2

TERMS AND DEFINITIONS ................................................................................................................................... 7

1 - INTRODUCTION................................................................................................................................................... 9

2 - AIRFOIL CREATION USING INVERSE MODELING...........................................................................12

2.1 INTRODUCTION....................................................................................................................................................12

2.2 CONFORMAL TRANSFORMATION......................................................................................................................12

2.3 VELOCITY PROFILE.............................................................................................................................................13

2.4 DESIGN COEFFICIENT OF LIFT ON AN AIRFOIL SEGMENT ...........................................................................16

3 -AEROFOIL USER’S INSTRUCTIONS ..........................................................................................................20

3.1 INTRODUCTION....................................................................................................................................................20

3.2 AIRFOIL DATA.....................................................................................................................................................21

3.2.1 Airfoil Data Files......................................................................................................................................22

3.2.2 Drawing Interchange Format.................................................................................................................23

3.2.3 Raw x, y, and z Data.................................................................................................................................24

3.2.4 Airfoil Data File Creation.......................................................................................................................25

3.3 AIRFOIL CREATION USING INVERSE MODELING............................................................................................34

3.3.1 Introduction...............................................................................................................................................34

3.3.2 Inverse Modeling......................................................................................................................................34

3.3.3 Example Inverse Modeling .....................................................................................................................46

3.4 CONTROL SURFACES...........................................................................................................................................51

3.5 MEAN LINE MODIFICATION ..............................................................................................................................54

3.6 COORDINATE SPACING.......................................................................................................................................57

3.7 SURFACE SMOOTHING........................................................................................................................................59

3.8 SAVING DATA......................................................................................................................................................61

3.9 VIEWING OPTIONS...............................................................................................................................................63

3.9.1 Velocity Display........................................................................................................................................65

3

3.9.2 Pressure Coefficient .................................................................................................................................66

3.9.3 Boundary Layer ........................................................................................................................................68

3.9.4 Vortex Strength .........................................................................................................................................69

3.9.5 Graphical Display....................................................................................................................................70

3.9.6 Drag Polar.................................................................................................................................................72

3.9.7 Velocity vs Surface Location ..................................................................................................................73

3.9.8 Tabulated Results .....................................................................................................................................74

3.10 OPTIONS .............................................................................................................................................................76

3.10.1 Multiple Airfoils......................................................................................................................................77

3.10.2 Reynolds Number....................................................................................................................................80

3.10.3 Constant Lift ............................................................................................................................................82

3.10.4 Wing Simulation .....................................................................................................................................86

3.10.5 Compressibility.......................................................................................................................................90

3.10.6 Surface Conditions.................................................................................................................................91

3.10.7 Angle Of Attack.......................................................................................................................................93

3.10.8 Coordinate Locations............................................................................................................................95

3.10.9 Renaming The Airfoil.............................................................................................................................96

3.910.10 Display Options..................................................................................................................................97

4 - RESULTS ................................................................................................................................................................98

APPENDIX 1 – REPRESENTATIVE RESULTS..............................................................................................99

REFERENCES:.........................................................................................................................................................107

4

Figure 1.1 – Airfoil Polygon..............................................................................................10

Figure 2.1a – Typical Velocity Profile, 0º Angle of Attack ..............................................13

Figure 2.1b – Typical Velocity Profile, 6º Angle of Attack..............................................14

Figure 2.2 – Arc Segments On The Transformation Circle ...............................................15

Figure 2.3 – Transformed Airfoil.......................................................................................15

Figure 2.4 – Inverse Model Velocity Profile .....................................................................17

Figure 2.5 – Inverse Model Airfoil ....................................................................................18

Figure 2.6 – Potential Flow Velocity Profile .....................................................................18

Table 2.1 – Design Velocity vs Potential Flow Solution...................................................19

Figure 3.1 – Airfoil Data....................................................................................................21

Figure 3.2 – NACA 4 and 5 Digit Airfoils ........................................................................26

Figure 3.3 – NACA 6 and 7 Series Airfoils.......................................................................29

Figure 3.4 – NACA 747A315............................................................................................31

Figure 3.5 – Oshkosh Airfoil Transformation...................................................................32

Figure 3.6 – Inverse Modeling...........................................................................................35

Figure 3.7 – Inverse Modeling, Front Segment .................................................................36

Figure 3.8 – Front Segment Parameters.............................................................................37

Figure 3.9 – Switching.......................................................................................................39

Figure 3.10 – Inverse Modeling, Pressure Recovery Region............................................40

Figure 3.11 – Pressure Recovery Parameters ....................................................................41

Figure 3.12 – Switching.....................................................................................................42

Figure 3.13 – Options ........................................................................................................43

Figure3.14 – File Options ..................................................................................................45

5

Figure 3.15 – Example Inverse Model...............................................................................46

Figure 3.16 – Pseudo NFL-0414F Lift vs Drag.................................................................48

Figure 3.17 – Pseudo NFL-0414F Lift, Drag, and Pitch vs Angle of Attack ....................49

Table 3.1 – Inverse Model Example ..................................................................................50

Figure 3.18 – Control Surfaces ..........................................................................................51

Figure 3.19 – Simple Hinges .............................................................................................52

Figure 3.20 – Frise Hinges.................................................................................................53

Figure 3.21 – Flaps ............................................................................................................53

Figure 3.22 – Mean Line Modification..............................................................................54

Figure 3.23 – Mean Line Modification..............................................................................56

Figure 3.24 – Coordinate Spacing......................................................................................57

Figure 3.25 – Surface Smoothing ......................................................................................59

Figure 3.26 – Saving Data .................................................................................................61

Figure 3.27 – Viewing Options..........................................................................................63

Figure 3.28 – Velocity Display..........................................................................................65

Figure 3.29a – Pressure Coefficient Display.....................................................................66

Figure 3.29b – Pressure Coefficient Display.....................................................................67

Figure 3.30 – Boundary Layer Display..............................................................................68

Figure 3.31 – Vortex Strength Display..............................................................................69

Figure 3.32 – Graphical Display: Lift, Drag, and Pitch vs Angle of Attack .....................70

Figure 3.33 – Drag Polar Display: Lift vs Drag.................................................................72

Figure 3.34 – Velocity vs Surface Location......................................................................73

Figure 3.35 – Tabulated Results ........................................................................................74

6

Figure 3.36 – Options ........................................................................................................76

Figure 3.37 – Multiple Airfoils..........................................................................................77

Figure 3.38 – Multiple Airfoil Selection............................................................................77

Figure 3.39 – Comparison Of Three Airfoils: Lift, Drag, and Pitch vs Angle of Attack ..78

Figure 3.40 – Comparison Of Three Airfoils: Lift vs Drag...............................................79

Figure 3.41 – Reynolds Number Calculator ......................................................................80

Figure 3.42 – Constant Lift................................................................................................82

Figure 3.43 – A Direct Comparison of Two Airfoils Using The Constant Lift Option,

Minimum Lift Coefficient = 0.15 At Reynolds Number = 8,000,000 Lift, Drag, and

Pitch vs Angle of Attack ............................................................................................84

Figure 3.44 – A Direct Comparison of Two Airfoils Using The Constant Lift Option,

Minimum Lift Coefficient = 0.15 At Reynolds Number = 8,000,000 Lift vs Drag..85

Figure 3.45 – Wing Simulation..........................................................................................86

Figure 3.46 – Wing Simulation Results: Lift, Drag, and Pitch vs Angle of Attack ..........89

Figure 3.47 – Compressibility............................................................................................90

Figure 3.48 – Surface Conditions ......................................................................................91

Figure 3.49 – Angle of Attack ...........................................................................................93

Figure 3.50 – Coordinate Locations...................................................................................95

Figure 3.51 – Renaming The Airfoil..................................................................................96

Figure 3.52 – Display Options ...........................................................................................97

7

Terms And Definitions

AerodynamicCenter

A point, near the quarter-chord location, about which the pitchmoment coefficient is constant.

Angle of Attack The angle of an airfoil, measured from the chord line, to the airflow.

Angle of Zero Lift The angle of attack, measured from the airfoil’s chord line, atwhich there is no lift.

Boundary Layer Air flowing along an airfoil does not have a uniform velocity. Themolecules of air that are in direct contact with the airfoil surfaceare not moving. At some small distance from the airfoil surface,the local air velocity is uniform. The thickness of the disturbedairflow is known as the boundary layer.

There are three basic types of boundary layers; laminar, turbulent,and separated. Laminar boundary layers are thin and create thelowest drag. At some point, a laminar boundary layer will becometurbulent. The thickness and drag increases more rapidly.Separation occurs when the boundary layer detaches from theairfoil surface and eddies are created. Boundary layer thicknessand drag increase very rapidly.

Boundary LayerTransition

The point at which the boundary layer is said to transition fromlaminar to turbulent or separated and from turbulent to separated.In AeroFoil, the transition points are given in dimensionless unitsof length, x / chord. A value of 0.0 represents the leading edge.1.0 represents the trailing edge.

