Resonance in swirling wakes and sloshing waves: non-normal and sublinear effects

Similarly to mechanical structures, stable flows can exhibit resonance when perturbed by an impulsive or harmonic forcing. Swirling wakes and sloshing waves belong to this kind of flows and manifest large energy response when excited close to their natural frequencies. Although these frequencies can be predicted by linear modal analysis, the full flow dynamics differs from the modal one because entailed by the mutual cooperation of the natural modes (non-normal effects) and dependent on the oscillation amplitude (nonlinear effects). In this thesis, the response of swirling wakes subjected to a harmonic forcing is studied numerically and theoretically. Direct numerical simulations show that a large variety of helical modes can be excited and amplified in trailing vortices when a harmonic inlet or volume forcing is imposed, with the appearance of higher wavenumber modes at higher frequency. The mode-selection mechanism is shown to be directly connected to the local stability properties of the flow, and is simultaneously investigated by a WKB approximation, in the framework of weakly non-parallel flows, and by the global resolvent approach. This analysis is then extended to the case of turbulent swirling flows to investigate the physical origin of the meandering oscillations of the hub vortex, that is observed in wind turbine wakes experiments. We show as this low frequency spectral component is the result of a convectively unstable single-helix structure that oscillates at a frequency equal to one third the rotational frequency of the wind turbine rotor. Consequently, an adjoint-based technique for the passive control of these helical instabilities is proposed. We then turn our attention towards the transient decay of sloshing waves affected by a viscous friction at the container’s wall, that exhibits a sublinear dependence in the interface velocity, i.e. a power law with an exponent smaller than one. This capillary effect is exacerbated in our experiment by placing a thin layer of foam on the liquid phase that act as a collection of air-liquid interfaces. In contrast to classical theory, we uncover the existence of a finite-time singularity in our system yielding the arrest of the sloshing oscillations in a finite time and we propose a minimal theoretical framework to capture this effect. Using first principles, we then study the effect of contact angle hysteresis on sloshing waves. We show asymptotically that, in contrast to viscous damping where the wave motion decays exponentially, the contact angle hysteresis acts as Coulomb solid friction yielding the damping rate induced by the motion of the liquid meniscus to increase at small amplitude, consistently with the experimental observation.


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