On the characterization of a synthetic jet actuator driven by a piezoelectric disk (volume A) - Unsteady gravitational liquid sheet flows (volume B)
Creators: name: Girfoglio, Michele
Publisher: name: Università degli Studi di Napoli Federico II
Year: 2015

description: The present PhD thesis consists of two volumes. The volume A is aimed at characterizing the frequency response of a synthetic jet actuator driven by a thin piezoelectric disk. A lumped element mathematical model of the operation of a synthetic jet actuator is both analytically and numerically investigated in order to obtain information about the frequency response of the device. The model considers the three basic elements of the actuator: the oscillating membrane, the cavity, the orifice. The dynamics of the diaphragm is described through the motion equation of a one-degree of freedom forced-damped spring-mass system, while the resonant cavity and orifice components are described by means of proper forms of the continuity and Bernoulli's unsteady equations. Direct numerical simulation of the non-linear governing equations system has been carried out by a home-made code written in MATLAB environment. From the analytical viewpoint, it is found that the device behaves as a two-coupled oscillators system; by neglecting the non-linear damping term of the acoustic oscillator and by solving numerically the relevant eigenvalues problem, it is possible to obtain an accurate estimate of the two resonant modes of the system; moreover, by making the further assumption of complete absence of damping effects, simple closed-form analytical formulas are given in order to predict the two modified peak frequencies, as functions of the uncoupled first-mode structural and Helmholtz resonance frequencies. These
predictions are well confirmed by numerical simulations of the fully non-linear equations. The model is also validated through systematic experimental tests carried out on three
devices, one with the membrane in brass and two in aluminum, having different mechanical and geometrical characteristics leading to an increasing coupling factor; it is found a very strict agreement between exit flow velocity measurements and analogous numerical data for any tested device, for different supply voltages. Finally, a comprehensive and detailed modelling to evaluate the efficiency of energy conversion of the device is developed. The contribution is original because the analysis is
based on the energy equations of the two coupled oscillators, the membrane and the acoustic one, which are directly derived from the corresponding motion equations. The modelling is validated against numerical as well as experimental investigations carried out on the device exhibiting the strongest coupling effect. The volume B addresses the unsteady global dynamics of a gravitational liquid sheet interacting with a one-sided adjacent air enclosure, typically referred to as nappe oscillation, under the assumptions of potential  flow and both in absence and in presence of surface tension effects. To the purpose of shedding physical insights, the investigation is carried out from both the dynamics and the energy aspects. An interesting re-formulation of the problem consists of recasting the nappe global behavior as a driven damped spring-mass oscillator, where the inertial effects are linked to the liquid sheet mass and the spring is represented by the equivalent stiffness of the air enclosure acting on the average displacement of the
compliant nappe centerline. The investigation is carried out by means of modal (i.e. time asymptotic) linear approach, which is corroborated by direct numerical simulations of the governing equation. In absence of surface tension effects, the modal analysis shows that the flow system is characterized by low-frequency and high-frequency oscillations, the former related to
the crossing time of the perturbations over the whole domain, the latter related to the spring-mass oscillator. The low-frequency oscillations, observed in real life systems, are produced by the (linear) combination of multiple modes. The flow system is characterized by short-time energy amplifications even in asymptotically stable configurations, that are confirmed by numerical simulations and justified by energy budget considerations. Strong analogies with the edge tones problem are encountered. In presence of surface tension effects, it is known that the basic nature of the global dynamics of gravitational liquid sheet flows depends crucially on the inlet Weber number. When the inlet Weber number is greater than unity, the local Weber number too is greater than unity at each streamwise location, so that the flow can be defined supersonic-like everywhere. When the inlet Weber number is lower than unity, there exists an initial region where the sheet flow is subsonic, up to the transonic location, downstream of which the flow becomes supersonic. From the theoretical viewpoint the problem is not straightforward, because it is known that the equation governing the evolution of small disturbances exhibits a singularity just at the transonic station, although the solution to the problem is not yet yielded. Preliminary physical insights of the sheet centerline sinuous modes show that the nappe dynamics features the propagation of two wave fronts both directed downstream or one downstream and the other one upstream depending on whether the flow is supersonic or subsonic, respectively. In every situation the inlet section is considered as a perfectly
reflecting boundary. As regards the other boundary condition, in supersonic flow it is of null centerline slope at the inlet; in subsonic regime it is imposed just at the singularity station, by enforcing the boundedness of the sheet slope there. An additional analytical investigation on the nature of the singularity, based on the Frobenius theory, predicts that the flow is unconditionally stable for any Weber number lower than unity and that the disturbance time growth rate results to be bounded above. The findings of the eigenvalues spectral analysis confirm all the theoretical predictions discussed above. They in turn closely agree with direct numerical simulations of the partial differential equation governing in the space-time domain the global evolution of the disturbances, starting from an initial gaussian-like shape, with particular attention to transonic and high Weber number regimes.