Bifurcation and chaos in zero-Prandtl-number convection

Content Provider	Semantic Scholar
Author	Pal, Pinaki Wahi, Pankaj Paul, Supriyo Verma, Mahendra K. Kumar, K. P. Harish Mishra, Pankaj Kumar
Copyright Year	2009
Abstract	We present a detailed bifurcation structure and associated flow patterns near the onset of zero-Prandtl-number Rayleigh-Bénard convection. We employ both direct numerical simulation and a low-dimensional model ensuring qualitative agreement between the two. Various flow patterns originate from a stationary square observed at a higher Rayleigh number through a series of bifurcations starting from a pitchfork followed by a Hopf and finally a homoclinic bifurcation as the Rayleigh number is reduced to the critical value. Homoclinic chaos, intermittency, and crises are observed near the onset. Copyright c © EPLA, 2009 Thermal convection is observed almost everywhere in the universe: industrial appliances, liquid metals, atmosphere, oceans, interiors of planets and stars, galaxies etc. An idealized version of convection called RayleighBénard convection (RBC) has been studied for almost a century and it is still an area of intense research [1]. The two most important parameters characterizing convection in RBC are the Rayleigh number, describing the vigour of buoyancy, and the Prandtl number, being the ratio of kinetic viscosity and thermal diffusivity. Solar [2] and geological flows [3] are considered to have very low Prandtl numbers, as do flows of liquid metals [4]. RBC exhibits a wide range of phenomena including instabilities, patterns, chaos, spatio-temporal chaos, and turbulence for different ranges of Rayleigh number and Prandtl number [1]. The origin of instabilities, chaos, and turbulence in convection is one of the major research topics of convection. Direct numerical simulation (DNS), due to its high dimensionality, generates realistic but excessively voluminous numerical outputs which obscure the underlying dynamics. Lower-dimensional projections lead to models which, if done improperly, lose the overall physics. In this letter, our aim is to unfold and discover the underlying physics of low-Prandtl-number flows [5] by examining the natural limit of zero Prandtl number (zero-P) [6–12]. This offers a dramatic simplification without sacrificing (a)Present address: Kabi Sukanta Mahavidyalaya Angus, Hooghly (WB), India. (b)E-mail: mkv@iitk.ac.in significant physics, as well as displays a fascinatingly rich dynamic behaviour. In particular, since zero-P flows are chaotic immediately upon initiation of convection, we adopt a nonstandard strategy of approaching this system from the post-bifurcation direction. Moreover, we attack the problem simultaneously with DNS (to ensure accuracy) as well as a low-dimensional model (to aid physical interpretation); and we stringently refine both the model and DNS until satisfactory agreement is obtained at all levels of observed behaviour. Our results show a diverse variety of both new and previously observed flow patterns. These flow patterns emerge as a consequence of various bifurcations ranging from pitchfork, Hopf, and homoclinic bifurcations to bifurcations involving double zero eigenvalues. Convection in an arbitrary geometry is quite complex, so researchers have focused on Rayleigh-Bénard convection wherein the convective flow is between two conducting parallel plates [1]. The fluid has kinematic viscosity ν, thermal diffusivity κ, and coefficient of volume expansion α. The top and bottom plates are separated by distance d, and they are maintained at temperatures T2 and T1, respectively, with T1 >T2. The convective flow in RBC is characterized by the Rayleigh number R= α(T1−T2)gd/νκ, where g is the acceleration due to gravity, and the Prandtl number P = ν/κ. Various instabilities, patterns, and chaos are observed for different ranges of R and P [1,6,13]. Transition to chaotic states through various routes have been reported in convection [14,15].
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Alternate Webpage(s)	http://home.iitk.ac.in/~mkv/Site/Publications_files/zero-P-epl.pdf
Language	English
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Subject Keyword	Anatomic bifurcation Arabic numeral 0 Behavior Bifurcation theory Chaos Chaos theory Coefficient Cognitive function: initiation Computer simulation Convection Copyright Crisis (dynamical systems) Direct numerical simulation Exhibits as Topic Flow Kinetics Level of detail Metals Numerical analysis Onset (audio) Projections and Predictions Rayleigh–Ritz method Role-based collaboration Secretin-KABI Stars, Celestial Stationary process Temporal Lobe Turbulence
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Resource Type	Article