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**Wahid Ghaly Mechanical and Industrial Engineering**

Shape Representation Wahid Ghaly Mechanical and Industrial Engineering NATO RTO AVT-167 Lecture Series October 26-27, 2009 Montreal, Canada

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**Outline Objectives and context**

Shape representation/parameterization options Compressor and turbine airfoil representation Turbine stage representation in 3D flow Summary

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Shape Representation Accurate, flexible and robust shape representation Most suitable representation for a given shape Least number of shape parameters that are directly related to the design parameters and are used as optimization variables Preferably a CAD-native parameterization Can the geometric representation make the optimization approach more efficient? Can it reduce the design problem complexity?

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**Intended applications**

Component level optimization, e.g. turbine or compressor Single and multiple blade rows, disciplines, objectives, single and multipoint Airfoils (2D) and blades (3D) profiles Global - low fidelity - representation Local/Global high fidelity representation

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**Low and high fidelity representations**

Global - low fidelity – representation Shape is represented by a few low order polynomials Change in any point on the curve affects the shape globally Local/Global high fidelity representation Shape is represented by a continuous curve with e.g. NURBS, B-splines, Bezier curves, …(Note that the 2nd and 3rd representations are subsets of NURBS)

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**Global – low fidelity - representation**

Turbine airfoil is represented by 5 Conic sections

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**Global – low fidelity - model**

E/TU-4

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**Global/Local high fidelity rep., NURBS**

C NURBS curve Pi Control points wi Weights Ni,p Basis function p degree of polynomial, (p=2 in this work) U Knot vector

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**Examples of C2 Continuity Curves**

DFVLR ETU-4 VKI

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**Shape optimization methodology**

Shape representation: Low order - global - representation High order representation, e.g. NURBS, B-Splines, Bezier Optimization method: Direct: GA, SA Indirect: Gradient/Newton-based, Control Theory-based Choice and computation of objective function: High fidelity simulations (CFD solver of your choice) Low fidelity using a surrogate model (ANN, RBF, wavelets)

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**Compressor airfoils in 2D flow**

Inviscid transonic case viscous subsonic cases

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**Geometric description and parameterization**

The airfoil shape is described by a camber line and a thickness distribution Camber line overall flow turning Thickness structural constraints They are parameterized using a high fidelity NURBS function with 11 control points for camber line, f(x), and 9 for thickness distribution, T(x). Y-coordinates of the control points are used as the design variables (17 points)

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**NACA Transonic compressor redesign**

Performance map shows Dh ~ 1.7% Original and redesigned compressor airfoils

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**NACA 65 subsonic compressor redesign**

Range of airfoil profiles explored in the design space Performance map shows Dh ~ 7% Original and redesigned compressor airfoils

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**A turbine airfoil profile in 2D flow**

Optimization is done successively on two geometric parameterizations: Starting from a global shape representation of the airfoil using the design parameters, optimization is carried out The resulting profile is used as input to a high fidelity shape representation so as to refine the profile locally

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**The original turbine airfoil**

Total pressure loss coeff. = % Adiabatic efficiency = % Pressure ratio (inlet/outlet) =1.518 Inlet flow angle = 57.4o Exit flow angle = o Corrected mass flow rate = 0.191 Note that this is a low subsonic turbine airfoil with over 91% adiabatic efficiency

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**Airfoil shape: global representation, MRATD**

MRATD model: Feature-based representation. By construction, it eliminates infeasible turbine airfoil shapes

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**Global-Shape Aerodynamic Optimization**

Objectives Improve efficiency Maintain or increase pressure ratio Constraints: Keep the same operating point Same rotor speed, inlet Pt, Tt, and exit Ps (CFD) Fixed corrected mass flow rate and flow angles (penalty terms added to the objective function) Design variables All parameters affecting the airfoil SS (6 in all) Original airfoil: ETU turbine profile MRATD (design) parameters Number of blades = 30 Radius = m Axial chord C = m Tangential chord = 78.19% Throat = 33.54% Unguided turning = 12o TE radius = 0.55% Inlet metal angle = 39.4o Exit metal angle = -66.0o SS Inlet wedge angle =15o PS Inlet wedge angle = 30o PS Outlet wedge angle =2.5o Maximum thickness = 26.86% Axial location of maximum thickness = 35% LE ellipse major diameter = 12.61% LE ellipse minor diameter =5.04%

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**Global-Shape Optimal profile (MRATD)**

6 design variables Dh = 0.4% Same pressure ratio, reduced mass flow rate and flow angles

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**Original vs. Optimal MRATD parameters**

MRATD Design parameters Original Optimal Tangential chord 0.031m m Throat m m Unguided turning 12° 9.95° SS inlet wedge angle 15° 14.83° Maximum thickness m 0.0122m LE ellipse minor diameter 0.002m m

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**Airfoil shape: local refinement, NURBS**

A close look at the curvature and pressure distributions helps to pinpoint regions where improvements can be made.

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**NURBS optimal vs. MRATD optimal profile**

Efficiency improved by an additional 0.165%, for the same pressure ratio, reduced mass flow rate and flow angles, using 6 NURBS control points.

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**Turbine blade profiles in 3D flow**

Geometry representation: 2D Airfoils: MRATD, B-splines and NURBS Hub-to-tip: stacking line going through the 2D airfoils 3D blade shape: obtained by skinning the stacked 2D airfoils, using compatible B-splines

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**CATIA-CFD integration**

NURBS and B-splines are CAD-native parameterizations can be directly integrated into the CAD system All blade features are extracted and updated into solid model during the optimization process using: CAD neutral packages, e.g. CARPI from MIT CATIA Application Program Interface (API) Note: MRATD can be integrated into CAD using e.g. CATIA-API

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**The Stacking Curve (or line)**

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**Quadratic Rational Bezier Curve (QRBC)**

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QRBC as Stacking Curve 27

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**Leaning the Stacking Curve**

Circumferential Direction 28

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**Sweeping the Stacking Curve**

Axial Direction 29

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**Bowing the Stacking Curve**

Circumferential Direction 30

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Design Variables Meridional Plane Circumferential Plane 31

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**Design Variables Design Variable QRBC Parameter Symbol 1. Sweep angle**

Axial coordinate of P2 b 2. Lean angle Circumferential coordinate of P2 a 3. Bowing shape in radial direction Radial coordinate of P1 g 4. Bowing shape in circumferential direction Circumferential coordinate of P1 q 5. Bowing intensity Weight of P1 w1 32

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**Single Stage Turbine (E/TU3)**

Low speed subsonic turbine (rpm) Flow coefficient 0.74 Stage loading 1.93 Stage P.R. = 2 Reaction 31% Reav = 2 Millions Stator solidity 1.56 Aspect ratio 0.57 Rotor solidity 1.5 Aspect ratio 0.904 33

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**Dh = 1.2% with 5 design variables**

Stage Optimization aso bso a ro bro wr htt Min. -30 -15 -5 -10 - Max. 10 5 20 15 3 Original 87.50 Optimum -29.5 -9.4 2.2 -9.7 0.05 88.56 Dh = 1.2% with 5 design variables Stator Rotor 34

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Summary Geometric representation can improve the efficiency of the optimization approach It can also reduce the design problem complexity by: reducing the number of design variables Eliminating infeasible blade profiles It is critical to pick the ‘right’ representation and the ‘right’ parameterization for a given shape

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Thank You

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