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

在研究项目范围内，开发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微分模拟和优化方法JAX-Fluids构成了该项目执行的基础。该软件包使得可以使已知的和建立的方法，用于在界面处的状态重构和用于所得到的黎曼问题的近似解，在遵守基本数学要求的同时更加灵活，并且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）流场的局部分析和三维特征。先前训练的卷积神经网络评估局部流场的三维特性（剪切、体积变化、平滑度、各向异性），并将此信息传递给下两个组件。2)用于计算细胞界面处的状态的重建过程将基于一组Harten型多项式，与经典方法相比，该多项式将不由代数函数加权，而是由专门针对流动物理过程训练的神经网络代替。隐式大涡模拟的成功方法使用不同离散方法的固有特性来隐式地获得SGS项，否则必须显式地确定这些SGS项，以便同时实现激波和湍流特性的最高精度。3)建立的数值通量函数，最初开发的无摩擦可压缩流的模拟，往往允许一个高的模拟质量和冲击波，但往往是太耗散的湍流结构的表示。此外，伽利略不变性通常不能得到保证。在该项目的背景下，再次使用神经网络，它结合了经典通量函数（如HLLC）的凸组合和耗散明显较少的方法（如ALDM），这些方法特别适用于各自的局部流场。在拟议的项目的特点，独创性和新奇是，网络的训练阶段是由整个流量求解器的自动微分。这里，优化所需的目标函数是逐点误差与来自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的方法相当的精度的结果，大大降低了空间和时间分辨率，从而大大减少了数值工作。

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.