Three-dimensional Rayleigh–Bénard convection of molten gallium in a rotating cuboid under the influence of a vertical magnetic field (original) (raw)

Abstract

The present work deals with magnetoconvection of molten gallium in a cuboid rotating about a vertical axis passing through its center. The governing equations are derived in a non-inertial frame of reference considering both centrifugal and Coriolis forces. A vertical magnetic field is applied through the center opposite to the direction of gravity. The cuboid is heated from below and cooled at top, while the remaining walls of the cuboid are thermally insulated. The modified Marker and Cell method is adopted for the numerical solution of the governing equations. The gradient dependent consistent hybrid upwinding scheme of second order is adopted for the discretization of the convective terms in the momentum equations. The operator splitting algorithm is used for the numerical treatment of the energy equation. The effects of cavity rotation and applied magnetic field on heat and momentum transport processes have been investigated. The uniform thorough mixing of fluids by rotation and regularization of flow by magnetic field are observed. The governing flow field and temperature distribution are shown graphically to elucidate the intricate physics of the phenomenon.

Loading...

Loading Preview

Sorry, preview is currently unavailable. You can download the paper by clicking the button above.

References (60)

  1. H. Ozoe, K. Okada, The effect of the direction of the external magnetic field on three-dimensional natural convection in a cubical enclosure, Int. J. Heat Mass Transfer 32 (1989) 1939-1954.
  2. J. Garandet, T. Alboussière, R. Moreau, Buoyancy-driven convection in a rectangular enclosure with a transverse magnetic field, Int. J. Heat Mass Transfer 35 (1992) 741-748.
  3. T. Alboussière, J. Garandet, R. Moreau, Buoyancy-driven convection with a uniform magnetic field. Part 1. Asymptotic analysis, J. Fluid Mech. 253 (1993) 545-563.
  4. T. Alboussière, J. Garandet, R. Moreau, Asymptotic analysis and symmetry in MHD convection, Phys. Fluids 8 (1996) 2215-2226.
  5. H. BenHadid, D. Henry, Numerical study of convection in the horizontal Bridgman configuration under the action of a constant magnetic field. Part 2. Three-dimensional flow, J. Fluid Mech. 333 (1997) 57-83.
  6. D. Hurle, Temperature oscillations in molten metals and their relationship to growth striae in melt-grown crystals, Philos. Mag. 13 (1966) 305-310.
  7. D. Hurle, E. Jakeman, C. Johnson, Convective temperature oscillation in molten gallium, J. Fluid Mech. 64 (1974) 565-576.
  8. K. Okada, H. Ozoe, Experimental heat transfer rate of natural convection of molten gallium suppressed under an external magnetic field in either X, Y, or Z direction, ASME J. Heat Transfer 114 (1992) 107-114.
  9. L. Davoust, M. Cowley, R. Moreau, R. Bolcato, Buoyancy-driven convection with a uniform magnetic field. Part 2. Experimental investigation, J. Fluid Mech. 400 (1999) 59-90.
  10. A. Juel, T. Mullin, H. BenHadid, D. Henry, Magnetohydrodynamic convection in molten gallium, J. Fluid Mech. 378 (1999) 97-118.
  11. A. Juel, T. Mullin, H. BenHadid, D. Henry, Three-dimensional free convection in molten gallium, J. Fluid Mech. 436 (2001) 267-281.
  12. B. Hof, A. Juel, T. Mullin, Magnetohydrodynamic damping of convective flows in molten gallium, J. Fluid Mech. 482 (2003) 163-179.
