FLOW AROUND A CYLINDER - istiarto

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– 4.16 – formulae which combine the convective and diffusive transports in a special way. Depending on the grid Peclet number Pe, being the ratio of the convective and diffusive conductance, Pe � q L �S where L is the nodal distance, either the upwind-scheme convection, central-difference diffusion, or combination of the two, is considered to transport any scalar quantity � across a cell face. The hybrid scheme (Spalding, 1972) uses the upwind scheme for large Peclet numbers (|Pe| ≥ 2) and central difference for small Peclet numbers (|Pe| < 2). According to this C D scheme, the total flux across the east face, Fe � Fe � Fe , is defined as follows: � for Pe e � q eL PE � eS e � �2 , only the convective transport is taken into account: n�n�1 � ��1 �� Fe � qe �E (4.35) � for �2 � Pe e � q eL PE � eS e � 0, a part of the diffusive transport is also taken into consideration: n�n�1 � ��1 Fe � qe �E � 1 � 0.5Pee �� e � Se ��1 ��1 � �� E � �P �� e Se LPE � �� n�� e �� e�� (4.36) � for 0 � Pe e � q eL PE � eS e � 2, a part of the diffusive transport is also taken into consideration: n�n�1 � ��1 Fe � qe �P � 1 � 0.5Pee �� e � Se ��1 ��1 � �� E � �P �� e Se LPE � �� n �� e �� e�� for Pe e � q eL PE � eS e � 2 , only the convective transport is taken into account: (4.37) n�n�1 � ��1 �� Fe � qe �P (4.38) The power-law scheme (Patankar, 1980, p. 90-91) sets the limiting value of Pe where the diffusion no longer affects the transport at Pe = 10, instead of Pe = 2 used in the hybrid scheme. � for Pe e � q eL PE � eS e � �10 : n�n�1 � ��1 �� Fe � qe �E (4.39) � for �10 � Pe e � q eL PE � eS e � 0: n�n�1 � ��1 Fe � qe �E � 1 � 0.1Pee �� 5 � e � Se ��1 ��1 � �� E � �P �� e Se LPE � �� n�� e �� (4.40) �� e��
� for 0 � Pe e � q eL PE � eS e � 10: n�n�1 � ��1 Fe � qe �P � 1 � 0.1Pee �� 5 � e � for Pe e � q eL PE � eS e �10 : – 4.17 – � Se ��1 ��1 � �� E � �P �� e Se LPE � �� n�� e �� (4.41) �� e�� n�n�1 � ��1 �� Fe � qe �P (4.42) Equations 4.35 to 4.42 can be combined into a compact form as follows (Patankar, 1980, pp. 94-95): F e n�n�1 � � max qe,0 �� P ��1 � �� E ��1 � � max �qe,0 � D ��e Se ��1 ��1 � �fe �� E � �P �� e Se LPE �� n �� e �� e�� (4.43) where f D is a factor that depends on the absolute value of grid Peclet number; it has a different form for the hybrid and power-law schemes as shown in Table 4.3. Table 4.3 Hybrid and power-law scheme diffusion factors. Arranging the terms in Eq. 4.43, one has: F e �� Scheme f D � f �Pe�� f�qL �S� Hybrid max�� 1� 0.5 Pe�, 0� Power-law max �1� 0.1Pe� 5 � , 0� n� n�1 �� D �e Se �� max�qe ,0�� fe �� LPE �� 1 �� D �e Se �� P � ��max� �qe ,0�� fe �� LPE �� 1 E �� D � �fe �� e Se implicit � �� n�� e �� e �� explicit and for the west face, the convective-diffusive flux reads: (4.43a)
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FLOW AROUND A CYLINDER IN A SCOURED
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- iii - Flow around a Cylinder in a
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- v - Écoulement à Surface Libre
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- vii - Acknowledgements I wish to
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- ix - Table of Contents Abstract i
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- xi - 4.4 Pressure-velocity coupli
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Chapter 1 - 1.i - 1 Introduction 1.
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1 Introduction 1.1 Flow around a cy
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- 1.3 - pitot tubes. Ahmed and Raja
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Chapter 2 - 2.i - 2 Flow Measuremen
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2 Flow Measurements Abstract - 2.1
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2.1 Measurement design - 2.3 - The
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- 2.5 - The water circuit is a clos
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- 2.7 - Consider a target ―i‖ m
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- 2.9 - Obtaining the velocity comp
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- 2.11 - Fig. 2.4 Cartesian and cyl
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- 2.13 - transducer were used, a fo
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- 2.15 - typically needed for this
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- 2.17 - 2.3.3 Comparison to other
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- 2.19 - Fig. 2.7 Scour hole geomet
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- 2.21 - Fig. 2.8 Distributions of
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u ��u �� u� � Du exp�
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- 2.25 - �o , and accordingly the
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- 2.27 - Measurements in zone A. Th
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- 2.29 - and 0.3 u �� u ��
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- 2.31 - Fig. 2.12b Cont’d.
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- 2.33 - Fig. 2.12d Cont’d.
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2.5.4 Measurements in the plane �
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- 2.37 - Fig. 2.13b Cont’d.
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- 2.39 - Fig. 2.13d Cont’d.
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- 2.41 - the solid boundary of the
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- 2.43 - Fig. 2.14b Cont’d.
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- 2.45 - Fig. 2.14d Cont’d.
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- 2.47 - Turbulence intensities in
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- 2.49 - Fig. 2.15a Vertical distri
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- 2.51 - Fig. 2.15c Cont’d.
