含有颗粒的湍流流动是自然环境和工程应用中广泛存在的流动形式。各向同性湍流作为自然湍流的局部近似，具有重要的研究价值。大涡模拟(Large-Eddy Simulation, LES) 是一种采用大尺度网格求解湍流瞬时量的湍流数值模拟方法，相比于直接数值模拟(Direct Numerical Simulation, DNS) 极大地节省了计算成本，被广泛应用于高雷诺数湍流数值模拟。LES 可以对湍流大尺度的运动进行解析，但缺失小尺度湍流，这对于计算流场瞬时量以及湍流中颗粒运动有较大的影响。

因此，针对LES中小尺度湍流脉动缺失的问题，本文首先在LES中研究了小尺度湍流的效应，然后建立了基于卷积神经网络的各向同性湍流大涡模拟超分辨率重构模型，恢复LES 中缺失的小尺度湍流。最后，对于LES中惯性颗粒运动预测的问题，将超分辨率重构模型迁移到各向同性湍流数值模拟程序中，恢复小尺度湍流对惯性颗粒运动的贡献。

本文主要创新性工作包括以下三个部分。

针对LES 中缺失的小尺度湍流问题，在DNS、滤波DNS (Filtered DNS, FDNS)以及LES 中对湍流统计量进行了比较和分析。通过湍流能谱、纵向速度梯度的概率密度函数以及涡量场二维切面云图等流场统计量，对DNS、FDNS 和LES的预测结果进行了误差的分析，研究了LES 中缺失的小尺度湍流的效应。

针对LES 缺失的小尺度湍流明显影响流场统计量计算的问题，采用多隐藏层的深度卷积神经网络结合元学习方法，构建了一个自监督学习的各向同性湍流大涡模拟超分辨率重构模型(MLDCNN)。模型通过元学习算法实现了自监督学习，无需采用人工滤波的湍流数据进行模型训练。在不同雷诺数的FDNS 和较高雷诺数下的不同网格分辨率的LES 中分别做了先验评估和后验验证。先验结果显示，MLDCNN 几乎精确地恢复了FDNS 流场的湍流能谱、纵向速度梯度的概率密度函数以及涡量场二维切面云图，表明DLDCNN 模型具有重构FDNS流场中被滤除湍流结构的能力。在后验验证中，MLDCNN 模型较为准确地重构了LES 的湍流能谱和纵向速度梯度的概率密度函数，较好地恢复了LES 中缺失的小尺度湍流。

针对LES 中缺失的小尺度湍流导致LES 无法准确预测湍流中惯性颗粒运动的问题，将MLDCNN 模型迁移到各向同性湍流数值模拟程序。首先采用DNS和LES 冻结流场计算惯性颗粒的径向分布函数和径向相对速度并与正常发展演进的DNS 和LES 流场的相关结果进行比较，讨论了冻结流场的影响。然后在LES 超分辨率重构的冻结流场中对惯性颗粒的径向分布函数和径向相对速度统计量进行了预测，恢复小尺度湍流对颗粒运动的贡献，提高LES 对惯性颗粒运动的预测精度。

Particle-laden turbulent flows are a common flow phenomenon in environmental  and engineering applications. Isotropic turbulence, as a local approximation of natural  turbulence, has an important research value. Large-eddy simulation (LES) is a numerical  simulation method that uses coarse grids to solve filtered Navier-Stokes equations.Compared with direct numerical simulation (DNS), LES saves a lot of calculation costs,  and is widely used in high Reynolds number turbulent flows. LES can analyze largescale  turbulent motion, but it lacks small-scale turbulence, which has a significant impact  on calculating the instantaneous flow field and particle motion in turbulence.

Therefore, in order to recover the missing small scale turbulence fluctuations in LES, this article first studied the effects of small scale turbulence in LES, and then established  a super-resolution reconstruction model based on convolutional neural networks  to recover the missing small scale turbulent flows in LES. Finally, for the prediction of  inertial particle motion in LES, the super-resolution reconstruction model is transferred  to an isotropic turbulence numerical simulation program to recover the contribution of  small-scale turbulent flows to inertial particle motions.

The main innovative contributions of this article include the following three parts.

We compared and analyzed statistics of turbulence in DNS, filtered DNS (FDNS),  and LES to address the effects of missing small-scale turbulence in LES. Based on the  turbulent energy spectrum, the probability density function of the longitudinal velocity  gradient, and the two-dimensional color contour of the vorticity field, the error analysis  of the prediction results of DNS, FDNS and LES is carried out, and the effects of the missing small-scale turbulent flows in LES are clarified.

In order to solve the problem that the absence of small-scale turbulent flows in LES  obviously affects the calculation of flow field statistics, a self-supervised learning superresolution  reconstruction model (MLDCNN) for large-eddy simulation of isotropic turbulence  was constructed by using the depth convolution neural network with multiple  hidden layers and the meta learning method. The model realizes self-supervised learning  through meta learning algorithm, and does not need to use manually filtered turbulent  data for model training. The 𝑎 𝑝𝑟𝑖𝑜𝑟𝑖 test and 𝑎 𝑝𝑜𝑠𝑡𝑒𝑟𝑖𝑜𝑟𝑖 validations are carried out  in FDNS at different Reynolds number and LES at different grid resolutions at higher Reynolds number. The 𝑎 𝑝𝑟𝑖𝑜𝑟𝑖 results show that the MLDCNN almost accurately recovers  the turbulent energy spectrum, the probability density function of the longitudinal  velocity gradient and the two-dimensional color contour of the vorticity field of  the FDNS flow field, which indicates that the DLDCNN model has the ability to reconstruct  the filtered turbulent structure in the FDNS flow field. In the 𝑎 𝑝𝑜𝑠𝑡𝑒𝑟𝑖𝑜𝑟𝑖 test, the MLDCNN model accurately reconstructed the turbulent energy spectrum and the probability  density function of the longitudinal velocity gradient of LES, and approximately  recovered the missing small-scale turbulence in LES.

To solve the problem that LES cannot accurately predict the inertial particle relative  movement in turbulence due to the lack of small-scale turbulent flows in LES, the MLDCNN model is transferred to the isotropic turbulent numerical simulation program. Firstly, the radial distribution function and radial relative velocity of inertial particles are  calculated by DNS and LES frozen flow field respectively and compared with the relevant  results of the normal evolving DNS and LES flow field, and the influence of frozen  flow field is discussed. Then the radial distribution function and radial relative velocity  statistics of inertial particles are predicted in the reconstructed LES frozen flow field, so  as to recover the contribution of small-scale turbulent flows to particle movement, and  improve the prediction accuracy of LES for inertial particle movement.