Previous Lattice Boltzmann Models Research Articles

Improved lattice Boltzmann model for moving contact-line with soluble surfactant

Modeling moving contact-line with surfactant has become a widely sought methodology due to its scientific relevance and extensive applications. Within the phase field framework, we present an improved lattice Boltzmann (LB) model for simulating moving contact-line dynamics with soluble surfactant. In this model, a LB equation is used to solve the Navier–Stokes equations, and another two LB equations are utilized to solve the two Cahn–Hilliard-like equations. The modified chemical potentials are incorporated in the LB model by using an equivalent variant of the free energy functional and the corresponding equilibrium distribution functions are also amended. These modifications could circumvent the degraded accuracy of previous LB models in capturing the interfacial behavior and surfactant distribution, and also improve the well-posedness of the LB model. In addition, a dynamic contact angle formulation is introduced to account for the surfactant effect on surface wettability and the resulting contact angle is further implemented in the LB model via a popular geometrical wetting approach. We comprehensively evaluate the numerical performance of the LB model by simulating some benchmark problems. It is found that the LB model achieves a higher accuracy than previous LB models in solving the phase field and surfactant profiles, and also numerical prediction of moving contact-line dynamics with surfactant shows good agreement with the analytical solution. Finally, the LB model is applied to investigate droplet shearing dynamics on solid substrate. The influences of capillary number and solid wetting property on droplet deformation and breakup are analyzed in detail.

Phase-field-based lattice Boltzmann model for simulating thermocapillary flows.

This paper proposes a simple and accurate lattice Boltzmann model for simulating thermocapillary flows, which can deal with the contrast between thermodynamic parameters. In this model, two lattice Boltzmann equationsare utilized to solve the conservative Allen-Cahn equationand the incompressible Navier-Stokes equations, while another lattice Boltzmann equationis used for solving the temperature field, where the collision term is delicately designed such that the influence of the contrast between thermodynamic parameters is incorporated. In contrast to the previous lattice Boltzmann models for thermocapillary flows, the most distinct feature of the current model is that the forcing term used in the present thermal lattice Boltzmann equationis not needed to calculate space derivatives of the heat capacitance or the order parameter, making the scheme much more straightforward and able to retain the main merits of the lattice Boltzmann method. The developed model is first validated by considering the thermocapillary flows in a heated microchannel with two superimposed planar fluids. It is then used to simulate the thermocapillary migration of a two-dimensional deformable droplet, and its accuracy is consistent with the theoretical prediction when the Marangoni number approaches zero. Finally, we numerically study the motion of two recalcitrant bubbles in a two-dimensional channel where the relationship between surface tension and temperature is assumed to be a parabolic function. It is observed that due to the competition between the inertia and thermal effects, the bubbles can move against the liquid's bulk motion and towards areas with low surface tension.

Mesoscale multi-component energy density transport for natural convective microchip cooling with solid–liquid–gas phase changes

A design of microchip cooling with natural convection driven multi-phase coolant flows is proposed by using thermoelectric modules as heat sinks. A new multi-component multi-phase (MCMP) thermal energy transport model is built for convective heat transfer with solid–liquid–gas phase changes. Different from previous lattice Boltzmann models based on temperature distributions, the present model is based on MCMP energy density distributions, which track sensible and latent heat transport for each phase. The local MCMP heat fluxes are related to mesoscale relaxation times and phase volume fractions. Transient solid–liquid–gas interfaces and local temperatures are obtained through the dynamical balance of mesoscale forces and fluxes among all phases together with MCMP enthalpy–temperature distributions. Comparisons of the on and off states of the heat sink modules show that the chip temperature can drop from about 300 to 110 °C, with the coolant liquid–gas phase change temperature of 80 °C at equivalent heat source and sink intensities of 100 W/cm2. Results also show that increasing heat sink intensity to 300 W/cm2 or decreasing gas–liquid phase change temperature to 60 °C can further decrease the chip temperature to 90 °C. The effects of sizes of chips, modules, and cooling units are also illustrated. It is observed that gas bubbles attached on chips impede heat transfer due to the low gas phase thermal conductivity, which means future improvements should also focus on removing or degrading attached gas bubbles. This model provides a quantitative tool for immersion chip cooling performances.

