Shan-Chen Model Research Articles

Pore-scale investigation on multiphase reactive transport for the conversion of levulinic acid to γ-valerolactone with Ru/C catalyst

Hydrogenation of levulinic acid (LA) into γ-valerolactone (GVL) with Ru/C catalytic system involves homogeneous/heterogeneous reactions, multiphase mass transfer and hydrogen dissolution. The understanding of these physiochemical phenomena is fundamental to propose the performance improvement strategies in such reaction system. In this study, a pore-scale multiphase reactive transport model based on the lattice Boltzmann method (LBM) is developed to simulate the conversion of LA to GVL over Ru/C catalyst. A Carnahan-Starling equation of state (CS EOS) modified Shan-Chen pseudopotential model is applied for gas–liquid flow. The model is validated by comparing the simulation predictions with existing experimental results. Simulation results show that the intra-particle diffusion distance of hydrogen is shorter than that of LA. The LA consumption rate is found to be determined by temperature and dissolved hydrogen when the average concentration of LA is higher than a critical value (about 0.5 mol/L). Notable negative effect of LA intra-particle mass transfer on the reaction has been observed for the LA average concentration lower than 0.5 mol/L. The reaction-generated water would result in a little inhibition on LA hydrogenation. To enhance the utilization of catalyst and reduce the local transport resistant, hollow spheres or coating catalyst are more effective for this system. We demonstrate the usefulness of the present pore-scale LB model in providing guidance for efficient hydrogenation LA into GVL.

Researches on leakage with lubricating oil using a composite multi-phase lattice Boltzmann method

In this paper, a novel model to investigate leakage of gaseous working fluid in pressured devices with lubricating oil was created with lattice Boltzmann method and Shan-Chen multi-phase model. A method to adapt actual pressure-density relation into the lattice via a self-adapting timestep and simplify the simulation of compressible fluid was developed. A model to simulate two-phased leakage with lubricating oil was created with a combination of Shan-Chen model and passive scalar model. The model can realize the phase distribution simulation in the leakage field without causing the pressure and the inter-phase interactions to overlap. This model is also able to be combined with other multi-phase models. After a group of preliminary tests of the model, the characteristics of phase distribution and leakage were investigated qualitatively. Five types of phase distribution in the simulation results were classified, which are: uniformed distribution, sphered drips, gas channel, blocked channel, and slug bubbles. The results of simulations show good conformance with actual leakage patterns. Preliminary discussions about the leakage features are made upon the results. However, these simulation results are only qualitative and cannot show the quantitative features in leakages. More experimental investigations should be carried out to realize correlations to the model.

Lattice Boltzmann simulation of low-Reynolds-number cavitating contracting-nozzle flow interacting with a moving valve

Investigating internal-injector cavitating flow dynamics is difficult but important. The interaction of nozzle cavitation with the moving needle valve dictates the fuel mass flow rate and therefore spray combustion performance and emissions. In the present study, a two-dimensional low-Reynolds-number cavitating contracting-nozzle flow interacting with a moving valve is simulated using the lattice Boltzmann (LB) method. The Bhatnagar–Gross–Krook algorithm coupled with the immersed boundary method and an improved pseudo-potential multiphase flow model are employed and further developed based on the open-source LB code PALABOS. The performance of the immersed boundary method is first verified in a case where an oscillating cylinder moves according to a sine function in water. In order to improve the pseudo-potential model on its limitation of the density ratio, so to be used in engineering multiphase flow, the Carnahan–Starling equation of state is incorporated together with the exact difference method force scheme and an upgraded interaction force term. The upgraded pseudo-potential model proves via validations to be effective in improving numerical stability at large density ratios. With a seamless cooperation of the improved Shan–Chen model and the immersed boundary method achieved in PALABOS, cavitation in a contracting nozzle is simulated for a whole cycle of the valve motion. Cavitation dynamics under different fuel mass flow rates is investigated. It is found that cavitation dynamics, including interface conditions, cavitation bubble distributions, and inside-bubble vapor-phase flow fields, is distinctly different when the flow path is widely open and completely shut by the valve.

Simulation flows with multiple phases and components via the radial basis functions-finite difference (RBF-FD) procedure: Shan-Chen model

The current paper describes a rapid and impressive numerical technique to simulate the flows with multiple phases and components based upon the Shan-Chen model. The Shan-Chen model is a generalization of the incompressible Navier-Stokes equation with the variable density. The local radial basis function-generated finite difference method or local RBF-FD approach has been developed to solve the considered model. Furthermore, to get acceptable results, an extra diffusion term has been added to the density equation with a small coefficient. A reduced order technique e.g. proper orthogonal decomposition (POD) idea has been utilized to achieve a fast numerical procedure and to decrease the elapsed CPU time. The proposed numerical experiments show the ability, efficiency and acceptable results of the presented numerical scheme.

