Tootfinder

A formal log(Re)-cost framework for the engineering turbulence problem
Jiaqi Li, Robert F. Kunz, George Huang, Xiang I. A. Yang
https://arxiv.org/abs/2607.20199 https://arxiv.org/pdf/2607.20199 https://arxiv.org/html/2607.20199
arXiv:2607.20199v1 Announce Type: new
Abstract: 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 $Re_\tau=1000$ corrects the wake-layer error of the baseline model and retains the improvement at $Re_\tau=5200$. 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.
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