Proposed Lattice Boltzmann Model Research Articles

Lattice Boltzmann model for incompressible flows through porous media with time-fractional effects

Anomalous transport has been commonly observed in the fluid flow through a complex porous medium, where the evolution exhibits a complex memory-like behavior. Classical integer-order models fail to depict anomalous transport phenomena, while fractional calculus has been proved effective in describing such behavior due to its ability to characterize long memory processes. In this paper, the time-fractional generalized Navier-Stokes (N-S) equations are formulated to model anomalous transport in porous media at the representative elementary volume (REV) scale by incorporating the Caputo fractional derivative of time into the widely employed generalized N-S equations. An innovative lattice Boltzmann (LB) model is presented for solving the time-fractional generalized N-S equations. We validate the proposed LB model by conducting a numerical example with analytical solutions, and observe a good agreement between the LB results and the analytical solutions. The proposed LB model is utilized for simulating Poiseuille flow, Couette flow, and cavity flow in porous media. Unlike the previous studies, which focus on the steady state of these flows, the present work focuses on the unsteady process from the initial state to the final steady state. It is found that time-fractional effects play a significant role in the unsteady processes of these flows. As the fractional order decreases, the flows undergo a slower evolution process and the impact of Darcy number on the unsteady process becomes increasingly noticeable. The Reynolds number has a significant impact on the velocity profile in the steady state, but it has relatively little effect on the duration of the unsteady process.

Simulation of CO2 dissolution reactions in saline aquifers using lattice Boltzmann method

CO2 dissolution reactions in saline aquifers during geologic carbon sequestration (GCS) involve multiple physicochemical processes and pose a challenge for both experimental and numerical techniques to capture reactive transport features and complex porous structures. Benefited from advanced models in the lattice Boltzmann (LB) community, namely, multiphase flow LB model and multicomponent transport LB model with chemical reactions, dynamics of CO2-water interfaces, concentration evolution of ions, and heterogeneous and homogeneous reactions could be readily captured by the proposed LB model to characterize CO2 dissolution reactions applicable to far-field applications. This LB model can not only emphasize CO2 dissolution processes at the scale of a few pores but also elaborate on the corresponding controlling mechanisms in porous media during GCS in saline aquifers. LB simulations herein indicate that both diffusion coefficient and reactive interfacial length may change the total concentration gradient at the CO2-water interface to further affect CO2 dissolution reactions. It is also confirmed that CO2 dissolution is a non-equilibrium process, which could not be neglected in reservoir-scale simulations. The effect of reactive interfacial length on CO2 dissolution is negligible due to a narrower range of contact angles in saline aquifers. Influenced by the structural complexity of porous media, the dissolution of CO2 clusters shows a strong heterogeneous characteristic evaluated by the pore size distribution. The evolution of pH in homogeneous reactions is estimated via mass action equations. Besides, the effects of partial pressure, formation temperature, and salinity, as three key factors in the practical application, are thoroughly probed. The high pressure is conducive for sequestration because of its greater sequestration efficiency, but the temperature has both positive and negative impacts on CO2 dissolution trapping and should be carefully evaluated in a certain saline aquifer before the operation. High ion strength would decrease the efficiency of dissolution trapping to some extent.

Modeling and simulation of nanofluid in low Reynolds numbers using two-phase Lattice Boltzmann method based on mixture model

In the present work, a novel two-phase scheme was implemented for modeling and simulation of Al2O3/water nanofluid within a rectangular channel with the Lattice Boltzmann method based on the mixture model. In this proposed model, a two-phase approach was adopted to consider the effect of the drift velocity of each phase. The simulations are performed in two cases at laminar flow regime and low Reynolds numbers (Re=0.5 in this case). In the first case, the results of the velocity distribution of the nanofluid through the channel were numerically validated with the finite element method results, and the predicted velocity profile at Reynolds numbers of 1, 10, and 50 showed good agreement with the velocities extracted from FEM. In the second case, the velocity distribution and particle volume fraction of the nanofluid were investigated at Reynolds numbers 0.5, 1, and 10. The results showed that considering the drift velocity in the modeling has remarkable effects only at low Reynolds numbers (Re=0.5 in this case), when the Reynolds number increases, the effect of drift velocity can be ignored. The proposed Lattice Boltzmann model is promisingly strong and suitable for engineering and scientific problems.

