Coins falling in water

When a coin falls in water, its trajectory is one of four types determined by its dimensionless moment of inertia $I^\ast$ and Reynolds number Re: (A) steady; (B) fluttering; (C) chaotic; or (D) tumbling. The dynamics induced by the interaction of the water with the surface of the coin, however, makes the exact landing site difficult to predict a priori. Here, we describe a carefully designed experiment in which a coin is dropped repeatedly in water, so that we can determine the probability density functions (pdf) associated with the landing positions for each of the four trajectory types, all of which are radially symmetric about the center-drop line. In the case of the steady mode, the pdf is approximately Gaussian distributed, with variances that are small, indicating that the coin is most likely to land at the center, right below the point it is dropped from. For the other falling modes, the center is one of the least likely landing sites. Indeed, the pdf's of the fluttering, chaotic and tumbling modes are characterized by a "dip" around the center. For the tumbling mode, the pdf is a ring configuration about the center-line, with a ring width that depends on the dimensionless parameters $I^\ast$ and Re and height from which the coin is dropped. For the chaotic mode, the pdf is generally a broadband distribution spread out radially symmetrically about the center-line. For the steady and fluttering modes, the coin never flips, so the coin lands with the same side up as was dropped. For the chaotic mode, the probability of heads or tails is close to 0.5. In the case of the tumbling mode, the probability of heads or tails based on the height of the drop which determines whether the coin flips an even or odd number of times during descent

Instability and trajectories of buoyancy-driven annular disks: a numerical study

We investigate the stability of the steady vertical path and the emerging trajectories of a buoyancy-driven annular disk as the diameter of its central hole is varied. The steady and axisymmetric wake associated with the steady vertical path of the disk, for small hole diameters, behaves similarly to the one past a permeable disk, with the detachment of the vortex ring due to the bleeding flow through the hole. However, as the hole diameter increases, a second recirculating vortex ring of opposite vorticity forms at the internal edge of the annulus. A further increase in the hole size leads to the shrinking of these recirculating regions until they disappear. The flow modifications induced by the hole influence the stability features of the steady and axisymmetric flow associated with the steady vertical path. The fluid-solid coupled problem shows a nonmonotonic behavior of the critical Reynolds number for the destabilization of the steady vertical path, for low values of the disk's moment of inertia. However, for very large holes, with dimension approximately more than half of the external diameter, a marked increase of the neutral stability threshold is observed. The nature of the primary instability changes as the hole size increases, with large (small) amplitude oscillations of the trajectory at intermediate (very small and large) internal diameters. We then illustrate results obtained with fully nonlinear simulations of the time-dependent dynamics, together with a comparison of the linear stability analysis results. Falling styles, typically described as steady, hula-hoop, fluttering, chaotic, and tumbling, are shown to emerge as attractors for the nonlinear dynamics of the coupled fluid-structure system. The presence of a central hole does not always reduce the falling Reynolds number, and it may cause the transition from tumbling towards fluttering, from fluttering to hula-hoop, and from hula-hoop to steady, hence promoting trajectories with smaller lateral deviations from the vertical path. The observed trajectories and patterns agree well with linear stability analysis results, in the vicinity of the threshold of instability

Kinematics and dynamics of freely rising spheroids at high Reynolds numbers

We experimentally investigate the effect of geometrical anisotropy for buoyant ellipsoidal particles rising in a still fluid. All other parameters, such as the Galileo number $Ga \approx 6000$ and the particle density ratio $\Gamma \approx 0.53$ are kept constant. The geometrical aspect ratio, $\chi$, of the particle is varied systematically from $\chi$ = 0.2 (oblate) to 5 (prolate). Based on tracking all degrees of particle motion, we identify six regimes characterised by distinct rise dynamics. Firstly, for $0.83 \le \chi \le 1.20$, increased rotational dynamics are observed and the particle flips over semi-regularly in a "tumbling"-like motion. Secondly, for oblate particles with $0.29 \le \chi \le 0.75$, planar regular "zig-zag" motion is observed, where the drag coefficient is independent of $\chi$. Thirdly, for the most extreme oblate geometries ($\chi \le 0.25$) a "flutter"-like behaviour is found, characterised by precession of the oscillation plane and an increase in the drag coefficient. For prolate geometries, we observed two coexisting oscillation modes that contribute to complex trajectories: the first is related to oscillations of the pointing vector and the second corresponds to a motion perpendicular to the particle's symmetry axis. We identify a "longitudinal" regime ($1.33 \le \chi \le 2.5$), where both modes are active and a different one, the "broadside"-regime ($3 \le \chi\le 4$), where only the second mode is present. Remarkably, for the most prolate particles ($\chi = 5$), we observe an entirely different "helical" rise with completely unique features.Comment: 46 pages, 20 figure

