DNS of rotating buoyancy- and surface tension-driven flow

International Journal of Heat and Mass Transfer

Volumn 51, Issue 25-26, 2008, Pages 6219-6234

  Raufeisen, A ^a,b   Breuer, M ^a   Botsch, T ^b   Delgado, A ^a  

^a UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

^b NUREMBERG INSTITUTE OF TECHNOLOGY   (Germany)

Author keywords

Buoyancy; Crystal growth; Czochralski; Direct numerical simulation; Large eddy simulation; Rayleigh B nard Marangoni convection; Rotation; Surface tension

Indexed keywords

BUOYANCY; CRYSTAL GROWTH; CRYSTAL GROWTH FROM MELT; DIRECT NUMERICAL SIMULATION; NATURAL CONVECTION; NUMERICAL MODELS; PRANDTL NUMBER; ROTATION; SURFACE TENSION; TURBULENCE;

DIMENSIONLESS NUMBER; FINITE VOLUME SCHEMES; LARGE EDDY SIMULATION TECHNIQUES; MARANGONI CONVECTION; SURFACE TENSION EFFECTS; SURFACE-TENSION-DRIVEN FLOW; THREE-DIMENSIONAL TURBULENCE; VELOCITY AND TEMPERATURE FIELDS;

LARGE EDDY SIMULATION;

EID: 55549110894     PISSN: 00179310     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.ijheatmasstransfer.2008.01.041     Document Type: Article

References (83)