INVESTIGATION ON DROPLET EVAPORATION CONSIDERING RADIATION HEAT TRANSFER BY A DEVELOPED LATTICE BOLTZMANN MODEL

Suspended droplet evaporation in high temperature is common seen and widely used in various industries. Different from the previous lattice Boltzmann model of liquid-vapor phase change, a newly developed model which can consider radiation heat transfer is proposed and used to simulate the suspended droplet evaporation process. This model is verified by the theoretical solutions and compared with the D<sup>2</sup> law. It is found that the deviation between simulated results and the D<sup>2</sup> line is much more obvious under high vapor temperature and small droplet size. And, the radiation heat-transfer ratio during droplet evaporation can reach 45.6% in our simulation, which indicates the importance of radiation heat transfer in droplet evaporation and the necessity of coupling radiation heat transfer into the liquid-vapor phase-change model under high temperature. Also, effects of extinction coefficient β and scattering albedo w on evaporation process are discussed by detailed analysis of streamlines and temperature distribution. Our model and results will benefit the understanding of radiation heat transfer in droplet evaporation process.

Phase-field lattice Boltzmann model for two-phase flows with large density ratio.

In this work, a lattice Boltzmann (LB) model based on the phase-field method is proposed for simulating large density ratio two-phase flows. An improved multiple-relaxation-time (MRT) LB equation is first developed to solve the conserved Allen-Cahn (AC) equation. By utilizing a nondiagonal relaxation matrix and modifying the equilibrium distribution function and discrete source term, the conserved AC equation can be correctly recovered by the proposed MRT LB equation with no deviation term. Therefore, the calculations of the temporal derivative term in the previous LB models are successfully avoided. Numerical tests demonstrate that satisfactory accuracy can be achieved by the present model to solve the conserved AC equation. What is more, the discrete force term of the MRT LB equation for the incompressible Navier-Stokes equations is also simplified and modified in the present work. An alternative scheme to calculate the gradient terms of the order parameter involved in the discrete force term through the nonequilibrium part of the distribution function is also developed. To validate the ability of the present LB model for simulating large density ratio two-phase flows, series of benchmarks, including two-phase Poiseuille flow, droplet impacting on thin liquid film, and planar Taylor bubble are simulated. It is found that the results predicted by the present LB model agree well with the analytical, numerical, and experimental results.

Consistent and Conservative Phase-Field Based Lattice Boltzmann Method for Incompressible Two-Phase Flows

In this work, we consider a general consistent and conservative phase-field model for the incompressible two-phase flows. In this model, not only the Cahn-Hilliard or Allen-Cahn equation can be adopted, but also the mass and the momentum fluxes in the Navier-Stokes equations are reformulated such that the consistency of reduction, consistency of mass and momentum transport, and the consistency of mass conservation are satisfied. We further develop a lattice Boltzmann (LB) method and show that through the direct Taylor expansion, the present LB method can correctly recover the consistent and conservative phase-field model. Additionally, if the divergence of the extra momentum flux is seen as a force term, the extra force in the present LB method would include another term which has not been considered in the previous LB models. To quantitatively evaluate the incompressibility and the consistency of the mass conservation, two statistical variables are introduced in the study of the deformation of a square droplet, and the results show that the present LB method is more accurate. The layered Poiseuille flow and a droplet spreading on an ideal wall are further investigated, and the numerical results are in good agreement with the analytical solutions. Finally, the problems of the Rayleigh-Taylor instability and dam break with the high Reynolds numbers and/or large density ratios are studied, and it is found that the present consistent and conservative LB method is robust for such complex two-phase flows.

A lattice Boltzmann model for the conjugate heat transfer

Conjugate heat transfer problems are of great importance in many applications, while the available lattice Boltzmann (LB) models have some limitations in the study of such problems with the heterogeneous media. In this paper, we develop a new LB model for the conjugate heat transfer problems where the evolution equation with a modified equilibrium distribution function and a source term is considered. Through the Chapman–Enksog analysis, the macroscopic temperature equation can be recovered correctly. In order to test the accuracy, efficiency and stability of this LB model, some simple but nontrivial benchmark problems, including the planar thermal Poiseuille flow with two immiscible fluids, the conjugate heat transfer between a stratified heterogeneous medium with variable thermophysical properties, the steady and unsteady-state planer thermal flows with the two immiscible fluids, the steady-state conjugate heat transfer problem with a vertical interface, and the natural convection in a square cavity filled with circular and square rods, are studied. It is found that the results obtained from the present LB model are in good agreement with the analytic solutions or some previous works. In addition, it is also demonstrated that the developed LB model has a second-order convergence rate in space, and is more stable than some previous LB models. Finally, it should be noted that in our work, there is no special treatment to the interface, and the continuity conditions at the interface can be satisfied automatically.

