof thecomponentarises
hornthefact thatit accountsfor mostof theeilectof interferenceon thespanloading. Thereasonfor thisis thatthe
pressurecoefficientsfor n=O are of invariablesign. The
effectof the&st Fouriercomponentis to reducethevelocity
inducednormalto thebody by thewing-aloneflowfieldto
zeroaveraged
moundthebodyfrom0=0 toO=r atanystreamw-iselocation.
.
1310
REPORT
125%NATIONAIJ
ADTTSORY
COMMITTEE
FORAERONAUTICS
o
.5
1.0
-1.5
2.0
-1.0
-.5
0
.5
0
.5
Lo
1.5
2.0
2.5
3.0
5%5
x/BO- r/a+ I
(a) n=O
(b) n=l
FrlmEE
12.—Interference
pmwuedistributions
of variousFouriercomponents;
wing-incidence
case.
.
4,0
INTERFEB13NCE
ATSUPF)RSOMC
SPEEDS
QUASI-CYIJNDRICW
THEORY
OFWINGBODY
UI
z
V&-r/a+I
(c) n=2
(d)n=3
FIGURE
12.—Concluded.
1311
1312
REPORT126%NATIONAL
ADVISORY
COMWTT13
E FORAERONAUTICS
For purpose9of comparisonwith the exaotredti for
n= O, some approximateresultshave been included in
-20
figure12(a). For valuesof ~~1 on thebody, theAckeret
-1.6
valueof P@(twicethe localstreamangledividedby ~) is a
close approximation
to the truepressurecoefhcient. This
is theresuhtof thefactsthatthepartof thebody affecting
theinterference
is effectivelyplanefor pointsneartheleading edge of the wing-bodyjunctureand that thereis no
variationof any quantitieswith 0 so that an approximate
two-dimensional
situationprevails. As ~ increasesbeyond
unityon thebody, thereis a rapiddecreasein thepressure
coefficientbelowthe Ackeretvaluedueto the effectof all
disturbamxs
in frontof thepointin questionasrepresented
by theintegralof equation(24).
In reference7, the followingapproximateresultswere
obtainedfor smallandlargevaluesof ~a~a+l for thepressurecoeilicient:
(53)
Four Fouriercomponents
Six Fouriercomponents
\
—
\
/
/
I
-1z
&P
fw
-.s
“42
m
-.4
I
o
.5
lo
I
1.5
20
x/p
25
Pi
30
,
3.s
40
FIGURE13.—!I’heomtiordpremum distribution
at wing-body
junotum
of combination
usingfour and six Fouriercompommts;
winginoidenco
case.
concernedisnotlargesothatthebodyiseffectivelya vertical
boundaryon whicha givendistributionof normalvelocity
is producingan interference
field. Supersonicwingtheory
appliedto thisconditiongivesfor thenetinterference
prweurecoefficient(ref.7).
BP. Ax when=0
iw ‘3@?a
ftla
(66)
(54)
It is clearthatthecalculatedresultscanbe jointedsmoothly
to this result. Usingthe resultof equation(66) onablm
satisfactoryresultsto be obtainedwith four Fouriercomponents.
The criticalregionin the convergenceof the solutionis
thatneartheleadingedgeof thewing-bodyjuncture. Tho
higherharmonicshavetheirmostimportanteffectnearhero
andrapidlydampdownstream
alongthebody. Emco moro
and more Fouriercomponentswould be requiredto got
For~a—~l <0.6 equation(53)is a good approximation
for
accuracyfor smallerand smallervaluesof ~
~a. However,
2n6 z - ‘l’+o
pP,=+– 32iw(4nl)(#”+r-2’)COS
1)24n+l 0‘a
T(2nt)2(4ni—
J
n= Oalthoughit isof littlevalueforMgheraderhqonics.
withtheresultof equation(55),thisestraworkis unneccsThereisageneraltendencyof POto approachauniformvalue~ W.
independent
of r as~~
1 becomeslarge,as shownby
Onepointof interestin figure13is thefact thatwhen&
Da a
equation(54). The dampingin thecharacteristic
direction,
althoughinitiallyinverselyproportionalto the squareroot
of r, is ultimatelyindependent
of r.
Pressure distribution
in junctureof wing-bodycombination,-By addingthe interferencepressurecoeiiicientsof
thevariousFouriercomponentsto thatfor the wingalone,
the prewm distributionfor the combinationis obtained.
Theadditionhasbeencarriedoutforthewing-bodyjuncture
usingfourFouriercomponentsandsixFouriercomponents,
and the resultsare presentedin figure13. The pressure
coefficientwithinterference
is lessin magnitudethan2, the
valuewithoutinterference,
showingthatsigniihnt lossesof
lift occurin thewing-bodyjuncture. A comparisonof the
resultsfor four componentsandsix componentsshowsthat
fourcomponents
givegoodover-aUaccuracyfor allvalueaof
ence of the oppositehalf-wingis felt in tho wing-body
juncture.
Pressure distributionon top meridianof wing-body
combination,-’l%epressuredistribution
on thetopmeridian
of thewing-bodycombination
isobtainedinthesamefashion
asthatat thewing-bodyjuncture,the dMerencebeingthat
the pressuresdue to the evennumberFouriercomponents
havethesamesignatthemeridianasatthejuncture,whereas
the odd numberedcomponentshave reversedsigns, The
pressuredistributions
basedon four and six Fouriercomponentsareshownin figure14.
Severalinterestingeffectsare exhibitedby the results,
~ greaterthan1. For smallvaluesof $ in thewing-body
The stepin the wing-alonepressureat ~a=l is effoctivoly
juncture,the curvatureof the body insofaraa the flow is
canceledby the interference
pressuresof the Fouriercom-
equalsapproximately
3, thepremurecoefficientincreasesin
magnitude.Thisis dueto thefact thatfor~> T theinflu-
.—. .. -—-—--.- —-— — .— —-— —-——
QUABl_tJXlANJJltlWLJ.I‘1’HMUltY UF WING-BUDY INTERFERENC.131
AT SUPERSONIC
SPEEDS
-20,
,-
I
I
,I
[ I
~Wkg ol&e J’
I
I
I I Ill
I
I
I
L&i-T
I.- \
-1,2
I
,,
,
I
I
1
-%
w
,
,
-8
-1.2
I--@
A
/
o
.4~
.4
,8
-20
I
/
-,4
1313
-1.6
Equohon156~
P
-
W&g alon’eplus
Q
w
-- four Fourier
Topmerldlan~
components
I
I
‘--Wing olone plus six
Fouriercomponents
o
;0
,2
;6
24
2S 3.2 56 40
1/
X//go
BP
~w
-.8
-.4
0
X/flu
FIGURE14,—’l’heoretical
pressure
distribution
on top of
combination
)?rGunEI
15.—The+metic.al
pressuredistribution
aotingon wingof
usingfourandsixFouriercomponents;
wing-inoidence
caae.
combination;
wing-incidence
case.
ponentsfrom~=1 to ~a=m/2,andfor~> r/2 thepressure
Sincethe r.~on of influenceof the body on the wingis
pa
of the Machlines
increasesrapidly and tends toward the two-dimensional confinedto the wingregiondownstream
value. The effectof the interference
pressurein canceling emanatingfrom the leadingedgeof the juncture,in front
areuniformat thetwo-dimensional
tlm effectof the wing aloneon the top of the body from of thislinethepressures
value,andbehindthelinethereis a decrease
inthemagnitude
3=1 to —=~/2 is to be expectedsince the wing of the of the pre9surecoefficient. If the body were a perfect
/3a
~a
extent,then
combination
crmhaveno effecton thetop of thebodyunless reflector,that is, a verticalwall of fits
therewould be no pressureloss. Howevm,the pr~we
? > ~/2, as lms been alreadypointedout. II an irdinite pulsesoriginatingon thewingareonlyin partreflectedby
~a–
numberof Fouriercomponents
hadbeentaken,thepressure thecircularbody. The eflkiencyof thebody as a reflector
is discussedsubsequently
in connectionwith spanloading.
coefficients
wouldbe identicallyzerofrom ~=0 to ~
The
tendency
of
thepressure
to increasein magnitudenear
pa
pa ’12”
theinboard
trailing
edgeis
dueto
the effectof theopposite
Thegeneralbehaviorin thisregardis evidenceof theplausiTV@
pan~
whichat
the
W@
j~cture
is felt downsticam
bilityof thecalculatedresults.
The tendencyof thepressures
to approachan asymptotic of thepoint~=r.
13a
value is also illustratedby figure 14. This asymptotic
Spanloading,-The
spanloaddistributions
for a rangeof
valuerepresented
by thesumof thewing-alone
pressure
plus
rectangukw
wing-body
combinations
withthe
body at zero
the asymptoticresultsfol the first Fouriercomponentis
angleof
attackcanbe
determined
f
rom
thepressure
distrigivenby thefollowingequation:
butionsof @ures 13, 14, and 15. Siucethe pressuredistributionsof iigure15 arein a formindependent
of Mach
(56) number,it is convenientto defhe a spanloadingwhichis
~ >2.4, the resultsof this equationaxe in good anintegralof thesedistributions.
‘or pa
agreement
withtheresultsof iigure14.
qawa
Someevidenceis furnishedfromthepressurecalculations
for thejunctureandtop of thebody concerningthenumber
ofFouriercomponents
necessary
foraccuracy. Comparisons
madein figures13and14showthataboutfourcomponents
aresticient andthattheadditionof twomoreis notworth
(57)
theextrawork.
Pressuredistribution
onwingof wing-bodycombination.— The quantityin thesquarebracketsis takento be thespan
The distributionof the pressureactingon the wingof the loading. If all distancesare takenin units of the body
combinationcanbe determined
in a mannersimilarto that radius,then‘(a” canbe set equalto unityin theformulas.
for the wing-bodyjunctureby addingto the wing-alone
Thepressureremdtsof figure15aiefor value9of themeiratio of 4 or less and for valuesof the
pressurethosedueto theFouriercomponents.Theresult- tive chord-radius
antpressuredistribution
for thewingbasedon fourFourier effectiveaspectratio of 2 or greater. Spanloadingsfor
4=J:=[~(59’)~@)]49
+“[:J(&Jdz]d,
‘%’[%(%w’~
●
componentsis shownin figure16. For smallvaluesof ~
pa
the higher-orderoscillationsin the pressurecoefficient.as
shownin iigure13havebeenignored,andthe curveshave
beenfairedthroughthem.
anycombination
of ~ (or c*) and&4in theserangescanbe
/3a
obtainedby integratingthe pressuredistributions.The
span loadingevaluationshave been made for c*=4 and
/?A22. Firstthespanloadingsdueto the.variousFourier
1314 -
FORAERONAUTICS
REPORT
125%NATIONAJJ
ADVISORY
COMMTl?PEE
componentsare discussed,and thenthe spanloadingsfor
theactualviing-bodycombinations
arepresented.
In figure16,thecontributions
to thespanloadingfor.the
first three?louriercomponentsare shown. For n=O the
pressure
fielddoesnot dependon O,beingtially~symmetric,
rmda constantloadingexistson the body. However,on
thewingasthespanwiae
distanceincreases
thereis a decxease
in thespanloading,dueprimarilyto decreasein thelength
of chord over whichthe interferencemes-sures
act. The
spanloadingdue to the iirstFourier~omponentcausesa
lossof lift everywhere
alongthespan.
1.6
With the techniquesof Laplacetransformtheory,it is
possibleto obtainasymptoticformulasfor thespanloadings
of the variousFouriercomponents.For the firstI’ourier
componentthefollowingasymptotic
resulthasbeenobtained
by the standardmethodsof Laplace transformtheory.
(SeeAppendixB.)
‘hen
; -’w
(68)
asymptoticresultfor the spanloadinggivenby this
equation,whencomparedwiththeresultsof the exactcalculationsinfigure16,isseentobeslightlylow. However,for
valuesof ~ greaterthan4thedifference
betweentheresults
@
decreases,and equation(58) thus providesa satisfactory
meansof extrapolating
theresultsof thepresentcalculations
for spanloadingto largervaluesof ~
The asymptoticresulthas alsobeen determinedfor tho
higher-orderFouriercomponentsas a matterof intarest,
Thespanloadingis
The
B
I\
n=l
-7
n=27
,
-.
8. .-
$
Q .3
L“
*
Wmg
Body
/
.l~
/
/
/
/
16@+*)(4”-’)’c
0
m.-C
-0
:
~ _z4
(,,,
7r(2n
!)z(4ns—1)2Jm-1
~
a
s
n .OT
‘\
-32
-40
\t
-4.80r–––––y.
20
ylo
?iO
4.0
5
I?mmm16.—Thwretical
spanloadingof varioua
Fouriercomponents
actingoncombination
ofbodyandrectangular
winghavingeffective
chord-radius
ratioof 4; wing-incidence
ease.
A comparison
of theresultsof iigure16forn=O andn=l
showsthat the first Fouriercomponentaccountsalmost
entirelyfor theeffectof interference
on thespanloadingof
“ thecombmation.For thebody thisfact is evenmoretrue
thanfor thewing. Thisfact is of considerable
importance
sinceit givesasimplemeansof extending
theliftandmoment
resultstolargervaluesof ~ thanthoseforwhichthepressure
distributions
havebem d~’ated. Also,it suggests
a simple
the adverseeilectsof interferenceon
m~~ of ~
lift aswillsubsequently
be pointedout.
Theresultsof equation(59)andthe exactsolutionfor n= 1
in iigure16bothcorroboratethefact thatthespanloadings
of allbutthetit Fouriercomponentarenegligible
for—
;a >4,
It is alsoto be notedthatthecontributionto theloadingof
theiirstcomponentgivenby equation(58)increama
without
limitasz+ m;whereasthespanloadingsof thohigher-order
components
arefinite.
To obtainthespanloadingfor thefamilyof combinations
-=4, it is neceswuyto considerthe loadingsof
‘or ‘bi& ~a
both the wing alone and the Fouriercomponents.Tbe
neceswuycalculations
havebeen carriedout, and tho span
loadingsfor the familyof combinations
basedon one and
fourFouriercomponentsarebothshownin figure17. The
loadingdueto the wingaloneis alsoshown. No effectof
wingtipshasbeenincluded. It is to be notedin figure17
that,whereastheloadingonthewingdueto itsownpressure
fieldis constant,thereis somelosson the body becauseof
thefact thatthepressurefieldof thewingaloneactson the
body only if x>~a sin 0. However,if an afterbodyis
included,someof tbe lift lost can be recovered. As has
alreadybeen pointedout, the pressuresdueto the first
Fouriercomponentsare positiveon the upperhalf of tho
wing-bodycombination
andproducealossof lift,asfigure17
shows. When the effectsof four Fouriercomponentsme
takeninto account,the net lift is slightlyhigherthanthat
for oneFouriercomponent,but the differenceis not signifi-
1315
INTERFER.EINCE
ATSUPERSONIC
SPEEDS
QUASI-CYLINDRICAL
THEORY
OFWTNG-BODY
theinteresting
fact that the body is somewhatlessthan50
percenteffectivein reflectionfor this particularfamilyof
configurations.
