AERODYNAMICS OF WRAP-AROUND 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 WRAP-AROUND 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 WRAP-AROUND 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 WRAP-AROUND 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 wrap-around fin. Evvard?s theory has been used to calculate the 
pressure loading due to angle-of-attack on a wrap-around 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 angle-of-attack, the application of Evvard?s theory can be extended to accurately 
compute the unique aerodynamic characteristic of wrap-around 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 Angle-of-Attack .................................................................................... 5 
2.2 Angle-of-Attack 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 Angle-of-Attack ................................................ 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: Wind-Tunnel Fin Geometries............................................................................... 6 
Table 2: Pressure Differential due to Angle-of-Attack..................................................... 20 
Table 3: Coefficient Uncertainty ...................................................................................... 30 
Table 4: Chord-wise Center-of-Pressure Non-Dimensionalized by L
REF
......................... 48 
ix 
LIST OF FIGURES 
Figure 1: Packaging Advantage of Wrap-Around 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: Three-Dimensional Surface Intersection........................................................... 14 
Figure 8: Zoning Verification at Mach 1.6 for 0.0-Degrees of Curvature........................ 14 
Figure 9: Zoning Verification at Mach 1.6 for 45.0-Degrees of Curvature...................... 15 
Figure 10: Zoning Verification at Mach 1.6 for 90.0-Degrees of Curvature.................... 15 
Figure 11: Zoning Verification at Mach 1.6 for 135.0-Degrees of Curvature.................. 15 
Figure 12: Zoning Verification at Mach 1.6 for 180.0-Degrees of Curvature.................. 16 
Figure 13: WAF on Splitter-Plate 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 Angle-of-Attack versus Mach Number ........................... 26 
Figure 19: Measured versus Correlated Induced Angle-of-Attack................................... 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 Wind-Tunnel  
LMMFC  Lockheed Martin Missile and Fire Control  
USAF   United States Air Force  
CFD   Computational Fluid Dynamics 
CAD   Computer-Aided Drafting 
APKWS  Advanced Precision Kill Weapon System  
BAT   Brilliant Anti-armor Technology  
CKEM   Compact Kinetic Energy Missile  
LOSAT  Line-of-Sight Antitank 
MLRS   Multiple Launch Rocket System 
TACAWS  The Army Combined Arms Weapons System  
WAF(s)  Wrap-Around 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 xy-Plane 
Dx   Incremental Chord-wise Length 
Dy   Incremental Span-wise Length 
Dz   Incremental Curve-wise Length  
b   Wingspan 
b/2   Fin semi-span 
L
REF
   Mean Chord Length  
S
REF
   Fin Plan-form Area, L
REF
 (b/2) 
AR   Fin Aspect Ratio, b
2
/(2.0 S
REF
) 
a   Angle-of-Attack 
a
INDUCED
  Induced Angle-of-Attack  
a
AERODYNAMIC
  Angle Between the Free-stream Mach number and the Fin Chord 
M, M
?
   Free-stream Mach number 
b   Compressibility Factor, 0.1
2
?M  
d   Fin Curvature slope angle 
L   Leading Edge Sweepback Angle 
?    Fin Curvature 
xii 
1. INTRODUCTION 
 Wrap-around fins (WAFs) are a family of fins that, when stowed, conform or 
?wrap-around? the surface of a cylindrical body. As a result of the packaging advantage 
WAFs have over planar fins, WAFs are prevalent on tube-launched missile and rocket 
systems. Several fielded missiles, rockets and munitions utilize WAFs for stability; 
among these systems are MLRS, TACAWS, APKWS, LOSAT, BAT, CKEM, Hydra-70, 
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 Wrap-Around 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 360-degrees 
divided by the number of fins. A majority of the systems utilizing WAFs have 4 fins; 
1 
2 
therefore, a WAF with 90-degrees of curvature is common. However, several 2.75-inch 
rockets are equipped with 3 WAFs for stability. Brilliant Anti-armor Technology (BAT) 
employs 4 overlapping WAFs for stability with a curvature angle of 180-degrees. 
 Wrap-around 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 angle-of-
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.
6-10
 Among these devices were 
WAFs, ringtails and flares. Limited data were collected on several WAF geometries on a 
splitter-plate and on a generic 4-inch diameter body with a 2-caliber secant ogive nose 
and an 8-caliber cylindrical after-body. The fins tested were limited to 90-degrees of 
curvature. 
 In terms of stability, the U.S. Army concluded that WAFs perform similarly to 
planar fins of equivalent projected plan-form shape. It was also noted that WAFs 
produced a substantial amount of rolling moment which varied with angle-of-attack 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 free-flight gun tests, wind-tunnel 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 fin-body junction angle. A majority of the testing was performed 
on a 2.22 aspect ratio rectangular fin with a thickness-to-chord ratio of 12.5-percent and a 
45-degree 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 body-fin 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 wrap-around 
fins on a splitter-plate at the Lockheed Martin Missile and Fire Control High Speed 
Wind-Tunnel (LMMFC HSWT) in Dallas, Texas in January of 2005 with the goal of 
developing a design methodology for wrap-around 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 angle-of-attack 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 
angle-of-attack when compared to a planar fin of the same projected plan-form 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 angle-of-attack. 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 angle-of-attack. 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 Angle-of-Attack 
 At the 2005 LMMFC HSWT, fin alone data was gathered via a splitter-plate 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 angle-of-attack 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 angle-of-attack. 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: Wind-Tunnel 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 
Semi-Span 
in.
LE Sweep 
Angle   
deg.
Projected 
Plan-Form 
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 Angle-of-Attack Dependence 
 In the late 1940?s, John Evvard 
11,12
 and others 
13,15,16
 solved the potential flow 
equations for a point-source 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 angle-of-
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 angle-of-
attack with curvature. Figure 3 illustrates the effect of curvature on the dividing Mach 
lines. 
x
y
? = 0.0-degrees ? = 180.0-degrees
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 plan-form fin geometry are used to define an 
array of 3-dimensional 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 span-wise and chord-wise panels, and a control point 
is positioned in the center of each control panel. With the chord-wise (x) and span-wise 
(y) coordinates of each control point known, the magnitude of the z-coordinate 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 z-component 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 y-direction, 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 z-component as: 
()
x
zy
22
0.1
tan
+?
=?                                                (8) 
y-axis
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: Three-Dimensional 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 three-dimensional CAD model. Figures 8 
through 12 show that the code results match the top-view 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 0.0-Degrees of Curvature 
14 
CODE RESULTS CAD RESULTS
y
x
y
x
 
