NUMERICAL METHODS FOR THE

NASA
TECHNICAL
NASA TN D-7713
NOTE
IpBm,
f,_
I
Z
I-.¢¢
.=¢
Z
;!)i. !_)>,,
NUMERICAL
THE
OF
METHODS
DESIGN
WINGS
by Harry
Langley
Hampton,
AND
AT
IV. Carlson
Research
Va.
FOR
ANALYSIS
SUPERSONIC
and
,
David
SPEEDS
S. Miller
Center
23665
7_'z6 .1_1_
NATIONAL AERONAUTICSAND SPACE ADMINISTRATION •
WASHINGTON, D C
• DECEMBER1974
1. Report
No.
2. Government
NASA
4. Title
Accession No.
3. Recipient's
Catalog No.
TN D-7713
and Subtitle
5. Report
NUMERICAL
METHODS
OF WINGS
AT
FOR
SUPERSONIC
THE
DESIGN
SPEEDS
7. Author(s)
Harry
W. Carlson
and David
NASA
Code
8. Performing Organization
Report No.
L-9542
S. Miller
Name and Address
Langley
Hampton,
Research
Va.
1974
6. Performing Organization
10. Work
9. Performing Organization
Date
December
AND ANALYSIS
Unit
No.
760-65-11-02
'11. Contract
Center
or Grant
No.
23665
13. Type of Report and Period Covered
12. Sponsoring
Agency
National
Name and Address
Aeronautics
Washington,
15. Supplementary
Technical
and Space
D.C.
Administration
14.
Sponsoring
Note
Agency
Code
20546
Notes
16. Abstract
In rather
extensive
arbitrary-planform
particularly
techniques
incorporated
subjected
17.
into
to wings
which
the
methods.
the
original
methods,
to a thorough
of numerical
at supersonic
in application
numerical
revised
employment
wings
review
with
overcome
wing
Supersonic
wing
Wing
twist
and
Drag
minimization
19. Security
certain
slightly
subsonic
the
major
In order
to provide
development
as well
in this
part
for
the
deficiencies
leading
of these
.and
analysis
have
been
revealed,
edges.
Recently
deficiencies
a self-contained
as the
design
more
have
devised
now been
description
recent
of
of the
revisions
have
been
report.
Key Words (Suggested by Author(s))
Supersonic
methods
speeds,
18. Distribution
Statement
Unclassified
analysis
-
Unlimited
design
camber
STAR
Classif. (of this report)
Unclassified
20.
Security
Classif. (of this page)
For
sale by the National
Technical
21.
No. of Pages
22. Price*
74
Unclassified
Information
Service, Springfield,
$4.25
Virginia
22151
Category
01
NUMERICAL
METHODS
OF WINGS
By Harry
FOR
AT
THE
DESIGN
SUPERSONIC
W. Carlson
Langley
and
ANALYSIS
SPEEDS
David
Research
AND
S. Miller
Center
SUMMARY
In rather
extensive
employment
of arbitrary-planform
wings
revealed,
in application
particularly
Recently
cies
devised
have
now
description
of the
revisions
have
arose
in the
leading
edge.
method
to suppress
ful
nine-point
patible
revised
disturbing
An aft-element
mined
formula.
methods
for
which
an optimized
submittal
to the
wing
evaluation
which
and
sometimes
formerly
loadings
to provide
and
edges.
deficien-
as the
more
recent
report.
irregularities
vicinity
incorporated
mode,
forces
been
a self-contained
as well
loadings
calculated
for
of a power-
with
reduced
more
small
and
that
wing
analysis
application
have
wing
that
of the
in the
required
between
analysis
of these
in combination
analysis
arose
part
immediate
been
improvements,
design
and
has
have
eliminated
in the
technique
These
in the
virtually
and
leading
major
in this
surface
oscillations
the
development
have
design
subsonic
In order
review
camber
sensing
pressure
discrepancies
original
method
of the
overcome
the
deficiencies
slightly
methods.
to a thorough
definition
summation
the
wing-design
smoothing
the
for
certain
with
which
methods,
subjected
to the
to wings
into
methods
speeds,
techniques
incorporated
been
Revisions
often
at supersonic
numerical
been
of numerical
but
forces
same
com-
deter-
shape
upon
program.
INTRODUCTION
Because
of its
rather
extensively
design
and
important
surfaces
all forces
available
delta
ple
or
for
on wings
for
arrow
planform,
certain
based
theory
and
shape.
problems,
for
evaluation
are
Although
example,
not
analysis
exact
definition
of loadings
directly
examples
and
applicable
been
employed
Descriptions
of
of results
obtained
1 to 4.
methods
in the
has
aircraft.
and
in references
of linearized-theory
of specified
solutions
given
theory
of supersonic
on linearized
minimization
and
linearized
analysis
are
application
drag
versatility,
and
problems
planforms
these
and
design
methods
to typical
One
lifting
in the
analysis
in application
simplicity
is in the
of pressure
design
of wing
loadings
and
over-
linearized-theory
solutions
are
of minimum-drag
surfaces
for
forces
on flat-plate
to the
complex
wing
wings
of sim-
planforms
and surface shapes often employed in real aircraft. Such limitations, however, have been
removed by the introduction of numerical methods for implementation of linearized theory
on high-speed digital computers.
Widely used computer-implemented numerical methods for the design and analysis
of wings with arbitrary planform which employ a rectangular grid system for representation of the wing lifting surface and simplified numerical techniques for evaluation of
linearized-theory integrals are presented in references 5, 6, and 7. These methods can
accommodate large numbers of wing elements (in the thousands) for the description of
rather complex planforms and the handling of intricate surface shapes. Reference 5
described a method for the design of wing camber surfaces to minimize drag at a given
lift coefficient through employment of an optimum combination of component loadings.
This was followed by a method (ref. 6) which employed the same basic formulations for
the evaluation of lifting pressures on flat-plate wings. The evaluation method was later
extended to cover the case of wings with arbitrary-surface shape as described in
reference 7.
Although the wing-design and analysis methods have for the most part been used
quite successfully for a number of years, certain deficiencies are known to exist. Most
notable is the tendency for solutions to be PoOrly behaved for wings whose leading edges
are only slightly subsonic. Recently, means of overcoming the major part of these
deficiencies have been devised and the results of the study are presented herein. In order
to provide a self-contained description of the revised methods, the original development
as well as the more recent revisions have been subjected to a thorough review.
In the design of wings with slightly subsonic leading edges, sporadic irregularities
were found in the definition of the camber surface in the immediate vicinity of the leading
edge. These irregularities could be removed by a manual alteration, but in fact were
more often ignored. A numerical procedure (programable for use on high-speed digital
computers) which approximates the strategy employed in manual elimination of irregularities has recently been devised and is now incorporated in the design method.
For the analysis method, especially in application to flat wings with near-sonic
leading edges, large oscillations in local pressure coefficients were known to exist from
the inception of the method. In the original method these oscillations were largely
eliminated by introduction of a powerful nine-point smoothing formula which operated
after an initial definition of unsmoothed pressure coefficients for all the wing elements.
The smoothing operation necessitated an extension of the wing grid system for four elements behind the actual wing trailing edge, and thus it effectively limited application of
the method to wings with supersonic trailing edges. For the particular case of a flat wing
with an exact sonic leading edge the oscillations became so severe that the only recourse
was to avoid that condition by considering either a slightly subsonic or slightly supersonic
2
leading edge. An aft-element sensing technique which will be described has now been
incorporated in the program to permit an integral smoothing and thus eliminate the
necessity for a separate terminal smoothing routine. This provision also extends
applicability of the method to wings with subsonic trailing edges.
There has also been a small but disturbing discrepancy between wing loadings and
forces determined for an optimized wing shape in the design mode and the loadings and
forces calculated for that same shape in the analysis mode. A part of that discrepancy is
resolved by employment of the previously discussed modifications. Other means of providing more accurate results in both modes so as to insure the proper correspondence are
to be discussed.
SYMBOLS
:
A(L,N)
leading-edge-element weighting factor for influence summations
A(L*,N*)
leading-edge-element weighting factor for force and moment summations
Ao
load
1
strength
factor
for
ith
loading
B(L,N)
trailing-edge-element
weighting
factor
for
influence
summations
B(L*,N*)
trailing-edge-element
weighting
factor
for
force
moment
b
wing
span
C(L,N)
wing
center-line
and
summations
element
or wing-tip-element
weighting
factor
for
influence
element
or wing-tip-element
weighting
factor
for
force
and
(eq.
(25))
summations
C(L*,N*)
wing
center-line
moment
summations
CD
drag
CD,ij
interference-drag
CL
lift
CL,d
coefficient
coefficient
between
ith
and
jth
specified
loadings
coefficient
design
lift
coefficient
3
C m
ACp
pitching-moment
lifting-pressure
local
C
cI
a m
coefficient
wing
mean
cd
coefficient
chord
aerodynamic
chord
section
drag
coefficient
section
lift
section
pitching-moment
(eq.
coefficient
(eq.
(16))
(15))
coefficient
L,N
designation
of influencing
elements
L*,N*
designation
of field-point
elements
wing
overall
M
Mach
R
influence
X
T
X a
Z c
length
function
value
(eq.
wing
distances
measured
distance
from
longitudinal
camber-surface
(3))
of influence
reference
smoothed
ZC_S
(17))
number
average
x,y,z
(eq.
function
within
a grid
element
(eqs.
(6),
(7),
and
(8))
area
in a Cartesian
wing
leading
distance
from
coordinate
edge
measured
leading
edge
z-ordinate
camber-surface
z-ordinate
system
(see
fig.
1)
in x-direction
of specified
area
loading
(see
fig.
8)
zr
camber-surface
angle
z-ordinate
of attack,
at wing-root
trailing
edge
deg
M M-2_I
A
wing
kL,km,_t z
Lagrange
leading-edge
sweepback
multipliers
for
lift,
angle,
deg
moment,
and
camber-surface
ordinates,
respectively
dummy
T
variables
designation
of integration
of a region
Mach
cone
from
the
for
of integration
point
x
and
bound
y,
respectively
by the
wing
planform
and
the
fore
(x,y)
Subscripts:
a,b
step
indices
ac
aerodynamic
center
C,F,T
aerodynamic
coefficients
and
the
i,j
ith
le
leading
max
maximum
min
minimum
n
te
and
totaled
jth
for
combination
specified
loading
number
of specified
loadings
trailing
edge
the
cambered
of cambered
wing,
and
the
flat
corresponding
wing,
flat
wing,
respectively
edge
5
NUMERICAL-CALCULATION METHODS
Camber Surface for a Given Loading
A typical wing planform described by a rectangular Cartesian coordinate system is
illustrated in figure 1. For a wing of zero thickness lying essentially in the z = 0 plane,
linearized
theory
for
lift
distribution
a specified
supersonic
flow
by the
defines
the
wing-surface
the
(x - _) ACp(_,_/)
is a slightly
influence
elements
with
over
the
the
shaded
wing
area
is,
arises
tion
9 and
also
(x,y)
of
y
the
for
Mach
cone
integral
the
to support
and
can be
value,
8.
