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

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3.2 Mean Flow Equations and their solution . . . . . . . . . . . . . . 70 3.3 RANS Turbulence models . . . . . . . . . . . . . . . . . . . . . . 72 3.3.1 High-Reynolds Numberk-ε Model . . . . . . . . . . . . . 72 3.3.2 Further modification in High-Rek-ε model for 3D simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.3.3 Low-Reynolds Numberk-ε model . . . . . . . . . . . . . 75 3.3.4 Approximation of turbulent heat fluxuiθ . . . . . . . . . 76 3.3.5 Low-Reynolds-Number Three and Four Equations model k-ε-θ 2 and k-ε-θ 2 -εθ . . . . . . . . . . . . . . . . . . . . . . 78 3.3.6 High-Reynolds-Number Three and Four Equations model k-ε-θ 2 and k-ε-θ 2 -εθ . . . . . . . . . . . . . . . . . . . . . . 80 3.3.7 Differential Stress Closure . . . . . . . . . . . . . . . . . . 80 3.3.8 Differential Stress Closure Used withθ 2 andεθ Transport Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4 Wall Functions 92 4.1 Conventional Wall Functions . . . . . . . . . . . . . . . . . . . . 92 4.1.1 Simple Hydraulic Wall Function . . . . . . . . . . . . . . 92 4.1.2 Standard Wall Function . . . . . . . . . . . . . . . . . . . 95 4.2 Analytical Wall Functions . . . . . . . . . . . . . . . . . . . . . . 98 4.2.1 Derivation of Analytical Wall Functions . . . . . . . . . . 98 4.2.2 Thermal Wall Function When the Near Wall Cell Is Thicker Than the Viscous Sublayer . . . . . . . . . . . . . . . . . . 100 4.2.3 Thermal Wall Function When the Near Wall Cell Is Thin- ner Than the Viscous Sublayer . . . . . . . . . . . . . . . 103 4.2.4 Hydrodynamic Wall Function When the Near Wall Cell Is Thicker Than the Viscous Sublayer . . . . . . . . . . . 104 4.2.5 Hydrodynamic Wall Function When Near Wall Cell Is Thinner Than Viscous Sublayer . . . . . . . . . . . . . . . 111 4.2.6 Convection and pressure gradient terms . . . . . . . . . 112 5 Numerical Implementation 116 5.1 TEAM Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 5.2 STREAM Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 5.3 Approximation of Convective and Diffusive terms . . . . . . . . 118 3
5.4 The velocity-pressure linkage scheme . . . . . . . . . . . . . . . 122 5.5 Sequence of solution steps . . . . . . . . . . . . . . . . . . . . . . 126 5.6 Time-dependent computations . . . . . . . . . . . . . . . . . . . 129 5.7 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 130 5.8 Simulation of inclined cavity . . . . . . . . . . . . . . . . . . . . 132 5.9 Implementation of buoyancy related terms in the STREAM code 133 5.10 Closing Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6 Inclined Cavity-2D simulations 134 6.1 Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.2 Test case description . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.3 Overview of 2-dimensional flow in the tall, differentially heated cavities at different angles of inclination . . . . . . . . . . . . . . 143 6.4 Inclined tall cavity -60 ◦ stable . . . . . . . . . . . . . . . . . . . . 157 6.4.1 Flow Pattern (including comparison with 90 ◦ case) . . . 157 6.4.2 k-ε predictions . . . . . . . . . . . . . . . . . . . . . . . . 161 6.4.3 RSM predictions . . . . . . . . . . . . . . . . . . . . . . . 167 6.4.4 Employment ofθ 2 andεθ transport equations . . . . . . . 173 6.5 Inclined tall cavity -60 ◦ unstable . . . . . . . . . . . . . . . . . . 184 6.5.1 Numerical predictions . . . . . . . . . . . . . . . . . . . . 184 6.6 Inclined tall cavity -5 ◦ stable . . . . . . . . . . . . . . . . . . . . 191 6.6.1 Test case description . . . . . . . . . . . . . . . . . . . . . 191 6.6.2 Flow pattern . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6.6.3 Comparisons ofk-ε using LRN, AWF and SWF with LES data within the inclined5 ◦ stable tall cavity . . . . . . . . 194 6.7 Closing remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7 Inclined Cavity-3D time-dependent simulations 200 7.1 Validation of STREAM . . . . . . . . . . . . . . . . . . . . . . . . 201 7.2 3D time dependent simulation of inclined tall cavity -15 ◦ unstable210 7.2.1 Test case description . . . . . . . . . . . . . . . . . . . . . 210 7.2.2 Flow Pattern . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.2.3 Grid for analytical wall functions . . . . . . . . . . . . . . 212 7.2.4 3D time dependent simulation by k-ε-AWF . . . . . . . . 215 7.2.5 Time-averaged data byk-ε-AWF . . . . . . . . . . . . . . 222 4
Page 1: The computation of turbulent natura
Page 6 and 7: 7.2.6 FFT analysis . . . . . . . .
