無基礎模型之資料驅動物理訊息機器學習紊流閉合方法

Abstract

在計算流體力學（Computational Fluid Dynamics）中，紊流建模始終是一項核心挑戰。其中，雷諾平均納維-斯托克斯方程（Reynolds-Averaged Navier-Stokes）因兼具計算效率與模擬精度，至今仍為工程應用中最常見的紊流模擬方法之一。然而，傳統RANS紊流閉合模型高度仰賴經驗性參數，在處理複雜流場時，其模擬準確性常受到限制。為提升模型表現，近年來「物理訊息機器學習」（Physics-Informed Machine Learning）逐漸興起，透過結合物理知識與機器學習架構，建立具備可解釋性與可靠性的資料驅動紊流模型。

本研究首次提出完全獨立於基準模型（傳統紊流閉合模型）的資料驅動紊流閉合框架，並且採多階段設計進行模型建構與整合。首先，透過張量基底神經網路（Tensor Basis Neural Network）結合局部流場特徵與幾何資訊（流函數與速度勢），預測各向異性雷諾應力張量，以強化模型對整體流場狀態的識別與泛化能力。其次，運用隨機森林回歸器（Random Forest Regressor）將預測的向異性雷諾應力張量轉化為固定形式輸出下的最適紊流黏滯係數。最後，透過退火模擬演算法（Simulated Annealing）逐步調控紊流黏滯係數，確保模型整合至RANS求解器時的計算收斂性與穩定性。

為驗證本架構之適用性，本研究選取三類具挑戰性的流場幾何進行模擬，包括波狀底床、平滑曲階與後向階梯等案例。結果顯示，本架構不僅具備良好的模型預測能力，亦能在無需依賴任何傳統基準模型的情況下，成功嵌入RANS求解器進行穩定模擬。綜合而言，本研究展現機器學習應用於紊流建模上的可行性與潛力，為將來紊流模擬領域開創全新發展方向。

One of the key problems in Computational Fluid Dynamics is still modeling turbulence, especially in Reynolds-Averaged Navier-Stokes simulations, which are used extensively in engineering for their balance between accuracy and computational efficiency. However, traditional RANS closures often rely on ad-hoc empirical parameters, limiting their accuracy and generalizability in complex flow regimes. To overcome these limitations, recent advances in Physics-Informed Machine Learning seek to integrate data-driven techniques with physical concepts, offering a promising pathway toward more reliable and interpretable turbulence models.

This study develops a fully standalone Machine Learning-based turbulence closure framework with a multi-stage architecture. First, a Tensor Basis Neural Network predicts the anisotropic Reynolds stress tensor using both local flow features and geometry-informed variables, specifically the stream function and velocity potential. These additional inputs help the model incorporate global flow information and improve generalization. Second, a Random Forest regressor translates the predicted anisotropic Reynolds stress tensors into an optimized eddy-viscosity field with a standardized output function. Third, a Simulated Annealing approach is used to gradually control the eddy-viscosity field in order to guarantee numerical stability and convergence during integration using the RANS solver.

In order to assess the efficacy and suitability of the suggested methodology, this study conducts simulation tests on three representative and complex flow configurations: a Wavy Bottom, a Smooth-Curved Step, and a Backward-Facing Step. The results demonstrate that the developed model delivers both stable and accurate predictions, while being fully integrated into the RANS solver without relying on any baseline models. In general, this study underscores the potential of combining physics-based insights with data-driven approaches to advance turbulence modeling in CFD applications.