Lattice Boltzmann Equation Model Research Articles

Well-balanced kinetic schemes for two-phase flows

At equilibrium state of a two-phase fluid system, the chemical potential is constant and the velocity vanishes. However, such equilibrium state usually cannot be captured by numerical methods due to discretization errors. Consequently, inconsistent thermodynamic interfacial properties due to non-constant chemical potential, and spurious velocities due to discrete force imbalance, are frequently encountered in numerical simulations. Therefore, numerical schemes with well-balanced properties, which can capture the equilibrium state at discrete level, are preferred for simulating two-phase flows. In this paper, we report the progress of developing well-balanced mesoscopic numerical methods based on kinetic theory, focusing on the lattice Boltzmann equation (LBE) method and the discrete unified gas-kinetic scheme (DUGKS). First, the discrete balance property of a free-energy LBE for single-component two-phase flows is analyzed to identify the structure of force imbalance. Then, a well-balanced LBE (WB-LBE) model which has the same algorithm structure as the standard one is reviewed. The WB-LBE is theoretically shown to be able to achieve the discrete equilibrium state, and the well-balance properties are confirmed numerically. Some extensions to other two-phase LBE models are discussed with enhanced numerical stability. The well-balanced DUGKS, which can use non-uniform meshes and exhibits better numerical stability for large density ratio systems is also reported.

Improved well-balanced free-energy lattice Boltzmann model for two-phase flow with high Reynolds number and large viscosity ratio

Spurious velocities and inaccurate density properties arising from the imbalance of discretized forces at discrete level are frequently observed in numerical simulation of multiphase flows based on lattice Boltzmann equation (LBE) models. In this paper, an improved well-balanced free-energy LBE model is proposed for two phase flows with high Reynolds numbers and large viscosity differences based on the well-balanced LBE [Guo et al., Phys. Fluids 33, 031709 (2021)]. To this end, a free parameter associated with the shear rate is introduced into the equilibrium distribution function. This results in a fluid viscosity that is dependent not only on the relaxation time but also on the introduced parameter. The relaxation time can be chosen to improve the numerical stability and accuracy, while the viscosity is mainly determined by the new parameter. To test the capability of the present model in capturing discrete equilibrium states, both one-dimensional flat interface and two-dimensional stationary droplet are simulated. Numerical results show that the present model is capable of eliminating spurious velocities and maintaining a constant chemical potential when the system reaches an equilibrium state. To further validate the performance of the present LBE for dynamic problems, both binary droplet collision and single bubble rising are performed, which demonstrates that the present model has the capability to deal with two phase flows with high Reynolds number and large viscosity ratio.

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.

Well-balanced lattice Boltzmann model for two-phase systems

The standard two-phase lattice Boltzmann equation (LBE) method usually fails to capture the physical equilibrium state (zero velocity and constant chemical potential). Consequently, spurious velocities and inconsistent thermodynamic density properties are frequently encountered in LBE simulations. In this work, based on a rigorous analysis of the discrete balance equation of LBE, the structure of force imbalance due to discretization errors is identified. Then, a well-balanced LBE model is proposed, which can achieve the discrete equilibrium state. The well-balance properties of the model are confirmed by simulating a flat interface problem and a droplet system.

Numerical simulation of mixed convection in a two-sided lid-driven square cavity with four inner cylinders based on diamond arrays

The hydrodynamics and thermal characteristics due to mixed convection in a vertical two-sided lid-driven differentially square cavity containing four hot cylinders in a diamond array are investigated by the lattice Boltzmann equation model. The moving walls of the cavity are cold while the others are adiabatic. The flow in the cavity is driven by both the temperature difference and the moving vertical walls. The influence of different flow governing parameters, including the direction of the moving walls (the left wall moves up and the right wall moves down (Case I), both the left and right walls are moving upward (Case II), both the left and right walls are moving downwards (Case III)), the distance between neighboring cylinders [Formula: see text] ([Formula: see text]), and the Richardson number [Formula: see text] ([Formula: see text]) on the fluid flow and heat transfer are investigated with the Reynolds number in the range of [Formula: see text], the Grashof number of [Formula: see text] and the Prandtl number of [Formula: see text]. Flow and thermal performances in the cavity are analyzed in detail by considering the streamlines and isotherms profiles, the average Nusselt number, as well as the total Nusselt number. It is found that the heat transfer efficiency is highest when [Formula: see text] for the cases of the walls moving in the opposite direction. When the walls move in the same directions, the heat transfer efficiency obtained by [Formula: see text] is maximum among the considered values of [Formula: see text]. On the other hand, compared with the cases of [Formula: see text] and [Formula: see text], the cylinder positions corresponding to the largest and the smallest Nusselt numbers are very sensitive to the moving direction of the walls for [Formula: see text]. Moreover, the results also show that in terms of the value of Nusselt number and the stability the case of both walls moving downwards works well. Besides, the effect of the distance between neighboring cylinders is also discussed, it is found that increasing or decreasing the spacing between cylinders could enhance heat transfer to different degrees for the range of [Formula: see text] number considered. Finally, the empirical relationships among [Formula: see text], [Formula: see text], and the spacing between the cylinders ([Formula: see text]) are given, and predictive results match with the computed values very well.

