Aerodynamic Characteristics of Wrap-Around Rocket Fins

American Institute of Aeronautics and Astronautics
1
Aerodynamic Characteristics of Wrap-Around Fins using
OpenVSP and Flightstream
Benjamin T. Bauldree1
Auburn University, Auburn, AL, 36382
The wrap-around fin (WAF) design involves enveloping normal, planar fins around the
body of the rocket. The planar fins take on the radius of the rocket, curling around the outer
surface until flush with the body. Upon launch, the fins are released from the initial position
and extended till at the final position. This fin configuration offers rewards in that the initial
fin position minimizes the overall volume of the rocket while also decreasing the possibility of
incurring damage on the fins themselves. However, the WAF design suffers from instabilities
in side moment and roll reversal through the transonic region. Studies have shown that the
Mach number and angle of attack have large influences on the roll moment of the missile,
especially during the transition from subsonic to supersonic flight, where the missiles often
undergo a reversal in the roll moment and an increase in roll rate at higher Mach numbers.
Further studies have attempted to alleviate various instabilities associated with the WAF
design, such as incorporating slots into the fins. Most studies involved physical specimens
undergoing physical testing. This paper researched the possibility of implementing two easily-
accessible software packages, OpenVSP and Flighstream. OpenVSP was used to model three
different missile configurations incorporating straight fins, wrap-around fins, and wrap-
around fins with slots. The meshed version of these models were imported into Flightstream
where multiple simulations were performed at Mach numbers ranging from 0.5 to 2 and
angles of attack ranging from zero to twenty. Simulations showed inaccuracies in data results
due primarily to inefficiencies in the meshing process of OpenVSP and the inability for the
vortex lattice method to account for supersonic effects. Both tools proved useful in terms of
producing flow solutions quickly and efficiently, competing easily against higher-fidelity (and
more expensive) computational fluid dynamic software packages; however, the tools are far
better equipped for subsonic tests with larger airframes such as aircraft or winged-missiles.
Nomenclature
A = geometrical influence coefficient
B = velocity requirements
b = span
CL = lift coefficient
CDi = induced drag coefficient
CDo = skin drag coefficient
CM = moment coefficient
Cp = pressure coefficient
Dinduced = induced drag
dl = differential length
dy = differential distance in the y-direction
h = orthogonal distance from source
Linduced = induced lift
r = distance from source
Vinduced = induced velocity
V¥ = freestream velocity
wy = downwash in the y-direction
1
Graduate Student, Department of Aerospace Engineering, Office 313, Davis Hall.

2
y = location along the y-axis
G = vorticity strength
r¥ = freestream density
q = angle between source direction and control point
I. Introduction
ockets have undergone a number of various iterations throughout history, starting with the earliest versions where
ancient Chinese tribes experimented with bamboo tubes filled with a flammable powder, similar to today’s
gunpowder, or the Congreve rocket of the 19th
century, for which Francis Scott Key most notably depicted in The
Star-Spangled Banner with the famous line “the rockets’ red glare”. It wasn’t until the early 1900s, with Tsiolkovsky’s
idea of using rockets to travel into space or Goddard’s A Method of Reaching Exetreme Altitudes, that rockets morphed
into a form we would recognize today [1], with the likes of the German V-2 (the first long-range missile) [2], the
Russian S-75 Dvina (the first successful surface-to-air missile) [3], and even the famous Saturn V, which delivered
the first people to the Moon. As engineers worked to build rockets larger, there were others doing the exact opposite.
By decreasing the size, rockets could become more mobile and easier to transport, increasing their usability. Instead
of being hauled by powerful vehicles, they could be carried on the backs of soldiers or under the wings of aircraft.
One particular design element that revolutionized the flexibility of missiles is the tube-launch system. This
particular system saw prominence in ground-launched and air-launched artillery. In these launching methods, the
missile(s) could be loaded into canisters, individually or in multitudes, and propelled quickly. However, aerodynamic
stabilizers had to be designed that could be stowed away while loaded within the tube launcher and deployed shortly
after exiting the launching device. The wrap-around fin (WAF) design was aimed at combatting that issue. The WAF,
shown in Figure 1, conformed to the body of the missile while in the launch-ready configuration. By collapsing the
stabilizers against the surface of the body, the overall volume of the missile was minimized, ultimately diminishing
the overall size of the transportation/launching system. [4]
Figure 1. The wrap-around fin design would conform to the circumference of the missile body in the undeployed
position (left) and, after exiting the tube, would deploy to a standard orthogonal position. [5]
The WAF design typically involves a geometry predicated on the shape and diameter of the missile. WAFs usually
envelope the circumference of the missile. From this, the curved span is then determined by dividing the
circumference of the body by the total number of fins, typically around three to four (the total number depending on
the size of the rocket and the stability required for flight). [5]
While demonstrating similarities to their planform counterparts, wrap-around fins often exuberated an
unpredictable lateral behavior. The design suffered from induced rolling moments at low angles of attack and would
exhibit total roll reversal when transiting through the transonic regime. [4] A study by the United States Army,
conducted between 1971 and 1976, showed that the WAF would induce a normal force towards the center of curvature
of the fins at subsonic speeds, but would produce a normal force away from the center of curvature at supersonic
speeds. Further studies by the United States Air Force determined that the cause of the roll reversal was due to the
leading edge of the fin producing a bow shock that would interact with the concave and convex sides in different
R

