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Numerical simulation of bubble rising in porous media using lattice Boltzmann method
Journal of Applied Physics
Abstract
Rising bubble systems in porous media exist in a variety of industrial processes. However, the flow characteristics of the issue are not well understood. In this work, the rising of bubble/bubbles through two types of porous structures, namely, in-line structured pore and staggered structured pore, are studied using a large density ratio lattice Boltzmann model. The effects of Eötvös number, pore shape, viscosity ratio, initial bubble number, and arrangement manner of the initial bubbles on the bubble deformation, bubble rising velocity, residual bubble mass, bubble perimeter, and the number of bubble breakups are investigated. It is found that as the Eötvös number increases, the bubbles are more easily broken during the process of passing through the porous media, the shapes of the sub-bubbles deviate from the original ones more and more, the bubble perimeter increases, and the difference between the bubble dynamics obtained by the in-line and staggered porous media decreases. Compared to the results of circular and rectangular pores, the bubble rising through the diamondoid pore has a more considerable deformation, which causes a slower rising speed. Furthermore, in the case that two bubbles are originally placed under the porous medium, the bubble deformation is greater and the bubble fracture times increase if the initial bubbles are aligned vertically. The findings of this work can contribute to the understanding of gas–liquid two-phase flow in porous media.
Porous media have been widely used for liquid-gas separation, benefiting from the strong capillary force generated from the micro/nanoscale pores. Understanding the flow characteristics in pore scale is significant for the design of porous structure. In this study, a numerical model was established to investigate the dynamics of a bubble penetrating through porous media at the pore scale. The two-phase interface was captured using the diffuse interface method. The influence of pore shape, width, and height on the bubble deformation, velocity, and critical pressure was investigated. For the same pore size, the largest bubble centroid velocity and the highest critical pressure exist in the quadrilateral pores compared rather than in the circular or triangular pores. As the pore width decreases, both the average velocity of the bubble centroid and the critical pressure increase. However, the critical pressure is independent of the pore height. As the pore height increases, the average velocity of the bubble centroid increases. A new correlation of the critical pressure for bubble penetration has been proposed, which is a function of the shape factor, the pore width, and the bubble diameter. The findings of this work can contribute to improving the design of porous media for two-phase separation.
In this paper, natural convection melting in a square cavity with gradient porous media is numerically studied at pore-scale level by adopting the lattice Boltzmann method. To generate the gradient porous media, a Monte Carlo technique based on the random sampling principle is used. The effects of several factors, such as Rayleigh number, gradient porosity structure, gradient direction, and particle diameters on natural convection melting are investigated in detail. Based on the numerical data, it is observed that the thermal performance of the gradient porous media always depends on the Rayleigh number and, specifically, as the Rayleigh number is set to 106, the total melting time obtained for the case of the negative gradient porous media is always shorter than the cases of positive gradient and uniform porous media. However, if the Rayleigh number is equal to 104, at which the heat transfer is dominated by the heat conduction, it is noted that the performance of the positive gradient porous media is better than the other cases. To have a better understand on this point, various simulations are also performed and we found that there usually exists a critical value of Rayleigh number to determine the thermal performance of the gradient porous media. Moreover, our numerical results also show that the influence of the particle diameter on the liquid fraction is insignificant as Rayleigh number is set to 104, while it has a great impact on the liquid fraction when Rayleigh number equals 106.
Sub-seabed geological CO2 storage is discussed as a climate mitigation strategy, but the impact of any leakage of stored CO2 into the marine environment is not well known. In this study, leakage from a CO2 storage reservoir through near-surface sediments was mimicked for low leakage rates in the North Sea. Field data were combined with laboratory experiments and transport-reaction modelling to estimate CO2 and mineral dissolution rates, and to assess the mobilization of metals in contact with CO2-rich fluids and their potential impact on the environment. We found that carbonate and silicate minerals reacted quickly with the dissolved CO2, increasing porewater alkalinity and neutralizing about 5% of the injected CO2. The release of Ca, Sr, Ba and Mn was mainly controlled by carbonate dissolution, while Fe, Li, B, Mg, and Si were released from silicate minerals, mainly from deeper sediment layers. No toxic metals were released from the sediments and overall the injected CO2 was only detected up to 1 m away from seabed CO2 bubble streams. Our results suggest that low leakage rates of CO2 over short timescales have minimal impact on the benthic environment. However, porewater composition and temperature are effective indicators for leakage detection, even at low CO2 leakage rates.
In this paper, the dynamic characteristics of the bubble in a broken confined domain are studied. The broken confined domain is composed of a solid wall and a plate that has a hole. The axisymmetric numerical model is established by combining the Eulerian finite-element method with volume of fluid (VOF) method, and is validated by comparing the results with those from an experiment. Then the influences of the wall distance, plate distance and size of the hole are analyzed. The results show that cavity-attraction jet caused by the hole and annular jet caused by the upper solid wall compete with each other to dominate the bubble dynamics. The cavity-attraction jet develops earlier, but slower. Thus, jet load in the bubble stage is mainly generated by the annular jet with a higher impact speed. Within a certain range, the closer the hole is to the bubble or the smaller the hole, the longer the pulsation period of the bubble will be.
