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Author: Grafiati

Published: 6 September 2023

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1

Venturi, Sara, Silvia Di Francesco, Martin Geier, and Piergiorgio Manciola. "Forcing for a Cascaded Lattice Boltzmann Shallow Water Model." Water 12, no. 2 (February 6, 2020): 439. http://dx.doi.org/10.3390/w12020439.

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This work compares three forcing schemes for a recently introduced cascaded lattice Boltzmann shallow water model: a basic scheme, a second-order scheme, and a centred scheme. Although the force is applied in the streaming step of the lattice Boltzmann model, the acceleration is also considered in the transformation to central moments. The model performance is tested for one and two dimensional benchmarks.

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2

Gao, Shangwen, Chengbin Zhang, Yingjuan Zhang, Qiang Chen, Bo Li, and Suchen Wu. "Revisiting a class of modified pseudopotential lattice Boltzmann models for single-component multiphase flows." Physics of Fluids 34, no. 5 (May 2022): 057103. http://dx.doi.org/10.1063/5.0088246.

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Since its emergence, the pseudopotential lattice Boltzmann (LB) method has been regarded as a straightforward and practical approach for simulating single-component multiphase flows. However, its original form always results in a thermodynamic inconsistency, which, thus, impedes its further application. Several strategies for modifying the force term have been proposed to eliminate this limitation. In this study, four typical and widely used improved schemes—Li's single-relaxation-time (SRT) scheme [Li et al., “Forcing scheme in pseudopotential lattice Boltzmann model for multiphase flows,” Phys. Rev. E 86, 016709 (2012)] and multiple-relaxation-times (MRT) scheme [Li et al., “Lattice Boltzmann modeling of multiphase flows at large density ratio with an improved pseudopotential model,” Phys. Rev. E 87, 053301 (2013)], Kupershtokh's SRT scheme [Kupershtokh et al., “On equations of state in a lattice Boltzmann method,” Comput. Math. Appl. 58, 965 (2009)], and Huang's MRT scheme [Huang and Wu, “Third-order analysis of pseudopotential lattice Boltzmann model for multiphase flow,” J. Comput. Phys. 327, 121 (2016)]—are systematically analyzed and intuitively compared after an extension of the MRT framework. The theoretical and numerical results both indicate that the former three schemes are specific forms of the last one, which thus help further understand the improvements of these pseudopotential LB models for achieving thermodynamic consistency. In addition, we modified the calculation of the additional source term in the LB evolution equation. Numerical results for stationary and moving droplets confirm the higher accuracy.

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3

Wang, Liang, Zhaoli Guo, Baochang Shi, and Chuguang Zheng. "Evaluation of Three Lattice Boltzmann Models for Particulate Flows." Communications in Computational Physics 13, no. 4 (April 2013): 1151–72. http://dx.doi.org/10.4208/cicp.160911.200412a.

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AbstractA comparative study is conducted to evaluate three types of lattice Boltzmann equation (LBE) models for fluid flows with finite-sized particles, including the lattice Bhatnagar-Gross-Krook (BGK) model, the model proposed by Ladd [Ladd AJC, J. Fluid Mech., 271, 285-310 (1994); Ladd AJC, J. Fluid Mech., 271, 311-339 (1994)], and the multiple-relaxation-time (MRT) model. The sedimentation of a circular particle in a two-dimensional infinite channel under gravity is used as the first test problem. The numerical results of the three LBE schemes are compared with the theoretical results and existing data. It is found that all of the three LBE schemes yield reasonable results in general, although the BGK scheme and Ladd’s scheme give some deviations in some cases. Our results also show that the MRT scheme can achieve a better numerical stability than the other two schemes. Regarding the computational efficiency, it is found that the BGK scheme is the most superior one, while the other two schemes are nearly identical. We also observe that the MRT scheme can unequivocally reduce the viscosity dependence of the wall correction factor in the simulations, which reveals the superior robustness of the MRT scheme. The superiority of the MRT scheme over the other two schemes is also confirmed by the simulation of the sedimentation of an elliptical particle.

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4

Xu, Kun, and Li-Shi Luo. "Connection Between Lattice-Boltzmann Equation and Beam Scheme." International Journal of Modern Physics C 09, no. 08 (December 1998): 1177–87. http://dx.doi.org/10.1142/s0129183198001072.

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In this paper we analyze and compare the lattice-Boltzmann equation with the beam scheme in detail. We notice the similarity and differences between the lattice Boltzmann equation and the beam scheme. We show that the accuracy of the lattice-Boltzmann equation is indeed second order in space. We discuss the advantages and limitations of the lattice-Boltzmann equation and the beam scheme. Based on our analysis, we propose an improved multi-dimensional beam scheme.

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5

Qiu, Ruofan, Rongqian Chen, and Yancheng You. "An implicit-explicit finite-difference lattice Boltzmann subgrid method on nonuniform meshes." International Journal of Modern Physics C 28, no. 04 (April 2017): 1750045. http://dx.doi.org/10.1142/s0129183117500450.

