運用張量基底類神經網路建立不可壓縮渠流雷諾應力預測模型

Modeling Reynolds stress in incompressible, turbulent channel flow using tensor basis neural network

洪晨晏(Chen-Yen Hung)

指導教授 ： 周逸儒

共同指導教授 ： 陳瑞琳(Ruey-Lin Chern)

國立臺灣大學/工學院/應用力學研究所/博士(2021年)

https://doi.org/10.6342/NTU202103867

摘要

本研究採用深度學習中類神經網路來預測雷諾應力。由於類神經網路可將多個參數進行迴歸分析，本研究利用此特性來建立資料驅動之紊流模式以解決紊流求解的閉合問題。本研究提出之資料驅動模式乃根據代數雷諾應力模型。本研究提出的模式中各神經網路均被訓練並學習來自速度梯度張量所蘊含的流場特徵。為了使模式結果符合伽利略不變性，本研究採用張量表現法來表示即將被預測的張量。為確保紊流特徵的數值範圍，本研究使用摩擦速度對紊流特徵進行無因次化，藉此取代利用特定統計特徵的數據歸一化之過程。如此便可使被預測的張量主要受到低階基底張量所主導，且確保資料驅動模式的收斂性。本研究亦根據不可壓縮性與雷諾異向應力之跡為零之特性對模式加入物理限制條件，此條件將用以修改反向傳播中的誤差計算方法。結果顯示，加入物理限制條件後，資料驅動模式相較於修改前的模式，提升了模式預測準確度。此外，本研究提出的模式考慮了擾動壓力項的貢獻，即壓力應變相關性張量，此張量在異向紊流中扮演了重要的角色。當加入壓力應變相關性張量作為模式輸入特徵後，結果顯示模式將可預測出更準確的雷諾異向應力。基於代數雷諾應力模型，本研究提出鏈結張量基底類神經網路來實現壓力應變相關性張量與雷諾異向應力間之遞迴關係。本研究採用兩階段訓練方法，使得鏈結張量基底類神經網路可在二式模式基礎上預測雷諾異向應力。對比未加入壓力應變相關性張量之模式，本研究提出之得鏈結張量基底類神經網路能預測出更準確的雷諾異向應力。在後續驗證中，本研究結果顯示，使用本研究提出之模式預測結果代入求解，可在渠流中得到準確的平均速度剖面之 預測結果。

關鍵字

雷諾應力模式 ； 代數雷諾應力模型 ； 壓力應變相關性張量 ； 張量基底類神經網路 ； 鏈結張量基底類神經網路

並列摘要

In this study, we apply the deep learning technique to predict the Reynolds stress tensor with neural networks. Because the neural network has perfect regression ability with numerous variables, it can be used in data-driven turbulence modelling with a large dataset to solve the closure problem. The proposed method using deep learning is based on the algebraic Reynolds stress model. Each deep neural network in the proposed architecture is trained to learn the characteristics from the velocity gradients tensors. Using the tensor representation, the model obeys the Galilean invariance. To ensure appropriate values for turbulent features, we use the friction velocity for the normalization, instead of the statistical properties of data in conventional models. This maintains the dominance of the lower-order terms in the equation for the final output tensor and guarantees its convergence. The physical constraints due to the incompressibility and the traceless characteristics of the Reynolds anisotropic stress are used to modify the error for back-propagation in training the TBNN-based model. Results show that using the physical constraint in the model improves the predictability for the TBNN-based model. Moreover, our model considers the contribution of pressure fluctuation, i.e., the pressure-strain-rate correlation, which plays a critical role in anisotropic turbulence. Using the pressure-strain-rate correlation as an input feature, results show that the proposed TBNN-based model gives better predictions for the Reynolds anisotropic stress. Based on algebraic Reynolds stress model, the recursive relationship between a pressure-strain-rate correlation and the Reynolds anisotropic stress is presented by using the proposed concatenated tensor basis neural network. We develop a two-stage training procedure which allows the proposed concatenated tensor basis neural network to predict the Reynolds anisotropic stress at the two-equation turbulence closure level. Compared to models that do not consider the effect of fluctuating pressure and fluctuating velocity, the proposed concatenated tensor basis neural network better predicts the Reynolds anisotropic stress. In a posterior evaluation, we show that the prediction of the mean velocity profiles for the channel flow case can be thus improved.

並列關鍵字

Reynolds stress modeling ； algebraic Reynolds stress model ； pressure-strain-rate correlation ； tensor basis neural network ； concatenated tensor basis neural networks

參考文獻

Bardina, J., Ferziger, J. H., and Reynolds, W. C. (1980). Improved subgrid scale models for large-eddy simulation. AIAA Journal, page 1357.

Bengio, Y., Simard, P., and Frasconi, P. (1994). Learning long-term dependencies with gradient descent is difficult. IEEE Transactions on Neural Networks, 5:157–166.

Fukami, K., Fukagata, K., and Taira, K. (2019). Super-resolution reconstruction of turbulent flows with machine learning. Journal of Fluid Mechanics, 870:106–120.

Gamahara, M. and Hattori, Y. (2017). Searching for turbulence models by artificial neural network. Physical Review Fluids, 2:054604.

Gatski, T. B. and Speziale, C. G. (1993). On explicit algebraic stress models for complex turbulent flows. Journal of Fluid Mechanics, 254:59–78.