AERODYNAMICS OF WRAPAROUND FINS IN SUPERSONIC FLOW
Except where reference is made to the work of others, the work described in this thesis is
my own or was done in collaboration with my advisory committee. This thesis does not
include proprietary or classified information.
__________________________________
Brett Landon Wilks
Certificate of Approval:
________________________ ________________________
Roy J. Hartfield, Jr. John E. Burkhalter, Chair
Associate Professor Professor Emeritus
Aerospace Engineering Aerospace Engineering
________________________ ________________________
Chris J. Roy Stephen L. McFarland
Assistant Professor Acting Dean
Aerospace Engineering Graduate School
AERODYNAMICS OF WRAPAROUND FINS IN SUPERSONIC FLOW
Brett Landon Wilks
A Thesis
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Master of Science
Auburn, Alabama
December 16, 2005
iii
AERODYNAMICS OF WRAPAROUND FINS IN SUPERSONIC FLOW
Brett L. Wilks
Permission is granted to Auburn University to make copies of this thesis at its discretion,
upon request of individuals or institutions and at their expense. The author reserves all
publication rights.
______________________________
Signature of Author
______________________________
Date of Graduation
iv
VITA
Brett Landon Wilks was born on March 28
th
, 1980 in Huntsville, Alabama. His
parents, Kenneth and Jackie Wilks, own a small tire business in Arab that was a
significant part of his career development. He started helping his father at an early age
and developed a strong interest in the mechanics of automobiles. It was that interest that
propelled him into engineering studies after graduating from Arab High School in 1998.
In December of 2002, he graduated Summa Cum Laude with a Bachelor of Science
degree in Mechanical Engineering from the University of Alabama in Huntsville. During
his undergraduate studies he began to work cooperatively with the United State Army for
the System Simulation and Development Directorate in the Aerodynamics Technology
Functional Area at Redstone Arsenal, Alabama. With growing interest in aerodynamics,
he began Graduate School at Auburn University in Aerospace Engineering in 2003 with a
focus on aerodynamics and propulsion. Upon course completion, he returned to
Huntsville to fulfill his cooperative education agreement with the United States Army and
to marry his girlfriend of seven years, Amanda Fischer Wilks. They are expecting their
first child in March 2006.
v
THESIS ABSTRACT
AERODYNAMICS OF WRAPAROUND FINS IN SUPERSONIC FLOW
Brett Landon Wilks
Master of Science, December 16, 2005
(B.S., University of Alabama in Huntsville, 2002)
73 Typed Pages
Directed By John E. Burkhalter
Existing supersonic fin theory has been modified to compute the pressure
distribution over a wraparound fin. Evvard?s theory has been used to calculate the
pressure loading due to angleofattack on a wraparound fin by including fin curvature
as a variable in the definition of the zones of influence. Evvard?s theory uses the
intersections of the fin surface and the Mach cones originating from the leading edge
discontinuities to split the fin surface into regions of influence. For a planar fin, the
intersections are linear; however, the intersections on a curved fin form curved lines. By
redefining the Mach lines to account for fin curvature and using an empirically derived
induced angleofattack, the application of Evvard?s theory can be extended to accurately
compute the unique aerodynamic characteristic of wraparound fins.
vi
ACKNOWLEDGEMENTS
The author would like to thank his wife, Amanda Wilks, and his parents, Kenneth
and Jackie Wilks, for their support, patience and encouragement. He would also like to
thank Mr. Richard Kretzschmar and Mr. Lamar Auman for being excellent mentors
during this research. Finally, he would like to thank the United States Army Aviation and
Missile Command for funding this research.