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In this paper, an implicit-explicit finite-difference lattice Boltzmann method with subgrid model on nonuniform meshes is proposed. The implicit-explicit Runge–Kutta scheme, which has good convergence rate, is used for the time discretization and a mixed difference scheme, which combines the upwind scheme with the central scheme, is adopted for the space discretization. Meanwhile, the standard Smagorinsky subgrid model is incorporated into the finite-difference lattice Boltzmann scheme. The effects of implicit-explicit Runge–Kutta scheme and nonuniform meshes of present lattice Boltzmann method are discussed through simulations of a two-dimensional lid-driven cavity flow on nonuniform meshes. Moreover, the comparison simulations of the present method and multiple relaxation time lattice Boltzmann subgrid method are conducted qualitatively and quantitatively.

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6

Wen, Mengke, Weidong Li, and Zhangyan Zhao. "A hybrid scheme coupling lattice Boltzmann method and finite-volume lattice Boltzmann method for steady incompressible flows." Physics of Fluids 34, no. 3 (March 2022): 037114. http://dx.doi.org/10.1063/5.0085370.

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We present a new hybrid method coupling the adaptive mesh refinement lattice Boltzmann method (AMRLBM) and the finite-volume lattice Boltzmann method (FVLBM) to improve both the simulation efficiency and adaptivity for steady incompressible flows with complex geometries. The present method makes use of the domain decomposition, in which the FVLBM sub-domain is applied to the region adjacent to the walls, and is coupled to the lattice Boltzmann method (LBM) sub-domain in the rest of the flow field to enhance the ability of the LBM to deal with irregular geometries without sacrificing the high efficiency and accuracy property of the LBM. In the LBM sub-domain, a cell-centered lattice structure-based AMRLBM is used and, in the FVLBM sub-domain, the gas-kinetic Bhatnagar–Gross–Krook (BGK) scheme-based FVLBM is adopted to reduce the numerical dissipation and enhance the efficiency of FVLBM. Moreover, not like the conventional LBM and Navier–Stokes equation solver-based hybrid schemes, the present hybrid scheme combines two kinds of lattice Boltzmann equation solvers, that is, AMRLBM and FVLBM, which makes the present scheme much simpler and better consistency than the conventional hybrid schemes. To assess the accuracy and efficacy of the proposed method, various benchmark studies, including the Kovasznay flow, the lid-driven cavity flow with Reynolds number [Formula: see text], 400, and 1000, and the steady flow past a cylinder with [Formula: see text] and 40, are also conducted. The numerical results show that the present scheme can be an efficient and reliable method for steady incompressible flows.

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7

SUGA, SHINSUKE. "STABILITY AND ACCURACY OF LATTICE BOLTZMANN SCHEMES FOR ANISOTROPIC ADVECTION-DIFFUSION EQUATIONS." International Journal of Modern Physics C 20, no. 04 (April 2009): 633–50. http://dx.doi.org/10.1142/s0129183109013856.

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The stability of the numerical schemes for anisotropic advection-diffusion equations derived from the lattice Boltzmann equation with the D2Q4 particle velocity model is analyzed through eigenvalue analysis of the amplification matrices of the scheme. Accuracy of the schemes is investigated by solving benchmark problems, and the LBM scheme is compared with traditional implicit schemes. Numerical experiments demonstrate that the LBM scheme produces stable numerical solutions close to the analytical solutions when the values of the relaxation parameters in x and y directions are greater than 1.9 and the Courant numbers satisfy the stability condition. Furthermore, the numerical solutions produced by the LBM scheme are more accurate than those of the Crank–Nicolson finite difference scheme for the case where the Courant numbers are set to be values close to the upper bound of the stability region of the scheme.

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8

Van Der Sman, R. G. M. "Lattice-Boltzmann Scheme for Natural Convection in Porous Media." International Journal of Modern Physics C 08, no. 04 (August 1997): 879–88. http://dx.doi.org/10.1142/s0129183197000758.

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A lattice-Boltzmann scheme for natural convection in porous media is developed and applied to the heat transfer problem of a 1000 kg potato packaging. The scheme has features new to the field of LB schemes. It is mapped on a orthorhombic lattice instead of the traditional cubic lattice. Furthermore the boundary conditions are formulated with a single paradigm based upon the particle fluxes. Our scheme is well able to reproduce (1) the analytical solutions of simple model problems and (2) the results from cooling down experiments with potato packagings.

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9

ZHENG, H. W., and C. SHU. "EVALUATION OF THE PERFORMANCE OF THE HYBRID LATTICE BOLTZMANN BASED NUMERICAL FLUX." International Journal of Modern Physics: Conference Series 42 (January 2016): 1660152. http://dx.doi.org/10.1142/s2010194516601526.

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It is well known that the numerical scheme is a key factor to the stability and accuracy of a Navier-Stokes solver. Recently, a new hybrid lattice Boltzmann numerical flux (HLBFS) is developed by Shu's group. It combines two different LBFS schemes by a switch function. It solves the Boltzmann equation instead of the Euler equation. In this article, the main object is to evaluate the ability of this HLBFS scheme by our in-house cell centered hybrid mesh based Navier-Stokes code. Its performance is examined by several widely-used bench-mark test cases. The comparisons on results between calculation and experiment are conducted. They show that the scheme can capture the shock wave as well as the resolving of boundary layer.

