An improved bounce-back scheme for complex boundary conditions in lattice Boltzmann method

In recent years, the lattice Boltzmann method (LBM) has been widely adopted to simulate various fluid systems, and the boundary treatment has been an active topic during the LBM development. In this paper, we present a novel approach to improve the bounce-back boundary treatment for moving surfaces with arbitrary configurations. We follow the framework originally proposed by Ladd [A.J.C. Ladd, Numerical simulations of particulate suspensions via a discretized Boltzman equation. Part 1. Theoretical foundation, Journal of Fluid Mechanics 271 (1994) 285–309]; however, the adjustment in the density distribution during the bouncing-back process at the boundary is calculated using the midpoint velocity inter-/extrapolated from the boundary and fluid velocities, instead of the real boundary velocity in the Ladd method. This modification ensures that the bouncing-back process and the density distribution adjustment both take place at a same location: the midpoint of a boundary lattice link, and thus removes the discrepancy of bouncing-back at the midpoint but density distribution adjustment at the boundary point in the original Ladd method. When compared with other existing boundary models, this method involves a simpler algorithm and exhibits a comparable or even better accuracy in describing flow field and flow-structure interaction, as demonstrated by several test simulations. Therefore, this boundary method could be considered as a competitive alternative for boundary treatment in LBM simulations, especially for particulate and porous flows with large fluid–solid interfacial areas.     

A curved no-slip boundary condition for the lattice Boltzmann method

We present a generalization of the no-slip boundary condition by Lätt et al. [J. Lätt, B. Chopard, O. Malaspinas, M. Deville, A. Michler, Straight velocity boundaries in the lattice Boltzmann method, Physical Review E 77 (5) (2008) 056703] from straight to curved geometries for the lattice Boltzmann Bhatnager–Gross–Krook method (LBGK). The boundary condition is based on a reconstruction of the populations from the density, velocity and rate of strain. For curved boundaries, the reconstruction reduces the question of accuracy to a technical issue of interpolation. We present a method of interpolation allowing a very accurate representation of the curved boundary. The resulting boundary condition is verified for three different test cases: Taylor–Couette flow in-between rotating cylinders, laminar flow around a cylinder and flow past an impulsively started cylinder, demonstrating its second order accuracy and low error constant. The present boundary is stable for relaxation frequencies close to two.     

A fluctuating lattice Boltzmann scheme for the one-dimensional KPZ equation

Based on the well-known mapping between the Burgers equation with noise and the Kardar–Parisi–Zhang (KPZ) equation for fluctuating interfaces, we develop a fluctuating lattice Boltzmann (LB) scheme for growth phenomena, as described by the KPZ formalism. A very simple LB-KPZ scheme is demonstrated in 1+1 spacetime dimensions, and is shown to reproduce the scaling exponents characterizing the growth of one-dimensional fluctuating interfaces.     

A lattice Boltzmann equation for diffusion

The formulation of lattice gas automata (LGA) for given partial differential equations is not straightforward and still requires some sort of magic. Lattice Boltzmann equation (LBE) models are much more flexible than LGA because of the freedom in choosing equilibrium distributions with free parameters which can be set after a multiscale expansion according to certain requirements. Here a LBE is presented for diffusion in an arbitrary number of dimensions. The model is probably the simplest LBE which can be formulated. It is shown that the resulting algorithm with relaxation parameter =1 is identical to an explicit finite-difference (EFD) formulation at its stability limit. Underrelaxation (0<<1) allows stable integration beyond the stability limit of EFD. The time step of the explicit LBE integration is limited by accuracy and not by stability requirements.     

Generalized Boltzmann equation for lattice gas automata

In this paper a theory is formulated that predicts velocity and spatial correlations between occupation numbers that occur in lattice gas automata violating semi-detailed balance. Starting from a coupled BBGKY hierarchy for then-particle distribution functions, cluster expansion techniques are used to derive approximate kinetic equations. In zeroth approximation the standard nonlnear Boltzmann equation is obtained; the next approximation yields the ring kinetic equation, similar to that for hard-sphere systems, describing the time evolution of pair correlations. The ring equation is solved to determine the (nonvanishing) pair correlation functions in equilibrium for two models that violate semidetailed balance. One is a model of interacting random walkers on a line, the other one is a two-dimensional fluid-type model on a triangular lattice. The numerical predictions agree very well with computer simulations.     

