Force on a body in a continuously stratified fluid. Part 2. Sphere

Abstract

In this paper the experimental study presented in Part 1 is extended to the three-dimensional case. The in-line force coefficients (added mass and damping) of a sphere oscillating horizontally in a uniformly stratified fluid of limited depth and in a smooth pycnocline are evaluated from Fourier-transforms of the experimental records of impulse response functions. The hydrodynamic loads in the three- and two-dimensional cases are shown to be essentially different, notably in the low-frequency limit, where the damping in the three-dimensional case is zero, while in the two-dimensional case it is maximized due to phenomena akin to blocking. The generalization of the experimental results for affinely similar geometries is discussed. It is found that, as the characteristic vertical extent of stratification decreases, the mean power of internal waves radiated by the oscillating sphere reduces and the maximum of the frequency spectrum of wave power shifts toward lower frequency, which is qualitatively similar to the effects observed in the two-dimensional case. Physically, horizontal stratified waveguides act as low-pass filters since internal waves with nearly vertical group-velocity vector cannot be effectively radiated from oscillating bodies.

Hydrodynamics of natural oscillations of neutrally buoyant bodies in a layer of continuously stratified fluid

“…the virtual mass tends to zero m 0 ≈ 0. This agrees qualitatively with the results [16] for a sphere with a forced oscillation frequency close to the buoyancy frequency.…”

“…Although the values of the oscillation period calculated with account for the dependence of the virtual mass on the oscillation frequency in a continuously stratified fluid lie closer to the values observed, the difference is still noticeable [16]. Since in this case the oscillation frequencies are greater than the buoyancy frequency, the virtual mass must be negative.…”

Oscillations of a cylinder piercing a linearly stratified fluid layer

“…The qualitative conclusions [10] concerning the proportionality of the damping coefficient to the square of the thickness for a relatively thin pycnocline is confirmed for the low-frequency limit of the damping coefficient. This is attributable to the analogy with the blocking effect in the slow horizontal motion of plane bodies.…”

Generation of internal gravity waves by a katabatic wind in an idealized alpine valley

“…Voisin (2007) showed theoretically that, when N is fixed and Ω varies from 0 to its upper bound N, the power of the waves radiated by an oscillating sphere is maximum for Ω/N ≃ 0.84 (and for ≃ 0.82 for an oscillating cylinder). These results have been verified experimentally (Ermanyuk & Gavrilov (2003)). Interestingly, when a mixed region is released at its equilibrium level (Wu (1969)) or above it (Cerasoli (1978)), in a constant N stratified fluid, the power spectrum of the radiated waves is peaked at a frequency Ω such that Ω/N ≃ 0.8 as well.…”