A uniformly asymptotic approximation for the development of shear dispersion
A uniformly asymptotic approximation for the development of shear dispersion
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DOI：
10.1017/s002211209600897x
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发表时间：
1996-12
影响因子：
3.7
通讯作者：
C. G. Phillips;S. R. Kaye
中科院分区：
文献类型：
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作者：
C. G. Phillips;S. R. Kaye
在本文中，我们考虑将扩散示踪剂物质引入包含流动流体的管子或管道后剪切分散的发展，重点是固定轴向位置处浓度随时间变化的特征。渐近结果是通过假设示踪剂引入点下游的距离（适当地无量纲化）很大而得出的。首先，我们考虑时间浓度变化的中心矩，包括它们对横向位置和示踪剂初始横向分布的依赖性。有限佩克莱数的矩用其对应的无限佩克莱数矩来表示，而后者是针对泊肃叶流明确给出的。然后，假设佩克莱特数是无限的，我们得出格林函数的近似解，表示示踪剂在管内任意点引入后的浓度。该解决方案以无量纲时间变量的三个数值评估函数表示，参数依赖于示踪剂释放点下游的距离。通过计算泊肃叶流中色散的近似解来说明该方法。与之前的近似不同，本解是一致渐近的，代表浓度分布的尾部以及近似高斯的中心部分；在这三个区域中，给出了更简单的近似解析形式。与以前的计算解决方案进行比较表明，即使在距示踪剂释放点很近的距离处，当前的近似值仍然相当准确。
In this paper we consider the development of shear dispersion following the introduction of a diffusing tracer substance into a tube or duct containing flowing fluid, with emphasis on the characterization of the temporal variation of concentration at a fixed axial position. Asymptotic results are derived by assuming that the distance downstream of the point of tracer introduction, appropriately non-dimensionalized, is large. First, we consider the central moments of the temporal concentration variation, including their dependence on transverse position and on the initial transverse distribution of tracer. The moments for finite Péclet number are expressed in terms of their infinite-Péclet-number counterparts, and the latter are given explicitly for Poiseuille flow. Then, assuming the Péclet number is infinite, we derive an approximate solution for the Green's function expressing tracer concentration following its introduction at an arbitrary point within the tube. The solution is expressed in terms of three numerically evaluated functions of a dimensionless time variable, with parametric dependence on the distance downstream of the point of tracer release. The method is illustrated by calculation of the approximate solution for dispersion in Poiseuille flow. Unlike previous approximations, the present solution is uniformly asymptotic and represents the tails of the concentration distribution as well as the approximately Gaussian central part; in these three regions, simpler analytic forms of the approximation are given. Comparison with previous computational solutions suggests the present approximation remains reasonably accurate even at quite short distances from the point where tracer is released.