Flow around a Wingmill device for energy extraction

The dynamics of a closed loop self controlled underwater oscillating foil device for energy extraction (a wingmill) is studied through numerical simulations. The viscous two and three dimensional flow around the foil was computed using OpenFOAM and a Lattice-Boltzmann Equation model, respectively. Heaving is driven by the computed hydrodynamic lift and a damper, that extracts energy, while pitching is driven by the hydrodynamic torque and a feedback control torque that leads the foil to a given angle of attack. Unlike most of the related work found in the literature, the heaving and pitching motion of the foil is not prescribed. Dimensional analysis suggests a compromise between the generator and control gains, so a parametric study was carried out. The effect of a three dimensional finite wingspan on the performance of the device, and the flow is compared with the two dimensional case. This fully coupled fluid-solid-body interaction configuration will allow for the system identification, control and optimization of energy harvesting devices in future studies.

Simulating flow in porous media using the lattice Boltzmann method: Intercomparison of single-node boundary schemes from benchmarking to application

Lattice Boltzmann method (LBM) is widely adopted in simulating flow and reactive transport in porous media due to its easy treatment of boundaries and high computational efficiency. There have been various boundary schemes developed in the LBM due to their vital roles when dealing with fluid-solid interface. In the present work, we aim to review and intercompare all the single-node boundary schemes (including the existing schemes and our newly-developed schemes) in the LBM through a series of numerical experiments from simple benchmarking cases to realistic applications. Both the accuracy and computational efficiency of all the schemes are evaluated and analyzed. Among the existing schemes, it is known that the bounce-back scheme (BB) is commonly employed at the boundaries in the LBM. However, the BB is of only first-order spatial accuracy when zero-velocity is not exactly located in the middle of the fluid and solid nodes, which cannot be guaranteed in porous media flows due to the complex pore structures. Therefore, fine grids are usually generated to obtain satisfactory results yet computationally expensive. Recently, a second-order single-node boundary scheme (SSN), which is more flexible than the BB while retaining the locality, has been proposed (Zhao et al, J. Comput. Phys., 329:1-15, 2017 Zhao and Yong (2017)). Unfortunately, the SSN fails to provide a viscosity-independent permeability, i.e., the predicted permeability is related to the fluid viscosity, which is unphysical. Based on the SSN, we first developed a magic second-order single-node boundary scheme (MSSN) to overcome this shortcoming based on the Chapman-Enskog analysis of the two-relaxation-time lattice Boltzmann equation (TRT-LBE) model. Comparative study of the BB, SSN/MSSN, and another single-node boundary scheme, i.e., the improved bounce-back scheme (IBB)/the magic improved bounce-back scheme (MIBB) is then performed both theoretically and numerically. Numerical simulations of porous media flows in both two and three dimensions confirm that the BB, MIBB, and MSSN can obtain the viscosity-independent permeability, while the IBB and SSN fail to do so. First of all, as benchmarking tests, flow through a 2D fracture is studied using different schemes, the results of which show that the IBB and SSN cannot provide viscosity-independent permeability while such drawbacks can be overcome by the BB/MIBB/MSSN. Furthermore, simulations of flows through a hexagonal array of cylinders and a 3D sphere packing demonstrate that the MSSN is more accurate than both the BB and MIBB. Finally, for porous media samples generated from the micro-CT images where the locations of the boundaries are not exactly known, the MSSN is more flexible to locate the positions of the fluid-solid interface than the BB and MIBB, which in turn can provide more accurate descriptions of the fluid flow. The current intercomparison study will benefit the community by providing comprehensive review and evaluation of the performance when using different single-node boundary schemes in the LBM.