Figure 9: Zoning Verification at Mach 1.6 for 45.0-Degrees of Curvature 
CODE RESULTS CAD RESULTS
y
x
y
x
 
Figure 10: Zoning Verification at Mach 1.6 for 90.0-Degrees of Curvature 
CODE RESULTS CAD RESULTS
y
x
y
x
 
Figure 11: Zoning Verification at Mach 1.6 for 135.0-Degrees of Curvature 
15 
CODE RESULTS CAD RESULTS
y
x
y
x
 
Figure 12: Zoning Verification at Mach 1.6 for 180.0-Degrees 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 angle-of-attack 
method is being utilized, the angle-of-attack (?) 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 Angle-of-Attack 
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 Angle-of-Attack 
 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 splitter-plate to minimize the appearance of shock waves upstream of the 
fins. Figure 13 shows one of the WAFs mounted on the splitter-plate 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 Splitter-Plate with Sign Convention 
 In order to obtain the relationship, the induced angle-of-attack of the three 
different aspect ratio families was plotted at different supersonic Mach numbers. A linear 
curve-fit was used to investigate a correlation between the angle of curvature and the 
induced angle-of-attack. 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 angle-of-attack. 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 second-order 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 angle-of-attack (?
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<Mach
()
5.12630.2
2630.2
?
?
??+=
Mach
bab
INDUCED
?                                (30) 
As seen in Figure 18 and 19, the correlation worked well. The induced angle-of-attack 
tends to have an inflection point near Mach 2.25. The dividing Mach number helps the 
correlation capture the inflection of the data. The overall performance of the correlation is 
seen in Figure 19. While the correlated induced angle-of-attack is not quite within the 
uncertainty of the measured induced angle-of-attack (+- 0.1-degree), the correlation was 
assumed to be adequate to proceed with the methodology. 
25 
1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
26 
1.6
1.8
Mach Number
I
nd
uc
e
d A
n
gl
e-
of
-A
t
t
ac
k
Calculated
Measured
 
Figure 18: Correlated Induced Angle-of-Attack versus Mach Number 
-0.5 0 0.5 1 1.5 2
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Calculated Induced Angle-of-Attack
M
eas
ur
ed I
n
duc
ed Angl
e-
of
-
A
t
t
ac
k
Calculated
Test Error
Ideal
 
