格子Boltzmann守恒型非平衡态的外推边界

针对非平衡态外推边界处理方法在某些条件下发生质量泄漏的问题,结合质量守恒定律对其进行修正,建立一种新的质量守恒型的边界处理格式.其基本思想是在边界处定义一个虚拟密度来修正平衡态分布函数,使得系统质量守恒.以重力作用下的密度场及磁流体模拟问题为例,利用该格式进行分析,讨论质量泄露问题的本质及各种因素的影响.通过对三种典型的流动:二维Poiseuille流、Couette流、Womersley流的模拟,验证该边界格式的计算精度和稳定性.     

格子Boltzmann方法中三维运动边界的统一模型

格子Boltzmann方法(LBM)中边界条件的处理很复杂,在现有的边界条件处理方法中,动力学格式能够精确满足宏观边界条件,但由于要解一个不定方程,必须引入附加假设确保方程非奇异.作为动力学格式和反弹格式的一种扩展,提出一种处理三维任意速度运动边界的统一模型,其中人口速度和固体壁面速度是该模型的特殊情形.给出用于三维15速度的表达式.为了检验该模型,模拟对角顶盖驱动三维空腔流,并将结果与有限差分法计算的结果进行比较,说明所提出的统一模型是合理可行的.     

格子Boltzmann方法中的曲边界处理

研究了格子Boltzmann方法中实现曲边界条件的3种格式,对它们的精度和稳定性进行了分析和比较.通过二维Poiseuille流和等边三角域上空腔流的模拟,讨论了这3种格式的数值精度和稳定性.     

Natural Convection in a Concentric Annulus: A Lattice Boltzmann Method Study with Boundary Condition-Enforced Immersed Boundary Method

In this paper, a boundary condition-enforced IBM is introduced into the LBM in order to satisfy the non-slip and temperature boundary conditions, and natural convections in a concentric isothermal annulus between a square outer cylinder and a circular inner cylinder are simulated. The obtained results show that the boundary condition-enforced method gives a better solution for the flow field and the complicated physics of the natural convections in the selected case is correctly captured. The calculated average Nusselt numbers agree well with the previous studies.     

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.     

The effect of surface roughness on rarefied gas flows by lattice Boltzmann method

This paper studies the roughness effect combining with effects of rarefaction and compressibility by a lattice Boltzmann model for rarefied gas flows at high Knudsen numbers. By discussing the effect of the tangential momentum accommodation coefficient on the rough boundary condition, the lattice Boltzmann simulations of nitrogen and helium flows are performed in a two-dimensional microchannel with rough boundaries. The surface roughness effects in the microchannel on the velocity field, the mass flow rate and the friction coefficient are studied and analysed. Numerical results for the two gases in micro scale show different characteristics from macroscopic flows and demonstrate the feasibility of the lattice Boltzmann model in rarefied gas dynamics.     

Boundary conditions for lattice Boltzmann simulations

A heuristic interpretation of no-slip boundary conditions for lattice Boltzmann and lattice gas simulations is developed. An improvement is suggested which consists of including the wall nodes in the collision operation.     

Gas Flow in Microchannels – A Lattice Boltzmann Method Approach

Gas flow in microchannels can often encounter tangential slip motion at the solid surface even under creeping flow conditions. To simulate low speed gas flows with Knudsen numbers extending into the transition regime, alternative methods to both the Navier–Stokes and direct simulation Monte Carlo approaches are needed that balance computational efficiency and simulation accuracy. The lattice Boltzmann method offers an approach that is particularly suitable for mesoscopic simulation where details of the molecular motion are not required. In this paper, the lattice Boltzmann method has been applied to gas flows with finite Knudsen number and the tangential momentum accommodation coefficient has been implemented to describe the gas-surface interactions. For fully-developed channel flows, the results of the present method are in excellent agreement with the analytical slip-flow solution of the Navier–Stokes equations, which are valid for Knudsen numbers less than 0.1. The present paper demonstrates that the lattice Boltzmann approach is a promising alternative simulation tool for the design of microfluidic devices.     

Lattice Boltzmann Simulation of Convection in a Porous Medium with Temperature Jump and Velocity Slip Boundary Conditions

To investigate the convection in a porous medium, a horizontal quiescent layer of one fluid saturating a porous medium heated from bottom is numerically studied using single lattice-Boltzmann method (LBM) and the generalized Navier-Stokes equation proposed by Nithiarasu et al. [P. Nithiarasu, K.M. Seetharamu, and T Sundararajan, Int. J. Heat Mass Trans. 40 (1997) 3955]. Due to the rarefaction, the boundary conditions are considered as both temperature jump and velocity slip. The computational results are validated against the analytical results, and excellent agreement has been obtained. The results have shown that the Rayleigh number is increased with increasing temperature jump, the stabilization effect of temperature is much more significant than that of velocity slip, and the computation stability of present model is better than that of Darcy and Brinkman models.     