Application of the pseudo-potential lattice Boltzmann model for simulating interaction of moving solids with liquids

Previous lattice Boltzmann model (LBM) studies on solid entry/exit problems are limited to the free-surface LB model in which effects of the surface tension and gas phase are neglected, the surface wettability cannot be adjusted, boundary conditions need to be applied on the interface, and the interface has to be tracked during time. In addition, conventional macroscopic models for simulating the phenomenon such as Volume of Fluid (VOF) and Constrained Interpolation Profile (CIP) have the same difficulties with the interface, besides the higher computational cost and time. Therefore, for the first time in this paper, a robust pseudo-potential based multi-phase LB model is coupled with moving boundary LB schemes for simulating liquid entry/exit of solids with the circular cylinder as a selected case study without losing generality. The current model has none of the free-surface LBM limitations and is also superior over the conventional models by automatic interface capturing and lower computational cost and time. Furthermore, the integrated model is capable of simulating the phenomenon at relatively high We and Re numbers and density ratios as high as the water/air one. Formation and propagation of the pressure wave in the case of liquid entry are shown and discussed. Cavity and subsequent pinch-off and jets formations for a hydrophobic surface are also captured and quantified. Effects of the We, Re, Fr, and impact velocity on the pinch-off time and depth, and velocity of subsequent jets are investigated, plotted, and discussed in details. Results show that the pinch-off time and depth are independent of the surface tension and liquid viscosity, but are increased linearly with the impact velocity. Furthermore, the velocity magnitude of both downward and upward jets after the pinch-off is increased with Re and We numbers and is decreased with Fr number.

A diffuse interface lattice Boltzmann model for thermocapillary flows with large density ratio and thermophysical parameters contrasts

A diffuse interface lattice Boltzmann model for thermocapillary flows with large density ratio and thermophysical parameters contrasts is developed in this paper. In this model, three distribution functions are used to describe the evolution of phase field, velocity field and temperature field. The conservative Allen-Cahn-based lattce Boltzmann equation which has good numerical stability in simulating multiphase flows at high density ratio is employed to capture the phase interface. The velocity-based lattice Boltzmann equation is utilized to capture the hydrodynamics with thermocapillary effect. In particular, a lattice Boltzmann scheme is proposed to solve the temperature field equation based on the diffuse interface concept, in which the source term is computed locally. Unlike previous lattice Boltzmann model for thermocapillary flows, the thermophysical parameters (heat capacitance and thermal conductivity) of two fluids are allowed to be different. The present model is validated by simulating several numerical examples. Numerical results indicate the reliability of proposed diffuse interface lattice Boltzmann model in simulating thermocapillary flows with large density ratio (up to 1000) and thermophysical parameters contrasts. Moreover, we also derive an unsteady solution for thermocapillary-driven flow in a heated microchannel with two superimposed planar fluids, which can be used to assess the numerical accuracy of numerical algorithm for thermocapillary flows.

Cole-Hopf Transformation Based Lattice Boltzmann Model for One-dimensional Burgers’ Equation**Supported by the National Natural Science Foundation of China under Grant No. 51576079

In this paper, we present a Cole-Hopf transformation based lattice Boltzmann (LB) model for solving one-dimensional Burgers’ equation, and compared to available LB models, the effect of nonlinear convection term can be eliminated. Through Chapman-Enskog analysis, it can be found that the converted diffusion equation based on the Cole-Hopf transformation can be recovered correctly from present LB model. Some numerical tests are also performed to validate the present LB model, and the numerical results show that, similar to previous LB models, the present model also has a second-order convergence rate in space, but it is more accurate than the previous ones.

