Motion of a single bubble rising in a countercurrent flow in a Hele-Shaw cell

We investigate experimentally the motion of isolated bubbles rising in a vertical Hele-Shaw cell in the presence of a downward flow. The bubbles are strongly flattened in the plane of the cell, their equivalent diameter d being large compared to the gap of the cell e . Furthermore, their dynamics is strongly influenced by the confinement which imposes thin liquid films between the bubble and the walls and strongly attenuates the flow perturbation in the liquid due to wall friction

Oscillatory motion and wake of a bubble rising in a thin-gap cell

We investigate the characteristics of the oscillatory motion and wake of confined bubbles freely rising in a thin-gap cell (h=3.1 mm width). Once the diameter d of the bubble in the plane of the cell is known, the mean vertical velocity of the bubble Vb is proportional to the gravitational velocity [...], where g is the gravitational acceleration. This velocity is used to build the Reynolds number [...] that characterizes the flow induced by the bubble in the surrounding liquid (of kinematic viscosity ν), and which determines at leading order the mean deformation of the bubble given by the aspect ratio χ of the ellipse equivalent to the bubble contour. We then show that in the reference frame associated with the bubble (having a fixed origin and axes corresponding to the minor and major axes of the equivalent ellipse) the characteristics of its oscillatory motion in the plane of the cell display remarkable properties in the range 1200<Re<3000 and h/d<0.4. In particular, the velocity of the bubble presents along its path an almost constant component along its minor axis (fluctuations in time of approximately 5 %), given by Va/Vb≃0.92 for all Re. The dimensionless amplitude of oscillation of the angular velocity is also constant for all Re, rd/Vb≃0.75, while that of the transverse velocity of the bubble (along its major axis) is given by Vt/Vb≃0.32χ, reaching values comparable to those of the axial velocity Va for the most deformed bubbles (χ≈3). Furthermore, the frequency f of oscillation scales with the inertial time scale based on the transverse velocity of the bubble Vt, corresponding to a constant Strouhal number St=[...]. Using high-frequency particle image velocimetry, we investigate in detail the properties of the wake associated with the oscillatory motion of sufficiently confined bubbles. We observe that vortex shedding occurs for a maximal transverse velocity Vt of the bubble, corresponding to a maximal drift angle of the bubble. Furthermore, the measured vorticity of the vortex at detachment corresponds to the estimation [...] of the vorticity produced at the bubble surface. Three stages then emerge concerning the evolution in time of the wake generated by the bubble. For one to two periods of oscillation Tx following the release of a vortex, a rapid decay of the vorticity of the released vortex is observed. Meanwhile, the released vortex located initially at a distance of approximately one diameter from the bubble centre moves outwards from the bubble path and expands. At intermediate times, the vortex street undergoes vortex pairing. When viscous effects become predominant at a time of the order of the viscous time scale τν=[...], the vortex street becomes frozen and decays exponentially in place

Mouvement et sillage de bulles isolées ou en interaction confinées entre deux plaques

