N73 - ,,_sa ,/5 NASA CR-2228

N?3
25045
N73 - ,,_sa ,/5
NASA
NASA
CONTRACTOR
CR-2228
REPORT
PART I
m
C_I
"
I
Z
AN
IMPROVED
ANALYSIS
IN
OF
SUBSONIC
Part
I-
by F.
A.
Langley
NATIONAL
THE
AERODYNAMIC
CONFIGURATIONS
AND
FLOW
and
SUPERSONIC
Application
Woodward
by
AEROPHYSICS
Bellevue,
FOR
WING-BODY-TAIL
Theory
Prepared
for
METHOD
RESEARCH
Wash.
CORPORATION
98009
Research
Center
AERONAUTICS
AND
SPACE ADMINISTRATION
°
WASHINGTON,
D.
C.
°
MAY
1973
AN
IMPROVED
OF
METHOD
FOR
THE
WING-BODY-TAIL
IN
PART
SUBSONIC
I
By
Analytical
AERODYNAMIC
ANALYSIS
CONFIGURATIONS
AND
THEORY
SUPERSONIC
AND
FLOW
APPLICATION
F. A.
Woodward
Methods,
Incorporated
SUMMARY
A new
method
sure
distribution
tail
combinations
A computer
program
calculations.
has
been
developed
for
calculating
and
aerodynamic
characteristics
of
in
subsonic
and
supersonic
potential
has
been
developed
to perform
the
the
preswing-bodyflow.
numerical
The
configuration
surface
is
subdivided
into
a large
number
of panels,
each
of which
contains
an
aerodynamic
singularity
distribution.
A constant
source
distribution
is used
on
the
body
panels,
and
a vortex
distribution
having
a linear
variation
in
the
streamwise
direction
is
used
on
the
wing
and
tail
panels.
The
normal
components
of velocity
induced
at
specified
control
points
by
each
singularity
distribution
are
calculated
and
make
up
the
coefficients
of
a system
of
linear
equations
relating
the
strengths
of
the
singularities
to the
magnitude
of
the
normal
velocities.
The
singularity
strengths
which
satisfy
the
boundary
condition
of tangential
flow
at the
control
points
for
a given
Mach
number
and
angle
of attack
are
determined
by
solving
this
system
of equations
using
an
iterative
procedure.
Once
the
singularity
strengths
are
known,
the
pressure
coefficients
are
calculated,
and
the
forces
and
moments
acting
on the
configuration
determined
by
numerical
integration.
Several
this
program
data.
Good
achieved.
examples
of pressure
are
presented,
and
correlation
between
distributions
calculated
compared
with
experimental
theory
and
experiment
has
by
been
TABLE OF CONTENTS
Page
SUMMARY..........................
1
INTRODUCTION .......................
2
LIST OF SYMBOLS ......................
3
AERODYNAMICTHEORY ....................
7
Description
Derivation
Derivation
of Method
.................
of the Incompressible
Velocity
of the Compressible
Velocity
Aerodynamic
Representation
The
Boundary
Calculation
Condition
of Pressures,
COMPUTER
PROGRAM
Program
Program
Operating
Program
Program
Components
Components
.
..............
Equations
Forces,
............
and
Moments
.....
.....................
7
7
.
.
.
34
43
47
57
59
Description
Structure
Instructions
Input
Data
Output
Data
.................
...................
.................
..................
..................
59
59
59
61
76
VERIFICATION
.................
78
EXPERIMENTAL
Isolated
Bodies
Isolated
Wing-Body
Wings
....................
Combinations
.................
82
87
........................
91
CONCLUSIONS
....................
APPENDIX
I:
Integration
APPENDIX
II:
Panel
APPENDIX
III:
Sample
REFERENCES
Procedures
Geometry
Case
Calculation
.................
........................
78
...........
Procedure
93
....
94
i00
125
iii
1.
Report
NASA
No.
4. Title
AN
2.
CR-2228,
and
Pt.
_THOD
CONFIGURATIONS
I
Accession
No.
Subtitle
IMPROVED
PART
Government
-
FOR
IN
THEORY
_
AERODYNAMIC
SUBSONIC
AND
AND
ANALYSIS
SUPERSONIC
OF
9.
A.
Report
Nameand
Research
Addre_
Date
May 197._
FLOW
mUnder
Corporation
Subcontract
Analytical
187
9320
Bellevue,
Washington
Sponsoring
Agency
6.
Performing
Organization
8.
Performing
Organization
10.
Work
No.
11.
Contract
98009
Name
and
by:
Aeronautics
Washington,
D.
15.
Supplementary
16.
Abstract
Report
A
new
computer
E.
Inc.
Bellevue,
Shoreland
Drive
Washington
or
Grant
No.
NASI-I0408
98004
13.
Type
of
Report
14.
Sponsoring
and
method
has
of
been
Administration
developed
has
been
a
wing
surface
vortex
and
to
is
panels.
points
by
each
singularity
linear
equations
relating
calculating
The
perform
having
a
normal
components
linear
are
strengths
a
constant
large
Report
Agency
Code
number
the
aerodynamic
potential
of
panels,
distribution
variation
in
the
velocity
calculated
of
and
supersonic
flow.
A
calculations.
source
of
distribution
and
numerical
into
A
pressure
subsonic
the
subdivided
distribution
the
the
in
distribution.
distribution
tall
for
combinations
developed
singularity
and
the
Space
wing-body-tail
configuration
panels,
and
is
make
up
to
the
of
which
used
on
the
streamwise
induced
singularities
each
direction
at
the
is
specified
used
control
coefficients
magnitude
contains
body
of
of
the
a
system
normal
velocities.
The
singularity
control
points
system
of
for
given
using
Several
numerical
satisfy
number
interative
are
the
and
boundary
angle
of
procedure.
calculated,
and
condition
attack
Once
the
forces
are
of
tangential
determined
the
singularity
and
moments
by
flow
strengths
acting
on
are
the
Key
Words
of
pressure
experimental
Potential
by
data.
distributions
Good
calculated
correlation
by
between
18.
Author(s))
Distribution
Lifting
Surface
this
theory
program
and
are
presented,
experiment
has
Distribution
Statement
Unclassified
Theory
Representation
Solution
of
19.
Classif.
Security
Linear
known,
configuration
Flow
Pressure
Vortex
(Suggested
the
this
achieved.
17.
at
solving
integration.
examples
with
which
Mach
an
coefficients
by
compared
strengths
a
equations
pressure
determined
Covered
20546
program
The
Period
Notes
aerodynamic
the
No.
Addr_s
and
C.
characteristics
on
Code
501-06-01-06
Methods,
S.
Unit
Contractor
National
an
No.
APPLICATION
Organization
Aerophysics
12.
5.
Catalog
Woodward
Performing
Box
Recipient's
WING-BODY-TAIL
7. Author(s)
F.
3.
I
Equations
(of this report)
20.
Unclassified
Security
Classif.
(of this
page)
21.
No.
Unclassified
*For
sale
by the
National
Technical
Information
of
Pages
128
Service,
Soringfield,
Virclinia
7_151
been
and
of
INTRODUCTION
A unified
approach
body-tail
configurations
originally
presented
in
to
the
aerodynamic
in subsonic
and
references
1 and
analysis
supersonic
2.
This
of wingflow
was
method
has
been
extended
by
the
introduction
of
several
new
aerodynamic
singularity
distributions
which
substantially
improve
its
capability
to
represent
arbitrary
shapes.
For
example,
the
new
method
permits
the
analysis
of non-circular
bodies,
provides
a more
accurate
representation
of rounded
wing
leading
edges,
and
allows
the
determination
of
wing
interference
effects
in
the
presence
of body
closure.
A computer
program
has
been
developed
to perform
the
numerical
calculations.
The
program
accepts
the
standard
geometry
input
format
currently
in use
at
the
Langley
Research
Center,
and
described
in
reference
3.
The
graphics
capability
of
the
program
of
reference
3 may
be
used
to
obtain
a visual
display
of
the
configuration
input
geometry.
In addition,
the
new
program
has
two
boundary
condition
options
available
for
determining
the
pressure
distribution
on
lifting
surfaces.
In
the
first
option,
the
aerodynamic
singularities
are
located
on
the
mean
plane
of
the
surface,
and
approximate
planar
boundary
conditions
applied
to determine
the
singularity
strengths.
In
the
second
option,
the
aerodynamic
singularities
are
located
on the
upper
and
lower
surfaces
of
the
lifting
component,
and
exact
surface
boundary
conditions
applied.
This
results
in a
more
accurate
pressure
distribution,
more
computer
time.
Surface
boundary
applied
in
the
determination
of
the
Part
describes
compares
isolated
contains
cluding
but
requires
conditions
body
pressure
I of
this
report
outlines
the
aerodynamic
theory,
the
input
requirements
of
the
computer
program,
and
the
program
output
with
experimental
data
for
several
wings,
bodies,
and
wing-body
combinations.
Part
II
a
a
detailed
complete
description
of
program
listing
the
and
The
by Mr.
tance
author
wishes
to
acknowledge
E. W.
Geller
to the
aerodynamic
given
by
Dr.
T.
S. Chow
in the
solution
ment
of
2
considerably
are
always
distribution.
techniques,
the
computer
and
by
program.
Mr.
D.
N.
computer
sample
program,
case.
in-
the
contributions
made
theory,
and
the
assisformulation
of
the
matrix
Bergman
in
the
develop-
LIST OF SYMBOLS
A consistent
a
set
of units
is
assumed throughout
Aerodynamic
influence
coefficient,
panel
inclination
angle
8, or wing
parameter
(_2
-
Matrix
of
aerodynamic
cross-sectional
area
b
Wing
span,
or
c
Panel
chord
C
Aerodynamic
d
Distance
or
body
influence
influence
major
axis
length,
body
slope
coefficients,
coefficient,
of
ellipse
or
reference
or
chord
or
wing
panel
length
of
control
point
from
singularity
point
from
wing
origin,
diameter
Diagonal
e
Distance
of
intersection
E
Off-diagonal
F,
G,
H
Velocity
distribution
I
Integral
expression
k
Supersonic
K
Kernel
function
Length
of
block
matrix
control
block
factor,
source
edge
or
Body
panel
slope
M
Mach
number,
n
Direction
cosine
of
panel
component
normal
to
panel
or
force,
or
tip
functions
scaling
line
panel
matrix
m
Normal
of
edge
coefficient
D
N
tangent
panel
report.
_i )
A
thickness
this
or
vortex,
iteration
or
body
number
length
dy/dx
pitching
number
moment
normal
of
vector,
aerodynamic
or
velocity
singularities
3
NW
Number
of
wing
and
NB
Number
of
body
singularities
q
Magnitude
r
Radial
R
Reynolds
s
Auxiliary
S
Wing
t
of
tail
velocity
singularities
at
control
point
distance
number
variable
reference
Auxiliary
thickness
area
velocity
T
Tangential
force
U,
Components
of
distribution
induced
function,
or
wing
velocity
Vw
W
V
Induced
X,
Cartesian
velocity
at
control
coordinates
of
point
points
Y,
z
Greek
Angle
of
6
Mach
number
Y
Ratio
of
singularity
attack
parameter,
specific
heats
strengths
Inclination
angle
&
Incremental
value
E
Minor
0
Inclination
Tangent
cosine
axis
(i
of
of
-
for
M2)
½
air,
or
panel
with
x
panel
with
x,y
aerodynamic
axis
ellipse
angle
of
of panel
sweepback
angle
of coordinate
transformation
plane
dx/dy,
or
direction
Sweepback
angle
Direction
cosines
Integration
of
coordinate
variables
along
transformation
x
and
y
axes
n
Ratio
of
circumference
to
diameter
of
Radial
distance
of
control
point
from
line
through
wing
panel
tip
intersection
Velocity
potential,
and
x axis
×
Integration
or
angle
between
variable
Velocity
component
normal
to
Subscripts
B
Body
base
Body
base
c
Wing
camber
D
Drag
i
Index
of
panel
J
Index
of
influencing
k
Index
of
panel
L
Lift
M
Pitching
max
Maximum
N
Normal
P
Pressure
t
Wing
moment
force
thickness
control
corner
point
panel
point
panel
circle
streamwise
velocity
vector
T
Tangential
W
Wing
Xf
Refer
Y,
z
6
to
force
x,
y,
z
axes
AERODYNAMICTHEORY
Description
of Method
The configuration
surface
is divided
into a large number
of panels,
each of which contains
an aerodynamic singularity
distribution.
A constant
source distribution
is used on the
body panels,
and a vortex
distribution
having a linear
variation in the streamwise
direction
is used on the wing and tail
panels.
A typical
configuration
panel subdivision
is shown on
Figure i.
Analytical
expressions
are derived
for the perturbation
velocity
field
induced by each panel singularity
distribution.
These expressions
are used to calculate
the coefficients
of a
system of linear
equations
relating
the magnitude of the normal
velocities
at the panel control
points
to the unknown singularity strengths.
The singularity
strengths
which satisfy
the
boundary condition
of tangential
flow at the control
points
for
a given Mach number and angle of attack
are determined
by solving this
system of equations
by an iterative
procedure.
The
pressure
coefficients
at panel control
points
are then calculated in terms of the perturbation
velocity
components,
and the
forces and moments acting on the configuration
obtained
by numerical
integration.
The following
perturbation
singularities,
tion
equations,
coefficients,
non-standard
are
given
in
Derivation
paragraphs
describe
the derivation
of
the
induced
by
the
aerodynamic
solution
of
the
boundary
condiused
to calculate
the
pressure
on
the
configuration.
Two
repeatedly
in
these
derivations
velocity
components
the
formation
and
and
the
procedure
forces,
and
moments
integrals
appearing
Appendix
I.
of
the
Incompressible
Velocity
Components
Formulas
for
the
perturbation
velocity
components
u,
v,
and
w induced
by
the
aerodynamic
singularity
distributions
in
incompressible
flow
are
derived
by
superposition
of elementary
line
sources
or vortices
located
in
the
plane
of
the
panel.
The
resulting
expressions
are
subsequently
transformed
by
Gothert's
rule
to
subsonic
obtain
the
compressible
and
supersonic
flow.
Elementary
induced
by
a distance
a
_
line
point
from
source.-
source
of
the
origin
v =
velocity
The
component
velocity
unit
strength
is given
by:
1
at
a
located
formulas
point
on
for
P(x,
the
y,
x
z)
axis
(1)
+ y2 + z
8
0
-4
4_
-4
4_
-4
O_
Figure
1 - Aerodynamic
Representation
0
-_1
_-_1
XE_
_0
The
source
velocity
is
and
the
field
directed
point
P.
along
the
line
joining
the
point
The
u, v,
and
w components
of velocity
at
the
point
P
induced
by
a unit
strength
line
source
coincident
with
the
axis
and
having
a length
_ is obtained
by
resolving
V into
x, y,
and
z components
and
integrating
with
respect
to
E.
geometry
is illustrated
on the
following
sketch:
x
its
The
_y
£
_V
X
sin
Z
w__
v
Y
u
=0J£V_
cos
1
_
[i
dE
(x-
= _0J_x-
_)
d_
(2)
_;2+ y2_ z2],_2
v=/ v
s,ncos
0
0
=
(x
-
_)2
+
y2
_
sin
8
dE
_)2
d_
+ y2
+
z2]
+
z21%_
4_ r 2
3/2
2
I"
W
0J_
V
sin
£
=
z
/
41T
[(X-
_
z
4nr 2
x__
d2
(4)
dz
0
9
where
r
= /y2
x I
=
dl
=_
z2
x
x2
=
The
+
+
tan-*
three
r2
x
r
-
x2
=
d2
= /(x
6
8
components
of
velocity
x
=
-
£
-
tan -I
satisfy
£)2
+
r2
zY
Laplace's
equation,
velocity
potential
since
_u
_v
_w
= °
and
of
u
the
line
The
a
_--_x'v
=
8__y, w
(x
The
that
respect
to
sin
_
i
4_r
There
i )
used
in
the
as
the
basis
this
report.
The
velocity
coincident
sin
-
=
[(x
_
6) 2
is
-
at
with
of
the
the
point
P
the
x axis
Biot-Savart's
more
induced
and
law
to
d_
+
y2
+
z2
normal
_)2 r +
y2
to
+
the
plane
z z] ½
and
containing
the
integrating
with
xl
1
dl
is no
axial
component
The
v and
w components
and
z components.
Thus,
u=
is
6,
V
vortex.
its
y
source
is
derived
velocity
vector
and
the
point
P.
Noting
_
£ is obtained
by
applying
the
vortex
and
integrating.
v ¼ 0/
axis
_-_z' where
Elementary
line
vortex.unit
strength
line
vortex
having
a length
each
element
of
x
=
source.
elementary
line
source
distributions
complex
by
=
0
(5)
of
are
velocity
obtained
induced
by
by
resolving
the
V
line
into
(6)
The
components
equation.
v = - V sin
e -
w
8
=
V
cos
notation
is
of velocity
The
elementary
of
the
more
complex
Care
must
be
taken
vortex
lines
form
vortex
theorem.
z
4zr
2
=-Y 4zr2
defined
can
[xz
d 2
xl]I
d
(7)
[ x2
d2
xl]
dl
(8)
following
equation
(4).
