Page 1
NASA Contractor
Report
2865
A GeneralizedVortex
Lattice
Method
forSubsonic and
Supersonic
Flow
Applications
LuisR.Miranda,Robert
D.Elliott,
and
William
M.
Baker
J
,8
CONTRACT
NAS
112972
DECEMBER1977
N/ /X
Page 2
Page 3
NASAContractor
Report
2865
A Generalized Vortex
Lattice
Method
forSubsonic
and
Supersonic
Flow
Applications
Luis
and
R.
William
Miranda,Robert
Baker
D.Elliott,
M.
LockheedCalifornia
Company
Burbank, California
Prepared
Langley
under
for
Research
Contract
Center
NAS112972
National Aeronautics
and Space Administration
Scientific
Information
end Technical
Office
1977
Page 4
Page 5
TABLE OFCONTENTS
LIST OFILLUSTRATIONS.........................
SUMMARY...............................
INTRODUCTION.............................
THEOP_TICALDISCUSSION........................
TheBasicEquations.......................
Extension
The
toSupersonicFlow.................
.................SkewedHorseshoe Vortex
Modeling
Modeling
Computation
of
of
Lifting
Fusiform
of Sideslip
Surfaces
Bodies
withThickness...........
...................
Effects.................
THEGENERALIZEDVORTEX LATTICEMETHOD.................
Description
Numeric
ofComputational Method ...............
8.1 Considerations....................
COMPARISONWITH OTHERTHEORIES ANDEXPERIMENTAL RESULTS........
CONCLUDINGREMARKS.........................
APPENDIXAUSER'S
FOR
MANUAL
SUBSONIC
FOR
AND
A GENERALIZED
SUPERSONIC
VORTEX
FLOW
LATTICEMETHOD
......APPLICATIONS
THE VORLAXCOM.VJTERPROGRAM....................
PRACTICALINPUTINSTRUCTIONS.....................
INPUTCARDIMAGEDESCRIPTION.....................
PROGRAMOUTPUT............................
RECOMMENDATIONSFORTHEEFFICIENTUSEOFTHEVORLAXPROGRAM....
ApPENDIXBCOMPLETE
GENERALIZED
SUPERSONIC
PROGRAMCOMPILATION
VORTEX
FLOWAPPLICATIONS
AND
METHOD
..............
EXECUTION
FOR
FORA
LATTICESUBSONICAND
HARDWAREANDSYSTEMS.......................
PROGRAM
sUPERSONIC
UNIFIEDVORTEXLATTICE
.....................
METHODFORSUBSONICAND
FLOW
Page
iv
i
i
3
3
5
9
14
15
17
18
18
2O
21
22
37
39
4o
46
61
93
97
99
i01
iii
Page 6
APPENDIX C WAVE DRAG TOVORLAX INPUTCONVERSION PROGRAM(WDTVOR). . .
SUMMARY...............................
INTRODUCTION............................
NOMENCLATURE............................
PROCEDURE.............................
PROGRAM WAVE DRAG TOVORLAZ INPUTCONVERSIONPROGRAM(WDTVOR)....
APPENDIX D VORLAX TOWAVE DRAG INFCTCONVERSIONPROGRAM(VORTWD). . .
SUMMARY..............................
INTRODUCTION............................
NOMENCLATURE ...........................
PROCEDURE ..............................
PROGRAM VORLAX TOWAVE DRAG INPUTCONVERSION PROGRAM(VORTWD)....
REFERENCES...........................
Page
249
251
251
253
253
279
327
329
329
33O
331
341
371
_.v
Page 7
Figure
I
2
3
4
5
6
7
8
9
10
Ii
12
13
14
AI
A2
A3
A4
A5
LISTOFILLUSTRATIONS
Definition
of
ofintegration
part .....................
regionsfor thecomputation
principal
Sweptbackhorseshoe vortex.................
Modeling ofthickwingwithhorseshoevortices.....
Vortexlatticecollocation.................
Modelingoffusiform bodywithhorseshoevortices ......
Conventional approachesforanalysisof winginsideslip..
Latticegeometryfor analysisof wing insideslip......
Generalized
configuration
vortexlattice modelofwingbody
.......................
Theoretical
aerodynamic
comparison
center
of arrowwing
................
liftslope and
location
Theoretical
factor
comparison
..........................
ofarrowwingdrsgduetolift
Theoreticalcomparisonofchordwiseloadingfordeltawing.
Theoretical
rectangular
comparison
wing
ofchordwiseloadingforsweptback
.....................
Comparison
wingbody
with
model
experimental
at Math
pressure
0.5 ...............
distributionon
=
Theoreticalcomparisonofpressuredistribution on
ellipsoids
flow
atzeroangleof attackinincompressible
.......................' ....
Controlsurface nomenclature...............
Modelingofthickwingwithhorseshoevortices.......
Vortexlatticecollocation .................
Modelingoffusiformbodywithhorseshoevortices......
VORLAXcasedatadecksetup .................
Page
23
24
25
26
27
28
29
3O
31
32
33
34
35
36
4l
42
44
45
47
V
Page 8
Figure
A6
A°7
CI
C2
C3
C4
C5
C6
C7
C8
C9
CIO
CII
DI
D2
D3
D4
Generalized
configuration
vortex latticemodel ofwingbody
.......................
Vortex
engine
lattice
fighter
panel representation
......................
fortwin
TypicalWAVEDRAG format dataset..............
Graphic representationof WAVEDRAGdataset .........
Listing of output datasetin VORLAX format .........
Result
dataset
ofconverting
back
circular
WAVE
fuselage
format
VORLAX
toDRAG .............
Plotoffigure C4dataset .................
Result
back
of
to
converting
WAVE
flat
format
fuselage
.................
dataset
DRAG
Plotof figureC6dataset ................
Converteddataset beforeeditingforVORLAYrun......
Converteddatasetafterediting for VORL_[ run....
Plotofdataseteditedforsubmittal toVO_qL_Y.......
Portionof outputfromsubmittaltoVORLAX.........
VORLAX datasetwith smarts cardsincluded ..........
WAVE DRAG formatdatasetresulting from conversion.....
Planformplotfrom figureD2dataset..........
IsometricplotfromfigureD2dataset...........
Page
49
95
257
259
26O
f
16o
26_3
269
27?
276
277
33_
336
338
339
vi
Page 9
AGENERALIZEDVORTEX
SUPERSONIC
LATTICE
FLOW
METHOD
APPLICATIONS
FORSUBSONIC
AND
LuisR.Miranda,
LockheedCalifornia
RobertD.Elliott,and William M.Baker
Company
SUMMARY
Avortexlattice
It
method
shown
applicable
that
tobothsubsonic
vortex
andsupersonic
is
flow
is described.isif thediscretelatticeconsidered
as
the
vorticityinduced
thevelocity
vortex
simulating
lattice
simulated
anapproximation
generalized
tothe surfacedistributed
part of an
field.
generatedby
methodvalidfor
nonzerothickness
elementsareincluded.
byadouble(biplanar)
vorticity,
yields
incorporation
vortex
flow.
surfacesand
Thicknesseffects
vortexlattice
then the
term
this
concept
to
term
the
for
with
components
fusiform
of
principal
velocity
field
integral
Theproper
discrete
supersonic
lifting
aresidual the
toof
thelines renders
techniques
bodies
present
latticeSpecial
fusiform
ofwinglike
layer,
vortex
are
andbodies
are
drical
described.
techniques
bythe
ispresented.
identified
Appendix
Appendix
transforming
represented
surfaces.
bya vortex
The
gridarranged
sideslip
onaseries of
the
application
of the
andexperimental
digital
program
executed
conversion
WaveDrag
concentrical
subject
cylin
analysis
considerations
discussed.
thisreport
This method
VORLAX.A
complete
Appendices
inputbetween
ofeffectsby method
of
is
Numerical
arealso
of
peculiar
summary
other
beenimplemented
manual
compilation
describe
VORLAX
tothethese
obtained
results
code
A comparison
theoretical
results
methodwith
has inacomputer
iscontained
arecontained
programs
programs.
asusers
Fortran
C and
forthe VORLAX
and
input
NASA
in
A.
