The computation of turbulent natural convection flows - Turbulence ...

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181 θ rms /ΔΘ θ rms /ΔΘ θ rms /ΔΘ θ rms /ΔΘ 0.12 0.1 0.08 0.06 0.04 0.02 0.12 Y=0.95 EXP RSM-θ 2 -ε θ -TCL 0 0 0.2 0.4 0.6 0.8 1 0.1 0.08 0.06 0.04 0.02 0.12 X 0 0 0.2 0.4 0.6 0.8 1 0.1 0.08 0.06 0.04 0.02 0.12 Y=0.7 X EXP RSM-θ 2 -ε θ -TCL 0 0 0.2 0.4 0.6 0.8 1 0.1 0.08 0.06 0.04 0.02 Y=0.5 X Y=0.1 X EXP RSM-θ 2 -ε θ -TCL EXP RSM-θ 2 -ε θ -TCL 00 0.2 0.4 0.6 0.8 1 θ rms /ΔΘ θ rms /ΔΘ θ rms /ΔΘ θ rms /ΔΘ 0.12 0.1 0.08 0.06 0.04 0.02 0.12 0.08 0.06 0.04 0.02 Y=0.95 EXP 4-equation EVM 0 0 0.2 0.4 0.6 0.8 1 0.1 0.12 0.08 0.06 0.04 0.02 X Y=0.7 0 0 0.2 0.4 0.6 0.8 1 0.1 0.12 X Y=0.5 X EXP 4-equation EVM EXP 4-equation EVM 0 0 0.2 0.4 0.6 0.8 1 0.1 0.08 0.06 0.04 0.02 Y=0.1 00 0.2 0.4 0.6 0.8 1 Figure 6.44 – Temperature fluctuations within vertical cavity using RSM-TCL(AWF) and 4-equations EVM. X EXP 4-equation EVM
Inclined Cavity-2D simulations 182 θ rms /ΔΘ θ rms /ΔΘ θ rms /ΔΘ θ rms /ΔΘ 0.12 0.1 0.08 0.06 0.04 0.02 0.12 Y=0.95 EXP RSM-θ 2 -ε θ -Basic 0 0 0.2 0.4 0.6 0.8 1 0.1 0.08 0.06 0.04 0.02 0.12 X 0 0 0.2 0.4 0.6 0.8 1 0.1 0.08 0.06 0.04 0.02 0.12 Y=0.7 EXP RSM-θ 2 -ε θ -Basic X 0 0 0.2 0.4 0.6 0.8 1 0.1 0.08 0.06 0.04 0.02 Y=0.5 Y=0.1 EXP RSM-θ 2 -ε θ -Basic X EXP RSM-θ 2 -ε θ -Basic 0 0 0.2 0.4 0.6 0.8 1 X θ rms /ΔΘ θ rms /ΔΘ θ rms /ΔΘ θ rms /ΔΘ 0.12 0.1 0.08 0.06 0.04 0.02 0.12 0.08 0.06 0.04 0.02 Y=0.95 EXP 4-equations-AWF 0 0 0.2 0.4 0.6 0.8 1 0.1 0.12 0.08 0.06 0.04 0.02 X Y=0.7 EXP 4-equations-AWF 0 0 0.2 0.4 0.6 0.8 1 0.1 0.12 Y=0.5 X EXP 4-equations-AWF 0 0 0.2 0.4 0.6 0.8 1 0.1 0.08 0.06 0.04 0.02 X EXP 4-equations-AWF 0 0 0.2 0.4 0.6 0.8 1 Figure 6.45 – Temperature fluctuations within 60 ◦ stable inclined cavity using RSM-basic(AWF) and 4-equations EVM. Y=0.1 X
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The computation of turbulent natura
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3.2 Mean Flow Equations and their s
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7.2.6 FFT analysis . . . . . . . .
