Airfoil Geometry Parameterization

Airfoil Geometry Parameterization through Shape Optimizer and Computational Fluid

Dynamics

Manas KhuranaThe Sir Lawrence Wackett Aerospace Centre

RMIT University Melbourne - Australia

46th AIAA Aerospace Sciences Meeting and Exhibit7th – 10th January, 2008

Grand Sierra Resort – Reno, Nevada

Presentation Outline

Introduction Role of UAVs Research Motivation & Goals

o Design of MM-UAV o Current Design Status

Direct Numerical Optimization Airfoil Geometry Shape Parameterisation

o Test Methodology & Results Flow Solver

o Selection, Validation & Results Analysis Optimization

o Airfoil Analysis

Summary / Conclusion Questions

www.airliners.net

I-view: www.defense-update.com

Introduction

Multi-Mission UAVs Cost Effective; Designed for Single Missions; Critical Issues and Challenges; Demand to Address a Broader Customer Base; Multi Mission UAV is a Promising Solution; and Provide Greater Mission Effectiveness

Research Motivation & Goals Project Goal - Design of a Multi-Mission UAV; and Research Goal – Intelligent Airfoil Optimisation

o Design Mission Segment Based Airfoilo Morphing Airfoils

Pegasus: www.NorthropGrumman.com

X-45: www.Boeing.com

RMIT University: Preliminary RC-MM-UAV Design Concept

Aerodynamic Optimisation

Design Methodology Direct Numerical Optimisation

o Geometrical Parameterization Model;

and

o Validation of Flow Solver

Coupling of the two Methods

Swarm Intelligence Optimization

Neural Networks DNO Computationally Demanding;

Development of an ANN within DNO;

and

Integrate Optimisation Algorithm within

the ANN Architecture

0 0.2 0.4 0.6 0.8 1-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

Geometric Representation Technique Features

Key Requirements Flexibility and Accuracy; Cover Wide Design Window with Few

Variables; Generate Smooth & Realistic Shapes; Provide Independent Geometry Control; Application of Constraints for Shape

Optimization; and Computationally Efficient

Approaches Discrete Approach; Shape Transformations: Conformal

Mapping; Polynomial Representations; and Shape Functions added to Base-Line Profile

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08Discrete Approach

x/c

y/c

Airfoil Shape Transformations

Conformal Mapping Approach Computationally In-Expensive; Joukowski & Kármán-Trefftz

Transformations; Transformation from Complex to -Plane;

and Five Shape Parameters

xc - Thickness yc - Camber towards leading edge xt - Thickness towards trailing edge yt - Camber towards trailing edge n - Trailing edge angle

Conformal Mapping Restrictions Limited Design Window; Divergent Trailing Edge Airfoils not

possible; and Failure to Capture Optimal Solution

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14Camber Variation

x/c

y/c

2

4

6

8

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-1.5

-1

-0.5

0

0.5

1

1.5Conformal Mapping Approach

Re(s)

Im(s

)

z

z'

Airfoil

Airfoil Shape Functions

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

1Bernstein Polynomials

x

y

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

NACA 0015 Analytic Function

n

iii

AirfoilInitiali

xfxyxy1

)()(),(

Introduction Analytical Approach; Control over Design Variables; Cover Large Design Window; Linearly Added to a Baseline Shape;

Participating Coefficient act as Design Variables (i); and

Optimization Study to Evaluate Parameters

Population & Shape Functions

n

iii

AirfoilInitiali

xfxyxy1

)()(),(

i

Optimization

Shape Function Convergence Criteria

Convergence Measure Requirements Flexibility & Accuracy; and

Library of Target Airfoils

Geometrical Convergence Process Specify Base & Target Airfoil;

Select Shape Function;

Model Upper & Lower Surfaces;

Design Variable Population Size (2:10);

Perturbation of Design Variables;

Record Fitness - Geometrical Difference

of Target and Approximated Section;

Aggregate of Total Fitness; and

Geometrical Fitness vs. Aerodynamic

Performance

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 0015Target 1: NASA LRN(1)-1007Target 2: NASA LS(1)-0417ModTarget 3: NASA NLF(1)-1015

