Kolmychkov V V, Shcheritsa O V, Mazhorova O S
Keldysh Institute of Applied Mathematics RAS, Miusskaya sq.4, Moscow, 125047, Russian Federation.
Phys Rev E. 2016 Dec;94(6-1):063118. doi: 10.1103/PhysRevE.94.063118. Epub 2016 Dec 29.
The paper deals with the hexagonal convective flow near the stability threshold in an internally heated fluid layer. In our previous numerical study of convection near the stability threshold in a square box with internal heat generation [Phys. Lett. A 377, 2111 (2013)]PYLAAG0375-960110.1016/j.physleta.2013.06.013 for a region of large horizontal extent, it has been shown that at small values of Prandtl number (Pr), convection sets in as a pattern of hexagonal cells with upward motion in the center (up-hexagons), whereas at large Pr, a stable flow pattern is formed by hexagonal cells with a downward motion in the center (down-hexagons). Here, we study axisymmetric convection in a cylinder as a model of motion in a single hexagonal cell. The radius of the cylinder matches the size of hexagons observed in our three-dimensional simulation. The lateral boundary of the cylinder is free and heat insulated. Horizontal bounding surfaces are rigid. The upper boundary is maintained at a constant temperature; the lower one is insulated. Two stable, steady-state motions with the upward and downward flow at the cylinder axis have been attained in calculations, irrespective of Pr. Cylindrical motion with the same direction of circulation as in the stable hexagons has a maximum temperature drop measured along the radius at the bottom of the cell. We suggest maximization of the temperature drop as a selection criterion, which determines the preferred state of motion in an internally heated fluid layer. This new selection principle is validated by the comparative analysis of the dominant nonlinear effects in low- and high-Prandtl number convection.
本文研究了内部加热流体层中接近稳定性阈值的六边形对流流动。在我们之前对具有内部热生成的方盒中接近稳定性阈值的对流进行的数值研究中[《物理快报A》377, 2111 (2013)]PYLAAG0375 - 960110.1016/j.physleta.2013.06.013，对于大水平范围的区域，已经表明在小普朗特数（Pr）时，对流以中心向上运动的六边形单元模式出现（上六边形），而在大Pr时，由中心向下运动的六边形单元形成稳定的流动模式（下六边形）。在此，我们研究圆柱中的轴对称对流，将其作为单个六边形单元中运动的模型。圆柱的半径与我们三维模拟中观察到的六边形尺寸相匹配。圆柱的侧向边界是自由且绝热的。水平边界表面是刚性的。上边界保持在恒定温度；下边界绝热。在计算中，无论Pr如何，都获得了两种稳定的稳态运动，在圆柱轴线上分别有向上和向下的流动。与稳定六边形中循环方向相同的圆柱运动在单元底部沿半径测量有最大温度降。我们提出将温度降最大化作为一种选择标准，它决定了内部加热流体层中运动的优选状态。通过对低普朗特数和高普朗特数对流中主导非线性效应的比较分析，验证了这一新的选择原则。
