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

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List of Figures 2.1 Comparison of experimental measurements and correlations[2] 50 2.2 Schematics of the flow geometry, (a) flat plate, (b) backward- facing step, (c) forward-facing step. . . . . . . . . . . . . . . . . . 51 2.3 Velocity profiles under aiding and opposing buoyancy force[3]. 52 2.4 Effect of aiding natural convection on a turbulent flow in a vertical tube. uε is the velocity at the border of the viscous layer,um is the mean velocity in the core of the flow.[3]. . . . . . . . . . . 52 2.5 Heat transfer data and correlations for the Rayleigh-Bénard configuration[4]. 53 2.6 Schematic view of the air filled cavity[5]. . . . . . . . . . . . . . 56 2.7 Boundary layer structure at Y=0.5 (vertical velocity profile near the hot wall)[6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.8 Vector plot of mean velocity profiles (a) Non-partitioned cavity (b) Partitioned cavity[7]. . . . . . . . . . . . . . . . . . . . . . . . 58 4.1 A low-Reynolds-number and wall function grids. . . . . . . . . 93 4.2 Turbulent velocity boundary layer. . . . . . . . . . . . . . . . . . 93 4.3 Schematic of near-wall cell. . . . . . . . . . . . . . . . . . . . . . 102 5.1 Grid and control volume arrangements for pressure and mean velocity storage in the TEAM code. . . . . . . . . . . . . . . . . . 117 5.2 Schematic of control volumes. . . . . . . . . . . . . . . . . . . . . 119 5.3 Calculation procedure . . . . . . . . . . . . . . . . . . . . . . . . 127 5.4 Schematic of the inclined cavity. . . . . . . . . . . . . . . . . . . . 132 6.1 100x270 Grid for low-Renumber-model computation of the tall cavity, (scale of the small side is enlarged by a factor of 7). . . . 136 6.2 40x220 Grid for high-Renumber-model computation of the tall cavity, (scale of the small side is enlarged by a factor of 7). . . . 136 7
6.3 80x180 Grid for low-Re-number computation of the inclined tall cavity at5 ◦ stable, (scale of the small side is enlarged by a factor of 7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.4 40x180 Grid for high-Re-number computation of the inclined tall cavity at 5 ◦ stable, (scale of the small side is enlarged by a factor of 7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 6.5 Schematic of inclined tall cavity and its thermal boundary con- dition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 6.6 V and Vrms profiles at , and off, the centreline of the 60 ◦ stable cavity, to illustrate the two-dimensionality of the flow, Cooper et al. [8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.7 Velocity profile in the 60 ◦ stable cavity at the centreline where Y=0.95, Cooper et al. [8] . . . . . . . . . . . . . . . . . . . . . . . 141 6.8 Temperature profiles at , and off, the centreline of the 60 ◦ unstable cavity, to illustrate the two-dimensionality of the flow, Cooper et al. [8] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 6.9 Wall-parallel velocity profiles in the 15 ◦ stable cavity along the spanwise direction where X=0.5, to illustrate the three-dimensionality of the flow, Cooper et al. [8] . . . . . . . . . . . . . . . . . . . . . 142 6.10 Wall-normal velocity profiles in the 15 ◦ unstable cavity along the spanwise direction where X=0.5, to illustrate the three-dimensionality of the flow, Cooper et al. [8] . . . . . . . . . . . . . . . . . . . . . 143 6.11 Temperature and stream lines within a 0 ◦ stable inclined cavity resulting from RSM-Basic (Ra = 0.86×10 6 ), (scale of the small side is enlarged by a factor of 7). . . . . . . . . . . . . . . . . . . 144 6.12 Temperature and stream lines within a 5 ◦ stable inclined cavity resulting from RSM-Basic (Ra = 0.86×10 6 ), (scale of the small side is enlarged by a factor of 7). . . . . . . . . . . . . . . . . . . 145 6.13 Temperature and stream lines within a10 ◦ stable inclined cavity resulting from RSM-Basic (Ra = 0.86×10 6 ), (scale of the small side is enlarged by a factor of 7). . . . . . . . . . . . . . . . . . . 146 6.14 Temperature and stream lines within a15 ◦ stable inclined cavity resulting from RSM-Basic (Ra = 0.86×10 6 ), (scale of the small side is enlarged by a factor of 7). . . . . . . . . . . . . . . . . . . 147 8
Page 1: The computation of turbulent natura
Page 4 and 5: 3.2 Mean Flow Equations and their s
Page 6 and 7: 7.2.6 FFT analysis . . . . . . . .
Page 10 and 11: 6.15 Temperature and stream lines w
Page 12 and 13: 6.45 Temperature fluctuations withi
Page 14 and 15: 7.9 rms velocity fluctuations compa
Page 16 and 17: 7.43 Temperature contours within x-
Page 18 and 19: 7.82 Mean velocity distributions re
Page 20 and 21: 7.119Time-averaged rms velocity flu
Page 22 and 23: 8.23 Contour plots of V, W andk at
Page 24 and 25: 8.49 Contour plots of V, W andk at
Page 26 and 27: For the inclined tall cavities, in
Page 28 and 29: Declaration No portion of the work
Page 30 and 31: Acknowledgements I would like to ex
Page 32 and 33: Nomenclature Acronyms AWF Analytica
Page 34 and 35: λ Thermal conductivity µ Molecula
Page 36 and 37: cl Equilibrium length scale constan
Page 38 and 39: yd Thickness of the dissipative lay
Page 40 and 41: Chapter 1 Introduction In this rese
Page 42 and 43: 41 need fine numerical resolution n
Page 44 and 45: 43 nonlinear discretization methods
Page 46 and 47: 45 the only way to achieve computat
Page 48 and 49: 47 elaborate expressions for modell
Page 50 and 51: Chapter 2 Literature Review In this
Page 52 and 53: 51 local Nusselt number occurs in t
Page 54 and 55: 53 Cavities and Enclosures Kirkpatr
Page 56 and 57: 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
Page 72 and 73:
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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Page 150 and 151:
Page 152 and 153:
Page 154 and 155:
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.
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
Page 182 and 183:
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
Page 196 and 197:
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
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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
Page 236 and 237:
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
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
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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.
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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
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
Page 278 and 279:
277 PSD (T) PSD (T) PSD (T) 1 0.8 0
Page 280 and 281:
279 PSD (T) PSD (T) PSD (T) 1 0.8 0
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281 cold wall temperatures are 53
Page 284 and 285:
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
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
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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.
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
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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
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
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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
Page 344 and 345:
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
Page 352 and 353:
351 Figure 8.7 - Contour plots of T
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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
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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
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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
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397 τ ∗ =240 τ ∗ =360 Figure
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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
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405 Z=0.69 τ ∗ =0 τ ∗ =120 τ
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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
Page 416 and 417:
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.
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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