Packed bed reactors have been widely applied in industrial production, such as for catalytic hydrogenation. Numerical simulations are essential for the design and scale-up of packed beds, especially direct numerical simulation (DNS) methods, such as the lattice-Boltzmann method (LBM), which are the focus of future researches. However, the large density difference between gas and liquid in packed beds often leads to numerical instability near phase interface when using LBM. In this paper, a lattice-Boltzmann (LB) model based on diffuse-interface phase-field is employed to simulate bubble rising in complex channels saturated with liquid, while the numerical problems caused by large liquid-to-gas density ratio are solved. Among them, the channel boundaries are constructed with regularly arranged circles and semicircles, and the bubbles pass through the channels accompanied by deformation, breakup, and coalescence behaviors. The phase-field LB model is found to exhibit good numerical stability and accuracy in handing the problem of the bubbles rising through the high-density liquid. The effects of channel structures, gas-liquid physical properties, and operating conditions on bubble deformation, motion velocity, and drag coefficient are simulated in detail. Moreover, different flow patterns are distinguished according to bubble behavior and are found to be associated with channel structure parameters, gravity Reynolds number (ReGr), and Eötvös number (Eo).
Foam has shown promising characteristics for soil remediation, especially as a blocking agent. In very permeable porous media, high groundwater velocity reduces the efficiency of in situ remediation by decreasing the contact time with contaminants. To use foam to reduce water velocity, this paper investigates its properties in very permeable porous media (> 4.10−10 m2). Alpha olefin sulfonate foam injection experiments were conducted in columns containing glass beads or sand, with varying permeabilities and injection techniques. Pressure and mass balance data were acquired to measure foam resistance factor and liquid saturation. The results show that foam pregeneration is mandatory for generating a highly viscous foam in highly permeable porous media. However, the permeability of the foam generator does not affect the results within the range studied. Pregeneration generated highly viscous foam, even in very high permeable porous media (> 10−9 m2). The influence of permeability on the apparent viscosity of foam in porous media was related to the shear-thinning foam behavior. These experimental results show that foam injection is a relevant method for water blocking and mobility control in highly permeable porous media.
The merits of CO2 capture and storage to the environmental stability of our world should not be underestimated as emissions of greenhouse gases cause serious problems. It represents the only technology that might rid our atmosphere of the main anthropogenic gas while allowing for the continuous use of the fossil fuels which still power today’s world. Underground storage of CO2 involves the injection of CO2 into suitable geological formations and the monitoring of the injected plume over time, to ensure containment. Over the last two or three decades, attention has been paid to technology developments of carbon capture and sequestration. Therefore, it is high time to look at the research done so far. In this regard, a high-level review article is required to provide an overview of the status of carbon capture and sequestration research. This article presents a review of CO2 storage technologies which includes a background of essential concepts in storage, the physical processes involved, modeling procedures and simulators used, capacity estimation, measuring monitoring and verification techniques, risks and challenges involved and field-/pilot-scale projects. It is expected that the present review paper will help the researchers to gain a quick knowledge of CO2 sequestration for future research in this field.
Gas bubbles can be naturally generated or intentionally introduced in sediments. Gas bubble migration and trapping affect the rate of gas emission into the atmosphere or modify the sediment properties such as hydraulic and mechanical properties. In this study, the migration and trapping of gas bubbles are simulated using the pore-network model extracted from the 3D X-ray image of in situ sediment. Two types of bubble size distribution (mono-sized and distributed-sized cases) are used in the simulation. The spatial and statistical bubble size distribution, residual gas saturation, and hydraulic conductivity reduction due to the bubble trapping are investigated. The results show that the bubble size distribution becomes wider during the gas bubble migration due to bubble coalescence for both mono-sized and distributed-sized cases. And the trapped bubble fraction and the residual gas saturation increase as the bubble size increases. The hydraulic conductivity is reduced as a result of the gas bubble trapping. The reduction in hydraulic conductivity is apparently observed as bubble size and the number of nucleation points increase.
Shapes and paths of an air bubble rising inside a liquid are investigated experimentally. About three hundred experiments are conducted in order to generate a phase plot in the Galilei and Eötvös numbers plane, which separates distinct regimes in terms of bubble behaviour. A wide range of the Galilei and Eötvös numbers are obtained by using aqueous glycerol solutions of different concentrations as the surrounding fluid and by varying the bubble size. The dynamics are investigated in terms of shapes, topological changes, and trajectories of the bubbles. Direct numerical simulations are conducted to study the bubble dynamics, which show excellent agreement with the experiments. To the best of our knowledge, this is the first time an experimentally obtained phase plot showing the distinct behaviour of an air bubble rising in a quiescent medium is reported for such a large range of Galilei and Eötvös numbers.
This paper describes the experiments designed to control bubble size during gas injection through porous media into liquid cross flow. A parametric study examined the effect of control variables on average bubble size and standard deviation. Results showed that for a given air and liquid flow rate, changing liquid channel height at the air injection site had the largest effect on bubble size and size distribution while varying porous media grade and electrolyte concentration had smaller, though significant, effects. In this study, the channel height was varied from 0.8 to 8 mm, porous media grade from 0.5 to 100 and salt concentration varied from zero to 3%. The resulting average bubble diameters were 0.085-2.5 mm.
