Volume 20, Issue 4, April 2008

The effect of solublesurfactants on the unsteady motion and deformation of a bubble rising in an otherwise quiescent liquid contained in an axisymmetric tube is computationally studied by using a finitedifference/fronttracking method. The unsteady incompressible flow equations are solved fully coupled with the evolution equations of bulk and interfacial surfactant concentrations. The surface tension is related to the interfacial surfactant concentration by a nonlinear equation of state. The nearly spherical, ellipsoidal, and dimpled ellipsoidalcap regimes of bubble motion are examined. It is found that the surfactant generally reduces the terminal velocity of the bubble but this reduction is most pronounced in the nearly spherical regime in which the bubble behaves similar to a solid sphere and its terminal velocity approaches that of an equivalent solid sphere. Effects of the elasticity number and the bulk and interfacial Peclet numbers are examined in the spherical and ellipsoidal regimes. It is found that the surface flow and interfacial surfactant concentration profiles exhibit the formation of a stagnant cap at the trailing end of the bubble in the ellipsoidal regime at low elasticity and high interfacial Peclet numbers. Bubble deformation is first reduced due to rigidifying effect of the surfactant but is then amplified when the elasticity number exceeds a critical value due to overall reduction in the surface tension.
 SPECIAL TOPIC: RECENT ADVANCES IN MULTIPHASE FLOWS: NUMERICAL AND EXPERIMENTAL


Report on the IUTAM Symposium on Recent Advances in Multiphase Flows: Numerical and Experimental (11–14 June 2007, Istanbul, Turkey)
View Description Hide Description  Particulate Flows

A weakly nonlocal anisotropic fluid model for inhomogeneous Stokesian suspensions
View Description Hide DescriptionA continuum model is proposed for a weakly inhomogeneous Stokesian suspensions, as an extension with minor amendments of a previous work on homogeneous suspensions [J. D. Goddard, J. Fluid Mech.568, 1 (2006)]. In the present model, stress and particle flux are given as invariant tensor functions of particle volume fraction , deformation rate , and secondrank anisotropytensor, in a form that is also linear in and the gradients of , , and . In contrast to models without history dependence, all nonlinear dependence of particle flux on arises from the evolution of . Detailed attention is paid to unsteady viscometric flow, where a contribution of streamline curvature to particle migration emerges as a natural consequence of tensorial gradients. The model predicts equal curvatureinduced fluxes in gradient and vorticity directions but there is an unexplained disagreement with recent experiments on Couette and torsional flows. A previously proposed corotational evolution equation for , with a twomode exponential relaxation, is employed to investigate the transient response following the reversal of shearing in sinusoidal and in steady shear. The model predicts roughly equal response for the two flows if sinusoidal strains are of order unity, which is consistent with some but not all experiments. The model for particle flux admits an asymmetric diffusiontensor which, owing to Stokesian reversibility, can become nonpositive upon abrupt reversal of shearing. This effect is diminished by nonStokesian response on short strain scales, which, although poorly understood, appears essential to elementary models without dependence on shear history. A synthesis is given of multipolar Stokesian resistance and the associated Stokesian dynamics, showing how these follow from a single grand resistance kernel. In addition to unifying and extending large literature on Stokesian resistance formulae, this provides some justification for the proposed continuum model and possible multipolar extensions.

Suspension properties at finite Reynolds number from simulated shear flow
View Description Hide DescriptionThis work examines the role of particlescale inertia in a monodisperse suspension of nonBrownian and neutrally buoyant spherical particles subjected to simpleshear flow. The dimensionless parameters governing the problem are the solidvolume fraction and the Reynolds number defined , where is the sphere radius, is the shear rate, and and are the viscosity and density of the fluid, respectively. Using numerical simulations in a wallbounded domain via the latticeBoltzmann method, the bulk rheological properties of relative viscosity, normal stress differences, and particle pressure are reported for and . The anisotropy in microstructure at finite Re is studied through the pair distribution function . Also presented are the probability density functions of particle velocity fluctuations in gradient and vorticity directions. Comparisons to low Reynolds number theory and simulations are provided wherever possible.

