Effects of a small magnetic field on homoclinic bifurcations in a low-Prandtl-number fluid (original) (raw)
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Homoclinic bifurcations in low-Prandtl-number Rayleigh-Bénard convection with uniform rotation
EPL (Europhysics Letters), 2013
We present the effects of small Coriolis force on homoclinic gluing and ungluing bifurcations in low-Prandtl-number(0.025 ≤ P r) rotating Rayleigh-Bénard system with stress-free top and bottom boundaries. We have performed direct numerical simulations for a a wide range of Taylor number (5 ≤ T a ≤ 50) and reduce Rayleigh number r (≤ 1.25). We observe homoclinic ungluing bifurcation, marked by the spontaneous breaking of a larger limit cycle into two possible set of limit cycles in the phase space, for lower values of T a. Two unglued limit cycles merge together for higher values of T a, as Ra is raised sufficiently. The range of T a for which both gluing and ungluing bifurcation can be seen depends on the Prandtl number P r. The variation of the bifurcation points with T a is also investigated. We also present a low-dimensional model which qualitatively captures the dynamics of the system near the homoclinic bifurcation points. The model is used to study the variation of the homoclinic bifurcation points with Prandtl number for different values of T a.
Physics of Fluids, 2020
We investigate the transitions near the onset of thermal convection in electrically conducting low Prandtl-number (Pr) fluids in the presence of rotation about a vertical axis and external horizontal magnetic field. Three-dimensional direct numerical simulations (DNSs) and low dimensional modeling are performed with the Rayleigh-Bénard convection system in the ranges 0 < Q ≤ 1000 and 0 < Ta ≤ 500 of the Chandrasekhar number (Q) and the Taylor number (Ta), respectively, for that purpose. For larger Q(≥32.7), DNSs show substantial enhancement of convective heat transport and only finite amplitude steady two dimensional roll patterns at the onset. On the other hand, for smaller Q(<32.7), very rich dynamics involving different stationary as well as time dependent patterns, including stationary two-dimensional rolls, cross rolls, and oscillatory cross rolls, are observed at the onset of convection. Our investigation uncovers the cause of enhancement of heat transport and the origin of different flow patterns at the onset. We establish that a first order transition to convection occurring at the onset is responsible for the enhancement of the heat transport there. Furthermore, as the Rayleigh number (Ra) is increased beyond the onset, subsequent transitions near it are also explored in detail for smaller Q, and these are found to be associated with a variety of bifurcations including subcritical/supercritical pitchfork, Hopf, imperfect pitchfork, imperfect gluing, and Neimark-Sacker.
EPL (Europhysics Letters), 2015
PACS 47.20.Bp-Buoyancy-driven flow instabilities PACS 47.52.+j-Chaos in fluid dynamics PACS 47.35.Tv-Magnetohydrodynamics in fluids Abstract-We investigate oscillatory instability and routes to chaos in Rayleigh-Bénard convection of electrically conducting fluids in presence of external horizontal magnetic field. Three dimensional direct numerical simulations (DNS) of the governing equations are performed for the investigation. DNS shows that oscillatory instability is inhibited by the magnetic field. The supercritical Rayleigh number for the onset of oscillation is found to scale with the Chandrasekhar number Q as Q α in DNS with α = 1.8 for low Prandtl numbers (Pr). Most interestingly, DNS shows Q dependent routes to chaos for low Prandtl number fluids like mercury (Pr = 0.025). For low Q, period doubling routes are observed, while, quasiperiodic routes are observed for high Q. The bifurcation structure associated with Q dependent routes to chaos is then understood by constructing a low dimensional model from the DNS data. The model also shows similar scaling laws as DNS. Bifurcation analysis of the low dimensional model shows that origin of different routes are associated with the bifurcation structure near the primary instability. These observations show similarity with the previous results of convection experiments performed with mercury.
Role of uniform horizontal magnetic field on convective flow
The European Physical Journal B, 2012
The effect of uniform magnetic field applied along a fixed horizontal direction in Rayleigh-Bénard convection in low-Prandtl-number fluids has been studied using a low dimensional model. The model shows the onset of convection (primary instability) in the form of two dimensional stationary rolls in the absence of magnetic field, when the Rayleigh number R is raised above a critical value Rc. The flow becomes three dimensional at slightly higher values of Rayleigh number via wavy instability. These wavy rolls become chaotic for slightly higher values of R in low-Prandtl-number (Pr) fluids. A uniform magnetic field along horizontal plane strongly affects all kinds of convective flows observed at higher values of R in its absence. As the magnetic field is raised above certain value, it orients the convective rolls in its own direction. Although the horizontal magnetic field does not change the threshold for the primary instability, it affects the threshold for secondary (wavy) instability. It inhibits the onset of wavy instability. The critical Rayleigh number Ro(Q, Pr) at the onset of wavy instability, which depends on Chandrasekhar's number Q and Pr, increases monotonically with Q for a fixed value of Pr. The dimensionless number Ro(Q, Pr)/(RcQPr) scales with Q as Q −1. A stronger magnetic field suppresses chaos and makes the flow two dimensional with roll pattern aligned along its direction.