1. Bénard H. Les tourbillons cellulaires dans une nappe liquide. Première partie: description générale des phénomènes. Rev. Gén. Sci. Pure Appl. 12 (1900) 1261-1271
2. Bénard H. Les tourbillons cellulaires dans une nappe liquide. Deuxième partie: procédé mécaniques et optiques d'examen. Lois numériques des phénomènes. Rev. Gén. Sci. Pure Appl. 12 (1900) 1309-1328
3. Rayleigh L. On convection currents in a horizontal layer of fluid, when the higher temperature is on the under side. Philos. Mag. (1916) 519-546
4. Koschmieder E. Bénard Cells and Taylor Vortices. first ed. Cambridge Monographs on Mechanics and Applied Mathematics (1993), Cambridge University Press, Cambridge
5. Getling A. Rayleigh-Bénard Convection (1998), World Scientific, Singapore
6. Thual O. Zero-Prandtl-number convection. J. Fluid Mech. 240 (1992) 229-258
7. Paul M., Chiam K., Cross M., Fischer P., and Greenside H. Pattern formation and dynamics in Rayleigh-Bénard convection: numerical simulations of experimentally realistic geometries. Physica D 184 (2003) 114-126
8. Cross M., and Hohenberg P. Pattern formation outside of equilibrium. Rev. Mod. Phys. 65 3 (1993) 851-1112
9. Chandrasekhar S. Hydrodynamic and Hydromagnetic Stability (1961), Oxford University Press, Oxford
10. Drazin P., and Reid W. Hydrodynamic Stability (1981), Cambridge University Press, Cambridge
11. Siggia E. High Rayleigh number convection. Ann. Rev. Fluid Mech. 26 (1994) 137-168
12. Grossmann S., and Lohse D. Scaling in thermal convection: a unifying theory. J. Fluid Mech. 407 (2000) 27-56
13. Hölling M., and Herwig H. Asymptotic analysis of heat transfer in turbulent Rayleigh-Bénard convection. Int. J. Heat Mass Transfer 49 (2006) 1129-1136
14. Niemela J., Skrbek L., Sreenivasan K., and Donnelly R. Turbulent convection at very high Rayleigh numbers. Nature 404 (2000) 837-840
15. Wu X., and Libchaber A. Scaling relations in thermal turbulence: the aspect-ratio dependence. Phys. Rev. A 45 (1992) 842-845
16. Belmonte A., Tilgner A., and Libchaber A. Temperature and velocity boundary layers in turbulent convection. Phys. Rev. E 50 (1994) 269-278
17. Cioni S., Ciliberto S., and Sommeria J. Strongly turbulent Rayleigh-Bénard convection in mercury: comparison with results at moderate Prandtl number. J. Fluid Mech. 335 (1997) 111-140
18. Verzicco R., and Camussi R. Transitional regimes of low-Prandtl thermal convection in a cylindrical cell. Phys. Fluids 9 (1997) 1287-1295
19. Verzicco R., and Camussi R. Prandtl number effects in convective turbulence. J. Fluid Mech. 383 (1999) 55-73
20. Kek V., and Müller U. Low Prandtl number convection in layers heated from below. Int. J. Heat Mass Transfer 36 (1993) 2795-2804
21. Glazier J., Segawa T., Naert A., and Sano M. Evidence against 'ultrahard' thermal turbulence at very high Rayleigh numbers. Nature 398 (1999) 307-310
22. Takeshita T., Segawa T., Glazier J., and Sano M. Thermal turbulence in mercury. Phys. Rev. Lett. 76 (1996) 1465-1468
23. Kerr R., and Herring J. Prandtl number dependence of Nusselt number in direct numerical simulations. J. Fluid Mech. 419 (2000) 325-344
24. Rossby H. A study of Bénard convection with and without rotation. J. Fluid Mech. 36 (1969) 309-335
25. Krishnamurthi R. Some further studies on the transition to turbulent convection. J. Fluid Mech. 60 (1973) 285-303
26. Clever R., and Busse F. Transition to time-dependent convection. J. Fluid Mech. 65 (1974) 625-645
27. Clever R., and Busse F. Convection at very low Prandtl numbers. Phys. Fluids A 2 (1990) 334-339
28. Block M. Surface tension as the cause of Bénard cells and surface deformation in a liquid film. Nature 178 (1956) 650-651
29. Pearson J. On convection cells induced by surface tension. J. Fluid Mech. 4 (1958) 489-500
30. Nield D. Surface tension and buoyancy effects in cellular convection. J. Fluid Mech. 19 (1964) 341-352
31. Davis S. Thermocapillary instabilities. Ann. Rev. Fluid Mech. 19 (1987) 403-435
32. Hashim I., and Wilson S. The onset of Bénard-Marangoni convection in a horizontal layer of fluid. Int. J. Eng. Sci. 37 (1999) 643-662
33. Thess A., and Bestehorn M. Planform selection of in Bénard-Marangoni convection: l hexagons versus g hexagons. Phys. Rev. 52 (1995) 6358-6367
