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

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379 8.5 Steady-state simulation (Ra = 3.1×10 13 ) In the previous sections, a horizontal penetration test case was investigated when Ra = 4.5 × 10 8 . In this Rayleigh number as it was shown before, the turbulence level was low. For instance, the ratio of turbulent and molecular viscosity was around 1.0 inside the tube. In this section a horizontal penetration test case is simulated by the k-ε-AWF when the Rayleigh number has the value of3.1×10 13 to study the effect of higher turbulence on the flow structure. Figures 8.32-8.34 show the flow in the central, X=0, longitudinal, y-z, plane. A comparison between the axial velocity contours at the two different Rayleigh numbers, Figures 8.6 and 8.33, shows that the increase in Rayleigh number causes the outward motion to be confined closer to the lower side of the annu- lus. Outside the penetration, the axial motion is similar to that at the lower Ra. Corresponding comparisons for the vertical velocity component show for this velocity component the Rayleigh number has little effect. Similar comparisons between Figures 8.7 and 8.34, show that increasing the Rayleigh number has an even weaker effect on the normalised temperature. Comparisons of the turbulent kinetic energy contours, Figures 8.7 and 8.34, show that again within the penetration, there is very little turbulence generation along the surface of the cold tube, and as also noted in the lower Ra case, most of the turbulence is generated by the downward motion along the vertical wall, below the penetration opening. The normalisedk levels, however, increase by an order of magnitude as the Rayleigh number increases from4.5×10 8 to3.1×10 13 . The spanwise, U, velocity comparisons, Figures 8.5 and 8.32, show very little effect of Rayleigh number, while contours of the pressure deviation from hydrostatic conditions, Figures 8.6 and 8.33, show that at the higher Rayleigh number within the penetration the positive deviation occupies a greater proportion of the lower half, but the overall variation is the same at both Ra values. Figures 8.35-8.36 show the contour plots of U, V, W, P ” , k and T inside the penetration, within cross-sectional planes normal to its axis, at locations Z=0.23, 0.46, 0.69 and 0.87. Comparisons of mean velocity contours, Figures 8.9 and 8.35, reveal a stronger Rayleigh number effect. Near the closed end of the penetration, Z of 0.23 and 0.46, the contours of the vertical velocity are
Annular horizontal penetration 380 more symmetric at the higher Rayleigh number, which suggests that there is less large-scale unsteadiness. The axial velocity contours indicate that at the higher Rayleigh number, the outward motion along the bottom side of the penetration is stronger, a trend which is consistent with the stronger buoyancy force at the higher Ra level. The normalised spanwise velocity component is generally lower at the higher Ra value, which is consistent with the presence of more symmetric conditions. The corresponding mean temperature comparisons, Figures 8.10 and 8.36, show that the overall features are the same at both Rayleigh numbers, but at the higher Ra value, the size of the lower temperature region at the bottom half of the cross-section is reduced. Since an increase in Ra has been shown to cause an increase in the flow entering the penetration, it is inevitable that the cold inner pipe will have a weaker effect on the temperature of the fluid within the penetration. The contour plots of the turbulent kinetic energy, Figure 8.36, confirm earlier finding, that within the penetration turbulence is generated around surface of the inner pipe and then it is con- vected downwards. Turbulence levels are also shown to increase closer to the exit of the penetration. The normalised deviation from the hydrostatic pressure is also not significantly affected by the change in Rayleigh number. Figure 8.36 shows turbulence contour plots inside the cavity at Z=0.23, 0.46, 0.69 and 0.87. As it was expected turbulence level is larger compared to lower Ra case. They also show that turbulence generated underneath the cold tube penetrated further down compared with the lower Ra case. Figure 8.36 shows contour plots of P ” and temperature. P ” implicates to deviation from the hydrostatic pressure. They show that the maximum deviation from the hydrostatic pressure occurs at the lower half of the tube which temperature is slightly reduced due to existence of the cold tube at the middle. The reason for slight temperature change is that the cooling power is not enough to make signifi- cant thermal changes inside the domain. Figures 8.37-8.39 show the contour plots of U, V, W, P ” , k and T outside the penetration for the higher Ra case. Outside the penetration, as seen in Figure 8.37, there is strong downward motion in the middle region, which
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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
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323 u rms /V 0 u rms /V 0 0.15 0.1
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325 T 1 0.8 0.6 0.4 0.2 Y=0.05 T 1
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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
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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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 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 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
Page 424 and 425: 423 stable and unstable configurati
Page 426 and 427: 425 penetration was concerned, the
Page 428 and 429: 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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