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

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373 Z=0.69 τ ∗ =0 τ ∗ =120 τ ∗ =240 τ ∗ =360 Z=0.87 τ ∗ =0 τ ∗ =120 τ ∗ =240 τ ∗ =360 Figure 8.26 – Vector plots at different time steps within cross-sections normal to the axis <strong>of</strong> the penetration, at Z=0.69 and 0.87, at Ra = 4.5×10 8 .
Annular horizontal penetration 374 Time-averaged Steady state Figure 8.27 – Contour plots <strong>of</strong> time-averaged and steady-state V within longitudinal, y-z, plane, at X=0, at Ra = 4.5×10 8 .
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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
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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.
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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
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263 V/V 0 V/V 0 V/V 0 0.4 0.2 0 -0.
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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
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277 PSD (T) PSD (T) PSD (T) 1 0.8 0
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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
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291 T T 1 0.8 0.6 0.4 0.2 Y=0.7 0 0
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293 V/V 0 0.8 0.6 0.4 0.2 0 -0.2 Y=
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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
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305 T 1 0.8 0.6 0.4 0.2 Y=0.05 EXP
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307 T 1 0.8 0.6 0.4 0.2 Y=0.05 T 1
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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-
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313 U/V 0 0.15 0.1 0.05 0 -0.05 -0.
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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
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 370 and 371: 369 τ ∗ =0 τ ∗ =120 τ ∗ =2
Page 372 and 373: 371 τ ∗ =0 τ ∗ =120 τ ∗ =2
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
Page 396 and 397: 395 total turbulent kinetic energy
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
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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.
Page 436 and 437:
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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