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List of Figures 2.1 Comparison of e
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6.15 Temperature and stream lines w
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6.45 Temperature fluctuations withi
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7.9 rms velocity fluctuations compa
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7.43 Temperature contours within x-
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7.82 Mean velocity distributions re
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7.119Time-averaged rms velocity flu
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8.23 Contour plots of V, W andk at
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8.49 Contour plots of V, W andk at
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For the inclined tall cavities, in
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Declaration No portion of the work
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Acknowledgements I would like to ex
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Nomenclature Acronyms AWF Analytica
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λ Thermal conductivity µ Molecula
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cl Equilibrium length scale constan
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yd Thickness of the dissipative lay
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Chapter 1 Introduction In this rese
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41 need fine numerical resolution n
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43 nonlinear discretization methods
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45 the only way to achieve computat
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47 elaborate expressions for modell
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Chapter 2 Literature Review In this
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51 local Nusselt number occurs in t
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53 Cavities and Enclosures Kirkpatr
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55 boundary conditions which avoid
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57 of cold wall. They showed that t
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59 their predictions and their expe
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61 and the fluid near the vertical
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63 term in the ε equation performe
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65 space and time. Then the full al
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67 with heated vertical walls. They
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69 horizontal axis. This tall cavit
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71 • Large eddy simulation (LES):
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73 ρuiuj = 2 3 ρkδij ∂Ui −
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75 3.3.3 Low-Reynolds Number k-ε m
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77 diffusivity model is rather simp
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79 ∂(ρεθ) ∂t + ∂(ρUjεθ)
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81 stresses can be derived from the
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83 One option is to derive and solv
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85 εθ = 2α ∂θ Basic Different
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87 c1 c2 c3 c1w c2w 1.8 0.6 0.5 0.5
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89 are defined as: A = 1− 9 8 (A2
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91 2/3εδij. At moderate Reynolds
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93 Figure 4.1 - A low-Reynolds-numb
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95 energy over the near-wall contro
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97 wall heat flux can be obtained f
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99 values. • The convection terms
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101 is: Θ1 = Prυ µυ + Prυbµ
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103 where Θwall = Θn − Prυ + P
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105 C = µ2 υ ρ 2 υ kP ∂(ρUU
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107 +A1y ∗ + by∗2 2 + bµy ∗
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109 where B2 = y ∗ C1 υ 2 y∗
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111 where ∂U2 ∂y∗ = C2y∗ +A
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113 and the other option is to calc
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115 explained. It might be noted th
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117 near-wall treatments available
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119 N n W w P e E s S Figure 5.2 -
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121 φw = φW + φP −φW 2 − φ
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123 The schemes are SIMPLE (“Semi
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125 Pe = P ∗ +P ′ ′ 1 +P 2 (5
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127 Update P ′ from equation 5.26
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129 5.6 Time-dependent computations
Page 132 and 133: 131 For the thermal boundary condit
Page 134 and 135: 133 gx = g cos(Ψ) (5.42) gy = −g
Page 136 and 137: 135 stable and unstable inclined ta
Page 138 and 139: 137 variations. In these calculatio
Page 140 and 141: 139 Rayleigh number Ra = βg∆ΘL3
Page 142 and 143: 141 V/V 0 0.2 0 -0.2 Y=0.95 EXP 0 0
Page 144 and 145: 143 U/V 0 0.2 0 -0.2 0 0.2 0.4 0.6
Page 146 and 147: 145 Temperature contours Stream lin
Page 154 and 155: 153 diagrams show that the LRN k-ε
Page 156 and 157: 155 2 k/V0 V/V 0 0.08 0.06 0.04 0.0
Page 158 and 159: 157 2 k/V0 2 k/V0 0.03 0.02 0.01 LR
Page 160 and 161: 159 Figure 6.26 - Mean velocity vec
Page 162 and 163: 161 6.4.2 k-ε predictions In Figur
Page 164 and 165: 163 In Figure 6.29, the Nusselt num
Page 166 and 167: 165 moves down from the top of the
Page 168 and 169: 167 v rms /V 0 v rms /V 0 v rms /V
Page 170 and 171: 169 T T T T 1 0.8 0.6 0.4 0.2 Y=0.9
Page 172 and 173: 171 2 uv/V0 2 uv/V0 2 uv/V0 2 uv/V0
Page 174 and 175: 173 Nu 20 15 10 5 Cold Wall 0 0 0.2
Page 176 and 177: 175 T T T T 1 0.8 0.6 0.4 Y=0.95 0.
Page 178 and 179: 177 V/V 0 V/V 0 0.4 0.2 0 -0.2 -0.4
Page 180 and 181: 179 In Figures 6.44-6.46, the rms t
Page 184 and 185: 183 θ rms /ΔΘ θ rms /ΔΘ θ rm
Page 186 and 187: 185 of the cavity. Figure 6.49 pres
Page 188 and 189: 187 T T 1 0.8 0.6 0.4 0.2 Y=0.1 0 0
Page 190 and 191: 189 θ rms /ΔΘ θ rms /ΔΘ 0.15
Page 192 and 193: 191 Nu 20 15 10 5 Cold Wall 0 0 0.2
Page 194 and 195: 193 Figure 6.58 - Stream traces and
Page 196 and 197: 195 and SWF treatments is compared
Page 198 and 199: 197 T T T T T 1 0.8 0.6 0.4 0.2 0 0
Page 200 and 201: 199 Nu 60 40 20 LES AWF LRN SWF 0 0
Page 202 and 203: 201 After validation of the STREAM
Page 204 and 205: 203 V/V 0 V/V 0 0.5 0 -0.5 0.6 0.4
Page 206 and 207: 205 T T 1 0.8 0.6 0.4 0.2 0 0 0.2 0
Page 208 and 209: 207 v rms /V 0 v rms /V 0 0.2 0.15
Page 210 and 211: 209 V/V 0 V/V 0 0.4 0.2 0 -0.2 -0.4
Page 212 and 213: 211 the distance between the hot an
Page 214 and 215: 213 1.5 Y 2 1 0.5 0 0 0.05 0.1 0.15
Page 216 and 217: 215 7.2.4 3D time dependent simulat
Page 218 and 219: 217 0.65 1.5 Y 0.5 Y 2 1 0.65 0.65
Page 220 and 221: 219 0.35 0.5 0.05 0.5 0.5 0.65 0.65
Page 222 and 223: 221 Y Y 2 1.5 1 0.5 0.35 0.5 0.5 0.
