Wavelet-based resolvent analysis of non-equilibrium channel flow subjected to a sudden transverse pressure gradient*

A better understanding of the influence of non-equilibrium effects on turbulence is important for the analysis of non-stationary turbulent flows, which are ubiquitous in engineering applications. In turbulent channel flow reacting to a strong spanwise pressure gradient imposed suddenly, the Reynolds shear stresses are counter-intuitively attenuated as the flow rotates. In this work, we apply a wavelet-based resolvent analysis framework to study this effect in more detail. Under the wavelet-based formulation, optimal forcing and response modes that encode both time and frequency information are computed for the Navier-Stokes equations linearized about a time-changing mean, Fourier-transformed in the homogeneous directions and wavelet-transformed in time. Prior to the imposition of the spanwise pressure gradient, we find that the optimal forcing exhibits a strong wall-normal component. The mode thus exploits the lift-up mechanism, which couples the wall-normal velocity component with the streamwise and spanwise components. After the onset of the pressure gradient, the wall-normal contribution of the optimal forcing decreases significantly, indicating that lift-up is no longer an efficient way of amplifying velocity perturbations. Finally, by considering the linearized wall-normal velocity-vorticity equations, we argue that it is the misalignment between the mean shear and the mean velocity profiles during the non-equilibrium portion of the flow which attenuates the lift-up term.