The
although
spanwise
the
regions
point
wing
to the
shown
This
integrand
_/= y
concept
when
of the
in section
found
generaliza3 of refer-
to be convergent
exist
by
and
general
at
explained
chordwise
as
extends
improper
of a more
generally
_-
(x,y)
a singularity
and
represents
from
of integration.
of nonconvergence
of the
integral
of being
region
according
is thus
derivative
field
plane
have
The
of integration
appearance
is discussed
integral
the
(1)
originating
region
from
not
treated
which
The
z = 0
does
8.
vortices
within
in the
d_/d_
of reference
the
_/= y
form
integral
surface,
gives
at
theory
in reference
which
fore
limiting
principal
on a wing
spans.
The
lifting-surface
Cauchy
and
singularity
the
Consequently,
points
ues
however,
(77a)
of horseshoe
chords
the
1.
of the
from
of the
ence
within
in figure
integrand
of equation
distribution
small
planform
because
z / 0.
form
vanishingly
divergent
that
modified
of a continuous
necessary
equation
az
which
slopes
if there
at
are
val-
integral
(x - _) ACp(_,_?)
{(x - _)2 _ fi2(y
is not
behind
single
a discontinuity
can
occur
the
loading
results
valued
at spanwise
_ = y.
in the
the
the
which
regions
of the
of this
conditions
can
leading-edge
for
These
remainder
purposes
Such
wing
stations
distribution.
over
For
at
_ 77)2
wing
study,
arise
sweep
(for
discontinuities
along
a streamwise
example,
appear
of nonconvergence,
at the
in spanwise
however,
line
wing
directly
apex)
derivatives
and
of
do not invalidate
surface.
equation
(1) will
be
rewritten
in the
form
az c
--_-(x,y)
= .l_
ACp(X,y)+
4_ _
R(x-_,y-r/)ACp(_,v/)d/_
T
d_
(2)
where
the
function
R
is defined
R(x-_,y-_/)
=
x_2(y
and
its
may
be thought
influence
of the
_? = y
has
of the
function
field
been
to zero,
nature
of this
upwash
influence
extending
of streamwise
change
positive
negative
limits,
attractive
that
the
by forward
may
shaped
surface
and
reductions
upwash
largely
for
tion,
has
been
values
difficulties
numerical
the
it is first
necessary
to introduce
sketch
used
is illustrative
numbers
assigned
element
of integration
element
associated
equal
to
to
and
and
and
N*
than
be
of lift
The
near
theory
however,
little
evidence
seen
supersonic
near
near
outboard
abrupt
A
the
the
edges
leading
edge
inclined
that
drag
wings
if the
of any
appreciable
speeds.
Mach
upwash
predicts
flat
the
leading
on a forwardly
for
of upwash
The
of the
at and
achieved
for
field,
vortices.
subsonic
Because
a drag.
tip
of
field
flow
of upwash
with
suriace
integration
a grid
the
in application
(d_,dfi_/).
with
x
indicated
L
wings
amount
downwash
trailing
The
singularity
of upwash.
values
in representing
in describing
only;
wing
angularity.
full
benefits
Unfortunately,
Mach
cone
supersonic-flow
limits,
this
is also
phenomena
by
techniques.
to replace
system
the
is set
limits,
efficiency.
also
realized;
large
the
can
In order
coordinate
cally
be
the
of finite-element
rather:
levels
can
with
responsible
means
thrust
suction
field,
wing,
large
arrow
an appreciable
these
suction
cambered
of the
a local
approaching
the
cone
upwash
ACp(x,y)
of the
to the
of the
negative
entirely
at
significance
upwash
remainder
Mach
of aerodynamic
elements
of leading-edge
same
The
standpoint
produce
at the
singularity
(2)
The
(_,77) to
A graphical
physical
in a strong
corresponds
(x,y).
definition
local
3.
felt
The
and particularly
and
the
is composed
also
in the
in figure
element.
to exist
twisted
define
at point
The
If in equation
presence
tips,
2.
The
it is applied
will
infinities
so as to create
of leading-edge
amount
lifting
element
field,
makes
created
be
the
is noted
upwash
from
the
point
in figure
surface.
its
loading
character.
is illustrated
to positive
singularity
It is this
cone
from
when
makes
local
at downstream
peculiar
-_(x,y)
_? = y
the
is shown
lifting
Oz
slope
from
lines
from
elemental
at
downstream
R
its
_ _/)2
relating
slope
understood
distribution
function
function
to illustrate
resultant
(3)
- _)2 _ _2(y
function
be better
the
_
necessary
influence
by a small
equal
the
the
expanded
may
produced
_ _7)2 _(x
of as an influence
in determining
representation
as
system
wing
more
identify
starred
immediately
is numerically
the
equal
to
of
L
field
the
in figure
elements
spaces
of the
over
as shown
grid
values
ahead
(2) by a numerical
superimposed
planform
many
N
The
in equation
summaCartesian
4.
would
(This
be employed.)
in the
grid
which
replace
and
N
identify
the
point
Ely, where
(x,y);
x
and
L*
BY
the
space
or
is numeritake
on only
7
integer values. The region of integration, originally bound by the wing leading edge and
the Mach lines, now consists of a set of grid elements approximating that region as shown
by the shaded area in the example of figure 4. Inclusion of partial as well as full grid
elements provides a better definition of the wing leading edge and tends to reduce any
irregularities that may arise in local surface slopes for elements in the vicinity of the
leading edge.
The contribution of each element of the wing (L,N) to the local slope at (x,/3y) may
be written as
OZ c
-_-(x,y)
Terms
detail
in this
in the
equation
R
R(x-_,y-7).
tions
the
value
the
_, as
(L*
the
d_
R
factor
has
expression
fi72
Jf_71
=
1
takes
= (L*
becomes
8
_y = N*,
_71
one
from
(4)
ACp(L,N)
will
be described
grid
R
_2(y
the
form
and
integral
may
be written
x -
_(L*
_72
in some
(5)
_ 7)2
insensitive
evaluation
of
to varia-
as
midpoint
- L + 0.5) 2 -_2(y
- L + 0.5)2f_(Y
function
dfi7
L + 0.5)
_ L + 0.5) 2 - f32(y
at the
= N + 0.5
of the
by numerical
to be relatively
_ 7)2_(L.
of
- L + 0.5)
the
Since,
(L*
value
element
element.
factor
the
the
, (x_)
_ 7)2_(x_
_)2 _ f_2(y
determined
_9_72
_71
= N - 0.5,
within
be found
(L*
with
C(L,N)
evaluation
value
_2(y
been
the
representing
R(L*-L,N*-N)
and
B(L,N)
in their
may
over
an approximation
- L + 0.5)
integration
factor
extends
R(L*-L,N*-N)
with
used
an average
of this
integration
integrand,
in
represents
1
_2
= A_ Afl_? "J_l
fi(L*-L,N*-N)
the
methods
A(L,N)
paragraphs.
term
The
R(L*-L,N*-N)
and
following
The
in which
= _
of the
d2_?
(6)
- 7) 2
element.
On
(7)
_ _?)2
- 7)
(see
fig.
4),
the
influence
factor
R(L*-L,N*-N)
= ¢(L* - L + 0.5)2 - (N* - N - 0.5)2
(L*
- L + 0.5)(N*
- N - 0.5)
{(L* - L + 0.5)2 - (N* - N + 0.5)2
(8)
(L* - L + 0.5)(N* - N + 0.5)
A graphical
representation
variations
in the
of the
y-
of the
balancing
values
all
introduces
no net
L*
- L = 0
that
element
that
be useful
of defining
The
elements
leading-edge
The
takes
The
values
shape.
of 0.5
This
in a later
fro'
single
a lifting
of equal
amounts
of the
element
which
section
that
a specified
A(L,N)
- N = 0
wing,
downwash
and thus
it is moving.
will
with
the
At
the
have
will
R
value
no influence
inverse
of
on
problem,
and
takes
factor
permits
which
a better
on values
from
allows
consideration
definition
of the
0 to 1 given
wing
by
L - Xle =
< 0)
- Xle
influencing-element
0 to 1 given
B(L,N)
: 0
B(L,N)
= 1 - (L-
B(L,N)
= 1
(9)
(0<L-Xle<l)
is a trailing-edge
=
weighting
factor
which
also
by
(L - Xte
Xte)
-> 1)
(0<,
('
is a wing-tip
1 given
N*
summation
surface.
process
factor
at
summation,
dealing
summation
The
in which
small
variations
a complete
and
an element
report
rather
spanwise
value
or
spanwise
the
drastic
the
of upwash
in the
of the
the
negative
medium
insures
Note
element,
in the
= 1
or
to the
5.
with
of elements,
(4) is a weighting
A(L,N)
term
set
in equation
= L - Xle
from
- L
term
A(L,N)
term
in figure
contrasted
that
one
fact,
: 0
on values
C(L,N)
consists
A(L,N)
B(L,N)
due
insures
is only
loadings
A(L,N)
L*
displacements
there
is zero.
will
of partial
which
is shown
L-direction
a given
This
vertical
where
factor
or
to be zero,
others.
field
x-
For
is found
the
a flow
itself,
in the
or N-direction.
R
produce
of this
factor
influencing-element
(lo)
<0)
weighting
factor
which
takes
on
by
9
1.o
Desired
values
element
of the
element.
The
within
a given
The
edge
are
obtained
pressure
from
loading
may
vary
point
(x,y)
of an element
mathematical
wing-surface
of the
coefficient
from
ACp(L,N)
formulas
element
modeling
slope
at that
point
elements
within
the
evaluated
to element
is fixed
of the
may
supersonic
be found
influencing
=
vertical
lines
L*In the
as used
in
IN*-
and
B(L,N)
IN* - N[
original
method
directly.
camber
that
direction
for
downstream
figure
are
6.
Values
portion
point
element
and
10
third
next
A
)
element.
element
g(L*-L,N*-L)
L:l+[Xle
]
(12)
absolute
value
whole-number
only when
for
Errors
surface
figure.
slopes
of the
part
As
to the
the
rear
enclosed
of the
shown,
first
followed
were
found
a forward
produces
on this
rear
(12)
symbols
previously,
of the
by
equation
the
slope
midpoint
(L*,N*)
quantity.
az/ax
edge
often
with
(especially
a smaller
error
rapidly
observation
their
and
as a dashed
calculated
of
8z/ax
in marked
solutions,
notable
opposite
successive
assumed
for
element.
over
By
is depicted
and
of that
definition
in the
for
(12)
arise.
analytic
elements
and
value
equation
in the
leading
element
extrapolation
a
by
irregularities
wing
to decrease
based
circular
as given
comparisons
were
to the
half
and
leading-edge
from
slope
that
of the
the
solid
is applicable
over
shown
process
As noted
surface
slopes
is small)
as small
(L*,N*)
to extend
occur
element.
of the
has
vicinity
A smoothing
of the
of
as
L=L*-[N*-N[
the
of the
calculated
factor
respectively,
upper
(L*+I,N*)
the
5) values
immediate
which
elements.
shown,
assumed
errors
weighting
The
ACp(L,N)
designate
experience
in the
it was
the
(ref.
