Machine-learning (ML) techniques provide a new and encouraging perspective for constructing turbulence models for Reynolds-averaged Navier-Stokes (RANS) simulations. In this study, an iterative ML-RANS computational framework is proposed that combines an ML algorithm with transport equations of a conventional turbulence model. This framework maintains a consistent procedure for obtaining the input features of an ML model in both the training and predicting stages, ensuring a built-in reproducibility. The effective form of the closure term is discussed to determine suitable target variables for the ML algorithm, and the multi-valued problem of existing constitutive theory is studied to establish a proper regression system for ML algorithms. The developed ML model is trained under a cross-case strategy with data from turbulent channel flows at three Reynolds numbers, and a posteriori simulations of channel flows show that the framework is able to predict both the mean flow field and turbulent variables accurately. Interpolation tests for the channel flow show the proposed framework can reliably predict flow features that lie between the minimum and maximum Reynolds numbers associated with the training data. A further test of the ML model in a flow over periodic hills also demonstrates improved performance compared with the k-$\omega$ SST model, even though the model is only trained with planar channel flow data.

机器学习（ML）技术为构建雷诺平均Navier-Stokes（RANS）模拟的湍流模型提供了令人鼓舞的新观点。在这项研究中，提出了一种迭代ML-RANS计算框架，该框架将ML算法与常规湍流模型的输运方程相结合。该框架维护了一致的过程，可在训练和预测阶段获得ML模型的输入特征，从而确保内置的可重复性。讨论了封闭项的有效形式，以确定适合ML算法的目标变量，并研究了现有本构理论的多值问题，以建立适合ML算法的回归系统。通道流动的后验模拟表明，该框架能够准确预测平均流场和湍流变量。通道流量的插值测试表明，所提出的框架可以可靠地预测与训练数据相关的最小和最大雷诺数之间的流量特征。在超过周期丘陵流的ML模型的进一步的试验还证明与相比改进的性能ķ -$\omega$ 即使仅使用平面通道流量数据训练该模型，也可以使用SST模型。