Characterization of the Fluctuating Reynolds-Stress

This paper presents a comprehensive investigation of fluctuating Reynolds- stress gradients in a self-similar adverse-pressure-gradient turbulent boundary layer (SS- APG-TBL) at the verge of separation. These gradients constitute the nonlinear forc- ing that drives the resolvent operator and are therefore essential for elucidating the spatial structure and spectral distribution of turbulence-production mechanisms. The forcing terms are extracted from time-resolved three-dimensional velocity fields obtained through direct numerical simulation of the SS-APG-TBL. Their spectral characteristics are analysed by integrating over the wall-normal coordinate and averaging in the stream- wise direction after normalisation with self-similar scaling variables. Based on one-point power spectral densities (auto-spectra), the three forcing components exhibit compa- rable amplitudes and similar spectral distributions. Peak forcing intensity occurs at λz/δ1 = 0.1 and ωδ1/Ue = 35, corresponding to higher temporal frequencies and shorter spanwise wavelengths than those associated with maximum amplification of the linear re- solvent operator. The spatial structure, characterised through the wall-normal–spanwise wavenumber (y–λz) spectrum at the peak frequency, indicates that the maximum forc- ing amplitude occurs at y/δ1 = 1.2, which lies farther from the wall than the peak in turbulent kinetic energy and coincides with the inflection point of the mean streamwise velocity profile. These findings provide new physical insight into the spatial organisation and spectral characteristics of nonlinear forcing in these flows, with direct implications for reduced-order modelling and potentially resolvent-informed flow control.