I- Reference100ding,
campletereflection
Referenceloading,no reflection
I/
\
‘.
\
‘.,s.Combination loodingfour Fouriercomponents
/
I
1
‘\~Comblnation
loodingone Fourier component
Lift,-For values of -&
4 the pressuredistributions
alreadypresentedaresufficientfor obtainingspanloadingor
lift on eitherthe wingor body for all combinations
having
sticiently largeaspectratiosto avoideffectsof thetipson
thewing-bodyinterference.Thisis thecasefor1%4.22.The
liftresultsarepresented
in termsof anondimensional
parameterkm,definedas theratioof thelift on tbe exposedhalfwingein combination(exclusiveof thaton thebody) to that
on the exposedhalf-wingsjoinedtogether.
(60)
/
For *>4
the value of kw can be obtainedby usingthe
asymptoticformof thespanloadinggiven‘byequation(58).
kw=l–
8F+&10’(a-NOg@+’)1
mC@(2PA—1)
C*+w
(61)
‘ Thevaluesof kwhavebeendetermined
fromthepressure
distributions
of figure15for valuesof&4
tion (61)for values4 of ~>4.
andfrom equa-
The eflectof the wingtips
hasbeentakeninto accountby utihzingreference20. The
resultsareshowninfigge 18whereinkwisgivenasafunction
y/o
I?mmm17.—Theareticnl
spanloadingfor combination
of bodyand
rectangular
winghavingeffectiveohord-zadius
ratioof 4; winginoidenco
caw.
cant. For most engineering
purpose9,one Fouriercompog the span loadingwhen
nent is sufficientfor determining
.
~>4.
f?a–
Someinsightintothemechanism
of wing-bodyinterference
can be gainedby comparingthe spanloadingfor the combinationwiththosefor tworeferenceloadings:(1) thecompletereflectioncasefor whichtheblanketedareaof thewing
actseffectivelyat&, and(2) theno-reflection
casefor which
theblanketedareaof thewingis supposedto aoteffectively
at zeroangleof attack. The spanloadingcorresponding
to
thefirstcaseof completereflectionof thewingpressure
pulses
by thebodyis,infact,thespanloadingmarked“wingalone”
in figure17. A comparison
of thiscurvewiththatbasedon
oneor fourFouriercomponents
showsthattheloadinggiven
on thoassumption
thatthewingblanketedareaisfullyeffective in lift is too optimistic. Underthe conditionsof the
secondreferenceloading,the solepurposeof the blanketed
meaisto supportliftgenerated
by thewings. A compfion
of thespanloadingfor thiscasewiththetrueloadingshows
thattheaverageloadon thebodyis wellpredicted,butthat
theloadingonthewingis under~timated...
A comparison
of
~ thetrueloadingwiththosefoi’thetworeferencecasesreveals
.
\\
.92
II
I
I
Y
b
‘ \2
.88
/
!II/
I
x
/
/
.
.84
I
I
.800
I
I
2
I
I
I
I
!
I
I I I
I
I
I
I
‘- -Asymptoticformu[o
‘hosed on one
-Fourier component,.
equation (61)
I
I
I
I
I Body ot zero armle of ottack I
12
10
4
8
6
Effective chord-rodius raha, c/~0
t
I
I
I
I
14
16
Fmcmn18.—Lifteffectiveness
forwingor controlsurfacein
combination
withbody.
of ~ for variouseffectiveaspectratiosof 2 andgreater. It
fla
shouldbeborneinmindthattheresultsof thefigurearefor a
combination
of bodyandrectangular
wingor anall-movable,
~rectangular
controlsurfacewithno gap. It is notedin the
figurethattheexactresultsfor ~a<4 canbe fairedinto the
10theasymptotfo
emalrtlcaf
mprssdom
forkrrsnd+lc arenot_
by
4In rofemm
vfrtneofenfncorrwtn
prmllndtonfmfnteaml.Tbrnadxnaron
nrnei-fcnlo
rrorinkwls
abto.ol mhzdcawt
amtivqb-ofe.
‘I’hepreekevalaesfuogfven
hIthk roIMrL Thesefor6A-2 areonchrrngd.
1316
FORAERONAUTICS
REPORT125%NATTONAL
ADVISORY
COMMITTEE
Thevaluesof ~ havebeendetermined
fromthe pressure
asymptoticresultsfor ~>4, therebyprovidinga dwign
chart for engineeringpurposesfor the entirerangeof ~
distributions
of figure15for valuesof &-4 andby equation
The curvesof figure18 illustratethe decreamof & as ~
(62)for valuesof &>4.
increasesat constanteffectiveaspectratio,andtheslowincreaseof k~ asthewingchordbecomesverylarge. Theloss
of lift is mostseriousfor &4=2, beingabout15percentin
theworstcase.
A practicalpointin connectionwiththelossof lift on the
wingdue to interference
is that this10= occursno matter
whatthebody angleof attack,eventhoughthecalculations
aremadefor aB=O. It occurseitherin thecaseof a wing
mountedon a body or in thecaseof a deflectedatl-movable
controlsurface. For wingswith sweptleadingedgesfor
whichallof thewingarealiesin theregionaffectedby the
interference,
ev~ largerlossesthanoccurwithrectangukw
wingsareto be anticipated.However,thelossof lift at the
designconditioncan, at leastin principle,be largelypreventedby design@ thefuselageso thatit conformE
to the
firstFouriercomponentin thewing-alone
flow-. Thiswould
involvecon~ting the fuselageabovethehorizontalplane
of symmetryinarotationally
symmetric
fashionandexpandinga likeamountbeneaththehorizontalplaneof symmetry.
Whetheror not sucha changewouldimprovethe lift-drag
ratiocanbeatbe determined
by experiment.
Center of pressu.re.-The cmter-of-pressurelocations
havebeencalculatedfor thesamerangeasthelift resultsof
@e 18. The center-of-pressure
locationin chordlengths
behindtheleadingedgearepresented
infigure19. Forlarge
valuesof c*, an asymptoticresulth= been calculatedfor
ZJCusingthemethodsof Laplacetransform
theoryandconsideringonlyoneFouriercomponent.
beentakeninto consideration.The exactresultsfor &4
%2
C
112
2U+$ log &
3f7A f?zi%[ z()
k. (l–+A
.
. ).
–; log (C*+U]
asti+co
(62)
The lossof lift nearthe tipshas
havebeenfairedintotheasymptoticresultsfor largevaluea
of ~ by dsahedcurvesto providean engineeringdesign
chartcoveringtheentirerangeof&
It is againmentioned
thatthischartis applicablebothto thewingof an airplane
or missileor to an all-movable,rectangularcontrolsurface
with no gap. The curvesof figure19 start at valuesof
‘w corresponding
to thosefor thewingaloneat~=0. As~
Z&
increaaes
for constant9A,thereis aforwardmovementof the
centerof pressure
becauseof thelossofliftduetointerference
whichis mostlyeffectiveon therearof thewing. For the
lowesteffectiveaspectratioof 2 thereis abouta 4-percent
forwardmovementof the centerof pressuredueto intsrforencein the extremecaae. For largeeffectiveaspectratios
theforwardmovementis not nearlyso large. As thovaluo
of ~ increaseafor constant/3A,there is an mymptotio
pa
approachof the centerof pressureback to the wing-alone
vrdue.
AN~EOF-A’ITACK
CASE
In figure4 (a) it is show-n
howtheflowfieldof a combinationcanbe builtup of a bodyaloneandtwowing-bodyflow
fields. The firstwing-bodyflowfield((2) of fig. 4 (a)) has
beensolvedin the precedingsection,andwe now solvethe
secondwing-bodyproblem((3) of fig. 4 (a)), The wingis
tiectivelytvvisted
sothattheslopeof itssurfaceis a“ m given
by equation(1). It shouldbe notedthattheproblomof the
combinationand body with a rectangularwing twisted
accordingto the second term of equation(1) has been
solvedby Bailey and Phinneyin reference11 usingthe
presenttheory. Theircalculationis restrictedto tho body
j’c
- 3.0
,2@):7
I
2.5
a=l
2.0
1.5I
Effective chord-radius ratio, c/~a
FmuEH
19.—Center
of pressure
forwingoi &ntrolsurfacp
in
cmnbinatfon
withbody..
~
-l.o L
o
Fmum
,.
.2
.4
.6
x
[.0
.8
.’
12
‘ ‘.:,
20.-Gmphi@ repmmntstitm
of velocityamplitudp
funotions;
angle+f-attack
case.
r,., ,,, .
1317
QUASI—CYHNDRICAL
THEORY
OFWING-BODY
INTERFERENOH
ATSUPERSONIC
SPEEDS
o
.5
1.0
I,5
2.0
2.5
3.00
.5
LO
L5
2.0
2.5
3.0
3.5
4.0
.T/@ - r/o + I
(a) n=O
of variousFouriercomponents;
angh+of-attaok
case.
l?IGUEI!Z1.—bterference premnre distributions
cmdis carriedout for downstream
distancesof 2Bafromthe
wing leadingedge. Actually,the resultsof reference11
representthe ditlerencebetweenthe angle-of-attackand
wing-incidenee
crisestreatedhereandarein agreement
with
tho presentresults. This agreementis, in effect,an independentcheck on the accuracyof the presentnumerical
resultsfor theinterference
pressuredistributions.
Wing-alonepotential,-Thetit step in the calculation
is to detmninethewing-alonepotential.Becausethewing
is twistedto conformto thebodyupwashfield,thisdeterminationis fairlytediousandhasbeentied outin Appendix
Cl The formof thewing-alonepotentialfoundin reference
11 is in agreementwith thosefoundhereinfor the wingincidenceandangle+f-attackcases.
Fourieramplitudes
of bodynormalvelooity.-Thevelocity
amplitudefunctionsfor the presentcase were computed
numerically
by performinga Fourieranalysisof the calculatedbodynormalveloci@distribution
at a numberof body
crosssections.b analyticaldetermination
wasmadeof the
velocityamplitudefunctionsby the authomof reference11
forvalueaof z<2. However,forz>2 thevelocityamplitude
functionsare saidby theseauthorsto lead to incomplete
413672-57-sa
ellipticintegrals,andno analyticaldetermination
wasmade,
A numerical
determin
ationhasbeenmadehereinfor0<x<4.
Thenumerieal
wdueaof thefj,(z) functionsfor thiseaseare
tabulatedin tableII andplottedin figure20for illustrative
purposw.
Interferencepressure.distributions,-The interference
pressuredistributionshave been calculatedby numerical
integrationusingequation(22). The resultsare shownin
&-me21. The interference
pressuredistributions
arevery
similsxto thosefor the wing-incidencecase, being about
twiceaslarge.
Pressuredistribution
in junctureof wing-bodycombination,-The pressuredistributionof the combinationis
obtainedby addingthe interference
pressurecoefficientsto
the pressurecodicients of the wing alone. The results,
usingfour andsixPz, components,areshownin figure22.
Thisfigureshowsthatfourcomponents
givea closeapproximationto thelinear-theory
valuefor x/@>l. At z/@= O,
the wing leadingedge,lineartheorywith Beakinupwash
theorygivesexactly/3P/cr=
–4.0. For the region3/f?a<l
the higherharmonicshave theirgreatestimportance,and
manycomponents
wouldbe neoessaryto getgoodaccuracy.
1318
mPORT 1252-NATIoNu ADmORY coMMITTEEFORAERONAUTICS
-2.0
-1.5
-Lo
m
N
m
g
<
c
Q
‘“5
o
.5
I .0
0
.5
Lo
15
2.0
x/~o -r/a+
(b)
I
m=l
Fmmw21.—Continued.
2.5
3.0
55
4.0
1319
QUASI-OYIJNDRICAL
TEEORYOFWINGBODY
~TERFERENCE
ATSUPERSONIC
SPEEDS
-15,
..-
,
-1,0
al
-.5
0
.5
I.OO
.5
Lo
L5
2.0
2.5
so
3.5
4.0
.@a -rla + I
(o) n=2
FIGURE
,
21.—Continued.
t
..
1320
REPORT125*NATIONAI.IADVISORYCOMMIT173E
FORAERONAUTICS
[email protected]+ 1
(d)n=3
FIGURE
21.—Concluded.
-4.0.
1
,
,
\\
-32
-24
&
F
-1.6
-.8
\v
,--’
,
!!
Six Fmmer components
/ \..- tl%ur Fourier comooner&-
m
I
I
Juncture,
\,
,
of the oppositehalf-wingreachingthe wing-bodyjuncture
at thispointas shownin the sketch.
It shouldnowbe notedthatthepressuredistribution
for
the-easeof thebodyat angleof attackwiththewingat zero
angleof incidence
isrepresented
by thesumof cases(1)and(3)
aa givenby figure4 (a). However,we are neglectingtho
contributionof case (1) becauseit is small. The contribution to the pressurecoefficientrepresentedby cam (1) is
thatdueto a yawedinfinitecylindersincewe areneglecting
noseeffects. This contribution,whichis clearlypresentin
front of the wing,is
P
(63)
— = (.YB(l
—4Cos%)
C#B
()
0
Lo
1.5
20
25
5
3.0
3.5
4.0
x//3o
Forthejunctureof thecombination(0=0°) thecontribution
FIQUREI
22.-Theoretical
pressure
distribution
atwing-body
junoture is about0.1for aB=zO and0.3for aB=60. At an=2° tho
of combination
usingfourandsix Fouriercomponents;
angleof- effectis thusnegligiblecomparedto P/aB of about4, andat
attaokU.
aB=6° therearedefinitenOdiU~ effectsthatm& fbprecise applicationof linear theoW inaccurate. For those
However,satisfacto~accuracycanbe obtainedby fairinga reasonsthe contributiongivenby equation(63) hnabeen
curvethroughthisregionsinceboth endpointsareknown. neglected. For the top andbottomof thebody thecontriOne item of interestin figure22 is the increa9ein the butionsare one-thirdof the foregoingand henceare also
magnitude
of ~P/aB
nearpoint1. Thisis dueto theinfluence negligible.
QUASI+~~CW
~ORY
Pressuredistribution
onbodyof wing-bodycombination.—
Tho pressuredistributionon the body is also obtainedby
addingthe interference
preaaure
coefficientsto the pressure
coefficients
dueto thewingalone. Theinterference
pressure
distribution
for anyvalueof o differshornthatin thewingbodyjuncture,0=0, onlyby a cos2m9factor. For example,
in thejuncturecos 2n0is always+1. Ontop of thebody,
i9=r/2,cos2n0alternates
between+1 and—1asn increasa.s.
OntheO=CT/4
meridiancos2n0hasvaluesof O,+1, and–1,
so thatwhenn is oddPZ.=O. Thepressuredistributions
on
thetop meridianof thebody andon the0=45° meridianof
thebodyareshownin figures23and24,respectively.
-4
-3
.
LLl
.-Wing aloneJ’
,!r
;’
‘
:
‘-
-
-
-
-
-
coeiiicients
wouldbe identicallyzerofromx/~a=Oto z/pa=
r/2. Thesameeflectsareexhibitedby figure24exceptthat
the wing-alonestep occurs at z/pa=&/2 and the Mach
helix intersectsthe meridianat x/~a=x/4, point 1. The
Mach helix from the oppositewing panel intersectsthe
mtidian at point 2 causingan additionalpresmwerise.