34. Proctor M. Inertial convection at low Prandtl number. J. Fluid Mech. 82 (1977) 97-114
35. Ben Hadid H., and Roux B. Buoyancy- and thermocapillary-driven flows in differentially heated cavities for low-Prandtl-number fluids. J. Fluid Mech. 235 (1992) 1-36
36. T. Boeck, Bénard-Marangoni convection at low Prandtl numbers, Ph.D. thesis, Technical University Ilmenau, Shaker-Verlag, Aachen, 2000.
37. Eckert K., Bestehorn M., and Thess A. Square cells in surface-tension-driven Bénard convection: experiment and theory. J. Fluid Mech. 358 (1998) 149-197
38. Scheel J., and Cross M. Scaling laws for rotating Rayleigh-Bénard convection. Phys. Rev. E 72 5 (2005) 1-10
39. Julien K., Legg S., McWilliams J., and Werne J. Hard turbulence in rotating Rayleigh-Bénard convection. Phys. Rev. E 53 (1996) R5557-R5560
40. Jou J., Kung K., and Hsu C. Effects of Coriolis force and surface tension on Bénard-Marangoni convective instability. Int. J. Heat Mass Transfer 40 (1997) 1447-1458
41. Chang F., and Chiang K. Oscillatory instability analysis of Bénard-Marangoni convection in a rotating fluid under a uniform magnetic field. Int. J. Heat Mass Transfer 41 (1998) 2667-2675
42. Ristorcelli J., and Lumley J. Instabilities, transition and turbulence in the Czochralski crystal melt. J. Cryst. Growth 116 (1992) 447-460
43. Müller G. Convective instabilities in melt growth configurations. J. Cryst. Growth 128 (1993) 26-36
44. Kakimoto K. Flow instability during crystal growth from the melt. Prog. Cryst. Growth Ch. 30 (1995) 191-215
45. Yi K., Kakimoto K., Eguchi M., Watanabe M., Shyo T., and Hibiya T. Spoke patterns on molten silicon in Czochralski system. J. Cryst. Growth 144 (1994) 20-28
46. Basu B., Enger S., Breuer M., and Durst F. Three-dimensional simulation of flow and thermal field in a Czochralski melt using a block-structured finite-volume method. J. Cryst. Growth 219 (2000) 123-143
47. Jones A. An experimental model of the flow in Czochralski growth. J. Cryst. Growth 61 (1983) 235-244
48. Tanaka M., Hasebe M., and Saito N. Pattern transition of temperature distribution at Czochralski silicon melt surface. J. Cryst. Growth 180 (1997) 487-496
49. Watanabe M., Eguchi M., Kakimoto K., and Hibiya T. Double-beam X-ray radiography system for three-dimensional flow visualization of molten silicon convection. J. Cryst. Growth 133 (1993) 23-28
50. Gräbner O., Müller G., Virbulis J., and Tomzig E. Effects of various magnetic field configurations on temperature distributions in Czochralski silicon melts. Micro. Eng. 56 (2001) 83-88
51. S. Enger, Numerical simulation of flow and heat transfer in Czochralski crucibles, Ph.D. thesis, Lehrstuhl für Strömungsmechanik, Universität Erlangen-Nürnberg, 2001.
52. Mihelčić M., Wingerath K., and Pirron C. Three-dimensional simulation of the Czochralski bulk flow. J. Cryst. Growth 69 (1984) 473-488
53. Xiao Q., and Derby J. Three-dimensional melt flows in Czochralski oxide growth: high resolution, massively, parallel, finite element computations. J. Cryst. Growth 152 (1995) 169-182
54. Enger S., Gräbner O., Müller G., Breuer M., and Durst F. Comparison of measurements and numerical simulations of melt convection in Czochralski crystal growth of silicon. J. Cryst. Growth 230 (2001) 135-142
55. Kumar V., Basu B., Enger S., Brenner G., and Durst F. Role of Marangoni convection in Si-Czochralski melts - Part II: 3D predictions with crystal rotation. J. Cryst. Growth 255 (2003) 27-39
56. Schäfer F., Kumar V., Breuer M., and Durst F. Visualization and computational steering of fluid motion in Czochralski crucibles during silicon crystal growth. Int. J. Comput. Fluid Dyn. 19 7 (2005) 501-515
57. Enger S., Basu B., Breuer M., and Durst F. Numerical study of three-dimensional mixed convection due to buoyancy and centrifugal force in an oxide melt for Czochralski growth. J. Cryst. Growth 219 (2000) 144-164
58. Basu B., Enger S., Breuer M., and Durst F. Effect of crystal rotation on the three-dimensional mixed convection in the oxide melt for Czochralski growth. J. Cryst. Growth 230 (2001) 148-154
59. Vizman D., Gräbner O., and Müller G. Three-dimensional numerical simulation of thermal convection in an industrial Czochralski melt: comparison to experimental results. J. Cryst. Growth 233 (2001) 687-698
