FLOW AROUND A CYLINDER - istiarto

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�� V ˆ h � �� ˆ �� 2 � ˆ �� u 1 �� u 2 2 1 2 � ˆ u 1 cos� �� T �� sin� T �� – 2.10 – The velocity direction, �V, is given by the relations below: �V � arcsin ˆ ��u 2 � u ˆ 2 cos� �� T �� V ˆ ��, if u ˆ 1 � 0 �� h sin� T �� V � 180 �� arcsin ˆ ��u 2 � u ˆ 2 cos� �� T �� V ˆ h sin� T �� , if u ˆ 2 � 0 (2.10) (2.11) The three relations, Eqs. 2.9, 2.10, and 2.11, are the ones required to describe the threedimensional instantaneous velocity of the moving target. It is, however, more convenient to describe the velocity by its components along the Cartesian or cylindrical coordinate system. The following section discusses the decomposition of the velocity into those components. Cartesian and cylindrical velocity components The Cartesian coordinate axes are defined as x, y, and z for the horizontal, transversal, and vertical directions, respectively. The cylindrical coordinate axes are defined as r, �� = � – 180°), and z for the radial, angular, and vertical directions, respectively (see Fig. 2.4). The origin of the two coordinate systems is defined at the center of the cylinder, at the original (uneroded) bed level. The decomposition of the velocity into its components along the Cartesian and cylindrical coordinate systems depends on the orientation of the ADVP with respect to the coordinate system. In the measurements, the instrument is positioned along radial planes around the cylinder. Fig. 2.4 shows a typical placement of the ADVP and the decomposition of the velocity into its components along the Cartesian and cylindrical coordinate systems. By using geometrical relationships, the Cartesian velocity components can be obtained from the following expressions: � � � � u ˆ � ˆ V h cos � � �R � � V v ˆ � ˆ V h sin � � �R � � V w ˆ � w ˆ 1 � w ˆ 2 where the definitions of the angles are given in Fig. 2.4. (2.12)
– 2.11 – Fig. 2.4 Cartesian and cylindrical coordinate systems and the decomposition of the measured instantaneous velocity into its components in these coordinate systems. The velocity components along the Cartesian and the cylindrical coordinate systems are interchangeable by using transformation functional expression as follows: ��u ˆ r �� u ˆ � �� w ˆ �� cos� sin � 0�� u ˆ �� sin� cos� 0 �� v ˆ �� 0 0 1�� w ˆ �� cos� �sin � 0�� u ˆ �� sin� �cos� 0 �� v ˆ �� 0 0 1�� w ˆ �� u ˆ �� v ˆ �� w ˆ �� cos� �sin� 0�� u ˆ r �� sin � cos� 0 �� u ˆ � �� 0 0 1�� w ˆ �� cos� sin � 0�� u ˆ r �� sin� �cos� 0 �� ˆ u � �� 0 0 1�� w ˆ �� (2.13a) (2.13b) where ˆ u r, ˆ u � , and ˆ w are the radial, angular, and vertical velocity components, respectively, along the cylindrical coordinate system, (r,�,z), whose origin is defined at the center of the cylinder. Time-averaged and fluctuating velocity components Obtaining the instantaneous velocity data over the measurement period, the statistical parameters can be found. Three quantities are computed, i.e. the mean (the time-averaged velocities), the variance (the squared values of the turbulent intensities), and the covariance (the Reynolds stresses).
Page 1 and 2: FLOW AROUND A CYLINDER IN A SCOURED
Page 3 and 4: - iii - Flow around a Cylinder in a
Page 5 and 6: - v - Écoulement à Surface Libre
Page 7 and 8: - vii - Acknowledgements I wish to
Page 9 and 10: - ix - Table of Contents Abstract i
Page 11 and 12: - xi - 4.4 Pressure-velocity coupli
Page 13 and 14: Chapter 1 - 1.i - 1 Introduction 1.
