airfoil geometry parameterization
DESCRIPTION
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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
TargetHicks-HennePARSEC
-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
TargetHicks-HennePARSEC
-5 0 5 10 15 200
50
100
150
( )
L/D
TargetHicks-HennePARSEC
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
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Acknowledgements
Viscovery Software GmbH [http://www.viscovery.net/]
Mr. Bernhard Kuchinka
Kindly provided a trial copy of Viscovery SOMine