3
manners. The shock would be focused towards the center of curvature on the concave side, generating large pressures;
the convex side produced a high pressure area near the joint between the fin and the body. The total net force
differential resulted in a direction away from the center of curvature. [5]
The majority of these studies were performed using an experimental approach due to the lack of computational
power at the time. But with the exponential growth of computational machines over the last few decades,
computational fluid dynamic software has become more prevalent and accessible. With this in mind, the goal of this
report is to utilize those available resources to demonstrate the feasibility in performing rudimentary CFD studies by
replicating the results from past studies involving WAF designs. The software packages that will be used are
OpenVSP, an open-source parametric aircraft geometry tool, and Flightstream, a surface vorticity flow solver.
II. Software Tools
A. OpenVSP
The three basic phases in aerodynamic vehicle design are conceptual, preliminary, and detailed. Many airframe
designers utilize computer-aided design (CAD) software packages from the very start, all the way through production.
While beneficial in many regards for larger groups whose design aspects stretch over the three main phases, this
approach can be excessive and unnecessary for many other groups whose activities are focused more prominently on
individual phases. In order to better fill this gap, the engineers at the Aeronautics Systems Analysis Brach at NASA
Langley Research Center developed a preliminary, conceptual aircraft designing tool known as Vehicle Sketch Pad
(VSP). [6]
VSP centers around the earliest design phase and focuses on allowing the user to make large geometry changes by
quickly and effortlessly changing a minimum set of parameters dictated as necessary for developing an early design
model for an airframe. Instead of using basic textual input or the more complex method of CAD, VSP uses a
parameterized geometry modeler that allows the user to interact with a graphical user interface (GUI) to better
visualize the model airframe. VSP also makes it possible to perform a general analysis of an airframe to determine
possible aerodynamic characteristics for better comparison with alternative configurations and designs. Once a design
has been settled upon, VSP allows the user to export the product into formats that can be further used in various flow
simulation solvers or CAD modeling software. [6]
B. Flightstream
Flightstream is a fast-predicting surface vorticity flow solver that implements surface mesh generation with a
vortex-lattice method compressible flow solver. The combination of these components decreases solution time
dramatically compared to traditional CFD solvers while maintaining the high-fidelity results desired by the user. [7]
1. Surface Mesh Generation
Flightstream utilizes an unstructured surface mesh in its potential flow solver. An analysis of the unstructured
mesh approach is best started by describing the structured mesh method. A structured surface mesh is typically defined
as the mapping of a surface along two numerical axes (called U-V mapping) with each face being bound by at least
four vertices, creating a quadrilateral shape based upon the underlying structure. [8] The structured grid has the
characteristic of knowing each vertices neighbor implicitly, such that for a point on a 2D plane located at (i,j), as
shown in Figure 2, neighbors would be located at (i+1,j), (i-1,j), etc. [9]
Figure 2. The mapping of a structured mesh. [10]

