Data- and physics-driven closure model of harmonic-balanced Navier-Stokes equations using turbulent DNS and experimental data

Coherent structures in turbulent flows often arise from instabilities during transition or from the turbulent mean flow. At realistic flow regimes, direct numerical simulation (DNS) of the Navier–Stokes equations remains computationally infeasible, even with exaflop-scale machines. Thus, modeling relies on lower-fidelity approaches that parameterize small-scale effects via subgrid models. However, in turbulent regimes, frequency-domain simulations of the Navier–Stokes equations offer a computationally efficient alternative to time-domain methods. The periodic solutions of Navier-Stokes equations are obtained through the Fourier–Galerkin method, yielding the Harmonic-Balanced Navier–Stokes (HBNS) equations. In this work, we develop physics-informed neural networks (PINNs) to model the nonlocal closure terms of frequency-truncated HBNS equations. By embedding HBNS as soft constraints in the loss function, we infer high-frequency corrections for a turbulent flow over a cylinder at Reynolds numbers Re = 3900 (DNS) and Re=5000 (experimental). We validate and benchmark our closure model to previously unseen Reynolds numbers. Furthermore, we systematically assess the model's accuracy by varying the number of retained temporal harmonics in the turbulent regimes.