Bifurcations in two-dimensional differentially heated cavity
Abstract
In this work, we propose a numerical analysis of a bidimensional instationary natural convection in a square cavity filled with air and inclined 45 degree versus to horizontal. The vertical walls are subjected to non-uniform temperatures while the horizontal walls are adiabatic. The equations based on the formulation vorticity-stream function are solved using the Alternating Directions Implicit scheme (ADI) and Gauss elimination method. We analyze the influence of Rayleigh number on the roads to chaos borrowed by the natural convection developed in this cavity, and we are looking for stable solutions representing the nonlinear dynamic system. A correlation between the Nusselt number and the Rayleigh number is proposed. We have analyzed the vicinity of the critical point. The transition of the point attractor to another limit cycle attractor is characterized by the Hopf bifurcation.References
Aklouche-Benouaguef, S., B. Zeghmati, K. Bouhadef (2014) Numerical simulation of chaotic natural convection in a differentiated closed square cavity. Numerical Heat Transfer, Part A 65(3): 229-246.
Behringer, R.P. (1985) Rayleigh-Benard convection and turbulence in liquid helium. Reviews of modern physics 57(3): 657.
Berge, P., Y. Pomeau & C. Vidal (1998) L’ordre dans le chaos : Vers une Approche Déterministe de la Turbulence. Paris: Hermann.
Bratsun, D. A., A. V. Zyuzgin & G. F. Putin (2003) Non-linear dynamics and pattern formation in a vertical fluid layer heated from the side. International journal of heat and fluid flow 24(6): 835-852.
D’Orazio, M.C.,C. Cianfrini & M. Corcione (2004) Rayleigh Benard convection in tall rectangular enclosures. International Journal Thermal Sciences 43(2): 136-144.
De Vahl Davis, G. (1983) Natural Convection of air in a square Cavity: A Benchmark Numerical Solution. International Journal for Numerical Methods in Fluids 3(3): 249-264.
Feigenbaum, M.J. (1980) The transition to aperiodic behavior in turbulent systems. Communications in mathematical physics 77(1): 65-86.
Ivey, G. N. (1984) Experiments on transient natural convection in a cavity. Journal of Fluid Mechanics 144: 389-401.
Jahnke, C., & F.E.C. Culick (1994). Application of bifurcation theory to the high-angle-of-attack dynamics of the F-14. Journal of Aircraft, 31(1), 26-34.
Manneville, P. & Y. Pomeau (1980) Different ways to turbulence in dissipative dynamical systems. Physica D: Nonlinear Phenomena, 1(2): 219-226.
Mizushima, J. & Y. Hara (2000) Routes to unicellular convection in a tilted rectangular cavity. Journal of the Physical Society of Japan, 69(8): 2371-2374.
Mukutmoni, D.D. & K.T. Yang (1993a) Rayleigh-Bénard Convection in a Small Aspect Ratio Enclosure: Part I-Bifurcation to Oscillatory Convection. Journal of Heat Transfer 115(2):360-366.
Mukutmoni, D. & K.T. Yang (1993b) Rayleigh- Bénard Convection in a Small Aspect Ratio Enclosure: Part II-Bifurcation to Chaos. Journal of Heat Transfer 115(2):367-375.
Ndamne, A. (1992) Etude expérimentale de la convection naturelle en cavité: de l'état stationnaire au chaos (Doctoral dissertation).
Paolucci, S., & D.R. Chenoweth (1989) Transition to chaos in a differentially heated vertical cavity. Journal of Fluid Mechanics 201 : 379-410.
Patterson, J.C. (1984) On the existence of an oscillatory approach to steady natural convection in cavities. Journal of heat transfer 106(1) : 104-108.
Wakitani, S. (1997). Development of multicellular solutions in natural convection in an air-filled vertical cavity. Journal of heat transfer. 119(1): 97-101.
Woods, L.C. (1954). A note on the numerical solution of fourth order differential equations. The Aeronautical Quarterly 5(4) : 176-184.
Behringer, R.P. (1985) Rayleigh-Benard convection and turbulence in liquid helium. Reviews of modern physics 57(3): 657.
Berge, P., Y. Pomeau & C. Vidal (1998) L’ordre dans le chaos : Vers une Approche Déterministe de la Turbulence. Paris: Hermann.
Bratsun, D. A., A. V. Zyuzgin & G. F. Putin (2003) Non-linear dynamics and pattern formation in a vertical fluid layer heated from the side. International journal of heat and fluid flow 24(6): 835-852.
D’Orazio, M.C.,C. Cianfrini & M. Corcione (2004) Rayleigh Benard convection in tall rectangular enclosures. International Journal Thermal Sciences 43(2): 136-144.
De Vahl Davis, G. (1983) Natural Convection of air in a square Cavity: A Benchmark Numerical Solution. International Journal for Numerical Methods in Fluids 3(3): 249-264.
Feigenbaum, M.J. (1980) The transition to aperiodic behavior in turbulent systems. Communications in mathematical physics 77(1): 65-86.
Ivey, G. N. (1984) Experiments on transient natural convection in a cavity. Journal of Fluid Mechanics 144: 389-401.
Jahnke, C., & F.E.C. Culick (1994). Application of bifurcation theory to the high-angle-of-attack dynamics of the F-14. Journal of Aircraft, 31(1), 26-34.
Manneville, P. & Y. Pomeau (1980) Different ways to turbulence in dissipative dynamical systems. Physica D: Nonlinear Phenomena, 1(2): 219-226.
Mizushima, J. & Y. Hara (2000) Routes to unicellular convection in a tilted rectangular cavity. Journal of the Physical Society of Japan, 69(8): 2371-2374.
Mukutmoni, D.D. & K.T. Yang (1993a) Rayleigh-Bénard Convection in a Small Aspect Ratio Enclosure: Part I-Bifurcation to Oscillatory Convection. Journal of Heat Transfer 115(2):360-366.
Mukutmoni, D. & K.T. Yang (1993b) Rayleigh- Bénard Convection in a Small Aspect Ratio Enclosure: Part II-Bifurcation to Chaos. Journal of Heat Transfer 115(2):367-375.
Ndamne, A. (1992) Etude expérimentale de la convection naturelle en cavité: de l'état stationnaire au chaos (Doctoral dissertation).
Paolucci, S., & D.R. Chenoweth (1989) Transition to chaos in a differentially heated vertical cavity. Journal of Fluid Mechanics 201 : 379-410.
Patterson, J.C. (1984) On the existence of an oscillatory approach to steady natural convection in cavities. Journal of heat transfer 106(1) : 104-108.
Wakitani, S. (1997). Development of multicellular solutions in natural convection in an air-filled vertical cavity. Journal of heat transfer. 119(1): 97-101.
Woods, L.C. (1954). A note on the numerical solution of fourth order differential equations. The Aeronautical Quarterly 5(4) : 176-184.
Published
2017-03-05
How to Cite
AKLOUCHE-BENOUAGUEF, Sabiha; ZEGHMATI, Belkacem.
Bifurcations in two-dimensional differentially heated cavity.
Journal of Applied Engineering Science & Technology, [S.l.], v. 3, n. 1, p. 7-11, mar. 2017.
ISSN 2571-9815.
Available at: <https://revues.univ-biskra.dz/index.php/jaest/article/view/1939>. Date accessed: 16 nov. 2024.
Issue
Section
Section B: Thermal, Mechanical and Materials Engineering
Keywords
Natural convection; Instability; Chaos; Bifurcation; Attractor; Phase trajectory
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