Square cells in surface-tension-driven Bénard convection: Experiment and theory

Journal of Fluid Mechanics

Volumn 356, Issue , 1998, Pages 155-197

  Eckert, Kerstin ^a   Bestehorn, Michael ^b   Thess, André ^a  

^a DRESDEN UNIVERSITY OF TECHNOLOGY   (Germany)

^b UNIVERSITY OF STUTTGART   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; COMPUTER SIMULATION; NUSSELT NUMBER; PERTURBATION TECHNIQUES; PRANDTL NUMBER; THERMODYNAMIC STABILITY; TRANSITION FLOW;

SQUARE CELLS; SURFACE TENSION DRIVEN BENARD CONVECTION;

HEAT CONVECTION;

CONVECTION; FREE SURFACE FLOW; MATHEMATICAL MODELING;

BENARD; CONVECTION; FLOW MODELLING;

EID: 0032501941     PISSN: 00221120     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0022112097007842     Document Type: Article

References (63)

1. BÉNARD, H. 1900 Les tourbillons cellulaires dans une nappe liquide. Rev. Gén. Sci. Pures Appl. 11, 1261-1271.
2. BESTEHORN, M. 1993 Phase and amplitude instabilities for Bénard-Marangoni-convection in fluid layers with large aspect ratio. Phys. Rev. E 48, 3622-3634.
3. BESTEHORN, M. 1996 Square patterns in Bénard-Marangoni convection. Phys. Rev. Lett. 76, 46-49.
4. BESTEHORN, M. FANTZ, M. FRIEDRICH, R. & HAKEN, H. 1993 Hexagonal and spiral patterns of thermal convection. Phys. Lett. A 174, 48-56.
5. BLOCK, M. J. 1956 Surface tension as the cause of Bénard cells and surface deformation in a liquid film. Nature 178, 650-651.
6. BOECK, TH. & THESS, A. 1997 Inertial Bénard-Marangoni convection. J. Fluid Mech. 350, 149-175.
7. BRAGARD, J. & LEBON, G. 1993 Non-linear Marangoni convection in a layer of finite depth. Europhys. Lett. 21, 831-836.
8. BUSSE, F. H. 1967 On the stability of two-dimensional convection in a layer heated from below. J. Math. Phys. 46, 140-149.
9. BUSSE, F. H. 1981 Transition to turbulence in Rayleigh-Bénard convection. In Hydrodynamic Instabilities and the Transition to Turbulence (ed. H. L. Swinney & J. P. Gollub). Springer.
10. BUSSE, F. H. & RIAHI, N. 1980 Nonlinear convection in a layer with nearly insulating boundaries. J. Fluid Mech. 96, 243-256.
11. BUSSE, F. H. & WHITEHEAD, J. A. 1971 Instabilities of convection rolls in a high Prandtl number fluid. J. Fluid Mech. 47, 305-320.
12. BUSSE, F. H. & WHITEHEAD, J. A. 1974 Oscillatory and collective instabilities in large Prandtl number convection. J. Fluid Mech. 66, 67-79.
13. CHANDRASEKHAR, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.
14. CERISIER, P., JAMOND, C., PANTALONI, J. & CHARMET, J. C. 1984 Déformation de la surface libre en convection de Bénard-Marangoni. J. Phys. Paris 45, 405-411.
15. CERISIER, P., JAMOND, C., PANTALONI, J. & PÉREZ-GARCÍA, C. 1987a Stability of roll and hexagonal patterns in Bénard-Marangoni convection. Phys. Fluids 30, 954-959.
16. CERISIER, P., PÉREZ-GARCÍA, C., JAMOND, C. & PANTALONI, J. 1987b Wavelength selection in Bénard-Marangoni convection. Phys. Rev. A 35, 1949-1952.
17. CERISIER, P., PÉREZ-GARCÍA, C. & OCCELLI, R. 1993 Evolution of induced patterns in surface tension-driven Bénard convection. Phys. Rev. E 47, 3316-3325.
18. CILIBERTO, S., PAMPALONI, E. & PÉREZ-GARCÍA, C. 1988 Competition between different symmetries in convective pattern. Phys. Rev. Lett. 61, 1198-1201.