  13. B. Hof, A. Juel, T. Mullin, Magnetohydrodynamic damping of oscillations in low-Prandtl-number convection, J. Fluid Mech. 545 (2005) 193-201.
  14. S. Kenjereš, K. Hanjalic ´, Numerical simulation of magnetic control of heat transfer in thermal convection, Int. J. Heat Fluid Flow 25 (2004) 559-568.
  15. E. Güray, H. Tarman, Thermal convection in the presence of a vertical magnetic field, Acta Mech. 194 (2007) 33-46.
  16. P. Niler, F. Bisshopp, On the influence of Coriolis force on onset of thermal convection, J. Fluid Mech. 22 (1965) 753-761.
  17. G. Veronis, Large-amplitude Bénard convection in a rotating fluid, J. Fluid Mech. 31 (1968) 113-139.
  18. G. Küppers, D. Lortz, Transition from laminar convection to thermal turbulence in a rotating fluid layer, J. Fluid Mech. 35 (1969) 609-620.
  19. H. Rossby, A study of Bénard convection with and without rotation, J. Fluid Mech. 36 (1969) 309-335.
  20. C. Hunter, N. Riahi, Nonlinear convection in a rotating fluid, J. Fluid Mech. 72 (1975) 433-454.
  21. R. Clever, F. Busse, Nonlinear properties of convection rolls in a horizontal layer rotating about a vertical axis, J. Fluid Mech. 94 (1979) 609-627.
  22. J. Hudson, D. Tang, S. Abell, Experiments on centrifugal driven thermal convection in a rotating cylinder, J. Fluid Mech. 86 (1978) 147-159.
  23. D. Tang, J. Hudson, Experiments on a rotating fluid heated from below, Int. J. Heat Mass Transfer 26 (1983) 943-949.
  24. K. Búhler, H. Oertal, Thermal cellular convection in rotating rectangular boxes, J. Fluid Mech. 114 (1982) 261-282.
  25. J. Chew, Computation of convective laminar flow in rotating cavities, J. Fluid Mech. 153 (1985) 339-360.
  26. F. Busse, A. Or, Convection in a rotating cylindrical annulus: thermal Rossby waves, J. Fluid Mech. 166 (1986) 173-187.
  27. A. Or, F. Busse, Convection in a rotating cylindrical annulus. Part 2. Transitions to asymmetric and vacillating flow, J. Fluid Mech. 174 (1987) 313-326.
  28. J. Pfotenhauer, J. Niemela, R. Donnelly, Stability and heat transfer of rotating criogens. Part 3. Effects of finite cylindrical geometry and rotation on the onset of convection, J. Fluid Mech. 175 (1987) 85-96.
  29. S. Condie, R. Griffiths, Convection in a rotating cavity: modeling ocean circulation, J. Fluid Mech. 207 (1989) 453-474.
  30. T. Lee, T. Lin, Transient three-dimensional convection of air in a differentially heated rotating cubic cavity, Int. J. Heat Mass Transfer 39 (1996) 1243-1255.
  31. J. Mandal, C. Sonawane, Simulation of flow inside differentially heated rotating cavity, Int. J. Numer. Meth. Heat Fluid Flow 23 (2013) 23-54.
  32. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publications, New York, 1961.
  33. Y. Nakagawa, Experiments on the instability of a layer of mercury heated from below and subject to the simultaneous action of a magnetic field and rotation, Proc. R. Soc. Lond. A 242 (1957) 81-88.
  34. A. Soward, Thermal and magnetically driven convection in a rapidly rotating fluid layer, J. Fluid Mech. 90 (1979) 669-684.
  35. J. Aurnou, P. Olson, Experiments on Rayleigh-Bénard convection, magnetoconvection and rotating magnetoconvection in liquid gallium, J. Fluid Mech. 430 (2001) 283-307.
  36. N. Gillet, D. Brito, D. Jault, H.-C. Nataf, Experimental and numerical studies of convection in a rapidly rotating spherical shell, J. Fluid Mech. 580 (2007) 83- 121.