Lattice Boltzmann Method for the Generalized Black-Scholes Equation

In this paper, an efficient lattice Boltzmann model for the generalized Black-Scholes equation governing option pricing is proposed. The Black-Scholes equation is firstly equivalently transformed into an initial value problem for a partial differential equation with a source term using the variable substitution and the derivative rules, respectively. Then, applying the multiscale Chapman-Enskog expansion, the amending function is expanded to second order to recover the convective and source terms of the macroscopic equation. The D1Q3 lattice Boltzmann model with spatial second-order accuracy is constructed, and the accuracy of the established D1Q5 model is greater than second order. The numerical simulation results demonstrate the effectiveness and numerical accuracy of the proposed models and indicate that our proposed models are suitable for solving the Black-Scholes equation. The proposed lattice Boltzmann model can also be applied to a class of partial differential equations with variable coefficients and source terms.

Axisymmetric lattice Boltzmann model for liquid flows with super-hydrophobic cylindrical surfaces

The lattice Boltzmann (LB) method has been extensively applied to the simulation of various fluid flows. Nonetheless, the standard LB model in rectangular coordinates faces some challenges when attempts are made to use it to simulate single-phase liquid flows involving super-hydrophobic cylindrical surfaces with a large slip length. Therefore, in this paper, boundary conditions for an axisymmetric LB model are proposed for the simulation of liquid slip at both convex and concave cylindrical surfaces. Based on the Navier slip boundary conditions, an equilibrium–nonequilibrium boundary scheme is developed to obtain the liquid slip in the azimuthal direction, with a combined bounce-back and specular-reflection boundary condition being developed to obtain the liquid slip in the axial direction. These boundary schemes are verified by simulating several established cases, namely, cylindrical Couette flow, harmonic oscillatory cylindrical Couette flow, Hagen–Poiseuille flow, and annular Poiseuille flow. The results are in excellent agreement with analytical results. Furthermore, some more complex flows involving super-hydrophobic cylindrical surfaces, such as stepwise oscillatory cylindrical Couette flow, are also simulated and studied using the proposed LB model. The axisymmetric LB model presented here overcomes the inability of standard LB models to accurately and efficiently simulate single-phase liquid flows involving super-hydrophobic cylindrical surfaces with a large slip length.

A study with the lattice Boltzmann method on the conversion efficiency of a packed-bed reactor with different oriented packed beads configurations

We consider packed-bed reactors with dielectric beads in a two-dimensional channel geometry, apply an electric field perpendicular to the walls, and explore numerically the sensitivity of reaction conversion efficiencies of a dissociation reaction on system parameters like shape, orientation, and size of the beads and porosity of packed-bed systems. We have developed a lattice Boltzmann (LB) model that allows for simultaneous simulation of the flow field, the electric field within fluid and (solid) beads, and transport of (charged) species, such as ions and reagents. It solves Navier–Stokes for the fluid flow and the concentration field for neutral and charged species by the advection–diffusion and Nernst–Planck equation, respectively, formulated in the LB framework. The model allows to compute electric field strengths in the fluid and in the beads, by solving the Poisson equation. The method is suitable for arbitrary geometries of the flow domain and does not require body-fitted meshes. Two important conclusions can be drawn. First, the proposed LB model enables simulation of a reactive electro-kinetic fluid in a reactor with dielectric packed beads of arbitrary shape, size, and orientation. The LB method is based on Cartesian meshes irrespective of the shape of the beads and is highly parallelizable and can be extended to three-dimensional packed-bed reactors. Second, we show that reactor conversion efficiency is sensitive to shape, orientation, and size of the beads and the porosity of the packed-bed reactor. Present observations will guide the parameter settings for the beads and packed-bed reactor of more realistic three-dimensional configurations.

A pseudopotential lattice Boltzmann model for simulating mass transfer around a rising bubble under real buoyancy effect

A pseudopotential multicomponent lattice Boltzmann (LB) model that can account for the real buoyancy effect is proposed to simulate the mass transfer process around a rising bubble. The density profiles at the equilibrium state are determined based on the hydrostatic condition and the zero diffusion flux condition (the balance of chemical potential). Compared with the LB models using effective buoyancy force, the proposed model has three advantages: (1) avoiding the unrealistic distribution of gas components within the bubble due to the upward effective buoyancy force, (2) removing the undesirable diffusion process due to the application of effective buoyancy force, and (3) considering the effect of the pressure gradient on the change of bubble size. In addition, Henry's law, which can be automatically recovered from the multicomponent LB equation, is re-interpreted from the perspective of the balance of chemical potential. Simulation results showed that the diffusion flux non-uniformly distributes over the surface of a rising bubble. The diffusion zone primarily occurs at the top and the lateral side of a rising bubble, whereas the diffusion transport just below the rising bubble is much less significant than its counterpart above the rising bubble. Various bubble shapes and their corresponding diffusion zones have been obtained. Moreover, the correlation between the Sherwood number and the Peclet number derived from the simulation results is consistent with those from previous numerical results. Thus, the proposed LB model is capable of conducting a quantitative analysis of the mass transfer around a rising bubble.

