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High Order Conservative Semi-Lagrangian Scheme for the bgk model of the Boltzmann Equation

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Communications in Computational Physics 2021年第1期29卷 1-56页

作者： Sebastiano Boscarino Seung-Yeon Cho Giovanni Russo Seok-Bae Yun Department of Mathematics and Computer Science University of Catania95125 CataniaItaly Department of Mathematics Sungkyunkwan UniversitySuwon 440-746Republic of Korea

In this paper,we present a conservative semi-Lagrangian finite-difference scheme for the bgk model.Classical semi-Lagrangian finite difference schemes,coupled with an L-stable treatment of the collision term,allow large time steps,for all the range of Knudsen number[17,27,30].Unfortunately,however,such schemes are not conservative.Lack of conservation is analyzed in detail,and two main sources are identified as its cause.First,when using classical continuous Maxwellian,conservation error is negligible only if velocity space is resolved with sufficiently large number of grid points.However,for a small number of grid points in velocity space such error is not negligible,because the parameters of the Maxwellian do not coincide with the discrete moments.Secondly,the non-linear reconstruction used to prevent oscillations destroys the translation invariance which is at the basis of the conservation properties of the scheme.As a consequence the schemes show a wrong shock speed in the limit of small Knudsen number.To treat the first problem and ensure machine precision conservation of mass,momentum and energy with a relatively small number of velocity grid points,we replace the continuous Maxwellian with the discrete Maxwellian introduced in[22].The second problem is treated by implementing a conservative correction procedure based on the flux difference form as in[26].In this way we can construct conservative semi-Lagrangian schemes which are Asymptotic Preserving(AP)for the underlying Euler limit,as the Knudsen number vanishes.The effectiveness of the proposed scheme is demonstrated by extensive numerical tests.

关键词： bgk model Boltzmann equation semi-Lagrangian scheme conservative correction discrete Maxwellian

A Local Velocity Grid Approach for bgk Equation

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Communications in Computational Physics 2014年第9期16卷 956-982页

作者： Florian Bernard Angelo Iollo Gabriella Puppo Department of Mechanical and Aerospace Engineering Politecnico di Torino10129 TorinoItaly. Univ.Bordeaux IMBUMR 5251F-33400 TalenceFrance.INRIAF-33400 TalenceFrance. Dip.di Scienza ed Alta Tecnologia Universita dell’InsubriaComoItaly

The solution of complex rarefied flows with the bgk equation and the Discrete Velocity Method(DVM)requires a large number of velocity grid points leading to significant computational costs.We propose an adaptive velocity grid approach exploiting the fact that locally in space,the distribution function is supported only by a sub-set of the global velocity grid.The velocity grid is adapted thanks to criteria based on local temperature,velocity and on the enforcement of mass conservation.Simulations in 1D and 2D are presented for different Knudsen numbers and compared to a global velocity grid bgk solution,showing the computational gain of the proposed approach.

关键词： Kinetic models bgk model discrete velocity method

Numerical Simulations of Rarefied Gases in Curved Channels: Thermal Creep, Circulating Flow, and Pumping Effect

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Communications in Computational Physics 2009年第10期6卷 919-954页

作者： Kazuo Aoki Pierre Degond Luc Mieussens Department of Mechanical Engineering and Science Graduate School of EngineeringKyoto UniversityKyoto 606-8501Japan Universite de Toulouse UPSINSAUT1UTMInstitut de Mathematiques de ToulouseF-31062 ToulouseFrance CNRS Institut de Mathematiques de Toulouse UMR 5219F-31062 ToulouseFrance Universite de Bordeaux Institut de Mathematiques de Bordeaux351cours de la Liberation33405 Talence cedexFrance

We present numerical simulations of a new system of micro-pump based on the thermal creep effect described by the kinetic theory of gases.This device is made of a simple smooth and curved channel with a periodic temperature distribution.Using the Boltzmann-bgk model as the governing equation for the gas flow,we develop a numerical method based on a deterministic finite volume scheme,implicit in time,with an implicit treatment of the boundary conditions.This method is comparatively less sensitive to the slow flow velocity than the usual Direct Simulation Monte Carlo method in case of long devices,and turns out to be accurate enough to compute macroscopic quantities like the pressure field in the channel.Our simulations show the ability of the device to produce a one-way flow that has a pumping effect.

关键词： Knudsen compressor thermal creep flow transpiration flow bgk model implicit scheme

A High-Order Accurate Gas-Kinetic Scheme for One-and Two-Dimensional Flow Simulation

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Communications in Computational Physics 2014年第4期15卷 911-943页

作者： Na Liu Huazhong Tang HEDPS CAPT&LMAMSchool of Mathematical SciencesPeking UniversityBeijing 100871P.R.China

