标题： 多尺度湍流合成：二维流体力学中的验证 显示英文标题

标题： Multiscale Turbulence Synthesis: Validation in 2D Hydrodynamics

摘要： 数值模拟可以跟踪流体运动在发展湍流中的演变过程。 然而，它们的运行成本相当高，尤其是在三维情况下。 在过去二十年中，生成模型已经出现，能够在计算成本相当于模拟几个时间步的情况下生成合成随机流动。 这些简化模型在特定情况下（如不可压缩三维流体力学或磁流体力学）定性地表现出湍流的一些特征，但通常在合成相干结构方面存在困难。 我们的目标是为各种控制偏微分方程生成具有真实物理特性的随机场（例如速度、密度、磁场等），并且相对于时间分辨模拟而言成本较低。 我们提出了一组应用于给定偏微分方程集的近似方法，并在最简单的框架中测试了我们方法的有效性：二维衰减不可压缩流体力学湍流。 我们将二维衰减模拟的结果与我们合成湍流的快照进行比较。 我们首先使用标准统计工具定量评估差异：功率谱、增量和结构函数。 我们的方法可以在大约三分之一的周转时间尺度内再现这些指标。 我们还考虑了最近开发的散射变换统计方法，能够有效地表征非高斯结构。 这揭示了更显著的差异，但可以通过自助法来弥补。 最后，一次合成所需的傅里叶变换数量随分辨率呈对数增长，而时间分辨模拟则呈线性增长。 我们设计了一种多尺度湍流合成（MuScaTS）方法，以高效地绕过昂贵的数值模拟，生成现实的瞬时场。 显示英文摘要

摘要： Numerical simulations can follow the evolution of fluid motions through the intricacies of developed turbulence. However, they are rather costly to run, especially in 3D. In the past two decades, generative models have emerged which produce synthetic random flows at a computational cost equivalent to no more than a few time-steps of a simulation. These simplified models qualitatively bear some characteristics of turbulent flows in specific contexts (incompressible 3D hydrodynamics or magnetohydrodynamics), but generally struggle with the synthesis of coherent structures. We aim at generating random fields (e.g. velocity, density, magnetic fields, etc.) with realistic physical properties for a large variety of governing partial differential equations and at a small cost relative to time-resolved simulations. We propose a set of approximations applied to given sets of partial differential equations, and test the validity of our method in the simplest framework: 2D decaying incompressible hydrodynamical turbulence. We compare results of 2D decaying simulations with snapshots of our synthetic turbulence. We assess quantitatively the difference first with standard statistical tools: power spectra, increments and structure functions. These indicators can be reproduced by our method during up to about a third of the turnover time scale. We also consider recently developed scattering transforms statistics, able to efficiently characterise non-Gaussian structures. This reveals more significant discrepancy, which can however be bridged by bootstrapping. Finally, the number of Fourier transforms necessary for one synthesis scales logarithmically in the resolution, compared to linearly for time-resolved simulations. We have designed a multiscale turbulence synthesis (MuScaTS) method to efficiently short-circuit costly numerical simulations to produce realistic instantaneous fields.