Page 8 and 9: List of Figures 2.1 Comparison of e
Page 10 and 11: 6.15 Temperature and stream lines w
Page 12 and 13: 6.45 Temperature fluctuations withi
Page 14 and 15: 7.9 rms velocity fluctuations compa
Page 16 and 17: 7.43 Temperature contours within x-
Page 18 and 19: 7.82 Mean velocity distributions re
Page 20 and 21: 7.119Time-averaged rms velocity flu
Page 22 and 23: 8.23 Contour plots of V, W andk at
Page 24 and 25: 8.49 Contour plots of V, W andk at
Page 26 and 27: For the inclined tall cavities, in
Page 28 and 29: Declaration No portion of the work
Page 30 and 31: Acknowledgements I would like to ex
Page 32 and 33: Nomenclature Acronyms AWF Analytica
Page 34 and 35: λ Thermal conductivity µ Molecula
Page 36 and 37: cl Equilibrium length scale constan
Page 38 and 39: yd Thickness of the dissipative lay
Page 40 and 41: Chapter 1 Introduction In this rese
Page 42 and 43: 41 need fine numerical resolution n
Page 44 and 45: 43 nonlinear discretization methods
Page 46 and 47: 45 the only way to achieve computat
Page 48 and 49: 47 elaborate expressions for modell
Page 50 and 51: Chapter 2 Literature Review In this
Page 52 and 53: 51 local Nusselt number occurs in t
Page 54 and 55:
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
Page 72 and 73:
71 • Large eddy simulation (LES):
Page 74 and 75:
73 ρuiuj = 2 3 ρkδij ∂Ui −
Page 76 and 77:
75 3.3.3 Low-Reynolds Number k-ε m
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77 diffusivity model is rather simp
Page 80 and 81:
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
Page 90 and 91:
89 are defined as: A = 1− 9 8 (A2
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91 2/3εδij. At moderate Reynolds
Page 94 and 95:
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
Page 102 and 103:
101 is: Θ1 = Prυ µυ + Prυbµ
Page 104 and 105:
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 ∗
Page 110 and 111:
109 where B2 = y ∗ C1 υ 2 y∗
Page 112 and 113:
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
Page 128 and 129:
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
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135 stable and unstable inclined ta
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137 variations. In these calculatio
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139 Rayleigh number Ra = βg∆ΘL3
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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 148 and 149:
Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
153 diagrams show that the LRN k-ε
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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
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161 6.4.2 k-ε predictions In Figur
Page 164 and 165:
163 In Figure 6.29, the Nusselt num
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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 182 and 183:
181 θ rms /ΔΘ θ rms /ΔΘ θ rm
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
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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
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201 After validation of the STREAM
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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
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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.
Page 232 and 233:
231 u rms /V 0 u rms /V 0 u rms /V
Page 234 and 235:
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
Page 238 and 239:
237 U/V 0 0.4 0.2 0 -0.2 -0.4 Y=0.9
Page 240 and 241:
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
Page 246 and 247:
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
Page 254 and 255:
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
Page 258 and 259:
257 Y 2 1.5 1 0.5 0.3 0.45 0.45 0.3
Page 260 and 261:
259 not sufficiently detailed to pr
Page 262 and 263:
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
Page 268 and 269:
267 v rms /V 0 v rms /V 0 v rms /V
Page 270 and 271:
269 v rms /V 0 v rms /V 0 0.25 0.2
Page 272 and 273:
271 X X X 0.15 0.1 0.05 0 0 0.1 0.2
Page 274 and 275:
273 1.5 Y 2 1 0.5 0 0 0.05 0.1 0.15
Page 276 and 277:
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
Page 282 and 283:
281 cold wall temperatures are 53
Page 284 and 285:
283 The reverse feature is observed
Page 286 and 287:
285 normal components in Figures 7.
Page 288 and 289:
287 V/V 0 V/V 0 0 -0.2 -0.4 -0.6 0
Page 290 and 291:
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
Page 298 and 299:
297 v rms /V 0 v rms /V 0 0.15 EXP
Page 300 and 301:
299 active sides, y-z, there are re
Page 302 and 303:
301 V/V 0 V/V 0 0.2 0 -0.2 -0.4 -0.
Page 304 and 305:
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=
Page 312 and 313:
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.
Page 318 and 319:
317 reproduce the measured temperat
Page 320 and 321:
319 T T T 1 0.8 0.6 0.4 0.2 Y=0.9 E
Page 322 and 323:
321 U/V 0 U/V 0 0.1 0.05 0 -0.05 -0
Page 324 and 325:
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.
Page 332 and 333:
331 V/V 0 V/V 0 0.6 0.4 0.2 0 -0.2
Page 334 and 335:
333 v rms /V 0 v rms /V 0 0.15 EXP
Page 336 and 337:
335 Nu Nu X 0.15 0.1 0.05 20 15 10
Page 338 and 339:
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
Page 342 and 343:
341 8.2 Grid Symmetry boundary cond
Page 344 and 345:
343 Figure 8.3 - Grid in x-y plane.
Page 346 and 347:
345 the diameter of the cold pipe i
Page 348 and 349:
347 downward motion close to the pe
Page 350 and 351:
349 Figure 8.5 - Contour plots of U
Page 352 and 353:
351 Figure 8.7 - Contour plots of T
Page 354 and 355:
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
Page 362 and 363:
361 8.4 Time-dependent simulation (
Page 364 and 365:
363 number, it is necessary to empl
Page 366 and 367:
365 τ ∗ =240 τ ∗ =360 Figure
Page 368 and 369:
367 τ ∗ =240 τ ∗ =360 Figure
Page 370 and 371:
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
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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
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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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