vii
TABLE OF CONTENTS
LIST OF TABLES...........................................................................................................viii
LIST OF FIGURES ........................................................................................................... ix
LIST OF SYMBOLS AND ACRONYMS....................................................................... xii
1. INTRODUCTION ...................................................................................................... 1
1.1 Historical Perspectives........................................................................................ 2
1.1.1 United States Army......................................................................................... 2
1.1.2 United States Air Force................................................................................... 3
1.2 Current Perspective............................................................................................. 4
2. METHODOLOGY ..................................................................................................... 5
2.1 Induced AngleofAttack .................................................................................... 5
2.2 AngleofAttack Dependence ............................................................................. 7
3. THEORY .................................................................................................................... 9
3.1 Curved Fin Geometry ......................................................................................... 9
3.2 Dividing Mach Lines ........................................................................................ 11
3.3 Pressure Differential in Each Region of Flow .................................................. 16
3.3.1 Region I......................................................................................................... 16
3.3.2 Region II ....................................................................................................... 17
3.3.3 Region III...................................................................................................... 18
3.3.4 Region IV...................................................................................................... 19
3.3.5 Region V ....................................................................................................... 19
3.4 Empirically Derived Induced AngleofAttack ................................................ 20
4. INTEGRATION OF THE PRESSURE DISTRIBUTION....................................... 27
5. RESULTS ................................................................................................................. 29
5.1 Comparison to Test Data .................................................................................. 29
5.1.1 Normal Force ................................................................................................ 30
5.1.2 Side Force ..................................................................................................... 36
5.1.3 Root Bending Moment.................................................................................. 42
5.1.4 Hinge Moment .............................................................................................. 48
5.2 Pressure Contour Plots...................................................................................... 54
6. LIMITATIONS......................................................................................................... 58
7. CONCLUSION......................................................................................................... 59
8. REFERENCES ......................................................................................................... 60
viii
LIST OF TABLES
Table 1: WindTunnel Fin Geometries............................................................................... 6
Table 2: Pressure Differential due to AngleofAttack..................................................... 20
Table 3: Coefficient Uncertainty ...................................................................................... 30
Table 4: Chordwise CenterofPressure NonDimensionalized by L
REF
......................... 48
ix
LIST OF FIGURES
Figure 1: Packaging Advantage of WrapAround Fins ...................................................... 1
Figure 2: Photo of Tested WAFs ........................................................................................ 6
Figure 3: Curvature Effect on Mach Lines at Mach 1.6 ..................................................... 8
Figure 4: Curved Fin Geometry........................................................................................ 10
Figure 5: Mach Cone  WAF Surface Intersection ........................................................... 11
Figure 6: Zoning Rules ..................................................................................................... 12
Figure 7: ThreeDimensional Surface Intersection........................................................... 14
Figure 8: Zoning Verification at Mach 1.6 for 0.0Degrees of Curvature........................ 14
Figure 9: Zoning Verification at Mach 1.6 for 45.0Degrees of Curvature...................... 15
Figure 10: Zoning Verification at Mach 1.6 for 90.0Degrees of Curvature.................... 15
Figure 11: Zoning Verification at Mach 1.6 for 135.0Degrees of Curvature.................. 15
Figure 12: Zoning Verification at Mach 1.6 for 180.0Degrees of Curvature.................. 16
Figure 13: WAF on SplitterPlate with Sign Convention................................................. 21
Figure 14: Curvature Effects for AR = 1.4118 ................................................................. 22
Figure 15: Curvature Effects for AR = 1.8824 ................................................................. 22
Figure 16: Curvature Effects for AR = 2.8333 ................................................................ 23
Figure 17: Aspect Ratio Dependence at Mach 2.25 ......................................................... 24