  37. M. Schrinner, K.-H. Rädler, D. Schmitt, M. Rheinhardt, U. Christensen, Mean- field concept and direct numerical simulations of rotating magnetoconvection and the geodynamo, Geophys. Astrophys. Fluid Dyn. 101 (2007) 81-116.
  38. W. Liu, Numerical study of the magnetorotational instability in Princeton MRI experiment, Astrophys. J. 684 (2008) 515-524.
  39. M. Rieutord, F. Rincon, The Sun's supergranulation, Living Rev. Solar Phys. 7 (2010) 1-82.
  40. B. Sreenivasan, C. Jones, Helicity generation and subcritical behaviour in rapidly rotating dynamos, J. Fluid Mech. 688 (2011) 5-30.
  41. D. Gubbins, B. Sreenivasan, J. Mound, S. Rost, Melting of the Earth's inner core, Nature 473 (2011) 361-364.
  42. A. Mikelson, Y. Karklin, Control of crystallization processes by means of magnetic fields, J. Cryst. Growth 52 (1981) 524-529.
  43. D. Kim, P. Adornato, R. Brown, Effect of vertical magnetic field on convection and segregation in vertical Bridgman crystal growth, J. Cryst. Growth 89 (1988) 339-356.
  44. V. Galindo, G. Gerbeth, W. Ammon, E. Tomzig, J. Virbulis, Crystal growth melt flow control by means of magnetic fields, Energy Convers. Manage. 43 (2002) 309-316.
  45. C. Lan, M. Liang, J. Chian, Influence of ampoule rotation on three-dimensional convection and segregation in Bridgman crystal growth under imperfect growth conditions, J. Cryst. Growth 212 (2000) 340-351.
  46. G. Sutton, A. Sherman, Engineering Magneto Hydrodynamics, Dover Publications, New York, 2006.
  47. C. Hirt, J. Cook, Calculating three dimensional flows around structures and over rough terrain, J. Comput. Phys. 10 (1972) 324-340.
  48. S. Kim, T. Benson, Comparison of the SMAC, PISO and iterative time advancing schemes for unsteady flows, Comput. Fluids 21 (1992) 435-454.
  49. S. Vanka, B.-J. Chen, W. Sha, A semi-implicit calculation procedure for flow described in body-fitted coordinate systems, Numer. Heat Transfer 3 (1980) 1- 19.
  50. A.J. Chorin, Numerical methods for solving incompressible viscous flow problems, J. Comput. Phys. 2 (1967) 12-26.
  51. K. Muralidhar, T. Sundararajan, Computational Fluid Flow and Heat Transfer, second ed., Narosa Publishing House Pvt. Ltd, New Delhi, 2004.
  52. A. Brandt, J. Dendy, H. Ruppel, The multi-grid method for semi-implicit hydrodynamic codes, J. Comput. Phys. 34 (1980) 348-370.
  53. R. Issa, Solution of the implicitly discretized fluid flow equations by operator- splitting, J. Comput. Phys. 62 (1985) 40-65.
  54. K. Muralidhar, M. Varghese, K. Pillai, Application of an operator splitting algorithm for advection-diffusion problems, Numer. Heat Transfer Part B 23 (1993) 99-114.
  55. P. Haldenwang, Resolution tridimensionnelle des equations de Navier-Stokes par methodes spectrales tchebycheff, these d'etat (Ph.D. thesis), Universite de Provence, Marseille, 1984.
  56. W. Hackbush, Iterative Solution of Large Sparse System of Equations, Springer- Verlag, New York, 1994.
  57. R. Gentry, R. Martin, B. Daly, An Eulerian differencing method for unsteady incompressible flow problems, J. Comput. Phys. 1 (1966) 87-118.
  58. G. Raithby, K. Torrance, Upstream-weighted differencing schemes and their applications to elliptic problems involving fluid flow, Comput. Fluids 2 (1974) 191-206.
  59. P. Roache, Computational Fluid Dynamics (revised printing), Hermosa Albuquerque, New Mexico, 1985.
  60. T. Tagawa, H. Ozoe, Enhancement of heat transfer rate by application of a static magnetic field during natural convection of liquid metal in a cube, ASME J. Heat Transfer 119 (1997) 265-271.