MULTI MISSION RE-CONFIGURABLE UAV – AIRFOIL SHAPE PARAMETERISATION STUDY

Multi Mission Re-Configurable UAV – Airfoil Shape Parameterisation Study

22nd

International Unmanned Air Vehicle Systems Conference – 16-18 April 2007

MULTI MISSION RE-CONFIGURABLE UAV – AIRFOIL SHAPE

PARAMETERISATION STUDY

Manas S. Khurana

#, Arvind K. Sinha, Hadi Winarto

The Sir Lawrence Wackett Centre for Aerospace Design Technology

RMIT University, GPO Box 2476v, Melbourne, Victoria 3001, Australia

Tel: +61 3 9645 4530 Fax: +61 3 9645 4534 #Email: [email protected]

Abstract

The paper outlines the methodology of shape parameterisation techniques to generate airfoils for design

and optimisation. The aim of this investigation is to examine the flexibility of two analytical shape

functions; the Hicks-Henne and Wagner Polynomials for airfoil shape parameterisation. A symmetrical

base airfoil is used as the starting point and an optimiser is used to duplicate three distinct target sections

as designed specifically for slow speed long endurance roles; a foreseen operational scenario for a Multi-

Mission UAV. The fitness function though a linear gradient search algorithm is formulated based on an

objective function of minimising the difference between base and target geometries is evaluated through a

high fidelity flow solver. The results indicate that the Hicks-Henne approximation is superior to Wagner

polynomials in duplicating the target airfoils; as observed through a lower fitness measurement, higher

computational efficiency and in the convergence of aerodynamic coefficients in comparison to actual

airfoil data.

Biography

Manas holds a Bachelor of Engineering Degree in Aerospace with Honours and also a Graduate Certificate

in Engineering Management. He is currently undertaking a PhD program in computational and applied

aerodynamic design of morphing wings. In addition to his candidature, Manas is also a Research Assistant

at the Wackett Aerospace Centre.

Arvind K. Sinha has a service record of 31 years, which includes Defence forces, industry and academic

institutions. He has several qualifications, scholarships, awards, industrial research projects, research

papers and public presentations to credit. He is presently the Director of The Sir Lawrence Wackett Centre

for Aerospace Design Technology, Royal Melbourne Institute of Technology, Melbourne, Australia.

Hadi Winarto is an Associate Professor at the School of Aerospace, Mechanical and Manufacturing Engineering. His research areas include applied and computational aerodynamic design with emphasis on software development, analysis of turbulent flows and thermo-fluid dynamics. Hadi further supervises students on their research at the Wackett Aerospace Centre.


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Nomenclature

iia β, = Hicks-Henne Peak

ƒapprox. = Approximated Airfoil

λi = Design Variables

ƒi(x) = Shape Functions

(L/D)max = Maximum Lift-to-Drag

ƒtarget = Target Airfoil

(x/c)i = Chordwise Position

ak = Scalar Step Length

CL = Coefficient of Lift

Configurable-Unmanned

Aerial Vehicle

CP = Coefficient of Pressure

Location Values

MM-RC-UAV = Multi-Mission- Re-

pk = Search Direction

Ratio

Re = Reynolds Number

t/c = Thickness-to-Chord Ratio

xk = Current Iteration

∆ƒ = Fitness Function

∆ƒmin = Objective Function

Introduction

Traditional concepts of developing platforms for

uni-mission requirements has led to a large fleet

of UAVs with inherent ‘issues and challenges’ of

operation and support [1]. A multi-mission

platform is needed to address the issues and

challenges. Development of pioneering design

aircraft concepts providing multi-role and multi-

mission capabilities has been acknowledged by

renowned operators and designers and is cost and

mission effective. Hence, the need to introduce

aerial platforms that addresses a wide client base

by encompassing civil and military mission

capabilities in a single platform require further

investigations.