However,
surface
of design-program
noted
system.
contributions
is expressed
)
C(L,N)
of the
trailing
N[ >= 1 + [Xle ]
observation
when
constant
element
vortex
by a summation
quantity,
and the brackets
in
[Xle 3
designate
the
The initial
summation
with respect
to
L
is made
used
or
of the
to be
of the
horseshoe
region
ACp(L*,N*)+
× A(L,N)
of the
space
centroid
is assumed
midspan
N=Nmin
were
to each
at the
but
at the
N=Nmax
oz
The
assigned
element.
control
by the
each
grid
/
(N
of lifting-pressure
(II)
the
values
contrast
in
distribution
line
in the
a given
field-
That
slope
front
half
from
the
to the
is
of the
second
original
value. A simple averaging of these two values is found to yield a much more reasonable
value of the first element slope. With the process carried out for successive downstream
elements a smoothed slope distribution as shown in the lower portion of figure 6 is
obtained. In equation form the smoothing procedure is expressed as
z c,s
(L*,N*)
ax
The
station
z-ordinates
of the
y = N*/f_
8z
_
- 1
2
may
0z c
+ -_-(L*
(L*,N*)
wing
be found
surface
+ 1,N*)
x = L*
at station
by a chordwise
- 1
2
summation
8zc
-(L*
ax
+ 0.5
for
of the
local
+ 2,N*)
a given
(13)
semispan
slopes
L*=I +_Xte_
"Dz
Zc,s(X,y)
c's
8x
=
(L*,N*)
A(L*,N*)
(14)
,.,:1+
Wing-section
ordinates
as a function
of the
lift,
and
pitching-moment
the
appendix)
chord
fraction
x'/c
may
be found
by linear
interpolation.
Section
station
drag,
y = N*/fi
(see
L*=l+[xte
coefficients
may
be
evaluated
at any
by the
selected
semispan
following
summations:
_
\
Cl = 1
>
ACp(L*,N*)
A(L*,N*)
(15)
B(L*,N*)
L*=l+[xte _
c d = -_-I
-_-_(L
Oz
*,N*)
_
ACp(L*,N*)
A(L*,N*)
B(L*,N*)
(16)
L*=l+_le]
\
Cm = 1c2
>
(L*)
ACp(L*,N*),
A(L*,N*)
B(L*,N*)
(17)
L*=I +_le_
The
weighting
A(L*,N*)
factor
which
term
takes
A(L*,N*)
= 0
A(L*,N*)
= L*
A(L*,N*)
= 1
in equations
on values
(14)
from
to (17)
0 to 1.5
is a leading-edge
given
field-point-element
by
L * _ Xle =
< 0)
- Xle + 0.5
0 < L* - Xle < I)
(18)
(,.
>,)
, _ Xle =
II
The
B(L*,N*)
weighting
The
the
term
factor
in equations
which
B(L*,N*)
= 0
B(L*,N*)
= 0.5
B(L*,N*)
= 1
lifting-pressure
loading
takes
formulas
(15),
on values
- (L*-
(16),
from
and
(17)
0 to 1.5
(0>
Xte 1
given
for
evaluated
at the
correspond
to the surface
slope
element
representation
employed
the
- Xte >= 0)
L*
- Xte
field-point
midspan
(19)
>-1)
t e <-1)
=
elements
of the
_Zc,s] ax defined
in camber-surface
field-point-element
by
(L*
L*-x
coefficient
is a trailing-edge
ACp(L*,N*)
trailing
edge
is obtained
of the
element
from
so as to
at that point.
Figure
7 illustrates
the
definition
and in force
and moment
determination.
Wing
spanwise
lift,
drag,
and
integrations
pitching-moment
of the
2 fb/2
CL = --_S _0
Cm
The
integrals
selected
found
=
are
set
The
2
_S
-
CD
2
wing
evaluated
through
S:
12
in the
(21)
(22)
of standard
corresponding
numerical
techniques
to integer
expressions
for
the
values
aerodynamic
of
applied
to a
N*.
coefficients
may
be
and
x L*= l+_te_
2_ _
leading-edge
described,
as follows:
a summation
N*=Nma
The
from
(20)
by means
stations
used
obtained
CmC2 dy
_0
area
are
CdC dy
fb/2
of spanwise
data
respectively,
c/c dy
fb/2
_0
_
section
coefficients,
and
the
_
trailing-edge
center-line
A(L*,N*)B(L*,N*)C(L*,N*)
grid-element
or wing-tip
fractions
grid-element
(23)
are
width
determined
is defined
as previously
by
C(T,*,N*)
:
C(L*,N*)
(N*:0)
= I
:
(N*:Nm )
Optimum
In reference
the problem
arrow
10 Lagrange's
Combination
method
of selecting a combination
and delta wings producing
planform,
(24)
(0 < N* < Nmax)
of Loadings
of undetermined
of component
multipliers has been applied to
loadings yielding a minimum
a given lift. The method
provided that the interference-drag
may
drag for
be used for wings
of any
coefficients are first determined.
By using
the nomenclature
of the present report, the drag coefficient of the interference between
any two loadings
i,j may
be expressed
N*:Nma
-2
CD,ij
= CD,ji
=
x L*=I+
as
[Xte]
2_?
ACP,
_(_---_I
j (L*,N*)
i(L*'N*)
A(L*,N*)
B(L*,N*)
C(L*,N*)
N*:0 L*:1+ id
N*=Nma
x L*-l+[xte
:s)
)
N*:0
and
may
be evaluated
In reference
problem
minimum
cal
drag
methods
z-ordinate
The
5 Lagrange's
subject
only
extended
at the
wing-root
lift
,
ACp,j(L*,N
A(L*,N*)
B(L*,N*)
(25)
C(L*,N*)
means.
method
to permit
trailing
of undetermined
of loadings
to a restraint
coefficient
8z
) (_a---_)i (L*,N*)
l+[Xld
a combination
were
total
L*-
by numerical
of selecting
]
multipliers
on arbitrary
on lift
planform
coefficient.
additional
constraints
was
applied
wings
In reference
on pitching
to the
to yield
11 the
moment
a
numeriand
the
edge.
resulting
from
n
wing-loading
distributions
is given
by
i=n
CL = _
i=l
where
factor
CL,iA
i
CL, i denotes
the lift
of the ith loading.
The
wing-loading
distributions
(26)
coefficient
of the ith
total pitching-moment
is given
loading
and
coefficient
A i is the load
resulting
from
strength
n
by
i=n
Cm = _
i=l
Cm, iAi
(27)
13
the pitching-moment
where
Cm, i denotes
load strength
factor
for the ith loading.
trailing
edge
resulting
from
n
coefficient
Similarly,
wing-loading
of the
the
ith
loading
z-ordinate
distributions
and
at the
is given
Ai
is the
wing-root
by
i=n
zr = _
i=l
where
face
If,
and
denotes
Zr,i
required
given
zr
multipliers
of each
the
to support
in addition
are
yields
(28)
Zr,i Ai
z-ordinate
the
imposed
wing-root
trailing
edge
on the
camber
sur-
ith loading.
to a given
the
at the
lift,
on the
following
set
the
constraints
of zero
drag-minimization
of equations
pitching
problem,
which
the
establishes
moment
(Cm
method
of Lagrange
the
relative
= 0)
strength
loading:
i=n
XLCL,
1 + _mCm,
1 + XzZr,
1 +
_
i=l
CD, liAi
= 0
CD,2iAi
= 0
i=n
_LCL,2
+ _mCm,
2 + _zZr,
2 +
i=l
i--n
_LCL,n
+ _mCm,n
+ _zZr,n
+ _
i=l
CD, niA i = 0
(29)
i=n
_
i=1
CL,iAi
= CL_ d
i=n
Cm,iAi
= 0
i=l
i=n
_
Zr,iA
i = zr
i=l
Machine-computing
thus,
the
determined
camber
techniques
surface
for
allow
the
an optimum
evaluation
combination
of the
weighting
of preselected
factors
loadings
Ai,
and,
may
be
as
i=n
Zc(X,y
14
) = _
i=l
Zc,i(x,y)A
i
(30)
The corresponding drag coefficient is
i=n
j--n
_
_
i=1
j=1
1
CD=2
The
numerical
planforms
loading
which
uniform,
versatility
important
straints
are
because
than
are
illustrated
just
one.
loading
it is found
solution
that
R(L*-L=0,N*-N=0)
the
for
linear
wings
three
wing-
In the
11,
five
present
additional
procedure
of loadings.
requirement
arbitrary
specified
drag-minimization
specified
with
spanwise.
combination
stringent
of satisfying
wing-loading
will
This
is partic-
three
con-
distributions
presently
to support
a specified
8.
for
a Given
Camber
Surface
the
camber
surface
required
for
field-point
= 0).
for
in reference
the
optimum
eight
in figure
numerical
so that
more
The
and
introduced
the
of the
surfaces
implemented
chordwise,
provided
Loading
In the
of camber
5 was
linear
in computing
rather
available
design
improvements
distributions
more
the
in reference
incorporates
wing-loading
ularly
for
presented
distributions:
method,
have
method
as first
(31)
CD, ijAiAj
Thus,
element
equation
has
(12)
no influence
can
Nmax
on itself
be rewritten
(e.g.,
as
L*-IN*-NI
\
4
+I
7
)
Nmin
x B(L,N)
and
the
lifting-pressure
surface
shape
order
thus,
have
been
tion.
not
provided
all
required
B(L,N),
and
calculations
obtained
the
C(L,N)
and
R
for
summation.
are
as
can
are
be determined
performed
is from
the
apex
within
the
fore
coefficients
thevalueof
for
ACp
ACp(L*,N*)
pressure
previously
Since
the
(32)
ACp(L,N)
distribution
of calculating
L*);
C(L,N)
Lle
no unknown
L*=
The
L
in the
rearward
Mach
pressure
and
for
N*=
as in the
sequence.
(i.e.,
increasing
from
any
coefficients
N
solution
The
values
element
arise
is zero,
for
of arbitrary-
proper
cone
influencing-element
defined
a wing
of
will
in the
summa-
ACp(L=L*,N=N*)
weighting
factors
a camber
surface
is
A(L,N),
for
a given
loading.
Theoretically,
the
midspan
sented
of the
in reference
ACp(L*,N*)
trailing
edge
6 provision
defined
of the
was
by equation
L*,N*
made
for
element.
determination
(32)
is the
In the
pressure
numerical
of an average
coefficient
method
value
at
preof
ACp
15
over the element. In spite of this averaging, however, there remained large oscillations
in pressure coefficient from element to element which were subduedby inclusion of a
powerful nine-point terminal smoothing formula. The terminal smoothing procedure took
place after an initial definition of unsmoothed pressures for the entire wing and necessitated an extension of the wing grid system for four elements behind the actual wing
trailing edge. This effectively limited application of the method to wings with supersonic
trailing edges. An aft-element sensing technique which permits an integral smoothing
has now been incorporated in the numerical method to eliminate the need for both
averaging and the terminal smoothing steps.
