Buoyancy-driven Bubbles Research Articles

Terminal shape and velocity of a rising bubble by phase-field-based incompressible Lattice Boltzmann model

This article describes the simulation of three-dimensional buoyancy-driven bubble rise using a phase-field-based incompressible Lattice Boltzmann model. The effect of the Cahn–Hilliard mobility parameter, which is the rate of diffusion relaxation from non-equilibrium toward equilibrium state of chemical potential, is evaluated in detail. In contrast with previous work that pursues a high density ratio of binary fluids in the hydrodynamic equation, we apply a large dynamic viscosity ratio, together with a matched density pair and a separate compensating gas phase buoyant force, and the numerical results fit previous experimental results well. Through analysis, it is noted that for cases with moderate Reynolds number, a large value of mobility keeps a relatively sharp interface, while smaller values of mobility would result in diffusive interfacial regions. Moreover, for cases with large Reynolds number, small bubbles at the tail tend to separate more easily when the value of mobility is larger. This article offers some potentially useful details for performing phase-field-based simulations.

Experimental measurement of oxygen mass transfer and bubble size distribution in an air–water multiphase Taylor–Couette vortex bioreactor

Experimental measurements of the volumetric liquid mass transfer and bubble size distribution in a vertically oriented semi-batch gas–liquid Taylor–Couette vortex reactor with radius ratio η=ri/ro=0.75 and aspect ratio Γ=h/(ro−ri)=40 were performed, and the results are presented for axial and azimuthal Reynolds number ranges of Rea=11.9–143 and ReΘ=0–3.5×104, respectively. Based on these data, power-law correlations are presented for the dimensionless Sauter mean diameter, bubble size distribution, bubble ellipticity, and volumetric mass transfer coefficient in terms of relevant parameters including the axial and azimuthal Reynolds numbers. The interaction between wall-driven Taylor vortices and the axial passage of buoyancy-driven gas bubbles leads to significantly different dependencies of the mass transfer coefficient on important operating parameters such as inner cylinder angular velocity and axial superficial gas velocity than has been observed in horizontally oriented gas–liquid Taylor vortex reactors. In general, the volumetric mass transfer coefficients in vertical Taylor vortex reactors have a weaker dependence upon both the axial and azimuthal Reynolds numbers and are smaller in magnitude than those observed in horizontal Taylor vortex reactors or in stirred tank reactors. These findings can be explained by differences in the size and spatial distribution of gas bubbles in the vertically oriented reactor in comparison with the other systems.

An efficient numerical method for simulating multiphase flows using a diffuse interface model

This paper presents a new diffuse interface model for multiphase incompressible immiscible fluid flows with surface tension and buoyancy effects. In the new model, we employ a new chemical potential that can eliminate spurious phases at binary interfaces, and consider a phase-dependent variable mobility to investigate the effect of the mobility on the fluid dynamics. We also significantly improve the computational efficiency of the numerical algorithm by adapting the recently developed scheme for the multiphase-field equation. To illustrate the robustness and accuracy of the diffuse interface model for surface tension- and buoyancy-dominant multi-component fluid flows, we perform numerical experiments, such as equilibrium phase-field profiles, the deformation of drops in shear flow, a pressure field distribution, the efficiency of the proposed scheme, a buoyancy-driven bubble in ambient fluids, and the mixing of a six-component mixture in a gravitational field. The numerical result obtained by the present model and solution algorithm is in good agreement with the analytical solution and, furthermore, we not only remove the spurious phase-field profiles, but also improve the computational efficiency of the numerical solver.

Experimental and numerical study of buoyancy-driven single bubble dynamics in a vertical Hele-Shaw cell

