Standard Lattice Boltzmann Method Research Articles

Improved axisymmetric lattice Boltzmann scheme

This paper proposes an improved lattice Boltzmann scheme for incompressible axisymmetric flows. The scheme has the following features. First, it is still within the framework of the standard lattice Boltzmann method using the single-particle density distribution function and consistent with the philosophy of the lattice Boltzmann method. Second, the source term of the scheme is simple and contains no velocity gradient terms. Owing to this feature, the scheme is easy to implement. In addition, the singularity problem at the axis can be appropriately handled without affecting an important advantage of the lattice Boltzmann method: the easy treatment of boundary conditions. The scheme is tested by simulating Hagen-Poiseuille flow, three-dimensional Womersley flow, Wheeler benchmark problem in crystal growth, and lid-driven rotational flow in cylindrical cavities. It is found that the numerical results agree well with the analytical solutions and/or the results reported in previous studies.

Rectangular lattice Boltzmann method

A set of rectangular lattice Boltzmann methods for fluid flows is developed. It is shown that reformulating local equilibrium distribution functions can result in the rectangular lattice Boltzmann models without the aid of an interpolation for shallow water equations, Navier-Stokes equations, and axisymmetric flow equations. In addition, schemes for correct incorporation of force terms into the models are proposed for simulations of flows involving forces in practice. The methods completely retain the innate kinetic features and the simple procedure of the standard lattice Boltzmann method with an additional benefit of being suitable for rectangular lattices at little extra computational cost. The methodology is illustrated and validated through its application to different flow problems, demonstrating the potential power of the models.

Lattice Boltzmann model for axisymmetric thermal flows

A thermal lattice Boltzmann (LB) model is presented for axisymmetric thermal flows in the incompressible limit. The model is based on the double-distribution-function LB method, which has attracted much attention since its emergence for its excellent numerical stability over the multispeed LB method. Compared with the existing axisymmetric thermal LB models, the present model is simpler and retains the inherent features of the standard LB method. Numerical simulations are carried out for the thermally developing laminar flows in circular ducts and the natural convection in an annulus between two coaxial vertical cylinders. The Nusselt number obtained from the simulations agrees well with the analytical solutions and/or the results reported in previous studies.

Entropic Lattice Boltzmann Method for high Reynolds number fluid flows

The Lattice Boltzmann Method (LBM) has emerged as a computationally efficient and increasingly popular numerical method for simulating complex fluid flow. However, the standard LBM has always suffered from severe numerical instability. In this paper, we mainly focus on Entropic Lattice Boltzmann Methods (ELBM) for high Reynolds number fluid flow. ELBM, which is derived from H-theorem, enhances the numerical stability of LBM. The compliance of ELBM with H-theorem makes ELBM much more stable than the standard LBM, but the computational cost increases because of solving a non-linear equation. We propose a new optimisation strategy to improve the efficiency of ELBM, and also implement some numerical tests to validate the computational stabilities and correctness.

Characteristic nonreflecting boundary conditions for open boundaries in lattice Boltzmann methods

A boundary condition for lattice Boltzmann methods, based on the movement of information through Euler characteristic directions, is developed. With respect to the similar conditions used in finite-difference or finite-volume implementations, some corrections are needed to compensate the isothermal compressible nature of standard lattice Boltzmann methods for fluid flow. The results show that the proposed method for inlets and outlets is highly nonreflecting, and mass conserving.

Steady state convergence acceleration of the generalized lattice Boltzmann equation with forcing term through preconditioning

Several applications exist in which lattice Boltzmann methods (LBM) are used to compute stationary states of fluid motions, particularly those driven or modulated by external forces. Standard LBM, being explicit time-marching in nature, requires a long time to attain steady state convergence, particularly at low Mach numbers due to the disparity in characteristic speeds of propagation of different quantities. In this paper, we present a preconditioned generalized lattice Boltzmann equation (GLBE) with forcing term to accelerate steady state convergence to flows driven by external forces. The use of multiple relaxation times in the GLBE allows enhancement of the numerical stability. Particular focus is given in preconditioning external forces, which can be spatially and temporally dependent. In particular, correct forms of moment projections of source/forcing terms are derived such that they recover preconditioned Navier–Stokes equations with non-uniform external forces. As an illustration, we solve an extended system with a preconditioned lattice kinetic equation for magnetic induction field at low magnetic Prandtl numbers, which imposes Lorentz forces on the flow of conducting fluids. Computational studies, particularly in three-dimensions, for canonical problems show that the number of time steps needed to reach steady state is reduced by orders of magnitude with preconditioning. In addition, the preconditioning approach resulted in significantly improved stability characteristics when compared with the corresponding single relaxation time formulation.

Analytic solution for a higher-order lattice Boltzmann method: Slip velocity and Knudsen layer

We present the analysis of a higher-order lattice Boltzmann (LB) method based on the fourth-order Gauss-Hermite quadrature, with emphasis on the slip velocity and the Knudsen layer. The exact solution of the slip velocity for the higher-order LB equation is obtained for Poiseuille flows with finite Knudsen numbers. Due to increased accuracy in velocity space discretization, the higher-order scheme gives much improved slip coefficients as compared with the standard LB method based on the third-order Gauss-Hermite quadrature. A multiple relaxation time model is investigated to show the effects of the relaxation times for higher-order moments on the slip phenomena.

