Despite their well-known limitations, Reynolds averaged Navier-Stokes (RANS) models remain the most commonly employed tool for modeling turbulent flows in engineering practice. RANS models are predicated on the solution of the RANS equations, but the RANS equations involve an unclosed term, the Reynolds stress tensor, which must be modeled. The Reynolds stress tensor is often modeled as an algebraic function of mean flow field variables and turbulence variables. This, however, introduces a discrepancy between the Reynolds stress tensor predicted by the Reynolds stress model and the exact Reynolds stress tensor. This discrepancy can result in inaccurate mean flow field predictions for complex flows of industrial relevance. In this paper, we introduce a data-informed approach for arriving at Reynolds stress models with improved predictive performance. Our approach relies on learning the components of the Reynolds stress discrepancy tensor associated with a given Reynolds stress model in the mean strain-rate tensor eigenframe. These components are typically smooth and hence simple to learn using state-of-the-art machine learning strategies and regression techniques. Our approach automatically yields Reynolds stress models that are symmetric, and it yields Reynolds stress models that are both Galilean and frame invariant provided the inputs are themselves Galilean and frame invariant. To arrive at computable models of the discrepancy tensor, we employ feed-forward neural networks and an input space spanning the integrity basis of the mean strain-rate tensor, the mean rotation-rate tensor, the mean pressure gradient, and the turbulent kinetic energy gradient, and we introduce a framework for dimensional reduction of the input space to further reduce computational cost. Numerical results illustrate the effectiveness of the proposed approach for data-informed Reynolds stress closure for a suite of turbulent flow problems of increasing complexity.

尽管存在众所周知的局限性，雷诺平均纳维-斯托克斯 (RANS) 模型仍然是工程实践中最常用的湍流建模工具。RANS 模型基于 RANS 方程的解，但 RANS 方程涉及一个非封闭项，即雷诺应力张量，必须对其进行建模。雷诺应力张量通常被建模为平均流场变量和湍流变量的代数函数。然而，这会导致雷诺应力模型预测的雷诺应力张量与精确的雷诺应力张量之间存在差异。这种差异会导致对工业相关的复杂流的平均流场预测不准确。在本文中，我们引入了一种基于数据的方法来获得具有改进预测性能的雷诺应力模型。我们的方法依赖于学习与平均应变率张量特征框架中给定雷诺应力模型相关的雷诺应力差异张量的分量。这些组件通常是平滑的，因此使用最先进的机器学习策略和回归技术很容易学习。我们的方法自动产生对称的雷诺应力模型，并且它产生的雷诺应力模型既是伽利略又是框架不变的，前提是输入本身是伽利略和框架不变的。为了获得差异张量的可计算模型，我们采用前馈神经网络和跨越平均应变率张量完整性基础的输入空间，平均旋转速率张量、平均压力梯度和湍流动能梯度，我们引入了输入空间降维的框架，以进一步降低计算成本。数值结果说明了所提出的基于数据的雷诺应力闭合方法对一系列复杂性不断增加的湍流问题的有效性。