4
An unstructured mesh, however, does not necessarily conform to a quadrilateral shape, but instead involves
irregular patterns, using either triangles or tetrahedrons as the shape base. [11] The advantages of employing the
unstructured method versus the structured method are [8]:
1. In order to convert a quadrilateral into a triangle, a facet (or vertice) must be “forced”; thus, a structured
triangle occupies more memory compared to an unstructured triangle.
2. A forced U-V mapping in highly curved or concaved areas will generate surplus shapes.
3. A quadrilateral shape requires “sanitization” in order to establish the normal direction of the surface.
Advantages of using the structured mesh approach do exist, though, especially with regards to potential-flow
solvers such as the vortex lattice method. Vorticity flow solvers require a face-edge approach which cannot be
achieved using an unstructured approach. But through the development and implementation of the Method of
Integrated Circulation, this requirement can be circumvented.
2. Method of Integrated Circulation
The method of integrated circulation is based on the idea of linearized aerodynamic effects. [8] This theory is
derived from the Laplace’s fundamental equation regarding flow over an arbitrary body, which states that for an
inviscid, incompressible flow, a flow-field solution can be generated by superimposing various elementary flows. [12]
In terms of vorticity strength, this takes on the following form:
𝛤 = 𝛤# + 𝛤% + ⋯ + 𝛤'
which states that the net circulation is equal to the sum of all smaller circulations. This can be applied further to three-
dimensional bodies with arbitrary surfaces by drawing a two-dimensional loop around the body perpendicular to the
direction of the flow and calculating the induced velocity at each edge that falls on the closed loop plane, which can
then be integrated for the net circulation
𝛤( = 𝑉*+,-./, 𝑑𝑙
2
3
[8].
At this point, should the body in question be of a two-dimensional shape, the Kutta-Joukowski theorem could be
applied to determine the lift per unit span. However, for a three-dimensional body, a multitude of 2D planes must be
generated around the body. Each plane exists as its own loop, which can be referred to as an Integrated Circulation
Loop (ICL). In this manner, the above steps can be evaluated around each loop to produce a result for the induced lift
and drag, which when summed together using the equations below, provide a solution for the body as a whole. [8]
𝐿*+,-./, = 𝜌6 𝑉6 𝛤7 𝑑𝑦
9
%
:
9
%
𝐷*+,-./, = 𝜌6 𝑤7 𝛤7 𝑑𝑦
9
%
:
9
%
𝑤7 =
1
4𝜋
1
𝑦 − 𝑦3
−𝑑𝛤7A
𝑑𝑦
𝑑𝑦3
9
%
:
9
%
3. Unstructured Surface Mesh Solver
In order to calculate the vorticity of a loop within an unstructured surface, the application of the method of
integrated circulation has to be molded to adapt to a triangular shape. This is easily accomplished by maintaining that
each edge of the facet, joined together by two vertices, is a finite line segment as shown in Figure 3 below, by which
the induced velocity along that line segment can be calculated. Once the induced velocity for each segment is
calculated, the total for the facet is simply the sum of all segments. Helmholtz’s laws, which state that the strength of
a vortex filament is constant along its length and that a vortex filament cannot end in a fluid, but must either extend
to the boundaries of the fluid or form a closed path [12], are met in this regard. [8]

5
Figure 3. An unstructured mesh surface facet showing the three line segments along with the vortex filament ring
directions. [8]
In order to calculate the induced velocity along the line segments that make up the facet, the Biot-Savart Law must
be implemented. The Biot-Savart Law, shown below, provides a means of calculating the induced velocity at some
point P with respect to a vortex filament, which in the case of the three-sided facet, is a single segment. [12]
𝑑𝑽 =
𝛤
4𝜋
𝒅𝒍 × 𝒓
𝒓H
Figure 4. The induced velocity at some point P located near a facet can be calculated based on the location of P with
respect to the individual vertices that make up the segment and the length of the segment itself. [8]
By knowing the geometry of the triangle, shown in Figure 4 above, and the location of point P with respect to the
two vertices that cap the end of the segment, the induced velocity generated by that single segment can be solved for
by integrating the Biot-Savart Law, as shown below.
𝑉* =
𝛤
4𝜋ℎ
sin 𝜃 𝑑𝜃
NO
NP
=
𝛤
4𝜋ℎ
cos 𝜃% − cos 𝜃#
To get the total velocity induced by an individual facet in the unstructured mesh, the induced velocities for each
segment that make up the facet need only be summed. [8]
The remaining question to be solved at this point is the unknown vorticity strength. Based on the above equation,
all components on the right-hand side, other than the vorticity strength, are geometrically dependent, meaning they
are known from the mesh generated for the body being analyzed and are a property of the facet in question. If the