The
three
also
be
shown
to satisfy
Laplace's
line
vortex
solution
is used
as
the
basis
vortex
distributions
derived
in this
report.
during
these
derivations
to ensure
that
all
closed
rings
and
thus
satisfy
Helmholtz's
Rotation
of
coordinates.In
the
following
applications,
the
line
source
or
vortex
coordinate
system
is
in
general
rotated
with
respect
to
the
reference
coordinate
system
of
the
panel.
Using
primed
coordinates
to
refer
to
the
rotated
line
source
or
vortex,
and
defining
I = tan
A to be
the
tangent
of
the
sweep
angle
of
the
rotated
system,
the
following
coordinate
transformations
apply:
x'
=
Ix + y
(i + 12)9
(9)
y,
=
ly - x
(i + 12)½
(i0)
z
(ii)
z ' =
in
The
geometry
the
following
of
the
sketch:
rotated
coordinate
A
system
is
illustrated
x,
P(x,y,z)
X
ii
The distance
d from the field
point to the origin
is
unchanged in this
transformation,
but the perpendicular
distance
of the point from the line source or vortex
is given
by
=f(x
r'
The
coordinate
-
velocity
system
=
(i
-
+
=
w
=
(i
are
transformed
into
the
reference
v'
12)%
Iv I +
v
+z 2
components
as
follows:
lu'
u
ly) 2
+
(12)
u l
12)½
(13)
w'
(14)
Constant
source
distribution
on
unswept
panel
with
streamwise
taper.The
velocity
components
induced
at a point
P by
a
constant
source
distribution
in the
plane
of
the
panel
are
derived
by
summing
the
influences
of
a series
of
elementary
line
sources
extending
across
the
panel
parallel
to
the
leading
edge.
The
geometry
of
the
elementary
line
source
located
a distance
_ from
the
leading
edge
and
having
a strength
dE
is
illustrated
in the
following
sketch:
b
1
(_,
_/line
sou_c y
m
_
_,
b
+
m2_)
i
2
c
4
P(x,
y,
z)
x
lies
In
in
left
end
12
the
the
of
following
derivation,
it
x, y plane.
The
distance
the
line
source
is
d I =
is
assumed
that
of
the
point
P
[(y
-
m1_)
2
+
(x
the
from
-
panel
the
_)2
+
z2]½
and
the
distance
from
the
right
end
of
the
line
source
is
d 2 =
[(y
- b - m2_) 2 +
(x - _)2
+ z2]½
.
The
panel
edge
slopes
m = dy/dx
may
be
arbitrary.
The
velocity
components
are
obtained by
applying
a 90
degree
coordinate
rotation
to
the
line
source
velocity
formulas
given
by
equations
(2)
- (4),
and
integrating
across
the
panel
chord
as
follows:
C
u
=
-v'
-i
_
=
/
(X
_x
-
_)
_)2 d_+
z 2 [__i__ dl
_
y
-
b
d2 -
m2_]
(15)
0
C
v=
u'
i/[i
=
1
d2
] d_
(16)
3
c
w
=
w,
=
-z
_w
Only
the
as the
second
translation.
/
0
(x
d_
_)2
-
first
integral
integral
may
For
the
same
only
at
the
lower
limit.
respond
to
the
influence
ing
edge.
Denoting
these
ul
=
__I
4n
_
[
+ m;m12)½
(i
sinh
-I
Y
-i
l
4_(i
wl
sinh
----
=
m12)
1
-,
4-_ tan
The
corners
origin
+
z2
[_[-
dlm_-
[-
bd2
The
resulting
velocity
components
of
the
inboard
corner
of
the
panel
results
by
the
subscript
one,
l
[(y
mix) x 2 + + mly
(I
-
+
corlead-
(18)
J
x
-I
z(x 2 + y2
-x(y
- mix)
(17)
m12)z2]½
]
½
m2_]
in each
formula
need
be
evaluated,
be
obtained
by
a simple
coordinate
reason
the
integrals
are
evaluated
. sinh_
(x2 + z2)½
V
+
[(y
+
+
-
mlx)
z2)½
mlzZ
+
2
m,y
+
(l
(19)
+
m12)z2]½
(20)
velocity
components
induced
by
the
remaining
three
are
obtained
by
applying
the
above
formulas
with
the
shifted
to
the
corner
under
consideration,
and
using
the
appropriate
edge
slope.
13
The
influence
of
the
influences
of
the
to
the
corner
numbers
the
complete
four
corners,
shown
on
the
panel
where
sketch.
is
obtained
by
the
subscripts
summing
refer
u
=
uI
-
u2
-
u 3
+
u_
(21)
v
=
v I
-
v 2
-
v 3 +
v_
(22)
w
=
w I
-
w 2
-
w 3 +
w_
(23)
The
velocity
components
given
by
equations
(18)
(20)
are
expressed
in terms
of
a coordinate
system
lying
in
the
plane
of
the
panel.
One
additional
rotation
of
coordinates
about
the
y
axis
is required
to obtain
the
formulas
used
in the
computer
program.
Referring
to the
following
sketch,
the
panel
coordinate
system
now
denoted
by
primes,
is
rotated
through
an angle
6 with
respect
to
the
unprimed
reference
coordinate
system.
The
reference
system
also
has
its
origin
at
the
inboard
corner
of the
panel
leading
edge,
but
the
x axis
is parallel
to the
body
reference
axis.
Z
!
X
!
_x
Defining
a
=
tan_,
x'
=
x
(i
y'
=
y
z'
14
the
=
coordinate
+
+
az
a2)½
transformations
are
(24)
(25)
z
-
ax
(i
+
a2)½
(26)
m
m' = (I + a2)½
Similarly,
the
u'
u
-
velocity
(27)
components
become:
aw'
=
(i
+
a2)½
[ mG
4_(i
v
=
w
=
v'
+
=
w'
(I
-G(I
+
+
-
H
-
aF
(28)
]
aZ)½
+ a2)½
47
(29)
au'
a2)½
1
=
+
4_(i
where
F
=
[F
a2)½
+
a(mG
tan
-1(z
-
ax)(x
-
z
mx)
+
y2
-
z(ay
+
=
sinh(I
=
x
+
2
+
(30)
z2)½
-
mz)
1
H
H)]
:
-x(y
G
-
+
sinh-
a2
i
+
m2)½
my
+
az
i
[(y
-
mx)
(ay
-
mz)
2
+
(z
-
ax)
2]½
y
(x2 + z2)½
Constant
taper.stant
derived
series
parallel
source
distribution
on
The
velocity
components
induced
source
distribution
in the
plane
in a similar
manner
by
summing
of
elementary
line
sources
to
the
leading
edge.
In
swept
panel
with
spanwise
at
a point
P by
a conof
a swept
panel
are
the
influences
of
a
extending
this
case,
across
the
panel
the
line
sources
15
are swept back by the angle A. The geometry of an elementary
line source located
a distance
_ from the leading edge, and
having strength
dE, is illustrated
on the following
sketch:
y=b
,
(_, 0)
_y
--line
source
c
(_
x
The
of
[(x
tance
d z =
panel
is assumed
the
point
P from
- _)2
+ y2
+ z2]½
of
the
line
where
I is
source
is
the
tangent
The
velocity
of
the
line
integrating
lu v u
P(x,
d
y,
+
Ib,
b)
z)
to
lie
in
the
x,
y plane.
The
disthe
left
end
of
the
line
source
is
and
the
distance
from
the
right
end
- _x
_f-t-_
components
are
source
velocity
across
the
panel
- _ leading
obtained
formulas
chord
Ib) 2
edge
+
(y - b) 2
sweepback
by
rotating
through
the
as
follows:
+ z2]½,
angle
A.
the
coordinates
angle
A, and
v'
=
(1 + 12)½
1
=
4Tr(l
+
12)½
I
o
+
x
- _ - ly
r2
/el[
[ 1(xL
1
1
d2
4z
_) + y
dz
l
1(x
d2
d_
(31)
16
IV
V
I
+
U I
=
(i
+
12)½
C
1
4_(i
+
/
0
12)%
I
1
1
d2
l(x-
W
----W
_-
Ib)d2
-
+
_
r 2
y-
-
I (x - _) + y
ly)
b l
dl
I
(32)
dE
I
z(l
where
grals
1(X
d z
+ 12)½
/Cd__
4_
0
r2
r2
=
(x
-
In order
to
are
divided
_
-
[ l(x-
ly) 2
obtain
by
(i
+
_)+
dl
(i
+
y-
12 )
the
results
+ 12)½
prior
l(x-
_-
Ib)+
d2
Y
-
b]
(33)
z2
in
standard
to
their
form
the
evaluation.
inteAs
before,
only
those
integrals
associated
with
the
inboard
edge
of the
panel
require
evaluation,
and
then
only
at
their
lower
limit.
The
resulting
velocity
components
correspond
to the
influence
of
the
inboard
corner
of
the
panel
leading
edge.
Denoting
these
results
ul
=
4_(I
Vl
=
z
_Z
by
-i
+ 12)½
[
(i
+
the
subscript
sinh-1
x12)½
[(x
sinh-1
_
-
one,
ly) Ix
2 +
+
[(x
sinh-
-
y(i
Ix
ly)+ 2 ¥ +
i
(y2
x
+
+
12)z
(I
+
]
z2)½
J
(34)
2]½
12)z
2]½
(35)
17
w I
i[
-
z x2+
y2+
tan -I
47
-xy
+
l(y 2
+
-
tan -I
z 2)
z]
-Y
(36)
The
velocity
components
induced
by
the
remaining
three
corners
are
obtained
by
applying
the
above
formulas
with
the
origin
shifted,
and
using
the
value
of
I corresponding
to
the
leading
or
trailing
edge.
The
influence
of
the
complete
panel
is
obtained
by
summing
the
influences
of
the
four
corners
as
indicated
by
equations
(21)
(23).
Linearly
var_in@
source
distribution
on
swept
panel
with
spanwise
taper.The
velocity
components
induced
by
a source
distribution
having
a linear
variation
in the
chordwise
direction
are
derived
in the
same
manner
as described
in the
preceding
section
for
the
constant
source
distribution.
In this
case,
however,
the
expressions
under
the
integral
signs
in
equations
(31)
(33)
are
multiplied
by
_ prior
to integration.
The
velocity
components
induced
by
the
inboard
corner
of
the
panel
leading
edge
are
given
below:
ul
_
-I
4_
+
I (i
x +
-
Y
_y
12)%
sinh-1
x
+
(y2
z[tan_
I
L
v I
18
=
1
-4_
_
sinh-
-
Iz
z(x 2
-xy
I (x
I
-
I
sinh-1
ly)
+
+
y2
[ tan -I
t
+
1(y
[
x
+
ly) 2 _x
+
+(i y +
12)z 2]%
z2)½
2
z2)½
+
z2)½
+
sinh-1
12)½
]
+
(37)
z 2)
1
(i
(y2
[(x -
x
+
[x 2
z(x2 + y2 + z2)½
-xy + l(y 2 + z 2)
[(x
+
-
y2
Ix
ly) 2
+
+
+
y
(I
+
12)z
2] ½
z2]½
(38)
W,
=
iI
_
(X
-
+
z
[(I
_
i
sinh-1
ly)
+
It
an -l
12)½
sinh
x
+
(y2
z(x2
-xy
-I
z 2) ½
+
+
[(x
]
y2
+ z2)½
1(y 2 + z 2)
-
Ix
+
y
ly) 2
+
(i
-
tan-
iz]
y
+
12)z2]½
(39)
1
The
velocity
components
induced
by
the
remaining
three
corners
are
obtained
by
applying
the
above
formulas
with
the
origin
shifted,
and
using
the
appropriate
value
of
I.
The
influence
of
the
complete
panel
is
obtained
by
summing
the
influences
of
the
four
corners
as
indicated
by
equations
(21)
(23).
Constant
vortex
distribution
on
swept
panel
with
spanwise
taper.The
velocity
components
induced
at
a point
P by
a constant
vortex
distribution
in the
plane
of
a swept
panel
are
derived
by
summing
the
influences
of
elementary
line
vortices
extending
across
the
panel
parallel
to
the
leading
edge,
and
concentrated
edge
vortices
extending
back
to infinity
from
the
panel
side
edges.
The
geometry
of
an elementary
line
vortex
located
a distance
_ from
the
leading
edge,
and
having
strength
d_,
is illustrated
on
the
following
sketch:
y=b
,
0
(_,
0)
--y
bound
_
line
vortex
P
trailing
y
/
X
vortices
/
_
19
The influence
of the bound vortices
are considered
first.
The distance
of the point P from the left
end of the vortex
is
dl = [(x - _) + y2 + z2]½, and the distance
from the right
end
of the vortex
is d 2 = [(x - _ - Ib) 2 + (y - b) 2 + z_ %, where
is the tangent of the leading
edge sweepback angle as before.
The velocity
components are obtained
by rotating
the coordinates
of the line vortex
velocity
formulas through the angle A, and
integrating
from the leading
edge to infinity
as follows:
u =;_
lu'
0J
- v'
;
z
47
K
d_
(40)
oo
0
v
=
-
lu
w
=
-i
4-_
(41)
oo
(x
-
_
-
ly)K
_)
+
y
d_
(42)
0
where
K
and
=
r 2
=
i__ [l(x
r2
L
(x
-
-
l(x
-
_
d I
_
-
-
Ib)
+
[
-
b]
J
d2
ly) 2
+
(i
+
12)z
2
Only
those
integrals
corresponding
to the
inboard
the
panel
require
evaluation,
since
the
outboard
edge
obtained
by
a coordinate
translation.
In this
case,
both
upper
and
lower
limits
of
the
integrals
must
be
edge
of
can
be
however,
evaluated
to
obtain
the
correct
results.
The
resulting
velocity
components
give
the
influence
of
a semi-infinite
region
bounded
by
the
leading
edge
and
the
x axis,
with
origin
at the
inboard
leading
edge
corner
of
the
panel.
They
are
identified
by
the
subscript
one.
u I
2O
=
_1
[ tan -z
z(x2
-xy
+
+
y2
_(y2
+
+
z2)½
z 2)
tan-I
z ]
-Y
(43)
v I = - lu
wl
[
1
4_
-
I
(44)
(i
I
sinh
It should
be noted
ed by
considering
edges
of the
panel
integral
approaches
+
12)%
sinh
-I (y2
+x
-I [(x
z2)½
-
that
the
last
the
influence
simultaneously
infinity.
-
I
Ix 2 + + y (i
ly)
log
(y2
+
+
12)z
1
z2)½
=
_0
= _[x
AW I _
Therefore,
-Y[
4n x
the
y2
+
[(x - x
_) 2
z2
(45)
term
of
equation
(45)
is obtainof
both
inboard
and
outboard
as
the
upper
limit
of
the
The
edge
vortex
contributes
only
to
the
v and
of velocity.
The
velocity
components
are
obtained
grating
equations
(7)
and
(8)
for
a line
vortex
of
length
with
respect
to
_, as
follows:
Avl
2] ½
_
+
y2
+
z
w components
by
inteinfinite
2]½]
J
+
(x2
+
y2
+ y2
z 2 + z2)½]
(46)
+
(x2
y2
(47)
velocity
+
+
y2
z2
+
z2)½]
components
induced
by
the
inboard
lead-
ing
edge
corner
of
the
panel
are
given
by
equation
(43),
the
sum
of
equations
(44)
and
(46),
and
the
sum
of equations
(45)
and
(47).
The
velocity
components
induced
by
the
remaining
three
corners
are
obtained
by
applying
these
equations
with
the
origin
fluence
fluences
(23).
shifted,
and
using
the
appropriate
of
the
complete
panel
is obtained
of
the
four
corners
as
indicated
value
of
I.
The
inby
summing
the
inby
equations
(21)
-
21
Linearly
varying
vortex
distribution
on
swept
panel
with
spanwise
taper.A vortex
distribution
is considered
which
has
a linear
variation
in the
chordwise
direction,
and
lies
within
the
triangular
region
bounded
by
the
panel
leading
and
trailing
edges
extended
to intersection,
and
the
panel
inboard
edge.
The
velocity
components
induced
at
a point
P by
this
vortex
distribution
are
derived
in three
steps.