The aft-element sensing technique involves the determination of preliminary
ACp
results for a given field-point element and for the element immediately following, combined with a subsequent fairing or smoothing of these preliminary results. The fairing is
applied to the velocity potential (i.e., the integral of the pressure) rather than to the
pressure itself because of the noticeably better behavior of the velocity potential in regard
to the absence of discontinuities. The procedure outlined in the following steps may be
clarified by reference to figure 9 which shows application of the technique to a typical
element:
(a) Calculate and retain temporarily preliminary
row with L* = Constant.
Designate
as
ACp,a(L*,N*).
(b) Calculate
following
previous
_Cp,
row
step
b(L*,
For
retain
temporarily
a final
AC
P
value
elements,
N*)
defined
= Ill
as
L*
elements,
defined
as
L*
ACp, a values
L* = Constant.
a fairing
In the evaluation
summation
may
=_
- Xle(N*)
for
the
obtained
in the
Designate
as
of integrated
preliminary
±Cp,
of
ACp, b(L*,N*),
be separated
into
a(L*,N*)
ACp,
< 1,
a(L,,N,
_ Z_Cp,b(L.,N.
- Xle(N*)
3
_Cp(L*,N*)
16
from
+ 1 A(L*,
+ A(L*,N*).]
N*)J
+ 21 [ 1 A(L*,N*)
+ A(L*,N*)J
other
values
results.
leading-edge
all
ACp
N*).
ACp(L*,
For
preliminary
with
L* = Constant
+ 1 by using
for contributions
from
the row with
(c) Calculate
ACp
and
ACp values for a given
)
)
(33)
> 1,
1
+_-ACp,b(L*,N*)
the influence-function--pressure-coefficient
two parts.
One part
consists
of all
(34)
the
spanwise
rows for which L*is
calculated
to avoid
preliminary
repetition
For
required
in the
that
of the
ACp
of forces
done
in the
The
lift
values.
be
evaluated
moments
The
calculation
loading
and
design
of
of only
first
all
from
part
may
Hence,
the
row
temporarily
row.
surface,
is little
at selected
which
following
camber
there
coefficients
L = 1
be retained
for
to a given
elements.
section
L*-
ACp(L*,N*)
corresponding
for
this
it is
advantage
span
in
stations
as
method.
coefficient
N*=Nma
ACp
subsequent
determination
evaluation
was
with
L _>2, and the second consists
may
be obtained
from
the
following
summation
over
all
elements:
x L*=Lte
7
C L = _-_
.
N*=0
The
+ 7
+ 1)J A(L*,N*)
B(L*,N*)
C(L*,N*)
L*=Lle
pitching-moment
(35)
coefficient
N*=Nma
Cm
ACp(L*)
about
x = 0
is
x L*=Lte
_S_
(L*)
N*=0
ACp(L*)
+-4 ACp(L*
+ 1
A(L*,N*)
B(L*,N*)
L*=LIe
x C(L*,N*)
The
drag
(36)
coefficient
may
N*=Nma
be expressed
as follows:
x L*=Lte
CD = _S
-2
__
ACp(L*)
N*=0
1 az c
+ _- --_--
This
relationship
suction"
only
force
for
the
The
tion
I_
not
of any
inclination
element
of the
employed
or
method
and
illustrated
wing
forces
of the
flow
normal
factors
method.
of computing
in figure
and
any
separated
in wing-loading
ular
-_
(L*)
A(L*, N*) B(L*, N*) C(L*, N*)
consider
weighting
wing-design
+
L*=Lle
(L*
does
+ _1 ACp(L*
to increase
the
total
10 was
effects
force
and
force
adopted
compatibility
of the
associated
to the
in equations
Figure
definition
contribution
the
and
moment
to provide
the
its
are
and
in the
accounts
rapid
descrip-
representation
determination.
coefficients
wing-design
exclusion
defined
element
moment
a more
"leading-edge-
wind.
(32) to (37)
in force
with
theoretical
with
relative
10 illustrates
and
(37)
as described
convergence
The
partic-
previously
of total
method.
17
The distribution of wing lift in the streamwise and spanwise direction may be
obtained from summations, taken row by row, of grid-element forces in the L- and
N-directions, respectively. These distributions are conveniently expressed as fractions
of total wing lift as follows:
For the streamwise lift distribution,
N*=Nmax
2
(Lift)L*
Total
and
for
panel
the
>
_
ACp(L*,N*)
A(L*,N*)
B(L*,N
_) C(L*,N
$)
N*=0
(38)
lift
_CLS
spanwise
lift
distribution,
N*
at a selected
value
on the
right-hand
wing
only,
L *= L t e
>
(Lift) N *
Total
The
camber
at zero
*,N*)
of equation
at unit
angle
(39)
the
evaluation
angle
of attack
(32).
of attack,
and
of loadings
as
moment
the
by calculating
may
for
specified
By repeating
and
characteristics
Lift
C(L*,N*)
L* =Lie
permits
form
coefficients.
B(L*,N*)
_CLS
--_--(L
azc
aerodynamic
A(L*,N*)
lift
method
surface
ACp(L*,N*)
be obtained
coefficients,
wings
by the
solution
with
element
for
a flat
interference-drag
over
a range
respectively,
an arbitrarily
surface
wing
warped
slopes
of the
same
coefficients,
of angles
may
of attack
be found
by
plan-
wing
and
lift
a direct
addition:
CL,T
= CL,
C + (CL,
F)a=I
a
(40)
Cm, --¢m,¢+
The
drag
drag
variation
pressures
on the
may
coefficient,
however,
with
acting
flat-wing
lift
for
on the
surface.
the
requires
flat
cambered
By using
consideration
wing,
and
wing
surface
and
the
method
shown
of the
an interference
by
drag
of the
drag
defined
cambered
wing
in reference
warped
the
by flat-wing
pressures
7, the
wing,
drag
acting
coefficient
be evaluated as
CD, T = CD, C + (CD, F-C
18
(41)
+ CD,
+(co,
c,_ c,,,
__"L]
F--)__T]
(42)
The interference-drag
terms employed in equation (42) are defined as follows:
For flat-wing pressures acting on the cambered wing surface,
N*=Nmax L*= Lte
CD, F-C
_S
N*=0
L*=Lle
1 _Zc
×
and
for
cambered
wing
+
I3 _(L*)
pressures
,]
aX--c(L*-
acting
on the
N*)
A(L*,
flat-wing
B(L*,
N*)
C(L*,
N*)
(43)
surface,
8z C
CD, C_F
= -CL,
-
CL, C(-0.01746)
ILLUSTRATIVE
EXAMPLES
c
(aXF)_=
1
Design
The
slopes
for
effect
and
ordinates
arrow
support
ence
of the
wings
with
a uniform
12.
subsonic
load
Slopes
spanwise
design-program
and
positions
corresponding
Adjacent
rather
highly
localized
irregularities
highly
swept
wings
are
the
Mach
line.
the
fore
cone
edges
the
erratic
Application
representative
is illustrated
is used
as
in figure
an example.
the
number
design
Note
this
line
the
region
of the
numerical
method
in the
loadings
An arrow
the
large
wing
local
three
midsemibetter
the
only
improvements
in
minor
Appreciable
the
can
improvements,
leading
for
edge
highly
be adequately
of integration
integration
techniques
and
an optimum
with
a leading-edge
ordinates
wings
represented
for
is narrow
The
of camber
approaches
swept
whereas
elements.
surface
wing
to illustrate
elements;
definition
for
For
is that
rectangular
the
refer-
to suppress.
where
and
from
to
is designed
method.
A
results
designed
position
near
shown
behavior
is broad
of rectangular
component
13.
_ cot
for
are
present
of
over
Mach
behavior
of the
of the
reason
method
A) there
value
integration
approaching
by a limited
use
of integration
numerical
represented
support
region
_ cot
0.8)
of chordwise
are
newer
and
results
stations
the
Program
linearized-theory
separated
of
the
0.6,
locations
largest
Apparently,
Mach
overcome
the
A = 0.4,
as a function
of camber-surface
respectively.
element
through
for
12,
definition
to adjacent
widely
values
(_ cot
exact
shown
which
(small
noted
straightforward
leading
than
definition
however,
with
are
on the
11 and
edges
compared
ordinates
span.
camber-surface
in figures
leading
are
Method
modifications
is illustrated
(44)
smoothing
combination
wing
and
is poorly
routine
in this
surfaces
by
helps
later
case.
designed
of those
sweep
of 70 ° at
called
for
near
to
loadings
M = 2.0
the
root
19
chord for all the loadings. Generally, these singularities occur whenever there is a
discontinuity in leading-edge sweep (the apex in this case). Camber-surface severity in
these regions can be minimized in a design problem by substitution of a smoothed leading
edge so that the transition from one leading-edge sweep to another takes place over several leading-edge elements. A limitation may also be placed on the allowable ordinate at
a specified location by exercising an available program option. There is no guarantee,
however, that ordinates exceeding this limit will not then arise at some other location.
An illustration of the effect of the design-method number of component loadings on
the optimum combination of loadings, on the camber surface, and on the drag-due-to-lift
factor is given in figure 14 for an ogee wing at M = 2.0. The changes
in optimized
loadings
appear
factor
(8.4
subtle.
to be relatively
percent).
In spite
loadings,
the
Changes
of the
use
concerning
followed.
Certain
three
degree
benefits.
present
technique,
should
approaches
refs.
13,
14,
and
of the
previous
sonic,
and
method.
supersonic
coefficients
are
shown
the
highly
present-method
than
technique.
results.
case
data
procedure
present
2O
the
These
for
are
for
given
with
component
caution.
Much
methods
have,
powerful
can
is yet
to
be implicitly
however,
in figure
with
results
15.
for
been
shown
for
linearized
arrow
to
with
sensing
operation
wings
for
subsonic,
lifting-pressure
one
theory
shown
aft-element
smoothing
Flat-wing
position
are
of the
terminal
of examples
of chordwise
numerical
increasing
employment
the
in a set
compared
the
afforded
swept
the
It will
edge,
of the
edges
a function
6),
results
more
leading
(ref.
improvement
more
curve,
is shown
to be rather
15.)
through
need
appear
semispan
(ref.
and
section.
16).
without
For
the
the
nine-point
operation.
The
for
as
results
method
smoothing
This
leading
Numerical-method
previous
the
in drag-due-to-lift
Method
method,
eliminates
surface
be approached
optimum-design
(See
change
with
linearized-theory
wing-evaluation
effectively
camber
improvement
Analysis
The
resultant
to which
restricted
appreciable
to the
corresponding
theoretical
than
the
compared
in the
predicted
of more
be learned
yield
large
wing
previous-method
is essential
This
appear
closely
data
to the
from
success
elimination
with
the
the
of the
of the
of scatter
rather
enough
for
results.
previous
for
closely
_ cot
about
the
does
Thus,
but
smoothing
apparent
A).
smoothing
oscillations
the
not
The
theoretical
terminal
near
the
linearized-theory
an uninformed
method
a final
of
of the
initial
Such
method.
previous
need
values
of some
approach
correct
be immediately
application
in spite
to be mild
not
(low
amount
that,
results
may
examples
results
however,
oscillations
method
a considerable
present-method
unsmoothed
newer
leading-edge
display
be noted,
to approximate
method.
by the
manual
appear
a terminal
is not
(and
fairing
to be the
smoothing
required
for
a corresponding
the
extension of the wing surface four elements behind the trailing edge) constitutes one of the
prime advantages of the new system because, as will be demonstrated later, it permits
consideration of subsonic trailing edges.
