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

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299 active sides, y-z, there are recirculation cells similar to those discussed in the previous section. Figures 7.89,7.94,7.95,7.96 and 7.97 show the temperature distributions within a number of longitudinal planes. These comparisons reveal that the imple- mentation of turbulent viscosity limit does not make significant changes to the temperature profiles. There are only slight changes near the top and bottom of the cavity. Figures 7.90, 7.98, 7.99 present the corresponding longitudinal velocity distributions within longitudinal, x-y, planes and Figures 7.102 and 7.103 the distribution of the longitudinal and normal velocities within the mid-plane plane parallel to the thermally active sides, x-z plane at Y=0.5. The comparisons show that introduction of the viscosity limiter, in addition to enabling the k-ε-AWF to return unstable flow conditions, leads to some improvements in the predicted time-averaged velocity field, but there still large deviations between the predicted and measured flow fields. Figures 7.92, 7.93, 7.100, 7.101, 7.104 and 7.105 show the corresponding comparisons for the corresponding rms velocity profiles. Improvements in the predicted rms are evident, most notably near the end walls, where the predicted levels are now significantly higher than those of k-ε-AWF predictions without the viscosity limiter, and close to the measured values. These predictive improvements are most likely to result from the fact that the predicted rms velocities now include both the modelled component from the solution of the transport equation for k, which represents the contribution of the small-scale motion, and the resolved component, obtained by post processing the instantaneous velocity and which represents the contribution of the large-scale unsteady motion. This is consistent with the fact that the largest improvements in the predicted rms velocity field is observed near the two end walls, where the predicted flow is most unstable. Finally in Figure 7.106, the time-averaged longitudinal distributions of the Nusselt number at four spanwise locations, Z=0.5, 0.625, 0.75, 0.865, predicted by the k-ε-AWF, are compared to the measured data. At all spanwise locations, the low Nu levels near the bottom end wall are under estimated, the higher levels near the top end wall are well predicted, but the predicted longitudinal variation in local Nusselt number from the bottom to the top end wall is non-monotonic. The local Nusselt number rises from zero at the bottom end
Inclined Cavity-3D time-dependent simulations 300 wall to either a maximum, or a plateau, then shows a dip, before rising again near the top end wall. The experimental data are not detailed enough but a local Nu variation like the one indicated by the k-ε-AWF predications is un- likely. The three-dimensional k-ε-AWF computations, thus return the correct levels for the Nusselt number, but not the correct variation. T T T 1 0.8 0.6 0.4 0.2 Y=0.9 EXP k-ε-AWF 0 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 X 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 T T 1 0.8 0.6 0.4 0.2 Y=0.7 0 0 0.2 0.4 0.6 0.8 1 1 0.8 0.6 0.4 0.2 X EXP k-ε-AWF X EXP k-ε-AWF Y=0.4 0 0 0.2 0.4 0.6 0.8 1 X X Figure 7.89 – Time-averaged temperature distributions resulting from k-ε-AWF with limiter, Z=0.5. T 1 0.8 0.6 0.4 0.2 Y=0.1 EXP k-ε-AWF 0 0 0.2 0.4 0.6 0.8 1
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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
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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.
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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
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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
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
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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
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
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
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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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