In the
first
step,
the
velocities
induced
by
a horseshoe
vortex
of
strength
_ d_ having
its
bound
segment
located
along
a radial
line
from
the
intersection
of
the
leading
and
trailing
edges
are
evaluated
and
integrated
across
segments
sketch.
the
panel
of
the
chord.
horseshoe
The
geometry
vortex
are
of
the
bound
shown
on
the
y=b
,
0
and
trailing
following
_Y
(_, O)
bound
vortex
c
"
d1_
/2
//_
X
The
origin.
bound
The
__//_
/_
vortices
¸
vortex
point
is
P
is
located
located
a
a
distance
distance
+ z 2] ½ from
the
inboard
end
of
the2vortex,
d2
= [(x
- _ - Ib) 2 +
(y - b) 2 + z ]½
from
In thfs
derivation,
the
slope
of
the
vortex
of
_,
I = 11 +
are
the
slopes
respectively.
angle
ponents
22
A,
trailing
_
d I
and
the
is
from
=
the
[(x
-
panel
_)2
a distance
outboard
a linear
+
y2
end.
function
a_/c,
where
a = 12
- 17,
b = c/a,
and
11 and
_2
of
the
leading
and
trazling
edges
of
the
panel,
The
line
vortex
formulas
are
rotated
through
the
as before,
of
the
bound
to
obtain
expressions
for
the
vortex
prior
to
integration.
velocity
com-
The velocity
components
are given
below
in integral
form:
c
z
u =
v
=-
(48)
K_
7r
/
lu
(49)
c
w
where
I
=
K
=
4_ci
=
11
+
l(x
/
o
(x
-
_ r2-
l(x
-
I[)K_
These
substitution
-
_)
+
y
_
-
=
(x
-
_
+
y
-
b
d2
-
integrals
in terms
ly) 2
are
of
-
+
-
(i
ay) 2
ay)
+
After
a lengthy
integration
ponents
induced
by
the
inboard
In
the
following
formulas,
the
leading
edge
c
4rip 2
z[
7
12)z
-
a2z
2
making
use
variable
allz2].
of
the
X-
- x
following
(51)
2
procedure,
edge
of
the
parameter
I
the
velocity
companel
are
obtained.
is redefined
as
the
slope.
z
U
+
evaluated
by
the
integration
11y)(c
(c
+
lb)
m
c[(x
= L
panel
(50)
a_/c
d I
r2
d_
--+
lad
C
(Cl
-
(
t
ax)as
-
(c
-
ay)e
2
]
G 1
c
i[
p2
(ci
-
ax)az
2
+
(c
-
ay) s
]
(52)
F
0
23
V =
-
(CA
W
=
-
(C
t
_.
-
ax)(c
- ay)t
- ay)u/p
+
(cA
- ax)azu/p
_)
where
2 - azt
-iI [ cx+
4,rrp 2
d
+ [_
(cA
2
- ay)(cA
- ay)s
+ 7
- ay)e 2 -
(C
+
(cA
- a_{,)]
J
p2
- ax)s
_ e__
p_2 [ (c
(54)
- ax) (c
2c
[(c
(53)
+
ae2z2]
- ax)az
(cX
2]
- y
+ ar--_2
G22c ]
G x
iic
- ax)as
(55)
F,
o
and
d
=
[(x
r 2
=
y2
p2
=
e2 =
(c
- _)2
+
(c
=
tan
r2]½
z 2
- ay) 2 +
(cA
=
+
-
a2z 2
ax) 2 +
- ay)(x
-
G x =
I
1
_ sinh-1
e
Ay)
2
(56)
(I - a_/c)r
(cA
c[[x-
+ alz
zd
-I
F
p2
Ay
+
2 - y(x
ax)(x
-
_)
(C - ay)_/C]
-
_)
+
y(c
2 +
-
z2[l
ay)
+
- az 2
(A
- a_/c2]]½
(57)
24
G
=
--1
sinh
X
--
(58)
r
It should
from
those
be
noted
defined
The
functions
axial
that
the
following
distribution
of
can
be
determined
velocity
u
for
z
functions
equation
FI,
GI
(30).
and
vorticity
corresponding
by
examining
the
=
0.
From
- c(x
equation
G2
differ
to these
behaviour
of
the
(51),
- ly)
U
4 (C
Along
Along
the
the
panel
trailing
x
vorticity
inversely
edge,
=
c
+
x
=
12y,
ly
and
therefore
/
z
y2
+
0.
ay)
distribution
as the
local
[
_ d_
4_C
=
(59)
-
is
seen
to vary
chord
spanwise.
linearly
The
contribution
of
the
trailing
vortex
originating
the
inboard
edge
of
the
panel
is
considered
next.
This
contributes
only
v and
w components
of
velocity,
which
tained
by
multiplying
equations
(7) and
(8)
for
a line
of
infinite
length
by
_, and
integrating.
The
results
follows:
_v =
u
therefore
-c
=
4(c
the
and
ay) 2
leading
edge
u
Thus,
wise,
-
1 +
Z2
[
[(X
-
-2
_)
+
y
+
z2]½
-z/x t
chord-
along
vortex
are
obvortex
are
as
]
1c
= 8-Jc" _-_ x - _ + [cx- _)_ + r_]_+ G_o
z[
4zr2
x
-
c
+
[(x
x
-
_
-
c)2
+
r2]½
]
(60)
Similarly,
Aw=
-y
8_C
+
I
x-
r2
4nrY 2
+
[(x
-
612
+
rZ] ½
I 1c
+
Gz
0
Ix
-
c
+
[(x
-
c) 2
+
r 23½]
(61)
25
The first
term in the braces gives the velocities
induced
by a pair of line vortices
of quadratic
strength
along the x
axis,
and the last term gives the velocities
induced by a
linearly
varying
vortex
from the panel trailing
edge.
The combination
gives the contribution
of a line vortex
of quadratic
strength
to the trailing
edge, followed
by a constant
vortex
of strength
c/2 extending
downstream in the wake.
A constant
vortex
of equal but opposite
strength
trails
downstream from the
outboard tip of the triangular
panel.
In the second step, the velocities
induced by a vortex
distribution
having a linear
variation
in both chordwise and
spanwise directions
is derived
and subtracted
from those given
above to obtain
the velocity
components corresponding
to a vortex
distribution
having a linear
variation
chordwise,
but remaining
constant
spanwise.
In this
step, the bound vortex
located
along
the radial
line from the intersection
of the panel leading
and
trailing
edges is given a linear
variation
in the spanwise direction prior
to performing
the chordwise
integration.
The linearly
varying
bound vortex
is made up by superimposing
a series of
horseshoe vortices
of strength
_d_dq with inboard edge located
at q, and outboard edge located
at b.
The geometry is illustrated
below:
y=q
0
y=b
_y
+
c
,
(
P
X
26
I
+
lb,
b)
The contribution
of the bound segment of this elementary
horseshoe vortex
is obtained
from the line vortex
formulas,
with
the origin
shifted
to the point
(_ + IH,_),
and the coordinates
rotated
through
the
angle
d I =
[(x - _ - I_) _ +
the
bound
vortex,
and
from
the
outboard
end.
in
integral
form:
(y
d2
A.
The
c
u
=
v
=
w
-
4_c
-
P
is
r2
0
-i
distance
dn
(62)
0
(63)
c
[
_ (x-
J
0
11
+
a_/c
a
=
I
-
1
K
=
2
1(x
_
-
ly)
d_
-
1
_
-
In)
+
y
1(x
-
-
_
d1
(x
(64)
dD
r2
0
=
=
a
inboard
end
of
- b)2
+ z2]½
are
given
below
lu
I
r2
located
from
the
Ib) 2 +
(y
components
b
K
4zc
where
point
_ _)2
+ z2]½
= [ (x - _ The
velocity
-
_
-
-
Ib)
+
y
-
b
d2
ly)2
+
(i
+
12)z
2
Only
the
first
term
in
the
K integral
requires
the
second
term
cancels
in
the
superposition
grating
this
with
respect
to n,
evaluation,
as
process.
Inte-
b
I
=
1(x
d
:
[(x
d
and
d 2
_)
+
0
=
where
-
is
-
y
dl -
(i
dn
+
12)_
(65)
d 2
the
_)2
same
+
y2
as
+
z2]½
previously
defined.
27
The integrals
(60)
-
z
u
(d
=
=
-
d 2)_
d_
(66)
c
4_c
v
(62) may now be written
o
r2
-lU
(67)
c
W
_-
These
by
equation
board
edge
the
panel
-i
4_c
/
o
(d
-
d2)(x
r 2
_
-
I[)_
d_
integrals
are
evaluated,
using
(51).
The
velocity
components
of
the
panel
are
given
below,
leading
edge
slope.
-c
z[x-
U
4_p
2
z[
p2
(cl
2(CI
-
ax)cs
-
-
ax)(y(c
e2(y(c
p2
-
(68)
the
substitution
given
induced
by
the
inwhere
I is redefined
as
-
ay)
ay)
-
-
az2)
az 2)
]
G2
-
zd
G z
c
i[
+
V
(cl
-
ax)cz
+
s(y(c-
ay)-
az2)]
(69)
FII
0
where
28
v
=
-
(cl
w
=
-
(c
c
t-
-
-
ax)
(c
ay)t
4_p
2
c
+
-i
[(Cl
+
-
ay)u/p
(cl
-
2
-
azt
ax)azu/p
2
ax)(y(c
-
(71)
az'
p2
-
(70)
ay)
-
az2)s
+
ce2z
2
2]
+
B
2c
G2
+
+
e2[
V
(y(c
-p2
(cl
-
-
ay)
-
ax)cs
az2)s
+
c(cl
e2
-
(y(c
-
]
-
ay)
ax) z 2
-
az 2)
GI
Iic
(72)
F
i
0
and
the
ing
equation
remaining
The
functions
distribution
velocity
equation
-cy
The
is
(x
-
axial
the
u
are
defined
follow-
vorticity
given
by
corresponding
the
value
of
to
u
for
z
these
new
=
From
0.
ly)
If
ay) 2
velocity
trailing
is
edge,
=
the
c
+
leading
edge,
and
12y,
the
ay)
new
axial
velocity
function
ay
=
4(c
-
the
=
ay)
4(c
combined
-
=
sF
2
I
by
a/c
+
z
[
e2G
--
4
functions
1
-
ay)
on
the
panel,
which
is
along
the
trailing
edge,
direction.
The
velocity
distribution
are
given
41Tp
multiplied
1
+
Thus,
is
from
the
original,
the
value
of u along
the
trailbe
constant.
This
can
be
seen
by
multiplying
by
a/c
and
subtracting
from
equation
(59).
The
-c
u
along
x
(73)
-
and
subtracted
ing
edge
will
equation
(73)
result
is:
U
zero
where
-cy
=
4(c
bution
stant
wise
vortex
variables
_---
4(c
along
of
functions
(68)
U
and
(55).
give
the
zero
along
and
varies
components
below:
I
-
(cA
-
desired
the
leading
linearly
corresponding
ax) G
2
II
c
vortex
distri-
edge,
in
the
to
conchordthis
(74)
0
v
=
-
(cl
w
=
-
(c
-
-
ax)(c
ay)t
-
+
ay)u/p
(cl
-
2-
(75)
azt
ax)azu/p
2
(76)
29
where :
cls[
t
4_P 2
V
( Y
+
e2Gl
arc 2
)G 2
-
(Cl
+
cA
-
F
ax)G2]
c
ax
d
1
(77)
Ic
0
and
the
remaining
ing
equation
ity
functions
simpler
than
The
functions
(55).
given
either
and
variables
are
defined
It
by
of
should
be
noted
that
the
final
velocequations
(74)
(77)
are
considerably
the
preceding
sets.
of
the
derivation
velocity
component
formulas
vortex
distribution
is completed
by
adding
the
the
wake.
Returning
to
the
sketch
on page
26,
that
the
elementary
horseshoe
vortices
generate
vortex
sheet
of
constant
strength.
This
vortex
utes
nent
(7)
only
to the
v and
w
of velocity
will
be
for
a line
vortex
of
components
derived
infinite
c
AV
_
d,
and
I
The
=
[(x
=
_
inner
AV
-
this
of
velocity.
The
first
by
integrating
length,
as
follows:
v compoequation
(Y
- _ _)2
(Y
-
[1
+
z2
+
z2] ½
+
x
-
_
1
-
)'q,]
(78)
0
-
+
f
d_
0
where
for
contribution
of
it
can
be
seen
a trailing
sheet
contrib-
b
f
--a---{z
4nc2
=
follow-
_
-
ln) 2
+
n) 2
a_/c
integral
is
tanc[
[
-a
4_c 2
evaluated
first,
giving
[(x -- _)+
2 + r2]%
-y(x
_r 2
i
_tan-_
z]
_
d_
0
-tan-1
=
-a
8-_ I [tan-Z
zt
z -y(x
(x-
(x
c
where
r
3O
=
u
(y2
and
+
t
z2)½,
-
ly)(c
(c
are
given
and
_
-
by
is
- _)2
_)
-
+
+
ay)
ay) 2
+
+
alz 2
a2z 2
equations
redefined
z ]
J
rIr2 2 ½
(74)
as
the
ul
c
(79)
0
and
(77)
leading
respectively,
edge
slope
_,.
The w component of velocity
by integrating
equation
(8) for
Here,
c
b
4zc2
where
d I
is
The
_ d_
0
0
defined
inner
and
is
I
=
11
evaluated
+
manner,
length.
1 + x - _d, - In ]
(y _ D)2 + z 2
above,
integral
is derived
in a similar
a line vortex
of infinite
(80)
a_/c.
first,
giving
c
Aw
=
sinh
4_c 2
-
(i
o
sinh
+
the
last
Aw=
[(x
½
-
_
y
+
l(x
-
ly) 2
+
_)
(i
+
12 )z 2] ½
x-
-I
+
r
Only
Thus
12)
-I
two
log
r]
integrals
a
4Trc 2
[I1-I
_
can
+I]
2
(81)
d_
be
evaluated
in
closed
form.
(82)
3
where
II
c
:
(i
+
I_
12)½
sinh-
i
[(x-
_-
Y
+ l(xlyi _ +
_)
(i +
(83)
d_
12)z21½
0
I2
'I
=
4
(3x
+
_)
[ d
-
(x
-
_)G2]
+
d2G2
1c
(84)
o
c2
I
-
log
a
(85)
r
2
where
I
equation
computer
It
=
11 +
(55).
a_/c,
and
Equation
d,
r,
and
G 2 are
(83)
is
integrated
defined
following
numerically
in
the
program.
should
been
multiplied
rectly
account
be
by
for
noted
-a/c
the
that
Av
prior
to
contribution
and
Aw
as
integration
of
the
derived
in
wake.
above
order
have
to
cor-
31
The velocity
components induced by a vortex
distribution
which has a linear
variation
in the chordwise direction,
and
remains constant
in the spanwise direction
have now been derived
for a triangular
region bounded by the panel leading
and trailing edges, and the inboard side edge.
In the third
step of
this
analysis,
these velocity
component formulas are combined
to give the influence
of a swept, tapered panel of arbitrary
span.
This is accomplished
by superimposing
two of these triangular regions having common outboard intersections
and equal
values of the leading
and trailing
edge slopes.
The superposition
process is illustrated
by the following
sketch:
y=b
y=c/a
y
a
C 2
C
32
=
12
-
11
The upper triangular
panel has a concentrated
vortex
of
strength
c I trailing
from the inboard edge, and a vortex
sheet
of strength
a behind the trailing
edge.
There is no concentrated
vortex
shed from the outboard tip,
since the circulation
around the trailing
vortex
sheet is equal and opposite
to that
of the concentrated
edge vortex.
A similar
vortex
pattern
is
shed by the second triangular
panel,
except that the concentrated
vortex
has a strength
c 2.
The
influence
of
a swept,
can
be
obtained
by
superimposing
indicated.
It
should
be
noted
trailing
if the
vortex
from
panel
sheet
tapered
panel
of
finite
span
b
the
two
triangular
panels
as
that
the
concentrated
vortices
the
edges
of
this
panel
is
tapered,
the
difference
in
the
wake.
The
vortex
are
of
unequal
being
made
distribution
up
on
strength
by
the
the
panel
is
zero
along
the
leading
edge,
and
varies
linearly
in the
chordwise
direction
to
a constant
value
along
the
trailing
edge.
The
axial
component
of velocity
u is given
by
equation
(74),
the
v
component
of
velocity
is
given
by
the
sum
of
equations
(60),
(75)
and
(81),
and
the
w component
of
velocity
is given
by
the
sum
of
equations
(61),
(76),
and
(82).
care
If the
influence
must
be
taken
in
of
a triangular
the
evaluation
panel
is
required,
of
equations
(74)
special
and
(77).
In this
case,
the
chord
of
the
outboard
panel
subtracted
in
the
superposition
process
is
zero,
and
two
terms
in
the
equations
become
indeterminate.
The
limiting
values
of
these
terms
are
given
below.
First,
the
function
[GI]
c =
0
_
2
are
the
GI
becomes:
2
lim
c÷0
where
and
1
2
log
(x
(x
-
slopes
fly) 2
X2y)2
of
+
+
the
(i
(i
panel
+
+
2
l_
)z
X2)2z2
leading
(86)
and
trailing
edges.
Second,
the
last
two
terms
in
the
expression
for
t
become:
C
lim
c+0
The
[i
c
remaining
(r2Ga
+
xd)]
=
(x 2
+
r2)½
(87)
0
terms
in
the
equations
are
unchanged.
33
Derivation
of the
Compressible
Velocity
Components
The compressible
velocity
components induced by the source
and vortex
distributions
are obtained by applying
Gothert's
rule to the incompressible
velocity
components derived
in the
previous
section.
The original
derivation
of Gothert's
rule
presented
in reference
4 considered
only compressible
subsonic
flows;
here the rule is extended to include
supersonic
flows
as well.
The extended rule states that the velocity
components
u, v, and w at a point P(x, y, z) in a compressible
flow
are
equal
to
the
real
parts
of
u i,
8vi
and
8wi,
where
ui,
vi
and
wi
are
the
incompressible
velocity
components
evaluated
at
a point
P(x,
BY,
8z),
and
8 =
(i - M2)½.
