Rayleigh-Bénard convection with uniform vertical magnetic field

Abstract

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 patterns for higher values of Ra consist of periodic, quasiperiodic, and chaotic wavy rolls above the onset of the secondary instability for η=1. In addition, stationary as well as time-dependent cross rolls are observed, as Ra is further raised. The ratio r(o)/Pr is independent of Q for smaller values of Q. The delay in the onset of the oscillatory instability is significantly reduced in a simulation box with η=2. We also observe inclined stationary rolls for smaller values of Q for η=2.

On oscillatory convection with the Cattaneo–Christov hyperbolic heat-flow model

“…Indeed, having investigated Cattaneo-Christov heat-flow effects in the classic Rayleigh-Bénard system, it is important to consider its role in other thermal convection systems, especially those which are known to yield oscillatory or non-stationary instability under the more usual Fourier law, such as magnetized or rotating fluids [26,28]. The author plans to address some of these problems in future publications.…”

Effects of a small magnetic field on homoclinic bifurcations in a low-Prandtl-number fluid

“…9) suggests that the expansions of the induced horizontal magnetic fields b 1 , b 2 should be similar to that of the vertical velocity v 3 , and the expansion of the induced vertical magnetic field b 3 should be similar to those of the horizontal velocities (v 1 and v 2 ). A consistent expansion for the induced magnetic field is possible by considering the boundaries to be electrically insulating 33 . This leads to the following conditions at the horizontal boundaries located at z = 0 and z = 1:…”

“…Thermal convection in a thin fluid layer in the presence of a uniform magnetic field, also known as RayleighBénard (RB) magnetoconvection, has been studied both experimentally [18][19][20][21][22][23][24][25] and theoretically [26][27][28][29][30][31][32][33][34] . The study of RB magnetoconvection is useful for geophysical as well as for astrophysical problems [35][36][37] .…”

Zero Prandtl-number rotating magnetoconvection

“…The effects of rotation and external magnetic field on the instabilities and bifurcation structures near the onset of convection when acted separately have recently been investigated in [55][56][57][58][59][60][61][62] and reported very rich dynamics including chaos at the onset of convection. However, instabilities and the associated bifurcation structures near the onset of RBC of electrically conducting fluids in presence of both rotation and uniform external magnetic field have not been studied so far.…”