sn
+=
THE
PRACTICAL
CALCULATION
CHARACTERISTICS
OF
OF
SLENDER
THE
AERODYNAMIC
FINNED
VEHICLES
by
James.
A Dissertation
and
Submitted
Architecture
Fulfillment
in Aerospace
of The
S. Barrowman
to the Faculty
Catholic
of the Requirements
of the
University
for
the Degree
Engineering.
March
Washington,
1967
D. C.
School
of Engineering
of America
in Partial
of Master
of Science
This dissertationwasapprovedby
Dr. ShuJ. Ying
Assistant Professorof SpaceScienceandAppliedPhysics
as Director andby
as Reader.
ii
Dr. Hsing-PingPao
ACKNOWLEDGMENTS
I would like to express my sincere appreciationto Dr. ShuJ. Ying andDr.
Hsing°PingPao of the Departmentof SpaceScienceandAppliedPhysicsfor their
helpful suggestionsconcerningthis dissertation, I would also like to expressmy
thanksto Mr. EdwardMayoinparticular andthe staff of GoddardSpaceFlight Center
in generalfor their helpfulnessandpatiencewhile preparingthis report.
iii
TABLE
OF
CONTENTS
Page
ACKNOWLEDGEMENTS
..............................................
iii
TABLE
..............................................
iv
OF
CONTENTS
LIST
OF
FIGURES
.................................................
v
LIST
OF
SYMBOLS
.................................................
vi
I.
SYNPOSIS
2.
INTRODUCTION
3.
NORMAL
4.
AXIAL
5.
COMPARISON
6.
CONCLUSIONS
..................................................
74
7.
REFERENCES
..................................................
75
APPENDIX
.....................................................
.................................................
FORCE
FORCE
A -
1
AERODYNAMICS
AERODYNAMICS
WITH
Exerpt
....................................
2
......................................
EXPERIMENTAL
of Aerodynamic
1
DATA
Theory
42
.............................
From
iv
Reference
62
1 ..................
77
OF
LIST
FIGURES
Figure
Page
3-i
Fin Dihedral Angle ..........................................
5
3-2
Fin Geometry
7
3-3
Proportionality Triangle for Chord
3-4
Fin Cant Angle .............................................
12
3-5
Forces
13
3-6
Roll Damping
3-7
Body
3-8
Approximate
3-9
Supersonic
3-10
Subsonic
3-11
Supersonic
3-12
Pitch Damping
4-1
Skin Friction Coefficient ......................................
44
4-2
Skin Friction With Roughness
45
4-3
.............................................
...............................
Due to Fin Cant .......................................
Parameters
Component
.....................................
Shapes .......................................
Tangent Body
....................................
I_¢r> ...........................................
XB(_}
8
14
19
22
33
............................................
34
×B_,r) ...........................................
38
Parameters
Supersonic C%
Variation
....................................
...................................
39
.....................................
49
4-4
Cylinder Drag Variation ......................................
51
4-5
Leading
52
4-6
Two
4-7
Nose
4-8
Three
5-1
Aerobee
350 ..............................................
63
5-2
Aerobee
350 Normal
64
5-3
Aerobee
350 Center of Pressure
5-4
Tomahawk
5-5
Tomahawk
5-6
Basic Finner ..............................................
69
5-7
Basic Finner Roll Damping
70
5-8
Aerobee
150A
.............................................
71
5-9
Aerobee
150A Drag Coefficient ..................................
72
5-10
Aerobee
150A Reynolds
73
Edge Drag ..........................................
Dimensional
Pressure
Base
Pressure
Drag at Mach
Dimensional
................................
54
One ................................
58
Base Pressure
...............................
Force Coefficient Derivative .....................
.................................
...............................................
Roll Forcing Moment
Coefficient Derivative
Moment
Number
.................
Coefficient Derivative ...............
.................................
V
61
65
67
68
LIST
Symbol
OF
SYMBOLS
Description
A
Area
AB
Base
Area
Base
Area
of One
Nose
Base
Area
Fin
AB r
Conical
Shoulder
or Boattail
Base
Area
AB s
Ar
Reference
AT
Planform
AwB
A WB^
Total
Area
Area
Body
Wetted
Wetted
Area
Nose
of One
Exposed
Fin
Nose
Afterbody
Area
of Booster
Wetted
Panel
Area
A WN
A WN^
Wetted
Area of Sustainer
and the Shoulder.
If there
der of the Sustainer.
Conical
Shoulder
or
Nose and the Cylindrical
Afterbody
Between
the Nose
is no Shoulder,
the Afterbody
is Taken
as the Remain-
Boattail
Wetted
Area
A Ws
A WsA
Wetted
the
Area
AR
Aspect
a
Ambient
B3
Third
CD
C DB
C%T
CD f
CDLT
CDp
CDTT
%.
of the
Shoulder
or
Ratio
Shoulder
of Exposed
Speed
Order
Irreversibility
Coefficient,
Base
Drag
Coefficient
Tail
Trailing
Edge
Drag
Skin
Friction
Drag
Coefficient
Tail
Leading
Edge
Dra_
Total
Drag
Tail
Thickness
Wave
Drag
the
of Sound
Busseman
Drag
and
Fin
Drag
Pressure
or Bottail
Boattail
D
Coefficient
Coefficient
Coefficient
Coefficient
Drag
Coefficient
Coefficient
vi
Coefficient
Portion
of the
Sustainer
Aft
of
Symbol
cf
Description
Incompressible
Skin
Compressible
Skin
Friction
Coefficient
Friction
Coefficient
Cf c
Rolling
CI
Moment
Coefficient,
_/qA r L r
(PL_
C,
Roll
Damping
Roll
Forcing
Moment
Coefficient
Derivative,
_c_/_
PL r
\TV]
_
2-_ : o
P
C_
C
C
Moment
Coefficient
Pitch
Moment
Coefficient,
Pitch
Damping
Moment
Derivative,
_/_ArL
_C_/_
,_ _ = 0
r
Coefficient
Derivative,
_cm/_
(_)
_
--qLr2V
= 0
q
%
Cp
C
CMA
Normal
Force
Coefficient,
Normal
Force
Coefficient
Pressure
Coefficient,
Cord
Length
Mean
Aerodynamic
Fin
Root
Fin
Tip
n/_A
Derivative,
_CN,/_
@_ = 0
5P '_
Chord
Chord
C r
C t
Chord
D
Aerodynamic
Drag
d
Diameter
d_
Diameter
of Nose
dS
Diameter
of Shoulder
or
F
Force
fB
Total
Fineness
Ratio
Body
Diederich's
f_
Equivalent
fN
Nose
Force
Base
Planform
Base
Correlation
F_neness
Fineness
Boattail
Ratio
Parameter
of a Truncated
for
Fins
Cone
Ratio
Fineness
Ratio
of the
Nose
and
Fineness
Ratio
of the
Shoulder
Nose
Afterbody
fN^
or
Boattail
fs^
h
Fin Trailing
Edge
K,k
Interference
Factors
KI
First
Busseman
K2
Second
Order
Order
Thickness
Busseman
and
General
Coefficient
Coefficient
vii
Constants
and
Shoulder
Afterbody
Symbol
Description
Third Order Busseman Coefficient
Length
Reference Length
Location of Shoulder or Bottail Measured from the Nose Tip
Fin Root Leading Edge Wedge Length
Nose Length
Total Body Length
Shoulder or Bottail Length
Location of Fin Leading Edge Intersection with Body, Measured from Nose Tip
Fin Root Trailing Edge Wedge Length
Roll Moment About Longitudinal Axis
Mach Number
Cotangent
rL
Pitch Moment About Center of Gravity
Number of Fins ( 3 or 4)
Normal Aerodynamic Force
Prandtle Factor, I/Prandtle Factor for Swept Fins, 1//1
- M~ cos2 rc
Roll Rate
Pitch Rate
Dynamic Pressure
Reynolds Number
Surface Roughness Height
Fin Leading Edge Radius
Body Radius at Tail
Exposed Fin Semispan Measured from Root Chord
Maximum Fin Thickness
Velocity
Body Volume
Downwash Velocity
Symbol
XCG
Description
Center
of Gravity
Distance
Location
Between
Fin
Measured
Root
Leading
from
Edge
Nose
and
Tip
the
Fin
Tip
Leading
Edge
XL T
Measured
Parallel
Longitudinal
Center
Y
Spanwise
Center
YMA
Spanwise
Location
Angle
to the
Root
of Pressure
Location
of Pressure
Location
of C_A Measured
Measured
Measured
from
the
from
from
Root
Nose
the
Tip
Longitudinal
Chord
of Attack
Supersonic,
- 1; Subsonic,
Midchord
Line
Sweep
Angle
Quarter
Chord
Sweep
Angle
VL
Leading
Edge
Sweep
Angle
rT
Trailing
Edge
Sweep
Angle
V1
First
C2
Second
Region
Boundary
Region
Sweep
Boundary
of Specific
_'_ - M2
Angle
Sweep
Angle
-y
Ratio
_L
Fin
Leading
Edge
Wedge
Half
Angle
in a Plane
Parallel
to the
Flow
Fin
Trailing
Edge
Wedge
Half
Angle
in a Plane
Parallel
to the
Flow
Incidence
A
of Surface
Fin
Dihedral
Fin
Taper
Mach
Heats
Ratio,
Ambient
Density
+
Tip
r t
ct
'C
r
Angle
Kinematic
s
in a Plane
Angle
Ambient
Nose
to Flow
Half
Viscosity
Angle
Ratio
r t
ix
Parallel
to the
Flow
Axis
Subscripts
B
Body
B(T)
Body
cr
Critical
N
Nose
i"
Root
S
Conical
T
Tail
T(B)
Tail
t
Tip
W
Wetted
1
Value
in the
Presence
of
the
Tail
Value
Shoulder
in the
for
or
Presence
one
Boattail
of the
Body
fin
X
1.00 SYNPOSIS
The
basic
objective
ic characteristics
guided
of slender
missiles.
derivative,
The
coefficient
coefficient,
CD.
made
to analyze
able
is a slender
The
fins
they
of pressure,
The
body
having
are
combined
Use
of established
effort
the
has
cases,
been
the
The
made
method
and
are
2.00
INTRODUCTION
2.10
BACKGROUND
In order
the
vehicles
mation;
turns
to study
but they
curacy
20,
both
is
relations
between
both
are
them
are
applic-
the
is accounted
fins
and
used
in deriving
available
to the
contempory
or simplify
complex
relations.
and the
that
the
for
body
the
are
equations.
engineer,
no
In many
results
the
theory
are
compared
predicts
to respective
aerodynamic
charac-
data.
or existing
Windtunnel
and
drag
No attempt
in mind.
vehicles
of a new
expensive
and
of work
time
developed
Jones,
application
though
they
to practical
reliability
speed
has
flight
vehicle
or flight
consuming.
armaments,
been
the
done
tests
Lacking
29,
later
22,
provide
calculations.
of their
slender
on the
a.number
technique
references
of conical
references
high
However,
reference
flow.
and
performance
the
damping
cq ; and the
flow.
to which
necessary,
theories
experimental
C_s; roll
derivative,
interference
readily
indicates
characteristics.
deal
Tsien,
Even
rocket
and
coefficient
applicable.
computer
comparison
of the
is exhaustive.
Morikawa,
cability
The
of rockets
reference
excellent
the
bombs,
force
the
can
engineer
prov{de
either
time
must
the
have
needed
or money
infor-
the
engineer
predictions.
A great
supersonic
and
are
day
of contributors
airships.
the
with
aerodynam-
derivative,
supersonic
methods
are
order
and
speed
normal
the
fins.
When
is a tool
higher
to sounding
percent
vehicle.
the
high
the
coefficient
or four
and
they
are
subsonic
semi-empirical
when
rockets,
configuration
three
of computing
coefficient
moment
general
computer
derived
aerodynamic
In this
Munk,
digital
data.
to theoretical
interest.
methods
is applied
ten
total
and
to linearize
flight
within
the
sounding
moment
both
separately;
theoretical
speed
equations
windtunnel
teristics
purely
high
analyzed
to form
Both
Since
for
region.
method
considered
damping
determined
transonic
body
as
forcing
C_p; pitch
are
such
_; roll
axisymmetric
subdivided.
is made
vehicles
a practical
characteristics
derivative,
and
is to provide
finned
Equations
the
are
thesis
aerodynamic
CN_ ; center
moment
when
of this
flow
of slender
modified
theory
3, 18 and
valuable
Their
results.
aerodynamics
of works
Munks
7, 8, 9, extended
to wing
17,
have
inherent
Those
Munk's
lift and
that
out and
theory
technique
and
to apply
work
drag
are
analyzed
many
of the
and
accuracy
The
of particular
applied
list
interest.
it to low speed
it to pointed
to finite
wings
projectiles
and
Nielsen,
wing-body
contributions
approximations
are
of considerable
configurations.
are
estimation.
assumptions
provide
vehicles
of such
stand
body
thoroughly
understanding,
finned
usually
also
Pitts,
Ferrari,
interference
have
effects.
limited
reduce
so complex
in
provided
applithe
acor
abstractthey cannotdirectly providethe desired aerodynamiccharacteristics. This thesisis an
attemptto form concrete,usefulexpressionsby applyingthe bestabstract theoreticaltreatments
as well as reliable empirical data.