Sincethe regionin which13P/a~=O
is bow-n andsince the
exactlineartheoryis wellapproximated
by fourcomponents
for largevalucaof z/~a,theoreticalcurvesof goodaccuracy
can be fairedhornfigures23 and24. The areaunderthe
highpeaksin the curvesnearz/Ba=rJ4wouldbecomeinfitesimal if an idinite numberof interferencepressure
componentswere taken.
Pressuredistribution
onwingof wing-bodycombination.—
For theregionin frontof theMachwavefromtheleading
edgeof thejuncture,the calculationof pressureceefiicients
is just a wing-aloneproblem. The pressurecoafiicients
in this regioncan thereforebe obtaineddirectlyfrom the
wing-alonepotentialasgivenin AppendixC. Theresultis
P=–2aB
A
I
A?+
I
Y/’t--l
1321
OF-Q-BODY INTERFERENCE
AT SUPERSONIC
SPEEDS
Four Fb@ercomponents
,
1 i I 1
SIX Fouriercomponents
I
[’
1
(&-yf1312
(64)
1
In theregionbehindthe Mach wavethe pr=ure coef6cientswereobtaineddirectlyfrom the WSJZ,r) functions,
I I I\/ I
aswasdoneon thebody. Theresultsof thesecalculations
for
the wingpressuredistributions
are shownin figure26
3.5
4.0
1.5
2,0
2.5
3.o
‘o
5
LO
and are to be comparedwith the pressuredistributions
of
x/p o
@e 15.
23.—Theoretical
p~u.m distribution
on top of combination
FmunE
Spanloading.-The spanloadingshavebeendetermined
usingfourandsixFouriercomponents;
angle+f-attaok
case.
by graphicalintegrationof the pressure-distribution
curves
of figulw 21 to 25. For a combinationwith a value of
-4
c/~a of 4 the span loadingsassociatedwith the.various
Fouriercomponentsme shownin figure26,whichis to be
-3
comparedwithfigure16 for the wing-incidence
case. The
magnitudesfor the n=O harmonicof the angle-of-attack
-2
caseare abouttwicethosefor thewing-incidence
case,but
g
otherwisetie two case9are similar. The span loading
includingwing-aloneand interferenceeilectsis shownin
-1
@ure 27, whichis to be compmedwith Qure 17. The
importmtdifferenceis notedthatthepeakspanloadingis
0
nearlyequalto theroot loadingin the angle-of-attack
case,
butis considerably
greaterthantherootloadingin thewing1 +. r
‘o
.5
1.0
1.5
20
25
3.0
3.5
-4.0
4.0
X/pa
Fmmm24.—Tlmoretioal
prwauredistribution
on 0=45°meridian
of
bodyof combination
usingfourandsixFouriercomponents;
angleof-attaokcase.
‘\
-2.4
flP
%
I
1
,-
rh =40
Severalinteresting
effectsareexhibitedby figures23 and
24. Thostepinthewing-alonepressureat %/pa=1in figure
-1.6
23 is effectivelycanceledby the interference
pressurehorn
T
‘\
,
x/j’3a=1to%/@a=r/2, andforx/fla>~/2thepressure
increases
ti=w
rapidly. The effectof theinterference
pressurein canceling
-.8
tho offcct of the wing aloneon the top of the body horn
+!$
x/pa=1 to x/fla=~/2is to be expectedsincethewingof the
0
20 25 30 3.5 4.0
.5
10
1.5
combinationcanhaveno effecton thebody in tint of the
Machhelix(point1of sketch)originating
attheleadingedge
of thewing-bodyjuncture. If an infinitenumberof com- FIWJZE
25.—lWoretiwd
p-me distribution
aotingon wing of
combination;
angle-of-attaok
can
ponents had been computed,the combinationpressure
1322
FORAERONAUTICS
REPORT125*NATIONALADTISORY
COMMFM’EE
.8
0
A
\
.8
~—Looding due to
-1.6
-24
$
-32
@‘12
;7 +.~
I
G -48
~
~,, %mb’inotlon loadi~gfour Fouriercomponents
!
100cilng~~m~ino+ion
one Fourier component
z -3.6
c
0
$
-64
-7.2
Body
-ac
4
Wing
-88
-9.6
04
y/o
1.0
20
y/o
3.0
4D
5
Fmmw26.—Theoretiml
spanloadingof variousFouriercomponents
spanloadingfor combination
of bodyand
actingoncombination
of bodyandrectangular
winghavingeffective l?mmm27.—Theoretical
rectangular
winghavingeffeotiveohord-radius
ratioof 4; mglo-ofchord-radius
ratioof 4; angle-of-attack
cam.
attaok
case.
incidencecase.. Because of tie cliflerencoin the sl.mpeof
tlm span loadings,a differenttrailingvortexpattern would
be nssociat~dwith each. ?No@ect of wing tips is included
in figures26and27.
reference21. The lift of the entirewing aloneincluding
the blanketedarea was so determined.The lift of lho
blanketedarea was then calculatedfrom the potonliml
Lift,-From the theoreticalwingpressuredistributions
of
functiongivenin AppendixC rmdsubtractedfromtho lift
tbe combinationthelift of thewingpanelsin thepresence of the entirewingaloneto get thelift of the panels. Tlw
of thebody canbe calculatedas a functionof PA andc/@.
loss of lift on the panelsdueto interference
as determined
To showhowthebody upwashis effectivein increasing
the by graphicalintegrationwas then subtractedto get ~~a.
lift of the wing, a factor KWhas been calculated. This The valuesof Km so calctiatedareshownin figure28 (a)
factorhasbeendefied as
asa functionof c/paandin figure28 (b) asa functionof a/s.
Figure28 (a) showsa largeeffectof IL4at constnntc/pa;
kc.
&.=()
(65) whereasfigure28 (b) showsa smalleffectof &l at constmt
&= y- ,
a/s.
In figure28 (b) theeffectof aspectratioon Kmat wfi..ecl
Here.&Cis thelift of thepandsin thepresenceof thebody
as
nnd.& is thelift of thewingpads joinedtogetherat angle valueof a/s is lessthantheprectilonof the calculations
area. For comparisonthe
of attacka.. In calculating& firstthelift of theexposed indicatedby the cross-hatched
theoryhnvebeen
panelsm partof theW@ alonemustbe calculated.This valuesof Kw calculatedfromslender-body
wasdoneby the use of revemibilitytheoremsdescribedin includedin thefigure
((33)
1323
ATSUPERSONIC
SPEEDS
THEORY
OFWEW+BOD NTERFERENCE
QUAsI=~HCW
The close agreementbetweenthe linear-theoryresultsfor
the presentcase and the slender-body-theory
resultsis
noteworthysincethe rectangularwingandbody combinationsconsideredherearenot slender. Thisresultsuggests
that slender-bodytheory can be used for calculatinglift
ratiosfor nonslender
coniigumtions.
20
18
III.COMPARISON
OFEXPERIMENT
ANDTHEORY
FOR
RECTANGULAR
WINGANDBODYCOMBINATION
APPARATUS
AND
PROCEDURE
b investigation
to evaluatethepresenttheorywasmade
in theAmes1-by 3-footsupersonic
windtunnel. Thiswind
tunnelwasequippedwitha flexible-plate
nozzlethatcould
be adjustedto give tes.ksection
Machnumbemfrom 1.2to
2.2. The pressuremeasurements
are obtainedas photographicrecordingsof a multiple-tubemanometerboard
usingdibutylphthalateasthefluid.
16
Kw
14
1.2
\
\
-h
Y
\
6.00
3-
—6
y/o
00...00
-
—2.58
. -1.92
_ I,02
0000.
I.*
(0)
eooooo
1,00
I
Effectivechord-r%diusrotio c/&
(Q)
4
3
x
\
O..
t’
.*OO
— 1.25
Effectof chord-radius
ratio.
FIGURE
28.—Lifteffectiveness
for rectangular
wingin combination
withbody;angle+f-attaok
case.
3.75
00000
—3.92
I 1.02
2,0
+3.00+
\’
—4.25 4
1.8
~rifice
,’
1.6
I
Presenttheory=,
25 flAS6
‘\
Kr,
{
tI +a
A
/
/
I .(
(b)
.2
.4
.6
Effechve rodius-semisponrotio, 0/S
(b) Effeotof radius-semispan
ratio.
28.—Concluded.
FIGURE
I
Fmurm
29.—Pressure
distribution
model(all dimensions
in inohes).
1,4
1.2
surfc-ce
.8
1.0
Thesting-supported
model,whichis diagramm
edinfigure
29,is a combination
consistingof a cylindricalbodywithan
ogivrdnose and a rectangular,wedge-shaped
wing. The
dimensions
of the modelaregivenin figure29. The wing
wasmade10 percentthickto minimizeaeroelasticeffects.
It wasmountedin thebodyby meansof a setof anglebloeka
whichenabledthe flat wingsurfaceconttig the oriiices
to be setat 0°, —1.9°,—3.8°,and—5.7°anglesof incidence
withrespectto the body centerline. The presureorifices
werealllocatedon theuppersurfaceof themodd. The47
ori.iiceswere distributedalong seven.manwisestationsin
orderto givea comparison
withtheoryfor thewingandthe
I body. Thelocationsof theofices aregivenin tableIII.
1324
REPORT125>NATIONAL ADVISORYCOMMT17EE
FORA13RONA’UTK!S
Sincethisinvestigation
requireda comparisonof thedata
forseveralMachnumbersandReynoldsnumbersatthesame
dues
of @j and&-,it wasnecessaryto set aB and &- accuratelyfor eachmeasurement.Thestingsupportby which
themodelwasmountedhadsufficientflexibilitythatit was
deemednecessaryto havemeansfor accuratelysettingthe
vrduesof ~ivand& undertunneloperatingconditions. The
valuesof& wereaccuratelysetby meansof angleblocksin
thebody. The angleof attackwasset by a specialimage
projectiondevice. A mirror-mi.sinsertedin the schlieren
systemso thatanimageof themodelwascastupona screen.
Withthewindoff, themodelwassetat thedesiredvalueof
aB andtheinclination
of themodelimagewasmmkedon the
screen. Withthetunnelinoperationatthedesiredpressure,
the angleof attackof the modelwas adjusteduntil the
inclinationof itaimagewaspdel to the calibrationline
made on the screenwith the wind off. To check this
method,a horizontalandverticalwiregridwasplacedonthe
tunnelwindowandschlieren
picturesweretakenof themodel
whilethe tunnelwasin operation. Thesepicturesshowed
thattheimageprojectiondeviceset @ to within+ 0.07°of
the desiredvalue. It was especiallynecessaryto set a~
accuratelyfor the smallanglesI%avoid largepercentage
errorsin theanglesetting.
Themodelangleof attackrangedfrom + 6° to —6°in 2°
increments,
andthewing-incidence
anglerangedhorn 0° to
–5.7° in 1.9° increments.The twt wasperformedat the
two Mach numbers1.48 and 2.00 and at the Reynolds
numbersof 0.6,1.2,and1.5million,basedon thewingchord.
Themodelwastestedfor allcombinations
of thesevaluesof
thefourparameters
investigated.
A completeset of datain theformof ~ for theReynolds
numbers0.6, 1.2, and 1.5X10aat ilf’=1.48 and for l?=
1.5X 10eat M=2.OQispresentedin tableIV. Thesevalues
of P are,for themostpart,averagesof tworeadings.
REDU~ON
AND
ACCURACY
OFDATA
All data are reducedto the coefficientform (p-pJ/gO.
Actuallythequanti@(p-pT)/q. was measured,
and subsequentcorrectionswereappliedto changethereferencestatic
pressuretopl (pIisthestaticpressureattheparticularorifice
in questionwhen @=&= 0°) and the referencedynamic
pressureto qo. Sincep, includestheeffectsof nosethiclme~
andstreamangle,usingpl as a referencepressure~es
theseeffectsandessentially
givesonly thepressuresdue to
theanglesettingsof themodel. The dynamicpressurewas
adjustedfrom q~to goon the basisof a previouspressure
wasnegligible
surveyof thetunnel. Thislatteradjustment
forfW=1.48andamountedto lessthana 3-percentcorrection
for iM=2.00. For the purposeof comparisonwith theory
thepressure
ccdicient @—pJ/goisreducedto theparameters
#IP/ciBfor iW=OO
andBPJiw for aB=OO.
Two typesof errorsenteredintothe experimental
investigation:systematicerrorsandrandomerrors. In thispaper
accuracywill be takenas the abili~ of the experimentto
givethetruevalueswithoutnoseeffector streamangleand,
hence,is a measureof thesystematicerrors. Preckionwill
be takenas the abilityto repeatthe dataand,hence,is a
measureof therandomerrorsin the experiment.
Severalfactorscontributedrandomerrors, The major
factorwastheerrorin theangle-of-attack
setting. Theuncertain ineachanglesettingwas+ 0.07°,buteachmeasurementwasdependent
upontwoanglesettings:thesettingfor
the conditionrepresented
andthe settingto determinethe
zero correction. This leads to a net uncertaintyof O.1°
whichwouldaccountfor a 5-percenterrorfor anglesof + 2°.
Most of theremainderof theuncertaintyin thedatais due
to thefactthatthereferencewallstaticpressure
inthetunnel
changedslightlyfrom run to run whilethe total prmsure
remainedconstant. Althoughthemagnitude
of thispressure
changewasquitesmall,it waslargeenoughcomparedto the
smallpressuredifferences
for the 2° anglesettingsto cmso
asmuchas a 3-percenterror. In additionto thesefactors,
betweenl-percentand 2-percentunctiinty wasobserwxl
in readingthe datafrom the manometer-board
pictures.
To detwmineexperimentally
the precisionof the dots, L
large number of repeat measurements
were taken and
compared. It was found that for ~B or iW=&2°, two
independent
detetinationsof ~P/uB or BP/& differedfrom
eachotherby +7 percentontheaverage.For aB or&= +4°
and~BoriW=&6°,theexperimentally
determined
precision
of /3P/aB and/3P/iw are &4 percentand &2 percent,respectively. Theprecisionin &/aB increases
withthemagnitude
of theanglebecausea largepartof therandomerroris due
to theanglesetting. Thelmownmajorexperimental
errors
are due to stream-angle
andbody-noseeffects. The effect
of these factors was not determined,but, as previously
described,correctionswereappliedto minimizetheireffect,
assuming
theeffectsdidnot varyappreciably
withangle-ofattacksettings. This assumptionshouldbe good for the
body-noseeffect. However,it is not necessarilya good
assumptionfor the stream-angleeffect since the stream
anglevarieswith verticallocationin the tunneland the
modelmovesapproximately
6 inchesin n verticaldirection
betweenaD= + 6° and aB= —6°. Since the stream-angle
correctionthatwasusedwasobtainedfortheffB=0°position
in the tunnel,data obtainedat aD=0° shouldhave no
appreciable
errordueto streamangle. For othervaluesof
a~, someerrordueto streamangleis possible.s
For thepurposesof thispaper,theimportantquestionis,
“How well does theory predictthe experimental
data?”
