The present investigation focuses on the application of deep neural network (DNN) models to predict the filtered density function (FDF) of mixture fraction in large eddy simulation (LES) of variable density mixing layers with conserved scalar mixing. A systematic training method is proposed to select the DNN-FDF model training sample size and architecture via learning curves, thereby reducing bias and variance. Two DNN-FDF models are developed: one trained on the FDFs generated from direct numerical simulation (DNS), and another trained with low-fidelity simulations in a zero-dimensional pairwise mixing stirred reactor (PMSR). The accuracy and consistency of both DNN-FDF models are established by comparing their predicted scalar filtered moments with those of conventional LES, in which the transport equations corresponding to these moments are directly solved. Further, DNN-FDF approach is shown to perform better than the widely used $\beta$-FDF method, particularly for multi-modal FDF shapes and higher variances. Additionally, DNN-FDF results are also assessed via comparison with data obtained by DNS and the transported FDF method. The latter involves LES simulations coupled with the Monte Carlo (MC) methods which directly account for the mixture fraction FDF. The DNN-FDF results compare favorably with those of DNS and transported FDF method. Furthermore, DNN-FDF models exhibit good predictive capabilities compared to filtered DNS for filtering of highly non-linear functions, highlighting their potential for applications in turbulent reacting flow simulations. Overall, the DNN-FDF approach offers a more accurate alternative to the conventional presumed FDF method for describing turbulent scalar transport in a cost-effective manner.

翻译：本研究聚焦于利用深度神经网络（DNN）模型预测大涡模拟（LES）中变密度混合层（含守恒标量混合）的混合物分数滤波密度函数（FDF）。提出了一种系统化训练方法，通过学习曲线选择DNN-FDF模型的训练样本规模与架构，从而降低偏差与方差。开发了两种DNN-FDF模型：一种基于直接数值模拟（DNS）生成的FDF进行训练，另一种利用零维成对混合搅拌反应器（PMSR）中的低保真度模拟数据进行训练。通过将两种DNN-FDF模型预测的标量滤波矩与传统LES（直接求解这些矩对应的输运方程）结果对比，验证了其精度与一致性。进一步表明，DNN-FDF方法优于广泛使用的$\beta$-FDF方法，尤其针对多模态FDF形状及高方差情形。此外，DNN-FDF结果还与DNS数据及输运FDF方法的结果进行了对比评估。后者涉及LES模拟耦合蒙特卡洛（MC）方法，直接描述混合物分数FDF。DNN-FDF结果与DNS及输运FDF方法的结果吻合良好。同时，在高度非线性函数的滤波中，DNN-FDF模型相较于滤波DNS展现出优异的预测能力，突显了其在湍流反应流模拟中的应用潜力。总体而言，DNN-FDF方法以高性价比方式为描述湍流标量输运提供了相较于传统假定FDF方法更精确的替代方案。