Use of the two-moment boundary condition in the problem of rarefied gas slip over a solid cylindrical surface

The slip velocity of a rarefied gas nonuniform in temperature and mass velocity is calculated for gas slip over the surface of a right circular cylinder. The calculation uses the two-moment boundary condition in an approximation linear in Knudsen number. Corrections to the slip velocity that are due to the interface curvature, volume temperature stresses, and nonuniform temperature distribution in the Knudsen layer are studied as func-tions of the accommodation coefficients in the first two moments of the distribution function. The Bhatnagar-Gross-Krook model of the Boltzmann kinetic equation is employed as the basic equation for the gas state.     

Lattice-Boltzmann Simulations of Fluid Flows in MEMS

The lattice Boltzmann model is a simplified kinetic method based on the particle distribution function. We use this method to simulate problems in MEMS, in which the velocity slip near the wall plays an important role. It is demonstrated that the lattice Boltzmann method can capture the fundamental behaviors in micro-channel flow, including velocity slip, nonlinear pressure drop along the channel and mass flow rate variation with Knudsen number. The Knudsen number dependence of the position of the vortex center and the pressure contour in micro-cavity flows is also demonstrated.     

A Lattice Boltzmann Kinetic Model for Microflow and Heat Transfer

In this paper, we propose a lattice Boltzmann BGK model for simulation of micro flows with heat transfer based on kinetic theory and the thermal lattice Boltzmann method (He et al., J. Comp. Phys. 146:282, 1998). The relaxation times are redefined in terms of the Knudsen number and a diffuse scattering boundary condition (DSBC) is adopted to consider the velocity slip and temperature jump at wall boundaries. To check validity and potential of the present model in modelling the micro flows, two two-dimensional micro flows including thermal Couette flow and thermal developing channel flow are simulated and numerical results obtained compare well with previous studies of the direct simulation Monte Carlo (DSMC), molecular dynamics (MD) approaches and the Maxwell theoretical analysis     

过渡区微尺度流动的有效黏性多松弛系数格子Boltzmann模拟

为提高采用二维九速离散速度模型的格子Boltzmann方法 (LBM)模拟微尺度流动中非线性现象的精度和效率,引入Dongari等提出的有效平均分子自由程对黏性进行修正(Dongari N,Zhang Y H,Reese J M2011 J.Fluids Eng.133 071101);并针对以往研究微尺度流动时采用边界处理格式含有离散误差的问题,采用多松弛系数格子Boltzmann方法结合二阶滑移边界条件,对微尺度Couette流动和周期性Poiseuille流动进行模拟,并将速度分布以及质量流量等模拟结果与直接模拟蒙特卡罗方法模拟数据、线性Boltzmann方程的数值解以及现有的LBM模型模拟结果进行对比.结果表明,相对于现有的LBM模型,引入新的修正函数所建立的有效黏性多松弛系数LBM模型有效提高了LBM模拟过渡区的微尺度流动中的非线性现象的能力.     

微尺度振荡Couette流的格子Boltzmann模拟

采用有效多松弛时间-格子Boltzmann方法(Effective MRT-LBM)数值模拟了微尺度条件下的振荡Couette和Poiseuille流动. 在微流动LBM中引入Knudsen边界层模型,对松弛时间进行修正. 模拟时平板或外力以正弦周期振动,Couette流中考虑了单平板振动、上下板同相振动这两类情况. 研究结果表明,修正后的MRT-LBM模型能有效用于这类非平衡的微尺度流动模拟;对于Couette流,随着Kn数的增大,壁面滑移效应变得越明显. St越大,板间速度剖面的非线性特性越剧烈;两板同相振荡时,若Kn,St均较小,板间流体受到平板拖动剪切的影响很小,板间速度几乎重叠在一起;在振荡Poiseuille流动中,St数增大到一定值时,相位滞后现象减弱;相对于Kn数,St数对振荡Couette 和Poiseuille流中不同位置处速度相位差的产生有较大影响. 关键词： 格子Boltzmann方法 有效MRT模型 Knudsen层 振荡流     

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.     