Figure 18: Correlated Induced AngleofAttack versus Mach Number ........................... 26
Figure 19: Measured versus Correlated Induced AngleofAttack................................... 26
Figure 20: Normal Force Comparison for AR = 1.4118 at Mach 1.5............................... 31
Figure 21: Normal Force Comparison for AR = 1.4118 at Mach 2.25............................. 31
Figure 22: Normal Force Comparison for AR = 1.4118 at Mach 3.0............................... 32
Figure 23: Normal Force Comparison for AR = 1.8824 at Mach 1.5............................... 32
Figure 24: Normal Force Comparison for AR = 1.8824 at Mach 2.25............................. 33
Figure 25: Normal Force Comparison for AR = 1.8824 at Mach 3.0............................... 33
x
Figure 26: Normal Force Comparison for AR = 2.8333 at Mach 1.5............................... 34
Figure 27: Normal Force Comparison for AR = 2.8333 at Mach 2.25............................. 34
Figure 28: Normal Force Comparison for AR = 2.8333 at Mach 3.0............................... 35
Figure 29: Normal Force for AR = 1.8824; L = 35
o
at Mach 1.5 .................................... 35
Figure 30: Side Force Comparison for AR = 1.4118 at Mach 1.5.................................... 37
Figure 31: Side Force Comparison for AR = 1.4118 at Mach 2.25.................................. 37
Figure 32: Side Force Comparison for AR = 1.4118 at Mach 3.0.................................... 38
Figure 33: Side Force Comparison for AR = 1.8824 at Mach 1.5.................................... 38
Figure 34: Side Force Comparison for AR = 1.8824 at Mach 2.25.................................. 39
Figure 35: Side Force Comparison for AR = 1.8824 at Mach 3.0.................................... 39
Figure 36: Side Force Comparison for AR = 2.8333 at Mach 1.5.................................... 40
Figure 37: Side Force Comparison for AR = 2.8333 at Mach 2.25.................................. 40
Figure 38: Side Force Comparison for AR = 2.8333 at Mach 3.0.................................... 41
Figure 39: Side Force for AR = 1.8824; L = 35
o
at Mach 1.5.......................................... 41
Figure 40: Root Bending Moment Comparison for AR = 1.4118 at Mach 1.5 ................ 43
Figure 41: Root Bending Moment Comparison for AR = 1.4118 at Mach 2.25 .............. 43
Figure 42: Root Bending Moment Comparison for AR = 1.4118 at Mach 3.0 ................ 44
Figure 43: Root Bending Moment Comparison for AR = 1.8824 at Mach 1.5 ................ 44
Figure 44: Root Bending Moment Comparison for AR = 1.8824 at Mach 2.25 .............. 45
Figure 45: Root Bending Moment Comparison for AR = 1.8824 at Mach 3.0 ................ 45
Figure 46: Root Bending Moment Comparison for AR = 2.8333 at Mach 1.5 ................ 46
Figure 47: Root Bending Moment Comparison for AR = 2.8333 at Mach 2.25 .............. 46
Figure 48: Root Bending Moment Comparison for AR = 2.8333 at Mach 3.0 ................ 47
Figure 49: Root Bending Moment for AR = 1.8824; L = 35
o
at Mach 1.5...................... 47
Figure 50: Hinge Moment about C
R
/2.0 Comparison for AR = 1.4118 at Mach 1.5 ....... 49
Figure 51: Hinge Moment about C
R
/2.0 Comparison for AR = 1.4118 at Mach 2.25 ..... 49
Figure 52: Hinge Moment about C
R
/2.0 Comparison for AR = 1.4118 at Mach 3.0 ....... 50
Figure 53: Hinge Moment about C
R
/2.0 Comparison for AR = 1.8824 at Mach 1.5 ....... 50
Figure 54: Hinge Moment about C
R
/2.0 Comparison for AR = 1.8824 at Mach 2.25 ..... 51
xi
Figure 55: Hinge Moment about C
R
/2.0 Comparison for AR = 1.8824 at Mach 3.0 ....... 51
Figure 56: Hinge Moment about C
R
/2.0 Comparison for AR = 2.8333 at Mach 1.5 ....... 52
Figure 57: Hinge Moment about C
R
/2.0 Comparison for AR = 2.8333 at Mach 2.25 ..... 52
Figure 58: Hinge Moment about CR/2.0 Comparison for AR = 2.8333 at Mach 3.0 ...... 53
Figure 59: Hinge Moment about C
R
/2.0 for AR = 1.8824; L = 35
o
at Mach 1.5 ............. 53
Figure 60: Pressure Contour for ? = 0.0; AR = 1.4118 at Mach 1.5................................. 55
Figure 61: Pressure Contour for ? = 0.0; AR = 1.4118 at Mach 3.0................................. 55
Figure 62: Pressure Contour for ? = 90.0; AR = 1.4118 at Mach 1.5............................... 55
Figure 63: Pressure Contour for ? = 90.0; AR = 1.4118 at Mach 3.0............................... 55
Figure 64: Pressure Contour for ? = 180.0; AR = 1.4118 at Mach 1.5............................. 56
Figure 65: Pressure Contour for ? = 180.0; AR = 1.4118 at Mach 3.0............................. 56
Figure 66: Pressure Contour for L = 35.0; ? = 0.0; AR = 1.8824 at Mach 1.5 ................ 56
Figure 67: Pressure Contour for L = 35.0; ? = 0.0; AR = 1.8824 at Mach 3.0 ................ 56
Figure 68: Pressure Contour for L = 35.0; ? = 90.0; AR = 1.8824 at Mach 1.5 .............. 57
Figure 69: Pressure Contour for L = 35.0; ? = 90.0; AR = 1.8824 at Mach 3.0 .............. 57
Figure 70: Pressure Contour for L = 35.0; ? = 180.0; AR = 1.8824 at Mach 1.5 ............ 57
Figure 71: Pressure Contour for L = 35.0; ? = 180.0; AR = 1.8824 at Mach 3.0 ............ 57
LIST OF SYMBOLS AND ACRONYMS
AMRDEC Army Missile Research, Development and Engineering Center
HSWT High Speed WindTunnel
LMMFC Lockheed Martin Missile and Fire Control
USAF United States Air Force
CFD Computational Fluid Dynamics
CAD ComputerAided Drafting
APKWS Advanced Precision Kill Weapon System
BAT Brilliant Antiarmor Technology
CKEM Compact Kinetic Energy Missile
LOSAT LineofSight Antitank
MLRS Multiple Launch Rocket System
TACAWS The Army Combined Arms Weapons System
WAF(s) WrapAround Fin(s)
C
N
Normal Force Coefficient
C
RBM
Root Bending Moment Coefficient
C
Y
Side Force Coefficient
C
HM
Hinge Moment Coefficient
DA
x,y
Incremental Fin Panel Surface Area Projected onto xyPlane
Dx Incremental Chordwise Length
Dy Incremental Spanwise Length
Dz Incremental Curvewise Length
b Wingspan
b/2 Fin semispan
L
REF
Mean Chord Length
S
REF
Fin Planform Area, L
REF
(b/2)
AR Fin Aspect Ratio, b
2
/(2.0 S
REF
)
a AngleofAttack
a
INDUCED
Induced AngleofAttack
a
AERODYNAMIC
Angle Between the Freestream Mach number and the Fin Chord
M, M
?