Page 224 and 225: 223 data. The comparisons show that
Page 226 and 227: 225 flow at this angle of inclinati
Page 228 and 229: 227 T T T 1 0.8 0.6 0.4 0.2 T X 1 0
Page 230 and 231: 229 U/V 0 U/V 0 U/V 0 0.4 0.2 0 -0.
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231 u rms /V 0 u rms /V 0 u rms /V
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233 u rms /V 0 u rms /V 0 0.15 0.1
Page 236 and 237:
235 T 1 0.8 0.6 0.4 0.2 T Y=0.5 X 1
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237 U/V 0 0.4 0.2 0 -0.2 -0.4 Y=0.9
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239 u rms /V 0 0.25 0.2 0.15 0.1 0.
Page 242 and 243:
241 Figure 7.34 - Time average vect
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243 Y 2 1.5 1 0.5 0.3 0.5 0.55 0.35
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245 PSD (T) PSD (T) PSD (T) 1 0.8 0
Page 248 and 249:
247 PSD (T) PSD (T) PSD (T) 1 0.8 0
Page 250 and 251:
249 PSD (T) PSD (T) PSD (T) 1 0.8 0
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251 the central, z=0.5, longitudina
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253 1.5 Y 2 1 0.5 Y 0 0 0.05 0.1 0.
Page 256 and 257:
255 τ * =550 τ * =495 τ * =440
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257 Y 2 1.5 1 0.5 0.3 0.45 0.45 0.3
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259 not sufficiently detailed to pr
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261 T 1 0.8 0.6 0.4 0.2 T Y=0.5 X 1
Page 264 and 265:
263 V/V 0 V/V 0 V/V 0 0.4 0.2 0 -0.
Page 266 and 267:
265 V/V 0 V/V 0 0.4 0.2 0 -0.2 -0.4
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267 v rms /V 0 v rms /V 0 v rms /V
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269 v rms /V 0 v rms /V 0 0.25 0.2
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271 X X X 0.15 0.1 0.05 0 0 0.1 0.2
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273 1.5 Y 2 1 0.5 0 0 0.05 0.1 0.15
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275 PSD (T) PSD (T) PSD (T) 1 0.8 0
Page 278 and 279:
277 PSD (T) PSD (T) PSD (T) 1 0.8 0
Page 280 and 281:
279 PSD (T) PSD (T) PSD (T) 1 0.8 0
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281 cold wall temperatures are 53
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283 The reverse feature is observed
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285 normal components in Figures 7.
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287 V/V 0 V/V 0 0 -0.2 -0.4 -0.6 0
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289 v rms /V 0 v rms /V 0 0.15 0.1
Page 292 and 293:
291 T T 1 0.8 0.6 0.4 0.2 Y=0.7 0 0
Page 294 and 295:
293 V/V 0 0.8 0.6 0.4 0.2 0 -0.2 Y=
Page 296 and 297:
295 V/V 0 V/V 0 0.6 0.4 0.2 0 -0.2
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297 v rms /V 0 v rms /V 0 0.15 EXP
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299 active sides, y-z, there are re
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301 V/V 0 V/V 0 0.2 0 -0.2 -0.4 -0.
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303 v rms /V 0 v rms /V 0 0.15 0.1
Page 306 and 307:
305 T 1 0.8 0.6 0.4 0.2 Y=0.05 EXP
Page 308 and 309:
307 T 1 0.8 0.6 0.4 0.2 Y=0.05 T 1
Page 310 and 311:
309 V/V 0 0.8 0.6 0.4 0.2 0 -0.2 Y=
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311 v rms /V 0 0.15 0.1 0.05 EXP k-
Page 314 and 315:
313 U/V 0 0.15 0.1 0.05 0 -0.05 -0.
Page 316 and 317:
315 u rms /V 0 0.15 0.1 0.05 0 0 0.