2.20 STATEMENTOF PROBLEM
Determinea set of expressionsfor the practical calculationof the aerodynamiccharacteristics
of slender,axisymmetricfinnedvehiclesat bothsubsonicandsupersonicspeeds.Thecharacteristics of interest are the normalforce coefficientderivative, CN_
; center of pressure,X; roll forcing
momentcoefficientderivative, C_; roll damping moment coefficient
derivative,
C; ; pitch damping
_p
moment
2.30
coefficient
GENERAL
METHOD
1.
Divide
2.
Analyze
the
3.
Analyze
wing-body
4.
Recombine
2.40
3.0
derivative,
vehicle
into
body
The
angle-of-attack
2.
The
flow
3.
The
vehicle
4.
The
nose
NORMAL
There
rate.
These
Subdivide
either
when
derivatives
that
separately.
total
vehicle
necessary.
solution.
zero.
body.
point.
AERODYNAMICS
of aerodynamic
acting
normal
by four
excitations
forcing
near
irrotational.
is a sharp
generated
a pitch
and
is a rigid
tip
a number
are
& V; a roll
Cy ; and
and tail.
tail
CD.
interference.
is steady
forces
fins
body
is very
FORCE
are
on the
SOLUTION
coefficient,
ASSUMPTIONS
1.
aerodynamic
drag
OF
and
to form
GENERAL
Cmq; and
produce
moment
damping
main
the
coefficient
moment
coefficient
to the
longitudinal
sources;
normal
axis
coefficient
derivative,
C4_; a roll
derivative,
are
vehicle.
angle-of-attack;
force
coefficient
of the
fin
cant
derivative,
directly
dependent
The
normal
angle,
roll
rate,
CN_, at a center
damping
moment
C q respectively.
The
coefficient
only
on the
forces
normal
acting
and
pitch
of pressure,
derivative,
force
,p
acting
on the
body
is due to the
angle
of attac_k.
3.10
TAIL
NORMAL
FORCE
AERODYNAMICS
3.11
TAIL
NORMAL
FORCE
COEFFICIENT
In subsonic
force
coefficient
flow,
the
semi-empirical
derivative
of a finite
is the normal
force
at a center
of pressure.
DERIVATIVE
method
thin
This
plate
of Diederich
as;
from
reference
2 gives
the
normal
Af
(3-1)
2
+Fp
+--
4
where;
%_0
= Normal
FD
Diederich's
=
According
Force
Coefficient
Planform
Derivative
of a two
Correlation
dimensional
airfoil.
Parameter.
to Diederich;
:
F D
By the
thin
airfoil
theory
of potential
AR
1
2---#%=o cos Fc
flow
(reference
(3-2)
26),
(3-3)
C__a 0 =
Correcting this for compressible
27
flow according to Gothert's rule (reference 26)
27
CN ° - 13
(3-4)
Thus;
J_
Substituting
equations
(3-4)
and
(3-5)
into
C-Na
This
as
is the
The
given
value
for
one
Third
order
where;
Kz = 2/z
(T
+
I)
M 4
4Z"
-
4,_ 2
simplifying,
27 _ (Af "Ar]
--
(3-6)
fin.
expansion
of the
Cpi
=
)
and
value
of (_)_
for one fin in supersonic
in Appendix
A. It is essentially
a strip
Busemann's
K2
(3-1)
(3-5)
Fc
FD - cos
:
KI
flow is computed
using the method
of reference
theory
which analyses
two dimensional
strips
compressible
_i
+ K2
2
"_i +
flow
K3
3
"Oi +
1,
using
equations.
B3V_
(3-7)
K3 =
(W + i) Ms + (2_
B3 =
1
(5,+ I) M 4 [(S - 37) M 4 + 4('>'
- 3) M s + 8] 4-'8
The
strip
values
that
are
intersected
by
1 is
that
it
a thickness
To
used.
the
Only
Both
rives
the
Figure
are
A three
of
(3) fin
force
and
analysed
effect
S) M_ + 10('y+
6/3 7
summed
the
fin
mach
fin
to
that
the
The
most
the
analysis.
fin
the
analyzed,
is
CN= of tWO fin
in
applied
advantage
effect
since
a plane
panels
a correction
distinct
vehicle
are
one
with
of fin
these
parallel
at very
of the
the
must
most
flow.
angles
those
method
dihedral
are
to the
small
to
strips
in reference
be
con-
commonly
Reference
13
de-
of attack.
angle, A.
configuration
derivative
vehicles
assuming
the dihedral
in
a multi-finned
finned
on
direction
cone.
distribution
four
dihedral
i) M4 - 12M 2 + 8
in a spanwise
tip
C.N_ of one
three
3-I defines
normal
then
includes
apply
sidered.
are
- 7W-
has
two
effective
panels
at
a 30 ° dihedral
angle.
Thus
the
tail
is,
(3-9)
Since
the
normal
dihedral
force
angle
derivative
of
the
two
effective
panels
equations
a four
(4)
fin
configuration
is
zero,
the
tail
is;
=)T
From
of
3-9
and
3-10
we
can
4
(4
generalize
for
N
fins)
three
(3-10)
and
four
fin
vehicles.
_y
Figure 3-1-Fin
Dihedral Angle (As Viewed From Rear)
Substituting
3-6
into
3-11,
for
subsonic
flow;
(c__) T
Nv PR (Af/A)
V/4+ f\cos
2 +
3.12
TAIL
CENTER
From
a two
the
by the
pressure
the
of pressure
above
argument,
line
and
is considered
of the
mean
is also
located
the
of its
the
mean
center
with
shows
coordinate.
the
To find
fin coordinate
this
function,
is defined
mach
number.
equation
that
the
fin edges
--
its
leading
edge.
chord
line.
chord
of the
is located
at the
treatment,
the
It remains
of
Thus,
on
By definifin.
There-
intersection
subsonic
of
center
to determine
the
system.
c 2 dy
The
of
length
all
relation
c
L-y
straight
(3-13)
generalized
=
and
chord
is set
is a function
up.
See figure
of the
form
spanwise
3-3.
ct
L* -s
(3-14)
a trapizoid.
From
the
first
two
terms
of
3-14;
c = c - Y---c,
•
From
of pressure
f0
_f
a proportionality
are
center
quarter
aerodynamic
In this
the
as;
c,_
L_
It is assumed
from
the
of pressure
chord.
1
3-2
chord
along
26,
chord.
CMA
Figure
reference
is located
fin
constant
chord
length
aerodynamic
aerodynamic
aerodynamic
flow,
along
subsonic
mean
potential
the
of pressure
to remain
mean
at 1/4
center
the
'f
Fc]
LOCATION
of subsonic
is located
fin,
chord
and position
The
theory
airfoil
center
quarter
PRESSURE
airfoil
dimensional
fore,
the
thin
dimensional
a three
tion,
the
OF
(3-12)
the
first
and
last
terms
of equation
3-14;
CtL* : c rL* -c
Simplifying,
(3-15)
L*
S
and defining;
C
_. =_L
C t
(3-16)
l
IMA
,QUARTER-CHORD
LINE
\
)-CHORD
\
\
i
Ict
MEAN
ct
AERODYNAMIC
CHORD
Figure
3-2-Fin
Geometry
LINE
cr
-,_-Y _-_
L*-Y
L*
Figure
3-3-Proportionality
Triangle
for Chord
we find;
s
Lo =
(3-17)
Substituting 3-17 into 3-15,
(3-18)
c = cr (1 + ky)
where,
I
k
)_-I
-
(3-19)
----
L*
s
Using 3-18 in 3-13;
r
=_ c2
c_^
fo'
(1
+ ky)
2 dy
(3-20)
Integrating
C 2
r
---
[(1 +ks) 3 - 1]
c_^
- Af 3k
CMA
= r--LA f
1 +ks
+3
s
(3-21)
Substituting 3:19 into 3-21 and simplifying;
1
C2
CM^ = 3
•
S
[}2 +_+1]
(3-22)
+c t) s
(3-23)
Af
But;
1
A,=_(%
thus;
2
c2
[\_ +_+1]
CMA
3
C r
+
C t
or,
(3-24)
CMA
To find
the
spanwise
location
of
%^,
3-
r
='3"
equate
_
Ct
3-24
r+Ct
C r
Cr
and
+ Ct
3-18
and
=c r
+ C t
solve
1+
for
y.
y
S
or
(3-25)
c
Y_^
After
much
(% +c,)
I'_
2cr
3(%
spanwise
subsonic
center
of pressure
ICr
+2Ct
coordinate
:
can
c
s
-c,
is then;
be seen
from
figure
3-1,
the
+_-
rt
+
longitudinal
(3-26)
1
YT
As
"1
algebra;
s
The
ct t)
+c
--Ct
(3-27)
J
coordinate
is;
1
= 17 + lu^ + _-cM^
(3-28)
But;
YMA
J'MA
Combining
equations
3-24,
3-26,
3-28,
T_r : 17 +Yx
The
fin
center
of pressure
t
=
YMA
tan
and
3-29;
c
t
;+ 2cc--_-j
in supersonic
flow
r L = --
5
+
Cr
is found
10
s
x t
+c
(3-29)
.
Cr
by the
+
(3-30)
Ct
method
of reference
1 in appendix
A.
3.13 TAIL ROLL FORCINGMOMENTCOEFFICIENTDERIVATIVE
A force arises ona fin at zero angleof attackif the fin is cantedat an anglewith the longitudinal axis. Seefigure 3-4. The cantangleis an effectiveangle-of-attack. As canbeseenin
figure 3-5, the total force resulting from cantingall the fins at the sameangleis zero onthree and
four fin configurations.However,a roll forcing momentaboutthe longitudinalaxis is produced.
Sincethe cantangleactsas aneffectiveangleof attackthe roll forcing momentis subsonicflow
may bewritten in terms of (C_)_;
a
The
values
of (C_),
coefficient
and VT are
is defined
found
using
I
equations
3-6
and
3-27
respectively.
The
rolling
moment
as,
(3-32)
C_ : _A L
thus;
using
equation
3-26;
N YT CN_ 8
'
Cj_:
L
(3-33)
r
The
roll
forcing
moment
coefficient
derivative
is defined
C4
thus;
the
subsonic
roll
forcing
coefficient
as;
c_
:
derivative
(3-34)
is,
7r
In supersonic
This
method
3.14
TAIL
When
panel
duced
The
ROLL
the
on the
position.
and
free
fin
about
local
C_:s is computed
using
the
moment
has
stream
local
the
forcing
MOMENT
a roll
velocity,
velocity
the
longitudinal
moment
total
axis
at
the
method
the
a local
a local
force
roll
and
of the
force
rate
11
them
in appendix
in a spanwise
A.
direction.
DERIVATIVE
is zero,
_ is,
1 as given
sums
local
angle-of-attack
resulting
the
strip
superposition
produces
opposes
of reference
of each
COEFFICIENT
produces
angle-of-attack
configurations
rolling
roll
DAMPING
vehicle
The
four
flow,
determines
(see
which
distribution
as
tangential
shown
figure
3-6)
velocity
is a function
along
the
and
is a roll
fin
of spanwise
fin.
previously.
of the
For
The
three
moment
damping
promoment.
V
J
Figure
3-4-Fin
Cant
12
Angle
-'-F
(A)
F"
VIEWS LOOKING
FORWARD
F
T/,
F
(e)
F"
Figure
3-5-Forces
Due
13
to
Fin
Cant
TOP
VIEW
,.y
•
I
X
F(()_a(()
F
_v
LOCAL
S+r
VIEW
t
FROM
Figure
ANGLE
REAR
3-6-Roll
Damping
14
Parameters
-OF-ATTACK
dt : (F(_)
The
local
angle
of attack
(3-36)
at 4 is,
(3-37)
But;
v(_) : - P _7
Thus;
(3-38)
For
v >>P,_ the
local
angle-of-attack
is near
zero,
thus;
_,(,_) - - --
(3-39)
V
In subsonic
compressible
flow
the
local
normal
CN_°
is defined
in equation
3-4.
on an airfoil
- %,0 h ,(=)
v(:)
Where
force
Knowing
c(_)
,1-_
3-18
is found
using
equation
3-4.
(3-40)
that,
y - _ - ,-
equation
strip
(3-41)
gives
c(_') - c r
1 ---I •
(: _ rt )
S
or
c(4)
:
'
.
s
s
1----_ -_
15
-_ r
(3-42)
Combining
equations
3-36,
3-39,
3-40,
and
3-42;
Simplifying;
d- 1
=
-
CNa0
_
pcr
(1
-
4)
(L •
VS
where
L ° is
defined
by
equation
3-17.
