FLOW AROUND A CYLINDER - istiarto

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4.4 Pressure-velocity coupling 4.4.1 SIMPLE algorithm – 4.24 – When solving the momentum equation for velocity, the pressure is unknown and an estimated value, p � , is firstly used instead. In general, the velocity that is obtained does not satisfy the continuity equation. A correction to the estimated pressure is added and a new solution is sought for the new velocity. This procedure is repeated until it gives pressure and velocity fields satisfying not only the momentum equation but also the continuity equation. An iterative solution procedure known as SIMPLE (Semi-Implicit Method for Pressure-Linked Equation) method (Patankar and Spalding, 1972) is widely used for this velocity-pressure computation. The method requires velocity and discharge at cell faces, which are not immediately available with the use of non-staggered grids in the present model. The interpolation technique of Rhie-and-Chow (Rhie and Chow, 1983) solves this problem. The technique gives interpolated velocity at cell faces from the nodal values. The standard SIMPLE algorithm is then used to perform the pressure correction. This section gives some details of the procedure, which follows the derivation given by Patankar (Patankar and Spalding, 1972; Versteeg and Malalasekera, 1995; Ferziger and Peric, 1997). In the iteration �� 1, the discretized momentum equation, Eq. 4.52 with � = u, v, w, can be rewritten as: a ˜ Pui,P �� 1 ��1 � anbui,nb nb � ˜ b � 1 � V �p P ��1 �� xi �� P (4.53) where the symbols u i and xi are used to denote the Cartesian components of the velocity and direction, ui � u, v, w and xi � x, y, z , respectively. Note that the pressure gradient �� in the above expression has intentionally been extracted from the source term, �� b , for a reason that will be evidenced later ( ˜ b in Eq. 4.53 is thus not exactly the same as that in Eq. 4.52). The coefficients a ˜ P , anb , and the source terms, ˜ b , are functions of the known variables either at the precedent iteration, �� , or time step, n. For practical solutions of Eq. 4.53, since there are only 3 equations for 4 unknowns, the pressure p is temporarily fixed at its initial value. The following system of equations is solved in the first stage: ˜ a P u i,P � � � � a nbui,nb nb � ˜ b � V P � �� p � �� xi �� P (4.54a) �� p� � p � (4.54b) The estimated pressure, p � , and the velocities obtained from this pressure, u � , v � ,w � , are of course to be corrected:
– 4.25 – �� u ��1 � c i � ui � ui (4.55a) �� p��1 � p � � p c (4.55b) c c where the corrections, u i and p , will result from the momentum equations combined with the continuity equation. The corrections are such that the velocity will satisfy the continuity and momentum equations when the iteration converges. It can therefore be said that Eqs. 4.54a,b become Eq. 4.53 when �� . In that case, having a sufficiently � � � �� 1 � ui ; and this is so for the coefficients large number of �� -iterations, we have u �� i and source terms. We can therefore obtain the relation between the pressure and velocity corrections by subtraction of Eqs. 4.54a,b from Eq. 4.53: ˜ a P u i,P � c c � a nbui,nb � nb � V P � �� p c �� xi �� P (4.56) Eq. 4.56 is a relation in which the corrections tend towards zero. A simplifying approximation can then be introduced by neglecting à priori the correction terms of the neighboring cells (Patankar and Spalding, 1972). The velocity correction thus reduces to: c 1 V P ui,P � � � ˜ a P �� p c �� x i �� P (4.57) �� Since the coefficient �� a P is derived in such a way that it is the same for all velocity u v w components, i.e. a ˜ P � a ˜ P � ˜ a P � a ˜ P , the above relation is valid for any velocity component at any point and thus also for the normal velocity component at a cell face. Writing for the east face, one has: c 1 un,e � � ��V ��˜ a �� p c �� n �� e �� e (4.58) The coefficient at the cell face, V �� a �e , is defined as the average value of those of the neighboring cell centers P and E: ��V �� a ˜ �� e 1 ��V �� 2 ��a ˜ P �� P V �� a ˜ �� P �� E �� E represents the coefficient ˜ (4.59) Note that a ˜ P a P of Eq. 4.52 written for the cell E, that is not the coefficient aE of Eq. 4.52 written for the cell P. Note also that the interpolation in Eq. 4.59 does not match the linear interpolation in Eq. 4.19 since this latter is not relevant for the volumes. Indeed the volume related to a cell for the east face is composed of the half volume of cell P and the half volume of cell E. Using the deferred-correction method as in the discretisation of the diffusive terms (see Sect. 4.3.6, Eq. 4.31) to compute the normal pressure gradient yields:
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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
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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
Page 97 and 98: 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: - 4.23 - b T � V P �t � n P
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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
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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
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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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