Some issues concerning largeeddy simulation of inertial particle dispersion in turbulent bounded flows
View Description Hide DescriptionThe problem of accurate Eulerian–Lagrangian modeling of inertial particle dispersion in largeeddy simulation(LES) of turbulent wallbounded flows is addressed. We run direct numerical simulation (DNS) of turbulent channel flow at shear Reynolds number and corresponding a priori and a posterioriLES on two coarser grids. For each flow field, we tracked swarms of particles with different inertia to examine the behavior of particle statistics, specifically focusing on particle preferential segregation and accumulation at the wall. Our object is to discuss the necessity of a closure model for the particle equations when using LES and we verify if the influence of the subgrid turbulence filtered by LES is an important effect on particle motion according to particle size. The results show that wellresolved LES gives particle velocity statistics in satisfactory agreement with DNS. However, independent of the grid, quantitatively inaccurate predictions are obtained for local particle preferential segregation, particularly in the nearwall region. Inaccuracies are observed for the entire range of particle size considered in this study, even when the particle response time is much larger than the flow time scales not resolved in LES. The satisfactory behavior of LES in reproducing particle velocity statistics is thus counterbalanced by the inaccurate representation of local segregation phenomena, indicating that closure models supplying the particle motion equation with an adequate rendering of the flow field might be needed. Finally, we remark that recovering the level of fluid and particle velocityfluctuations in the particle equations does not ensure a quantitative replica of the subgrid turbulence effects, thus implying that accurate subgrid closure models for particles may require information also proportional to the higherorder moments of the velocityfluctuations.

Inclusion of heat transfer computations for particle laden flows
View Description Hide DescriptionA newly developed direct numerical simulation method has been used to study the dynamics of nonisothermal cylindrical particles in particulate flows. The momentum and energy transfer equations are solved to compute the effects of heat transfer in the sedimentation of particles. Among the effects examined is the drag force on nonisothermal particles, which we found strongly depends on the Reynolds and Grashof numbers. It was observed that heat advection between hotter particles and fluid causes the drag coefficient of particles to significantly increase at relatively low Reynolds numbers. For Grashof number of 100, the drag enhancement effect diminishes when the Reynolds number exceeds 50. On the contrary, heat advection with colder particles reduces the drag coefficient for low and medium Reynolds number for Grashof number of . We used this numerical method to study the problem of a pair of hot particles settling in a container at different Grashof numbers. In isothermal cases, such a pair of particles would undergo the wellknown draftingkissingtumbling (DKT) motion. However, it was observed that the buoyancy currents induced by the hotter particles reverse the DKT motion of the particles or suppress it altogether. Finally, the sedimentation of a circular cluster of 172 particles in an enclosure at two different Grashof numbers was studied and the main features of the results are presented.

Experimental techniques for multiphase flows
View Description Hide DescriptionThis review discusses experimental techniques that provide an accurate spatial and temporal measurement of the fields used to describe multiphase systems for a wide range of concentrations, velocities, and chemical constituents. Five methods are discussed: magnetic resonance imaging(MRI), ultrasonic pulsed Doppler velocimetry (UPDV), electrical impedance tomography (EIT), xray radiography, and neutron radiography. All of the techniques are capable of measuring the distribution of solids in suspensions. The most versatile technique is MRI, which can be used for spatially resolved measurements of concentration, velocity, chemical constituents, and diffusivity. The ability to measure concentration allows for the study of sedimentation and shearinduced migration. Onedimensional and twodimensional velocity profiles have been measured with suspensions,emulsions, and a range of other complex liquids. Chemical shift MRI can discriminate between different constituents in an emulsion where diffusivity measurements allow the particle size to be determined. UPDV is an alternative technique for velocity measurement. There are some limitations regarding the ability to map complex flow fields as a result of the attenuation of the ultrasonic wave in concentrated systems that have high viscosities or where multiple scattering effects may be present. When combined with measurements of the pressure drop, both MRI and UPDV can provide local values of viscosity in pipe flow. EIT is a low cost means of measuring concentration profiles and has been used to study shearinduced migration in pipe flow. Both xray and neutron radiographes are used to image structures in flowing suspensions, but both require highly specialized facilities.
 Bubbly Flows

Effect of bubble deformability in turbulent bubbly upflow in a vertical channel
View Description Hide DescriptionAs bubbles rising in a vertical channel with upflow become bigger, it is well known that the void fraction distribution changes in a fundamental way, from a wall peak for small bubbles to a maximum void fraction at the channel center for larger bubbles. Here, we use direct numerical simulations of buoyant bubbles in a turbulent flow to show that it is not the size of the bubbles that matters, but their deformability.