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10

LALLEMAND, PIERRE, and LI-SHI LUO. "HYBRID FINITE-DIFFERENCE THERMAL LATTICE BOLTZMANN EQUATION." International Journal of Modern Physics B 17, no. 01n02 (January 20, 2003): 41–47. http://dx.doi.org/10.1142/s0217979203017060.

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We analyze the acoustic and thermal properties of athermal and thermal lattice Boltzmann equation (LBE) in 2D and show that the numerical instability in the thermal lattice Boltzmann equation (TLBE) is related to the algebraic coupling among different modes of the linearized evolution operator. We propose a hybrid finite-difference (FD) thermal lattice Boltzmann equation (TLBE). The hybrid FD-TLBE scheme is far superior over the existing thermal LBE schemes in terms of numerical stability. We point out that the lattice BGK equation is incompatible with the multiple relaxation time model.

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11

Dubois, François, and Pierre Lallemand. "On Triangular Lattice Boltzmann Schemes for Scalar Problems." Communications in Computational Physics 13, no. 3 (March 2013): 649–70. http://dx.doi.org/10.4208/cicp.381011.270112s.

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AbstractWe propose to extend the d’Humieres version of the lattice Boltzmann scheme to triangular meshes. We use Bravais lattices or more general lattices with the property that the degree of each internal vertex is supposed to be constant. On such meshes, it is possible to define the lattice Boltzmann scheme as a discrete particle method, without need of finite volume formulation or Delaunay-Voronoi hypothesis for the lattice. We test this idea for the heat equation and perform an asymptotic analysis with the Taylor expansion method for two schemes named D2T4 and D2T7. The results show a convergence up to second order accuracy and set new questions concerning a possible super-convergence.

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12

Hsu, C. T., S. W. Chiang, and K. F. Sin. "A Novel Dynamic Quadrature Scheme for Solving Boltzmann Equation with Discrete Ordinate and Lattice Boltzmann Methods." Communications in Computational Physics 11, no. 4 (April 2012): 1397–414. http://dx.doi.org/10.4208/cicp.150510.150511s.

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AbstractThe Boltzmann equation (BE) for gas flows is a time-dependent nonlinear differential-integral equation in 6 dimensions. The current simplified practice is to linearize the collision integral in BE by the BGK model using Maxwellian equilibrium distribution and to approximate the moment integrals by the discrete ordinate method (DOM) using a finite set of velocity quadrature points. Such simplification reduces the dimensions from 6 to 3, and leads to a set of linearized discrete BEs. The main difficulty of the currently used (conventional) numerical procedures occurs when the mean velocity and the variation of temperature are large that requires an extremely large number of quadrature points. In this paper, a novel dynamic scheme that requires only a small number of quadrature points is proposed. This is achieved by a velocity-coordinate transformation consisting of Galilean translation and thermal normalization so that the transformed velocity space is independent of mean velocity and temperature. This enables the efficient implementation of Gaussian-Hermite quadrature. The velocity quadrature points in the new velocity space are fixed while the correspondent quadrature points in the physical space change from time to time and from position to position. By this dynamic nature in the physical space, this new quadrature scheme is termed as the dynamic quadrature scheme (DQS). The DQS was implemented to the DOM and the lattice Boltzmann method (LBM). These new methods with DQS are therefore termed as the dynamic discrete ordinate method (DDOM) and the dynamic lattice Boltzmann method (DLBM), respectively. The new DDOM and DLBM have been tested and validated with several testing problems. Of the same accuracy in numerical results, the proposed schemes are much faster than the conventional schemes. Furthermore, the new DLBM have effectively removed the incompressible and isothermal restrictions encountered by the conventional LBM.

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13

Ghidaoui, Mohamed S., and Nanzhou Li. "Generalized Boltzmann equation for shallow water flows." Journal of Hydroinformatics 5, no. 1 (January 1, 2003): 1–10. http://dx.doi.org/10.2166/hydro.2003.0001.

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This invited review paper introduces the Boltzmann-based approach for the numerical modelling of surface water flows to hydroinformaticians. The paper draws upon earlier work by our group as well as others. This review formulates the generalized Boltzmann equation for 1D and 2D shallow water flows and shows that the statistical moments of these generalized equations provide the classical continuity and momentum equations in shallow waters. The connection between the generalized Boltzmann equation and classical shallow water equations provides a framework for formulating new computational approaches to surface water flows. To illustrate, a first-order explicit scheme based on the generalized Boltzmann equation for 1D shallow waters in frictionless and horizontal channels is formulated. The resulting scheme is applied to the classical dam break problem. Comparison with the analytical solution shows that the Boltzmann-based scheme is highly accurate and free of spurious oscillations, illustrating the potential of the method for surface water problems and other applications.

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14

SOFONEA, VICTOR, and ROBERT F. SEKERKA. "DIFFUSIVITY OF TWO-COMPONENT ISOTHERMAL FINITE DIFFERENCE LATTICE BOLTZMANN MODELS." International Journal of Modern Physics C 16, no. 07 (July 2005): 1075–90. http://dx.doi.org/10.1142/s0129183105007741.

