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166 René A. Van den Braembussche Only 2-level designs (every variable can take two values) and one centralpoint run (every value is at the center of the allowed range) are considered for the database used in following examples. The high and low value of each design variable are located at 75% and 25% of the parameter range, respectively. The central point is the mid value (50%). The range is defined by the designer based on his experience about feasible geometries and mechanical constraints. 6.4 Single Point Optimization of Turbine Blade The optimizing design procedure has been successfully tested on a large number of designs and will be illustrated here by the design of a 2D highly loaded axial turbine blade section. 6.4.1 2D Blade Geometry Definition The geometry definition, described here, makes use of Bézier curves to define the camber line and the blade suction and pressure side relative to the camber line. The camber line is defined by 3 control points (Fig. 6.13.a). The first one coincides with the leading edge. The second one is at the trailing edge. Its position relative to the leading edge is defined by the stagger angle (γ) and axial chord length Lref. A third control point is located at the intersection between the tangent to the camber at leading and trailing edge. It is defined by the angles β1m and β2m with the axial direction. The suction side is defined by a Bézier curve starting at the leading edge and ending at the tangent to a circle of radius Rte, defining the trailing edge thickness. The camber line is divided in a number of intervals which can be of equal or variable length using a stretching factor (Fig. 6.13.b). The lengths Tss(1), Tss(2), Tss(3), Tss(4), Tss(5), Tss(6) and Tss(7) measured in the direction perpendicular to the camber line determine the position of the 7 Bézier control points of the suction side. The first Bézier control point coincides with the leading edge (start of the camber line) (Fig. 6.14.a) and the third point is the one defined by the parameter Tss(1). The second Bézier control point is defined by imposing that the suction side starts perpendicularly to the camber line at the leading edge and that the suction side curvature radius at the leading edge equals Rle. This condition defines the length Tss(0). δte is the wedge angle between suction and pressure side at the trailing edge (Fig. 6.14.b). A Bézier control point is defined as the intersection of the
6 Numerical Optimization for Advanced Turbomachinery Design 167 Tss(3) Tss(2) Tss(1) Tss(0) + β1m γ Lref Tss(4) Tss(5) Bezier control points Bezier curve(=camber line) (a) Camber line (b) β 2m Tss(6) Bezier polygon Tss(7) Suction side Tss(8) Fig. 6.13 Geometry model: (a) definition of the camber line and (b) the suction side tangent at the trailing edge circle and the line perpendicular to the camber line at point 8. The pressure side is defined in a way similar to the suction side. Imposing the same Rle as on the suction side assures a continuous curvature at the leading edge. Figure 6.15 demonstrates the capabilities of the method to represent the large variety of blades encountered in axial turbines.
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Optimization and Computational Flui
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Editors: Prof. Dr.-Ing. Dominique T
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vi Preface Our first research proje
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viii Contents 2.4.1 GoverningEquati
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x Contents 6.2.3 Parameterization..
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xii Contents 9.4.1 Multi-objectiveO
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xiv List of Contributors Gábor JAN
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Part I Generalities and methods
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4 Dominique Thévenin 12 10 8 6 4 2
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6 Dominique Thévenin Objective 10
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8 Dominique Thévenin Objective 10
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10 Dominique Thévenin Parameter 2
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12 Dominique Thévenin three-dimens
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14 Dominique Thévenin Let us come
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16 Dominique Thévenin References 1
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18 Gábor Janiga 2.1 Introduction D
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20 Gábor Janiga y Tinlet = 293 K 0
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22 Gábor Janiga y [mm] 25 20 15 10
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24 Gábor Janiga Table 2.1 Number o
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26 Gábor Janiga A dominates the in
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28 Gábor Janiga be shown later, th
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30 Gábor Janiga INPUT FILE Gambit
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34 Gábor Janiga ∆T ∆P Position
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36 Gábor Janiga ∆P [mPa] 2.2 2.0
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38 Gábor Janiga Table 2.3 Speed-up
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40 Gábor Janiga For all computatio
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42 Gábor Janiga y [mm] 4 2 0 -1 0
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44 Gábor Janiga Air mass flow-rate
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46 Gábor Janiga Temperature variat
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50 Gábor Janiga The modified k-ω
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52 Gábor Janiga Error for Re∗ =
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54 Gábor Janiga Error for Re∗ 20
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56 Gábor Janiga References 1. Ali,
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58 Gábor Janiga 43. Janiga, G., Th
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Chapter 3 Mathematical Aspects of C
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3 Mathematical Aspects of CFD-based
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Chapter 4 Adjoint Methods for Shape
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4 Adjoint Methods for Shape Optimiz
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Part II Specific Applications of CF
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112 Nicolas R. Gauger Nomenclature
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114 Nicolas R. Gauger Fig. 5.1 Cost
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Page 206 and 207: 192 Laurent Dumas 7.1 Introducing A
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Page 226 and 227: 212 Laurent Dumas Fig. 7.13 Blades
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Chapter 8 Multi-objective Optimizat
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8 Multi-objective Optimization in C
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8 Multi-objective Optimizatio Fig.
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Chapter 9 CFD-based Optimization fo
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9 CFD-based optimization for paperm
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292 Index heat exchanger, 222, 224,
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