Buoyancy effects on added mass in density-stratified fluids

International audienceIn the presence of stable density stratification, owing to buoyancy, fluid motion gives rise to internal gravity waves which redistribute momentum and energy through the fluid. As a result, the added mass of moving bodies is modified and becomes anisotropic and frequency-dependent. The influence of these modifications on the definition itself of added mass and on its relation to hydrodynamic pressure, impulse, energy and to the dipole strength of the bodies is discussed. Coefficients of added mass are calculated explicitly for the small oscillations of spheres and circular cylinders. Implications for energy radiation and for the motion of floats are considered. In the first case the existence of a maximum at a frequency of oscillation equal to a fixed fraction 0.8 of the buoyancy frequency, practically independent of the direction of oscillation, is pointed out together with possible inferences for turbulent motion. In the second case classical results by Larsen on neutrally buoyant spheres and cylinders are recovered

Added mass for motion in density-stratified fluids

International audienceIn a density-stratified fluid, owing to the generation of internal gravity waves, the added mass of a moving body becomes anisotropic and frequency-dependent. Two definitions of added mass, equivalent in a homogeneous fluid, become different: one based on the impulse of the fluid, and the other on the pressure on the body. We start by deriving a Kirchhoff-Helmholtz integral for internal waves, extending earlier attempts, and deduce from it the relation between the two definitions. We then calculate the motion of the fluid for oscillating spheres and cylinders, extending earlier results, and deduce from it the coefficients of added mass, consistent with experiment. At frequencies large compared with the buoyancy frequency the coefficients take their values in a homogeneous fluid (1/2 for the sphere and 1 for the cylinder); at frequencies smaller than the buoyancy frequency they become complex, of real part associated with added mass in the usual sense and imaginary part associated with wave drag. These variations are related by Fourier transformation to the free oscillations, consistent with experiment, of the bodies displaced from their neutrally buoyant levels then released

Influence of the multipole order of the source on the decay of an inertial wave beam in a rotating fluid

We analyze theoretically and experimentally the far-field viscous decay of a two-dimensional inertial wave beam emitted by a harmonic line source in a rotating fluid. By identifying the relevant conserved quantities along the wave beam, we show how the beam structure and decay exponent are governed by the multipole order of the source. Two wavemakers are considered experimentally, a pulsating and an oscillating cylinder, aiming to produce a monopole and a dipole source, respectively. The relevant conserved quantity which discriminates between these two sources is the instantaneous flowrate along the wave beam, which is non-zero for the monopole and zero for the dipole. For each source the beam structure and decay exponent, measured using particle image velocimetry, are in good agreement with the predictions

Internal waves and boundary layer for an oscillating disc in a stratified fluid

International audienceInternal or baroclinic tides, namely internal waves generated in the ocean by the oscillation of the barotropic tide over bottom topography, exhibit a complex pattern of primary wave beams and secondary beams resulting from the interaction of the primary beams with themselves and with the boundary layer at the topography. In this context, the problem of an oscillating horizontal disc, however remote it may be from oceanic configurations, recently gained visibility as the only problem for which a full analytical solution may be found including both waves and the boundary layer (Davis & Llewellyn Smith 2010). To date, all available approaches used an approximate free-slip condition at the topography instead of the actual no-slip condition, thus eliminating the boundary layer. In this communication, we examine the relation between those approaches and compare them with original high-resolution experimental measurements. Specifically, we consider the vertical heaving oscillations of a horizontal circular disc and compare fully inviscid investigations where both the generation and propagation of the waves are inviscid – using either orthogonal curvilinear coordinates (Sarma & Krishna 1972), boundary integrals (Gabov & Pletner 1988) or eigenfunction expansions (Martin & Llewellyn Smith 2011, 2012) – with fully viscous investigations where both generation and propagation are viscous – using either the actual disc oscillating in free space (Davis & Llewellyn Smith 2010) or a fictitious baffled disc oscillating through an aperture in a horizontal plane (Chashechkin, Vasil'ev & Bardakov 2004; Bardakov, Vasil'ev & Chashechkin 2007). We discuss the relevance of an intermediate model where propagation is viscous but generation inviscid (as used recently for a sphere by Voisin, Ermanyuk & Flór 2011), as a function of the Reynolds-Stokes number. At all but very high values of this number (of order $10^5$ or more say), it appears that the presence of the boundary layer must be taken into account for accurate prediction of the waves