Applying
÷
equation
rt
_
3-32
_)
_2
(3-43)
d_
provides
the
local
roll
moment
coefficient.
CNa° Pc r (1 -k)
(3-44)
dC,_ -: -
Integrating
over
the
exposed
A L
VS
(L"
÷ r t - f')
52 d_
fin,
C£ : k
/'[ S÷rt
J:
[(L
•
÷
rt )
-
4)
f]
_
z2
(3-45)
d_
;t
where;
pc
k
=
r
(1
(3-46)
ArL_
Performing
the
ks
k and
2
[(1
pc
roll
damping
+ 34)
s 2 + 4(1
+ 24)
sr t + 6(1
_ k)
(3-47)
r_]
simplifying;
C_ :
The
VS
integration;
CL : 12(1->0
Reintroducing
CNa 0
-
moment
r S
CNa 0
12 _ A¢ L r V
coefficient
[(1
derivative
+ 34)
s 2 + 4(1
+ 2k)
defined
as;
is
sr t +6(I
+_
r2]
(3-48)
(3-49)
\2V/IPLr
o
I-re °
16
Then,from 3-48,
crSCsa°[(1 +3k) s2
+4(1
+2>0
sr t +6(1
÷k)
(3-50)
r_]
6L2r Ar
However,
CN_° must
parameter.
From
moment
be
the
coefficient
corrected
for
results
derivative
for
is,
order
to
determine
angle-of-attack
appendix
To
insure
The
that
the
of
resulting
C_
is
using
3-1,
3-5,
the
and
Diederich
3-6,
the
correlation
subsonic
roll
damping
S
r
6L
distribution
A.
N fins
effects
equations
(3-51)
×--p
In
dimensional
3.11,
Nc
C;
three
of section
2
r
l
roll
damping
equation
rolling
truely
moment
3-37
moment
a linear
is
is
coefficient
introduced
divided
by
the
value
derivative,
derivative
into
PL
of
the
2v
%.x
in
supersonic
equations
to form
must
the
be
of
reference
roll
damping
near
zero.
flow,
the
1 in
derivative.
To
provide
_p
this
condition
we
assume
PL
(3
--:k
52)
2V
where
k is
any
defined
constant;
and,
k<<l
From
equations
3-37
and
3-41:
max
Using
equation
t Rll -I
3-52,
-_ tan
For
the
slender,
-
finned
vehicles
under
(3-53)
-I
consideration;
2(s _ r)
--:,4
L
Thus
mRx
- tan -_ (4k)
17
(3-54)
To satisfy the criterion,
:: tan
equation3-54dictatesthat,
k <_10
-a
Using
the
equality,
the
angle-
of-attack
variation
becomes;
.002
_(y)
This
variation
termined.
is
The
introduced
into
corresponding
the
equations
C_ _ is
found
. r t)
(3-55
L
of reference
1 and
a value
of the
damping
C
is
de-
by,
C;
For
(y
-
C_
N fins;
C/
(3-56
= IO00NC_
p
3.20
BODY
NORMAL
FORCE
COEFFICIENT
3.21
BODY
NORMAL
FORCE
DERIVATIVE
The
axisymmetric
cylinders
3-7.
and
It is
allows
will
conical
be
considered
since
For
a slender
state
that
running
there
over
assumption
of a slender
frustrums.
required
integration
not
body
the
entire
in this
the
axially
normal
load
no
force
symmetric
nose
A(×)
angle-of-attack
is
the
is
local
constant
shape
of the
From
coefficient
body
generally
is
usually
at
body.
conical
the
interfaces
The
effect
a static
derivative
in
a combination
subsonic
stability
of
flow,
or
tangent
of
these
a nose
shape,
ogival.
See
components.
compressibility
point
increases
of
of viewi
near
references
roach
15 and
is
figure
This
in
,this
circular
subsonic
flow
a conservative
one.
4 derive
the
steady
as,
n(x)
Where
is
discontinuities
length
analysis.
normal
vehicle
The
be
DERIVATIVE
crossectional
in
area
:'
= _,Y__x
of the
[A(x)
body.
×,
18
w(x)]
The
(3-57)
rigid
body
downwash
is
a function
of the
(a)
NOSE
CONE
OGIVE
(b)
CYLINDRICAL
SECTIONS
(
(c) CONICAL
FRUSTRUMS
SHOULDER
BOATTAIL
Figure
3-7-Body
Component
19
Shapes
For
angles-of-attack
near
zero
w:Va
Then
equation
3-57
becomes,
n(x)
From
the
definition
of normal
force
= _V2 a dA(x)
dx
coefficient
derivative,
Cn(x ) _ n(x)
qA
Applying
equation
3-60
_
1
n(x)
over
the
(3-60)
_V_A
to 3-59,
2_ dA(x)
Cn(x) : A
dx
Integrating
(3-59)
length
of the
(3-61)
body,
%=a_
d-_dx
._ °dA
__-2 _ fo '°da
(3-62)
Thus,
2_2
Cn : _-
By definition
of normal
force
3-63
(3-63)
derivative,
c_
equation
[A(Io) - A(O)]-
:
(3-64)
provides,
2
cN : _
Immediately
equation
ent of the body
shape
[AO0) - AC0)]
(3-65)
3-65 indicates
that
C_ is independent
of the body shape
as long as the independas long as the integration
of equation
3-62
is valid
over the body.
The
2O
bodycomponentsbeinganalysedare slender and haveno discontinuities in their crossectional
areaor its derivatives. Thus,equation3-65 maybe appliedto them. Sincethe nosetip is assumed
to bea sharppoint,
A(0). 0
Thus,the bodynormal force coefficientderivativein subsonicflow is;
Cu :B
_)
Reference
derivative
of pointed
theory
are
27 presents
approximates
each
bodies
the
analyzed
a very
method
of revolution
body
using
useful
by a set
for
of tangent
an exponential
series
Po, C, and
at the
pressure
leading
S° are
edge
constants
of the
distribution
for
frustrum
conical
(S0s
for
the
while
normal
force
second
order
See
figure
frustrums.
2
s
___
+Sl
The
pressure
2
Pc = pressure
= angle
( )2 = indicates
3o8.
expansion
The
s_+...)
s
frustrums
(3-67)
By applying
retaining
only
the
the
boundary
first
series
conditions
term
the
becomes;
(3-68)
where;
-- surface
shock
distribution
P = Pe - ('Pc -P2 ) e-_
_ =
coefficient
3
+S
frustrum.
at infinity
the
flow.
expansion
a particular
and
(3-66)
determining
in supersonic
P = - Po + C e
where
= 2 _ABN
<p -_os
_
coordinate
on a cone
between
the
having
surface
value
and
is taken
the
same
slope
longitudinal
at the
as
the
frustrum
axis
front
edge
21
of the
current
frustrum.
TANGENT
BODY
Figure
3-8-Approximate
Tangent
22
Body
The
notation
and
that
ized
of
reference
reference.
8 is
The
characteristic
used
value
equation
and
of (_P';:,s)
in axially
_P - k _
_s
T_s
defined
to
2 is
determined
symmetric
cos
1
_
simplify
correlation
by
an
between
approximate
this
solution
presentation
of
the
general-
flow,
_P
-,k
_C 1
(_
cos/a
sin_,sin_)
(3-69)
r
where
2)'p
sin
c,
The
2_
= First
Characteristic
= Mach
Angle
r = local
radius
, -
of
ratio
solution
Coordinate
specific
heats
is;
3P
B2
_')l
- sin
(3-70)
where;
"Tp.M 2
B -
2(M
2 -
1")
)
M
The
, ('21--)M_
_21
J
2{_-_
( )]
indicates
the
value
is
taken
at the
trailing
edge
of
the
preceeding
( )3
indicates
the
value
is
taken
at the
trailing
edge
of
the
current
value
of
(_P/.s)_
is
just
(:_P
':_s) 3
of the
previous
over
applying
the
equations
tangent
body
3-68,
is
3-70,
determined.
and
3-71
The
to
local
frustrum.
frustrum.
- P3
By
frustrum.
each
_P
element
nondimensional
23
in
succession,
normal
the
load
pressure
is
given
distribution
by;
A = n(x)
2
-
CL_ d
if"
TM_
d(P/P_)
--
JO
cos ¢d¢
(3-72)
da
where
4_ = Circumferential
d = Local
The
diameter.
derivative
to angle
Angle.
of the
of attack;
pressure
and
then
distribution
applying
d(P/P_)
-da
Using
3-73
in 3-72
and
is evaluated
physical
d(Pc/P
:
(1
-
by differentiating
restraints
to simplify
_)
X 2 d(P
e -v)
equation
the
3-68
with
1 /P_)
(3-73)
+ e -v
da
X1
respect
result.
da
simplifying,
(3-74)
X1
Where
"tcx"
indicates
by successive
initial
cone
Where
"v"
derivative
the
application
value
of an equivalent
of equation
cone.
The
3-74
to the
tangent
at the
initial
equivalent
running
body
load
elements.
distribution
The
is determined
value
of :xl, for
the
is just,
and
"tcv"
are
the
is determined
values
by the
longitudinal
integration
cone
of the
vertex.
running
The
load
normal
force
coefficient
distribution.
(3-76)
(CNT)B =_BB
A r dx
where;
AB = Body
1 = Body
The
initial
presented
cone
Base
Area.
Length.
and
in tables
equivalent
3-1
thru
cone
characteristics
are
3-3.
24
obtained
from
references
24 and
25; and
are
0
0
0
o
.,.d
r/l
,.....-i
i
I
..Q
0
0
p..
cxl
t_
0
0
12..--i
0
...-i
0
0
_Z_
_-
0
___
_
_
O0
_-
_
"_
O_
_
_
,-_
_0
_
_"xl
_
_
¢Q
_
_
_1_
• --_
o
LO
O0
[ _--
__-
_"
_
_.1_
_
_
¢_
O_
._
O_
_--_
_
_
•
_
•
25
_'-
_
O_
¢Q
"_
_
0
_
L_
_0
L_
I_
_
_t_
'_
[_"
_
_
_
_
:o
_
_1_
_-
_
C_1
,-_
o
'_ _
_
_1_
p-
_
_
¢Q
_
_o
,_
,g
_-
_
o
o_
C
0
c_
©
c_
,
0
D--
'_
L""
_
O_
0
_
_
O0
_
_
_D
_0 _
_1 _
_'_
Lt'_
co _
'_
"C'4
,_
_
_
_
o
_'-
_
_
_1"_
_
_
_
'_
L,_
_.
,_
.-, o_
_ -_
_
0
o_ ._
,_
o
_
.-,o
_° _
o_ _
o
_
0
,_
•
0
$
,_ _0 _; o
o
_- _. o
_- _
_;
_
0
_
o
•
o
_
_o
_
_.
,,_
_
_
o
_
o
26
0
0
t_
o
._
©
0
0
I
cO
A
0
0
co
¢xl
0
•
0o _
._ o0
_o _
•-_
•
_.
o_
0
._
_
_
¢_1
,--4
N
_
-
0
N
L_
u-_
CO
_
_
_.
,_
_o
_
_
,_
co
_
0
_
_l
co
o
0
¢o
_
0
_
_
_
_
_
_
_
_
tt_
_
u')
_
_
_-
o
_- _- _
_
_
_-
_
0
0
_o _
0
o
_
_
_- _
Ob
cO
_1
o
C_l
o
o
0
0
O_
o
o
_
_
_
o
0
0
_1 _
o
0
_
o
o
0
_o ._
t _"
o
0
o0 o
_.
o
27
co
_
0
o
3.22
BODY
As
be
tire
no
CENTER
in the
OF
subsonic
PRESSURE
body
discontinuities
normal
LOCATION
between
the
force
body
coefficient
component
derivative
shapes
in
analysis,
order
to
it
allow
is
required
integration
that
over
there
the
en-
body.
By definition,
the
pitching
moment
of the
local
aerodynamic
force
about
the
front
of the
body
(x = 0) is;
5('x_
The
pitch
moment
coefficient
is
defined
: xn(x)
(3-77)
as;
Cm : --5
A
(3-78)
L
thus,
_ xn(x)
_ Ar L
c
(3-79)
Since,
(3-80)
n
% -_A
equation
(3-79)
becomes
x %(x)
Cm(x)
The
value
equation
of the
subsonic
local
normal
force
L
(3-81)
coefficient
is
given
in equation
3-61.
Using
it in
3-81.