A detail market survey on Australian mission

requirement provided an operational and design

window for the development of UAV with multi-

mission capabilities [2]. Researchers at the Sir

Lawrence Wackett Centre for Aerospace

Technology have examined the prospect of

developing a Re-Configurable Multi-Mission

Unmanned Aerial Vehicle (MM-RC-UAV)

design concept for the identified class of UAVs (

Table 1). The concept proposes flexibility of

multiple payload configurations. Initial

investigations of platform concepts examined the

prospect of introducing wing and fuselage

extensions to address the disparate requirements

of payload and flight performance. The optimal

operational requirements though achievable, it is

a major design and manufacturing challenge.

Table 1: Australian UAV Market Survey

Technology

Classification

Missions

Support UAV

� High-Altitude Long Endurance

(HALE);

� High-Altitude Long Operation

(HALO); and

� Medium Altitude Medium

Endurance (MAME)

Combat UAV

� Unmanned Combat Aerial

Vehicle (UCAV-HL) – High

Altitude Long Endurance;

� Unmanned Combat Aerial

Vehicle (UCAV-MM) – Medium

Altitude Medium Endurance;

The mission requirements cover long endurance

sorties at medium altitudes over large distances.

This results in high fuel requirements and fuel

storage in wings will result in thick airfoil

sections. The reconnaissance and surveillance

component of the mission is at low speeds.

Excessive viscous affects are prominent at low

Mach and Reynolds number. Thus a constrained

optimisation model to maximise the glide ratio to

overcome the high drag properties associated

with low speed operations is needed in the

design. Traditional uni-mission UAVs have

limited performance capability and operations

outside the intended design spectrum lead to sub-

optimal flight performance. A revolutionary

design concept to address this limitation is

foreseen for future operations.

Wing extensions as proposed earlier, was

considered in the form of morphing wings to

address the requirements of long endurance.

Morphing wings will enhance the operational

performance with inbuilt flexibility of wing

shape to achieve the desired aerodynamic

performance. Development of an intelligent

airfoil optimisation model is needed to provide

the framework for a more detail wing design.

Design of unique airfoils that are best suited for

each flight segment of the mission profile is to

be established. In this paper a geometrical

methodology for the development of an airfoil

optimisation model is presented.

The research is part of an overall effort of

developing morphing airfoils for MM-RC-


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UAVs. The first section of the paper presents an

overview of the design process required to

design and optimise airfoils. The effectiveness of

the proposed geometrical methods and the

methodology required to test the robustness is

presented. The operational status of the adopted

optimisation tool is presented and correlation

with the formulated objective function is

introduced.

The second part of the paper adopts the proposed

methodology from the first section and the

testing process is initiated. Measurement of

geometrical differences and the iterations

required for solution convergence are recorded

and an aerodynamic analysis is presented to

equate the fitness function to lift, drag and

coefficient of pressure performance.

The final section summarises the major findings

of the investigation and a brief outline of the

proposed research roadmap in the design of

morphing airfoils for MM-RC-UAV is

presented. Recommendations for future design

modifications are also discussed.

Airfoil Shape Representation

General Airfoil Design Review Direct airfoil optimisation is composed of two

branches; a) establishment of flow solver to

compute the aerodynamic forces and b)

development of a geometrical shape

parameterisation method as integrated to an

optimizer (global or gradient) to find a desirable

shape based on user defined constraints and

objectives.

A study based on the above mentioned

methodology has been attempted in limited

capacity. The first stage provided an indication

as to the validity of the adopted flow solver. A

Low-Speed Airfoil Section as developed by

NASA for which experimental data was made

available [3] was simulated within Fluent CFD

package. The κ-ω turbulence model computed

lift and drag that was within 3% and 10% of

wind tunnel data for a linear angle of attack

range of (0°-7°) [4]. Experiment Mach and

Reynolds number conditions were applied within

CFD which also match the cruise phase for a

foreseen Airborne Chemical Detection sortie,

hence providing a case study for which the

analysis could be based around.

The second stage involved utilising conformal

mapping technique for airfoil shape

representation. Minimisation of design variables,

λi within the overall optimisation routine is a

requirement to ease the overall computational

expense. Kármán-Trefftz transformations were

initially deemed appropriate for airfoil shape

parameterisation since three design variables

could be used to generate a family of airfoils.