Quantifying microbubble clustering in turbulent flow from singlepoint measurements
View Description Hide DescriptionSinglepoint hotwire measurements in the bulk of a turbulent channel have been performed in order to detect and quantify the phenomenon of preferential bubble accumulation. We show that statistical analysis of the bubbleprobe collidingtime series can give a robust method for investigation of clustering in the bulk regions of a turbulent flow where, due to the opacity of the flow, no imaging technique can be employed. We demonstrate that microbubbles in a developed turbulent flow, where the Kolmogorovlength scale is , display preferential concentration in small scale structures with a typical statistical signature ranging from the dissipative range, , up to the low inertial range . A comparison to Eulerian–Lagrangian numerical simulations is also presented to further support our proposed way to characterize clustering from temporal time series at a fixed position.

Numerical investigation of threedimensional cloud cavitation with special emphasis on collapse induced shock dynamics
View Description Hide DescriptionThe aim of the present investigation is to model and analyze compressible threedimensional (3D) cavitating liquid flows with special emphasis on the detection of shock formation and propagation. We recently developed the conservative finite volume method CATUM(Cavitation Technische Universität München), which enables us to simulate unsteady 3D liquid flows with phase transition at all Mach numbers. The compressible formulation of the governing equations together with the thermodynamic closure relations are solved by a modified Riemann approach by using time steps down to nanoseconds. This high temporal resolution is necessary to resolve the wavedynamics that leads to acoustic cavitation as well as to detect regions of instantaneous high pressure loads. The proposed twophase model based on the integral average properties of thermodynamic quantities is first validated against the solution of the Rayleigh–Plesset equation for the collapse of a single bubble. The computational fluid dynamics tool CATUM is then applied to the numerical simulation of the highly unsteady twophase flow around a 3D twisted hydrofoil. This specific hydrofoil allows a detailed study of sheet and cloud cavitation structures related to 3D shock dynamics emerging from collapsing vapor regions. The time dependent development of vapor clouds, their shedding mechanism, and the resulting unsteady variation of lift and drag are discussed in detail. We identify instantaneous local pressure peaks of the order of , which are thought to be responsible for the erosive damage of the surface of the hydrofoil.

Numerical study on the shearinduced lift force acting on a spherical bubble in aqueous surfactant solutions
View Description Hide DescriptionA single bubble motion in aqueous surfactantsolutions is discussed in this paper. We focus on the change of the lift force acting on a bubble in a linear shear flow under the condition that the bubble surface is contaminated by surfactantadsorption which leads the Marangoni effect. With an increase of Langmuir number which corresponds to a decrease of desorption rate constant of surfactant, the lift force acting on a spherical bubble decreases from the value of a clean bubble to near zero. This reduction is significantly related to a nonaxisymmetric distribution of pressure on the surface. Comparing the present results with those of our previous simulations using an axisymmetric stagnant cap model, the lift coefficients in the present simulations show larger values than those of the stagnant cap model. This is due to a nonaxisymmetric distribution of surface concentration. This asymmetry of the distribution enhances the asymmetry of the surface pressure distribution, which ends up the larger shearinduced lift force than that of the axisymmetric stagnant cap model.

Shape oscillations on bubbles rising in clean and in tap water
View Description Hide DescriptionThis paper deals with air bubbles rising in purified water in the range of equivalent diameters where surface oscillations appear on the interface. The shape of the bubbles including these capillary distortions is recorded by taking a large number of high speed pictures for each spiraling or zigzagging bubble trajectory. In analogy with surface harmonics, the oscillations are indicated as (2,0) axisymmetric and with wavelength equal to the distance from pole to pole and (2,2) nonaxisymmetric and with wavelength equal to onehalf of the length of the equator. In the second series of experiments, the phenomena in the wakes of rising bubbles are made visible by using Schlieren optics, which are applicable because a temperature gradient is applied to the water. The frequencies of vortex shedding correspond to the (2,0) mode of surface oscillation, whereas in other works reported in the literature, they correspond to twice the frequency of the spiraling or zigzagging bubble paths. By measurements and by analysis, it is shown here that the latter is due to contamination of surfactants.
 Drops and Drop Flows