For wing subsonic leading-edge sweep angles in the range of much practical interest
(values of /3cot A from 0.6 to 1.0), the present method is superior in predicting pressures. In general, unsmoothed pressures from the newer method give a better representation of exact linearized theory than do the smoothed pressures of the older system.
The improvement for the sonic leading-edge case (fi cot A = 1.0)is
particularly
impressive.
With
and,
the
thus,
the
wing
the
advantages
the
previous
is shown
shown
100 percent
of the
2000
elements.
problem
results
and
they
with
cover
the
the
methods
newer
method
Correlation
method,
discrepancy
observed
17.
but
It is seen
that
that
neither
may
be immaterial,
in experimental
each
wing
where
17.
a distinct
pressure
numerical
the
the
A
method
method
stations
/3 cot
present
present
previous
contained
at 25,
from
is
50,
and
0.4 to 1.6.
approximately
method
generally
better
out that
gives
a poor
handling
of
previous-method
a trailing-edge
slopes
for
with
is seen
the
extension
linearized-theory
to be very
particular
For
case
other
little
of
sweep
values
difference
/3 cot
A = 1.0
angles,
both
results.
results
with
18.
Theoretical
method
its
edges.
oscillations
however,
case
overcome
of the
from
for
of
require
analytic
in figure
In any
which
present
the
advantage.
the
data
from
be pointed
There
except
with
the
overcome
trailing
differences
adequate.
method
shown
so that
lift-curve
methods,
is given
data
values
which
of numerical-method
planform
that
that
are
It might
in figure
well
Note
sonic
conditions.
with
pressures
offers
reasonably
newer
wings
do not
of subsonic
two numerical
double-delta
reference
smoothed
and
instances
method.
16 is shown
agree
are
and
present
the
chosen
of numerical-method
reference
where
isolated
16.
the
minor
representation
distributions
of delta
there
consideration
Correlation
between
the
only
preclude
from
are
the
suppressed,
Now,
only
to be
leading-edge
figure
pressure
were
Although
are
appears
with
in figure
of the
length
dimensions
correlation,
the
overall
avoided.
there
Either
be completely
other.
lifting-pressure-distribution
side
Spanwise
not
to be
as any
angles
near-sonic
is afforded
right-hand
on the. left.
program
sweep
associated
and
had
as well
methods.
of the
method
on the
Wing
previous
could
condition
leading-edge
at subsonic
appraisal
oscillations
to be handled
to be no disadvantages
Another
and
and
divergent
leading-edge
seems
supersonic
present
appear
obvious
sonic
condition
For
between
method
exactly
leading-edge
there
previous
linearized
are
reproduces
because
more
the
such
theory
results
were
subdued
for
a more
obtained
in the
complex
from
present
pressure
discontinuities.
discontinuities
have
not
This
been
investigations.
21
Application of the present numerical method to prediction of pressure distributions
on a flat-plate wing with a subsonic trailing edge is illustrated in figure 19. Numericalmethod results (new method only, previous method not applicable) are compared with
theory from reference 18. The Kutta condition, vanishing ACp at the trailing edge, is
seen to be met. However, again the pressure discontinuities are not properly represented. A better approximation may be obtained by increasing the number of elements
and decreasing their size, but the jump will continue to be represented by a more gradual
variation over a number of elements.
A final example of the application of the wing-evaluation method to flat-plate wings
treats an arbitrary planform of the ogee type (fig. 20). The numerical methods were
designed with application to just such arbitrary planforms as an objective; however,
because theoretical solutions are not available for arbitrary planforms, verification of
the methods was accomplished for the simpler planforms previously discussed. The data
of figure 20 show a somewhat smaller degree of ACp oscillation for the present method;
otherwise, the results are quite similar and appear to be in reasonable agreement.
Methods in Combination
In an airplane-design project it is often desirable to use the design method and the
evaluation method in combination. Because design-method results can yield camber
surfaces too severe for incorporation in practical airplanes, these surfaces are often
modified and use is then made of the evaluation method to assess the effect of the modification. This procedure may be misleading, however, if there is not a sufficient degree
of correspondence between the two methods. One test of this correspondence is to submit
a design-method surface directly to the evaluation-method program and to compare
drag-due-to-lift factors. In one instance reported in reference 11 a difference in dragdue-to-lift factor of as much as 7 percent was found. Use of the design-method smoothing
procedure, the evaluation aft-element sensing technique, and appropriate treatment of
numerical integrations has been found to reduce this discrepancy considerably. This
improvement is illustrated in figure 21. The previously mentioned example from reference 11 has been used to make the comparison. At the left of the figure, data from
reference 11 using the previous design and analysis methods have been repeated.
Results for the same example when performed with the present methods are shown at the
right. The results show an appreciable improvement for one of the most severe discrepancies encountered.
A more detailed comparison of design- and analysis-method results is shown in
figures 22 and 23. Again, a camber surface from the design program has been submitted
directly to the evaluation program. The first example is that of a clipped-tip delta wing
with three component loadings at a Mach number of 2. A more complex ogee planform
22
and a seven-term loading is considered in the second example. The delta wing was
represented by 2104 program elements and the ogee by 2387.
From figure 22 a comparison can be made of the design-method pressure distribution for an optimum combination of loadings and the pressure distribution evaluated for
that surface by the analysis method. For both examples, evaluation-method pressures
nearly duplicate the design pressures except in the immediate vicinity of the leading edge.
From figure 23 a comparison can be made of the design-method spanwise loading
distribution and the loading distribution calculated by the analysis method. For the
simpler case of three loadings on a delta wing, the loading distribution appears to be
faithfully reproduced. For the seven-loading ogee example, discrepancies are more
obvious. Much of the difficulty lies in drag-distribution peak in the vicinity of the root
chord. Suchpeaks can occur wherever there are discontinuities in the wing leading-edge
sweep. Thus, care must be exercised to provide closely spaced design-method computation stations in these regions. The integrated forces show the lack of a complete
agreement between the design and analysis methods. Nevertheless, the discrepancies
are relatively small and well within the ability of linearized-theory methods to account
for real-world aerodynamic phenomena.
In order to illustrate convergence characteristics of the methods, the designmethodmanalysis-method correlation for the previous delta-wing example was
repeated a number of times with various element arrays being used to represent the
wing. In figure 24, force data and aerodynamic center are shown as a function of the
number of elements. Inset sketches illustrate the planform representation for different
numbers of elements. The dotted line simply indicates a constant level (for reference
purposes) to which the results appear to be converging. Converged results consistent
with the validity of linearized theory seem to be attained with about 300 to 1000 elements.
Although the correspondence of the design and analysis methods has been improved,
essentially identical results are not obtained within reasonable computational times.
Therefore, care must be taken in the conduct of trade or sensitivity studies in which the
effects of relatively small changes in wing-design parameters are to be evaluated.
Either method could be used in the prediction of trends (for example, the variation of
drag-due-to-lift factors with sweep angle); however, any intermixing of results should
be avoided.
An indication of the computational time requirements as well as of the convergence
characteristics for typical applications of the present design and analysis methods is
given in figure 25. The clipped-tip delta wing with _ cot A = 0.836
was used for the
examples.
convergence.
these
factors
Drag-due-to-lift
factor
In the
of the
camber
to the
complete
are
design
applicable
ACD/_CL
2
was
surface
lift-drag
and
taken
in the
polar.
as the
quantity
evaluation
In the
case
used
of the
of the
to judge
flat
wing,
evaluation
23
of a specified camber surface, lift and drag were evaluated for the condition corresponding to a specified design lift coefficient, and thus the factor is applicable for only
one point on the lift-drag polar. The maximum number of wing elements employed was
selected to give indications of a converged solution for all projects. Computational
times shown here do not include the time required to place the program in core storage,
a time which varies considerably from one system to another. These calculations were
performed on the Control Data Corporation (CDC) 6600 computer.
Results shown herein indicate that, in instances where estimates of overall force
characteristics are sufficient as in conceptual design projects, adequate results can be
obtained in remarkably short times. Such a capability should be of use in the selection
of candidate configurations from large numbers of possible combinations of geometric
design variables. Detailed camber-surface descriptions and pressure distributions, of
course, require a better planform representation and considerably greater computational
times.
In the computer programs which now implement the numerical methods, emphasis
was placed on the development of straight-forward logic closely associated with the physics and mathematics of the problem; little attention was given to advancedprograming
strategies.
CONCLUDING REMARKS
In rather extensive employment of numerical methods for the design and analysis of
arbitrary-planform wings at supersonic speeds, certain deficiencies have been uncovered.
Recently, means of overcoming the major part of these deficiencies have been devised
and are now incorporated into the methods. In order to provide a self-contained description of the revised methods, the original development as well as the more recent revisions have been subjected to a thorough review in this report.
Revisions to the wing-design method have virtually eliminated irregularities that
often arose in the definition of the camber surface in the immediate vicinity of the wing
leading edge. An aft-element sensing technique has been incorporated into the analysis
method to suppress pressure oscillations which formerly required application of a powerful nine-point smoothing formula. Elimination of the need for the smoothing formula and
for the associated four-element trailing-edge extension now permits the handling of subsonic trailing edges. These improvements, in combination with more compatible summation methods in the design and analysis mode, have reduced small but disturbing
discrepancies which sometimes arose between wing loadings and forces determined for an
optimized wing and loadings and forces calculated for that same shape upon submittal to
the evaluation program.
24
Examples have been presented to illustrate changes in program results brought
about by the modifications and to show correlation with exact linearized-theory methods
where applicable. Application of the methods to sample problems indicates that, in
instances where estimates of overall force characteristics are sufficient, as in conceptual
design projects, adequate results can be obtained in remarkably short times (Central
Processing Unit CDC 6600 computer times measured in seconds). Detailed cambersurface descriptions and pressure distributions require considerably greater com )utational times.
Langley Research Center,
National Aeronautics and SpaceAdministration,
Hampton, Va., August 12, 1974.
25
APPENDIX
COMPUTER-PROGRAM
DESCRIPTIONS
Wing-Design
The numerical
the method
method
for the CDC
The wing-planform
and/or units.
the element
semispan
grid elements
N's
and
L's
50/_ SPAN/XMAX,
combination
6600 computer
data may
Reduction to program
planform,
both
for the definition of camber
for the selection of an optimum
and programed
arrangement
NON
surfaces for given loadings and
of loadings have been combined
(Langley program
be submitted
A4411).
to the program
scale is accomplished
NON
must
For a given
by selection of the number
and by the choice of design Mach
Thus,
in any convenient scale
by built-in logic.
is uniquely determined
is limited to 100.
whichever
Method
number.