Development of a new correlation to calculate permeability for flows with high Knudsen number

Flows with high Knudsen number play a prominent role in many engineering applications. The present study is an effort toward the simulation of flow with high Knudsen number using modified lattice Boltzmann method(LBM) through a porous medium in a channel. The effect of collision between molecules and solid walls, which is required to accurately simulate transition flow regime, is taken into account using a modified relaxation time. Slip velocity on the wall, which is another significant difficulty in simulating transition flow regime, is captured using the slip reflection boundary condition(SRBC). The geometry of porous medium is considered as in-line and staggered. The results are in good agreement with previous works. A new correlation is obtained between permeability, Knudsen number and porosity for flows in transition flow regimes.     

Analytical solution of axi-symmetrical lattice Boltzmann model for cylindrical Couette flows

Analytical solution for the axi-symmetrical lattice Boltzmann model is obtained for the low-Mach number cylindrical Couette flows. In the hydrodynamic limit, the present solution is in excellent agreement with the result of the Navier–Stokes equation. Since the kinetic boundary condition is used, the present analytical solution using nine discrete velocities can describe flows with the Knudsen number up to 0.1. Meanwhile, the comparison with the simulation data obtained by the direct simulation Monte Carlo method shows that higher-order lattice Boltzmann models with more discrete velocities are needed for highly rarefied flows.     

Variable slip coefficient in binary lattice Boltzmann models

We present a new method in order to obtain variable slip coefficient in binary lattice Boltzmann models to simulate gaseous flows. We present the Boundary layer theory. We study both the single-and multi-fluid BGK-type models as well. The boundary slip and the Knudsen layer are analyzed in detail. Benchmark simulations are carried out in order to compare the analytical derivation with the numerical results. Excellent agreement is found between the two analytical formalism and the numerical simulations.      

Rarefied molecular gas flow near the rotating cylindrical surface

An exact analytic solution of the problem of the right circular cylinder in a rarefied molecular gas is constructed in the isothermal approximation. An expression for the velocity of a rarefied molecular gas entrained by the cylinder rotated therein is obtained in the regime of a flow with slip accounting for the second-order correction in terms of the Knudsen number. A generalization of the BGK model of the Boltzmann kinetic equation accounting for the rotational degrees of freedom of gas molecules is used as the governing equation, and the diffuse reflection model is used as a microscopic boundary condition on the cylinder surface. The given approach is shown to enable the consideration of the gas flow dependence on the Prandtl number and the gas temperature.     

微圆管进口区气体流动与换热特性研究

对微圆管进口区运用一阶速度滑移和温度跳跃边界,考察了Kn、动量调和及热调和系数对流动与换热特性的影响机理和规律.模拟结果表明:流动进口段长度随Kn增加而增加,但随动量调和系数减小而减小;热进口段长度随Kn增加而增加,但随动量调和系数及热调和系数减小而减小;Nu数随Kn增加及热调和系数减小而减小,但随动量调和系数减小而增加.     

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.     

Discrete velocity modelling of gaseous mixture flows in MEMS

The need of developing advanced micro-electro-mechanical systems (MEMS) has motivated the study of fluid-thermal flows in devices with micro-scale geometries. In many MEMS applications the Knudsen number varies in the range from 10⁻² to 10². This flow regime can be treated neither as a continuum nor as a free molecular flow. In order to describe these flows it is necessary to implement the Boltzmann equation (BE) or simplified kinetic model equations.The aim of the present work is to propose an efficient methodology for solving internal flows of binary gaseous mixtures in rectangular channels due to small pressure gradients over the whole range of the Knudsen number. The complicated collision integral term of the BE is substituted by the kinetic model proposed by McCormack for gaseous mixtures. The discrete velocity method is implemented to solve in an iterative manner the system of the kinetic equations. Even more the required computational effort is significantly reduced, by accelerating the convergence rate of the iteration scheme. This is achieved by formulating a set of moment equations, which are solved jointly with the transport equations.The velocity profiles and the flow rates of three different binary mixtures (He–Ar, Ne–Ar and He–Xe) in 2D micro-channels of various height to width ratios are calculated. The whole formulation becomes very efficient and can be implemented as an alternative methodology to the classical method of solving the Navier–Stokes equations with slip boundary conditions, which in any case is restricted by the hydrodynamic regime.     

