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276 J. Hämäläinen, T. Hämäläinen, E. Madetoja, H. Ruotsalainen z, CD Fig. 9.7 The coordinate frame of the full 3D model y x, MD 9.3.4 Validation of the Depth-averaged Navier-Stokes Equations The reduced model, that is, the depth-averaged Navier-Stokes equations, has been validated by comparing the results to a complete CFD model which includes the headbox slice channel and its free jet in 3D. The coordinate frame of the model is shown in the sketch in Fig. 9.7. The x-axis follows the main flow direction, while the z-axis lies in the span-wise direction. The 3D CFD model utilized has been validated using wind tunnel experiments [31]. The fiber orientation angle profile is mostly determined by the CD velocity component, because variations in the MD velocity are only of the order of one per mille. Thus, the CD velocity is the most important component in the validation. The CD velocity profiles at the slice opening for Step 1 are presented in Fig. 9.8 (Step 1 is the diagnostics step determining flow rate profiles inside the headbox which creates the measured fiber orientation profile). The whole CD velocity field is presented in Fig. 9.9 to illustrate how the cross-directional flows develop in the slice channel. As can be seen in Fig. 9.8, the prediction given by HOCS Fiber agrees well with the result calculated using the whole 3D model, even though the reduced model does not take into account the effect of the jet. After the diagnostics step, the fiber orientation is optimized. The CD profiles obtained from Step 2 are presented in Fig. 9.10. As can be seen, the difference between the reduced and the full model is now slightly bigger than in Step 1, but both models predict very small CD velocities resulting in optimal fiber orientation angle profile.
9 CFD-based optimization for papermaking 277 CD Velocity (m/s) CD Position (m) HOCS 3D CFD Fig. 9.8 Comparison of CD velocity for the diagnostics (Step 1) between HOCS Fiber and the three-dimensional CFD model Fig. 9.9 CD velocity field (Step 1) in the slice channel predicted by the 3D model The validation shows that the depth-averaged equations are sufficiently accurate for use in solving industrial fiber orientation trouble-shooting problems. The reduced 2D model is solved by using a stabilized, upwinding Petrov- Galerkin finite element method [3, 5]. One CFD solution takes only a few seconds on a laptop computer for a typical finite element mesh consisting of 5,000 elements. The CPU time is only 1/1000 compared to the full 3D model (2-3 million elements, turbulence model and free jet). HOCS Fiber can perform both optimization steps, that is, both the diagnostics and optimal control steps, in a couple of minutes. If the full 3D model had been
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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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206 Laurent Dumas Table 7.3 Compari
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208 Laurent Dumas Fig. 7.9 General
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210 Laurent Dumas Table 7.4 Four ex
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212 Laurent Dumas Fig. 7.13 Blades
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214 Laurent Dumas Table 7.5 Converg
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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 Optimization in C
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8 Multi-objective Optimizatio Fig.
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