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Diffusion equations are derived for an isothermal lattice Boltzmann model with two components. The first-order upwind finite difference scheme is used to solve the evolution equations for the distribution functions. When using this scheme, the numerical diffusivity, which is a spurious diffusivity in addition to the physical diffusivity, is proportional to the lattice spacing and significantly exceeds the physical value of the diffusivity if the number of lattice nodes per unit length is too small. Flux limiter schemes are introduced to overcome this problem. Empirical analysis of the results of flux limiter schemes shows that the numerical diffusivity is very small and depends quadratically on the lattice spacing.

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15

Succi, S., and P. Vergari. "A Lattice Boltzmann Scheme for Semiconductor Dynamics." VLSI Design 6, no. 1-4 (January 1, 1998): 137–40. http://dx.doi.org/10.1155/1998/54940.

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16

WANG, Y., Y. L. HE, T. S. ZHAO, G. H. TANG, and W. Q. TAO. "IMPLICIT-EXPLICIT FINITE-DIFFERENCE LATTICE BOLTZMANN METHOD FOR COMPRESSIBLE FLOWS." International Journal of Modern Physics C 18, no. 12 (December 2007): 1961–83. http://dx.doi.org/10.1142/s0129183107011868.

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We propose an implicit-explicit finite-difference lattice Boltzmann method for compressible flows in this work. The implicit-explicit Runge–Kutta scheme, which solves the relaxation term of the discrete velocity Boltzmann equation implicitly and other terms explicitly, is adopted for the time discretization. Owing to the characteristic of the collision invariants in the lattice Boltzmann method, the implicitness can be completely eliminated, and thus no iteration is needed in practice. In this fashion, problems (no matter stiff or not) can be integrated quickly with large Courant–Friedriche–Lewy numbers. As a result, with our implicit-explicit finite-difference scheme the computational convergence rate can be significantly improved compared with previous finite-difference and standard lattice Boltzmann methods. Numerical simulations of the Riemann problem, Taylor vortex flow, Couette flow, and oscillatory compressible flows with shock waves show that our implicit-explicit finite-difference lattice Boltzmann method is accurate and efficient. In addition, it is demonstrated that with the proposed scheme non-uniform meshes can also be implemented with ease.

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17

STRUCKMEIER, JENS, and KONRAD STEINER. "SECOND-ORDER SCHEME FOR THE SPATIALLY HOMOGENEOUS BOLTZMANN EQUATION WITH MAXWELLIAN MOLECULES." Mathematical Models and Methods in Applied Sciences 06, no. 01 (February 1996): 137–47. http://dx.doi.org/10.1142/s0218202596000080.

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In the standard approach particle methods for the Boltzmann equation are obtained using an explicit time discretization of the spatially homogeneous Boltzmann equation. This kind of discretization leads to a restriction on the discretization parameter as well as on the differential cross-section in the case of the general Boltzmann equation. Recently, construction of an implicit particle scheme for the Boltzmann equation with Maxwellian molecules was shown. This paper combines both approaches using a linear combination of explicit and implicit discretizations. It is shown that the new method leads to a second-order particle method when using an equiweighting of explicit and implicit discretization.

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18

Wang, Dongmin, Gaoshuai Lin, Yugang Zhao, and Ming Gao. "Effects of Numerical Schemes of Contact Angle on Simulating Condensation Heat Transfer in a Subcooled Microcavity by Pseudopotential Lattice Boltzmann Model." Energies 16, no. 6 (March 10, 2023): 2622. http://dx.doi.org/10.3390/en16062622.

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Various numerical schemes of contact angle are widely used in pseudopotential lattice Boltzmann model to simulate substrate contact angle in condensation. In this study, effects of numerical schemes of contact angle on condensation nucleation and heat transfer simulation are clarified for the first time. The three numerical schemes are pseudopotential-based contact angle scheme, pseudopotential-based contact angle scheme with a ghost fluid layer constructed on the substrate with weighted average density of surrounding fluid nodes, and the geometric formulation scheme. It is found that the subcooling condition destabilizes algorithm of pseudopotential-based contact angle scheme. However, with a ghost fluid layer constructed on the substrate or using geometric formulation scheme, the algorithm becomes stable. The subcooling condition also decreases the simulated contact angle magnitude compared with that under an isothermal condition. The fluid density variation near a microcavity wall simulated by pseudopotential-based contact angle scheme plays the role of the condensation nucleus and triggers “condensation nucleation”. However, with a ghost fluid layer constructed on the substrate or using geometric formulation scheme, the simulated fluid density distribution near the wall is uniform so that no condensation nucleus appears in the microcavity. Thus, “condensation nucleation” cannot occur spontaneously in the microcavity unless a thin liquid film is initialized as a nucleus in the microcavity. The heat flux at the microcavity wall is unphysical during the “condensation nucleation” process, but it becomes reasonable with a liquid film formed in the microcavity. As a whole, it is recommended to use pseudopotential-based contact angle scheme with a ghost fluid layer constructed on the substrate or use the geometric formulation scheme to simulate condensation under subcooling conditions. This study provides guidelines for choosing the desirable numerical schemes of contact angle in condensation simulation by pseudopotential lattice Boltzmann model so that more efficient strategies for condensation heat transfer enhancement can be obtained from numerical simulations.