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.

Thermodynamic-consistent multiple-relaxation-time lattice Boltzmann equation model for two-phase hydrocarbon fluids with Peng-Robinson equation of state

In this work, a multiple-relaxation-time (MRT) lattice Boltzmann (LB) equation model with Beam-Warming (B-W) scheme is proposed to simulate a multi-phase fluid system with Peng-Robinson (P-R) equation of state (EOS). The mathematical model of the multi-phase fluid flow is derived based on the NVT-based framework, where the Helmholtz free energy of P-R EOS is introduced. The nonideal force in the multi-phase flow is directly computed from the free energy in order to obtain a more compact formulation of hydrodynamic equations, which is termed as potential form. The MRT-LB model is developed based on the potential form of hydrodynamic equations, which can eliminate spurious currents effectively. In addition, to capture the tiny nonconvex perturbation from the linear trend of P-R model precisely, the B-W scheme is utilized in the present MRT-LB model, which gives rise to an adjustable Courant-Friedrichs-Lewy (CFL) number. Also, the second order accuracy can be naturally achieved by this scheme without any other requirements and numerical boundary conditions. In the numerical experiments, a realistic hydrocarbon component (isobutane) in three dimensional space is simulated by the proposed MRT-LB model. Numerical results show that the magnitude of spurious currents can be significantly reduced by the present MRT-LB model. In addition, our numerical predictions of surface tension agree well with the experimental data, which verify the effectiveness of the proposed MRT-LB model.

Preconditioned multiple-relaxation-time lattice Boltzmann equation model for incompressible flow in porous media

An improved preconditioned multiple-relaxation-time lattice Boltzmann equation model for incompressible flow (IPMRT-LBE) in porous media is proposed. Motivated by previous LBE models [Guo et al., Phys. Rev. E 70, 066706 (2004); Premnath et al., J. Comput. Phys. 228, 746 (2009); Guo et al., J. Comput. Phys. 165, 288 (2000)], the current model is demonstrated to have the advantages of accurate implementation of the no-slip boundary condition, reducing the compressible effect as well as fast convergence rate compared with standard LBE models. To validate the IPMRT-LBE model, flows in two- and three-dimensional synthetic porous media (square array of cylinders and body-centered cubic array of spheres) are simulated. The results show that the current model can predict the macroscopic property (such as permeability) accurately with significantly accelerated convergence rate. Furthermore, simulations of flow through a three-dimensional sandpack confirm the applicability and advantages of the IPMRT-LBE model.

A discrete unified gas-kinetic scheme for immiscible two-phase flows

In this work, a discrete unified gas-kinetic scheme (DUGKS) is proposed for two-phase flows. In the framework of DUGKS, two kinetic equations are used to solve the quasi-incompressible phase-field governing equations (Yang and Guo, 2016), one of the kinetic models is developed for the Chan-Hilliard (CH) equation and the other is for the Navier-Stokes equations. The DUGKS can correctly recover the quasi-incompressible phase-field governing equations through the Chapman-Enskog analysis. Unlike previous phase-field-based lattice Boltzmann equation (LBE) models, the Courant-Friedricks-Lewy (CFL) condition in DUGKS is adjustable which can increase numerical stability. Furthermore, with the finite-volume formulation the model can be easily implemented on non-uniform meshes which can improve numerical precision. The proposed model is validated by simulating several benchmark problems, including a stationary drop, the layered Poiseuille flow, a rising bubble in the liquid and the Rayleigh-Taylor instability and some comparisons with the quasi-incompressible LBE model are presented. Numerical results show that the present model is capable of dealing with a wider range of viscosity and density ratios than the quasi-incompressible LBE model. In particular, the numerical accuracy can be improved by using non-uniform meshes. Overall, the present model is a promising tool for numerical simulation of two-phase flows.