A phase-field-based lattice Boltzmann model for multiphase flows involving N immiscible incompressible fluids

In this work, an efficient and accurate lattice Boltzmann (LB) model is developed based on phase-field theory to study multiphase flows involving N (N≥2) immiscible incompressible fluids. In this model, a reduction-consistent physical formulation including a volume-fraction-dependent mobility in the Cahn–Hilliard (C–H) equations is adopted. Usually, the effect of cross-diffusion makes it difficult to solve such equations directly with the classic LB method. To avoid requiring a special treatment on the cross-diffusion terms of the chemical potential gradients, the proposed LB model introduces some non-diagonal collision operators. In addition, the proper auxiliary source terms are constructed to ensure the correct macroscopic equations. Through a direct Taylor expansion, the C–H equations are recovered from the present LB model. Finally, four classical problems including static droplets, the spreading of a liquid lens between two phases, the Kelvin–Helmholtz instability, and the dynamics of droplets in a four-phase system are used to demonstrate the capability of the LB model. The numerical results show that the present model satisfies the reduction-consistent property and produces physically accurate results.

Lattice Boltzmann model for multidimensional polarized radiative transfer: theory and application

Polarimetric observations of planets are providing increasing details of the three-dimensional (3D) atmospheric structure. The one-dimensional plane-parallel approximation model neglects horizontally polarized radiative transfer. Multidimensional polarized radiative transfer models, especially 3D models, are required to contain the horizontal polarization mechanism. Here, we propose a lattice Boltzmann (LB) model for multidimensional polarized radiative transfer, which enables a simple solution of the multidimensional vector radiative transfer equation (VRTE) by performing collision and streaming processes. Through the Chapman–Enskog analysis, we rigorously derive the multi-dimensional VRTE from the proposed LB model. 2D and 3D numerical tests demonstrate that the proposed LB model is effective and accurate for simulating multidimensional polarized radiative transfer. Furthermore, we apply the proposed LB model to investigate the effects of multiple scattering on radiation intensity and degree of polarization in a 3D case and find that multiple scattering enhances the radiation intensity but dampens the degree of polarization throughout almost the whole angular space in multidimensional polarized radiative transfer. This work is expected to provide a simple and effective mesoscopic tool for multidimensional polarized radiative transfer.

A Novel Lattice Boltzmann Model for Fourth Order Nonlinear Partial Differential Equations

In this paper, a novel lattice Boltzmann (LB) equation model is proposed to solve the fourth order nonlinear partial differential equation (NPDE). Different from existing LB models, a source distribution function is introduced to remove some unwanted terms in the nonlinear part of the equation. Hereby, the equilibrium distribution function is designed to follow the rule of Chapman–Enskog (C–E) analysis. Through the C–E procedure, the fourth order NPDE can be recovered perfectly from the proposed LB model. A series of numerical experiments have been carried out to solve some widely studied fourth order NPDEs, including the Kuramoto–Sivashinsky equation, Cahn–Hilliard equation with double-well potential and a fourth order diffuse interface model with Peng–Robinson equation of state. Numerical results show that the performance of the present LB model is much better than other existing LB models.

A lattice Boltzmann model for the coupled cross-diffusion-fluid system

In this paper, we propose a lattice Boltzmann (LB) model for the generalized coupled cross-diffusion-fluid system. Through the direct Taylor expansion method, the proposed LB model can correctly recover the macroscopic equations. The cross diffusion terms in the coupled system are modeled by introducing additional collision operators, which can be used to avoid special treatments for the gradient terms. In addition, the auxiliary source terms are constructed properly such that the numerical diffusion caused by the convection can be eliminated. We adopt the developed LB model to study two important systems, i.e., the coupled chemotaxis-fluid system and the double-diffusive convection system with Soret and Dufour effects. We first test the present LB model through considering a steady-state case of coupled chemotaxis-fluid system, then we analyze the influences of some physical parameters on the formation of sinking plumes. Finally, the double-diffusive natural convection system with Soret and Dufour effects is also studied, and the numerical results agree well with some previous works.