Camber A line mid-way between the upper and lower surfaces of an airfoil.

Chord The distance from the leading edge to the trailing edge of theairfoil. The chord line is a straight line between the leading andtrailing edge and is used as a standard reference in all airfoils.

Drag Coefficient A measure of an airfoil’s resistance to air flow. Total wing drag =dynamic pressure x Wing Area x Drag Coefficient

Dynamic Pressure The amount of energy in the air flow. Dynamic pressure = ½ x airdensity x velocity2

Free StreamVelocity

The velocity of air flow, undisturbed by the airfoil.

8

Ideal Flow Also known as potential flow. The airflow that would exist if airhad no viscosity.

Inviscid Having no viscosity. An ideal fluid.

Lift Coefficient A measure of the amount of lifting force that an airfoil cangenerate. Total wing lift = dynamic pressure x Wing area x LiftCoefficient

Mean Line Also known as the camber line. The mid-point of an airfoilthickness profile.

Pitch Coefficient A measure of the twisting force that an airfoil will generate inflight. A negative value implies that the leading edge of the airfoilwill be twisting downward. Unless otherwise specified, the pitchcoefficient is referenced to a point located on the airfoil chord, 25percent back from the leading edge. This point is known as thequarter-chord.

Total Wing Pitch = dynamic pressure x Wing area x Chord lengthx Pitch Coefficient

Potential Flow Also known as ideal flow. The airflow that would exist if air hadno viscosity.

Reynolds Number A dimensionless number that is the ratio of a fluid’s inertial forcesto its viscous forces. Re = chord length x velocity / kinematicviscosity

Stagnation Point A point at the leading edge of an airfoil where the airflow divides.Any molecule of air that is above the stagnation point will passover the top surface and any that is below the stagnation point willpass under the bottom surface.

Stall The point of maximum coefficient of lift. The lift coefficient willincrease as the angle of attack is increased up to its maximumvalue. After the stall, the coefficient of lift will decrease withincreasing angle of attack.

Thickness Profile The distance between the upper and lower surfaces of the airfoil.The thickness profile is independent of the chamber line.Combining a thickness profile with a chamber line is one methodof deriving an airfoil.

Viscosity The measure of a fluid’s resistance to shear. Oil has a higherviscosity than water, which has a much higher viscosity than air.

9

1 - Introduction

In the early days of aviation, airfoils and wings were designed by eye; the builder would

model his wing after anything that could fly. This lead to some interesting designs that

were more like bats or birds or any other creature that could raise itself off of the ground.

Designs that worked were copied and modified, those that did not have become

interesting curiosities. As the art and knowledge of airfoil design improved,

mathematical tools became the most important part of the process.

A mathematical model known as the theory of thin wing sections was first developed in

the 20’s and refined throughout the 30’s1. Researchers of the era discovered that many of

the characteristics of airfoils could be determined from the shape of the airfoil’s camber

line. Predictions could be made of the chordwise load distribution, angle of zero lift,

pitch coefficient, and location of the aerodynamic center.

In the early 30’s, NACA researchers discovered that the successful airfoils of the time

had nearly the same basic shape when the camber line was straightened. Airfoils such as

the Clark Y and Göttingen 389 have virtually the same thickness profile2. These

researchers developed a relatively simple equation that could be combined numerically

with an equation for a camber line and the resulting airfoil would have predicable

characteristics. These became the well-known NACA 4 digit series of airfoils. Work

continued by the development of additional camber line equations to create the NACA 5

digit series. The best know examples are the NACA 23012 and 23015, which are still in

service today.

10

The researchers knew that certain thickness profiles would result in desired

characteristics such as low drag or high lift and these profiles could be combined with the

mean line to create a new airfoil with predictable behavior. By the early 40’s, the

methodology was refined to the point that a good approximation of specific properties of

an airfoil could be calculated. The new airfoil would then be tested in a wind tunnel to

verify the calculations.

Boundary layer calculations now allow accurate predictions to be made of airfoil

performance characteristics. Inverse modeling is used to specifically tailor an airfoil to

optimize its performance to the desired conditions.

In the AeroFoil program, an airfoil is approximated by a finite number of line segments.

The segments are assumed to be a distribution of vortices, whose strengths vary linearly

along the length of each panel. The line segments form a closed polygon that is in the

shape of an airfoil.

Figure 1.1 – Airfoil Polygon

In fluid mechanics, a vortex is a simple mathematical model of a whirlpool or tornado. A

series of vortices can be combined with a uniform flow field in such a way as to

11

approximate the ideal flow around a body. In the case of an airfoil, a set of equations is

created that result in an ideal airflow on the surface. It will have a stagnation point near

the leading edge, flow smoothly along the upper and lower surface, and recombine at the

trailing edge. Integration of the forces along the airfoil surface gives the lift and pitch

coefficient that would occur if the air had no viscosity. This ideal flow is then used in a

set of boundary layer calculations that estimate viscous effects that cause drag and

changes in lift and pitch coefficient as the angle of attack is increased.

The viscous effects are dependent upon airspeed, altitude, temperature, and airfoil chord

length. These parameters are combined into a dimensionless value known as the

Reynolds Number. The calculational results of AeroFoil have been compared to

experimental data for a variety of airfoils over a range of Reynolds Numbers that is

typical of those seen in general aviation. As can be seen in the appendix, the results

compare very favorably with experimental data.

12

2 - Airfoil Creation Using Inverse Modeling

2.1 Introduction

Traditional airfoil design used one of several different methods.

• A thickness profile could be combined with a camber line

• Camber lines could be altered based on theoretical models

• The top surface of one airfoil could be combined with the bottom surface of another

• “eyeball” engineering judgement could be used to create something brand new

Inverse modeling defines a desired velocity profile and then calculates the airfoil that

would have the profile. The mathematics used to generate the airfoil is relatively

complex, but as implemented in AeroFoil, the process can be automated and made very

easy.

2.2 Conformal Transformation

A conformal transformation is a mapping operation that transfers one plane to another

plane without distortion of infinitesimally small regions on either plane. This means that

an infinitely small observer on the surface of some body in space will not know that the

body has been changed. In the case of an airfoil, a circle is transformed into an airfoil

shape. The best known example in aerodynamics is the Joukowski airfoil. Using

elementary fluid flow and the principle of superposition, an exact solution for the flow

around a circle, with or without circulation, may be found 3. The Joukowski

transformation maps the solution into an airfoil shape that is an exact solution to the

inviscid flow around that airfoil shape.

13

In inverse modeling, conformal mapping is used to transfer a desired velocity profile onto

a transformation circle in the computational plane. The transformation circle is then

mapped into the real plane. The resulting airfoil will have the desired velocity profile

when it is placed in a flow field.

2.3 Velocity Profile

The velocity profile along the surface of an airfoil will have two distinct regions. From

the leading edge back to some point, surface velocity must increase to a value greater

than the free stream velocity. From this point back to the trailing edge, known as the

pressure recovery region, the velocity must decrease to a value less than the free stream

velocity. In addition, there is one point near the leading edge at which the surface

velocity is zero. This is known as the stagnation point. Any fluid particle that is above

the stagnation point will pass over the top surface of the airfoil and any particle below the

stagnation point will pass under the airfoil. The stagnation point, shape of the front

surface velocity, and location of the beginning of the pressure recovery region will

change as the angle of attack is changed.

Figure 2.1a – Typical Velocity Profile, 0º Angle of Attack

14

Figure 2.1b – Typical Velocity Profile, 6º Angle of Attack

In Figures 2.1a and 2.1b, the velocity along the upper surface is shown in blue and along

the lower surface in red.

The method used in AeroFoil is based on Eppler’s inverse design4. The mathematical

operations of the AeroFoil program are invisible to the user, but the information given in

the following paragraphs is provided as background so the design process may be more

easily understood.

The transformation circle is divided into arc segments. The segments begin at the

stagnation point on the transformation circle and progress along the top and bottom

surfaces until reaching the pressure recovery region. In AeroFoil, there are seven arc

segments on both the upper and lower surfaces of the transformation circle.

15

Figure 2.2 – Arc Segments On The Transformation Circle

Figure 2.3 – Transformed Airfoil

Considering the upper surface of the transformation circle, the first six arc segments at

the front are assigned a design value of angle of attack at which the fluid velocity will be

constant. If the airfoil’s angle of attack is less than the design value, the surface velocity

along that segment will be increasing. If the airfoil’s angle of attack is greater than the

upper 1

upper 2

upper 3

upper 4

upper pressure recovery

lower 1

lower 2

lower pressure recovery

a

a

16

design value, the surface velocity along that segment will be decreasing. The design

value of angle of attack has a direct effect on the boundary layer formation over the arc

segment. When the fluid velocity over a segment is increasing, the pressure gradient over

that segment is favorable for laminar flow. When the surface velocity over a segment is

decreasing, the boundary layer will become thicker and separation is more likely to

occur.