In this work, an interface-capturing lattice Boltzmann equation (LBE) model is proposed for two-phase flows. In the model, a Lax-Wendroff propagation scheme and a properly chosen equilibrium distribution function are employed. The Lax-Wendroff scheme is used to provide an adjustable Courant-Friedrichs-Lewy (CFL) number, and the equilibrium distribution is presented to remove the dependence of the relaxation time on the CFL number. As a result, the interface can be captured accurately by decreasing the CFL number. A theoretical expression is derived for the chemical potential gradient by solving the LBE directly for a two-phase system with a flat interface. The result shows that the gradient of the chemical potential is proportional to the square of the CFL number, which explains why the proposed model is able to capture the interface naturally with a small CFL number, and why large interface error exists in the standard LBE model. Numerical tests, including a one-dimensional flat interface problem, a two-dimensional circular droplet problem, and a three-dimensional spherical droplet problem, demonstrate that the proposed LBE model performs well and can capture a sharp interface with a suitable CFL number.
The flow in the wake of single and two interacting air bubbles freely rising in water is studied experimentally using digital-particle-image-velocimetry in combination with high-speed recording. The experiments focus on ellipsoidal bubbles of diameter of about 0.4–0.8 cm which show spiraling, zigzagging, and rocking motion during their rise in water, which was seeded with small tracer particles for flow visualization. Under counterflow conditions in the vertical channel, the bubbles are retained in the center of the observation region, which allows the wake oscillations and bubble interaction to be observed over several successive periods. By simultaneous diffuse illumination in addition to the light sheet, we were able to record both the path and shape oscillations of the bubble, as well as the wake structure in a horizontal and vertical cross section. The results show that the zigzagging motion is coupled to a regular generation and discharge of alternate oppositely oriented hairpin-like vortex structures. Associated with the wake oscillation, the bubble experiences a strong asymmetric deformation in the equatorial plane at the inversion points of the zigzag path. The zigzag motion is superimposed on a small lateral drift of the bubble, which implies the existence of a net lift force. This is explained by the observed different strength of the hairpin vortices in the zig and zag path; a seemingly familiar phenomenon was found in recent numerical results of the sphere wake flow. For spiraling bubbles the wake is approximately steady to an observer moving with the bubble. It consists of a twisted pair of streamwise vortex filaments which are wound in a helical path and are attached to the bubble base at an asymmetrical position. The minor axis of the bubble is tilted in the tangential plane as well as in the radial plane toward the spiral center. Due to the pressure field induced by the asymmetrically attached wake two components of the lift force exist, one that causes the lateral motion and the other a centripetal force that keeps the bubble on a circular path. A mechanism is proposed to explain the reason for one bubble to spiral or to zigzag. Experiments with two simultaneous released bubbles show that bubble interaction is strongly triggered by the wake dynamics. Once a bubble is captured in the wake of a rocking bubble, it accelerates and rises via successive jumps until they collide. The jumps are explained by the upwards induction effect of the ring-like heads of the hairpin vortices being shed from the leading bubble. The final collision and repulsion thereafter abruptly enlarges the wake for a short moment, which is suggested to be one major contribution to the amplification of turbulence production in bubbly flows.
At the microscale level, it is impossible to obtain a completely smooth wall surface, and the effect of surface roughness may be a main factor responsible for some different characteristics between fluid flow in the microchannels and that in conventional size channels. In the present work, the lattice Boltzmann method is applied to investigate the gaseous flow in a microchannel with surface roughness which is modeled by an array of rectangular modules. The effects of relative surface roughness, roughness distribution, and rarefaction on gaseous flow are studied, but the compressibility effect is neglected since the Mach number is less than 0.2. It was shown that the surface roughness had an important influence on friction factor and mass flow rate. In particular, this effect becomes more significant with the decrease of the Knudsen number. This is because the rarefaction reduces the interaction between the gas molecules and the channel walls, which results in reduction of the surface roughness effect.
The capability of modeling and simulating complex interfacial dynamics of multiphase flows has been recognized as one of the main advantages of the lattice Boltzmann equation (LBE). A basic feature of two-phase LBE models, i.e., the force balance condition at the discrete lattice level of LBE, is investigated in this work. An explicit force-balance formulation is derived for a flat interface by analyzing the two-dimensional nine-velocity (D2Q9) two-phase LBE model without invoking the Chapman-Enskog expansion. The result suggests that generally the balance between the interaction force and the pressure does not hold exactly on the discrete lattice due to numerical errors. It is also shown that such force imbalance can lead to some artificial velocities in the vicinity of phase interface.
