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

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117 near-wall treatments available in the present version of the TEAM code are low-Re-number turbulence models, the standard wall function and the analytical wall function, as described in Chapter 4. Figure 5.1 – Grid and control volume arrangements for pressure and mean velocity storage in the TEAM code. The grid arrangement adopted in the TEAM code is a fully staggered pattern, as shown in Figure 5.1. This arrangement is used in order to avoid chequer- board effects in the velocity and pressure fields. The method for grid generation in the TEAM code involves first generating the scalar control volume boundaries and then placing the scalar nodes at the centre of each control volume. The velocity nodes are located at the mid-point of the faces of the scalar control volumes. This method of grid generation is convenient, especially when wall boundaries of the geometry are located on faces of the scalar control volumes. 5.2 STREAM Code The STREAM code has been employed in this research to simulate the three dimensional 15 ◦ stable and unstable tilted tall cavities and annular horizontal penetration. STREAM stands for “Simulation of Turbulent Reynolds-averaged Equations for All Mach numbers”. This code [63] was developed to simulate two or three dimensional flows. It is a fully elliptic code and uses block structured non-orthogonal body fitted grids and the finite volume method. It can analyze the flow under both steady state and transient conditions. The
Numerical Implementation 118 STREAM code employs the SIMPLE algorithm for velocity-pressure linkage. The grid arrangement adopted in the STREAM code is a collocated grid [64]. This means that both scalar and velocity variables are stored at the centre of control volumes. Regarding the discretization of convective terms, QUICK, UPWIND [65] and UMIST [66] schemes are included in the STREAM code. The STREAM code is also designed so that it can employ different near wall treatments, and again the available choices in the present version are low-Re- number models, the standard wall function and the analytical wall function. A wide range of RANS turbulence models is available in the STREAM code. The method for grid generation in the STREAM code involves first reading the scalar control volume boundaries from external files and then placing the nodes at the centres of the control volumes. 5.3 Approximation of Convective and Diffusive terms To study fluid flow problems, the effects of convection and diffusion must be accounted for. The general form of a steady convection-diffusion transport equation for a quantity φ is: div(ρuφ) = div(Γgradφ)+Sφ (5.1) In the case of steady one-dimensional convection and diffusion in the ab- sence of sources, the above equation reduces to: whilst the continuity equation is: d d (ρuφ) = Γ dx dx dφ dx d(ρu) dx (5.2) = 0 (5.3) after integration of equation 5.2 over the control volume around node P (Figure 5.2):
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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
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 110 and 111: 109 where B2 = y ∗ C1 υ 2 y∗
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 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
Page 160 and 161: 159 Figure 6.26 - Mean velocity vec
Page 162 and 163: 161 6.4.2 k-ε predictions In Figur
Page 164 and 165: 163 In Figure 6.29, the Nusselt num
Page 166 and 167: 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
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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
Page 238 and 239:
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.
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
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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
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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
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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
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
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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
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=
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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.
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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
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321 U/V 0 U/V 0 0.1 0.05 0 -0.05 -0
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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
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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
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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
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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
Page 350 and 351:
349 Figure 8.5 - Contour plots of U
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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
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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
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
Page 396 and 397:
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
Page 432 and 433:
431 [28] R. Boudjemadi, V. Maupu, D
Page 434 and 435:
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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