Dynamics of zero-Prandtl number convection near onset (original) (raw)
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Bifurcation and chaos in zero-Prandtl-number convection
EPL (Europhysics Letters), 2009
We present the 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. Global chaos, intermittency, and crises are observed near the onset.
Dynamics of zero-Prandtl number convection near the onset
2010
In this paper we present various convective states of zero-Prandtl number Rayleigh-Bénard convection using direct numerical simulations (DNS) and a 27-mode lowdimensional model containing the energetic modes of DNS. The origin of these convective states have been explained using bifurcation analysis. The system is chaotic at the onset itself with three coexisting chaotic attractors that are born at two codimension-2 bifurcation points. One of the bifurcation points with a single zero eigenvalue and a complex pair (0, ±iω) generates chaotic attractors and associated periodic, quasiperiodic, and phase-locked states that are related to the wavy rolls observed in experiments and simulations. The frequency of the wavy rolls are in general agreement with ω of the above eigenvalue of the stability matrix. The other bifurcation point with a double zero eigenvalue produces the other set of chaotic attractors and ordered states such as squares, asymmetric squares, oscillating asymmetric squares, relaxation oscillations with intermediate squares, some of which are common to the 13-mode model of Pal et al.
Patterns and bifurcations in low–Prandtl-number Rayleigh-Bénard convection
Europhysics Letters (epl), 2010
We present a detailed bifurcation structure and associated flow patterns for low-Prandtl number ($P=0.0002, 0.002, 0.005, 0.02$) Rayleigh-B\'{e}nard convection near its onset. We use both direct numerical simulations and a 30-mode low-dimensional model for this study. We observe that low-Prandtl number (low-P) convection exhibits similar patterns and chaos as zero-P convection \cite{pal:2009}, namely squares, asymmetric squares, oscillating asymmetric squares, relaxation oscillations, and chaos. At the onset of convection, low-P convective flows have stationary 2D rolls and associated stationary and oscillatory asymmetric squares in contrast to zero-P convection where chaos appears at the onset itself. The range of Rayleigh number for which stationary 2D rolls exist decreases rapidly with decreasing Prandtl number. Our results are in qualitative agreement with results reported earlier.
Bifurcations and chaos in large-Prandtl number Rayleigh�B�nard convection
Int J Non Linear Mech, 2011
A low-dimensional model of large Prandtl-number (P) Rayleigh Bénard convection is constructed using some of the important modes of pseudospectral direct numerical simulations. A detailed bifurcation analysis of the low-dimensional model for P = 6.8 and aspect ratio of 2 √ 2 reveals a rich instability and chaos picture: steady rolls, time-periodicity, quasiperiodicity, phase locking, chaos, and crisis. Bifurcation analysis also reveals multiple co-existing attractors, and a window with time-periodicity after chaos. The results of the low-dimensional model matches quite closely with some of the past simulations and experimental results where they observe chaos in RBC through quasiperiodicity and phase locking.
BIFURCATION ANALYSIS OF THE FLOW PATTERNS IN TWO-DIMENSIONAL RAYLEIGH–BÉNARD CONVECTION
International Journal of Bifurcation and Chaos, 2012
We investigate the origin of various convective patterns for Prandtl number P = 6.8 (for water at room temperature) using bifurcation diagrams that are constructed using direct numerical simulations (DNS) of Rayleigh-Bénard convection (RBC). Several complex flow patterns resulting from normal bifurcations as well as various instances of "crises" have been observed in the DNS. "Crises" play vital roles in determining various convective flow patterns. After a transition of conduction state to convective roll states, we observe time-periodic and quasiperiodic rolls through Hopf and Neimark-Sacker bifurcations at r 80 and r 500 respectively (where r is the normalized Rayleigh number). The system becomes chaotic at r 750, and the size of the chaotic attractor increases at r 840 through an "attractor-merging crisis" which results in traveling chaotic rolls. For 846 ≤ r ≤ 849, stable fixed points and a chaotic attractor coexist as a result of an inverse subcritical Hopf bifurcation. Subsequently the chaotic attractor disappears through a "boundary crisis" and only stable fixed points remain. These fixed points later become periodic and chaotic through another set of bifurcations which ultimately leads to turbulence. As a function of Rayleigh number, |W 101 | ∼ (r − 1) 0.62 and |θ 101 | ∼ (r − 1) −0.34 (velocity and temperature Fourier coefficient for (1, 0, 1) mode). However the Nusselt number scales as (r − 1) 0.33 .
