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Transcript of Computational Analysis
Evan Kontras, Kyle Gould, Davide Maffeo
MAE 4440/7440 Aerodynamics
University of Missouri
Department of Mechanical and Aerospace Engineering
NACA Airfoil Evaluation
Abstract
Aerodynamic analysis has been conducted for the wingtip airfoil of the A10 Thunderbolt. The
NACA 4212 airfoil was analyzed using three separate methods. Results from each method are
compared to gain an understanding of the capabilities and limitations of each method. To begin,
analytic tools and hand calculations were used to obtain a simple solution for the lift coefficient
over a range of different angles of attack. Next, a 2D panel method was implemented using
Matlab to compute the lift and drag coefficients over a range of angles of attack. Finally, the
Ansys Workbench commercial computational fluid dynamics (CFD) program, Fluent, was used
to similarly obtain and plot lift and drag coefficients. Both analytic tools and the panel method
assume inviscid flow and therefore detailed drag computations, which are largely based on
viscous effects are not accounted for. Using Fluent, three separate cases of fluid flow were
analyzed. Inviscid flow, laminar flow, and turbulent flow were all simulated and the results
compared. The general procedure for each of the three methods used is presented, and the results
compared and discussed.
INTRODUCTION
Developed in the early 1970’s by Fairchild Republic, the A10 Thunderbolt is a strait
wing close air support aircraft used by the United States Air Force to combat ground vehicles
such as tanks and armored vehicles. Designed around a 30 mm automatic cannon (GAU8
Avenger), the A10 has an unrivaled capability to quickly and effectively destroy armored targets
on the ground. The A10 ‘s strait wing design limits its aerial characteristics, and as such it is not
known as a fighter plane. The wingtip cross section is designated by the NACA 4212 airfoil, as
shown below in Fig. 1.
Figure 1. Profile of the NACA 4212 Airfoil.
-0.05
3E-16
0.05
0.1
0.15
0.2
0.25
0.3
-0.2 0 0.2 0.4 0.6 0.8 1 1.2
Dis
tan
ce (m
)
Distance (m)
NACA 4212 Airfoil Profile
Y Upper
Y Lower
Camber
Using three separate methods, the lift and drag coefficients of the NACA 4212 airfoil are
computed over a range of angles of attack. Though the A10 is a well proven aircraft and its flight
capabilities are not in question, the main goal of this analysis is to compare each method and
determine how the aerodynamic properties differ between them. Using analytic tools and the
assumption that the airflow over the airfoil is inviscid, hand calculations were first done to obtain
a general knowledge and an expectation for the aerodynamic properties using the other methods.
A 2D panel method was then used, also assuming inviscid flow. Central to the panel method is
the segmenting of the airfoil into separate pieces connected by strait lines. The coefficient of
pressure is calculated for each segment, and all segments then summed to obtain the overall
properties of the airfoil. In the hopes of obtaining an even more accurate description of the
aerodynamic properties, the Ansys Workbench CFD program, Fluent, was used as well. As both
the panel method and analytic tools do not account for viscous effects, inviscid flow was first
used in the Fluent simulation for comparison. To obtain a more complete description of the
aerodynamic properties of the NACA 4212, both laminar and turbulent flows were then
simulated.
PROCEDURE
ANALYTIC TOOLS
Central to the analytical approach to determining the aerodynamic properties of an airfoil
is the assumption that the fluid flow is inviscid. The process used is a generalization of the
method used for analyzing a symmetric airfoil. Unlike a symmetric airfoil, the NACA 4212 is a
cambered airfoil. When the slope of the camber line for this airfoil is considered, the term
(1)
becomes non-zero. After obtaining the geometric information for the NACA 4212, x and y
coordinates were plotted along with the camber line in Excel. A polynomial trend line was then
fit to the camber line coordinates, to obtain the camber line as a function of the x coordinate
alone, as shown below.
x 12.3594x5 36.4845x4 41.052x3 21.8421x2 5.58x 0.5357
(2)
For our purpose, we needed an expression for the vortex strength in the following integral.