6
induced velocity is defined as a controlled variable dictated by boundary conditions, then it to is known. From this, a
system of equations can be generated that allow for the vorticity strength to be solved for:
𝑉S = 𝛤S
1
4𝜋ℎ*
cos 𝜃%* − cos 𝜃#*
'TUVWUXYT
*Z#
= 𝛤S 𝐴,]
𝐴#,# ⋯ 𝐴#,'
⋮ ⋱ ⋮
𝐴',# ⋯ 𝐴','
𝛤#
⋮
𝛤'
=
𝐵#
⋮
𝐵'
𝛤#
⋮
𝛤'
=
𝐴#,# ⋯ 𝐴#,'
⋮ ⋱ ⋮
𝐴',# ⋯ 𝐴','
:#
𝐵#
⋮
𝐵'
where N is the total number of faces that make up the mesh, A is defined as the geometrical influence coefficient
matrix, and B is the velocity requirements defined by the simulation parameters. [8]
III. Test Models
At the beginning of this project, the major goal was to measure the effects of the wrap-around fin design using
available software (OpenVSP and Flightstream) in hopes of matching the same instabilities measured in past studies
of the issue. Once the data had been gathered, additional simulations were to be done with the same design but with
slots integrated into the fins, as originally proposed by Abate and Winchenbach (detailed in Aerodynamics of Missiles
with Slotted Fin Configurations). These results would be compared with Abate’s and Winchenbach’s in hopes that
the results correlated well and would provide further validation of Abate’s and Winchenbach’s findings, the
Flightstream software, and provide more confidence in the use of simple, fast-predictor flow-solvers as opposed to
exhausting large monetary sources through physical tests or consuming large amounts of computational energy and
time by using ultra-high fidelity CFD products.
The models used in this study included the following characteristics [13], as depicted in Figure 5:
• 10-caliber ogive-cylinder-tail configuration
• 2.5-caliber tangent ogive nose
• 7.5-caliber length, right circular cylinder
• WAFs in the open position
• Fin curvature and length designed to fold around the body of the missile
Figure 5. Geometric description of the wrap-around fin model. [13]

7
A. Straight-fin Model
OpenVSP was used to build a model that mimicked that described above as closely as
possible. The first step in the modeling process was to design a missile with the same body
shape and size, but without the wrap-around fins. Instead, the fins were modeled as straight, but
with the same span as the WAF design (which in this case, would be equal to one-quarter the
circumference of the missile body). The results for this model are shown below in Figure 6.
OpenVSP offers a feature called CFD Mesh, in which an unstructured mesh is built off the user’s parametric
model. In terms of this rocket, many challenges arose from using this method. In the first iteration of the straight-
winged missile, the fins did not connect through the body of the missile. Instead, they attached to the outer surface of
the body. When the CFD mesh was generated within OpenVSP, the fins were ignored and only the body of the missile
was recognized. To overcome this issue, the fins were instead modeled as wings that stretched through the body of
the missile. The other problem resulted from the mesher ignoring flat plate airfoils, again resulting in only the body
of the missile being meshed. Instead of modeling the fins as flat plates, as had been done in the physical models in
which this missile was designed with respect to, the fins were modeled as very thin airfoils (specifically, a NACA
0001, based on NACA naming conventions). In this way, the entire missile model was meshed and can be viewed in
Figure 7 below.
Figure 7. Meshed model of a straight-finned missile.
Figure 6. Three-dimensional model of a missile with normal fins, developed in OpenVSP.

8
B. Wrap-Around Fin Model
The wrap-around fin missile was modified from the existing straight-fin missile. The “wings” that made up the
four fins were each broken into three sections. The bottom section stayed orthogonal to the missile body while the
other two sections were angled at approximately thirty degrees towards the horizontal plane in which the wing was
normal to. This method of “bending” the fins was the only available method into implement a curve due to the limited
geometrical techniques allowed within OpenVSP. The final design is shown in Figure 8 below.
The WAF design suffered from issues with mesh generation as well. The mesh generator was unable to connect
all vertices, specifically around the joints of the fins and the body, causing the meshed model to not be “water tight”.
Ultimately, in order to have the model mesh properly, the fins were moved one centimeter towards the nose and were
extended outwards by one-quarter of a centimeter away from the body. The fully meshed, WAF implemented missile
is shown in Figure 9.
Figure 9. Meshed model of a wrap-around fin missile.
A comparison of the physical characteristics of the model from Abate and Winchenbach with that modeled in
OpenVSP is given below in Table 1.
Figure 8. A three-dimensional view of a missile with wrap-around fins, developed in OpenVSP.