Pairwise interaction of capsules in simple shear flow: Threedimensional effects
View Description Hide DescriptionWe use a boundary integral formulation to investigate the collision of two identical capsules in simple shear flow. Each capsule consists of a viscous liquid drop enclosed by an elastic membrane. The hydrodynamic interaction is characterized by an irreversible crossflow displacement after the capsules have crossed each other, as also observed for rough spheres or drops. This deflection would cause a shearinduced dispersion in a dilute suspension of capsules. After a previous work devoted to the interaction of two capsules located in the same shear plane, here, we focus on the more general case of capsules also shifted in the vorticity direction. For sufficiently spaced trajectories, the capsules exhibit negative deflections which displace them to closer streamlines after they have crossed. Furthermore, the crossflow displacement occurs mainly in the vorticity direction. Both phenomena are in contradiction with the conclusions drawn for a pair of drops, and are, therefore, attributed to the elastic nature of the capsule interface.

The mechanism of surfactant effects on drop coalescence
View Description Hide DescriptionWe utilize numerical solutions, based on a boundaryintegral scheme, to investigate the mechanisms by which surfactant influences the coalescence of a pair of equal size drops that undergo a headon collision in a biaxial linear flow. It is known that the addition of surfactant inhibits coalescence in the sense that the time required for film drainage to the point of film rupture is significantly increased. Although there is a direct effect on the rate of film drainage due to Marangoni effects within the thin film, we find that an equally important effect is due to the fact that the hydrodynamic force pushing the drops together is increased, hence causing the film to be more strongly deformed into a dimpled configuration that slows the film drainage process.

Squeezing of a periodic emulsion through a cubic lattice of spheres
View Description Hide DescriptionSqueezing of a periodic, highly concentrated emulsion of deformable drops through a dense, simple cubic array of solid spherical particles at zero Reynolds number is simulated by considering one drop in a periodic cell. The particles are rigidly held in space. The drops with nondeformed diameter comparable with the particle size (and considerably larger than the interparticle constrictions) squeeze under a specified average pressure gradient. This three dimensional problem serves as a useful prototype model of dropsolid interaction for emulsionflow through granular materials. The solution allows us to study permeabilities for both phases in detail and determine the critical conditions when the drop phase flow stops due to blockage in the pores by capillary forces. The algorithm employs a boundaryintegral formulation with periodic Green’s function, Hebeker representation for solidparticle contributions, and recent desingularization tools [A. Z. Zinchenko and R. H. Davis, J. Fluid Mech.564, 227 (2006)] to alleviate difficulties with lubrication. Calculations are challenging in that tens of thousands of boundary elements per surface and 10 000–20 000 time steps are required for nearcritical squeezing conditions, and the use of multipole acceleration is crucial to make such simulations feasible. The results are presented for 36% and 50% concentrated emulsions flowing through an array of almost packed particles, at droptomedium viscosity ratios of 1 and 4. Scaling for the squeezing time of the drop phase at nearcriticial capillary numbers is extracted from the calculations. For all the simulated cases, the drops move, on average, faster than the continuous phase.

Mixing and evaporation of liquid droplets injected into an air stream flowing at all speeds
View Description Hide DescriptionThis paper deals with the formulation, implementation, and testing of three numerical techniques based on (i) a full multiphase approach, (ii) a multisizegroup (MUSIG) approach, and (iii) a heterogeneous MUSIG (HMUSIG) approach for the prediction of mixing and evaporation of liquiddroplets injected into a stream of air. The numerical procedures are formulated following an Eulerian approach, within a pressurebased fully conservative finite volume method equally applicable in the subsonic, transonic, and supersonic regimes, for the discrete and continuous phases. The twoequation turbulencemodel is used to account for the droplet and gas turbulence with modifications to account for compressibility at high speeds. The performances of the various methods are compared by solving for two configurations involving streamwise and crossstream sprayings into subsonic and supersonic streams. Results, which are displayed in the form of droplet velocity vectors, contour plots, and axial profiles, indicate that solutions obtained by the various techniques exhibit a similar behavior. Differences in values are relatively small with the largest being associated with droplet volume fractions and vapor mass fraction in the gas phase. This is attributed to the fact that with MUSIG and HMUSIG, no droplet diameter equation is solved and the diameter of the various droplet phases is held constant, as opposed to the full multiphase approach.