Figure 19: Measured versus Correlated Induced Angle-of-Attack
4. INTEGRATION OF THE PRESSURE DISTRIBUTION 
 Using the induced angle-of-attack in conjunction with the modified form of 
Evvard?s theory results in a method of predicting the induced and angle-of-attack 
dependent pressure loading on a WAF. In order to obtain the force and moment 
coefficients, the pressure differentials must be numerically integrated over the WAF 
surface. 
 The aerodynamic loads exerted on the fin are a product of the differential pressure 
coefficient and the appropriate fin area. The normal force and side force are based on the 
area of the fin projected onto the xy-plane and the xz-plane, respectively. The moments 
are a summation of the product of the incremental forces and their respective distances to 
the reference location. The root bending moment is referenced about the root, and the 
hinge moment is referenced about the leading edge. The forces and moments can be 
expressed as: 
yxA
yx
???=?
,
                                                        (31) 
?
=
???=
NP
i i
yxP
REF
N
AC
S
C
i1
,
1
                                            (32) 
i
NP
i i
yxiP
REFREF
HM
xAC
LS
C
?
=
????
?
=
1
,
1
                                    (33) 
?
=
?????=
NP
i
iiP
REF
Y
zxC
S
C
i1
1
                                             (34) 
27 
i
NP
i
iiP
REF
Y
dy
dz
yxC
S
C
i
?
= ?
?
?
?
?
?
?
?
??????=
1
1
                                     (35) 
?
=
????=
i
i
i
yxP
REF
Y
AC
S
C
i1
,
tan?
NP1
                                      (36) 
[]
??
==
?????+????
?
=
NP
i
ii
i
yxP
NP
i
iyxP
REFREF
RBM
zACyAC
LS
C
iii 1
,
1
,
tan
1
?              (37) 
28 
29 
5. RESULTS 
 The results of applying the induced angle-of-attack to Evvard?s theory modified 
for fin curvature show good agreement at both positive and negative angles-of-attack. 
The results are presented in two segments: 1.) selective plots comparing theoretical 
results and test data as a function of angle-of-attack and 2.) pressure differential contours 
at 5-degrees of incidence on selective fins. The normal force, side force, root bending 
moment and hinge moment will be compared to test data with respect to angle-of-attack. 
Since the focus is on the pressure driven forces and moments, the axial force and 
associated axial moment will be neglected in the results. 
5.1 Comparison to Test Data 
 Each plot contains data for various angles of curvature from both the 
methodology and from the splitter-plate test data. The theoretical solution is represented 
as a line (dotted, dashed and solid). The test results are represented as various symbols 
denoted in the legend on each chart along with the word ?Test?. The test data represents 
both viscous and inviscid effects; while, the theoretical solution only models the inviscid 
effects. One source of discrepancy between the theoretical solution and the test results is 
the thickness of the fins. Theoretically, the fins are infinitely thin. In addition to a finite 
thickness, the test fins have an increased thickness near the root for required structural 
properties (Figure 2). The test data uncertainty as quoted by LMMFC HSWT can be seen 
in Table 3 and is shown on select coefficients in Figures 20 through 59. In most cases, the 
size of the error band is smaller than the data symbol. 
Table 3: Coefficient Uncertainty 
0.00250.00550.00200.0055 
uCRBMuCHMuCYuCN
 
? ? ? ?
5.1.1 Normal Force 
 Figures 20 through 29 show the comparison of the theoretical normal force and 
the test results. Overall, the addition of the curvature term in the zoning laws allows 
Evvard?s theory to model the changes in the normal force slope extremely well (within 
the uncertainty of the test data). The normal force slope of the moderate aspect ratio (AR 
= 1.4118 and AR = 1.8824) fins are modeled slightly better than the extreme aspect ratio 
fins. For the higher aspect ratio cases (AR = 2.8333), Figures 26 through 28, the normal 
force slope appears to be slightly different at positive versus negative angles-of-of attack. 
Evvard?s theory works well at Mach 1.5, but the accuracy seems to improve with 
increasing Mach number. As for the induced normal force, the empirical fit for the 
induced angle-of-attack allows the normal force to match the test data at low angles-of-
attack. The largest discrepancy in the low incidence normal force modeling is seen in 
Figure 20 for a fin curvature of 180-degrees. Even though leading edge sweep angle was 
not accounted for in the empirical fit, Figure 29 shows good agreement between theory 
and test data for a leading edge sweep angle of 35-degrees. 
30 
-0.5
0
0.5
-10 0 10
Angle-of-Attack (degrees)
C
N
? = 0 ? = 0 Test
? = 45 ? = 45 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
? = 180 ? = 180 Test
 