Lattice Boltzmann modeling of melting of phase change materials in porous media with conducting fins

Based on the averaged energy equation in terms of enthalpy, a modified lattice Boltzmann (LB) model is proposed for simulating melting of phase change materials in porous media with a conducting fin. Different from previous LB models, the present model incorporates the total enthalpy and a free parameter into the equilibrium distribution function, and thus makes it suitable for modeling conjugate heat transfer and has high computational efficiency by avoiding iteration procedure for phase change. To ensure the numerical stability, the multiple-relaxation-time collision scheme is adopted in the model. The model is validated by three benchmark problems. It is found that the numerical results are in good agreement with the analytical solutions and numerical results. In addition, the effect of adding the conducting fin on the melting processes in a porous cavity is investigated. The numerical results indicate that the melting heat transfer can be further enhanced by adding a conducting fin into the porous medium. The melting speed increases as the length of the fin increases and the heat capacity of the fin decreases. However, the vertical position of the fin has no remarkable impact on the melting speed.

An efficient phase-field-based multiple-relaxation-time lattice Boltzmann model for three-dimensional multiphase flows

In this paper, an efficient phase-field-based lattice Boltzmann (LB) model with multiple-relaxation-time (MRT) collision operator is developed for the simulation of three-dimensional multiphase flows. This model is an extension of our previous two-dimensional model (Liang et al., 2014) to the three dimensions using the D3Q7 (seven discrete velocities in three dimensions) lattice for the Cahn–Hilliard equation and the D3Q15 lattice for the Navier–Stokes equations. Due to the less lattice-velocity directions used, the computational efficiency can be significantly improved in the study of three-dimensional multiphase flows, and simultaneously the present model can recover the Cahn–Hilliard equation and the Navier–Stokes equations correctly through the Chapman–Enskog procedure. We compare the present MRT model with its single-relaxation-time version and the previous LB model by using two benchmark interface-tracking problems, and numerical results show that the present MRT model can achieve a significant improvement in the accuracy and stability of the interface capturing. The developed model can also be able to deal with multiphase fluids with high viscosity ratio, which is demonstrated by the simulation of the layered Poiseuille flow and Rayleigh–Taylor instability at various viscosity ratios. The numerical results are found to be in good agreement with the analytical solutions or some available results. In addition, it is also found that the instability induces a more complex structure of the interface at a low viscosity.

Comparative study of the lattice Boltzmann models for Allen-Cahn and Cahn-Hilliard equations.

In this paper, a comparative study of the lattice Boltzmann (LB) models for the Allen-Cahn (A-C) and Cahn-Hilliard (C-H) equations is conducted. To this end, a new LB model for the A-C equation is first proposed, where the equilibrium distribution function and the source term distribution function are delicately designed to recover the A-C equation correctly. The gradient term in this model can be computed by the nonequilibrium part of the distribution function such that the collision process can be implemented locally. Then a detailed numerical study on several classical problems is performed to give a comparison between the present model for the A-C equation and the recently developed LB model [H. Liang et al., Phys. Rev. E 89, 053320 (2014)PLEEE81539-375510.1103/PhysRevE.89.053320] for the C-H equation in terms of tracking the interface of two-phase flow. The results show that the present LB model for the A-C equation is more accurate and more stable, and also has a second-order convergence rate in space, while the convergence rate of the previous LB model for the C-H equation is only about 1.5.

Solitary wave of the Korteweg-de Vries equation based on lattice Boltzmann model with three conservation laws

In this paper, a new lattice Boltzmann model for the Korteweg-de Vries (KdV) equation is proposed. By using the Chapman-Enskog expansion and the multi-scale time expansion, a series of partial differential equations in different time scales and several higher- order moments of equilibrium distribution functions are obtained. In order to make the scheme obey the three conservation laws of the KdV equation, two equilibrium distribution functions are used and a correlation between the first conservation law and the second conservation law is constructed. In numerical examples, the numerical results of the KdV equation obtained by this scheme are compared with those results obtained by the previous lattice Boltzmann model. Numerical experiments demonstrate this scheme can be used to reduce the truncation error of the lattice Boltzmann scheme and preserve the three conservation laws.