Evolution of Multiphase Lattice Boltzmann Method: A Review

With the simplicity and robustness of the lattice Boltzmann method (LBM), it is getting more attention nowadays in the multiphase flow applications. Different models of multiphase flow studies are available in LBM. Theory and applications of lattice Boltzmann multiphase models R–K color gradient, Shan–Chen (SC), Free energy (FE) and He–Chen–Zhang (HCZ) approaches have been discussed in this review. The methodologies of these methods have been explained in details. Besides, the advantages and limitations of these models have been explained by considering the application field. Based on the applications in the multiphase flows, recent developments in these models have been explained. The major applications of LBM in multiphase flows are flow through porous media, cavitation phenomenon, bubble rise or droplet fall, Rayleigh–Taylor instabilities, coalescence of bubbles or droplets, wettability and contact angles. An effort has been made to explain the methodologies for the multiphase LBM. Modified R–K color gradient model is more accurate and less efficient. SC model is more efficient and less accurate. FE model is numerically more accurate and less efficient. HCZ model more efficient and more accurate. The efficiency of modified R–K color gradient and free energy models almost similar.

Lattice Boltzmann simulations of droplet dynamics in two-phase separation with temperature field

This paper adds a temperature field based on the Shan–Chen model and constructs a new model. The two-phase separation, fluid flow, and heat transfer characteristics under the temperature field were studied by using this model. The performance of the three processes of collision, interface opening, and coalescence experienced by droplet formation was analyzed in detail. The results show that the velocity and temperature on the liquid film of the droplet are symmetric with respect to the central position of the liquid film. Moreover, the droplet velocity is also symmetric about the center of the droplet, which provides a theoretical basis for the droplet to maintain stability. By changing the wall temperature difference, the temperature distribution formula in the square cavity is proposed, which is highly consistent with the simulated value, and the maximum error is 10.1%. The proposed new model makes a meaningful supplement to the improvement of two-phase separation.

Pore-scale investigation on the thermochemical process in uniform and gradient porous media considering immiscible phase

In the present study, the complex thermochemical process in uniform and gradient porous media is investigated by the thermochemical dissolution model based on the lattice Boltzmann method considering immiscible phase. The two immiscible phase flow, solute transport and heat transfer are solved by Shan-Chen multicomponent lattice Boltzmann model, mass transport lattice Boltzmann model and multicomponent thermal lattice Boltzmann model, respectively. The immiscible phase, undissolved solid component and porosity gradient effects on the thermochemical process are investigated. For the solid, the dissolvable component is considered and updated by Volume of Pixel method. The results show that the immiscible phase and its initial distribution considerably affect the solid dissolution, acid transport and the average temperature in porous media. The undissolved solid component and the porosity gradient influence the thermochemical process significantly.

Simulation of high-viscosity-ratio multicomponent fluid flow using a pseudopotential model based on the nonorthogonal central-moments lattice Boltzmann method.

In this research, the development of a pseudopotential multicomponent model with the capability of simulating high-viscosity-ratio flows is discussed and examined. The proposed method is developed based on the non-orthogonal central moments model in the lattice Boltzmann method, and the exact difference model (EDM) is used to apply the intercomponent interaction force. In contrast to the original Shan-Chen model, in which the applying force has the viscosity-dependent error term, the error term of this model does not depend on the viscosity. A GPU parallel cuda code has been developed and is employed to study the proposed method. Different cases are considered to evaluate the ability of the model, including the Laplace test, a static droplet, and a two-component concurrent channel flow. Also, wetting and nonwetting relative permeabilities for flows with dynamic viscosity ratios between 0.0002 and 5000 are predicted. Numerical results are compared with those of available analytical solutions. Very good agreement between these results are observed. The model has the capability of simulating multicomponent flows with very low kinematic viscosities of the order of 10^{-5} and dynamic viscosity ratios of up to an order of 10^{4}, which is a much wider range compared with that of existing pseudopotential models. Furthermore, the results showed that the parallel processing on GPU significantly accelerated computations. The present parallel performance evaluation shows that the cuda parallel can achieve about 41 times improvement than the CPU serial implementation. The aforementioned enhancement increases the flexibility of the multicomponent lattice Boltzmann method and its applicability to a broader spectrum of engineering applications.

