FLOW AROUND A CYLINDER - istiarto

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�� z-momentum: – 4.40 – ��1 �b � bP uP � � B Fn,z � � F �� t, z �� b � �� t�2 ue n,x � 2 ve n, y � we n,z�e n,z � �t,wall �ue t,x � v et,y�e t, z � Sb �P �n,P �� b � � �� P � �t �e n,z en,z�� t,wall et,z et,z � � B � S b � �� n,P �� b P�B ��1 wP (4.91c) Wall function for the k equation. The wall function is used to evaluate the source term of the k equation, G � �� The turbulent kinetic-energy production, G, for cells neighboring the wall is not evaluated by Eq. 4.12, but is directly obtained from Eq. 4.83. Since the velocity has been known when solving the k equation, this information can be used in calculating the wall shear-stress term in the G equation, Eq. 4.83. The turbulent kinetic-energy dissipation, �, is evaluated by Eq. 4.84; this contains a non-linear term in k which is then linearized. The following steps apply: � set all coefficients related to the contribution from the boundary node B to zero: C D C D aB � aB � 0, a P � � � �a B P� � b B D � � � 0 B � compute the source, that is the energy production and dissipation, and linearise the source: m�1 b � bP kP � �G P � �P�V P �� m 1 4 m 1 2 c� kP � � � c� 3 4 m kP � � �� nt �� V P V P �� b n,P � �n,P �� b b P 1 2 m�1 k (4.92) P where the magnitude of the shear stress is evaluated with the velocity already computed from the momentum equations (see Eq. 4.82): m �� nt �� b 1 � �� nt,b � c �� 1 4 m � k �� ln E � n �� m � � �� t,wall �� n �� P 1 2 �� P m�1 Vt,P m � � �� t,wall �� n �� m�1 m�1 m �1 �uP et,x � vP et,y � w P et,z � �� P m�1 VP � �� e t Note that k P m�1 is unknown for this computation step, while uP m�1 , vP m�1 , and w P m �1 are already fixed.
– 4.41 – Wall function for the � equation. The turbulent kinetic-energy dissipation for cells neighboring the wall is defined by Eq. 4.84. This can be easily implemented as follows: � set all coefficients related to the neighboring cells to zero: a E � a W � a N � a S � a T � a B � 0 � set the coefficient at cell P to unity: aP = 1 � set the source terms by (see Eq. 4.84): c m�1 � b � �P � � � 3 4 m�1 kP � � n,P 3 2 m�1 where kP is already computed in the previous step (4.93) Wall function and pressure correction. Since the discharge across the wall is zero, the p coefficient of the boundary node B in Eq. 4.68 is set to zero: aB � 0 . The pressure correction at the boundary node is obtained by direct extrapolation from the cell center P: c c pB � pP . 4.5.5 Symmetry boundary At the symmetry plane, for example at the east face (see Fig. 4.9), the convective transport across the plane and the shear stress along the plane are zero. These properties make the velocity at E be easily obtained from the projection of the velocity at P to the plane. For the scalar variables, k and �, an approximation is used by extrapolating the values at P to the boundary E. The following expressions thus apply at symmetry boundaries: Fig. 4.9 Symmetry boundary.
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FLOW AROUND A CYLINDER IN A SCOURED
Page 3 and 4:
- 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
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3.2.2 Flow pattern around the cylin
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- 3.9 - Fig. 3.2 Cont’d.
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Flow in the plane � = 45° and 90
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- 3.13 - Fig. 3.4 Contours of the m
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3.2.4 Vertical velocity (downward f
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- 3.17 - flow may undergo a rotatin
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- 3.19 - Fig. 3.7 Cont’d.
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- 3.21 - A similar observation, in
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- 3.23 - Fig. 3.9 Measured turbulen
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The following is to be observed: -
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- 3.27 - Fig. 3.10 Cont’d.
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- 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 and 140: - 4.15 - n,��1 during a time st
Page 141 and 142: � for 0 � Pe e � q eL PE �
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
Page 155 and 156: - 4.31 - The above relation holds a
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: - 4.39 - boundary node B are then a
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
Page 175 and 176: - 4.51 - ��
Page 177 and 178: 4.7 Summary - 4.53 - Development of
Page 179 and 180: - 4.55 - B, �B [-] constant and w
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
Page 191 and 192: - 5.7 - (i) Yulistiyanto data (ii)
Page 193 and 194: - 5.9 - ( p � � 1 [mm]), but th
Page 195 and 196: - 5.11 - Fig. 5.4 Model test result
Page 197 and 198: - 5.13 - within a greater number of
Page 199 and 200: - 5.15 - Fig. 5.6 Computational his
Page 201 and 202: - 5.17 - 5.5 Simulation of flow aro
Page 203 and 204: - 5.19 - Fig. 5.9 Definition sketch
Page 205 and 206: - 5.21 - Fig. 5.10 Computed (lines)
Page 207 and 208: - 5.23 - Fig. 5.11 Computed and mea
Page 209 and 210: - 5.25 - 1.2U ∞ of the measuremen
Page 211 and 212: Water-surface profile - 5.27 - The
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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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