Buoyancy-driven single bubble behaviour in a vertical Hele-Shaw cell with various gap Reynolds numbers Re(h/d)2 has been studied. Two gap thicknesses, h = 0.5 mm (Re(h/d)2 = 1.0–8.5) and 1 mm (Re(h/d)2 = 6.0–50) were used to represent low and high gap Reynolds number flow. Periodic shape oscillation and path vibration were observed once the gap Reynolds number exceeds the critical value of 8.5. The bubble behaviour was also numerically simulated by taking a two-dimensional volume of fluid method coupled with a continuum surface force model and a wall friction model in the commercial computational fluid dynamics package Fluent. By adjusting the viscous resistance values, the bubble dynamics in the two gap thicknesses can be simulated. For the main flow properties including shape, path, terminal velocity, horizontal vibration, and shape oscillation, good agreement is obtained between experiment and simulation. The estimated terminal velocity is 10%–50% higher than the observed one when the bubble diameter d ≤ 5 mm, h = 0.5 mm and 9% higher when d ≤ 18 mm, h = 1.0 mm. The estimated oscillation frequency is 50% higher than the observed value. Three-dimensional effects and spurious vortices are most likely the reason for this inaccuracy. The simulation confirms that the thin liquid films between gas bubbles and the cell walls have a limited effect on the bubble dynamics.

Simulations of soluble surfactants in 3D multiphase flow

A finite-difference/front-tracking method is developed for simulations of soluble surfactants in 3D multiphase flows. The interfacial and bulk surfactant concentration evolution equations are solved fully coupled with the incompressible Navier–Stokes equations. A non-linear equation of state is used to relate interfacial surface tension to surfactant concentration at the interface. Simple test cases are designed to validate different parts of the numerical algorithm and the computational results are found to be in a good agreement with the analytical solutions. The numerical algorithm is parallelized using a domain-decomposition method. It is then applied to study the effects of soluble surfactants on the motion of buoyancy-driven bubbles in a straight square channel in nearly undeformable (spherical) and deformable (ellipsoidal) regimes. Finally the method is used to examine the effects of soluble surfactants on the lateral migration of bubbles in a pressure-driven channel flow. It is found that surfactant-induced Marangoni stresses counteract the shear-induced lift force and can reverse the lateral bubble migration completely, i.e., the contaminated bubble drifts away from the channel wall and stabilizes at the center of the channel when the surfactant-induced Marangoni stresses are sufficiently large.

A study of gas bubbles in liquid mercury in a vertical Hele-Shaw cell

High-quality observations of mesoscopic gas bubbles in liquid metal are vital for a further development of pyrometallurgical gas injection reactors. However, the opacity of metals enforces the use of indirect imaging techniques with limited temporal or spatial resolution. In addition, accurate interface tracking requires tomography which further complicates the design of a high-temperature experimental setup. In this paper, an alternative approach is suggested that circumvents these two main restrictions. By injecting gas in a thin layer of liquid metal entrapped between two flat and closely spaced plates, bubbles in a Hele-Shaw flow regime are generated. The resulting quasi-2D multiphase flow phenomena can be fully captured from a single point of view and, when using a non-wetted transparent plate material, the bubbles can be observed directly. The feasibility of this approach is demonstrated by observations on buoyancy-driven nitrogen bubbles in liquid mercury in a vertical Hele-Shaw cell. By using a moving high-speed camera to make continuous close up recordings of individual bubbles, the position and geometry of these bubbles are quantified with a high resolution along their entire path. After a thorough evaluation of the experimental accuracy, this information is used for a detailed analysis of the bubble expansion along the path. While the observed bubble growth is mainly caused by the hydrostatic pressure gradient, a careful assessment of the volume variations for smaller bubbles shows that an accurate bubble description should account for significant dynamic pressure variations that seem to be largely regime dependent.

Phase-field-based lattice Boltzmann model for incompressible binary fluid systems with density and viscosity contrasts

A lattice Boltzmann model (LBM) is proposed based on the phase-field theory to simulate incompressible binary fluids with density and viscosity contrasts. Unlike many existing diffuse interface models which are limited to density matched binary fluids, the proposed model is capable of dealing with binary fluids with moderate density ratios. A new strategy for projecting the phase field to the viscosity field is proposed on the basis of thecontinuity of viscosity flux. The new LBM utilizes two lattice Boltzmann equations (LBEs): one for the interface tracking and the other for solving the hydrodynamic properties. The LBE for interface tracking can recover the Chan-Hilliard equation without any additional terms; while the LBE for hydrodynamic properties can recover the exact form of the divergence-free incompressible Navier-Stokes equations avoiding spurious interfacial forces. A series of 2D and 3D benchmark tests have been conducted for validation, which include a rigid-body rotation, stationary and moving droplets, a spinodal decomposition, a buoyancy-driven bubbly flow, a layered Poiseuille flow, and the Rayleigh-Taylor instability. It is shown that the proposed method can track the interface with high accuracy and stability and can significantly and systematically reduce the parasitic current across the interface. Comparisons with momentum-based models indicate that the newly proposed velocity-based model can better satisfy the incompressible condition in the flow fields, and eliminate or reduce the velocity fluctuations in the higher-pressure-gradient region and, therefore, achieve a better numerical stability. In addition, the test of a layered Poiseuille flow demonstrates that the proposed scheme for mixture viscosity performs significantly better than the traditional mixture viscosity methods.