0.58c

0.2c

0.45c

0.010c

Camber Location

0.0220.3c17%Target 2: NASA LS(1)-0417Mod

0.060.4c7%Target 1: NASA LRN(1)-1007

00.3c12%Base: NACA 0012

0.4c

Thickness Location

0.047

Max. Camber

15%Target 3: NASA NLF(1)-1015

t/cAirfoil

0.58c

0.2c

0.45c

0.010c

Camber Location

0.0220.3c17%Target 2: NASA LS(1)-0417Mod

0.060.4c7%Target 1: NASA LRN(1)-1007

00.3c12%Base: NACA 0012

0.4c

Thickness Location

0.047

Max. Camber

15%Target 3: NASA NLF(1)-1015

t/cAirfoil])/()/([.argmin iapproxiett

cxfcxfabsf

Intelligent Search Agent – Particle Swarm Optimization

Swarm Approach Models Natural Flocks and Movement

of Swarms; Quick, Efficient and Simple

Implementation; Ideal for Non-Convex Discontinuous

Problems; Solution Governed by Position of

Particle within N-dimensional Space; Each Particle Records Personal

Fitness – pbest;

Best Global Fitness – gbest;

Velocity & Position Updates based on Global Search Pattern; and

Convergence – Particles Unite at Common Location

J. Kennedy and R. Eberhart, "Particle Swarm Optimization“, presented at IEEE International Conference on Neural Networks, 1995.

Algorithm1. Initialise Particle Swarm2. Initialise Particle Velocities3. Evaluate Fitness of Each Particle4. Update according to:

i. Velocity Updateii. Position Update

5. Repeat until Convergence Satisfied

Algorithm1. Initialise Particle Swarm2. Initialise Particle Velocities3. Evaluate Fitness of Each Particle4. Update according to:

i. Velocity Updateii. Position Update

5. Repeat until Convergence Satisfied

Particle Swarm Optimization Set Up

PSO Structure / Inputs Definition Velocity Update:

Position Update:

SP

SO

o 0.1-10% of NDIM

o c1 = 2

o c2 = 2

0.1-10% of NDIM

‘w’ Facilitates Global Search ‘w’ Facilitates Local Search

Determine ‘pull’ of pbest & gbest

c1 – Personal Experience

c2 – Swarm Experience

A-P

SO

o 0.1-10% of NDIMMaximum Velocity

Inertia Weight (w):

o c1 = 2

o c2 = 2

Scaling Factors Cognitive & Social

(c1 & c2)

;42

2w

2

21 cc where

ijij

ij

bestbest

bestij

gp

pxijISA

ijISAij ew

1

11

Standard vs. Adaptive PSO

kxPrandckxPrandckvwkv igiiii 211

11 kvkxkx iii

Particle Swarm Optimizer Search Agents

Particle Swarm Optimizer - Function Test

1

1

221

2 1)(100)(n

iiii xxxxf

nixi ,...,2,1,100100

0)(),1,...,1( ** xfX

-10-5

05

10

-10

-5

0

5

100

5

10

15

x 105

x

Rosenbrock Function

y

z

3015 ix

Definition:

Search Domain:

Initialization Range:

Global Minima (Fitness):

Velocity Fitness Fitness

Low Velocity = Low Fitness

Particle Swarm Optimizer - Function Test

Definition:

Search Domain:

Initialization Range:

Global Minima (Fitness):

0)(),1,...,1( ** xfX

n

iii xxnxf

1

)sin(9829.418)(

nixi ,...,2,1,500500

500250 ix

Velocity Fitness Fitness

Low Velocity = Low Fitness

Shape Parameterization Results

Summary of Results Measure of Geometrical Difference Hicks-Henne Most Favorable Legendre Polynomials

Computationally Not Viable Aerodynamic Coefficients

Convergence

10

1

2

3

4

5

6

7

8

Shape Function

Co

st

Magnitude of Cost Function

BernsteinHicks-HenneLegendreNACAWagner

Geometrical Convergence Plots / Animations

sHicks-Henne Geometrical

Convergence

s Bernstein Geometrical Convergence

Aerodynamic Convergence Plots / Animations

sHicks-Henne Aerodynamic

Convergence

s Bernstein Aerodynamic Convergence

Shape Functions Limitations

Polynomial Function Limitation Local Shape Information; No Direct Geometry Relationship; NURBS Require Many Control Points; and Lead to Undulating Curves