Permeability sets the linear path instability of buoyancy-driven disks

The prediction of trajectories of buoyancy-driven objects immersed in a viscous fluid is a key problem in fluid dynamics. Simple-shaped objects, such as disks, present a great variety of trajectories, ranging from zig-zag to tumbling and chaotic motions. Yet, similar studies are lacking when the object is permeable. We perform a linear stability analysis of the steady vertical path of a thin permeable disk, whose flow through the microstructure is modelled via a stress-jump model based on homogenization theory. The relative velocity of the flow associated with the vertical steady path presents a recirculation region detached from the body, which shrinks and eventually disappears as the disk becomes more permeable. In analogy with the solid disk, one non-oscillatory and several oscillatory modes are identified and found to destabilize the fluid-solid coupled system away from its straight trajectory. Permeability progressively filters out the wake dynamics in the instability of the steady vertical path. Modes dominated by wake oscillations are first stabilized, followed by those characterized by weaker, or absent, wake oscillations, in which the wake is typically a tilting induced by the disk inclined trajectory. For sufficiently large permeabilities, the disk first undergoes a non-oscillatory divergence instability, which is expected to lead to a steady oblique path with a constant disk inclination, in the nonlinear regime. A further permeability increase reduces the unstable range of all modes until quenching of all linear instabilities

Flapping vortex dynamics of two coupled side-by-side flexible plates submerged in the wake of a square cylinder

The flapping vortex dynamics of two flexible plates submerged side-by-side in the wake of a square cylinder are investigated through a two-way fluid–structure interaction (FSI) simulation. The gap between the two plates can stabilize wakes, lengthen vortex formation, elongate vortices, suppress vortex shedding, and decrease hydrodynamic forces. The numerical results indicate that the two flexible plates can exhibit four distinct modes of coupled motion: out-of-phase flapping, in-phase flapping, transition flapping, and decoupled flapping, depending on the gap spacing. Additionally, it is discovered that each of the four coupling modes has a unique pattern of vortex development. The findings of this study should proved valuable in the design of FSI-based piezoelectric energy harvesters utilizing cylinder–plate systems

Investigation of single drop particle scavenging using an ultrasonically levitated drop

Airborne particulate, known as aerosols, produced by both natural and anthropogenic means, have significant health and environmental impacts. Therefore understanding the produc-tion and removal of these particles is of critical importance. The main thrust of this thesis research is concerned with improving the understanding of removal of particulates via interaction with falling liquid drops, known as wet deposition. This process occurs naturally within rain and can be imposed in industrial applications with wet scrubbers. Therefore improved models for wet scavenging have applications in both climatology and pollution control. To perform this study, first the performance of existing models for wet deposition was investigated. Models for drop scavenging of aerosols via inertial impaction proposed by Slinn and by Calvert were compared with published experimental measurements. A parametric study was performed on the residual of the model predictions from the measurements to identify dimensionless groups not included in these models, which might increase model performance. The study found that two dimensionless groups, the relative Stokes number, Stkr and the drop Reynolds number Re, are both well correlated with the residual of these models. They are included in modified versions of both of these models to provide better performance. That these two dimensionless groups improve model performance suggests that an inertial mechanism and an advective mechanism not accounted for in the existing models play some role in aerosol scavenging in the inertial regime. These findings were experimentally investigated to identify more specifically these mecha-nisms. To do this, single drop particle scavenging was experimentally measured using an ultrasonic levitation technique. This technique enabled measurements of scavenging efficiency, E, for individ-ual drops, and allowed for control of drop axis ratio, α, drop shape oscillations, and Re independently from drop diameter. This allowed for more controlled manipulation of the drop wakes in both at-tached and vortex shedding regimes. Non-evaporating drops were used which resulted in essentially zero temperature and vapor concentration difference between the drop surface and the surrounding air, virtually eliminating the possibility of confounding phoretic effects. Plots of E versus Stokes number, Stk, were found to depend on α. These plots became independent of α when Stk was calculated using the Sauter mean diameter (as opposed to the equivolume diameter). Furthermore, E was shown to be insensitive to both Re and drop shape oscillations, suggesting that wake effects do not have a measurable impact on E. Finally, a method was developed to relate models of E for spherical drops (the assumed shape in existing scavenging model predictions) to E for arbitrarily deformed drops, such as those occurring in rain. Of note, these are the first measurements of droplet scavenging obtained using ultrasonic levitation. Finally, as drop scavenging is heavily dependent on particle size, a novel technique was identified and explored for improving aerosol sizing measurements. To do this, experiments were carried out in an impactor where the distance between the impactor nozzle and the impactor plate was small, much less than the typically used one nozzle diameter separation. The aerosol deposition patterns in this impactor were investigated for aerosols in the 3µm to 15µm diameter range. Ring-shaped deposition patterns were observed where the internal diameter and thickness of the rings were a function of the particle diameter. Specifically, the inner diameter and ring thickness were correlated to the Stokes number, Stk; the ring diameter decreased with Stk, and the ring thickness increased with Stk. At Stk ∼ 0.4 the ring closed up, leaving a mostly uniform disk deposition pattern. These ring patterns do not appear to correspond to patterns previously described in the literature, and an order of magnitude analysis shows that this is an inertially dominated process. Though this method was not used for particle sizing in this thesis research, it is possible that further development of this approach will result in a more advanced particle sizing tool for aerosol science research