The number
of
of
be less than 100 or less than
is smaller.
The user has the option of supplying wing leading- and trailing-edge coordinates as
tabular entries for a full series of successive
element
locations (y = (b/2)N/NON)
or definition points.
provide the necessary
scale.
composed
In the latter case, linear interpolation methods
The second option simplifies the handling of more
of straight-line segments.
stations corresponding
a linear chordwise
to selected integer values of
wise stations selected may
increased
computational
in program
A numerical
obtain wing lift,drag, and pitching-moment
This technique is simpler
employed
and is more
evaluation program.
N
When
of
The number
of span-
N's, but at the expense
of
trapezoidal-integration technique is used to
the spanwise-section
data.
to the integration techniques used in the wing-
it is desirable to reduce
of spanwise
computational
stations, care must
with complex
leading-edge
breakpoints or regions of rapid curvature
spanwise
but only at spanwise
(JBYS).
coefficients from
directly comparable
selection, especially for wings
closely spaced
conventional plan-
in application than the linked cubic formulation previously
of a relatively small number
leading-edge
as exemplified
loading is applied.
be as large as the number
time.
to
A similar option is provided for the descrip-
Surface slopes are not calculated for every wing element,
time by employment
be exercised
shapes.
in their
In the vicinity of
it is necessary
stations than for other locations because
shapes that are often called for.
26
are exercised
full set of leading- and trailing-edge x-ordinates
tion of the specified area to which
to program-
or as tabular entries for a selected series of break
The first option is appropriate for wings with continuous curvature
by the ogee type.
forms
span stations corresponding
to have more
of the severe
surface
APPENDIX - Continued
By employment
may
of selector
be considered
the
moment
the
first
and/or
three
additional
primary
and
stations
units
JBYS
chord
by
additional
program
results,
subject
selection
are
given
fractions.
characteristics
Additional
printout
data
factor
surface
has
programed
Wing-planform
data
method.
for
the
wing-design
full
set
of leadingThe
wing
locations
may
and
the
behind
moment)
set
form
to the
The
used
numerical
controlled
number
may
purpose
of the
area
are
given
of the
as
desired.
Lift,
drag,
reference
area
are
to the
program
user
has
that
at least
one
to an optimum
in any
selected
leading
for
and
as wing
the
desired
spanwise
edge
as well
in terms
of
aerodynamic
optimized
coefficients
wing
for
number
data
for
all
design.
the
to be
for
program
same
set
span
as
the
described
planform
by a
of breakpoints.
of ordinates
stations.
or
camber
A4410).
manner
of defining
a set
a given
at specified
A factor
ordinates
(RATIO)
in parametric
definition.
wing
as
an array
of rectangular
semispan
very
be discussed
coefficients
as
ordinates
10) or
a reference
in the
of selected
of desired
than
(Langley
option
is supplied
a set
of the
the
loading
or by a selected
wing-planform
(less
grid
large
(greater
elements
elements
NON.
than
50) depending
is
This
on the
later.
area
and
a corresponding
expressed
in terms
span
of the
allow
arbitrary
the
reference
units.
the
planform,
a function
wing
given
computer
nondimensionalized
as in program
same
the
for
as will
results,
with
expressed
of pressure
ordinates
chord
small
input
the
6600
definition
calculation
Program
wing
CDC
trailing-edge
of the
aerodynamic
as well
the
representation
be very
Additional
resultant
for
submitted
in the
by selection
of applying
Method
determination
Again,
to convert
scale
option
it is believed
at the
of interference-drag
the
are
of local
be employed
be
characteristics
for
camber-surface
in percent
available
pairings.
method
been
the
corresponding
Ordinates
Wing-Analysis
numerical
has
problem
may
RATIO.
pitching
loading-distribution--camber-surface
The
loadings
matter,
surface
restraints,
aQrodynamic
include
also
optimization
camber
of distance
and
eight
restraint.
of the
drag,
user
a practical
in any
the
in terms
of the
The
As
to certain
Section
(lift,
process.
be included
each
of loadings
combination
restraint.
should
for
any
optimization
z-ordinate
loadings
combination
local
the
loading
The
scale
in the
codes
pressure
are
of standard
and
obtained
coefficients
tabulated
percent-chord
pitching-moment
for
both
the
for
for
each
the
camber
of the
program
stations
for
coefficients
cambered
and
surface
elements
selected
for
the
flat
wing
and
and
semispan
program
from
for
area
a flat
are
also
stations
if
and
a
for
program
summations.
27
APPENDIX
Interference-drag
pressures
spanwise
28
are
used
lift-distribution
coefficients
in the
definition
data
are
= Concluded
between
the
of tabulated
also
provided.
flat
and
lift-drag
cambered
polar
wing
data.
surfaces
Streamwise
and
and
REFERENCES
i. Baals,
Donald D.; Robins,
A. Warner;
Integration of Supersonic
pp. 385-394.
2. Carlson,
Harry W.;
and Harris,
dynamic Analysis.
1970, pp. 639-658.
3. Bonner,
E.:
Aircraft.
Analytic Methods
Expanding
F.A."
A Unified System
in Supersonic
5. Carlson,
NASA
Method
Harry
Camber
1964.
W.;
CR-2228,
and Middleton,
for the Aerodynamic
Wilbur
TN
7. Middleton,
D-2570,
Wilbur
Optimizing
Wings
D.; and Carlson,
the Flat-Plate Pressure
NASA
Harry
Aerodynamic
Heaslet, Max.
in Linearized
NACA
2252.)
9. Mangler,
K.W."
Improper
2424, British R.A.E.,
10. Grant,
Frederick
Supersonic
C.:
Wing.
Wing
Warren
NACA
and
W."
Method
for the Design of
Planforms.
A Numerical
NASA
Method
Wings
TN
D-2341,
for Calculating
of Arbitrary
Planform.
NACA
Numerical
Method
of Estimating
of Arbitrary
Planform
NASA
and
Wings.
1965, pp. 261-265.
NACA
B.:
Integrals and Integral
Rep. 1054, 1951.
Combination
Rep. 1275, 1956.
A Method
RM
W."
(Supersedes
Rep. No. Aero.
1951.
Wing
TN
of Lift Loadings
(Supersedes
S.: Numerical
Camber
for Least Drag
NACA
Method
TN
on a
3533.)
for Design
of
With Constraints on Pitching Moment
D-7097,
for the Design
Specified Flight Characteristics
(Supersedes
A Numerical
Characteristics
Theory.
Proper
Supersonic
A.:
Analysis of Wing-Body-
Integrals in Theoretical Aerodynamics.
June
The
Surface Deformation.
12. Tucker,
SP-228,
Aircraft Design.
Pt. I - Theory
A.; and Fuller, Franklyn
11. Sorrells, Russell B.; and Miller, David
Minimum-Drag
D.:
Distributions on Supersonic
D.; and Carlson,
Equations
TN
NASA
Aero-
1965.
Supersonic
Harvard;
Flow.
With Arbitrary
Harry
J. Aircraft, vol. 2, no. 4, July-Aug.
8. Lomax,
of Supersonic
Pt. I, 1973.
Surfaces of Supersonic
6. Middleton, Wilbur
Design
1971, pp. 347-353.
Tail Configurations in Subsonic and Supersonic
Application.
Aerodynamic
in Aircraft Aerodynamics,
Role of Potential Theory
An Improved
Roy V., Jr.:
J. Aircraft, vol. 7, no. 5, Sept.-Oct. 1970,
Roy V., Jr."
J. Aircraft, vol. 8, no. 5, May
4. Woodward,
and Harris,
and
1972.
of Sweptback
at Supersonic
Speeds.
Wings
NACA
Warped
To Produce
Rep. 1226, 1955.
L51F08.)
29
13. Carlson,
Harry
of Highly
NASA
Swept
TM
14. Robins,
Aerodynamic
Arrow
X-332,
Nov.-Dec.
15. McLean,
Employing
Odell
Aerodynamics
1966,
Francis
dynamic
Wings
Morris,
in the
Characteristics
pp.
E.;
and
To
17-19,
1963,
NASA
and
Stewart,
H.J.:
17. Cohen,
Lift
Doris;
and
NACA
18. Cohen,
Wings
Speeds.
and
Drag
TN
Doris:
With
J.
Harris,
Roy
2.05
of Twist
V.,
Vehicles.
of Thin,
W.:
of a Series
and
Jr.:
Camber.
Recent
J. Aircraft,
Application
Supersonic
Research
vol.
TM
X-905,
Studies
1963,
Aerodynamic
Sci.,
Morris
Sweptback
D.:
Wings
of Wing
Performance.
Feasibility
Aeronaut.
Friedman,
2959,
and
Harry
Improve
September
Supersonic
A:;
Carlson,
on Supersonic-Transport
E.;
Degrees
Number
3,
no.
6,
573-577.
Interference
A.
Various
of Supersonic
Conference
16. Puckett,
at Mach
1960.
A. Warner;
Results
3O
W.:
vol.
pp.
no.
Aero-
and
of NASA
Supporting
Research
165-176.
10,
Theoretical
With
and
Proceedings
Performance
14,
Warp
of Delta
Oct.
1947,
Investigation
Increased
Sweep
Wings
pp.
567-578.
of the
Near
at
Supersonic
the
Root.
1953.
Formulas
Leading
for
Edges
the
Supersonic
Behind
the
Loading,
Mach
Lines.
Lift
and
NACA
Drag
Rep.
of Flat
1050,
Swept-Back
1951.
-
Z
4
#Y
l
x
5
M ==_=_>
2
BY,P_7
0
-l
-2
-5
-4
0
x,_
Figure
1.-
Cartesian
coordinate
system.
31
¢z
Airflow
Figure
2.-
Graphical
representation
of the
influence
factor
Ro
Figure
3.-
Airflow
Graphical
element
representation
Lifting
Upwosh
of the
upwash
produced
by a lifting
element.
L_1
]
2
5
4
L
5
6
"7
8
10
9
4
4
3
5
2
2
0
N*,N
-I
-2
-5
-4
-4
0
I
I
I
I
I
I
I
I
I
I
1
2
3
4
5
6
7
8
9
10
x,,C
Figure
34
4.-
Grid
system
used
in numerical
solution.
¢._
-2
-t
0_
Figure
5.-
Numerical
J
representation
J
of the
influence
factor
R
(the
R
function).
_--Wing
leading
edge
N_
i
IS
Origina
I-_--]
I
I I
L___J
_
calculated
Assumed
L......
L-.._
distribution
L-
_Zc,s
ax
va ues
(L.)
'
..O I
_F-_-_--Extrapolation
from
L*+I
and
L* +2
\
c)z
ax
_
Averaged
i
L__J..___
value
for
L*
L__.
Original
value
for
L*
....
az
ax
--_/
il i}
I
1
/
I
6.-
Illustration
i
Smoothed
--l__t__
I
L
Figure
c)x
I
_zc's
(L*)
I
4_
of the
application
in
36
I
distribution,
the
of
wing-design
the
surface-slope
method.
smoothing
technique
_x
_Z
ACp
I
!