Freestream Mach number
b Compressibility Factor, 0.1
2
?M
d Fin Curvature slope angle
L Leading Edge Sweepback Angle
? Fin Curvature
xii
1. INTRODUCTION
Wraparound fins (WAFs) are a family of fins that, when stowed, conform or
?wraparound? the surface of a cylindrical body. As a result of the packaging advantage
WAFs have over planar fins, WAFs are prevalent on tubelaunched missile and rocket
systems. Several fielded missiles, rockets and munitions utilize WAFs for stability;
among these systems are MLRS, TACAWS, APKWS, LOSAT, BAT, CKEM, Hydra70,
and variants of the Zuni rocket. Figure 1 shows a set of 4 WAFs both stowed around the
body of a rocket and deployed.
Figure 1: Packaging Advantage of WrapAround Fins
The geometry of a WAF is typically determined by the diameter of the missile
and the number of fins. The curved span of the WAF is typically the missile
circumference divided by the number of fins and the angle of curvature is 360degrees
divided by the number of fins. A majority of the systems utilizing WAFs have 4 fins;
1
2
therefore, a WAF with 90degrees of curvature is common. However, several 2.75inch
rockets are equipped with 3 WAFs for stability. Brilliant Antiarmor Technology (BAT)
employs 4 overlapping WAFs for stability with a curvature angle of 180degrees.
Wraparound fins, however, do come with aerodynamic peculiarities. Systems
equipped with WAFs exhibit significant rolling moments at zero incidence. The
?induced? rolling moment is documented as a function of Mach number and angleof
attack.
1.1 Historical Perspectives
1.1.1 United States Army
A series of tests were conducted between 1971 and 1976 by the Aeroballistics
Directorate of U.S. Army Missile Research, Development and Engineering Center
(AMRDEC) to identify alternative stabilizing devices.
610
Among these devices were
WAFs, ringtails and flares. Limited data were collected on several WAF geometries on a
splitterplate and on a generic 4inch diameter body with a 2caliber secant ogive nose
and an 8caliber cylindrical afterbody. The fins tested were limited to 90degrees of
curvature.
In terms of stability, the U.S. Army concluded that WAFs perform similarly to
planar fins of equivalent projected planform shape. It was also noted that WAFs
produced a substantial amount of rolling moment which varied with angleofattack and
Mach number. These variations in rolling moment could possibly lead to significant
dynamic problems including Magnus instability and roll rate variations during ballistic
flight if not compensated for correctly. Furthermore, the rolling moment was found to be
3
a strong function of Mach number as the direction of the rolling moment changed near
Mach 1.0. In supersonic flow, the fins produced an induced normal force away from the
center of curvature at zero incidence. Conversely, the fins produced an induced normal
force toward the center of curvature in subsonic flow at zero incidence.
1.1.2 United States Air Force
In the late 1980?s the U. S. Air Force (USAF) began investigating the cause of the
low incidence rolling moment generated by their tube launched missile systems equipped
with WAFs.
1,14,18
The USAF used several techniques to investigate the flow field near a
WAF including freeflight gun tests, windtunnel tests (with and without the aid of
pressure sensitive paint), and computational fluid dynamics (CFD). The USAF also
investigated several methods of reducing the magnitude of the induced roll by slotting
WAFs and altering the finbody junction angle. A majority of the testing was performed
on a 2.22 aspect ratio rectangular fin with a thicknesstochord ratio of 12.5percent and a
45degree leading edge wedge angle. Interest was focused between Mach 2.15 and Mach
3.83.
According to the USAF studies, the leading edge of the fin causes a bow shock
that interacts with the convex and concave sides of the fin much differently. On the
concave side of the fin, the shock is focused near the center of curvature causing a region
of relatively high pressure which diminishes as the shock becomes more acute at higher
Mach numbers. The convex side of the fin shows a small region of high relative pressure
near the bodyfin juncture that intensifies as the Mach number increases. The result is a
net force away from the center of curvature which decreases with Mach number.
4
1.2 Current Perspective
The U.S. Army Aviation and Missile Command tested a series of wraparound
fins on a splitterplate at the Lockheed Martin Missile and Fire Control High Speed
WindTunnel (LMMFC HSWT) in Dallas, Texas in January of 2005 with the goal of
developing a design methodology for wraparound fins. The test data for the WAF show
two notable features. The more notable feature is an induced normal force on the WAF at
zero incidence which leads to an induced rolling moment when the fins are used on a
missile system. The second difference is a slight increase in the normal force slope with
respect to angleofattack with increasing curvature. Since there is only a slight change in
the normal force slope, it appears that the fin curvature effectively generates an induced
angleofattack when compared to a planar fin of the same projected planform shape.