60. Kumar V., Basu B., Enger S., Brenner G., and Durst F. Role of Marangoni convection in Si-Czochralski melts - Part I: 3D predictions without crystal. J. Cryst. Growth 253 (2003) 142-154
61. Kishida Y., and Okazawa K. Geostrophic turbulence in CZ silicon crucible. J. Cryst. Growth 198/199 (1999) 135-140
62. Kishida Y., Tamaki T., Okazawa K., and Ohashi W. Geostrophic turbulence in CZ silicon melt under CUSP magnetic field. J. Cryst. Growth 273 (2005) 329-339
63. Watanabe M., Vizman D., Friedrich J., and Müller G. Large modification of crystal-melt interface shape during Si crystal growth by using electromagnetic Czochralski method (EMCZ). J. Cryst. Growth 292 (2006) 252-256
64. C. Wagner, Turbulente Transportvorgänge in idealisierten Czochralski-Kristallzüchtungsanordnungen, Habilitationsschrift, Lehrstuhl für Fluidmechanik, Technische Universität München, 2003.
65. Wagner C., and Friedrich R. Direct numerical simulation of momentum and heat transport in idealized Czochralski crystal growth configurations. Int. J. Heat Fluid Flow 25 (2004) 431-443
66. Raufeisen A., Botsch T., Franz E., Kumar V., Breuer M., and Durst F. Prediction of flow and heat transfer in a Czochralski crucible using LES with interface tracking. In: Guenther O., Khujadze W., Frewer O., and Peinke (Eds). Progress in Turbulence 2 (2007), Springer-Verlag, Berlin, Heidelberg, New York
67. Raufeisen A., Botsch T., Franz E., Kumar V., Breuer M., and Durst F. Large eddy simulation of flow and heat transfer with interface prediction in a Czochralski configuration for crystal growth. Euromech Colloquium 469 (2005), Dresden, Germany
68. Breuer M., and Rodi W. Large-eddy simulation of complex turbulent flows of practical interest. In: Hirschel E. (Ed). Flow Simulation with High-Performance Computers II. Notes on Numer. Fluid Mech vol. 52 (1996), Vieweg-Verlag, Braunschweig 258-274
69. Breuer M. Large eddy simulation of the sub-critical flow past a circular cylinder: numerical and modeling aspects. Int. J. Numer. Meth. Fluids 28 (1998) 1281-1302
70. Breuer M. A challenging test case for large eddy simulation: high Reynolds number circular cylinder flow. Int. J. Heat Fluid Flow 21 5 (2000) 648-654
71. M. Breuer, Direkte Numerische Simulation und Large-Eddy Simulation turbulenter Strömungen auf Hochleistungsrechnern, Habilitationsschrift, Universität Erlangen-Nürnberg, Berichte aus der Strömungstechnik, ISBN 3-8265-9958-6, Shaker-Verlag, Aachen, 2002.
72. Ferziger J., and Perić M. Computational Methods for Fluid Dynamics. third ed. (2002), Springer-Verlag
73. Stone H.L. Iterative solution of implicit approximations of multidimensional partial differential equations. SIAM J. Numer. Anal. 5 530-558 (1968) 530-558
74. Rhie C., and Chow W. A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation. AIAA J. 21 (1983) 1525-1532
75. Kolmogorov A. The local structure of turbulence in an incompressible fluid at very high reynolds numbers. Dokl. Akad. Nauk SSSR 30 (1941) 299-303
76. Kolmogorov A. Dissipation of energy in the locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32 (1941) 16-18
77. Pope S. Turbulent Flows (2000), Cambridge University Press
78. Grötzbach G. Spatial resolution requirements for direct numerical simulation of the Rayleigh-Bénard convection. J. Comput. Phys. 49 (1983) 241-264
79. Fein J., and Pfeffer R. An experimental study of the effects of Prandtl number on thermal convection in a rotating, differentially heated cylindrical annulus of fluids. J. Fluid Mech. 75 (1976) 81-112
80. Kakimoto K., Watanabe M., Eguchi M., and Hibiya T. The baroclinic flow instability in rotating silicon melt. J. Cryst. Growth 128 (1993) 288-292
81. Hide R., and Fowlis W. Thermal convection in a rotating annulus of liquid: effect of viscosity on the transition between axisymmetric and non-axisymmetric flow regimes. J. Atmospher. Sci. 22 (1965) 541-558
82. Jones A. The temperature field of a model Czochralski melt. J. Cryst. Growth 69 (1984) 165-172
83. Kumar V., Biswas G., Brenner G., and Durst F. Effect of thermocapillary convection in an industrial Czochralski crucible: numerical simulation. Int. J. Heat Mass Transfer 46 (2003) 1641-1652