As an inherent physical phenomenon of the contact-line motion, the contact angle hysteresis (CAH) has an important effect on dynamic behaviors of bubble/bubbles from a solid surface. In this study, boiling heat transfer considering the CAH is investigated by employing a phase change lattice Boltzmann (LB) model. Effects of the CAH on the bubble/bubbles dynamics during the boiling process and the boiling curves for both the hydrophilic and hydrophobic hysteresis windows are studied in detail. It is found that the bubble base, bubble height and phase change rate increase as the increase of the width of hysteresis window. On the other hand, the pinning of three-phase contact line is observed during the bubble growth processes. And the pinning times increase with the contact angle hysteresis. For the hydrophilic hysteresis window, the bubble completely departs from the heater if a small range of the hysteresis window is considered, while a residual bubble is left when the bubble departs from the heater in the cases of the hysteresis window larger than an approximate critical value of 45°. For the hydrophobic hysteresis window, a residual bubble can be observed regardless of the level of the contact angle hysteresis. The numerical results also indicate that the CAH affects the position of bubbles coalescence. The bubbles merge above the heater for the hydrophilic hysteresis window with a small level of hysteresis. While the bubbles merge on the heater for the hydrophilic hysteresis window with a large level of hysteresis as well as for the hydrophobic hysteresis window. Moreover, as the increase of the width of the hysteresis window, the critical heat flux (CHF) as well as the Leidenfrost temperature (the minimum temperature for film boiling) decrease, and the transition boiling regimes become shorter. Finally, we present the fitting equations between the width of the hysteresis window and the CHF/Leidenfrost temperature. The results show that the CHF and the Leidenfrost temperature decrease linearly as the width of the hysteresis window increases.
This paper reports our numerical investigation on the bubble dynamics of two adjacent bubbles formed on the heated surface as the liquid pool is subjected to induced vibrations caused by oscillating solid bodies in periodic motion. The modeling involves 2D simulations of the entire ebullition cycle comprising of bubble nucleation, growth, coalescence, and departure by employing a combination of multiple relaxation time based lattice Boltzmann method with the finite difference method based thermal model. The numerical results throw insight into the different processes pertaining to bubble growth in the two systems, viz., the quiescent system (QS) and the system with oscillating solid bodies (OSBS). These include the bubble growth rate, vapor bridge formation, subsequent coalescence, and movement of three-phase contact lines. It is observed that the induced vibrations in the liquid pool leads to earlier nucleation and growth of the bubbles, and higher bubble departure frequency (f∗) due to additional forces acting on the bubble, which at one instance helps the two adjacent bubbles to coalesce and at the following instance pulls the coalesced bubble off the solid surface. A force balance analysis is presented to explain the evolution of the adjacent bubbles and their interactions. A sensitivity study is conducted to investigate the effects of unequal sizes of nucleation sites, unequal surface superheat (Ja), and distance between the nucleation sites. In all these cases, multiple bubbles are seen to form on the heated elements, which subsequently coalesce with each other and depart in a single ebullition cycle in OSBS, whereas only two initially formed bubbles are seen to merge and depart in a single ebullition cycle in QS. Subsequently, a sensitivity study is conducted to investigate the effects of surface wettability, and it is found that for a given surface superheat (Ja) and configuration of nucleation sites, f* reduces after a threshold value of wetting angle (θ∗) in QS while it reaches a maximum in OSBS before coming down. It is further observed that if the hydrophobicity of the surface is increased from θ* = 1.0 to 1.1 in OSBS, f∗ remains high until a threshold Ja, beyond which it reduces drastically due to a higher rate of bubble generation compared to detachment.
In this paper, the saturated pool boiling heat transfer from a heated circular surface is investigated numerically based on the popular lattice Boltzmann phase-change method. The crucial interfacial phenomena, including the bubble growth, departure, coalescence, and rising under different wall superheats, wettabilities, and heater sizes are studied, and the corresponding heat transfer characteristics within these bubble dynamics are also revealed. The numerical experiments indicate that at low wall superheats the bubble nucleation occurs only at the top of the heated circle. With the increase in the wall superheat, the sides and the bottom of the heated circle also become nucleate sites. As the wall superheats continue to increase, more and more nucleate sites are activated so that the heated circle is wrapped by the vapor film early in the boiling process. And the average heat flux varies periodically with time corresponding to bubble dynamics in nucleation boiling regime. It reaches a steady state of film boiling regime after the stable vapor film is generated from the heated circular. It is also found that hydrophobic surface is conducive to the onset of boiling. However, it leads to a low critical heat flux and incurs film boiling at a lower wall superheat, which is also observed in both experimental and numerical studies. On the other hand, the boiling regimes can undergo the transition from nucleate boiling regime to film boiling regime at the same wall superheat with the increase in the heater size. In addition, under various wall superheats and wettabilities, the bubble departure diameters from the circular surface are smaller than those from the flat surface, while the bubble departure periods from the circular surface are less than those from the flat surface one. Finally, the saturated boiling curves from the circular surface for different wall wettabilities, heater sizes, liquid–vapor density ratios, and heating modes are also achieved.
Bubble dynamics and two-phase flow phenomena are closely related to the performance of proton exchange membrane electrolyzer cells (PEMECs). This paper reports an in-situ study of the oxygen bubble behavior and associated multiphase evolutions in the anode side of PEMECs with titanium (Ti) felt liquid gas diffusion layers (LGDLs) by a high-speed visualization system. The micro oxygen bubble dynamics was captured and analyzed at different locations and virous operating conditions. The results show that the bubble detachment frequency and detachment diameter greatly increase with the operating current density. Additionally, they are significantly impacted by the local pore structure and morphology of Ti felt LGDLs. In the flow channels, there exist only several discrete micro bubbles at a low current density (0.04 A/cm²) and a large flow velocity (133 mm/s). At a current density (0.2 A/cm²) and a flow velocity (67 mm/s), a number of gas slugs are formed in the follow channels, in addition to discrete micro bubbles. At a high current density (1 A/cm²) and a flow velocity (67 mm/s), more bubbles appear in the channel, and the flow field is dominated by slug or annular flows. These investigations can help to better understand the two-phase flow and bubble detachment mechanism, and provide a foundation for electrochemical reaction, multiphase flow studies and optimize the design of gas diffusion layers and flow fields for PEMECs in the future.
