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Dynamical and chaotic behaviors of natural convection flow in semi-annular cylindrical domains using energy-conserving low-order spectral models

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  • Khodakaram-Tafti, Amin
  • Emdad, Homayoun
  • Mahzoon, Mojtaba

Abstract

This paper presents a comprehensive theoretical study on the dynamical behavior of natural convection flow in the confined region between horizontal half-cylinders. For this purpose, a low-order spectral model with three modes will produce for the fluid flow system using the Galerkin technique. It proved that the generated model is physically meaningful, as it conserves energy in the dissipationless limit and has bounded solutions in the phase space. Analytical procedures indicate that the system has three stationary points and the onset of instability in the flow is when the Rayleigh number reaches a critical value. With an appropriate Lyapunov function is proved that for the Rayleigh numbers below the critical value, the flow is globally stable. As the Rayleigh number gets higher and reaches a fixed value, a Hopf bifurcation occurs, and chaotic motion appears in the system. The critical and Hopf Rayleigh numbers relation are derived parametrically based on dynamical system theories. Also, numerical simulations will carry on the presented low-order model. Different dynamical behaviors of this flow and its transition from regular to chaotic motion are explained, with phase portraits and velocity-temperature diagrams obtained by numerical solutions. This parametric study can pave the way for future researchers to determine at around values of critical parameters should an experiment or direct numerical simulation be performed to have more accurate data without resorting to tests at all operating conditions.

Suggested Citation

  • Khodakaram-Tafti, Amin & Emdad, Homayoun & Mahzoon, Mojtaba, 2022. "Dynamical and chaotic behaviors of natural convection flow in semi-annular cylindrical domains using energy-conserving low-order spectral models," Applied Mathematics and Computation, Elsevier, vol. 433(C).
    • Handle: RePEc:eee:apmaco:v:433:y:2022:i:c:s0096300322004891
    DOI: 10.1016/j.amc.2022.127415
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    References listed on IDEAS

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    1. Siddheshwar, P.G. & B. N., Shivakumar & Zhao, Yi & C., Kanchana, 2020. "Rayleigh-Bénard convection in a newtonian liquid bounded by rigid isothermal boundaries," Applied Mathematics and Computation, Elsevier, vol. 371(C).
    2. Roy, D. & Musielak, Z.E., 2007. "Generalized Lorenz models and their routes to chaos. II. Energy-conserving horizontal mode truncations," Chaos, Solitons & Fractals, Elsevier, vol. 31(3), pages 747-756.
    3. Ahmed, Hamdi E. & Ahmed, Mirghani I., 2017. "Thermal performance of annulus with its applications; A review," Renewable and Sustainable Energy Reviews, Elsevier, vol. 71(C), pages 170-190.
    4. Zhang, Fuchen, 2015. "On a model of the dynamical systems describing convective fluid motion in rotating cavity," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 873-882.
    5. Roy, D. & Musielak, Z.E., 2007. "Generalized Lorenz models and their routes to chaos. I. Energy-conserving vertical mode truncations," Chaos, Solitons & Fractals, Elsevier, vol. 32(3), pages 1038-1052.
    6. Roy, D. & Musielak, Z.E., 2007. "Generalized Lorenz models and their routes to chaos. III. Energy-conserving horizontal and vertical mode truncations," Chaos, Solitons & Fractals, Elsevier, vol. 33(3), pages 1064-1070.
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