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317 reproduce the measured temperat
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319 T T T 1 0.8 0.6 0.4 0.2 Y=0.9 E
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321 U/V 0 U/V 0 0.1 0.05 0 -0.05 -0
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323 u rms /V 0 u rms /V 0 0.15 0.1
Page 326 and 327:
325 T 1 0.8 0.6 0.4 0.2 Y=0.05 T 1
Page 328 and 329:
327 V/V 0 0.4 0.2 0 -0.2 -0.4 -0.6
Page 330 and 331:
329 v rms /V 0 0.15 0.1 0.05 0 0 0.
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331 V/V 0 V/V 0 0.6 0.4 0.2 0 -0.2
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333 v rms /V 0 v rms /V 0 0.15 EXP
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335 Nu Nu X 0.15 0.1 0.05 20 15 10
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337 1.5 Y 2 1 0.5 0 0 0.1 0.2 0.3 Z
Page 340 and 341:
Chapter 8 Annular horizontal penetr
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341 8.2 Grid Symmetry boundary cond
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343 Figure 8.3 - Grid in x-y plane.
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345 the diameter of the cold pipe i
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347 downward motion close to the pe
Page 350 and 351:
349 Figure 8.5 - Contour plots of U
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351 Figure 8.7 - Contour plots of T
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353 Z=0.23 Z=0.46 Z=0.69 Z=0.87 Z=0
Page 356 and 357:
355 Z=0.88 Z=0.92 Z=1.0 Z=0.88 Z=0.
Page 358 and 359:
357 Z=0.88 Z=0.92 Z=1.0 Z=0.88 Z=0.
Page 360 and 361:
359 X=0.25 X=0.5 X=0.25 X=0.5 Figur
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361 8.4 Time-dependent simulation (
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363 number, it is necessary to empl
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365 τ ∗ =240 τ ∗ =360 Figure
Page 368 and 369:
367 τ ∗ =240 τ ∗ =360 Figure
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369 τ ∗ =0 τ ∗ =120 τ ∗ =2
Page 372 and 373:
371 τ ∗ =0 τ ∗ =120 τ ∗ =2
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373 Z=0.69 τ ∗ =0 τ ∗ =120 τ
Page 376 and 377:
375 Time-averaged Steady state Figu
Page 378 and 379:
377 Time-averaged Z=0.23 Z=0.46 Z=0
Page 380 and 381:
379 8.5 Steady-state simulation (Ra
Page 382 and 383:
381 reduces towards the sides and a
Page 384 and 385:
383 Figure 8.33 - Contour plots of
Page 386 and 387:
385 Z=0.23 Z=0.46 Z=0.69 Z=0.87 Z=0
Page 388 and 389:
387 Z=0.88 Z=0.92 Z=1.0 Z=0.88 Z=0.
Page 390 and 391:
389 Z=0.88 Z=0.92 Z=1.0 Z=0.88 Z=0.
Page 392 and 393:
391 Y=0.22 (Inlet) Y=-0.11 Y=0.22 (
Page 394 and 395:
393 The contours of the instantaneo
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395 total turbulent kinetic energy
Page 398 and 399:
397 τ ∗ =240 τ ∗ =360 Figure
Page 400 and 401:
399 τ ∗ =240 τ ∗ =360 Figure
Page 402 and 403:
401 τ ∗ =0 τ ∗ =300 τ ∗ =6
Page 404 and 405:
403 τ ∗ =0 τ ∗ =300 τ ∗ =6
Page 406 and 407:
405 Z=0.69 τ ∗ =0 τ ∗ =120 τ
Page 408 and 409:
407 Time-averaged Steady state Figu
Page 410 and 411:
409 Time-averaged Z=0.23 Z=0.46 Z=0
Page 412 and 413:
411 8.7 Closing remarks In this cha
Page 414 and 415:
Chapter 9 Conclusions and Future Wo
Page 416 and 417:
415 60 ◦ Stable Case Buoyancy-dri
Page 418 and 419:
417 using the k-ε-AWF are in close
Page 420 and 421:
419 the spanwise direction, suggest
Page 422 and 423:
421 results of the 3D numerical sim
Page 424 and 425:
423 stable and unstable configurati
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425 penetration was concerned, the
Page 428 and 429:
427 convection within rectangular c
Page 430 and 431:
429 [8] D. Cooper, T. Craft, K. Est
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431 [28] R. Boudjemadi, V. Maupu, D
Page 434 and 435:
433 [47] D. Naot, A. Shavit, and M.
Page 436 and 437:
435 [68] K. Hanjalić. Achievements
Page 438:
437 T V/V 0 v rms /V 0 1 0.8 0.6 0.
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