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- 2.53 - Reynolds stresses in the p
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- 2.55 - Fig. 2.16c Cont’d.
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2.6 Summary and conclusions - 2.57
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- 2.59 - Rolland, T. (1994). Develo
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Chapter 3 - 3.i - 3 Elaboration of
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- 3.1 - 3 Elaboration of the measur
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3.1 Introduction - 3.3 - Presented
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- 3.5 - Fig. 3.1 Measured velocity
Page 89 and 90: 3.2.2 Flow pattern around the cylin
Page 91 and 92: - 3.9 - Fig. 3.2 Cont’d.
Page 93 and 94: Flow in the plane � = 45° and 90
Page 95 and 96: - 3.13 - Fig. 3.4 Contours of the m
Page 97 and 98: 3.2.4 Vertical velocity (downward f
Page 99 and 100: - 3.17 - flow may undergo a rotatin
Page 103 and 104: - 3.21 - A similar observation, in
Page 105 and 106: - 3.23 - Fig. 3.9 Measured turbulen
Page 107 and 108: The following is to be observed: -
Page 111 and 112: - 3.29 - 3.4 Bed shear-stresses alo
Page 113 and 114: x [cm] - 3.31 - Table 3.1 Estimated
Page 115 and 116: - 3.33 - Fig. 3.12 Estimated bed sh
Page 117 and 118: - 3.35 - (see Fig. 3.10 and Fig. 3.
Page 119 and 120: - 3.37 - Fig. 3.14 Axonometric pres
Page 121 and 122: - 3.39 - U, U∞ [m/s] sectional av
Page 123 and 124: Chapter 4 - 4.i - 4 Numerical Model
Page 125 and 126: - 4.1 - 4 Numerical Model Developme
Page 127 and 128: �u �t �v �t �w �t �
Page 129 and 130: - 4.5 - The volume integrals of the
Page 131 and 132: - 4.7 - Fig. 4.1 Overall iterative
Page 133 and 134: - 4.9 - increase the computer stora
Page 135 and 136: - 4.11 - �� 1 ��
Page 137 and 138: - 4.13 - convective transport throu
Page 139: - 4.15 - n,��1 during a time st
Page 143 and 144: 4.3.8 Source terms - 4.19 - The sou
Page 145 and 146: 2D � � � x �� b 2D �
Page 147 and 148: - 4.23 - b T � V P �t � n P
Page 149 and 150: - 4.25 - �� u ��1 � c i
Page 151 and 152: - 4.27 - Next, we need to define th
Page 153 and 154: 4.5 Boundary conditions 4.5.1 Bound
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Page 157 and 158: - 4.33 - � extrapolate all variab
Page 159 and 160: G � � t 2 �V t - 4.35 - 2 �
Page 161 and 162: with � �� nt,b � ��
Page 163 and 164: - 4.39 - boundary node B are then a
Page 165 and 166: - 4.41 - Wall function for the �
Page 167 and 168: k and � equations - 4.43 - � se
Page 169 and 170: �� z-momentum: - 4.45 - ��1
Page 171 and 172: - 4.47 - � reconstruct the mesh,
Page 173 and 174: - 4.49 - Fig. 4.13 The structure of
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Page 177 and 178: 4.7 Summary - 4.53 - Development of
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Page 181 and 182: - 4.57 - �, � 1 , � p [-] und
Page 183 and 184: Chapter 5 - 5.i - 5 Numerical Simul
Page 185 and 186: 5 Numerical Simulation Abstract - 5
Page 187 and 188: 5.2 Experimental data - 5.3 - 5.2.1
Page 189 and 190: - 5.5 - computational nodes. The ot
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- 5.7 - (i) Yulistiyanto data (ii)
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- 5.9 - ( p � � 1 [mm]), but th
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- 5.11 - Fig. 5.4 Model test result
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- 5.13 - within a greater number of
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- 5.15 - Fig. 5.6 Computational his
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- 5.17 - 5.5 Simulation of flow aro
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- 5.19 - Fig. 5.9 Definition sketch
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- 5.21 - Fig. 5.10 Computed (lines)
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- 5.23 - Fig. 5.11 Computed and mea
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- 5.25 - 1.2U ∞ of the measuremen
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Water-surface profile - 5.27 - The
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- 5.29 - using a Joukowski transfor
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- 5.31 - 5.6.3 Results of the simul
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Velocity fields around the cylinder
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- 5.35 - Fig. 5.18 Computed and mea
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- 5.37 - than the measured one. The
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- 5.39 - Production and dissipation
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- 5.41 - Production of turbulent ki
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- 5.43 - If the above approach were
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- 5.45 - and surface profiles (see
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- 5.47 - Fig. 5.24 Surface boundary
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- 5.49 - S [m 2 ] surface area. So,
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Chapter 6 - 6.i - 6 Summary and Con
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- 6.1 - 6 Summary and Conclusions 6
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- 6.3 - The spatial distributions o
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- 6.5 - � Laboratory measurements
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- 6.7 - Fig. 6.1 Measured and compu
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IDENTITE - 7.1 - Curriculum Vitæ N
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- iii - Appendices Appendix A: Expe
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A Experimental Data - A.1 - The exp
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B Numerical Model B.1 Program struc
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- B.3 -
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B.2 Input data file - B.5 - The inp
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- B.7 - #31 imon1, jmon1, kmon1 thr
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BLO for blocked cells, INL for inle
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