Consistent lattice Boltzmann model for multicomponent mixtures

Abstract

Study of pore-scale coke combustion in porous media using lattice Boltzmann method

In-situ combustion (ISC) for heavy oil recovery is a high-risk process and thus calls for a deep understanding of the coke combustion front. However, simulations for describing pore-scale coke combustion are challenging because of complicated combustion dynamics. This study proposes a multiple-relaxation-time (MRT) lattice Boltzmann (LB) method to simulate solid coke combustion in porous media at the pore scale. The MRT LB model can enhance numerical stability compared with the single-relaxation-time counterpart. This new LB model offers two major advances over existing LB models. First, thermal expansion effects are considered via temperature-dependent density, satisfying the low Mach number flow condition. Second, conjugate heat transfer and species conservation at the coke-fluid interface are modelled by a source term and a bounce-back scheme, respectively, without resorting to iterative methods. Simulations of pore-scale coke combustion are conducted to validate the superior performance of the developed LB model. The results show that this model can reproduce coke combustion dynamics and capture variations in isotherms from the fluid phase to the solid phase at the pore scale. In contrast, without considering effects of thermal expansion or conjugate heat transfer as in previous models, simulations would underestimate the burning temperature and even fail to predict instability caused by a high driving force. Moreover, using the proposed LB model, a parametric study is performed to assess the impact of the inlet air and porous structure on coke combustion. The findings suggest that inlet air temperature and driving force should be controlled within certain ranges, otherwise their influences may become negligible or even negative. Large porosity accelerates coke combustion and leads to a desired combustion front. Heterogeneous coke distribution affects pore-scale combustion details but not the overall combustion intensity, while a random matrix structure may influence combustion dynamics both locally and globally. As a whole, this study develops a reliable LB model to investigate pore-scale coke combustion, which contributes to advancing the knowledge base for heavy oil recovery using the ISC process.

Three-dimensional simulation of reservoir temperature and pollutant transport by the lattice Boltzmann method

Predicting the three-dimensional (3D) transport processes of reservoir temperature and pollutants is essential for water environmental protection and restoration, and introducing the lattice Boltzmann (LB) method into this prediction is necessary because of its simple algorithm, straightforward implementation of boundary conditions, and high computation efficiency. In this paper, a triple-distribution function (TDF) LB model for flow-temperature-concentration coupling simulations is introduced. Some essential techniques for implementing this method in 3D reservoirs are also described, including the treatment of water surface fluctuation, the consideration of surface heat exchange, and the hardware acceleration using the graphics processing unit (GPU). Two cases verified the proposed model, and then, the temporal-spatial variations of flow, temperature, and pollutants in the upper reservoir of a pumped-storage power station during both pumping and generating modes were analyzed to demonstrate its applicability. In the reservoir, the water forms several circulations, the cold water from the inlet flows as an undercurrent firstly, and then spread laterally, and the spreading of pollutants directly relates to the flow velocity. The results of flow, temperature, and concentration fields in different working conditions are consistent with model tests and physical laws, which shows good prospects of the proposed LB model.

Lattice Boltzmann method for simulation of shale gas flow in kerogen nano-pores considering temperature dependent adsorption

Due to the combined action of gas adsorption, surface diffusion and slippage, classical simulation approaches based on Darcy's law may not be appropriate for simulating shale gas flow in shale. In this work, a novel lattice Boltzmann (LB) model is proposed to study shale gas flow in a kerogen pore by introducing temperature dependent thickness of adsorption layer. The surface diffusion, which is caused by gradient of adsorption density, is considered as the slippage velocity on the surface of adsorption layer. The proposed LB model was adopted to simulate shale gas flow in a nano-pore. The results show that adsorption can significantly decrease the permeability of nano-pores. Surface diffusion improves the gas movement in nano-pores at lower pressure. With the decrease of pore size, the adsorbed layer had more impacts on gas permeability. Increasing temperature improves gas flow ability in nano-pores when pore size is less than 10nm. [Received: December 18, 2017; Accepted: May 2, 2018]

Investigation of open channel flow with unsubmerged rigid vegetation by the lattice Boltzmann method

Aquatic vegetation can significantly affect flow structure, sediment transport, bed scour and water quality in rivers, lakes, reservoirs and open channels. In this study, the lattice Boltzmann method is applied for performing the two dimensional numerical simulation of the flow structure in a flume with rigid vegetation. A multi-relaxation time model is applied to improve the stability of the numerical scheme for flow with high Reynolds number. The vegetation induced drag force is added in lattice Boltzmann equation model with the algorithm of multi-relaxation time in order to improve the simulation accuracy. Numerical simulations are performed for a wide range of flow and vegetation conditions and are validated by comparing with the laboratory experiments. Analysis of the simulated and experimentally measured flow field shows that the numerical simulation can satisfactorily reproduce the laboratory experiments, indicating that the proposed lattice Boltzmann model has high accuracy for simulating flow-vegetation interaction in open channel.

