Transient growth and nonlinear breakdown of wavelet-based resolvent modes in turbulent channel flow

Abstract

We study the effectiveness of the time-localised principal resolvent forcing mode at actuating the near wall cycle of turbulence. The mode is restricted to a wavelet pulse and computed from an SVD of the windowed wavelet-based resolvent operator so that it produces the largest amplification via the linearised Navier-Stokes equations. We then inject this time-localised mode into the turbulent minimal flow unit at different intensities, and measure the instantaneous deviation of the system's response from the optimal resolvent response mode. This is possible under the new formulation, which enables the modes to represent transient trajectories. For the most energetic spatial wave numbers in the minimal flow unit -- constant in the streamwise direction and once-periodic in the spanwise direction -- the forcing mode takes the shape of streamwise rolls and produces a response mode in the form of streamwise streaks that transiently grow and decay. For initial times and close to the wall, the DNS response matches the principal response mode well, but due to nonlinearities, the response across all forcing intensities decays prematurely, and a higher forcing intensity leads to faster energy decay. The principal forcing mode still leads to significant energy amplification and is more effective than a randomly-generated forcing structure and the second suboptimal resolvent forcing mode at amplifying the near-wall streaks. We compute the nonlinear energy transfer to secondary modes and observe that the breakdown of the actuated mode proceeds similarly across all forcing intensities: in the near-wall region, the induced streak forks into a structure twice periodic in the spanwise direction; in the outer region, the streak breaks up into a structure that is once periodic in the streamwise direction. In both regions, spanwise gradients account for the dominant share of nonlinear energy transfer.