Lattice Boltzmann simulation of immiscible displacement in the cavity with different channel configurations

In this work, the immiscible displacement in a cavity with different channel configurations is studied using an improved pseudo-potential lattice Boltzmann equation (LBE) model. This model overcomes the drawback of the dependence of the fluid properties on the grid size, which exists in the original pseudo-potential LBE model. The approach is first validated by the Laplace law. Then, it is employed to study the immiscible displacement process. The influences of different factors, such as the surface wettability, the distance between the gas cavity and liquid cavity and the surface roughness of the channel are investigated. Numerical results show that the displacement efficiency increases and the displacement time decreases with the increase of the surface contact angle. On the other hand, the displacement efficiency increases with increasing distance between the gas cavity and the liquid cavity at first and finally reaches a constant value. As for the surface roughness, two structures (a semicircular cavity and a semicircular bulge) are studied. The comprehensive results show that although the displacement processes for both the structures depend on the surface wettability, they present quite different behaviors. Specially, for the roughness structure constituted by the semicircular cavity, the displacement efficiency decreases and displacement time increases evidently with the size of the semicircular cavity for the small contact angle. The trend slows down as the increase of the contact angle. Once the contact angle exceeds a certain value, the size of the semicircular cavity almost has no influence on the displacement process. While for the roughness structure of a semicircular bulge, the displacement efficiency increases with the size of bulge first and then it decreases for the small contact angle. The displacement efficiency increases first and finally reaches a constant for the large contact angle. The results also show that the displacement time has an extreme value in these cases for the small contact angles.

A kinetic theory based thermal lattice Boltzmann equation model

A thermal lattice Boltzmann equation (LBE) model within the framework of double distribution function (DDF) method is proposed from the continuous DDF Boltzmann equation, which has a clear physical significance. Since the discrete velocity set in present LBE model is not space filled, a Lax–Wendroff scheme is applied to solve the evolution equations by which the spatial interpolation of two distribution functions is overcome. To validate the model, some classical numerical tests include thermal Couette flow and natural convection flow are simulated, and the results agree well with the analytic solutions and other numerical results, which showed that the present model had the ability to describe the thermal fluid flow phenomena.

Localized lattice Boltzmann equation model for simulating miscible viscous displacement in porous media

A localized lattice Boltzmann equation (LBE) model for simulating the miscible viscous displacement in porous media is proposed. The Darcy’s law for flow and the convection–diffusion equation (CDE) describing the transport of solute are solved numerically by the present model. To ensure the local implementation of the collision process in this model, the pressure and concentration gradients are computed from the moments of the nonequilibrium distribution functions, which are of second-order accuracy. Consequently, the advantages of the lattice Boltzmann method (LBM) are retained. The model is validated with a stable displacement problem, and is employed to study the viscous fingering instability that occurs in the process of the miscible viscous displacement. The results agree well with previous studies. Furthermore, although the present model is an explicit scheme, it is interesting to find that it is capable of simulating the viscous displacement over a wide range of Péclet (Pe) numbers, indicating the superior stability of this model.

Lattice Boltzmann method for binary fluids based on mass-conserving quasi-incompressible phase-field theory.

In this paper, a lattice Boltzmann equation (LBE) model is proposed for binary fluids based on a quasi-incompressible phase-field model [J. Shen et al., Commun. Comput. Phys. 13, 1045 (2013)10.4208/cicp.300711.160212a]. Compared with the other incompressible LBE models based on the incompressible phase-field theory, the quasi-incompressible model conserves mass locally. A series of numerical simulations are performed to validate the proposed model, and comparisons with an incompressible LBE model [H. Liang et al., Phys. Rev. E 89, 053320 (2014)PLEEE81539-375510.1103/PhysRevE.89.053320] are also carried out. It is shown that the proposed model can track the interface accurately. As the stationary droplet and rising bubble problems, the quasi-incompressible LBE gives nearly the same predictions as the incompressible model, but the compressible effect in the present model plays a significant role in the phase separation problem. Therefore, in general cases the present mass-conserving model should be adopted.