Thermal equation of state for lattice Boltzmann gases

The Galilean invariance and the induced thermo-hydrodynamics of the lattice Boltzmann Bhatnagar--Gross--Krook model are proposed together with their rigorous theoretical background. From the viewpoint of group invariance, recovering the Galilean invariance for the isothermal lattice Boltzmann Bhatnagar--Gross--Krook equation (LBGKE) induces a new natural thermal-dynamical system, which is compatible with the elementary statistical thermodynamics.     

A consistent lattice Boltzmann equation with baroclinic coupling for mixtures

We propose a consistent lattice Boltzmann equation (LBE) with baroclinic coupling between species and mixture dynamics to model the active scalar dynamics in multi-species mixtures. The proposed LBE model is directly derived from the linearized Boltzmann equations for mixtures and it has the following two distinctive features. First, it uses the multiple-relaxation-time collision model so that it has the flexibility of independent Reynolds and Schmidt numbers, and better numerical stability. Second, it satisfies the indifferentiability principle therefore leads to a set of consistent hydrodynamic equations for barycentric velocity for mixtures. The proposed LBE model is validated through simulations of decaying homogeneous isotropic turbulence in three dimensions. We simulate both the active and passive scalar dynamics in decaying turbulence for mixtures. We also compute various statistical quantities and their decay exponents in decaying turbulence. Our results agree well with existing results for both scalar dynamics and decaying turbulence.     

Accuracy analysis of high-order lattice Boltzmann models for rarefied gas flows

In this work, we have theoretically analyzed and numerically evaluated the accuracy of high-order lattice Boltzmann (LB) models for capturing non-equilibrium effects in rarefied gas flows. In the incompressible limit, the LB equation is shown to be able to reduce to the linearized Bhatnagar–Gross–Krook (BGK) equation. Therefore, when the same Gauss–Hermite quadrature is used, LB method closely resembles the discrete velocity method (DVM). In addition, the order of Hermite expansion for the equilibrium distribution function is found not to be directly correlated with the approximation order in terms of the Knudsen number to the BGK equation for incompressible flows. Meanwhile, we have numerically evaluated the LB models for a standing-shear-wave problem, which is designed specifically for assessing model accuracy by excluding the influence of gas molecule/surface interactions at wall boundaries. The numerical simulation results confirm that the high-order terms in the discrete equilibrium distribution function play a negligible role in capturing non-equilibrium effect for low-speed flows. By contrast, appropriate Gauss–Hermite quadrature has the most significant effect on whether LB models can describe the essential flow physics of rarefied gas accurately. Our simulation results, where the effect of wall/gas interactions is excluded, can lead to conclusion on the LB modeling capability that the models with higher-order quadratures provide more accurate results. For the same order Gauss–Hermite quadrature, the exact abscissae will also modestly influence numerical accuracy. Using the same Gauss–Hermite quadrature, the numerical results of both LB and DVM methods are in excellent agreement for flows across a broad range of the Knudsen numbers, which confirms that the LB simulation is similar to the DVM process. Therefore, LB method can offer flexible models suitable for simulating continuum flows at the Navier–Stokes level and rarefied gas flows at the linearized Boltzmann model equation level.     

Advanced lattice Boltzmann scheme for high-Reynolds-number magneto-hydrodynamic flows

Is the lattice Boltzmann method suitable to investigate numerically high-Reynolds-number magneto-hydrodynamic (MHD) flows? It is shown that a standard approach based on the Bhatnagar–Gross–Krook (BGK) collision operator rapidly yields unstable simulations as the Reynolds number increases. In order to circumvent this limitation, it is here suggested to address the collision procedure in the space of central moments for the fluid dynamics. Therefore, an hybrid lattice Boltzmann scheme is introduced, which couples a central-moment scheme for the velocity with a BGK scheme for the space-and-time evolution of the magnetic field. This method outperforms the standard approach in terms of stability, allowing us to simulate high-Reynolds-number MHD flows with non-unitary Prandtl number while maintaining accuracy and physical consistency.     