B.
Acasein
Duseful for
theand
INTRODUCTION
Theseveralversionsorvariations ofthevortexlatticemethodthat are
presently
tools
urations.
available
the
The
have proven
analysis
the
tobe
and
very
design
isdue
practical
of
in
andversatile
andnonplanar
part to
theoretical
config
relative
foraerodynamic
success
planar
greatof method the
simplicity
within
most
subsonic
vortex
presented
supersonic
derived
of
limitations
work
flowapplication.
latticetheory
herein
Mach
in thenext
thenumerical techniques
thebasic
lattice
The
supersonic
thedirect
The
sectionstarting
involved,
theory,
methods
applicability
flow
extension
equations
from
and tothehigh
obtained.
concentrated
techniques
ignored.
lattice
extended
order vector
accuracy,
the
the
of oftheresults
to
the
largely
vortex
this
first
But
ofonvortex appearshave
basic
on
ofof
tohasbeen
of
Themethod
to allows
numbers.
techniques
application
equations
allowing
the
are
governing
case
sonic
inviscid
skewedhorseshoe
horseshoe.
compressibleflow.
with
They
special
arethen
attention
appliedtothe
to
particular
thesuperofa vortex given
Page 10
In the following
the simulation
presented.
native, with somewhatreduced computational requirements, to the methodof
quadrilateralvortex rings (refs.1 and 2).
volume effects makespossible the computation of the surface pressure distri
bution on wingbody configurations.The fact that this
having to resort to additional types of singularities,
resultsin a simpler digitalcomputer code.
theoreticaldiscussion,the basic argumentsinvolved in
by vortex lattice
represents an alter
of thickness and volume effects
This particularmodeling of the aboveeffects
elements are
The simulation of thickness and
can be donewithout
such as sources,
2
Page 11
THEORETICAL
TheBasic Equations
DISCUSSION
Wardhas shown, (ref.
of an inviscid
equations:
3), that the smallperturbation,
compressible fluidis governed by the three first
linearized
order vector
flow
v = ®% V..Q,=
v
(l)
on
functions
perturbation
•is a
with
theassumption
of
that
point
the
whose
with
vorticity
position
orthogonal
tensor
with the
[2 j
_and
vector
Cartesian
that
freestream
thesource
R.
components
orthogonal
direction
intensity
The vector_
Q are
is
and
coordinates
form
known
the
velocity
isthe
w, u,v,and
constant
xaxis
symmetrical
aligned
for Cartesian
hasthethe
i
 MO0
q'=0 __(2)
00
where M_is the freestreamMachnumber. If82= IM2,thenthevector_
has
by
denotes
the
flow
the
Robinson
components_
(ref.
thetotal
fluiddensity,
= 82u T
called
vector,
can
+ v _+wk.Thisvector
current
+u)
irrotational
was
velocity".
+ v_
firstintroduced
If_
and
homentropic
4),
velocity
then
whoit the
i.e.,
shown
"reduced
_= (u_
that
T + w_,
and
p
itbe for
p _= p_ u_+p_ + higherorderterms(3)
where
e.g.,
directly
the
u_
subscript
= u_
related
_indicates the
a
value
linear
mass
ofthequantityat upstream
vector_
infinity,
is i.Therefore,
tothe
to approximation,
flux
the
perturbationas follows:
(4)
The
sourcefree
secondequation
flows
of(1),
0),
i.e.,
is
the
conserved
continuitycondition,showsthat for
(Q=waquantity.
Ward has integrated
to resort
solutions
ornegative
into
the
to
for
(supersonic
single
three
auxiliary
(R),
firstorder
potential
vectorequations directly
obtainedwithout
two
sonic
bined
havingan
V
_anctlon.
whether
Thesetwo
thefollowing
He
different
flow),
formally
depending
flow).
on 82 is positive
solutions
convention
(sub
com
used:
can be
a expression if is
K=2 for_2> 0
K=1 for
B2
< 01/2
R, = Realpartof{(XXl)2+_2_yyl)2+ (ZSl)2]}
3
Page 12
= Finite
and 6).
part of integralas defined by Hadamard(refs.5
The resulting solution for the perturbationvelocity _ at the point
whoseposition vector isR1= x 11+ Ylj+ Zlk, isgiven by
V( ) =  2_" RB dS
S
iR
R3
 RI
2_K
S
dS
+ 2__ dV2wK R3
VV
< dV(5)
This formula
S.
determines
vector
Furthermore,
Vand
the
is
value
the
it
within
ofV withinthe region
the
for
dependence
V bounded
V)
by
the
to
only
surface
the
those
The
S.
_ unit
is understood
outward (from
that
region
supersonic
normal
flowsurface
parts ofS lyingthe domain of(Mach
forecone) of thepoint RI aretobe includedi_n theintegration.
Forsourcefree(Q_O)_ irrotational(_0)flow, equation(5)reduces
to
2_K
i 82]JRRI
Rf dS (6)
B
n wVdS +
ss
This isarelationbetweenVinsideS and thevaluesofn.w and_ <v
onS,butthesetwo quantities cannotbe specifiedindependently onS.
Todetermine
of
surface
of
thesourcefree,
(6),assume
body,
radius
irrotational
that the
anytrailing
enclosing
flowabout
coincides
that
thewhole
anarbitrary
with
have,
flow
body
B
wetted
asphere
bymeans equation
of
infinite
surface
wake
body
S the
andthewith it maywith
about theand field
it,namely,S= SB+S W+ S•
This
and
surface
Viinternal
S divides
to
the
Applying
spaceinto
equation
two regions,
(6)
Veexternal
Ve and
tothe
since body,it.tobothVi,
theintegralsoverS_convergetozero, thefollowing expression isobtained:
4
Page 13
= l
2wK
A _(R)V1
R_
L
{Nx AV(R)} ×_ dS
(7)
•dS 2wK
R3j
S B+ SW SB+ SW
where
positive
N= Hi=
from
he
the
is
interior
theunit normal
the
to the body,
of
or
body,
wakeasthe
We
case
 wi,and
maybe,
toexterior the A _
A q
the
as
= Ve
corresponding
representing
 vi.Herethesubscripts
S.
designate
first
ofa
thevalues
integral
distribution
of
can
of
the
be
surface
quantities
considered
on
face ofThe surface
source thecontributiondensity
•A _,
distribution
whilethe second
of
surfaceintegral
N
gives
A V.
thecontribution ofa vorti
citysurfacedensity×
If theboundary
toboth
condition
external
ofzeromass fluxthroughthe surface SB+ SW
isappliedandinternalflows
• 0 _e=_ • (P_o + P_)=0
N.
:N.(8)
then
uniquely
thecondition
determined
N. A w=0exists over SB+ SW, andthe flow field is
by
x RRI
SB+ Sw
where_(R) = N × A Visthesurfacevorticitydensity.