A lattice Boltzmann model for multi-component two-phase gas-liquid flow with realistic fluid properties

Multi-component multi-phase flows are of significant interests in nature and engineering problems of different fields. Modeling the phenomena involved in multi-phase flows is challenging, attributed to the complexity in simulating phase interface dynamics and diffusion processes. Owing to its kinetic nature, lattice Boltzmann (LB) method emerges as an attractive computational approach, in dealing with complicated fluid flow problems and microstructure geometries with effectiveness of parallelized processing. However, some critical drawbacks of basic multi-phase LB models, such as the low density and kinematic viscosity ratios, thermodynamic inconsistency, and dependence of surface tension and density distribution on relaxation times, limit its application in realistic multi-component multi-phase systems. Based on the original pseudopotential model and progresses in single-component multi-phase model, a multi-component LB model was proposed to study the two-phase gas-liquid flow with realistic fluid properties. The importance of improved model is in simultaneously realizing the realistic fluid flow characteristics for multi-component two-phase system, including high density and viscosity ratios, good thermodynamic consistency, independently tunable surface tension and appropriate two-phase boundaries. The proposed LB model is validated with Laplace law and visualization experiment results, and the effects of surface tension, gas flow velocity and wall wettability on dynamic behaviors of fluid flow are investigated.

Lattice Boltzmann model for high-order nonlinear partial differential equations.

In this paper, a general lattice Boltzmann (LB) model is proposed for the high-order nonlinear partial differential equation with the form ∂_{t}ϕ+∑_{k=1}^{m}α_{k}∂_{x}^{k}Π_{k}(ϕ)=0 (1≤k≤m≤6), α_{k} are constant coefficients, Π_{k}(ϕ) are some known differential functions of ϕ. As some special cases of the high-order nonlinear partial differential equation, the classical (m)KdV equation, KdV-Burgers equation, K(n,n)-Burgers equation, Kuramoto-Sivashinsky equation, and Kawahara equation can be solved by the present LB model. Compared to the available LB models, the most distinct characteristic of the present model is to introduce some suitable auxiliary moments such that the correct moments of equilibrium distribution function can be achieved. In addition, we also conducted a detailed Chapman-Enskog analysis, and found that the high-order nonlinear partial differential equation can be correctly recovered from the proposed LB model. Finally, a large number of simulations are performed, and it is found that the numerical results agree with the analytical solutions, and usually the present model is also more accurate than the existing LB models [H. Lai and C. Ma, Sci. China Ser. G 52, 1053 (2009)1672-179910.1007/s11433-009-0149-3; H. Lai and C. Ma, Phys. A (Amsterdam) 388, 1405 (2009)PHYADX0378-437110.1016/j.physa.2009.01.005] for high-order nonlinear partial differential equations.

Study on the micro-scale effect in the micro–nano organic pores of a shale reservoir

The micro-scale effect occurs because of the gas flows in shale reservoirs containing small organic pores. In this study, based on a lattice Boltzmann model incorporating the bounce-back and specular-reflection boundary conditions, the gas flow through two parallel plates driven by differential pressures is simulated, which in turn verifies the model. Considering the effects of the slippage, surface diffusion, and adsorption, the gas flow in the organic channels is simulated based on the proposed lattice Boltzmann model. It is shown that the micro-scale effect is significant in the gas flows in the micro-scale channels. The compression effect leads to a nonlinear distribution of pressures along the centreline of the flow channels, and the nonlinearity increases with an increase in the pressure difference between the two sides of the flow channel. In case of gas flows in organic pores, the adsorption effect decreases the free-gas velocities, and with a decrease in pore sizes, the average free-gas velocity is reduced owing to the increased adsorption effect. The slippage effect and surface diffusion have a significant impact on the free-gas velocity of the mass flow. Compared to the slippage effect, surface diffusion contributes more to the mass flow of gas in organic pores. An increase in the Knudsen number intensifies the slippage effect and surface diffusion and enhances the mass flow.

Lattice Boltzmann solution of the transient Boltzmann transport equation in radiative and neutron transport.

Applications of the transient Boltzmann transport equation (BTE) have undergone much investigation, such as radiative heat transfer and neutron transport. This paper provides a lattice Boltzmann model to efficiently resolve the multidimensional transient BTE. For a higher angular resolution, enough transport directions are considered while the transient BTE in each direction is treated as a conservation law equation and solved independently. Both macroscopic equations recovered from a Chapman-Enskog expansion and simulated results of typical benchmark problems show not only the second-order accuracy but also the flexibility and applicability of the proposed lattice Boltzmann model. This approach may contribute a powerful technique for the parallel simulation of large-scale engineering and some alternative perspectives for solving the nonlinear transport problem further.