19. CLOOT, A. & LEBON, G. 1984 A nonlinear analysis of the Bénard-Marangoni problem. J. Fluid Mech. 145, 447-469.
20. DAUBY, P. C. & LEBON, G. 1996 Bénard-Marangoni instability in rigid rectangular containers. J. Fluid Mech. 329, 25-64.
21. DAVIS, S. H. 1987 Thermocapillary instabilities. Ann. Rev. Fluid Mech. 19, 347-359.
22. DAVIS, S. H., MÜLLER, U. & DIETSCHE, C. 1984 Pattern selection in single-component systems coupling Bénard convection and solidification. J. Fluid Mech. 144, 133-151.
23. ECKERT, K. 1997 Von hexagonalen zu quadratischen Konvektionszellen: Experimentelle Untersuchungen zur oberflächenspannungsgetriebenen Bénard-Konvektion. PhD dissertation, Ottovon-Guericke-University of Magdeburg.
24. GOLLUB, J. P. & MCCARRIAR, A. R. 1982 Convection in Fourier space. Phys. Rev. A 26, 3470-3476.
25. GOLOVIN, A. A., NEPOMNYASHCHY, A. A. & PISMEN, L. M. 1997 Nonlinear evolution and secondary instabilities of Marangoni convection in a liquid-gas system with deformable interface. J. Fluid Mech. 341, 317-341.
26. HADJI, L., SCHELL, M. & RIAHI, D. N. 1990 Interfacial pattern formation in the presence of solidification and thermal convection. Phys. Rev. A 41, 863-873.
27. KARCHER, CH. & MÜLLER, U. 1995 Convection in a porous medium with solidification. Fluid Dyn. Res. 15, 25-42.
28. KITTEL, CH. 1966 Introduction to Solid State Physics. John Wiley & Sons.
29. KOSCHMIEDER, E. L. 1967 On convection under an air surface. J. Fluid Mech. 30, 9-15.
30. KOSCHMIEDER, E. L. 1991 The wavelength of supercritical surface-tension-driven Bénard convection. Eur. J. Mech. E/Fluids 10, 233-237.
31. KOSCHMIEDER, E. L. & BIGGERSTAFF, M. I. 1986 Onset of surface-tension-driven Bénard convection. J. Fluid Mech. 167, 49-64.
32. KOSCHMIEDER, E. L. & PALLAS, S. G. 1974 A sensor for heat transfer measurements. Rev. Sci. Instrum. 45, 1164-1165.
33. KOSCHMIEDER, E. L. & PRAHL, S. A. 1990 Surface-tension-driven Benard convection in small containers. J. Fluid Mech. 215, 571-583.
34. KOSCHMIEDER, E. L. & SWITZER, D. W. 1992 The wavenumber of supercritical surface-tension-driven Bénard convection. J. Fluid Mech. 240, 533-548.
35. KOSCHMIEDER, E. L. 1993 Bénard Cells and Taylor Vortices. Cambridge University Press.
36. KRISHNAMURTI, R. 1970 On the transition to turbulent convection. Part 2. The transition to time-dependent flow. J. Fluid Mech. 42, 309-320.
37. LE GAL, P., POCHEAU, A. & CROQUETTE, V. 1985 Square versus roll pattern at convective threshold. Phys. Rev. Lett. 54, 2501-2504.
38. MALKUS, W. V. R. & VERONIS, G. 1958 Finite amplitude cellular convection. J. Fluid Mech. 4, 225-260.
39. MERZKIRCH, W. 1974 Flow Visualization. Academic.
40. MINDLIN, G. B., ONDARÇUHU, T., MANCINI, H. L., PÉREZ-GARCÍA, C. & GARCIMARTÍN, A. 1994 Comparison of data from Bénard-Marangoni convection in a square container with a model based on symmetry arguments. Intl J. Bifurcation Chaos 4, 1121-1133.
41. MORRIS, S. W., BODENSCHATZ, E., CANNELL, D. S. & AHLERS, G. 1993 Spiral defect chaos in large aspect ratio Rayleigh-Bénard convection. Phys. Rev. Lett. 71, 2026-2029.