Numerical Investigation of Binary Fluid Flow through Propped Fractures by Lattice Boltzmann Method

Abstract In this paper, a pore-scale multiphase, multicomponent Shan–Chen model with proper pressure boundary scheme is developed and implemented to model binary flows in porous media, which enable...

Comparison and evaluation of Shan–Chen model and most commonly used equations of state in multiphase lattice Boltzmann method

A method to couple the Equation of State (EoS) with the Shan–Chen model (SC) is proposed in this paper. The difference between previously proposed approaches and the one developed here is in the manner of determining parameters of the EoS, which are a function of the critical properties of a substance. The modified relationships of the parameter that controls the strength of the inter-particle force as a function of temperature are developed for each EoS. These modified relationships eliminate the difference between pressures calculated by the SC model and each EoS. A comparison between the SC model and different popular EoSs is presented based on the proposed approach. Four models—the SC model, van der Waals EoS, Redlich–Kwong EoS, and Carnahan and Starling' EoS—are evaluated and compared, and the sign of the strength-controlling parameter (G) is discussed. The effects of the range of coexistence density, an arbitrary constant (ρo), temperature, and G on surface tension are discussed. The results of simulations show that the SC model sometimes yields negative values of surface tension when ρo is set below 1.5. Therefore, ρo should be set to 1 or ≥ 1.5.

Pore-scale investigation on reactive flow in porous media with immiscible phase using lattice Boltzmann method

Reactive flow happens during carbonate rock acidification. In the present study, carbonate rock dissolution process is investigated by lattice Boltzmann method at pore scale. The two immiscible fluids flow, solute transport and heat transfer are solved by the Shan-Chen multiphase multicomponent lattice Boltzmann model, mass transport lattice Boltzmann model and multicomponent thermal lattice Boltzmann model, respectively. The solid phase is updated by Volume of Pixel method. The results show that the immiscible phase has negative effect on dissolution, while positive effect on acid penetration. The immiscible phase effect is influenced by the inlet velocity, porous media porosity, surface wettability and temperature. The increasing inlet velocity and temperature accelerate the dissolution rate, but decrease the degree of immiscible phase suppression on dissolution. The surface wettability impact on dissolution process is influenced by the immiscible phase distribution. The porosity affects the degree of immiscible phase suppression on dissolution process slightly.

Discretization limits of lattice‐Boltzmann methods for studying immiscible two‐phase flow in porous media

SummaryDigital images of porous media often include features approaching the image resolution length scale. The behavior of numerical methods at low resolution is therefore important even for well‐resolved systems. We study the behavior of the Shan‐Chen (SC) and Rothman‐Keller (RK) multicomponent lattice‐Boltzmann models in situations where the fluid‐fluid interfacial radius of curvature and/or the feature size of the medium approaches the discrete unit size of the computational grid. Various simple, small‐scale test geometries are considered, and a drainage test is also performed in a Bentheimer sandstone sample. We find that both RK and SC models show very high ultimate limits: in ideal conditions the models can simulate static fluid configuration with acceptable accuracy in tubes as small as three lattice units across for RK model (six lattice units for SC model) and with an interfacial radius of curvature of two lattice units for RK and SC models. However, the stability of the models is affected when operating in these extreme discrete limits: in certain circumstances the models exhibit behaviors ranging from loss of accuracy to numerical instability. We discuss the circumstances where these behaviors occur and the ramifications for larger‐scale fluid displacement simulations in porous media, along with strategies to mitigate the most severe effects. Overall we find that the RK model, with modern enhancements, exhibits fewer instabilities and is more suitable for systems of low fluid‐fluid miscibility. The shortcomings of the SC model seem to arise predominantly from the high, strongly pressure‐dependent miscibility of the two fluid components.

Benchmark cases for a multi-component Lattice–Boltzmann method in hydrostatic conditions

Hydrostatic properties of partially saturated granular materials at the pore scale are evaluated by the lattice Boltzmann method (LBM) using Palabos implementation of the multi-component multiphase Shan-Chen model. Benchmark cases are presented to quantify the discretization errors and the sensitivity to geometrical and physical properties. This work offers practical guidelines to design LBM simulations of multiphase problems in porous media. Namely, a solid walls retraction procedure is proposed to reduce discretization errors significantly, leading to quadratic convergence. On this basis the equilibrium shapes of pendular bridges simulated numerically are in good agreement with the Young-Laplace equation. Likewise, entry capillary pressure and meniscus profiles in tubes of various cross-sectional shapes are in agreement with analytical predictions. The main points of this article are summarized as:•Benchmark cases for a multi-component Lattice-Boltzmann method are illustrated to be a guideline to calibrate the method in hydrostatic conditions.•A wall retraction procedure is introduced to minimize discretization errors.