Figure 20: Normal Force Comparison for AR = 1.4118 at Mach 1.5 
-0.4
0
0.4
-10 0 10
Angle-of-Attack (degrees)
C
N
? = 0 ? = 0 Test
? = 45 ? = 45 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
? = 180 ? = 180 Test
 
Figure 21: Normal Force Comparison for AR = 1.4118 at Mach 2.25 
31 
-0.3
0
0.3
-10 0 10
Angle-of-Attack (degrees)
C
N
? = 0 ? = 0 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
? = 180 ? = 180 Test
 
Figure 22: Normal Force Comparison for AR = 1.4118 at Mach 3.0 
-0.6
0
0.6
-10 0 10
Angle-of-Attack (degrees)
C
N
? = 0 ? = 0 Test
? = 90 ? = 90 Test
 
Figure 23: Normal Force Comparison for AR = 1.8824 at Mach 1.5 
32 
-0.4
0
0.4
-10 0 10
Angle-of-Attack (degrees)
C
N
? = 0 ? = 0 Test
? = 90 ? = 90 Test
 
Figure 24: Normal Force Comparison for AR = 1.8824 at Mach 2.25 
-0.3
0
0.3
-10 0 10
Angle-of-Attack (degrees)
C
N
? = 90 ? = 90 Test
 
Figure 25: Normal Force Comparison for AR = 1.8824 at Mach 3.0 
33 
-0.6
0
0.6
-10 0 10
Angle-of-Attack (degrees)
C
N
? = 0 ? = 0 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
 
Figure 26: Normal Force Comparison for AR = 2.8333 at Mach 1.5 
-0.4
0
0.4
-10 0 10
Angle-of-Attack (degrees)
C
N
? = 0 ? = 0 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
 
Figure 27: Normal Force Comparison for AR = 2.8333 at Mach 2.25 
34 
-0.3
0
0.3
-10 0 10
Angle-of-Attack (degrees)
C
N
? = 90 ? = 90 Test
? = 135 ? = 135 Test
 
Figure 28: Normal Force Comparison for AR = 2.8333 at Mach 3.0 
-0.6
0
0.6
-10 0 10
Angle-of-Attack (degrees)
C
N
? = 0 ? = 0 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
 
Figure 29: Normal Force for AR = 1.8824; L = 35
o
 at Mach 1.5 
35 
36 
5.1.2 Side Force 
 The side force of the fin, as seen in Figure 13, is the force that is directed from the 
fin tip to the root. The pressure contour plots (Figures 60 through 71) can be used to 
visualize the manner in which the pressure exerts a side force on the fin. A curved fin has 
surface area in the xz-plane; therefore, the pressure differential generates a side force on a 
WAF. 
 The theoretical solution for all fins and conditions shows a pushing (directed from 
tip to root) side force at positive angles-of-attack and a pulling (directed from root to tip) 
side force at negative angles-of-attack. The test data shows that a pulling force is 
dominant even at positive angles-of-attack. As seen in Figure 2, the tested fins were 
thickened near the root for structural rigidity which alters the area of the fin in the xz-
plane. In addition to the thickened root, the splitter-plate also is suspected to influence the 
side force. For finite thickness fin in supersonic flow, the leading edge of the fins will 
produce a shock wave. On the convex side of the tested WAFs near root, the shock wave 
is most likely reflected by the splitter-plate forming a region of high pressure. A region of 
high pressure in this location would generate a pulling side force. Again, the theory does 
not account for fin thickness. However, the side force order of magnitude and data trend 
is modeled well. Figures 30 through 39 show the side force coefficient comparison of the 
theory to the test data. The theory typically over-predicts the magnitude of side force at 
positive angles-of-attack, but matches the magnitude well at negative angles-of-attack. 
The computed side force of the swept fin matches test data remarkably better than the 
rectangular fins, Figure 39. 
-0.2
0
0.2
-10 0 10
Angle-of-Attack (degrees)
C
Y
? = 0 ? = 0 Test
? = 45 ? = 45 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
? = 180 ? = 180 Test
 
Figure 30: Side Force Comparison for AR = 1.4118 at Mach 1.5 
-0.2
0
0.2
-10 0 10
Angle-of-Attack (degrees)
C
Y
? = 0 ? = 0 Test
? = 45 ? = 45 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
? = 180 ? = 180 Test
 
Figure 31: Side Force Comparison for AR = 1.4118 at Mach 2.25 
37 
-0.2
0
0.2
-10 0 10
Angle-of-Attack (degrees)
C
Y
? = 0 ? = 0 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
? = 180 ? = 180 Test
 
