The Bénard problem for a rarefied gas: Formation of steady flow patterns and stability of array of rolls

Physics of Fluids

Volumn 9, Issue 12, 1997, Pages 3898-3914

  Sone, Yoshio ^a   Aoki, Kazuo ^a   Sugimoto, Hiroshi ^a  

^a KYOTO UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000812618     PISSN: 10706631     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.869489     Document Type: Article

References (44)

1. H. Bénard, "Les tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en régime permanent," Ann. Chim. Phys. 23, 62 (1901).
2. Lord Rayleigh, "On convective currents in a horizontal layer of fluid when the higher temperature is on the under side," Philos. Mag. 32, 529 (1916).
3. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Dover, New York, 1981), Chap. II.
4. J. S. Turner, Bouyancy Effects in Fluids (Cambridge U.P., Cambridge, 1973).
5. F. H. Busse and R. M. Clever, "Instabilities of convection rolls in a fluid of moderate Prandtl number," J. Fluid Mech. 91, 319 (1979).
6. J. Mizushima and K. Fujimura, "Higher harmonic resonance of two-dimensional distributions in Rayleigh-Bénard convection," J. Fluid Mech. 234, 651 (1992).
7. E. L. Koschmieder, Bénard cells out Taylor vortices (Cambridge U.P., Cambridge, 1993), Part I.
8. C. Burges and S. Zaleski, "Buoyant mixtures of cellular automaton gases," Complex Syst. 1, 31 (1987).
9. A. Puhl, M. Malek Mansour, and M. Mareschal, "Quantitative comparison of molecular dynamics with hydrodynamics in Rayleigh-Bénard convection," Phys. Rev. A 40, 1999 (1989).
10. C. Cercignani and S. Stefanov, "Bénard's instability in kinetic theory," Transp. Theory Stat. Phys. 21, 371 (1992).
11. S. Stefanov and C. Cercignani, "Monte Carlo simulation of Bénard's instability in a rarefied gas," Eur. J. Mech. B 11, 543 (1992).
12. G. A. Bird, Molecular Gas Dynamics (Oxford U.P., Oxford, 1976).
13. G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows (Oxford U.P. Oxford, 1994).
14. Y. Sone, K. Aoki, H. Sugimoto, and H. Motohashi, "The Bénard problem of rarefied gas dynamics," in Rarefied Gas Dynamics, edited by J. Harvey and G. Lord (Oxford U. P., Oxford, 1995), Vol. 1, p. 135.
15. P. L. Bhatnagar, E. P. Gross, and M. Krook, "A model for collision process in gases. I. Small amplitude processes in charged and neutral one-componenl systems," Phys. Rev. 94, 511 (1954).
16. P. Welander, "On the temperature jump in a rarefied gas," Ark. Fys. 7, 507 (1954).
17. M. N. Kogan, "On the equation of motion of a rarefied gas," Appl. Math. Mech. 22, 597 (1954).
18. T. Watanabe, H. Kaburaki, and M. Yokokawa, "Simulation of a two-dimensional Rayleigh-Bénard system using the direct simulation Monte Carlo method," Phys. Rev. E 49, 4060 (1994).
19. K. Yoshimura and T. Abe, ''Rarefaction effect on the Rayleigh-Bénard instability," AIAA Paper, 95-2055 (1995).
20. E. Goldshtein and T. Elperin, "Convective instabilities in rerefied gases by direct simulation Monte Carlo method," J. Thermophys. Heat Trans. 10, 250 (1996).
21. C. K. Chu, "Kinetic-theoretic description of the formation of a shock wave," Phys. Fluids 8, 12 (1965).
22. Y. Sone, K. Aoki, and I. Yamashita, "A study of unsteady strong condensation on a plane condensed phase with special interest in formation of steady profile," in Rarefied Gas Dynamics, edited by V. Boffi and C. Cercignani (Teubner, Stuttgart, 1986), Vol. 2, 323.
23. Y. Sone and H. Sugimoto, "Strong evaporation from a plane condensed phase," in Adiabatic Waves in Liquid-Vapor Systems, edited by G. E. A. Meier and P. A. Thompson (Springer, Berlin, 1990), p. 293.
24. K. Aoki, Y. Sone, and T. Yamada, "Numerical analysis of gas flows condensing on its plane condensed phase on the basis of kinetic theory," Phys. Fluids A 2, 1867 (1990).
25. K. Aoki, K. Nishino, Y. Sone, and H. Sugimoto, "Numerical analysis of steady flows of a gas condensing on or evaporating from its plane condensed phase on the basis of kinetic theory: Effect of gas motion along the consed phase," Phys. Fluids A 3, 2260 (1991).
26. Y. Sone and S. Takata, "Discontinuity of the velocity distribution function in a rarefied gas around a convex body and the S layer at the bottom of the Knudsen layer," Transp. Theory Stat. Phys. 21, 501 (1992).
27. Y. Sone and K. Aoki, Molecular Gas Dynamics (Asakura, Tokyo, 1994) (in Japanese).