This paper develops a high-order accurate gas-kinetic scheme in the framework of the finite volume method for the one-and two-dimensional flow simulations,which is an extension of the third-order accurate gas-kinetic scheme[Q.B.Li,K.Xu,and S.Fu,J.Comput.Phys.,229(2010),6715-6731]and the second-order accurate gas-kinetic scheme[K.Xu,J.Comput.Phys.,171(2001),289-335].It is formed by two parts:quartic polynomial reconstruction of the macroscopic variables and fourth-order accurate flux evolution.The first part reconstructs a piecewise cell-center based quartic polynomial and a cell-vertex based quartic polynomial according to the“initial”cell average approximation of macroscopic variables to recover locally the non-equilibrium and equilibrium single particle velocity distribution functions around the cell interface.It is in view of the fact that all macroscopic variables become moments of a single particle velocity distribution function in the gas-kinetic theory.The generalized moment limiter is employed there to suppress the possible numerical oscillation.In the second part,the macroscopic flux at the cell interface is evolved in fourth-order accuracy by means of the simple particle transport mechanism in the microscopic level,i.***.free transport and the Bhatnagar-Gross-Krook(bgk)collisions.In other words,the fourth-order flux evolution is based on the solution(i.***.the particle velocity distribution function)of the bgk model for the Boltzmann equation.Several 1D and 2D test problems are numerically solved by using the proposed high-order accurate gas-kinetic scheme.By comparing with the exact solutions or the numerical solutions obtained the secondorder or third-order accurate gas-kinetic scheme,the computations demonstrate that our scheme is effective and accurate for simulating invisid and viscous fluid flows,and the accuracy of the high-order GKS depends on the choice of the(numerical)collision time.

关键词： Gas-kinetic scheme initial reconstruction bgk model Boltzmann equation velocity distribution function

A Three-Dimensional Gas-Kinetic bgk Scheme for Simulating Flows in Rotating Machinery

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Advances in Applied Mathematics and Mechanics 2019年第1期11卷 168-196页

作者： Di Zhou Zhiliang Lu Tongqing Guo Department of Aerodynamics College of Aerospace EngineeringNanjing University of Aeronautics and AstronauticsQinhuai DistrictNanjing 210016JiangsuChina

This paper focuses on the development and application of a threedimensional gas-kinetic Bhatnagar-Gross-Krook(bgk)method for the viscous flows in rotating machinery.For such flows,a rotating frame of reference is usually used in formulating the Navier-Stokes(N-S)equations,and there are two major concerns in constructing the corresponding bgk model.One is the change of the convective velocities in the N-S equations,which can be reflected through modification of the gas streaming velocity.The other one is the necessity to account for the effect of the additional Coriolis and centrifugal forces.Here,a specifically-designed acceleration term is added into the modified Boltzmann equation so that the source effects can be naturally included into the gas evolution process and the resulted fluxes.Under the finitevolume framework,the constructed bgk model is locally solved at each cell interface and then the numerical fluxes can be evaluated.When employing the bgk scheme,it is sometimes found that the calculated spatial derivatives of the initial and equilibrium distribution functions are sensitive to the mesh quality especially in complex rotating flow applications,which may significantly influence flux evaluation.Therefore,an improved approach for computing these slopes is adopted,through which the modeling capability for viscous flows is enhanced.For validation,several numerical examples are presented.The computed results show that the present method can be well applied to a wide range of flows in rotating machinery with favorable accuracy.

关键词： Gas-kinetic scheme bgk model non-inertial reference frame acceleration term rotating machinery

KINETIC model FOR PARTICLE PHASE IN TURBULENT CONCENTRATED SEDIMENT-LADEN FLOWS

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KINETIC MODEL FOR PARTICLE PHASE IN TURBULENT CONCENTRATED S...

第九次河流泥沙国际学术大会

作者： Xudong FU and Guangqian WANGKey Lab for Water and Sediment Sciences of the Ministry of Education Department of Hydraulic Engineering Tsinghua University Beijing 100084 China.Gordon HUANG EVSE Faculty of Engineering University of Regina Regina SK S4S 0A2 Canada

＜正＞Formulating various underlying mechanisms is essential for modeling sediment-transport in turbulent concentrated flows. In this study, a kinetic model allowing for collisional dissipation, correlation between two colliding particles, and certain shear rate was developed. A comparison with experimental data of solid-fluid mixtures in a relatively concentrated simple shear flow shows a good agreement. The varying importance of the phasic stresses with concentration shows that a generalized model like the present one should be applied to flows with C ranging from 0.2 to 0.5;outside this intermediate range of C, previous rapid-granular-flow or low-concentration models may be practical alternatives. The C→0 asymptote occurs at a relatively high shear rate, for which low collisional dissipation of the particle phase in the dilute limit （C→0） can be responsible. No consideration of friction forces among particles lowers the applicability of the present model to extremely concentrated flows.

关键词： Kinetic model Fokker-Planck operator bgk model Simple shear flow Sediment Diffusion Collision Asymptote

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bgk模型稳定性分析及其改进

高等学校计算数学学报 2017年第2期39卷 108-116页

作者：王立海李春光景何仿杨程北方民族大学数学与信息科学学院北方民族大学土木工程学院北方民族大学数值计算与工程应用研究所

1引言格子Boltzmann方法相对于传统的计算流体力学,具有计算简单、节省内存、边界条件容易处理、适合并行计算等优点,在许多传统模拟方法难以胜任的领域,如多孔介质流[1]、多相流、电动力学[2]、磁流体[3]、非牛顿流体[4]、颗粒流、血液... 详细信息

1引言格子Boltzmann方法相对于传统的计算流体力学,具有计算简单、节省内存、边界条件容易处理、适合并行计算等优点,在许多传统模拟方法难以胜任的领域,如多孔介质流[1]、多相流、电动力学[2]、磁流体[3]、非牛顿流体[4]、颗粒流、血液流[5]等,都可以进行有效的数值模拟.然而,Boltzmann方程中的碰撞算子非常复杂,阻碍了该方法的应用.

关键词： lattice Boltzmann method bgk model D2Q9model numerical simulation

Preformance of comparsion of kinetic models for molecular gases in the modelling of normal shock waves and thermal creep flow

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Preformance of comparsion of kinetic models for molecular ga...