Figure 32: Side Force Comparison for AR = 1.4118 at Mach 3.0 
-0.2
0
0.2
-10 0 10
Angle-of-Attack (degrees)
C
Y
? = 0 ? = 0 Test
? = 90 ? = 90 Test
 
Figure 33: Side Force Comparison for AR = 1.8824 at Mach 1.5 
38 
-0.2
0
0.2
-10 0 10
Angle-of-Attack (degrees)
C
Y
? = 0 ? = 0 Test
? = 90 ? = 90 Test
 
Figure 34: Side Force Comparison for AR = 1.8824 at Mach 2.25 
-0.2
0
0.2
-10 0 10
Angle-of-Attack (degrees)
C
Y
? = 90 ? = 90 Test
 
Figure 35: Side Force Comparison for AR = 1.8824 at Mach 3.0 
39 
-0.2
0
0.2
-10 0 10
Angle-of-Attack (degrees)
C
Y
? = 0 ? = 0 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
 
Figure 36: Side Force Comparison for AR = 2.8333 at Mach 1.5 
-0.2
0
0.2
-10 0 10
Angle-of-Attack (degrees)
C
Y
? = 0 ? = 0 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
 
Figure 37: Side Force Comparison for AR = 2.8333 at Mach 2.25 
40 
-0.2
0
0.2
-10 0 10
Angle-of-Attack (degrees)
C
Y
? = 90 ? = 90 Test
? = 135 ? = 135 Test
 
Figure 38: Side Force Comparison for AR = 2.8333 at Mach 3.0 
-0.2
0
0.2
-10 0 10
Angle-of-Attack (degrees)
C
Y
? = 0 ? = 0 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
 
Figure 39: Side Force for AR = 1.8824; L = 35
o
 at Mach 1.5 
41 
42 
5.1.3 Root Bending Moment 
 The span-wise center-of-pressure of the fins is modeled well with the theory as 
reflected in the root bending moment comparisons seen in Figures 40 through 49. The 
root bending moment is a combination of the normal and side forces with their respective 
moment arms (equation 37). While the root bending moment is modeled accurately at 
low incidence, the theory tends to under-predict the root bending moment at higher 
angles-of-attack. The root thickness is most likely moving the span-wise center-of-
pressure of the tested fins towards the fin tip thus increasing the root bending moment. At 
negative angles-of-attack, the root bending moment is less sensitive to fin curvature. 
-0.2
0
0.2
-10 0 10
Angle-of-Attack (degrees)
C
RBM
? = 0 ? = 0 Test
? = 45 ? = 45 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
? = 180 ? = 180 Test
 
Figure 40: Root Bending Moment Comparison for AR = 1.4118 at Mach 1.5 
-0.15
0
0.15
-10 0 10
Angle-of-Attack (degrees)
C
RB
M
? = 0 ? = 0 Test
? = 45 ? = 45 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
? = 180 ? = 180 Test
 
Figure 41: Root Bending Moment Comparison for AR = 1.4118 at Mach 2.25 
43 
-0.1
0
0.1
-10 0 10
Angle-of-Attack (degrees)
C
RBM
? = 0 ? = 0 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
? = 180 ? = 180 Test
 
Figure 42: Root Bending Moment Comparison for AR = 1.4118 at Mach 3.0 
-0.25
0
0.25
-10 0 10
Angle-of-Attack (degrees)
C
RBM
? = 0 ? = 0 Test
? = 90 ? = 90 Test
 
Figure 43: Root Bending Moment Comparison for AR = 1.8824 at Mach 1.5 
44 
-0.2
0
0.2
-10 0 10
Angle-of-Attack (degrees)
C
RBM
? = 0 ? = 0 Test
? = 90 ? = 90 Test
 
Figure 44: Root Bending Moment Comparison for AR = 1.8824 at Mach 2.25 
-0.15
0
0.15
-10 0 10
Angle-of-Attack (degrees)
C
RBM
? = 90 ? = 90 Test
 
Figure 45: Root Bending Moment Comparison for AR = 1.8824 at Mach 3.0 
45 
-0.5
0
0.5
-10 0 10
Angle-of-Attack (degrees)
C
RBM
? = 0 ? = 0 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
 
Figure 46: Root Bending Moment Comparison for AR = 2.8333 at Mach 1.5 
-0.3
0
0.3
-10 0 10
Angle-of-Attack (degrees)
C
RBM
? = 0 ? = 0 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
 
Figure 47: Root Bending Moment Comparison for AR = 2.8333 at Mach 2.25 
46 
-0.2
0
0.2
-10 0 10
Angle-of-Attack (degrees)
C
RBM
? = 90 ? = 90 Test
? = 135 ? = 135 Test
 