In subsonic
flow,
this
rule
agrees
exactly
with
that
given
by
Gothert
if each
of the
compressible
velocity
components
are
divided
by
the
constant
B2.
In supersonic
flow,
the
compressible
velocity
components
become
complex
functions,
and
care
must
be
taken
to
extract
the
real
parts
of
these
functions
in order
to obtain
the
correct
results.
However,
this
procedure
evaluating
the
velocity
a straightforward
method
fields
corresponding
to
tion.
A
forming
simple
example
the
velocity
along
the
equations
that
d I =
terms
are
is generally
much
simpler
than
formally
components
by
integration,
and
provides
for
obtaining
the
supersonic
velocity
any
existing
incompressible
flow
solu-
of the
components
extended
induced
x axis.
The
velocity
(2)
(4) are
unchanged
(x 2 + B2r2)½
and
d2 =
real
in
subsonic
flow;
rule
by
is obtained
by
transa line
source
located
component
formulas
given
by
this
transformation,
((x - _)2
+ B2r2)½.
Both
but
in
supersonic
flow,
by
except
these
both
are
imaginary
ahead
of
the
Mach
cone
from
the
origin,
d I is
real
but
d 2 is imaginary
between
the
Mach
cone
from
the
origin
and
the
rear
Mach
cone,
and
both
are
real
behind
the
rear
Mach
cone.
Thus
the
velocity
components
are
zero
ahead
of
the
Mach
cone
from
the
origin,
and
the
finite
length
of
the
source
has
no
influence
on
the
velocity
field
except
within
the
rear
Mach
cone.
Considerable
advantage
is
taken
of this
ability
to
correctly
define
ity
component
basic
sented
the
regions
formulas
in
of
influence
the
following
of each
term
applications.
The
compressible
velocity
components
singularity
distributions
used
in
in
the
following
sections.
Constant
source
distribution
on
for
each
this
method
unswept
panel
in
veloc-
of the
five
are
pre-
with
wise
taper.The
incompressible
velocity
components
for
source
distribution
are
given
by
equations
(28)
(30).
corresponding
compressible
velocity
components
are:
34
the
streamthis
The
u --
-k
+ B2a2-_
4_(i
/aF_
(B2mG
_
(88)
v = _a
2)%
4_
----- G
(89)
W=
k
4_(i
+
S2aZ)_
IF
+
a(B2mG
_
H)]
(90)
where
F
=
tan-l
-x(y
_
(z rex)
a×)
d
o2_.
S z (ay
-
mz)
(91)
G
-
1
e
sinh-1
x___'
Br'
(92)
H
=
_
Sinh-1
By
(×2 + B2 z2)%
(93)
and
B2
=
1
-
M2
k
=
f
d
=
e
-- [i
d'
=
r'
i for
2 for
M _ 1
M > I
(x2 + S2r2)½
+ 82(a 2 + m2)]%
de
= [(y
_ rex)2
+
(z - a×)2
+
x'
=
sonic
Velocity
The
COnstant
x
+
82 (my
k
Components.
gives
the
+
B2 (ay
- mz)2]½
az)
COrrect
Scaling
factor
for
the
SUper-
35
In supersonic
flow,
the real parts of the functions
F, G,
and H must be determined.
The function
F is zero everywhere
ahead of the Mach cone from the origin,
except for panels
having supersonic
side edges, when F = ± _ within
the "twodimensional"
region bounded by the Mach waves from the side
edge and the Mach cone from the origin.
The boundaries
of the
two-dimensional
region are given in the following
sketch,
which
shows the traces of the Mach waves and Mach cone from the origin
on a plane perpendicular
to the x axis.
_
zT
1
=
-
e 2
ae)x+
my
X
az
t
z
=
+
a2
Imz
+
-
ayle'
m 2
ax
_
y
\
\
\
(m
+
1
the
tion
The
function
G takes
relative
sweepback
on
in
logarithmic
form,
x'
G
or
G
=
=
_
e
+
log
several
the
side
different
edges.
ae')x
-
e2
forms
depending
Expressing
the
d'
for
e2
>
0
for
e2
=
0
for
e2
<
0
and
-
8'r'
--d
X I
1
or
36
G
=
--COS
e'
--I
X'
B'r'
8'r'
<
x'
<
B'r'
on
func-
or
G
=
+
--_
e'
for
e 2
and
x
<
>
0,
my
x'
+
_< -B'r'
az
+
a2
or
G
where
B'
=
e'
=
Finally,
=
in
H
=
0
lay
+
[-i
-
1)%
-
B 2 (a 2
supersonic
8'
+
flow,
tan -Id
mzle'
m 2
elsewhere
(M 2
-
(94)
m 2)] ½
the
for
function
x
>
H
becomes:
8r
B'y
or
H
=
0
elsewhere
Constant
source
distribution
_.The
incompressible
velocity
distribution
are
given
by equations
ing
compressible
velocity
components
(95)
on
swept
panel
with
spanwise
components
for
this
source
(34)
(36).
The
correspondare:
-kG I
u
-
v
=
-4_
FI
IG I
k[
w-
where
(96)
4n
4_
=
F 1
-
-
tan_1
GI
F
2
z
-xy
F 2
(97)
=
tan_
I
=
1
-- sinh-1
e
+
]
(98)
d
(99)
Ir 2
z
-Y
(100)
Ix
+
82y
ly) 2
+
(82
(i01)
8[(x-
+
12)Z2]½
37
G2 = sinh_ I 8rx
and
(102)
1
for
M
_< 1
2
for
M
>
k
d
=
(x 2
+
82r2)
e 2
=
(32
+
1. 2
r 2
=
y2
+
Z2
1
½
In
supersonic
flow,
the
real
parts
of the
functions
Fz,
F 2 , G z and
G 2 must
be
determined.
The
function
F 2 is always
real,
and
can
be
dropped
from
equation
(98)
without
affecting
the
results
since
the
contributions
from
the
four
corners
of
the
panel
ahead
of
always
the
Mach
cancel.
cone
The
from
the
function
origin,
F z is
except
zero
everywhere
for
panels
having
supersonic
leading
edges,
when
F I = ± w within
the
"twodimensional"
region
bounded
by
the
Mach
waves
from
the
leading
edge
and
the
Mach
cone
from
the
origin.
The
boundaries
of
the
two-dimensional
region
for
this
case
are
shown
on
the
following
sketch:
f
f
y = ix/8 ,2
/
x =
/
x
Xy
/
/
\
J
38
/
/
--,
_-y + e'lzl
The
function
on
the
relative
the
function
in
1
e
GI
or
GI
or
G I =
or
G
=
=
log
+
where
8'
=
(M 2
e'
=
(-82
d'
=
or
function
different
panel
leading
for
e2
>
0
for
e2
=
0
for
e2
<
0
and
-8'r'
for
e2
and
x
forms
depending
edge.
Expressing
x w
8'r'
-e'
G
The
x'
+ d'
8'r'
1
--cos
e'
or
1
several
of
the
form,
d
_
i
=
G I takes
sweepback
logarithmic
0
<
<
>
0,
ly
x'
<
8'r'
x'
<
-8'r'
+
e'Iz
I
elsewhere
-
i)½
-
12) ½
(103)
x'
=
Ix
r'
=
[ (x
+
82y
-
ly) 2
+
e2z2]½
ed
G2
G2
=
log
G2
=
0
becomes:
x + d
-_
8'r
for
x
>
8r
elsewhere
(104)
Linearly
varyin_
source
distribution
on
swept
panel
spanwise
taper.The
incompressible
velocity
components
source
distribution
are
given
by
equations
(37)
(39).
corresponding
compressible
velocity
components
are:
u-
v
-k
4_
-
[ (x-
k[
4_
(x
ly)G
-
I
ly)(IG,
+
yG 2
-
-
G 2)
z(F I
+
x
-F
+
d
with
for
this
The
2 )]
-
Iz(F
(105)
1
-
F
2
]
)
(106)
39
W
where
=
the
_
in
the
-
functions
by
equations
of
functions
and
the
real
(104).
The
by
d
from
(x
ly)
"te o-
(F 1
FI,
F2,
GI,
G2,
d,
e,
k,
(107)
and
r
are
defined
(99)
(102).
In
supersonic
flow,
the
behaviour
F I and
F 2 is described
following
equation
(102)
parts
of
G I and
G 2 are
given
by
equations
(103)and
sum
(x + d)
appearing
in equation
(106)
is
replaced
supersonic
origin.
Constant
flow,
vortex
and
distribution
_.The
incompressible
vortex
distribution
are
corresponding
compressible
U
=
k
4-"_
v
=
-lu
w
=-
IF
is
I
_
real
on
only
swept
within
the
panel
Mach
with
cone
spanwise
velocity
components
for
the
given
by
equations
(43)
- (45).
velocity
components
are:
bound
The
Fa ]
(108)
(109)
GI
4_
-
IG2
]
(ii0)
where
the
functions
F I, F 2, G,,
G z, d,
e,
k,
and
r are
defined
by
equations
(99)
(102).
In supersonic
flow,
the
behaviour
of
the
functions
F I and
F 2 is described
following
equation
(102),
and
the
real
parts
of
G I and
G 2 are
given
by
equations
(103)
and
(104).
The
is
given
contribution
of
Au
=
0
edge
vortices
in
compressible
flow
Av
=
kz
4_
Aw
=
-
(iii)
x
ky
4_
In supersonic
flow,
equations
is replaced
cone
from
the
origin.
4O
the
by
+ d
r2
(112)
x
(113)
+ d
r 2
the
sum
by
d,
(x + d)
appearing
and
is
real
only
in
the
above
within
the
Mach
Linearly
varyin@
vortex
distribution
on
swept
panel
with
spanwise
taper.The
incompressible
velocity
components
for
the
bound
vortex
distribution
are
given
by
equations
(74)
(77).
The
corresponding
compressible
velocity
components
are:
-k
p2
I sF
+
1
u
-
V
=
-
(Cl
W
=
-
(C
t
-
ck
_ s
4_P z ( V
z [e2G
-
1
(Cl
-
ax)G2]
0
-
-
ax)(c
ay)t
-
+
ay)u/p
(cl
-
2
-
ax)azu/p
azt
(115)
2
(ii6)
i
where
(114)
Ic
2
ze
[e2G
-
i
(cl
-
ax)G
2]
p
F
2
1
C
ar2)G
2
C
where
k
+
1
for
M
(
1
2
for
M
>
1
-
1
+
82r2]½
+
a2z
(cl
-
ax)
d
(117)
____
a
=
I
d
=
[(x-
r 2
=
p2
=
(c
e2
=
(cl
s
=
(c
y2
2
_)2
+
Z 2
-
ay) 2
-
-
ax) 2
ay)(x
2
+
82p 2
-
ly)
+
alz 2
41
and
zd
-I
F
=
-y(x-
_)
+
(I-
(cl
GI
(118)
tan
1
_
1
-
a_/c)r
ax)(x
2
-
_)
+
82[y(c
-
ay)
-
az 2]
sinh-1
8c[[x
e
-
ly
+
(C
-
ay)_/c]
2
+
z2182
+
(I
+
a_/c)
2]]½
(119)
G
=
--I
sinh
X
--
(120)
8r
In
described
and
in
G 2
supersonic
following
are
flow,
the
equation
given
x'
=
(cl
r'
=
c[[x
d'
=
e
e'
:
[-
-
by
equations
ax)(x
-
ly
behaviour
of
function
(102),
and
the
real
-
+
_)
(c
(103)
+
-
82[y(c
ay)_/c]
2
and
(104),
-
ay)
-
+
z2182
F I
parts
is
of
G I
where
az 2]
+
(I
+
a_/c)
2]] ½
d
(cl
Finally,
compressible
Au
the
=
ax)
2
_
82p2]½
contribution
flow
is given
of
by
the
vortex
sheet
in
the
wake
(121)
0
I
Av
-a
_
=
I k(F
I
-
F 2)
zt
c
C
(x
-
ly)
(c
-
ay)
+
alz 2
)
(122)
+
where
u
and
Aw
where
Ii,
the
Gothert
42
t
=
12
(c
are
2k-_a
4_c
-
given
[II
and
13 are
transformation
-
ay) 2
+
a2z 2
by
equations
I2
+
u
(114)
and
(117)
respectively.
I3 ]
given
by
equations
applied.
(123)
(83)
-
(85),
with
Aerodynamic
Representation
The source and vortex distributions
derived
in the preceding sections
provide the basis for the aerodynamic
representation
of the configuration.
The strengths
of these singularities
are determined by satisfying
the boundary condition
of
tangential
flow at panel control
points
for given Mach number
and angle of attack.
In general,
the control
points
are
located
at the panel centroids,
except where noted below. The
body is represented
by constant
source distributions
on surface
panels,
but two optional
methods are available
to represent
the wing and tail
surfaces.(Here,
tail
surface
implies
any
horizontal
or vertical
tail
or canard surface.)
Planar boundary condition
option.In this option,
the
panels are located
in the mean plane of the wing or tail
sur,
faces.
Linearly
varying
source distributions
are used to
simulate
the airfoil
thickness,
and linearly
varying
vortex
distributions
are used to simulate
the effects
of camber, twist,
and incidence.
The slope of the airfoil
thickness
distribution
is approximated by linear
segments between the panel leading
and trailing
edges.
This linear
distribution
is constructed
by superimposing
a series of triangular
source distributions
extending
over two
adjacent
panels.
The strength
of the triangular
source distribution
is determined
by the slope of the thickness
distribution
at the intermediate
panel edge, as illustrated
below:
X
/
dz t
dx
x
Chordwise
slope
thickness
distributions
and
x
x
i-i
X
i
Triangular
distribution
i+l
source
43
The same method is used to approximate
the chordwise vortex
distribution
on the wing.
In this
case, the strengths
of the
vortex
distributions
are determined
by satisfying
the boundary
condition
that the resultant
normal velocity
is zero at panel
control
points.
A typical
chordwise vortex distribution
is
shown below:
x
Chordwise
vortex
distribution
In subsonic
flow,
or
if the
trailing
edge
is
swept
behind
the
Mach
line
in
supersonic
flow,
the
Kutta
condition
implies
that
the
vorticity
goes
to
zero
along
the
trailing
edge.
In
this
case,
the
control
points
are
located
at
the
panel
centroids.
If the
trailing
edge
lies
ahead
of
the
Mach
line
in
supersonic
flow,
an
additional
vortex
singularity
is added
at
the
trailing
edge,
as
indicated
by
the
dashed
line
in
the
above
sketch.
In this
case,
an additional
control
point
is
added
on
the
trailing
edge
of
the
wing,
and
the
intermediate
control
points
adjusted
and
the
trailing
on or
is swept
point
is
located
ing
edge
control
linearly
between
the
leading
edge
control
point
edge.
If
the
leading
edge
of
the
wing
lies
behind
the
Mach
line,
the
leading
edge
control
at
the
centroid
as before,
otherwise
the
leadpoint
is
located
on
the
wing
leading
edge.
In the
non-planar
boundary
condition
option,
the
panels
are
located
on
the
upper
and
lower
surfaces
of
the
wing
and
tail,
and
linear
vortex
distributions
alone
are
used
to
simulate
both
lift
and
thickness
effects.
The
upper
and
lower
surface
vortex
distributions
are
similar
to
those
described
above,
44
and
the
two
vortex
sheets
are
joined
together
at
the
leading
edge by equating the vortex
strengths
of the leading
edge panels.
The resulting
continuous
distribution
of vorticity
around the chord is illustrated
below.
T
T
In subsonic flow,
the non-planar
boundary condition
option
presents the problem that one more control
point exists
than the
number of vortex
distributions
if the Kutta condition
is enforced at the trailing
edge.
An additional
source or vortex
distribution
must be included
to make the resulting
system of
equations
determinate.
One way to resolve
this problem is to
introduce
an additional
pair of trailing
edge vortices
having
equal and opposite
strength,
as indicated
by the dashed line on
the above sketch.
Another method is to add an internal
line
source at some point in the interior
of the airfoil.
In either
method, the strength
of the additional
line source or vortex
approaches
zero and has small effect
on the final
solution.
The second method is recommended, however, since the first
tends to generate an ill-conditioned
system of equations
for
airfoils
with small trailing
edge angles.
In supersonic
flow,
a similar
problem exists
if the trailing edge lies on or is swept behind the Mach line.
If the
trailing
edge lies
ahead of the Mach line,
additional
trailing
edge vortex
singularities
must be added on the upper and lower
45
surfaces,
and the strengths
of these determined
by satisfying
the boundary conditions
at additional
control
points on the
trailing
edge.
The remaining
vortex
strengths
are determined
as described
above.
If the leading
edge lies
ahead of the Mach
line,
the vortex
distributions
on the upper and lower surfaces
of the airfoil
are determined
independently
using control
points
located
on the panel leading
and trailing
edges.
The influence
of the trailing
vortices
in the wake is included in the velocity
component formulas
derived
for the
constant
and linearly
varying
vortex
distributions.
Since the
wake is assumed to lie in the plane of the panel in these derivations,
the wake vortices
must be rotated
at the leading
edge
of each downstream panel to follow
the contour of the upper or
lower surface
of the wing to the trailing
edge.
The paths of
the trailing
edge vortices
are illustrated
on the following
sketch.
