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

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315 u rms /V 0 0.15 0.1 0.05 0 0 0.2 0.4 0.6 0.8 1 Z u rms /V 0 Y=0.9 EXP k-ε-AWF 0.15 0.1 0.05 Y=0.1 u rms /V 0 0.15 0.1 0.05 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 EXP k-ε-AWF Z Figure 7.105 – Time-averaged rms velocity fluctuations resulting fromk-ε-AWF with limiter, X=0.5. Z Y=0.5
Inclined Cavity-3D time-dependent simulations 316 Nu Nu 20 15 10 5 0 0 0.2 0.4 0.6 0.8 1 20 15 10 5 Cold Wall Y EXP k-ε-AWF Z=0.5 Cold Wall Z=0.625 EXP k-ε-AWF 0 0 0.2 0.4 0.6 0.8 1 Y Nu Nu 20 15 10 5 Cold Wall Z=0.75 0 0 0.2 0.4 0.6 0.8 1 20 15 10 5 Y Y EXP k-ε-AWF Cold Wall Z=0.875 EXP k-ε-AWF 0 0 0.2 0.4 0.6 0.8 1 Figure 7.106 – Nusselt distributions resulting from k-ε-AWF with limiter. 7.3.4 Simulation by RSM-Basic Since neither of the two versions of the k-ε tested has been as successful in reproducing the three-dimensional flow features of the 15 ◦ stable cavity, as was the case for the 15 ◦ unstable angle, the next logical step was to introduce the Basic Reynolds stress model, RSM, again with the analytical wall function. In this section, the flow simulation by RSM-Basic using AWF is compared with the experimental data. Even though the computation was time-dependent as well as three-dimensional, the solution reached the steady-state conditions in the middle of the cavity, with some unsteadiness near the end walls. For this reason all the comparisons presented in this section, involve time-averaged predictions. Figures 7.107, 7.112, 7.113 and 7.115 present the mean temperature distri- bution within longitudinal planes. The comparisons show that, in common with the k-ε, the Basic RSM, apart from some minor discrepancies, is able to
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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.
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 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 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 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 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
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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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