Page 15 and 16: 1 Introduction 1.1 Flow around a cy
Page 17 and 18: - 1.3 - pitot tubes. Ahmed and Raja
Page 19 and 20: Chapter 2 - 2.i - 2 Flow Measuremen
Page 21 and 22: 2 Flow Measurements Abstract - 2.1
Page 23 and 24: 2.1 Measurement design - 2.3 - The
Page 25 and 26: - 2.5 - The water circuit is a clos
Page 27 and 28: - 2.7 - Consider a target ―i‖ m
Page 29: - 2.9 - Obtaining the velocity comp
Page 33 and 34: - 2.13 - transducer were used, a fo
Page 35 and 36: - 2.15 - typically needed for this
Page 37 and 38: - 2.17 - 2.3.3 Comparison to other
Page 39 and 40: - 2.19 - Fig. 2.7 Scour hole geomet
Page 41 and 42: - 2.21 - Fig. 2.8 Distributions of
Page 43 and 44: u ��u �� u� � Du exp�
Page 45 and 46: - 2.25 - �o , and accordingly the
Page 47 and 48: - 2.27 - Measurements in zone A. Th
Page 49 and 50: - 2.29 - and 0.3 u �� u ��
Page 51 and 52: - 2.31 - Fig. 2.12b Cont’d.
Page 53 and 54: - 2.33 - Fig. 2.12d Cont’d.
Page 55 and 56: 2.5.4 Measurements in the plane �
Page 61 and 62: - 2.41 - the solid boundary of the
Page 67 and 68: - 2.47 - Turbulence intensities in
Page 69 and 70: - 2.49 - Fig. 2.15a Vertical distri
Page 71 and 72: - 2.51 - Fig. 2.15c Cont’d.
Page 73 and 74: - 2.53 - Reynolds stresses in the p
Page 75 and 76: - 2.55 - Fig. 2.16c Cont’d.
Page 77 and 78: 2.6 Summary and conclusions - 2.57
Page 79 and 80: - 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
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x [cm] - 3.31 - Table 3.1 Estimated
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- 3.33 - Fig. 3.12 Estimated bed sh
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- 3.35 - (see Fig. 3.10 and Fig. 3.
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- 3.37 - Fig. 3.14 Axonometric pres
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- 3.39 - U, U∞ [m/s] sectional av
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Chapter 4 - 4.i - 4 Numerical Model
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- 4.1 - 4 Numerical Model Developme
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�u �t �v �t �w �t �
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- 4.5 - The volume integrals of the
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- 4.7 - Fig. 4.1 Overall iterative
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- 4.9 - increase the computer stora
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- 4.11 - �� 1 ��
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- 4.13 - convective transport throu
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- 4.15 - n,��1 during a time st
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� for 0 � Pe e � q eL PE �
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4.3.8 Source terms - 4.19 - The sou
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2D � � � x �� b 2D �
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- 4.23 - b T � V P �t � n P
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- 4.25 - �� u ��1 � c i
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- 4.27 - Next, we need to define th
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4.5 Boundary conditions 4.5.1 Bound
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- 4.31 - The above relation holds a
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- 4.33 - � extrapolate all variab
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G � � t 2 �V t - 4.35 - 2 �
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with � �� nt,b � ��
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- 4.39 - boundary node B are then a
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- 4.41 - Wall function for the �
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k and � equations - 4.43 - � se
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�� z-momentum: - 4.45 - ��1
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- 4.47 - � reconstruct the mesh,
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- 4.49 - Fig. 4.13 The structure of
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- 4.51 - ��
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4.7 Summary - 4.53 - Development of
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- 4.55 - B, �B [-] constant and w
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- 4.57 - �, � 1 , � p [-] und
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Chapter 5 - 5.i - 5 Numerical Simul
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5 Numerical Simulation Abstract - 5
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5.2 Experimental data - 5.3 - 5.2.1
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- 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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FLOW AROUND A CYLINDER IN A SCOURED
Page 249 and 250:
- 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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FLOW AROUND A CYLINDER IN A SCOURED
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