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

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285 normal components in Figures 7.87 and 7.88 also for the most part confirm the conclusions of the earlier comparisons. Near the upper and lower end walls, the fluctuation levels are under-estimated and elsewhere predicted core levels are close to those measured and now some of the features of the spanwise variation are also reproduced. Clearly in contrast to the15 ◦ unstable angle, thek-ε-AWF, even though it re- produces the main features of the temperature field, it has failed to predict the three-dimensionality of this flow. This predictive failure may be caused by the fact that the three-dimensional flow features are not as strong as for the15 ◦ unstable angle of inclination. In order to explore ways of improving the URANS predictions, the subsequent sections present comparisons obtained first with a version of the k-ε-AWF which includes a limiter for the turbulent viscosity based on realisability criteria and then with the Basic Reynolds Stress Closure.
Inclined Cavity-3D time-dependent simulations 286 T T T 1 0.8 0.6 0.4 0.2 T X 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 Y=0.7 Y=0.9 EXP k-ε-AWF 0 0 0.2 0.4 0.6 0.8 1 EXP k-ε-AWF 0 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 Y=0.3 Y=0.5 EXP k-ε-AWF X EXP k-ε-AWF 0 0 0.2 0.4 0.6 0.8 1 X T T T 1 0.8 0.6 0.4 0.2 Y=0.6 0 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 X X EXP k-ε-AWF EXP k-ε-AWF Y=0.4 0 0 0.2 0.4 0.6 0.8 1 0 0 0.2 0.4 0.6 0.8 1 X X Figure 7.72 – Temperature distributions resulting from k-ε-AWF, Z=0.5. 1 0.8 0.6 0.4 0.2 Y=0.1 EXP k-ε-AWF
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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
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131 For the thermal boundary condit
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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
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143 U/V 0 0.2 0 -0.2 0 0.2 0.4 0.6
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145 Temperature contours Stream lin
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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
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157 2 k/V0 2 k/V0 0.03 0.02 0.01 LR
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159 Figure 6.26 - Mean velocity vec
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161 6.4.2 k-ε predictions In Figur
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163 In Figure 6.29, the Nusselt num
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165 moves down from the top of the
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167 v rms /V 0 v rms /V 0 v rms /V
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169 T T T T 1 0.8 0.6 0.4 0.2 Y=0.9
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171 2 uv/V0 2 uv/V0 2 uv/V0 2 uv/V0
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173 Nu 20 15 10 5 Cold Wall 0 0 0.2
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175 T T T T 1 0.8 0.6 0.4 Y=0.95 0.
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177 V/V 0 V/V 0 0.4 0.2 0 -0.2 -0.4
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179 In Figures 6.44-6.46, the rms t
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181 θ rms /ΔΘ θ rms /ΔΘ θ rm
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183 θ rms /ΔΘ θ rms /ΔΘ θ rm
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185 of the cavity. Figure 6.49 pres
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187 T T 1 0.8 0.6 0.4 0.2 Y=0.1 0 0
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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
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193 Figure 6.58 - Stream traces and
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195 and SWF treatments is compared
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197 T T T T T 1 0.8 0.6 0.4 0.2 0 0
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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
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205 T T 1 0.8 0.6 0.4 0.2 0 0 0.2 0
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207 v rms /V 0 v rms /V 0 0.2 0.15
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209 V/V 0 V/V 0 0.4 0.2 0 -0.2 -0.4
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211 the distance between the hot an
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213 1.5 Y 2 1 0.5 0 0 0.05 0.1 0.15
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215 7.2.4 3D time dependent simulat
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217 0.65 1.5 Y 0.5 Y 2 1 0.65 0.65
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219 0.35 0.5 0.05 0.5 0.5 0.65 0.65
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221 Y Y 2 1.5 1 0.5 0.35 0.5 0.5 0.
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223 data. The comparisons show that
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225 flow at this angle of inclinati
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227 T T T 1 0.8 0.6 0.4 0.2 T X 1 0
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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
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
Page 244 and 245: 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 252 and 253: 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 282 and 283: 281 cold wall temperatures are 53
Page 284 and 285: 283 The reverse feature is observed
Page 288 and 289: 287 V/V 0 V/V 0 0 -0.2 -0.4 -0.6 0
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
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Page 302 and 303: 301 V/V 0 V/V 0 0.2 0 -0.2 -0.4 -0.
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
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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
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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
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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
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355 Z=0.88 Z=0.92 Z=1.0 Z=0.88 Z=0.
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357 Z=0.88 Z=0.92 Z=1.0 Z=0.88 Z=0.
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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
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367 τ ∗ =240 τ ∗ =360 Figure
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369 τ ∗ =0 τ ∗ =120 τ ∗ =2
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371 τ ∗ =0 τ ∗ =120 τ ∗ =2
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373 Z=0.69 τ ∗ =0 τ ∗ =120 τ
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375 Time-averaged Steady state Figu
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377 Time-averaged Z=0.23 Z=0.46 Z=0
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379 8.5 Steady-state simulation (Ra
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381 reduces towards the sides and a
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383 Figure 8.33 - Contour plots of
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385 Z=0.23 Z=0.46 Z=0.69 Z=0.87 Z=0
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387 Z=0.88 Z=0.92 Z=1.0 Z=0.88 Z=0.
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389 Z=0.88 Z=0.92 Z=1.0 Z=0.88 Z=0.
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391 Y=0.22 (Inlet) Y=-0.11 Y=0.22 (
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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
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399 τ ∗ =240 τ ∗ =360 Figure
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401 τ ∗ =0 τ ∗ =300 τ ∗ =6
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403 τ ∗ =0 τ ∗ =300 τ ∗ =6
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405 Z=0.69 τ ∗ =0 τ ∗ =120 τ
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407 Time-averaged Steady state Figu
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409 Time-averaged Z=0.23 Z=0.46 Z=0
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411 8.7 Closing remarks In this cha
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Chapter 9 Conclusions and Future Wo
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415 60 ◦ Stable Case Buoyancy-dri
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417 using the k-ε-AWF are in close
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419 the spanwise direction, suggest
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421 results of the 3D numerical sim
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423 stable and unstable configurati
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425 penetration was concerned, the
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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
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433 [47] D. Naot, A. Shavit, and M.
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435 [68] K. Hanjalić. Achievements
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437 T V/V 0 v rms /V 0 1 0.8 0.6 0.
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