The mapping technique was concluded to be

insufficient due to the limited design space as

control over important airfoil regions was not

possible. Results from the second stage of the

analysis indicated the need to perform a

parameterisation study to determine a suitable

geometrical method for airfoil development. The

paper addresses this issue and forms an

underlying framework for a more exhaustive

study on the optimisation of morphing airfoils.

A base airfoil is a requirement for airfoil

optimisation and is treated differently based on

the optimisation approach adopted. A gradient

method requires an initial section and the design

variables are varied to examine a range of

candidate solutions, until an optimum solution is

obtained. A global method, including Genetic

Algorithms and Simulated Annealing does not

directly require an initial starting point. Instead

design variables based on an initially defined

section are varied simultaneously such that a

large population of solutions can be examined

for applicability based on user defined objectives

and constraints. Thus airfoil parameterisation

method must accurately duplicate a base airfoil

for future optimisation cycles and the following

sections describe the methodology undertaken.

Airfoil Parameterisation Methodology An airfoil shape can be generated by many

methods. Airfoil design through an intelligent

optimiser requires a shape function to generate

and test possible solutions. A parameterisation

approach that can examine a large population of

candidate airfoils during the global search with

minimal computational expense is a requirement

in multi-objective optimisation. Different

approaches include Discrete, Polynomial, Spline

and Analytical [5].

There are not set guidelines that govern the use

of one method over the other; instead a

parameterisation study must be formulated to

determine a suitable geometrical method based

on user defined constraints and objectives. The

test requires duplicating a set of target airfoils


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with an initial starting section. To test the

flexibility of the design space within the different

geometrical solutions, a sample of target airfoils

with unique geometrical features must be

adopted. A symmetrical NACA 0015 airfoil is

used as the base section. The selection of target

sections was based on the requirement of

obtaining airfoils that provided performance

characteristics that matched the proposed MM-

RC-UAV. Slow speed, long endurance sorties is

a major design requirement and the following

airfoils were selected based on this requirement

as presented in Figure 1:

� NASA LRN(1)-1007; Based on the

methodology of avoiding laminar flow

separation at low Reynolds number, where

the affects of viscosity, thus drag rise is

considerable. The design goal was to

maximise the lift-to-drag ratio (L/D)max at

lift coefficients, CL of 1.0 based for a cruise

angle of attack of 4° [6]. The airfoil contour

was designed through an inverse code such

that an attached laminar boundary-layer as

far as 60% chord is achievable [6].

� The LS(1)-0417; Modified 17% Thick Low

Speed Section with 2% chord ratio. This

design was based on the requirement of

reducing the pitching moment coefficient

CM,, whilst maximising the lift-drag ratio at

climb. Experimental tests have indicated a

reduction of CM,, at cruise lift coefficient CL,

of 0.40, with increases in lift-drag ratio. The

CLmax was further increased slightly within a

Reynolds number range of 2-4 million [3].

� NASA Natural Laminar Flow (NLF) (1)-

1015 airfoil; Design methodology aims at

maintaining favorable pressure gradient thus

achieving laminar boundary layer over large

segments (≥30% chord) [7]. The design is

based on the requirement of attaining

minimum drag at slow speeds for long

endurance sorties and a high thickness-to-

chord (t/c) ratio is employed for fuel

requirements.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

-0.05

0

0.05

0.1

0.15

Comparison of Airfoil Shape Configuration for Geometrical Shape Parameterisation

x/c

y/c

Base: NACA 0015

Target 1: NASA LRN(1)-1007

Target 2: NASA LS(1)-0417Mod

Target 3: NASA NLF(1)-1015

Figure 1: Comparison of Airfoil Shape for

Geometrical Parameterisation

As with any parameterisation method, the

variable population size defines the flexibility

and robustness of the model. Generally by

prescribing additional variables, the design

window increases and the model is likely to

represent additional shapes. Conversely, the

model becomes computationally expensive and

is inefficient for an exhaustive optimisation

exercise. Thus, a compromise between the

design variables and the design window must be

established before using the method for shape

optimisation.