Lattice Boltzmann modeling of phonon transport

A novel lattice Boltzmann scheme is proposed for phonon transport based on the phonon Boltzmann equation. Through the Chapman-Enskog expansion, the phonon lattice Boltzmann equation under the gray relaxation time approximation recovers the classical Fourier's law in the diffusive limit. The numerical parameters in the lattice Boltzmann model are therefore rigorously correlated to the bulk material properties. The new scheme does not only eliminate the fictitious phonon speed in the diagonal direction of a square lattice system in the previous lattice Boltzmann models, but also displays very robust performances in predicting both temperature and heat flux distributions consistent with analytical solutions for diverse numerical cases, including steady-state and transient, macroscale and microscale, one-dimensional and multi-dimensional phonon heat transport. This method may provide a powerful numerical tool for deep studies of nonlinear and nonlocal heat transports in nanosystems.

An alternative lattice Boltzmann model for three-dimensional incompressible flow

In this work, an alternative lattice Boltzmann (LB) model for three-dimensional (3D) incompressible flow is proposed. The equilibrium distribution function (EDF) of the present model is directly derived in accordance with the incompressibility conditions by applying the Hermite expansion. Moreover, an alternative formula for pressure computation is designed from the second order moment of the distribution function. The present 3D LB model inherits the advantageous features of Guo’s LB model: the density is a constant, the fluid pressure is independent of density and the Navier–Stokes (N–S) equations for incompressible flow can be derived. Two benchmark tests, flow over a backward-facing step and the lid-driven cavity flow, are applied to validate the present model. Accurate results for these tests are obtained with the present model, and further comparisons with the previous LB models (the standard LB model, the He–Luo model and Guo’s LB model) demonstrate that the present model provides better accuracy in the region of high deviatoric stress and such advantage is further enhanced by using the D3Q27 lattice.

Relativistic lattice Boltzmann model with improved dissipation

We develop a relativistic lattice Boltzmann (LB) model, providing a more accurate description of dissipative phenomena in relativistic hydrodynamics than previously available with existing LB schemes. The procedure applies to the ultrarelativistic regime, in which the kinetic energy (temperature) far exceeds the rest mass energy, although the extension to massive particles and/or low temperatures is conceptually straightforward. In order to improve the description of dissipative effects, the Maxwell-J\"uttner distribution is expanded in a basis of orthonormal polynomials, so as to correctly recover the third-order moment of the distribution function. In addition, a time dilatation is also applied, in order to preserve the compatibility of the scheme with a Cartesian cubic lattice. To the purpose of comparing the present LB model with previous ones, the time transformation is also applied to a lattice model which recovers terms up to second order, namely up to the energy-momentum tensor. The approach is validated through quantitative comparison between the second- and third-order schemes with Boltzmann approach multiparton scattering (the solution of the full relativistic Boltzmann equation) for moderately high viscosity and velocities, and also with previous LB models in the literature. Excellent agreement with BAMPS and more accurate results than previous relativistic lattice Boltzmann models are reported.

A lattice Boltzmann model for the Korteweg–de Vries equation with two conservation laws

A lattice Boltzmann model for the Korteweg–de Vries (KdV) equation is presented by using the higher-order moment method. In contrast to the previous lattice Boltzmann model to the KdV equation, our method has higher-order accuracy. Two key steps in the development of this model are the addition of a momentum conservation condition, and the construction of a correlation between the first conservation law and the second conservation law. The numerical example shows the higher-order moment method can be used to raise the truncation error of the lattice Boltzmann scheme.

A higher-order moment method of the lattice Boltzmann model for the Korteweg–de Vries equation

In this paper, a lattice Boltzmann model for the Korteweg–de Vries (KdV) equation with higher-order accuracy of truncation error is presented by using the higher-order moment method. In contrast to the previous lattice Boltzmann model, our method has a wide flexibility to select equilibrium distribution function. The higher-order moment method bases on so-called a series of lattice Boltzmann equation obtained by using multi-scale technique and Chapman–Enskog expansion. We can also control the stability of the scheme by modulating some special moments to design the dispersion term and the dissipation term. The numerical example shows the higher-order moment method can be used to raise the accuracy of truncation error of the lattice Boltzmann scheme.

Evaluation of the external force term in the discrete Boltzmann equation

A representation of the forcing term in the Boltzmann equation based on a Hermite expansion of the Boltzmann distribution function in velocity phase space is derived. Based on this description of the forcing term, a systematic comparison of previous lattice Boltzmann models describing a nonideal gas behavior is given.