Accurate simulation of transition from the laminar to the turbulent flow is of great importance in industrial applications. In the present work, the framework of field inversion and machine learning has been applied to improve the four-equation k–ω–γ–A_r transition model. The low-speed transitional flows past two airfoils were numerically simulated. Based on the experimental transition locations, the regularizing ensemble Kalman filtering (EnKF) was performed to obtain the distributions of space-varied correction terms for the first mode time scale in the transitional flows over a natural-laminar-flow (NLF) airfoil, NLF(1)-0416. Then, two machine learning methods, random forest (RF) and artificial neutral network (NN), were adopted to construct the mapping from the mean flow variables to the correction terms. Finally, the learned models were embedded into the original solver. The results show that the regularizing EnKF can efficiently obtain the posterior distribution of the correction terms only by the transition locations. Meanwhile, both the RF- and NN-augmented transition models can predict more accurate transition locations past NLF(1)-0416 at both interpolated and extrapolated angles of attack. Moreover, the RF-augmented model can predict more accurate transitional flows on both the windward and leeward sides of NACA0012 at the same angle of attack. It indicates that the discrepancies within the model are learned and reduced. The modified model has good applicability and generalization ability. Furthermore, by analyzing the relative importance of the features in the RF model, it is found that the streamwise pressure gradient plays the most important role in the physical information and interpretation of the learned model.

中文翻译：

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通过场反转和机器学习框架改进k–ω–γ–Artransition模型

从层流到湍流过渡的准确模拟在工业应用中非常重要。在目前的工作中，已经应用了场反演和机器学习的框架来改进四方程k–ω–γ–A _r过渡模型。数值模拟了经过两个机翼的低速过渡流。基于实验过渡位置，进行了正则化集合卡尔曼滤波（EnKF），以获取自然层流（NLF）翼型NLF过渡流中第一模式时间标度的空间变量校正项的分布（1）-0416。然后，采用随机森林（RF）和人工神经网络（NN）这两种机器学习方法来构建从平均流量变量到校正项的映射。最后，将学习到的模型嵌入到原始求解器中。结果表明，正则化的EnKF只能通过过渡位置有效地获得校正项的后验分布。与此同时，RF和NN增强的过渡模型都可以预测内插和外插攻角下超过NLF（1）-0416的更准确的过渡位置。此外，RF增强模型可以在相同的迎角下预测NACA0012的上风侧和下风侧的更准确的过渡流。它表明该模型内的差异已得到学习并得到了减少。改进后的模型具有良好的适用性和泛化能力。此外，通过分析RF模型中特征的相对重要性，发现沿流方向的压力梯度在物理信息和学习模型的解释中起着最重要的作用。RF增强模型可以在NACA0012的迎风面和背风面以相同的迎角同时预测更准确的过渡流。它表明该模型内的差异已得到学习并得到了减少。改进后的模型具有良好的适用性和泛化能力。此外，通过分析RF模型中特征的相对重要性，发现沿流方向的压力梯度在物理信息和学习模型的解释中起着最重要的作用。RF增强模型可以在相同的迎角下预测NACA0012的上风侧和下风侧的更准确的过渡流。它表明该模型内的差异已得到学习并得到了减少。改进后的模型具有良好的适用性和泛化能力。此外，通过分析RF模型中特征的相对重要性，发现沿流方向的压力梯度在物理信息和学习模型的解释中起着最重要的作用。