Direct comparisons
betweenlinmr theoryand experinmnt
will be madeonly for aB=&2° and iW=—l.f10dat~, In
iigure30experimental
pressure
distributions
inthewing-body
junctureobtainedfromtwoindependent
measuremmts
with
&-——1.9°and aB=OOare shown togetherwith a faired
curve of their averagevalues. The +7-percentlimit of
precisionabout the averagevalue is representedby the
dottedlines. The &o showsthat the theoreticalvalue
generallylies betweenthesedottedlines,and thereforetho
theorypredictstheexperimental
valueswithintheprecision
of thedatain thisexample.
GRNERAL
PHYSICAL
PRINCIPLR9
Beforethediscussion
of theresultsof theinvestigation
in
detail,it is wellto givefirsta generalphysicaldescription
of the effectsto be expected. Figures31 and 32 show
~Astremn+nglo
andprmure
oluymmotry
~Y Oftie ~ _
~ ti vcrtlml
Pfnno
fndfmtdthatstream-angfe
variation
caosedthemagnftudo
of thoos@montalvaluca
of
BP/aBtoIM4pa’a!nthfghon thoavamge.
1325
QUASI—CYLINDRICAL
THEORY
OFWJIW-BODY
lNTERFERENOll
ATSUPERSONIC
SPEEDS
-4
TT=T’7
❑o
-3
BP
G-2
to curlaroundthebody untiltheystrikethewingpanelat
points3, wherepart of the pressuredisturbancecontinues
along the wing and part of it is reflectedalong another
Mach helixon the body, causinga furtherincreasain the
magnitudeof thepressurecoefficients
at points4. Another
pressuredisturbanceoriginatesat the trailingedgeof the
wing-bodyjuncturethatcausesthedecrease
inthemagnitude
of the pressurecoefficientsnoted at points 5 of the two
figures.
Two independentmeasurements
Fcuredaverogeof two measurements
* 770 llmlt of preclslonon averoqe
o
-1
–—
o
I
5
1.0
1.5
20
2.5
3.0
3.5
x/k20
Intersectionof Mati
IYMeSwith surfoce
of combinotim
v
I’XCHJZE
30.—Comparison
between
twoindependent
readings
ofprmum
distribution
in wiag-bodyjunoture;a~=O, iW=l.90, ~=L@
R=l.fixloo.
‘---—
intersectionof Moth
coneswilh surface
of combination
I
32.—Ie0metrio
drawingof pressuredistribution
aotingon
combination
of bodyandrectangular
wing;wing-inoidence
cairn.
I?murm
drawingof
FIGUnE31.—Ieometrio
premuredistribution
aotingon
combination
of bodyandreotangulsr
wing;angle+f+ttack
case.
qualitatively
thepressuredistributions
to be mqectedon a
rectangularwing and body combinationfor the angl=fattackcaseandthewin@ncidencecase,respectively.The
chordwisevariationsof the coe5cient,BP/aB or f9PIim, are
shownfor fivesttitionsby theshadedareas.eThesefigures
show that-Mach coma emanatingfrom the wing-body
juncturedeterminethe pointsat whichthe variouseffects
of wing-bodyinterference
arefelt. Onthe cylindricsJ
body
tlmpressurecoefficientis zero in front of the Mach helix
originatingat the leadingedgeof the wing-bodyjuncture.
Thebodypressurecoefficients
herearetakenm zerobecause
the effectsof crossflowon the body pressuresare very
small,asshownin connectionwithequation(63). However,
asshownby thetwostationson thebody,thepressurerises
abruptlybehindthisMach helix,point 1, in both figures.
ThoMachheliceafromthetwowingpanelscrossthe0=7/2
stationsimultaneously
so thatthereis onlyonelargeincrease
in the magnitudeof the pressurecoefficient.TheseMpch
holiceacrossthe 0=3r/4 stationat two differentpointsso
that beyondpoint 1 thereis a secondaryincreasein the
pressure
coefficients
atpoint2. TheseMachhelicescontinue
dlstrfbutlon
slmm
forthe0m3r/4
station
onthebody k Identfrd to the
eThoprcssare
P~
dfstrfbntlon
fortho8../4 st.ailon
duototie symme- oft~ m~~
413072—57—84
Onthewingof the combination
thepressurecoefficientis
thesameasthatfor a wingalonein froritof theMachwave
hornthewing-bodyjuncture,exceptthatwhenthebodyis
at anangleof attackthebodyupwasheffectivelytwiststhe
wing in a mannersuchthat a.=a.(1 +aa/#). Figure31
showsthiseffectof body upwashalongtheleadingedgeof
the wing wherethe pressureewdlkientdecreasesas y/a
increascabecauseof the effectivetwist of the wing. The
importanceof body upwashem be seenby comparingthe
pressuredistribution
alongtheleadingedgein figure31with
thatin figure32. Thepressurecoeflieient
at thewing-body
juncturein figure31is twicethatin figure32wherethereis
no body upwash. The pressurecoefficientat any given
spanwisestationremainsnearlyconstantbetweenthewing
leadingedgeandtheMachwavehornthewing-bodyjuncture. Behindthe Machwave,interference
from thewingbody juncturecausesthe pressurecoefficientto decreasein
magnitudeasshownin thetwofigures.
=mf3 OFANGLE
OFAmA~
Comparisonsbetweentheory and experimentfor the
angle-of-attack
case are madein figures33 for data at a
Reynoldsnumberof 1.5x10e and Mach numbersof 1.48
and2.00with&-=OOandctB= +2° and *6”.
Pressuredistribution
in junctureof wing-bodycombination,-A comparisonbetweenlineartheoryand experiment
for the pressuredistributionin the wing-bodyjunctureis
madein figures33 (a) and33 (b) for both Machnumbers.
The sketchcashowthe pertimmtMachlinesandthe span-
___
1326
REPORT125*I?A~ONAL ADVISORY
COMbfTITE
E FORAERONAUTICS
-6
Q -6”
U -2”
0 2“
-5
y/u=1.0,\(ija
.
-8~6.
//A
/&~
6°
O
4
w
m
Q
\,
\
-3
0
H
\ \
,.
#t
-2
Y
6
~Present theory
+ &
0
0’
u
-
W
(0)
<“
d
L-
—
0
?
-1
-6
-5
y/a x1.02
L
,/
g4 \
x,
,
,/’
-2
“o
,
k
‘Present
I
0°
shownin figure33 for valuesof y/a of 1.92,2.68,and3,92.
For ikf=l.48 body upwashcausedthe shockwave to be
detachedfrom the wingin the wing-bodyjunctureso that
no calculationof tho spreadcould be made there. For
.J4=2.00it wasfoundthatnearthewing-bodyjuncturotho
predictedspreadin fiP/a~
between—6°and+6° wasnbout
twicethe experimental
spread;whereasfor y/a greaterthan
about1.5theexperimental
spreadwasfairlywellpredicted.
This differencebetweenshock-expansion
theory and tho
experimental
datain thewing-bodyjunctureisprobablyduo
to the combinationof severalthings. First,nearthewiugbodyjuncturethebodyupwashismodifiedby viscouseft’octs.
Second,thetheoreticalspreadwascalculatedat theleading
edgeof thewing,andthisvaluewasamumedto applyrearwardto theiirstorifice. Thisassumption
is probablygood
beyondy/a= 1.5 wherethe chordwisechangesin prcssurcr
aresmallback to thefirstorifice,but, in the juncture,tho
changesin the chordwisedirectionarelargenearthowing
leadingedgeso that this assumptionis probublyinvalid.
Third,thecontribution
of thebodycrossflowfieldpreviously
mentionedis present(eq. (63)).
Anotherphenomenonnot predictedby lineartheoryis
shownby figure33 (a). Thelineartheorypredictsthatthe
g
w
:
0
0
I
I
&l
-=!
.
-,
.
&
\
9‘
\
r
-4-
I
ToP_-
d
thetiry
t ?
-
-2
(b)
.5
mertd!on
1
I
\ ,@
\/\x
/
-3
w
Lo
1.5
20
25
3.0
3.5
X437
(a) M=l.48, y/a=l.02
(b) M=2.00,lda=l.m
l?muan33.—Premure
distributions
duetoangleofattack;R=l.5X 10E.
wise location of the ori6ees.7 The experimentaldata points
from the wing surffLcO
on which a comprwsionoccurs (gegative angleof attack) arerepresentedby flaggedsymbols,and
the data points from the surface on which an expansion
occurs (positive angle of attack) are representedby unflaggedsymbols. The @ures show that the theorypredicts
the magnitudeof ~P/a~about5 percentbelowthe average
of the aD=&2° experimental
vahmsat -ii= 1.48andabout
15 percentbelow experimental
valuesat J4=2.00. The
chordwisevariationis wellpredictedby thetheory.
Lineartheorypredictsthattheparameter~P/a~is independentof angleof attack. Actuallyit is not, and the
nonlineaxeffectsof angleof attackcausea spreadin the
data. It is possibleto evaluateapproximately
thevariation
in the parameter/3P/a~
with angleof attackat the wing
leadingedge. IWsttheupwashjustin frontof theleading
edgewas calculatedusingequation(1) whichis basedon
lineartheory. Then the pressurecoefficientsat the wing
leadingedgewerecomputedusingshock-expansion
theory.
Thevaluesof ~P/a~for aB= —6°and+6° so calculatedare
7Thelw9U0110fthsSAfwhffQ!a L9cdyqodftauveksnse.thocdddlomwem
nmd9
USfW~*-w@fou
~}
~fi b =nrnPti
that thsrom no Id Mach nnnwr
theI@@ s&0ofthowiru. TOSfMpWy
theskotchesj
theBkc!hhdfms
vruhtionbehind
onthobwlyars
represenbxl
asstralghtlfnsa
g
‘
‘i@ 0.
o’
\
d
‘+,
? -:
o
-1
‘ ‘Present theory
w
8 ; ~ 0:’ ‘
Q-6°
a -2°
o 2°
O 6°
,
{
f
FL
6
Y
0
Q{
(c)
I
-4
Top_7
meridmn,
I
1
\
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‘x
//
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.M
w
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; ‘\ ;-
Q,
@’
T
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6
e
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d
%
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-
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{
@
(d)
~.8
o
:
Presenttheory
o
.8
1.6
2.4
3.2
Mb
(o) itf=1.48,topmeridian.
(d) iW=2.00,topmeridian.
Fmmm33.—Continued.
4.o
4.8
1327
QU~I-Cfi~RICAL mORY OFWING-BODY
INTERFERENCE
ATSUPERSONIC
SPDEDS
-
J!ii!b;
,,;~-:-~
8=45°
merld!on
,/
-3
‘@ G
\
0d
d,
8
~2
Q
a
0
O
I
-1
Present theory:
-6°
-2”
2°
6°
i? ~
0-
[e)
@+
Q+
@IT
I
+@
-4
-4
-3
@j’/
/’7\
~ 2
@
j
b ‘:
-1
-3
-
~~
RP
w
r-
c
&
o
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e
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x
:
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PresenttheoryJ
(f)
!8
d
0_ : /
1\
o
Y
I--@
42!
“0=’”25
--j/ o“
8=45°
mendlan
o
i
!
v
_
v
0
~
TrL1--l4’Jl
Y 0’
/
c1
\o
-
d
‘--R esent theory
-1
(D~
,8
I
I
I
1.6
@L
n
2.4
1
(h)
1
32
4.0
4.8
0
Mb
5
1.0
1.5
2.0
25
3.0
3.5
v!2a
(g) M=l.48, v/a=l.25
(h) M=2.00,~la=l.25
(e) ~=1.48, 0=45°meridian.
(f) M=2.00,0=45°meridian.
FIGURE
33.—Continued.
fiGmm33.—Ckmtinued.
lvfachhelixfrom the oppositewingpariel(seesketch)should
intersect the tig-body juncture at point 1, causing am
incrensoin the magnitudeof DP/uB. Thiseffectis observed
Q)
ffB= +2°, particularly
at lW=1.48. However,nonlinear
effectsdueto ~Bcausea hingespreadbetweenthe dtttafor
aB=+60 and.aB=—60. All the effectspredictedto occur
experimentally
for negativevaluesof aB in front of point on the body in the sectionof thereport“GeneralJ?hysical
1 ratherthanexactlyat point 1. The re’hsonis that for I?rinciples”are observedmperimentally,
but not exactly
negativevaluesof aBa eompreasion
occurson the orificed at the pointspredictedbemuseof nonlineareffects. The
surfacoreducingthe local Mach numberfrom the free- pressurerisepredictedat point1 of figures33 (c) and33 (d)
stmmmMach number,thusincreasingthe Mach angleand ocoursprematurely
andis lessabruptthanexpectedfor all
causingthe Machhelixto shiftforward. The resultis the anglesof attackbecauseof thebound~ layeron thebody.
spreadof thedatashow-n
in figure33 (a) nearpoint1. This The variationin local Mach numbercausesthe Mach
effectis not shownby figure33 (b) becausetheMachhelix helicesto shift forwardfor the negativeanglesof attack
liesmorerearwardfor i14=2.00so thatthe oriiicesdo not as discussedin thesectiontreatingthewing-bodyjuncture.
extendto theMrLch
helixasshownby thesketch.
Theincreasein themagnitudeof f?P/aBexpectedat point2,
Figures33 (a) and33 (b) showthatMachnumberhasno x/pa=3ir/2,actuallyocoursat aboutz/@=4 for ~E=—2°.
eUectuponthemagnitudeof thehigher-order
spreaddueto The deoreasein magnitudeof &/a* that is e.spectedat
rmgleof attackor upon the chordwisevariationof BP/aB,
point 3 actudy occurs at about x/Ba=4.Ofor aB=—6°.
but on the averagethe magnitudeof BP/aB is about 10 Forthepositiveanglesof attacktheMachhelicesareshifted
percenthigherfor 31=2.00thanfor M= 1.48.
rearward
sothattheseeffectsarenotobservedexperimentally
Pressuredistribution
ontopmeridianof bodyof wing-body in therangeof z//3ameasured.
Figures33 (c) and 33 (d) show that, in general,the
oombination,—
A comparisonbetweenthelineartheoryand
experiment
for thepressuredistribution
on thetopmeridian itf= 1.48dataarepredictedbetterby the theorythanare
of the body ie madein figures33 (c) and 33 (d). These the M=2.00 data. For i14=2.00thereis an unexpectedly
figuresshowthmttheoryandexperiment
arein goodaccord largepre8surecoefficientin front of point 1 for negative
for
1328
-5
REPORT125%NATIONAL
ADVISORY
CO~
I
I
I
I
-5
CY-6”
a -2”
-G
+9
2“
0
r
#
r’
-2
6“
%3‘“.,
.
I
4
I
J’
‘
O(r
1’ !’
J’
[
I
I
I
Nonlineor theory
—
(shock-exponslon)
a
=-6” with upwosh
-4 — /B
‘aB= 6“ with upwosh
,
t
t 1
,, ;
!’
ol~
-3 ; .’ w
1
Nonllneortheory
— (shock-exponslon)
r - --aB=-& with upwosh
-4 + t - Ie-OBX@with UpWOSh-
-3
@P
T
o
,/
FOBAERONAUTICS
A
w
:
*
‘.
y/u=L9>~
+2
\
. y/o .
