Local oscillations, traveling waves, and chaos in Rayleigh-Bénard convection (original) (raw)

Pattern dynamics in a Bénard-Marangoni convection experiment

Physica D: Nonlinear Phenomena, 1996

In this paper we report a B~nard-Marangoni convection experiment in a small aspect ratio square cell, at a high Rayleigh number. In the time dependent convection regime, low dimensional chaotic behaviour is identified for a variable measured from the pattern. A dynamical system is proposed to model the behaviour of this variable, and predictions induced by this model on the dynamics of the temperature are experimentally verified. The comparison between the experimental data and the one obtained from the model relies on the topological properties (braid type) of unstable periodic orbits.

Chaotic dynamics in two-dimensional Rayleigh-B\'enard convection

2010

We investigate the origin of various convective patterns using bifurcation diagrams that are constructed using direct numerical simulations. We perform two-dimensional pseudospectral simulations for a Prandtl number 6.8 fluid that is confined in a box with aspect ratio Γ = 2 √ 2. Steady convective rolls are born from the conduction state through a pitchfork bifurcation at r = 1, where r is the reduced Rayleigh number. These fixed points bifurcate successively to time-periodic and quasiperiodic rolls through Hopf and Neimark-Sacker bifurcations at r ≃ 80 and r ≃ 500 respectively. The system becomes chaotic at r ≃ 750 through a quasiperiodic route to chaos. The size of the chaotic attractor increases at r ≃ 840 through an "attractor-merging crisis" which also results in travelling chaotic rolls. We also observe coexistence of stable fixed points and a chaotic attractor for 846 ≤ r ≤ 849 as a result of a subcritical Hopf bifurcation. Subsequently the chaotic attractor disappears through a "boundary crisis" and only stable fixed points remain. Later these fixed points become periodic and chaotic through another set of bifurcations which ultimately leads to turbulence. PACS numbers: 47.20.Bp, 47.27.ek, 47.52.+j

Pattern dynamics near inverse homoclinic bifurcation in fluids

Physical Review E

We report for the first time the pattern dynamics in the vicinity of an inverse homoclinic bifurcation in an extended dissipative system. We observe, in direct numerical simulations of three dimensional Rayleigh-Bénard convection, a spontaneous breaking of a competition of two mutually perpendicular sets of oscillating cross rolls to one of two possible sets of oscillating cross rolls as the Rayleigh number is raised above a critical value. The time period of the cross-roll patterns diverges, and shows scaling behavior near the bifurcation point. This is an example of a transition from nonlocal to local pattern dynamics near an inverse homoclinic bifurcation. We also present a simple four-mode model that captures the pattern dynamics quite well. PACS numbers: 47.20.Ky, 47.55.pb, 47.20.Bp Extended dissipative systems driven away from thermodynamic equilibrium often form patterns, if the driving force exceeds a critical value . Competing instabilities may lead to interesting pattern dynamics, which helps in understanding the underlying instability mechanism. Several patterns are observed in continuum mechanical systems, such as Rayleigh-Bénard systems [2], Bénard-Marangoni systems [3], magnetohydrodynamics [4], ferrofluids [5], binary fluids [6], granular materials [7] under shaking, biological systems [8], etc. Symmetries and dissipation play a very significant role in pattern selection in such systems . The selection of a pattern is a consequence of at least one broken symmetry of the system. Unbroken symmetries often introduce multiple patterns, which may lead to a transition from local to global pattern dynamics. The gluing [10] of two limit cycles on two sides of a saddle point in the phase space of a given system is an example of a local to nonlocal bifurcation. It occurs when two limit cycles simultaneously become homoclinic orbits of the same saddle point. This phenomenon has been recently observed in a variety of systems including liquid crystals , fluid dynamical systems [12], biological systems [13], optical systems , and electrical circuits , and is a topic of current research. The pattern dynamics in the vicinity of a homoclinic bifurcation has, however, not been investigated in a fluid dynamical system.

Dynamics of fluctuations in a fluid below the onset of Rayleigh-Bénard convection

Physical Review E, 2004

We present experimental data and their theoretical interpretation for the decay rates of temperature fluctuations in a thin layer of a fluid heated from below and confined between parallel horizontal plates. The measurements were made with the mean temperature of the layer corresponding to the critical isochore of sulfur hexafluoride above but near the critical point where fluctuations are exceptionally strong. They cover a wide range of temperature gradients below the onset of Rayleigh-Bénard convection, and span wave numbers on both sides of the critical value for this onset. The decay rates were determined from experimental shadowgraph images of the fluctuations at several camera exposure times. We present a theoretical expression for an exposure-time-dependent structure factor which is needed for the data analysis. As the onset of convection is approached, the data reveal the critical slowing-down associated with the bifurcation. Theoretical predictions for the decay rates as a function of the wave number and temperature gradient are presented and compared with the experimental data. Quantitative agreement is obtained if allowance is made for some uncertainty in the small spacing between the plates, and when an empirical estimate is employed for the influence of symmetric deviations from the Oberbeck-Boussinesq approximation which are to be expected in a fluid with its density at the mean temperature located on the critical isochore.