Derivation of Slip boundary conditions for the Navier-Stokes system from the Boltzmann equation

Rarefied gas flow behavior is usually described by the Boltzmann equation, the Navier-Stokes system being valid when the gas is less rarefied. Slip boundary conditions for the Navier-Stokes equations are derived in a rigorous and systematic way from the boundary condition at the kinetic level (Boltzmann equation). These slip conditions are explicitly written in terms of asymptotic behavior of some linear half-space problems. The validity of this analysis is established in the simple case of the Couette flow, for which it is proved that the right boundary conditions are obtained.     

Simulation of Cardiac Flow under the Septal Defect Based on Lattice Boltzmann Method

In this paper, the lattice Boltzmann method was used to simulate the cardiac flow in children with aseptal defect. The inner wall model of the heart was reconstructed from 210 computed tomography scans. By simulating and comparing the cardiac flow field, the pressure field, the blood oxygen content, and the distribution of entropy generation before and after an operation, the effects of septal defect on pulmonary hypertension(PH), cyanosis, and heart load were analyzed in detail. It is found that the atrial septal defect(ASD) of the child we analyzed had a great influence on the blood oxygen content in the pulmonary artery, which leads to lower efficiency of oxygen binding in the lungs and increases the burden on the heart. At the same time, it also significantly enhanced the entropy generation rate of the cardiac flow, which also leads to a higher heart load. However, the main cause of PH is not ASD, but ventricular septal defect (VSD). Meanwhile, it significantly reduced the blood oxygen content in the brachiocephalic trunk, but rarely affects the blood oxygen contents in the downstream left common carotid artery, left subclavian artery, and descending aorta are not significantly affected by VSD. It causes severe cyanosis on the face and lips.     

用Lattice Boltzmann方法模拟1/4圆腔内的定常层流运动

分析格子Boltzmann方法中二阶精度的曲线边界处理方法.应用格子Boltzmann方法及其边界处理方法模拟1/4圆腔内的定常层流运动,引入流线图和等涡线图分析流场随Re数的变化.并且发现当Re数在10~100区间内变化时,随着Re数的增大,顺时针旋转流场的涡心位置偏离x轴的角度逐渐减小,而逆时针旋转流场的涡心位置偏离x轴的角度却越来越大.     

Lattice Boltzmann Model for Free Surface Flow for Modeling Foaming

We present a 2D- and 3D-lattice Boltzmann model for the treatment of free surface flows including gas diffusion. Interface advection and related boundary conditions are based on the idea of the lattice Boltzmann equation. The fluid dynamic boundary conditions are approximated by using the mass and momentum fluxes across the interface, which do not require explicit calculation of gradients. A similar procedure is applied to fulfill the diffusion boundary condition. Simple verification tests demonstrate the correctness of the algorithms. 2D- and 3D-foam evolution examples demonstrate the potential of the method.     

A Discrete Effect of the Thermal Lattice BGK Model

In this article, a discrete effect in the thermal Lattice BGK two-speed model is studied. These effects are due to the non-equilibrium state in the particle distribution function, and the non-equilibrium occurs near walls. The mechanism of the LBM counterpart of the thermal creep flow, which appears due to the temperature gradient of the boundary in rarefied gases, is clarified analytically and numerical calculations are performed for some cases. A technique for eliminating this effect is also shown.     

Convergence of a Distributional Monte Carlo Method for the Boltzmann Equation

Direct Simulation Monte Carlo (DSMC) methods for the Boltzmann equation employ a point measure approximation to the distribution function, as simulated particles may possess only a single velocity. This representation limits the method to converge only weakly to the solution of the Boltzmann equation. Utilizing kernel density estimation we have developed a stochastic Boltzmann solver which possesses strong convergence for bounded and $L^\infty$ solutions of the Boltzmann equation. This is facilitated by distributing the velocity of each simulated particle instead of using the point measure approximation inherent to DSMC. We propose that the development of a distributional method which incorporates distributed velocities in collision selection and modeling should improve convergence and potentially result in a substantial reduction of the variance in comparison to DSMC methods. Toward this end, we also report initial findings of modeling collisions distributionally using the Bhatnagar-Gross-Krook collision operator.     

