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Mesures résolues en temps et en espace d'ondes à la surface de l'eau : Application aux modes piégés

Abstract : This thesis represents a contribution to the understanding of certain water wave phenomena of interest in current research. A number of significant results have been obtained by means of a new free-surface measuring technique that has been developed for such studies. This optical profilometric technique consists in projecting a sinusoidal-profile fringe pattern of known characteristics onto the free surface and in observing the projected images from a different direction. The surface deformation, as well as the perspective, introduce a local frequency modulation of the fringe pattern. Analysis of the deformed image and its comparison with a reference image allow for the reconstruction of the free-surface deformation. In particular, this technique presents the advantage of determining the surface's profile from only one image, which allows the study of highly unstationary surface flows. The high spatio-temporal resolution achieved allow, for the first time, the exploration of a vast variety of water wave phenomena. Two major experimental studies on surface waves have been carried out during the course of this thesis. In the first one, we focused on the study of water wave resonances around a circular cylinder of radius a placed symmetrically between the parallel walls of a waveguide of width 2d (trapped modes). The relevant dimensionless parameters in this case are the frequency of waves kd (k being the wavenumber), and a/d, the aspect ratio between the cylinder and the waveguide. In the framework of this study, several values of the aspect ratio have been explored. This work provides the first complete experimental characterization of trapped modes in the frequency space, as well as a detailed analysis of their spatial structure. This caracterization has been obtained by decomposing the free surface deformation field in harmonics of the driving frequency, which has allowed us to evaluate the relative contribution of linear and non-linear modes. Our results show that the linear component is dominant in our experiences, therefore validating the theoretical approaches based on the linear theory of water waves. A decomposition of the linear deformation field in terms of the natural symmetries of the problem enables us, for the first time, to provide experimental evidence of the spatial structure of trapped modes. These manifest in the form of non-propagative oscillations of the free surface, antisymmetric with respect to the longitudinal axis of the waveguide, confined to the vicinity of the cylinder. Two different types of trapped modes have been observed: either symmetric or antisymmetric with respect to a line perpendicular to the walls passing through the center of the cylinder. While the first type of trapped mode is always present, the second type has only been observed in the case of the largest aspect ratios. Our results regarding the spatial structure of the trapped modes confirm the theoretical predictions arising from a multipole expansion method. The frequency characterization of the trapped modes has been obtained by the analysis of the problem in the far field. By introducing reflection and transmission coefficients for the antisymmetric perturbations inside the waveguide, we were able to build resonance curves for every value of the aspect ratio a/d considered. On this curves, the occurrence trapped modes is evidenced by the presence of one or two resonance peaks. A marked asymmetry is observed on these curves, which cannot be properly described by the classical Breit-Wigner shape. This asymmetry has been also found in a complementary numerical study. In order to describe adequately this behaviour, we have proposed a model which takes into account the proximity to the waveguide's threashold for propagation. This model allowed us to reproduce the asymmetry of the resonance curves and was successfully validated with the experimental results. Finally, all the experimental results are summarized on master curve, depicting the dependence of the trappedmode frequency kd with the aspect ratio a/d. This curve is composed, as expected, by two branches, corresponding to the two types of trapped modes observed. This results show an excellent agreement with the predictions available in the literature. The second experimental study conducted in the frame of this thesis regards the turbulence of bending waves in a thin elastic plate. In this case, fully space-time resolved measurements of the plate deformation have been employed to determine, for the first time, the three-dimensional energy spectrum of wave turbulence. Analysis of this spectrumshows the presence of a turbulent energy cascade: low wavenumbers are characterized by a strong anisotropy associated to the forcing, the isotropy being recovered at large wavenumbers. Moreover, analysis of the three-dimensional spectrum leads to the observation that the energy is mainly concentrated in the vicinity of a 2D surface, representing a weakly non-linear dispersion relation. This experimental result confirms the persistence of the spatio-temporal structure of waves comprising the turbulent cascade. Our experimental approach for wave turbulence revealed also the principal characteristics of the weakly coupled waves that can be usefully compared with the predictions of weak turbulence theory. This study confirms and quantifies the weakly non-linear behaviour of the waves in the turbulent cascade. Furthermore, our results confirmed that the scaling law in the supplied power is the same for the energy spectrum. We have shown a bon accord between experimental results and weak turbulence theory. Two other preliminary studies are brieflymentioned, regarding the time-reversal of waterwaves and the spatio-temporal evolution of the free surface after the impact of a drop.
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Pablo Cobelli. Mesures résolues en temps et en espace d'ondes à la surface de l'eau : Application aux modes piégés. Mécanique des fluides [physics.class-ph]. Université Paris-Diderot - Paris VII, 2009. Français. ⟨pastel-00555647⟩

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