(9)
ExtensiontoSupersonic Flow
Inorderto extend
it
the
essential
applicationof thevortex
the
latticemethod
element
to
of supersonicflow, istoconsiderfundamentalthe
method,
integral
the
expression
vortexfilament,
(9)
asa
of
numerical
a real
approximation
physical
scheme
The
tothe
instead entity.velocity field
generated
process,
bya
result
vortexfilament
being
canbeobtained bya straightforwardlimiting
the
c
(lo)
Page 14
where['=limV.8
7_oo
6o
8 is a
classical
dimension
vortex
normal
lattice
toV,andd_is thed_stance
only
element
subsonic
along
flow,
Y.
the
In
vorti
the
method,applicable to
city
is
fields
distribution
replaced
are
over
suitable
thebody
arrangement
determined
and thewake,
vortex
i.e.,
filaments
(lO).
overthe
whose
procedure
surfaceSB+ SW,
bya ofvelocity
iseverywherebyequation Thisno
longer
to
done
appropriate
back
in the
forsupersonic
(9)and
flow.
derive
For
an
this
approximation
latter case,
to
it is
This
necessary
isgotoequation
following.
toit.
Ifthe
composed
thesurface
equation
surfaceSB + SW,
large
which
number
density
defines
of
V
thebody
flat
and
area
approximately
equation:
itswake,
elements
isconsidered
T over
constant,
as
which
then
beingof
vorticity
can
adiscrete
can
by the
beassumed
following(9)beapproximated
N
f_
 2nK
/ RR1
R3 9(RI ) =
J=l Tj
where
whose
crete
N
position
area
isthetotal
vector
be
number
is R1
of
is
discrete
not
by the
area
of
elements
Tj,
value
T.Whenthe point
partthe
theorem
integralover
follows:
this dis
canapproximatedmean as
R3_ CjS
where
is
means
be
whose
discrete
ofthe
order
close
Cjisa line
normal
velocity
for
per
T,
hype
in Tj
to
parallel
Cj,
field
outside
length
integral
to the
integral
Just
to theaverage
arc
adiscrete
by some
But
(_i)
=R1 at
for
distance
direction
length
of Vinrj,
Cj.
80"
Thi@
can
line
of
adistance
that
approximated
strength
and d_is the element
vorticity
mean
if the
has an
some
thiscase,
_.
along
patchthe inducedby
T j
_T.
rj
points
unit
the
due
the
located
of discrete
point
inherent
point
consider
Asindicated
vortex
part
singularity
within
a
is
in
yj
equatlon
that
expression
above
R1is the
area
Cauchy
to
to
factR _.
point
in
In
evaluate
R1butT bya figure
l, thearea ofintegration inr is dividedintotworegions,A r_¢ andAe.
Obviously,
finite
Thus,
the
concept
integraloverA __ehasno Cauchytype
perform
singularity,
indicated
Hadamard's
partbeing sufficientto theintegration.
6
Page 15
_xdS=lim
R3_e..o
T
+ lim
_,.O
A eA
T_
R_RI
R3_= lira I(_)+ V_ _<
_.._O
C
d_ (13)
Thelast integral
contribution
the
inequation
whose
(13) represents
presents
I(e)
the
no
conventional
difficulty.
that,
discrete
In ordervortex
to
lineevaluation
denoteddetermine integration byassumeforsimplicity,
the
determined
coordinate system
the
iscentered
area
at the
Then,
point
if
RI, and that
the
the
modulus
xy plane
of
is
bydiscreteT. y denotes y,
A
,sinA xcosA )2 dxdy
Y!X2 _ B2(2+ e2_
(14)
where
in
Ais the
B2
angle
8_
between
> 0
theyaxis
flow).
andthe
The
direction
components
of
of
the
the
vorticity
vector • , and=(supersonic
cross
been
carried
the
product
left
out,
vorticity
_ ×(RRI)
ofequation
theywill
direction
=_
(14)
x Rwhich
because,
The
through
are
when
A¢
(I+B)e
notnormal
the
isbounded
and
to theplaneof_ have
is outlimitoperation
bya
the
¢o
parallelvanish.
going
arealine
intersection
to
x byofthe
Mach
the
forecone
integration
from thepoint
respect
(o,
to
o, s) with
performed
the zplane,
first,
consequently, if
withx is
f,t F
 xdxdy
ix2_B2(y2 + 2)}3/2}
ty(I+B)e
(_)
wheret= tanA , andkl,_2 arethevalues ofycorrespondim_ totheinter
sectionof theline x=ty (l+B)¢with thehyperbolax B_y 2+ e2"Let
= e2(l+RB)_2(l+B)_ty_(B2_t 2) y2,then the finitepart ofthex
integrationyields
l(e)
=
'Y cosA+
ty (ty(1+B)_)
B2(y2+ )Jf
z
_ dy
47i
Page 16
= 7 cosA_B2_(I+B)_y(B2t2)y2dy
(16)
B2
J
22
y
+
¢
Since
is
the
¢ is a very
be
sm_llquantity,
small,
equation
the
therefore,
(16)can
variationofy in theinterval
within
mean
(kl,42)
ingoing
last
to
integrand
equally and,the quantity
by
brackets
andof bereplaceda valuetaken
outside
it will
interval,
denotes
of
vary
theintegral
from
then
a mean
sign.
y
increase
ofy,
The same
through
toeo
can be
isnot
finite
y
written
trueoftheterm
in the
this
1/__
integration
in mind,
since
co for= 41,govalues
With andagain for= _.
as
and if
valueI(¢)
I(¢)=Y CosA
B2
B2_(l+B)gt_
,,,2
y
 (B2t 2)
2
¢
_2 ___
J
dy
(17)
+
_
But
the
41,
polynomial
_2 arethe
denoted
rootsoftyc
=
B_y2+2, i.e.,they aretheroots of
by _.Thus
_=_2(l_2B)_2(l+B)_ty (B2_t2) y2 = B__t2 " _(_l_y)(y__)(18)
Introducing
the
thisexpression
value
for ^/_ into
obtained:
(17), andtaking thelimit
¢o, followingfor I(¢)is
I(o)= lira I(¢)
¢ .o
=Y cosA
B2
_2
dy
B2t 2_ _(41Y)(Y _)
41
(19)
The
variable
integralappearing inequation
is
(19) can
be
beeasilyevaluated bycom
plexmethods; itsvaluefoundto
41
dy
_i 41Y) (Y _)
= (20)
8
Page 17
Thecontribution
T,induced
given
of
the
the
vorticity
inherentsingularity
patch
to
dented
thevelocity
herein
field,
byw*, within
therefore
by T, andis
by
._2
2w
lim I(_)
o
_cosA_B 2 t2(21)
_2
This contribution
meaning
Mach lines.
and(12),
is
B2 >
is
perpendicular
t2,
expression
thevortex
tothe
the
taken
plane
vortex
in
method
ofT,and
are
ithas only
in physical
of the
(ll)
wheni.e.,when linesswept
with
to
front
It
makes
(21),
lattice
conjunction
applicable
equations
supersonicthat flow.