Eppler’s formulation of the velocity profile in the pressure recovery region is based upon

a set of mathematical equations that creates a curve approximating typical airfoil profiles.

There is a limited ability to alter the shape of the curves at the beginning of the pressure

recovery region and at the trailing edge. Curve parameters are altered in a non-intuitive

way to both adjust the shape of the velocity profile in the pressure recovery region and to

cause the velocity profile to close at the trailing edge. In AeroFoil, the shape of the

pressure recovery region is altered in a highly intuitive way that directly controls the

boundary layer formation and subsequent lift, pitch coefficient, and drag of the final

airfoil.

2.4 Design Coefficient Of Lift On An Airfoil Segment

The mathematics of conformal transformation requires that the design value of segments

of the transformation circle be in radians, measured from the angle of zero lift. During

the design process, this concept is not “user friendly” but basic airfoil theory does allow

something different. The theoretical slope of a lift curve for any airfoil is 2p per radian.

17

In simple terms, as the angle of attack of a theoretical airfoil is increased by one degree,

the coefficient of lift increases by 0.10966.

In AeroFoil, the design value of each segment is expressed in terms of the coefficient of

lift on that segment. Along the upper surface, if the overall coefficient of lift of the

airfoil is less than the design value for a segment, then the velocity along that upper

surface segment will be increasing. The opposite is true on the lower surface. If the

overall coefficient of lift of the airfoil is greater than the design value of that lower

segment, then the velocity along that lower surface segment will be increasing.

The design coefficient of lift in AeroFoil is only an approximation. Real airfoils do not

always have a lift curve slope that agrees with the theoretical models. As the camber of

the airfoil increases, the lift curve slope tends to flatten out slightly. Using a coefficient

of lift as the design parameter is a simpler concept than using a design angle of attack,

measured from the angle of zero lift.

As an example of the inverse model, consider the following velocity profile, shown at a 0

degree angle of attack:

Figure 2.4 – Inverse Model Velocity Profile

18

The resulting airfoil is:

Figure 2.5 – Inverse Model Airfoil

The potential flow around this airfoil is:

Figure 2.6 – Potential Flow Velocity Profile

By inspection, it can be seen that the potential flow solution around the derived airfoil is

virtually identical to the design velocity profile.

19

A comparison of the airfoil parameters shows an equally good agreement:

DesignVelocityProfile

PotentialFlow

Solution

Lift Coefficient 0.340 0.316

Pitch Coefficient -0.077 -0.074

Angle of zero lift -2.89 -2.663

Table 2.1 – Design Velocity vs Potential Flow Solution

The parameters used to create this airfoil are given in Section 3.3.3

The velocity profile must meet certain restrictions in order to be valid. This valid

solution is in the form of a Fourier series. The first three coefficients of the Fourier series

must be satisfied in order to obtain a valid solution; therefore, three constraint equations

must be met. After the user changes a parameter, the remaining parameters are adjusted

until the constraint equations are satisfied.

Two of the three constraint equations are satisfied by iteration on the magnitude of the

velocity profile and the difference between the magnitude of the velocities on the upper

and lower surface. The third constraint equation is satisfied by iterating on either the

"curvature" of the leading edge or the relative magnitude of the trailing edge velocity.

20

3 -AeroFoil User’s Instructions

3.1 Introduction

AeroFoil is an airfoil design and analysis program written in Visual Basic using the

vortex panel method to calculate ideal flow around an airfoil at different angles of attack.

The ideal flow is used in a series of integral boundary layer equations to calculate the

viscous effects on drag, lift, and airfoil pitching coefficient. Direct comparisons of up to

three airfoils at a time may be performed. Calculations are performed at either constant

velocity, as in a wind tunnel, or in one of two simulations of the actual conditions of an

airfoil in flight. The airfoil is defined by readily available coordinate data, created using

NACA methods, a simple transformation model, or generated from a user-defined

velocity profile through what is known as an inverse design.

Results show an excellent comparison to published wind tunnel data. When compared

with XFoil Version 6.94, AeroFoil has a higher degree of accuracy, particularly in the

estimation of post-stall behavior and the maximum coefficient of lift and angle of attack

at which the maximum will occur.

21

3.2 Airfoil Data

Figure 3.1 – Airfoil Data

Calculations begin after selection of an airfoil. Airfoil coordinate data may be obtained

in one of six ways:

1. From existing data files,

2. Imported in the Drawing Interchange Format,

3. Imported as raw x, y, and x data

4. Generated by the NACA airfoil thickness and mean line equations,

5. Generated from a variation of the Joukowski transformation, or

6. Created by an inverse model where the used defines a velocity profile and the

program calculates the required airfoil.

22

The airfoil coordinate data may be manipulated by the addition of a control surface,

smoothing, addition, or deletion of data points, and modification of the airfoil’s mean

line.

3.2.1 Airfoil Data Files

Two different data file formats are possible. The first format requires an airfoil

name/description in the first line and is followed by x and y coordinates. A maximum of

250 coordinates is allowed. The coordinates begin at the trailing edge of the airfoil,

continue along the surface, and terminate at the trailing edge. The coordinates may be

either top or bottom surface first. Data input is free format but must not contain any

blank lines. The coordinates are dimensionless and in terms of x/chord and y/chord. An

example of this format in bottom surface order is:

NACA 747a315 1.00000 0.00000 0.94996 -0.00405 . . . . . . . . . . . . . . . . . . . . . . 0.95004 0.00481 1.00000 0.00000

The second data file type has a mixed upper and lower surface. This format requires an

airfoil name/description in the first line. The second line contains the number of

coordinate points that describe the top surface and the bottom surface. The third line is

blank. Coordinate data pairs then follow. The top surface is given first, followed by the

bottom surface. Data files that are created from some drafting or design program

packages may include z coordinates. If z coordinates are included in the data file, they

are ignored. The surface coordinates begin at the leading edge and continue to the

23

trailing edge. A blank line separates the top surface coordinates from those of the

bottom. Data input is free format. An example of this data format is:

NASA/LANGLEY NLF 0414F AIRFOIL 43.0 40.0

0.0000000 0.0000000 0.0002094 0.0032797 . . . . . . . . . . . . . . . . . . . . 0.9979467 0.0003898 1.0000000 0.0001959

0.0000000 0.0000000 0.0002217 -.0021140 . . . . . . . . . . . . . . . . . . . . 0.9955088 -.0000158 1.0000000 -0.0001959

In both cases, any z coordinate information that may be present is ignored.

An extensive file of airfoil coordinates is maintained by Dr. Michael S. Selig of the

Applied Aerodynamics Group at the University of Illinois at Urbana-Champaign.

Additional information may be found at:

http://amber.aae.uiuc.edu/~m-selig/ads/coord_database.html

3.2.2 Drawing Interchange Format

The Drawing Interchange Format (.dxf) is a standard file format used to transfer

graphical information between a variety of different software packages. It is commonly

24

used in drafting and solid modeling programs. AeroFoil has a limited capability to

import airfoils from .dxf files.

After selection of a .dxf file, AeroFoil will examine the file for drawing entities that are

formatted in either the 2dPolyline or Line format. If any are found, the data is scaled to a

maximum x/c of 1.0 and rotated into the x/y plane. The drawing entity must begin at the

airfoil trailing edge, continue along the surface, and terminate at the trailing edge. It may

have a maximum of 250 coordinate points.

The file may contain multiple airfoils in the 2dPolyline format and each valid example

will be displayed in sequence. Any file that contains multiple examples of airfoils (or

any other drawing object) in the Line format will be rejected.

The selected airfoil must be named before beginning calculations.

3.2.3 Raw x, y, and z Data

A drawing or solid model software package may have the capability to save airfoil

coordinate data in an x, y, z coordinate system.

After selection of a file, AeroFoil will scale the data to a maximum x/c of 1.0 and rotate it

into the x/y plane. The data must begin at the airfoil trailing edge, continue along the

surface, and terminate at the trailing edge. It may have a maximum of 250 coordinate

points. The file may not contain any non-numeric values.

25

The selected airfoil must be named before beginning calculations.

3.2.4 Airfoil Data File Creation

Airfoil coordinate data may be automatically created using NACA thickness profiles that

are mathematically combined with a camber mean line5 or through a simple numerical

transformation that is related to the Joukowski airfoil6.

NACA airfoils developed in the 1930’s and 40’s use a consistent naming convention that

is based on the airfoil thickness and shape of the camber line. In the 4-digit airfoils, the

first integer indicates the maximum value ordinate of the mean line in percent of chord.

The second integer indicates the location of that maximum measured from the leading

edge in percent of chord and the last two integers represent the maximum thickness of the

airfoil in percent. A NACA 4212 has a 4 percent camber that whose maximum value is

located at 20 percent of chord and is 12 percent thick.

In the NACA 5-digit airfoils, the naming convention is based on the theoretical design

characteristics of the airfoil. The first integer is the amount of camber in terms of the

magnitude of the design lift coefficient. The second and third integers indicate twice the

distance from the leading edge to the location of the maximum camber. The last two

integers represent the thickness in percent of chord. A NACA 23015 has a design lift

coefficient of 0.2, the location of the maximum chamber is at 15 percent behind the

leading edge, and the airfoil is 15 percent thick.