Figure 48: Root Bending Moment Comparison for AR = 2.8333 at Mach 3.0 
-0.25
0
0.25
-10 0 10
Angle-of-Attack (degrees)
C
RBM
? = 0 ? = 0 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
 
Figure 49: Root Bending Moment for AR = 1.8824; L = 35
o
 at Mach 1.5 
47 
5.1.4 Hinge Moment 
 The hinge moment data are presented in Figures 50 through 59 about the mid root 
chord since fins are typically hinged about their theoretical center-of-pressure. As a 
result, the hinge moments are very small, and the differences between theoretical and test 
results are visually amplified. Presenting the data about the mid root chord gives a better 
indication of the chord-wise center-of-pressure accuracy. The test results indicate that the 
center-of-pressure is more forward than the theoretical results. Since the test articles were 
not infinitely thin, a shock wave from the leading edge is suspected to decrease the Mach 
number over the fin which would shift the center-of-pressure forward. As seen in Table 4, 
the chord-wise center-of-pressure of the fin is modeled within 5 to 10-percent relative 
error of the test results. As a result of matching both the normal force coefficient and the 
chord-wise center-of-pressure reasonably well, the hinge moment comparison yields 
reasonable results. 
Table 4: Chord-wise Center-of-Pressure Non-Dimensionalized by L
REF 
 
48 
-0.1
0
0.1
-10 0 10
Angle-of-Attack (degrees)
C
HM
? = 0 ? = 0 Test
? = 45 ? = 45 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
? = 180 ? = 180 Test
 
Figure 50: Hinge Moment about C
R
/2.0 Comparison for AR = 1.4118 at Mach 1.5 
-0.04
0
0.04
-10 0 10
Angle-of-Attack (degrees)
C
HM
? = 0 ? = 0 Test
? = 45 ? = 45 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
? = 180 ? = 180 Test
 
Figure 51: Hinge Moment about C
R
/2.0 Comparison for AR = 1.4118 at Mach 2.25 
49 
-0.04
0
0.04
-10 0 10
Angle-of-Attack (degrees)
C
HM
? = 0 ? = 0 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
? = 180 ? = 180 Test
 
Figure 52: Hinge Moment about C
R
/2.0 Comparison for AR = 1.4118 at Mach 3.0 
-0.1
0
0.1
-10 0 10
Angle-of-Attack (degrees)
C
HM
? = 0 ? = 0 Test
? = 90 ? = 90 Test
 
Figure 53: Hinge Moment about C
R
/2.0 Comparison for AR = 1.8824 at Mach 1.5 
50 
-0.04
0
0.04
-10 0 10
Angle-of-Attack (degrees)
C
HM
? = 0 ? = 0 Test
? = 90 ? = 90 Test
 
Figure 54: Hinge Moment about C
R
/2.0 Comparison for AR = 1.8824 at Mach 2.25 
-0.04
0
0.04
-10 0 10
Angle-of-Attack (degrees)
C
HM
? = 90 ? = 90 Test
 
Figure 55: Hinge Moment about C
R
/2.0 Comparison for AR = 1.8824 at Mach 3.0 
51 
-0.1
0
0.1
-10 0 10
Angle-of-Attack (degrees)
C
HM
? = 0 ? = 0 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
 
Figure 56: Hinge Moment about C
R
/2.0 Comparison for AR = 2.8333 at Mach 1.5 
-0.1
0
0.1
-10 0 10
Angle-of-Attack (degrees)
C
HM
? = 0 ? = 0 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
 
Figure 57: Hinge Moment about C
R
/2.0 Comparison for AR = 2.8333 at Mach 2.25 
52 
-0.1
0
0.1
-10 0 10
Angle-of-Attack (degrees)
C
HM
? = 90 ? = 90 Test
? = 135 ? = 135 Test
 
Figure 58: Hinge Moment about CR/2.0 Comparison for AR = 2.8333 at Mach 3.0 
-0.1
0
0.1
-10 0 10
Angle-of-Attack (degrees)
C
HM
? = 0 ? = 0 Test
? = 90 ? = 90 Test
? = 135 ? = 135 Test
 
