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Modeling Reynolds stress anisotropy invariants via machine learning

基于机器学**的雷诺应力各向异性不变量建模

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Abstract

The presentation and modeling of turbulence anisotropy are crucial for studying large-scale turbulence structures and constructing turbulence models. However, accurately capturing anisotropic Reynolds stresses often relies on expensive direct numerical simulations (DNS). Recently, a hot topic in data-driven turbulence modeling is how to acquire accurate Reynolds stresses by the Reynolds-averaged Navier-Stokes (RANS) simulation and a limited amount of data from DNS. Many existing studies use mean flow characteristics as the input features of machine learning models to predict high-fidelity Reynolds stresses, but these approaches still lack robust generalization capabilities. In this paper, a deep neural network (DNN) is employed to build a model, map** from tensor invariants of RANS mean flow features to the anisotropy invariants of high-fidelity Reynolds stresses. From the aspects of tensor analysis and input-output feature design, we try to enhance the generalization of the model while preserving invariance. A functional framework of Reynolds stress anisotropy invariants is derived theoretically. Complete irreducible invariants are then constructed from a tensor group, serving as alternative input features for DNN. Additionally, we propose a feature selection method based on the Fourier transform of periodic flows. The results demonstrate that the data-driven model achieves a high level of accuracy in predicting turbulence anisotropy of flows over periodic hills and converging-diverging channels. Moreover, the well-trained model exhibits strong generalization capabilities concerning various shapes and higher Reynolds numbers. This approach can also provide valuable insights for feature selection and data generation for data-driven turbulence models.

摘要

湍流各向异性的表示和建模对于研究大尺度湍流结构和建立湍流模型至关重要. 然而, 准确计算雷诺应力的各向异性往往需要昂贵的直接数值模拟(DNS). **年来, 数据驱动湍流建模的一个热点是如何利用雷诺**均模拟方法(RANS)的数据和少量的DNS数据建立高精度的雷诺应力模型. 许多现有的研究实现了从**均流场特征到高精度的雷诺应力的预测模型, 但这些方法仍然缺乏**泛化能力. 本文采用深度神经网络(DNN), 以RANS的张量不变量作为输入映射高精度雷诺应力的各向异性不变量. 从张量分析和特征设计两方面, 在保持模型不变性的同时增**模型的泛化能力. 从理论上推导了雷诺应力各向异性不变量的泛函框架. 然后构造了完备不可约的 不变量, 作为DNN的备选输入特征. 此外, 我们提出了一种基于周期流动傅里叶变换的特征选择方法. 结果表明, 数据驱动模型在预测周期性流动和收缩扩张管道流动的湍流各向异性方面具有较高的精度, 而且训练的模型对外形和雷诺数具有很**的泛化能力. 本文的方法还可以为数据驱动湍流建模的特征选择和数据生成提供新的解决思路.

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Data availability

The data that support the findings of this study are openly available in Kaggle at https://doi.org/10.34740/kaggle/dsv/2637500, Ref. [51], and within Refs. [36,46–49].

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 92152301).

Author information

Authors and Affiliations

  1. School of Aeronautics, Northwestern Polytechnical University, **’an, 710072, China

    **anglin Shan  (单湘淋), Xuxiang Sun  (孙旭翔), Wenbo Cao  (曹文博) & Weiwei Zhang  (张伟伟)

  2. Department of Engineering Mechanics, Zhejiang University, Hangzhou, 310027, China

    Zhenhua **a  (夏振华)

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  1. **anglin Shan  (单湘淋)
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  2. Xuxiang Sun  (孙旭翔)
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  5. Zhenhua **a  (夏振华)
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Contributions

Author contributions Weiwei Zhang and **anglin Shan designed the research. Weiwei Zhang acquired the financial support for the project leading to this publication. **anglin Shan investigated the methodology, designed the computational framework, constructed the model, and wrote the manuscript. Xuxiang Sun and Wenbo Cao help to formulate overarching research goals and analyze the results. Zhenhua **a and Weiwei Zhang helped organize the manuscript, revised, and edited the final version.

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Correspondence to Weiwei Zhang  (张伟伟).

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Shan, X., Sun, X., Cao, W. et al. Modeling Reynolds stress anisotropy invariants via machine learning. Acta Mech. Sin. 40, 323629 (2024). https://doi.org/10.1007/s10409-024-23629-x

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