A Multigrid-Solver for the Discrete Boltzmann Equation

This paper introduces a nonlinear multigrid solution approach for the discrete Boltzmann equation discretized by an implicit second-order Finite Difference scheme. For simplicity we restrict the discussion to the stationary case. A numerical example shows the drastically improved efficiency in comparison to the widely used Lattice–Bathnagar–Gross–Krook (LBGK) approach.     

A Boundary Condition-Implemented Immersed Boundary-Lattice Boltzmann Method and Its Application for Simulation of Flows Around a Circular Cylinder

A boundary condition-implemented immersed boundary-lattice Boltzmann method (IB-LBM) is presented in this work. The present approach is an improvement to the conventional IB-LBM. In the conventional IB-LBM, the no-slip boundary condition is only approximately satisfied. As a result, there is flow penetration to the solid boundary. Another drawback of conventional IB-LBM is the use of Dirac delta function interpolation, which only has the first order of accuracy. In this work, the no-slip boundary condition is directly implemented, and used to correct the velocity at two adjacent mesh points from both sides of the boundary point. The velocity correction is made through the second-order polynomial interpolation rather than the first-order delta function interpolation. Obviously, the two drawbacks of conventional IB-LBM are removed in the present study. Another important contribution of this paper is to present a simple way to compute the hydrodynamic forces on the boundary from Newton's second law. To validate the proposed method, the two-dimensional vortex decaying problem and incompressible flow over a circular cylinder are simulated. As shown in the present results, the flow penetration problem is eliminated, and the obtained results compare very well with available data in the literature.     

可压流体Rayleigh-Taylor不稳定性的离散Boltzmann模拟

使用离散Boltzmann模型模拟了可压流体系统中多模初始情况下的Rayleigh-Taylor不稳定性.该离散Boltzmann模型等效于一个Navier-Stokes模型外加一个关于热动非平衡行为的粗粒化模型.通过模拟Riemann问题:Sod激波管、冲击波碰撞和热Couette流问题验证模型的有效性,所得数值结果与解析解一致.利用该模型对界面间断随机多模初始扰动的可压Rayleigh-Taylor不稳定性进行数值模拟研究,得到不稳定性界面演化过程的基本图像.由于黏性和热传导共同作用,一开始扰动界面被"抹平",演化较慢;随着模式互相耦合而减少,演化开始加速,并经历非线性小扰动阶段和不规则非线性阶段,而后发展成典型的"蘑菇状",后期进入湍流混合阶段.由于扰动模式的耦合与发展,轻重流体的重力势能、压缩能与动能相互转化,系统先是趋于热动平衡态,而后偏离热动平衡态以线性形式增长,接着再次趋于热动平衡态,最后慢慢远离热动平衡态.     

Hydrodynamic and Thermodynamic Nonequilibrium Effects around Shock Waves: Based on a Discrete Boltzmann Method

A shock wave that is characterized by sharp physical gradients always draws the medium out of equilibrium. In this work, both hydrodynamic and thermodynamic nonequilibrium effects around the shock wave are investigated using a discrete Boltzmann model. Via Chapman–Enskog analysis, the local equilibrium and nonequilibrium velocity distribution functions in one-, two-, and three-dimensional velocity space are recovered across the shock wave. Besides, the absolute and relative deviation degrees are defined in order to describe the departure of the fluid system from the equilibrium state. The local and global nonequilibrium effects, nonorganized energy, and nonorganized energy flux are also investigated. Moreover, the impacts of the relaxation frequency, Mach number, thermal conductivity, viscosity, and the specific heat ratio on the nonequilibrium behaviours around shock waves are studied. This work is helpful for a deeper understanding of the fine structures of shock wave and nonequilibrium statistical mechanics.     

Continuous and Discrete Adjoint Approach Based on Lattice Boltzmann Method in Aerodynamic Optimization Part I: Mathematical Derivation of Adjoint Lattice Boltzmann Equations

The significance of flow optimization utilizing the lattice Boltzmann (LB) method becomes obvious regarding its advantages as a novel flow field solution method compared to the other conventional computational fluid dynamics techniques. These unique characteristics of the LB method form the main idea of its application to optimization problems. In this research, for the first time, both continuous and discrete adjoint equations were extracted based on the LB method using a general procedure with low implementation cost. The proposed approach could be performed similarly for any optimization problem with the corresponding cost function and design variables vector. Moreover, this approach was not limited to flow fields and could be employed for steady as well as unsteady flows. Initially, the continuous and discrete adjoint LB equations and the cost function gradient vector were derived mathematically in detail using the continuous and discrete LB equations in space and time, respectively. Meanwhile, new adjoint concepts in lattice space were introduced. Finally, the analytical evaluation of the adjoint distribution functions and the cost function gradients was carried out.