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168 René A. Van den Braembussche Suction side b Tss(1) Tss(0)= 9 Rle * b 10 (a) Tss(2) Camber 8 Camber line Rte Tss(8) Suction side Fig. 6.14 Geometry model: (a) detailed view of the leading edge (b) and trailing edge Fig. 6.15 Typical turbine blade sections generated by a 17-parameter model 6.4.2 Penalty for Non-optimum Mach Number Distribution PMach The characteristics of the optimum Mach number M distribution for 2D turbine blades are well known and allow the definition of penalties for a non optimum Mach number distribution. These penalties can also be derived from experimental correlations. (b) δ te
6 Numerical Optimization for Advanced Turbomachinery Design 169 Mis M2 M1 M2 M1 .. . . . . . . . . M11 M10 M9 M9 M10 M11 . . . . . . . . . M19 M20 M20 M19 S Smax Fig. 6.16 Parametric representation of the Mach number distribution The OF used in a turbine blade optimization [15] increases when there is a high probability of early transition, laminar or turbulent separation or poor off-design performances. It is a sum of penalties based on the Mach number in 20 points on each side of the blade (Fig. 6.16). The values may be defined from an NS calculation or predicted by the ANN. • Penalty on the local slope of the Mach number distribution: a minimum positive slope of the Mach number distribution is required on the front part of the suction side in order to avoid deterioration of the performance at off-design incidence angle. • Penalty on the second derivative of the Mach number distribution: the optimization process may result in a wavy Mach number distribution because of local changes in curvature radius of the blade surface. This may have a small impact on blade losses if the boundary layer is already turbulent but can deteriorate the off-design performances of the blade. One therefore penalizes the Mach number distribution for which the second derivative changes sign, i.e., with an inflection point in the suction side Mach number distribution. • Penalty on the deceleration: it is important to limit the deceleration on the front part of the blade pressure side in order to avoid separation at negative incidence angles. This penalty is proportional to the difference between the first maximum Mach number found on the pressure side (starting from the stagnation point) and the minimum Mach number along the pressure side. It is also well known that the deceleration on the second half of the suction side has an important influence on the losses and in case of low Reynolds number may lead to flow separation.
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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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116 Nicolas R. Gauger and the four
Page 132 and 133: 118 Nicolas R. Gauger CL := 1 � C
Page 134 and 135: 120 Nicolas R. Gauger the cost func
Page 136 and 137: 122 Nicolas R. Gauger Table 5.1 An
Page 138 and 139: 124 Nicolas R. Gauger The main adva
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Page 150 and 151: 136 Nicolas R. Gauger drag, lift, s
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Page 154 and 155: 140 Nicolas R. Gauger the presence
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Page 206 and 207: 192 Laurent Dumas 7.1 Introducing A
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Page 212 and 213: 198 Laurent Dumas Fig. 7.5 Numerica
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Page 218 and 219: 204 Laurent Dumas • if ˜ J(x ng
Page 220 and 221: 206 Laurent Dumas Table 7.3 Compari
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Page 224 and 225: 210 Laurent Dumas Table 7.4 Four ex
Page 226 and 227: 212 Laurent Dumas Fig. 7.13 Blades
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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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