Role of uniform horizontal magnetic field on convective flow (original) (raw)
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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.
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.
Effect of a vertical magnetic field on turbulent Rayleigh-Bénard convection
Physical review, 2000
The effect of a vertical uniform magnetic field on Rayleigh-Bénard convection is investigated experimentally. We confirm that the threshold of convection is in agreement with linear stability theory up to a Chandrasekhar number QӍ4ϫ10 6 , higher than in previous experiments. We characterize two convective regimes influenced by MHD effects. In the first one, the Nusselt number Nu proportional to the Rayleigh number Ra, which can be interpreted as a condition of marginal stability for the thermal boundary layer. For higher Ra, a second regime NuϳRa 0.43 is obtained.
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.
Dynamics of flow reversals in the presence of a vertical magnetic field (a)
EPL (Europhysics Letters), 2021
We investigate the dynamics of flow reversals close to the onset of Rayleigh-Bénard convection (RBC) of electrically conducting low Prandtl number (Pr) Boussinesq fluids in the presence of an external vertical magnetic field. The investigation is carried out by performing three-dimensional (3D) direct numerical simulations (DNS) of the related mathematical model considering the quasi-static approximation. The numerical investigation reveals a rich dynamics of the flow reversals close to the onset of convection. Three qualitatively different kinds of flow reversals, namely chaotic, intermittent and periodic are identified and all are found to occur in the presence of weak magnetic field, measured in terms of the Chandrasekhar number (Q). Explored ranges of the Prandtl number (Pr), the Rayleigh number (Ra) and the Chandrasekhar number are 0 < Pr ≤ 0.03, 6.5 × 10 2 ≤ Ra ≤ 7 × 10 3 and 0 < Q ≤ 100, respectively. Out of the three different types of flow reversals, intermittent flow reversal is new and has not been reported so far in convective system to the best of our knowledge. The probability density function of the time interval of occurrences of chaotic flow reversals is found to decay exponentially. The heat transfer properties of the system are also numerically investigated and it is observed that magnetic field facilitates the heat transfer in the considered ranges of the parameter space. focus article
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.
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.
Journal of Fluid Mechanics, 2009
The paper deals with the numerical investigation of the steady convective flow in a horizontal channel of rectangular cross-section subjected to a uniform longitudinal temperature gradient imposed at the walls. It is shown that at zero Prandtl number the solution of the problem corresponds to a plane-parallel flow along the channel axis. In this case, the fluid moves in the direction of the imposed temperature gradient in the upper part of the channel and in the opposite direction in the lower part. At non-zero values of the Prandtl number, such solution does not exist. At any small values of Pr all three components of the flow velocity differ from zero and in the channel cross-section four vortices develop. The direction of these vortices is such that the fluid moves from the centre to the periphery in the vertical direction and returns to the centre in the horizontal direction. The stability of these convective flows (uniform along the channel axis) with regard to small three-dimensional perturbations periodical in the direction of the channel axis is studied. It is shown that at low values of the Prandtl number the basic state loses its stability due to the steady hydrodynamic mode related to the development of vortices at the boundary of the counter flows. The growth of the Prandtl number results in the strong stabilization of this instability mode and, beyond a certain value of the Prandtl number depending on the crosssection aspect ratio, a new steady hydrodynamic instability mode becomes the most dangerous. This mode is characterized by the localization of the perturbations near the sidewalls of the channel. At still higher values of the Prandtl number, the spiral perturbations (rolls with axis parallel to the temperature gradient) become the most dangerous modes, at first the oscillatory spiral perturbations and then the Rayleightype steady spiral perturbations. The influence of the channel width on these different instabilities is also emphasized.
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.
Transition to chaos and magnetic field generation in rotating Rayleigh-B\'enard convection
arXiv (Cornell University), 2024
Hydrodynamic and magnetohydrodynamic convective attractors in threedimensional rotating Rayleigh-Bénard convection are studied numerically by varying the Taylor and Rayleigh numbers as control parameters. First, an analysis of hydrodynamic attractors and their bifurcations is conducted, where routes to chaos via quasiperiodicity are identified. Second, the behaviour of the magnetohydrodynamic system is investigated by introducing a seed magnetic field and measuring its growth or decay as a function of the Taylor number, while keeping the Rayleigh number fixed. Analysis of the attractors shows that rotation has a significant impact on magnetic field generation in Rayleigh-Bénard convection, with the critical magnetic Prandtl number changing nonmonotonically with the rotation rate. It is argued that a nonhysteretic blowout bifurcation with on-off intermittency is responsible for the transitions to dynamo.