Impact of boundary conditions on Rayleigh-Bénard convection: stability, heat transfer and chaos

Abstract

The paper compares the results of Rayleigh-Bénard convection problem for rigid-rigid-isothermal, rigid-free-isothermal and free-free isothermal boundaries. A minimal Fourier-Galerkin expansion yields the generalized-Lorenz-model whose scaled version is reduced to the Stuart-Landau-model using the multiscale-method. Nusselt number is estimated for both steady and unsteady regimes. Regular, chaotic, and periodic natures of the solution are studied using the Hopf-Rayleigh-number and by means of a bifurcation diagram. The linear and weakly-nonlinear-analyses reveal that the onset of regular and chaotic motions in the case of rigid-free-isothermal boundaries happens later than that of free-free isothermal boundaries but earlier than rigid-rigid-isothermal boundaries. It is shown that the scaled-Lorenz-model possesses all the features of the classical-Lorenz-model. Beyond the value of the Hopf-Rayleigh-number, we observe chaotic-motion between two consecutive spells of periodic motion. It is found that one can also have a window of periodicity for all three boundaries.

KEYWORDS:

• Rayleigh-Bénard
• Lorenz-model
• Runge–Kutta
• Chaos
• Hopf-bifurcation

Acknowledgments

Author Noor Arshika S is thankful for the financial support given by the Centre for Research, CHRIST (Deemed to be University). Author P.G. Siddheshwar and Sameena Tarannum hereby acknowledge the support of CHRIST (Deemed to be University) for their research. Author Kanchana C is thankful to Universidad de Tarapacá for financial support. The authors immensely thank Dr Om P Suthar of MNIT, Jaipur, Rajasthan, India, and his Ph.D. student, Miss Tanya Rastogi, for having helped us with the DNS part of the paper. The authors are grateful to the Reviewers for their most useful comments that helped us refine our paper to the present form.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Author contributions

Authors Noor Arshika S and P.G. Siddheshwar contributed to the mathematical formulation, result analysis and revision of the manuscript. Sameena Tarannum contributed to the mathematical formulation and revision of the manuscript. Kanchana C contributed to the mathematical formulation and critical revision of the paper. All authors approved the final version of the paper.