Generally, researchers have found a population

size of 14-18 variables as adequate for airfoil

shape modeling [8-14]. The choice of

parameterisation method and the corresponding

variable size is dependent on the application

within which the intended airfoil is to operate.

An independent study must be performed to

examine the suitability of a geometrical model.

In this paper, a total of 14 variables are used; 7

for suction and pressure sides to prescribe the

lifting surface. By maintaining a constant

variable size between the two methods, a solid

conclusion as to the flexibility and robustness of

the models for application in the design of MM-

RC-UAV airfoils can be made.

Effectiveness of Airfoil Parameterisation

Methodology

The effectiveness of the parameterisation scheme

is measured in two folds:

a) Firstly a geometrical comparison between

the target and approximated airfoil is made

through a fitness function evaluation ∆ƒ, in

equation 2. The objective function then


22nd


becomes minimisation of ∆ƒmin between the

two geometrical sets.

Position Chordwise)/(

and Airfoil; edApproximat

Airfoil;Target

:

))/()/((

approx.

target

.arg

=

=

=

−∑=∆

i

iapproxiett

cx

f

f

where

cxfcxfabsf

(Eqn 1)

b) The magnitude of ∆ƒmin is then measured

from an aerodynamic view point through

CFD based on an established and verified

test domain [4] in Figure 2. Lift, drag and

pressure coefficient data was computed and

compared with target section to define the

magnitude of ∆ƒmin in aerodynamic

coefficients.

Figure 2: CFD Computational Domain

� Analytical Approach – Shape Functions

An analytical approach by Hicks-Henne [15] can

be used to develop an airfoil/wing body for

optimisation purposes. Methods include the

Hicks-Henne, Wagner, Legendre, Bernstein and

NACA normal modes. The analytical technique

operates by adding a finite sum of closed shape

functions, ƒi(x) to an initially specified airfoil

shape in to generate a target section in equation

2. The design variables, act as multipliers to

ƒi(x) and determine the contribution of each

function to the final shape [5].

∑=

+=7

1

)()(),(

i

iiairfoilinitiali xfxyxy λλ

Functions Shape)(

and Variables;Design

:where

i

=

=

xf i

λ

(Eqn 2)

� Hicks-Henne Method

Hicks-Henne shape functions were designed to

gain control over the airfoils leading edge region

in the design of supercritical airfoils [15].

Adverse pressure gradients greatly affect the

onset of stall at the leading edge. As a result,

shape functions that could directly alter the shape

such that a geometrical relationship between

pressure recovery performance and the layout of

the leading edge could be made were a

requirement.

The parameterisation technique was applied

separately by de-coupling both the upper and

lower surfaces. The approximated curves were

then integrated to form a final shape for analysis.

The Hicks-Henne shape functions in equation 2

are defined as and presented in Figure 3 :

)(sin)( ii xxfa

iβπ= (Eqn 3)

]99.0,90.0,70.0,55.0,40.0,15.0,05.0[

and ; ]3,4,5,5.3,5.2,5.2,2[

functions shape ing correspond of

ValuesLocation Peak Adjusted Priori and

:where

=

=

=

i

i

ii

p

a

a β

)ln(

)5.0ln(

i

ip

=β (Eqn 4)

� Wagner Polynomials

The Wagner functions were introduced for the

purposes of computing the chordwise pressure

loads over a finite wing [16]. The functions were

derived such that they are linearly independent to

the ‘Pseudo orthonormal property” for a given

weighting function [16], thus suitable for airfoil

shape parameterisation. The Wager shape

functions are defined in equations 5-6 and

presented in Figure 5 [8]:

−

+=

2sin

)sin()(

21

θ

π

θθxf

(Eqn 5)

1kfor ])1sin[()sin(

)( >−

+=π

θ

π

θ k

k

kxf k

(Eqn 6)

Where:

)(sin2 1 x−=θ

(Eqn 7)