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19

Peng, Yong, Bo Wang, and Yunfei Mao. "Study on Force Schemes in Pseudopotential Lattice Boltzmann Model for Two-Phase Flows." Mathematical Problems in Engineering 2018 (2018): 1–9. http://dx.doi.org/10.1155/2018/6496379.

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Multiphase flows are very important in industrial application. In present study, the force schemes in the pseudopotential LBM for two-phase flows have been compared in detail and the force schemes include Shan-Chen, EDM, MED, and Guo’s schemes. Numerical simulations confirm that all four schemes are consistent with the Laplace law. For Shan-Chen scheme, the smaller τ is, the smaller the surface tension is. However, for other schemes, τ has no effect on surface tension. When 0.6<τ≤1, the achieved density ratio will reduce as τ reduces. During this range of τ, the maximum density ratio of EDM scheme will be greater than that of other schemes. For a constant T, the curves of the maximum spurious currents (u′) has a minimum value which is corresponding to τ′ except for EDM schemes. In the region of τ′<τ≤1, u′ will reduce as τ decreases. On the other hand, in the area of τ≤τ′, u′ will increase as τ decreases. However, for EDM scheme, u′ will increase as τ increases.

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20

XUAN, Yimin. "Application of lattice Boltzmann scheme to nanofluids." Science in China Series E 47, no. 2 (2004): 129. http://dx.doi.org/10.1360/03ye0163.

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21

Dubois, François, Pierre Lallemand, and Mahdi Tekitek. "On a superconvergent lattice Boltzmann boundary scheme." Computers & Mathematics with Applications 59, no. 7 (April 2010): 2141–49. http://dx.doi.org/10.1016/j.camwa.2009.08.055.

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22

Succi, S., M. Vergassola, and R. Benzi. "Lattice Boltzmann scheme for two-dimensional magnetohydrodynamics." Physical Review A 43, no. 8 (April 1, 1991): 4521–24. http://dx.doi.org/10.1103/physreva.43.4521.

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23

Ma, Huifang, Bin Wu, Ying Wang, Hao Ren, Wanshun Jiang, Mingming Tang, and Wenyue Guo. "A Novel Lattice Boltzmann Scheme with Single Extended Force Term for Electromagnetic Wave Propagating in One-Dimensional Plasma Medium." Electronics 11, no. 6 (March 10, 2022): 882. http://dx.doi.org/10.3390/electronics11060882.

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A one-dimensional plasma medium is playing a crucial role in modern sensing device design, which can benefit significantly from numerical electromagnetic wave simulation. In this study, we introduce a novel lattice Boltzmann scheme with a single extended force term for electromagnetic wave propagation in a one-dimensional plasma medium. This method is developed by reconstructing the solution to the macroscopic Maxwell’s equations recovered from the lattice Boltzmann equation. The final formulation of the lattice Boltzmann scheme involves only the equilibrium and one non-equilibrium force term. Among them, the former is calculated from the macroscopic electromagnetic variables, and the latter is evaluated from the dispersive effect. Thus, the proposed lattice Boltzmann scheme directly tracks the evolution of macroscopic electromagnetic variables, which yields lower memory costs and facilitates the implementation of physical boundary conditions. Detailed conduction is carried out based on the Chapman–Enskog expansion technique to prove the mathematical consistency between the proposed lattice Boltzmann scheme and Maxwell’s equations. Based on the proposed method, we present electromagnetic pulse propagating behaviors in nondispersive media and the response of a one-dimensional plasma slab to incident electromagnetic waves that span regions above and below the plasma frequency ωp, and further investigate the optical properties of a one-dimensional plasma photonic crystal with periodic thin layers of plasma with different layer thicknesses to verify the stability, accuracy, and flexibility of the proposed method.

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24

ZHOU, J. G. "AN ELASTIC-COLLISION SCHEME FOR LATTICE BOLTZMANN METHODS." International Journal of Modern Physics C 12, no. 03 (March 2001): 387–401. http://dx.doi.org/10.1142/s0129183101001833.

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An elastic-collision scheme is developed to achieve slip and semi-slip boundary conditions for lattice Boltzmann methods. Like the bounce-back scheme, the proposed scheme is efficient, robust and generally suitable for flows in arbitrary complex geometries. It involves an equivalent level of computation effort to the bounce-back scheme. The new scheme is verified by predicting wind-driven circulating flows in a dish-shaped basin and a flow in a strongly bent channel, showing good agreement with analytical solutions and experimental data. The capability of the scheme for simulating flows through multiple bodies has also been demonstrated.

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TAKADA, NAOKI, AKIO TOMIYAMA, and SHIGEO HOSOKAWA. "LATTICE BOLTZMANN SIMULATION OF INTERFACIAL DEFORMATION." International Journal of Modern Physics B 17, no. 01n02 (January 20, 2003): 179–82. http://dx.doi.org/10.1142/s0217979203017308.

