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.     

Kinetic theory based lattice Boltzmann equation with viscous dissipation and pressure work for axisymmetric thermal flows

A lattice Boltzmann equation (LBE) for axisymmetric thermal flows is proposed. The model is derived from the kinetic theory which exhibits several features that distinguish it from other previous LBE models. First, the present thermal LBE model is derived from the continuous Boltzmann equation, which has a solid foundation and clear physical significance; Second, the model can recover the energy equation with the viscous dissipation term and work of pressure which are usually ignored by traditional methods and the existing thermal LBE models; Finally, unlike the existing thermal LBE models, no velocity and temperature gradients appear in the force terms which are easy to realize in the present model. The model is validated by thermal flow in a pipe, thermal buoyancy-driven flow, and swirling flow in vertical cylinder by rotating the top and bottom walls. It is found that the numerical results agreed excellently with analytical solution or other numerical results.     

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.     

Study of the 1D lattice Boltzmann shallow water equation and its coupling to build a canal network

The D1Q3 lattice Boltzmann (LB) shallow water equation is analyzed in detail and compared with other numerical schemes. Analytical results are derived and used to discuss the accuracy and stability of the model. We show how such D1Q3 LB models for canal reaches may be easily coupled with various hydraulic interconnection structures to build models of complex irrigation networks.     

关于分离变量法的一个注记

从B-cklund变换出发，利用Cole-Hopf变换，一些有重要物理意义的(2+1)维非线性演化方程(组)被简化为具有两个分别关于(x,t),(y,t)的任意函数的单个线性偏微分方程.通过对该方程解的研究，获得了原方程一些新的精确解.其中，一些近年来被广泛研究的由分离变量法所获得的解也被重新得到. 关键词： B-cklund变换 Cole-Hopf变换 分离变量法 精确解     

A study on the inclusion of body forces in the lattice Boltzmann BGK equation to recover steady-state hydrodynamics

When the lattice Boltzmann (LB) method is used to solve hydrodynamic problems containing a body force term varying in space and/or time, its modelling at the mesoscopic scale must be verified in terms of consistency in order to avoid the appearance of non-hydrodynamic error terms at the macroscopic scale. In the present work it is shown that the modelling of spatially varying steady body force terms in the LB equation must be different from the time-dependent case, when a steady-state flow solution is sought. For that, the Chapman-Enskog analysis is used to derive the LB body force model for the LB BGK equations in a steady-state flow problem. The theoretical findings are supported by numerical tests performed on two different 2D steady-state laminar flows driven by spatially varying body forces with known analytical solutions.     

Solving Integrable Broer-Kaup Equations in （2＋1）-Dimensional Spaces via an Improved Variable Separation Approach

Starting from Baecklund transformation and using Cole-Hopf transformation, we reduce the integrable Broer-Kaup equations in (2 1)-dimensional spaces to a simple linear evolution equation with two arbitrary functions of {x, t} and {y, t} in this paper. And we can obtain some new solutions of the original equations by investigating the simple nonlinear evolution equation, which include the solutions obtained by the variable separation approach.     

Solving Integrable Broer-Kaup Equations in (2+1)-Dimensional Spaces via an Improved Variable Separation Approach

Starting from Backlund transformation and using Cole-Hopf transformation, we reduce the integrable Broer-Kaup equations in (2 1)-dimensional spaces to a simple linear evolution equation with two arbitrary functions of {x,t} and {y,t} in this paper. And we can obtain some new solutions of the original equations by investigating the simple nonlinear evolution equation, which include the solutions obtained by the variable separation approach.     

Stochastic Burgers and KPZ Equations from Particle Systems

We consider two strictly related models: a solid on solid interface growth model and the weakly asymmetric exclusion process, both on the one dimensional lattice. It has been proven that, in the diffusive scaling limit, the density field of the weakly asymmetric exclusion process evolves according to the Burgers equation and the fluctuation field converges to a generalized Ornstein-Uhlenbeck process. We analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the (non linear) Burgers equation with a random noise on the density current. For the solid on solid model, we prove that the fluctuation field of the interface profile, if suitably rescaled, converges to the Kardar–Parisi–Zhang equation. This provides a microscopic justification of the so called kinetic roughening, i.e. the non Gaussian fluctuations in some non-equilibrium processes. Our main tool is the Cole-Hopf transformation and its microscopic version. We also develop a mathematical theory for the macroscopic equations. Received: 24 October 1995/Accepted: 9 July 1996     

Erratum and Addendum to “A note on Burgers' equation with time delay: Instability via finite-time blow-up” [Phys. Lett. A 372 (2008) 6363]

I point out and examine an error in my derivation of the instability result presented in [P.M. Jordan, Phys. Lett. A 372 (2008) 6363]. Based on this analysis, I then show that the Cole-Hopf transformation can, in fact, be used to obtain an exact, time-harmonic plane wave-type solution to a form of Burgers' equation with time delay, provided the frequency assumes a particular value.     