0
w
~ Presentthmry
w
o
-1
(1)
0+ @~q
0
-5
1
1 i
1
Nonllnmrtheory
— [Shock-exponslon)
,---~=-6°w!th upwosh
-4 d ;- ,r
‘B. &w,th upwosh
1’ r’
v
,
w
/t
I
@
%
-3 J :
8 ,,
/
,
j3P
0
!?
f<
,
,(
Y
T
J
,!
1
-e
@/
I+
0°
6°
.
@ “.,
,
9
‘.
y,,o=1.92’>
BP
-J/
G
-2
!
‘-2 J
[P resent thmry
-1
:
~
0
I
I
y/o =
2,5
,
0
,
‘%.
0
$
(Y
&
8
‘.. o
.....
- -- Present theory
-1
(J)
0
I
I
1 I
Nonllneor theory ‘
— (shock-expons”@
,aB=-& with upwosh
4 — .
,aB= 6“ ~“th upwosh
I
; ‘
Q
w
-3 ;; “
, ,
6
-5
1
(3+ Q+ q
5
10
CD @
(1)
15
m%
20
2.5
3.0
35
(i)M=l.4S,y/u=l.02
(j) iM=2.00,
V/a=l.92
FIcimm
.%.-Continued.
00
.5
ID
1.5
2.0
2.5
?
3.0
:
)
X/p a
(k) M=l.4S, p/a=2.ELt3
(1) iM=2.00,
y/a=2.6S
Fmurm&L—Continued.
is felt from thewing-bodyjunctureso that the theomticnl
pressuredistribution
forawingaloneinthebodyupwashfield
isusedin thisregion. Figures33 (g) to 33(n)showthaton
the averagethe wing-alonetheorypredictsmagnitudesof
@’/aB about6 percentbelowthemeasurements
for a~= A2°
due to the bound~-layer conditionon the body andwill for M=l.48 andabout12percentbelowthemeruwremcmts
be discussedin detailin the sectiondealingwithReynolds for M=2.00. The spreadin the data betweena~= +6°
numbertiect.
and aD=—60is fairlywell predictedby shock-expansion
I?ressuxedistributionon 0=45° meridianof body of theoryfor y/a gnmterthanabout1.6 (figy.33 (i) to 33(n)),
wing-bodycombination,-Acomparisonbetweenthe linear At y/a= 1.25thepredictedspread(notshown)is too large,
theoryandexperiment
for the pressuredistributionon the justasfor thewing-bodyjuncture.
0=45° meridianof the body is madein iigures33 (e) and
Someof thginterference
effectsdiscussedin thesectionof
33 (f). Essentially
thesameeffectsareshownonthismerid- the reportentitled“GeneralPh~ical Principles”me illu5
ianason thetopmeridian.
tratedinfigures33(g) to 33(n). Theinterference
effectfrom
Justasfor thetopmeridianof thebodytheexpwimentis, the oppositawingpanelis observedin figure33 (g) where,
in general,betterpredictedby thetheoryfor M= 1.48 than justin frontof point1,thesamespreadin thedataoccursm
forikf=2.00,andthesameboundary-layer
effectsareevident in the wing-bodyjuncture. Accordingto lineartheorythe
nearpoint1 for ilI=2.00.
disturbanceoriginatingat the nearerwing-bodyjunotum
Pressuredistribution
onwingof wing-bodycombination,— Bhouldbe felt at point2 of figures33(i) to 33 (m),andthe
E.sperirnental
chordwisepressuredistributions
on the wing magnitude
of pP/CYBshouldbeginto decreasefromthewingareshownin figures33 (g) to 33 (n) for the four spanwise done valuethere. Thesefiguresshowthatthe magnitude
or&e stationsy/a=l.25, 1.92,2.58,and3.92. In frontof
of point2. They
of BP/aB doesdecressein theneighborhood
theMachconefkomthewing-bodyjunctureno interference dso showthat,in general,the an= +6° and the @= —6°
rmgleaof attack. The predictedpressurecoefficientdue tQ
crossflowis only about 0.1 in unitsof the ordinateandhence
does not account for the observed effect at M=2.00. For
@= —2° ttnd~=2.()(), &/a= dips slightlynearpOiIlt1 and
thenrisesand overshootsthe aB= —6°data. Thiseflectis
1329
QUASI—CYLINDRICAL
THEORYOFWING-BODYINTERFERENCE
AT SUPERSONIC
SPEEDS
1111
0 -6”
Nonlmeortheory
(shock-expons,on)
-4 — - /RB*-60w,th upwosh- —
/ /a8. 6° with dpwosh
I/
; ./
a -2” y/u=39
o 2°
0 6°
>, .,,
@’
/
I
rK!%Hw@=-Jx%?)&
-6°
@
Theexperimental
andtheoretical
resultsfor thespanloading
distributionon the wingandbody of the combinationare
presentedin figure34. No accounthas beent~kenof tip
\ 0;
‘m
20
PJ’‘i
i
1
I
I
1
1
! /t 1
!
If
w
<s
@
I
I
1%
I
=12
@co
QU
u
0
I
I
-
I
I
i
%G
10=3.92 . ~L,
e;
,/
~.
:
A
Q&+$
/’
<>
8#
ti
da
I
p
I
I
I
-4
+
(0)
,
j
‘Q
0
\
/’
/’
I
I
1
‘k
B@
fp
,
CD
@l
I
I
t
o
y/L7=o
1,
I
I
Sodypving
~
i
20
I
I
i
E!E!
I
1
16
6
LO
8
,
,
-–Present theory
e
15
20
?
25
W3a
(m) df=l.48, 9/a=3.92
(n) M=2.00,yia=8.02
30
3.5
I
\
-T
+(3
.
I
y Present theory
*Q
(n)
5
A-
2“
6“
$“
Q
I
,,,
Nonlmeortheor~
(shock-expons,on)
-4 — - -aB.-E?wllhupwostl
[ ~aB.ewtth upwosh
t
/ 1’
o’
Q
,,
o’
&j3 Jr”
d
0
0
0
i’
,
v
o
-2 >’
/
I
1
LPresenttheory
I
-1
o
0
o
1
[m)
-5
o
O
I
‘Present theory
1
-1
I
I
I
o
‘=’
~8
\
R12
$$
y
\
\
\
t
6
n
~
\ll
o
\
-y/a=o
8
I
I
I
Fmmm33.—Conoluded.
4
1
1
I
I
I
I
1
I
/
“/
/
datacome togetherin the neighborhoodof point 2. This
Rnfiv!Wkm
@@
convergence
is dueto a variationin thelocalMachnumber
@
CDC3
witha~, Thisis shownby the sketchin figure33 (j) where
3
5
6
2
4
I
the disturbance
fromthewing-bodyjunctureis firstfelt at
o
ym
point 3 for a~= —6°,wherwieit is iirstfelt at point4 for
(a) M=l.48
Since
themagnitudeof ~P/aB
beginsto decrwse
aD= +6°.
(b) M=2.00
m soon as thisdisturbance
is felt, the magnitudeof @’/aB
due to angle of attaok;
beginsto decreaseat a smallervalueof z/flafor a==– 6° Fmmm M.-Span load distributions
R= L6X1OO.
thunfor aB= + 6°, thus causingthe convergenceobserved.
The sketchesin figures33 (k) and 33 (m) showthatthe effects in calculatingthe span loading because the twist of
disturbance
fromthewingtipshouldalsocausethea~=+ 6° the wing makea a determinationof these effecti a ~cult
andaD—
–– 6° datato cometogetherbeyondpoint6 in these wing problem. The theory is thus valid only inboard of
point 2. If an approximateansweris needed,the Busemann
figures. The figuresshow that the data not only come
togetherbutactuallycrossoverandreverseorderjustbeyond tip solution (ref. 20) can be joined onto the span loading at
point 2. l?igure34 showsthat the theoryis generallyabout
point6.
10 percent below experiment. This result is not surprising
The only significanteffect of Mach numbershownby
in view of the comparisonsbetween the experimentaland
figures33(g) to 33(n)is theapproximately
10-percent-larger theoreticalpressuredistributionsof figure33. Of particular
wdueaof @’/aD for d!f=2.00thanfor ~= 1.48. Nearly40 interestis the fact that, in general,the higher+rder &&3rpercentof thisdifference
maybe dueto differences
in stream encesduetc a~thatwereso largefor thepressure-distrlmtion
unglein thewindtunnelfor thetwoMachnumbers.
resultsarenegligiblefor the spanloading distribution. The
Spanload distribution.-spanloadingis definedfor both only exceptionis on thetop of thebody, g/a=O, andM=!2.00,
where the effects of boundq-layer and shock-waveinterthebodyandthewingaatheintegral(seeeq. (57))
m
1330
FORAERONAUTICS
REPORT125%NATIONAJJ
ADVISORY
COMMITTEE
actionarelarge. The explanationfor the independencefrom
aB is that the higher+rder effects on the top surface are
compensatedfor by higher+rdertiecti of the samemagnitude on the lower surfaceso that the net loading per unit
angleis very nearlyindependentof angleof attack.
EFFECT
OFWING-INCIDENCE
ANGLE
Comparisonis made between theory and experimentfor
thefig-incidence casefor dah takenat a Reynoldsnumber
of 1.5X106imd Mach numbersof 1.48and 2.00 tith aBeOO
and~T= —1.9° and —5.7°. It will be remarnberedfrom the
sectionon the accuracyof data that thereis no appreciable
error due to stremnangle for the wing-incidencecase, and
the comparisonbetween experimentand theory reflectsthis
fact.
fw
111111
u -1.90
Q -5.7”
r-11
4
-3
+j2
-1
0
I
-4
y/o .1.027
,1 .7”
-4
-3
“:
-3
.Nonhrmr theo
BP - #r “. (shock-exponsi{
T
J’
iw=-5.7”
I
I
I
I
-2 <
Illllmrr-1
‘.
-1
M
/
W“”
-2
I
/I
I
I
I
I
BP
-F
-1
I
<Presenttheory
0
(o)
@
o
-5
-’.[
y/u=lo2;
-4
L(D
/’0=
i
43
/
,
Nonlineartheory
+ tr .(shock-exponsion)
-1
/# -5.7°
Q
x/&
(o) M=l.48, topmeridian.
(d) i$f=2.00,topmeridian.
Fmmm35.—Continued.
35 (a) and 35 (b) show that the spread predicted & this
mannercanaccountfor theexperimental
results. Thoprematureincrease
in
the
magnitude
o
f
L@/iwnearpoint
1 is
-2 \
&
dueto theeffectof theoppositewingpanelandvariationof
>
.
the local Mach numberas discussedin the anglo-of-attnclc
@
f
No significanteffectof Mach numberwas found
section.
-1
1
i
on theparameter19P/&.
L Presenttheory
Pressuredistribution
ontopmeridianofbodyofwing-body
(b)
combination.-Acomparisonbetweenthelineartheoryand
1.5
o
.5
M
LO
20
experiment
for thepressuredistribution
on thetop meridian
xl~a
of
the
body
is
madein
figures35
(c)
md
35 (d). Tlmso
(a) M=l.4S, ~/a=l.02
figures
showthattheoryandexperiment
arein
good accord
(b) iu=2.oo, v/a=l.02
for {W=–1.9°. However,nonlineareffectsdueto i~ causo
FIGURE35.—Pressuredistributionsdue to wing incidence; R= 1.5X 105
muchlargerclHerences
betweentheoryand experiment
for
.
Pressuredistribution
in wing-bodyjunctures.-Thelinear i~= –5.7°. This is consistentwith the angle+f-ottaok
theoryandesperimentnl
pressuredistributions
in thewing- casewherethe higherardereffectdue to aB m8 hwgofor
body junctureare comparedin figures35 (a) and 35 (b). negativeanglesof attack.
All of theeffectsobservedfor theangle-of-attack
casecluo
The symbolsin thefiguresareflaggedto be consistentwith
to
disturbance
fromthewingarealsoshown
to
occurfor
t,ho
theuseof flaggedsymbolsfor negativeangle-of-attack
data.
casein figures35 (c) and35 (d). Thopoths
The figuresshowthatthe experimental
valueaare about5 wing-incidence
aspredictedby lineartheoryareshown
percentbelowthosepredictedby the theoryfor iW=—1.9°. of thesedisturbances
Themagnitude
of thenonlineareilectsdueto&is predicted on the sketch,and the positionsat whichthe effectsnro
at the lendingedge by shock-expansion
theory. Figures expectedto occurareshownon thenbscissn.
.-.
. . .— ---
. . . . -—---
. . —..
-4
----
1331
INTERFDRENCE
AT SUPERSONIC
SPEEDS
iw
iw
u -1.90
e -5;7”
w -5.7”
/(
& ‘.
/
-3
r--Nonlinwr theory
+ -(shock-exponsion)
i~ -5.7”
;
0’
$j2
0°
y/O
=125
-5.7”
0°
,
*
J’
w
-1
o’
_
c‘
d
-
+
T
//
d !
d
—
d
,,
~ Present theo ry
o
(e)
(9)
Q~
0+
@
Q
OJ
I
-4
- 0=45”
i merldlan
-3
Y/O= 1.25
,/
Y\@
~
,
+
,
/
/
_
~
~- 2
/
-1
o
~
+
-Nonlinear the~ry
-(shock-expansion)
/
J’
>
0°
8
>
d
d
o“
/’
iw=-5.7°
F
‘PresentIheary
2**
/
&
g ; f-.,— +
//
e
t
“Present theory
(f)
J.8
o
@,
8
1.6
ok
24
X/@
(h) @
3.2
4.0
4.8
(o) M= 1.48, 0=45° meridian.
(f) 31=2.00,0=45°meridian.
~GWRE
35.—Continued.
The only significanteffectof Mach numberapparentin
figures35 (c) and 35 (d) is the largerboundary-layer
and
shock-waveinteractionfor 1W=2.00thanfor M= 1.48near
point 1. The .M=2.00 experimental
data for &= – 1.9°
dip andthenovershootat thispoint. Thisphenomenon
is
discussedin more detailin the sectionof the report on
Royuoldsnumbereffect.
Pressuredistributionon 0=46° meridian of body of
wing-bodycombination,-Linemtheoryis comparedwith
cxTerimrmt
al resultsfor the pressuredistributionon the
8=45° meridianof the body of the combinationin figures
36 (o) and36 (f). The effects’shownby thefigureareconsistentwith thoseshownfor the angl~f-attack case and
for thewing-incidence
caseon thetopmeridianof thebody.
Pressuredistribution
onwingof wing-bodyoombination,—
A comparison
betweenlineartheoryandexperiment
for the
pressuredistribution
alongseveralspanwise
stationsismade
in figures35 (g) to 35 (n). The experimental
data (figs.
