Leaves falling in air and bubbles rising in water provide daily examples of nonstraight paths associated with the buoyancy-driven motion of a body in a fluid. Such paths are relevant to a large variety of applicative fields such as mechanical engineering, aerodynamics, meteorology, and the biomechanics of plants and insect flight. Although the problem has attracted attention for ages, it is only recently that the tremendous progress in the development of experimental and computational techniques and the emergence of new theoretical concepts have led to a better understanding of the underlying physical mechanisms. This review attempts to bring together the main recent experimental, computational, and theoretical advances obtained on this fascinating subject. To this end it describes the first steps of the transition in the wake of a fixed body and its connection with the onset and development of the path instability of moving bodies. Then it analyzes the kinematics and dynamics of various types of bodies along typical nonstraight paths and how the corresponding information can be used to build low-dimensional predictive models.

Wake-Induced Oscillatory Paths of Bodies Freely Rising or Falling in Fluids

This review article addresses the dynamics and control of low-frequency unsteadiness, as observed in some aerodynamic applications. It presents a coherent and rigorous linearized approach, which enables both to describe the dynamics of commonly encountered open-flows and to design open-loop and closed-loop control strategies, in view of suppressing or delaying instabilities. The approach is global in the sense that both cross-stream and streamwise directions are discretized in the evolution operator. New light will therefore be shed on the streamwise properties of open-flows. In the case of oscillator flows, the unsteadiness is due to the existence of unstable global modes, i.e., unstable eigenfunctions of the linearized Navier-Stokes operator. The influence of nonlinearities on the dynamics is studied by deriving nonlinear amplitude equations, which accurately describe the dynamics of the flow in the vicinity of the bifurcation threshold. These equations also enable us to analyze the mean flow induced by the nonlinearities as well as the stability properties of this flow. The open-loop control of unsteadiness is then studied by a sensitivity analysis of the eigenvalues with respect to base-flow modifications. With this approach, we manage to a priori identify regions of the flow where a small control cylinder suppresses unsteadiness. Then, a closed-loop control approach was implemented for the case of an unstable open-cavity flow. We have combined model reduction techniques and optimal control theory to stabilize the unstable eigenvalues. Various reduced-order-models based on global modes, proper orthogonal decomposition modes, and balanced modes were tested and evaluated according to their ability to reproduce the input-output behavior between the actuator and the sensor. Finally, we consider the case of noise-amplifiers, such as boundary-layer flows and jets, which are stable when viewed in a global framework. The importance of the singular value decomposition of the global resolvent will be highlighted in order to understand the frequency selection process in such flows. © 2010 by ASME.

/pdf/dynamics-and-control-of-global-instabilities-in-open-flows-a-2wer6a89l5.pdf

Dynamics and Control of Global Instabilities in Open-Flows: A Linearized Approach

In this paper, we study the effect of the Reynolds number (Re) on the dynamics and vortex formation modes of spheres rising or falling freely through a fluid (where Re = 100–15000). Since the oscillation of freely falling spheres was first reported by Newton (University of California Press, 3rd edn, 1726, translated in 1999), the fundamental question of whether a sphere will vibrate, as it rises or falls, has been the subject of a number of investigations, and it is clear that the mass ratio m* (defined as the relative density of the sphere compared to the fluid) is an important parameter to define when vibration occurs. Although all rising spheres (m* 1) always move without vibration. However, in contrast with previous studies, we discover that a whole regime of buoyant spheres rise through a fluid without vibration. It is only when one passes below a critical value of the mass ratio, that the sphere suddenly begins to vibrate periodically and vigorously in a zigzag trajectory within a vertical plane. The critical mass is nearly constant over two ranges of Reynolds number (m*crit ≈ 0.4 for Re = 260–1550 and m*crit ≈ 0.6 for Re > 1550). We do not observe helical or spiral trajectories, or indeed chaotic types of trajectory, unless the experiments are conducted in disturbed background fluid. The wakes for spheres moving rectilinearly are comparable with wakes of non-vibrating spheres. We find that these wakes comprise single-sided and double-sided periodic sequences of vortex rings, which we define as the ‘R’ and ‘2R’ modes. However, in the zigzag regime, we discover a new ‘4R’ mode, in which four vortex rings are created per cycle of oscillation. We find a number of changes to occur at a Reynolds number of 1550, and we suggest the possibility of a resonance between the shear layer instability and the vortex shedding (loop) instability. From this study, ensuring minimal background disturbances, we have been able to present a new regime map of dynamics and vortex wake modes as a function of the mass ratio and Reynolds number {m*, Re}, as well as a reasonable collapse of the drag measurements, as a function of Re, onto principally two curves, one for the vibrating regime and one for the rectilinear trajectories.

