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We performed numerical simulations of a homogeneous swarm of bubbles rising at large Reynolds number, $Re=760$ , with volume fractions ranging from 1 % to 10 %. We consider a simplified model in which the interfaces are not resolved, but which allows us to simulate flows with a large number of bubbles and to emphasize the interactions between bubble wakes. The liquid phase is described by solving, on an Eulerian grid, the Navier–Stokes equations, including sources of momentum which model the effect of the bubbles. The dynamics of each bubble is determined within the Lagrangian framework by solving an equation of motion involving the hydrodynamic forces exerted by the fluid accounting for the correction of the fictitious self-interaction of a bubble with its own wake. The comparison with experiments shows that this coarse-grained simulations approach can reliably describe the dynamics of the resolved flow scales. We use conditional averaging to characterize the mean bubble wakes and obtain in particular the typical shear imposed by the rising bubbles. On the basis of the spectral decomposition of the energy budget, we observe that the flow is dominated by production at large scales and by dissipation at small scales and we rule out the presence of an intermediate range in which the production and dissipation are locally in balance. We propose that the $k^{-3}$ subrange of the energy spectra results from the mean shear rate imposed by the bubbles, which controls the rate of return to isotropy. [ABSTRACT FROM AUTHOR]

A stochastic model is presented for the probability density function (p.d.f.) of the liquid velocity fluctuations generated by high-Reynolds-number rising bubbles. It considers three elementary sources of fluctuations: the potential flow disturbance around each bubble; the average bubble wakes, which are assumed to decay exponentially; and the turbulent agitation resulting from the flow instability, which is assumed to be isotropic, homogeneously distributed all over the flow and statistically independent of the two others. The model reproduces well and explains the characteristics of the experimental p.d.f.s: exponential tails, asymmetry of vertical fluctuations and evolution with the gas volume fraction. The model involves two a priori unknown parameters: the volume of the wake and the velocity scale of the turbulent agitation. Because some parts of the probability functions depend only on a single contribution, these two parameters can be uniquely and independently determined from experimental p.d.f.s. This defines an objective method to separate the various kinds of fluctuations and allows one to determine the contribution of each of them to the total agitation. [ABSTRACT FROM AUTHOR]

This work reports an experimental investigation of the dispersion of a low-diffusive dye within a homogeneous swarm of high-Reynolds-number rising bubbles at gas volume fractions ${\it\alpha}$ ranging from 1 % to 13 %. The capture and transport of dye within bubble wakes is found to be negligible and the mixing turns out to result from the bubble-induced turbulence. It is described well by a regular diffusion process. The diffusion coefficient corresponding to the vertical direction is larger than that corresponding to the horizontal direction, owing to the larger intensity of the liquid fluctuations in the vertical direction. Two regimes of diffusion have been identified. At low gas volume fraction, the diffusion time scale is given by the correlation time of the bubble-induced turbulence and the diffusion coefficients increase roughly as ${\it\alpha}^{0.4}$. At large gas volume fraction, the diffusion time scale is imposed by the time interval between two bubbles and the diffusion coefficients become almost independent of ${\it\alpha}$. The transition between the two regimes occurs sooner in the horizontal direction ($1\,\%\leqslant {\it\alpha}\leqslant 3\,\%$) than in the vertical direction ($3\,\%\leqslant {\it\alpha}\leqslant 6\,\%$). Physical models based on the hydrodynamic properties of the bubble swarm are introduced and guidelines for practical applications are suggested. [ABSTRACT FROM AUTHOR]

The present work focuses on the collective effect on both bubble dynamics and mass transfer in a dense homogeneous bubble swarm for gas volume fractions α up to 30%. The experimental investigation is carried out with air bubbles rising in a square column filled with water. Bubble size and shape are determined by means of a high-speed camera equipped with a telecentric lens. Gas volume fraction and bubble velocity are measured by using a dual-tip optical probe. The combination of these two techniques allows us to determine the interfacial area between the gas and the liquid. The transfer of oxygen from the bubbles to the water is measured from the time evolution of the concentration of oxygen dissolved in water, which is obtained by means of the gassing-out method. Concerning the bubble dynamics, the average vertical velocity is observed to decrease with α in agreement with previous experimental and numerical investigations, while the bubble agitation turns out to be weakly dependent on α. Concerning mass transfer, the Sherwood number is found to be very close to that of a single bubble rising at the same Reynolds number, provided the latter is based on the average vertical bubble velocity, which accounts for the effect of the gas volume fraction on the bubble rise velocity. This conclusion is valid for situations where the diffusion coefficient of the gas in the liquid is very low (high Péclet number) and the dissolved gas is well mixed at the scale of the bubble. It is understood by considering that the transfer occurs at the front part of the bubbles through a diffusion layer which is very thin compared with all flow length scales and where the flow remains similar to that of a single rising bubble. [ABSTRACT FROM AUTHOR]