C (x)
Integrating
-
over
the
length
of
the
2ax
dA(x)
= Ar L_
dx
(3-82)
body,
C
_
m
28
A L
m
10X
28
(dA(x_
\--_x
dx
1
(3-83)
This canbe integratedby parts;
u : x
du
dv
= dx-----------v
(dA(x)_
\-_--x
/
:
dx
= A(x)
giving;
(3-84)
A L
But,
by
definition,
the
integral
term
is
the
oA(Io)-
body
volume,
C = 23
m
Ar Lr
By
definition,
the
pitch
moment
coefficient
A(x)
dx
therefore;
rtoA(10)
L
derivative
_ VB]
(3-85)
is;
(3-86)
O_
Then;
from
equation
[a:O
3-85,
(3-87)
(Cm:)s:A-_L
The
center
of pressure
is
determined
[10A(I0)-VBI
by,
(3-88)
Using
equations
3-65
and
3-87
in 3-88
and
simplifying
yields,
V B
Re : I o - __
%
29
(3-89)
The
volume
do not merit
readily
nose
formulae
repetition
available
shape
and
for cones,
conical
in this paper.
is given
is determined
frustrums
However,
herein
the volume
for the readers
in reference
and
circular
cylinders
formula
for a tangent
convenience.
The
are
volume
well
ogive
known
and
is not
of a tangent
ogive
12 as,
dN A(L)
(')
-
(3-90)
- 1
where;
L N
fN
In supersonic
flow, the method
of reference
27 determines
C
=---2-_
m
Where
flow
is
\(x)
is
found
then
determined
using
equation
by
the
of
A
the pitch moment
rx
coefficient
derivative
dx
(3-91)
AB d
3-74.
use
:--
The
center
equations
3-76
of pressure
and
3-91
location
in equation
of
the
body
in
supersonic
3-88,
(3-92)
\%/
3.30
TAIL-BODY
The
INTERFERENCE
effect of interference
of a slender,
finned vehicle
EFFECTS
flow
can
between
be expressed
the tail and
in terms
(C_,,)T< B} :
The
interference
factors
are
determined
by
(C,_,)
applying
vehicle.
3O
on the normal
of interference
(%')r
(C_)T(B):
the body
T
factors
force
aeTodynamics
(reference
23).
Kr(s)
(3-93)
kT(B)
(3-95)
slender
body
theory
to the
flow
over
the
total
as,
3.31 TAIL
IN THE
PRESENCE
OF
THE
BODY
Because the fins under consideration are of low aspect ratio, the value of KT(B} is available
from references 7 and 17.
(3-96)
Reference
23 determines
that the longitudinal tailcenter-of-pressure
is independent of the inter-
ference flow.
Xr_B> : XT
3.32
BODY
The
fins.
IN THE
only
PRESENCE
portion
Subsonically,
of the
slender
OF
THE
body
affected
body
theory,
(3-97)
TAIL
by interference
reference
flow
is the
portion
between
and
behind
the
3, determines;
(3-98)
where
KT(_) is given
by equation
be strongly
affected
by the fin tip
3-96.
mach
In supersonic
flow, the interference
cone.
The fin tip mach
cone intersects
(1)
fl _ (1 + _,) _
Until
this
mach
cone
in the
form
the
result
condition
is reached
impinges
on the
of curves.
of this
curve,
equation
body,
Figure
3-9
the
3-98
function
shows
the
is still
+1
valid
quite
of the
can
(3-99)
in supersonic
becomes
31
on the body
body when,
> 4
variation
K_(r_ ;
effect
the
complex
corrected
flow.
and
Once
is more
interference
the
easily
fin tip
presented
factor.
Using
Cu)r_(r)
By
of the
the
extension
of the
center-of-pressure
fin
lifting
location
line
of
: K'B(T):(1
into
the
the
tail
• ')
body,
(,_ - 1)
(3-100)
reference
interference
23
flow
over
derives
the
the
subsonic
variation
body.
--1
XB(T) = IT + "4_
Where,
_i_=
s > r and
0.
values
Slender
of
_. and
slender
body
curve.
Figure
pressure
;i_
_>4.
rt
•
f
v_(r)
the
Y--BeT) curves
3_=
0.
3-10
shows
the
interference
full
body
are
of
gives
the
body
interference
s
r
is
applied
-_B(T
the
In
by
I/4
(3-101)
subsonically
curves
found
tanF
.;_values
variation.
is
- r,l/
2
between
(Cm:JB(T )
--d+l
pitch
_
rt
interpolated
body
z
"7
,
intermediate
'_(r)
the
_(B(T)
23
-
theory
at
the
--
Reference
s
s
interpolated
subsonic
over
1
obtained
shape
flow
cosh
rs2)f_'--_
"-_ ,
are
The
r 2
slender
body
at
values
s
-
In addition,
sweep
value
of the
c
only
lifting
is
for
line
assumed
supersonic
applying
to find
the
at
For
other
and
similar
flow,
equation
_ = 1.
values
XB(T)
to
the
the
_ = 1
center-of-
3-92;
(3-102)
v
)
moment
derivative
as
derived
from
slender
body
theory;
-
8 (m_)3/2
dy
x cos -I
x
y
This
is
for
fins
with
supersonic
leading
(C
%
ma
= _
(T)
edges,
m_ > 1.
4,2m
A r _,'-_ ._32m
For
subsonic
foX?
leading
dy
2
-
1
32
v
dx
m)r_
-
y
edges,
02
< 1 and,
(3-104)
#_rn T
.,L
5 ,X
6
I
QO
2
(/94)(_+X)
(_+i)
>4
RADIAN ON ORDINATE
4
SCALE
q_
0.8X
,_.
0.6
( /
©
0,4
0,8
1.2
1.6
2.0
2 Brr
(CrlT
Figure3-O-SupersonicKB(T)
33
2.4
2,8
3.2
3.6
4.0
0.3
i
.....
EXTR APOLATION
T =O
04\
_6"
0.2
f
f
o.2,,,,,
/
_°
/
0.6 i
I'
O.l
0
(33
0.3
0.4
IX
,.E
rr
23
u3
u)
uJ
r'r"
(3_
I.J_
0
LIJ
I-Z
I.iJ
£b
\
O.2\
"r=O
x.
-wF ---..--. --. -
0.2
0.6 '2
.
O.I
0
I
0.3
_k_T=0,0.2,04,
J
0.2
0.6 --
0.1
2
Q) NOLEAOING-EDGE
SWEEP, X=O.
3
EFFECTIVE
4
5
ASPECT RATIO, _.,4R
(b) NO LEADING-EDGE
Figure
3-10-Subsonic
34
6
SWEEP, X =_--.(c)NO
XB(T)
7
8
LEADING-EDGE SWEEP, X=I
0.4
0.2
r- 0.6
h
A
0.3
k
\0.4
0.2
0
0.1
0
(d)
0.4
r =0.6
'_'----,"f
\
_,
16...,
,f
0
\ 0.4
\02
\0
(e)
0.3
\
0.2
r =0,0.2,0.4,0.6
0.1
0
(d)NO MIDCHORD
2
3
EFFECTIVE
4
ASPECT
5
RATIO,
SWEEP, X=O. (e)NO MIDCHORDSWEEP,
Figure
3-10
(Continued)-Subsonic
35
6
/_AR
I
k = -_-- (f)NO
XB(T)
7
MIDCHORD
8
SWEEP, k=l.
0.6
r =0.6
0.5
.ram.
_"0.4
0.4
\
0.2
0.3
0.2
0
(g)
0.4
"S_y'--
m
r =o6//I
J,.
o0.3
/
/
r
0.4
0.2
IX
J
/
0
_0.2
W
U_
o 0.1
nW
I--Z
W
o
h_
0
0.5
/
0.2
r :0, 0.2,0.4,0.6
0.1
0
(glNOTRAILING-EDGE
2
SWEEP, k=O. (h)
3
EFFECTIVE
4
ASPECT
NO TRAILING-EDGE
Figure
3-10
(Continued)-Subsonic
36
5
6
7
RATIO,/_.,_
I
SWEEP, X=--_. (ilNOTRAILINGEDGE
XB(T)
8
SWEEP, k=l.
Wherethe
coordinate
system
3-100
along
with either
3.33
CANTED
TAIL
The
has
3-103
IN
been
or
THE
in figure
in equation
PRESENCE
effect of fin cant at zero
at an angle-of-attack
given
3-104
OF
THE
angle-of-attack
alone. Reference
_(T_
3-2.
3-102.
is determined
The
result
by using
is shown
equation
in figure
3-11.
BODY
is determined
23 determines
in a manner
the corresponding
similar
interference
to the effect
factor,
[-
1
kT(B)
J_2
(T _ 1) 2
_72
,,,_-2
?2
. 1,)2
_2 _ 1
-r2 (T - 1) 2
2:'(_
1)
T(_
- 1)
2
,
(72
+ 1)2
(sin-,
T2 - I I
4(T
+ 1)
sin-,
('r2
- 1_
8
in (T2 + I_
* CT
---5
- I_
_--5_/J
No
center
since
only
3.40
TOTAL
The
of
the
pressure
roll
value
moment
NORMAL
need
derivative
FORCE
total normal
force
be
considered
is
considered.
for
the
canted
fin
in
(3-105)
the
presence
of
the
body
AERODYNAMICS
coefficient
derivative
is the sum
of all the components.
(3-106)
CN_
The
overall
center
of pressure
XB
CN_:
+
(B)
=
is the solution
(CN=_
; B(T)
of the moment
+XT(B)(CN_(B
)
sum
of the components.
+XB(r)(CN_(T)
(3-107)
The
3.41
value
PITCH
When
the
fins.
of
fins
total
zero.
DAMPING
the
on the
See
V is
figure
MOMENT
vehicle
free
The
has
stream
3-12.
The
an
angular
total
cz
value
is
COEFFICIENT
pitch
determined
from
equation
3-95.
DERIVATIVE
velocity,
the
superposition
velocity
produces
a local
angle
moment
produced
by
induced
the
37
of
of
attack
force
which
is;
the
local
is
pitch
constant
velocity
across
at
the
0.7
m#
I
=0.2
|
rn
LJ
_c
Go
Go
,,,
05
CC
EL
I,
0
rr
i,i
0.4
Z
(b)
0.3
0
4
).8
EFFECTIVE
1.2
1.6
BODY-DIAMETER,ROOT-CHORDRATIO
Figure
3-11
-Supersoni
38
c XB
(T)
2.0
2.4
C--;-
2.8
L
x CG
•
XT(B)
OVERALL
PARAMETERS
F
1
LOCAL
ANGLE
V
Figure 3-12-:Pitch Damping Parameters
39
-OF-ATTACK
in
(3-108)
FC.x]
where;
L_×
=
XTCB)
-
XCG
But,
(3-109)
and,
'
'(v)
a = l:an-
For
V >>v,
v
(3-110)
v
the
local
pitch
velocity
is,
(3-111)
v : q(Ax_
Combining
equation 3-108, 3-109, 3-110, and 3-111 and simplifying;
(3-112)
T(B)
Knowing
the
definition
of pitch
moment
coefficient,
equation
3-78,
c°:1%.,'_-/(v)
The
pitch
damping
moment
coefficient
derivative
is defined
(3-113)
as,
(3-114)
k2V/lqcr:0
2V
Thus, using equation 3-113 in 3-114.
4O
(3-115)
Equation3-123is equallyapplicableto three andfour fin configurationssincethe tail normal force
derivativeis correctedfor dihedraleffects.
3.42 ROLL DAMPINGIN THE PRESENCE
OF THE BODY
Anytail bodyinterferencefactor maybe expressedas the ratio,
s÷r
fv
,
%(_)
c(f')
d_
K:
(3-116)
s+r
_
(See
c(f_
df
figure
6) where;
:_,(O
-- the
angle-of-attack
variation
over
the
fin
in the
• 0(f)
= the
angle-of-attack
variation
over
the
fin
alone.
%(0
a tail-body
Reference
23 gives
the
value
of
for
a i (O
where
angle
%(_)
% is the
angle
of attack
at the
body
presence
combination
of the
at an angle
body
of attack,
_, as;
r_
: a + f2 %
surface.
For
(3-117)
the
case
of the
tail-body
combination
at an
of attack,
% = _ : %(0
However,
for
a roiling
vehicle
at a zero
angle
of attack,
equation
3-39
gives;
Pf
: -_-
%(0
and
Pr
V
%-
(3-118)
Then, equations 3-39, 3-117, and 3-118 give,
=i (O = - V
41
+
(3-119)
Therefore,the roll dampinginterferencefactor, kR_B)
, becomes
;
kR(B)
-_
f+
c(,f)d{
(3-120)
:
_+r
jr P
Simplifying;
+rs
c q-(5)d#
f_
kR(B)
Integrating
equation
3-121
using
the
chord
:
1
+
variation
given
The
roll
damping
moment
coefficient
(z + 1) (_ -X)
2
derivative
(C_p)T(B)
kR(B) is given
Where
4.00 AXIAL
The
axis
friction
only
FORCE
by equation
and pressure
yields,
111
(3-122)
(C,c,p)
then
be expressed
as,
(3-123)
kR(B)
3-122.
AERODYNAMICS
coefficient
of the vehicle
3-42
(1 -s)
(r a - 1)
3(_ - i)
can
=
by equation
1 -L
T-1
v-L
kR(B) = 1 +
(3-121)
s÷r
dependent
is the
drag
distribution.
on the
coefficient.
