Symmetry-based Turbulence Modelling for Engineering Applications

对称性是所有物理理论的核心，例如经典、量子力学和相对论，因为它们反映了基本物理的公理性质。对于Navier-Stokes湍流，申请人将这一性质扩展到包括统计对称性，这是对湍流的非高斯统计和间歇性关键属性的么正度量。他从多点关联方程的无穷序列和Lundgren-Novikov-Monin概率密度函数族严格地发展了这些公式。在一些出版物中，申请人能够证明这些对称性是所有湍流标度定律的公理基础，最近他将其推广到任意时刻的标度定律，例如对数区域。然而，在湍流模型的发展中，对称性在很大程度上是在不知不觉中被使用的，但至少自20世纪40年代以来，随着每一类新的模型，模型中都包括了更多的对称性。从20世纪70年代起，最重要的模型包括经典力学的所有对称性，即伽利略群。然而，这显然不适用于统计对称性，到目前为止，任何湍流模型中都没有包括统计对称性。它的核心工作假设是，所有对称性的显式和严格实施，即经典对称性和统计对称性，将导致模型预测精度的显著提高。这将在多方程湍流模型和雷诺应力输运模型中实施。其基本原理是，由于新的湍流模型包括所有中心对称性，一阶矩和更高阶矩的所有标度律都基于这些中心对称性，因此模型也准确地表示了这些对称性。标度律通常只描述湍流中的子区域，但其非常精确的建模意味着更复杂的流动被更精确地建模。无限小形式的对称性机制允许对模型方程进行优雅而严格的推导，由此首次发表的初步工作表明，如预期的那样，某些模型自由度被保留，但特别是模型参数不能从对称性中确定。对于新湍流模型的最终参数确定，选择了机器学习(ML)方法，该方法允许并行使用不同湍流模型的数据。这些模型是在惠普精确的内部代码BOSS中实现的，该代码BOSS基于不连续伽辽金方法，允许非常精确地分离数值误差和模型误差。在BOSS中也实现了ML概念，从而实现了高效的实现。

Symmetries are at the heart of all physical theories, such as classical and quantum mechanics and relativity, as they reflect the axiomatic properties of the underlying physics. For Navier-Stokes turbulence, this property was extended by the applicant to include statistical symmetries, which are a unitary measure of non-Gaussian statistics and intermittency - key properties of turbulence. He rigorously developed these from the infinite sequence of the multi-point correlation equation and the Lundgren-Novikov-Monin probability density function hierarchy. In a number of publications, the applicant was able to show that these symmetries are the axiomatic basis of all turbulent scaling laws and recently he extended this to scaling laws for arbitrary moments, e.g. for the log-region. In turbulence model development, however, symmetries have been used largely unknowingly, but at least it has been achieved that since the 1940s, with each new class of models, further symmetries have been included in the models. From the 1970s onwards, the most important models included all symmetries of classical mechanics, i.e. the Galilean group. However, this explicitly did not apply to the statistical symmetries, which have not been included in any turbulence model so far. It is the central working hypothesis that the explicit and rigorous implementation of all symmetries, i.e. the classical as well as the statistical symmetries, will lead to a significant improvement of the accuracy of the model prediction. This is to be implemented in a multi-equation turbulence model and for a Reynolds stress transport model. The rationale is that, because the new turbulence models include all central symmetries on which all scaling laws for the first and higher moments are based, these are also accurately represented by the model. Scaling laws usually describe only subregions in a turbulent flow, but their very accurate modelling implies that even more complex flows are modelled much more precisely. The machinery of symmetries in infinitesimal form allows an elegant and rigorous derivation of the model equations, whereby a first published preliminary work has shown that, as expected, certain model freedoms are retained, but in particular model parameters cannot be determined from the symmetries. For the final parameter determination of the new turbulence models a machine-learning (ML) approach is chosen, which allows the parallel use of data of different turbulent model flows. The models are implemented in the hp-accurate in-house code BoSSS, which, based on the discontinuous Galerkin method, allows a very precise separation of numerical and model errors. The ML concept is also implemented within BoSSS, which allows an efficient implementation.