46
The Boundary
Condition
Equations
A system of linear
equations
is established
which relates
the magnitude of the velocity
normal to the surface
at each
panel control
point to the aerodynamic
singularity
strengths.
The geometrical
relationship
between each influencing
panel and
control
point is required
to evaluate
the coefficients
of this
system of equations.
Wing
and
body
panel
geometry.A typical
panel
subdivision
of
a configuration
which
includes
a wing,
body,
and
tail
is
illustrated
on
figure
1 (page
8).
A reference
coordinate
system
is established
with
origin
at or
near
the
nose
of
the
body,
having
its
x axis
on
the
center
line
and
parallel
to
the
body
axis,
and
a vertical
z axis.
Since
symmetry
about
the
xz
plane
is
assumed
throughout
this
analysis,
panels
are
located
only
on
the
positive
y
(right
hand)
side
of
the
configuration.
The
body
panel
corners
are
defined
by
the
intersections
a series
of
planes
normal
to the
x axis,
and
longitudinal
ian
lines.
A maximum
of
30 rings
of
panels
may
be
used,
containing
up
to 20
rows
of
panels
around
the
circumference.
of
merideach
The
top
to
body
panels
are
numbered
of
each
ring,
starting
in
with
The
wing
and
tail
surface
intersections
of
a series
of
x axis,
and
lines
of constant
columns
of panels
all
other
horizontal
may
sequence
from
the
the
forward
ring.
panel
vertical
percent
be
used,
or
vertical
corners
planes
chord.
including
tail
or
each
column
may
contain
up
to
30 rows
of
tail
panels
are
numbered
in sequence
from
the
trailing
edge
of each
column,
starting
column
of
the
wing.
bottom
the
are
defined
by
the
parallel
to
the
A maximum
of
20
those
canard
on
the
wing
surfaces,
and
and
panels.
The
wing
the
leading
edge
with
the
inboard
and
to
For
each
panel,
the
corner
point
coordinates,
centroid
coordinates,
inclination
angles,
area,
and
chord
length
through
the
centroid
are
calculated,
using
the
method
outlined
in
Appendix
II.
It should
be
noted
that
the
panel
inclination
angles
$ and
8 are
related
to the
direction
cosines
of
the
normal
as follows:
nx
=
-
sin
ny
=
-
cos
nz
=
cos
6
6
cos
sin
8
(124)
8
47
A primed system of coordinates
is introduced,
at corner point k of panel j, and inclined
at the
originating
angle 8.3
with respect
to the xy plane.
For body panels,
the x' axis
is parallel
to the reference
x axis,
and the y' axis lies
in
the plane of the panel through the leading
edge.
The panel is
inclined
at the angle 6j to the x'y'
plane.
For wing panels,
the x' axis lies
in the plane of the panel along the inboard
side edge, and is perpendicular
to the y axis.
The z' axis is
normal to the panel.
In this case, the x' axis is inclined
at
the angle _j to the x axis.
The geometry of the wing and body
panels,
trated
Z
and panel
corner
point
numbering
convention,
is
illus-
below:
!
Z
Z v
j'
L
v
X !
x
y!
y!
2
4
4
1
X v
2
Body
panel
Wing
panel
The
control
point
of
a panel
is defined
as
that
point
on
the
panel
where
the
boundary
conditions
are
satisfied,
and
each
panel
has
a unique
control
point
associated
with
it.
The
control
point
of panel
i is normally
located
at
the
panel
centroid.
Exceptions
to this
rule
exist
for
wing
or
tail
surfaces
using
48
planar boundary conditions,
as described
in the previous
section.
The coordinates
of the control
point are given in
terms of the primed system originating
at corner k of panel
j as follows:
For body panels,
condition
option,
and wing panels
x'
=
AX
y'
=
Ay
cos
e
z_
=
Az
cos
O.
3
Ax
=
xi
-
xk
Ay
=
Yi
-
Yk
Az
=
zi
-
zk
using
the planar
boundary
l
i
where
For
wing
panels
using
the
J
+
Az
sin
e
-
Ay
sin
O.
]
non-planar
IIAx
+
12AY
+
13Az
'
Yi
=
_
+
_
+
_
z_i
=
_IAx
+
_2AY
+
_3 Az
Az
are
Ax
2 Ay
(125)
boundary
x i' =
I
J
3
condition
option,
Az
(126)
where
and
Ax,
Ay
and
defined
above,
-n
nz
l
=
I
_1
k
(nx2
=
+
nz2)½
nx
=
0
3
(nx2
2
_2
=
3
ny
(n x
2
) ½
+
nz2)½
n z
z
2
+
=
+
-nyn
-nxny
_I
x
nz2
_2
:
(nx
nz2)½
3
(nx 2
+
nz 2 ) ½
(127)
49
The
direction
with
noted
n
n x,
and
Y
n
z
are
given
by
equation
(124)
subscript
j applied
to the angles
8 and 6.
It should
that
equations
(126)
reduce
to (125)
for 6. = 0.
3
The
opposite
(126)
cosines
coordinates
side
of the
with
by
calculate
=
- Yi
panel
of
xz
the image
plane
are
- Yk"
symmetry
The
of control
point
given
by equations
image
control
point
be
i on the
(125)
or
is
used
to
effects.
Calculation
of the normal
velocity
at the control
points.The
resultant
velocity
normal
to panel
i at the control
point
is the sum of the normal
component
of the free
stream
velocity
vector
and the normal
velocities
induced
by the panel
singularity
distributions.
In the
following
analysis,
the free
stream
velocity vector
is assumed
to have
unit magnitude,
and lie in the xy
plane
at an angle
_ to the x axis.
The component
of the velocity
vector
normal
to panel
i is
=
sin
_
cos
i
axes
8
cos
_
i
The three
at control
- cos
_
sin
i
6
(128)
i
components
of velocity
point
i are given
by
parallel
to
the
following
the reference
equations:
N
Aui
=
_
[ (11 u_.
_3
+ _ i v'.
l]
+ _ iw_.)
_3
+(Ix u_lj
+
+ _lwij
-'
j=l
(129)
-'
_xvij
'JIIYj
N
:X
+.v j÷
j=l
(130)
U2
ij
j
j
N
Aw i =
I:[<
13uij
+
_3vij
+
u3wij)
j=l
(131)
5O
where
1],
is
the
total
is
the
strength
number
of
of
the
are
the
three
components
at
control
point
i by
v_.
13,
primed
coordinate
singularities
.th
3
singularity
of
panel
system
velocity
j,
given
induced
in
the
of
panel
j
of
j
of
velocity
given
in
panel
j
induced
the
I,
are
W_.
l]
U w
,
°
13,
are
the
three
components
at
image
point
i by
panel
primed
coordinate
system
V w
13,
_'.
13
and
the
coefficients
by
equations
with
panel
of
(127),
the
where
transformation
the
direction
v,
are
given
associated
j.
The
normal
component
of velocity
the
panel
singularity
distributions
above
velocity
components
as
follows:
A_.
=
i
_
l
The
coefficients
(127),
except
panel
U,
cosines
Au.
l
+
_
2
_i,
that
Av.
l
+
_
is
at panel
i induced
given
in terms
of
(132)
Aw
3
v2,
and
_s
the
direction
by
the
i
are
also
cosines
given
by
equation
are
associated
with
i.
Combining
equations
velocity
at control
point
n.
=
1
_.
+
(128)
i is
and
(132),
the
resultant
normal
A_.
1
1
N
=
_.
l
+
_
(133)
a..y.
l]
]
j=l
where
from
the
aerodynamic
equations
(129)
For
body
panels,
condition
option,
the
axis,
and
6. = 0.
In
]
-
influence
(132).
coefficient
a
can
be
obtained
ij
and
wing
panels
using
the
planar
boundary
x'
axis
is parallel
to the
reference
x
this
case,
the
normal
and
tangential
51
velocity
at control
point
i can be written
N
w:
1
lj
cos
(8.3
-8.) 1
+
v'.
13
sin
(8 j
-8
i)
w'jl
cos
(8.3
+
+
V_.
13
sin
(8 j
+
8i)]y.j 3
[v_ lj
cos
(8.-8
3
13
sin
(8.3
-
8i )
cos
(8
w_
lj
sin
(8
+
O
j=l
+
8.)
1
(134)
N
v"l
=
_
i )-w_.
j=l
-V_.
+
13
and
axes
the
at
three
control
8
j
components
point
of
are:
i
)
+
i
velocity
j
parallel
to
)ly
i J
j
the
(135)
reference
N
Au i
=
_
(u'..13+
uqj)Yj
j=l
AV i
=
v[
cos
8i
-
w_
sin
8i
_w.
=
w['
i
cos
8
+
v['
1
sin
8.
1
1
Then,
from
_.
=
1
i
equation
w?
1
cos
_.
(132)
-
du.
1
Formation
of
condition
equations
(136)
sin
_.
1
(137)
1
the
boundary
are
obtained
condition
equations.by
setting
n.
= 0
in
The
boundary
equation
1
(133).
The
complete
set
N
of
equations
N
=
i=lj=l
in
matrix
[Aij]
52
be
written:
N
_aijYj
Alternatively,
may
- _
_i
(138)
i=l
notation,
{Yj}
=-
I_i}
(139)
where A.. is the matrix of aerodynamic
influence
13
and _.1 is given by equation
(128).
In general,
coefficients,
the matrix
is
subdivided
into four partitions
in order to simplify
the
calculation
procedures.
The first
partition,
ABB, gives the
influence
of the body panels on the body control
points,
the
second, ABW, gives the influence
of the wing panels on the body
control
points,
the third,
AWB, gives the influence
of the body
panels on the wing control
points,
and the fourth,
AWW, gives
the influence
of the wing panels on the wing control
points.
Equation
(139) is rewritten
in terms of these four partitions
below.
[w If 1
The
subscripts
present
program,
W
and
the
B refer
maximum
The
right
side
of
modified
if the
planar
In this
case,
the
slope
tan
where
_
(14o)
to wing
or body
panels.
order
of
each
partition
In
the
is
600.
the
boundary
condition
equations
is
boundary
condition
option
is
selected.
of the
wing
surface
is
given
by:
= ( I_
dzc
dx
i
+
dzt
dx
(141)
/i
dz c
is
dx
dz__it
is
the
slope
of
the
wing
camber
surface.
the
slope
of
the
wing
thickness
distribution.
dx
The
positive
sign
to
the
In addition,
sign
lower
the
applies
to
the
surface.
normal
velocity
N
A_
i
=
_a
bij
NW
=
cos
is
at
surface,
control
the
point
i
negative
is
given
by
NW
i3
.y
j
+
cos
e _
j =i
where
upper
bi
j
(dzt
dx
(142)
1
/j
j =i
_i ( wijc°s
the
number
vij sin@
Oi
-
of
wing
i
-
u ij tan
_i )
panels,
53
and
are
the
point
panel
W0
velocity
i
j
by
components
the
source
induced
at
distribution
on
control
wing
0
13
The
second
the
projection
of
the
wing.
term
in
equation
of
the
free
stream
Combining
n
i
equations
=
cos
+
_
L[sin
_
ij y j
+cos
i
_aN
(128)
cos
(142)
is
velocity
and
e
Setting
n.
1
=
0,
cos
_b NW
j =i
in
by
cos
_,
the
plane
(142)
-
i
multiplied
vector
_
tan
_
i
]
(143)
i j (dzt)
d-_-
j
j =i
the
new
N
boundary
condition
N
equations
are:
N
(144)
_aijYj
=
i=lj=l
where
_
i
=
cos
-
_
cos
i
_
mi
i=l
[cos
_
tan
_
-
i
sin
_
cos
8
i
]
NW.(dzt
1
_
bi3\a-_-lj
_
(145)
j=l
On
for
the
wing,
tan
panels
lying
6.
l
is
in
the
dzt
dx
So
by
plane
6
cos
i
equation
of
=
the
(141).
Furthermore,
wing,
NW(dzti___
bij
dx
j=l
lj
that
i
=
For
non-coplanar
used.
54
given
cos
6 i [cos
wing
_
or
dzc
dx
tail
li
-
sin
segments,
_
cos
(146)
8i]
equation
(145)
must
be
Solution
of
the
boundary
condition
equations.Several
methods
could
be
employed
to
solve
the
boundary
condition
equations
for
the
unknown
source
and
vortex
strengths.
For
example,
equation
(139)
could
be
solved
by
direct
inversion,
even
though
this
is generally
impractical
for
dense
matrices
of
orders
up
to
1200.
On
the
other
hand,
the
partitioned
matrix
of
equation
(140)
can
be
solved
using
the
method
described
in
reference
i, which
requires
the
inversion
of
only
the
diagonal
partitions,
having
a maximum
order
of 600,
together
with
matrix
multiplications
of
the
off-diagonal
partitions.
A rapidly
convergent
iteration
scheme
for
solving
large
order
systems
of equations
has
been
reported
in
reference
5.
In
this
method,
as
applied
in
this
report,
the
partitions
are
further
subdivided
into
smaller
blocks,
with
no block
exceeding
order
60.
The
matrix
elements
in each
block
are
carefully
chosen
to
represent
some
well
defined
feature
of
the
original
configuration.
For
example,
a body
block
represents
the
influence
of one
ring
of panels
around
the
body,
while
a wing
block
represents
the
influence
of
one
chordwise
column
of wing
panels.
For
wings
using
the
non-planar
boundary
condition
option,
the
block
size
corresponds
to
the
total
number
of
panels
on
the
upper
and
lower
surface
of
the
column.
The
initial
iteration
calculates
the
source
and
vortex
strengths
corresponding
to each
block
in
isolation.
For
this
step,
only
the
diagonal
blocks
are
present
in
the
aerodynamic
matrix.
Once
the
initial
approximation
to the
source
and
vortex
strengths
is determined,
the
interference
effect
of
each
block
on
all
the
others
is
calculated
by matrix
multiplication.
The
incremental
normal
velocities
obtained
are
subtracted
from
the
normal
velocities
specified
by
the
boundary
conditions.
This
process
is repeated
15 to
20
times,
or
until
the
residual
interference
velocities
are
small
enough
to
ensure
that
convergence
has
occured.
At
present
the
computer
program
repeats
the
iteration
a fixed
number
of
times,
namely
15.
The
consisting
are
To
procedure
of
nine
designated
solve
where
yj
is
illustrated
blocks.
The
, the
specified
Aij
A..
z3
yj
=
=
below
unknown
normal
for
an
aerodynamic
matrix
singularity
strengths
velocities
_
.
i
_.l
AIi
Al 2
AI 3
A21
A22
A23
A31
A32
A33
55
Put
A = D
+
E
D
=
Therefore
[D
+
or
First
All
0
0
A22
0
0
E]
A
+
A
A2z
0
{ Y}
=
{_ }
{Y}
=
D-z[ ,_
A_ I
=
E
-
E{y
Second
Similarly,
approximation
k th
{yl
I
:
approximation:
{._}k= D-_{_ - A_k-_ }
Note'that
56
D -z
=
0
32
}}
A -z
ii
0
0
0
A -z
22
0
0
0
A_3
1
1
A23
A
A31
33
approximation:
Calculate
A
12
Calculation
of Pressures,
Forces,
and Moments
Once the strengths
of the aerodynamic
singularities
have
been determined,
the three components of velocity
at a point i
can be determined
as follows:
where
_u
1
,
Av.
1
u.1
=
_u.
v.1
=
_v.1
w.
1
=
_w.
1
+
are
given
and
6w.
1
l
+
cos
_
(147)
(148)
sin
_
(149)
by
equations
(129)
-
(131)
If the
planar
boundary
condition
option
has
been
selected,
the
incremental
velocity
components
induced
by
the
wing
thickness
distribution
must
also
be
calculated
and
added
to
the
above
equations.
the
exact
where
The
pressure
coefficient
isentropic
formula
Cp i
_
YM
2-2
q2
i
=
u 2
i
=
0,
=
1
For
M
IE
+
1
+
Y-12
v 2
i
+
w2
i
M2 (i
is
-
ql)
then
1
-
calculated
1
using
1
(150)
2
CPi
The
forces
-
(151)
qi
and
moments
acting
on
the
then
be
calculated
by
numerical
integration.
tangential
force,
and
pitching
moment
about
coordinates
of
panel
i are
given
by:
N.l
=-
Ti
=
AiCPiCOS
AiCPi
sin
8.cos
l
$i
6i
configuration
can
The
normal
force,
the
origin
of
(152)
(153)
57
M. = N.x. - T.z.
l
1 i
i i
where
A
i
8
x
is
the
panel
(154)
area
l
8.
l
are the panel inclination
by equation
(124)
l
z.
l
are the
point
coordinates
angles,
of the
panel
defined
control
The total
force and moment coefficients
acting on the configuration
are obtained
by summing the panel forces
and
moments on both sides of the plane of symmetry.
N
1
CN
S
2Nl
(155)
2T. i
(156)
2M i
(157)
i=l
N
S1
CT
_
i=l
N
CM
1
S_
-
Z
i=l
Finally,
the
lift
and
C
drag
=
C
L
C
=
The
computer
program
acting
on
the
body,
the
complete
configuration.