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This study describes the numerical simulations of two-phase interfacial deformations using the binary fluid (BF) model in the lattice Boltzmann method (LBM), where a macroscopic fluid involves mesoscopic particles repeating collisions and propagations and an interface is reproduced in a self-organizing way by repulsive interaction between different kinds of particles. Schemes for the BF model are proposed to simulate motions of immiscible two phases with different mass densities. For higher Reynolds number, the finite difference-based lattice Boltzmann scheme is applied to the kinetic equations of particles, which include convection terms to reduce the diffusivity of each phase volume. In addition, two parameters are introduced into the BF model to adjust surface tension and interfacial thickness independently. The numerical results of three-dimensional bubble motion under gravity and two-dimensional droplet deformation under shear stress indicate that the lattice-Boltzmann BF model with the proposed schemes would be applicable to simulating interfacial dynamics of immiscible two-phase fluids.

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26

ZHOU, JIAN GUO. "LATTICE BOLTZMANN SIMULATIONS OF DISCONTINUOUS FLOWS." International Journal of Modern Physics C 18, no. 01 (January 2007): 1–14. http://dx.doi.org/10.1142/s0129183107010280.

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The lattice Boltzmann model for the shallow water equations (LABSWE) is applied to the simulation of certain discontinuous flows. Curved boundaries are treated efficiently, using either the elastic-collision scheme for slip and semi-slip boundary conditions or the bounce-back scheme for no-slip conditions. The force term is accurately determined by means of the centred scheme. Simulations are presented of a small pulse-like perturbation of the still water surface, a dam break, and a surge wave interaction with a circular cylinder. The results agree well with predictions from alternative high-resolution Riemann solver based methods, demonstrating the capability of LABSWE to predict shallow water flows containing discontinuities.

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27

Delouei, A. Amiri, M. Nazari, M. H. Kayhani, and S. Succi. "Immersed Boundary – Thermal Lattice Boltzmann Methods for Non-Newtonian Flows Over a Heated Cylinder: A Comparative Study." Communications in Computational Physics 18, no. 2 (July 30, 2015): 489–515. http://dx.doi.org/10.4208/cicp.060414.220115a.

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AbstractIn this study, we compare different diffuse and sharp interface schemes of direct-forcing immersed boundary — thermal lattice Boltzmann method (IB-TLBM) for non-Newtonian flow over a heated circular cylinder. Both effects of the discrete lattice and the body force on the momentum and energy equations are considered, by applying the split-forcing Lattice Boltzmann equations. A new technique based on predetermined parameters of direct forcing IB-TLBM is presented for computing the Nusselt number. The study covers both steady and unsteady regimes (20<Re<80) in the power-law index range of 0.6<n<1.4, encompassing both shear-thinning and shear-thickening non-Newtonian fluids. The numerical scheme, hydrodynamic approach and thermal parameters of different interface schemes are compared in both steady and unsteady cases. It is found that the sharp interface scheme is a suitable and possibly competitive method for thermal-IBM in terms of accuracy and computational cost.

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28

Li, Qiaojie, Zhoushun Zheng, Shuang Wang, and Jiankang Liu. "A Multilevel Finite Difference Scheme for One-Dimensional Burgers Equation Derived from the Lattice Boltzmann Method." Journal of Applied Mathematics 2012 (2012): 1–13. http://dx.doi.org/10.1155/2012/925920.

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An explicit finite difference scheme for one-dimensional Burgers equation is derived from the lattice Boltzmann method. The system of the lattice Boltzmann equations for the distribution of the fictitious particles is rewritten as a three-level finite difference equation. The scheme is monotonic and satisfies maximum value principle; therefore, the stability is proved. Numerical solutions have been compared with the exact solutions reported in previous studies. TheL2, L∞and Root-Mean-Square (RMS) errors in the solutions show that the scheme is accurate and effective.

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BERNASCHI, MASSIMO, and SAURO SUCCI. "ACCELERATED LATTICE BOLTZMANN SCHEME FOR STEADY-STATE FLOWS." International Journal of Modern Physics B 17, no. 01n02 (January 20, 2003): 1–7. http://dx.doi.org/10.1142/s021797920301700x.

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JI, C. Z., C. SHU, and N. ZHAO. "A LATTICE BOLTZMANN METHOD-BASED FLUX SOLVER AND ITS APPLICATION TO SOLVE SHOCK TUBE PROBLEM." Modern Physics Letters B 23, no. 03 (January 30, 2009): 313–16. http://dx.doi.org/10.1142/s021798490901828x.

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This paper presents an approach, which combines the conventional finite volume method (FVM) with the lattice Boltzmann Method (LBM), to simulate compressible flows. Similar to the Godunov scheme, in the present approach, LBM is used to evaluate the flux at the interface for local Riemann problem when solving Euler/Navier-Stokes (N-S) equations by FVM. Two kinds of popular compressible Lattice Boltzmann models are applied in the new scheme, and some numerical experiments are performed to validate the proposed approach. From the sharper shock profile and higher computational efficiency, numerical results demonstrate that the proposed scheme is superior to the conventional Godunov scheme. It is expected that the proposed scheme has a potential to become an efficient flux solver in solving compressible Euler/N-S equations.

