Stochastic theory and direct numerical simulations of the relative motion of high-inertia particle pairs in isotropic turbulence
Stochastic theory and direct numerical simulations of the relative motion of high-inertia particle pairs in isotropic turbulence
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各向同性湍流中高惯性粒子对相对运动的随机理论和直接数值模拟
DOI：
10.1017/jfm.2016.859
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发表时间：
2017
影响因子：
3.7
通讯作者：
Koch, Donald L.
中科院分区：
文献类型：
--
作者：
Dhariwal, Rohit;Rani, Sarma L.;Koch, Donald L.
采用直接数值模拟（DNS）和基于概率密度函数（PDF）动力学模型的Langevin模拟（LS）研究了各向同性湍流中单分散高惯性粒子对的相对速度和位置。在先前的研究中（拉尼等人，流体力学杂志，第756卷，2014年，pp. 870-902），作者开发了一种随机理论，涉及在单分散颗粒对的PDF方程中的扩散率张量的高斯托克斯数的极限中导出闭合。扩散率包含的时间积分的欧拉两个时间相关的流体相对速度看到的对，几乎是静止的。两个时间的相关性分析解决了通过近似的时间变化的流体相对速度看到一对发生主要是由于平流的小涡过去的对大规模的涡流。因此，两个扩散率表达式得到的基础上是否对质量中心保持固定的流动时间尺度，或响应于积分尺度涡移动。在目前的研究中，定量分析（拉尼等人。2014）随机理论是通过比较使用LS获得的对统计数据与来自DNS的对统计数据来执行的。LS包括对分离和相对速度的朗之万方程的演化，这在统计上等价于求解经典的Fokker-Planck形式的对PDF方程。使用三种封闭形式的扩散率进行粒子对分散的朗之万模拟-即包含所见流体相对速度的欧拉两次相关性的时间积分和两个分析扩散率表达式的封闭形式。在第一种封闭形式中，两个时间的相关性计算使用DNS的强制各向同性湍流与固定颗粒负载。这两种解析闭合形式的优点是，它们可以使用与DNS谱密切匹配的湍流能谱模型进行评估。这三个扩散系数进行了分析，以量化的影响，在推导它们的近似。对相对运动统计从三套朗之万模拟的结果进行了比较，从DNS的（移动）粒子负载的强制各向同性湍流。第一个封闭形式，涉及欧拉两个时间的流体相对速度的相关性，表现出最好的协议与DNS结果的PDF。
The relative velocities and positions of monodisperse high-inertia particle pairs in isotropic turbulence are studied using direct numerical simulations (DNS), as well as Langevin simulations (LS) based on a probability density function (PDF) kinetic model for pair relative motion. In a prior study (Rani et al., J. Fluid Mech., vol. 756, 2014, pp. 870–902), the authors developed a stochastic theory that involved deriving closures in the limit of high Stokes number for the diffusivity tensor in the PDF equation for monodisperse particle pairs. The diffusivity contained the time integral of the Eulerian two-time correlation of fluid relative velocities seen by pairs that are nearly stationary. The two-time correlation was analytically resolved through the approximation that the temporal change in the fluid relative velocities seen by a pair occurs principally due to the advection of smaller eddies past the pair by large-scale eddies. Accordingly, two diffusivity expressions were obtained based on whether the pair centre of mass remained fixed during flow time scales, or moved in response to integral-scale eddies. In the current study, a quantitative analysis of the (Rani et al. 2014) stochastic theory is performed through a comparison of the pair statistics obtained using LS with those from DNS. LS consist of evolving the Langevin equations for pair separation and relative velocity, which is statistically equivalent to solving the classical Fokker–Planck form of the pair PDF equation. Langevin simulations of particle-pair dispersion were performed using three closure forms of the diffusivity – i.e. the one containing the time integral of the Eulerian two-time correlation of the seen fluid relative velocities and the two analytical diffusivity expressions. In the first closure form, the two-time correlation was computed using DNS of forced isotropic turbulence laden with stationary particles. The two analytical closure forms have the advantage that they can be evaluated using a model for the turbulence energy spectrum that closely matched the DNS spectrum. The three diffusivities are analysed to quantify the effects of the approximations made in deriving them. Pair relative-motion statistics obtained from the three sets of Langevin simulations are compared with the results from the DNS of (moving) particle-laden forced isotropic turbulence for . The first closure form, involving the Eulerian two-time correlation of fluid relative velocities, showed the best agreement with the DNS results for the PDFs.
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