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Aerodynamics of a Morphing Airfoil

Adam Niksch

Senior, Aerospace Engineering

Texas A&M University

The purpose of this research is to develop a MATLAB model which can calculate the aerodynamics, such

as lift, drag, and pressure, on an airfoil as it changes shape. This model needs to be computationally

efficient and should have reasonable accuracy. A constant strength doublet panel code, which takes

thickness and camber distribution as inputs, is used to model the aerodynamics. This code has the ability

for thickness and camber to change, which effectively allows the airfoil to morph.

INTRODUCTION/BACKGROUND

The purpose of this research is to develop a

computational model with which to

calculate the aerodynamics on a morphing

airfoil. One of the major requirements forthis model is that it be computationally

efficient. Some error is allowable, but themodel must be reasonably accurate and must

produce the correct shapes for lift and drag

plots.

There are many possible ways to implement

this model. One way is to use thin airfoil

theory. The drawbacks to using thin airfoiltheory are the assumptions that must be

made, namely that the airfoil must be thin.Another possible approach is to useconformal mapping, which is an exact

method using complex variables. A third

possibility is to use a panel method.

However, high order panel methods can bevery computationally inefficient.

During this research, both conformalmapping and a doublet panel code were

attempted and each had its advantages and

disadvantages. The doublet panel code waschosen in the end because it works for bothsymmetric and asymmetric airfoils without

too much modification required. This panel

code is able to accurately predict thepressure distribution on any airfoil. The

panel code did have some problems, but it

proved to be the best option in the end.

EXPERIMENTAL SETUP

A MATLAB code is developed to model the

aerodynamic changes an airfoil would

encounter as it changes its shape. This code

accepts thickness and camber distributionfor the airfoil as inputs, not specified points

as in some panel codes. This gives the userthe ability to easily change these parameters,

which causes the airfoil to morph.

This MATLAB code is a powerful tool in

that it allows thickness, camber, chord, and

angle-of-attack all to vary. If a simple for

loop were to be placed around the functionwith all of these parameters varying, the

code could simulate morphing the shape ofthe airfoil using 4 degrees of freedom. Thereason this code is so versatile is that it was

designed to be a subprogram in a larger

reinforcement learning program, which is abranch of artificial intelligence. With the

combination of these two programs running,

an aircraft will learn how to change theshape of its own airfoil for control purposes

rather than use current control mechanisms.

One of the assumptions made when writingthis code is that the flow is incompressible.

This assumption is valid because currentinterests lie in the realm of micro air

vehicles, which fly at speeds less than Mach

0.3.


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Since the final model uses a panel method to

calculate the aerodynamics, it is verysensitive to the grid, or location of the

panels, and the number of panels created.

The grid must be a sinusoidal spaced grid in

the x direction, which puts more points atthe trailing edge of the airfoil. This is

necessary because many aerodynamicchanges occur near the trailing edge. If the

number of panels used were to decrease, the

accuracy of the model would also decrease.The code also assumes inviscid flow, and

thus it is only valid for the linear range of

airfoils, or angles-of-attack prior to stall.

RESEARCH PLAN

Since thin airfoil theory required too many

assumptions be placed on the airfoil, it was

almost immediately decided not to use thatapproach. The first attempted model was a

MATLAB code which used conformal

mapping to calculate the aerodynamics. Asymmetric airfoil is modeled and only its

thickness is allowed to change. The model

produced very accurate results when

compared to wind tunnel data in Ref. 1. Thebasic equations for conformal mapping are

listed below in equations 1-3 [3].

However, once camber was integrated into

the model, the process became extremely

complex and required complicatednumerical analysis. Conformal mapping was

no longer a feasible approach given the

scope of this research and was abandoned.

A constant strength doublet panel method

replaced conformal mapping as themodeling tool. This code required no

changes to be made if the airfoil is

asymmetric. Since it is a low order panelmethod, it is computationally efficient. The

equations for the velocity on each panel are

listed as equations 4 and 5 [3] where x and zare in the local panel coordinate system.

Since these equations require panel

coordinates, a transformation from theglobal coordinate system to the local panel

coordinate system must be made. This

transformation is listed as equation 6 [3].

The panel code is based on the nopenetration condition, which states that the

flow cannot cross the solid boundary of the

airfoil, thus the velocity normal to the

surface is 0 in the global coordinate system.Equation 7 is used to transform the

velocities from equations 4 and 5 into the

global coordinate system. [3].

Now it is possible to solve for the doublet

strengths using equations 4-7. These doubletstrengths can be used to find the tangential

velocities at each point. Once the tangential

velocities are calculated, the pressurecoefficient can be calculated using

Bernoullis equation which, when modified,

produces equation 8 [2].

The pressure coefficient can be broken up

into normal and axial forces using simpleintegration. These forces can also further be

broken up into lift and drag using simpletrigonometry. These equations are listed asequations 9-12.

RESULTS

The first airfoil to be modeled using the

constant strength doublet panel method wasa NACA 0012. This airfoil was chosen first

because of its symmetric shape, which

makes finding errors within the code easier.There is also proven experimental data for

the NACA 0012 as can be seen in Ref. 1.