5
2. METHODOLOGY
In order to develop a design methodology for WAFs, the effect of curvature on
the pressure loading of a WAF must be understood. The pressure sensitive paint results
presented in Reference 14 show the pressure loading of a WAF at zero incidence is
similar to the pressure loading of a planar fin at an angleofattack. The pressure loading
has distinct divisions that appear much like Mach lines. The interior of the WAF has a
fairly constant pressure and the tip of the fin has a much lower pressure. The pressure
loading is similar to the results obtained from Evvard?s theory for a planar fin at a non
zero incidence. Therefore, it is reasonable to assume that the pressure loading of a WAF
can be estimated with Evvard?s theory with the addition of an induced angleofattack. In
addition to obtaining the normal force and hinge moment of the fin, the geometry of the
WAF can then be used to obtain the side force and root bending moment from the
pressure distribution.
2.1 Induced AngleofAttack
At the 2005 LMMFC HSWT, fin alone data was gathered via a splitterplate for
three different aspect ratio rectangular fins with various curvature. The fins were attached
to a six component balance; therefore, a complete force and moment data set was
gathered. The zero normal force angleofattack of each tested fin was derived from the
test data, and a correlation dependent on Mach number, aspect ratio and fin curvature
was formulated for the induced angleofattack. The geometry of the fins tested is
tabulated in Table 1 and a photo of the test fins can be seen in Figure 2.
Table 1: WindTunnel Fin Geometries
Cfg
Root
Chord
in.
Tip
Chord
in.
Reference
Length
in.
Reference
Area
in.
2
Curvature
Angle deg.
Curvature
Radius in.
Aspect
Ratio
Taper
Ratio
Exposed
SemiSpan
in.
LE Sweep
Angle
deg.
Projected
PlanForm
Area in.
2
Wetted Plan
Form Area
in.
2
cr ct L
ref
S
ref ?
RAR? b/2 ? S
p
S
w
F010 4.2500 4.2500 4.2500 12.75 0.0 ? 1.4118 1.0 3.000 0.0 12.7500 12.7500
F012 " " " " 45.0 3.9197 1.4118 " " 0.0 " 13.0837
F014 " " " " 90.0 2.1213 1.4118 " " 0.0 " 14.1616
F016 " " " " 135.0 1.6236 1.4118 " " 0.0 " 16.2584
F018 " " " " 180.0 1.5000 1.4118 " " 0.0 " 20.0277
F020 3.0000 3.0000 3.0000 12.75 0.0 ? 2.8333 1.0 4.250 0.0 12.7500 12.7500
F024 " " " " 90.0 3.0052 2.8333 " " 0.0 " 14.1617
F026 " " " " 135.0 2.3001 2.8333 " " 0.0 " 16.2584
F030 4.9100 2.4550 3.6825 12.75 0.0 ? 1.8824 0.5 3.466 35.0 12.7635 12.7635
F034 " " " " 90.0 2.4508 1.8824 " " 35.0 " 14.1765
F036 " " " " 135.0 1.8758 1.8824 " " 35.0 " 16.2757
F040 3.6825 3.6825 3.6825 12.75 0.0 ? 1.8824 1.0 3.466 0.0 12.7635 12.7635
F044 " " " " 90.0 2.4508 1.8824 " " 0.0 " 14.1765
F010F010 F012 F014 F016F016 F018
F020F020 F026F026F024
F040 F044F044
Figure 2: Photo of Tested WAFs
6
7
2.2 AngleofAttack Dependence
In the late 1940?s, John Evvard
11,12
and others
13,15,16
solved the potential flow
equations for a pointsource distribution over a planar fin in supersonic flow. In order to
utilize Evvard?s solution, the fin is divided into regions of similar disturbance types
governed by the Mach lines emanating from leading edge discontinuities. An additional
region can form on swept fins when the Mach line originating from the root leading edge
discontinuity is reflected by the fin tip (Region V in Figure 3). Each region consists of
one or more of the three fundamental disturbance types: infinite fin, triangular fin and fin
tip. The potential flow solution applicable to each region is used to determine the
pressure differential of the upper and lower surface of the fin as a function of angleof
attack.
Since the regions of flow are defined by the intersection of the Mach cones and
the fin surface, curvature can have a significant effect on the zoning of the fin surface.
While a Mach cone intersects a planar fin with a linear Mach line, the intersection of a
Mach cone and a WAF produces a curved Mach line. As the curvature increases, the area
of the fin in the region that creates the largest pressure differential, Region I, also
increases. The result is an increase in the normal force slope with respect to angleof
attack with curvature. Figure 3 illustrates the effect of curvature on the dividing Mach
lines.
x
y
? = 0.0degrees ? = 180.0degrees
x
y
M
? M
?