35 (k) and35 (1))showthat,in generil,&/& for their=
– 1.9° datais constantandnearlyequalto –2 in frontof
thohf.nchcone. BehindtheMachconethetheorygenerally
predictsvaluesabout5 percentabovetheexperimental
data
for im=—1.9°. The higher-ordereffectsdue to & cause
.5
1.0
1.5
20
25
3.0
3.5
.%%
(g) 31=1.48,I//a=l.25
(h) M=2.00, ~/a=l.25
FIGURE35.—Continued.
largerdifferences
betweenlineartheoryand experiment
for
iw= —5.7°. The figuresshow that these differencesare
wellpredictedby shock-expansion
theory. The effectsdue
to the influenceof the Mach wavesare the sameas those
discussedfor the angbof-attackcase. Thereis no eiTect
of Machnumberevidenton thewingof thewing-bodycombinationotherthanthatpredictedby lineartheory.
Span load distribution,-A comparisonbetween the
theoreticaland experimental
resultsfor spanload distributionon thewingandbody of thecombinationis madein
figure36 for &= –1.9°. The decreasein the spanloading
dueto thewingtip wascalculatedby themathodof Busemann(ref.20). In part (a) of figure36,interference
from
both thebody andthewingtip is feltbetweenpoints1 and
2, butinpart(b) no interference
is feltbetweenpoints1 and
2, and the spanloadingis that of a two-dimensional
wing
alone.
Figure36showsthat,in general,the experiment
is 5 percent lower than the linear-theoryprediction. Site all
pressuremeasurements
for the wing-incidencecase were
made for negativevalueaof &, the experimental
values
usedin this figurewere obtainedby doublingthe values
of 19P/iT obtainedfor iW=—1.9°ratherthanby considering
1332
-5
Q-
y/a=
/< /’
l-9~” Q~
-4
.:- - Nonlineor theory
)
– ( sho:k- expansion
,
,
IW=-5.7”
,,’
+
o
fi;3
G
-2
FORA13RONAU!HCS
REPO~ 125%NAT’IONUADVISORY
CO~
w- —
L
w -5.7”
u -1.9”
o“
.
-5-
%
iw
0 -1.9°
Q -5.7”
;W
4
,rNonllneortheory
-3 ~(shock-exponsion)—
ifl-5.7”
BP
G
/
I
J
c‘
u
c“
d
-2
<f
i,
-
-1
-
Present theory
-1
w
@
-%- .
,/
/
Present theory>’
#
“
f
“
lY—
<~
I
(k)
(i)
0
B+
P
0
P
-5
-5,
y/o =
-4
“2--
.- +,
w’
‘=
I
Nonlinear theory
:
)
-3, : 1 (sho$k-exponsicm
/w.-5.7”
P
/
[W J
w
~
-2’
I
‘,
:
-1
I
I
I
--Present theory
0 (i)
.5
Q
1.0
I
&Q-!O
I
,rNonhneartheory
~shcck-exponsion)
iw. -5,7°
&3 /
J
u
a
@
0’
-2
u
Q
I
y/a= 2,5s
4
~
“
,/’
1
o)
1.5
,+
/
PresenttheoryJ’
-1
-
2.0
2.5
3.0
3.5
x/’&
fi) M= I.48, Idsz=l-92
(j M=2.00, I//u=l.92
I?mJEE 35.—ContinuecL
two surfacesas for the angle-of-attackcase. Since this
increasesthenonlinesreffectsof & ratherthanmhimking
them,only the i~= —1.9° data (for whichthe nonlinear
effects are small) ware plotted. However, de praent
methodis applicableto the predictionof the net span
loadingfor largervalueaof& becausethenonlineareilects
on the upperandlowersurfacestendto canceleachother,
asdown for theangle-of-attack
case.
EFFECT
OFREYNOLDS
NUMBER
The primaryeffectof Reynoldsnumberin thisinvestigationwason thebody. Reynoldsnumberwasfoundto have
no significant
effecton thepressuredistribution
on thewing
of the combinationfor the range-investigated.Figure37
showstheboundary-layer
condition,x observedin schlieren
pictures,on top of thebodyat thepointof intersection
with
the Mach wave from the leadingedge of the wing-body
juncture for R=O.6 and 1.5X108. The transitionand
separation
regionsshownin figure37indicateapproximately
therangesof a~andiwin whichtheboundarylayerchangw
from laminarto turbulentor separatedflow at the Mach
wavefromthewing-bodyjuncture. In lamiParandturbulentregionstheflowremainslaminaror turbulentacrossthe
‘o
.5
Lo
P
P
L5
2.0
.2.5
30
:
X/&
(k) M=l.4S, ~/a=2.6S
(1)M=2.00,~/a=2.68
~GlJE13
35.—Continued.
Mach wave. Someof the Reynoldsnumbereffectshown
by&we 37maybe dueto changea
in theturbulence
levelof
thewindtunnel.
It is to be expectedthatdataobtainedfor severalarsglo
combinations
withinanyoneof theregionsshowninfigure37
wouldshowno significantdifferences
dueto viscouscdfects,
but thatthesedatawoulddiilerfromdatain otherregions.
For example,for M= 1.48 and R=O.6X 106the dato for
U*=—z” with &—_
—0° should dMer from the data for
~~=—6°with&=OObecausetransitionoccursat theshock
wavefor thelattercasebut notfor theformer. Thatthere
is shownin iiggire38 wherethepressuredisis a d.ifEerence
tributionson top of the body for thesetwo conditionsaro
compared. In front of the shockwavetheflowis laminar
for bothanglesof attackso thatthereis no difference
in tlm
two setsof data. However,for ff3=—6° transitionoccurs
at theshockwaveandthepressurerisesaspredicted,whilo
——2°laminarflowpersistsbehindthepointat which
for ~3—
theshockis expectedandthepressure
riseoccursmuchlater
thanpredicted. Jn fact, the pressurerise doesno~occur
untilthetransitionpointshownin thefigureisreached,and
thenit tendsto overshoot. Thisphenomenon
of thedolnyod
pressurerisewasobservedto occurwheneverlaminarflow
QUASI-CYLINDRICAL
THEORY
OFTTTNQ-BODY
INTERFDRENoE
AT SUPERSONIC
SPEEDS
20
-5”
-4
16
Nonhneortheory
- (shock-expans,on)
-3
~ -57° -
/
BP
~
j
~~
Q
,
I
r’
L
‘-Present theory
*$
]2 .
~~~s
c
‘>’
* 1
d
-2
o
a - 1.90
o -57”
: 0°
*O
-5.7°
;*
/
\
c>~
r
-1 -
4
(m)
P
o
(cl)
-5
❑
1
I
I
1
I
I
\
/
Tip
)~
solution”
\
\
\
I
1
@~
1
I
I
I
‘is
~
w
t
d
/
\
~ Presenf theory
I2
I
%1%*
~n
~’
8
:Presenttheory-~
,
I/
I
I
n
~
-
.
0 -. .
%*
1
-1
I
I
I
‘
,J%
\
I
1
16
~
\
(n)
o
,
I(Z)O
/*- ,.@
L.
~~
d
,,’
‘\\
\.
,/
~‘CQTIbirwd
effects of
- tips ondinterference
I
I
1
o
e=> !’5.7°
/“
-2 ~
x,
i
,+0”
I
1
I
20
y/O392 >,~~,~ ~
~3
1
I
1
1300g
o
I
\
, - Nonlmeortheory
f -(shock-expormon)
/w=-570-
-1.9°I
I
I
I
1
I Present theory
r
1
/
1
/
-J :I
@+
P
-4
‘L—-yjo=o
Iw ;
/w
y/O
.392
1333
,@
.5
1.0
L5
+0
2.0
4
25
30
3.5
X/@
(m)dfEM3,v/a=3.92
(n) M=2.00,v/a=3.92
FrQURE
36.—C!onoluded.
pmsistedbeyondthepointat whicha shockwavefromtbe
wingwas predictedto exist. Whentie d.kturbance
from
tlmwingis anexpansion
wave,thepressure-coeilk.ient
curve9
rise approximately
as predicted,regardlessof the type of
bounda~ layer. The conditionsfor which this delayed
pressurerisewasobservedto occurareshownby thedotted
aremin figure37. Two otherexampkaof thisphenomenon
may be seennearpoints1 of iigures33 (d) and35 (d) for
——2°,&=OOandaB=OO,
iW=—1.9°,respectively.
an—
In figure39,thepressuredistributions
on top of thebody
mecomparedfor threeReynoldsnumbers. It isshownthat
data for the two highestReynoldsnumbem,R=l.2 and
1.6X 10°,agreewell,whilethe datafor thelowestReynolds
numberdifFerfromthosefor thehigherReynoldsnumbers.
COMPARISON
WITHTHRORYPROMOTHRRSOURCR9
Thethreetheoriesforwhichnumerical
resultsareavailable
am comparedin figure40. The theorydueto I?errariwas
obtainedby cross-plotting
from a &ure in reference22 so
thatthecurveshownis onlyappro-te.
The thmretic~
curvedue to Morikawais obtainedfrom tabulatedrcmdts
given in reference4. The experimentaldata
region was
determined
by theextremevrduesobtainedfor aB= %2° for
!
I
I
:
Tip Sotutkm’
/
/
( (b) :ody~Wly
n
1
+@
2
3
)’@
‘,
\
4
5
6
(a) M=l.43
(b) M=2.00
FIGURE
f36.-SpFLn
load distributions
dueto wing inoidanca;
R=1.5X106.
Machnumbers1.48and2.00. Fromthisfigureit appears
thateitherthetheoryof Morikawaor thepresent@en-y can
be usedto predictthepressuredistribution
in thejuncture
theoqrpre&ct9
of a wing-bodycombination.l?sITfi’s
valuesthataresomewhat
lowattheleadingedgeof thewing>
butit appearsthatif numerical
resultswereavailable
beyond
z]pa=0.7, they would lie within the experimental
mnge.
For a morecompletecomparisonof the theork of Ferrari
andNjelsen,seereferences
9 and23. Exceptfor thepresent
theory,no numericalresultsfor thepressuredistribution
on
thebodywereavailablefor comparison.
CONCLUSIONS
A theoryof wing-bodyinterference
for supersonicspeeds
hw beendeveloped. The liheorywasappJiedto the calculationof theseparateeffectsof bodyangIeof attackandwing
incidenceon the prwsuredistributionsactingon a rectangularwingandbody combmation.on thebasisof comPmiaon betweenthe theoreticalpredictionsand experimental
1334
FORAERONAUTICS
REPORT
125%NAI?1ONMJ
ADV180RY
COMMI’ITElll
Laminar
111]111]]
Turbulent
~:;:~
-;. . . . . . .
~~~
Trcr.siticm
~///~ .
Seporotion
-4
cf R=l.5xi06
d
L2X106
& 0.6x106
Delayedpressure ise region
o
#
&3
$
0
v
u’
d
w
-2
0’
Presenttheory— -,
-1
w
Y
!8
0
W
a
1
r!
l.’
d
.8
x/b
24
3.2
4,0
4.8
l?mmm39.—Effect of Reynoldsnumberon Pressure
distribution
on
topmeridian
ofbody;~= 1.48, C@=~6°, andiW=OO.
(b)
.5
Lo
t
t
1.5 20
,
25
1
,
30
35
1
40
x/~o
FrGuan40.—Comparison
amongseveraltheOretiOSl
caloulatione
of
pressures
in thewing-bodyjunctureof the combination;
anglo-ofattackcase.
(c)
Wing-incidenceongle,;W deg
(b) M=l.48, R=1.5x1O$
(d) df=2.00,R= 1.5x 10’
R=O.6Xl@
(C)iif=2.t)o, R=O.6X
NY
(a) M=l.Q
FIGURE37’.-BoundaIlayerer condition on top meridian of body
point of intereeotionwith iUaohhelix from wing.
-4
u -2”
0 -6°
at
c
-3
Q
w
u
w
0’
d
Presen! theory-Y,
lj2
d
o’
.
-1
@
o
!.8
r
P
@
o
t
t.-Tronsdton
point
fora’-2°
Trongtlon..j
point
fora~~-6”
o
.8
1.6
x@o
24
3.2
4.0
4.8
l%um 38.-E!Teot of transition
positionon pressure
distribution
on
topmeridian
of body;M=l.48, R=O.6X10$,andiW=OO.
measurements,the followingconclusionsare drawn:
1. The presenttheory predicts the pressuredistributions
due to tig incidenceabout 5 percenthigh for anglesup to
2°. However, the pressure distribution due to angle of
attackispredictedabout5 percentlow for M= 1.48andabout
10percentlowatilZ=2.00for anglesbetween+2° and—2°,
2. Nonlineareflectadue to angleof attack and wingincidenceanglearelarge. On thewingthe differencefrom
lineartheorydue to nonlineareffectsof anglecan be predictedby shock-expansion
theory,exceptnearthewing-body
juncturefor theangle-of-attack
case.
3. Spanloadingwas shownto be predictedwithin+10
percentforboththebodyandthewing. Thepredictedsprm
loadingsarehighfor thewing-incidence
caseandlowfor tho
angle-of-attack
cnae.
4. For the angle-of-attack
case,the pressurecoefficicmts
on the wingare experimentally
about5 percenthigherfor
M=2.00 thanfor i14=l.48,whenreducedto a forxnthatis
theoreticallyindependentof lMach number. Otherwise
lMachnumberhasno importanteffect.
5. ViiccmsefFectsareimportintonly on the body whore
tbe shockwavefrom thewingcnuseslargeboundmy-lmyer
andshock-waveinteractions
for someangleconditions.
Amm bBoN-4ww LABORATORY
NATIONAL
ADVISORY
ComnmmnFORAIQRONAUTICS
MoF~~ FIELD,CALIF.,Jan.4,196.4
APPENDIXA
DECOMPOSITION
OFBOUNDARY
CONDITIONS
OFWING-BODY
COMBINA~ON
A detailedanalysisof theboundaryconditionsfor a wingbody combinationis now carriedout for the following
conditions:
1, Thewingis a flatplatein thez=O plane.
2. Thebodyis aninfinitecylinder,ther=l cylinder.
3. Thelendingedgesof thewingaresupersonic.
4, The Mnchnumberis-@
Considera wing-bodycombinationcorresponding
to figure
4 (u) and shownin greaterdetailin figure41 (a). The
pohmtidsfor theflowmustsatisfyseveralconditions:
1. It mustbe a solutionof thewaveequation.
2, It mustproduceno flownormalto solidboundaries.
3. It must.produceno upstream-moving
disturbancea.s
mdon thebody
%+); ~=~
I!hefirst step in the decomposition
is to break yainto a
?otentialdueto flow alongthe z axisandone alongthe z
ti in accordancewiththesuperposition
principle
Potential
Pa
v
()
ap
z-u+
+d
(b)
ttttt
Potential:
h
Va*
“
#f
+
%
+
av
~
~on body:
() r=
+h
(c)
andcrossflow.
(a) Pamdlel
(b) Parallel
flow.
(o) Crossflow
FI~IJRE
41.—Decomysition
of boundary
ccmditiona.
If pais thepotentialfor the completeflowaboutthewingbody combination,then the boundaryconditionon the
wingis
ape
2=()
—=
- irvq
Z)z
p.