9
Table 1. Physical Properties Comparison
A&W’s Model OpenVSP Model
Diameter, cm 1.91 1.91
Mass, gm 126.7 126.7
Ix, gm-cm2
66.7 55.7
Iy, gm-cm2
3922 3320
Length, cm 19.1 19.1
C.G., percent from nose 48.7 52.4
C. Slotted Wrap-Around Fin Model
Attempts were made to modify the WAF design to implement slots into the fins. Because OpenVSP does not
allow freeform or edge building, fins could not be designed with a gap located in the center. OpenVSP does, however,
have a “negative volume” function within its geometry options. Using the design elements for the slots prescribed in
Figure 5, a box was made and interlaced with each individual fin of the WAF model. The box created a void in the
center of each fin that was one-half the span and approximately one-third the chord (40% of the chord due to CFD
meshing failures when at one-third). This void was intended to remove approximately one-sixth of the surface area
of each fin. [13] The resulting CFD mesh is shown in Figure 10 below.
Figure 10. Mesh of wrap-around fin model with slots.
IV. Data Analysis
The analysis of the WAF design on a missile followed three primary paths. First, simulation parameters had to be
chosen. To acquire a thorough design analysis required runs at multiple Mach numbers and angles of attack. The run
parameters chosen consisted of Mach numbers 0.5, 1.0, 1.5, and 2.0. At each Mach number, the angle of attack would
be varied from zero degrees to twenty degrees in five degree increments. Standard atmospheric conditions were set
for the environmental characteristics.
Each missile design would be simulated within Flightstream at each of the conditions described above. The missile
simulation order would proceed in terms of complexity, with the straight fin being the first to undergo the analysis,
the WAF design being second, and the WAF with slots being third. The key data results that were gathered at the end
of each run include the lift coefficient, the induced drag coefficient, the skin friction drag coefficient, and the roll
moment coefficient.
A. Flightstream Results

10
Upon running the analysis, an error was encountered with the WAFs with slots. The Flightstream program returned
an error of insufficient physical memory on the machine being used. This was assumed to be attributed to the number
of vertices used to build the mesh. The mesh growth ratio had to be decreased in order to complete the build for the
slotted fins due to multiple program crashes by OpenVSP when attempting the mesh with the default growth ratio.
This decreased growth ratio tripled the number of vertices required to complete the mesh. Because of this, no data
was collected for the WAF design with slots.
Below, in Figure 11 through Figure 14, are the graphs making a comparison of the data gathered from Flightstream
for straight fins and wrap-around fins. These runs are comparing various coefficients at a zero angle of attack through
a Mach number regime of 0.5-2.0. The data is available in tabular format in Appendix A: Flightstream Data.
Figure 12. Induced drag coefficient for straight fins and WAFs over a range of Mach numbers.
0.001
0.01
0.1
1
0 0.5 1 1.5 2 2.5
CDi
Mach
Straight Fins WAFs
Figure 11. Lift coefficient for straight fins and WAFs over a range of Mach numbers.
0
0.05
0.1
0.15
0.2
0 0.5 1 1.5 2 2.5
CL
Mach
Straight Fins WAFs

11
Figure 13. Skin drag coefficient for straight fins and WAFs over a range of Mach numbers.
Figure 14. Roll moment coefficient for straight fins and WAFs over a range of Mach numbers.
The lift coefficient values in Error! Reference source not found. show little difference between the straight fin
and the WAF. The trends are relatively similar to other studies and are agree with intuition fairly well since at the
same angle of attack, the force generated by the fins should be minuscule. However, both do show great disparities
in terms of the values themselves when compared with past reports. [14] In fact, none of the above graphs agree with
past studies when comparing coefficient values, except for Figure 13, which shows many similarities in terms of
general skin friction trends (the degrading trend as Mach number increases), but values are still far too low.
The graphs below provide a comparison of the data trends of the straight fin and the WAF over various angles of
attack. The graphical lines depict the average values of the coefficients over a Mach number regime of 0.5-2.0. An
average was plotted due to the diminutive difference between the values at different Mach numbers. The values
themselves can be found in Appendix A: Flightstream Data.
0
0.05
0.1
0.15
0.2
0 0.5 1 1.5 2 2.5
CDo
Mach
Straight Fins WAFs
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0 0.5 1 1.5 2 2.5
CM (roll)
Mach
Straight Fins WAFs

