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.