TheSkewedHorseshoe Vortex
Velocityfields
of
due
due tocomplex vortex
simple
line
by three
leg_
vortex
in
curves
vortex
vortex
rectilinear
figure
segment
theformulation
vortex
can
geometries.
beobtained
segments:
Therefore,
of constant,
of
lattices.
begeneratedby the linear
thesuperposition
velocity
the corresponding
skewed segment,
thevelocity
is thefundamental
coefficients
fields
to
fields
and
field
induced
a horseshoe
induced
twotrailing
due to
building
complex
by For
by
instance,
the addition
a transverse
determination
arbitrary,
influence
Choosing
fieldcanof
2. the
but
of
a line
block
sweep
aerodynamic
of threedimensionalacoordinate
system suchthat thevortexline liesinthe planez =O,theconventionaldiscrete
vortex
Ls given,
contribution
in
to the
components,
velocity at a point
the
whosecoordinates
expressions
are(Xo, Yo' Zo)
Cartesian byfollowing
u=+2."'_ (X_Xo)2+ _2( (y_yo)2+ Zo2) 3/2
,v =  rZ° 22 K
F
,82_C
w=+ 2wK
I (xx°)2+ _2( (y_yo)2+ Zo2)} 3/2
(XXo)dy(yyo)dx
I(xx°)2 ÷ _2((y_yo)2+Zo2)I 3/2
(22)
9
Page 18
where
length,
satisfies
rrepresents
and
the
thecirculationper
be
unit
carried
lengthof
along
discrete
that
vortex
part
line
C whichtheintegrations
conditions
aretoout of
)2 _2)2 2).
(x Xo+((Y Yo + Zo>0
and
xx<0 ifM >i.
o
Forthe transverse
(22)
integrations
By defining
leg
expressed
carried
thefollowing
ofthe horseshoe
asa
between
auxiliary
vortex,the
y,
coordinate
i.e.,
y= s
x appearing
and
= +s,
in
indicated
figure
equationscanbefunction
the
ofx= ty,the
outlimits
variables
andy
2.
+ ts x2=x  ts
XI= XO0
Yl= Yo+sY2= Yos
X*= x° ty °
The
transverse
resultingformulas
rectilinear
giving
vortex
the
filament
velocitycomponents
bewritten
induced
as
by theskewed
canfollows:
u=+ r___&z .
2_K
1I tXl
2
+ 132y1
_2
*2(t2+_) Zo 2
,_,/X+)_/X2 + _
x+( 2+z222
i Ylo
]
(Y2 +Zo)
V_m
W
rz
2_K 2 + (t%_2)z°2
.t "ItXl+ _2yl
_/xl2 +_2 (yl%Zo 2)_/x22÷_2 (y2%Zo 2)
F
*I txl
.
+
rx _2yl
2_K *2
2
[ _/
_2
(yl2+Zo 2
x+(t2+_2) Zox2+
1
tx 2 + _2y 2
(23)
tx2
2
+ _2y2""
,,2,2
+ #
]
llfX2 2,
tY 2+Z o)
In theabove
are
xy
xaxis
expressions,
measured
plane
and
the
respect
coordinate
vortex
coordinates
to
(Xo, Yo,Zo)
the
of thereceiving
field
segment,
fined
point with
the
the
the
system
midpoint
coinciding
of rectilinear
withthe
vortex
dethe
the
ofplane
byitself.
lO
F
Page 19
Thecaseofarectilinear
since
be parallel
vortex
trailing
tothe
segmentparallel
ofa
Since
to thexaxis
vortex
equations
(t = _ ) is
are gen
(22)
of
erally
become
specialimportance
to
thelegshorseshoe
dy assumed xaxis.=0,
u=0
FZo'_2_
A
dx
v { 122 21Y21 (2 I
(XXo+((YYo) + Zo
W
F(YY°2trK ),82_{ dx}3
(XXo)2+ fl2@((yyo)2+ Zo2/2
If
yields
_he vortexsegment extends fromx= xitox= xf,the aboveintegra
tion
u=0
_ZO
2wK
LIjXX,
((yoy) + zo )
 01
V=o _2
22
(XoXl) _ +
%
+ _2((yo_y)2+ Z° 2)
XO xf
2
(Xo_Xf)2 + f12((yo_y) + Zo )
1 (2_)
Cyoy)2+Zo 2
{, x °
2
xf
((yoy)
] Yo
2
y
÷ (XoXf)+I_2+ Zo2) (yoy)z2
0
Foraconventional
infinity,
horseshoe
(25)
vortex
would
whosetrailing
contribution
legsstretch
ofthe
to down
legstreamequationsgive theport with
thefollowing substitutions
x.=_ ifM_<I
i
•=x 4_2 ((Yo+S) 2 +z 2)
O
i,fM. >I
XIO
xf ts
yS
ii
Page 20
Likewise, the contribution
by introducing
from the starboard trailing
values into equations (25)
leg can be computed
the following
x.=
+ts
1
xf=_ ifM_<I
_ __1322)
xf=x° ((YoS)2 + z°
ifM® >l
y=
4"8
Combining
field
theseresults
a
to
withequations
vortex
xaxis
finite part
(23),the
of
formulas
a skewed
defining
segment
the
and
obtained.
the fol
flow
trailing
Keeping
lowing
induced
legs
inmind
notation
by discrete
the
consisting
(theskewedhorseshoe
concept,
two
parallel
Hadamard's
vortex)
introducing
are
and after
2
txl+_Yl
F1=
_/Xl
2 + _2(yl2+Zo
2)
2
tx2+ _ Y2
F 2= __
_x222 )
+(y2 2+zo
_2
x1
G1=
_/2 _22 2)
Xl+(Yl+ Zo
+C; (M®<'I:C= i;M.>I:C =O)
(26)
x 2
G2=+C;
I2 _22 2)
x2+ (Y2+Zo
(M_<i:C= i; M. >i: C = 0)
then, thehorseshoe vortex inducedfield formulascanbeexpressed as follows:
12
Page 21
rZo
u (xo, yo, Zo)=+ 2__x*2 + (t 2 + _2) z 2
o
v (xo, Yo' Zo)=+FI
[ 
(FI
F 2 )t
2wK Z°*2_22
x + (t2+) z°
(Fz  F2)
G1
+
22
Yl+Zo
Q2I
22 _ (27
Y2+ Zo
I
x
1
F (FI F2)Yl Y2
w (Xo, Yo'Zo)2_K*2" t 22+2 2 GI22G2
+(+132) z Yl+z Y2+z
ooo
Thefinite partconceptdetermines thevalueofthe constantC appearingin
thedefinitionofG1 and G2.
A notablesimplification of equations (27) occursforsupersonicflow
when
zo
identically;
the
0.
receiving
First,
point
values
(Xo,Yo'
the
z
axialwash
expression
) is inthe
and
becomes
plane
sidewash,
ofthe
u
horseshoe,
andv,
namely
o
= the of
upwash
vanish
secondly,the
w (xo, Yo' 0)
rIFIF2 G1 G2)
(28
Equation
format.
Gfunctions
fication
Fand
(28)
But
isapplicable
the
null
upwash
the
to both
flow
the
subsonic
case,
finite
Introducing
version
andsupersonic
fact
concept
thecorresponding
equation
flow
constant
further
values
inits present
C of
simpli
of
for
becomes
the
supersonicthe
part
thatthe
allows
the
dueto
ofequation.
er)anded
the
Gfunctions, of (28)is
2wK
+
w(xo, Yo' O)
tx I +2]
Yl tx2
,.
