多边形和光滑截面管道中被动标量输运的质量分布和偏斜度
Mass distribution and skewness for passive scalar transport in pipes with polygonal and smooth cross-sections
摘要：
我们扩展我们以前的结果表征的加载特性的扩散被动标量平流层流剪切流在管道和通道更一般的横截面形状，包括正多边形和平滑的拐角管道源自变形的椭圆形。对于三角形的情况，短时间偏斜度被精确地计算为正，而长时间渐近性显示它为负。蒙特卡罗模拟证实了这些预测，并记录了符号变化的时间尺度。有趣的是，等边三角形是唯一具有这种性质的正多边形，所有其他多边形在任何时候都具有正偏度，尽管由于缺乏这种几何形状的封闭形式流解，这无法在有限时间内证明。或者，封闭形式的流动解决方案，可以构造椭圆的平滑变形，并说明如何在时间上的多个符号切换的可能性是无关的域角。精确的条件有关的中位数和偏度的平均值，保证当偏度的符号意味着前（后）加载性能的发展示踪剂分布沿管道。短时间和长时间渐近性证实了这一条件，并且蒙特卡罗模拟始终验证这一点。
英文摘要：
We extend our previous results characterizing the loading properties of a diffusing passive scalar advected by a laminar shear flow in ducts and channels to more general cross-sectional shapes, including regular polygons and smoothed corner ducts originating from deformations of ellipses. For the case of the triangle, short time skewness is calculated exactly to be positive, while long-time asymptotics shows it to be negative. Monte-Carlo simulations confirm these predictions, and document the time scale for sign change. Interestingly, the equilateral triangle is the only regular polygon with this property, all other polygons possess positive skewness at all times, although this cannot cannot be proved on finite times due to the lack of closed form flow solutions for such geometries. Alternatively, closed form flow solutions can be constructed for smooth deformations of ellipses, and illustrate how the possibility of multiple sign switching in time is unrelated to domain corners. Exact conditions relating the median and the skewness to the mean are developed which guarantee when the sign for the skewness implies front (back) loading properties of the evolving tracer distribution along the pipe. Short and long time asymptotics confirm this condition, and Monte-Carlo simulations verify this at all times.
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