A model for Rayleigh-Bénard magnetoconvection (original) (raw)
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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. ...
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.
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.
Benard convection in a non-linear magnetic fluid
Acta Mechanica, 1990
This work examines the convective instability of a horizontal layer of magnetohydrodynamic fluid of variable permeability when subjected to a non-vertical magnetic field. We use a model proposed by P. H. Roberts [9] in the context of neutron stars but the results obtained are aso relevant to the area of ferromagnetic fluids. The presence of the variable permeability has no effect on the development of instabilities through the mechanism of stationary convection but influences the threshold of overstable convection which is often the preferred mechanism in non-terrestrial applications. In the context of ferromagnetic fluids, both stationary and overstable instability can be expected to be realisable possibilities. R----where g is the acceleration of gravity, fl the uniform adverse temperature gradient, d the depth of the layer and oc, v and ~ are respectively the coefficients of volume expansion, kinematic viscosity and thermal diffusivity. If the Rayleigh number for any layer exceeds some critical value then the fluid layer would be unstable if heated from below.
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.
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.
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.
Rayleigh Bénard convection: dynamics and structure in the physical space
Communications in Mathematical Sciences, 2007
The main objective of this article is part of a research program to link the dynamics of fluid flows with the structure and its transitions in the physical spaces. As a prototype of problem and to demonstrate the main ideas, we study the two-dimensional Rayleigh-Bénard convection. The analysis is based on two recently developed nonlinear theories: geometric theory for incompressible flows [10] and the bifurcation and stability theory for nonlinear dynamical systems (both finite and infinite dimensional) . We have shown in [8] that the Rayleigh-Bénard problem bifurcates from the basic state to an attractor AR when the Rayleigh number R crosses the first critical Rayleigh number Rc for all physically sound boundary conditions, regardless of the multiplicity of the eigenvalue Rc for the linear problem. In this article, in addition to a classification of the bifurcated attractor AR, the structure and its transitions of the solutions in the physical space is classified, leading to the existence and stability of two different flows structures: pure rolls and rolls separated by a cross the channel flow. It appears that the structure with rolls separated by a cross channel flow has not been carefully examined although it has been observed in other physical contexts such as the Branstator-Kushnir waves in the atmospheric dynamics . R = gαβ κν h 4 exceeds a certain critical value, where g is the acceleration due to gravity, α the coefficient of thermal expansion of the fluid, β = |dT /dz| = (T 0 −T 1 )/h the vertical temperature gradient withT 0 the temperature on the lower 1991 Mathematics Subject Classification. 35Q, 76, 86.
Chaos, Solitons & Fractals, 2018
In this article, we study the dynamic transitions of a low-dimensional dynamical system for the Rayleigh-Bénard convection subject to a vertically applied magnetic field. Our analysis follows the dynamical phase transition theory for dissipative dynamical systems based on the principle of exchange of stability and the center manifold reduction. We find that, as the Rayleigh number increases, the system undergoes two successive transitions: the first one is a well-known pitchfork bifurcation, whereas the second one is structurally more complex and can be of different type depending on the system parameters. More precisely, for large magnetic field, the second transition is of continuous type and gives to a stable limit cycle; on the other hand, for low magnetic field or small height-to-width aspect ratio, a jump transition occurs where an unstable periodic orbit eventually collides with the stable steady state, leading to the loss of stability at the critical Rayleigh number. Finally, numerical results are presented to corroborate the analytic predictions.
Rayleigh–Bénard convection in the presence of a radial ramp of the Rayleigh number
Journal of Statistical Mechanics: Theory and Experiment, 2006
We present experimental results for pattern formation in a thin horizontal fluid layer heated from below. The fluid was SF 6 at a pressure of 20.0 bar with a Prandtl number of 0.87. The cylindrical sample had an interior section of uniform spacing d = d 0 for radii r < r 0 and a ramp d(r) for r > r 0. For Rayleigh numbers R 0 > R c in the interior, straight or slightly curved rolls with an average wavenumber k s =k c + k 1 ε 0 (ε 0 ≡ R 0 /R c − 1) with k 1 0.8 were selected. The critical wavenumberk c depended sensitively on the cell spacing. For the largestk c the patterns were skewed-varicose unstable and dislocation pairs were generated repeatedly in the interior and for all ε. For slightly smaller k c time-independent rolls were stable for ε 0.15, but for larger ε the skewedvaricose instability was encountered near the sample centre and dislocation pairs were formed repeatedly for all samples. When stationary rolls were stable, their slight curvature and the width of their wavenumber distribution slowly increased with ε. This led to a complicated shape and overall broadening of the structure factor S(k). For ε 0.05 the inverse width ξ 2 of S(k) was roughly constant and presumably limited by the finite sample size, but for larger ε we found ξ 2 ∝ ε −0.5 .