+_2y2
_gi
x+ Yl22
l
+ Y22
xl/Y 1x2/Y 2
._Xl2 _22
+ Yl
,_x22_12
+ y22
(29
13
Page 22
Since inx=Xo iof tYo (xI _=Xl _yl)tYl%/x12= x2+ _2 ty ],yl 2 therea[rrangementand (x2 of Jx22equation + _2(2i) Y22 factorsI_ tY2) i
finallyreduces theupwash formulato
l2+_222I
w(x, yo O)=Fi _XlYl__x22 + _
Y2
Y22
o 2_K " _x Yl(30 )
When
is
the
approximated
with
fieldpoint (Xo,
the
Yo' O) is
horseshoe
upwash
within the
vortex,
by
distributed
e.g.,
equation
vorticity
thecontrol
(30) has
patch
point which
associated
bydiscrete
the thehorseshoe, giventobe
complemented
upwash
inthe
lumped
ship
bythe
as
distribution
given
occupied
discrete
F ofequation
duetothe generalized
If _x
vorticity
legof
Fofequations
principal
thedistance,
¥
circulation
part ofthe
integral,
xdirection,
into
between
byequation
by the
(21). ismeasured
hasbeen
therelation
and(30)
distributed
vortex
andthe
which
the
the
transverse_
(21) (27)is
F=_cosA6x (31)
Modelingof LiftingSurfaces withThickness
The
(ref.
method
l)
ofquadrilateral
provides
bodiesusing
vortex
computing
vortex
ringsplaced
surface
only.
on the actualbody sur
face
of
a way
discrete
of the
lines
pressure
Numerical
distribution
difficultiesarbitrary
may
sharp
nearly
computer
or biplanar,
with
approximation
theclassical
occurwhen the
edges
direction.
and
sheet
as
to the
lifting
above method
to
is applied
close
alternative
handle
horseshoe
schematically
location
surfacetheory
totheanalysis
two
requiring
consists
model
3. This
ofairfoils
surfaces
somewhat
in using
lifting
constitutes
similar
cambered
with
of
less
double,
trailing
parallel
storage
duethe
An
to
proximity ofvortex
approach,
easier
swept
shown
true
numerically,
vortices
infigure
thesingularities,
approximation
a
oftoasurface
thickness,an
of in
sheet.
nature to
ofa
All the
corresponding
swepthorseshoe
to
vortices,
given
andtheir
upper
boundary
orlower,
condition
are
control
in apoints,a surface,located
same
gap
bution.
value
plane.
which
The upperandlower
chordwise
not
the
surface
average
sensitive
full maximum
lattice
of
to
chordwise
planes
airfoil
magnitude
thickness
areseparated
thickness
of
bya
represents
Theresults
between
the
are
the
the
distri
gap;
the
toothis
of
any
onehalf to airfoil
has
maximum
been found
thickness.
to be
adequate,
Furthermore,
thepreferred
the gap
value
vary
being
in
two
direction
thirds ofthe
canthenormalto
the
chordwise
xaxis to allowfor spanwise
or spacing,
thickness
of
taper.
transverse
Onthe
elements
otherhand,
of
the
horsedistribution, thethe
shoe vorticeshavea significantinfluence ontheaccuracy ofthe computed
Page 23
surface pressure distribution.
numberof horseshoe vortices,
spacedaccording to the
For greater accuracy, for a given chordwise
the transverse legs have to be longitudinally
cosine distributionlaw
[(2J l) l
v =c1 cos n_
xj x °2
(32)
where
of
chord
number
x_ xorepresents
leg of
through
horseshoe
the distance
horseshoe
midpoints
per
from
vortex,
a
theleading
is
chordwise
chordwise
edge
length
strip,
control
tothe
the
and
point
midpoint
local
N is
location
the Swept
running
of
the Jthcthe of
theof given
The
the
vorticesstrip.
corresponding tothisdistribution ofvortexelements isgiven by
C=c1 cos_
xj

x °
The
chordwise
collocation
controlpoints
strip
of
arelocated
4).
lattice
along
has
the centerline,
(ref.
defined
ormidpoint
the
(32)
line,of the
(fig.
the
Lan
elements,
shown7) thatchordwise 'cosine'
(33),byequationsand
greatly
His
distributions
sented
improvethe
directly
wings
accuracyofthecomputation
tothe
by
of theeffects
of
lattice
duetolift.
resultsare extendable
with
computation
thebiplanar
surface pressure
ofthickness scheme pre
herein.
Thesmallperturbationboundarycondition
is applied
_' =m_
the actual
Theuse
atthe
where
control
_,
surface.
small
points.
and
In equation
thedirection
(34)
boundary
(34),n: _ +m_
the
<<I
+n_,
normal
my
with
and
to+nk,m,n are
Equation
cosines
that
of
airfoil
the
implies
condition
J_ul
consistent
+ nw_.
ofperturbationis the
presentbiplanar approachto thesimulation of thickwings.
(34)
Modeling ofFusiformBodies
The modeling
concentrical
effects,
carried
auxiliary
distribution
without camber.
approximated
horseshoe
body (which
possible
offusiform
vortex
computation
To define
identical
actual
Thecrosssectional
polygon
vortices.The
bydefinition
cross sections
bodies
lattice
with
if
the
lattice,
crosssectional
witha
shape
sidesdetermine
ofthe
rectilinear
body)define
horseshoe
simulation
surface
it
vortices
of
pressure
necessary
shape
baricentric
thisauxiliary
the transverse
polygon and
(zerocamber)
asetof
requires
volume
distribution,
to consider
longitudinal
line,
body
legs
axisof
and internal
radialplanes
a
special
ment
to
an
thethedisplace
and
out.
body,
to
the ofare
first
area
be this
in
body,
is
and
thestraight
of
i.e.,
is
of
the
then
bya whose
vertices
is
ofthe
the
the auxiliary
to all
whichin
15
Page 24
the
(fig.
corresponding
that the
The axial
vortex
The boundary
face,
byequation
boundtrailing
As
legs
body
of the horseshoe
section
allowed
vertices must
ofthe crosssectional
orpolygonal
conditioncontrol
the bisector
(33).
vortices lie
along
parallel
its
but
sa_rleset
thatdetermine
cosine
onthe
theirlongitudinal
totheaxis
the
constraint
planes.
transverse
equation
body
spacing
5).the
polygon
cross
is
changes
change
always
shapelength,
with
ofradial
the
law of
auxiliary
toaccordingly,
in the
planes
followsthe
are located
planes, with
the
polygonal
spacing
elements,
lie
rings,
points
(B2).
sur
givenand in radial
The
mass
boundar_ _ condition
fluxequation.
tobe satisfiedat thesecontrol pointsisthe
zero
whereallthecomponents ofthe scalarproduct_ . K= _2_u+ my + nw
are tobe retained. Thus,equatio_(35)isa higher orderconditionthan
equation
framework
fore,
matical
found
quisite
(34).
of
can
derivation.
that the
for
The
linearized
only be
useof this
theory,
higher
is
by
order
mathematically
results
treatment
orexact
of the
boundarycondition,
consistent.
thanby
fusiform
boundaryconditions
surfacepressure
within the
a not There
mathe it justified
!n the
higher
its rather
of
a strict
present
order,
bodies,it has
is
been
re use
accurate
ofa
the determinationdistribution.
The
equation
that
flow
sequently,
factthat
(_5)j.requires
smallperturbations
I),the
theboundary
thevector _:instead
elaboration.
. n_
_dentical
eqL_tion
of V,appears
First,
Furthermore,
theperturbation
(3_)is
in the
should
lefthand
pointed
incompressible
velocity
withthe
member
ofsome itbeout
for
(8
wV n'.
to
for
= vector_is _.Con
conditionconsistentcon
tinuity
higher
dition
remembered
equation,
order
is applied
that
_.w = O_toa
flow.
firstorder
But
to
the
forcompressible
higher
flow,
boundary
it
not
andto any
forincompressible
incompressible
thissolution
when
iinearized
conservation
a ordercon
flowa solution,
of _,
should
of V,
be
satisfiesi.e.,
. _= O. Thus, thehigherorder
perturbation
boundary condition
massflux,
should
vector
involve
as
the
equation reducedcurrentvelocity,or w, in
(35),ratherthan theperturbationvelocityvector_.
