Multi-relaxation-time lattice Boltzmann model for incompressible flow

doi:10.1016/j.physleta.2006.07.074

Physics Letters A

Volume 359, Issue 6, 11 December 2006, Pages 564-572

https://doi.org/10.1016/j.physleta.2006.07.074 Get rights and content

Abstract

In this Letter an incompressible MRT-LB model has been proposed. The equilibria in momentum space are derived from an earlier incompressible LBGK model by Guo et al. Through the Chapman–Enskog expansion the incompressible Navier–Stokes equations can be recovered without artificial compressible effects. The steady Poiseuille flow, the driven cavity flow and the double shear flow have been carried on by the incompressible MRT-LB model. The numerical simulation results agree well with the analytical solutions or the existing results. It is found that the incompressible MRT-LB model shows better numerical stability.

Introduction

The lattice Boltzmann Bhatnagar–Gross–Krook (LBGK) method also called single-relaxation-time (SRT) LB method is an innovative numerical method based on kinetic theory to simulate various hydrodynamics [1], [2], [3]. From a computational viewpoint, the notable advantages of LBM are its intrinsic parallelism of algorithm, simplicity of programming, and ease of incorporating microscopic interactions. It has been successfully used in varies of complex fluid systems, such as multiphase fluids [4], [5], [6], [7], suspensions in fluid [8], and magneto-hydrodynamics [9], [10]. However, despite the notable success of the LBM, some shortcomings of the LBGK model are apparent. For instance, the method may lead to numerical instability when the dimensionless relaxation time τ is close to 0.5. One way to overcome these shortcomings of the LBGK model is to use a multi-relaxation-time (MRT) version [11], [12], [13], [14], [15], [16], [17] which nevertheless retains the simplicity and computational efficiency of the LBGK model.

The MRT-LB model is of better numerical stability and has more degrees of freedom than the commonly used SRT-LB model [18]. The main idea of the MRT-LB model is that the advection is mapped onto the momentum space by a linear transformation and the flux is still finished in the velocity space. Most of the existing MRT-LB models are constructed for the compressible Navier–Stokes (NS) equations in the low Mach number limit, and it is well understood that “compressible” error exits for simulating incompressible fluid flows. The recovered macroscopic equations from the existing MRT-LB model is the approximate incompressible NS equations through the Chapman–Enskog (CE) expansion. Considering the significance of the incompressible NS equations in theory and applications, it is necessary to establish an MRT-LB model which can exactly model the incompressible NS equations.

Following the method of MRT-LB model, an incompressible MRT-LB (IMRT-LB) model has been proposed in this Letter based on the equilibrium distribution functions (called GEDF) in velocity space by Guo et al. [19]. Through the CE expansion the incompressible NS equations can be recovered exactly without artificial compressible effects. The steady Poiseuille flow, the driven cavity flow and the double shear flow have been carried on by the incompressible MRT-LB model. The simulating results agree well with the analytical solutions or other benchmark data. Furthermore, the numerical results indicate that the present IMRT-LB model exhibits better numerical stability compared to the LBGK model proposed in [19].

Section snippets

2D incompressible SRT-LB model

In 2D model, space is discretized into a square lattice, and there are nine discrete velocities given by $e_{i} = {\begin{matrix} (0, 0), i = 0, \\ (\cos [(i - 1) π / 2], \sin [(i - 1) π / 2]) c, i = 1 – 4, \\ (\cos [(i - 1) π / 2 + π / 4], \sin [(i - 1) π / 2 + π / 4]) \sqrt{2} c, \\ i = 5 – 8, \end{matrix}$ where $c = δ_{x} / δ_{t}$ is the particle velocity and $δ_{x}$ and $δ_{t}$ are the lattice grid spacing and time step, respectively. From here on we shall use the units of $δ_{x} = δ_{t} = 1$ such that all the relevant quantities are dimensionless.

The lattice Boltzmann equation (LBE) reads $f_{α} (x + e_{α} δ_{t}, t + δ_{t}) - f_{α} (x, t) = Ω_{α},$ where $f_{α} (x, t)$

Incompressible MRT-LB model

The multi-relaxation-time lattice Boltzmann equation reads $f_{α} (x + e_{α} δ_{t}, t + δ_{t}) - f_{α} (x, t) = - S_{α i} (f_{i} (x, t) - f_{i}^{eq} (x, t)),$ where $f_{α}^{eq}$ is GEDF and S is collision matrix. The nine eigenvalues of S are all between 0 and 2 so as to maintain linear stability and the separation of scales, which means that the relaxation times of non-conserved quantities are much faster than the hydrodynamic time scales. The LBGK model is the special case in which the nine relaxation times are all equal, and the collision matrix $S = \frac{1}{τ} I$

Numerical simulation

In this section, we take the simulations of steady Poiseuille flow, the driven cavity flow and the double shear flow. In all the simulations, the non-equilibrium extrapolation scheme [20] is used to treat the boundary conditions.

