A formal log(Re)-cost framework for the engineering turbulence problem

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.