Mass and Moment of Inertia Govern the Transition in the Dynamics and Wakes of Freely Rising and Falling Cylinders

Abstract

In this Letter, we study the motion and wake-patterns of freely rising and falling cylinders in quiescent fluid. We show that the amplitude of oscillation and the overall system-dynamics are intricately linked to two parameters: the particle's mass-density relative to the fluid m * ≡ ρp/ρ f and its relative moment-of-inertia I * ≡ Ip/I f . This supersedes the current understanding that a critical mass density (m * ≈ 0.54) alone triggers the sudden onset of vigorous vibrations. Using over 144 combinations of m * and I * , we comprehensively map out the parameter space covering very heavy (m * > 10) to very buoyant (m * < 0.1) particles. The entire data collapses into two scaling regimes demarcated by a transitional Strouhal number, Stt ≈ 0.17. Stt separates a mass-dominated regime from a regime dominated by the particle's moment of inertia. A shift from one regime to the other also marks a gradual transition in the wake-shedding pattern: from the classical 2S (2-Single) vortex mode to a 2P (2-Pairs) vortex mode. Thus, auto-rotation can have a significant influence on the trajectories and wakes of freely rising isotropic bodies.Path-instabilities are a common observation in the dynamics of buoyant and heavy particles. Common examples are the fluttering of falling leaves and disks, and the cork-screw and spiral trajectories of air-bubbles rising in water [1,2,3]. The oscillatory dynamics of such particles can vary a lot depending on the particle's size, shape and its inertia, and the surrounding flow properties. This can be important in a variety of fields ranging from sediment transport and fluidization to multiphase particleand bubble-column reactors [4,5,6].Examples of organisms that exploit path-instabilities are plants and aquatic animals. These often make use of passive appendages attached to their bodies (plumed seeds, barbs, tails, and protrusions) to generate locomotion [7,8,9]. Recently, Lācis et al. [7] demonstrated that the interaction between the wake of a falling bluff body and a protrusion clamped to its rear end can generate a sidewards drift by means of a symmetry-breaking instability similar to that of an inverted pendulum. Such kinds of passive interactions are advantageous to locomotion, since no energy needs to be spent by the animal. Instead, the energy can be extracted through fluid-structure interaction.

Kinematics and wake of freely falling cylinders at moderate Reynolds numbers

“…For the same Re d , the Strouhal number values they computed were thus lower for cylinders of fixed spanwise length. Finally, it can be pointed out that Namkoong et al (2008) observed numerically that for 66 < Re d < 185 the degrees of freedom in rotation and translation of an infinitely long (i.e. two-dimensional) cylinder lead to a slight decrease in the frequency of vortex shedding and of the associated oscillatory motion of the cylinder, with typically St BvK 0.17 for ρ = 1.01 and Re d = 156, corresponding to the trend observed here for lower Re d ( figure 19).…”

“…The problem of an infinitely long circular cylinder free to move in any direction perpendicular to its axis is a two-dimensional counterpart to the case of the sphere, and its characteristic dimension D is the cylinder diameter. To investigate the coupling between the wake instability and path oscillations, Namkoong, Yoo & Choi (2008) performed numerical simulations of the two-dimensional motion of a freely falling/rising infinitely long circular cylinder in an infinite fluid, but considered also situations of partially restrained motion in translation and rotation of the body. They focused on density ratios 0.5 < ρ < 4 and Reynolds numbers between 65 and 185, corresponding to an unsteady wake exhibiting periodic vortex shedding in the case of a fixed cylinder.…”

“…In the latter case, they recorded displacement oscillations of amplitude of the order of D in the transverse direction at a frequency approximately 1.3 times smaller (and 0.3D in the vertical direction at twice this frequency), in association with a wake structure comprising two vortex pairs generated per cycle. In the same configuration, Mathai et al (2017) investigated numerically the effect of both ρ and I * = I p /I f , the particle moment of inertia I p relative to that of the fluid I f , by varying them separately in the range [0.1, 10], thus covering heavy (ρ 10) to light (ρ 0.1) cylinders. They separate for Ar 220 (the corresponding Re is not specified) a regime of cylinder oscillation with weak to intermediate displacement amplitudes, 0 < A < 0.3D, dominated by the effect of the parameter ρ, from a regime dominated by the effect of the moment of inertia and the rotational degree of freedom, corresponding to higher amplitudes, 0.4D < A < 0.75D.…”

Dynamics of heavy and buoyant underwater pendulums

“…It is well known that added mass coefficients m * a for oscillating cylinders depend on a variety of parameters, including the frequency of oscillation, distance to boundaries, free surfaces, etc. Most studies have focused on the case of cylinders near a free surface and for small oscillations (Dong 1978;Konstantinidis 2013;Koo & Kim 2015;Tatsuno & Bearman 1990;Mathai et al 2017Mathai et al , 2018. However, in complex situations with relative motions, curved trajectories and unsteady three-dimensional wakes with flow separation, m * a can deviate significantly from the two-dimensional potential flow added mass coefficient.…”

Bubbly and Buoyant Particle–Laden Turbulent Flows

“…Although the mean forces on the particle can still be approximated, the instantaneous drag and lift can no longer be described using simplified coefficients. Furthermore, owing to the lightness of the particle, this regime paves way for a strong coupling between the unsteady wake dynamics and the particle motion, often resulting in vigorous path instabilities (Brücker 1999;Ern et al 2012;Mathai et al 2016bMathai et al , 2017Mathai et al , 2018bMougin and Magnaudet 2006). As reported in Risso (2017), there is, today, compelling evidence that the wakes and dynamics of isolated buoyant particles are remarkably robust to turbulent perturbations (Ford and Loth 1998).…”