No unified model is available yet to explain the dynamics of laser-induced cavitation bubbles during laser ablation of solid targets in liquids, when an extremely high capillary number is achieved (>100), i.e., when the viscous forces strongly contribute to the friction. By investigating laser-induced bubbles on gold and yttrium-iron-garnet targets as a function of the liquid viscosity, using a nanosecond laser and an ultrafast shadowgraph imaging setup, we give a deeper insight into what determines the bubble dynamics. We find that the competition between the viscous forces and the surface tension (capillary number Ca), on the one hand, and the competition between the viscous forces and inertia (Reynolds number Re), on the other hand, are both key factors. Increasing the viscous forces, and hereby Ca up to 100 has an impact on the bubble shape and results in a very pronounced rim, which separates the bubble in a spherical cap driven by inertia and an interlayer. The temporal evolution of the footprint radius of the interlayer can be addressed in the framework of the inertiocapillary regime. For an intermediate viscosity, the thickness of the interlayer is consistent with a boundary layer equation. Interestingly, our data cannot be interpreted with simplified hydrodynamic (Cox–Voinov) or molecular-kinetic theory models, highlighting the originality of the dynamics reported when extremely high capillary numbers are achieved. Upon bubble collapse, spherical persistent microbubbles are created and partly dispersed in water, whereas the high-viscous polyalphaolefines lead to long-standing oblate persistent bubbles sticking to the target’s surface, independent of the ablated target. Overall, liquid’s viscosity determines laser ablation-induced cavitation.
In this paper, we develop a conservative phase-field method for interface-capturing among N (N≥2) immiscible fluids, the evolution of the fluid-fluid interface is captured by conservative Allen-Cahn equation (CACE), and the interface force of N immiscible fluids is incorporated to Navier-Stokes equation (NSE) by chemical potential form. Accordingly, we propose a lattice Boltzmann equation (LBE) method for solving N (N≥2) immiscible incompressible NSE and CACE at high density and viscosity contrasts. Numerical simulations including stationary droplets, Rayleigh-Taylor instability, spreading of liquid lenses, and spinodal decompositions are carried out to show the accuracy and capability of present LBE, and the results show that the predictions by use of the present LBE agree well with the analytical solutions and/or other numerical results.
The porous electrodes (Ni, Cu) with 110 pores per inch (PPI) are adopted in water electrolysis for hydrogen production under normal-to-electrode magnetic field. The result shows that the voltage drop between electrodes can be reduced up to 2.5% under 0.9 T field, and the electric energy efficiency is improved correspondingly. Based on the numerical simulation method, the micro-magnetohydrodynamic (micro-MHD) convection induced by Lorentz force within the porous structure is found. The results showed that although the apparent current direction is parallel to magnetic field outside the porous electrode, the electric field may be distorted within the porous structure, and the Lorentz force is involved near the rib of the micro structure where the current is not parallel to the magnetic field any more. Micro-MHD plays the role of strengthening the mass transfer and facilitating bubble to eject from the porous structure, which results in the cell voltage decreasing. The combined application of porous electrode and magnetic field should be potential to further improve energy efficiency of water electrolysis for hydrogen production.
The dynamics behavior of a bubble passing through a bifurcated microchannel is studied numerically by using the lattice Boltzmann model. The effects of channel wettability, the viscosity ratio, the capillary number (Ca), the initial bubble size, and the flow flux ratio on the interface dynamic behavior, breakup mechanism, and residual mass of the bubble through the bifurcated microchannel are studied systematically. The simulation result indicates that these factors have significant influence on the bubble motion behavior. The bubble splits into two sub-bubbles and flow out of the channel completely when the channel surface is hydrophilic. However, some mass residuals of the bubble are observed when the channel surface is hydrophobic and the residual mass increases with the contact angle. On the other hand, as the viscous ratio of gas-liquid increases, the bubble is more likely to break up and to flow out of the channel. In addition, for the case of low capillary number and small bubble size, the bubble cannot break up, so it finally strands in the main channel. Besides, as capillary number increases, the flow flux ratio required for the bubble to flow out of subchannels increases. Eventually, we establish the relation for the critical flow flux ratio Qc as Qc=0.604e13.44Ca and Qc=1.985e5.53Ca to describe whether the bubble breaks up or not for different bubble radii.