Thebody
taken
which
camber,
into
are
in
which
acco_mt
was
in
in
eliminated
the
equation
condition
This scheme
provided
inthe definition
the
of the
cosines
effect
spatial
approximation
camber
auxiliary
body,
and
is
of
cambered
large.
iscomputation
(35).
of direction
the
inthe
fair
body
_,m,
n,implicit
the
elements.
bodies
Therefore,
ignored
give
amount
ofcamber
placement
to
is not
represented
the horseshoe
boundarybut
will
the
a
offusiform thattoo
16
Page 25
ComputationofSideslip Effects
Theaerodynamics
approaches
ofanisolated
on
of
vector,
wing
coordinate
wind axes,
figure
in sideslip canbeanalyzed by two
approach,
aligned
problem
different
the
with
depending
consists
velocity
thesystem
longitudinal
This
chosen.Inone
being
of
coordinate
the
system the
6.
axis
freestreamformulation the
is
this
zero
known
known as theskewedwing
will
The
skewed
approach,
dominant
approach,
and
effects
also
a firstorder
sideslip,
schematically
basedon
solution
even
obtained
for
figure
formulation
within
case
and
framework
dihedral.
as
give
other
freestream
the of theof
shown
is
in6
theapproach,abodyaxis
of
adequate
sideslip
within
the problem
for
for
the
andthe
dihedral,
zero
of
corresponding first
to
To
order
produce
compute
solution,
the
the sideslip
formulation,
though it
effects
may be
large
low
framework
will failsignificant of
ordihedral.
skewed
effects
is
correctly
a freestreamitnecessary to
solve
turbation
than
On the
partialdifferential
velocities.
required
otherhand,
equations
implies
solution
containing
a much
of the
ofthe
second
involved
order
order terms of the
procedure
equations
anything
per
This
the
application
more
first
skewedwing
computational
perturbation
approach
thatfor
the
(i)
tomore
complex
configuration
complicated.
thananisolated
with
wing
fuselage,
insideslip,
and
suchasmight be
geometrically
thecase witha
wing,nacelles, becomesvery
Theapproach
above,
adoptedherein
with
isacombination
objective
ofthetwo
reasonably
approaches men
tioned formulatedthe ofobtaining accurate
sideslip
of the
it is
vortex
ments
the
lattice,
invariant
The
sumed
which
tionality
invariant
effects
geometrical
assumed
wake
thatmodel
wake are
containing
in
freelegs
to extend
is proportional
factors
in
usingonlyafirst orderperturbation
in the
representing
freeelements
considered
elements,
transverseand
system, thechordwise
lattice arenot
downstream infinity
to theangles
areunity,thenthe
axes.
solution butwithoutall
complications
thevortex
of both
rigid surfaces
socalled
both
a bodyaxis
of the
to
inherent
lattice
bound
are
free
skewedwing
the
or
bound,
7.
or
legs being
force
toa predetermined
andsideslip.
portionof
approach. Basically
its
fila
constitute
ofthe
segments,
parallelto
ratherthey
direction
Ifthe propor
latticewould
that configuration
legs;
and those
Thebound
chordwise,
and
consists andthe vortex
that
portion the figure
trailing, is
xaxis.
as
the
areactually
parallel
ofattack
free
free,
the be
wind
After thecirculationstrengths of theabove lattice geometry are solved
for
tribution
under the
is
appropriate
computed
boundary
accordance
conditions,
with
thepressure
order
coefficient
expression
dis
inthe higher
2
cIU_ u + v_ v) (36)
p2
q_
where U_andV_arethecomponents alongthex andybodyaxes of thefree
stream
components
ofthe
velocityvector
denoted
approximation
of
by
modulus
uand
q_; the
usual.
corresponding
The
perturbation
equation
velocity
instead arev, as useof(36)
linear
17
Page 26
2
Cp2U_u (37)
q_
is
rolling
required forthe
due
correct
tosideslip
computation,
of
within
wing.
the present
This
framework,
due to
ofthe
thatmomenta planar is the fact
the
freestream
tribute
order
of second
rolling
order
boundtrailing
flow
the
in
order,
moment
fora planar,
legs, being defined
according
This
namely,
included
since
planar,
inbody
to
axes,
theorem
is
Eventhough
computation
quantity
are not
of
lined
Kutta,
up
they
the
with the
andtherefore,
force.
(36),
must
to sideslip,
ornearly
the con
tonormal
equation
it
due
contribution
V_
in
this
wing.
represented bysecond
term v.
the
this
of
contribution
thedifferential
isof the
is
be
itselfsecond
THE GENERALIZEDVORTEX LATTICEMETHOD
Description ofComputational Method
The
vorticityinduced
representing
polygonal
latticegeometry
ofsideslip
knownas
inwhatfollows,
fouritemsdiscussed
residual
thickness
for
consisting
effects,
generalized
has
inthepreceding
w* for
the
of
bound
section, i.e.,theinclusion
biplanar
concentrical
thespecial
the analysis
procedure
method,
program
of
the
for
term supersonic
ofa
fusiform
andfree
ina
(GVL)
Fortran
flow, the
of
and
for
scheme
effects,
simulation
of both
been
vortex
codified
usevortex
bodies,
elements
computational
method.
IVcomputer
grid
cylinders the
have implemented
lattice
in
herein
theThis outlined
(VORLAX).beena
The
legs
basic
has
element
both
infinity
legs
ofthemethod
free
arbitrary,
laidout
is theswept
The
predetermined,
proper
horseshoe
latter
vortex whose
trail
whereas
surfaces
trail
to
the
ing
downstream
bound
boundand
any
segments.segments
direction
control
may
in
are
but
trailing onthecylindrical in
a
matically
direction which
the
is paralleltothe
wingbody
x bodyaxis.
configuration
Figure8
within
illustrates sche
representation ofa the context
of
the
the
lattice
present method. In
cosine
thisillustration,
distribution
the streamwise
equation
arrangement
(32),but
of
chord followsthe law, both
wise
to
there
trailing
isdetermined
and
either
corresponds
legs
spanwisedistributions
thecosine
a control
thehorseshoe;
equation
of vortex
the
which
the
ifthe
lines
spacing.
placed
canbe independently
each
between
of
distribution
specified
vortex
bound
be oforofequalTohorseshoe
point ismidway
location
the
controlof
by
longitudinal
cosine
thepoint
been(33)chordwise has
chosen,
required
otherwise
by
it is locatedhalfway betweenthetransverse legs, as
quarterchord?threequarterchordrule.
Thedirectionof floatationof thewake vortexfilaments is definedby
the
the
The
values
constant.
two
angle
proportionality
being
angles
of
_vand6_
and
constants
thesideslip
shown
the
infigure
being
part
constant,
8, the formerbeing
to
input,
for
proportional
sideslip
the recommended
theangle
to
attack,latter
are
proportional
theprogram
and0 or
the angle.
of
0 for0.5of attack
18
Page 27
Thevelocity
(27)
field
above
induced
constants
by the
are
elementary
both
horseshoe
and
vortex
somewhat
is
more
given by
equationswhenzero,by compli
cated
either
presented
ofequations
expressions
one
here,
which
of
takeintoaccountthe
parameters
easily
kinks inthe trailing
nonzero.
through
legs
Though
application
when
notor both the
expressions
wakefloatation
can
are
these be derivedthe
(23).
When
is
the
being
by
influence
evaluated,
equation
induced
the
bya horseshoevortex
the
normalwash
uponits own
principal
freestream
control
point
as
contribution
included
from
the
generalized part,
given (21),is in if theis
supersonic•
formula,
Furthermore,
equation
for
used
thesupersonic
whenever
case,
receiving
thesimplified
point
downwash
the (30),isthe isin same
planeof the inducinghorseshoe.