Evaluation of equations of state in multiphase lattice Boltzmann method with considering surface wettability effects

A two-dimensional lattice Boltzmann method (LBM) is applied to investigate the use of various equations of state (EoS) for the simulation of liquid–vapor two-phase flow systems with considering the wetting properties, namely the hydrophilic and hydrophobic characteristics, for solid surfaces. The pseudo-potential single-component multiphase Shan–Chen model is used to resolve inter-particle interactions and phase change between the liquid and its vapor. Several EoSs, including the Redlich–Kwong (R–K), Carnahan-Starling (C–S), and Peng–Robinson (P–R) in comparison with the Shan–Chen (S–C) model are considered to study their effects on the numerical simulation results in terms of density ratios, spurious velocities and the contact angle of the two-phase flow with the solid wall. Accuracy and performance of the multiphase LBM by incorporating various EoSs are examined by solving two-phase flow systems at different conditions. Herein, three test cases considered are an equilibrium state of a droplet suspended in the vapor phase, a liquid droplet located on the solid surface, and a liquid droplet motion through a grooved channel with different wetting conditions. The results obtained demonstrate that implementation of the wall boundary condition with the wettability effects significantly impacts the numerical stability of the LBM with the EoSs employed for simulation of liquid–vapor flow problems, particularly at high-density ratio. Simulation of the equilibrium state of a droplet on a surface with considering wettability effects shows that the S–C model, R–K, P–R and C–S EoSs are stable for the maximum density ratio up to ρl∕ρv=74.6, 78.9, 4904.4 and 147, respectively. It is defined that the parasitic currents do not increase significantly due to imposing the wetting condition on the solid wall, however, the numerical solutions with considering wettability effects are more sensitive at high-density ratios. The present study demonstrates that the P–R EoS is more stable for simulation of high-density ratio liquid–vapor systems with reasonable spurious currents in the interfacial region for the flow problems with the periodic computational domain. However, with considering the wetting wall boundary condition, the C–S EoS produces less spurious velocity in the interface region, which leads to more precise and stable numerical simulations in comparison with the other EoSs applied for the equilibrium state of a liquid droplet on the solid surface. The results obtained also demonstrate the capability of the multiphase LBM for predicting practical flow characteristics with different EoSs implemented.

Numerical study of water–air distribution in unsaturated soil by using lattice Boltzmann method

Identification of water–air distribution at meso-scale in unsaturated soil is of great significance in the field of soil studies. In order to better understand the water–air distribution in soil pore, a meso-scale numerical model was developed: (1) a stochastic generation method was developed for the generation of mesoscopic soil structure, (2) the lattice Boltzmann method was selected for the numerical solutions of multiphase flow partial differential equation (PDE) in soil pore, (3) the water–air interface was tracked by using Shan–Chen model. The generation of soil structure and the solution of governing equations were completed by Open Source Lattice Boltzmann Code (OpenLB). The simulated water–air distribution shapes were almost the same with those reported in literature, indicating that the model developed in this study can be well used to evolve the water–air interface formation. Water–air​ distributions in soil pore at different porosities, wettabilities and saturations were detailed.

Simulation study on bubble motion in capillaries based on lattice boltzmann method

The lattice Boltzmann method with mesoscopic properties can conveniently describe the interaction of multiphase molecules and has wide application prospects in the field of multiphase flow. In this paper, the improved Shan-Chen pseudo-potential multiphase model in lattice Boltzmann method was used to simulate the process of bubble passing through stenotic capillaries during the pathogenesis of decompression sickness, and the velocity variation of the fluid in the process of flow was studied. According to the research results, it can be concluded that: (1) in the direct channel, the velocity of the fluid slows down with the increase of the gas composition, and the clogging can cause a more obvious trend of deceleration; (2) in the narrow channel, the fluid velocity changes abruptly when the gas enters and leaves the narrow area, and with the increase of the gas composition, the velocity change tends to be stable when the gas can completely fill the narrow area. This research provides a theoretical basis for further understanding the pathogenesis of decompression sickness.