Since
aerodynamic
Drag
both
forces
mechanisms,
42
forces
arise
acting
from
and
parallel
two
their
basic
to the
longitudinal
mechanisms,
interactions,
are
skin
extremely
complex,mostpractical theoreticaldrag computationsmustbe temperedwith statistical and
empirical data. The majority of the equationsderivedin this sectionare either theoretical generalizationsof statistical-empirical dataor direct curve fits of experimentalresults.
4.10 SKINFRICTIONDRAG
The
the
skin
friction
dynamic
coefficient
pressure
and
is defined
the
wetted
as the
area
of the
skin
friction
drag
of a plane
surface
divided
by
surface.
_friction
Cf
4.11
INCOMPRESSIBLE
The
number,
and
skin
FLOW
friction
see
figure
coefficient
4-1.
by theoretical
SKIN
For
boundary
for
(4-1)
FRICTION'COEFFICIENT
incompressible
laminar
layer
_A.
flow,
flow
their
can
be expressed
has
been
relation
considerations,
reference
in terms
established
of the
both
reynolds
experimentally
6.
1.328
Cf
For
reynolds
then
turbulent.
the
experimental
work
Hama
numbers
of von
formed
greater
No good
data
Karman
than
about
theoretical
for
a simpler
skin
experimental
formula
have
which
and
form
The
of the
transitional
increment
has
curve
the
may
found
flow
applied
the
(R of)
agrees
with
becomes
for turbulent
Schoenherr
tog
transitional
flow.
the
turbulent
Figure
theoretical
flow
and
4-1
boundary
: 24_____22
0
equation
(4-3)
4-3
within
±2%.
R
be approximated
-
5.6)
2
by subtracting
an increment
from
equation
form
k
on experimental
transitional
curves
data
are
shown
layer
(4-4)
log
ACr
Based
shows
relationship;
1
(3.46
4.4.
layer
established
Cf :
The
boundary
been
friction.
data
(4-2)
_/-'_
5 × l0 s the
solutions
turbulent
to the
:
Prandtl
indicated
on figure
4-1.
that
(4-5)
=
k = 1700.
Combining
43
This
equations
fact,
4-4
as
and
well
4-5,
as
other
possible
0
I
I
I
[
I
44
/
0
./
o
o
o
o
"G
,:.,z
.E
LI-
I
"7,
,i
45
o
o
d
f
6
o
d
o
_o
0
O_
o
o
_o
o
rr
ii
u
_,
ii
Table
Approximate
Type
Surfaces
Surface
like
that
and
Optimum
Planed
Paint
in aircraft-mass
Steel
plating
cement
Surface
with
Incorrectly
Natural
Raw
asphalt-type
Average
metal
sprayed
surface
wooden
production
surface
Dip- galvanized
coating
surface
aircraft
of cast
paint
iron
boards
concrete
0.004
0.5
0.02
2
0.1
5
0.2
15
0.6
20
1
50
2
50
2
100
4
150
6
200
8
250
10
500
20
1000
40
surfaces
- bare
Smooth
0.1
surfaces
boards
surface
46
R s in mils
0
surfaces
paint-sprayed
of
0
.
glass
sheet-metal
wooden
Heights
Approximate
microns
of a "mirror"
polished
Aircraft-type
4-1
Roughness
Surfaces
of Surface
of average
Finished
Surface
Physical
Cf:
forR
1
1700
(3.46 log R - 5.6)2 - _
(4-6)
> 5 × 105 .
The
above
sublayer.
considerations
For
4-2.
The
ness
is reached.
are
appreciable
"smooth"
for
rough
curve
surface
surfaces
is followed
until
roughness
turbulent
skin
the
critical
reynolds
reynolds
independent
numbers
higher
of reynolds
than
number,
the
critical
reference
that
are
friction
submerged
in the
coefficient
number
laminar
is shown
corresponding
in figure
to the
value,
the
(4-7)
skin
friction
coefficient
can
be considered
6.
Cf = 0.032
The
flow
is considered
trips
transition
mate
average
4.12
COMPRESSIBLE
to be turbulent
even
at low
roughness
In compressible
remains
subsonic
unchanged
while
the
the
numbers.
for
SUBSONIC
(4-8)
throughout
reynolds
heights
rough-
IR \-i 039
= Sl (__)
\'_r/
(R•)_
For
sizes
the
reynolds
Table
physical
4-1
number
is a list
range
from
since
reference
the
roughness
6 of the
approxi-
surfaces.
VARIATION
flow
reference
"smooth"
6 indicates
turbulent
c,
value
that
varies
the
laminar
skin
friction
coefficient
as,
= c, (i - .o9 M2)
(4-9)
c
and
the
change
for
turbulent
flow
with
roughness
is,
Cfc = Cf (1 - .12 M2)
4.13
SUPERSONIC
Theoretical
friction
VARIATION
solutions
coefficient
in reference
16 indicate
been
the
variation
the
supersonic
variation
of the
Laminar
confirmed
of "smooth"
by experimental
turbulent
Cf
=
¢
has
that
skin
is;
Cf
This
(4-10)
(1 + .045
data,
(4-11)
m2) I/4
reference
Cfc in supersonic
6.
flow
47
Experimental
indicates
a strong
data,
reference
dependence
6, for
on the
wall
heattransfer as well as machnumber,figure 4-3. The wall temperaturealternatives shownare
the twopossibleextremes. Theactualconditionwouldprobablybeclosestto the recoverytemperature. The curvefit of figure 4-3 is,
Cf
Cf
=
(4-12)
c
where
k = .15 for
According
no heat
transfer
to reference
and
10, the
(1 + k M_)-ss
k = .0512
turbulent
for
skin
cooled
wall.
friction
with
roughness
drops
off according
to the
relation,
Cf
Cf
However,
flow with
4.14
(4-13)
=
c
I + .18 Ms
..
this equation
must be used with care.
It is apparent
roughness
should
never
be less than the corresponding
DRAG
Once
value
smooth
of c%
value.
calculated
for
skin
area,
friction
is found
coefficient
is determined,
the
skin
friction
drag
coefficient,
as
drag
the
fins,
the
coefficient
wetted
area
of each
fin
is twice
it's
body
skin
(4-14)
planform
:2NCf
ft
reference
on
area.
Thus,
the
tail
skin
friction
is,
CD
The
based
by,
C% = cfc (_)
For
the
COEFFICIENT
the
reference
the
friction
coefficient
must
c
be corrected
(AT_
\%1
for
(4-15)
the
fact
that
the
body
is not a flat
plate,
6.
A
CDfn =
4.20
PRESSURE
4.21 TAIL
Cfc Ar
(4-16)
DRAG
PRESSURE
The pressure
+
DRAG
drag on a fin in subsonic flow arises from three basic contributions: a finite
radius leading edge; thick trailingedge; and the overall fin thickness.
48
I,I
.i
,_-II
1.0
0.8
Twall=Tomb
0.6
{,3
=Trec.
0.4
)-
0.2
I0
0
Figure
4-3-Supersonic
49
Cf¢ Variation
II
has
A finite
radius
no base
drag.
an infinite
leading
edge
Figure
circular
4-4
cylinder
ference
between
these
plotted
in figure
4-5.
Subsonically
can be considered
from
reference
perpendicular
two
curves
up to M = .9 this
.9 < M < 1, the
curve
6 shows
to the
is the
drag
curve
is fitted
(1
AC D --
Between
-
is essentially
the
curve
may
flow
flow
theory)
M2) --417
and
the
of the
variation
of the
1
-
(M - .9)
be approximated
reference
variation
in cross
leading
total
of its
edge.
flow
that
effectively
drag
coefficient
base
drag.
This
difference
of
The
difis
by,
(M < .9)
(4-17)
(.9 < M _<1)
(4-18)
by a high
---+-.502
6) the
order
+--.0231
.1095
M2
cross
cylinder
linear.
ACD : 1.214
From
the
coefficient
ACD : 1 - 1.5
Supersonically,
a circular
M4
drag
reference
6.
(M> 1)
(4-19)
M6
coefficient
CD_"
polynomial,
= COS
3 F
= 2 LLE
rL
of a cylinder
inclined
to the
flow
is,
(4-20)
c1
The
frontal
area
of a typical
leading
edge
is;
ALE
the
length
can be expressed
in terms
of the
span,
S
LLE
cos
F L
thus;
ALE
2
A r
A r cos
S rL
(4-21)
From
equations
4-20
and
4-21,
the
total
leading
edge
FL
drag
of N fins
becomes,
(4-22)
CDLv = 2N (_--_-rrL)cos 2 FL (ACD)
where
ACo is obtained
The
rected
trailing
skin
friction
using
edge
drag
equations
is two
coefficient,
4-17
dimensional
reference
thru
4-19.
base
5, for
drag
incompressible
0.135
CDBT
and
=
50
_fB
can
be expressed
in terms
of the
cor-
flow.
(4-23)
I
I
1
l
1
[
i
'
2.5
]
1
CYLINDERS
1
"
ARC,
-
o AVA,
• NACA
BETWEEN
TUNNEL
WIRES
i TUNNEL
TUNNEL---
+ NACA,FLIGHT
2.O
--
-
÷
WALLS
TESTS
,, BRITISH,
TEST
STANTON
A DVL_SUBSONIC-----
"..
i'_
_.'
_t_,a +
-- _-...-; --.=..--_=j
-.=,e-..t_,
--,_;_
;
_e_A "_r
JUNKERS_TUNNEL
• NACA
"_.%
/DRAG
(38.
" ",..,_;_
%
•
/TOTAL
K)
SUBSONIC---
-
_ ,i._..
BASE
I.O
DRAG
/_ SUBSONIC
CYLINDER
.
SUBSONIC
CYLINDER
-
TRANSONIC
o
i
0.5
-
%
/I.43/M
_- SUBSONIC
z
BASE
D._G"'_,\/
SPHERE
,k NAVORD,
I
0
0.1
O. 2
0.3
_
0.4
I
0.6
MACH
[ J
0.8
I
I
-r_-_-7=-_-
2
:3
4
NUMBER
Figure
4-4-Cylinder
Drag Variation
51
6
I
8
I0
CYLINDER
SUPERSONIC
(CORRECTED)
SPHERE
J
Q
cJ
J
Lt_
Lr_
J
c_
I
Q
O
52
m
--
O
_
f_
cJ
O
co
f_
ta_
W
_J
L_
LL
where;
C%=2Ce(c)
For
subsonic
reference
compressible
flow,
this
may
be
(4-24)
corrected
using
the
prandtl
factor
for
swept
wings,
6,
1
Pf =
(4-25)
1 -M 2 cos 2 F c
However,
in
order
to
maintain
a finite
value
at
M2 cos 2 F c :
the
prandtl
factor
is
corrected
1
to,
1
P; -
(4-26)
K - M2 cos 2 F;
where
equation
K is a constant
4-24
becomes,
to be
determined
from
the
value
CfB:2Cf
After
correcting
to the
reference
area,
the
CDBT at M = 1.
of
Also,
for
compressible
(C )
subsonic
(4-27)
trailing
.135
edge
drag
coefficient
for
N fins
(4-28)
,/,< _ M2
figure
11
4-6.
pression
gives
This
which
a curve
is
for
the
two
the
base
drag
coefficient
within
the
cross
hatching
essentially
remains
well
is,
N ('a'Bft
\At/
Reference
flow,
dimensional
base
ro
pressure
refered
coefficient
to
the
base
at
area.
supersonic
An
analytic
speeds
ex-
is,
1 - .52M -t't9
M2
But,
this
must
be
corrected
for
possible
boattailing
of
the
fin,
1 - .52 M -_ .19
+ 18 Cf
53
reference
6.
(4-29)
M2
-0.80
,
|
-0.72
-0.64
-0.56
Z
Lul
-o.48
1.1_
i1
I.,iJ
0
0
-040
UJ
Od
u_
cO
"'
--0.32
(]C
(3_
L_
O9
m
--0.24
--0.16
NOT E :
REFERENCE
NUMBERS
ARE THOSE OF REFERENCE
II
-0.08
0
2
5
FREE-STREAM
Figure
4-6-Two
Dimensional
54
MACH NUMBER,
Base
Pressure
4
Moo
For
N fins,
In order
the
supersonic
to determine
trailing
the
edge
constant
drag
coefficient,
I<, equations
'135N
4-28
the
superscript
* indicates
corrected
4-30
,
values.
t
is used
This
the
method
remaining
From
sonic
in equation
flow
4-28
to evaluate
of maintaining
finite
the
values
set
equal
reference
area
is then,
at _1 = 1.
i
Solving
[
cos 2
18C
223
for
subsonic
of the
K,
. 4.02 C_ \l_7]j
Cr
This
are
to the
N(1 - 52)(Au_/\A/
_c
sonic
t<
and
QA_)
-
where
as
C_
]+
compressible
prandtl
(4-31)
factor
trailing
at mach
edge
one
drag.
is used
throughout
derivations.
reference
can
6, the
be expressed
drag
due to thickness
of two dimensional
airfoil
in compressible
sub-
as,
(4-32)
For
a three
dimensional
fin,
based
CDTT
on the
:c+
reference
area
this
becomes,
+(/, _M_
co,_i
55
(4-33)
Correctingit for a fin sweepandintroducingthe correction constantin the prandtl factor,
(4-34)
The value of the thickness drag of finitefins at mach
reference 6. A curve fitof the result is,
CDT
T. = 1.1S \cr]
one has been determined
and presented in
"
1.61 + _f -
¢('_ - 1.43) 2 + .578
-.