26

The naming convention of NACA 6-series of airfoils is a combination of the location of

maximum thickness, the type of mean line used to generate the chamber line, and the

thickness ratio. A NACA 63-415 will have its maximum thickness at approximately 30

percent of chord, the design lift coefficient is 0.4, and the airfoil is 15 percent thick.

3.2.2.1 NACA 4 and 5 Digit Airfoils

Figure 3.2 – NACA 4 and 5 Digit Airfoils

NACA 4 and 5 digit wing sections combine a thickness distribution with a camber line.

In the early 30’s, NACA found that the successful airfoils of the time had very similar

thickness distributions after their camber lines were straightened. The available data was

used to create an equation for a thickness distribution that was similar in turn to those

airfoils. That equation is:

y = +/- t (0.29690 x0.5 - 0.12600 x - 0.35160 x2 + 0.28430 x3 - 0.10150 x4) / 0.20

27

where t is the thickness expressed as a fraction of the chord.

The camber line for a 4-digit airfoil is given by:

yc = m (2 p x - x2) / p2 forward of the maximum ordinate, and

yc = m [(1-2 p) + 2 p x - x2] / (1 - p)2 aft of the maximum ordinate

where m is the maximum ordinate of the mean line expressed as a fraction of the chord

and p is the chordwise location of the maximum ordinate.

The camber line for a 5-digit airfoil is given by:

yc = k1 [x3 - 3 m x2 + m2 (3 - m) x] / 6 from x = 0 to x = m, and

yc = k1 m3 (1 - x) from x = m to x = 1

Values of m were chosen to give five positions of p, the maximum camber, of 0.05c,

0.10c, 0.15c, 0.20c, and 0.25c, where c in the airfoil chord. Values of k1 were calculated

to result in a design lift coefficient of 0.3.

Mean Line

Designation p m k1

210 0.05 0.0580 361.4

220 0.10 0.1260 51.64

28

230 0.15 0.2025 15.957

240 0.20 0.2900 6.643

250 0.25 0.3910 3.230

The mean line and thickness distribution are combined to give the airfoil shape by the

following:

xu = x – y Sin ?

yu = yc + y Cos ?

xl = x + y Sin ?

yl = yc - y Cos ?

where xu and yu are the upper surface coordinates, xl and yl are the lower surface

coordinates, and ? is the angle between the mean line and the airfoil chord.

Various modification of the mean line, leading edge radius of curvature, and trailing edge

angle were made as a part of the NACA research; however, these are not implemented in

AeroFoil.

The slider bars seen in Figure 3.2 are used to change the thickness and camber line

parameters. Camber line slider bars are only used for the four digit airfoils, five digit

airfoil camber lines are the NACA default values.

29

3.2.2.2 NACA 6 and 7 Series Airfoils

Figure 3.3 – NACA 6 and 7 Series Airfoils

The NACA 6 series airfoils were developed based on work in the late 30's and 40's to

provide laminar flow over a large portion of the airfoil surface. Thickness distributions

were calculated to obtain a pressure distribution that would allow laminar flow and delay

the transition to turbulent flow. This resulted in significantly lower section drag

coefficients for the six series when compared to earlier airfoils.

The thickness distributions were generally given in tabular form for thickness values of

0.06c, 0.08c, 0.09c, 0.10c, 0.12c, 0.15c, 0.18c, and 0.21c, where c is the airfoil chord.

Thickness distributions based on other maximum thickness use linear interpolation to

30

scale the profile. The 67- and 747- are later developments of this work and are only

available for a 15 percent thickness.

Mean lines are based on the theory of thin wing sections, developed in the 20's. In this

theory, a wing section thickness was assumed to be replaced by, and equivalent to, its

mean line. The mean line was in turn composed of a chordwise distribution of vortices

and a related chordwise distribution of load. Solving for the vortex strength resulted in

several simple relationships between lift, angle of attack, and pitch coefficient.

The NACA mean lines are calculated based on a design lift coefficient and a desired

chordwise load distribution. The mean line designation corresponds to a load distribution

that is uniform for x/c = 0 to x/c = a and linearly decreasing to zero from x/c = a to x/c =

1.

The mean line equations can be combined to alter the mean line's shape and

corresponding behavior. For example, the NACA 747A315 airfoil is the 747- thickness

distribution, cambered with two mean lines, a = 0.4 Cl = 0.763 and a = 0.7 Cl = -0.463.

The two equations combine algebraically and result in a mean line that has a slight

trailing edge reflex. The NACA 747A415 includes a third mean line of a = 1.0 Cl =

0.100.

31

Figure 3.4 – NACA 747A315

The calculated values shown in Figure 3.3 for ideal angle of attack, ideal pitch

coefficient, and angle of zero lift provide an estimate of the wing section properties,

based only on the shape of the mean line. The mathematical model developed by

Theodorsen7 is used to estimate the angle of zero lift, pitch coefficient, and design lift

coefficient that will result from the mean line.

The mean line and thickness distributions are controlled by the slider bars and option

buttons shown in Figure 3.3 and combined to give the airfoil shape in the same manner as

with the NACA 4 and 5 digit series airfoils. A maximum of three mean lines may be

combined together to create the final shape.

32

3.2.2.3 Oshkosh Airfoil Transformation

Figure 3.5 – Oshkosh Airfoil Transformation

The Oshkosh airfoil is a modification of the Joukowski airfoil transformation, developed

by R. T. Jones in the 80's. The Joukowski airfoil is based on a conformal transformation

of a circle with circulatory flow combined with a uniform flow field into an airfoil shape.

This results in an exact solution to ideal fluid flow around an airfoil type shape. The

Oshkosh airfoil first transforms the circle into an oval, and then transforms the oval into

an airfoil. Selection of the transformation parameters can give an airfoil that is similar to

a NACA 6 series.

The five parameters, Xc, Xt, Yc, Yt, and D control the airfoil transformation. If Xt, Yt,

and D are set to zero, the result is a Joukowski airfoil.

33

The blue and red lines represent the air velocity over the upper and lower surfaces

respectively and will vary with the angle of attack. The top five slider bars shown in

Figure 3.5 control the airfoil design parameters Xc, Xt, Yc, Yt, and D. The bottom slider

bar alters the angle of attack and subsequent changes in the surface velocity.

34

3.3 Airfoil Creation Using Inverse Modeling

3.3.1 Introduction

Inverse modeling is a powerful tool used to design modern airfoils. The designer defines

a velocity profile that will meet a set of desired characteristics such as boundary layer

growth, lift coefficient, and pitching coefficient and then an airfoil is created which will

give the desired velocity profile. The velocity profile may be altered as necessary to

modify the airfoil characteristics until the design criteria is met.8 In AeroFoil, the

creation of a velocity profile is performed in a highly intuitive way, with estimates of lift

coefficient and pitching coefficient being automatically updated and displayed at any

desired angle of attack.

3.3.2 Inverse Modeling

Inverse modeling uses a mathematical method that is similar to Joukowski conformal

mapping, except that the velocity distribution is specified on the surface of the conformal

mapping circle. The solution is in the form of a Fourier series. The first three

coefficients of the Fourier series must be satisfied in order to obtain a valid solution;

therefore, three constraint equations must be met. After the user changes a parameter, the

remaining parameters are adjusted until the constraint equations are satisfied.

Two of the three constraint equations are satisfied by iteration on the relative magnitude

of the velocity profile and the difference between the magnitude of the velocities on the

upper and lower surface.

35

The third constraint equation is satisfied by iterating on either the "curvature" of the

leading edge or the relative magnitude of the trailing edge velocity. This is a user-

selected option.

The airfoil is divided into front segments and the rear portion that is known as the

pressure recovery region.

Figure 3.6 – Inverse Modeling

The user begins modification of the front segments or the pressure recovery region on

either the upper surface or lower surface by the option buttons shown in Figure 3.6. The

velocity profile on the upper surface is shown in blue and on the lower surface in red.

36

3.3.2.1 Front Segments

Figure 3.7 – Inverse Modeling, Front Segment

The front surface is divided into six different segments on both the upper and lower

surfaces. The locations of the segment endpoints are shown by the large circles and may

be adjusted by the horizontal slider bar. The front of the first segment begins at the

leading edge of the airfoil and the front of each subsequent segment begins at the rear of

the preceding segment. The horizontal slider bar shown in Figure 3.8 will adjust the rear

location of the selected segment.

37

Figure 3.8 – Front Segment Parameters

The design value for the coefficient of lift for each segment represents an approximation

of the coefficient of lift at which the surface velocity will be constant. These values are

adjusted by the vertical slider bar shown in Figure 3.8. The horizontal slider bar alters

the chordwise location of the end of the selected segment.