PARSEC Airfoil Representation 6th Order Polynomial;

Eleven Variables Equations Developed as a Function of

Airfoil Geometry; and Direct Geometry Relationship

H. Sobieczky, “Parametric Airfoil and Wings“, in: Notes on Numerical Fluid Mechanics, Vol. 68, pp. 71-88, 1998

10

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Shape Functions

Fitn

ess

Mag

nitu

de

BernsteinHicks-HennePARSECLegendreNACAWagner

Fitness Magnitude of Shape Functions2

16

1

n

nnPARSEC XaZ

PARSEC Airfoils

PARSEC Aerodynamic Convergence Convergence to Target Lift Curve Slope Convergence to Target Drag Polar

Convergence to Target Moment Convergence to Target L/D

-5 0 5 10 15 200.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

( )

CL

TargetHicks-HennePARSEC

0 0.02 0.04 0.06 0.08 0.1 0.120.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

CD

CL


-5 0 5 10 15 20-0.11

-0.1

-0.09

-0.08

-0.07

-0.06

-0.05

-0.04

-0.03

( )

CM


-5 0 5 10 15 200

50

100

150

( )

L/D


PARSEC Design Variables Definition

Effect of YUP on PARSEC Airfoil Aerodynamics Lift Coefficient Drag Coefficient Moment Coefficient Lift-to-Drag Ratio

Effect of YUP on PARSEC Airfoil GeometryYUP Nose Radiust/c Camber

Low YUP = Good CD Performance

Shape Function Modifications Airfoil Surface Bumps

Aerodynamic Performance Improvements; Rough Airfoils Outperform Smooth Sections at Low Re; Control Flow Separation; Passive & Active Methods for Bypass Transition; Reduction in Turbulence Intensity; and Bumps Delay Separation Point

Shape Functions - Further Developments Local Curvature Control; Roughness in Line with Boundary Layer Height; and Control over Non-Linear Flow Features

Airfoil Surface Bumps to Assist Flow Reattachment

Source: A. Santhanakrishnan and J. Jacob, “Effect of Regular Surface

Perturbations on Flow Over an Airfoil”, - University of Kentucky, AIAA-2005-5145

Ideal Surface

Bumpy Surface

Flow Solver – Computational Fluid Dynamics

Laminar Turbulent

6,000Maximum Iteration Count

1.0 x 10-6Residual Solution Convergence

0.32Flow Mach Number

Turbulence Intensity = 0.5%; Viscosity Ratio = 5

Turbulence Intensity = 2%; Viscosity Ratio = 20

Boundary Conditions:InletPressure Outlet

Air as an Ideal GasFlow Medium

6.0 x 106Reynolds Number

- & SA Turbulence ModelingViscous Model

Second Order UpwindDiscretization Scheme

1.055Wall Cell Intervals

96,000Total Mesh Size (approx.)

Segregated Implicit Formulation of RANS

Energy Equations also Solved

Solver

1Wall y+ Range (approx.)

80Circumferential Lines

100Radial Lines

2D Structured C-TypeMesh

6,000Maximum Iteration Count

1.0 x 10-6Residual Solution Convergence

0.32Flow Mach Number

Turbulence Intensity = 0.5%; Viscosity Ratio = 5

Turbulence Intensity = 2%; Viscosity Ratio = 20

Boundary Conditions:InletPressure Outlet

Air as an Ideal GasFlow Medium

6.0 x 106Reynolds Number

- & SA Turbulence ModelingViscous Model

Second Order UpwindDiscretization Scheme

1.055Wall Cell Intervals

96,000Total Mesh Size (approx.)

Segregated Implicit Formulation of RANS

Energy Equations also Solved

Solver

1Wall y+ Range (approx.)

