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186 René A. Van den Braembussche Fig. 6.32 Blade lean versus stress and efficiency for database and optimization geometries The blade lean is positive in the direction of rotation (Fig. 6.7). It results from the integration of the β distribution at hub and shroud when the trailing edge rake is limited to 45.0 ◦ . It is often limited to a small arbitrary value for stress reasons. Iteration 49 on Table 6.5 shows that it can be as low as −15.0 ◦ without exceeding the maximum stress limit. Figure 6.32 confirms that minimum stresses are observed around −15.0 ◦ . Several geometries with good efficiency are found for lean angles between −40.0 ◦ to −5.0 ◦ . The decreasing efficiency for lean angles larger than −5.0 ◦ suggests that a limited negative lean may have a favorable effect on performance. Figure 6.33 shows the impact of the leading edge blade height on the stress and efficiency. The radius at the LE shroud (see Fig. 6.22) can vary between 6.5and7.5 mm, resulting in a blade height of 4.25 and 5.25 mm, respectively. Values at 4.5 mmand5.0 mm are database samples. Shortening the blades lowers the stresses but the database samples suggest a small drop in efficiency. This explains the difficulty in maintaining a high efficiency when reducing the stress. However, the optimized geometries have shorter vanes and show a high efficiency. This indicates that the efficiency also
6 Numerical Optimization for Advanced Turbomachinery Design 187 Fig. 6.33 Blade leading edge height versus stress and efficiency depends on an optimum choice of other parameters. Although they have a less pronounced influence on stress and efficiency, a correct definition of their value is needed to reach the optimum. This illustrates the strongly coupled nature of the design problem and the need for an optimization tool. 6.7 Conclusions It has been shown how a two-level optimization technique, an adequate parameter selection for the GA, the use of DOE for the definition of the database and an optimized learning technique for the ANN can considerably decrease the computational effort required by evolutionary theories. The proposed procedure is a self-learning system that makes full use of the expertise gained during previous designs. The automated design method can be used with any flow solver and does not require the definition of a target pressure or Mach number distribution.
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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
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118 Nicolas R. Gauger CL := 1 � C
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120 Nicolas R. Gauger the cost func
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122 Nicolas R. Gauger Table 5.1 An
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124 Nicolas R. Gauger The main adva
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126 Nicolas R. Gauger Spalart-Allma
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128 Nicolas R. Gauger (a) (b) Fig.
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134 Nicolas R. Gauger Fig. 5.10 Plo
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Page 206 and 207: 192 Laurent Dumas 7.1 Introducing A
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8 Multi-objective Optimization in C
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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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