Conclusion

The multi-relaxation-time lattice Boltzmann model is of better numerical stability and has attract much attention in recent years. The basic idea is to take the advection in momentum space and finish the flux in velocity space. The equilibria in momentum space are closed to the particle equilibrium functions. The previous MRT-LB models, the choice of the equilibrium distribution functions leads to some compressible errors in simulating incompressible flows. In this Letter the incompressible

Acknowledgements

Authors would like to thank Prof. Zhaoli Guo for helpful discussions. This work is supported by the National Natural Science Foundation of China (Grant No: 70271609).

References (26)

R. Benzi et al.
Phys. Rep.
(1992)
T. Inamuro et al.
Comput. Phys. Commun.
(2000)
I. Ginzburg
Adv. Water Res.
(2005)
I. Ginzburg
Adv. Water Res.
(2005)
Z.-L. Guo et al.
J. Comput. Phys.
(2000)
U. Ghia et al.
J. Comput. Phys.
(1982)
S. Hou et al.
J. Comput. Phys.
(1995)
H. Yu et al.
Comput. Fluids
(2006)
H. Yu et al.
J. Comput. Phys.
(2005)
D.L. Brown et al.
J. Comput. Phys.
(1995)

Y. Qian et al.

Annu. Rev. Comput. Phys.

(1995)

S. Chen et al.

Annu. Rev. Fluid Mech.

(1998)

X. Shan et al.

Phys. Rev. E

(1994)

Cited by (103)