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.
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
Physics of Fluids, 2013
We present the results of direct numerical simulations of flow patterns in a low-Prandtl-number (P r = 0.1) fluid above the onset of oscillatory convection in a Rayleigh-Bénard system rotating uniformly about a vertical axis. Simulations were carried out in a periodic box with thermally conducting and stress-free top and bottom surfaces. We considered a rectangular box (L x ×L y ×1) and a wide range of Taylor numbers (750 ≤ T a ≤ 5000) for the purpose. The horizontal aspect ratio η = L y /L x of the box was varied from 0.5 to 10. The primary instability appeared in the form of two-dimensional standing waves for shorter boxes (0.5 ≤ η < 1 and 1 < η < 2). The flow patterns observed in boxes with η = 1 and η = 2 were different from those with η < 1 and 1 < η < 2. We observed a competition between two sets of mutually perpendicular rolls at the primary instability in a square cell (η = 1) for T a < 2700, but observed a set of parallel rolls in the form of standing waves for T a ≥ 2700. The three-dimensional convection was quasiperiodic or chaotic for 750 ≤ T a < 2700, and then bifurcated into a two-dimensional periodic flow for T a ≥ 2700. The convective structures consisted of the appearance and disappearance of straight rolls, rhombic patterns, and wavy rolls inclined at an angle φ = π 2 − arctan (η −1) with the straight rolls.
Zero-Prandtl-number convection with slow rotation
Physics of Fluids, 2014
We present the results of our investigations of the primary instability and the flow patterns near onset in zero-Prandtl-number Rayleigh-Bénard convection with uniform rotation about a vertical axis. The investigations are carried out using direct numerical simulations of the hydrodynamic equations with stress-free horizontal boundaries in rectangular boxes of size (2π/k x) × (2π/k y) × 1 for different values of the ratio η = k x /k y. The primary instability is found to depend on η and T a. Wavy rolls are observed at the primary instability for smaller values of η (1/ √ 3 ≤ η ≤ 2 except at η = 1) and for smaller values of T a. We observed Küppers-Lortz (KL) type patterns at the primary instability for η = 1/ √ 3 and T a ≥ 40. The fluid patterns are found to exhibit the phenomenon of bursting, as observed in experiments [Bajaj et al. Phys. Rev. E 65, 056309 (2002)]. Periodic wavy rolls are observed at onset for smaller values of T a, while KL-type patterns are observed for T a ≥ 100 for η = √ 3. In case of η = 2, wavy rolls are observed for smaller values of T a and KL-type patterns are observed for 25 ≤ T a ≤ 575. Quasi-periodically varying patterns are observed in the oscillatory regime (T a > 575). The behavior is quite different at η = 1. A time dependent competition between two sets of mutually perpendicular rolls is observed at onset for all values of T a in this case. Fluid patterns are found to burst periodically as well as chaotically in time. It involved a homoclinic bifurcation. We have also made a couple of low-dimensional models to investigate bifurcations for η = 1, which is used to investigate the sequence of bifurcations.
Rayleigh-Bénard convection with rotation at small Prandtl numbers
Physical Review E, 2002
This paper reviews results from and future prospects for experimental studies of Rayleigh-Bénard convection with rotation about a vertical axis. At dimensionless rotation rates 0 ≤ Ω ≤ 20 and for Prandtl numbers σ 1, Küppers-Lortz-unstable patterns offered a unique opportunity to study spatio-temporal chaos immediately above a supercritical bifurcation where weakly-nonlinear theories in the form of Ginzburg-Landau (GL) or Swift-Hohenberg (SH) equations can be expected to be valid. However, the dependence of the time and length scales of the chaotic state on ≡ ∆T /∆T c −1 was found to be different from the expected dependence based on the structure of GL equations. For Ω > ∼ 70 and 0.7 < ∼ σ < ∼ 5 patterns were found to be cellular near onset with local four-fold coordination. They differ from the theoretically expected Küppers-Lortz-unstable state. Stable as well as intermittent defect-free rotating square lattices exist in this parameter range. Smaller Prandtl numbers (0.16 < ∼ σ < ∼ 0.7) can only be reached in mixtures of gases. These fluids are expected to offer rich future opportunities for the study of a line of tricritical bifurcations, of supercritical Hopf bifurcations to standing waves, of a line of codimension-two points, and of a codimension-three point.