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: 19 apr. 2024.
Issue
Section
Section B: Thermal, Mechanical and Materials Engineering
Keywords
Natural convection; Instability; Chaos; Bifurcation; Attractor; Phase trajectory
J. Appl. Eng. Sci. Technol. (JAEST - ISSN 2352-9873) is a peer-reviewed quarterly journal dedicated to the applied engineering sciences and technology. The JAEST provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge.
There is no submission or publication fee for papers published in the JAEST.
Authors who publish in the JAEST agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in the JAEST.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in the JAEST.
- Authors are permitted to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access). Any such posting made before acceptance and publication of the Work shall be updated upon publication to include a reference to the JAEST and a link to the online abstract for the final published Work in the Journal.