12
Figure 15. Lift coefficient for straight fins and WAFs over a range of angle of attacks.
Figure 16. Induced drag coefficient for straight fins and WAFs over a range of angle of attacks.
Figure 17. Skin drag coefficient for straight fins and WAFs over a range of angle of attacks.
0
0.5
1
1.5
2
0 5 10 15 20 25
CL
AOA (deg)
Straight Fin Wrap-around Fins
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20 25
CDi
AOA (deg)
Straight Fins WAFs
0.146
0.148
0.15
0.152
0.154
0.156
0 5 10 15 20 25
CDo
AOA (deg)
Straight Fins WAFs

13
Figure 18. Roll moment coefficient for straight fins and WAFs over a range of angle of attacks.
The above graphs depicting characteristics of both the straight fins and the WAFs show relatively correct trends.
The lift coefficient is reasonable and validating of other studies. The lifting characteristics of the WAF are essentially
the same as the straight fin design, with a marginal difference in terms of total magnitude. [14] The induced drag
coefficient has a large drop at five degrees angle of attack for the WAFs only, whereas the straight fins stay relatively
constant and very low, almost near zero. This seems counterintuitive because the induced drag should gradually
decrease as the angle of attack increases and as the airflow moving over the body and fins begins to separate. The
skin friction gradually increases as expected, with the magnitude of the WAFs being greater than the straight fins due
to the WAFs having a greater angle of attack. The moment coefficient seems to speak to the fact that there is supposed
to be an increase in roll due to the fins being curved. The trends relatively match other studies with slight differences
in the values however.
B. AERODSN Results
An additional part of the project involved comparing the results gathered from the software tools used and
comparing them with data from a legacy, preliminary design tool and aerodynamic characteristic solver for missiles
known as AERODSN. AERODSN is best implemented with simple missile configurations, such as those with fins,
wings, or both, over a Mach range from zero to four. It assumes a cylindrical body shape, but can solve using two
different nose styles, cone and blunted ogive. [15]
Because AERODSN only accepts rudimentary missile designs, the wrap-around fin could not be simulated at
compared. However, a straight fin design as described previously was simulated and the results are provided below.
Additionally, because the nose can only be conical or a blunted ogive, the blunted ogive was chosen as it best
represented the overall ogive shape of the nose.
The first set of runs were based on the actual size and mass of the modeled missile above (see Table 1). However,
errors were produced by AERODSN that misconstrued data radically. In order to solve this problem, the missile was
increased in size three orders of magnitude. This does not affect the data in that AERODSN produces only
coefficients, which because the body shape and mass properties remained constant, does not change the overall
outcome.
The following plots were produced from the AERODSN data. Only the lift coefficients for a straight-fin and WAF
were able to be compared because AERODSN outputs total drag and moment coefficients, as opposed to skin and
induced drag or roll moment coefficients that were produced from Flightstream data. Plots are still given for the drag
and moment coefficient data from AERODSN to allow the reader to make a comparison of the overall data trends
produced in comparison with those from Flightstream.
-0.15
-0.1
-0.05
0
0.05
0 5 10 15 20 25
CM (roll)
AOA (deg)
Straight Fins WAFs

14
Figure 19. Lift coefficient comparison of Flightstream and AERODSN results.
As you can tell from Figure 19, the lift coefficient trends align reasonable well with the low Mach number results
from AERODSN (taking into account the lines for the straight and wrap-around fins are averages of the coefficient
over the total Mach number regime due to the minute difference in the values produced). The largest difference in the
trends are that the results from Flightstream seem linear in nature, opposed to AERODSN which maintains a nonlinear
trend set. This could be primarily due to the fact that AERODSN is purposed for supersonic Mach numbers, whereas
Flightstream is not.
Figure 20. Drag coefficients produced using AERODSN.
The drag coefficient results from AERODSN, shown in Figure 20, show reasonable results with a nonlinear
progression starting around 0.3 at low angles of attack to greater than 1 at higher angles of attack. This is due to the
increase in expose of surface area as the nose rises with respect to the direction of the freestream flow. At low angles
of attack, at simple estimation can be made that approximately the surface area of the base of the nose cone is exposed
to the freestream flow. As the angle of attack increases, the exposed area from the body of the missile begins to
increases, resulting in a larger drag coefficient.
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
0 5 10 15 20 25
CL
AOA (deg)
Straight Fin Wrap-around Fins AERODSN, M = 0.5
AERODSN, M = 1.0 AERODSN, M = 1.5 AERODSN, M = 2.0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 5 10 15 20 25
CD
AOA (deg)
M = 0.5 M = 1.0 M = 1.5 M = 2.0

