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

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NATO RTO AVT Outline Objectives and context Shape representation/parameterization options Compressor and turbine airfoil representation Turbine stage representation in 3D flow Summary

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NATO RTO AVT 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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NATO RTO AVT 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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NATO RTO AVT 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 2 nd and 3 rd representations are subsets of NURBS)

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NATO RTO AVT Global – low fidelity - representation Turbine airfoil is represented by 5 Conic sections

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NATO RTO AVT Global – low fidelity - model E/TU-4

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NATO RTO AVT Global/Local high fidelity rep., NURBS –CNURBS curve –P i Control points –w i Weights –N i,p Basis function –p degree of polynomial, (p=2 in this work) –UKnot vector

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NATO RTO AVT Examples of C 2 Continuity Curves DFVLR VKI ETU-4

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NATO RTO AVT Shape optimization methodology –Shape representation –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 –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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NATO RTO AVT Compressor airfoils in 2D flow Inviscid transonic caseviscous subsonic cases

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NATO RTO AVT 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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NATO RTO AVT NACA Transonic compressor redesign Performance map shows ~ 1.7% Original and redesigned compressor airfoils

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NATO RTO AVT NACA 65 subsonic compressor redesign Performance map shows ~ 7% Range of airfoil profiles explored in the design space Original and redesigned compressor airfoils

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

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NATO RTO AVT Airfoil shape: global representation, MRATD MRATD model: Feature-based representation. By construction, it eliminates infeasible turbine airfoil shapes

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NATO RTO AVT Global-Shape Aerodynamic Optimization Objectives –Improve efficiency –Maintain or increase pressure ratio Constraints: Keep the same operating point –Same rotor speed, inlet P t, T t, and exit P s (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 1.Number of blades = 30 2.Radius = m 3.Axial chord C = m 4.Tangential chord = 78.19% 5.Throat = 33.54% 6.Unguided turning = 12 o 7.TE radius = 0.55% 8.Inlet metal angle = 39.4 o 9.Exit metal angle = o 10.SS Inlet wedge angle =15 o 11.PS Inlet wedge angle = 30 o 12.PS Outlet wedge angle =2.5 o 13.Maximum thickness = 26.86% 14.Axial location of maximum thickness = 35% 15.LE ellipse major diameter = 12.61% 16.LE ellipse minor diameter =5.04%

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NATO RTO AVT Global-Shape Optimal profile (MRATD) 6 design variables = 0.4% Same pressure ratio, reduced mass flow rate and flow angles

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NATO RTO AVT Original vs. Optimal MRATD parameters MRATD Design parameters OriginalOptimal Tangential chord0.031m m Throat m m Unguided turning12°9.95° SS inlet wedge angle15°14.83° Maximum thickness m0.0122m LE ellipse minor diameter0.002m m

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NATO RTO AVT 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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NATO RTO AVT 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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NATO RTO AVT 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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NATO RTO AVT 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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NATO RTO AVT The Stacking Curve (or line)

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NATO RTO AVT Quadratic Rational Bezier Curve (QRBC)

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NATO RTO AVT QRBC as Stacking Curve

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NATO RTO AVT Leaning the Stacking Curve Circumferential Direction

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NATO RTO AVT Sweeping the Stacking Curve Axial Direction

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NATO RTO AVT Bowing the Stacking Curve Circumferential Direction

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NATO RTO AVT Circumferential Plane Meridional Plane Design Variables

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NATO RTO AVT Design VariableQRBC ParameterSymbol 1. Sweep angleAxial coordinate of P 2 2. Lean angleCircumferential coordinate of P 2 3. Bowing shape in radial direction Radial coordinate of P 1 4. Bowing shape in circumferential direction Circumferential coordinate of P 1 5. Bowing intensityWeight of P 1 w1w1 Design Variables

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NATO RTO AVT Stator solidity 1.56 Aspect ratio 0.57 Rotor solidity 1.5 Aspect ratio Single Stage Turbine (E/TU3) Low speed subsonic turbine 7800 (rpm) Flow coefficient 0.74 Stage loading 1.93 Stage P.R. = 2 Reaction 31% Re av = 2 Millions

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NATO RTO AVT Stage Optimization = 1.2% with 5 design variables Stator Rotor s o r o wrwr tt Min Max Original Optimum

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NATO RTO AVT 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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NATO RTO AVT Thank You

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