FLOW AROUND A CYLINDER - istiarto

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– 4.38 – �� Knowing Vt,Pthe unit vector �� e t can be easily computed: �� e t � Vt Vt �� P Implementation of the wall function (4.88) Wall function for the momentum equation. Since there is no discharge across the wall, the convective flux does not exist across the wall face; the only flux is due to the diffusion. In the discretized momentum equation, Eq. 4.52, the diffusion term is not evaluated by Eq. 4.33, but by evaluating this term as a normal force (per unit mass) acting on the wall. Similarly the wall shear-stress, Eq. 4.82, is also transformed as a shear force. Both forces are considered as a source term and are evaluated at iteration �� , which then linearized such that the velocities at the cell center P become unknown variables. The force due to the normal stress acting on the wall (see Fig. 4.8) can be computed as: � ��Fn �� nn �� b � �� Sb � ��t 2 b �V �� n �� n � Sb � ��t 2 �� b �V �� n �� n �� e n � Sb � ��t 2 �� P V �� n �� n �� e n � �� e n�� P S b (4.89) �� The negative sign is necessary since the normal stress, �� nn , is in the negative direction of Vn (see Fig. 4.8). All terms are evaluated explicitly, that is the velocity is from the �� th iteration, the kinetic energy is from the mth iteration, and the geometry is from the nth time iteration. The force due to the shear stress acting on the wall is obtained from Eq. 4.82, but its form is modified to allow easy computation of the turbulent viscosity later on. �� Ft �� b �� nt �� Sb � � �� b c 1 4 m 1 2 �� k � � �� n �� Vt �� e t �� Sb �� ln�E �n� �� n �� P �� P �� (1) (2) The terms grouped in the first bracket on the right-hand side of the above expression have together the dimension of a viscosity, thus can be considered as the wall turbulentviscosity, � t,wall: ��Ft �� b � � �� Vt �� t,wall �� Sb , with �t,wall � n ��P c �� 1 2 � � � � � 1 4 k m ln E � n � � �� n �� P (4.90) Both forces are added to the source term of the boundary cell P, as the contribution from the wall boundary node B. The other coefficients related to the contribution from the
– 4.39 – boundary node B are then assigned to zero. The following steps are used in evaluating the coefficients in Eq. 4.52: � set all coefficients related to the contribution of the boundary node B to zero: C D C D aB � aB � 0, �a P� � �a B P� � b B D � � � 0, and b B 2D � � � 0 B �� compute �� e t using Eqs. 4.87 and 4.88 � compute the forces due to the normal and shear stresses as source terms and linearise the source: �� x-momentum: ��1 �b � bP uP � � B Fn,x � � F �� t, x �� b � �� t�ue n,x � 2v en,y � 2we n,z�e n,x � �t,wall �ve t, y � wet,z �e t, x � Sb �P �� n,P �� b � � �� P � �t �e n, x en,x �� t,wall et,x et,x � � B � S b � �� n,P �� y-momentum: �bP�B ��1 uP ��1 �b � bP uP � � B Fn,y � � F � �� t, y�� b � Sb � �� t�2 ue n,x � v en,y � 2we n,z�e n,y � �t,wall �u et, x � we t,z�e t, y �P � � B � Sb �� t �e n, y en,y�� t,wall �e t,y et,y��P � �� n,P �� b P�B ��1 vP (4.91a) (4.91b) �n,P �� b
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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
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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.
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 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: with � �� nt,b � ��
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
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Page 173 and 174: - 4.49 - Fig. 4.13 The structure of
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Page 177 and 178: 4.7 Summary - 4.53 - Development of
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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
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Page 199 and 200: - 5.15 - Fig. 5.6 Computational his
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Page 203 and 204: - 5.19 - Fig. 5.9 Definition sketch
Page 205 and 206: - 5.21 - Fig. 5.10 Computed (lines)
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Page 209 and 210: - 5.25 - 1.2U ∞ of the measuremen
Page 211 and 212: 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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