On the upper surface at any coefficient of lift less than this design value, the surface

velocity will be increasing. As the coefficient of lift is increased above the design value,

surface velocity will begin to decrease. At this point, the attached boundary layer will

begin to thicken and transition to turbulent or separated flow will occur. Conversely, on

a bottom surface segment at any coefficient of lift greater than its design value, the

surface velocity on the segment will be increasing.

The "Design Cl" is only an approximation. In a theoretical airfoil, the slope of the lift

curve as a function of angle of attack is constant. In theory, the coefficient of lift will

38

increase 0.10966 as the angle of attack increases 1 degree. In real airfoils, the slope is

affected by the shape of the camber line.

In general, the magnitude of the design lift coefficient for each segment should progress

from a maximum at the leading edge and decrease toward the trailing edge. This is the

default and is signified by the "Continuous Slope" option selection. In this case,

AeroFoil will adjust the design lift coefficients of segments both upstream and

downstream of the segment that is being modified. If the "Continuous Slope" option is

not selected the upstream and downstream adjustment is not made.

The "Create" button shown in Figure 3.8 will cause the calculations to be run several

times until a final, fully converged, result is obtained. The “Restore” button will return

the velocity profile to the most recent fully converged case.

The right-most vertical slider bar shown in Figure 3.8 alters the angle of attack. The

velocity profile is re-drawn and estimated lift coefficient and pitching coefficient are re-

calculated and displayed automatically.

39

Figure 3.9 – Switching

The switching option shown in Figure 3.9 is used to alternate between the upper and

lower surface or change to the Pressure Recovery section of the inverse model.

40

3.3.2.2 Pressure Recovery Region

Figure 3.10 – Inverse Modeling, Pressure Recovery Region

The rear portion of the airfoil is an area known as the pressure recovery region. Surface

velocity must decrease to some value less than the free stream velocity. The beginning of

the pressure recovery region is the end of the final front segment.

41

Figure 3.11 – Pressure Recovery Parameters

The magnitude and slope of the beginning of the pressure recovery region are controlled

by the first two slider bars shown in Figure 3.11. The magnitude and slope of the trailing

edge are controlled by the second two slider bars. The relative velocity magnitudes are

shown in units of dimensional-less velocity - v/V. The magnitude of the slope is shown

in units of dimensional-less velocity divided by dimensional-less length - v/V divided by

x/c.

Within certain constraints, the magnitude of the trailing edge velocity may be adjusted.

The magnitude of the trailing edge velocity is shown and in units of dimensionless

velocity - v/V.

If the "Trailing Edge Adjustment" option is selected and the magnitude of the trailing

edge velocity is altered, then the slope of the trailing edge velocity on the selected surface

is adjusted. If this adjustment continues beyond the limits set on the slope of the trailing

42

edge velocity, then the leading edge design coefficient of lift is altered to satisfy the

constraint equations.

If the "Leading Edge Adjustment" option is selected, the leading edge design coefficient

of lift is altered to satisfy the constraint equations. The most reliable results are generally

obtained if the "Trailing Edge" option is selected.

The "Create" button shown in Figure 3.11 will cause the calculations to be run several

times until a final, fully converged, result is obtained. The “Restore” button will return

the velocity profile to the most recent fully converged case.

The right-most vertical slider bar shown in Figure 3.11 alters the angle of attack. The

velocity profile is re-drawn and estimated lift coefficient and pitching coefficient are re-

calculated and displayed automatically.

Figure 3.12 – Switching

43

The switching option shown in Figure 3.12 is used to alternate between the upper and

lower surfaces or change to the Front Segment section of the inverse model.

3.3.2.3 Options

Figure 3.13 – Options

The default selection is “trailing edge adjustment”. With this option, the third constraint

equation is satisfied by iterating on the relative magnitude of the trailing edge velocity.

This option generally gives the most reliable results. If the “trailing edge adjustment”

option is selected and the magnitude of the trailing edge velocity is altered, then the slope

of the trailing edge velocity on the selected surface is adjusted. If this adjustment

continues beyond the limits set on the slope of the trailing edge velocity, then the leading

edge design lift coefficient is altered to satisfy the convergence.

If the “leading edge adjustment” option is selected, the third constraint equation is

satisfied by iterating on the design lift coefficient at the leading edge.

44

If the "continuous display" option is selected, a new airfoil is displayed after every

change in the velocity parameters. This is satisfactory on fast computers but it can be

disabled on slower machines. In this option, the velocity profile is continuously re-

calculated but the resulting airfoil is not. The calculated airfoil may be seen by

depressing the “Create” button.

When selecting the symmetric case, the velocity parameters for the lower surface are set

equal and opposite to those of the upper surface.

In general, the magnitude of the design lift coefficient for each segment should progress

from a maximum at the leading edge and decrease toward the trailing edge. This is the

default and is signified by the "continuous slope" option selection. In this case, AeroFoil

will adjust the design lift coefficients of segments both upstream and downstream of the

segment that is being modified. If the "continuous slope" option is not selected the

upstream and downstream adjustment is not made.

As the parameters are changed, a partially converged solution to the equations is

obtained. If the changes are relatively small, the partially converged solution will

become more accurate as the changes are made. Each time the “Create” button is pushed,

a fully converged solution is obtained. Prior to ending the creation portion of AeroFoil,

the user should press the "Create" button one last time. This will ensure that fully

converged airfoil is used in subsequent analysis.

45

The “Restore” button will return all parameters to those in effect at the last time the

“Create” button was pressed.

In certain cases, an invalid result can be achieved. Generally, this would appear to be an

airfoil that has a figure "8" shape, where the lower surface at the trailing edge is above

the upper surface. When this happens, AeroFoil will display a warning message and

attempt to restore the parameters back to the last valid airfoil.

Discontinuities in the velocity profile may be seen in the region of the leading edge and

may be removed by selective adjustment of the design coefficient of lift on either the

upper or the lower surface. In general, at a high angle of attack, the lower surface leading

edge should be adjusted and at a negative angle of attack, the upper surface leading edge

should be adjusted.

Figure3.14 – File Options

Parameters for the inverse model may be saved in data files.

46

3.3.3 Example Inverse Modeling

The example airfoil “Pseudo NFL-0414F” was created to approximate a NLF-0414F.

The creation process involved a visual comparison between the velocity profile from a

NLF-0414F and the velocity profile generated during the inverse model. No attempt was

made to exactly duplicate any specific portion of the airfoil or the velocity profile.

Figure 3.15 – Example Inverse Model

47

On the upper surface, the minimum design lift coefficient is 0.500, with a gradual

increase towards the leading edge. At any coefficient of lift less than approximately

0.500, the upper surface will exhibit laminar airflow. The turbulent transition begins in

the pressure recovery region near the trailing edge. The increase in segment design lift

coefficient towards the leading edge prevents an abrupt transition to separated airflow as

the angle of attack increases.

The maximum design lift coefficient along the lower surface is 0.200. At any coefficient

of lift greater than approximately 0.200, the lower surface will exhibit laminar airflow to

the beginning of the pressure recovery region. The design lift coefficient on the lower

surface at the leading edge was chosen to provide a smooth transition around the leading

edge at high angles of attack.

48

Figure 3.16 – Pseudo NFL-0414F Lift vs Drag

In this example, at any coefficient of lift between approximately 0.200 and 0.500, the

upper and lower surface airflow will be laminar over a significant portion of the airfoil.

This effect can be seen in the Figure 3.16.

49

Figure 3.17 – Pseudo NFL-0414F Lift, Drag, and Pitch vs Angle of Attack

The location and shape of the pressure recovery region controls the boundary layer

growth as well as the pitch coefficient of the airfoil. Subtle changes in the entry and exit

slope of the pressure recovery can be used to tailor the rate of change of the boundary

layer development and consequent drag and stall behavior. The relative placement of the

upper and lower surface pressure recovery regions will directly affect the airfoil pitch

coefficient and lift coefficient.

50

The parameters used to create this airfoil are:

Upper Surface Lower Surface

Front Segments:

Approximate Chord

Location (x/chord)

Design Cl Approximate Chord

Location (x/chord)

Design Cl

leading edge – 0.001 2.225 leading edge – 0.005 -8.000

0.001 – 0.005 0.725 0.005 – 0.011 -3.775

0.005 – 0.019 0.550 0.011 – 0.019 0.000

0.019 – 0.140 0.500 0.019 – 0.043 0.200

0.140 – 0.750 0.500 0.043 – 0.750 0.200

0.750 – 0.860 -4.475 0.750 – 0.808 4.725

Pressure Recovery:

0.9957 Entry Magnitude 0.9218

-2.143 Entry Slope -1.240

-4.757 Trailing Edge Slope -10.260

Trailing Edge Magnitude 0.842

Table 3.1 – Inverse Model Example

It can be seen in the example that the design coefficient of lift in the sixth segments of

both the upper and lower surfaces are distinctly different than might be expected. These

values are used to change the shape and magnitude of the beginning of the pressure

recovery region.