In this paper, a novel lattice Boltzmann (LB) model based on the Allen-Cahn phase-field theory is proposed for simulating axisymmetric multiphase flows. The most striking feature of the model is that it enables to handle multiphase flows with large density ratio, which are unavailable in all previous axisymmetric LB models. The present model utilizes two LB evolution equations, one of which is used to solve fluid interface, and another is adopted to solve hydrodynamic properties. To simulate axisymmetric multiphase flows effectively, the appropriate source term and equilibrium distribution function are introduced into the LB equation for interface tracking, and simultaneously, a simple and efficient forcing distribution function is also delicately designed in the LB equation for hydrodynamic properties. Unlike many existing LB models, the source and forcing terms of the model arising from the axisymmetric effect include no additional gradients, and consequently, the present model contains only one non-local phase field variable, which in this regard is much simpler. In addition, to enhance the model’s numerical stability, an advanced multiple-relaxation-time (MRT) model is also applied for the collision operator. We further conducted the Chapman-Enskog analysis to demonstrate the consistencies of our present MRT-LB model with the axisymmetric Allen-Cahn equation and hydrodynamic equations. A series of numerical examples, including static droplet, oscillation of a viscous droplet, breakup of a liquid thread, and bubble rising in a continuous phase, are used to test the performance of the proposed model. It is found that the present model can generate relatively small spurious velocities and can capture interfacial dynamics with higher accuracy than the previously improved axisymmetric LB model. Besides, it is also found that our present numerical results show excellent agreement with analytical solutions or available experimental data for a wide range of density ratios, which highlights the strengths of the proposed model.
Rising bubble systems are used and investigated in a wide variety of industrial applications. However, the influence of strong confinement in rectangular flow regions has received little attention. An experimental study is undertaken here on a flow channel that allows the passage of bubbles from a region that can be modelled as two parallel plates into a region of rectangular confinement. The effect of a co-flow of a water/glycerol mixture on bubble size and rising velocity in the two confined regions for a wide variety of size ranges is investigated using particle shadow velocimetry. In the parallel plate region, as bubbles become larger in size, their terminal velocity increases due to the relatively higher buoyancy force and negligible effects of the confining geometry, compared to smaller bubble sizes. On entering the rectangular confinement, however, bubbles of relatively large size decelerate to a much lower terminal velocity due to the drag force expressed by the confining walls. Available models in the literature for predicting bubble terminal velocity through circular tubes and parallel plates were evaluated and showed poor predictive performance. To address this gap, a semi-empirical model for the bubble terminal velocity in a rectangular geometry is developed, based on the experimental data, to predict this motion. This model includes the effect of bubble size, fluid medium properties, net co-flow, and confinement geometry. The curious phenomenon of the threshold size of a bubble, which maintains a constant velocity through both geometries, is then predicted using the model.
In this paper, we present a simple and accurate lattice Boltzmann (LB) model for immiscible two-phase flows, which is able to deal with large density contrasts. This model utilizes two LB equations, one of which is used to solve the conservative Allen-Cahn equation, and the other is adopted to solve the incompressible Navier-Stokes equations. A novel forcing distribution function is elaborately designed in the LB equation for the Navier-Stokes equations, which make it much simpler than the existing LB models. In addition, the proposed model can achieve superior numerical accuracy compared with previous Allen-Cahn type of LB models. Several benchmark two-phase problems, including static droplet, layered Poiseuille flow, and Spinodal decomposition are simulated to validate the present LB model. It is found that the present model can achieve relatively small spurious velocities in the LB community, and the obtained numerical results also show good agreement with the analytical solutions or some available results. At last, we use the present model to investigate the droplet impact on a thin liquid film with a large density ratio of 1000 and the Reynolds number ranging from 20 to 500. The fascinating phenomenon of droplet splashing is successfully reproduced by the present model and the numerically predicted spreading radius exhibits to obey the power law reported in the literature.
The rise and shape oscillations of bubbles in a homogeneous, isotropic turbulent flow are studied numerically and compared to bubbles rising in quiescent liquid. The disturbances are generated by applying a pseudo-spectral forcing method on a fully periodic domain where a body force is randomly distributed in Fourier space at small wave-numbers. This produces velocity fluctuations at large length scales while the smallest length scales evolve naturally as a solution of the Navier–Stokes equations. Simulations of various cases considering a less deformable bubble ( ) and a more deformable bubble ( ) were carried out. The simulations were performed for clean bubbles without a surface tension gradient. The forcing parameters were chosen such that the bubble size is about equal and half the integral length scale. The mean rise Reynolds number ranged from 58 to 94, the ratio of the isotropic liquid velocity fluctuation to the bubble rise velocity varied from 0.01 to 0.12 and Stokes numbers ranging from 0.3 to 13.2, depending on the characteristic time scales of the liquid flow, were computed. The results for bubbles rising in a liquid with imposed velocity fluctuations revealed a reduction of the bubble rise velocity of up to 38% and an increase of the bubble velocity fluctuations, mainly caused by an increase of the lateral bubble motion. While minor changes in the average deformation for less deformable bubbles were found, the ellipticity of deformable bubbles increased up to 9.8%. The fluctuations of the orientation angle, as well as the angle of motion, were also increased. The characteristic frequencies of path oscillation and the frequencies of the shape and orientation angle were determined. Besides the amplification of dominant frequencies it was found that the frequency range expanded to lower and higher frequencies for simulations with forced velocity fluctuations.