Implicit–explicit finite‐difference lattice Boltzmann method with viscid compressible model for gas oscillating patterns in a resonator

AbstractDifficulties for the conventional computational fluid dynamics and the standard lattice Boltzmann method (LBM) to study the gas oscillating patterns in a resonator have been discussed. In light of the recent progresses in the LBM world, we are now able to deal with the compressibility and non‐linear shock wave effects in the resonator. A lattice Boltzmann model for viscid compressible flows is introduced firstly. Then, the Boltzmann equation with the Bhatnagar–Gross–Krook approximation is solved by the finite‐difference method with a third‐order implicit–explicit (IMEX) Runge–Kutta scheme for time discretization, and a fifth‐order weighted essentially non‐oscillatory (WENO) scheme for space discretization. Numerical results obtained in this study agree quantitatively with both experimental data available and those using conventional numerical methods. Moreover, with the IMEX finite‐difference LBM (FDLBM), the computational convergence rate can be significantly improved compared with the previous FDLBM and standard LBM. This study can also be applied for simulating some more complex phenomena in a thermoacoustics engine. Copyright © 2008 John Wiley & Sons, Ltd.

Accuracy of higher-order lattice Boltzmann methods for microscale flows with finite Knudsen numbers

Accuracy of the lattice Boltzmann (LB) method for microscale flows with finite Knudsen numbers is investigated. We employ up to the eleventh-order Gauss–Hermite quadrature for the lattice velocities and diffuse-scattering boundary condition for fluid–wall interactions. Detailed comparisons with the direct simulation Monte Carlo (DSMC) method and the linearized Boltzmann equation are made for planar Couette and Poiseuille flows. All higher-order LB methods considered here give improved results as compared with the standard LB method. The accuracy of the LB hierarchy, however, does not monotonically increase with the order of the Gauss–Hermite quadrature at moderate and large Knudsen numbers. The results also show the sensitivity to a quadrature chosen, even when the Gauss–Hermite quadratures have the same order of formal accuracy. Among the schemes investigated here, D2Q16 is the most efficient method and offers a quantitative prediction in the slip and transition regimes. The higher-order LB methods predict the Knudsen layer up to Kn = O ( 0.1 ) . The Knudsen layer, however, rapidly disappears when the Knudsen number approaches unity due to a finite number of the lattice velocities, while it is still present for Kn = O ( 1 ) in the Boltzmann equation. It is also found that the higher-order LB methods adopted here do not capture the asymptotic behavior of the Boltzmann equation at large Knudsen numbers.

Multispeed models in off-lattice Boltzmann simulations

The lattice Boltzmann method is a highly promising approach to the simulation of complex flows. Here, we realize recently proposed multispeed lattice Boltzmann models [S. Chikatamarla, Phys. Rev. Lett. 97 190601 (2006)] by exploiting the flexibility offered by off-lattice Boltzmann methods. The approach is based on the general characteristic-based algorithm for off-lattice Boltzmann simulations that preserves all appealing properties of the standard lattice Boltzmann method while extending the method to unstructured grids. We show that the use of multispeed models indeed gives rise to major improvements in accuracy. The suggested approach thus renders truly large-scale off-lattice Boltzmann computations practical.

Slip velocity and Knudsen layer in the lattice Boltzmann method for microscale flows

We present mesoscopic fluid-wall interaction models for lattice Boltzmann (LB) model simulations of microscale flows. The exact solution of the slip velocity for the LB equation with the Bhatnagar-Gross-Krook collision operator is obtained for Poiseuille flow at finite Knudsen numbers. With a consistent definition of the Knudsen number, the slip coefficients of the LB equation with the standard D2Q9 scheme are found to be slightly larger than those of the Boltzmann equation with the same boundary condition, which makes the standard LB method remain quantitatively accurate only for small Knudsen numbers. By modifying the nonequilibrium energy flux or introducing the effective relaxation time, the LB method is analytically shown to reproduce the slip phenomena up to second order in the Knudsen number. For the standard LB method, the Knudsen layer is captured only with modification of the relaxation dynamics such as in the effective relaxation time model.

Analysis of lattice Boltzmann nodes initialisation in moving boundary problems

The lattice Boltzmann method (LBM) is a numerical scheme based on a fixed grid. Dealing with moving boundary problems, proper routines are needed to initialise the variables at the new nodes, created by the variations of the computational fluid domain. We use asymptotic analysis to investigate the problem and possible solutions. A simple algorithm is proposed, able to achieve the same accuracy as the standard LBM. The theoretical predictions are tested on simple benchmarks.

IMPLICIT-EXPLICIT FINITE-DIFFERENCE LATTICE BOLTZMANN METHOD FOR COMPRESSIBLE FLOWS

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.