Theoretical and numerical studies of non-equilibrium slip effects on a catalytic surface

A new concentration slip model to describe the rarefied gas effect on the species transport in microscale chemical reactors was derived from the approximate solution of the Boltzmann equation. The present model is more general and recovers the existing models in the limiting cases. The analytical results showed that the concentration slip is dominated by two different mechanisms, the reaction induced concentration slip (RIC) and the temperature slip induced concentration slip (TIC). The magnitude of RIC slip is proportional to the product of the Damköhler number and Knudsen number. The impact of the velocity, concentration and temperature slips on the coupling between the surface catalytic reactions and the homogeneous gas phase reactions was examined using the detailed chemistry of hydrogen and methane within a wide range of accommodation coefficients in a two-dimensional microscale chemical reactor. The results showed that the impact of reaction induced concentration slip (RIC) effects on catalytic reactions strongly depends on the Damköhler number, the Knudsen number and the surface accommodation coefficient. It was found that the TIC slip had a strong effect on the fuel oxidation rates and the RIC slip dramatically changed the mass fraction distributions of radicals, especially when the mass accommodation coefficients were far less than unity.     

Simulation of microchannel flow using the lattice Boltzmann method

For microchannel flow simulation, the slip boundary model is very important to guarantee the accuracy of the solution. In this paper, a new slip model, the Langmuir slip model, instead of the popularly used Maxwell slip model, is incorporated into the lattice Boltzmann (LB) method through the non-equilibrium extrapolation scheme to simulate the rarefied gas flow. Its feasibility and accuracy are examined by simulations of microchannel flow. Although, for simplicity, in this paper our recently developed LB model is used to solve the flow field, this does not prevent the present boundary scheme from easily incorporating other LB models, for example the more advanced collision model with multiple relaxation times. In addition, the existing non-equilibrium extrapolation LB boundary scheme for macroscopic flows can be recovered naturally from the present scheme when the Knudsen number .     

Construction of the mass velocity profile for a rarefied molecular gas entrained by a rotating sphere taking into account thermophysical parameters of the gas

The problem of a sphere rotating in a molecular gas is solved in the isothermal approximation. The expression for the velocity of a rarefied molecular gas entrained by a sphere rotating in it is derived for sliding flow conditions taking into account the second-order correction in the Knudsen number. A generalization of the Boltzmann kinetic equation in the BGK model to the case of rotational degrees of freedom of gas molecules is used as the basic equation. The diffusive reflection model is employed as the microscopic boundary condition on the surface of the sphere. It is shown that this approach makes it possible to take into account the dependence of the gas velocity on the Prandtl number and gas temperature.     

微通道内气体流动的格子Boltzmann方法研究

采用格子Boltzmann方法模拟了微通道在滑移区内不同Knudsen数下的微气体Poiseuille流,分析了微气体流动的速度分布以及流量与压降的关系,并给出了相对滑移长度和Poiseuille数随Knudsen数的变化特性。研究结果表明,微气体Poiseuille流的速度轮廓呈抛物线分布,但是边界速度大于0,出现速...     

Slip Boundary Conditions for the Compressible Navier–Stokes Equations

The slip boundary conditions for the compressible Navier–Stokes equations are derived systematically from the Boltzmann equation on the basis of the Chapman–Enskog solution of the Boltzmann equation and the analysis of the Knudsen layer adjacent to the boundary. The resulting formulas of the slip boundary conditions are summarized with explicit values of the slip coefficients for hard-sphere molecules as well as the Bhatnagar–Gross–Krook model. These formulas, which can be applied to specific problems immediately, help to prevent the use of often used slip boundary conditions that are either incorrect or without theoretical basis.     

Simulation of micro- and nano-scale flows via the lattice Boltzmann method

We use the lattice Boltzmann method (LBM) for analysis of high and moderate Knudsen number phenomena. Simulation results are presented for microscale Couette and Poiseuille flows. The slip velocity, nonlinear pressure drop, and mass flow rate are compared with previous numerical results and/or experimental data. The Knudsen minimum is successfully predicted for the first time within the LBM framework. These results validate the usage of the LBM based commercial, arbitrary geometry code PowerFLOW for simulating nanoscale problems.     

Gas kinetic algorithm for flows in Poiseuille-like microchannels using Boltzmann model equation

~~Gas kinetic algorithm for flows in Poiseuille-like microchannels using Boltzmann model equation1. Feynman, R., There's plenty of room at the bottom, Journal of Microelectromechanical Systems, 1992, 1: 60 -66. 2. Piekos, E. S., Breuer, K. S., Numerical modeling of micromechanical devices using the direct simulation Monte Carlo method, Transactions of the ASME, Journal of Fluids Engineering, 1996, 118: 464-469. 3. Beskok, A., Karniadakis, G. E., Trimmer, W., Rarefaction and …