Discovery of Nonlinear Flow Physics Using Optimal Modal Momentum and Energy Budget Analysis*

We propose an orthogonal modal decomposition method for data-driven energy budget analysis, which identifies pairs of modes that represent the acceleration and convective terms of the spectral Navier-Stokes equations. These modes are triadically consistent and optimally correlated. The method extracts coherent structures involved in three-wave interactions by focusing on spectral momentum transfer, optimizing third-order space-time flow statistics. It distinguishes between two interacting components: a catalyst and a momentum donor, which together contribute to a tertiary component, the recipient. The resulting modes maximize the covariance between the projections of the convective and recipient terms onto their respective modes. This method extends bispectral mode decomposition (BMD) by incorporating the exact quadratic nonlinearity of the Navier-Stokes equations. Similar to classical proper orthogonal decomposition (POD), it provides ranked, orthonormal bases for the convective and recipient terms that are jointly optimal. Applications include numerical data of a canonical unsteady cylinder wake and experimental data of a turbulent wind turbine wake by Biswas & Buxton (2024, JFM).