2D and 3D problems of internal-wave radiation by a body oscillating in a uniformly stratified fluid

Abstract : Internal waves play important role in the dynamics of atmosphere and ocean, being responsible for significant transport of momentum and energy through stratified fluids. Radiation and diffraction of internal waves significantly affect the motions of deep submersibles, submerged buoys and moored structures. In the case of constant vertical density gradient (uniform stratification) theoretical description of internal-wave generation by an oscillating body is complicated by non-trivial dispersion relation. The energy radiated by a vibrating cylinder is known to be spread into four wave beams inclined at angle θ to the vertical, with the cylinder at the center of this pattern known as 'St. Andrew cross' wave. In 3D case the internal waves are confined within a double cone. The dispersion relation for internal waves in a uniformly stratified fluid relates angle θ with the frequency of oscillations. Since the wave length does not appear in the dispersion relation, the spatial structure of internal waves has non-trivial dependence on the body geometry, direction and frequency of oscillations, and the fluid viscosity. Another complication is that linear theory of ideal uniformly stratified fluid predicts infinite displacements of fluid particles along the lines tangent to the oscillating body and inclined at angle θ to the vertical. With fluid viscosity taken into account, realistic estimate of the cross-beam distribution of the displacements of fluid particles has been obtained in the literature. However, that solution does not fulfill the no-slip condition on the body surface and therefore neglects the effect of the boundary layers. When the amplitude of oscillations and the thickness of the boundary layer are small compared to the size of the oscillating body, the solution has been confirmed experimentally. Analysis of hydrodynamic loads has been experimentally confirmed in and extended to the bodies of arbitrary shape. When the amplitude of oscillations, the thickness of the boundary layer and the size of the oscillating body are comparable quantities, we may expect strong non-linear interaction between the boundary-generated vorticity and wave radiation and generation of secondary mass-transfer currents. This issue is addressed in the present study where in 2D case we consider large-amplitude horizontal, vertical and circular oscillations of a circular cylinder. Also, we consider the duration of the transient processes of internal-wave beams formation after the start-up of the motion. Theory considers only the steady-state harmonic oscillations of a body and yields no information on the transients. As the main experimental tool in 2D case we use synthetic schlieren technique. In the 3D case syntetic schlieren data processing requires a tomographic inversion. In the case of axisymmetric waves generated by vertical oscillations of a sphere the technique is described in the literature and extended to 3D cases. In the present work we consider internal waves generated by horizontal oscillations of a sphere as a generic example of a 3D problem and compare the results of measurements with theoretical estimates.
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Evgeny V. Ermanyuk, N. V. Gavrilov, Jan-Bert Flór, Bruno Voisin. 2D and 3D problems of internal-wave radiation by a body oscillating in a uniformly stratified fluid. 23rd International Workshop on Water Waves and Floating Bodies, Apr 2008, Jeju, South Korea. pp.45-48. ⟨hal-00581853v2⟩

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