Design, fabrication, and modeling of a low-cost droplet on demand device for lab-on-chip applications

In many recent biomedical and other microfluidic engineering applications, generation and processing of a drop for a specific application has gained importance. The droplet-on-demand (DOD) is the important component of the microfluidic systems. In this work, the efficient low-cost PCB based design and fabrication method along with the analysis of a flow focused DOD system is explored. Modeling such multiscale and multiphysics problem is a challenging task. As a numerical design tool for such simulations, an improved multiphase Shan-Chen model has been used with 2D-lattice Boltzmann method. The numerical technique is validated with the experimental results. The effect of orifice geometry on droplet characteristics such as jetting and time interval of droplets has been studied in detail. Results show that the decrease in the orifice width leads to the reduction in the jetting effect and the droplet area. With the obtained result, one can tune the droplet area and time by choosing the orifice width and length for a particular size of the drop.

Pore-Scale Simulations of Simultaneous Steady-State Two-Phase Flow Dynamics Using a Lattice Boltzmann Model: Interfacial Area, Capillary Pressure and Relative Permeability

The dynamics of simultaneous flow of immiscible two-phase fluids at the steady state in the capillary force-dominated regime were investigated. It was described by the key state variables, including interfacial area between fluids, capillary pressure and relative permeability, as functions of fluid saturation. Based on the two-dimensional pore-scale simulations using the Shan-Chen multi-component lattice Boltzmann model (SCMC-LBM), the steady-state interfacial area, capillary pressure and relative permeability versus saturation relationships subjected to flow conditions, such as initial fluid distribution, saturation history and magnitude of hydraulic gradient, were examined. Also, the intrinsic differences in the interfacial area and capillary pressure versus saturation curves between at dynamic equilibrium in the steady-state infiltration and at quasi-static equilibrium in the transient displacement were explored. As the SCMC-LBM simulations revealed, the relative permeabilities of both fluids were insensitive to initial fluid distribution or saturation history because of the simultaneous steady-state flow dynamics, but dependent on the magnitude of hydraulic gradient due to the existence of a threshold hydraulic gradient. The hysteresis behaviours of interfacial area-saturation and capillary pressure–saturation curves were captured in the transient displacement but absent in the steady-state infiltration, and the unique interfacial area-capillary pressure–saturation surface at dynamic equilibrium did not tend to overlap with the one at quasi-static equilibrium. The hysteretic capillary pressure behaviour for these two flow patterns was of great theoretical and practical significances.

Modeling condensation on structured surfaces using lattice Boltzmann method

A single component, two-phase lattice Boltzmann model based on the Shan-Chen pseudopotential model is coupled to a continuum-based thermal energy solver. The model is verified using the Stefan problem and Nusselt’s falling film. We then use the model to simulate 2D drop-wise condensation of a saturated vapor on hydrophobic structured surfaces. We report the normalized condensed mass and condensation rate as we vary the spacing between structures, numerical interface thickness, and structure topology. For a non-dimensional spacing of s=6.4, a primary droplet engulfs each post after a non-dimensional thermal diffusion time of t∗≈1.1 (square), t∗≈0.59 (semi-circular), and t∗≈2.38 (equilateral). The growth of this thermally resistant primary droplet is fed by coalescing satellite droplets characterized by low heat transfer resistance. These small satellite droplets continuously nucleate, grow and merge into the primary droplet resulting in condensation rates four to seven times greater than the film-wise condensation rate. These small droplets only appear to form when the ratio of numerical interface thickness w to structure length L is sufficiently small (w/L<0.1).

Numerical Study on Bubble Motion in Pore Structure under Microgravity Using the Lattice Boltzmann Method

This paper reports on a study of bubble motion in a regularly spaced pore structure under microgravity using the lattice Boltzmann method (LBM). Initially, a single bubble’s motion is derived; then the simulation is extended for two-bubble dynamics. By considering a gas-liquid two-phase flow, the interaction force (namely the fluid-fluid cohesive force and fluid-solid adhesive force) is derived using the Shan-Chen model with multicomponent single relaxation. This determines the interaction forces governing a single bubble’s dynamics in the porous structure. Then the primary parameters that influence bubble motion are studied, such as the bubble’s diameter, distance between adjacent cells, arrangement between cells, and the effects of flow-field characteristics on single-bubble motion. The simulation developed for single-bubble motion provides the basis for studying a two-bubble system’s motion trajectory and coalescence behavior. By employing the aforementioned analysis, the proposed approach optimizes porous media’s structural parameters under microgravity, resulting in increased bubble movement speed and an enhanced two-phase flow in porous media.