A typical process in many industrial applications is rising bubble dynamic in viscous liquids like two-phase reactors. Examining the physical behavior of bubbles may improve the understanding of systems regarding design and operation. This study focused on the splitting of bubbles resulting from their impact on solid obstacles. Fragmentation of the bubbles appears in many applications such as lab on a chip in small scale or slug bubbly flow moving upward in a tube in large scales. Using a new index-function model in Lattice Boltzmann technique proposed by “He”, we simulated the deformation and motion of a bubble in different regimes, through which, we accurately captured a sharp interface between the two phases. We extended the aforementioned technique from 2D to 3D modelling of buoyancy-driven motion of a single bubble in quiescent viscous liquid. It was demonstrated that there is a reasonable agreement in terms of terminal rising velocity as well as bubble shape. This was found by comparing it with other available experimental and numerical results in different regimes through varying the two non-dimensional numbers (Eotvos and Morton) to characterize the fluid regime behind the rising bubble. In addition, by applying Bounce-Back no slip boundary condition to the surface area of the tubes with circular cross sections, we simulated the impact of the bubble during its upward motion. Changing the distance between the tubes and their corresponding diameters causes different shapes in the bubbles. Our simulation demonstrated that the 3D model based on index-function model of LBM is a suitable tool for 3D numerical simulation of rising bubbles in the presence of obstacles.
The dynamics of a toroidal bubble near a solid wall for a large part of stand-off parameters γ (γ=d/Rmax, d is the distance between the solid wall and the bubble centre at the moment of formation and Rmax is the maximum bubble radius) have been extensively studied, but some mechanics of a toroidal bubble are not completely clear, especially for the small stand-off parameters γ ≤ 0.8. In the present study, on the basis of the finite volume method, the Navier-Stokes equations with inviscid and incompressible assumption are directly solved using a staggered grid on the fixed grid. The dynamics of the toroidal bubble near the solid for different stand-off parameters (γ = 0.4, 0.6, 0.8, and 0.97, respectively) are simulated by a front tracking method. Initial conditions of numerical simulation are estimated through the Rayleigh-Plesset equation, based on the maximum size and collapse time of a spark-generated bubble. One of the numerical results is compared with a spark-generated bubble experiment, showing that the results between them are favorable with regard to both the bubble shape history and translational motion of the bubble. The numerical results for the different stand-off parameters, including the change process of the water layer, the development process of the splash flow and radial flow, the splitting phenomenon of the toroidal bubble, and the trend of pressure on the center of the solid wall, are discussed, where some new phenomena are discovered.
This article presents a critical review of the theory and applications of a multiphase model in the community of the lattice Boltzmann method (LBM), the pseudopotential model proposed by Shan and Chen (1993) [4], which has been successfully applied to a wide range of multiphase flow problems during the past two decades. The first part of the review begins with a description of the LBM and the original pseudopotential model. The distinct features and the limitations of the original model are described in detail. Then various enhancements necessary to improve the pseudopotential model in terms of decreasing the spurious currents, obtaining high density/viscosity ratio, reducing thermodynamic inconsistency, unraveling the coupling between surface tension and equations of state (EOS), and unraveling the coupling between viscosity and surface tension, are reviewed. Then the fluid–solid interactions are presented and schemes to obtain different contact angles are discussed. The final section of this part focuses on the multi-component multiphase pseudopotential model. The second part of this review describes fruitful applications of this model to various multiphase flows. Coupling of this model with other models for more complicated multiple physicochemical processes are also introduced in this part.
A newly developed two-phase mixture model is applied, in conjunction with a control-volume-based finite difference method, to numerically investigate boiling with thermal convection in a porous layer heated from below. The numerical procedure employs a fixed grid and avoids tracking explicitly the moving interface between the liquid and two-phase regions. Numerical results are obtained to shed light on the intricate interactions between boiling and natural convection as well as to explain experimental observations. Four distinct flow patterns that were observed in previous experiments are predicted. A quantitative comparison of the predicted and measured vapor volume fraction in the porous bed shows good agreement. The numerical results also agree with published linear stability results. In addition, the present study documents the effects of important parameters such as Rayleigh number, bottom heat flux, and aspect ratio.
a b s t r a c t Dynamics of a single rising gas bubble is studied using a Lattice Boltzmann Method (LBM) based on the Cahn–Hilliard diffuse interface approach. The bubble rises due to gravitational force. However, deformation and velocity of the bubble depend on the balance of other forces produced by surface tension, inertia, and viscosity. Depending on the primary forces acting on the system, bubble dynamics can be classified into dif-ferent regimes. These regimes are achieved computationally by systematically changing the values of Mor-ton number (Mo) and Bond number (Bo) within the following ranges ð1 Â 10 À5 < Mo < 3 Â 10 4 Þ and ð1 < Bo < 1 Â 10 3 Þ. Terminal shape and Reynolds number (Re) are interactive quantities that depend on size of bubble, surface tension, viscosity, and density of surrounding fluid. Accurate simulation of terminal shape and Re for each regime could be satisfactorily predicted and simulated, since they are also functions of Mo and Bo. Results are compared with previous experimental results.