Path and wake of cylinders falling in a liquid at rest or in a bubble swarm towards the hydrodynamical modeling of ebullated bed reactors

The origin of this PhD thesis lies in the study of Ebullated Bed Reactors (EBRs). These chemical reactors are very active research topics in chemical processes, notably thanks to their usage in heavy oil processing. Many complex phenomena take place within EBRs, and make their design and optimization difficult. In fluid mechanics, a lot of physical mechanisms present in EBRs are active fields of study (three-phase flow, fluid-body interaction...). Hence, in the present work, a study of the mechanisms participating in the hydrodynamics of an EBR with cylindrical catalysts is performed. In a first part, the impact of the catalyst anisotropy on its fall is investigated. In order to gain insight on the effect of the body anisotropy on its fall dynamics, we investigate experimentally the free fall of a solid cylinder in a fluid at rest. The sensitivity to two dimensionless parameters, the Archimedes number (Ar) and the aspect ratio of the cylinder (L/d) is examined. Experiments are conducted with two orthogonal cameras, and advanced image processing techniques are developed in order to measure the position and orientation of the cylinder in 3D. Within the range of parameters studied (200 < Ar < 1100, 2 < L/d < 20), the cylinders adopt different types of falling motion. Two main types of paths are observed, the first one is a rectilinear fall of the cylinder that keeps its axis horizontal, and the second one is a fluttering oscillatory motion. Other more complex types of motion are observed and discussed. The fluttering motion of the cylinder is analyzed in details. On top of the study of the body motion, the cylinder wake is also visualized and characterized. A large number of particles are present at the same time inside an EBRs (about 40% of the mass). Interactions between multiple objects have a strong impact on the motion of each individual particle, but are very complex. In a first approximation, we take into account the presence of numerous particles by introducing a confined medium. We study experimentally the fall of a single cylinder in a confined vertical thin-gap cell, where the cylinders are free to move in only two directions. The cylinder elongation ratio (3<L/d<40) and density ratio ( c / f = 1.16, 2.70, 4.50) are the two parameters of interest. The Archimedes number of the cylinder lies within the same range as in the unconfined medium, and the two main modes of motion of the cylinder are a rectilinear motion, and a fluttering one. However, for the same parameters (Ar,L/d), the motion of the cylinder in the confined cell is strongly different in form to that in the unconfined medium. We also studied the interaction between a freely falling cylinder and a rising swarm of bubbles. This investigation was performed experimentally, in the confined cell used in the second part. Cylinders of various density ratio ( c / f = 1.16, 2.70, 4.50) and elongation ratio (3<L/d<20) are released in a bubble swarm of gas volume fraction between 2% and 5%. The cylinder motion is greatly modified by the bubble swarm. Several mechanisms of interaction between the cylinder and the bubbles are identified (direct contact, interactions with fluid perturbations...), and their effect is characterized. We perform a statistical analysis of the cylinder motion in the swarm, and compare it to results in the confined fluid at rest. The cylinder density ratio and elongation ratio both play an important role in its motion in the bubble swarm. Conditional statistics allow us to further investigate the effect of the contact between the cylinder and a bubble, and of the cylinder orientation in the swarm. Finally, the dispersion of the cylinder motion in the swarm is characterized. A major effect of the bubble swarm is to increase, through bubble-cylinder contacts, the probability of the cylinder to be in nearly vertical orientations. This drastically changes the kinematics of the cylinder as compared to its motion in the fluid at res

Elastically-bounded flapping plates for flow-induced energy harvesting

This work concerns a novel concept for energy harvesting (EH) from fluid flows, based on the aeroelastic flutter of elastically-bounded plates immersed in laminar flow. The resulting flapping motions are investigated in order to support the development of centimetric-size EH devices exploiting low wind velocities, with potential application in the autonomous powering of low-power wireless sensor networks used, e.g., for remote environmental monitoring. The problem is studied combining three-dimensional direct numerical simulations exploiting a state-of-the-art immersed boundary method, wind-tunnel experiments on prototypal EH devices, and a reduced-order phenomenological model based on a set of ordinary differential equations. Three key features of the aeroelastic system are investigated: (i) we identify the critical condition for self-sustained flapping using a simple balance between characteristic timescales involved in the problem; (ii) we explore postcritical regimes characterized by regular limit-cycle oscillations, highlighting how to maximize their amplitude and/or frequency and in turns the potential energy extraction; (iii) we consider arrays of multiple devices, revealing for certain arrangements a constructive interference effect that leads to significant performance improvements. These findings lead to an improved characterization of the system and can be useful for the optimal design of EH devices. Moreover, we outline future research directions with the ultimate goal of realizing high-performance networks of numerous harvesters in real-world environmental conditions