1
2
x
0
d dx
dz
(3)
where is the angle of attack, and is the distance from the leading edge of the airfoil, given
below. Though this integral is difficult to solve, the assumptions corresponding to the following
equations allow for simplification, using the substitution variable .
c
2 1 cos
(4)
x c
2 1 cos
0
(5)
d c
2
(6)
2 A0
1 cos
sin Ansin n
i n
(7)
For this study, coefficients from zero to two were needed. The first three Fourier’s coefficients
are then given by
(8)
A1
2
dx
dy
0
0 cos
0 d
0 0.3439909
(9)
A2
2
dx
dy
0
0 cos 2
0 d
0 0.184326
(10)
Once the first three Fourier’s coefficients were found, the lift coefficient can be calculated using
the following.
2c A0
2A1
(11)
cl 2 A0
A1
2
(12)
Results were then computed over a range of angles of attack using Excel. Hand calculations can
also be found in the appendix.
2D PANEL METHOD
Panel methods are a technique to solve incompressible potential flow over thick 2D and
3D geometries. For a 2D analysis as was done for this report, the geometry of the body being
analyzed is segmented into piecewise strait line segments. Each line represents a boundary
element, and vortex sheets are placed along the segment to act as the boundary around the airfoil,
giving rise to circulation, and hence lift. For an airfoil generating lift, in general the upper
surface is characterized by clockwise rotating vortices while the lower surface is characterized
by counter-clockwise rotating vortices. If there are more clockwise rotating vortices than
counter-clockwise rotating vortices, there is a net clockwise circulation around the airfoil,
creating lift. For each line segment along the airfoil, there is a vortex sheet of strength
0ds0
(13)
where is the length of the line segment. Each line is defined by its end points, and by a
control point located at the segments midpoint as shown in Fig. 2 below. At this control point,
the boundary condition
constant
(14)
is applied.
Figure 2. Schematic of Panel Approximation.
To ensure that the flow velocity is tangential to the airfoil surface, it is treated as a streamline
and assumed that no flow occurs through the surface. The stream function is an addition of the
effects due the uniform free stream flow velocity and the effects due to the vortices on each
panel. Using the definition of the velocity components based on the stream function,
y u
x v
(15)
the free stream function is given by the following.
u y v x
(16)
The stream function of a counter-clockwise vortex of radius r and strength is given by
2 ln r
(17)
where the radial and tangential components of velocity are shown below, respectively.
vr
1
r
0 v
r
2 r
(18)
For each individual line segment, the stream function can be written in terms of the differential
line length and strength as
0ds0
2 ln r r0
(19)
where is simply calculated using the following.
(20)
By integrating over the entire airfoil surface, the stream function for all infinitesimal vortices at
each control point can be obtained.
0ln r r0 ds0
2
(21)
Adding the free stream and vortex effects, the equation used to obtain the circulation for each
line segment is given by,
u y v x
0ln r r0 ds0
2
(22)
where C is a constant. This integral equation is subject to the constraint that the vortex strength
on the upper and lower surface must be the same at the trailing edge, commonly called the Kutta
Condition. The unknowns are the vortex strength on each panel and the value of the stream
function, C. Once the vortex strength is obtained, the coefficient of pressure is calculated using
cp 1
02
2
(23)
where is the free stream velocity.
To implement these equations in Matlab to obtain a solution for the aerodynamic properties of
the airfoil, Eq. (22) is written in terms of two indices. The airfoil is divided into N panels, of
which each is numbered j, where j 1,2,…N . On each panel it is assumed that is constant,
therefore the vortex strength is indexed also, The control points for each line segment are
also denoted by an index i, where i 1,2,…N . The integral Eq. (22), is then written as follows.
u yi v xi 0,j
2 ln r i r 0 j
N
j 1
ds0 0
(24)
The index i refers to the control point at which the equation is applied, the index j refers to the
line segment over which the line integral is evaluated. After obtaining the coefficient of pressure
the coefficient of normal force is computed by integrating the difference between the upper and
lower surface pressure coefficients as,
cn
1
cp,l cp,u
T
ds
(25)
from which the coefficients of lift and drag are resolved using
cl cn,y cos cn,x sin
(26)
cd cn,y sin cn,x cos
(27)
where is the angle of attack of the airfoil. The geometry of the airfoil being analyzed is opened
from a text file, containing x and y coordinates of each end point for all line segments. This
geometry information was obtained for the NACA 4212 from the University of Illinois online
database.