An improved numerical algorithm for front tracking method is developed to simulate the rising of a bubble in quiescent viscous liquid due to buoyancy. In the new numerical algorithm, volume correction is introduced to conserve the bubble volume while tracking the bubble’s rising and deforming, and volume flux conservation based SIMPLE algorithm is adopted to solve the Navier–Stokes equation for fluid flow using finite volume method. The new front tracking algorithm is validated systematically by simulating single bubble rising and deforming in quiescent viscous liquid under different flow regimes. The simulation results are compared with the experimental measurement in terms of terminal bubble shape and velocity. The simulation results demonstrate that the new algorithm is robust in the flow regimes with larger ranges of Reynolds number (Re < 200), Bond number (Bo < 200), density ratio (ρl/ρb < 1000) and viscosity ratio (μl/μb < 500). The new front tracking algorithm is also applied to investigate bubble rising and deforming behaviour in the various flow regimes of “air bubble/water solution” system under effects of Reynolds number, Bond number, density ratio, viscosity ratio as well as the bubble initial shape, which have been explored previously by experiments. The predicted bubble shape and terminal velocity agree well with the experimental results. Hence, the new modelling algorithm expands the conventional front tracking method to more realistic and wider applications.
A method to simulate unsteady multi-fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described. The flow field is discretized by a conservative finite difference approximation on a stationary grid, and the interface is explicitly represented by a separate, unstructured grid that moves through the stationary grid. Since the interface deforms continuously, it is necessary to restructure its grid as the calculations proceed. In addition to keeping the density and viscosity stratification sharp, the tracked interface provides a natural way to include surface tension effects. Both two- and three-dimensional, full numerical simulations of bubble motion are presented.
A general interface tracking method based on the phase-field equation is presented. The zero phase-field contour is used to implicitly track the sharp interface on a fixed grid. The phase-field propagation equation is derived from an interface advection equation by expressing the interface normal and curvature in terms of a hyperbolic tangent phase-field profile across the interface. In addition to normal interface motion driven by a given interface speed or by interface curvature, interface advection by an arbitrary external velocity field is also considered. In the absence of curvature-driven interface motion, a previously developed counter term is used in the phase-field equation to cancel out such motion. Various modifications of the phase-field equation, including nonlinear preconditioning, are also investigated. The accuracy of the present method is demonstrated in several numerical examples for a variety of interface motions and shapes that include singularities, such as sharp corners and topology changes. Good convergence with respect to the grid spacing is obtained. Mass conservation is achieved without the use of separate re-initialization schemes or Lagrangian marker particles. Similarities with and differences to other interface tracking approaches are emphasized.
A lattice Boltzmann method for two-phase immiscible fluids with large density ratios is proposed. The difficulty in the treatment of large density ratio is resolved by using the projection method. The method can simulate two-phase fluid flows with the density ratio up to 1000. The method is applied to the simulations of a single rising bubble in liquid and many bubbles rising in a square duct. The terminal shapes and the terminal Reynolds numbers of the single bubble for various Morton and Eötvös numbers are in good agreement with available experimental data. The complicated unsteady structures of the interface and the flow field are illustrated in many bubbles rising in a square duct.
In this paper a conservative phase-field method based on the work of Sun and Beckermann [Y. Sun, C. Beckermann, Sharp interface tracking using the phase-field equation, J. Comput. Phys. 220 (2007) 626–653] for solving the two- and three-dimensional two-phase incompressible Navier–Stokes equations is proposed. The present method can preserve the total mass as the Cahn–Hilliard equation, but the calculation and implementation are much simpler than that. The dispersion-relation-preserving schemes are utilized for the advection terms while the Helmholtz smoother is applied to compute the surface-tension force term. To verify the proposed method, several benchmarks are examined and shown to have good agreements with previous results. It also shows that the satisfactions of mass conservations are guaranteed.
We propose the lattice BGK models, as an alternative to lattice gases or the lattice Boltzmann equation, to obtain an efficient numerical scheme for the simulation of fluid dynamics. With a properly chosen equilibrium distribution, the Navier-Stokes equation is obtained from the kinetic BGK equation at the second-order of approximation. Compared to lattice gases, the present model is noise-free, has Galileian invariance and a velocity-independent pressure. It involves a relaxation parameter that influences the stability of the new scheme. Numerical simulations are shown to confirm the speed of sound and the shear viscosity.
A new and very general technique for simulating solid-fluid suspensions has been described in a previous paper (Part I); the most important feature of the new method is that the computational cost scales with the number of particles. In this paper (Part II), extensive numerical tests of the method are described; for creeping flows, both with and without Brownian motion, and at finite Reynolds numbers. Hydrodynamic interactions, transport coefficients, and the short-time dynamics of random dispersions of up to 1024 colloidal particles have been simulated. Comment: Text and figures in uuencode-tar-compressed postcript Email tony_ladd@llnl.gov
A new and very general technique for simulating solid-fluid suspensions is described; its most important feature is that the computational cost scales linearly with the number of particles. The method combines Newtonian dynamics of the solid particles with a discretized Boltzmann equation for the fluid phase; the many-body hydrodynamic interactions are fully accounted for, both in the creeping-flow regime and at higher Reynolds numbers. Brownian motion of the solid particles arises spontaneously from stochastic fluctuations in the fluid stress tensor, rather than from random forces or displacements applied directly to the particles. In this paper, the theoretical foundations of the technique are laid out, illustrated by simple analytical and numerical examples; in the companion paper, extensive numerical tests of the method, for stationary flows, time-dependent flows, and finite Reynolds number flows, are reported. Comment: Text and figures in uuencode-tar-compressed postcript Email tony_ladd@llnl.gov
Effect of the pore geometry on pressure distribution within a bubble penetrating a single pore
- S Ansari
- D S Nobes
Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation
Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results