Ce travail de recherche s'intéresse à la dynamique de bulles en ascension à grand nombre de Reynolds dans un liquide fortement confiné entre deux plaques (cellule de Hele-Shaw). Dans le régime étudié, les trajectoires des bulles et leurs déformations sont contenues dans le plan de la cellule. La dynamique essentiellement bidimensionnelle favorise en particulier l'observation d'interactions entre bulles. Cette étude expérimentale comprend donc deux volets : l'analyse de la dynamique d'une bulle isolée et de son sillage et celle des interactions hydrodynamiques entre deux bulles. La cinématique des bulles (forme, trajectoire et vitesse) est mesurée à partir de visualisations par ombroscopie sur une large gamme de tailles caractérisées par un diamètre équivalent dans le plan, noté d. La dynamique des sillages est quant à elle étudiée par Vélocimétrie par Image de Particules (PIV) à Haute Fréquence. Concernant l'étude de la bulle isolée, nous avons exploré la situation où les bulles montent dans un liquide au repos et celle où elles sont soumises à un écoulement descendant à contre-courant. En liquide au repos, pour des bulles de taille suffisante qui ne sont pas mobiles dans l'interstice d'épaisseur e nous avons montré que la vitesse moyenne d'ascension Vb est proportionnelle à (e/d)⅙ √gd, et que le nombre de Reynolds défini par Re=Vbd/v fixe la déformation des bulles. De plus des lois d'échelle simples sont obtenues dans la gamme 1200≤Re≤3000 et e/d ≤ 0,4 pour les grandeurs décrivant les oscillations de trajectoire dans le repère de la bulle. Par ailleurs, des mesures de vitesse nous ont permis de caractériser la structure du sillage associé aux oscillations de trajectoire de la bulle. Nous avons tout d'abord étudié en détail les caractéristiques du détachement tourbillonnaire. Ces mesures de vitesse dans les sillages ont également mis en évidence l'existence de deux dynamiques distinctes sur deux échelles de temps nettement séparées : la période d'oscillation de la bulle et le temps visqueux défini à partir de e. En écoulement à contre-courant, un résultat intéressant consiste en la disparition de la phase intermédiaire d'appariement tourbillonnaire dans l'allée de von Karman de bulles oscillantes pour la plus importante des vitesses du contre-écoulement. La caractérisation de la cinématique des bulles isolées et des perturbations de vitesse qu'elles induisent dans le liquide a permis d'aboutir à des lois d'échelle suffisamment robustes pour pouvoir prédire leur comportement instationnaire simplement à partir de leur taille. Cette connaissance s'avère cruciale dans l'analyse des interactions entre deux bulles pour explorer les écarts de leur comportement cinématique par rapport au cas isolé. Les expériences d'interaction entre deux bulles consistent à injecter deux bulles successives et à observer leur mouvement ainsi que celui qu'elles induisent dans la phase liquide. Le suivi des bulles par ombroscopie permet de distinguer plusieurs modes d'interaction entre les bulles : attraction horizontale, entrainement vertical, éjection du sillage ou rebond, contournement, positionnement préférentiel et coalescence. Certains mécanismes d'interaction ont été plus spécifiquement étudiés à l'aide de mesures par vélocimétrie. Nous avons ainsi pu quantifier l'effet du sillage de la première bulle sur la deuxième, et notamment caractériser l'interaction bulle - tourbillon. ABSTRACT : We study the dynamics of bubbles rising in a liquid confined in a thin-gap cell (Hele-Shaw cell of thickness e). In the regime investigated corresponding to high Reynolds numbers, bubble paths and deformations occur in the plane of the cell. This two-dimensional dynamics facilitates the observation of bubbles interaction. The aim of the investigation is twofold: the analysis of the coupling between the motion of an isolated oscillating bubble and its wake, and the analysis of the hydrodynamical interactions between two bubbles. Bubble motions (shape, trajectory and velocity) are measured from visualizations using shadowgraphy for a large range of bubble sizes characterized by their in-plane equivalent diameter d. The behaviour of the wake is explored using High Frequency Particle Image Velocimetry (HF PIV). We investigated the kinematics of an isolated bubble when its size d increases. We showed that the mean vertical velocity of the bubble Vb is proportional to (e/d)⅙ √gd, and that the Reynolds number Re=Vbd/v determines its mean deformation. Simple scaling laws were then obtained in the range 1200≤Re≤3000 and e/d ≤ 0,4 for all the quantities describing the path oscillations of the bubble in its reference frame. Moreover, measurements of the liquid velocity allowed us to characterize the structure of the wake associated to the oscillating bubbles. We first investigated in detail the characteristics of vortex shedding. We then showed that the time evolution of the bubble wake depends on two contrasted time scales. The first corresponds to short times on the order of the period of oscillation and the second to the effect of wall friction becoming predominant for times comparable to the viscous time scale based on the gap thickness e. In the presence of a sufficiently strong counterflow, we observed the disappearance of the intermediate phase of vortex pairing in the wake of an isolated oscillating bubble. The characterization of the bubble kinematics and of the bubble-induced velocity perturbation in the liquid phase for the isolated bubble provided scaling laws robust enough to predict their periodic motion. This knowledge is fundamental for the discussion of hydrodynamical interactions, allowing us to discuss the kinematics of interacting bubbles as compared to their kinematics as isolated bubbles. Experiments consisted in the injection of two successive bubbles in the cell, the observation of their motions and the measurement of the perturbations induced in the liquid phase. Visualizations of the bubbles motions allowed us to observe several types of interactions: horizontal attraction, vertical entrainment, ejection or bouncing, preferential positioning, and coalescence. Some mechanisms occurring during interaction have been more precisely studied using HF PIV, in particular the effect of the wake of the leading bubble on the trailing bubble, and the associated bubble-vortex interaction