58
be
calculated
output.
cos
_
-
C
N
D
moment
may
an
optional
coefficients
C
are:
sin
(158)
cos
(159)
T
sin
_
+
N
C
T
computes
the
forces
wing
and
tail
surfaces,
In
addition,
section
for
the
wing
and
tail
and
moment
and
the
forces
and
surfaces
as
COMPUTER
Program
PROGRAM
Description
A computer
program
has
been
developed
to
calculate
the
pressure
distribution
and
aerodynamic
characteristics
of
wingbody-tail
combinations
in subsonic
and
supersonic
flow.
The
program
is written
in CDC
FORTRAN
IV,
version
2.3
for
a SCOPE
3.0
operating
system
and
library
file.
It
is designed
for
the
CDC
and
eral
6000
series
operates
in
disc
files
of
computers,
occupies
70,000
(octal)
words,
OVERLAY
mode.
The
program
requires
five
periphin
addition
to the
input
and
output
files.
Program
Structure
The
Figure
calls
SOLVE.
CONFIG,
while
LINVEL,
overlay
structure
of
the
program
is
illustrated
on
2.
The
main
overlay
program
is designated
USSAERO,
and
the
three
primary
overlay
programs
GEOM,
VELCMP,
and
In turn,
GEOM
calls
seven
secondary
overlay
programs
NEWORD,
WNGPAN,
NEWRAD,
BODPAN,
NUTORD,
and
TALPAN,
VELCMP
calls
three
secondary
overlay
programs
BODVEL,
and
WNGVEL.
The
complete
19
subroutines.
routine
are
given
program
Detailed
in
Part
tions
give
the
purpose
the
method
used,
and
and
constants.
of
list
Operating
The
program
deck
and
sequence:
job
card,
system
program
deck,
end-of-record
file
card.
The
input
data
section.
consists
of
14 overlay
programs
and
descriptions
of
each
program
and
subII of
this
report.
These
descripthe
the
program
names
of
or
subroutine,
the
principal
outline
variables
Instructions
data
deck
are
loaded
in the
following
control
cards,
end-of-record
card,
card,
input
data
deck,
and
end-ofdeck
is
described
in
the
following
59
OVERLAY(0,0)
USSAERO
OVERLAY(3,0 )
SOLVE
OVERLAY(2,0)
VELCMP
OVERLAY( 1,0 )
GEOM
INVERT
PARTIN
DIAGIN
ITRATE
PRESS
FORMOM
TRAP
PANEL
DERIV
DERIVl
DERIV2
CUBIC2
COMCU
SCAMP4
OVERLAY(2,1)
BODVEL
SORPAN
OVERLAY(2,2 )
LINVEL
CONFIG
OVERLAY
(i, 1 )
SORVEL
VORVEL
NEWORD
OVERLAY(I,2)
WNGPAN
OVERLAY(I,3)
I
I
OVERLAY( 2,3 )
WNGVEL
VORPAN
TRANS
NEWRAD
OVERLAY(I,4)
I
BODPAN
OVERLAY
(i, 5 )
NUTORD
OVERLAY
(i, 6) 1
TALPAN
OVERLAY(I,7)
I
Figure
6O
2 - Program Overlay
Structure
Program Input
Data
The input to this program consists
of two basic parts,
namely, the numerical
description
of the configuration
geometry
as described
in reference
3, and an auxiliary
data set specifying
the singularity
paneling
scheme, program options,
Mach
number, and angle of attack.
The program input is illustrated
by the sample case presented
in Appendix III.
Description
of configuration
_eometry
configuration
is defined
to be
symmetrical
therefore
only
one
side
of
the
configuration
The
convention
used
in this
program
is
to
the
configuration
located
on
the
positive
plane.
The
number
of
input
cards
depends
ponents
used
to describe
the
configuration,
detail
used
to describe
each
component.
Card
tifying
1
1-3
Identification.-
information
Card
2
each
punched
may
be
used
lowing:
Columns
-
in
columns
Card
1
input
cards.The
about
the
xz plane,
need
be
described.
present
that
half
of
y side
of the
xz
on
the
number
of
comand
the
amount
of
contains
any
desired
iden-
1-80.
-
Control
integers.Card
2 contains
24
integers,
right
justified
in a 3-column
field.
Columns
73-80
in
any
desired
manner.
Card
2 contains
the
fol-
Variable
J0
Value
0
1
4-6
Jl
0
1
-i
7-9
J2
0
1
-i
Description
No
reference
area
Reference
area
to
No wing
Cambered
Uncambered
be
read
data
wing
data
to be
read
wing
data
to be
read
No
fuselage
data
Data
for
arbitrarily
shaped
fuselage
to be
read
Data
for
circular
fuselage
read
(With
J6=0,
fuselage
be
cambered.
With
J6=-l,
lage
will
be
symmetrical
plane.
uration
With
will
J6=l,
entire
be
symmetrical
to be
will
fusewith
xyconfigwith
xy-plane)
10-12
J3
No
pod
(nacelle)
data
Pod
(nacelle)
data
to
be
read
61
Columns
Variable
13-15
J4
16-18
J5
19-21
J6
Value
0
1
NWAF
No fin
(vertical
tail)
data
Fin (vertical
tail)
data to be
read
No
canard
(horizontal
Canard
(horizontal
be
read
A cambered
fuselage
2-20
Number
to
25-27
NWAFOR
if
circular
J2
is
tail)
data
tail)
data
to
or
arbitrary
nonzero
Complete
configuration
is
symmetrical
with
respect
to
xy-plane,
which
implies
an
uncambered
circular
fuselage
if
there
is
a fuselage
Uncambered
circular
fuselage
with
J2 nonzero
-i
22-24
Description
3-30
of
describe
Number
of
airfoil
the
sections
used
wing
ordinates
used
to
define
each
wing
airfoil
section.
If
the
value
of NWAFOR
is
input
with
a
negative
sign,
the
program
will
expect
to read
lower
surface
ordinates
also
28-30
NFUS
1-4
31-33
NRADX(i)
3-30
34-36
NFORX(1 )
2-30
Number
NRADX(2 )
3-30
40-42
NFORX(2 )
2-30
43-45
NRADX( 3)
3-30
62
fuselage
segments
Number
of
half-section
segment.
the
program
number
of
points
used
to represent
of
first
fuselage
If
fuselage
is circular,
computes
the
indicated
yand
z-ordinates
Number
stations
lage
37-39
of
of
for
first
fuse-
segment
Same
as
fuselage
NRADX(1),
segment
but
for
second
Same
as
fuselage
NFORX(1),
segment
but
for
second
Same
as
fuselage
NRADX(1),
segment
but
for
third
Columns
Variable
Value
Description
46-48
NFORX(3)
2-30
Same as NFORX(1),
fuselage
segment
but
for
third
49-51
NRADX(4)
3-30
Same as NRADX(1),
fuselage
segment
but
for
fourth
52-54
NFORX(4)
2-30
Same as NFORX(1),
fuselage
segment
but
for
fourth
55-57
NP
0-9
Number of pods described
58-60
NPODOR
4-30
Number of stations
at which
radii
are to be specified
61-63
NF
0-6
Number of fins
to be described
64-66
NFINOR
3-10
Number of ordinates
used to
describe
each fin
(vertical
tail)
airfoil
section
67-69
NCAN
0-2
Number of canards
(horizontal
tails)
to be described
70-72
NCANOR
3-10
Number or ordinates
used to define
each canard (horizontal
tail)
airfoil
section.
If the value of
NCANORis input with a negative
sign,
the program will
expect to
read lower surface ordinates
also,
otherwise
the airfoil
is assumed
to be symmetrical
(vertical
pod
tails)
Cards
3, 4,
. . . - remainin@
input
data
cards.The
remaining
input
data
cards
contain
a detailed
description
of
each
component
of
the
configuration.
Each
card
contains
up
to
I0 values,
each
value
punched
in a 7-column
field
with
a decimal
point
and
may
be
identified
in columns
73-80.
The
cards
are
arranged
in
the
following
order:
reference
area,
wing
data
cards,
fuselage
data
cards,
pod
data
cards,
fin
(vertical
tail)
data
cards,
and
canard
(horizontal
tail)
data
cards.
in
Reference
columns
1-7
tains
Wing
the
area
and
card:
may
be
data
cards:
locations
in
The
reference
identified
as
The
first
wing
percent
chord
area
REFA
value
is
in
columns
punched
73-80.
data
card
(or
cards)
at which
the
ordinates
conof
63
all the wing airfoils
ly NWAFORlocations
identified
the
last
are to be specified.
in percent chord given.
in columns
location
in
73-80
percent
by
the
chord
symbol
given
There will
Each card
be exact-
XAFJ
where
J
on that
card.
may
be
denotes
The
next
wing
data
cards
(there
will
be
NWAF
cards)
each
contain
four
numbers
which
give
the
origin
and
chord
length
of
each
of
the
wing
airfoils
that
is
to be
specified.
The
card
representing
the
most
inboard
airfoil
is
given
first,
followed
by
the
cards
for
successive
airfoils.
These
cards
contain
the
following:
Columns
Contents
1-7
x-ordinate
of
airfoil
leading
edge
8-14
y-ordinate
of
airfoil
leading
edge
15-21
z-ordinate
of
airfoil
leading
edge
22-28
airfoil
73-80
card
identification,
WAFORGJ
where
J denotes
the
paricular
airfoil,
thus
WAFORGI
denotes
the
most
inboard
airfoil
data
values
If
a
cards
of
cambered
is the
delta
z
streamwise
chord
length
wing
has
been
specified,
the
next
set
of wing
mean
camber
line
cards.
There
will
be NWAFOR
referenced
to the
z-ordinate
of
the
airfoil
leading
edge,
each
value
corresponding
to
a specified
percent
chord
location
on
the
airfoil.
These
cards
are
arranged
in
the
order
which
begins
with
the
most
inboard
airfoil
and
proceeds
outboard.
Each
card
may
be
identified
in
columns
73-80
as TZORDJ
where
J denotes
the
particular
airfoil.
Note
that
the
z-ordinates
are
dimensional.
Next
are
the
wing
ordinate
cards.
There
will
be NWAFOR
values
of
half-thickness
specified
for
each
airfoil
expressed
as
percent
chord.
These
cards
are
arranged
in the
order
which
begins
with
the
most
inboard
airfoil
and
proceeds
outboard.
Each
card
may
be
identified
in
columns
73-80
as WAFORDJ
where
J denotes
the
particular
airfoil.
Fuselage
the
x
There
in
of
64
data
cards:
values
of
the
fuselage
will
be NFORX(1)
values
columns
the
last
73-80
by
fuselage
The
first
stations
and
the
card
(or
cards)
of
the
first
cards
may
be
the
symbol
XFUSJ
where
station
given
on that
J denotes
card.
specifies
segment.
identified
the
number
If the fuselage
is circular,
the next card (or cards) gives
the fuselage
cross sectional
areas, and may be identified
in
columns 73-80 by the symbol FUSARDJ where J denotes the number
of the last
fuselage
station
given on that card.
If the fuselage is of arbitrary
shape, NRADX(1) values of the y-ordinates
for a half-section
are given and identified
in columns 73-80
as YJ where J is the station
number.
Following
the y-ordinates
are the NRADX(1) values of the corresponding
z-ordinates
for
the half-section
identified
in columns 73-80 as ZJ where J is
the station
number.
Each station
will
have a set of y and z,
and the convention
of ordering
the ordinates
from bottom to
top is observed.
For each fuselage
segment a new set of cards as described
must be provided.
The segment descriptions
should be given in
order of increasing
values of x.
Pod data cards:
The first
the location
of the origin
fies
contains
the
pod (nacelle)
of the first
data
pod.
card speciThe card
following:
Columns
Contents
1-7
x-ordinate
of
origin
of
first
pod
8-14
y-ordinate
of
origin
of
first
pod
15-21
z-ordinate
of
origin
of
first
pod
73-80
card
where
The
ordinates,
of
be
the
zero
next
pod
input
data
referenced
to
the
pod
radii
and
the
cards
may
where
J
For
must
gram
cate
(or cards)
origin,
at
to be
specified.
x value
is the
identified
in
denotes
the
pod
each
additional
columns
of
Fin
data
describe
a
contains
the
zero
The
length
73-80
PODORGJ
pod
number
contains
the
xwhich
NPODOR
values
first
x value
of
the
pod.
by
the
symbol
must
These
XPODJ
number.
pod,
new
be
provided.
Only
single
pods
assumes
that
if
the
y-ordinate
is located
symmetrically
with
y-ordinate
to
be
are
last
card
pod
identification,
J denotes
the
implies
cards:
Exactly
fin
(vertical
a
PODORG,
XPOD,
and
PODR
cards
are
described
but
the
prois not
zero
an exact
duplirespect
to
the
xz-plane,
a
single
pod.
three
tail).
data
The
input
first
cards
are
used
fin
data
card
following:
65
Columns
Contents
1-7
x-ordinate
on inboard
leading
edge
airfoil
8-14
y-ordinate
leading
15-21
z-ordinate
leading
chord
29-35
x-ordinate
43-49
airfoil
of
inboard
airfoil
length
leading
edge
y-ordinate
leading
edge
z-ordinate
leading
x
inboard
edge
22-28
36-42
of
edge
of
inboard
of
outboard
airfoil
of
outboard
airfoil
of
outboard
airfoil
edge
50-56
chord
73-80
card
identification,
J denotes
the
fin
The
expressed
second
in
length
of
outboard
J
The
card
denotes
The
third
fin
input
data
the
fin
airfoil
half-thickness
Since
the
fin
airfoil
must
be
on
the
positive
y side
of
the
The
card
identification
FINORDJ
where
J denotes
the
fin
number.
may
the
be
identified
fin
number.
airfoil
FINORGJ
number
fin
input
data
card
contains
NFINOR
percent
chord
at which
the
fin
airfoil
are
to be
specified.
73-80
as
XFINJ
where
airfoil
in
where
values
of
ordinates
columns
card
contains
NFINOR
values
of
expressed
in percent
chord.
symmetrical,
only
the
ordinates
fin
chord
plane
are
specified.
may
be
given
in columns
73-80
For
each
fin,
new
FINORG,
XFIN,
and
FINORD
cards
must
be
provided.
Only
single
fins
are
described
but
the
program
assumes
that
if the
y-ordinate
is not
zero
an exact
duplicate
is
located
symmetrically
with
respect
to
the
xz-plane,
a yordinate
of
zero
implies
a single
fin.
Canard
data
cards:
airfoil
is
symmetrical,
a canard,
and
the
input
fin.
If,
however,
the
66
If
the
canard
(or
exactly
three
cards
is given
in
the
same
canard
airfoil
is not
horizontal
tail)
are
used
to describe
manner
as
for
a
symmetrical
(indicated
by a negative
value of NCANOR), a fourth
canard
input data card will
be required
to give the lower ordinates.
The information
presented
on the first
canard input data card
is as follows:
Columns
Contents
1-7
x-ordinate
of inboard
leading
edge
airfoil
8-14
y-ordinate
of inboard
leading
edge
airfoil
15-21
z-ordinate
of inboard
leading
edge
airfoil
22-28
chord
29-35
x-ordinate
of outboard
leading
edge
airfoil
36-42
y-ordinate
of outboard
leading
edge
airfoil
43-49
z-ordinate
of outboard
leading
edge
airfoil
50-56
chord
73-80
card
where
The
second
canard
input
x expressed
in percent
chord
nates
are
to be
specified.
columns
73-80
as
XCANJ
where
the
card.
The
be
punched
must
may
the
the
be
identified
canard
number.
lower
ordinates
length
of
identification,
J
canard,
new
airfoil
of outboard
denotes
the
in
card
contains
expressed
in
columns
73-80
If the
canard
are
presented
airfoil
CANORGJ
canard
CANORG,
XCAN,
NCANOR
percent
number
values
chord.
as
CANORDJ
airfoil
is
on
a second
program
expects
both
upper
and
lower
as
positive
values
in percent
chord.
For
another
be
provided.
inboard
data
card
contains
NCANOR
values
at which
the
canard
airfoil
ordiThe
card
may
be
identified
in
J denotes
the
canard
number.
The
third
canard
input
data
canard
airfoil
half-thickness
This
card
J denotes
metrical,
length
and
of
where
not
symCANORD
ordinates
CANORD
of
to
cards
67
Description
Card
identifying
Card
planar
Non
of Auxiliary
i.i
- Identification.information
in
columns
Input
Cards
Card
i.i
1-80.
1.2
- Boundary
condition
boundary
conditions
are
contains
and
control
point
always
applied
on
however
card
1.2
permits
the
selection
of boundary
apply
on
a wing,
fin
(vertical
tail),
or
canard
This
card
also
selects
the
output
print
options.
tains
the
following:
Columns
Variable
1-3
LINBC
Value
0
any
desired
definition.a body,
conditions
(horizontal
This
card
to
tail).
con-
Description
Control
points
on
surface
fin
(vertical
tail),
and
(horizontal
tail).
This
ferred
to
as
ary
condition
of wing,
canard
is re-
the
nonplaner
option.
bound-
Control
points
in plane
of wing,
fin
(vertical
tail),
and
canard
(horizontal
tail).
