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

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109 where B2 = y ∗ C1 υ 2 y∗ C2 υ − + α by∗2 υ (Θwall −Θref)+ 2 Prυy∗ υ Cth1y 3µυ ∗ υ 4 +Ath1 (R+Θwall −Θref) − b y α ∗ υ − PrυCth2y ∗ ∗ υ yυ 1 + − 2µυααt 2 α M 1 +y α αt ∗ υ y − bµ ∗3 + bµ 6 {[C1 +b(Θwall −Θref)]} y ∗3 υ 6 + bbµRµy ∗ υ α b Prυy ∗ υ µυ +A1y ∗ υ Cth1 5 y∗ υ +Ath1 1−bµ A2 = (C1 −C2)y ∗ υ Cth1y∗ N = C2 α + b + b y∗2 2 + b y∗2 2 Prυy + b ∗2 υ 2µυ Prυy − bbµ ∗3 υ 3µυ y ∗ υ 2 3 +Ath1 Cth1y ∗ A1 = y ∗ n − 1 α −y∗ υ lnYn 4 +Ath1 − bµy ∗ υ 5 + PrυCth2 2µυαt −Rµy ∗ υ Cth1 6 y∗ υ +Ath1 y ∗2 υ + M αt +A1 −Ry ∗ υ (4.65) (4.66) µUn −N lnYn α +y∗ (4.67) y∗2 υ −bµ 2 + (C1 −C2) y α ∗ υlnYn + (R+Θwall −Θref)fb1 + PrυCth2 fb2 µυαt Θwall −Θref + Prυy∗ υ Cth1y∗ 4 +Ath1 2 −y∗ υfb3 3µυ Prυ Cth1y µυα ∗ 3 +Ath1 lnYn − 2R y∗ υαlnYn Y − bµ ∗3 6 [C1 +b(Θwall −Θref)] − bbµ − bbµ Prυy ∗4 υ 6µυ Prυy ∗3 υ 3µυα Cth1 5 y∗ υ +Ath1 Cth1 4 y∗ υ +Ath1 C1 2 y∗ C2 υ − α +M fb4 y −bµ ∗ υ Cth1 5 6 y∗ υ +Ath1 lnYn −bbµRµfbµ y ∗ υ (4.68)
Wall Functions 110 fbµ = 1 y α ∗ n −y ∗ 1 υ − α −y∗ υ lnYn fb2 = 1 y α ∗2 n 2 −y∗ 1 n α −y∗ 1 υ + α −y∗ 2 υ lnYn fb3 = 1 y α ∗ 1 n − α − y∗ υ lnYn − 2 1 ∗ yυ 2 2 1 + α fb4 = 1 YTn (lnYTn −1)−y α αt ∗ 1 n + α lnYn − 1− α Qn + αt 1 +y αt ∗ υ The integration of Qn = y ∗ n y ∗ υ fbµ = 1 y α ∗ n −y∗ 1 υ − α lnYn YTn = [1+αt(y ∗ n −y∗ υ )] Yn = [1+α(y ∗ n −y∗ υ )] lnYT Y dy∗ , appearing in fb4, can not analytically be determined. Therefore, a series of numerical integrations are calculated in the range of 0.7 ≤ Pr ≤ 10 and the following expression is obtained to approximate Qn as: Qn = 2(y∗ n −y∗ υ)Pr 0.26 υ 1+ y∗ n −Y∗ υ Then the wall shear stress can be calculated as: τwall = − ρυ √ kP µυ A1 (4.69) (4.70) The average production of turbulence kinetic energy within the near wall control volumes is now calculated as: Pk = 1 yn √ ρυ kP µυ y ∗ n y ∗ υ µυα(y ∗ −y ∗ υ) ∂U2 ∂y∗ 2 dy ∗ (4.71)
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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
Page 60 and 61: 59 their predictions and their expe
Page 62 and 63: 61 and the fluid near the vertical
Page 64 and 65: 63 term in the ε equation performe
Page 66 and 67: 65 space and time. Then the full al
Page 68 and 69: 67 with heated vertical walls. They
Page 70 and 71: 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
Page 78 and 79: 77 diffusivity model is rather simp
Page 80 and 81: 79 ∂(ρεθ) ∂t + ∂(ρUjεθ)
Page 82 and 83: 81 stresses can be derived from the
Page 84 and 85: 83 One option is to derive and solv
Page 86 and 87: 85 εθ = 2α ∂θ Basic Different
Page 88 and 89: 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
Page 92 and 93: 91 2/3εδij. At moderate Reynolds
Page 94 and 95: 93 Figure 4.1 - A low-Reynolds-numb
Page 96 and 97: 95 energy over the near-wall contro
Page 98 and 99: 97 wall heat flux can be obtained f
Page 100 and 101: 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
Page 106 and 107: 105 C = µ2 υ ρ 2 υ kP ∂(ρUU
Page 108 and 109: 107 +A1y ∗ + by∗2 2 + bµy ∗
Page 112 and 113: 111 where ∂U2 ∂y∗ = C2y∗ +A
Page 114 and 115: 113 and the other option is to calc
Page 116 and 117: 115 explained. It might be noted th
Page 118 and 119: 117 near-wall treatments available
Page 120 and 121: 119 N n W w P e E s S Figure 5.2 -
Page 122 and 123: 121 φw = φW + φP −φW 2 − φ
Page 124 and 125: 123 The schemes are SIMPLE (“Semi
Page 126 and 127: 125 Pe = P ∗ +P ′ ′ 1 +P 2 (5
Page 128 and 129: 127 Update P ′ from equation 5.26
Page 130 and 131: 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
Page 136 and 137: 135 stable and unstable inclined ta
Page 138 and 139: 137 variations. In these calculatio
Page 140 and 141: 139 Rayleigh number Ra = βg∆ΘL3
Page 142 and 143: 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 154 and 155: 153 diagrams show that the LRN k-ε
Page 156 and 157: 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
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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.
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
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181 θ rms /ΔΘ θ rms /ΔΘ θ rm
Page 184 and 185:
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
Page 222 and 223:
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
Page 230 and 231:
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
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
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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
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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
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
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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
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
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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.
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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 (
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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
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369 τ ∗ =0 τ ∗ =120 τ ∗ =2
Page 372 and 373:
371 τ ∗ =0 τ ∗ =120 τ ∗ =2
Page 374 and 375:
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
Page 398 and 399:
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
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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
Page 430 and 431:
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
Page 438:
437 T V/V 0 v rms /V 0 1 0.8 0.6 0.
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