Directional effect of a magnetic field on oscillatory low-Prandtl-number convection
Physics of Fluids, 2008
The directional effect of a magnetic field on the onset of oscillatory convection is studied numerically in a confined three-dimensional cavity of relative dimensions 4:2:1 ͑length:width:height͒ filled with mercury and subject to a horizontal temperature gradient. The magnetic field suppresses the oscillations most effectively when it is applied in the vertical direction, and is the least efficient when applied in the longitudinal direction ͑parallel to the temperature gradient͒. In all cases, however, exponential growths of the critical Grashof number, Gr c ͑Gr, ratio of buoyancy to viscous dissipation forces͒ with the Hartmann number ͑Ha, ratio of magnetic to viscous dissipation forces͒ are obtained. Insight into the damping mechanism is gained from the fluctuating kinetic energy budget associated with the time-periodic disturbances at threshold. The kinetic energy produced by the vertical shear of the longitudinal basic flow dominates the oscillatory transition, and when a magnetic field is applied, it increases in order to balance the stabilizing magnetic energy. Moreover, subtle changes in the spatial distribution of this shear energy are at the origin of the exponential growth of Gr c . The destabilizing effect of the velocity fluctuations strongly decreases when Ha is increased ͑due to the decay of the velocity fluctuations in the bulk accompanied by the appearance of steep gradients localized in the Hartmann layers͒, so that an increase of the shear of the basic flow at Gr c is required in order to sustain the instability. This yields an increase in Gr c , which is reinforced by the fact that the shear of the basic flow naturally decreases at constant Gr with the increase of Ha, particularly when the magnetic field is applied in the vertical direction. For transverse and longitudinal fields, the decay of the velocity fluctuations is combined with an increase of the shear energy term due to a sustained growth in stabilizing magnetic energy with Ha.
A model for Rayleigh-Bénard magnetoconvection
The European Physical Journal B, 2015
A model for three-dimensional Rayleigh-Bénard convection in low-Prandtl-number fluids near onset with rigid horizontal boundaries in the presence of a uniform vertical magnetic field is constructed and analyzed in detail. The kinetic energy K, the convective entropy Φ and the convective heat flux (N u − 1) show scaling behaviour with ǫ = r − 1 near onset of convection, where r is the reduced Rayleigh number. The model is also used to investigate various magneto-convective structures close to the onset. Straight rolls, which appear at the primary instability, become unstable with increase in r and bifurcate to three-dimensional structures. The straight rolls become periodically varying wavy rolls or quasiperiodically varying structures in time with increase in r depending on the values of Prandtl number P r. They become irregular in time, with increase in r. These standing wave solutions bifurcate first to periodic and then quasiperiodic traveling wave solutions, as r is raised further. The variations of the critical Rayleigh number Raos and the frequency ωos at the onset of the secondary instability with P r are also studied for different values of Chandrasekhar's number Q.
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 .
Overstable rotating convection in the presence of a vertical magnetic field
Physics of Fluids
Recently, Banerjee et al. [Phys. Rev. E 102, 013107 (2020)] investigated overstable rotating convection in the presence of an external horizontal magnetic field and reported a rich bifurcation structure near the onset. However, the bifurcation structure near the onset of overstable rotating convection in the presence of a vertical magnetic field has not been explored yet. We address the issue here by performing three dimensional direct numerical simulations and low-dimensional modeling of the system using a Rayleigh–Bénard convection model. The control parameters, namely, the Taylor number (Ta), the Chandrasekhar number (Q), and the Prandtl number (Pr) are varied in the ranges 750≤Ta≤106, 0<Q≤103, and 0<Pr≤0.5. Our investigation reveals two qualitatively different onset scenarios including bistability (coexistence of subcritical and supercritical convections). Analysis of the low-dimensional model shows that a supercritical Hopf bifurcation is responsible for the supercritical...
Rayleigh-Bénard convection with uniform vertical magnetic field
Physical review. E, Statistical, nonlinear, and soft matter physics, 2014
We present the results of direct numerical simulations of Rayleigh-Bénard convection in the presence of a uniform vertical magnetic field near instability onset. We have done simulations in boxes with square as well as rectangular cross sections in the horizontal plane. We have considered the horizontal aspect ratio η=L(y)/L(x)=1 and 2. The onset of the primary and secondary instabilities are strongly suppressed in the presence of the vertical magnetic field for η=1. The Nusselt number Nu scales with the Rayleigh number Ra close to the primary instability as [{Ra-Ra(c)(Q)}/Ra(c)(Q)](0.91), where Ra(c)(Q) is the threshold for onset of stationary convection at a given value of the Chandrasekhar number Q. Nu also scales with Ra/Q as (Ra/Q)(μ). The exponent μ varies in the range 0.39≤μ≤0.57 for Ra/Q≥25. The primary instability is stationary as predicted by Chandrasekhar. The secondary instability is temporally periodic for Pr=0.1 but quasiperiodic for Pr=0.025 for moderate values of Q. ...
Convective Instability and Pattern Formation in Magnetic Fluids
Journal of Mathematical Analysis and Applications, 1997
A nonlinear convective instability in a layer of magnetic fluid is investigated in the presence of an applied magnetic field and temperature gradient. The stability of steady state patterns resulting from the convective instability is discussed using bifurcation theory. Rolls are found to be stable on both the square and hexagonal lattices.