22nd


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Hicks-Henne Shape Functions

x

y

Figure 3: Hicks-Henne Shape Functions

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Wagner Shape Functions

x

y

Figure 4: Wagner Shape Functions

Line Search Optimisation

The optimiser is implemented to minimise the

objective function from equation 1, at each (x/c)i

location in the form ).(y ,,21,min nyyf…

∆ The yi

ordinates are a function of the design variables

λi, and can be represented

by ].,,[ 72,1 λλλ …=iy A total of 14 variables

are used in the study; 7 for upper and lower

surfaces. The limits of λ1-7 are defined with a set

of side constraints to minimise the design

window, thus the computational expense in the

form UpperLower717171 −−− ≤≤ λλλ , where a two sets of

constraints apply to both the suction and pressure

sides.

The objective function and constraints are linear

functions of the design variables and gradient

information is required to direct the search

towards a minimum. A higher order Quasi-

Newton method is used to develop this

information after each iteration. An optimal

solution is obtained when the partial derivative

of the objective function remains unchanged

over a 30 iteration count through equation 8 [17].

0)( ** ≈+=∇ cHxxf (Eqn 8)

ectorConstant v c ;Matrix Hessian H

where;

x

:is solution x optimal thewhere

1*

*

==

−= −cH

(Eqn 9)

The Quasi-Newton method is appropriate since

the search direction is updated through the

Hessian matrix by approximating the behavior of

ƒ(x) and ∇ƒ(x) after each iteration K, through

developed update formulas of Broyden, Fletcher,

Goldfarb and Shanno (BFGS) [17].

The gradient information is not supplied within

the optimiser instead a finite differences method

is used. As a starting point, 7 variables are

randomly generated in line with the side

constraints as stated previously. The variables

are then perturbed and the rate of change of the

objective function is established to direct the

search path towards the minimum solution. After

each iteration the line search algorithm is used to

compute the search direction with a scalar step

length in equation 10 [17]:

kkkk paxx +=+1 (Eqn 10)

DirectionSearch p

and Length; StepScalar Positivea

Iteration;Current

k

k

=

=

=kx

where

A descent direction is required to guarantee a

decreasing objective function in equation 11

[17]:

)(.1

kkk xfHp ∇−= − (Eqn 11)

Wolfe conditions are applied to ensure the step

length is appropriate to decrease the objective

function adequately along the line

,kkk pax + until the termination convergence

criteria of constant gradient computation is

obtained [17].

Results and Discussion

Based on the developed methodology, two major

points of consideration are used as a means of

measure for objective function in equation 1. The

speed on convergence through the number of


22nd


iterations required to achieve the termination

criteria and the value of ∆ƒmin at the termination

point is established for the two methods. The

optimal solution of λ1-7 for both the upper and

lower surfaces is presented in Appendix A. From

visual inspection, the six plots a-f, indicate

excellent convergence to target sections.

Inspection of time and fitness convergence is

established in Table 2, with favorable properties

shaded to examine the effectiveness of the two

models in detail. The graphical layout of Table 2

is also presented in Appendix B - Figure 6(a-c).

Table 2: Evaluation of Fitness and Iteration

Count Convergence

LRN(1)-1007

Hicks

Henne

Wagner Hicks

Henne

Wagner Airfoil

Surface

Fitness No. of Iterations

Pressure 0.0104 0.028 36 22

Suction 0.009 0.011 36 51

LS(1)-0417 Mod

Pressure 0.0173 0.026 34 28

Suction 0.0148 0.021 52 57

NLF(1)-1015

Pressure 0.0154 0.020 44 64

Suction 0.009 0.0118 59 50

Geometrical convergence data in Table 2 and

Figure 6 indicates fluctuating results in terms of

the speeds required for solution convergence

between the two methods tested. Both shape

functions in equations 3, 5-6 use sine waves

which are superimposed to define the lifting

surface. The shape function amplitudes in the

Hicks-Henne were constant for both the pressure

and suction sides, thus no calculations within the

optimiser were required. The amplitudes with the

Wagner solution are a function of x/c from

equation 7 and were computed for each x/c

station, thus involved an additional

computational process. The NASA NLF(1)-1015

has the largest trailing edge angle of the three

target sections selected, and Wagner functions

become computationally expensive in modeling

this part of the geometry. The fitness value is

relatively higher than the Hicks-Henne method

for the same airfoil.