Local second-order boundary methods for lattice Boltzmann models

A new way to implement solid obstacles in lattice Boltzmann models is presented. The unknown populations at the boundary nodes are derived from the locally known populations with the help of a second-order Chapman-Enskog expansion and Dirichlet boundary conditions with a given momentum. Steady flows near a flat wall, arbitrarily inclined with respect to the lattice links, are then obtained with a third-order error. In particular, Couette and Poiseuille flows are exactly recovered without the Knudsen layers produced for inclined walls by the bounce back condition.     

一种滑移区气体流动的格子Boltzmann曲边界处理新格式

针对滑移区复杂气-固边界存在速度滑移现象,提出了一种基于格子Boltzmann方法的非平衡态外推与有限差分相结合的曲边界处理新格式.该格式具有可考虑实际物理边界与网格线偏移量的优势,较传统half-way DBB(diffusive bounce-back)格式更能准确反映实际边界情况,同时还可获取壁面处气体宏观量及其法向梯度等信息.采用本文所提曲边界处理格式模拟分析了滑移区气体平直/倾斜微通道Poiseuille流、微圆柱绕流和同心微圆柱面旋转Couette流问题.研究结果表明,采用曲边界处理新格式所得结果与理论值以及文献结果符合良好,适用于滑移区气体流动的复杂边界处理,且比half-way DBB格式具有更高的精度,较修正DBB格式具有更好的适应性.     

Asymptotic analysis of the lattice Boltzmann equation

In this article we analyze the lattice Boltzmann equation (LBE) by using the asymptotic expansion technique. We first relate the LBE to the finite discrete-velocity model (FDVM) of the Boltzmann equation with the diffusive scaling. The analysis of this model directly leads to the incompressible Navier–Stokes equations, as opposed to the compressible Navier–Stokes equations obtained by the Chapman–Enskog analysis with convective scaling. We also apply the asymptotic analysis directly to the fully discrete LBE, as opposed to the usual practice of analyzing a continuous equation obtained through the Taylor-expansion of the LBE. This leads to a consistency analysis which provides order-by-order information about the numerical solution of the LBE. The asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities, such as vorticity. It also justifies the use of Richardson’s extrapolation method. As an example, a two-dimensional Taylor-vortex flow is used to validate our analysis. The numerical results agree very well with our analytic predictions.     

A lattice Boltzmann model for the generalized Burgers-Huxley equation

In this paper we develop a lattice Boltzmann model for the generalized Burgers-Huxley equation (GBHE). By choosing the proper time and space scales and applying the Chapman-Enskog expansion, the governing equation is recovered correctly from the lattice Boltzmann equation, and the local equilibrium distribution functions are obtained. Excellent agreement with the exact solution is observed, and better numerical accuracy is obtained than the available numerical result. The results indicate the present model is satisfactory and efficient. The method can also be applied to the generalized Burgers-Fisher equation and be extended to multidimensional cases.     

A mass-conserved multiphase lattice Boltzmann method based on high-order difference

The Z–S–C multiphase lattice Boltzmann model [Zheng, Shu, and Chew(ZSC), J. Comput. Phys. 218, 353(2006)]is favored due to its good stability, high efficiency, and large density ratio. However, in terms of mass conservation, this model is not satisfactory during the simulation computations. In this paper, a mass correction is introduced into the ZSC model to make up the mass leakage, while a high-order difference is used to calculate the gradient of the order parameter to improve the accuracy. To verify the improved model, several three-dimensional multiphase flow simulations are carried out,including a bubble in a stationary flow, the merging of two bubbles, and the bubble rising under buoyancy. The numerical simulations show that the results from the present model are in good agreement with those from previous experiments and simulations. The present model not only retains the good properties of the original ZSC model, but also achieves the mass conservation and higher accuracy.     

