Basak Arnab, Raveendran Rohit, Kumar Krishna
Department of Physics, Indian Institute of Technology, Kharagpur-721302, India.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Sep;90(3):033002. doi: 10.1103/PhysRevE.90.033002. Epub 2014 Sep 5.
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), 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.
我们展示了在接近不稳定起始点的均匀垂直磁场存在下瑞利 - 贝纳德对流的直接数值模拟结果。我们在水平面上具有正方形以及矩形横截面的盒子中进行了模拟。我们考虑了水平纵横比η = L(y)/L(x)=1和2。对于η = 1，在垂直磁场存在的情况下，一次和二次不稳定的起始被强烈抑制。努塞尔数Nu在接近一次不稳定时与瑞利数Ra的标度关系为{Ra - Ra(c)(Q)}/Ra(c)(Q)，其中Ra(c)(Q)是在给定钱德拉塞卡数Q值下稳态对流起始的阈值。Nu也与Ra/Q的标度关系为(Ra/Q)(μ)。对于Ra/Q≥25，指数μ在0.39≤μ≤0.57范围内变化。如钱德拉塞卡所预测的，一次不稳定是稳态的。对于Pr = 0.1，二次不稳定是时间周期性的，但对于Pr = 0.025且Q为中等值时是准周期性的。对于η = 1，在二次不稳定起始之上，更高Ra值的对流模式由周期性、准周期性和混沌波状卷组成。此外，随着Ra进一步升高，观察到了稳态以及与时间相关的交叉卷。对于较小的Q值，r(o)/Pr与Q无关。在η = 2的模拟盒子中，振荡不稳定起始的延迟显著减小。对于η = 2且Q值较小时，我们还观察到了倾斜的稳态卷。