Scaling parameters of bubbles and drops; interpretation and case study with air in silicon oil

Buoyancy-driven bubbles and drops are deformable and exhibit different boundaries from mobile to immobile. Owing to the complexity of these flows, the literature knows various scaling parameters for their description. This paper determines a set of parameters on the basis of the force balance equations. It identifies the Archimedes number as the leading parameter while the Reynolds number is considered a flow variable. This ordering scheme of parameters is applied to a set of new data on bubbles rising in five different silicon oils. Data representation and limiting cases come out consistently. The use of the Archimedes number leads even to a successful simple model for the normalized rise velocity.

Numerical Simulation of Buoyancy-driven Bubble Motion Using Level Set Method

This paper discusses the numerical simulation of buoyancy-driven bubbles in various shape regimes using level set method. The effects of both inertia and viscous forces are considered in the mathematical model. Surface tension force at the fluid interface is implemented through the CSF model of Brackbill et al. [2]. The incompressible Navier-Stokes equations were solved on a finite volume based staggered grid using an explicit projection method. Although the level set method offers an elegant way of computing multiphase flows, it is often plagued by mass conservation problems. In the present work, the level set method is augmented with a volume-reinitialization scheme, which helps preserve the individual fluid masses within gas and liquid phases. This enabled us to compute the motion of bubbles in all important shape regimes on a relatively coarse grid.

Viscosity effects on the behavior of a rising bubble

In the incompressible fluid flow regime, without taking consideration of surface tension effects, the viscosity effects on the behavior of an initially spherical buoyancy-driven bubble rising in an infinite and initially stationary liquid are investigated numerically by the Volume Of Fluid (VOF) method. The ratio of the gas density to the liquid density is taken as 0.001, and the gas viscosity to the liquid viscosity is 0.01, which is close to the case of an air bubble rising in water. It is found by numerical experiments that there exist two critical Reynolds numbers Re 1 and Re 2, which are in between 30 and 50 and in between 10 and 20, respectively. As Re > Re 1 the bubble will have the transition to toroidal form, and the toroidal bubble will break down into two toroidal bubbles. In this case viscosity will damp the development of the liquid jet and delay the formation of the toroidal bubble. As Re< Re 1 the transition will not happen. As Re 2 < Re < Re 1, the bubble will split from its rim into a toroidal bubble and a spherical cap-like bubble, and as Re< Re 2 the splitting will not occur and the bubble can finally reach a stationary shape. With the decrease of the Reynolds number, the stationary shape changes from spherical-cap bubble with skirt to dimpled peach-like bubble. Before the bubble reaches its stationary shape the vortex structure in the flow field varies with time. The vortex structure corresponding to bubble stationary shape varies with the Reynolds number. It is also found that there exists another critical Reynolds number Re * which is in between Re 1 and Re 2, and as Re < Re *, after the bubble rises in an accelerating manner for a moment, it will rise with an almost constant speed, and the speed increases with increasing Reynolds number. As Re > Re *, it will not rise with a constant speed. The mechanism of the above phenomena has been analyzed theoretically and numerically.