15
Figure 21. Moment coefficients produced using AERODSN.
The moment coefficient shows somewhat similar trends to those produced by Flightstream; however, the values
vary dramatically. The negative progression of the moment coefficient is a result of the change in the center of
pressure as the nose becomes more inclined to the freestream flow. As the nose pitches up or down, the missile
experiences a moment in the opposite direction. This opposing moment attempts to bring the nose back down to a
static position. This can only occur, though, if the center of pressure is located aft of the center of gravity. Should the
center of pressure be located forward of the center of gravity, the missile will be unstable and will begin to tumble
with any disturbance.
V. Conclusion
The goal of this project was to use two available aerodynamic software packages, OpenVSP and Flightstream, and
attempt to analyze the effects of wrap-around fins versus normal fins (straight fins) in hopes of matching similarities
produced by past studies. Further, the fins would be given slots in order to see if the stability with regards to the
rolling moment was diminished in any manner, validating past studies using physical specimens could be mimicked.
By doing this, the two software packages could also be validated as useful in terms of aerodynamic testing with regards
to missiles with a wrap-around fin design, providing a secondary measure of testing validation, opposed to traditional
computational fluid dynamics or live fire testing.
Results for the project varied in terms of implementation and results produced. First, OpenVSP proved useful as
an intuitive model builder and mesh generator. Simple models of airplanes, missiles, etc., could easily be developed
by a novice user. However, the meshing generator proved cumbersome at times and partially reliable. Many
geometries would cause the entire program to crash. Producing a water tight mesh often consisted of simple trial and
error. Flightstream provided a number of aerodynamic properties with minimal setup and practice. Whereas most
flow-solvers are extremely complex, outdated, and require weeks of setup, Flightstream was simple and relatively
easy to learn. Its results seem to be unaffected by various changes in flow parameters, specifically with Mach numbers.
An increase in Mach number would often give no change to various coefficients. This could be attributed to
Flightstream being primarily designed for subsonic flows; however, for many tests with varied Mach numbers, there
were still little change within the subsonic region.
The results provided by both programs were varied and dissimilar from other studies using similar setups. Though
this could be attributed to differences in model design and experimental setups, trends should stay relatively consistent.
This was not the case for some of the tests, where trends differed greatly from what would be expected.
-35
-30
-25
-20
-15
-10
-5
0
0 5 10 15 20 25
CM
AOA (deg)
M = 0.5 M = 1.0 M = 1.5 M = 2.0

16
Appendix A: Flightstream Data
STRAIGHT FINS WRAP-AROUND FINS
CL CL
Mach 0 5 10 15 20 Mach 0 5 10 15 20
0.5 0.0773 0.3842 0.7036 1.0241 1.3417 0.5 0.1473 0.5422 0.9352 1.3229 1.7007
1 0.071 0.3737 0.6952 1.0171 1.3358 1 0.1473 0.5422 0.9352 1.3229 1.7007
1.5 0.0655 0.3671 0.6895 1.0123 1.3314 1.5 0.1473 0.5422 0.9352 1.3229 1.7007
2 0.0611 0.3624 0.6768 1.0056 1.326 2 0.1473 0.5422 0.9352 1.3229 1.7007
average 0.068725 0.37185 0.691275 1.014775 1.333725 average 0.1473 0.5422 0.9352 1.3229 1.7007

Cdi Cdi
Mach 0 5 10 15 20 Mach 0 5 10 15 20
0.5 0.0083 0.0141 0.0249 0.0344 0.0431 0.5 0.2323 0.008 0.2194 0.481 0.7876
1 0.0082 0.0137 0.0242 0.0336 0.0423 1 0.2323 0.008 0.2194 0.481 0.7876
1.5 0.008 0.0135 0.0238 0.0331 0.0417 1.5 0.2323 0.008 0.2194 0.481 0.7876
2 0.0079 0.0133 0.0233 0.0325 0.041 2 0.2323 0.008 0.2194 0.481 0.7876
average 0.0081 0.01365 0.02405 0.0334 0.042025 average 0.2323 0.008 0.2194 0.481 0.7876

Cdo Cdo
Mach 0 5 10 15 20 Mach 0 5 10 15 20
0.5 0.1686 0.1688 0.1691 0.1693 0.1695 0.5 0.1754 0.1759 0.1763 0.1766 0.177
1 0.15 0.1502 0.1504 0.1506 0.1508 1 0.1556 0.156 0.1564 0.1567 0.157
1.5 0.1406 0.1408 0.141 0.1412 0.1413 1.5 0.1455 0.1458 0.1461 0.1464 0.1467
2 0.1337 0.1337 0.1339 0.1339 0.134 2 0.1388 0.1391 0.1394 0.1395 0.1396
average 0.148225 0.148375 0.1486 0.14875 0.1489 average 0.153825 0.1542 0.15455 0.1548 0.155075