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31

Zhang, Raoyang, Chenghai Sun, Yanbing Li, Rajani Satti, Richard Shock, James Hoch, and Hudong Chen. "Lattice Boltzmann Approach for Local Reference Frames." Communications in Computational Physics 9, no. 5 (May 2011): 1193–205. http://dx.doi.org/10.4208/cicp.021109.111110s.

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AbstractIn this paper we present a generalized lattice Boltzmann based approach for sliding-mesh local reference frame. This scheme exactly conserves hydrodynamic fluxes across local reference frame interface. The accuracy and robustness of our scheme are demonstrated by benchmark validations.

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Schwarzmeier, Christoph, and Ulrich Rüde. "Comparison of refilling schemes in the free-surface lattice Boltzmann method." AIP Advances 12, no. 11 (November 1, 2022): 115324. http://dx.doi.org/10.1063/5.0131159.

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Simulating mobile liquid–gas interfaces with the free-surface lattice Boltzmann method (FSLBM) requires frequent re-initialization of fluid flow information in computational cells that convert from gas to liquid. The corresponding algorithm, here referred to as the refilling scheme, is crucial for the successful application of the FSLBM in terms of accuracy and numerical stability. This study compares five refilling schemes that extract information from the surrounding liquid and interface cells by averaging, extrapolating, or assuming one of the three different equilibrium states. Six numerical experiments were performed, covering a broad spectrum of possible scenarios. These include a standing gravity wave, a rectangular and cylindrical dam break, a Taylor bubble, a drop impact into liquid, and a bubbly plane Poiseuille flow. In some simulations, the averaging, extrapolation, and one equilibrium-based scheme were numerically unstable. Overall, the results have shown that the simplest equilibrium-based scheme should be preferred in terms of numerical stability, computational cost, accuracy, and ease of implementation.

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Che Sidik, Nor Azwadi, and Aman Ali Khan. "Simulation of Flow over a Cavity Using Multi-Relaxation Time Thermal Lattice Boltzmann Method." Applied Mechanics and Materials 554 (June 2014): 296–300. http://dx.doi.org/10.4028/www.scientific.net/amm.554.296.

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This article provides numerically study of the multi-relaxation time thermal lattice Boltzmann method (LBM) for compute the flow and isotherm characteristics in the bottom heated cavity located o n a floor of horizontal channel . A double-distribution function (DFF) was coupled with MRT thermal LBM to study the effects of various grashof number (Gr), Reynolds number (Re) and Aspect Ratio (AR) on the flow and isotherm characteristic. The results we re compared with the conventional single-relaxation time lattice Boltzmann scheme and benchmark solution for such flow configuration. The results of the numer ical simulation indicate that multi-relaxation time thermal lattice Boltzmann scheme demonstrated good agreement, which supports its validity in computing fluid flow problem.

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NOR AZWADI, C. S., and T. TANAHASHI. "SIMPLIFIED THERMAL LATTICE BOLTZMANN IN INCOMPRESSIBLE LIMIT." International Journal of Modern Physics B 20, no. 17 (July 10, 2006): 2437–49. http://dx.doi.org/10.1142/s0217979206034789.

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In this paper, an incompressible thermohydrodynamics for the lattice Boltzmann scheme is developed. The basic idea is to solve the velocity field and the temperature field using two different distribution functions. A derivation of the lattice Boltzmann scheme from the continuous Boltzmann equation is discussed in detail. By using the same procedure as in the derivation of the discretised density distribution function, we found that a new lattice of four-velocity model for internal energy density distribution function can be developed where the viscous and compressive heating effects are negligible. This model is validated by the numerical simulation of the porous plate couette flow problem where the analytical solution exists and the natural convection flows in a square cavity.

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MILLER, W. "CRYSTAL GROWTH KINETICS AND FLUID FLOW." International Journal of Modern Physics B 17, no. 01n02 (January 20, 2003): 227–30. http://dx.doi.org/10.1142/s0217979203017394.

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A new type of a lattice phase-field model is developed and coupled with the lattice Boltzmann method to compute the soldification influenced by convection. Two methods of treating the solid-fluid interaction within the lattice Boltzmann scheme are tested.

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Enders, Peter. "Historical Prospective: Boltzmann’s versus Planck’s State Counting—Why Boltzmann Did Not Arrive at Planck’s Distribution Law." Journal of Thermodynamics 2016 (January 27, 2016): 1–13. http://dx.doi.org/10.1155/2016/9137926.

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Why does Planck (1900), referring to Boltzmann’s 1877 probabilistic treatment, obtain his quantum distribution function while Boltzmann did not? To answer this question, both treatments are compared on the basis of Boltzmann’s 1868 three-level scheme (configuration—occupation—occupancy). Some calculations by Planck (1900, 1901, and 1913) and Einstein (1907) are also sketched. For obtaining a quantum distribution, it is crucial to stick with a discrete energy spectrum and to make the limit transitions to infinity at the right place. For correct state counting, the concept of interchangeability of particles is superior to that of indistinguishability.

