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

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343 Figure 8.3 – Grid in x-y plane. Figure 8.4 – Grid inside the tube.
Annular horizontal penetration 344 8.3 Steady-state simulation (Ra = 4.5×10 8 ) This section presents the results of steady state simulation of horizontal annular open penetration. Figures 8.5-8.8 show the contour plots of U, V, W, P ” , T,k andµt within the central, X=0, longitudinal, y-z, plane, inside and outside of the penetration. The positive axial velocity (W) denotes velocity from right to left, thus coming out of the penetration, and positive vertical velocity (V) denotes upward motion. It can be seen, from the V contours of Figure 8.5 that the flow enters to the computational domain at the inlet and moves down. The downward flow outside the penetration becomes faster along the vertical wall, below the open end of the penetration. The axial velocity outside the penetration is considerably lower than the vertical. The contours of both the axial and vertical velocity suggest that within the annular penetration there are irregular flow structures, which is consistent with flow unsteadiness. Axial flow is confined to the penetration and the region just outside it. Over the top half of the penetration, above the cold central core, the axial motion is predomi- nately inward, towards the closed end. Over the lower half of the penetration, the axial flow direction mostly reversed, with predominantly outward flow, towards the open end of the penetration. The contour plot of the vertical velocity component shows that in the penetration, as already pointed out, within the central vertical plane an irregular flow pattern is observed, with frequent changes in the direction of the vertical velocity along the axial direction. This observed flow behaviour is consistent with fluid within the annular penetration being cooled by the central core, then due to its higher density falling first out of the cavity and then dropping out of the flow domain at the lower boundary and then hot fluid being drawn into the flow domain from the top and then being drawn into the horizontal penetration to replace the cooled fluid. The spanwise, U, velocity component, as can be seen in Figure 8.5, remains low throughout. The mean temperature contours of Figure 8.7, which show that the fluid at the lower half of the penetration is at a lower temperature than the inlet temperature and that some of it spills out, are consistent with those for the axial and vertical velocity components. It is also worth noting that the fluid temperature drops by only a fraction of the differ- ence between the hot inlet and the cold pipe temperatures. The reason is that
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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
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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Page 302 and 303: 301 V/V 0 V/V 0 0.2 0 -0.2 -0.4 -0.
Page 304 and 305: 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 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
Page 366 and 367: 365 τ ∗ =240 τ ∗ =360 Figure
Page 368 and 369: 367 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 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 (
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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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