The influence of turbulence on the settling velocity of small particles remains an inconclusive research subject. Both enhanced and retarded particle settling compared to quiescent settling have been reported in previous literature, and several theories have been proposed. Among the mechanisms that account for reduced settling velocities in turbulence, the loitering effect is an important one that generally exists in various conditions. This study focuses on an analytical prediction of reduced particle settling velocities due to the loitering effect. By considering the velocity autocorrelation function as a step function analogous to the free path theory, and using a modified integral time scale to capture the essence of the loitering effect, an analytical expression is derived for predicting the reduction in particle settling velocities in turbulent flows. Calculation results of the expression are then examined by comparing with the results of a random walk model, direct numerical simulations that conditionally captured the reduction of particle settling velocities due to the loitering effect, and several representative experiments. Major possible influencing factors on the prediction of the analytical expression and applicable conditions of the expression are then further discussed. The proposed analytical expression is shown to be suitable for predicting the reduced settling velocities of small particles with relatively weak inertia in turbulent flows and could provide a reasonable explanation for reported cases in which small to moderate reductions in particle settling velocities were observed.

中文翻译：

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预测湍流中小颗粒沉降速度降低的解析表达式

湍流对小颗粒沉降速度的影响仍然没有定论。在以前的文献中已经报道了与静态沉降相比增强和延迟的粒子沉降，并且已经提出了几种理论。在引起湍流沉降速度降低的机制中，游荡效应是重要的一种，通常在各种条件下都存在。这项研究着重于分析分析预测由于游荡效应而降低的颗粒沉降速度。通过将速度自相关函数视为类似于自由路径理论的阶跃函数，并使用修正的积分时间标度来捕获游荡效应的本质，推导了一个解析表达式，用于预测湍流中颗粒沉降速度的降低。然后，通过与随机游走模型的结果进行比较，直接数值模拟（有条件地捕获由于游荡效应而导致的颗粒沉降速度降低）以及几个代表性实验，来检查表达式的计算结果。然后进一步讨论了预测分析表达式的主要可能影响因素和表达式的适用条件。