This
is
referred
to
ary
condition
4-6
THICK
7-9
PRINT
0
Do
not
matrix
1
Calculate
if LINBC
as
the
planar
option.
calculate
0
Print
forces
1
Print
out
wise
loads
canards
=
wing
1
wing
out
the
pressures
and
moments
option
on the
0
negative
output
LINBC,
integers.
68
value
indicated
of
and
the
the
spanfins,
and
and
the
velocsource
and
the
steps
Print
the
axial
print
adds
for
options
THICK,
and
PRINT
THICK
is
not
used
and
wing,
matrix
vortex
strengths
Print
out
option
2 and
the
iterative
solution
and
A
the
thickness
thickness
Print
out
option
1
ity
components
and
to
bound-
out
normal
the
1-4.
are
punched
if
LINBC
=
option
3
velocity
panel
as
0.
and
matrices
geometry
right
in
print
justified
out
Card
control
right
Columns
1-3
4-6
2.1
-
integers.justified
Variable
K0
K1
Revised
configuration
The
integers
K2
10-12
K3
13-15
K4
card
2.1
Value
No
reference
Reference
0
1
No wing
data
Wing
data
to
sharp
leading
Wing
data
to
round
leading
0
1
3
punched
as
lengths
length
data
to
be
read,
edge
be
read,
edge
be
read
wing
has
a
wing
has
a
No body
data
Body
data
follows
Not
0
1
description
are
Description
0
1
3
7-9
paneling
contents
of
as
follows:
used
No
fin
(vertical
tail)
data
Fin
(vertical
tail)
data
to
read,
fin
has
a sharp
leading
be
edge
Fin
(vertical
read,
fin
has
be
tail)
data
to
a round
leading
edge
16 -18
K5
0
1
3
No
canard
(horizontal
tail)
data
Canard
(horizontal
tail)
data
to
be
read,
canard
has
a sharp
leading
edge
Canard
(horizontal
tail)
data
to
be
read,
leading
19-21
K6
Not
22-24
KWAF
Number
define
canard
edge
KWAFOR
a
round
used
of wing
sections
the
inboard
and
panel
edges.
panel
edges
the
geometry
25-27
has
Number
of
define
the
edges
KWAFOR
defined
of
=
If KWAF
are
defined
input
ordinates
leading
used
to
outboard
=
used
and
0,
by
the
NWAF
in
to
trailing
the
wing
panels.
If
0, the
panel
edges
are
by
NWAFOR
in the
geometry
input
69
Columns
Variable
Value
Description
28-30
KFUS
31-33
KRADX(i)
0,
3-20
Number of meridian
lines
used to
define
panel edges on first
body
segment.
There are three options
for defining
the panel edges.
If KRADX(1) = 0, the meridian
lines
are defined by NRADX(1) in
the geometry input.
If KRADX(1)
is positive,
the meridian
lines
are calculated
at KRADX(1) equally
spaced PHIKs.
If KRADX(1) is
negative,
the meridian
lines
are
calculated
at specified
values
of PHIK
34-36
KFORX(i)
0,
2-30
Number of axial
stations
used to
define
leading
and trailing
edges
of panels on first
body segment.
If KFORX(1) = 0, the panel edges
are defined by NFORX(1) in the
geometry input
37-39
KRADX(2)
0,
3-20
Same as KRADX(1),
body segment
but
for
second
40-42
KFORX(2)
0,
2-30
Same as KFORX(1),
body segment
but
for
second
43-45
KRADX(3)
0,
3-20
Same as KRADX(1),
body segment
but
for
third
46-48
KFORX(3)
0,
2-30
Same as KFORX(1),
body segment
but
for
third
49-51
KRADX(4)
0,
3-20
Same as KRADX(1),
body segment
but
for
fourth
52-54
KFORX(4)
0,
2-30
Same as KFORX(1),
body segment
but
for
fourth
The number of fuselage
segments.
The program sets KFUS = NFUS
The program is restricted
to 600 body singularity
panels.
For this program there is an additional
restriction
that the
total
number of singularity
panels in the axial
direction
on the
body (fuselage)
cannot exceed 30.
The arbitrary
body (fuselage)
capability
of this program is limited
to those shapes for which
the radius
is a single-valued
function
of PHIK for each cross
section
of the body.
7O
Card
description
punched
as
Columns
1-3
2.2
-
Additional
revised
configuration
control
integers.The
right
justified
integers
Variable
0,
2-20
card
2.2
are
Description
Value
KF(1)
paneling
contents
of
as
follows:
Number
define
of
the
fin
sections
inboard
and
used
to
outboard
panel
edges
on
the
first
fin.
If KF(1)
= 0,
the
root
and
tip
chords
define
the
panel
edges
4-6
7-9
i0-12
13-15
16-18
19-21
22-24
25-27
28-30
31-33
34-36
KFINOR
(i)
KF (2)
KFINOR
0,
2-20
(2)
KF(3)
KFINOR
KF
(3)
(4)
0,
3-30
0,
2-20
(5)
KF(6)
KFINOR
0,
3-30
0,
2-20
KF (5)
KFINOR
0,
3-30
0,
2-20
(4)
KFINOR
0,
3-30
0,
3-30
0,
2-20
(6)
0,
3-30
Number
define
of ordinates
the
leading
used
to
and
trailing
edges
first
of
the
fin.
fin
panels
If KFINOR(1)
on
=
the
0,
the
panel
edges
are
by
NFINOR
for
second
but
for
for
third
but
for
for
fourth
but
for
for
fifth
but
for
for
sixth
but
for
Same
as
defined
for
KF(1),
for
KFINOR(1),
but
fin
Same
as
second
Same
fin
as
for
KF(1),
as
for
KFINOR(1),
but
fin
Same
third
Same
fin
as
for
KF(1),
as
for
KFINOR(1),
but
fin
Same
fourth
Same
fin
as
for
KF(1),
as
for
KFINOR(1),
but
fin
Same
fifth
Same
fin
as
for
KF(1),
as
for
KFINOR(1),
but
fin
Same
sixth
fin
71
Columns
Variable
37-39
KCAN( 1 )
Value
Description
Number
define
of
canard
the
inboard
sections
used
and
outboard
to
panel
edges
on
the
first
canard.
If KCAN(1)
= 0,
the
root
tip
chords
define
the
panel
edges.
If KCAN(N)
negative,
no vortex
sheets
carry
through
the
body
and
concentrated
vortices
are
shed
from
the
inboard
canard
or
tail
40-42
KCANOR(1 )
Number
the
the
of
leading
first
the
panel
NCANOR
43-45
KCAN(2 )
46-48
KCANOR(2 )
KCAN(3)
52-54
KCANOR(3)
,
,
3-30
55-57
KCAN(4)
0
,
2-20
58-60
KCANOR( 4)
61-63
KCAN(5 )
0
,
KCANOR(5)
,
3-30
67-69
KCAN(6)
0
!
2-20
70-72
KCANOR(6)
0!
3-30
72
of
the
used
to
define
and
canard.
trailing
edges
of
If KCANOR(1)=0,
edges
are
defined
but
Same
as
for
KCANOR(1),
second
canard
Same
third
as
for
canard
KCAN(1),
Same
third
as for
canard
KCANOR(1),
Same
as
for
KCAN(1),
fourth
canard
Same
as
for
KCANOR(1),
fourth
canard
2-20
64-66
ordinates
Same
as
for
KCAN(1),
second
canard
3-30
49-51
edge
surface
Same
fifth
as
for
canard
KCAN(1),
Same
fifth
as
for
canard
KCANOR(1),
Same
sixth
as
for
canard
KCAN(1),
Same
sixth
as
for
canard
KCANOR(1),
by
for
but
but
for
for
but
but
for
for
but
but
for
for
but
but
for
for
but
for
panels
The program is restricted
on the wing-fin-canard
to a total
combination.
of 600 singularity
For this program there is an additional
restriction
the total
number of singularity
panels in the spanwise
on the wing-fin-canard
combination
cannot exceed 20.
that
direction
Cards
3,
4,
. . . - remainin_
input
data
cards.The
remaining
input
data
cards
contain
a detailed
description
of
the
singularity
paneling
of each
component
of
the
configuration.
Each
card
contains
up to
i0 values,
each
value
punched
in
a
7-column
field
with
a decimal
point
and
may
be
identified
in
columns
73-80.
The
cards
are
arranged
in
the
following
order:
reference
lengths,
wing
data
cards,
fin
(vertical
tail)
data
cards,
canard
(horizontal
tail)
data
cards,
fuselage
(body)
data
cards,
and
finally
Mach
number
and
angle
of
attack
case
cards.
Note
that
the
present
program
will
not
handle
a pod
and
therefore
there
are
no
pod
panel
inputs.
However,
if the
geometry
input
contains
a pod
description
it will
be
read
and
ignored.
in
Reference
length
columns
73-80
and
Columns
card:
contains
This
the
card
may
following:
Variable
be
identified
as
REFL
Description
i- 7
REFA
Wing
reference
the
reference
the
value
of
input
8-14
REFB
Wing
value
semispan.
of
1.0
reference
area.
If
REFA
= 0,
area
is
defined
by
REFA
in
the
geometry
If REFB
= 0,
used
for
the
is
a
semispan
15-21
REFC
Wing
reference
a value
of
1.0
reference
chord
22-28
REFD
Body
(fuselage)
reference
diameter.
If REFD
= 0,
a value
of
1.0
is
used
for
the
reference
diameter
29-35
REFL
Body
(fuselage)
reference
length.
If REFL
= 0,
a value
of
1.0
is
used
for
the
reference
length
36-42
REFX
X
coordinate
of
moment
center
43-49
REFZ
Z
coordinate
of
moment
center
chord.
is used
If REFC
for
the
=
0,
73
Wing data cards:
The first
wing data card is the wing
leading
edge radius
card and is required
only when K1 = 3.
This card contains
NWAFvalues of leading
edge radius expressed
in percent
chord.
It may be identified
in columns 73-80 as
RHOJ where J denotes the number of the last radius
given on
that card.
Next is the wing panel leading
edge card.
This card contains
KWAFORvalues of wing panel leading
edge locations
expressed in percent
chord.
This card may be identified
in columns
73-80 as XAFKJ where J denotes the last location
in percent chord
given on that card.
Omit if KWAFOR= 0.
The last wing data card gives the wing panel side edge data.
This card contains
KWAF values of the y ordinate
of the panel
inboard edges.
This card may be identified
in columns 73-80 as
YKJ where J denotes the last y ordinate
on that card.
These
values are arranged in the order which begins with the most
inboard panel edge and proceeds outboard.
Omit if KWAF = 0.
Fin (vertical
tail)
data cards:
The first
fin data card
is the fin leading
edge radius
card and is required
only when
K4 = 3. This card contains
NF values of leading
edge radius
expressed
in percent
chord, one value for each fin.
It may be
identified
in columns 73-80 as RHOFIN.
Next is the fin panel leading
edge card for the first
fin.
This card contains
KFINOR(1) values of fin panel leading
edge
locations
expressed
in percent chord.
This card may be identified in columns 73-80 as XFINKJ where J denotes the fin number.
Repeat this card for each fin.
The last
fin data card gives the fin panel side edge data
for the first
fin.
This card contains
KF(1) values of the
z ordinate
of the panel inboard edges.
This card may be identified in columns 73-80 as ZFINKJ where J denotes the fin number.
These values are arranged in the order that begins with the most
inboard panel edge and proceeds outboard.
Repeat this card for
each fin.
Canard (horizontal
tail)
data cards:
The first
canard data
card is the canard leading
edge radius card and is required
only
when K5 = 3. This card contains
NCAN values of leading edge
radius expressed in percent chord, one value for each canard.
It may be identified
in columns 73-80 as RHOCAN.
Next is the canard panel leading
edge card for the
canard.
This card contains
KCANOR(1) values of canard
leading
may
be
edge
locations
identified
in
canard
number.
74
Repeat
expressed
in percent
columns
73-80
as XCANKJ
this
card
for
each
chord.
where
canard.
J
first
panel
This
card
denotes
the
The
data for
of the y
identified
number.
the most
card for
last canard data card gives the canard panel side edge
the first
canard.
This card contains
KCAN(1) values
ordinate
of the panel inboard edges.
This card may be
in columns 73-80 as YCANKJ where J denotes the canard
These values are arranged in the order that begins with
inboard panel edge and proceeds outboard.
Repeat this
each canard.
Fuselage
(body) data cards:
The first
body card is the
body meridian
angle card.
This card contains
KRADX(1) values
of body meridian
angle expressed in degrees and may be identified in columns 73-80 as PHIKJ where J denotes the body segment
number.
The convention
is observed that PHIK = 0. at the bottom of the body and PHIK = 180. at the top of the body.
Omit
unless KRADX(1) is negative.
Repeat this card for each fuselage segment.
The second body card is the body axial
station
card.
This
card contains
KFORX(1) values of the x ordinate
of the body
axial
stations
and may be identified
in columns 73-80 as XFUSKJ
where J denotes the body segment number.
Omit if KFORX(1) = 0.
Repeat
this
Mach
identified
card
for
number
and
in columns
Columns
each
angle
73-80
fuselage
segment.
of
attack
as MALPHA
card:
and
Variable
i- 7
This
contains
card
may
be
the
following:
Description
MACH
The
subsonic
Mach
number
(includ-
ing
the
value
MACH
= 0.)
or
the
supersonic
Mach
number
at which
it
is desired
to
calculate
the
aerodynamic
8-14
ALPHA
A
for
with
series
of
Mach
The
angle
of
attack
expressed
degrees
at which
it
is desired
calculate
the
aerodynamic
data
number
the
same
geometry
may
the
desired
values.
A
of
the
such
a
value
present
terminal
of
MACH
case.
card.
=
data
-i.
and
be
on
Geometry
angle
of
calculated
this
cards
attack
by
card
for
combinations
repeating
signifies
a
new
in
to
this
the
case
card
termination
can
follow
75
Program Output
Data
All output is processed by a standard
132 characters-perline printer.
The output from each run is always preceded by
a complete list
of the input data cards.
The amount and type
of the remaining
output depend on the PRINT option
selected,
the number of panels used, and whether the configuration
being
analyzed is an isolated
wing, an isolated
body, or a complete
wing-body-tail
combination.
The program output options
are
described
below:
PRINT = 0
The program prints
the case description,
Mach
number and angle of attack,
followed
by a table
listing
the panel number, control
point coordinates (both dimensional
and non-dimensional),
pressure
coefficient,
normal force,
axial
force,
and pitching
moment.
Separate tables are printed
for the body and wing panels,
noting
that any
tail,
fin or canard panels are included
with the
wing output.
If the planar boundary condition
option has been selected,
the results
for the
wing upper surface
are given in one table,
followed by a separate
table giving
the results
for
the wing lower surface.
Additional
tables giving
the total
coefficients
on the body, the wing and
the complete configuration
follow
the pressure
coefficient
tables.
These include
the reference
area, reference
span and reference
chord, the
normal force,
axial
force,
pitching
moment, lift,
and drag coefficients,
and the center of pressure
of the component.
PRINT = 1
In addition
to the output described
for PRINT = 0,
the program prints
out additional
tables giving
the normal force,
axial
force,
pitching
moment,
lift
and drag coefficients,
and the center of
pressure
of each column of panels on the wing and
tail
surfaces.
In addition,
the indices
of the
first
and last panel in the column are listed,
together
with the span, chord and origin
of the
column.
PRINT = 2
In addition
to the output described
for PRINT = i,
the program prints
out tables
listing
the panel
number, the source or vortex
strength
of that
panel,
and the axial
velocity
u, lateral
velocity
v, and vertical
velocity
w at the panel control
point.
The normal velocity
is also calculated
for
76
body panels.
Separate tables
are printed
for
the body and wing panels,
noting
again that any
tail,
fin,
or canard panels are included
with
the wing output.
If the planar boundary condition
option has been selected,
separate
tables
are given for the wing upper and lower surfaces.
PRINT = 3
In addition
the
the
at
PRINT
=
4
to the
program
source
each
step
In addition
the
program
normal
ments
prints
number
output
described
of
the
to
the
prints
iterative
solution
by
number
printing
PRINT
= 2,
and
number,
obtained
procedure.
output
described
for
PRINT
=
out
tables
of
the
axial
and
velocity
components
which
of the
aerodynamic
matrices.
out
the
matrix
row
number,
of elements
in
that
row.
four
matrix
option
is
for
prints
out
the
iteration
and
vortex
strength
arrays
partitions
selected,
each
will
of
3,
make
up
the
eleThe
program
and
gives
the
A maximum
of
be
printed
which
is
and
its
influence
description
the
velocity
component
tables.
if
this
identified
prior
to
If a negative
value
of PRINT
is
selected,
the
program
prints
all
the
information
described
above
for
the
positive
values,
together
with
the
complete
panel
geometry
description
of
the
configuration
following
the
list
of
input
cards.
This
consists
of
tables
giving
the
wing
panel
corner
points,
control
points,
inclination
angles,
areas,
and
chords.
If the
configuration
has
a horizontal
tail,
fin
or
canard,
additional
tables
are
printed
giving
the
same
information
as
listed
above
for
the
wing.
panel
angles
Finally,
if
corner
points,
are
listed.
The
sented
in
program
Appendix
the
configuration
control
points,
output
III.
is
illustrated
includes
areas,
by
a
and
the
body,
the
inclination
sample
case
body
pre-
77
EXPERIMENTAL VERIFICATION
Several examples of pressure
distributions
calculated
by
the program are presented
in this section,
and compared with
experimental
data.
