Tian Yifeng, Woodward Michael, Stepanov Mikhail, Fryer Chris, Hyett Criston, Livescu Daniel, Chertkov Michael
Information Sciences Group, Computer, Computational and Statistical Sciences Division (CCS-3), Los Alamos National Laboratory, Los Alamos, NM 87545.
Graduate Interdisciplinary Program in Applied Mathematics and Department of Mathematics, University of Arizona, Tucson, AZ 85721.
Proc Natl Acad Sci U S A. 2023 Aug 22;120(34):e2213638120. doi: 10.1073/pnas.2213638120. Epub 2023 Aug 16.
High-Reynolds number homogeneous isotropic turbulence (HIT) is fully described within the Navier-Stokes (NS) equations, which are notoriously difficult to solve numerically. Engineers, interested primarily in describing turbulence at a reduced range of resolved scales, have designed heuristics, known as large eddy simulation (LES). LES is described in terms of the temporally evolving Eulerian velocity field defined over a spatial grid with the mean-spacing correspondent to the resolved scale. This classic Eulerian LES depends on assumptions about effects of subgrid scales on the resolved scales. Here, we take an alternative approach and design LES heuristics stated in terms of Lagrangian particles moving with the flow. Our Lagrangian LES, thus L-LES, is described by equations generalizing the weakly compressible smoothed particle hydrodynamics formulation with extended parametric and functional freedom, which is then resolved via Machine Learning training on Lagrangian data from direct numerical simulations of the NS equations. The L-LES model includes physics-informed parameterization and functional form, by combining physics-based parameters and physics-inspired Neural Networks to describe the evolution of turbulence within the resolved range of scales. The subgrid-scale contributions are modeled separately with physical constraints to account for the effects from unresolved scales. We build the resulting model under the differentiable programming framework to facilitate efficient training. We experiment with loss functions of different types, including physics-informed ones accounting for statistics of Lagrangian particles. We show that our L-LES model is capable of reproducing Eulerian and unique Lagrangian turbulence structures and statistics over a range of turbulent Mach numbers.
高雷诺数均匀各向同性湍流（HIT）可以完全由纳维-斯托克斯（NS）方程描述，但这些方程在数值求解上非常困难。工程师们主要关注在有限的解析尺度范围内描述湍流，他们设计了一种启发式方法，即大涡模拟（LES）。LES是根据在空间网格上定义的随时间演化的欧拉速度场来描述的，其平均间距对应于解析尺度。这种经典的欧拉LES依赖于关于亚网格尺度对解析尺度影响的假设。在这里，我们采用另一种方法，设计基于随流运动的拉格朗日粒子的LES启发式方法。我们的拉格朗日LES，即L-LES，由推广弱可压缩平滑粒子流体动力学公式的方程描述，具有扩展的参数和函数自由度，然后通过对NS方程直接数值模拟的拉格朗日数据进行机器学习训练来求解。L-LES模型包括基于物理的参数化和函数形式，通过结合基于物理的参数和受物理启发的神经网络来描述解析尺度范围内湍流的演化。亚网格尺度的贡献通过物理约束单独建模，以考虑未解析尺度的影响。我们在可微编程框架下构建最终模型，以促进高效训练。我们用不同类型的损失函数进行实验，包括考虑拉格朗日粒子统计的基于物理的损失函数。我们表明，我们的L-LES模型能够在一系列湍流马赫数范围内再现欧拉和独特的拉格朗日湍流结构及统计特性。