Steady and unsteady shear flows of a viscoplastic fluid in a cylindrical Couette cell

Abstract Yield stress fluid flows in Couette cells have been widely studied in the last decades for their intriguingly exhibiting phenomena. In this paper, we use a {PIV} technique to investigate the axisymmetric flow and rheological properties of a Carbopol gel in a relatively wide cylindrical Couette device. Carbopol gel is known to exhibit viscoplastic behavior and is often described using a Herschel–Bulkley law, which is characterized by a plastic yield stress τ y and a shear-dependent nonlinear viscosity. In some cases, the elasticity of the material has to be accounted for to understand the whole dynamics of the system, in particular for unsteady flows as observed in the present study. Two set of experiments are conducted here in order to highlight these different rheological behaviors and the resulting dynamics: (i) a steady shear configuration and (ii) an unsteady shear configuration, in which the angular velocity of the inner cylinder is either constant or time dependent ( sin profile), respectively. In the steady configuration, a simple optimization model, based on the Herschel–Bulkley law, is developed to extract the rheological parameters of the viscoplastic contribution of the gel from the steady velocity fields. Results are shown to be in good agreements with rheological parameters obtained from a standard rheometer. On the other hand, the elastic contribution of the material is highlighted in the unsteady shear configuration, for which a spatio-temporal transition between solid-elastic and fluid behaviors is observed. Different models are proposed to describe the dynamics of the unsteady flow. First, quasi-steady state models allow to predict both the fluid shear zone close to the inner cylinder and the elastic deformation of the material as long as their contributions can be decoupled in space and in time. For more complex dynamics, i.e. when the flow becomes strongly unsteady, an elasto-viscoplastic model is developed to describe the flow dynamics. It is shown to quantitatively reproduce the experimental measurements. Finally, an elastic wave model is derived to describe an elastic front propagating from the inner cylinder to the outer one, and observed at every half forcing period. The front velocity is thus shown to scale on the phase velocity of an elastic wave in a deformable solid

Interaction of two oscillating bubbles rising in a thin gap

We investigate experimentally different mechanisms of hydrodynamic interaction between two oscillating bubbles rising in a liquid at rest confined in a thin gap cell. In order to understand the relation between the motion of a single bubble and its wake as well as the hydrodynamic interaction between two oscillating bubbles, we characterize the kinematics of the bubbles interaction using high speed cameras and we measure the associated liquid velocity by High-Frequency PIV. The motion of the trailing bubble is modified while passing through the unstable wake of the leading bubble: depending on the relative size of the bubbles and their relative position, we can observe horizontal attraction to the center of the leading bubble's wake, vertical entrainment, ejection of the second bubble by a vortex of the wake or a sequential combination of the three mechanisms

Mouvement, déformation et sillage de bulles oscillantes confinées entre deux plaques

Nous présentons une étude expérimentale sur le mouvement et le sillage d'une bulle oscillante en ascension à grand nombre de Reynolds dans un liquide fortement confiné entre deux plaques. Les différentes lois d'échelles obtenues pour la vitesse, la déformation et les caractéristiques du mouvement oscillatoire (fréquences, amplitudes...) des bulles sur une assez large gamme correspondant à des nombres de Reynolds compris entre 1 ; et 3 ; seront exposées. Puis nous présenterons l'évolution du sillage des bulles notamment à travers l'analyse de la trajectoire et de l'intensité des tourbillons générés derrière les bulles

Interaction of two oscillating bubbles rising in a thin-gap cell: vertical entrainment and interaction with vortices

We present an exploratory study of the hydrodynamical interaction between two bubbles rising at high Reynolds numbers in a thin-gap cell. When they are isolated, the bubbles exhibit oscillatory motions and develop an unsteady wake with periodic release of vortices. Experiments combine bubble tracking and measurements of the liquid velocity field through volumetric time-resolved particle image velocimetry. This enabled us to analyse the kinematics of the bubbles during their interaction in relationship with the liquid flow field induced by their motion and governing their behaviour. We first investigate how the kinematics of a bubble, already submitted to the intrinsic instability of its path and wake, is modified by the interaction, i.e. by the presence of a liquid flow field generated by the companion bubble. Two main effects are highlighted in association with (i) the role of the ascending flow generated by the leading bubble, and of its spatial evolution, leading to a slowly varying vertical entrainment of the trailing bubble, and (ii) the role of the vortices released by the leading bubble inducing strong localized horizontal deviations on a bubble in line or in oblique positioning. In the latter case, two major scenarios are identified: deviations of the trailing bubble towards the wake centre line (centring in the wake) or away from it (ejection from the wake). We also show that a regular succession of ejections and re-alignments events may take place (cyclic alternation of ejections and centrings). The analysis is built on the knowledge of the behaviour of isolated bubbles, which is used as the basis for comparison to characterize the effect of the interaction, for the modelling of the vertical entrainment, and for the definition of a criteria on a dimensionless parameter characterizing the ability of a vortex to drive the bubble motion. In turn, we investigate the effect of a bubble passage in the liquid flow field generated by the companion bubble, highlighting the destruction or reinforcement of vortices. We show in particular that both effects can occur without a significant impact on the bubble kinematics

Model of collective fish behavior with hydrodynamic interactions

Fish schooling is often modeled with self-propelled particles subject to phenomenological behavioral rules. Although fish are known to sense and exploit flow features, these models usually neglect hydrodynamics. Here, we propose a novel model that couples behavioral rules with far-field hydrodynamic interactions. We show that (1) a new "collective turning" phase emerges, (2) on average, individuals swim faster thanks to the fluid, and (3) the flow enhances behavioral noise. The results of this model suggest that hydrodynamic effects should be considered to fully understand the collective dynamics of fish

Motion and wake of isolated or interacting bubbles rising in a thin gap cell

Ce travail de recherche s'intéresse à la dynamique de bulles en ascension à grand nombre de Reynolds dans un liquide fortement confiné entre deux plaques (cellule de Hele-Shaw). Dans le régime étudié, les trajectoires des bulles et leurs déformations sont contenues dans le plan de la cellule. La dynamique essentiellement bidimensionnelle favorise en particulier l'observation d'interactions entre bulles. Cette étude expérimentale comprend donc deux volets : l'analyse de la dynamique d'une bulle isolée et de son sillage et celle des interactions hydrodynamiques entre deux bulles. La cinématique des bulles (forme, trajectoire et vitesse) est mesurée à partir de visualisations par ombroscopie sur une large gamme de tailles caractérisées par un diamètre équivalent dans le plan, noté d. La dynamique des sillages est quant à elle étudiée par Vélocimétrie par Image de Particules (PIV) à Haute Fréquence. Concernant l'étude de la bulle isolée, nous avons exploré la situation où les bulles montent dans un liquide au repos et celle où elles sont soumises à un écoulement descendant à contre-courant. En liquide au repos, pour des bulles de taille suffisante qui ne sont pas mobiles dans l'interstice d'épaisseur e nous avons montré que la vitesse moyenne d'ascension Vb est proportionnelle à (e/d)⅙ √gd, et que le nombre de Reynolds défini par Re=Vbd/v fixe la déformation des bulles. De plus des lois d'échelle simples sont obtenues dans la gamme 1200≤Re≤3000 et e/d ≤ 0,4 pour les grandeurs décrivant les oscillations de trajectoire dans le repère de la bulle. Par ailleurs, des mesures de vitesse nous ont permis de caractériser la structure du sillage associé aux oscillations de trajectoire de la bulle. Nous avons tout d'abord étudié en détail les caractéristiques du détachement tourbillonnaire. Ces mesures de vitesse dans les sillages ont également mis en évidence l'existence de deux dynamiques distinctes sur deux échelles de temps nettement séparées : la période d'oscillation de la bulle et le temps visqueux défini à partir de e. En écoulement à contre-courant, un résultat intéressant consiste en la disparition de la phase intermédiaire d'appariement tourbillonnaire dans l'allée de von Karman de bulles oscillantes pour la plus importante des vitesses du contre-écoulement. La caractérisation de la cinématique des bulles isolées et des perturbations de vitesse qu'elles induisent dans le liquide a permis d'aboutir à des lois d'échelle suffisamment robustes pour pouvoir prédire leur comportement instationnaire simplement à partir de leur taille. Cette connaissance s'avère cruciale dans l'analyse des interactions entre deux bulles pour explorer les écarts de leur comportement cinématique par rapport au cas isolé. Les expériences d'interaction entre deux bulles consistent à injecter deux bulles successives et à observer leur mouvement ainsi que celui qu'elles induisent dans la phase liquide. Le suivi des bulles par ombroscopie permet de distinguer plusieurs modes d'interaction entre les bulles : attraction horizontale, entrainement vertical, éjection du sillage ou rebond, contournement, positionnement préférentiel et coalescence. Certains mécanismes d'interaction ont été plus spécifiquement étudiés à l'aide de mesures par vélocimétrie. Nous avons ainsi pu quantifier l'effet du sillage de la première bulle sur la deuxième, et notamment caractériser l'interaction bulle - tourbillon.We study the dynamics of bubbles rising in a liquid confined in a thin-gap cell (Hele-Shaw cell of thickness e). In the regime investigated corresponding to high Reynolds numbers, bubble paths and deformations occur in the plane of the cell. This two-dimensional dynamics facilitates the observation of bubbles interaction. The aim of the investigation is twofold: the analysis of the coupling between the motion of an isolated oscillating bubble and its wake, and the analysis of the hydrodynamical interactions between two bubbles. Bubble motions (shape, trajectory and velocity) are measured from visualizations using shadowgraphy for a large range of bubble sizes characterized by their in-plane equivalent diameter d. The behaviour of the wake is explored using High Frequency Particle Image Velocimetry (HF PIV). We investigated the kinematics of an isolated bubble when its size d increases. We showed that the mean vertical velocity of the bubble Vb is proportional to (e/d)⅙ √gd, and that the Reynolds number Re=Vbd/v determines its mean deformation. Simple scaling laws were then obtained in the range 1200≤Re≤3000 and e/d ≤ 0,4 for all the quantities describing the path oscillations of the bubble in its reference frame. Moreover, measurements of the liquid velocity allowed us to characterize the structure of the wake associated to the oscillating bubbles. We first investigated in detail the characteristics of vortex shedding. We then showed that the time evolution of the bubble wake depends on two contrasted time scales. The first corresponds to short times on the order of the period of oscillation and the second to the effect of wall friction becoming predominant for times comparable to the viscous time scale based on the gap thickness e. In the presence of a sufficiently strong counterflow, we observed the disappearance of the intermediate phase of vortex pairing in the wake of an isolated oscillating bubble. The characterization of the bubble kinematics and of the bubble-induced velocity perturbation in the liquid phase for the isolated bubble provided scaling laws robust enough to predict their periodic motion. This knowledge is fundamental for the discussion of hydrodynamical interactions, allowing us to discuss the kinematics of interacting bubbles as compared to their kinematics as isolated bubbles. Experiments consisted in the injection of two successive bubbles in the cell, the observation of their motions and the measurement of the perturbations induced in the liquid phase. Visualizations of the bubbles motions allowed us to observe several types of interactions: horizontal attraction, vertical entrainment, ejection or bouncing, preferential positioning, and coalescence. Some mechanisms occurring during interaction have been more precisely studied using HF PIV, in particular the effect of the wake of the leading bubble on the trailing bubble, and the associated bubble-vortex interaction

Dynamique d'une bulle en ascension dans un écoulement à contre courant en cellule de Hele-Shaw

Nous présentons une étude expérimentale de la dynamique de bulles isolées en ascension dans un contre-écoulement dans une cellule de Hele-Shaw verticale