CFD ANALYSIS USING FLUENT
The Ansys Workbench is a powerful engineering simulation software suite. The
computational fluid dynamics module named Fluent, utilizes a finite element method to solve the
fundamental equations governing fluid flows. The first step in any Fluent analysis is to create the
fluid domain, and the geometry of the object of study. For this report, the geometry of the NACA
4212 airfoil was imported from an online data base as a simple text file. The x and y coordinates
were placed and merged, creating the 2D profile of the airfoil. A box was then created around the
airfoil, to be the fluid volume, with a circular shape on the side corresponding to the airfoil’s
leading edge. A Boolean subtraction was performed to fully designate the airfoil as a solid body,
and the outer geometry as the fluid domain. After appropriately naming the edges of the
geometry, edge sizing control was used to create a C shaped mesh around the airfoil, with a
rectangular mesh from the trailing edge and behind, as shown in Fig. 3 and 4.
Figure 3. Fluent Airfoil Mesh.
Figure 4. NACA 4212 Meshed in Fluent.
With the mesh generated, the computation/solution module of Fluent was launched. Velocity,
pressure, and wall boundary conditions were imposed. Critical to this analysis was to observe
three different cases for the fluid simulation. To begin, inviscid flow was selected. The
simulation was run with a free stream velocity of 10
over a range of different angles of attack.
The angle of attack was controlled by modifying the x and y components of the free stream
velocity appropriately. It was expected that the inviscid flow results be similar to those found
using both analytic tools and the 2D panel method, as these were also only applied for inviscid
flow. To obtain results that account for viscous effects, a laminar flow condition was then
selected, and the simulation recalculated. Accounting for viscous effects, it was expected that
stall would be observed beyond some angle of attack unlike the case for inviscid flow. However,
simulating laminar flow did not account for turbulence phenomenon, therefore a third and final
flow condition was selected. A k-epsilon k_ε turbulent flow condition was specified, which
assumes an isotropy of turbulence where by the normal stresses are equal. Upon defining the
appropriate characteristic length, density, and flow velocity, Fluent’s built in force calculations
were selected to calculate the lift and drag coefficients for all three flow conditions, over a range
of angles of attack. Having the lift coefficients output to the display window, the values were
recorded and plotted with the corresponding angles of attack using Excel.
RESULTS
ANALYTIC
Using the equations outlined in the analytical procedure section along with Excel, a plot
of the coefficient of lift over 13 separate angles of attack was created, as shown below in Fig. 3.
Figure 3. Coefficient of Lift vs. Angle of Attack Using Analytical Method.
y = 0.1097x + 0.8124
0
1
2
3
4
5
0 5 10 15 20 25 30 35
Co
eff
icie
nt
of
Lift
, Cl
Angle of Attack (Deg.)
Analytical Method: Cl vs Angle of Attack
2D PANEL METHOD
A total of nine separate text files were created containing the airfoil’s x and y
coordinates, as well as specifying the angle of attack to be used in the calculation. An example
text file can be found in the appendix. The Matlab m-file Panel_Method.m was run using 9
separate files, corresponding to the 9 angles of attack that were analyzed. The coefficients of lift
and drag, which were output by the m-file were recorded and plotted with the corresponding
angles of attack using Excel. Plots of coefficient of lift and drag, as well as the lift/drag ratio are
shown below as functions of the angle of attack.
Figure 4. Coefficient of Lift vs. Angle of Attack for NACA 4212 Using Panel Method.
Figure 5. Coefficient of Drag vs. Angle of Attack for NACA 4212 Using Panel Method.
y = -4E-05x3 + 0.0022x2 + 0.1166x - 1.0915
-2
-1
0
1
2
3
4
5
6
0 10 20 30 40 50 60 70
Co
effi
cien
t o
f lif
t, C
l
Angle of Attack (Deg)
Panel Method: Cl vs. Angle of Attack
y = 0.0016x2 + 0.0144x - 0.1055
-2
0
2
4
6
8
0 10 20 30 40 50 60 70
Co
effi
cie
nt
of
dra
g, C
d
Angle of Attack (Deg)
Panel Method: Cd vs. Angle of Attack
Figure 6. Lift to Drag Ratio (Cl/Cd) for NACA 4212 Using Panel Method.
FLUENT
Printing the coefficients of lift and drag directly from Fluent made plotting these values
vs. angle of attack strait forward. The capabilities of Fluent were also used to create contour plots
of static pressure. To better display the characteristics of the fluid flow, a plot of velocity vectors
was also created. For simplicity and conciseness, only one angle of attack, 15 degrees, was
selected to create all the plots for display as follows. Coefficients of lift and drag for each type of
flow are shown as well.
y = 5E-07x5 - 1E-04x4 + 0.0082x3 - 0.3124x2 + 5.4475x - 31.202
-35
-30
-25
-20
-15
-10
-5
0
5
10
0 10 20 30 40 50 60 70Li
ft t
o D
rag
Rat
io, C
l/C
d
Angle of Attack (Deg.)
