FLOW AROUND A CYLINDER - istiarto

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– 4.26 – c 1 ��V un,e � � ��˜ a �� c c ��pE � pP �� e �� LPE �� impl un,e : implicit 1 ��V ��a ˜ �� p �� e c �� n �� e �pc �� e�� no un,e : explicit old (4.60) in which the terms in the second square bracket are due to non-orthogonality of the cell and are evaluated explicitly. From the starred-velocity and the velocity correction, the normal velocity component can be computed: � c � impl no un,e � un,e � un,e � un,e � un,e � un,e (4.61) Here the velocity correction due to the cell non-orthogonality is written separately. Whilst the velocity correction is obtainable from Eq. 4.60, the starred-velocity unfortunately is not directly available at the cell face. Interpolating the starred velocity at the neighboring cell centers to get the cell face value would result in the decoupling of the velocity from the pressure that causes an oscillation of the solution. A remedy to this problem is to use the so-called Rhie-and-Chow interpolation technique (Rhie and Chow, 1983) which replaces the interpolated pressure gradient at a cell face with the one computed from the pressure at the immediate neighboring cell centers. As can be seen in the relation below, writing the equivalent of Eq. 4.54a at the east face by simply interpolating these starred velocities would result in velocities that have no direct relation with the pressure difference between P and E: � �u n� � e ��˜ �� ˜ a P � b � a nbun,nb �� 1 ��V �� p ��˜ a �� e � �� n �� e e (4.62) In order to relate the velocity at the east face back to the pressure difference between P and E, a correction is given to this interpolated velocity: � un,e � �� ˜ �� ˜ a P � b � anbun,nb �� e 1 ��V �� p � ��a ˜ �� e � �� 1 ��V �� p � �� n �� a ˜ �� e e �� n �� e p �� E �� pP L PE �� (4.63) This expression can be seen as the velocities at the cell centers P and E interpolated to the east face and corrected by a factor due to the interpolation: � � RC � �un � � u e n,e (4.64) u n,e where the overbar term is obtained from linear interpolation of the velocities in cell centers P and E. Using this expression to substitute the starred velocity in Eq. 4.61 gives: � RC impl no un,e � �u n� � u e n,e � un,e � un,e (4.65)
– 4.27 – Next, we need to define the equation of the pressure correction, which is carried out by using the continuity equation. The discretized continuity equation can be obtained by writing the governing equation, Eq. 4.15, with � = 1, � = 0, and R = 0 that gives: � � cf � � � qcf � un,cf Scf cf� ewnstb cf�ewnstb V �d �� S � V �d S �� S cf�ewnstb � � � � 0 (4.66) Using Eq. 4.65 with the development of the implicit term in Eq. 4.60, and inserting the result to the above continuity equation, one gets the pressure correction equation of the form: � 1 ��V �� a ˜ � 1 ��V �� a ˜ � Se c c �� pE � pP�� e LPE 1 � Ss c c �� p S � pP�� s LPS 1 � � q � cf�ewnstb where: q � � � � � u cf cf RC 1 ��V �� qcf � � ��a ˜ q cf ��V �� ˜ a ��V �� a ˜ RC no � � � q cf cf � qcf � � Sw c c �� pW � pP�� w LPW 1 � St c c �� pT � pP�� t LPT 1 � � 0 � � � �Scf,x ��v cf�Scf,y ��w cf�Scf,z �� cf no 1 ��V �� ˜ a �� cf �p � �� n �� cf �� p c �� n �� cf � p nb � � � pP L nbP � �pc �� cf After arranging the terms, one has: � � �� Scf old �� S cf Sn c c �� pN � pP� n L PN ��V �� a ˜ ��V �� ˜ a Sb c c �� p B � pP� b LPB interpolated discharge correction due to cell - face interpolation correction due to non - orthogonal terms (4.67) p c p c aP pP � � anb pnb nb � b p (4.68) where: p 1 � � ��V ��˜ a �� a E p 1 aS � � ��V �� a ˜ a P S e �� e LPE S s �� s LPS p 1 , aW � � ��V ��˜ a �� p 1 , aT � � ��V �� ˜ a � , b nb p � � q � � cf p p � � anb S w �� w L PW S t �� t L PT p 1 , aN � � ��V � ��a ˜ �� p 1 , aB � � ��V �� a ˜ RC no � � � q cf cf � qcf � �. S n �� n LPN S b �� w LPB , ,
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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
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- vii - Acknowledgements I wish to
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- ix - Table of Contents Abstract i
Page 11 and 12:
- 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
Page 25 and 26:
- 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
Page 41 and 42:
- 2.21 - Fig. 2.8 Distributions of
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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 ��
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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
Page 63 and 64:
- 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
Page 99 and 100: - 3.17 - flow may undergo a rotatin
Page 101 and 102: - 3.19 - Fig. 3.7 Cont’d.
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 109 and 110: - 3.27 - Fig. 3.10 Cont’d.
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 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: - 4.25 - �� u ��1 � c i
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 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
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
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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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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 - istiarto

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