51

3.4 Control Surfaces

Control surfaces and flaps may be simulated using the option buttons and slider bars.

Gaps that will exist in a real world airfoil are not modeled.

Figure 3.18 – Control Surfaces

Simple hinge : The simple hinge model places the hinge on the top surface, the bottom

surface, or mid-way between the surfaces. The hinge location is shown by a small blue

circle. Coordinate points are automatically added and removed as necessary. Additional

points are placed along a circular arc whose center is located at the hinge point and is

tangent to the opposite surface.

52

Figure 3.19 – Simple Hinges

Frise hinge: The characteristic of a Frise hinge is that a portion of the control surface will

extend below the bottom surface of the airfoil when the trailing edge is deflected

upwards. This is used to help control adverse yaw. Frise ailerons may have a hinge

located on the upper surface, the lower surface, or at some point below the chord line.

The hinge location is shown by a small blue circle.

In the case of a top-surface Frise hinge, a front break point (on the lower surface) must be

specified. As the location of the front break point is moved forwards, the amount of

discontinuity on the lower surface increases.

In the other cases, the front break point is located on the circle whose center is at the

hinge axis and is tangent at the upper surface.

53

Figure 3.20 – Frise Hinges

Flap: Points are added as necessary along a circular arc with its center at the hinge

location and tangent to the surfaces.

Figure 3.21 – Flaps

54

3.5 Mean Line Modification

Figure 3.22 – Mean Line Modification

Modification of a mean line is an older method of altering an airfoil in an attempt to

change its characteristics. The current airfoil data is de-composed into a thickness

distribution and a mean line. The calculated mean line may then modified with the

NACA mean line equations, or the thickness profile may be combined with an entirely

new mean line. The mean line and thickness distribution is combined to give the airfoil

shape in the same manner as with the NACA 4, 5, and 6 digit series airfoils.

55

A mathematical model known as the theory of thin wing sections was first developed in

the 20’s and refined throughout the 30’s. Researchers of the era discovered that most of

the characteristics of airfoils could be determined from the shape of the mean line. In this

model, a series of small vortices were placed along a line that represented an airfoil’s

mean line. The strength of each vortex is adjusted mathematically until the resulting

airflow is a streamline along the surface of the line. The mathematical model developed

by Theodorsen is used to estimate the angle of zero lift, pitch coefficient, and design lift

coefficient that will result from the mean line.

The calculation must first ensure that there are an identical number of data points on both

the top and bottom surfaces. In a modern airfoil, there are typically more data points

defining the top surface than the bottom. The de-composition routine adds coordinates as

necessary to the bottom surface and recalculates their location using a spline fit.

The user may specify which type of mean line equation will be used. In the example

shown in Figure 3.18, the NACA 6 digit type is selected. When the 4 digit type is

selected, the slider bar labels change to those shown in Figure 3.23.

56

Figure 3.23 – Mean Line Modification

In certain cases, the airfoil coordinate spacing and curvature cause errors in the

calculations. The minimum allowable spacing can be changed to remove these errors,

but there is a corresponding loss of coordinate points.

57

3.6 Coordinate Spacing

Figure 3.24 – Coordinate Spacing

Spacing of the surface coordinates can cause a significant change in the validity of the

calculations. Boundary layer development and transition from laminar flow to turbulent

flow or separation is highly dependent upon both the fluid velocity and velocity gradient.

Sparse coordinate data, especially near the leading edge, will exhibit unrealistic velocity

gradients and consequent flow transitions.

In general, additional coordinate data will result in a more realistic velocity gradient;

however, location of the coordinates is also important. Best results are obtained by

58

grouping the coordinates more closely at the leading edge. The gradient at the leading

edge is highest, especially at higher angle of attack. The calculation of boundary layer

thickness and consequent viscous effects are dependent upon the point at which

separation occurs. Separation begins at the trailing edge and progresses towards the

leading edge as the angle of attack is increased.

Spline calculations are used to find airfoil coordinates in one of two ways. The initial

airfoil coordinates are used to generate spline-fit coefficients that are then used to find the

new values of the coordinates.

In the "Uniform Spacing" method, coordinates along the leading edge are found at values

of x/c given by:

x/c = (1 + Cos ?) / 2

(125 <= ? <= 235 degrees)

which corresponds to the front 20 percent of the airfoil. The rest of the points are evenly

spaced as a function of x/c.

The other option is to double the number of points. In this option, the original data points

are not altered in any way. Points are added mid-way between each pair.

As a rule, even the sparsest set of data may be expanded to the next finer set of values.

This routine does not include any smoothing algorithm. Sparse coordinate data that is

expanded too far can produce a less than perfect result.

59

3.7 Surface Smoothing

Figure 3.25 – Surface Smoothing

The coordinate data of some airfoils, especially those from older sources, are not very

smooth. Local surface discontinuities may be visible and result in a velocity profile that

has localized discontinuities. The technique used in AeroFoil to smooth the data is a

series of comparisons using spline fits.

Each coordinate (except the two closest to the trailing edge and the one at the leading

edge) are compared with the results of a spline fit using the two coordinates on either side

of that point. For example, a spline curve fit is performed using the i+2, i+1, i-1, and i-2

60

coordinates. A new value of Yi is calculated at the Xi location. The difference between

the given and the calculated coordinates represents an error value. With each successive

computational pass, the coordinate pair with the largest error is adjusted by five percent

of the difference.

When the error is sufficiently small or after 500 passes through the data, a new airfoil is

overlaid on the original. Those locations that have changed are shown in red as in Figure

3.21. The user has the option of accepting or rejecting the changes. If the changes are

accepted, then the calculations may be repeated.

Airfoil coordinate locations are shown as small black hash marks and the range of the

smoothing calculations is shown by the large blue circles as shown in Figure 3.25.

61

3.8 Saving Data

Figure 3.26 – Saving Data

Airfoil coordinate data may be saved in one of four formats:

• Mixed format. The first line contains an airfoil name/description. The second line

contains the number of coordinate points that describe the top surface and the bottom

surface. The third line is blank. Coordinate data pairs then follow. The top surface is

given first, followed by the bottom surface. The surface coordinates begin at the

leading edge and continue to the trailing edge. A blank line separates the top surface

coordinates from those of the bottom.

• Bottom surface first. The first line contains an airfoil name/description. Subsequent

lines contain the x and y coordinate pairs. The coordinates begin at the trailing edge

of the airfoil, continue along the bottom surface, and terminate at the trailing edge.

• Top surface first. The first line contains an airfoil name/description. Subsequent

lines contain the x and y coordinate pairs. The coordinates begin at the trailing edge

of the airfoil, continue along the top surface, and terminate at the trailing edge.

• Drawing Interchange Format (.dxf)

62

Calculated results may be saved in an ASCII text file and the current graphical display

saved as a bit map image. A default file name is created using the airfoil name. If the

airfoil name contains any characters that are not valid in a file name, an error message is

displayed and the save option is terminated.

63

3.9 Viewing Options

Figure 3.27 – Viewing Options

64

The Heading on all of the graphical displays will be identical and show the following:

Line Number:

1. The airfoil name

2. Reynolds Number and Mach Number. The Mach number is shown only if theCompressibility option is selected.

3. Reynolds Number range is shown only if either the Constant Lift or Wing Simulationoption is selected.

4. Surface roughness factor is shown only if a non-smooth surface roughness is selectedin the Surface Conditions option.

5. The location of a forced transition from laminar to turbulent flow is shown only ifselected in the Surface Conditions option.

6. The Angle of Attack of the display in degrees.

7. Lift, Drag, and Pitch Coefficients at the indicated Angle of Attack.

8. The Angle of Zero Lift in degrees.

9. Location of the Aerodynamic Center and the Pitch Moment Coefficient about theAerodynamic Center.

65

3.9.1 Velocity Display

Figure 3.28 – Velocity Display

Blue represents the velocity along the upper surface and red along the lower surface. The

point of stagnation is shown as a gray circle near the leading edge. If the viscous option

has been selected, the approximate transition points are shown on both the velocity and

on the surface of the airfoil. As shown in Figure 3.28, the A/A slider bar adjusts the

angle of attack and the Display Scale slider bar increases or decreases the relative size of

the chosen parameter. The display is a linear interpolation of the data at the selected

angle of attack.

66

3.9.2 Pressure Coefficient

The pressure coefficient is defined as:

Cp = 1- [Surface Velocity / Free Stream Velocity]2

Pressure Coefficient vs Surface

Blue represents regions of low pressure (velocity greater than the free stream conditions)

and red represents high pressure (velocity less than the free stream conditions). Green

represents a pressure coefficient of 0.0.

Figure 3.29a – Pressure Coefficient Display

Pressure Coefficient vs Chord

Blue represents the Pressure Coefficient along the upper surface and red along the lower

surface. If the viscous option has been selected, the approximate transition points are

shown. The transition from laminar to turbulent flow is shown by the first blue or red

circle. The point of separation (if any) is shown by the next blue or red circle.