The lattice Boltzmann equation on irregular lattices

A general framework to extend the lattice Boltzmann equation to arbitrary lattice geometries is presented and numerically demonstrated for the case of a two-dimensional Poiseuille flow. The new scheme considerably extends the range of applicability of the Boltzmann method to problems requiring the use of nonuniform grids.     

A high-order Monte Carlo algorithm for the direct simulation of Boltzmann equation

A high-order algorithm of the direct simulation Monte Carlo (DSMC) method, H-DSMC, has been developed to simulate rarefied flow regimes. The mth order Taylor series expansion has been employed to obtain a more generalized form of the time discretization for the collision part of Boltzmann equation. In the purposed algorithm, the higher order collision terms are introduced as well as higher order terms in the time step of the probabilistic coefficients. These newly implemented higher order terms improve the accuracy and efficiency of the solution and enhance the convergence rate quite significantly. Comparison between results of the classic DSMC method and the H-DSMC method shows the promising performance of the introduced technique.     

A simple enthalpy-based lattice Boltzmann scheme for complicated thermal systems

To extend the lattice Boltzmann (LB) method to describe the applicable energy systems, the first key step is to build a suitable thermal LB model and corresponding boundary treatments. There are two main shortcomings in the existing related works: either some additional energy source terms are inconvenient to be naturally incorporated or the implementation of non-Dirichlet-type thermal boundary conditions is extremely difficult and sometimes impossible in them for complicated thermal systems, which restrict their applicability to only a few special classes of problems. In order to overcome these drawbacks by a simple way, in this paper a thermal LB model and corresponding boundary treatments are constructed based on the total enthalpy. The specific benefits due to the introduction of the total enthalpy are analyzed and it is found that the numerical results obtained by the present scheme agree well with the analytical solutions and/or the data reported in previous studies.     

Numerical study on the gas-kinetic high-order schemes for solving Boltzmann model equation

The high-order compact finite difference technique is introduced to solve the Boltzmann model equation, and the gas-kinetic high-order schemes are developed to simulate the different kinetic model equations such as the BGK model, the Shakhov model and the Ellipsoidal Statistical (ES) model in this paper. The methods are tested for the one-dimensional unsteady shock-tube problems with various Knudsen numbers, the inner flows of normal shock wave for different Mach numbers, and the two-dimensional flows past ...     

Derivation of lattice Boltzmann equation via analytical characteristic integral

A lattice Boltzmann(LB) theory, the analytical characteristic integral(ACI) LB theory, is proposed in this paper.ACI LB theory takes the Bhatnagar–Gross–Krook(BGK)-Boltzmann equation as the exact kinetic equation behind Navier–Stokes continuum and momentum equations and constructs an LB equation by rigorously integrating the BGK-Boltzmann equation along characteristics. It is a general theory, supporting most existing LB equations including the standard lattice BGK(LBGK) equation inherited from lattice-gas automata, whose theoretical foundation had been questioned. ACI LB theory also indicates that the characteristic parameter of an LB equation is collision number, depicting the particle-interaction intensity in the time span of the LB equation, instead of the traditionally assumed relaxation time, and the over-relaxation time problem is merely a manifestation of the temporal evolution of equilibrium distribution along characteristics under high collision number, irrelevant to particle kinetics. In ACI LB theory, the temporal evolution of equilibrium distribution along characteristics is the determinant of LB method accuracy and we numerically prove this.     

Global solutions of the Boltzmann equation on a lattice

The nonlinear Boltzmann equation with a discretized spatial variable is studied in a Banach space of absolutely integrable functions of the velocity variables. Conservation laws and positivity are utilized to extend weak local solutions to a global solution. This is shown to be a strong solution by analytic semigroup techniques.Supported by National Science Foundation Grant ENG-7515882.