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Dubois, François. "Third order equivalent equation of lattice Boltzmann scheme." Discrete and Continuous Dynamical Systems 23, no. 1/2 (September 2008): 221–48. http://dx.doi.org/10.3934/dcds.2009.23.221.

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Krivovichev, G. V. "On the finite-element-based lattice Boltzmann scheme." Applied Mathematical Sciences 8 (2014): 1605–20. http://dx.doi.org/10.12988/ams.2014.4138.

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Halliday, I., L. A. Hammond, and C. M. Care. "Enhanced closure scheme for lattice Boltzmann equation hydrodynamics." Journal of Physics A: Mathematical and General 35, no. 12 (March 15, 2002): L157—L166. http://dx.doi.org/10.1088/0305-4470/35/12/102.

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SETA, Takeshi, Koji KONO, Daniel MARTINEZ, and Shiyi CHEN. "Lattice Boltzmann Scheme for Simulating Two-Phase Flows." JSME International Journal Series B 43, no. 2 (2000): 305–13. http://dx.doi.org/10.1299/jsmeb.43.305.

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Ho, Jeng-Rong, Chun-Pao Kuo, Wen-Shu Jiaung, and Cherng-Jyh Twu. "LATTICE BOLTZMANN SCHEME FOR HYPERBOLIC HEAT CONDUCTION EQUATION." Numerical Heat Transfer, Part B: Fundamentals 41, no. 6 (June 2002): 591–607. http://dx.doi.org/10.1080/10407790190053798.

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SETA, Takeshi, Koji KONO, Daniel MARTINEZ, and Shiyi CHEN. "Lattice Boltzmann Scheme for Simulating Two-Phase Flows." Transactions of the Japan Society of Mechanical Engineers Series B 65, no. 634 (1999): 1955–63. http://dx.doi.org/10.1299/kikaib.65.1955.

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Tian, Zhi-Wei, Chun Zou, Hong-Juan Liu, Zhao-Li Guo, Zhao-Hui Liu, and Chu-Guang Zheng. "Lattice Boltzmann scheme for simulating thermal micro-flow." Physica A: Statistical Mechanics and its Applications 385, no. 1 (November 2007): 59–68. http://dx.doi.org/10.1016/j.physa.2007.01.021.

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Alvarez-Ramírez, José, Francisco J. Valdés-Parada, and J. Alberto Ochoa-Tapia. "A lattice-Boltzmann scheme for Cattaneo’s diffusion equation." Physica A: Statistical Mechanics and its Applications 387, no. 7 (March 2008): 1475–84. http://dx.doi.org/10.1016/j.physa.2007.10.051.

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Chen, Sheng, Zhaohui Liu, Zhiwei Tian, Baochang Shi, and Chuguang Zheng. "A simple lattice Boltzmann scheme for combustion simulation." Computers & Mathematics with Applications 55, no. 7 (April 2008): 1424–32. http://dx.doi.org/10.1016/j.camwa.2007.08.020.

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Stiebler, Maik, Jonas Tölke, and Manfred Krafczyk. "Advection–diffusion lattice Boltzmann scheme for hierarchical grids." Computers & Mathematics with Applications 55, no. 7 (April 2008): 1576–84. http://dx.doi.org/10.1016/j.camwa.2007.08.024.

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Xie, Wenjun. "An axisymmetric multiple-relaxation-time lattice Boltzmann scheme." Journal of Computational Physics 281 (January 2015): 55–66. http://dx.doi.org/10.1016/j.jcp.2014.10.019.

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48

Babovsky, H., and H. Neunzert. "On a simulation scheme for the Boltzmann equation." Mathematical Methods in the Applied Sciences 8, no. 1 (1986): 223–33. http://dx.doi.org/10.1002/mma.1670080114.

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van der Sman, R. G. M., and M. H. Ernst. "Convection-Diffusion Lattice Boltzmann Scheme for Irregular Lattices." Journal of Computational Physics 160, no. 2 (May 2000): 766–82. http://dx.doi.org/10.1006/jcph.2000.6491.

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PAN, X. F., AIGUO XU, GUANGCAI ZHANG, and SONG JIANG. "LATTICE BOLTZMANN APPROACH TO HIGH-SPEED COMPRESSIBLE FLOWS." International Journal of Modern Physics C 18, no. 11 (November 2007): 1747–64. http://dx.doi.org/10.1142/s0129183107011716.

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We present an improved lattice Boltzmann model for high-speed compressible flows. The model is composed of a discrete-velocity model by Kataoka and Tsutahara15 and an appropriate finite-difference scheme combined with an additional dissipation term. With the dissipation term parameters in the model can be flexibly chosen so that the von Neumann stability condition is satisfied. The influence of the various model parameters on the numerical stability is analyzed and some reference values of parameter are suggested. The new scheme works for both subsonic and supersonic flows with a Mach number up to 30 (or higher), which is validated by well-known benchmark tests. Simulations on Riemann problems with very high ratios (1000:1) of pressure and density also show good accuracy and stability. Successful recovering of regular and double Mach shock reflections shows the potential application of the lattice Boltzmann model to fluid systems where non-equilibrium processes are intrinsic. The new scheme for stability can be easily extended to other lattice Boltzmann models.

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