FLOW AROUND A CYLINDER - istiarto

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F w �� – 4.18 – n� n�1 �� D �w Sw �� 1 �� D �w Sw �� 1 � ��max�qw ,0�� fw �� P � ��max� �qw,0�� fw �� W �� LPW �� LPW �� D � �fw �� w Sw implicit � �� n�� w �� w �� explicit (4.43b) Shorter notations is used to write the expression of the convective-diffusive flux, for n�n�1 C D ��1 C D ��1 D example Eq. 4.43a maybe rewritten as F �� e ��aP � aP�� P ��a E � a E�� E � bE , in such a way that by summing up the convective-diffusive fluxes across the six faces of cell P, one obtains: F n�n�1 � a E �� C D ��1 C D ��1 C D ��1 C D � � aE�� E ��aW � aW�� W ��aN � aN�� N ��aS � aS �� S C D ��1 C D ��1 C D ��1 D �a T � a T�� T ��aB � aB�� B ��aP � a P�� P � b ��1 � (4.44) where the coefficients indicate contribution of convective-diffusive terms from neighboring cells as presented in Table 4.4. The diffusive-correction terms, being evaluated explicitly, are known from the previous iteration and are included in the independent coefficient b D . Table 4.4 Coefficients of the discretized convective-diffusive equations. Convective terms Diffusive terms �� a C � E � � max �qe ,0 �� a C W � �max �qw �� a C N � �max �qn �� a C S � � max �qs �� a C T � � max �qt �� a C B � � max �qb C C aP � �a nb �� nb�EWNSTB � � �� , 0� �� ,0� �� ,0� �� ,0� �� ,0� �� a D D � E � �f �e Se LPE a D D � W � �f �w Sw L PW a D D � N � �f �n Sn LPN a D D � S � �f �s Ss LPS a D D � T � �f �t St LPT a D D � B � �f �b Sb LPB � � � � qcf D aP � � D �anb cf�ewnstb nb�EWNSTB — b D � � f D � � S cf�ewnstb �� cf �� n �� cf � �� cf ��
4.3.8 Source terms – 4.19 – The source terms may consist of scalar quantities, first derivatives, or second derivatives of a scalar quantity. The source, in addition, may also include the time integration term, the diffusion correction, and known variables from the boundary conditions. Sometimes a term initially considered as a source takes advantage to be expressed as a function of the unknown variable at the cell center, such as cells next to a boundary. In that case, the term is linearized which gives (Versteeg and Malalasekera, 1995, p. 87): �� R dV �� V ��1 � R V P � b � bP �P (4.45) in which b includes all known quantities (either constants, prescribed, or known from previous iteration) and b P is the coefficient of the unknown variable at P. Scalar source terms The source term containing scalar quantities, b S , comes from the gravity accelerations, g x , g y , g z , or the turbulent energy production and dissipation, G and �. The scalar source at P represents the average value of those quantities in the cell being considered. It is known and thus is considered as a constant. Therefore, the source terms containing scalar quantities can be easily evaluated according to the following expression: bS � � � dV V �� P V P (4.46) In applying the above relation to the source terms of the k equation coming from the turbulent energy production, G, some approximations are needed. Writing Eq. 4.46 for G, we have: �� G dV � G V P V P Since the G term contains non-linear gradient terms (see Eq. 4.12), the above relation implies that these terms are evaluated individually (that is by using Eq. 4.20). This means that the integral of these terms are computed in the following fashion (an example is given here for the �u �x � � 2 term): 2 ��u �� dV � V ��x �� u �� V �� P � �x 1 �� 2 �� P V P �� V 2 �u �x dV �� This approximation, of course, will be inaccurate when the velocity gradient is important. Nevertheless, this method is selected for its easiness to implement. In the ��equation, the source term is linearized for the term containing � 2 in the following form: V P
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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
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 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: � for 0 � Pe e � q eL PE �
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 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)
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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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