Cos Cq?=vCosq?+
O
o
=
o
+V sin
ff,
O
+
o
O
~d
+
y,
v=
o
+
17
o=
o
+
o
Yb
=
ap
z Z-D+on wing:—imV=
()
ttHt
VaB
+
The flow conditionsat infinityand the prescribednormal
velocitiesat the combinationsurfacealso obey the superpositionprinciple.
The next stepsin the decompositionare to resolverfb
and PCinto potentialsthat can easilybe computed. The
decomposition
of vbintowing-alone
andbody-aloneproblems
isillustrated
in figure41 (b).
+’,
+
pb
on wing: —iIvv = —-’iryv+
ap
on body:
a~ r=l
(-)
“
=
Vsina.=
z
‘&
(M)
b
(Al)
$Stnmthe snrfwsonwM*tbb&
amdltbmsm
osivenm P8mIIolto
thezaxi&
Inordwtodtfferontfrdo
npstream
fromdownsban.
It Isneccsmy to IWO this mndltlon
o–
o =5f2@
n-o
‘&v
Cos2n&5f2n(z) Cos2n8
n-o
The potentialP, due to the wing aloneat incidenceir
producwa velocityfieldnormalto the r= 1 surfnceto be
occupiedby thebody. Thenormalvelocityfieldis decomposedinto a Fourierseries. Sincethe wingleadingedges
are supemonic,we can considerthe flow above the z= O
planealone. To preservethewing-alone
boundarycondition
whenpdis added,wemustconfineourselvesto cosineterms,
and,becauseof a verticalplaneof symmetry,wemustretain
onlycosinetermsof evenmultiplesof 6’in theFourierseries
To counteractthe distortionof ther= 1 surfacedueto the
wingalone,a body withoppositedistortionis addedin the
formof rfd.
1335
1336
REPORT125%NATIONALADVTSORY
COMMITTEEFORAERONAUTICS
The decompositionof pe into three componentsis convenient;a componentprassociated
withcylindricalcressflow, n
componentPOdue to a distortedbody alone,and a componentm dueto a twistedwingalone.
Potential:
P. =
+
Qf
0=0
ap
on wing
O =aBV
ap on cylinder: O =
z
() ,.~
+
~h
o
+
o
o
–
+0+0
awv=aBv
()z *..+
Pr
1
(b)
+
1+
o
The crossflowassociated
withq~causesanupwashdistributionin the z=O planewhichrequiresan equalandopposite
twistedwing to counteractit. Againthe r= 1 surfaceis
distortedby the wing-aloneflowfield,and a cylinderwith
oppositedistortionisintroducedin theformof POto counterI
act thedistortion.
VaB
+nsoh(z)
~s m-n~fw
1+$
()
~s z~
Two convenient casea in the wing-body interforonco
problemare differentiated;the wing-incidencewe in which
,aE=() but & #O and the @e-of-attack case in which
case“isropremntcd
iw=O but aB#O. The wing-incidence
by m andtheangle-of-attack
caseby PC.
APPENDIXB
FORFIRSTHARMONIC;WING-INCIDENCE
ASYMPTOTIC
SPANLOADING
CASE
The span loadingfor the first harmonicas given by
equation(57)is
Theseintagralsaregivenin termsof AngerandWeberfunctionsasgivenon page310of reference24.
JX2-%9’G%)
Jorsin0 cos(i$sin0)d8=T~1(b)
J0rsin0sin (1ssin0)&=7rJ1(z%)
(En)
Let us takea=l, ~=1, and Pu= –P~=Po; thenthe span
loadiruz
is
-2JX3*T%J%’*%“2)
(Weber funotionl
(Angerfunotion)
Thevalueof Fe(s)is then
The methodof calculatingthe asymptoticformulafor the
F.(s)=+V [E,(@+iJ,(i@]
@7)
spanloadingis tit to expandthe Laplacetransformin a
series abouttheoriginandthento taketheinversetransform
For smallvaluesof s theAngerandWeberfunctionshave
termby term.
theexpansion
Fromequation(17)
‘[-2JWAI=[%)I[WI
‘
‘3)
It is nownecessaryto findtheseriesabouttheoriginfor the
twopartsof thehmxfonnof equation(B3).
Fromequation(47)thereis obtained
m%)=:
[l+o(i.s)q
1
.J,(@ ~; [Zk+o(iloal
so that
(B8)
(lx))
Fo(8)=L~.(z)]=L (: J*lZ ‘&vein e de)
(B4)
The ratioof Besselfunctionsgivenin equation(B3) has
an expansionaroundthe originwhiohis a doublyinfinite
seriesof productsof powersofs andlogs.
‘0(8)=wJme-”dzJb’’shod
‘%JN--ILO’-=”
()
F’0(8)=%V
J
Lf
1
(B5)
Ko(@ = ‘(’Y+lOg@)-lOg 8+0(87+0(~10g8)
1 8log : +0(8)+0(@log8)
Z’(8) x –;–~
“sinfhos(hsine)do+i‘Sinesin(tisinqd
0
0
(B6)
‘(w)
~:(8)
‘8
(’Y-l-log
r/2)+8log 8+0(89+0(8’
10& 8)
(B1O)
QU~I=~~CAL
~ORY
OFWTNQ-BODY
INTERFE~NCD
ATSUPERSONIC
SPEEDS
Fromequations(B9)and(B1O)we get
‘[-21(3”1
=(*)(~q[:+0(8?]
1337
Takingthe inversetransformwith the help of reference25,
page 282, we obtain the desiredresult.
[8(7+log7’/2)+
(III’)
8log8+0(93lo&s)]
SOthat
For anyMachnumberandbodyradius
-’J1’2)’G)”:@(*
“2”
‘[-21(2)
’’ZI=:[(’+1:TP)+
1
logs n—-~ log8+0(8lo&8)
8
(ml)
(B13)
APPENDIXc
DETERMINATION
OFWING-ALONE
POTENTIAL
FORANGLE-OF-A’ITACE
CASE
Tho iimt step in calculatingthe potentialfor the wing of thcaewing sectionsand the results added together to
alonewill be to set up a mathematical
model. Sincethe obtain the potentialfor the entirewing alone. Thus,
reposedwingof thecombination
operatesinthebodyupwash
fieldwhicheffectivelytwiststhewing,thewing-alone
model0
isconsidered
tobe twistedin themannerpredictedby Beskin
Sincethe wing may be consideredto be composedof an
upwaahtheoryfor y2 a
infinitenumberof flat,rectangular
wings,the expression
““=+-$)=”’C+3
(
,(.W,,]+ -zcos-&pcosh-y&+
(cl)
The concealedwing may be extendedthroughthe body
r@on in anymanner$ut,sinmequation(Cl) @v@ ~w=’&
at both wing-bodyjunctures,it is takenas a flat plateat
Qngleof attack‘2~B(seefig.42).
I
-4
S::::n ~
Left bolf- wing+
1
-3
1
-2
-1
0
y/o
i
1
Right half-wing
I
3
2
I
4
(C3)
hornreference6 for thevelocitypotentialof a flat, rectangularwingwillbe usedasthebasicrelationfor the calculations. Equation(C3) gives the velocitypotentialat any
point (z,v,z) due to a flat, rectangukm
wing at angleof
attackaw, terminatingat V= O,and extendingto m along
thepositivey axis.l”
Sincethe twistedwingwasshownto be equivalentto a
baaicflat-platewingat angleof attack2“’ plusan infinite
numberof modifyii flabplatewings(seefig. 43), the potentialof therighthalf-wingmaybe writtenas
f%VE=d2”Bd/-l)+$’
FmSJRE
42.-fWqMof wingalonewitheffeotivetwistproducedby
bodyupmshfield;angle-of-attaok
case.
db,v-m)
(C4)
Q~=2aB+fial+Aa2-7
1
Thetwistingof thewingis accomplished
by superimposing
a seriesof flat-platewingsupon a basicflakplah wing at
a~=2aB(seefig.43). Each of the superimposed
wingsis
at an incremental
angleof attack,andeachsuccessive
wing
terminatesat a valueof y greaterthanthe previousone.
Asthemincremental
valuesof abecomeinfinitesimally
mmll,
1
a~=2(fB
1
theresultingpotentialapproachesthat of a wing with the
aw=2aB+Aal
l,L–
I
I
I
i
twist defied by equation(Cl).
I
I
g thewing-filonepotential,
For thepurposesof determining
i::J
the wingis consideredto be composedof the threeparts
Basic wing
Twls!edwing
shownin figure42: the right exposedbalking, the left
exposedhalf-wing,and the wing sectioninsidethe body. I?mmm4.3.-Formation
of infinite
of twistedwingby superposition
The perturbation
velocitypotentialis determinedfor each
number
of Satplates.
(-.<-
~
1130tha
andpnmtnkenmaolty.
-1~’fflr:
NThetopandbottom
wfng:~ arefl mndatiMWmdmti
1338
FORAERONAUTICS
REPORT125%NATIONMJADVISORYCOMMTI’PDE
The &t termon therightin equation(C4)is thepotential
dueto the basicexposedhalf-wingwhichterminates
at the
wing-bodyjuncture,y= 1, and is at angleof attack2aB.
The secondtermis thepotentialof theiVmodifyingwings
eachat angleof attackAafand terminating
at y= qiwhere
l<qi S OJ. Since equation(C3) is homogeneouswith respectto a, equation(~) maybe written
%VE=dzaB!v—l)+$’
9(1 ,?J-@a;
z
(C5)
Fromequation(Cl)
da= ——
2? dY
(C6)
FIQUWS
44.-Uppervalueof y included
intheforoMacheonoomonot
ingfromPI;p=l.
coneemanating
fromthearbitrarypointPIfor19=1tlmt
Therefore,
%R=d2~B,Y-1)-zaB
~9“q(l,y—q)$
J
Z12=21*+(q.~J~
(C7)
Therefore,theupperlimitof integration
is
wherethelimitsof integrationaredetermined
by therange
(08
FY1+4Z=7
of y on thewingincludedin theforeMachconeoriginating
from the point for whichWE is beingdetermined.From Thelowerlimitof integrationis at thewing-bodyjuncture
figure44 it is apparentfromthe equationof thefore Mach y=l. Fromequations(C3), (C7),and(C8),
Cmryingout theintegrationandcombiningtermsgives
plane,thepotentialfortheotherhalf-wingissimplyobtmine
by replacingy by –y in equation(C1O).
(C12
%VL=wR(%—?/,
z)
6+=)
cOsh-’,@+q-x
Cosh.,
(y’+&) ,&–y’-zs
(z+Y+)cOs-’
Z’-y(y-l)-z’+
,-
‘d’-l)
J(y-1)2+ z?&=2
1}
*CR.,. =
&&4)+
{(c@&,+c
z
n-
.
wings terminatesat the wing-bodyjunctureat y= —1,
figure42,andintends(throughthebody)indefinitely
in the
positivedirection. The otherwingterminates
at theother
wing-bodyjuncture,y= +1, and also extendsindefinitely
in the positivey direction. The d.iilerencebetweenthe
potentialsof thesetwo wingsis the potentialof the wing
sectionin thebody
(Cll)
The expressionfor q (aW,y) is given by equation(C3).
Since the model is symmetricalabout the verticaly=O
—’e
)
(Clo) G~+y—2
Equation ((210)gives thepotentialdue to the exposedright
half-wing. To thismustbe addedthe potentialsdue to the
othertwo wing sections.The potentialdue to the sectionof
thewing in the body regionis simplythe differencebetween
the potentirdsof two flat wings at a~=2aB.Oneof these
wB=P(~B,Y+l)—9(~B,Y—1)
Combiningequations(C1O),((211),and (C12) gives th(
potentialdueto theentirewingalone.
cosh-1
e~-
&ti+2)
(Y
“
cosh-’ Ji&7
–’(y-l)
+&--#-z
X(y+l)
[ Cos-k=mwe+cos-kmw=
x/J
(y’+”) ~~2
~sh_, z’-y(y-l)–
m
)
cosh_l
[
1+
&–y(y+l)–&
w–
‘l+2”(z-z)}
(013;
Investigationof ~ as givenby equation(013) revorh
thatthereare threeregionson the body in whichthe ron.
part of this espreaaion
assumesdiflerentforms, A fourt,l
region,regionIV, is entirelyon thewingandis, thereforo
not necesswyfor determmm
“ “ g thenormalvelocitydistribu
tion on the body. Theseregionsare determined
by three
—
1339
– INTDRFERENCFI
AT SUPERSONIC
SPEEDS
QUASI-CYIJNDRIOAL
THEORY OF WING-BODY
($:2+Y+2
)
co&-’
~~~+(+z)
4/+1)
—z(?J-l)
[
1+
cos-’4@=Fw4=+c0s-14
mww=
~sh_, &–&–y(y-l)
(?/’+
.9%3=7=
[
coah_,3?-&-y(y+l)
,-
4==–
]-2(.-3,.}
(C14)
RegionII:
.—
1
*
+-
,w=-- r -. CoS-’;~)[
L+
+2
G
Y–2) cosh-l~&-––
(&-z)
(Y’+a&FICWRB
45.—Intersootion
of oharaotetiloMaohconeswithz=Oplane
showing
corresponding
regions.
clmracteristic Mach cones. One of the Mach cones originates on the body axis at z= O,and the other two originate
at the leading edges of the two wing-body junctures (see
fig, 45). The expressionsfor the realpart of ~ in the three
regionson the body are:
Region I:
VCYB
VW. .7
—x
{[
cOs-ld~2.w+
“(Y-l)
co&_, &–&–y(y–l)_x=_zlT
1
(CM)
RegionIll:
Va
PIY=—~ –~ ~o,-1*_ —
3
(~+y–2)
cosh-’ d+2–
*z)cos-’
–(y–l)+coa-l
COS-l
-
J!!&
–
1
(Y’+aJ7+&–&
~~-+‘*(Y-l)
_, &–&–y(y-g_2(z_z)T
m
Cos
1
(C16)
1340
FORAERONAU!ITCS
REPORT125&NA~oNAL ADVISORY
CO&IWTT13E
REFERENCES
13. Pitt-g,WifliamC., Nielsen,Jack N., and Gionfriddo,MnuricoP..
1. Spreiter,JohnR.: The AerodynamicForceson SlenderPlane-and
ComparisonBetweenTheoryand Experimentfor Interference
Cruciform-Wingand Body Combination. NACA Rep. 962,
PressureFieldBetweenWingand Body at SupersonicSpeode
1950. (FormerlyNACA TN’s 1662and 1897.)
NACA TN 3128,1954.
2. Browne,S.H., Friedman,L., andHodes,I.: A Wing-BodyProblem
14. Tsien,H.: SupersonicFlow Overan InolinedBody of Revolution
in a SupersonicConicalFlow. Jour.Aero. Sci., vol. 15,no. 8,
Jour.Aero.S&, vol. 6, no.12, 1938,pp. 480-483.
Aug. 1948,pp. 443-452.
15.
Lagerstroq
P. A., andVan Dyke, MiltonD.: GeneralConsidera
3. Ferrari,Carlo:InterferenceBetweenWingandBodyat Supemonio
tions About Planarand NonplanarLiftingSystems. DouglaE
SpeedE-Theoryand NumericalApplication. Jour.Aero. Sci.,
AiroraftCo., Rep. No. SM-13432,June1949.