(4-35)
where
Solving 4-34 for K,
CDTT
K=_os_r
c + -
Equation 4-35 is used in 4-36 to determine
termine
the subsonic tailthickness drag.
t
A t
___t5
(4-36)
K. The result is used in the following equation to deFor N fins, equation 4-34 is;
30
4N
CDTT
At supersonic
equation
drag,
4-22.
C%r,
The
speeds
the
The
trailing
is computed
overall
drag
=
effect
on the
tail
Cf
cos
cos 2 7¢
r" +
c
(4-37)
(_-K
of a finite
edge
by the
t
radius
leading
_M2
edge
cos2
is found
drag
is calculated
using
method
of reference
1 in appendix
is the
sum
of the
different
equation
4-30.
by using
The
equation
thickness
4-19
in
or wave
A.
contributions.
In subsonic
flow
(4-38)
(CDT)T=CDfT+CDLT+CDBT+CDTT
56
At
supersonic
speeds,
CDT)T = CDf T ÷ CDLT ÷ CDBT + CDwT
Equation
4-38
4-39
uses
4.22
BODY
The
ing
uses
the
equations
results
4-15,
4-22,
PRESSURE
body
equations
4-30,
and
be
broken
4-15,
results
4-22,
4-28,
of
appendix
into
two
and
4-37
respectively.
Equation
A respectively.
DRAG
pressure
to reference
from
(4-39)
drag
6, the
may
forbody
drag
in
down
subsonic
contributions,
compressible
flow
for
may
be
body
and
base.
Accord-
written,
6 AWB C%
:
C D
(4-40)
p
Again
applying
the
correction
constant
to
the
prandtl
factor,
6 AWB C c
=
C D
p
From
experimental
figure
the
4-7.
In the
following
data,
range
reference
of values
19,
that
(4-41)
f3 A
the
forbody
are
of
drag
interest,
of both
1 <_ fB _ 10,
8s
D
cone
(C'p)D
ogive
equation
values
these
ogives
curves
and
cones
may
are
be
fitted
known,
by
equations.
i:.pl
Solving
M2) "6
(K-
4-41
for
(fN
+
(fN
+
:
"T)
(4-42)
1"29
.88
(4-43)
1")2'22
K;
K=I
_-6 AwB C_]
57
s/3
(4-44)
0
[
d
I
d
I
d
I
6
I
d
I
o
I
z
58
0
0
d_
I'o
0
Z
0
o
_E
a
D_
i
I.L
Then
K
The
subsonic
tion
4-41.
proper.
The
27,
is
as
compressible
For
the
AwN^
value
explained
integrated
forbody
nose
supersonic
6.82
!
AwB C_ (fN
6.82
__
.
Awe C_ (fN + 1)2"22ts/3
drag
fN^ should
determined
3.21
each
-
pressure
and
is
in section
around
=1+
by
of this
element
be
and
the
is
used.
The
over
:
by
For
axial
the
4
CDp
found
(4-46)
equations
a conical
shoulder
shock-expansion
component
length
P sin
(4-45)
s/3
using
second-order
thesis.
then
+ "731"29]
of the
,, d:
of
the
4-45
or
Aws^
and
4-46
method,
pressure
fs^
in equaare
reference
distribution
body.
d×
(4-47)
)_]2
where
P
comes
from
_, = the
_
local
equation
angle
= Circumferential
The
Base
drag
3-69
between
the
surface
and
the
longitudinal
axis
Angle.
of a body
of
revolution
in
incompressible
flow
is
presented
in reference
5 as,
(4-48)
where,
CfB = Cf
Correcting
this
to
the
reference
area
and
CDB
for
subsonic
:
compressibility.
(4-49)
59
Introducing
the
correction
constant
into
the
prandtl
factor
term.
0.29
(4-50)
(K -
From
reference
pressure
11, for
coefficient
no fins
at mach
near
the
body
base,
with
:
figure
The
thickness
results
of reference
11 indicate
mach
that
number.
equation
equation
subsonic
4-50
4-51
for
find
reference
the
base
_ 0.30
PB is roughly
in 4-52
over
11 and
11.
The
an initial
exponential,
the
a linear
function
of fin
trailing
edge
(4-51)
K,
compressible
References
ponential
body
0.185
and
(4-52)
+
simplifying
1
K :1 +
from
the
185+115
K=I
The
thickness,
Therefore,
c;B=E
Using
of zero
4-8.
at a constant
Solving
fins
h /c r = 0.1,
P_
See
or for
one is
P_
At machone
M23
body
6 both
free
range
free
base
present
flight
drag
is computed
curves
test
curve
and an inverse
flight
curve
of the
by using
base
is most
square
in figure
(4-53)
drag
4-8
60
may
the
4-53
in supersonic
representative
over
one.
equation
be fitted
flow.
of the
remainder
in 4-50.
of the
in terms
Figure
variation.
mach
4-8
is
It is an exnumbers.
of the value
at mach
To
-036
-----
FREE-FLIGHT
0
- 032
OF
ESTIMATE
B,
REF
\ _..-"W,THF,NS
FLAGGED
-0.24
ESTIMATE
LL
WITH
A /
_'_
FINS
MODEL
12, R :10
WIND-TUNNEL
--0.28
013
(3.
TESTS
FINNED
TO
12OxlO
DATA,WITH
0
REF.
26
0
REE
26
0
LANGLEY
SYMBOLS,
6
FINS:
9"SST
NO
FINS
¢
%_%_"
_
-0.20
D
L.)
W
rr
--0.16
(..f)
qJ
rr
CL
LLI -0.12
(.'3
(13
PRESENT
-0.08
ESTIMATE,
NO FINS, Mo_'M,:,_
I
- 0.04
NOTE: REFERENCE
1
0
1.0
1.2
NUMBERS
ARE THOSE OF REFERENCE
I
1.4
I
1.6
1,8
2.0
2,2
FREE-STREAM
Figure
4-8-Three
MACH
Dimensional
61
I
2.4
2.6
NUMBER,
Base
II.
I
Pressure
M
2.8
3.0
3.2
3.4
CBB :: CDB I0.88
where
M
variation
is the point
where
+ .12e
"J'58(M-11 ]
the two variations
(I _M_Mr,.)
(4-54)
Also, reference 5 gives the inverse square
cross.
as,
CDB
(M > Mcr3
M2
(4-55)
If it is assumed the ratio of the value of CoB at .M=M_rto C" remains constant for all curves, (inspection of the curves in reference 6 indicates this is a good assumption) then the value of the critical
roach number
can be found,
M
The value of C*
is given by equation 4-51.
z
0.892
(4-56)
__
In either subsonic or supersonic flow,
DB
COt)
B
For
subsonic
or 55 are
values,
equation
4-41
and
4-50
=
are
CDP
+ CDBB
used.
(4-57)
In supersonic
flow
equations
4-47
and
4.23 TOTAL
DRAG
The drag of the total vehicle is the sum
of the tailand body contributions.
(4-58)
:(%1,
+
No interference
5.00
4-54
used.
COMPARISON
drag
effects
WITH
are
considered.
EXPERIMENTAL
DATA
In an attempt to obtain as broad a range as possible without being voluminous,
tic is compaired
to the data from a different configuration. In some
each characteris-
cases, the choice of configuration
was dictated by the availabilityof the appropriate data.
5.10 NORMAL
FORCE
COEFFICIENT
The windtunnel data of the Aerobee
Figures
5-2 and 5-3 show
DERIVATIVE
AND
CENTER
OF
PRESSURE
350 sounding rocket, figure 5-1, are given in reference 14.
the results of the present
62
method
as well as data points from
63
I
"T
u'3
+_
u-
33.4
33_
52-
30
THEORETICAL
X
WIND
TUNNEL
A r = 381 in 2
26
22-CN
a[
20--
×
14-
X
x
12 --
X
IO--
f
I
2
I
3
I
4
I
5
Mm
Figure
5-2-Aerobee
350
Normal
64
Force
Derivative
i
i
I
6
7
8
410 --
4OO
THEORETICAL
x
390
WIND
TUNNEL
A r = 381 in 2
L r = 22
in.
x
38O
370
Xcp
360
350
340
330
--
320
-I"
0
I
2
I
3
I
4
I
5
Moo
Figure
5-3-Aerobee
350
65
Center
of
Pressure
I
i
I
6
7
8
reference
14.
deviation
of the
windtunnel
5.20
The
tests
ROLL
FORCING
forcing
vehicle
are
unpublished
Figure
5-5
shows
accuracy
of the
windtunnel
subsonic
theory
that
Reference
only
coefficient
time,
The
data
Unfortunately,
is 7.83_
The
the
maximum
of accuracy
and
of the
degree
over
provided
result
deviation
It is apparent,
at M = 1.5
test
and
of the
range
of mach
rocket,
by the
theoretical
5%.
a broad
sounding
figure
coordinator,
windtunnel
theoretical
both
from
numbers
5- 4.
the
for this
reference
data.
C.
that are
The data
28.
No data
is 7.54%.
windtunnel
are
The
data
and
the
is no good.
.DERIVATIVE
roll
the theoretical
been
the
value
a full
between
have
data
tomahawk
average
is about
for
authors.
derivative
but
the theoretical
CN, values
is 1.18%.
DERIVATIVE
stage
between
21 presents
config-uration
values
6%.
of the single
region.
DAMPING
theoretical
COEFFICIENT
comparison
subsonic
ROLL
those
at this
the
in the
5°30
as about
moment
are
of the
of pressure
MOMENT
to the author
available
derivation
center
is indicated
The only roll
available
maximum
theoretical
which
the
C:
set
of data
damping
range
of the
is 5.68%.
for
the
moment
data
basic
finner
coefficient
is relatively
Reference
test
small.
21 does
rocket,
derivative
figure
5-6.
are
available
data
Figure
not indicate
5-7
the
shows
degree
It is the
the
to the
comparison
of accuracy
of its
.p
data.
No subsonic
data
can
be found
for
Ca •
-p
5.40
DRAG
COEFFICIENT
Aerobee
150A,
figure
5-8,
drag coefficient
results
are
the test conditions,
the test
This
curve
highly
lent
used
finished.
full
efficient
5.50
was
surface
is 8.52%.
PITCH
There
4-1,
roughness
Again
DAMPING
are
table
no C
coefficient
compared
reynolds
in generating
From
scale
drag
data
to the
numbers
the
the
theoretical
data.
appropriate
data
available
from
reference
windtunnel
data in fig-ure 5-9.
were used to set up a Re vs.
is .1 mils.
no subsonic
are
The
surface
The
are
surface
roughness
maximum
29.
In order
curve,
M
of the
The
to duplicate
figure
5-10.
windtunnel
is about
.01
deviation
of the
taken
in windtunnel
theoretical
model
mils.
The
theoretical
is
equiva-
drag
co-
available.
DERIVATIVE
experimental
data
available.
The
data
experiments
are
mq
combination
angle-of-attack.
pitch
damping
of the
pitch
There
derivative
damping
derivative
is no way to separate
and
the
a time
two
can be attempted.
66
log derivative
effects.
Therefore,
due
to the
rate
of change
no comparison
of the
of
a
-
_g
oo
Z
Ld
t'-X
hi
C:3
£0
m
w
Fx
t.l,J
a
E3
F
i,
--.d
--(
0'_
I_
d
0
_1
%
d
6'7
i
o_
£0
<_lU-I
I-0
z
o3
c)
n"
.J
LLJ
I
o
I-'I
7,
&
50
m
45
THEORETICAL
x
40
WIND
TUNNEL
A r -- 64 7 in 2
Lr=
9
in.
55
30
cz 8
25x
20-
x
15-
x
I0-
x
5
0
I
I
I
2
I
.5
I
4
I
5
M
Figure
S-S-Tomahawk
Roll
68
Forcing
Derivative
I
6
I
7
I
8
°1
ll
o
J
o
r
i
I
t
69
c_
0
(._
Z
w
-_c/)
c_z
LL_"
O w
o_c_
QOW
I !
.u
uZ
I
_o
-45
--
-40
A r = 786
in 2
L r = I0 in.
-55
X
-30
czp
-25
-20
-15
THEORETICAL
X
-I0
0
I
0.5
WIND
TUNNEL
i
I
1.0
1.5
I
i
I
2.0
2.5
5.0
M
Figure
5-7-Basic
Finner
Roll
7O
Damping
Derivative
0
!