The effect of soluble surfactant on the transient motion of a buoyancydriven bubble
View Description Hide DescriptionThe effect of solublesurfactants on the unsteady motion and deformation of a bubble rising in an otherwise quiescent liquid contained in an axisymmetric tube is computationally studied by using a finitedifference/fronttracking method. The unsteady incompressible flow equations are solved fully coupled with the evolution equations of bulk and interfacial surfactant concentrations. The surface tension is related to the interfacial surfactant concentration by a nonlinear equation of state. The nearly spherical, ellipsoidal, and dimpled ellipsoidalcap regimes of bubble motion are examined. It is found that the surfactant generally reduces the terminal velocity of the bubble but this reduction is most pronounced in the nearly spherical regime in which the bubble behaves similar to a solid sphere and its terminal velocity approaches that of an equivalent solid sphere. Effects of the elasticity number and the bulk and interfacial Peclet numbers are examined in the spherical and ellipsoidal regimes. It is found that the surface flow and interfacial surfactant concentration profiles exhibit the formation of a stagnant cap at the trailing end of the bubble in the ellipsoidal regime at low elasticity and high interfacial Peclet numbers. Bubble deformation is first reduced due to rigidifying effect of the surfactant but is then amplified when the elasticity number exceeds a critical value due to overall reduction in the surface tension.
 Bubbles

Interacting twodimensional bubbles and droplets in a yieldstress fluid
View Description Hide DescriptionWe study the buoyancydriven motion of twodimensional bubbles and droplets in a Bingham fluid using a regularization method. The finiteelement computations are carried out using the method of level sets to track the interface. We find that multiple bubbles and droplets can move in a body force field under conditions where a single bubble or droplet with the same physical properties would be unable to overcome the integrated yield stress and would be trapped. The finite yielded region around a single bubble or droplet in a Bingham fluid causes a backflow, resulting in unyielded “ears” that rotate and exchange material points with the yielded fluid to maintain a fixed position on the equatorial plane as the bubble rises or the droplet falls. The backflow flattens the tail of the trailing bubble or droplet in a pair and, at a sufficiently high level of interfacial tension, causes a splitting of the tail and the creation of a cusp. Three bubbles in a triangular configuration interact in a manner that is qualitatively predictable by considering pair interactions. Despite important differences in detail, the general shape evolution of bubbles and droplets in a Bingham fluid is similar to that in a Newtonian liquid when time scales are considered on a comparable basis.

Statistical equilibrium of bubble oscillations in dilute bubbly flows
View Description Hide DescriptionThe problem of predicting the moments of the distribution of bubble radius in bubbly flows is considered. The particular case where bubble oscillations occur due to a rapid (impulsive or step change) change in pressure is analyzed, and it is mathematically shown that in this case, inviscid bubble oscillations reach a stationary statistical equilibrium, whereby phase cancellations among bubbles with different sizes lead to timeinvariant values of the statistics. It is also shown that at statistical equilibrium, moments of the bubble radius may be computed using the periodaveraged bubble radius in place of the instantaneous one. For sufficiently broad distributions of bubble equilibrium (or initial) radius, it is demonstrated that bubble statistics reach equilibrium on a time scale that is fast compared to physical damping of bubble oscillations due to viscosity,heat transfer, and liquid compressibility. The periodaveraged bubble radius may then be used to predict the slow changes in the moments caused by the damping. A benefit is that period averaging gives a much smoother integrand, and accurate statistics can be obtained by tracking as few as five bubbles from the broad distribution. The periodaveraged formula may therefore prove useful in reducing computational effort in models of dilute bubbly flow wherein bubbles are forced by shock waves or other rapid pressure changes, for which, at present, the strong effects caused by a distribution in bubble size can only be accurately predicted by tracking thousands of bubbles. Some challenges associated with extending the results to more general (nonimpulsive) forcing and strong twoway coupled bubbly flows are briefly discussed.

Cylindrical bubble dynamics: Exact and direct numerical simulation results
View Description Hide DescriptionThe axially symmetric collapse and growth of a cylindrical bubble are considered. A universal law of cylindrical bubble dynamics is obtained when the pressure at the boundary and that inside the bubble satisfy certain conditions. Both vapor bubbles having constant pressure and gas bubbles obeying the isothermal law are considered. Moreover, for gas bubbles, the energy equation within the bubble is considered in the uniform pressure approximation for the effect of heat conduction through the bubble wall, and an exact particular solution, leading to explicit gas pressure and gas temperature expressions, is obtained. Finally, the findings are validated by direct numerical simulations by using the front tracking/finite volume method.
