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.



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-



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



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



becomes minimisation of ∆ƒmin between the two geometrical sets.

Position Chordwise)/(

and Airfoil; edApproximat

Airfoil;Target :

))/()/((

approx.

target

.arg

=

=

=

−∑=∆

i

iapproxiett

cx

f

fwhere

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 xxf ai

βπ= (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

a

a

β

β

)ln()5.0ln(

ii p=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()( 2

1θ

πθθ

xf (Eqn 5)

1kfor ])1sin[()sin()( >−

+=π

θπθ k

kkxf k

(Eqn 6)

Where: )(sin2 1 x−=θ

(Eqn 7)



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

1Hicks-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

0.5Wagner 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 K∆ The yi ordinates are a function of the design variables λi, and can be represented by ].,,[ 72,1 λλλ K=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 UpperLower

717171 −−− ≤≤ 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 Hwhere;x

:is solution x optimal thewhere1*

*

==

−= − 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 pand Length; StepScalar Positivea

Iteration;Current

k

k

===kx

where

A descent direction is required to guarantee a decreasing objective function in equation 11 [17]:

)(.1kkk 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



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.



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.



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

0.2Hicks-Henne: NASA LRN(1)-1007 Parameterisation

x/c

y/c

NASA LRN(1)-1007Hicks-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 ModHicks-Henne Approximation

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)-1015Hicks-Henne Approximation

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)-1007Wagner 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 ModWagner Functions Approximation

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)-1015Wagner Functions Approximation

f) Wagner Parameterisation of NASA NLF(1)-

1015 Figure 5: Airfoil Geometry Parameterisation Comparison through Hicks-Henne & Wagner

Functions



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 UpperHicks-Henne UpperWagner LowerHicks-Henne Lower

a) NASA LRN(1)-1007: Evaluation of ∆ƒ

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


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


c) NASA NLF(1)-1015: Evaluation of ∆ƒ 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)-1007Hicks-HenneWagner

a) NASA LRN(1)-1007: Comparison of CP Distribution; Re=6.0e6; Mach=0.32; α=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 ModHicks-HenneWagner

b) NASA LS(1)-0417 Mod: Comparison of CP Distribution; Re=6.0e6; Mach=0.32; α=2°

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)-1015Hicks-HenneWagner

c) NASA NLF(1)-1015: Comparison of CP Distribution; Re=6.0e6; Mach=0.32; α=0° Figure 7: Comparison of Coefficient of

Pressure Distribution



References [1] Sinha, A. K., "A Systems Approach to

the Issues and Challenges of UAV Systems," 1-5, 2001.

[2] Wong, D. K. C., "Aerospace Industry Opportunities in Australia - Unmanned Aerial Vehicles (UAVs) - Are They Ready This Time? Are We?," 13, 1997.

[3] McGhee, R. J. & Beasley, W. D., 1981, "Wind-Tunnel Results for a Modified 17-Percent-Thick Low-Speed Airfoil Section," NASA, Virginia, NASA Technical Paper

[4] Khurana, M., Sinha, A. & Winarto, H., "Multi-Mission Re-Configurable UAV - Airfoil Analysis through Shape Transformation and Computational Fluid Dynamics," Twelfth Australian International Aerospace Congress - Second Australasian Unmanned Air Vehicles Conference, 1-22, 2007.

[5] Samareh, J. A., "A Survey of Shape Parameterization Techniques," CEAS/AIAA/ICASE/NASA Langely International Forum on Aeroelasticity and Structural Dynamics, 333-343, 1999.

[6] Evangelista, R., et al., "Design and Wind Tunnel Test of a High Performance Low Reynolds Number Airfoil," AIAA Applied Aerodynamics Conference, pp. 175-185, 1987.

[7] Somer, D. M., 1981, "Design and Experimental Results for a Natural-Laminar-Flow Airfoil for General Aviation Applications," NASA Langley Research Centre, Hampton, Virginia

[8] Namgoog, H., 2005, 'Airfoil Optimization of Morphing Aircraft', PhD thesis, Purdue, Indiana.

[9] Gallart, M. S., 2002, 'Development of a Design Tool for Aerodynamic Shape Optimization of Airfoils', Master of Applied Science thesis, University of Victoria.

[10] Fuhrmann, H., "Design Optimisation of a Class of Low Reynolds, High Mach Number Airfoils For Use in the Martian Atmosphere," 23rd AIAA Applied Aerodynamics Conference, 2005.

[11] Hager, J. O., Eyi, S. & Lee, K. D., "Two-Point Transonic Design Using Optimization for Improved Off-Design

Performance," Journal of Aircraft, vol. 31, No. 5, pp. pp 1143-1147, 1994.

[12] Hua, J., et al., "Optimization of Long-Endurance Airfoils," 21st AIAA Applied Aerodynamics Conference, pp 1-7, 2003.

[13] Barrett, T. R., Bressloff, N. W. & Keane, A. J., "Airfoil Shape Design and Optimization Using Multifidelity Analysis and Embedded Inverse Design," AIAA Journal, vol. 44, No. 9, pp. 2051-2060, 2006.

[14] Painchaud-Ouellet, S., et al., "Airfoil Shape Optimization Using a Nonuniform Rational B-Splines Parameterization Under Thickness Constraint," AIAA Journal, vol. 44, No. 10, pp. 2170-2178, 2006.

[15] Hicks, R. M. & Henne, P. A., "Wing Design By Numerical Optimisation," AIAA Aircraft Systems & Technology Meeting, pp 8, 1977.

[16] Ramamoorthy, P., Dwarakanath, G. S. & Narayana, C. L., 1969, "Wagner Functions," National Aeronautical Laboratory, Bangalore

[17] Mathworks, "MATLAB," 7.0.4.365 (R14) Service Pack 2 ed. Natick, Massachusetts, 2005.

multi mission re-configurable uav – airfoil shape parameterisation study

Documents