Data-driven closure of the harmonic-balanced Navier-Stokes equations in the frequency domain

The Fourier-Galerkin method is employed to calculate the multifrequency and multiscale asymptotic nonlinear flow response in the frequency domain, by expanding the solution as Fourier series. The resulting equations are known as the harmonic-balanced Navier-Stokes (HBNS) equations. Although near the threshold of transition a small number of harmonics suffice to achieve convergence, further away the computational cost becomes intractable because energy is transferred to higher harmonics, which can no longer be neglected. In this study, we propose a data-driven framework to model the residual (nonlocal closure) terms for the frequency-truncated HBNS equations. By splitting the sought solution into low-frequency (resolved) and high-frequency (unresolved) harmonics, we systematically express the low-frequency residual as a function of the resolved frequency harmonics only. A consistent deep learning architecture, which parameterizes the residual function, is designed and trained using high-fidelity results near the thresholds of transition for two-dimensional (2D) cylinder flow. We show that our proposed framework achieves low generalization error by predicting accurately the coarse-grained residual for unseen Reynolds numbers, and significantly reduces the computational cost by solving accurately for the coarse-grained dynamics.