80Circumferential Lines

100Radial Lines

2D Structured C-TypeMesh

Flow Solver Validation – Case 1: NASA LS(1)0417 Mod

-5 0 5 10 15 20-0.5

0

0.5

1

1.5

2

2.5

3Fixed Boundary Layer Transition: Lift Curve Slope

( )

CL

Exp

CFD

0.008 0.01 0.012 0.014 0.016 0.018 0.020.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2Fixed Boundary Layer Transition: Drag Polar

CD

CL

ExpCFD

Validation Data

CP Agreement at AOA 10;

Lift & Drag Convergence over Linear

AOA;

Lift 2% ; Drag 5%;

Solution Divergence at Stall; and

Fluid Separation Zone Effectively

Captures Boundary Layer Transition0 0.1 0.2 0.3 0.4 0.5 0.6

-4

-3

-2

-1

0

1

2

Fixed Transition CP Distribution Comparison: Re=6.0e6, Mach=0.32

x/c

CP

Exp

CFD

Flow Solver Validation – Case 2: NACA 0012

Validation Data

CP Agreement at AOA 11;

Lift & Drag Convergence over Linear

AOA;

Lift 5% ; Drag 7%;

Solution Divergence at Stall; and

Fluid Separation Zone Effectively

Captures Boundary Layer Transition

0 2 4 6 8 10 12 14-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6NACA 0012 - Fixed Boundary Layer Transition Lift Curve Slope

( )

CL

Exp.CFD

0 0.1 0.2 0.3 0.4 0.5 0.6

-6

-5

-4

-3

-2

-1

0

1

2

x/c

CP

NACA 0012 - Fixed Boundary Layer Transition CP Distribution: Re = 6.0e6, Mach 0.35

Exp.CFD

0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.022-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6NACA 0012 - Fixed Boundary Layer Transition Drag Polar

CD

CL

Exp.

CFD

Sample Optimization Run

Objective Function = 2 CL = 0.40 Minimize CD

Optimizer Inputs Final Solution Swarm Size = 20 Particles rLE = [0.001 , 0.04] 0.0368 YTE = [-0.02 , 0.02] 0.0127 Teg = [-2.0 , -25] -19.5 TEW = [3.0 , 40.0] 29.10 XUP = [0.30 , 0.60] 0.4581 YUP = [0.07 , 0.12] 0.0926 YXXU = [-1.0 , 0.2] -0.2791 XL = [0.20 , 0.60] 0.5120 YL = [-0.12 , -0.07] -0.1083 YXXL = [0.2 , 1.20] 0.6949

Results t/c = 20% CL = 0.4057 CD = 0.0069 Total Iterations = 29

0 5 10 15 20 25 300

0.002

0.004

0.006

0.008

0.01

0.012

0.014Optimization History Plot

Optimization History Plot

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25Airfoil Optimization

x/c

y/c

Final Airfoil Shape

Aerodynamic Coefficient Database – Artificial Neural Networks

Artificial Neural Networks – Airfoil Training Database Geometrical Inputs;

Aerodynamic Coefficient/s Output/s; Set-up of Transfer Function within the Hidden Layer; and Output RMS Evaluation

Coefficient of Lift NN Structure Coefficient of Drag NN Structure Coefficient of Moment NN Structure

R. Greenman and K. Roth “Minimizing Computational Data Requirements for Multi-Element Airfoils

Using Neural Networks“, in: Journal of Aircraft, Vol. 36, No. 5, pp. 777-784 September-October 1999

Coupling of ANN & Swarm Algorithm

Conclusion

Geometry Parameterisation Method Six Shape Functions Tested;

Particle Swarm Optimizer Validated / Utilized;

SOMs for Design Variable Definition; and

PARSEC Method for Shape Representation

Flow Solver RANS Solver with Structured C-Grid;

Transition Points Integrated;

Acceptable Solution Agreement; and

Transition Modeling and DES for High-Lift

Flows

Airfoil Optimization Direct PSO Computationally Demanding; and

ANN to Reduce Computational Data

www.cosmosmagazine.comwww.mathworks.com

Acknowledgements

Viscovery Software GmbH [http://www.viscovery.net/]

Mr. Bernhard Kuchinka

Kindly provided a trial copy of Viscovery SOMine

Airfoil Geometry Parameterization

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