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Figure 1: Plot of Cp for a NACA 0012 at zero

angle of attack

After accurately calculating the pressure

distribution on a symmetric airfoil, an

asymmetric airfoil was chosen next,

specifically a NACA 4412. As with theNACA 0012, there is proven experimental

data with which to compare the results of the

code. This allowed for quick and easyverification of the codes operation.

Figure 2: Plot of Cp for a NACA 4412 at zero

angle of attack

Figures 1 and 2 show that the code

accurately models both symmetric andasymmetric airfoil shapes. The pressure

distributions predict no lift on the symmetricairfoil and positive lift on the cambered

airfoil, which agrees with accepted theory

and proven experiments on airfoils [1].

Figure 3: Lift vs. Angle of Attack for a NACA

0012

Figure 3 checks model accuracy. The modelwas compared to proven experimental airfoil

data taken from Ref. 1. Since the code is

only able to model the linear range of theairfoil, the angles-of-attack tested ranged

from -5 to 5 degrees. When compared to

those from Ref. 1, there was some

inaccuracy, but it was within tolerance forthis problem.

An .avi file for Windows Media Player was

also created in order to better understandhow the code works. In this video, the user

is able to watch as the airfoil starts as aNACA 0006, then morphs into a NACA4418. After the airfoil has successfully

morphed into a NACA 4418, it morphs back

to its original NACA 0006 shape. The videoalso shows the airfoil sweeping through a 5

degree angle-of-attack as it changes its

shape. The user is also able to observe howthe pressure distribution changes across the

airfoil as it changes shape and angle-of-attack.

SUMMARY AND DISCUSSION

The original objectives of this research wereto model the aerodynamics of an airfoil as it

changes its shape. The model was required


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to be computationally efficient and have a

reasonable amount of accuracy. Theconstant strength doublet panel method

accomplishes these objectives. The

MATLAB code takes a negligible amount of

time to run using as many as 400 panelsalong the airfoil. The code produces

reasonably accurate results.

One unexpected problem which arose

concerned the trailing edge of the airfoil. Inorder to accurately model the trailing edge,

the Kutta Condition ought to be enforced by

inserting a wake panel. The problem was

discovered when rather high values for thepressure coefficient were seen. It was

determined that the best way to deal withthis problem was to remove these unrealisticvalues. Once these values were removed, the

results became much more accurate.

CONCLUSIONS

The doublet panel code is able to efficientlymodel the aerodynamic changes on an airfoil

as it changes shape. It is also more accurate

than expected for airfoils within the linearrange.

One of the main things learned from thisresearch is conformal mapping is not a

realistic option for morphing airfoils. The

process becomes extremely complex and

therefore becomes very inefficient.

Also, if the final two or three values from Cp

were removed, then the overall accuracy ofthe model increased.

This code is developed primarily to be usedas a tool for future research. Structural

effects will be added to the model later using

basic Euler-Bernoulli beam theory so themodel can also predict how the morphing

process will affect the physical object. This

model will also be used as the basis for the

morphing 3-D finite delta wing in the future.Also, as previously mentioned, a

reinforcement learning program will use this

model to learn the optimal airfoil shapes for

a range of flight conditions.

ACKNOWLEDGEMENTS

The author wishes to thank Dr. John Valasek

and Amanda Lampton of the Aerospace

Engineering Department at Texas A&MUniversity for their guidance and assistance

throughout this research. The author also

acknowledges the Texas A&M FlightSimulation Laboratory for all of their help

and assistance.

This Research Experience for

Undergraduates Site is sponsored by theNational Science Foundation Grant No.

0453578, the Air Force Office of Scientific

Research, U.S. Air Force, Department of

Defense and NASA Cooperative AgreementNo. NCC1-02038.

REFERENCES

[1] Abbot, Ira and Von Doenhoff, Albert.

Theory of Wing Sections. Dover

Publications, 1959.

[2] Anderson, John D. Fundamentals ofAerodynamics Third Edition. McGraw-Hill,

2001.

[3] Katz, Joseph, and Plotkin, Allen. Low-

Speed Aerodynamics Second Edition.Cambridge University Press, 2001.


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APPENDIX A - LIST OF EQUATIONS

1

z x i y

A B i C

A zz

= +

= +

= +

2 2 2 2

1 2

1 2

2 2 2 2

1 2

2 ( ) ( )

2 ( ) ( )

p

p

z zu

x x z x x z

x x x xw

x x z x x z

=

+ +

=

+ +

0

0

cos( ) sin( )

sin( ) cos( )

i i

i ip

x xx

z z z

=

cos( ) sin( )

sin( ) cos( )

pi i

i i p

uu

w w

=

2 2

21p u wC

V

+=

( )0

0

1

1

c o s ( ) s in ( )

s in ( ) c o s ( )

lo w e r u p p e r

u p p e r lo w er

c

n p p

cu p p e r l o w e r

a p p

l n a

d n a

C C C d xc

d y d yC C C d x

c d x d x

C C C

C C C

=

=

=

= +

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(10)

(9)

(11)

(12)

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