28. Y. Sone, "Continuum gas dynamics in the light of kinetic theory and the new features of rarefied gas flows," in Rarefied Gas Dynamics, edited by C. Shen (Peking U.P., Beijing, 1997), p. 3.
29. 2≡0.
30. K. Aoki, Y. Sone, K. Nishino, and H. Sugimoto, "Numerical analysis of unsteady motion of a rarefied gas caused by sudden changes of wall temperature with special interest in the propagation of a discontinuity in the velocity distribution function," in Rarefied Gas Dynamics, edited by A. E. Beylich (VCH, Weinheim, 1991), p. 222.
31. H. Sugimoto and Y. Sone, "Namerical analysis of steady flows of a gas evaporating from its cylindrical condensed phase on the basis of kinetic theory," Phys. Fluids A 4, 419 (1992).
32. Y. Sone and H. Sugimoto, "Kinetic theory analysis of steady evaporating flows from a spherical condensed phase into a vacuum," Phys. Fluids A 5, 1491 (1993); Erratum, Phys. Fluids 7, 2096 (1995).
33. Y. Sone and H. Sugimoto, "Evaporation of a rarefied gas from a cylindrical condensed phase into a vacuum," Phys. Fluids 7, 2072 (1995).
34. S. Takata, Y. Sone, and K. Aoka, "Numerical analysis of a uniform flow of a rarefied gas past a sphere on the basis of the Boltzmann equation for hard-sphere molecules," Phys. Fluids A 5, 716 (1993).
35. Y. Sone, S. Takata, and M. Wakabayashi, "Numerical analysis of a rarefied gas flow past a volatile particle using the Boltzmann equation for hard-sphere molecules," Phys. Fluids 6, 1914 (1994).
36. S. Takata and Y. Sone, "Flow induced around a sphere with a non-uniform surface temperature in a rarefied gas, with application to the drag and thermal force problems of a spherical particle with an arbitrary thermal conductivity," Eur. J. Mech. B 14, 487 (1995).
37. This is just for reference. The temperature and density variations are assumed to be small in the Boussinesq approximation; thus the situation is considerably different from ours. In some DSMC computations the stability boundary deviates largely from Ra=1700 curve even for small Kn. According to our DSMC computation in a different stability problem, the limit of stable region depends largely on the cell size and the number of particles in it in the computation of conventional size; it is also noted that between stable and unstable regions there is a grey zone where the decision is difficult.
38. In the classical fluid dynamics the problem is studied only on the basis of the Boussinesq approximation of the Navier-Stokes equation, except a special example under a different condition, e.g., Ref. 9. Thus, the stability range in the framework of the full Navier-Stokes equation is not clear.
39. 4 (e.g., for the case Kn=0.02 and α=0.9).
40. Each of these computations requires considerable time (50-170 days on HP 9000 735 computer). Thus, one may misunderstand its final state unless one patiently examines the variation of the speed of profile deformation (cf. Refs. 23-25). In view of such a long computing time for accurate computation, the computation with coarser time-step is convenient to get a first guess.
41. ε=0.01.
42. 2 are also studied for several cases. The results are consistent with the results in Table II.
43. Our discussion on the range of stable solutions is as follows: for the double-roll solutions, their stability is examined; for the others (the stationary and single-roll solutions), the information of the range known by the computation in the preceding sections is tried to be extended by constructing new stable solutions. The behavior at the intermediate points is judged by comparison of the behavior of surrounding points (e.g., the speed of convergence). Incidentally, a note on the range of the stationary solution in Figs. 1-3 should be made. As mentioned in Sec. II, a (stable or unstable) stationary solution exists for the whole range of the parameters. In Sec. IV A, the stress is put on the range where a nonstationary solution exists, and the extension of the stability range of a stationary solution into the range of the other solutions, with the aid of small perturbations, is not systematically studied.
44. If the stretched function is an approximate solution with errors, some perturbation is required to destroy its symmetry. 45 It is a matter of course that the possibility of very weak or very small scale rolls, such as Moffatt's vortex [H. K. Moffatt, "Viscous and resistive eddies near a sharp corner," J. Fluid Mech. 18, 1 (1964)] in a corner, beyond the accuracy of the numerical computation cannot be excluded.