Mechanical and Civil Engineering Seminar: PhD Thesis Defense

Abstract:

In this work, we develop a wavelet-based formulation of resolvent analysis in order to extend the method to transient phenomena and non-stationary flows. We apply this method in two ways: first, to analyze turbulent systems that were not previously amenable to traditional resolvent analysis, and second, to probe the limits of the resolvent forcing modes' "optimality" in a nonlinear simulation as well as investigate the mechanisms that suppress their effectiveness. In wavelet-based resolvent analysis, the Navier-Stokes equations are linearized about a mean profile, Fourier-transformed in the homogeneous directions, and wavelet-transformed in time. The nonlinear terms are represented as forcing terms acting on the system, and a maximally perturbing forcing mode and the response it produces are then computed for this linear system. The wavelet formulation enables the forcing and response modes to represent transient trajectories. By windowing the wavelet-based resolvent operator, we can also compute optimal forcing modes restricted to a time-localized pulse along with their transient response, which can act as a benchmark when studying the response of fully nonlinear systems.

For the first application of the method, we use the windowing approach to study bursting in channel flow. The optimal response mode grows and decays in time scales that match turbulent data, and we show that this optimal burst exploits the Orr mechanism. We also study channel flow subjected to a spanwise pressure gradient. The corresponding resolvent modes mirror the mean flow and gradually realign themselves according to the new flow conditions. More interestingly, they exhibit a collapse of the lift-up mechanism during this realignment, which offers an explanation to the depletion of tangential Reynolds stresses in the turbulent system.

For the second application of the method, we inject time-localized resolvent forcing modes for the minimal flow unit into a simulation of the system, at different intensities. The principal resolvent forcing mode is much more effective than a randomly generated forcing structure at amplifying the near-wall streak. For initial times and close to the wall, the turbulent minimal flow unit matches the principal response mode well, but due to nonlinear effects, the response decays prematurely. By computing the nonlinear energy transfer to secondary scales, we find that the breakdown of the actuated mode proceeds similarly across all forcing intensities: in the near-wall region, the induced streak forks into two branches, while in the outer region, the streak breaks up in the streamwise direction. In both regions, spanwise gradients account for the dominant share of nonlinear energy transfer.ng known behaviors to frameworks for exploring new phenomena.