Kaup—Kupershmidt方程的对称代数

利用Ｍｉｕｒａ变换和Ｃｏｌｅ－Ｈｏｐｆ变换，证明了Ｋａｕｐ－Ｋｕｐｅｒｓｈｍｉｄｔ方程的１０族时间无关对称构成一个非Ａｂｅｌ李代数，与１０族对称相应的１０族方程具有共同的强对称。１０族可积模型的时间无关对称及其构成的李代数亦被给定。     

A Consistent Grid Coupling Method for Lattice-Boltzmann Schemes

A method of coupling grids of different mesh size is developed for classical Lattice-Boltzmann (LB) algorithms on uniform grids. The approach is based on an asymptotic analysis revealing suitable quantities equalized along the grid interfaces for exchanging information between the subgrids. In contrast to other couplings the method works without overlap zones. Moreover the grid velocity (Mach number) is not kept constant, as the time step depends not linearly but quadratically on the grid spacing. To illustrate the basic idea we use a simple LB algorithm solving the advection-diffusion equation. The proposed grid coupling is validated by numerical convergence studies indicating, that the coupling does not affect the second-order convergence behavior of the LB algorithm which is observed on uniform grids. In order to demonstrate its principal applicability to other LB models, the coupling is generalized to the standard D2P9 model for (Navier-)Stokes flow and tested numerically. As we use analytic tools different from the Chapman-Enskog expansion, the theoretical background material is given in two appendices. In particular, the results of numerical experiments are confirmed with a consistency analysis.     

INFINITELY MANY SYMMETRIES OF THE BILINEAR KADOMTSEV-PETVIASHVILI EQUATION

A simple formula for symmetries of the bilinear Kadomtsev- Petviashvili equation (BKPE) is given by using a direct method presented by Lou. The symmetry algebra of the BKPE is not isomorphic to those of the usual KP equation though these equations are related by the well known Cole-Hopf transformation.     

Finite-Difference Lattice Boltzmann Scheme for High-Speed Compressible Flow: Two-Dimensional Case

Lattice Boltzmann (LB) modeling of high-speed compressible flows has long been attempted by various authors. One common weakness of most of previous models is the instability problem when the Mach number of the flow is large. In this paper we present a finite-difference LB model, which works for flows with flexible ratios of specific heats and a wide range of Mach number, from $0$ to 30 or higher. Besides the discrete-velocity-model by Watari [Physica A 382 (2007) 502], a modified Lax--Wendroff finite difference scheme and an artificial viscosity are introduced. The combination of the finite-difference scheme and the adding of artificial viscosity must find a balance of numerical stability versus accuracy. The proposed model is validated by recovering results of some well-known benchmark tests: shock tubes and shock reflections. The new model may be used to track shock waves and/or to study the non-equilibrium procedure in the transition between the regular and Mach reflections of shock waves, etc.     

A note on Burgers' equation with time delay: Instability via finite-time blow-up

Burgers' equation with time delay is considered. Using the Cole-Hopf transformation, the exact solution of this nonlinear partial differential equation (PDE) is determined in the context of a (seemingly) well-posed initial-boundary value problem (IBVP) involving homogeneous Dirichlet data. The solution obtained, however, is shown to exhibit a delay-induced instability, suffering blow-up in finite-time.     

Kaup-Kupershmidt方程的对称代数

利用Ｍｉｕｒａ变换和Ｃｏｌｅ－Ｈｏｐｆ变换，证明了Ｋａｕｐ－Ｋｕｐｅｒｓｈｍｉｄｔ方程的１０族时间无关对称构成一个非Ａｂｅｌ李代数。与１０族对称相应的１０族方程具有共同的强对称，１０族可积模型的时间无关对称及其构成的李代数亦被给定。     

The stochastic Burgers Equation

We study Burgers Equation perturbed by a white noise in space and time. We prove the existence of solutions by showing that the Cole-Hopf transformation is meaningful also in the stochastic case. The problem is thus reduced to the anaylsis of a linear equation with multiplicativehalf white noise. An explicit solution of the latter is constructed through a generalized Feynman-Kac formula. Typical properties of the trajectories are then discussed. A technical result, concerning the regularizing effect of the convolution with the heat kernel, is proved for stochastic integrals.     

An enhanced beam model for constrained layer damping and a parameter study of damping contribution

An enhanced analytical model is presented based on an extension of previous models for constrained layer damping (CLD) in beam-like structures. Most existing CLD models are based on the assumption that shear deformation in the core layer is the only source of damping in the structure. However, previous research has shown that other types of deformation in the core layer, such as deformations from longitudinal extension and transverse compression, can also be important. In the enhanced analytical model developed here, shear, extension, and compression deformations are all included. This model can be used to predict the natural frequencies and modal loss factors. The numerical study shows that compared to other models, this enhanced model is accurate in predicting the dynamic characteristics. As a result, the model can be accepted as a general computation model. With all three types of damping included and the formulation used here, it is possible to study the impact of the structure's geometry and boundary conditions on the relative contribution of each type of damping. To that end, the relative contributions in the frequency domain for a few sample cases are presented.     

基于Lattice Boltzmann方法的圆柱绕流大涡模拟

Lattice Boltzmann(LB)方法是近年来出现的一种流体计算的新方法,在流体力学及相关领域取得了很大的成功。但LB方法在模拟湍流流动时常常引起计算的不稳定。本文基于一种结合大涡模型的LB方法对圆柱绕流问题进行的模拟,并与其他文献的实验结果和计算结果进行对比。计算表明:这种混合LB方法能获得比较满意的结果。     

等离子体模型碰撞项的比较

本文在两种碰撞模型(BGK模型和LB模型)下,计算了碰撞对等离子体色散方程的影响。碰撞的主要作用是使色散函数Z(ξ)变为广义色散函数W(ξ,ν)。分析比较了两种模型下广义色散函数的性质。讨论了碰撞对低频漂移波(局域模)和离声波稳定性的影响。分析表明,LB模型不仅在物理上更为合理,而且在计算及表达形式上也比BGK模型更为简明。 关键词：