A simple lattice Boltzmann scheme for combustion simulation

A simple lattice Boltzmann model is developed for two-dimensional combustion simulations. In this model the time step and the fluid particle speed can be adjusted dynamically, depending on the “particle characteristic temperature”. The algorithm is still a simple process of hopping from one grid point to the next, the same as the standard lattice Boltzmann method. Therefore the outstanding advantages of the standard lattice Boltzmann method are retained in this model besides better numerical stability. Excellent agreement between the present results and other numerical or experimental data shows that this scheme is an efficient numerical method for practical combustion simulations.

A novel coupled lattice Boltzmann model for low Mach number combustion simulation

A novel coupled lattice Boltzmann model is developed for two- and three-dimensional low Mach number combustion simulations, in which the fluid density can bear sharp changes. Different to the hybrid lattice Boltzmann scheme [O. Filippova, D. Hanel, A novel lattice BGK approach for low Mach number combustion, J. Comput. Phys. 158 (2000) 139], this scheme is strictly pure lattice Boltzmann style (i.e., we solve the flow, temperature, and concentration fields using the lattice Boltzmann method only); different to the non-coupled lattice Boltzmann scheme [K. Yamamoto, X. He, G.D. Doolen, Simulation of combustion field with lattice Boltzmann method, J. Stat. Phys. 107 (2002) 367], the fluid density in this model is coupled directly with the temperature. In this model the time step and the fluid particle speed can be adjusted dynamically, depending on the “particle characteristic temperature”. And the algorithm is still a simple process of hopping from one grid point to the next, the same as the standard lattice Boltzmann method. Therefore the outstanding advantages of the standard lattice Boltzmann method are retained in this model besides better numerical stability. Excellent agreement between the present results and other numerical or experimental data shows that this scheme is an efficient numerical method for practical combustion simulations.

A NEW NUMERICAL APPROACH FOR FIRE SIMULATION

It is an urgent task to adopt/develop new numerical methods in fire research because of the intrinsic disadvantages of traditional numerical methods, such as in complicated geometries treatment, computational efficiency on parallel computers. The outstanding features of the lattice Boltzmann method (LBM) has emerged which is a very promising alternative in this field. Unfortunately, until recently the LBM could not be employed in fire research due to its inherent defects in combustion simulation. In order to extend the LBM into fire research, a novel LB model for fire simulation is designed in this study. Besides the intrinsic advantages of the standard LBM, this model shows improved numerical stability and can cover temperature ratio of more than one order of magnitude. The model is validated through a benchmark test "coflow methane-air diffusion flame". Furthermore, because little attention has been paid to the effect of the interaction between the inlet boundary and the interior of the flow field on stability of the computation and the quality of the solution, therefore in this study we also discuss this problem in detail to bridge this gap.

Extended volumetric scheme for lattice Boltzmann models

An extended volumetric scheme is proposed for lattice Boltzmann (LB) models. This scheme is particularly desirable for multispeed LB models due to its removal of nonlocal advection. It recovers the same macroscopic hydrodynamics as the standard lattice Boltzmann method without any further treatments. This scheme achieves an effectively reduced Courant-Friedrichs-Lewy number so that numerical stability is significantly enhanced for high Reynolds number simulations while maintaining the same order of numerical accuracy.

Optimization Strategies for the Entropic Lattice Boltzmann Method

The entropic formulation of the lattice Boltzmann method (LBM) features enhanced numerical stability due to its compliance with the Boltzmann H-theorem. This stability comes at the price of some computational overhead, associated with the need of adjusting the local relaxation time of the standard LBM in such a way as to secure compliance with the H-theorem. In this paper, we discuss a number of possible optimization strategies to reduce the computational overhead of entropic LBMs.

Two-dimensional interpolation-supplemented and Taylor-series expansion-based lattice Boltzmann method and its application

Two-dimensional interpolation-supplemented and Taylor-series expansion-based lattice Boltzmann method (ITLBM) is proposed to simulate flows in the non-uniform grids. The proposed method is based on the standard lattice Boltzmann method, interpolation-supplemented and Taylor-series expansion. The final formulation can be used on any mesh structure and lattice Boltzmann model. Numerical test of a two-dimensional channel flow around a square cylinder has been studied. The computational efficient and recirculation length at Re=1, 15 are obtained. Comparing the results from the ITLBM with those from the standard LBM, it has been concluded that the proposed method has good prospects in the hydrodynamic and aerodynamic engineering applications.

Application of Taylor series expansion and Least-squares-based lattice Boltzmann method to simulate turbulent flows

Lattice Boltzmann method (LBM) has become an alternative method of computing a variety of fluid flows, ranging from low Reynolds number laminar flows to highly turbulent flows. For turbulent flows, non-uniform grids are preferred. Taylor series expansion- and least-squares-based LBM (TLLBM) is an effective and convenient way to extend standard LBM to be used on arbitrary meshes. In order to show its ability to solve turbulent flows, we combine it with k–ω and S-A turbulence models. To validate these combinations, the benchmark problems of the turbulent channel flow and the turbulent flow over a backward facing step at Re = 44 000 are simulated. Our results compare well with the analytical solution and the experimental results of Kim et al. [22]. This shows that the combination of TLLBM with turbulence model can solve turbulent flows effectively.