The effect of Reynolds number on the dynamics and wakes of freely rising and falling spheres

The turbulent wake past parallelepiped bodies with a rectangular blunt trailing edge of height H and width W is investigated in wall proximity: various aspect ratios H* = H/W ∈ [0.51, 1.63] and ground clearances C* = C/W ∈ [0, 1.00] are explored at a Reynolds number of 4.5 × 104 based on the body width W. Base pressure measurements and particle image velocimetry show that the close wake often undergoes antisymmetric instabilities that can be either in the lateral direction (parallel to the wall) or in the vertical direction (normal to the wall). The instantaneous wake presents preferred states with high degrees of asymmetry; in some configurations, topology shifts are observed after long time scales Tl ∼ 103W/U0 leading to bistable behaviors. The effect of ground proximity is thoroughly studied for H* = 0.74 corresponding to the reference Ahmed geometry and for a case with a height dimension larger than its width H* = 1/0.74 = 1.34. When C* > 0.08, it is found that the Ahmed body is bistable in the latera...

Bi-stability in the turbulent wake past parallelepiped bodies with various aspect ratios and wall effects

Fluid turbulence is commonly associated with stronger drag, greater heat transfer, and more efficient mixing than in laminar flows. In many natural and industrial settings, turbulent liquid flows c...

Bubbly and Buoyant Particle-Laden Turbulent Flows

We consider the generic problem of wake instabilities past fixed axisymmetric bodies, and focus on the extreme cases of a sphere and a flat disk. Numerical results reveal that the wakes of these two bodies evolve differently as the Reynolds number is increased. Especially, two new vortex shedding modes are identified behind a disk. To interpret these results, we introduce a model based on the theory of mode interactions in presence of O(2) symmetry. This model, which was initially developed for the Taylor-Couette system, allows us to explain the structural differences observed in the evolution of the two types of wakes and to accurately predict the evolution of the lift force.

/pdf/bifurcations-and-symmetry-breaking-in-the-wake-of-3glmwwgqi6.pdf

Bifurcations and symmetry breaking in the wake of axisymmetric bodies

We numerically investigate the dynamics of thin disks falling under gravity in a viscous fluid medium at rest at infinity. Varying independently the density and thickness of the disk reveals the influence of the disk aspect ratio which, contrary to previous belief, is found to be highly significant as it may completely change the route to non-vertical paths as well as the boundaries between the various path regimes. The transition from the straight vertical path to the planar fluttering regime is found to exhibit complex dynamics: a bistable behaviour of the system is detected within some parameter range and several intermediate regimes are observed in which, although the wake is unstable, the path barely deviates from vertical. By varying independently the body-to-fluid inertia ratio and the relative magnitude of inertial and viscous effects over a significant range, we set up a comprehensive map of the corresponding styles of path followed by an infinitely thin disk. We observe the four types of planar regimes already reported in experiments but also identify two additional fully three-dimensional regimes in which the body experiences a slow horizontal precession superimposed onto zigzagging or tumbling motions.

/pdf/falling-styles-of-disks-56gyeogqgl.pdf

Falling styles of disks

The dynamics of light spheres rising freely under buoyancy in a large expanse of viscous fluid at rest at infinity is investigated numerically. For this purpose, the computational approach developed by Mougin \& Magnaudet (Int. J. Multiphase Flow, 28:1837-1851, 2002) is improved to account for the instantaneous viscous loads induced by the translational and rotational sphere accelerations, which play a crucial role in the dynamics of very light spheres. A comprehensive map of the rise regimes encountered up to Reynolds numbers (based on the sphere diameter and mean rise velocity) of the order of $10^3$ is set up by varying independently the body-to-fluid density ratio and the relative magnitude of inertial and viscous effects in about $250$ distinct combinations. These computations confirm or reveal the presence of several distinct periodic regions on the route to chaos, most of which only exist within a finite range of the sphere relative density and Reynolds number. The wake structure is analyzed in these various regimes, evidencing the existence of markedly different shedding modes according to the style of path. The variations of the drag force with the flow parameters is also examined, revealing that only one of the styles of path specific to very light spheres yields a non-standard drag behaviour, with drag coefficients significantly larger than those measured on a fixed sphere under equivalent conditions. The outcomes of this investigation are compared with available experimental and numerical results, evidencing points of consensus and disagreement.

/pdf/path-oscillations-and-enhanced-drag-of-light-rising-spheres-dkmgm9ti2j.pdf

Path oscillations and enhanced drag of light rising spheres

A series of benchmarks based on the physical situation of "phase inversion" between two immiscible liquids is presented. These benchmarks aim at progressing toward the direct numerical simulation of two-phase flows. Several CFD codes developed in French laboratories and using either Volume-of-Fluid or Level-Set interface tracking methods are used to provide physical solutions of the benchmarks, convergence studies and code comparisons. Two typical configurations are retained, with integral scale Reynolds numbers of 13.700 and 433.000, respectively. The physics of the problem are probed through macroscopic quantities such as potential and kinetic energies, or enstrophy. In addition, scaling laws for the temporal decay of the kinetic energy are derived to check the physical relevance of the simulations. Finally the droplet size distribution is probed. Additional test problems are also reported to estimate the influence of viscous effects in the vicinity of the interface.

pdf/a-phase-inversion-benchmark-for-multiscale-multiphase-flows-5ak6848c4i.pdf

A phase inversion benchmark for multiscale multiphase flows

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Papers

Bifurcations and symmetry breaking in the wake of axisymmetric bodies

Falling styles of disks

Falling styles of disks

Path oscillations and enhanced drag of light rising spheres

A phase inversion benchmark for multiscale multiphase flows