Bifurcations and chaos in single-roll natural convection with low Prandtl number

Physics of Fluids, 2005

Convective flows of a small Prandtl number fluid contained in a two-dimensional cavity subject to a lateral thermal gradient are numerically studied by using different techniques. The aspect ratio ͑length to height͒ is kept at around 2. This value is found optimal to make the flow most unstable while keeping the basic single-roll structure. Two cases of thermal boundary conditions on the horizontal plates are considered: perfectly conducting and adiabatic. For increasing Rayleigh numbers we find a transition from steady flow to periodic oscillations through a supercritical Hopf bifurcation that maintains the centrosymmetry of the basic circulation. For a Rayleigh number of about ten times that of the Hopf bifurcation the system initiates a complex scenario of bifurcations. In the conductive case these include a quasiperiodic route to chaos. In the adiabatic one the dynamics is dominated by the interaction of two Neimark-Sacker bifurcations of the basic periodic solutions, leading to the stable coexistence of three incommensurate frequencies, and finally to chaos. In all cases, the complex time-dependent behavior does not break the basic, single-roll structure.

Spatially quasiperiodic convection and temporal chaos in two-layer thermocapillary instabilities

Physical Review E, 1996

This paper describes an amplitude equation analysis of the interactions between waves with wave number k 1 ͑and phase speed c /k 1 ͒ and stationary convection with wave number k 2. These two modes may bifurcate almost simultaneously from the conductive state of a two-layer Bénard system, when the ratio of layer thicknesses is near a particular value ͑codimension-2 singularity͒. When k 2 2k 1 ͑nonresonant case͒ and the first bifurcation occurs for steady convection, a secondary bifurcation to a spatially quasiperiodic and time-periodic mixed mode is obtained when increasing the driving gradient. No stable small-amplitude solution exists when the Hopf bifurcation is the first one. The occurrence of either of these two possibilities depends on the thickness ratio. When k 2 ϭ2k 1 ͑resonant case͒, the system presents a much wider variety of dynamical behaviors, including quasiperiodic relaxation oscillations and temporal chaos. The discussion of the resonant system concentrates on a scenario of transition to chaos consisting of an infinite sequence of ''period-doubling'' homoclinic bifurcations of stable periodic orbits, for which the left-right symmetry of the convective system plays an essential role. For increasing constraint, a reverse cascade is observed, for which quadratic nonlinearities in the Ginzburg-Landau equations are shown to entirely determine the dynamics ͑cubic and higherorder terms may be neglected near the codimension-2 point͒. ͓S1063-651X͑96͒09207-0͔

Transition to chaos in a differentially heated vertical cavity

Journal of Fluid Mechanics, 1989

We investigate numerically the transition from laminar to chaotic flow of a Boussinesq fluid with Pr = 0.71 in two-dimensional closed, differentially heated, vertical cavities having aspect ratios near unity. The cavities have rigid conducting sidewalls, and rigid insulating top and bottom walls. The physical nature of the resulting flow is a function of the aspect ratio and Rayleigh number. It is shown that an oscillatory approach to steady-state, oscillatory instabilities, quasi-periodic flow, and chaotic flow exist for the flow regimes investigated. We find t h a t for aspect ratios of approximately three or larger the the first transition from steady-state is due to instability of the sidewall boundary layers, while for smaller aspect ratios, but larger than i, it is due to internal waves near the departing corners. For both instabilities we obtain the critical Rayleigh number as a function of aspect ratio and write expressions relating the fundamental frequencies of the oscillatory flow to the Rayleigh number and aspect ratio. When Ra is increased significantly above the first critical value, the flow becomes complex since both types of instabilities can be present. With a further increase in Rayleigh number the flow becomes chaotic and eventually turbulent. The above results are illustrated for different Rayleigh numbers and aspect ratios using time histories, spectral analysis, and streamlines at different values of time.

Quantifying spatiotemporal chaos in Rayleigh-Bénard convection

Physical Review E, 2012

Using large-scale parallel numerical simulations we explore spatiotemporal chaos in Rayleigh-Bénard convection in a cylindrical domain with experimentally relevant boundary conditions. We use the variation of the spectrum of Lyapunov exponents and the leading order Lyapunov vector with system parameters to quantify states of high-dimensional chaos in fluid convection. We explore the relationship between the time dynamics of the spectrum of Lyapunov exponents and the pattern dynamics. For chaotic dynamics we find that all of the Lyapunov exponents are positively correlated with the leading order Lyapunov exponent and we quantify the details of their response to the dynamics of defects. The leading order Lyapunov vector is used to identify topological features of the fluid patterns that contribute significantly to the chaotic dynamics. Our results show a transition from boundary dominated dynamics to bulk dominated dynamics as the system size is increased. The spectrum of Lyapunov exponents is used to compute the variation of the fractal dimension with system parameters to quantify how the underlying high-dimensional strange attractor accommodates a range of different chaotic dynamics.