Modeling and simulating turbulent secondary flow in a streamwise-rotating channel with LLM-guided symbolic regression and physics-informed neural networks

Abstract Turbulent secondary flow in a streamwise-rotating channel is dominated by Taylor–Görtler (TG) vortices whose direct numerical simulation is expensive. This paper develops a two-stage artificial-intelligence framework that learns the closure and simulates the flow. In the first stage, the LGF (LLM-guided Graph-Feynman) equation discovery engine targets the single decisive anisotropy component, the secondary-flow covariance 𝑏23 = 〈𝑣𝑤〉/2𝑘: twenty discovery runs (four feature pools of rotation-variant tensor channels, two LGF operating arms, and PySR at three seeds) are scored against DNS statistics of five rotation numbers 𝑅𝑜𝜏 = 7.5–150. The guided search buys stability: the eight LGF runs hold the case-normalized error within 0.051–0.137 at 13–17 nodes, whereas the twelve PySR runs spread over 0.042–0.334 with seed-level variation up to eightfold at about 60% more nodes. The four selected closures (0.042–0.055, five to seven constants each) share one skeleton: a saturating function of the local rotation number, balancing Coriolis production against isotropization, crossed with a wall function of the wall distance. In the second stage they are carried through the mean-spanwise-profile balance into a physics-informed spectral network (PINN) that reconstructs the time-averaged TG-vortex field at 𝑅𝑜𝜏 = 7.5, 30 and 75 in minutes on a desktop CPU. All four reproduce the two-layer counter-rotating roll cells with secondary-flow amplitudes within 10% of the DNS despite mean-profile errors of order the profile itself: the reconstruction is robust to O(1) profile errors, identifying stress production and viscosity, not mean-flow advection, as the governing balance and completing a “discover–close–simulate” pipeline with compact learned closures.