Internal waves and boundary layers in a density-stratified fluid

International audienceThe topic of internal gravity wave generation by oscillating bodies in density-stratified fluids has recently gained prominence owing to the importance of oceanic internal tides, generated by the oscillation of the barotropic tide over bottom topography, in the dynamics of the Earth-Moon system. To assess the role of the boundary condition at the body (in the ocean at the topography), we consider the only problem for which a full viscous solution is known: the oscillating circular disk. The inviscid theory, the full viscous theory (i.e. with viscous effects on both wave propagation and generation) and the partial viscous theory (i.e. with viscous effects on propagation alone) are compared with low-resolution conductimetric measurements from the literature and with new and original high-resolution PIV measurements. Viscous effects on propagation are required for prediction of the wave profiles; generation becomes free-slip at Stokes number over a million, and remains no-slip otherwise

Internal wave-vorticity coupling for an oscillating disk

International audienceIn a density-stratified fluid, viscosity couples internal waves with vertical vorticity. So far this coupling used to be neglected in analytical studies and only the viscous attenuation and spreading of the waves was taken into account, except in a very recent study of the oscillations of a horizontal circular disk (Davis & Llewellyn Smith, JFM 2010). We investigate the relations between the previous analytical approaches of the disk, considering either inviscid or viscous propagation of the waves and either free- or no-slip conditions at the disk, and compare their output with an original approach based on the boundary integral method. In particular, the role of the Stokes number is clarified. The analytical predictions are compared with contact measurements for vertical oscillations (Bardakov, Vasil'ev & Chashechkin, Fluid Dynamics 2007) and with original PIV measurements and visualizations for both vertical and horizontal oscillations

Monochromatic internal waves: From theory to experiment

An analytical theory is presented for the generation of small-amplitude three-dimensional monochromatic internal waves by an oscillating object. The theory expresses the structure of the waves in terms of the size of the object, the viscosity of the fluid and the time elapsed since the start-up; to this, it adds near-field effect and modification of the added mass of the object by the stratification. Comparison with experiment allows assessment of the relative importance of all five effects

Internal-wave radiation by a horizontally oscillating body in a uniformly stratified fluid

International audienceIn this experimental-theoretical study we consider the waves emitted by a horizontally oscillating sphere in a linearly stratified fluid. In contrast to former investigations, the thus generated wave pattern is a-symmetric and three-dimensional. We consider large and small amplitude horizontal oscillations for different size spheres. The spatial structure of internal waves has a non-trivial dependence on the body geometry, direction and frequency of oscillations. The flowfield is measured quantitatively, using an alternative version of the synthetic schlieren technique. In addition we exploit the technique to visualise internal waves with fluorescein dye planes used by Hopfinger et al (Exp. in Fluids, 11, 1991) to measure the displacement field of the internal waves. For the theory a uniformly stratified viscous Boussinesq fluid of infinite extent is considered, with small viscosity and the boundary layer on the body surface neglected. For small amplitude oscillations, the comparison with the theory is good, with the near-field theory being in very good agreement with the experimental results and the far field theory slightly overestimating the wave amplitude

Internal wave structure emitted by a horizontally oscillating sphere

International audienceAn oscillating body in a stratified fluid generates a double cone-shaped internal-wave pattern, the 3D analogue of the classic St.Andrew-cross. For sufficiently low frequency and large amplitude oscillations, higher-order wave harmonics may be generated along with the fundamental one. We present an experimental study of the 3D structure of first- and second-order wave fields emitted by a horizontally oscillating sphere. In contrast to the axisymmetric wave pattern found for a vertically oscillating sphere, for horizontal oscillations, the first- and higher-order-harmonic waves have different distributions of wave amplitudes in the azimuthal direction. The amplitude of the first-order waves is shown to follow the cosine dependence on the azimuthal angle, in accordance with theoretical predictions. The azimuthal distribution of the amplitude of the second-order waves gives evidence of a quadrupolar distribution, with four preferential directions of wave radiation in a horizontal plane, along the direction of oscillation and normal to it. Noteworthy is that the amplitudes of these second-order waves may exceed the amplitude of first-order waves

First and second harmonic internal waves from a horizontally oscillating sphere

International audienceA horizontally oscillating sphere in a density-stratified fluid is studied experimentally and theoretically, as a paradigm of the generation of three-dimensional internal tides by supercritical topography. The experiments implement a novel technique for the measurement of the spatial structure of internal wave fields, based on horizontal fluorescent dye planes and a mobile vertical laser sheet; they are compared with an original linear theory. Spectral analysis reveals the presence of two harmonics, namely a first harmonics at the fundamental frequency and a second harmonics at twice this frequency. The first harmonics has a dipolar structure, an amplitude varying linearly with the amplitude of oscillation, and is quantitatively described by the theory. The second harmonics is present at amplitudes of oscillation higher than one tenth of the sphere radius and has a quadrupolar structure. Its amplitude varies quadratically with the amplitude of oscillation, and may exceed the amplitude of the first harmonics. At frequencies smaller than half the buoyancy frequency, the second harmonics is evanescent and confined to the vicinity of the sphere; at frequencies larger than half the buoyancy frequency, it propagates away