Surface tension effects on the behaviour of a rising bubble driven by buoyancy force

In the inviscid and incompressible fluid flow regime, surface tension effects on the behaviour of an initially spherical buoyancy-driven bubble rising in an infinite and initially stationary liquid are investigated numerically by a volume of fluid (VOF) method. The ratio of the gas density to the liquid density is 0.001, which is close to the case of an air bubble rising in water. It is found by numerical experiment that there exist four critical Weber numbers We1, We2, We3 and We4, which distinguish five different kinds of bubble behaviours. It is also found that when 1 ≤ We < We2, the bubble will finally reach a steady shape, and in this case after it rises acceleratedly for a moment, it will rise with an almost constant speed, and the lower the Weber number is, the higher the speed is. When We > We2, the bubble will not reach a steady shape, and in this case it will not rise with a constant speed. The mechanism of the above phenomena has been analysed theoretically and numerically.

NUMERICAL INVESTIGATION OF THE INTERACTION MECHANISM OF TWO BUBBLES

In the frame of inviscid and incompressible fluids without taking into consideration of surface tension effects, the axisymmetric evolution of two buoyancy-driven bubbles in an infinite and initially stationary liquid are investigated numerically by VOF method. The numerical experiments are performed for two bubbles with same size, with the following one being half of the leading one, and with the leading one being half of the following one, and for different bubble distances. The ratio of gas density to liquid density is 0.001. It is found by numerical experiment that when the distance between the two bubbles is greater than or equal to one and half of the bigger bubble radius, the interaction is very weak and the two bubbles evolute like isolated ones rising in an infinite liquid. When the two bubbles come closer, the leading bubble itself evolutes like an isolated one rising in an infinite liquid. However, due to the smaller distance between the two bubbles, large pressure gradient forms in the liquid region near the top of the following bubble, which causes the upward stretch of its top part no matter what sizes of the two bubbles are. When the distance between the two bubbles is less than or equal to three-tenth of the bigger bubble radius, before the liquid jet behind the leading bubble fully developed, the top part of the following bubble has already been sucked into the leading one, giving a pear-like shape. Soon after the following bubble merges with the leading one. It is also found that for the smaller bubble, its transition to toroidal is always faster than that of the bigger one because of its smaller size. The mechanism of the above phenomena has been analyzed numerically.

Lattice-Boltzmann simulation of finite Reynolds number buoyancy-driven bubbly flows in periodic and wall-bounded domains

A lattice-Boltzmann method is used to probe the structure and average properties of suspensions of monodisperse, spherical, noncoalescing bubbles rising due to buoyancy with Reynolds numbers based on the bubble terminal velocities of 5.4 and 20. Unbounded suspensions subject to periodic boundary conditions exhibit a microstructure with a strong tendency toward horizontal alignment of bubble pairs even at volume fractions of as high as 0.2. This microstructure leads to a mean rise velocity whose dependence on the bubble volume fraction is not well fitted by a standard power-law function. Simulations with bounding vertical walls exhibit a deficit of bubbles near each wall and a peak of volume fraction approximately one bubble diameter from the wall. We attribute this structure to the effects of a repulsive wall-induced force and a lift force associated with the liquid flow driven by the variation in the buoyancy force with horizontal position. Weaker peaks of bubble volume fraction extend into the bulk of the suspension and these peaks are separated by a distance equal to the peak in the pair distribution function for bubble pairs in an unbounded fluid. This suggests that the layering is a result of hydrodynamic bubble-bubble interactions.

The effect of soluble surfactant on the transient motion of a buoyancy-driven bubble

The effect of soluble surfactants 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 finite-difference/front-tracking 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 ellipsoidal-cap 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.

A front-tracking method for computation of interfacial flows with soluble surfactants

A finite-difference/front-tracking method is developed for computations of interfacial flows with soluble surfactants. The method is designed to solve the evolution equations of the interfacial and bulk surfactant concentrations together with the incompressible Navier–Stokes equations using a non-linear equation of state that relates interfacial surface tension to surfactant concentration at the interface. The method is validated for simple test cases and the computational results are found to be in a good agreement with the analytical solutions. The method is then applied to study the cleavage of drop by surfactant—a problem proposed as a model for cytokinesis [H.P. Greenspan, On the dynamics of cell cleavage, J. Theor. Biol. 65(1) (1977) 79; H.P. Greenspan, On fluid-mechanical simulations of cell division and movement, J. Theor. Biol., 70(1) (1978) 125]. Finally the method is used to model the effects of soluble surfactants on the motion of buoyancy-driven bubbles in a circular tube and the results are found to be in a good agreement with available experimental data.