I
'
[
_
Figure
/
)_-io
_ _
//
/--
_
I
L,L*
I
Assumed
7.- Illustration
\_----.
I
value,
distribution,
Zc,s
.__
I
L
.
)
pressure
(L*)
.
Ox
63Z
ACp
I
.//F
/-_
i
Force
and surface
_'_
method.
coefficient
for the wing-design
of element
]
aZc,s
distribution
distribution
values
_Cp(L)
ACp(L*)
values,---_--(
Assumed
Calculated
0
Field-point
Assumed
loading
definition
Specified
j---Calculated
I
O/
_-
//_
/--
Surface
distribution,
loading
63Zc,s
,
slope
k_
I
I
representation
1
I
I
Calculated
volues,---_---(L
.
• Assumed
distribution
Assumed
Specified
determination
.
)
ACp(L*)
¢.o
co
ACp
oc
8.-
Illustration
(x') 2
chordwise
constant
Figure
ACp
Quadratic
ACp
Uniform
oc x'(x'-c)
loadings
chordwise
of component
ACp
Parabolic
x'
chordwise
ACp oc
Linear
for
ACp
the
oc
')
of wing
(x')2(t.Sc-x
chordwise
design
Cubic
Iy I
spanwise
ACp oc
Linear
camber
_
surfaces.
ACp
Specified
xo
area
y
2
spanwise
ACp _
Quadratic
_/_
l
&Cp for
Aft-element
elements
Figure
L*
9.-
Illustration
ACp
I
ACp [
of the
F-
_-
Fairing
of the
&Cp(L*)
wing-evaluation
application
L*
_Cp
of
I
method.
I
ummation
sensing
ACp
results
(L*, N*),
technique
&Cp, o (L*,
N*)
for (L" + 1, N*)
for
edge
in the
N*)
Z_Cp,b (L*,
&Cp
ACp
with
leading
elements
Preliminary
I_=L*
preliminary
(L* + 1, N*),
preliminary
preliminary
for
for
_
I
-_sWing
F--_I
I
I
I
---I----_
aft-element
of
Preliminary _Cp
Preliminary
&Cp
i _-_ - Integration
._-_'_--
//
-_-_---_
_--Final
i ///f_-
i
_]_
(L*,N*)
sensing
1-Final
ACp for (L*-I,
N*)
....... o- Pr,eliminary
&Cp for
inal
all
e00e
0
I
c_z
_"c rL. _
c)X _ J
ACp
£i#tre
10.-
____4_
I
]
Ittustration
L,L*
Calculated
/---Assumed
K3-
,
for
of etement
the
slope
wing-evaluation
surface
values, ACp(L*)
distribution,
, ......
L._
i
_--__.
Force
method.
L*
distribution,
representation
L*)
distribution
_x
O_Zc
values, Z_Cp(L")
Assumed
.... Calculated
_'_-._-
values,
- Assumed
_----Calculated
determination
and pressure-coefficient
_x
values,
c3Z
Calculated
J
Ox
,
surface
- Assumed surf?ce
- Specified
definition
_Z
__t
Loading
az
c
_L,d
C
z _x
Oz
#CE,dz JT
ctx
'_PCL,d !
•
Figure
0
_.,l
0
"1
-]L
"
0
c _z "
I
t
Present
.4
camber-surface
II.-
.2
Previous
and
x'/_
.6
I
slopes
i
(a)
.8
previous
method
I
required
to
design-method
support
# cot A = 0.4.
1.0
.448
.483
.517
b_
!
i
.2
for
a uniform
results
0
the
.6
I
method
definition
x'/c
load.
.4
1
Present
of
.8
I
theory
methods
linearized
Numerical
Exact
I
1.0
.448
.483
.517
_
az
/_CL,d.7 c_x
c
1
0
!
I
.2
I
•
0
it-
o._.
I
.2
:L
.4
I
Previous
x'/c
.6
l
method
I
l
_ cot
11.-
Figure
i.O
(b)
.8
Continued.
A = 0.6.
.457
.483
.514
Y
b/2
I
0
_"_
I
.2
__
I
,4
x'/c
.6
I
__,__
method
Present
theory
methods
lineerized
Numericol
Exoct
I
.8
I
1.0
•457
.483
.514
-b/2
az
#x
az
o_x
c
/_CL,d.Z
c
/_CL,dZ
--.I
1
"
0
.i
.2
0
I
0
_L
I
•2
.2
L
,t
"
.4
I
.6
I
method
X'/C
Previous
I
Figure
(c)
.8
I
11.-
_cotA=
1.0
0.8.
Concluded.
.463
.487
.512
b_
1
0
S_A
"1
.2
.4
1
Present
X_'C
.6
I
method
methods
linearized
Numerical
Exact
.8
I
theory
I
t.O
.463
.487
.512
b/2
-.08
0
- .08
,BCL,d Z_.O 4
Z
.O4
--.08
,SCL,d l -. 04
Z
Xl/C
1
(a)
I
l
o m
Figure
I
Present
.4
camber-surface
12.-
I
.2
I
0
I
and
.6
ordinates
previous
.8
required
results
for
I
.4
load.
of
I
.8
I
.6
definition
X'/C
the
a uniform
I
.2_
I
0
_,_ju,___<,
to support
design-method
# cot A = 0.4.
1.0
.448
.485
I
1.0
.4 48
.483
,517
.5t7
0
/_CL,d Z -.04
Z
Y
b/2
method
Present
.O4
method
theory
methods
lineorized
Numericol
Exoct
Y
b/2
Previous
/
¢.yl
/_CL, dI
Z
0
-.04
- .02
0
-.04.
-.02
-,04
I
.2
I
0
I
.4
x'/c
I
.6
I
I
/3 cot
12.-
Figure
1.0
(b)
.8
Continued.
A = 0.6.
.457
.483
.514
0
,8CL,d Z -. 02
Z
Y
method
.02
Previous
I
0
I
.2
.4
I
Present
Xl/C
.6
I
method
.8
I
theory
methods
linearized
Numericol
Exact
I
t,O
,45;7
.483
.514
Y
b/2
0
0
-.02
/gCL, d Z -. 0 i
Z
,0t
-.02
-.01
0
,01
-.02
I'
.2
I
0
#CL,
dZ_ot
o,____
z
•
_.
.4
t
Previous
X'/C
-.02
,6
I
-o3
-o
method
12.-
Figure
Concluded.
A = 0.8.
.463
•487
# cot
1,0
I
"m
Y
b/2
.5t2
(c)
.8
i
"'"
I½
J
I
0
t
.2
.4
[
Present
X_/C
.6
I
method
methods
lineorized
Numerical
Exact
.8
I
theory
I
t .0
.465
.487
.5i2
Y
b/2
ZC,S
.CL,
Zc,$
d
CL,dZ
ZC)S
_
CL,d [
13.-
Figure
I
.25
0
,2
•
T
0
_.4
0
.2
._
-'2
0
I
.50
I
.75
0
I
Y
b/2
.50
wing
solutions
.25
I
spanwise
t-
Lineor
on an arrow
Numerical-method
Y
b/2
Uni form
with
for
I
0
A=
camber
.75
.25
I
Y
b/2
70 °.
M=
I
.75
to support
2.0.
.50
1
chordwise
surfaces
f
Linear
0
I
I
.25
various
)
Optimum
loadings
Y
b/2
I
.50
I
.75
.75
.50
.25
x/z
co
CL, d Z
CL,d z
ACp
.6
.8
.8
t.O
I
t.O
/h
t.0
2
CL, d
combinations
Numerical-method
Y
b/2
I
.6"
x/Z
.5
14.-
.4
,4
I
•2
.2
xlt
Loodings,
AC D=.580
0
J
Figure
--.6
--.4
.2
0
.2
-1
0
1
2
3
3
....
0
xlZ
.2
.2
5
of loadings
.6
f
.5
for
8A
an ogee
camber
.6
x/_
Y
b/2
for
.4
.4
Loadings,
AC D :.565
......80
1.00
,60
.40
.20
solutions
--
.......
m_
--
wing.
surfaces
t,0
I
O
x/t
2
CL, d
I
.4
//]
M = 2.0.
I
optimum
Y
b/2
.5
.6
x/Z
,8
1.0
I
t.O
AC D.6= .348 x/zCL, d
Load ngs,
to support
0
.2
//_
7
2
_D
,8hCp
(2
-.08
0
O8
16
24
Figure
0
+
I
15.-
.2
°O
.4
I
x'/c
.6
I
Pressure-smoothing
•
Without
method
I
methods.
i
I ,0
Flat-arrow
characteristics
(a)
.8
cot A = 0.4.
smoothing
+ Terminal smoothing
(9 pt formula)
•
Previous
.345
b
I
wings.
of present
0
f
o,+
i
.2
and
I
.4
I
.8
I
smoothing
evaluation
xYc
,6
• Integral
method
previous
Present
theory
methods
linearized
Numerical
Exact
I
,1..0
0
C[
pACp
-.08
0
.08
,.t6
,24
I
.4
I
.2
t
0
-
Without
smoothing
method
X'/C
.6
I
(b)
.8
I
Figure
+ Terminal smoothing
(9 pt formula)
•
Previous
j___J_--J
J
J
1.0
l
15.-
Continued.
I
0
f
# cot A = 0.6.
- _
o
_, -'l-
.2
I
.4
i
x'/c
theory
.6
I
smoothing
method
• Integral
Present
methods
linearized
Numerical
Exact
I
.8
I
1.0
I--L
/3ACp
-.08
0
.O8
.24j
0
++
.2
•
I
.4
x'/c
.6
I
I
(c)
.8
-
Figure
Terminal
smoothing
(9 pt formula-)
+
I
Without
smoothing
method
•
Previous
//
I
= 0.8.
0
15.- Continued.
#cotA
t.0
I
•541 b
.,+
1
.2
I
.4
theory
X'/C
.6
I
smoothing
method
• Integral
Present
methods
I inearized
Numerical
Exact
J
.8
I
l .0
/_ACp
-.t6
-.08
0
.08
.16
.24
T
.2
0
+
.4
t
X'/C
I
.6
smoothing
formula)
+ Terminal
(9 pt
smoothing
method
Without
•
Previous
I
+
•
I
+
+4-
•
,
(d)
,8
I
4-
•
Figure
4-ff
•
_
_L
1.0
J
-4-
•
,525 b
J
15.-
1.0.
i
.2
.4
l
• Integral
Present
theory
X'/C
1
.8
I
.6
smoothing
method
methods
I nearized
Numerical
I
t
• ,+
Exact
0
Continued.
#cotA=
+
J
I
t.O
BACp
0
.O8
.16
.24
0
I
.2
1
.4
I
x'/c
°6
I
Without
Terminal
smoothing
(9 pt formula)
+
smoothing
method
•
Previous
(e)
!
Figure
.8
I
15.-
_cotA=
1.0
333
Continued.
1.2.