42. MOSES, E. & STEINBERG, V. 1986 Competing patterns in a convective binary mixture. Phys. Rev. Lett. 57, 2018-2021.
43. NIELD, D. A. 1964 Surface tension and buoyancy effects in cellular convection. J. Fluid Mech. 19, 341-352.
44. NITSCHKE-ECKERT, K. & THESS, A. 1995 Secondary instability in surface-tension-driven Bénard convection. Phys. Rev. E 52, R5772-R5775.
45. OLIVER, D. S. 1983 Planform of convection with strongly temperature-dependent viscosity. Geophys. Astrophys. Dyn. 27, 73-85.
46. ONDARÇUHU, T., MILLÁN-RODRÍGUES, J., MANCINI, H. L., GARCIMARTÍN, A. & PÉREZ-GARCÍA, C. 1993 Bénard-Marangoni-convective pattern in small cylindrical layers. Phys. Rev. E 48, 1051-1057.
47. PALMER, H. J. & BERG, J. C. 1971 Convective instabilities in liquid pools heated from below. J. Fluid Mech. 47, 779-787.
48. PAMPALONI, E., PÉREZ-GARCÍA, C., ALBAVETTI, L. & CILIBERTO, S. 1992 Transition from hexagons to rolls in convection in fluids under non-Boussinesq conditions. J. Fluid Mech. 243, 393-416.
49. PARMENTIER, P. M., REGNIER, V. C. & LEBON, G. 1996 Non-linear analysis of coupled gravitional and capillary thermoconvection in thin liquid layers. Phys. Rev. E 54, 411-423.
50. PEARSON, J. R. A. 1958 On convection cells induced by surface tension. J. Fluid Mech. 4, 489-500.
51. REGNIER, V., DAUBY, P. C., PARMENTIER, P. & LEBON, G. 1997 Square cells in gravitional and capillary thermoconvection. Phys. Rev. E 55, 6860-6865.
52. SCANLON, J. W. & SEGEL, L. A. 1967 Finite-amplitude cellular convection induced by surface tension. J. Fluid Mech. 30, 149-162.
53. SCHATZ, M. F., VAN-HOOK, S. J., MCCORMICK, W. D., SWIFT, J. B. & SWINNEY, H. 1995 Onset of surface-tension-driven Bénard convection. Phys. Rev. Lett. 75, 1938-1941.
54. SIGGIA, E. D. 1994 High Rayleigh number convection. Ann. Rev. Fluid Mech. 26, 137-168.
55. SIMANOVSKII, I. B. & NEPOMNYASHCHY, A. A. 1993 Convective Instabilities in Systems with Interface. Gordon and Breach.
56. THESS, A. & BESTEHORN, M. 1995 Planform selection in Bénard-Marangoni-convection: 1-hexagons versus g-hexagons. Phys. Rev. E 52, 6358-6367.
57. THESS, A. & ORSZAG, S. A. 1995 Surface-tension-driven Bénard convection at infinite Prandtl number. J. Fluid Mech. 283, 201-230.
58. THIELE, U. & ECKERT, K. 1997 Stochastic geometry of polygonal networks - an alternative approach to the hexagon-square-transition in Bénard convection. Phys. Rev. E (submitted).
59. VELARDE, M. G. & CASTILLO, J. L. 1982 Transport and reactive phenomena leading to interfacial instability. In Convective Transport and Instability Phenomena (ed. J. Zierep, H. Oertel, jr. & G. Braun).
60. WALDEN, R. W. & AHLERS, G. 1981 Non-Boussinesq and penetrative convection in a cylindrical cell. J. Fluid Mech. 109, 89-114.
61. WHITE, D. B. 1988 The planforms and onset of convection with a temperature-dependent viscosity. J. Fluid Mech. 191, 247-286.
62. WHITEHEAD, J. A. & PARSONS, B. 1978 Observations of convection at Rayleigh numbers up to 760 000 in a fluid with large Prandtl number. Geophys. Astrophys. Fluid Dyn. 9, 201-217.
63. YU, S.-T., JIANG, B.-N. & DUH, J. C. 1996 Three-dimensional simulation of Marangoni-Bénard convection in small containers by least-square finite element method. AIAA Paper 96-0863.