Figure 3: Curvature Effect on Mach Lines at Mach 1.6
8
9
3. THEORY
The theoretical modifications required to obtain the pressure loading on a WAF
surface begin with geometry. In order to apply Evvard?s theory, the fin of interest must
be divided into incremental surface panels with a control point in the center of each
panel. The curvature angle and projected planform fin geometry are used to define an
array of 3dimensional control points and the local surface slopes at each control point.
The fin geometry and the flow conditions are then used to define the Mach lines. Once
the control points are zoned based on their position relative to the Mach lines, Evvard?s
theory is used to determine the pressure differential at each control point. Finally, the
incremental panel area, the local surface slope and the differential pressure coefficient are
used to determine the normal force, hinge moment, side force and root bending moment
coefficients of the fin.
3.1 Curved Fin Geometry
Defining the geometry of the WAF surface is the basis of the analysis. The fin is
divided into the desired number of spanwise and chordwise panels, and a control point
is positioned in the center of each control panel. With the chordwise (x) and spanwise
(y) coordinates of each control point known, the magnitude of the zcoordinate is
determined based on the curvature of the fin. Figure 4 shows the basic nomenclature that
will be used to describe the geometry of a WAF.
z
o
y
o
R
y
z
?
y
max
?A
?z
?y
?
x (into page)
Center of
Curvature
Figure 4: Curved Fin Geometry
The center of curvature of the fin is defined by:
0.2
max
y
y
o
= (1)
()
0.2
sin
?
o
y
R = (2)
22
oo
yRz ??= (3)
Once the center of curvature is known, the zcomponent of the fin surface can be
obtained from the equation of a circle with center y
o
, z
o
.
10
()
oo
zyyRz +??=
2
2
(4)
Furthermore, the local surface slope of each control point will be used to obtain the
incremental panel area on which the pressure differential acts to produce a force on the
fin in the ydirection, i.e. side force. The surface slope angle at each control point is
defined below:
()
()
?
?
?
?
?
?
?
?
??
??
=
?
?
?
?
?
?
=
??
2
0
2
11
tantan
yyR
yy
dy
dz
o
? (5)
3.2 Dividing Mach Lines
The dividing Mach lines of a WAF are derived from the intersection of the Mach
cone originating at the fin tips and the fin surface. Figure 5 illustrates the intersection of
the two surfaces showing the coordinates that are referenced in equations 6 through 12.
z 1
.0

y
x
?
(0,0,0)
y
x
z
M
?
Figure 5: Mach Cone  WAF Surface Intersection
For a planar fin, the intersection of the Mach cone emanating from the fin tip and the
surface is simply a line defined by:
11
x
y?
=
0.1
tan? (6)
where
( )
M
1
sin
1?
=? (7)
As seen from Figure 5, the line describing the intersection of the Mach cone and a WAF
surface can be redefined to include the zcomponent as:
()
x
zy
22
0.1
tan
+?
=? (8)
yaxis
x
a
x
i
s
x
1
x
2
x
3
?
1.0
M
?
?
x
a
x
i
s
x
a
x
i
s
Figure 6: Zoning Rules
12
The Mach cone boundaries and their reflection lines are used to divide the fin into as
many as five regions of flow. The regions shown in Figure 6 can be defined as:
Region 1: x < x
1
and x < x
2
Region 2: x > x
1
but x < x
2
Region 3: x > x
2
but x < x
1
Region 4: x > x
1
and x > x
2
but x < x
3
Region 5: x > x
3
In order to finalize the new zoning laws, x
1
, x
2
and x
3
must be defined as a function of y
and z.
tan
1
0.1
2
?
? =?= M (9)
22
1
zyx +=? (10)
()
2
2
2
0.1tan zyx +?+?= ? (11)
()
2
2
3
0.1 zyx +?+= ?? (12)
The Mach cones and WAF surface intersections are represented in Figure 7 by three
dimensional surfaces.
13
M
?
Figure 7: ThreeDimensional Surface Intersection
In order to validate the equations used to zone the control points on the fin, the results at
Mach 1.6 for a rectangular fin with a chord of 4.25 inches and a span of 3.0 inches at
various angles of curvature are compared to the threedimensional CAD model. Figures 8
through 12 show that the code results match the topview of the CAD model seen in
Figure 7 for various angles of curvature.