_0
--
0
W
Or)
0
o
+
w
T
m
t,-
c_
(M
±
71
-
0
t
t<
¢_I-w
if-,
r0
t
_._j
I
z
0
i
(D
w
(;9
0
nO
C]
D
0
r_
32
or)
to
z
u_
w
t-o9
0
if)
Z
U_
rr
i11
Z
o')
9
i
itl
U-
0.6
THEORETICAL
0.5
WIND TUNNEL
A r = 225
in 2
0.4
CD
X
0
I
2
I
:5
I
4
I
5
M
Figure
S-9-Aerobee
150A Drag Coefficient
72
I
6
I
7
I
8
9
X-VALUE
USED
WIND
•
8
IN
TUNNEL
TESTS
(REFERENCE
THE
x
IS
SHAPE
FAIRED
WITH
7
OF
BY
THE
CURVE
COMPARISON
NOMINAL
REYNOLDS
VARIATION.
x
29)
FLIGHT
NUMBER
i
0
X
n.x
4
x
\
X
X
0
0
I
I
I
I
3
4
5
6
M
Figure 5-10-Aerobee
150A Reynolds Number
73
7
8
6.00
CONCLUSIONS
On the
sented
percent
Cmq , the
indicate
basis
of the
comparisons
in this dissertation
of experimental
in section
predicts
the
results
between
results
of section
that the theoretical
5-1,
C
coupled
is also
values
mach
5, the
method
presented
in this
dissertation
of CN_ , x, C_,
¢t , and c__ to well
P
2 and 8. Although
no comparison
with the
accurate
dependence
of
to about
10%.
Cmq on (_)T<B_
pre-
within
ten (10)
can be made for
and
X_<B), tend
to
mq
The
lack
of subsonic
accuracy
of the
subsonic
variations
the
supersonic
In general,
the
aerodynamic
present
experimental
method
of all
the
data
at subsonic
compared
prevents
any concrete
spbeds.
However,
characteristics
are
the
quite
conclusions
figures
reasonably
to be drawn
in section
valued
about
5 show
with
that
respect
values.
then,
the
method
characteristics
presented
in this
dissertation
oI slender
finned
vehicles.
74
can
be used
to confidently
predict
the
the
to
7.00 REFERENCES
1.
Barrowman,
of Fins
2.
Diederich,
tics
3.
J. S.,
FIN
at Supersonic
F. W.,
o[ Swept
Ferrari,
A Plan
Wings.
C.,
a Computer
Speeds.
Form
J. A. S., p. 317,
Goldstein,
S., Lectures
5.
Horner,
S. F.,
Base
6.
Horner,
S. F.,
Fluid-Dynamic
7.
Jones,
R. T.,
Speed
of Sound.
Drag
R. T.,
Theory
9.
Jones,
R. T.,
and
10.
11.
12.
Liepmann,
Rough
Surfaces.
Love,
E. S., Base
13.
Goddard,
J.A.S.,
Pressure
and
Mayo,
Cone
Cylinder
E.
Mayo,
E.,
721.2
Files,
E. E.,
McNerney,
J. D.,
California,
1963.
15.
Miles,
16.
Moore,
J. W.,
J. A. S., p. 229,
17.
Morikawa,
G.,
18.
Morikawa,
G. K.,
California
Inst.
Drag
by the
and
Numerical
New York,
1960.
October,
1950.
Author.
Midland
Park,
N. J.,
Wings
at Speeds
Below
and
at Supersonic
F. E.,
Note
o,1 the
Mach
October
Fins
1958.
Above
the
Ogive
Speeds.
NACAReport
Princeton
Number
Univ.
Effects
1284,
Press,
Upon
1956.
Princeton,
the
Skin
1960.
Friction
of
1957.
Speeds
Having
on Two-Dimensional
Turbulent
Cylinder
Boundary
Geometric
Airfoils
Layers.
and
Mass
and
NACA
on Bodies
TN-3819,
Characteristics.
1957.
Memo
1965.
the
Effects
of Fin
Dihedral
on Vehicle
Stability,
Memo
1967.
350 Windtunnel
Test
Flow.
G. B. W., Jansen
Section
Analysis,
E.,
12.4
Space
General
A.R.D.C.,
Compressible
Corporation,
Baltimore,
Boundary
E1Monte,
1955.
Layer-Laminar
Solution,
1952.
Supersonic
The
Speeds-Theory
J. A. S., p. 622,
Theory,
Supersonic
April,
Characteris-
1946.
Determining
Aerobee
Yound,
Edges.
Wing
and
Characteristics
Aerodynamic
Publishers.
Pointed
at Supersonic
for
at Supersonic
Interscience
Published
NASA-GSFC,
Unsteady
L. L.,
Body
Trailing
NASA-GSFC,
Files,
and
Speed
Without
Equations
721.2
Certain
Correlating
High
p. 784,
With
to Code
14.
D.,
of Revolution
to Code
Thick
of Wing-Body
Cohen,
H. W.,
for
Mechanics.
835,
Aerodynamic
1948.
Drag.
Rep.
the
1966.
of Low-Aspect-Ratio
NACA
Jones,
June
and
Properties
8.
Wing
on Fluid
Calculating
X-721-66-162,
1951.
Between
4.
for
Parameter
NACATN-2335,
Interference
Application.
Program
NASA-GSFC
Wing-Body
Wing-Body
of Tech.,
Lift.
Problem
J. A. S., Vol.
for
1949.
75
Linearized
18,
No.
4, April,
Supersonic
Flow.
1951,
pp.
217-228.
PhD Thesis,
19. Morris, D. N., A Summaryof the SupersonicPressureDragof Bodiesof Revolution. J. A. S.,
p. 622,July 1961.
20. Murk, M. M., TheAerodynamicForces onAirship Hulls. NASAReport184,1924.
21. Nicolaides,J. D., Missile Flight andAstrodyuamics,BuWepsTN-100A,Washington,D. C., 1959.
22. Nielsen,J. N., andPitts, W. C., Wing-BodyInterferenceat SupersonicSpeedswith an Application to Combinationswith RectangularWings. NACATN-2677,1952.
23. Pitts, W. C., Nielsen,J. N., andKaattari; G. E., Lift andCenterof Pressureof Wing-BodyTail Combinationsat Subsonic,Transonic,andSupersonicSpeeds.NACAReport1307,1959.
24. Sims,J. L., Tablesfor SupersonicFlowAroundRight Circular Conesat SmallAngles-of-Attack.
NASASP-3007,1964.
25. Sims,J. L., Tablesfor SupersonicFlowAroundRight Circular Conesat Zero Angle-of-Attack,
NASASP-3004,1964.
26. Shapiro,A. H., The DynamicsandThermodynamicsof CompressibleFluid Flow, Vol. II,
RonaldPress,NewYork, 1954.
27. Syverston,C. A., andDennis,D. H., A Second-OrderShock-Expansion
MethodApplicableto
Bodiesof RevolutionNearZero Lift, NACAReport1328,1957.
28. Thomas,C. G., Unpublished
WindtunnelDataon a .25ScaleTomahawkSounding
RocketTaken
at LRC UnitaryandARDCTunnelB, October1966.
29. Thomas,E. S., WindtunnelTests of the Aerobee150A. Aerojet-GeneralCorporation,Azusa,
California, April 1960.
30. Tsien, H. S., SupersonicFlowOveran InclinedBodyof Revolution,JAS,p. 480,Vol. 5, 1938.
76
X-712-66-162
APPENDIXA
FIN
A COMPUTERPROGRAMFOR CALCULATINGTHE AERODYNAMIC
CHARACTERISTICS
OF FINS AT SUPERSONIC
SPEEDS
(Exerpt of TheoreticalTreatment)
JamesS. Barrowman
SpacecraftIntegrationandSounding
RocketDivision
April 1966
Goddard Space Flight Center
Greenbelt, Maryland
77
LIST
Reference
SYMBOLS
area
Length
CDW
OF
of
airfoil
Wave
drag
Lift
coefficient
strip
chord
coefficient
: "'_L
qA
Fl
qA
C L
CL_
dC_/a_ (_ = O)
Ct
Rolling
C
ac,/_I _ ( 3 = o)
Mr
moment
Pitching
Ca
coefficient
moment
-
coefficient
q AL
-
Mp
q AL
C
Local
CP
pressure
coefficient
i
Cr
Length
d
Component
1
of
Portion
d w
Lift
Fl
chord
of
of
Drag
F a
root
force
force
(t
ii
parallel
behind
t
to
L
Reference
1L
Length
of airfoil-strip
Length
of root
Length
of
Length
of airfoil-strip
Length
of
fin-tip
freestream
Mach
cone
to freestream)
(normal
Iv.terference
1L
the
(parallel
K
to the
freestream)
factor
length
leading-edge
airfoil
region
leading-edge
chord
region
chord
,
r
1r
1T
center
root
region
and
of center-region
trailing-edge
airfoil
chord
region
trailing-edge
region
chord
chord
r
iw
Distance
between
the
leading
edge
and
the
strip
Length
Freestream
M
Pitching
of
local
region
Mach
moment
surface
number
about
the reference
axis
P
78
intersection
of the
Mach
cone
with
the
airfoil
LIST
Mr
Rolling
moment
n
Number
Local
t
of
static
P_
Ambient
ct_
Freestream
root
(Continued)
chord
1
normal
to the
freestream
pressure
static
pressure
dynamic
pressure
dw/d _ ratio
I"
t
s
Semispan
xL
Distance
length
from
the
reference
Center-of-pressure
XP
the
SYMBOLS
of strips
Component
P
about
OF
i
Distance
streamwise
axis
coordinate
from the reference
direction
V
Center-of-pressure
Y
Distance
_'_y
Width
of airfoil
strip
Angle
of attack
(degree)
axis
coordinate
from
the
root
chord
to the
airfoil-strip
measured
from
to the
forward
measured
from
in a spanwise
leading
the
reference
region
the
root
chord
direction
FL
Leading-edge
sweep
angle
rT
Trailing-edge
sweep
angle
Pt' P2
Ragion
Ratio
boundaries
of specific
heats
angles
for
air
= 1.4
S
Fin
_L
Leading-edge
wedge
half-angle
in the
streamwise
direction
Trailing-edge
wedge
half-angle
in the
streamwise
direction
_T
cant
sweep
angle
Inclination
of a local
Mach
semivertex
cone
surface
to the
freestream
angle
SUBSCRIPTS
B
Body
alone
B(T)
Body
in the
i
Local
region
number
J
Airfoil
strip
number
presence
of the
tail
79
in a streamwise
direction
axis
boundary
. _12 - 1
j)
edge
of a local
region
in a
LIST
OF
SYMBOLS
SUBSCRIPTS
o
Value at _ = 0
T
Tail alone
Tail in the presence
of the body
80
(Continued)
FIN
A COMPUTERPROGRAMFORCALCULATINGTHE AERODYNAMIC
CHARACTERISTICS
OF FINSAT SUPERSONIC
SPEEDS
BACKGROUND
Calculating
steps
the
in calculating
at supersonic
lar
aerodynamic
the
speeds
characteristics
overall
for
the
of the
aerodynamics
greater
fins
of a sounding
of a sounding
portion
oi_ their
rocket.
flight,
the
rocket
Because
regime
is one
sounding
at these
of the
basic
rockets
speeds
fly
is of particu-
interest.
Methods
are
previously
lacking
herent
used
in accuracy
in using
them
By using
rapidly
assumptions
inherent
and
functions
of angle
fin with
of attack
(
nearly
aerodynamics
many
solve
desired.
only
The
Supersonic
form
basic
The
equations.
Airfoil
chord
plane
symmetry,
at a given
•
Pitching
•
Center-of-pressure
coordinate
measured
from
the
reference
•
Center-of-pressure
coordinate
measured
from
the
root
FIN also
defined
computes
Lift
coefficient
•
Wave
drag
•
Pitching
coefficient,
in the
c
Rolling
the
in reference
Mach
number,
FIN
computes
program
as
slope,
at zero
moment
functions
m
is equivalent
function
of Mach
to the
number
CL
angle
coefficient
of attack,
slope,
CD%
C
moment
coefficient
slope,
c,
81
the
axis,
chord,
fin cant
of >.
m
•
are
amenable
CD_
C, is considereda
•
as described
may
CL
moment
coefficient
to be most
one
) the following:
coefficient,
following:
4)
in-
ob-
to pro1.