On leakage and seepage of CO2 from geologic storage sites into surface water

Geologic carbon sequestration is the capture of anthropogenic carbon dioxide (CO2) and its storage in deep geologic formations. The processes of CO2 seepage into surface water after migration through water-saturated sediments are reviewed. Natural CO2 and CH4 fluxes are pervasive in surface-water environments and are good analogues to potential leakage and seepage of CO2. Buoyancy-driven bubble rise in surface water reaches a maximum velocity of approximately 30 cm s−1. CO2 rise in saturated porous media tends to occur as channel flow rather than bubble flow. A comparison of ebullition versus dispersive gas transport for CO2 and CH4 shows that bubble flow will dominate over dispersion in surface water. Gaseous CO2 solubility in variable-salinity waters decreases as pressure decreases leading to greater likelihood of ebullition and bubble flow in surface water as CO2 migrates upward.

Nonlinear Photon Bubbles Driven by Buoyancy

We derive an analytic model for nonlinear "photon bubble" wave trains driven by buoyancy forces in magnetized, radiation pressure-dominated atmospheres. Continuous, periodic wave solutions exist when radiative diffusion is slow compared to the dynamical timescale of the atmosphere. We identify these waves with the saturation of a linear instability discovered by Arons; therefore, these wave trains should develop spontaneously. The buoyancy-driven waves are physically distinct from a second family of photon bubbles discovered by Gammie, which evolve into trains of gas pressure-dominated shocks as they become nonlinear. Like the gas pressure-driven shock trains, buoyancy-driven photon bubbles can exhibit very large density contrasts, which greatly enhance the flow of radiation through the atmosphere. However, steady state solutions for buoyancy-driven photon bubbles exist only when an extra source of radiation is added to the energy equation, in the form of a flux divergence. We argue that this term is required to compensate for the radiation flux lost via the bubbles, which increases with height. We speculate that an atmosphere subject to buoyancy-driven photon bubbles, but lacking this compensating energy source, would lose pressure support and collapse on a timescale much shorter than the radiative diffusion time in the equivalent homogeneous atmosphere.

Type Ia Supernova Explosion: Gravitationally Confined Detonation

We present a new mechanism for Type Ia supernova explosions in massive white dwarfs. The scenario follows from relaxing assumptions of symmetry and involves a detonation born near the stellar surface. The explosion begins with an essentially central ignition of a deflagration that results in the formation of a buoyancy-driven bubble of hot material that reaches the stellar surface at supersonic speeds. The bubble breakout laterally accelerates fuel-rich outer stellar layers. This material, confined by gravity to the white dwarf, races along the stellar surface and is focused at the location opposite to the point of the bubble breakout. These streams of nuclear fuel carry enough mass and energy to trigger a detonation just above the stellar surface that will incinerate the white dwarf and result in an energetic explosion. The stellar expansion following the deflagration redistributes mass in a way that ensures production of intermediate-mass and iron group elements with ejecta having a strongly layered structure and a mild amount of asymmetry following from the early deflagration phase. This asymmetry, combined with the amount of stellar expansion determined by details of the evolution (principally the energetics of deflagration, timing of detonation, and structure of the progenitor), can be expected to create a family of mildly diverse Type Ia supernova explosions.

Numerical Methods for Multiple Inviscid Interfaces in Creeping Flows

We present new, highly accurate, and efficient methods for computing the motion of a large number of two-dimensional closed interfaces in a slow viscous flow. The interfacial velocity is found through the solution to an integral equation whose analytic formulation is based on complex-variable theory for the biharmonic equation. The numerical methods for solving the integral equations are spectrally accurate and employ a fast multipole-based iterative solution procedure, which requires only O(N) operations where N is the number of nodes in the discretization of the interface. The interface is described spectrally, and we use evolution equations that preserve equal arclength spacing of the marker points. We assume that the fluid on one side of the interface is inviscid and we discuss two different physical phenomena: bubble dynamics and interfacial motion driven by surface tension (viscous sintering). Applications from buoyancy-driven bubble interactions, the motion of polydispersed bubbles in an extensional flow, and the removal of void spaces through viscous sintering are considered and we present large-scale, fully resolved simulations involving O(100) closed interfaces.