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256 Marco Manzan, Enrico Nobile, Stefano Pieri and Francesco Pinto Channel (a) f =0.417 Nu = 9.90 Channel (b) f =0.427 Nu = 9.96 Channel (c) f =0.436 Nu = 10.04 Fig. 8.18 Selected NURBS-based channels: fluid-dynamic (left) and thermal (right) fields. These three channels correspond to designs marked in Fig. 8.14; see also Fig. 8.19 8.8.4 Three-dimensional Analysis This part of the work is a proof-of-concept in 3D applications. A true optimization has not been performed on NURBS channels but, due to time limitation and computing resources available, a parametric analysis has been done in order to verify the applicability of the method to 3D problems. For the same reasons, the Reynolds number has been reduced from 200 to 100, in order to guarantee an adequate level of numerical accuracy. Therefore, the 2D Pareto fronts have been recomputed at this reduced value of the Reynolds number, in order to compare the performances of the 2D channels with the 3D ones. These have been obtained, as indicatedinFig.8.2,byextrusion, at different angles, of selected 2D channels. In Figs. 8.21(a) and 8.21(b), the results obtained are presented. In particular, the results for the simpler channels with linear piecewise walls are shown in Fig. 8.21(a), while the objective functions for the NURBS-based channels are shown in Fig. 8.21(b). From the figures it is evident that, for both type of channels, the Nusselt number in
8 Multi-objective Optimization in Convective Heat Transfer 257 Channel (d) f =0.539 Nu = 10.87 Channel (e) f =0.577 Nu = 10.92 Channel (f) f =0.587 Nu = 11.13 Fig. 8.19 Selected NURBS-based channels: fluid-dynamic (left) and thermal (right) fields. These three channels correspond to designs marked in Fig. 8.14; see also Fig. 8.18 Table 8.6 Variation of friction factor and Nusselt number versus extrusion angle for one design f/f0 Nu/Nu0 Extrusion angle 3.807 1.307 0 ◦ 3.932 1.366 20 ◦ 3.937 1.508 30 ◦ 3.909 1.530 40 ◦ general tends to increase, without a sensible increment in the friction factor, for the 3D channels when the extrusion gives rise to a 3D flow pattern. This trend is due to the presence of secondary motions that enhance the heat transfer at the walls. In Table 8.6, the results for one of the NURBS profile extrusions are reported, and the positive effect of secondary motions
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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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136 Nicolas R. Gauger drag, lift, s
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138 Nicolas R. Gauger solution of t
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140 Nicolas R. Gauger the presence
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142 Nicolas R. Gauger Fig. 5.16 Con
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144 Nicolas R. Gauger optimization
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Chapter 6 Numerical Optimization fo
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6 Numerical Optimization for Advanc
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192 Laurent Dumas 7.1 Introducing A
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194 Laurent Dumas C Fig. 7.2 Wake f
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196 Laurent Dumas Table 7.1 Example
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198 Laurent Dumas Fig. 7.5 Numerica
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200 Laurent Dumas 7.3.1.1 Genetic A
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202 Laurent Dumas START Random init
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204 Laurent Dumas • if ˜ J(x ng
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