CM roll CM roll
Mach 0 5 10 15 20 Mach 0 5 10 15 20
0.5 0.0194 0.0199 0.0197 0.0193 0.0187 0.5 -0.0617 0.0852 -0.0984 0.1068 -0.1126
1 0.0197 0.0201 0.0199 0.0196 0.019 1 -0.0617 0.0852 -0.0984 0.1068 -0.1126
1.5 0.0198 0.0201 0.0201 0.0197 0.0192 1.5 -0.0617 0.0852 -0.0984 0.1068 -0.1126
2 0.0198 0.0202 0.0207 0.02 0.0194 2 -0.0617 0.0852 -0.0984 0.1068 -0.1126
average 0.019675 0.020075 0.0201 0.01965 0.019075 average -0.0617 0.0852 -0.0984 0.1068 -0.1126

17
Appendix B: AERODSN Data
Mach AoA CN CA CM XCP CL CD
0.5 0 0 0.35786 0 0 0 0.35786
0.5 5 0.41072 0.35514 -2.16151 11.00677 0.378204596 0.389585192
0.5 10 0.90462 0.34707 -4.8247 11.14178 0.830608717 0.498882841
0.5 15 1.57433 0.33389 -8.54827 11.32589 1.434268915 0.729979561
0.5 20 2.51463 0.316 -13.90388 11.51578 2.25490089 1.156996981
1 0 0 0.39798 0 0 0 0.39798
1 5 0.59472 0.39047 -3.82191 13.22936 0.558425208 0.440817407
1 10 1.31977 0.36828 -8.46335 13.20332 1.235768577 0.591860655
1 15 2.32958 0.33242 -14.89541 13.16762 2.164164859 0.924032734
1 20 3.80187 0.28449 -24.23573 13.13066 3.475287874 1.567649276
1.5 0 0 0.37088 0 0 0 0.37088
1.5 5 0.66372 0.36742 -4.35419 13.48513 0.629171582 0.423868866
1.5 10 1.5147 0.35716 -9.95028 13.50204 1.42966812 0.614758832
1.5 15 2.77537 0.34049 -18.26339 13.52378 2.592676264 1.047206698
1.5 20 4.45905 0.31802 -29.38184 13.54047 4.081367135 1.823925967
2 0 0 0.33296 0 0 0 0.33296
2 5 0.59304 0.33165 -3.64163 12.68364 0.561878102 0.382074813
2 10 1.43509 0.32774 -9.03943 12.98583 1.356376305 0.571961656
2 15 2.6945 0.32128 -17.38061 13.27528 2.519533756 1.007720566
2 20 4.39503 0.31239 -28.83282 13.48521 4.023133587 1.796739368
The following equations were used to calculate CL and CD:
𝐶2 = 𝐶' cos 𝛼 − 𝐶c sin 𝛼
𝐶d = 𝐶' sin 𝛼 + 𝐶c cos 𝛼

18
Appendix C: Graphical Output
Figure 22. Pressure coefficient distribution at Mach 0.5, AoA 0.
Figure 23. Vorticity magnitude distribution at Mach 0.5, AoA 0.

19
Figure 25. Vorticity distribution at Mach 2.0, AoA 0.

20

21

22
References
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Moffett Field, 1983.
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[10] A. Chatterjee, "Novel Multi-block Strategy for CAD Tools for Microfluics Type Applications," Advances in
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[11] D. J. Mavriplis, "Mesh Generation and Adaptivity for Complex Geometries and Flows," in Handbook of
Computational Fluid Mechanics, Academic Press, 1996.
[12] J. D. Anderson, Fundamentals of Aerodynamics, 5th Edition ed., McGraw-Hill, 2011.
[13] G. L. Abate and G. L. Winchenbach, "Aerodynamics of missiles with Slotted Fin Configurations," 1991.
[14] C. W. Dahlke, "The Aerodynamic Characteristics of Wrap-Around Fins at Mach Numbers of 0.3 to 3.0," 1977.
[15] G. A. Sanders and W. D. Washington, "Computer Program for Estimating Stability Derivatives of Missile
Configurations - Users Manual," 1982.

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