CODE RESULTS CAD RESULTS
y
x
y
x
Figure 8: Zoning Verification at Mach 1.6 for 0.0Degrees of Curvature
14
CODE RESULTS CAD RESULTS
y
x
y
x
Figure 9: Zoning Verification at Mach 1.6 for 45.0Degrees of Curvature
CODE RESULTS CAD RESULTS
y
x
y
x
Figure 10: Zoning Verification at Mach 1.6 for 90.0Degrees of Curvature
CODE RESULTS CAD RESULTS
y
x
y
x
Figure 11: Zoning Verification at Mach 1.6 for 135.0Degrees of Curvature
15
CODE RESULTS CAD RESULTS
y
x
y
x
Figure 12: Zoning Verification at Mach 1.6 for 180.0Degrees of Curvature
3.3 Pressure Differential in Each Region of Flow
Now that the fin has been divided into zones based on regions of influence, the
pressure differential between the upper and lower surface can be evaluated based on the
types of disturbances that affect each region of the fin. Since the potential equation for a
fin in supersonic flow is described by an ordinary second order differential equation, the
laws of superposition apply. Therefore, the pressure differential in each region of the fin
is a summation of each upstream disturbance type. Since an induced angleofattack
method is being utilized, the angleofattack (?) seen in the Equations 14 through 25 can
be equated to:
INDUCEDCAERODYNAMI
??? += (13)
3.3.1 Region I
Region I is the fundamental portion of the fin which lies outside both Mach cones;
therefore, control points within Region I are only exposed to infinite fin (airfoil) type
16
disturbances. From linearized supersonic flow theory, the pressure coefficient on the
upper surface of a flat plate is given by:
1
2
2
,
?
=
M
C
lowerp
?
(14)
and
1
2
2
,
?
?=
M
C
upperp
?
(15)
Differencing the lower and upper pressure coefficients yield a differential pressure
coefficient of:
1
4
2
?
=?
M
C
p
?
(16)
The pressure differential in Region I using Evvard?s theory is based on linearized theory;
however, the leading edge sweep angle is included such that:
??
=?
22
,
tan
4
?
?
Ip
C . (17)
3.3.2 Region II
Region II is located within the interior Mach cone that is produced by the leading
edge discontinuity at the root of a swept fin; therefore, it is referred to in text as the
triangular fin region. Since there is no discontinuity at the root of a rectangular fin (L =
0), the triangular fin term is null, and Equation 18 reduces to Equation 17. Appropriately,
the triangular fin effect increases with sweep angle. The pressure differential in Region II
is defined as:
17
?
?
?
?
?
?
??
??
+
+?
+?
??
=?
??
tan
tan
cos
tan
tan
cos
tan
4
11
22
,
T
T
T
T
C
IIp
?
?
?
?
??
?
(18)
where
x
y
T ?= . (19)
3.3.3 Region III
Region III is located within the exterior Mach cone that is produced by the
leading edge fin tip; therefore, it is referred to in text as the fin tip region. Since a
pressure differential cannot be maintained at the tip of a fin, the potential flow equation is
solved with a boundary condition imposed such that the pressure differential at the tip of
the fin is zero. Region III is downstream of Region I; therefore, the tip effect is an
addition to the infinite fin solution. Since the tip effect uses the tip of the fin as a
reference, a coordinate system is defined at the leading edge fin tip such that:
??= tanxx
tip
(20)
0.1?= yy
tip
(21)
With the tip coordinates defined, the pressure differential coefficient due to the fin tip
disturbances can be written as:
( )[ ]
?
?
?
?
?
?
?
?
??
?++?
??
?=?
?
tan
tan2
cos
tan
4
1
22
,
tiptip
tiptip
tipp
yx
yx
C
?
??
?
(22)
The pressure differential in Region III can be expressed as:
tippIpIIIp
CCC
,,,
?+?=? (23)
18
3.3.4 Region IV
Region IV is the area within the interior and exterior Mach cones; therefore,
Region IV is affected by infinite fin disturbances, triangular fin disturbances and fin tip
disturbances. Since each of these types of disturbances have been defined, the pressure
differential in Region IV is simply:
tippIIpIVp
CCC
,,,
?+?=? (24)
3.3.5 Region V
In some swept fin cases, the Mach cone originating from the root leading edge
discontinuity intersects the fin tip; in which case, an addition Mach cone is created with
an origin at the fin tip intersection. Thus, the fifth fin region is formed within Region IV
designated as Region V. Region V is the result of a combination of Region IV
disturbances with an additional tip effect to yield a pressure differential defined as:
( )
()
?
?
?
?
?
?
?
?
?++
?+???
??
=?
?
tan2
tan22tan
cos
tan
4
1
22
,
tiptip
tiptip
Vp
yx
yx
C
?
??
?
(25)
A summary of the pressure coefficients for each region in presented in Table 2.
19
Table 2: Pressure Differential due to AngleofAttack
Region Region
Conditional
Pressure Coefficient Differential
I
x < x
1
and
x < x
2
??
=?
22
,
tan
4
?
?
Ip
C
II
x > x
1
but
x < x
2
?