VOL15,no. 6, June19@ pp. 317-336.
16. Ward,G. N.: The ApproximateExternaland IntarnolFlow Pact
4 Morikmva,GeorgeK.: The Wiig-Ikdy Problemfor Linearized
a Quasi-~lindricalTubeMovingat SupersonicSpeeds. Quart.
SupersonicFlow. Calif.Inst. of Tech. DoctJ3rd This, 1949.
Jour. Mech. and App. Math.j vol. 1, pt. 2, June 1948,pp.
GALCIT,Jet PropulsionLab.PR&l16.
226-246.
5. Bolton-Shaw,B. W.: Wiig-Body Interferenceat Supersonic 17. Fraenk+ L. E.: On the OperationalForm of the Linomized
Speede-RwtangularWiig at Inoidenceon a Body at Zero
Equationof SupersonicFlow. Jour.Aero. SOL,vol. 20, no. 0,
Incidence. EngliehElectriuCo. Rep. LA. t. 039.
1953,ReadersForum,pp. 647-648.
18. Mersnwq W. A.: NumerioalCalculationof Certain Inverso
6, Nieken,Jaokh’., andMatteson,FrederiokH-: CalculativeMethod
LaplaceTransform. (R6mun6)Vol. 2 of Proo. Intl. Cong,of
for Estimatingthe InterferencePremre Fieldat ZeroLift on a
Mathematioiane,
Amsterdam,1954.
SymmetricalSwep&BaokWiig Mountedon a CircularCylin19. Lamb,Horace:Hydrodynamics. SixthEd., Dover Pubfloatione
dricalBody. NACA RM A9E19,1949.
194S,p. 527.
7. Nieke~ Jack N.: SupersonicWing-Body Interference. Calif.
20. Busemann,Adolf:Intlnitecimal
ConicalSupemonfo
Flow. NACIA
Inst.of Tech. Doctmtd Thwia, 1951.
TM 1100,1947.
8. von Hhmin, Theodore,and Moore, Norton B.: R&dance of
SlenderBodiesMovingWithSupersonicVelocities,WithSpecial 21. Alden,HenryL., andSchindel,LeonH.: The Calculationof Wing
Lift and Momentsin NonuniformSupersonicFlows. Moteor
Referenceto Projectiles. Amer. Sea. Moth. Engm.,VOL64,
Rep. No. 53, M. I. T., May 1950.
Dec. 1932,pp. 303-310.
22. Cramer,R. H.: InterferenceBetweenWingandBodyat Superaonio
9. PMnney,R. E.: Wing-BodyInterference. ProgrewReportNo.4.
Speeds. PartV—PhaeeI Wind-TunnelTestsCorrelatedWith
Universityof Miohigan,
Eng.Res.Inst.,ProjeotM937,April1952.
theLinearTheory. timell Aero.Lab.Rep.No. CAL/CJM-507
10. hTielsen,
JackN., andPitts, Wii
C.: Wiig-Body Interference
Dee. 195o.
at SupersonicSpeedsWith an Applicationto Combination
23.Lawrence,H. R., and Flax, A. H.: Wing-BodyInterferenceat
‘iWthRectanguhuWings. NACA TN 2677,1952.
Subemicand SupersonicSpeeds. Surveyand New Devolop11. Bailey, H. E., and Phhmey, R. E.: Wing-Body Interference.
menti. Paperpresentedat the AnnualSummerMeetingof the
FinalReport. Part I. TheoreticalInvestigation. University
I. A. S., La Angeles,Jdy 15-17,1953.
of hfichiq Eng.I&. Inst., 1937–1–F,Jan.1964.
12. Bailey, H. E., and Phinney, R. E.: Wing-Body Interferonca 24. Watsoq G. N.: A Treaticoon the Theory of BesselFunotion.e
SecondEd. 194+ MaoMifIan.
FinalReport. Part II. ExperimentalInvestigationof CylindricalModeL Universityof hfichigan. Eng. Res. Inst., Feb.
2S. Cardaw,H. S., andJaeger,J. C.: OperationalMethodsin Applied
1954.
Mathematical.Sewnd Edition1947,OxfordUnivemityPreaa.
1341
QUASI-CYLINDRICAL
THEORYOF-WING-BODY
IXTDRFERENC13
AT SUPERSONKJ
SPEEDS
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0 1.1 1.2 1.3
x
(a) TV.(z,r)O <2.<2
CHARTI.—vdurs%
Of ~z&r) functions.
1.4
1.5
!.6
1.7 1.8
1.9 20
1342
.
REPORT125*NA’JXONALADVISORYCOMMJ!ITEE
FORAERONAUTICS
)
x
b) Wo(z,r)2 S z <4
CHAFLT
I.—Continued.
QUASI-CYLINDRICAL
TBEORYOFWINGBODYINTERFERENCE
AT SUPERSONIC
SPEEDS
x
(0) W,(z,r) O < x <2
CHART
I.—Contiued.
1343
1344
REPORT125*NATIONAL ADVISORYCOMMITTEE
FORAERONAUTICS
.,
x
(d) W,(z,r) 2< z <4
CHAETI.-Continued.
QUASI-CYLWRICALTBEORYOFWINGBODYINTERFERENCE
AT SUPERSONIC
SPEEDS
x
1345
F
1246
REPORT125%NAmoNALADTILSORY
COMMITTEE
FORADRONAUITCS
x
(f) W,(%r)2 s z <4
CHAET
l.—Contiiued.
QUASI-CYLINDRICAL
THEORY
OFW2NGBODY
INTERFERQNOll
ATSUPERSONIC
SPEEDS
(g) W,(z,r) o ; z <1.6
CHART
l.—Concluded.
1347
REPORT126*NATIONAL ADVISORYCOMMITTED
FORAERONAUTICS
TABLEI.—VALUES
OF A&(z)
z
o
.05
.1
. ;5
.25
.3
.35
.40
.45
.:0
.7
.75
.8
.85
.:5
M.(z)
—w
-----------
–L 321
H
40
–. 787
–~ 7;6
1.334
L 155
.735
.279
–. 119
–. 417
–. 595
:: f33;
-------------------------------
----------.131
-----------
-----.----.831
-------..-
---------------------
---------------------
-----------
-----------
-----------
-----------
-----------
1-----------
1-----------
–. 538
–. 166
.304
–. 378
.450
___________----------–. 292
.902
.560
245
–: 040
–. 253
–. 417
–. ::;
-------------------I 1 —:
.457
.235
.340
.406
.433
.420
.306
.123
–. 062
–. 197
–. 256
–. 239
–. 162
–: :);
–. 386
-----------l-----------1
‘. 434j
---------.
–. 224
----------
H
3.5
3_6
3.7
–: 5;1
.405
—m
-----------
-----------
::
L6
1.7
1.8
U
&o
3.1
3.2
M,(x)
-----------
–. 868
----------
H
M,(s)
-----------
i:
1.2
1.3
i:
21
2.2
23
24
25
I M,(z)
I
–. 180
----------
–. 147
–. 122
---------–. 102
---------–. 086
----------–. 073
----------–. 062
----------–. 053
-----------
–. 045
-----------
–. 038
-----------
–. 031
-----------
–. 026
-----------
–. 022
-----------
–. 018
-----------
–. 016
392
. ----------
.294
. ----------
.188
–. 412
–. 337
–. 231
/l:
095
.088
:163
204
----------.005
:211
197
----------–. 058
:164
--------------------.069
–. 100
020
----------–. 123
–: 022
–. 053
----------–. 073
–. 126
-----------
----------
–. 119
---------–. 103
---------–. 083
---------–. 061
----, -----–. 040
---------–. 022
---------–. 007
-----------
–.
–.
–.
–.
–.
–.
080 ~
071
060
038
019
002
.011
021
:027
027
:025
.022
.015
J
.
12
NOTEthatM*=@)+—-‘A-1
- asz+O.
.
.105
.134
.121
.085
.036
–. 013
–. 053
–. 063
–. 059
–. 042
–. :;;
.022
.030
.028
.019
.008
–. 002
–. 011
–. 014
–. 103
1349
QUASI+YLIIWRICAL
THEORY
OFWINQ-BODY
INTERFERENCE
ATSUPERSONIC
SPEEDS
TABLE 1L—VEL0CIT% AMPLITUDE FUNCTIONS
Angleof attaokcase,j,n(cc)funotions
z
_&
&
o
o
VCCB
o
.02
. l);
:08
, 10
.12
: it
:;:
.22
.24
.26
.28
.30
.32
,34
.36
,38
.40
.42
.44
::!
.50
.52
.64
: :;
: :;
.64
.66
:%
.72
;;:
.78
.80
.82
.84
:E
.90
.92
. Q4
. 9(3
.$38
;?
i2
L3
L4
L5
L6
L7
L8
ii
24
28
3.2
3.6
4.0
.001
.003
.006
: ::;
.014
.019
. 02!
.029
.032
--------.050
------.068
.079
.089
------.111
------.136
.151
------------:M
------.299
.208
------.308
------.363
------------.426
.467
------------------.687
------.667
------------.812
------------1.013
L 121
1.375
L 386
1.393
1.402
1.410
L 415
L 421
L424
L 425
L 424
L 421
L 384
L 356
L 337
L 323
L 314
413072-ci7+5
Wing-inuidence
oasq~ix(z)functions
f4
Kw
vaB
o
w
o
.003
.003
:
.008
.007
.008
.012
.011
.013
.017
.015
.019
.022
.020
.026
.029
.025
.034
.037
.031
.043
.047
.037
.052
.057
.044
.062
.068
.062
. ----------------- . --------.083
.097
.063
------------------.103
.127
.067
.112
143
.067
:161
.120
.065
------------------.133
.199
.051
------------------.140
.026
.139
.%
.008
------------------------------------.315
–. 065
:
–. 086
.333
------------------–. 14s
.088
.869
–. 175
.388
.069
------------------.016
–. 236
.420
------------------–. 052
–. 286
.450
------------. 461
------------------–. 181
–. 322
.476
–. 232
–. 323
. 478
------------------------------------------------------–. 212
–. 462
.456
------------------–. 109
.411
–. 552
-----.------------------------------.178
–. 673
.280
------------------------------------–. 623
.397
.004
–. 510
.378
–. 250
–. 053
–. 025
–. 669
–. 047
–. 032
–. 651
–. 045
-.040
–. 631
–. 045
–. 045
–. 613
–. 048
–. 047
–. 592
–. 055
–. 045
–. 572
–. 064
–. 040
–. 650
–. 075
–. 035
–. 627
–. o&
–. 033
–. 503
–. 088
–. 036
–. 479
–. 087
–. 037
–. 459
–. 085
–. 036
–. 434
–. 085
–. 036
–. 429
–. 086
–. 036
–. 427
–. 086
–. 036
–. 426
–. 085
–. 036
–. 425
M
o
o
.003
: %;
.013
.017
.022
.027
.033
.037
.040
.004
.007
.012
.016
.020
.022
.024
.025
.024
.023
---------
. ---------
-------
-------
.038
.026
.017
.007
------–. 023
------–.
068
–. 090
.014
–. 007
–. 022
–. 041
-------
–. 080
-------
–. 106
–. 114
----------–. 146
-------------
-------
-------
-------
-------
------–. 142
-------------
-------
-------------------
-------------------
–. 159
–. 178
–. 182
–. 176
–. 012
.075
.235
------.223
------------.093
------------–. 263
–. 340
–. 0268
–. 028?3
–. 0230
–. 0192
–. 0156
–. 0166
–. 0199
–. 0224
–. 0206
–. 0195
–. 0203
–. 0202
–. 0202
–. 0202
–. 0202
–. 0202
—. 121
—. 114
–. 082
–. 054
.047
.119
-------------
.184
.163
.027
-------
–. 170
-------------
–. 234
-------------
.062
–.
–.
–.
–.
–.
–.
–.
–.
–.
–.
–.
–.
–.
–.
–.
–.
ZL
0078
0108
0118
0152
0146
0122
0115
0135
0127
0129
0129
0129
0129
0129
0129
o
.0001
.0005
.0011
: %::
.0046
.0063
.0082
.0104
.0129
.0156
.0186
.0219
.0255
.0293
.0335
.0379
.0427
.0478
.0531
o
.0003
.0010
.0023
.0040
.0063
.0091
.0123
.0160
.0201
o
.0003
.0010
.0023
.0040
.0061
.0087
.0116
: %?
. 02i7
.0253
.0289
.0324
.0366
.0395
.0412
.0432
.0445
.0450
.0447
.0433
: EM:
.0966
.1043
.1119
.1195
.1269
.1342
.1411
.1477
.1539
.1596
.1646
.1688
.1722
.1746
.1757
.1755
.1737
.1701
.1644
.1562
.1452
.1308
.1125
.0894
.0603
.0235
–. 0237
–. 0865
–. 1777
–. 4244
::%
.0320
.0254
.0173
.0074
–. 0041
–. 0174
–. 0326
–. 0497
–. 0686
–. 0893
–. 1118
–. 1359
–. 1616
–. 1886
–. 2164
–. 2451
–. 2738
–. 3021
–. 3292
–. 3539
–. 3751
–. 3908
–. 3983
–. 3939
–. 3704
–. 3121
–. 0849
*
*
v
.0928
.1008
.1092
.1180
.1273
.1371
.1475
.1584
.1689
.1820
.1948
.2084
.2229
. ~~
.2722
.2912
.3118
.3842
.3591
.3871
.4194
.4584
o
.0002
.0010
.0022
.0038
: %;:
.0104
.01.28
.0151
.0171
.0187
.0197
.0201
.0196
.0181
:W
.0065
–. 0001
–. 0081
–. 0177
–. 0282
–. 0410
:: ;3;
–. 0829
–. 0985
–. 1142
—. 1297
–. 1446
—. 1583
–. 1703
–. 1808
–. 1870
–. 1897
–. 1898
–. 1838
–. 1739
–. 1565
–. 1360
–. 1063
–. 0716
–. 0312
: %E
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.1727
.1605
–. 0364
w
1350
FORAERONAUTICS
RDPORT125*NA!ITONALADVZSORY
COMMITI’DE
TABLE JII.-ORIFICE LOCATIONSON wING AND BODY OF WING-BODY COMBIATATION
~imensions
ininok measo.red
fromwingleadingedge]
1a= O.75inch.
.,
. .<
..
. .
,,
+.-
.,
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. .
. —— -..
TABLE 1T.—
BM1.On
-h-
–1.!Y
1
.-
#/I-o,lm . .. .. ..
a 431
a
..4
i
.149
j.m
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:$9
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U 3HEi
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.414
7Lm ..411
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—
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I 43
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.
.314
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:’21
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.ml
.
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1.
.!456
.=
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.911
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.11
. 1$?
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. WI
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.=
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1
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()
TABLE IV,-PRE%3UlZJ3
00EFFIOIJ3NlW~
-Continucxi
TABLE IV.—PRESSUlW COEFFICIENTS
&
(-)
-4%nlinwd
‘8
I ,/#=o.mm . ..–...1
ml
mgl
: