Interaction between two spherical bubbles rising in a viscous liquid

The three-dimensional flow around two spherical bubbles moving in a viscous fluid is studied numerically by solving the full Navier-Stokes equations. The study considers the interaction between two bubbles for moderate Reynolds numbers (50 ≤ Re ≤ 500, Re being based on the bubble diameter) and for positions described by the separation S (2.5 ≤ S ≤ 10, S being the distance between the bubble centres normalized by the bubble radius) and the angle θ (0o ≤ θ ≤ 90o ) formed between the line of centre and the direction perpendicular to the direction of the motion. We provide a general description of the interaction extending the results obtained for two bubbles moving side by side (θ = 0o ) by Legendre, Magnaudet & Mougin 2003 (J. Fluid Mech., 497,133-166) and for two bubbles moving in line (θ = 90o ) by Yuan & Prosperetti 1994 (J. Fluid Mech., 278, 325-349). Simple models based on physical arguments are given for the drag and lift forces experienced by each bubble. The interaction is the combination of three effects: a potential effect, a viscous correction (Moore correction) and a significant wake effect observed on both the drag and the transverse force of the second bubble when located in the wake of the first one

A numerical investigation of horizontal viscous gravity currents

We study numerically the viscous phase of horizontal gravity currents of immiscible fluids in the lock-exchange configuration. A numerical technique capable of dealing with stiff density gradients is used, allowing us to mimic high-Schmidt-number situations similar to those encountered in most laboratory experiments. Plane two-dimensional computations with no-slip boundary conditions are run so as to compare numerical predictions with the ‘short reservoir’ solution of Huppert (J. Fluid Mech., vol. 121, 1982, pp. 43–58), which predicts the front position lf to evolve as t1/5, and the ‘long reservoir’ solution of Gratton & Minotti (J. Fluid Mech., vol. 210, 1990, pp. 155–182) which predicts a diffusive evolution of the distance travelled by the front xf ~ t1/2. In line with dimensional arguments, computations indicate that the self-similar power law governing the front position is selected by the flow Reynolds number and the initial volume of the released heavy fluid. We derive and validate a criterion predicting which type of viscous regime immediately succeeds the slumping phase. The computations also reveal that, under certain conditions, two different viscous regimes may appear successively during the life of a given current. Effects of sidewalls are examined through three-dimensional computations and are found to affect the transition time between the slumping phase and the viscous regime. In the various situations we consider, we make use of a force balance to estimate the transition time at which the viscous regime sets in and show that the corresponding prediction compares well with the computational results

Effects of channel geometry on buoyancy-driven mixing

The evolution of the concentration and flow fields resulting from the gravitational mixing of two interpenetrating miscible fluids placed in a tilted tube or channel is studied by using direct numerical simulation. Three-dimensional (3D) geometries, including a cylindrical tube and a square channel, are considered as well as a purely two-dimensional (2D) channel. Striking differences between the 2D and 3D geometries are observed during the long-time evolution of the flow. We show that these differences are due to those existing between the 2D and 3D dynamics of the vorticity field. More precisely, in two dimensions, the strong coherence and long persistence of vortices enable them to periodically cut the channels of pure fluid that feed the front. In contrast, in 3D geometries, the weaker coherence of the vortical motions makes the segregational effect due to the transverse component of buoyancy strong enough to preserve a fluid channel near the front of each current. This results in three different regimes for the front velocity (depending on the tilt angle), which is in agreement with the results of a recent experimental investigation. The evolution of the front topology and the relation between the front velocity and the concentration jump across the front are investigated in planar and cylindrical geometries and highlight the differences between 2D and 3D mixing dynamics

Turbulence-induced secondary motion in a buoyancy-driven flow in a circular pipe.

We analyze the results of a direct numerical simulation of the turbulent buoyancy-driven flow that sets in after two miscible fluids of slightly different densities have been initially superimposed in an unstable configuration in an inclined circular pipe closed at both ends. In the central region located midway between the end walls, where the flow is fully developed, the resulting mean flow is found to exhibit nonzero secondary velocity components in the tube cross section. We present a detailed analysis of the generation mechanism of this secondary flow which turns out to be due to the combined effect of the lateral wall and the shear-induced anisotropy between the transverse components of the turbulent velocity fluctuations

Analytical and numerical computations of the van der Waals force in complex geometries: Application to the filtration of colloidal particles.

Particle capture during the filtration of colloidal dispersions depends on a complex balance between repulsive forces, such as hydrodynamic or electrostatic effects, and attractive forces, amongst them the van der Waals interaction forces. Satisfactory expressions for the latter are thus required in complex geometries. Exact expressions for the geometrical factor involved in the van der Waals interaction energy based on Hamaker's additivity hypothesis are derived for a sphere in interaction with a square wedge, a semi-infinite or finite slit, a semi-infinite slab, a 2D pillar, a rectangular rod, a corner and a rectangular channel. A numerical tool based on an adaptive mesh refinement strategy is presented and used to validate the analytical results. The analytical result for a sphere/wedge system is used to assess the domain of applicability of the sphere/plane model in the vicinity of the edge. The interaction between a sphere and a cylindrical pore in a plate of finite thickness is then simulated and the range of validity of the sphere/wedge system as a model of the sphere/pore system is deduced from the numerical results

On the relative impact of subgrid-scale modelling and conjugate heat transfer in LES of hot jets in cross-flow over cold plates

This work describes numerical simulations of a hot jet in cross-flow with applications to anti-ice systems of aircraft engine nacelles. Numerical results are compared with experimental measurements obtained at ONERA to evaluate the performances of LES in this industrial context. The combination of complex geometries requiring unstructured meshes and high Reynolds number does not allow the resolution of boundary layers so that wall models must be employed. In this framework, the relative influence of subgrid-scale modelling and conjugate heat transfer in LESs of aerothermal flows is evaluated. After a general overview of the transverse jet simulation results, a LES coupled with a heat transfer solver in the walls is used to show that thermal boundary conditions at the wall have more influence on the results than subgrid scale models. Coupling fluid flow and heat transfer in solids simulations is the only method to specify their respective thermal boundary conditions

A Three-Step Scenario Involved in Particle Capture on a Pore Edge

A scenario is proposed to describe the capture of a spherical particle around a cylindrical pore. This geometry, “ideal” as far as the problem of particle capture on a filtration membrane is concerned, is clearly relevant in view of the pore-scale geometry of nucleopore or microsieve filtration membranes, and also of some microfluidic systems used to perform fluid−particle separation. The present scenario consists of three successive steps: particle deposition on the membrane away from the pore, subsequent reentrainment of some of the deposited particles by rolling on the membrane surface, and final arrest by a stabilizing van der Waals torque when the particle rolls over the pore edge. A modeling of these three steps requires the hydrodynamic and physicochemical particle−membrane interactions to be detailed close to the singular pore edge region and raises questions concerning the role of particle surface roughness. The relevance and robustness of such of a scenario for rough micrometer-sized latex particles is emphasized and comparisons are made with existing experimental data

Fast, robust evaluation of the equation of state of suspensions of charge-stabilized colloidal spheres

Increasing demand is appearing for the fast, robust prediction of the equation of state of colloidal suspensions, notably with a view to using it as input data to calculate transport coefficients in complex flow solvers. This is also of interest in rheological studies, industrial screening tests of new formulations, and the real-time interpretation of osmotic compression experiments, for example. For charge-stabilized spherical particles, the osmotic pressure can be computed with standard liquid theories. However, this calculation can sometimes be lengthy and/or unstable under some physicochemical conditions, a drawback that precludes its use in multiscale flow simulators. As a simple, fast, and robust replacement, the literature reports estimations of the osmotic pressure that have been built by adding the Carnahan−Starling and the cell model pressures (CSCM model). The first contribution is intended to account for colloid−colloid contacts, and the second, for electrostatic effects. This approximation has not yet been thoroughly tested. In this work, the CSCM is evaluated by comparison with data from experiments on silica particles, Monte Carlo simulations, and solutions of the accurate Rogers−Young integral equation scheme with a hard-sphere Yukawa potential obtained from the extrapolated point-charge renormalization method for a wide range of volume fractions, surface charge densities, and interaction ranges. We find that the CSCM is indeed perfectly adequate in the electrostatically concentrated regime, where it can be used from vanishingly small to high surface charge because there is error cancellation between the Carnahan−Starling and cell model contributions at intermediate charge. The CSCM is thus a nice extension of the cell model to liquid-like dense suspensions, which should find application in the domains mentioned above. However, it fails for dilute suspensions with strong electrostatics. In this case, we show that, and explain why, perturbation methods and the rescaled mean spherical approximation are good alternatives in terms of precision, ease of implementation, computational cost, and robustnes

Modeling the Electrostatics of Hollow Shell Suspensions: Ion Distribution, Pair Interactions, and Many-Body Effects

Electrostatic interactions play a key role in hollow shell suspensions as they determine their structure, stability,thermodynamics, and rheology and also the loading capacity of small charged species for nanoreservoir applications. In this work, fast, reliable modeling strategies aimed at predicting the electrostatics of hollow shells for one, two, and many colloids are proposed and validated. The electrostatic potential inside and outside a hollow shell with a finite thickness and a specific permittivity is determined analytically in the Debye-Hückel (DH) limit. An expression for the interaction potential between two such hollow shells is then derived and validated numerically. It follows a classical Yukawa form with an effective charge depending on the shell geometry, permittivity, and inner and outer surface charge densities. The predictions of the Ornstein-Zernike (OZ) equation with this pair potential to determine equations of state are then evaluated by comparison to results obtained with a Brownian dynamics algorithm coupled to the resolution of the linearized Poisson-Boltzmann and Laplace equations PB-BD simulations). The OZ equation based on the DLVO-like potential performs very well in the dilute regime as expected, but also quite well, and more surprisingly, in the concentrated regime in which full spheres exhibit significant many-body effects. These effects are shown to vanish for shells with small thickness and high permittivity. For highly charged hollow shells, we propose and validate a charge renormalization procedure. Finally, using PB-BD simulations, we show that the cell model predicts the ion distribution inside and outside hollow shells accurately in both electrostatically dilute and concentrated suspensions. We then determine the shell loading capacity as a function of salt concentration, volume fraction, and surface charge density for nanoreservoir applications such as drug delivery, sensing, or smart coatings

Mélange gravitationnel de fluides en géométrie confinée

Ce travail basé sur la simulation numérique directe porte sur le mélange en géométrie confinée de deux fluides miscibles de densités différentes. Le mouvement des fluides est induit par la gravité. Différentes géométries sont étudiées : un tube cylindrique, un canal de section carrée et un écoulement purement bidimensionnel. Les simulations numériques confirment pleinement les résultats expérimentaux de Séon et al. en tube cylindrique, avec notamment la mise en évidence de trois régimes différents suivant l’inclinaison du tube. La comparaison des géométries montre que les écoulements tridimensionnels en tube et en canal inclinés présentent des comportements similaires tandis que le « modèle » bidimensionnel est incapable de donner des informations pertinentes sur un écoulement réel tridimensionnel tant au niveau quantitatif qu’au niveau phénoménologique. Une attention particulière est portée à l’analyse conjointe du champ de concentration et de la dynamique tourbillonnaire sous-jacente qui permet d’expliquer plusieurs aspects subtils de la dynamique du mélange