This paper presents a neural-network-based turbulence modeling approach for transonic flows based on the ensemble Kalman method. The approach adopts a tensor-basis neural network for the Reynolds-stress representation, with modified inputs to consider fluid compressibility. The normalization of input features is also investigated to avoid feature collapsing in the presence of shock waves. Moreover, the turbulent heat flux is accordingly estimated with the neural-network-based turbulence model based on the gradient diffusion hypothesis. The ensemble Kalman method is used to train the neural network with the experimental data in velocity and wall pressure due to its derivative-free nature. The proposed framework is tested in two canonical configurations, that is, two-dimensional transonic flows over the RAE2822 airfoils and three-dimensional transonic flows over the ONERA M6 wings. Numerical results demonstrate the capability of the proposed method in learning accurate turbulence models for external transonic flows.
National Natural Science Foundation of China's Basic Science Center Program for Multiscale Problems in Nonlinear Mechanics [11988102]
; National Natural Science Foundation of China [12102439, 12102435]
; China Postdoctoral Science Foundation [2021M703290, 2021M690154]
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