AI 中文总结
流体工程中湍流问题面临以低成本获准确预测的挑战，本文提出多保真度、物理约束、数据驱动框架，通过增强Spalart - Allmaras模型，在不同雷诺数下训练和测试，实现\(O(\log(Re))\)形式成本，有强大外推能力。
AI 中文摘要
在流体工程中，湍流问题长期以来一直是一个挑战，即要以可承受的计算成本获得工程量的准确预测。从计算复杂性来看，实用算法的成本增长不应超过\(O(N)\)，其中\(N\)表示问题规模。对于湍流，问题规模可近似为动态相关尺度的数量，进而由雷诺数\(Re\)表示。我们提出了一个多保真度、物理约束、数据驱动的框架，旨在在所陈述的假设下满足这一标准。我们通过场反演和机器学习增强了Spalart - Allmaras模型，使用了保留壁面定律的约束公式。该模型在低雷诺数下进行训练，此时高保真数据成本可承受，然后在更高雷诺数下部署。对于壁面边界流中的平均流对齐网格、固定展向分辨率以及稳定求解器成本与网格点数成线性关系的情况，低保真度的雷诺平均 Navier - Stokes（RANS）预测尺度为\(O(\log(Re))\)。高保真计算和学习阶段相对于目标雷诺数各贡献\(O(Re^0)\)，总体形式成本为\(O(\log(Re))\)。在平面通道流中，在\(Re_\tau = 1000\)下训练的模型纠正了基线模型的尾流层误差，并在\(Re_\tau = 5200\)时保持改进。在周期性山丘流中，在\(Re_b = 5600\)下训练的模型在\(Re_b = 10595\)、\(19000\)和\(37000\)下进行测试。约束公式随着雷诺数增加保留了分离和恢复行为，在所有测试中产生最低的均方根误差，并且误差几乎与雷诺数无关，表明具有强大的外推能力。
英文摘要
In fluid engineering, the turbulence problem is the longstanding challenge of obtaining accurate predictions of engineering quantities at affordable computational cost. Viewed through computational complexity, a practical algorithm requires cost growth no worse than $O(N)$, where $N$ denotes problem size. For turbulent flows, the problem size may be approximated by the number of dynamically relevant scales and hence by the Reynolds number $Re$. We propose a multi-fidelity, physics-constrained, data-driven framework designed to meet this criterion under stated assumptions. We augment the Spalart--Allmaras model through field inversion and machine learning using a constrained formulation that preserves the law of the wall. The model is trained at a low Reynolds number, where high-fidelity data are affordable, and deployed at higher Reynolds numbers. For a mean-flow-aligned grid in a wall-bounded flow, fixed spanwise resolution, and steady-solver cost linear in grid-point count, the low-fidelity RANS prediction scales as $O(\log(Re))$. The high-fidelity calculation and learning stage each contribute $O(Re^0)$ relative to the target Reynolds number, giving an overall formal cost of $O(\log(Re))$. In plane channel flow, a model trained at $Re_τ=1000$ corrects the wake-layer error of the baseline model and retains the improvement at $Re_τ=5200$. In the periodic hill, a model trained at $Re_b=5600$ is tested at $Re_b=10595$, $19000$, and $37000$. The constrained formulation preserves separation and recovery behavior as Reynolds number increases, yields the lowest root-mean-square error across all tests, and exhibits nearly Reynolds-number-independent error, indicating robust extrapolation.
Comments6 pages, 6 figures; to be presented at the 14th International Symposium on Turbulence and Shear Flow Phenomena (TSFP14), Heidelberg, Germany, July 28-31, 2026