Lattice Boltzmann method for conjugate heat and mass transfer with interfacial jump conditions

In this work we propose an interface treatment for conjugate heat and mass transfer with discontinuities or jumps of temperature (concentration) and/or heat (mass) flux at the interface using the lattice Boltzmann equation (LBE) method. The present interface treatment is based on the second-order accurate boundary condition treatments for Dirichlet and Neumann problems (Li et al., 2013) and second-order accurate interface treatment for standard conjugate heat and mass transfer with the continuity of temperature (concentration) and flux at the interface (Li et al., 2014). The interfacial jump conditions are intrinsically satisfied in the present treatment without iterative computations that are typically needed in conventional finite-difference or finite-volume methods. The interfacial temperature (concentration) values and the fluxes into the adjacent domains are conveniently obtained from the microscopic distribution functions in the LBE model without finite-difference approximations. Since the local intersection link fraction is included in the present treatment, the interfacial geometry is preserved and the present interface schemes are capable of handling curved interfaces. The numerical accuracy and convergence of the present interface schemes are verified with several numerical tests, including (i) one-dimensional (1D) steady diffusion within a two-solid slab; the slab is either aligned with the lattice velocity vector or with an inclination angle, (ii) 2D steady diffusion in a circular domain of two concentric solids, (iii) 3D steady diffusion in a spherical domain of two concentric solids, and (iv) 2D steady convection–diffusion in a channel. The two adjacent domains have different thermal (mass) transport properties and specific temperature (concentration) and flux jump conditions are imposed at the interface in each of those tests. It is verified that the present interface treatment for jump conditions does not introduce additional errors compared to the case without jumps; and second-order accuracy in space is obtained for the interior temperature (concentration) field, the interfacial temperature (concentration) values and interfacial fluxes for straight interfaces aligned with the lattice velocity vector in both diffusion and convection–diffusion problems. The effects of inclined and curved interfacial geometry on the order-of-accuracy of the LBE results are also investigated and the results are compared with previous findings.

Multiple-relaxation-time lattice Boltzmann model for binary mixtures of nonideal fluids based on the Enskog kinetic theory

In this paper, a lattice Boltzmann equation (LBE) model with multiple-relaxation-time (MRT) collision operator is developed based on the Enskog theory for isothermal nonideal mixtures, which is an extension of the previous single relaxation time (SRT) LBE model (Guo and Zhao in Phys Rev E 68:035302, 2003). The present MRT-LBE model overcomes some inherent defects of the original SRT-LBE model such as the fixed Schmidt number and limited viscosity ratio. It is also interestingly shown that the widely used Shan-Chen (SC) model, which is constructed heuristically based on the pseudo-potential concept, can also be regarded as a special case of the present model, and thus putting a solid foundation for this well-accepted multiphase LBE model. A series of numerical simulations, including the static droplet and layered co-current flow, are conducted to test the applicability of the present model for immiscible fluids with different Schmidt numbers and large viscosity ratio, which may be difficult for the original SRT-LBE model and the SC model.

Lattice Boltzmann equation method for the Cahn-Hilliard equation.

In this paper a lattice Boltzmann equation (LBE) method is designed that is different from the previous LBE for the Cahn-Hilliard equation (CHE). The starting point of the present CHE LBE model is from the kinetic theory and the work of Lee and Liu [T. Lee and L. Liu, J. Comput. Phys. 229, 8045 (2010)]; however, because the CHE does not conserve the mass locally, a modified equilibrium density distribution function is introduced to treat the diffusion term in the CHE. Numerical simulations including layered Poiseuille flow, static droplet, and Rayleigh-Taylor instability have been conducted to validate the model. The results show that the predictions of the present LBE agree well with the analytical solution and other numerical results.

A hybrid phase field multiple relaxation time lattice Boltzmann method for the incompressible multiphase flow with large density contrast

AbstractA hybrid phase field multiple relaxation time lattice Boltzmann method (LBM) is presented in this paper for simulation of multiphase flows with large density contrast. In the present method, the flow field is solved by a lattice Boltzmann equation. Concurrently, the interface of two fluids is captured by solving the macroscopic Cahn‐Hilliard equation using the upwind scheme. To be specific, for simulation of the flow field, an lattice Boltzmann equation (LBE) model developed in Shao et al. (Physical Review E, 89 (2014), 033309) for consideration of density contrast in the momentum equation is used. Moreover, in the present work, the multiple relaxation time collision operator is applied to this LBE to enable simulation of problems with large viscosity contrast or high Reynolds number. For the interface capturing, instead of solving another set of LBE as in many phase field LBMs, the macroscopic Cahn‐Hilliard equation is directly solved by using a weighted essentially non‐oscillatory scheme. In this way, the present hybrid phase field LBM shares full advantages of the phase field LBM while enhancing numerical stability. The ability of the present method to simulate multiphase flow problems with large density contrast is demonstrated by several numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.

Interface-capturing lattice Boltzmann equation model for two-phase flows.

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.