?
?
?
?
?
??
??
+
+?
+?
??
=?
??
tan
tan
cos
tan
tan
cos
tan
4
11
22
,
T
T
T
T
C
IIp
?
?
?
?
??
?
III
x > x
2
but
x < x
1
tippIpIIIp
CCC
,,,
?+?=?
IV
x > x
1
and
x > x
2
but
x < x
3
tippIIpIVp
CCC
,,,
?+?=?
V
x > x
3
( )
()
?
?
?
?
?
?
?
?
?++
?+???
??
=?
?
tan2
tan22tan
cos
tan
4
1
22
,
tiptip
tiptip
Vp
yx
yx
C
?
??
?
3.4 Empirically Derived Induced AngleofAttack
In order to develop an empirical expression to describe the induced forces and
moments generated by fin curvature, the test data collected at the January 2005 LMMFC
HSWT was thoroughly analyzed to find a correlation. In this particular test, the fins were
mounted on a splitterplate to minimize the appearance of shock waves upstream of the
fins. Figure 13 shows one of the WAFs mounted on the splitterplate along with the test
sign convention.
20
21
C
Y
C
NW
C
A
C
AM
C
RBM
C
HM
M
?
Figure 13: WAF on SplitterPlate with Sign Convention
In order to obtain the relationship, the induced angleofattack of the three
different aspect ratio families was plotted at different supersonic Mach numbers. A linear
curvefit was used to investigate a correlation between the angle of curvature and the
induced angleofattack. Figures 14 through 16 show the linear relationship of the three
aspect ratio fins.
MACH 1.5 MACH 2.0 MACH 2.25 MACH 2.5 MACH 3.0
y = 0.4642x
y = 0.4918x
y = 0.4172x
y = 0.3757x
y = 0.2767x
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2
CURVATURE, ? (RADIANS)
I
N
DU
CED AN
GL
E
O
F

A
T
T
A
C
K
(
D
E
G
GR
EE
S)
Figure 14: Curvature Effects for AR = 1.4118
MACH 1.5 MACH 2.0 MACH 2.25 MACH 2.5 MACH 3.0
y = 0.2845x
y = 0.2827x
y = 0.2978x
y = 0.2883x
y = 0.227x
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2
CURVATURE, ? (RADIANS)
I
N
D
U
C
ED AN
GL
E
O
F

A
T
T
A
C
K
(
D
EGG
R
EE
S)
Figure 15: Curvature Effects for AR = 1.8824
22
MACH 1.5 MACH 2.0 MACH 2.25 MACH 2.5 MACH 3.0
y = 0.148x
y = 0.0801x
y = 0.1543x
y = 0.2787x
y = 0.2961x
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2
CURVATURE, ? (RADIANS)
I
N
DU
CE
D AN
GL
E

OF

A
T
T
A
C
K
(
D
E
G
GR
E
E
S
)
Figure 16: Curvature Effects for AR = 2.8333
The slopes from the linear fits (Figures 14 through 16) were then plotted with
respect to aspect ratio at each Mach number. While the resulting plots varied with Mach
number, the aspect ratio effects were matched well with a power series expression for
each Mach number. Figure 17 shows the aspect ratio dependence of the slope at Mach
2.25.
23
y = 1.3176x
2.6376
R
2
= 0.9854
0
0.1
0.2
0.3
0.4
0.5
0.6
11.522.5
Aspect Ratio, AR
a
I
NDU
CED
/
?
3
Figure 17: Aspect Ratio Dependence at Mach 2.25
24
Up to this point in the analysis, only a small amount of the collected data had
been analyzed to find a correlation. Therefore, the next step was to include all the data in
the correlation. The results from the test data were tabulated with respect to curvature,
aspect ratio, Mach number, and induced angleofattack. In order to capture the Mach
dependency, a genetic algorithm was used to find three different power series
relationships that could be applied piecewise. A dividing Mach number, also a genetic
algorithm variable, would be used to capture any inflection in the data. The three power
series relationships are related using a linear interpolation based on Mach number about
the dividing Mach number. The induced angles from the test were compared to the
correlation, and the genetic algorithm was used to minimize the root squared sum of the
differences between the correlation and the test data. A secondorder polynomial scheme
was also investigated with less success. The power series constants and dividing Mach
number chosen by the genetic algorithm and the associated equations are presented
below. In equations 26 through 30, the induced angleofattack (?
INDUCED
) is represented
in degrees and curvature angle (?) is represented in radians.
1709.0
2425.0 ARa ??= ? (26)
4181.2
1583.1
?
??= ARb ? (27)
4323.0
4346.0
?
??= ARc ? (28)
For 2630.2?Mach
()
2630.20.3
0.3
?
?
??+=
Mach
cbc
INDUCED
? (29)
For 2630.2