The fitness function convergence indicates that

the Hicks-Henne method is superior as it

provides a lower fitness for all the test cases in

comparison to Wagner polynomials. Airfoil

performance is highly sensitive to geometrical

changes and drag and coefficient of pressure

performance can provide a strong indication as to

the differences between target and approximated

sections and quantify the meaning of ∆ƒmin from

an aerodynamic view point. Evaluation of

aerodynamic coefficients forms the overall

testing methodology and is presented in Table 3.

As expected the lower fitness value for the

Hicks-Henne method correlates with computed

aerodynamic data. Lift and drag performance

match target coefficients as shaded in Table 3

and major solution deviations are present within

the Wagner Polynomials.

Table 3: Comparison of Aerodynamic

Coefficients

Method Actual Hicks-Henne Wagner

LRN(1)-1007

CL 0.636 0.646 0.636

CD 0.0132 0.0132 0.0140

LS(1)-0417 Mod

CL 0.685 0.690 0.76

CD 0.0116 0.0115 0.0118

NLF(1)-1015

CL 0.817 0.873 0.951

CD 0.0118 0.012 0.0130

The CP distribution plots in Figure 7(a-c) -

Appendix B provide an additional measurement

tool. The Wagner solution indicates significant

deviations at the trailing edge region thus

pressure recovery is not accurately captured

Figure 7(b-c). The Wagner function fails to

capture the trailing edge angles precisely and a

mismatch in CP is obtained as a result. The

Hicks-Henne provides excellent solution

agreement across the entire test phase. The CP

distribution shape closely matches target solution

as expected due to close proximity of CL

between the target and approximated data in

Table 3. Pressure recovery and expansion of

flow about the leading edge is captured across all

the test sections.

From the analysis performed, the Hicks-Henne

method is seen to provide superior results in

duplicating the performance of slow speed, long

endurance airfoils in comparison to Wagner

polynomials. Even though the convergence rate

is slightly higher in some cases within the Hicks-

Henne method in Table 2, the lower fitness value

across the entire testing envelope justifies the use

of this method for airfoil shape parameterisation.


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Conclusion

Comparison of two geometric shape methods is

performed in the paper for airfoil shape

parameterisation designed to replicate candidate

airfoils for operation within the RC-MM-UAV

project. The effectiveness of the proposed shape

method with 14 variables is used through a

gradient based search algorithm to minimise the

geometrical differences between target and

approximated sections. The magnitude of the

fitness function was evaluated within a CFD

domain under the condition of slow speed, long

endurance flight at Mach and Reynolds number

of 0.32 and 6.0e6 respectively to equate ∆ƒ with

the aerodynamic coefficients.

The Hicks-Henne method provided excellent

solution agreement with lower fitness for all test

cases examined. Lift, drag and pressure

coefficient data was closely matched by target

solutions. The slightly longer computational

process associated with the Hicks-Henne method

does not justify the use of Wagner polynomials

due to large solution disagreement.

The peak functions iia β, , within the Hicks-

Henne method were constant for all the test cases

presented in equation 3. Future optimisation

routines could focus on integrating these

functions within the overall optimisation cycle

such that an independent set of iia β, values can

be obtained for each target surface.

A single side constraint was applied to all the 14

design variables and future applications could

focus on investigating the affect of each function

on the aerodynamic performance such that

realistic constraints can be denoted to each

variable. This has the potential of further

lowering the computational expense as the

optimiser will avoid searching outside an

unnecessary design window. The affect of

variable size can further be analysed by repeating

the proposed methodology and observing

solution convergence. Finally, the application of

Legendre, Bernstein and NACA normal modes

can also be performed to determine the

suitability of these methods for airfoil

parameterisation in the design of MM-RC-UAV.


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Appendix A:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

-0.05

0

0.05

0.1

0.15

0.2

Hicks-Henne: NASA LRN(1)-1007 Parameterisation

x/c

y/c

NASA LRN(1)-1007

Hicks-Henne Approximation

a) Hicks-Henne Parameterisation of NASA

LRN(1)-1007

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4Hicks-Henne: NASA LS(1)-0417 Mod Parameterisation

x/c

y/c

NASA LS(1)-0417 Mod


b) Hicks-Henne Parameterisation of NASA

LS(1)-0417 Mod

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3Hicks-Henne: NASA NLF(1)-1015 Parameterisation

x/c

y/c

NASA NLF(1)-1015


c) Wagner Parameterisation of NASA NLF(1)-

1015

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.1

-0.05

0

0.05

0.1

0.15

0.2Wagner Functions: NASA LRN(1)-1007 Parameterisation

x/c

y/c

NASA LRN(1)-1007

Wagner Functions Approximation

d) Wagner Parameterisation of NASA LRN(1)-

1007

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4Wagner Functions: NASA LS(1)-0417 Mod Parameterisation

x/c

y/c

NASA LS(1)-0417 Mod


e) Wagner Parameterisation of NASA LS(1)-

0417 Mod

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

0.3Wagner Functions: NASA NLF(1)-1015 Parameterisation

x/c

y/c

NASA NLF(1)-1015


f) Wagner Parameterisation of NASA NLF(1)-

1015

Figure 5: Airfoil Geometry Parameterisation Comparison through Hicks-Henne & Wagner

Functions


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Appendix B:

20 25 30 35 40 45 50 550.005

0.01

0.015

0.02

0.025

0.03

Iterations

∆(f

)

NASA LRN(1)-1007: Convergence of Fitness Function

Wagner Upper

Hicks-Henne Upper

Wagner Lower

Hicks-Henne Lower

a) NASA LRN(1)-1007: Evaluation of ∆ƒƒƒƒ

through Wagner and Hicks-Henne Functions

25 30 35 40 45 50 55 600.014

0.016

0.018

0.02

0.022

0.024

0.026

0.028

Iterations

∆(f

)

NASA LS(1)-0417 Mod: Convergence of Fitness Function

Wagner Upper

Hicks-Henne Upper

Wagner Lower

Hicks-Henne Lower

b) NASA LS(1)-0417 Mod: Evaluation of ∆ƒƒƒƒ

through Wagner and Hicks-Henne Functions

40 45 50 55 60 650.008

0.01

0.012

0.014

0.016

0.018

0.02

0.022

0.024

Iterations

∆(f

)

NASA NLF(1)-1015: Convergence of Fitness Function

Wagner Upper

Hicks-Henne Upper

Wagner Lower

Hicks-Henne Lower

c) NASA NLF(1)-1015: Evaluation of ∆ƒƒƒƒ through

Wagner and Hicks-Henne Functions

Figure 6: Analysis of Fitness Function

Convergence

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1.5

-1

-0.5

0

0.5

1

NASA LRN(1)-1007: Comparison of Coefficient of Pressure Distribution

x/c

CP

NASA LRN(1)-1007

Hicks-Henne

Wagner

a) NASA LRN(1)-1007: Comparison of CP

Distribution; Re=6.0e6; Mach=0.32; αααα=0°°°°

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1.5

-1

-0.5

0

0.5

1

NASA LS(1)-0417 Mod: Comparison of Coefficient of Pressure Distribution

x/c

CP

NASA LS(1)-0417 Mod

Hicks-Henne

Wagner

b) NASA LS(1)-0417 Mod: Comparison of CP


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

NASA NLF(1)-1015: Comparison of Coefficient of Pressure Distribution

x/c

CP

NASA NLF(1)-1015

Hicks-Henne

Wagner

c) NASA NLF(1)-1015: Comparison of CP


Figure 7: Comparison of Coefficient of

Pressure Distribution


22nd


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MULTI MISSION RE-CONFIGURABLE UAV – AIRFOIL SHAPE PARAMETERISATION STUDY

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