Solution of the non-linear equations of cellular convection and heat transport
Solution of the non-linear equations of cellular convection and heat transport
复制标题
细胞对流和热传输非线性方程的求解
DOI：
10.1017/s0022112061000408
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发表时间：
1961
影响因子：
3.7
通讯作者：
H. Kuo
中科院分区：
文献类型：
--
作者：
H. Kuo
通过将因变量展开为正交函数的级数，将这些函数的系数展开为参数η的幂级数，得到了元胞对流非线性方程组的解。在选择展开参数η时，要使它对瑞利数的所有有限值都保持小于1。结果表明，所得到的解在很大的温度范围内是有效的，且收敛速度快。该解提供了层流范围内对流热输运作为温差函数的定量理论。结果还表明，当实际瑞利数大于临界瑞利数的两倍时，在流体层中间形成一层等温（气体介质中的绝热渗透）平均温度层。随着实际瑞利数的增加，边界层的厚度也随之增加，同时边界层的温度梯度也随之增大，从而实现热输运的增加。结果进一步表明，大的温度梯度集中在冷下降流接近下界和暖上升流接近上界的区域。研究还表明，这些上升和下降的气流以蘑菇状的模式展开，这是孤立热气泡对流的特征，但从未被认为是有限细胞对流的形式。最近的光学观测表明，这是温度场最常见的形式。该解给出的热输运符合1.24指数幂律，与观测到的层流1.25指数幂律非常接近。
By expanding the dependent variables in series of orthogonal functions on the one hand, and expanding the coefficients of these functions in power series of a parameter η on the other hand, a solution has been obtained for the system of non-linear equations of cellular convection. The expansion parameter η is chosen in such a way as to make it remain less than 1 for all finite values of the Rayleigh number. The solution so obtained is found to be valid for a large range of the imposed temperature difference, and converges rapidly. This solution provides a quantitative theory for the convective heat transport as a function of the temperature difference in the range of laminar flow. The solution also reveals that when the actual Rayleigh number is greater than twice the critical Rayleigh number, a layer of isothermal (adiabatic lapserate in a gas medium) mean temperature develops in the middle of the fluid layer. The thickness of this layer increases as the actual Rayleigh number increases, and at the same time the temperature gradient increases in the boundary layer so that an increase in the heat transport is accomplished. The solution reveals further that the large temperture gradients are concentrated in the region where the cold descending current approaches the lower boundary and where the warm ascending current approaches the upper boundary. It is also shown that these ascending and descending currents spread out in mushroom-like patterns, a feature characteristic of the convection of isolated hot bubbles, but one which never has been considered as the form for finite cellular convection. Recent optical observations indicate that this is the most common form of the temperature field. The heat transport given by this solution fits a power law of exponent 1.24, which is very close to the observed power law of exponent 1.25 for laminar flow.