67

Figure 3.29b – Pressure Coefficient Display

The displayed results as a function of the angle of attack are a linear interpolation of the

calculated values. The Display Scale slider may be used to increase or decrease the

relative size of the display.

68

3.9.3 Boundary Layer

Figure 3.30 – Boundary Layer Display

Boundary layer thickness based on Eppler's H32 method.

The airfoil surface is shown in black. The laminar flow region is so thin that it is not

shown. Turbulent flow is shown in blue and separated flow (if any) in red. When the

Display Scale slider bar is in its default position, the boundary layer thickness is shown in

its true scale. The maximum display is three times actual thickness.

69

3.9.4 Vortex Strength

Figure 3.31 – Vortex Strength Display

Two-dimensional flow of an ideal fluid over an airfoil may be approximated by assuming

the airfoil is made up of a set of "panels", where each panel consists of a linear

distribution of vortices. The unknown vortex strengths at the panel endpoints are

calculated by a series of simultaneous equations that satisfy several boundary conditions

and yield a fluid flow that is tangent to the airfoil surface at the middle of each panel.

Blue represents a negative vortex (counter-clockwise rotation) and red a positive

(clockwise rotation).

70

3.9.5 Graphical Display

Figure 3.32 – Graphical Display: Lift, Drag, and Pitch vs Angle of Attack

The lift coefficient is displayed in blue, the pitch coefficient in red, and if the Viscous

option is selected, the drag coefficient in green.

71

The pitch coefficient is about the airfoil's quarter chord location, x/c = 0.25 and y/c = 0.0.

In the case of inviscid calculations, the lift coefficient and pitch coefficient are

approximately linear.

The left vertical axis is the lift coefficient. The lower right vertical axis is the pitch

coefficient and the upper right vertical axis is the drag coefficient. The horizontal axis is

the angle of attack in degrees.

Calculated results that are significantly after the point of stall may not be valid. In this

case, the angle of attack used for the calculations may be adjusted as necessary.

72

3.9.6 Drag Polar

Figure 3.33 – Drag Polar Display: Lift vs Drag

The horizontal axis is the airfoil's drag coefficient and the vertical axis is the lift

coefficient.

73

3.9.7 Velocity vs Surface Location

Figure 3.34 – Velocity vs Surface Location

The horizontal axis is the airfoil chord in terms of percent x/c. The vertical axis is

velocity in terms of surface velocity / free stream velocity. Red represents high angle of

attack and blue represents low angle of attack.

74

3.9.8 Tabulated Results

Figure 3.35 – Tabulated Results

Tabulated data has the following information:

• Angle of Zero Lift – The angle of attack in degrees, referenced to the airfoil chord

line, at which no lift is generated.

• Aerodynamic Center – A point about which the pitch coefficient is constant. The

location is given in terms of x/c and y/c, where x/c = 0, y/c = 0 is the leading edge

and x/c = 1, y/c = 0 is the trailing edge

• Pitch coefficient about the aerodynamic center.

• If any the Surface Conditions options have been selected, their values are shown.

• If the Compressibility Option is selected, the Mach number is shown. If either the

Constant Lift or Wing simulation is selected, the Mach Number range is shown.

75

• Reynolds Number – When viscous effects are included, Reynolds number is

displayed. If either the Constant Lift or Wing simulation is selected, the Reynolds

number is shown at each angle of attack.

• Angle of Attack – The angle of attack in degrees referenced to the airfoil chord line.

• Lift Coefficient

• Pitch Coefficient

• If viscous effects are included:

• Drag Coefficient

• Lift to drag ratio

• Chord location, in terms of x/c, of transition points from:

• Laminar flow to turbulent flow

• Turbulent flow to separation

76

3.10 Options

Figure 3.36 – Options

The default is inviscid (no boundary layer calculations) and is shown by the check mark

seen in Figure 3.36. In the viscous case, boundary layer calculations are performed and

adjustments made to the lift and pitch coefficient values based on the viscous effects.

77

3.10.1 Multiple Airfoils

Figure 3.37 – Multiple Airfoils

Up to three airfoils may be analyzed at one time. The addition or deletion of an airfoil is

controlled in the “options” drop-down menu as seen in Figure 3.37. Display of the

selected airfoil is by either the drop-down menu or the radio buttons shown in Figure

3.38.

Figure 3.38 – Multiple Airfoil Selection

78

Figure 3.39 – Comparison Of Three Airfoils: Lift, Drag, and Pitch vs Angle of Attack

79

Figure 3.40 – Comparison Of Three Airfoils: Lift vs Drag

80

3.10.2 Reynolds Number

Figure 3.41 – Reynolds Number Calculator

The viscous calculations depend upon the characteristics of the fluid. The user may

select a specific Reynolds number in order to replicate published wind tunnel data, or as

an option, select the free stream velocity, altitude, and chord to calculate the Reynolds

number. All of the airfoil and atmospheric parameters are controlled by the slider bars

shown in Figure 3.41.

The air parameters are given by the following relationships9:

? = 0.00237689 (1 - 0.000006875347 Altitude)4.2561

81

Temperature = 59 - 0.00356616 Altitude

Viscosity = [(232.8999 (Temperature + 459.7)1.5) / (Temperature + 675)] 10-10

Mach1 = [2403.0835 (59 - 0.00356616 Altitude + 459.688)] ½

Density is in slugs/ft3, altitude is in feet, temperature in degrees F, and viscosity in lb-

sec/ft2

82

3.10.3 Constant Lift

Figure 3.42 – Constant Lift

For an airplane in stable flight, angle of attack determines the airspeed. Boundary layer

development and maximum lift coefficient is highly dependent upon the Reynolds

number, which is a function of the airspeed, airfoil length, air density, and air viscosity.

The Constant Lift option assumes the value given by 0.5 * ? * V2 * Lift Coefficient is

constant as a function of the angle of attack.

83

The program user must specify a minimum and maximum velocity range over which the

calculation will be performed using the slider bars shown in Figure 3.42, the airfoil

chord, and the air parameters.

AeroFoil will calculate the angle of attack at the specified minimum velocity through

iteration. The lowest angle of attack is then calculated that will give the same section lift

at the maximum velocity. The range of angle of attack is sub-divided into equal

segments. At each new angle of attack, the process is repeated such that the section lift

remains constant.

84

Figure 3.43 – A Direct Comparison of Two Airfoils Using The Constant Lift Option,

Minimum Lift Coefficient = 0.15 At Reynolds Number = 8,000,000

Lift, Drag, and Pitch vs Angle of Attack

85

Figure 3.44 – A Direct Comparison of Two Airfoils Using The Constant Lift Option,

Minimum Lift Coefficient = 0.15 At Reynolds Number = 8,000,000

Lift vs Drag

86

3.10.4 Wing Simulation

Figure 3.45 – Wing Simulation

For an airplane in stable flight, angle of attack determines the airspeed. Boundary layer

development and maximum lift coefficient is highly dependent upon the Reynolds

number, which is a function of the airspeed, airfoil length, air density, and air viscosity.

In the case of a tapered wing, the Reynolds number and subsequent airfoil characteristics

vary over the span of the wing.

From the equation:

87

Gross Weight = 0.5 * ? * V2 * Lift Coefficient * Wing Area

the range of airspeed, and hence, Reynolds number may be calculated.

AeroFoil first calculates the minimum angle of attack using the maximum airspeed. The

highest angle of attack and minimum airspeed is then found through iteration. The

highest angle of attack in this case is assumed to occur when the airfoil at the wing root

reaches its maximum coefficient of lift. The range of angle of attack is then sub-divided

into equal segments and the airfoil characteristics are found at each.

Washout represents a difference in the angle of attack between the root chord and tip

chord. A positive value indicates that the tip is at a lower angle of attack than the root.

This is frequently used in aircraft design to ensure that the onset of stall will occur at the

wing root and progress towards the tip as the angle of attack is increased.

Re-arranging the first equation gives:

Cl = Gross Weight / (0.5 * ? * V2 * Wing Area)

Assuming a semi-elliptic span loading along the wing, then at the wing root:

Cl root = (4 * Cl * Wing Area) / (p * Root Chord * Span)

88

The mean aerodynamic chord (MAC) represents the average airfoil along the wing's

span. In the preliminary calculations of aircraft stability, the wing is assumed to act as if

all forces are acting on the MAC. It is found by:

MAC = 2 * [Root + Tip - (Root * Tip)/(Root + Tip)] / 3

If coordinate data for only one airfoil is given, AeroFoil assumes that airfoil is used

throughout the wing. If two airfoils are specified, airfoil #1 is used for the root and

airfoil #2 is the tip. The MAC is then found by linear interpolation between the two. If

three airfoils are specified, airfoil #1 is the root, airfoil #2 is the MAC, and airfoil #3 is

the tip.

Figure 3.46 shows the results of the wing analysis using the parameters given in Figure

3.45. In this example, the wing root is a NACA 65-318 with a 48-inch chord. The tip is

a NACA 65-415 with a 36-inch chord and 1 degree of washout relative to the root chord.

