Utilizing intra-triad energy conservation for phase reconstruction from low-order statistics*

We present a formulation for analytically determining phase information from the magnitude of the Fourier modes, or the mode shapes and amplitudes, in turbulent channel flow. This is done by utilizing the fact that nonlinear terms in the Navier-Stokes equations are energy conserving within a set of triadically consistent wavenumbers, an observation originally made by Schmid and Henningson (2001) and explored by Barthel (2022). The proposed algorithm successfully predicts the trends in phase shift between a set of triads, relative to an unknown reference mode, showing agreement within 15% when compared to DNS results. This work has broader implications in the context of utilizing the scalability of low-order moments, such as the power spectrum, while still retaining relative phase information, which is generally only present in higher order statistics. Two-point and space-time correlations provide some phase information; however, their computation requires knowledge of the full velocity field. The power spectrum lacks any directional information but holds the advantage that it can be computed via scaling from a lower Reynolds number spectrum. The reconstruction of relative phase shifts from the power spectrum has the potential to reintroduce directional information to low-order statistics, allowing for more complete and computationally accessible representations of high Reynolds number turbulent flows.