Navier–Stokes simulations of the agitation generated by a homogeneous swarm of high-Reynolds-number rising bubbles are performed. The bubbles are modelled by fixed momentum sources of finite size randomly distributed in a uniform flow. The mesh grid is regular with a spacing close to the bubble size. This allows us to simulate a swarm of a few thousand bubbles in a computational domain of a hundred bubble diameters, which corresponds to a gas volume fraction $\alpha $ from 0.6 % to 4 %. The small-scale disturbances close to the bubbles are not resolved but the wakes are correctly described from a distance of a few diameters. This simple model reproduces well all the statistical properties of the vertical velocity fluctuations measured in previous experiments: scaling as ${\alpha }^{0. 4} $, self-similar probability density functions and power spectral density including a subrange evolving as the power $- 3$ of the wavenumber $k$. It can therefore be concluded that bubble-induced agitation mainly results from wake interactions. Considering the flow in a frame that is fixed relative to the bubbles, the combined use of both time and spatial averaging makes it possible to distinguish two contributions to the liquid fluctuations. The first is the spatial fluctuations that are the consequence of the bubble mean wakes. The second corresponds to the temporal fluctuations that are the result of the development of a flow instability. Note that the latter is not due to the destabilization of individual bubble wakes, since a computation with a single bubble leads to a steady flow. It is a collective instability of the randomly distributed bubble wakes. The spectrum of the time fluctuations shows a peak around a frequency ${f}_{cwi} $, which is independent of $\alpha $. From the present results it is possible to determine the origin of the overall properties of the total fluctuations observed in the experiments. The scaling of the velocity fluctuation as ${\alpha }^{0. 4} $ is a combination of the scalings of the spatial and temporal fluctuations, which are different from each other. As the time fluctuations are symmetric in the vertical direction, the asymmetry of the probability density function of the vertical velocity comes from that of the spatial fluctuations. Both contributions exhibit a ${k}^{- 3} $ spectral behaviour around the same range of wavenumbers, which explains why it is observed regardless of the nature of the dominant contribution. [ABSTRACT FROM PUBLISHER]

We report an experimental analysis of path and shape oscillations of an air bubble of diameter $d$ rising in water at high Reynolds number $\mathit{Re}$ in a vertical Hele-Shaw cell of width $h$. Liquid velocity perturbations induced by the relative movement have also been investigated to analyse the coupling between the bubble motion and the wake dynamics. The confinement ratio $h/ d$ is less than unity so that the bubble is flattened between the walls of the cell. As the bubble diameter is increased, the Archimedes and the Bond numbers increase within the ranges $10\leq \mathit{Ar}\leq 1{0}^{4} $ and $6\ensuremath{\times} 1{0}^{\ensuremath{-} 3} \leq Bo\leq 140$. Mean shapes become more and more elongated. They first evolve from in-plane circles to ellipses, then to complicated shapes without fore–aft symmetry and finally to semi-circular-capped bubbles. The scaling law $\mathit{Re}= 0. 5\mathit{Ar}$ is valid for a large range of $\mathit{Ar}$, however, indicating that the liquid films between the bubble and the walls do not contribute significantly to the drag force exerted on the bubble. The coupling between wake dynamics, bubble path and shape oscillations evolves and a succession of different regimes of oscillations is observed. The rectilinear bubble motion becomes unstable from a critical value ${\mathit{Ar}}_{1} $ through an Hopf bifurcation while the bubble shape is still circular. The amplitude of path oscillations first grows as $\mathit{Ar}$ increases above ${\mathit{Ar}}_{1} $ but then surprisingly decreases beyond a second Archimedes number ${\mathit{Ar}}_{2} $. This phenomenon, observed for steady ellipsoidal shape with moderate eccentricity, can be explained by the rapid attenuation of bubble wakes caused by the confinement. Shape oscillations around a significantly elongated mean shape start for $\mathit{Ar}\geq {\mathit{Ar}}_{3} $. The wake structure progressively evolves due to changes in the bubble shape. After the break-up of the fore–aft symmetry, a fourth regime involving complicated shape oscillations is then observed for $\mathit{Ar}\geq {\mathit{Ar}}_{4} $. Vortex shedding disappears and unsteady attached vortices coupled to shape oscillations trigger path oscillations of moderate amplitude. Path and shape oscillations finally decrease when $\mathit{Ar}$ is further increased. For $\mathit{Ar}\geq {\mathit{Ar}}_{5} $, capped bubbles followed by a steady wake rise on a straight path. [ABSTRACT FROM PUBLISHER]