Solidification shrinkage and shrinkage-induced melt convection and their relation with solute segregation in binary alloys
2022, Computational Materials Science
Shrinkage flow of melt in the mushy zone are closely related to solidification cracking. Here, the phase field (PF) and Lattice Boltzmann Method (LBM) used to predict the dendrite morphology, solute precipitation and diffusion with different solid and liquid densities. A novel species conservation equation considering density changes was constructed to model convective and diffusive of solutes. We demonstrated that solidification shrinkage and shrinkage-flow can strongly affect the dendritic morphology and solute segregation. When solid density is considered, the predicted outline of dendritic decrease significantly, while the solute diffusion layer and the solute concentration at the front of dendrite increase with solid density. For multi-order solidification, solidification shrinkage cause larger solute segregation in the liquid. Then studied direction solidification with shrinkage flow in crystal growth and observed the effect of continuous feeding of melt to the liquid channel. The solute concentration in the channel increases with the orientation angle and solid density. The results from this work have application prospects in the simulation of solidification cracking.
Coupled influence of wettability alteration and geometry on two-phase flow in porous media
2021, Advances in Water Resources
The objective of this paper is to investigate the interplay between wettability alteration and domain geometry in influencing the pore-scale displacement and trapping mechanisms of two-phase flow in porous media. A new wettability alteration algorithm is introduced to quantify and track the spatial and temporal variability of the wettability state of every pore. Simulations in an initially oil-wet domain are conducted using the lattice Boltzmann model while utilizing the wettability alteration algorithm to capture the new wettability states. The simulations are then resumed under the new generated wettability states to investigate the coupled impacts of wettability alteration and domain topology on the resulting pore-scale displacement behavior. We observe varying degrees of improvements in the oil phase fractional flow as the wettability state is varied towards water-wet conditions influenced by the domain geometry, capillary number, and phase saturation. Significant heterogeneity, increased surface roughness, and the prevalence of narrower pore spaces in the rock sample in comparison to the spherepack increase the percentage of trapped oil ganglia. As a result, wettability alteration results in varying degrees of oil remobilization events, followed by coalescence with the main flow path, especially at higher capillary numbers. Moreover, oil phase fractional flow at higher oil saturation becomes influenced primarily by the increased phase connectivity, and improvements due to wettability alteration become less significant. Similar displacement mechanisms are observed in both domains, controlled primarily by the wettability distribution and the local pore geometry. However, the resulting flow pathways, which dictate the overall flow behavior, are controlled by the frequency of various displacement mechanisms and are a function of the domain geometry. The observations in this study highlight the complex interactions between wettability heterogeneity and pore geometry in influencing pore-scale two-phase flow. The wettability alteration algorithm introduced can be used to optimize pore-scale displacement processes by providing a framework to track the locations and the relative degree of wettability alteration, enable automatic wettability alteration, and detect trapped ganglia without the need for post-processing.
A novel LBM-DEM based pore-scale thermal-hydro-mechanical model for the fracture propagation process
2021, Computers and Geotechnics
Citation Excerpt :
In this study, since the multiple relaxation time (MRT) collision operator can increase the stability of LBM (Krüger et al., 2016; Mezrhab et al., 2010), the incompressible MRT-LBM model is used to solve the Navier-Stokes equation, and the thermal MRT-LBM model is adopted for the heat transport equation. In the models, two different distribution functions are used to express the fluid flow and heat transfer respectively, named the two-population approach in the literature (Du et al., 2006; Krüger et al., 2016). The incompressible MRT model is derived from the D2G9 model (Du et al., 2006), and the discrete velocity e and weighting factor ω are shown in Fig. 3.
Fracture propagation in rocks is a complex thermal-hydro-mechanical (THM) phenomenon present in geotechnics, such as enhanced geothermal systems, CO₂ sequestration, and shale gas exploitation. Numerical simulation is an essential way to study its mechanism. In this study, a pore-scale model combined with the lattice Boltzmann method (LBM) and the discrete element method (DEM) is developed, in which the comprehensive THM coupling phenomena are considered, consisting of convective heat transport, conjugate heat transfer, hydrodynamic force, and thermal strain. In addition, to couple the temperature and anisotropic thermal expansion of rock in LBM-DEM, a novel method based on the deformation of sphero-triangle particle is presented, where the deformation is calculated using the principle of minimum potential energy. The main features of the proposed THM model are that it can accurately calculate the fluid flow in fractures and the temperature in both fractures and DEM particles, and it takes full consideration of multiple anisotropic properties in rock. After validations, the THM fracturing mechanism is preliminarily investigated using the proposed model.
Enhancement of solar pond stability performance using an external magnetic field
2021, Energy Conversion and Management
A solar pond is a simple and reliable system that collects and stores solar energy for applications with low-grade heat supply. The main obstacle to the long-term stable operation of a solar pond is interface erosion induced by double-diffusive convection. In this study, an active method of using an external magnetic field is proposed to repress the intense convection region and improve its corresponding operating stability. A two-dimensional transient model is developed and solved using the lattice Boltzmann method with multiple-relaxation-time collisions. The double-diffusive convection, variation of solar energy absorption to depth, and changes in solution electrical conductivity are considered in the model. The fluid flow, heat, and mass transfer characteristics were investigated under continuous high illumination with or without an external magnetic field. When magnetic control is exerted on a solar pond, the decrease in the nonconvective zone thickness caused by interface erosion changes from 14.75% to 0. The state of the solar pond is transformed from a thermally unstable state to a theoretically stable state after 35 h of continuous high illumination. Further, the external magnetic field can delay concentration homogenization and improve the heat storage performance of a lower convective zone. A Hartmann number above 56.67 is recommended to enhance the stability of the solar pond. This research sheds new light on methods to improve the long-term stability of solar ponds.
Discrete unified gas kinetic scheme for incompressible Navier-Stokes equations
2021, Computers and Mathematics with Applications
The discrete unified gas kinetic scheme (DUGKS) combines the advantages of both the unified gas kinetic scheme (UGKS) and the lattice Boltzmann method. It can adopt the flexible meshes, meanwhile, the flux calculation is simple. However, the original DUGKS is proposed for the compressible flows. When we try to solve a problem governed by the incompressible Navier-Stokes (N-S) equations, the original DUGKS may bring some undesirable errors because of the compressible effect. To eliminate the compressible effect, the DUGKS for incompressible N-S equations is developed in this work. In addition, the Chapman-Enskog analysis ensures that the present DUGKS can solve the incompressible N-S equations exactly, meanwhile, a new non-extrapolation scheme is adopted to treat the Dirichlet boundary conditions. To test the present DUGKS for incompressible N-S equations, four problems are adopted. The first one is a periodic problem driven by an external force, which is used to test the influences of Courant–Friedrichs–Lewy condition number and the $M a c h$ number (Ma). Besides, some comparisons between the present DUGKS and some available results are also conducted. The second problem is Womersley flow, it is also used to test the influence of Ma, and the results show that the compressible effect is reduced obviously. Then, the two-dimensional lid-driven cavity flow is considered. In these simulations, the Reynolds number is varied from 400 to 1000000 to illustrate the accuracy, stability and efficiency of the present DUGKS. Finally, the numerical solutions of the three-dimensional lid-driven cavity flow suggest that the present DUGKS is suitable for the three-dimensional problems.
A lattice Boltzmann model for the viscous shallow water equations with source terms
2021, Journal of Hydrology
In this paper, a lattice Boltzmann (LB) model is proposed for a class of viscous shallow water equations, in which a second-order moment of the source term is applied to recover the viscosity in the governing equation and eliminate the additional errors generated during the Chapman-Enskog analysis. There are three different schemes based on different treatments of the source term. Through numerical simulations of several classical benchmark problems, we find that the second-order moment of the source term can not only improve the accuracy of the LB model, but also ensure the conservation of the system. In addition, the influence of rainfall intensity on shallow water flow is also taken into account, and the results show that present LB model can also accurately study such problems as overland flows.

View all citing articles on Scopus

¹: Present address: Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, PR China.

View full text

Multi-relaxation-time lattice Boltzmann model for incompressible flow

Abstract

Introduction

Section snippets

2D incompressible SRT-LB model

Incompressible MRT-LB model

Numerical simulation

Conclusion

Acknowledgements

Phys. Rep.

Comput. Phys. Commun.

Adv. Water Res.

Adv. Water Res.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

Comput. Fluids

J. Comput. Phys.

J. Comput. Phys.

Annu. Rev. Comput. Phys.

Annu. Rev. Fluid Mech.

Phys. Rev. E