3D stability analysis of Rayleigh–Bénard convection of a liquid metal layer in the presence of a magnetic field—effect of wall electrical conductivity
3D stability analysis of Rayleigh–Bénard convection of a liquid metal layer in the presence of a magnetic field—effect of wall electrical conductivity
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DOI：
10.1088/0169-5983/46/5/055507
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发表时间：
2014-06
影响因子：
1.5
通讯作者：
D. Dimopoulos;N. Pelekasis
中科院分区：
文献类型：
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作者：
D. Dimopoulos;N. Pelekasis
研究了矩形截面液态金属层在平行于截面水平方向的强磁场作用下的Rayleigh-Bénard稳定性。后者比垂直方向长得多，横截面呈大的纵横比。侧壁被视为高导电性。进行了线性稳定性分析，允许沿纵向发展的三维不稳定性。在考虑洞壁电导率的情况下，采用有限元方法对稳定性分析公式进行离散化。Arnoldi方法给出了问题的主要特征值和特征向量。为了便于数值解在大Hartmann数Ha时的并行执行，沿横截面的水平方向采用了区域分解。随着哈特曼数的增加，实本征值作为主要的不稳定本征模出现，标志着热对流的开始，其核心的主要涡度分量与磁场的方向一致。它的波长沿层的纵向约为其高度的两倍，并且随着Ha的增加而增加。对于大的Ha，得到了临界Grashof，并且它的规模与Ha 2相似，这意味着浮力和洛伦茨力之间的平衡。对于导电良好的侧壁，新出现的流型的性质由哈特曼层和哈特曼层的综合电导率CH+Ha−1决定。当考虑导电性差的哈特曼壁时，临界本征解由定义良好的哈特曼层和边层决定，CH≪1。由于哈特曼壁附近的电磁泵送，侧层的特征是流体在磁场方向上快速运动。增加哈特曼壁的电导率被认为可以延迟热对流的开始，同时在临界情况下保持上述标度。此外，对于导电和绝缘哈特曼墙以及所考察的整个Ha数范围，由于核心中的对流运动，没有发展出明确定义的准二维结构的趋势。将上述发现与以前的实验研究联系起来，表明当Gr进一步增加到临界值以上时，驻波和行波的开始。
Rayleigh–Bénard stability of a liquid metal layer of rectangular cross section is examined in the presence of a strong magnetic field that is aligned with the horizontal direction of the cross section. The latter is much longer than the vertical direction and the cross section assumes a large aspect ratio. The side walls are treated as highly conducting. Linear stability analysis is performed allowing for three-dimensional instabilities that develop along the longitudinal direction. The finite element methodology is employed for the discretization of the stability analysis formulation while accounting for the electrical conductivity of the cavity walls. The Arnoldi method provides the dominant eigenvalues and eigenvectors of the problem. In order to facilitate parallel implementation of the numerical solution at large Hartmann numbers, Ha, domain decomposition is employed along the horizontal direction of the cross section. As the Hartmann number increases a real eigenvalue emerges as the dominant unstable eigenmode, signifying the onset of thermal convection, whose major vorticity component in the core of the layer is aligned with the direction of the magnetic field. Its wavelength along the longitudinal direction of the layer is on the order of twice its height and increases as Ha increases. The critical Grashof was obtained for large Ha and it was seen to scale like Ha 2 signifying the balance between buoyancy and Lorentz forces. For well conducting side walls, the nature of the emerging flow pattern is determined by the combined conductivity of Hartmann walls and Hartmann layers, cH + Ha −1. When poor conducting Hartmann walls are considered, cH ≪ 1, the critical eigensolution is characterized by well defined Hartmann and side layers. The side layers are characterized by fast fluid motion in the magnetic field direction as a result of the electromagnetic pumping in the vicinity of the Hartmann walls. Increasing the electrical conductivity of the Hartmann walls was seen to delay the onset of thermal convection, while retaining the above scaling at criticality. Furthermore, for both conducting and insulating Hartmann walls and the entire range of Ha numbers that was examined, there was no tendency for a well defined quasi two-dimensional structure to develop owing to the convective motion in the core. A connection is made between the above findings and previous experimental investigations indicating the onset of standing waves followed by travelling waves as Gr is further increased beyond its critical value.