Lift to Drag Ratio
INVISCID FLOW
Figure 7. Pressure Contours at 15 Degree Angle of Attack, Inviscid Flow.
Figure 8. Velocity Vectors at 15 Degree Angle of Attack, Inviscid Flow.
Figure 9. Coefficient of Lift vs. Angle of Attack, Inviscid Flow.
Figure 10. Coefficient of Drag vs. Angle of Attack, Inviscid Flow.
y = -0.0004x2 + 0.0438x + 0.4854
00.20.40.60.8
11.21.41.61.8
2
0 10 20 30 40 50 60
Co
eff
icie
nt
of
Lift
, Cl
Angle of Attack (Deg.)
Coefficient of Lift, Inviscid Flow
y = -5E-06x2 + 0.0025x + 0.0276
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0.16
0 10 20 30 40 50 60Co
effi
cien
t o
f D
rag,
Cd
Angle of Attack (Deg.)
Coefficient of Drag, Inviscid Flow
LAMINAR FLOW
Figure 11. Pressure Contours at 15 Degree Angle of Attack, Laminar Flow.
Figure 12. Velocity Vectors at 15 Degree Angle of Attack, Laminar Flow.
Figure 13. Coefficient of Lift vs. Angle of Attack, Laminar Flow.
Figure 14. Coefficient of Drag vs. Angle of Attack, Laminar Flow.
y = -0.0005x2 + 0.0535x + 0.2306
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 10 20 30 40 50 60
Ce
ffic
ien
t o
f Li
ft, C
l
Angle of Attack (Deg.)
Coefficient of Lift, Laminar Flow
y = 1E-05x2 + 0.001x + 0.032
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 10 20 30 40 50 60
Co
effi
cien
t o
f D
rag,
Cd
Angle of Attack (Deg.)
Coefficient of Drag, Laminar Flow
TURBULENT FLOW (K-EPSILON)
Figure 15. Pressure Contours at 15 Degree Angle of Attack, Turbulent Flow.
Figure 16. Velocity Vectors at 15 Degree Angle of Attack, Turbulent Flow.
Figure 17. Coefficient of Lift vs. Angle of Attack, Turbulent Flow.
Figure 18. Coefficient of Drag vs. Angle of Attack, Turbulent Flow.
y = -2E-06x4 + 0.0002x3 - 0.0053x2 + 0.1081x + 0.3418
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
0 10 20 30 40 50
AxC
oe
ffic
ien
t o
f Li
ft, C
l
Angle of Attack (Deg.)
Coefficient of Lift, Turbulent Flow
y = -8E-06x3 + 0.0005x2 - 0.007x + 0.0684
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
0 10 20 30 40 50
Co
effi
cien
t o
f D
rag,
Cd
Angle of Attack (Deg.)
Coefficient of Drag, Turbulent Flow
CONCLUSIONS
Comparing the three methods used for aerodynamic analysis, some advantages and
disadvantages have been determined. Although the analytic method took the least amount of
time, and it did account for the airfoil shape based on the camber line coordinates, it proved to be
a more general solution when compared to the other two methods. As it does not account for
viscous effects, it is not applicable for any situation where the stall point of the airfoil is of
concern. The linear trend line for the coefficient of lift vs. angle of attack is likely not accurate
much beyond 15-20 degrees. The panel method was determined to be more accurate, but seemed
limited in its application to more complex airfoils. Two other airfoil geometries (NACA 6716
and 6724) were analyzed using the panel method m-file and the results deemed inaccurate and
nonsensical. However, for less radical airfoils, the panel method does a good job of accounting
for more complex phenomenon by allowing the airfoil to be segmented into many small pieces
for analysis. Finally, the expectation of Fluent to be the most accurate and easiest to implement
for complex airfoils was shown not to be true. Although Fluent can easily account for the viscous
effects of laminar and turbulent flow, much like the panel method it proved difficult to use with
more complicated airfoil shapes. The computational power of the program was realized, but it
was this complexity that made Fluent hard for the user to accurately obtain aerodynamic
analysis. The most time was spent using Fluent, but after getting more familiar with the program
satisfactory results for all three flow conditions were obtained.