Figure 59: Hinge Moment about C
R
/2.0 for AR = 1.8824; L = 35
o
 at Mach 1.5 
53 
54 
5.2 Pressure Contour Plots 
 Pressure contour plots are presented in Figures 60 through 71 for 5-degrees angle-
of-attack to show the effects of fin curvature and Mach number. The pressure contour 
plots can give another dimension to the aerodynamic assessment of WAFs. The centers-
of-pressures can be visualized and area of high pressure differential can be identified for 
possible fin redesign. The figures are arranged to show the effects of Mach number on 
each row and the effects of fin curvature on each column. 
 The center-of-pressure moves aft and toward the fin tip as the Mach number 
increases. The area of higher pressure differential (Region I) enlarges with increasing fin 
curvature. Potential performance enhancements could be generated based on analysis of 
the pressure contour plots such as clipping the trailing edge fin tip to decrease the fin 
surface area while retaining the fin region which produces a majority of the stabilizing 
force. 
 Fin curvature also increases the area of the fin in the xz-plane which creates a 
higher side force. Since fins are typically used in sets, the net fin side forces are 
cancelled; however, the side force can generate a rolling moment when the center-of-
pressure is located at a non-zero z-coordinate. While the induced fin normal force 
generates a majority of the induced rolling moment on a missile equipped with WAFs, 
the fin side force contributions must be known to accurately model the overall missile 
rolling moment. 
 
 
Figure 60: Pressure Contour for ? = 0.0; 
AR = 1.4118 at Mach 1.5 
 
Figure 62: Pressure Contour for ? = 
90.0; AR = 1.4118 at Mach 1.5 
 
Figure 61: Pressure Contour for ? = 0.0; 
AR = 1.4118 at Mach 3.0 
 
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 
 
Figure 66: Pressure Contour for L = 
35.0; ? = 0.0; AR = 1.8824 at Mach 1.5 
 
Figure 65: Pressure Contour for ? = 
180.0; AR = 1.4118 at Mach 3.0 
 
Figure 67: Pressure Contour for L = 
35.0; ? = 0.0; AR = 1.8824 at Mach 3.0 
56 
 
 
Figure 69: Pressure Contour for L = 
35.0; ? = 90.0; AR = 1.8824 at Mach 3.0 
Figure 68: Pressure Contour for L = 
35.0; ? = 90.0; AR = 1.8824 at Mach 1.5 
  
Figure 70: Pressure Contour for L = 
35.0; ? = 180.0; AR = 1.8824 at Mach 
1.5 
Figure 71: Pressure Contour for L = 
35.0; ? = 180.0; AR = 1.8824 at Mach 
3.0 
57 
58 
6. LIMITATIONS 
 While several fin parameters and flow conditions were used to present a 
generalized method of estimating the pressure driven forces and moments of a WAF, the 
methodology has Mach number, aspect ratio, leading edge sweep angle, and thickness-to-
chord ratio limits based the tested fin parameters. The empirical fit for induced angle-of-
attack is based on data collected on rectangular fins from Mach 1.5 to Mach 3.0 with 
aspect ratios ranging from 1.4118 to 2.8333. The thickness-to-chord ratio for all the fins 
varied linearly from 3-percent at the root to 1.5-percent at the tip (with the exception of 
the thickened root). The method compares well to wind-tunnel results for these fins; 
however, no proof exists that this correlation applies to fins with features outside the test 
envelope. The methodology to compute the pressure differential is derived from potential 
flow theory for an infinitely thin fin; therefore, the results from the method will represent 
a thin fin with no boundary layer or shock waves that will never stall. With these 
limitations stated, the method will provide a reasonable design envelop for typical WAF 
applications in supersonic flow. 
59 
7. CONCLUSION 
 A new method has been developed to obtain the pressure loading of a curved or 
wrap-around fin with modifications to existing supersonic fin theory and the aid of recent 
wind-tunnel test data. The theoretical results show agreement to wind-tunnel test data for 
the low incidence forces and moments that have puzzled aerodynamicist for years. The 
method provides an expedient analysis tool that has been developed to cover a broad 
range of fin parameters that can be used for roll tailoring or roll minimization on missile 
systems requiring the use of wrap-around fins. 
60 
8. REFERENCES 
1. Abate, G., ?Aerodynamic Research of Wrap Around Fin Missile Configurations 
and Alternative Wrap Around Fin Designs?, WL-TR-94-7015, Wright Laboratory 
Armament Directorate, Eglin Air Force Base, Florida, February 1994. 
 
2. Ailor, W. H., III, ?Supersonic Aerodynamic Characteristics of a Wrap Around 
Folding Fin?, IHTR 337, Naval Ordnance Station, Indian Head, Maryland, June 
1971. 
 
3. Auman, L. M., ?The Aerodynamic Characteristics of Production MLRS Wrap-
Around Fins?, Technical Report RD-SS-92-10, US Army Missile Command, 
Redstone Arsenal, Alabama, August 1992. 
 