The
system
horseshoe
of
boundary
either
overrelaxation
i.e.,Purcell's
i.e.,
equationsis
to achievea
multiplication
handled by proper
vortexvelocity field is
the
used
unknown
points.
procedure,
a vector
9).
ofanalysis,
distribution
thisinvolves
i.e., design
boundarycondition
to generate
vortex
This
known
orthogonalization
the inverse
the
(surface
a straightforward
andanalysis,
equations.
the
strengths
linear
as controlled
coefficients
of
appropriate
solved
successive
nique,
desired,
a linearequations
condition
GaussSeidel
relating
at the
iterative
(ref. 8),
Vector method
ordesign
computethe
surface
process.Mixed
grouping of
to the
iscontrol system
bya
or
(ref.
instead
slope
loading;
cases,
the
by tech
is If process
synthesis
used
specified
linear
warp)
system
required
of
to
matrix
areeasily
The pressure
components,
three
coefficients
the
ways,
arecomputed
being
interms ofthe
according
perturbation
velocity
one
computation
as
carriedout toeither
of possible follows:
la
If
on
the
both
surface
sides
under
(zero
consideration
thickness
isassumed
and
wetted
configuration
bythe flow
sidepanel) the
slip
based
ACp
angle
on
2y
is zero,
local
then
value
a net
of
loading
spanwise
coefficient
vorticity,
is computed
namely,the
cosA;
the
=
1
When
coefficient
theconfiguration
of
sideslip angle
surface
is not
is
zero, thenet
through
loading
azerothicknesscalculated the
useof thehigher orderexpression(36); and
o
When
under
only,
coefficient
surface
consideration
theisentropic
pressure coefficients
assumed
flow relationship
offreestream
arecomputed,
bythe
giving
Machnumber
i.e.,
on
pressure
andlocaltofree
thepanel
is wettedflow
the
oneside
interms
streamvelocity ratio isresortedto.
Theforce and momentcoefficientsarecalculated by numericalintegration
of
edge
program
thepressure
forces.
user,
coefficient
cosine
the computation
distribution
chordwise
of
with
spacing
due account
is
suction
beinggiven
by
zero
to
VORLAX
the
Iflattice
the
specified the
thicknessleadingedge of
19
Page 28
panels
to
field
is
forces
reference
of the
is carriedout
is
according
made
in
toLan's
by
report.
the
by
not
being
procedure
thegeneralized
Ifequal
leading
the technique
nearlyas
significantly
(ref.7), whose application
velocity
spacing
to
Hancock
themagnitude
supersonic
formulas
specified,
and
flow
presented
thecontribution
momentsis
i0; this
leadingedge
possible
this
vortexinduced
chordwise
suction
indicated
accurate as
underestimated.
lattice
singularity
by
Lan's,
ofedge the
calculated
approach
suction
in
is
The
asymmetrical
about
through
are treated
flowbeing
pointto
VORLAXcomputer
cases
orall
input
bythe
defined
the rotation
program
well
three
reference
has
configurations
axes,parallel
center.
of assuming
valuesof
center.
thecapability of
steady
coordinate
state
onset
ratesand
analyzing symmetrical
angular
axes,
rotation
flow, this
distance
and
tion
asas in
the
state rota
anyof togoing
the moment
subterfuge
by the
Steady
nonuniform
angular
angular cases
a onset
thethe of field
Groundproximity
is
then
the
ground
downstream
effects are analyzed
theground
stream
the arrangement.
it is assumed
parallel
by
plane;
which
themethod
the
coincides
this
trailing
planeof
ofimages,
around
with
i.e.,
airplane
theground
a configu
wake
the
configuration
andits
plane
ration
floats
mirrored
contains
symmetry
proximity,
infinity
about flowthe
image
to
in
to
a surface
due ofIn
the
modeling of
that
the
vorticity
ground.tothe
NumericalConsiderations
Atsupersonic
becomes
generated
surface
part
fieldpoints
velocity
Asimple
thecharacteristic
Mach
large
the
itself,
numbers,
in
transverse
the
This
theplane
behaved
treat
surfaces
the
very
leg
velocity
close
of
velocity
behavior
thehorseshoe.
thevicinity
numerical
theequation
induced
proximity
horseshoe.
correctly
ofthe
by
of
a
the
At
vanishes,
discrete
envelope
the
horseshoe
ofvortex
cones
envelope
finite
for
very
by
theMach
thecharacteristic
due
fieldoccurs
planar case,
characteristic
consists of
induced
singular
to the
only
the
surface..
defining
concept. velocity
Forthe
the
offof
in field
procedure
is wellof
to this
by
singularity
(XXl)2CB_(YYl+(ZZl
(38)
where
It
these
eter
C isa numerical
found
are
constant
this
insensitive
whosevalue
yields
reasonable
isgreater
satisfactory
than, butcloseto,
that
i.
hasbeen
results
C.
that
quite
procedureresults,
of
and
param to variationsthe
Another
in
point
lines
numerical
planar
is close
(sonic
problem,
case
toa transverse
vortex),
peculiar
point
to thesupersonic
plane
leg swept
lines
horseshoe
horseshoe)
parallel
in
vortex,
when
to
of
exists
field
Mach
the(field in theofthe
exactly
the
the
and
vortex
vortexwhiletheimmediately front
2O
Page 29
behind this
problem can be handled by replacing
sonic vortex with the averaging equation
sonic vortex are subsonic and supersonic, respectively.
the boundary condition
Thi_:
ecumti_:_n for such
YI*I + 2 Yi*  YI*+I : 0
(39)
where YI* is the circulation
¥I*i and YI*+I are the respective
adjacent subsonic and supersonic vortices.
strength of the critical
circulation
horseshoe vortex,
values for the foremudaft
_ud
The axialwash induced velocity
of the surface pressure distribution,
conditionfor fusiform bodies.
generating vorticity
conventional discrete
inthe closevicinity
and in theconcentrical
in thecomputation
becomesunacceptable.
technique, similar
niqueconsistsof
horseshoe as thesummation
tributed,according
the vorticityrepresented
thepointat which
fieldregion surrounding
component(u) is meededfor the com!_utation
and for the formulation
Whenthe fieldpoint is not too close to the
element, the axialwash is adequately
horseshoe vortexrepresentation.
ofthegeneratingelement, as
cylindricallattices of the
the axialwash due tothediscretization
This problemissolved byresorting
theonepresented inreference
computing theaxialwashinduced
ofseveral transverse
toan interdigitation scheme,
bythe sing3ediscrete
the axialwash valueis required
the originaldiscrete vortex.
of the bcundary
described
Butif
occur
present
by
point
biplanar
the
vorticity
vortexsplitting
Briefly,this
transverse
longitudinally
theregion
vortex.'?his is
lieswithim
the
this
in
is
maythe
method,
of
toa
error
ofthe
to]I.
the
tech
of byleg
redis
that
_]one only
given
a
legs
over contains
if
anear
COMPARISONWITHOTHERTHEORIES ANDEXPERImeNTAL RESULTS
Conical
linearized
These
numerical
andvery
thecomputed
Only some
flow
supersonic
results
techniques.
goodagreement
aerodynamic
typical
theory provides
flow,
beused
This
between
load
a body
some
bench
been
itand
distribution
arepresented
of exactresults, withinthecontext
offor
as
simple
mark
rather
conical
and
threedimensional
casesto
extensively
flowtheory
all force
this report,
configurations.
theaccuracy
theCVI, method,
been observed
moment coefficients.
figures9
exact can evaluateof
has donefor
hasin
and
comparisonsin through 12.