The wing span is 26 feet, gross weight is 1500 pounds, and the maximum airspeed is 200

miles per hour. The Mean Aerodynamic Chord is a linear interpolation of the root and tip

airfoils. The Reynolds number on the root chord ranges between 7.44 x 106 and 2.65 x

106. The Reynolds number on the tip chord ranges between 5.58 x 106 and 2.25 x 106.

89

Figure 3.46 – Wing Simulation Results: Lift, Drag, and Pitch vs Angle of Attack

90

3.10.5 Compressibility

Figure 3.47 – Compressibility

When airspeed is sufficiently fast, the compressibility of the air can alter the performance

characteristics of an airfoil. This effect may be evaluated by selecting the

"compressibility" option. AeroFoil uses the Karman-Tsien10 correction. This will alter

the surface velocity, which will change the lift, pitch coefficient, and drag coefficient to a

degree.

91

3.10.6 Surface Conditions

Figure 3.48 – Surface Conditions

The transition from laminar to turbulent flow is assumed to occur when the following

condition has been reached:

ln(Reynoldsd2) >= 18.4 H32 - 21.74 - 0.3 R

where:

Reynoldsd2 is the Reynolds number of the boundary layer momentum thickness

H32 is the ratio of boundary layer's energy thickness, d3, to its momentum thickness, d2

and

R is a roughness factor

92

A roughness factor of 0 is a natural transition on a smooth surface with little or no free-

stream turbulence. A roughness factor of 4 corresponds approximately to a NACA

standard roughness. (slider bar aligned over the NACA label)

It is also possible to specify a location on the surface where the laminar to turbulent

transition will occur. If the flow is laminar at the specified chord location (in units of

x/c), then the calculation switches to the turbulent flow equations.

A chord location of 1.0 will result in a natural transition.

93

3.10.7 Angle Of Attack

Figure 3.49 – Angle of Attack

AeroFoil performs the calculations over a range between –8 and +32 degrees. A

minimum of nine and a maximum of thirty-one calculations are performed. The default

is nine sets of calculations ranging between –8 and +22 degrees. The minimum range of

display is between 0 and +12 degrees. Results that are significantly beyond the stall

94

point may not be valid. The user may adjust the range and number of calculation as

necessary.

The maximum or minimum angle of attack is adjusted with the slider bars shown in

Figure 3.49 and the appropriate option button. Spacing is uniform over the selected

range. The “Reset” option returns the selection back to its default of –8 to +22 degrees.

The minimum allowable range is four degrees.

95

3.10.8 Coordinate Locations

Figure 3.50 – Coordinate Locations

Coordinate locations that are used to define the airfoil may be displayed.

96

3.10.9 Renaming The Airfoil

Figure 3.51 – Renaming The Airfoil

The selected airfoil may be renamed as necessary.

97

3.910.10 Display Options

Figure 3.52 – Display Options

The various lines, location circles, and font sizes used in the graphical displays and

printed output may be altered using the radio buttons shown in Figure 3.52. The vertical

location of the airfoil on each display may be altered with the slider bar. This is

especially useful with either thick airfoils or those with control surfaces.

98

4 - Results

A comparison between the results of AeroFoil and published wind tunnel data shows an

excellent agreement. The results include equivalent analysis from the computer program

XFoil Version 6.9411. It can easily be seen that AeroFoil more closely models the

behavior at and beyond the stall condition than XFoil. Representative examples of four

airfoils are shown in Appendix 1. These airfoils are typical of those in use by general

aviation; they are: NACA 65-41512, NACA 747A31513, NASA GA(W)-114, and NLF-

(1)0215F

The data files used for both the AeroFoil and XFoil calculations were created using the

published coordinate data. Fifty-one coordinate points were used for the NACA 65-415

and NACA 747A315 airfoils. Seventy-five coordinate points were used for the GA(W)-1

airfoil and sixty-one for the NLF-(1)0215F. No effort was made to smooth the data.

XFoil analysis was performed with default values of Ncr = 9 and a mach number of 0.1.

The airfoil coordinate data was expanded to 140 points through use of the PANE

command.

99

Appendix 1 – Representative Results

NACA 65-415 Re: 3,000,000

-0.5

0

0.5

1

1.5

2

-8 -4 0 4 8 12 16 20 24

Angle Of Attack

Lif

t C

oef

fici

ent

-0.2

0

0.2

0.4

0.6

0.8

Pit

ch M

om

ent

Lift: AeroFoil Lift: NACA Lift: XFoil

Pitch: XFoil Pitch: AeroFoil Pitch: NACA

100

NACA 65-415 Re: 3,000,000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0.016

0.018

0.020

0.022

0.024

0.026

0.028

-1 -0.5 0 0.5 1 1.5 2

Lift Coefficient

Dra

g C

oef

fici

ent

AeroFoil NACA XFoil

101

NACA 747A315 Re: 6,000,000

-0.8

-0.4

0

0.4

0.8

1.2

1.6

2

-8 -4 0 4 8 12 16 20 24

Angle Of Attack

Lif

t C

oef

fici

ent

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

Pit

ch M

om

ent

Lift: AeroFoil Lift: NACA Lift: XFoil

Pitch: XFoil Pitch: AeroFoil Pitch: NACA

102

NACA 747A315 Re: 6,000,000

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

0.024

0.026

0.028

-1 -0.5 0 0.5 1 1.5

Lift Coefficient

Dra

g C

oef

ficie

nt

AeroFoil NACA XFoil

103

NASA GA(W)-1 Re: 5,700,000

-0.8

-0.4

0

0.4

0.8

1.2

1.6

2

2.4

-8 -4 0 4 8 12 16 20 24

Angle Of Attack

Lif

t C

oef

fici

ent

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Pit

ch M

om

ent

Lift: AeroFoil Lift: NASA Lift: XFoil

Pitch: XFoil Pitch: AeroFoil Pitch: NASA

104

NASA GA(W)-1 Re: 5,700,000

0.004

0.008

0.012

0.016

0.02

0.024

0.028

0.032

-0.5 0 0.5 1 1.5 2

Lift Coefficient

Dra

g C

oef

fici

ent

AeroFoil NASA XFoil

105

NLF-(1)0215F Re: 9,000,000

-0.5

0

0.5

1

1.5

2

-8 -4 0 4 8 12 16 20 24

Angle Of Attack

Lif

t C

oef

fici

ent

-0.25

0

0.25

0.5

0.75

1

Pitc

h M

om

ent

Lift: AeroFoil Lift: NASA Lift: XFoil

Pitch: XFoil Pitch: AeroFoil Pitch: NASA

106

NLF-(1)0215F Re: 9,000,000

0.004

0.008

0.012

0.016

0.02

0.024

0.028

-0.5 0 0.5 1 1.5 2

Lift Coefficient

Dra

g C

oef

fici

ent

AeroFoil NASA XFoil

107

References:

1 Abbott, I. H. and Von Doenhoff, A. E., Theory of Wing Sections, Dover Publications,Inc., pages 65-75

2 Abbott, I. H. and Von Doenhoff, A. E., Theory of Wing Sections, Dover Publications,Inc., pages 113-116

3 Currie, I. G., Fundamental Mechanics of Fluids, McGraw-Hill Inc., pages 105-115

4 Eppler, et al, A Computer Program For the Design and Analysis of Low-Speed Airfoils,NASA-TM-80210, 1980

5 Abbott, I. H. and Von Doenhoff, A. E., Theory of Wing Sections, Dover Publications,Inc, 1959, PAGES 111-123

6 Jones, R.T. and McWilliams, R., Design Your Own Airfoil, The Oshkosh AirfoilProgram , Sport Aviation, January 1984

7 Abbott, I. H. and Von Doehhoff, A. E., Theory of Wing Sections, Dover Publications,Inc, 1959, page 70

8 Gosalarathnam, A. and Selig, M. S., Low Speed NLF Airfoils: A Case Study in InverseAirfoil Design, AIAA 99-0399, 37th AIAA Aerospace Sciences Meeting, January 11-14,1999

9 Roncz, J., “Questions and Answers”, Sport Aviation, June 1990

10 Stevens, W. A., et al, Mathematical Model For Two-Dimensional Multi-ComponentAirfoils In Viscous Flow, NASA Contractor Report NASA CR-1843, July 1971

11 XFoil 6.94, Drela, M., MIT Aero & Astro, and Harold Youngren, Aerocraft, Inc.,December 200112 Abbott, I. H. and Von Doehhoff, A. E., Theory of Wing Sections, Dover Publications,Inc, 1959, pages 628 and 629

13 Abbott, I. H. and Von Doehhoff, A. E., Theory of Wing Sections, Dover Publications,Inc, 1959, pages 684 and 685

14 McGhee, R. J. and Beasley, W.D., Low-Speed Aerodynamic Characteristics Of A 17-Percent Thick Airfoil Section Designed For General Aviation Applications, NASA TND-7428, December 1973