The examples include
isolated
bodies,
isolated
wings, and wing-body
combinations
in both
subsonic
and
supersonic
flow.
Isolated
Bodies
O_ive-cylinder
body
with
boattail
in subsonic
flow.The
theoretical
pressure
distribution
calculated
for
this
body
at
M = .40
and
_ = 0 degrees
is presented
on
Figure
3.
The
experimental
data
has
been
obtained
from
reference
6, which
also
contains
experiment
the
blunt
12 degree
additional
for
this
base
and
cone
aft
comparisons
between
the
present
theory
and
body
at M = .61
and
.83.
In
this
example,
sting
were
replaced
by
an
arbitrarily
chosen
of the
boattail
region
in
an
attempt
to
sim-
ulate
the
flow
separation
ment
between
the
theory
of
the
body.
retical
having
Haack-Adams
pressure
A
body
with
distribution
/A
base
and
=
.532
region
behind
experiment
base
and
in
supersonic
calculated
for
_/d
max
the
body.
is achieved
=
i0
is
a
Good
over
agreemost
flow.The
Haack-Adams
presented
on
theobody
Figure
max
4,
for
M = 2.01
and
e = 0 degrees.
The
experimental
data
for
this
body
is obtained
from
reference
7, which
also
gives
the
pressure
distribution
calculated
by
characteristics
theory.
The
present
method
agrees
closely
with
the
experimental
data
and
the
characteristics
theory
for
this
body.
Elliptic
cone
in
supersonic
flow.The
theoretical
pressure
distribution
on
an
elliptic
cone
is
compared
with
experimental
data
on
Figure
5,
for
M = 1.89
and
e = 0 and
6 degrees.
The
experimental
data
was
obtained
from
reference
8.
Again,
the
theory
agrees
well
with
experiment
except
near
the
leading
edge
on
the
lower
over-estimated.
78
surface,
where
the
positive
pressure
is
slightly
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81
Isolated
Two-dimensional
distribution
on
a
degrees
is compared
airfoil
NACA
64A010
with
the
Wings
in
subsonic
airfoil
at
experimental
flow.The
pressure
M = .167
and
e = 8
data
from
reference
on
Figure
option
was
6.
In
this
example,
the
surface
boundary
utilized
in the
theoretical
calculations.
agreement
surface
with
the
of
experiment
airfoil,
is excellent
indicating
that
on both
viscous
condition
The
upper
and
lower
effects
are
small.
The
potential
flow
solution
obtained
by
the
present
method
also
agrees
closely
with
that
given
by
the
viscous
flow
solution
presented
in reference
i0
for
this
airfoil,
except
for
a small
region
near
the
trailing
edge.
In general,
potential
flow
theory
tends
to
over-estimate
the
negative
pressure
peaks
in
two-dimensional
flow.
Figure
7 compares
the
results
of
the
present
program
with
the
exact
incompressible
pressure
distribution
around
a 10 percent
thick
Karman-Trefftz
airfoil.
Here,
the
program
results
agree
closely
with
the
exact
solution,
and
give
considerable
confidence
in
the
capability
of
the
present
method
to
reproduce
theoretical
two-dimensional
flows.
Variable
sweep
butions
calculated
section,
72
degrees
angles,
are
compared
win@
in
subsonic
flow.on
a variable
sweep
wing
inboard
sweep,
and
two
with
experimental
data
The
pressure
having
a NACA
outboard
wing
from
reference
distri64A006
sweep
ii
at M =
.23
and
e = 10.5
degrees
on
Figure
8.
In this
example,
the
boundary
conditions
are
applied
in
the
plane
of
the
wing.
The
theory
agrees
reasonably
well
with
experiment
at the
root
and
at
the
mid-span
break
point.
The
pressure
distribution
is
less
accurate
near
the
wing
tip,
although
the
net
loading
appears
to be
approximately
correct.
The
agreement
between
and
experiment
is considered
to be
acceptable,
considering
relatively
high
angle
of
attack
chosen
for
this
comparison.
theory
the
Cambered
arrow
win_
in bu_ersonic
flow.The
pressure
distributions
calculated
on
a cambered
and
twisted
arrow
wing
having
a 3 percent
circular
arc
section
and
70 degrees
sweepback
are
compared
with
experimental
data
from
reference
12
at
M = 2.01
and
e = 4 degrees
on Figure
conditions
are
applied
in the
plane
can
be
seen
to
agree
reasonably
well
entire
wing,
except
in
the
immediate
edge.
82
9.
Here,
the
boundary
of the
wing;
and
the
theory
with
experiment
over
the
vicinity
of the
leading
9
<
NACA
64A010
AIRFOIL
6.0
M =
.167
e =
8°
R =
4.1
× I0 _
5.0
Theory
Q
Experiment
4.0
- Cp
3.0
2.0
1.0
0.0
- 1.0
Figilre
6 -
Pressure
Pistribution
on
Two-dimensional
Airfoil
83
KARMAN-TREFFTZAIRFOIL
2.0
t/c
= 10%
M = 0.
2.5% Camber
1.6
-Cp
_ = 4°
Exact
1.2
O
Present
Theory
Method
.8
.4
--°4
8
Figure
84
7 -
Incompressible
Pressure
Distribution
on Two-dimensional
Airfoil
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Wing-Body
Combinations
O@ive-cylinder
body
with
swept
win 9 in supersonic
flow.The
planform
of
this
simple
wing-body
combination,
and
the
paneling
scheme
used
in
the
aerodynamic
representation
are
shown
on
Figure
i0.
The
wing
has
a NACA
65A004
section,
is
centrally
mounted
on
the
body,
and
the
quarter-chord
line
is
swept
back
45 degrees.
The
ogival
nose
is
3.5
body
diameters
in
length.
The
pressure
distribution
on
the
wing
is compared
with
experimental
data
from
reference
13
for
M = 2.01
and
_ = 5
degrees
on
Figure
ii.
The
theoretical
curve
was
calculated
using
the
planar
boundary
condition
option.
The
agreement
between
theory
and
experiment
is reasonably
good,
except
near
the
wing
leading
edge.
Pa:t
of
this
discrepancy
is
probably
due
to
shock
wave
detachment
ahead
of
the
round
leading
edge
of
the
airfoil
for
this
supersonic
leading
edge
circulation
to develop
around
the
leading
calculations
assume
an attached
Mach
wave
leading
edges,
prohibiting
the
development
flow.
The
lines
12.
iment
pressure
distribution
on
of
the
body
are
compared
with
In this
example
the
agreement
is
extremely
good.
the
wing,
allowing
a small
edge.
The
theoretical
along
supersonic
of
any
circulatory
upper
and
experimental
between
theory
lower
data
and
meridian
on Figure
exper-
87
88
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CONCLUSIONS
The
aerodynamic
analysis
method
described
in this
report
has
been
developed
to succeed
the
earlier
methods
reported
in
references
1 and
2.
Considerable
progress
has
been
made
in the
achievement
of this
goal,
but
additional
work
remains
if
the
full
potential
of
the
new
method
is to be
realized.
Several
promising
described
areas
briefly
for
the
below.
future
development
of
this
method
are
Program
refinements.Increased
geometrical
capability
is
required
to permit
the
analysis
of engine
pods,
nacelles,
or
fairings.
In addition,
improved
programming
techniques,
including
far
field
approximations
to
the
aerodynamic
singularity
distributions
would
be
desirable
to
reduce
the
time
required
to
calculate
the
matrix
of
aerodynamic
influence
coefficients.
Finally,
the
extension
of
the
force
and
moment
subroutine
to
include
the
calculation
of
additional
aerodynamic
stability
derivatives
would
be
valuable.
Development
of
computer
program
is
sure
distribution,
The
development
of
increase
its
range
problem
of determining
which
will
generate
the
presence
of
an
drag
and
interference
based
on
the
design
the
pro@ram
as a design
tool.The
present
restricted
to
the
determination
of
the
presforces
and
moments
on
given
configurations.
the
program
as a design
tool
would
greatly
of application.
For
example,
the
important
the
wing
camber
and
twist
distribution
favorable
surface
pressure
distributions
in
arbitrary
body,
or
for
minimizing
pressure
effects
could
be
included
in this
program
procedures
described
in
reference
i.
Development
of
leading
ed@e
vortex
model.The
use
of
linearly
varying
vortex
distributions
to
represent
the
circulatory
flow
around
lifting
wings
permits
the
Kutta
condition
to
be
imposed
along
leading
edges,
as well
as trailing
edges.
Using
this
flow
model,
the
vortex
sheet
from
leading
edge
panels
can
be
modified
to
trail
downstream
from
the
leading
edge
or
wing
tip
to
simulate
a separated
flow,
and
provide
a first
approximation
to the
lift
distribution
on wings
at high
angles
of
attack.
Application
of
the
method
to
the
analysis
of
transonic
flows.The
non-linear
effects
of
transonic
flow
can
be
approximated
by
using
the
local
Mach
nun_er
calculated
from
the
potential
flow
solution
to redefine
the
regions
of
influence
and
the
magnitude
of the
velocity
field
induced
by
the
aerodynamic
singularities.
An
iterative
solution
of
the
boundary
condition
91
equations
can
then
be
established,
in which
the
coefficients
of
the
equations
depend
on
the
local
Mach
number
distribution
of
the
preceding
step.
The
iterative
procedure
would
be
continued
until
a convergent
pressure
distribution
is obtained.
Analytical
Bellevue,
December
92
Methods,
Washington
31,
1972
Incorporated
APPENDIX I
Integration
The velocity
may
all
tables,
component
be
reduced
to
forms
and
two
non-standard
J
(v2
=
The
+
e2)[av
[
J2
method
i.
integrals
appearing
integrals
used
to
results
dv
2
(V 2
+
e2)[_v
2
evaluate
these
are
summarized
J2
Y_
-
y_
+
b2e
2
y_
+
b2e
2
a
real
-
c)y
2
+
2bv
+
c]½
=
tan
tanh-
-
b2e
2
2
i
e[av
sinh
-
be
root
y[av
-I
=
y2F
non-zero
and
G
c]_J
is
given
in
refer-
_2
=
(ae 2
F
+
=
z
is
2bv
integrals
below:
Y
Y
+
vdv
J
J
where
appearing
in the
text
in
standard
integral
given
below:
=
1
The
ence
Procedures
2
of
=
+
G ]
0,
2by
72
-
bv
vy
+
e_
+
2bv
(ae 2
-y2)v2
equation
_
be/y
=
+
C]½
+
C]½
vY
-x
the
+
+
e_
(c
-
_a)ea]½
93
APPENDIX II
Panel
Geometry
Calculation
Procedure
The analytical
procedure presented
here follows
closely
the method first
developed in reference
14. A quadrilateral
surface
element is described
by four corner points,
not necessarily
lying
in the same plane,
as shown in the sketch.
Note
that the numbering convention
of the corner points differs
from that used in the preceding
text.
The quadrilateral
element
is approximated
by a planar panel as follows:
2
I
\J
/
/
The
fied
coordinates
by
their
vectors
We
may
cross
T
in the
subscripts.
and
1
T
2
reference
The
of
system
are
the
diagonal
identi-
are
T ix
=
x 3
-
x i
T ly
=
Y 3
-
Yl
T iz
=
z3
-
z
T2x
=
x _
-
x2
T 2y
=
Y 4
-
Y2
T 2z
=
z_
-
z2
by
taking
of
the
now
product
obtain
a
vector
N
(and
diagonal
its
components)
vectors.
N
94
coordinate
components
=
T
2
×
T
1
the
N = T T
- T T
x
2y Iz
ly 2z
N
=
T
T
y
Ix
N
=
T
as N
unit
unit
normal
divided
normal).
by
-
2x
ly
vector,
n,
to
its
length
own
T
T
2x
T
z
The
2z
T
T
Ix
the
lZ
_y
plane
N
of
the
(direction
element
cosines
is
of
taken
outward
Nx
nx
-
N
ny
N
NZ
where
nz
-
N
N
= [ N 2
x
+
N 2
y
+
N2z]½
The
plane
of
the
element
is now
completely
determined
if
a
point
in this
plane
is specified.
This
point
is
taken
as
the
point
whose
coordinates,
_,
y,
z are
the
averages
of
the
coordinates
of
the
four
input
points.
1
Y
=
41
[Yl
l[z
Now
the
element
input
along
points
will
the
normal
+
x2
+
x
+
x
]
+
Y2
+
Y3
+
y
]
+z
+z
be
projected
vector.
The
+z]
into
the
resulting
plane
points
of
the
are
the
95
corner
are
points
of
equidistant
d
=
=
quadrilateral
the
plane,
Inx(X
The
coordinates
ordinate
system
x'
k
the
from
-
x I)
+
ny(y
of
the
corner
are
given
by
x
+
(I)
-
k+* n
k
and
-
element.
The
this
distance
yl)
points
+
in
nz(Z
the
-
input
is
points
zl)I
reference
co-
d
x
k+l
|
Now
the
requires
vectors,
Yk
=
Yk
+
(-1)
z'
k
=
z
+
(i)
-
k
nYd
k+1 n
k
=
i,
2,
3,
4
d
z
element
coordinate
the
components
of
one
of which
points
system
must
be
constructed.
three
mutually
perpendicular
along
each
of
the
coordinate
This
unit
axis
of the
system,
and
also
the
coordinates
of
the
origin
of
the
coordinate
system.
All
these
quantities
must
be
given
in
terms
of the
reference
coordinate
system.
The
unit
normal
vector
is
taken
as
one
of
the
unit
vectors,
so two
perpendicular
unit
vectors
unit
in
vectors
the
plane
t
of
and
t
1
by
its
own
length
the
element
The
2"
T
1
t
,
i.e.,
ix
TIX
T
t
1
TIY
ly
T 1
TIZ
lZ
T
1
96
t
needed.
is
1
-
t
vector
are
taken
Denote
as
T
these
divided
1
Ti
where
The
are
vector
t2
is
= [T ix2 +T ,y2 +T_]%
lZ
defined
t
by
=
n
2x
=
n
2y
=
The
vector
t
is
×
-
n
-
that
its
components
t
n
lZ
t
y
vector
so
ly
x
ly
t I,
t
z
t
unit
n
ix
x
the
-
t
n
2z
n
Iz
z
t
=
t
y
t
t
ix
parallel
to
the
x
or
_
axis
1
of
the
element
coordinate
system,
while
t
is
parallel
to
the
2
y or
_
ordinate
The
axis,
and
system.
corner
ordinate
points
n
points
is
parallel
are
now
system
based
have
coordinates
system.
with
this
on
to
plane
of
the
element
the
element,
coordinate
which
defines
is
a multiple
coordinate
n
in
the
by
_
they
into
a
of
axis
the
0.
of
this
element
co-
co-
as origin.
reference
These
coordinate
coordinate
Because
system
they
lie
zero
Also,
the
x or
_ axis
of
the
"diagonal"
and
the
coordinate
_
point
in
the
have
system.
or
element
, _ ,
k
k
z
or
because
_
the
in
coordinate
vector
the
element
coordinate
vector
from
point
1 to
n
are
equal.
In
the
I
coordinate
z
transformed
the
average
x',
y',
z'
k
k
k
Their
coordinates
origin
are
denoted
the
the
in
t
I
,
system,
3,
the
(_,n)
3
system,
the
corner
points
of
the
element
are:
!
_k
=
n
= t
k
t
ix
(x
-
x.')
K
+
t
ly
(_- x') + t
2x
k
(y
-
yk )
+
t
_z
(F- y') + t
2y
k
(z
-
z_)
(_- z')
2z
k
97
These corner
quadrilateral
I
points
are taken as the corners
as illustrated
in the following
of a plane
sketch.
(_3 '
1
n3)
A
(g_,,
n 4)
The
origin
of
the
element
coordinate
system
is
to
the
centroid
of
the
area
of
the
quadrilateral.
average
point
as origin
the
coordinates
of
the
element
coordinate
system
are:
_
o
-
1
3
1
-
T]
T]
[_
,+
(q
,
-
q
2
)
+
_
2
(D
now
transferred
With
the
centroid
in the
_
-
q
1
) ]
2
1
T]
0
3
__
These
are
subtracted
from
the
coordinates
of
the
corner
points
in
the
element
coordinate
system
based
on
the
average
point
as
origin
to
obtain
the
coordinates
of
the
corner
points
in the
element
coordinate
system
based
on
the
centroid
as origin.
Accordingly,
these
latter
coordinates
are
_k
=
_k
-
_o
k
nk
98
=
nk
-
n o
=
I,
2,
3,
4
Since the centroid
is
element,
its coordinates
are
required.
These
the
in
=x+t
Y0
=
y
+
tl y_ 0
=
z
+
t
0
area
A
IX
of
=
the
lZ
the
1
-- (_
2
3
_
_
0
coordinate
2
0
+
t2y_
0
+
t2z_
of the
system
0
quadrilateral
-
point
are
+tx_
0
control
reference
coordinates
X0
z
Finally,
to be used as the
_
)(_
l
is
2
_
)
4
99
APPENDIX III
SAMPLE CASE
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OFFICE:
1973-739-027/31