Finally,
of
the
paper
capability
is
of computing surface pressure
and
distributions bythe
method thisillustratedin figures13 14.
Page 30
CONCLUDINGREMARKS
The present vortex lattice
has the capability
and supersonic Machnumbersfor arbitrary
been found to be a very useful preliminary
aircraftconfigurations
supersonic flight are considered.
namely, the computation of the surface warp required to achieve a given load
distribution. Correlationwith experimental data and with results
theoriesshowsa good agreementnot only in the overall
coefficientsdue to lift, but also in the distribution
method, in the form of a computer program,
the aerodynamicload distribution
nonplanar configurations.
design tool,
whosemission requirements involve both subsonic and
It is also capable of the inverse process,
to calculateat subsonic
It
when
has
particularly
from other
force and moment
of the load coefficients.
The schemesshownfor the simulation
which allow the computation of surface pressure distribution
vortex latticesingularities,
though experience in
this respect
of thickness and volumeeffects,
by using only
purposes,appear adequate for most practical
is somewhatlimited.
Thetreatment
order
is not
analysis
caution
ofsideslip
as
cases
necessary
complicated
configurations
numerical
bythe
for
present
the
asthe
insideslip
that
methoddoes notrequire
approach, higher
and
Yet
and
solutions,
asgeometrically
of complex
due tothe
is skewed
skewedwing
freestream
it
the
formulation.
requires
from the
still care
anomalies may result inter
action
the
among aircraft
trailing
components,
legs of
such asa
vortices.
horizontal tail or a body,and
"free"the horseshoe
Additional
that
include
capabilities
enhance
following:
thatcan
of
be added
method
to
as
the
a preliminary
presentcomputer
design
code,
tooland would
the
the valuethe
Incorporation
multipliers
forminimum
of anoptimization
gradient
underspecified
algorithmbasedeither
the
on Lagrange
warpor
drag
ona method, todesign surface
constraints.
Application
adequate
with
of thetechnique
for
attention
ofreference ii,
jet
to
or
exhaust
supersonic
ofsome other
technique,
particular
the simulation
its
of effects,
flow.toextension
Introduction
geometry
i.e.,
lattice
the development
iterative
ofa design
to
both
of
such
procedure for thecalculation
pressure
The
is
when
of the
required
synthesis
simulation
achievea given
and
lifting
design
surface
thickness.
surface
procedure
distribution,
biplanar
wellsuited
combined
of camber
thick
a
vortex
for
an
a
ofwith
scheme.
22
Page 31
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AR= 6.95
AC/4= 35 °
I
I
I
I
048
15
b/2_
= 0.124
= 0.334
T/C = 12.4%
T/C
= 10.2%
T/ =0.55&1.0T/C=9.0%
0.8
0.6
0.4
0.2
Cpo
0.2
0.4
0.6
1.0
0.8
0.6
0.4
Cp0.2
0
0.2
0.4
0.6
A_
_ 7 = 0.15
. ,_,,..
"
U.S.....
L.S.
,
....._.. o,..o. _._
EXP
z_
o
LII
GVLt
!
,=,
Jl=!
,
G,2°
=
0,90
A
_.o_ o,,4,. _
GVL
'
, ,',,
EXP
U.S.
L.S.o
0.20.4 0.60.81.0
x/c x/c
Figure 13.Comparison
onwingbody
with experimental
modelat
pressure
0.5.
distribution
Mach=
35
Page 44
0.6
LID=
0.4
0
Cpo.2
L/D= 8
•_EXACT
FI..OW
[] GVL
POTENTIAL
oMETHOD
0.20.4 0.60.8 1.0
X/L
36
Figure
14.Theoretical comparisonofpressure
ellipsoids
flow.
atzero angleofattack
distributionon
inincompressible
Page 45
APPENDIXA
USER 'SMANUAL
FOR
AGENERALIZED
SUBSONIC
VORTEX
SUPERSONIC
LATTICE
FLOW
METHOD
APPLICATIONSFOR AND
37
Page 46
Page 47
THEVORLAX COMPUTERPROGRAM
A computer program has been developed for the aerodynamic analysis and
design of arbitraryaircraftconfigurations
This computer program, herein identified
FORTRAN IV for use on the CDC6600, the IBM 360, and the IBM 370 digital
computer systems. A complete compilation
is contained in Appendix B.Twoauxiliary
Appendices C and D, provide for input data transformation
format (Reference 12) to VORLAX format (Appendix C) and for transformation
from VORLAX to NASA WaveDrag format (Appendix D).
in subsonic and supersonic flow.
as VORLAX, has been codified in
and executed case in CDCFORTRAN IV
interfaceprograms, treated in
from NASA WaveDrag
The VORLAX program is based on a generalized vortex lattice
which extends the applicability
range of problems than has heretofore
configuration is represented by a threedimensional,
vortex lattice;the basic element of the lattice
vortex whoseinduced velocity formulas have been generalized for subsonic
and supersonic flow. Thickness effects
or sandwich) latticearrangement.Fusiform bodies can be modelled by a con
centric cylindricallattice of polygonal cross sections.
capabilitiesof the program include the following:
(GVL) method
of vortex lattice
been considered.
_echniques to a broader
In this program, the
generally nonplanar,
is the skewedhorseshoe
can be simulated by a double (biplanar
The computational
•
•
@ Surface warp (camber and twist)
input pressure distribution.
• Longitudinal/lateral
•Groundand wall (wind tunnel interference)
• Flow field survey.
• Symmetrical/as_etrical
Surface pressure or net load coefficient
Aerodynamic force and momentcoefficients.
distribution.
design in order to achieve
stabilityderivatives.
effects.
configurationsand/or flightconditions.
The limitations
based on inviscid
of the VORLAX program are characteristic
linearizedpotential
of methods
flow theory, as follows:
•
•
•
•
•
Attached flow.
Small perturbation
Flow entirely
Straight
Rigid vortex wake.
flow.
subsonic or supersonic (no mixed transonic
Machlines.
flow).
39
Page 48
PRACTICALINPUTINSTRUCTIONS
Indefining
X
directed
axis
point
symmetrical
components
a configuration
is assumed.
upward,
tostarboard.
the XZ
andasymmetrical
only
for
The
Xaxis
The
In
the
XZ
program
plane
pointing
origin
general,
components,
elements
input,
the
in the
the
theconfiguration
and in
need
a master
centerline
downstream
system
frame
plane,
direction;
any
can be
the sym
of
thereference
Zaxis
the
venient
upof
metrical
Y Zis
and the
Y pointsofcan be con
madein plane.
defining
bespecified. thestarboard
The
20
configuration
ofthese
to beinput
be
is divided
symmetrical
intoa
components
set ofmajor
being
panels;
countedup to panels caninput,
only
and
Complex
once.
with
For instance,
lofting
and
a wing
between
nonlinear
with
the
changes
straight
root
leading
tip,
twist
andtrailing
a major
sections
edges,
panel.
are
linear
planforms,
and
in
constitutes
and airfoil
described
program
elementary
isgenerated
bythe
distribution
spaced
bydefining
then
panels
internally.
user,two
so
distribution.
morethan
each
chordwise
The
being
in
one
major
panel for a given
into
wing.
number
The
smaller
mesh
is specified
or cosine
(2)the
computer
will subdivide
spaced
panel
spanwise,
and
a of
andi.e.,a finerlattice
chordwise
available:
airfoil
spanwise
(i)
wing
spacing
semicircle options
well
the
theory,known andandequally
Up to2000 elementary
Any
specification
systemof
require more
significant
panelscanbe
of
used
length
inthe
and
length,
largest
of
definition
area
but
length
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