Implicit LES of high Mach and high Reynolds number compressible turbulent flows enhanced by multidimensional flow field information using optimized flux functions and targeted reconstruction procedures due to machine-learned nonlinear neural operators

在本研究项目范围内，发展了高雷诺数下可压缩气体流动隐式大涡模拟的新方法。新开发的基于保守有限体积法的自动可微模拟和优化方法JAX-Fluid构成了该项目执行的基础。该程序包使已知的和已建立的用于界面上的状态重建和所产生的黎曼问题的近似解的方法变得更加灵活，同时遵守基本的数学要求，并特别针对基于物理的特性来优化它们。为此目的设想了三个主要组成部分：1)对流场进行局部分析，并从三维角度对其进行表征。以前训练的卷积神经网络评估局部流场的三维属性(剪切、体积变化、平滑度、各向异性)，并将该信息传递给下两个组件。2)计算单元界面状态的重建过程将基于一组Harten型多项式，与经典方法不同，该多项式不会用代数函数加权，而是由专门针对流动-物理过程训练的神经网络来取代。隐式大涡模拟的成功方法是利用不同离散化方法的固有性质来隐式获得SGS项，否则必须显式确定SGS项，以便同时获得激波和湍流特性的最高精度。3)已建立的数值通量函数最初是为模拟无摩擦可压缩流动而开发的，通常具有较高的模拟质量和激波，但对于湍流结构的表示而言，往往耗散太大。此外，伽利略不变性通常不能得到保证。在本项目的背景下，再次使用神经网络，它结合了经典通量函数(如HLLC)和耗散明显较少的方法(如ALDM)的凸组合，这些方法特别适合于各自的局部流场。该方案的特点、独创性和新颖性在于，网络的训练阶段是通过自动区分整个流解算器来执行的。这里，优化所需的目标函数是点状误差与来自DNS和高分辨率LES的低通滤波参考数据的组合，以及它们的光谱特性。该项目的目标是获得与最先进方法相当的精度，同时显著降低空间和时间分辨率，从而显著减少数值工作量。

Within the scope of the research project, novel approaches for the implicit large eddy simulation of compressible gas flows at high Reynolds numbers are developed. The newly developed automatically differentiable simulation and optimization method JAX-Fluids based on a conservative finite volume method forms the basis for the execution of the project. This package makes it possible to make known and established methods for state reconstruction at the interface and for the approximate solution of the resulting Riemann problem much more flexible while complying with basic mathematical requirements, and to optimize them specifically for physically based characteristics. Three main components are envisaged for this purpose: 1) The flow field is locally analyzed and characterized in three dimensions. A previously trained Convolutional Neural Network evaluates the three-dimensional properties of the local flow field (shear, volume change, smoothness, anisotropy) and passes this information to the next two components. 2) The reconstruction procedure for calculating the states at the cell interface will be based on a set of Harten-type polynomials, which, in contrast to classical methods, will not be weighted by algebraic functions but will be replaced by a neural network specially trained for flow-physical processes. Successful approaches to implicit large eddy simulation use inherent properties of different discretization methods to implicitly obtain the SGS terms that would otherwise have to be explicitly determined in order to simultaneously achieve highest precision on shock waves as well as on turbulence characteristics. 3) Established numerical flux functions, originally developed for the simulation of frictionless compressible flows, often allow a high simulation quality and shock waves, but are often much too dissipative for the representation of turbulent structures. In addition, Galilean invariance can usually not be guaranteed. In the context of this project, a neural network is used again, which combines the convex combination of classical flux functions such as HLLC and significantly less dissipative approaches such as ALDM, which are particularly suitable for the respective local flow field. The peculiarity, originality and novelty in the proposed project is that the training phase of the networks is performed by automatic differentiation of the entire flow solver. Here, the objective function required for optimization is a combination of point-wise errors versus low-pass filtered reference data from DNS and high-resolution LES, as well as their spectral characteristics. The goal of the project is to obtain results of comparable precision to state-of-the-art methods with significantly reduced spatial and temporal resolution and thus significantly reduced numerical effort.