4. Auman, L. M., ?Aerodynamic Characteristics of a Guided MLRS Rocket?, 
Technical Report RD-SS-98-4, U.S. Army Aviation and Missile Command, 
Redstone Arsenal, Alabama, December 1997. 
 
5. Butler, R. W., ?Evaluation of the Pressure Distribution on the Fins of a 
Wraparound Fin Model at Subsonic and Transonic Mach Numbers?, AEDC-TR-
71-273, Air Force Systems Command, Arnold Air Force Station, Tennessee, 
December 1971. 
 
6. Dahlke, C. W., ?Aerodynamics of Wrap-Around Fins: A Survey of the 
Literature?, RD-TR-71-1, U.S. Army Missile Command, Redstone Arsenal, 
Alabama, March 1971. 
 
7. Dahlke, C. W. and L. D. Flowers, ?The Aerodynamic Characteristics of Wrap-
Around Fins, Including Fold Angle, at Mach Numbers from 0.5 to 1.3?, Report 
No. RD-TR-75-19, U.S. Army Missile Command, Redstone Arsenal, Alabama, 
December 1974. 
 
8. Dahlke, C. W., ?The Aerodynamic Characteristics of Wrap-Around fins at Mach 
Numbers of 0.3 to 3.0? , Technical Report RD-77-4, U.S. Army Missile 
Command, Redstone Arsenal, Alabama, October 1976. 
 
61 
9. Dahlke, C. W., R. A. Deep and V. Oskay, ?Techniques for Roll Tailoring for  
Missiles with Wrap-Around Fins?, AIAA-83-0463, American Institute of 
Aeronautics and Astronautics 21
st
 Aerospace Science Meeting, Reno, Nevada, 
January 1983. 
 
10. Dahlke, C. W. and L. D. Flowers, ?The Aerodynamic Characteristics of Wrap-
Around Fins, Including Fold Angle, at Mach Numbers from 0.5 to 1.3?, Technical 
Report RD-75-19, U.S. Army Missile Command, Redstone Arsenal, Alabama, 
December 1974. 
 
11. Evvard, J. C., ?Distribution of Wave Drag and Lift in the Vicinity of Wing Tips at 
Supersonic Speeds?, NACA Technical Note No. 1382, Flight Propulsion 
Research Laboratory, Cleveland, Ohio, July 1947. 
 
12. Evvard, J. C., ?Theoretical Distribution of Lift on Thin Wings at Supersonic 
Speeds?, NACA Technical Note No. 1585, Flight Propulsion Research 
Laboratory, Cleveland, Ohio, May 1948. 
 
13. Harmon, S. M. and Isabella Jeffreys, ?Theoretical Lift and Damping in Roll of 
Thin Wings with Arbitrary Sweep and Taper at Supersonic Speeds?, NACA 
Technical Note No. 2114, Langley Aeronautical Laboratory, Langley Air Force 
Base, Virginia, May 1950. 
 
14. McIntyre, T. C., R. D. W. Bowersox, and L. P. Goss, ?Effects of Mach Number 
on Supersonic Wraparound Fin Aerodynamics?, Journal of Spacecraft and 
Rockets, Vol. 35, No. 6, November-December 1998. 
 
15. Mirels, Harold, ?Theoretical Wave Drag and Lift of Thin Supersonic Ring 
Airfoils?, NACA Technical Note No. 1678, Flight Propulsion Laboratory, 
Cleveland, Ohio, August 1948. 
 
16. Moeckel, W. E. and J. C. Evvard, ?Load Distribution due to steady Roll and Pitch 
for Thin Wings at Supersonic Speeds?, NACA Technical Note No. 1689, Flight 
Propulsion Research Laboratory, Cleveland, Ohio, August 1948. 
 
17. Robinson, M. L. and C. E. Fenton, ?Static Aerodynamic Characteristics of a 
Wrap-Around Fin Configuration with Small Rolling Moments at Low Incidence?, 
WRE-Technical Note-527 (WR&D), Australian Defence Scientific Service, 
Salisbury, South Australia, November 1971. 
 
18. Tilmann, C. P., R. E. Huffman, Jr., T. A. Bulter, and R. D. W. Bowersox, 
?Characterization of the Flow Structure in the Vicinity of a Wrap-Around Fin at 
Supersonic Speeds?, American Institute of Aeronautics and Astronautics 34
th
 
Aerospace Science Meeting, AIAA 96-0190, Reno, Nevada, January 1996