This paper reports on an experimental study of the motion of freely rising axisym- metric rigid bodies in a low-viscosity fluid. We consider flat cylinders with height h smaller than the diameter d and density ρb close to the density ρf of the fluid. We have investigated the role of the Reynolds number based on the mean rise velocity um in the range 80 ≤ Re = umd/ν ≤ 330 and that of the aspect ratio in the range 1.5 ≤ χ = d/h ≤ 20. Beyond a critical Reynolds number, Rec, which depends on the aspect ratio, both the body velocity and the orientation start to oscillate periodically. The body motion is observed to be essentially two-dimensional. Its description is particularly simple in the coordinate system rotating with the body and having its origin fixed in the laboratory; the axial velocity is then found to be constant whereas the rotation and the lateral velocity are described well by two harmonic functions of time having the same angular frequency, ω. In parallel, direct numerical simulations of the flow around fixed bodies were carried out. They allowed us to determine (i) the threshold, Recf1(χ), of the primary regular bifurcation that causes the breaking of the axial symmetry of the wake as well as (ii) the threshold, Recf2(χ), and frequency, ωf, of the secondary Hopf bifurcation leading to wake oscillations. As χ increases, i.e. the body becomes thinner, the critical Reynolds numbers, Recf1 and Recf2, decrease. Introducing a Reynolds number Re* based on the velocity in the recirculating wake makes it possible to obtain thresholds $\hbox{\it Re}^*_{cf1}$ and $\hbox{\it Re}^*_{cf2}$ that are independent of χ. Comparison with fixed bodies allowed us to clarify the role of the body shape. The oscillations of thick moving bodies (χ < 6) are essentially triggered by the wake instability observed for a fixed body: Rec(χ) is equal to Recf1(χ) and ω is close to ωf. However, in the range 6 ≤ χ ≤ 10 the flow corrections induced by the translation and rotation of freely moving bodies are found to be able to delay the onset of wake oscillations, causing Rec to increase strongly with χ. An analysis of the evolution of the parameters characterizing the motion in the rotating frame reveals that the constant axial velocity scales with the gravitational velocity based on the body thickness, $\sqrt{((\rho_f-\rho_b)/\rho_f)\,gh}$, while the relevant length and velocity scales for the oscillations are the body diameter d and the gravitational velocity based on d, $\sqrt{((\rho_f-\rho_b)/\rho_f)\,gd}$, respectively. Using this scaling, the dimensionless amplitudes and frequency of the body's oscillations are found to depend only on the modified Reynolds number, Re*; they no longer depend on the body shape.

The zigzag path of light flat cylinders rising freely in water at rest is investigated experimentally. The body dynamics are described in terms of the orientation of the cylinder axis relative to its path as a function of its aspect ratio and Reynolds number. We show that for thick bodies the inclinations of both the axis and the velocity oscillate almost in phase, whereas for thin bodies they are rather in quadrature. Most notably, for bodies of intermediate aspect ratio, the phase difference evolves continuously between these two limits when the body flatness changes. In contrast, this phase difference is nearly independent of the Reynolds number. Comparison with potential flow theory shows that these features can only be explained by taking into account vortical effects.

The forces and torques governing the planar zigzag motion of thick, slightly buoyant disks rising freely in a liquid at rest are determined by applying the generalized Kirchhoff equations to experimental measurements of the body motion performed for a single body-to-fluid density ratio ρs/ρf ≈ 1. The evolution of the amplitude and phase of the various contributions is discussed as a function of the two control parameters, i.e. the body aspect ratio (the diameter-to-thickness ratio χ = d/h ranges from 2 to 10) and the Reynolds number (100 < Re < 330), Re being based on the rise velocity and diameter of the body. The body oscillatory behaviour is found to be governed by the force balance along the transverse direction and the torque balance. In the transverse direction, the wake-induced force is mainly balanced by two forces that depend on the body inclination, i.e. the inertia force generated by the body rotation and the transverse component of the buoyancy force. The torque balance is dominated by the wake-induced torque and the restoring added-mass torque generated by the transverse velocity component. The results show a major influence of the aspect ratio on the relative magnitude and phase of the various contributions to the hydrodynamic loads. The vortical transverse force scales as fo = (ρf − ρs)ghπd2 whereas the vortical torque involves two contributions, one scaling as fod and the other as f1d with f1 = χfo. Using this normalization, the amplitudes and phases of the vortical loads are made independent of the aspect ratio, the amplitudes evolving as (Re/Rec1 − 1)1/2, where Rec1 is the threshold of the first instability of the wake behind the corresponding body held fixed in a uniform stream.

We describe a dynamical model that predicts the zigzag motion of disks and oblate spheroids moving freely in a viscous liquid over a continuous range of aspect ratios and Reynolds numbers. This model combines the generalized Kirchhoff equations to describe the linear and angular momentum balances for the fluid-body system with a dynamical model for the wake-induced force and torque that incorporates the main characteristics of the wake dynamics deduced from previous experimental observations. The resulting model is shown to be able to reproduce quantitatively the oscillatory paths measured experimentally.

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On observe souvent dans la nature que l'ascension ou la chute d'un corps peut présenter des mouvements oscillatoires (spirale, zigzag) ou plus désordonnés. Nous nous sommes penchés sur les causes des instabilités du mouvement d'un corps en ascension sous l'effet de la gravité, dans un fluide au repos. Nous avons conduit uneétude expérimentale des mouvements oscillatoires de corps légers montant librement dans un fluide peu visqueux. Des résultats originaux concernant la cinématique de cylindres minces sont présentés ici pour une large gamme de nombres d'Archimède (flottabilité sur effets visqueux) et du rapport de forme (diamètre surépaisseur). Nous avons analysé les oscillations de la vitesse et de l'orientation des cylindres (fréquences, amplitudes et différences de phases), ce qui a mis enévidence l'effet crucial du rapport de forme dans le couplage entre la translation et la rotation. Mots clés :Corps mobile / oscillations auto-entretenues / instabilité de trajectoire / sillage Abstract -Oscillatory motions of freely-moving bodies rising in a low-viscous fluid: the role of the aspect ratio. In many situations, freely falling or rising particles exhibit oscillatory motions: spiral, zigzag or tumbling. We are concerned with the causes of these path instabilities for particles moving under the effect of buoyancy in a fluid otherwise at rest. The oscillatory motion of light particles rising freely in a slightly viscous fluid was investigated experimentally. Original results concerning the kinematics of flat cylinders are reported for a wide range of the Archimedes number (buoyancy vs. viscous effects) and aspect ratio (diameter-to-height ratio). In this paper, we focus on the particle velocity and orientation oscillations (frequencies, amplitudes and phase differences), underlining the crucial effect of the body aspect ratio on the complex coupling between translation and rotation.

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