Rayleigh-B\'enard convection with uniform vertical magnetic field

TL;DR

This paper presents direct numerical simulations of Rayleigh-Bénard convection under a vertical magnetic field, revealing how magnetic suppression affects instability thresholds, flow patterns, and scaling laws near onset.

Contribution

It provides new insights into the effects of a uniform vertical magnetic field on convection onset, flow regimes, and scaling behaviors in different aspect ratio geometries.

Findings

Magnetic field suppresses primary and secondary instabilities.

Nusselt number scales with Rayleigh number near onset.

Flow patterns include periodic, quasiperiodic, and chaotic rolls.

Abstract

We present the results of direct numerical simulations of Rayleigh-B\'enard 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 horizontal aspect ratio $\eta = L_y/L_x =1$ and $2$. The onset of the primary and secondary instabilities are strongly suppressed in the presence of the vertical magnetic field for $\eta =1$. The Nusselt number $\mathrm{Nu}$ scales with Rayleigh number $\mathrm{Ra}$ close to the primary instability as $[\mathrm{\{Ra - Ra_c (Q)\}/Ra_c (Q)} ]^{0.91}$, where $\mathrm{Ra_c (Q)}$ is the threshold for onset of stationary convection at a given value of the Chandrasekhar number $\mathrm{Q}$. $\mathrm{Nu}$ also scales with $\mathrm{Ra/Q}$ as $(\mathrm{Ra/Q})^{\mu}$. The exponent $\mu$ varies in the range $0.39 \le \mu \le 0.57$ for $\mathrm{Ra/Q} \ge 25$. The primary instability is stationary as predicted by Chandrasekhar. The secondary instability is temporally periodic for $\mathrm{Pr}=0.1$ but quasiperiodic for $\mathrm{Pr} = 0.025$ for moderate values of $\mathrm{Q}$. Convective patterns for higher values of $\mathrm{Ra}$ consist of periodic, quasiperiodic and chaotic wavy rolls above onset of the secondary instability for $\eta=1$. In addition, stationary as well as time dependent cross-rolls are observed, as $\mathrm{Ra}$ is further raised. The ratio $r_{\circ}/\mathrm{Pr}$ is independent of $\mathrm{Q}$ for smaller values of $\mathrm{Q}$. The delay in onset of the oscillatory instability is significantly reduced in a simulation box with $\eta =2$. We also observe inclined stationary rolls for smaller values of $\mathrm{Q}$ for $\eta =2$.