0
o,+
.2
I
• Integral
Present
.4
I
Xt/C
I
.8
A
I
theory
.6
smoothing
method
methods
linearized
Numerical
Exact
I
t.O
/3ACp
Cl
-.08
.08
,.16
,24
0
I
.2
0
-
q
.4
x'/c
.6
I
Terminal
smoothing
(9 pt formula)
+
I
Without
smoothing
method
•
Previous
I
/3 cot
15.-
Figure
t.0
(f)
.8
t
F
0
@
Continued.
A = 1.4.
• 333 b
I
;
,,+
I
.2
@
theory
x'/c
.6
I
smoothing
method
I
.4
• Integral
Present
methods
linearized
Numerical
Exact
I
.8
I
1.0
¢jrl
/_ACp
C_
-.08
.08
_t6
.24
I
.2
|
0
+
Without
smoothing
method
.4
I
x'/C
.6
I
+ Terminal smoothing
(9 pt formula)
•
Previous
I
(g)
Figure
.8
I
15.-
I
1.6.
0
Concluded.
_cotA=
t .0
+,+
.2
I
• Integral
Present
x'/c
i
.6
smoothing
method
i
.4
theory
methods
linearized
Numerical
Exact
i
.8
i
I 0
7/I
\_-.
zzj,___i.
x,,
1.00
Previous
Present
method
_ACp
.t6
-
.08
-
method
x/t =:25
.16
Exact
/_ACp
cz
.08
-
linearized
Numerical
\
theory
methods
x/Z = .50
0
.t6
/_ACp
-(_
r-
.08
x/t = t.O
B
]___
- i .0
-.5
0
.5
,0
Y
b/2
(a)
Figure
56
16.-
Present
and
flat
delta
wings.
for
left-hand
panel.
previous
Present
method
# cot
evaluation-method
shown
for
A = 0.4.
results
right-hand
for
wing
pressure
panel;
distributions
previous
method,
on
1.00
Previous
method
,SACp
(_
Present
.08
method
x/Z=.25
_
Ii
_ACp
(2
I'll
Exoct
lineorized
Numerical
theory
methods
x/Z =.50
II
0
.16
-
x/Z = ! .0
0
I
-i
.0
I
-.5
0
.5
1.0
Y
b/2
(b)
Figure
fi cot
16.-
A = 0.6.
Continued.
57
.......
xlz
. .25
....
Previous
Present
method
/gACp
oz
.t6
-
.08
-
1.00
method
x/t =.25
0
,t6
/3ACp
O/
-
.08
-
Exact
inearized
Numer
cal
x/Z = .50
0 L
/3ACp
0l
.16
-
.08
x/Z : I.O
.0
(c)
Figure
58
_ cot
16.-
A = 0.8.
Continued.
methods
theory
.25
x/Z
.50
.00
/
Previous
\
method
/gACp
(2
Present
.16
-
.08
-
method
x/! =.25
Exact
linearized
Numerical
theory
methods
x/Z =.50
.16
,BACp.
c_
_ t1_
08
_eeoo
x/Z = t.O
"_hl'" ...............
0
I
-1.o
I
I
I
-15
0
.5
t.0
Y
b/2
(d)
Figure
fl cot
A = 1.0.
16.-
Continued.
59
/
t.O
/
Previous
Present
method
method
.t6
/3ACp
a
.08
x/! =.25
_
.16
Exact
Numerical
D
,%
x/Z =.5o
_
.16
_ACp
c_
.08
I
t
0
--°5
1.0
.5
Y
°
b/2
(e)
Figure
6O
/3cotA=
16.'
1.2.
Continued.
linearized
methods
theory
/
Previous
\
Present
method
t .00
method
.16
/3_Cp
(2
\ j
.08
I
x/I=.25
0 L
.t6_Exact
/3ACp
(2
.08
ineor ized
Numerical
theory
methods
x/Z =.50
0
,16-
PACp
(2
.08
x/t =1,0
0
i
-1.0
t
0
-.5
i
.5
i
t.O
Y
b/2
(f)
Figure
/3 cot
16.-
A = 1.4.
Continued.
61
1.00
Previous method
Present
method
.16
/_ACp
ol
.08
\ /
x/Z=.25
0
.16
Exact
/3ACp
linearized
Numerical
.O8
methods
x/Z : .50
0
.16
/gACp
Q_
:;v_=_:±c_c_:=_::v:
..........
ee
i
x/z : t.O
0
- t .0
--.5
.5
0
Y
b/2
(g)
Figure
62
_ cot
A = 1.6.
16.-
Concluded.
t .0
theory
Figure
0
.O2
.O4
.O6
.O8
0
17.-
Present
.4
I
.8
I
method
and
I
1.2
previous
/_ cot A
Previous
I
flat
delta
wings.
evaluation-method
1,6
0
results
I
I
.8
lift-curve
t .2
I
@
slopes
method
,8 cot A
Present
for
.4
theory
methods
linearized
Numerical
Exact
of
@
t .6
I
@
///
\\
/__--I-)-/_-
I--'_Previous
!--
method
ACp
a
.4_
-_,v/--_----_°
Present
I
.16
-
.08
-
x/l
method
= .45
i
i
oL
Exoct
Numericol
,.16
ACp
a
lineorized
theory
methods
.08
........
-_,
x/'f='90
0
"t.O
0
-.5
[
I
.5
1.0
Y
b/2
(a)
Figure
fiat
18.-
double-delta
method,
64
Present
for
and
wings.
left-hand
previous
Present
panel.
M
= v_.
evaluation-method
method
results
shown
for
for
right-hand
pressure
wing
distributions
panel;
previous
on
----1.0
Previous
method
Present
method
.t6
ACp
a
.O8
xlC=.50
0
Exact
•
.t6
ACp
a
lineorized
Numerical
theory
methods
.O8
I
I
0
.5
J
i .0
Y
b/2
(b)
Figure
M = 1.667.
18.-
Concluded.
65
Y
b/2
.10
ACp
a
.05
b/2
Y
_..o
le
I
0
_"'- _--_
.95
.10
ACp
• 05
0
-75
aCp
O_
0
.I0 F
a " .05
ACp
Present
I
0
.25
i
i
0
.5
Exact
numerical
linearized
method
theory
r
t .0
x:,'c
Figure
66
19.-
Present
method
results
ing and
trailing
evaluation-method
for
pressure
edges.
results
distributions
M = Vf2.
and
linearized-theory--lift-cancellation-
on a flat
untapered
wing
with
subsonic
lead-
\
/
/
/
/
/
•
/
/
/
/
/
/
/
/
/
!
/
Previous
....
method
.90
Present
method
o8i i
ACp
.04
•
(
o•
•',....,,..,.
x/Z = .30
0 L-
.O8
ACp
a
".
.04
o_
°•e
_
"*'e.,_.we...,o,.•,o.o,,_
x/! = .60
'''°
J
0
f
ACp
C_
.o8!
•
0o
.04 !
x/Z = .90
J
0 L
-t
1
.0
I
I
0
-.5
]
.5
t.O
Y
b/2
Figure
20.-
a flat
method,
Present
ogee
and
wing.
for
left-hand
previous
M = 2.0.
evaluation-method
Present
method
results
shown
for
for
pressure
right-hand
distributions
wing
panel;
on
previous
panel.
67
oo
Figure
wing
21.-
0
surface.
computation
40
Example
I
methods
from
of design-method
20
I
a........._..____
Previous
Sponwise
Correlation
0
.2
.4
.6
.8
1.0
ACp
/
z
/
reference
Evaluation
factors
with
Spanwise
20
I
present
40
and
I
methods
previous
results
computation
Present
evaluation-method
both
with
0
M =5.5
8 Ioadings
and
moment
restraint
11 treated
drag-to-lift
stations
I
60
a Design
s Evaluation
Design
methods.
for
stations
the
60
I
Design
same
Evaluation
Design
ACp__
Evaluation
::3 _
z __:_
_ ACp
M =2.0
:5 Ioadings
no restraints
.9
1.5
ACp
1.0
CL,d
.5
xlZ =.5
0
1.5
Design
ACp 1.0
CL,d
.5
x/Z =.6
Evaluation
0
ACp
1.0
CL, d .5
x/Z : .9
0
I
0
I
.2
_
I
,4"
t
,6
I
,8
1 °0
Y
b/2
(a) Delta
Figure
22.-
Correlation
wing;
three
of design-method
method
results
loadings;
specified
for
the
same
M = 2.0.
pressure
wing
distribution
with
evaluation-
surface.
69
Design
Evaluation
z
ACp
X
xlz
//.
.....
ACp
CL.,d
Design
ACp
t
CL,d
0
x/Z = .6
Evaluation
2
ACp
!
CL, d
0
x/! = 9
-!
I
0
I
.'_
I
.4
I
.6
.8
I
1..0
Y
b/2
(b)
Ogee
wing;
Figure
70
method
seven
22.-
loadings;
Concluded.
M
= 2.0.
method
Evoluotion
Design
ACp
z
ACp
/
\
M =2.0
"X
3 Ioadings
no
restroints
\\
.O8
cz _
.04
0
.O12
CL
----
.008
•
_C D
_CD/CL
Design
• 1000
•00393
.593
Evoluotion
. tO00
•00384
.384
2
O04
0
|
0
.2
I
I
I
.4
.6
.8
I
1.0
Y
b/2
(a) Delta
Figure
23.-
Correlation
wing;
of design-method
method
results
three
loadings;
spanwise
for
the
M = 2.0.
loading
same
distributions
with
evaluation-
surface.
71
Evoluotion
Design
ACp
z
M =2.0
\\
7
No
Ioodings
restraints
.08
CZ +
.04
0
CL
CD
Design
.tO00
.00578
.B78
Evaluotion
.0976
.00552
.570
.01 2
•
• OO8
Cd
c
z
.O04
0
I
0
.2
.4
.6
.8
1.0
Y
b/2
(b) Ogee
wing;
Figure
72
seven
loadings;
23.- Concluded.
M
= 2.0.
CD/CL z
Design
3
__
Evaluation
Ioadings
No Mrestraint
= 2.0
Design
Evaluation
.>
.ii
CL
.I 0
.O9
AC D
.004 _ ......
_
............
L
.003
.5O
AC---_°.40
CL 2
f .......
_2
....
$
• ...........
.30
Xoc
--
,o[
.65
.................
..
.........
".........
- ......=_..-_..
............
.60
t 0
t O0
t 000
Number
Figure
24.the
Correlation
same
wing
of designsurface.
and
Modified
of
elements
evaluation-method
delta
10000
wing;
aerodynamic
M = 2.0;
three
coefficients
for
loadings.
73
b_
O1
I
2
Figure
25.-
Time,
seconds
computer)
of
t
Y
elements
1000
/
I
for
typical
I
I
10
I
Number
of
of the
present
time
design
I
I
and
iO000
requirements
elements
I
Y
1000
/
,I
wing
I
I
flat
100
Evaluate
computational
applications
and
tO000
surface
characteristics
1O0
10
Number
I
I
cumber
I
Convergence
i
lO
1O0
1000
0
.1
,2
.5
Design
analysis
(Central
i0
I
I
of
methods.
I
Unit
I
I
CDC
lO000
wing
elements
iO00
I
cambered
Processing
Number
i O0
]
Evaluate
6600