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

More documents

Recommendations

Info

111 where ∂U2 ∂y∗ = C2y∗ +A2 µυY + + − b µυY b M YT (lnYT −1) µυY αt ∗ bbµRµy µυY (R+Θwall −Θref)y ∗ + prυ µυαt Cth2y ∗ ∗ y 2 −y∗ υ (4.72) The value ofPk is calculated by numerical integration within the near-wall cell in this study. The buoyancy generation term in thek transport equation is: Gk = −ρβgiuiθ where, using an eddy diffusivity model for the turbulent heat fluxes: uiθ = − νt ∂Θ Prt ∂xi (4.73) Two ways have been tested in this research to calculate the buoyancy generation term. One way is to calculate the temperature gradient in equation 4.73 from the nodal values and the other way is to calculate a cell-averaged generation term from the AWF equations as: Gk = ρυ √ kP µυ y∗ n 1 giβµυα ∆y Prt y∗ υ (y ∗ −y ∗ υ) ∂Θ2 dy∗ ∂y∗ Comparisons show that both methods produce the same temperature dis- tribution and Nusselt number predictions, although the cell-average temperature gradient is used in this study. 4.2.5 Hydrodynamic Wall Function When Near Wall Cell Is Thinner Than Viscous Sublayer In the situation that the viscous sublayer is thicker than the near wall control volume, the following equations are simplified compared to the previous
Wall Functions 112 section by setting a zero value to µt. A1 = µnUn −N y ∗ n N = Cy∗ n 2 +by∗2 n 2 − bµC y∗3 n 6 −bbµ y∗3 n 6 + bb 2 Prny µ ∗5 n Cth 30µn 6 y∗ n +Ath and the wall shear stress equation is: y∗2 n −bµ 2 Θwall −Θref + Prny∗ n Cthy∗ n 4 +Ath 3µn (Θwall −Θref)+ Prny ∗ n τwall = − ρn kp µn Cth 5 y∗ n +Ath Then the average production of turbulence kinetic energy is: µn A1 (4.74) (4.75) (4.76) Pk = 0 (4.77) 4.2.6 Convection and pressure gradient terms The evaluation of the convective and pressure gradient terms is a very im- portant feature of the AWF, especially in the case of the thermal convective terms. These terms in the momentum and energy equations respectively are: C = µ2 υ ρ 2 υ kP Cth = µ2 υ ρ 2 υkP ∂(ρUU) ∂x ∂(ρUΘ) ∂x + ∂(ρVU) ∂y + dP dx + ∂(ρVΘ) ∂y When the thickness of the zeroµt sublayer is less than that of the near wall control volume, there are two regions in the near wall control volume. One is the zero µt sublayer and the other is the region beyond it. Therefore there are two strategies that could be adopted to calculate the convective terms. One option is to calculate a single convection term for the entire control volume
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 8 and 9:
List of Figures 2.1 Comparison of e
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
Page 58 and 59:
57 of cold wall. They showed that t
Page 60 and 61:
59 their predictions and their expe
Page 62 and 63: 61 and the fluid near the vertical
Page 64 and 65: 63 term in the ε equation performe
Page 66 and 67: 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 114 and 115: 113 and the other option is to calc
Page 116 and 117: 115 explained. It might be noted th
Page 118 and 119: 117 near-wall treatments available
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
Page 168 and 169:
167 v rms /V 0 v rms /V 0 v rms /V
Page 170 and 171:
169 T T T T 1 0.8 0.6 0.4 0.2 Y=0.9
Page 172 and 173:
171 2 uv/V0 2 uv/V0 2 uv/V0 2 uv/V0
Page 174 and 175:
173 Nu 20 15 10 5 Cold Wall 0 0 0.2
Page 176 and 177:
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
Page 186 and 187:
185 of the cavity. Figure 6.49 pres
Page 188 and 189:
187 T T 1 0.8 0.6 0.4 0.2 Y=0.1 0 0
Page 190 and 191:
189 θ rms /ΔΘ θ rms /ΔΘ 0.15
Page 192 and 193:
191 Nu 20 15 10 5 Cold Wall 0 0 0.2
Page 194 and 195:
193 Figure 6.58 - Stream traces and
Page 196 and 197:
195 and SWF treatments is compared
Page 198 and 199:
197 T T T T T 1 0.8 0.6 0.4 0.2 0 0
Page 200 and 201:
199 Nu 60 40 20 LES AWF LRN SWF 0 0
Page 202 and 203:
201 After validation of the STREAM
Page 204 and 205:
203 V/V 0 V/V 0 0.5 0 -0.5 0.6 0.4
Page 206 and 207:
205 T T 1 0.8 0.6 0.4 0.2 0 0 0.2 0
Page 208 and 209:
207 v rms /V 0 v rms /V 0 0.2 0.15
Page 210 and 211:
209 V/V 0 V/V 0 0.4 0.2 0 -0.2 -0.4
Page 212 and 213:
211 the distance between the hot an
Page 214 and 215:
213 1.5 Y 2 1 0.5 0 0 0.05 0.1 0.15
Page 216 and 217:
215 7.2.4 3D time dependent simulat
Page 218 and 219:
217 0.65 1.5 Y 0.5 Y 2 1 0.65 0.65
Page 220 and 221:
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.
Page 224 and 225:
223 data. The comparisons show that
Page 226 and 227:
225 flow at this angle of inclinati
Page 228 and 229:
227 T T T 1 0.8 0.6 0.4 0.2 T X 1 0
Page 230 and 231:
229 U/V 0 U/V 0 U/V 0 0.4 0.2 0 -0.
Page 232 and 233:
231 u rms /V 0 u rms /V 0 u rms /V
Page 234 and 235:
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
Page 238 and 239:
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
Page 244 and 245:
243 Y 2 1.5 1 0.5 0.3 0.5 0.55 0.35
Page 246 and 247:
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
Page 252 and 253:
251 the central, z=0.5, longitudina
Page 254 and 255:
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
Page 258 and 259:
257 Y 2 1.5 1 0.5 0.3 0.45 0.45 0.3
Page 260 and 261:
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
Page 268 and 269:
267 v rms /V 0 v rms /V 0 v rms /V
Page 270 and 271:
269 v rms /V 0 v rms /V 0 0.25 0.2
Page 272 and 273:
271 X X X 0.15 0.1 0.05 0 0 0.1 0.2
Page 274 and 275:
273 1.5 Y 2 1 0.5 0 0 0.05 0.1 0.15
Page 276 and 277:
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
Page 282 and 283:
281 cold wall temperatures are 53
Page 284 and 285:
283 The reverse feature is observed
Page 286 and 287:
285 normal components in Figures 7.
Page 288 and 289:
287 V/V 0 V/V 0 0 -0.2 -0.4 -0.6 0
Page 290 and 291:
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
Page 298 and 299:
297 v rms /V 0 v rms /V 0 0.15 EXP
Page 300 and 301:
299 active sides, y-z, there are re
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 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
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 (
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
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
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.
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?