FLOW AROUND A CYLINDER - istiarto

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– 3.30 – The bed shear-stress, � o , obtained by the three methods applied to the measured data is presented in Table 3.1 and Table 3.2, and is plotted in Fig. 3.12; shown also in the figure is the one obtained from the numerical simulation. Evaluation is only done for the plane of symmetry upstream and downstream of the cylinder. The following is to be remarked: � In the approach region, x ≤ –45 [cm], all three methods render rather similar results, being 0.6 ≤ � o [Pa] ≤ 1.0. The bed shear-stress in the approach region was earlier obtained as being � o,� � 0.70 [Pa] or u �,� � 0.0265 [m s]. � In the scour hole upstream of the cylinder, –45 ≤ x [cm] ≤ –10, the agreement is less convincing. It appears that the data using the velocity measurements, Eq. 3.5, seem to be more reliable; evaluation of the measurement of the velocity, Vt , is more objective than the one of the shear-stress, ��u �� w ��, at the bed. This is also evidenced by the numerical simulation, Eq 4.82, which depends on the velocity close to the bed. The global method, Eq. 3.7, would be simple, but applies essentially to uniform flow and does not take into account the flow reversal close to the bed; thus, the sign change is not respected. Note that the negative value indicates flow reversal at the bed. � Behind the cylinder, 10 ≤ x [cm] ≤ 80, the three methods render surprisingly similar results, being 0 ≤ � o [Pa] ≤ 0.5, thus almost constant over the entire length measured. The measured bed shear-stress is slightly larger than the one in the upstream part of the scour hole; this was also observed by Melville and Raudkivi (1977) and Dey, (1997). On leaving the scour hole, the shear stress has about the same (absolute) value as the one in the upstream part of the scour hole; there is a weak tendency of an increase as leaving the scour hole. This is also observed from the numerical simulation result. � In both regions, upstream and downstream of the cylinder, the critical shear-stress, �o,cr � 1.36 [Pa], as calculated from the Shields diagram (for uniform sand with � d50 = 2.1 [mm]), �cr = 0.04 [–], was not exceeded. This is in agreement with eye observations. Thus the experiment is well a clear-water scour run, where the capacity of transport in the approach flow and in the scour hole is zero. The observation of the bed shear-stress variation as depicted in Fig. 3.12 reveals that the (high) bed shear-stress in the approach flow is gradually reduced on entering the scour hole, where it remains rather low on approaching the cylinder. Measurements by Melville and Raudkivi (1977) show a similar trend.
x [cm] – 3.31 – Table 3.1 Estimated bed shear-stresses based on the vertical distributions of the measured velocities and Reynolds stresses upstream of the cylinder. U [m/s] U/U ∞ [–] � o,1 [Pa] � o,1 � o,cr [–] � o,2 [Pa] � o,3 [Pa] u �,1 [m/s] u �,2 [m/s] u �,3 [m/s] -10 0.091 0.201 -0.010 -0.007 -1.800 0.040 -0.003 -0.042 0.006 -11 0.095 0.212 -0.102 -0.075 -1.800 0.045 -0.010 -0.042 0.007 -12 0.114 0.253 -0.047 -0.035 -1.400 0.064 -0.007 -0.037 0.008 -13 0.121 0.268 -0.161 -0.118 -1.200 0.071 -0.013 -0.035 0.008 -14 0.125 0.278 -0.246 -0.181 -1.200 0.076 -0.016 -0.035 0.009 -15 0.139 0.309 -0.303 -0.223 -0.900 0.095 -0.017 -0.030 0.010 -16 0.143 0.318 -0.071 -0.052 -0.600 0.100 -0.008 -0.024 0.010 -18 0.156 0.346 -0.080 -0.059 -0.300 0.119 -0.009 -0.017 0.011 -20 0.268 0.596 -0.196 -0.144 0.300 0.353 -0.014 0.017 0.019 -22 0.281 0.625 -0.223 -0.164 0.300 0.388 -0.015 0.017 0.020 -24 0.297 0.659 -0.205 -0.151 0.400 0.432 -0.014 0.020 0.021 -26 0.312 0.693 -0.134 -0.098 0.300 0.476 -0.012 0.017 0.022 -28 0.324 0.720 -0.076 -0.056 0.400 0.514 -0.009 0.020 0.023 -30 0.343 0.763 -0.030 -0.022 0.700 0.577 -0.005 0.026 0.024 -32 0.352 0.782 -0.019 -0.014 0.300 0.607 -0.004 0.017 0.025 -34 0.360 0.801 -0.163 -0.120 1.000 0.636 -0.013 0.032 0.025 -36 0.387 0.861 -0.084 -0.062 0.900 0.735 -0.009 0.030 0.027 -38 0.404 0.897 -0.097 -0.071 0.700 0.798 -0.010 0.026 0.028 -40 0.429 0.952 -0.072 -0.053 0.600 0.900 -0.008 0.024 0.030 -42 0.449 0.997 -0.053 -0.039 0.600 0.986 -0.007 0.024 0.031 -44 0.467 1.037 0.246 0.181 0.600 1.068 0.016 0.024 0.033 -47 0.449 0.998 0.855 0.629 0.600 0.988 0.029 0.024 0.031 -50 0.466 1.035 0.718 0.528 0.600 1.063 0.027 0.024 0.033 -60 0.454 1.009 1.031 0.758 0.600 1.010 0.032 0.024 0.032 -70 0.451 1.002 0.866 0.637 0.700 0.997 0.029 0.026 0.032 -80 0.457 1.016 1.366 1.004 0.700 1.024 0.037 0.026 0.032 Notes: U = local depth-averaged velocity U∞ = 0.45 [m/s] = approach velocity � o,1 � �� t �V t �n � �� t V t �n � t �1.3 �10 �5 [m 2 /s 2 ] = measured eddy viscosity in the approach flow V t = velocity measured closest to the bed, z ≈ 4 [mm], projected on a plane parallel to the bed � n = normal distance between V t and the bed �o,2 � �� u �� w �� cos� bed � � 2 �o,3 � ��u �,3� 2 � � U g C 2 � ��0.07 U� 2 , with C = 44 [m1/2 /s] = Chezy coefficient � o,cr = 1.36 [Pa] = critical shear-stress according to the Shields criterion for d50 = 2.1 [mm], or � cr � = 0.04.
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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.
Page 61 and 62: - 2.41 - the solid boundary of the
Page 63 and 64: - 2.43 - Fig. 2.14b Cont’d.
Page 65 and 66: - 2.45 - Fig. 2.14d Cont’d.
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
Page 81 and 82: Chapter 3 - 3.i - 3 Elaboration of
Page 83 and 84: - 3.1 - 3 Elaboration of the measur
Page 85 and 86: 3.1 Introduction - 3.3 - Presented
Page 87 and 88: - 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: - 3.29 - 3.4 Bed shear-stresses alo
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 � ��
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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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- 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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