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

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361 8.4 Time-dependent simulation (Ra = 4.5×10 8 ) Figures 8.17-8.31 show the solution of 3D time dependent simulation of the horizontal penetration test case by k-ε-AWF. In the steady-state simulation, it has been observed that results are not symmetrical, specially inside the tube away from the open end of the tube. That is why time dependent calculations are employed for this simulation to obtain a better understanding of how flow develops in this horizontal penetration test case. In the time dependent calculations, the normalized time step is set equal to 0.083,normalized by D2 V0 (where D2 is the diameter of the tube and V0 = √ gβ∆ΘD2) and the implicit scheme is employed for time descritisation. In each time step, it is set to iterate 20 times unless it gets to convergence criteria (all residuals below 10 −3 ). Figures 8.17-8.20 show velocity contours within the central (X=0) longitu- dinal plane, inside and outside the penetration. They show that downward flow which separates from the cold tube is not uniform all along the tube. The shape of the downward flow changes with time. The component of velocity along the tube (W/V0) is time dependent as well. The contours show that the flow generally moves into the tube from the upper part of the tube and moves out from the lower portion of the tube but the shape of the flow changes with time. Figures 8.21-8.26 show vector plots and contour plots of V, W and k within the penetration at cross-sections normal to its axis, at locations Z=0.23, 0.46, 0.69 and 0.87, at different time steps. Both sets of mean velocity contours show that over most of the penetration (Z = 0.23 to 0.79) there are strong oscillations. By the exit, however, these oscillations appear to have died down. This is also confirmed by flow animation videos. It thus appears that as the buoyancy effect weakens and the velocities become lower, inside the penetration, flow conditions become more unstable. The corresponding contour plots of the instantaneous turbulent kinetic energy and also the plots of the instantaneous velocity vectors also confirm the oscillatory nature of the flow within the penetration.
Annular horizontal penetration 362 One question that consequently arises, is how influential these instabilities are on the flow development, which in turn leads to the question of whether it is necessary to simulate these flows as three-dimensional and time-dependent. To address these questions, time-averaged predictions (resulting from the time- dependent computations) are compared with those obtained in the steady state computations. Time-averaging computations were carried out over a normalised period of time equal to 1000. This time period is larger than the time-scale calculated based on the largest length in the computational domain and V0. First, Figures 8.27 and 8.28 compare time-averaged and steady-state contours of the vertical and axial velocities respectively. The vertical velocity comparisons show differences in the flow structures within the penetration, with the steady-state computations returning a more fragmented flow field. The steady state computations also return a stronger downward motion along the vertical wall, below the open end of the penetration. The corresponding comparisons for the axial velocity in Figure 8.28, indicate that the time-averaged axial flow inside the penetration is stronger than that of the steady-state computations. Figures 8.29 and 8.30 show similar comparisons, involving the same velocity components, but for cross-sectional planes normal to the axis of the penetration, at locations Z=0,23, 0.46, 0.69 and 0.87 within the penetration. The vertical velocity comparisons, Figure 8.29, show that there are noticeable differences in the two predicted flow fields. In contrast to those of the steady-state predictions, the time-averaged contours of the vertical velocity, are symmetric over the entire length of the penetration. Even at the exit, Z=0.87, where both approaches result in symmetric conditions, the two sets of contours have some differences. The corresponding comparisons of the axial velocity contours, Figure 8.30, are perhaps of greater significance. They confirm that the time- dependent simulation returns a stronger inward flow within the penetration than the steady state simulation. This is of course an important characteris- tic of this flow and these comparisons suggest that, at least for this Rayleigh
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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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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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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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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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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
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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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Page 366 and 367: 365 τ ∗ =240 τ ∗ =360 Figure
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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 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
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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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