LIMITATIONS
Wave
moment
to accuracy
found
Theory
approximations
equations,
restrictions
theory
•
, as
and
1, 2, 3, and
solution.
theoretical
Lift
drag
(references
assumptions
closed
•
Since
coefficient,
Order
AND
a supersonic
as
basic
fin
of the
or
to numerically
the
Second
CAPABILITIES
Given
form
as accurately
in deriving
supersonic
because
a closed
computer
in Busemann's
PROGRAM
calculating
applicability
to reach
a high-speed
tainanswers
gramming
and
for
angle
(:)
at
_ = 0, the
rolling
as
The
•
Center-of-pressure
coordinate
measured
from
the
rgference
•
Center-of-pressure
coordinate
measured
from
the
root
range
program
of -. is from
allows
increment
for
of the
Since
FIN has
reference
references
with
in this
is applied
3.
by the
user,
points,
the
number
with
a four-finned
at zero
user
angle
angle
in increments
of attack,
of attack,
the
x0
Y0
of 1 degree.
specifying
to small
the
portion
The
special
fin-tip
Mach
is applied
The
range
and
attention
streamwise
airfoil
Mach
strip.
cone
cone
This
10.3
and
the
applicability
strips,
and
is accounted
may
or
values
restrictions
of CL, CDw,
strips
by applying
not
the
in this
are
technique
may
in
However,
shown
for
two
outlined
10.28.
the
correction-factor
correction
output
to the
to Figures
enlarges
the
values
of C_ and C_ are for
only to a single typical
fin.
subject
slightly
fin-tip
of each
vehicle,
fins; the output
x0, and T 0 apply
Theory
program
In addition,
necessary
1 and
use
Airfoil
192 to 241,
direction.
to the
for
Order
terms
theory
spanwise
designed
Second
1, pages
factor
of 100 Mach
C_ are for apair
of identical
fins.
The values
of _,'_,
third-order
Busemann's
_ma_' as determined
chord
at zero
number.
been
Cm, CL_, CDwo, and
pairs
of perpendicular
of the
to
a maximum
Mach
Busemann's
zero
axis
use
reference.
summed
in a
a correction
was
be included
obtained
at the
from
user's
discretion.
Figx_re
of the
1 shows
shapes
type
is given
in the
AERODYNAMIC
the
•
of the
implies
leading
modified
blunt
•
The
•
Each
The
to which
on usage.
Order
FIN
rockets
is applicable.
and
A listing
Airfoil
shock
region
local
airfoil
Theory
surface
low angles
edge
double
base
basic
of hns
on sounding
These
missiles.
The
configurations
program
of FIN in FORTRAN
input
IV is given
cover
pattern
all
for
in Appendix
each
A.
has
been
applied
to two-dimensional
airfoil
strips
assumptions:
All parts
The
section
Second
following
This
types
used
THEORY
Busemann's
with
the
normally
are
in supersonic
flow
and
make
small
angles
with
the
flow.
of attack.
is sharp.
wedge.
The
For
trailing
a single
edge
wedge
must
and
be sharp
a modified
for
single
a double
wedge,
wedge
the
fin or
effect
a
of the
is neglected.
waves
are
of flow
pressure
all
over
attached.
the
surface
coefficient
acts
equation
independently
of the
expansion:
82
Busemann
of the
others.
theory
is a third-order
series
(a]
Modified
Double
-_--_(c)Modified
Wedge
(b)
t-Single
Wedge
----------'---------1
Wedge
Figure
Double
(d)
1-Fin
Configurations
83
Single
Wedge
where
2
(2a)
K I
K
2
4
_4
(2b)
I)M _
(2 2
_ 7)
-5)M 6 _ 10(,
* I)M 4 -
12M 2 * 8
K_
6"77
(2c)
If the
flow-over
compressive
the
shock
surface
wave
region
anywhere
inclined
at
upstream
in
the
, , t)M* [(s
K*
v_
to
the
flow,
flow
the
- 3,)M_
has
been
value
+ 4(_
preceded
of K"
by
a single
is:
- 3)M_ , s]
(3)
48
If the
K" = 0.
flow
%
equation
The
index,
i,
The
Mach
cone
by
within
length
center
the
is
the
pressure
cone,
local
inclination
K" term
indicates
Mach
the
the
of the
1 represent
tively.
region
is
upstream
of
(Figure
2).
the
Maeh
of that
portion.
of pressure
the
first-,
the
for
into
By
the
surface.
of
this
means,
equations
and,
hence,
the
E,
of
is
of
obtained
is
for
terms
of
region
respec-
the
shock.
The
the
local
bow
including
airfoil
pressure
taken
the
moment
three
surface
region,
of
portion
pitching
first
flow,
calculated.
airfoil
to the
each
are
local
being
intersection
applied
pressure
shock-free
the
through
is
to a typical
at the
1/2
center
the
on
rise
coefficient
applied
The
expansive
pressures
pressure
portions
factor
only
Essentially,
pressure
1 is
two
contained
third-order
irreversible
which
A correction
(_,),
has
edge
and
equation
region
cone.
question
leading
third-order
of
the
in
second-,
surface
coefficient
dividing
region
local
strip
the
fin-tip
and
the
coefficient
as
lift
the
for
midpoint
(F_),
drag
the
of
(Fd_),
(Mp_):
F|
:
Cv
d
1 -
(4)
Fd,
=
Cp
n
I -
(5)
½d i
1 -
ri
+ --7
+_'-
i
:
r
(6)
r_
2
%,
:
84
r,
l
_I
(7)
_X
STRIP
Iw
I
MACH
\CONE
Iw
\
_,
:-
I
\
Xp i
AIRFOIL
\
STRIP
REFERENCE
AXIS
Y
REGION
BOUNDARIES
d
Figure
2-Mach
Cone Correction
85
Geometry
The airfoil-strip
lift
the local characteristics.
(Fij)
, drag
(Fd),J
and
pitching
moment
(Mv J ) totals
are
then
computed
by summing
6
Fl
:
_
Fl
(8)
F d
(9)
i=1
5
V_
F d •
J
l=l
6
Z
Mpi
(I0)
Mp
t=l
The
wise
strip
rolling
position,
moment
about
the
root
chord
total
from
the
airfoil
lift
and
strip
span-
y:
:
Mrj
The
is calculated
characteristics
characteristics
in a spanwise
over
the
entire
(ii)
YFIj
fin are
then
computed
by summing
the
airfoil
strip
direction.
(12)
(13)
Mp
M%
= _
(14)
}:1
(15)
The
dynamic
forces
pressure
and
moments
(q_)
and
computed
strip
width
in equations
(Ay).
4 through
Thus,
the
86
associated
15 are
actually
coefficients
divided
are:
by the
ambient
2F,
CL
:
C_w
=
Ca
:
CI
:
":_.y
A
(2
fins)
(2
fins)
(2
fins)
(16)
2F d • Ey
A
(17)
2 M p • _, y
AL
(18)
4M r • _y
The
center-of-pressure
appropriate
lift
AL
coordinates
are
(4
calculated
(19)
fins)
from
the
respective
moments
and
the
forces.
M
:
o
F, cos
(20)
M
_(
:
_
(21)
F t
To obtain
C,
and
gree).
the
Cz are
The
linear
calculated
slope
values
slope
coefficients
at a = 1 ° and are
per
radian
are
near
_ = 0 at a given
taken
as the
slope
by the
proper
obtained
Mach
values
number,
the
ct_ , c,
conversion
values
and
C_
of radians
(all
of CL,
per
de-
to degrees.
Thus:
CL-_
_r
zero
angle-of-attack
CONFIGURATION
The
root
airfoil
user
AND
indicates
leading
and
values
_
Cl 8
rad.
1_0
--(cJi_
the
shape
trailing
-
of CDw, )(, and
GEOMETRY
_ are
simply
_
= 1
a
=
I
)
(23>
= ,)
(24)
their
respective
values
at a = 0.
CONSIDERATIONS
of the
edge
CL
% ,
tad.
The
(22)
180
rad.
fin
region
configuration
chord
lengths
87
by specifying
(1L,
and
the
IT,),
root
leading
chord
and
length
trailing
(C r ),
edge
sweep
(S)
angles
(Figure
(F u and
3).
leading
edge
lengths
(l L and
The
V t ), region
program
in the
I t)
boundary
then
streamwise
direction
as
of
functions
sweep
angles
the
distance
computes
the
(X L),
chord
tan[
y
C
:
The
width
of
the
airfoil
strip
is
of
c cos
T
tan
y(tan;'
1 -
tanF'L)
determined
fin
semispan
axis
and
airfoil
the
length
to the
airfoil
airfoil-strip
region
chord
strip.
(25)
L)
" Cr
(26)
+ 1L
(27)
(28)
m- ta T )+
from
Zy
(C),
the
the
reference
._
y (tan[
1T : y(ta.
V 2 ), and
the
length
coordinate
:
1L
from
airfoil
spanwise
XL
(T 1 and
the
number
of
strips
(n)
desired
by
the
user.
S
--
:
(29)
n
The
general
airfoil
(Figures
lb,
crossection,
lengths
wise
strip
crossection
lc,
is
assumed
is
the
and
which
are
crossection
configuration
ld)
has
determined
direction,
_L
and
are
special
six
separate
by
specifying
to be
modified
cases
symmetrical
double
of this
basic
flow
regions.
As
the
leading-
and
about
wedge
shape.
can
be
the
(Figure
Figure
seen
wedge
line.
The
4 shows
in this
trailing-edge
chord
la).
The
other
in
the
l
:
I 4
general
half
angles
region
in
stream-
=--
the
L
v
cos
L
(30)
1T
13
12
program
the
surface
_T"
1 1
The
types
detail
figure,
most
three
computes
the
[6
Is
:
local-surface
(31)
r
COs
:
T
C -
inclination
l
lL -
angles
(32)
1T
(_i)
as
functions
of
_:
(33a)
_ L
_;2
:
-
,
T)3
:
-
*'T
(33b)
(33c)
74
-
L + "_
:
88
':
(33d)
S
1 _ -_
/'_
12
14
SECTION
Figure
A-A
3-Fin
TYPICAL
and Airfoil
89
AIRFOIL
Geometry
c
IM
L
--i
J
_
j
_'2
mf
J
J
'L
"4
/
J
Figure
4-Airfoil
Angles
go
!
From
these,
the
streamwise
and
"_s :
'
%
-'
normal
(38e)
+o
'T
components
d
=
1
n
:
1
cos
(38f)
of the
_,
(i
":i
_i
region
I
-
1
" 6)
lengths
are
calculated.
6)
(34)
The
distance
of the
forward
"sin
local-region
::
boundaries
from
the
reference
axis
are
then
determined
as:
xp,
×p_
::
x L _ {t,
(35b)
Xp_
:
xL + dz + d2
(35c)
XL _ d_
(35d)
XL _ (t4 * d s
(35e)
×P6
The
MACH
CONE
Mach-cone
an airfoil
angle
distance
strip
I w from
the
(_,) is a direct
at a spanwise
leading
function
:
by trigonometric
identities
direction
I w is greater
correction
than
or equal
is necessary.
Mach
(1)
y, the
number.
Arcsin
intersection
(1)
of the
(36)
strip
and
the
Mach
cone
is a
edge.
and
:
the
(S
use
1w
If
of the
Arctan
1w
But,
:
CORRECTION
a,
For
(35a)
xp2
Xps
FIN-TIP
xL
to the chord
If i w is such
-
y)tan_
L
of equation
:
(S
length,
-
y)
÷
tan(90
37,
(tan,
there
this
,.
is,
that
C " 1w Z C-
91
lT
+
(37)
--)
may
be reduced
to:
(38)
)
of course,
no actual
intersection
and no
thenthe ratio of the portion of Ir falling behindthe Machconeto Ir is:
c- 1
W
r 3
:
r 6
--
(39)
lT
AIso,
r I
If 1, is such
:
r 2
:
the
r 5
:
(40)
0
that
C-
then
1- 4
corresponding
ratio
for
the
middle
lv
_ 1w
regions
is
Cr2
:
IL
_
1,
1T
rs
(4t)
|M
Also,
r 3
r 5
],
r
r 4
0
(42a)
and,
If 1w is such
I
that
C-
then
the
ratio
(42b)
for the
leading
edge
region
lT
1
1
is
I L
r
" 0
I
-
1w
(43)
1L
r 4
Also,
r2
These
ratios
center
of pressure.
is taken
are
as half
the
values
used
In these
the value
r 3
in equations
equations,
the
of the CP calculated
r s
r 5
4, 5, and
6 for
pressure
coefficient
by equation
92
(44)
1.0
1.
calculating
in the
the
portion
local
lift,
behind
drag,
the
and
Mach
cone
REFERENCES
1. Hilton,W. F., High SpeedAerodynamics.Longmans,GreenandCo. NewYork, 1951.
2. Pitts, W. C., Nielsen,J. N. andKaattari, G. E., Lift andCenterof Pressureof Wind-Body-Tail
Combinations
at Subsonic,Transonic,andSupersonicSpeeds.NACAReport1307,1959.
3. Shapiro,A. H., The DynamicsandThermodynamics
of CompressibleFluid Flow. Vol. H.
RonaldPress. NewYork, 1954.
4. Harmon,S.M. andJeffreys,I., TheoreticalLift andDampingin Roll of Thin Wingswith Arbitrary Sweep and Taper at Supersonic
Speeds
Supersonic
Leading
and Trailing
Edges.
NACA
TN2114.
5.
Thomas,
6.
McNerney,
May
1950.
E. S., Windtunnel
J. D.,
Aerobee
Tests
of the
350 Windtunnel
Aerobee
150A.
Test
Analysis.
Aerojet
Space
General
General
Corp.,
Corp.
1960.
January_
1963.
¢
93