The spatial distribution, the velocity statistics and the dispersion of the gas phase have been investigated experimentally in a homogeneous swarm of bubbles confined within a thin gap. In the considered flow regime, the bubbles rise on oscillatory paths while keeping a constant shape. They are followed by unstable wakes which are strongly attenuated due to wall friction. According to the direction that is considered, the physical mechanisms are totally different. In the vertical direction, the entrainment by the wakes controls the bubble agitation, causing the velocity variance and the dispersion coefficient to increase almost linearly with the gas volume fraction. In the horizontal direction, path oscillations are the major cause of bubble agitation, leading to a constant velocity variance. The horizontal dispersion, which is lower than that in the vertical direction, is again observed to increase almost linearly with the gas volume fraction. It is however not directly due to regular path oscillations, which are unable to generate a net deviation over a whole period, but results from bubble interactions which cause a loss of the bubble velocity time correlation. [ABSTRACT FROM PUBLISHER]

Inertial shape oscillations of heptane drops rising in water are investigated experimentally. Diameters from 0.59 to 3.52 mm are considered, corresponding to a regime where the rising motion should not affect shape oscillations for pure immiscible fluids. The interface, however, turns out to be contaminated. The drag coefficient is considerably increased compared to that of a clean drop due to the well-known Marangoni effect resulting from a gradient of surfactant concentration generated by the fluid motion along the interface. Thanks to the decomposition of the shape into spherical harmonics, the eigenfrequencies and the damping rates of oscillation modes $n= 2$, 3, 4 and 5 have been measured. Frequencies are not affected by contamination, while damping rates are increased by a considerable amount that depends neither on drop instantaneous velocity nor on diameter. This augmentation, however, depends on the mode number: it is maximum for mode two (multiplied by 2.4) and then relaxes towards the value of a clean drop as $n$ increases. A previous similar investigation of a drop attached to a capillary has not revealed such an increase of the damping rates, indicating that the coupling between rising motion and surface contamination is responsible for this effect. [ABSTRACT FROM AUTHOR]

An experimental investigation of the flow generated by a homogeneous population of bubbles rising in water is reported for three different bubble diameters (d =1.6, 2.1 and 2.5 mm) and moderate gas volume fractions (0.005⩽a⩽0.1). The Reynolds numbers, Re=V0d/?, based on the rise velocity V0 of a single bubble range between 500 and 800. Velocity statistics of both the bubbles and the liquid phase are determined within the homogeneous bubble swarm by means of optical probes and laser Doppler anemometry. Also, the decaying agitation that takes place in the liquid just after the passage of the bubble swarm is investigated from high-speed particle image velocimetry measurements. Concerning the bubbles, the average velocity is found to evolve as V0α-0.1 whereas the velocity fluctuations are observed to be almost independent of α. Concerning the liquid fluctuations, the probability density functions adopt a self-similar behaviour when the gas volume fraction is varied, the characteristic velocity scaling as V0α0.4. The spectra of horizontal and vertical liquid velocity fluctuations are obtained with a resolution of 0.6 mm. The integral length scale Λ is found to be proportional to V0² /g or equivalently to d/Cd0, where g is the gravity acceleration and Cd0 the drag coefficient of a single rising bubble. Normalized by using Λ, the spectra are independent on both the bubble diameter and the volume fraction. At large scales, the spectral energy density evolves as the power -3 of the wavenumber. This range starts approximately from Λ and is followed for scales smaller than Λ/4 by a classic -5/3 power law. Although the Kolmogorov microscale is smaller than the measurement resolution, the dissipation rate is however obtained from the decay of the kinetic energy after the passage of the bubbles. It is found to scale as α0.9V0³ /Λ. The major characteristics of the agitation are thus expressed as functions of the characteristics of a single rising bubble. Altogether, these results provide a rather complete description of the bubble-induced turbulence. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]