复杂微通道内流体流动的格子Boltzmann模拟
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摘要
近年来,自然科学及工程技术发展的一个重要趋势就是朝着微型化方向发展,尤其是微机电系统(Micro Electro Mechanical Systems, MEMS)技术的飞速发展极大地推动了这一研究热潮。由于在微尺度条件下,惯性力变得无关紧要,而表面张力、粘性力和静电力成为主导。因此,流体在微尺度下的传输规律与宏观尺度下不尽相同。对微尺度下流体流动的理论与实验研究也成为相关领域研究的前沿热点之一。
格子Boltzmann方法(Lattice Boltzmann Method, LBM)是近几十年来发展起来的一种计算流体力学新方法。LBM基于分子动理论,具有清晰的物理背景。此外,格子Boltzmann方法还具有并行性,以及边界条件处理简单、程序易于实施等优点。目前除了在一般的流体力学问题中得到了成功的应用外,LBM在多孔介质、多相流、化学反应扩散、渗流、粒子悬浮流、磁流体力学等相关领域也得到了比较成功的应用。近几年,一些研究学者尝试将LBM用于模拟微通道内流体流动的模拟仿真,并且取得了一些卓有成效的成果。
本论文在前人的研究基础上,利用LBM对滑移区的微尺度流动进行了研究。首先,从LBM模拟微尺度流动的两个关键点——模型中Knudsen数(Kn)的引入和滑移边界条件的处理方法展开讨论。对于前者,本文依据分子运动论中的相关定义和结论,通过粘性系数的关系式,将Kn数引入到模型中；而对滑移边界的处理上,采用了无滑移的反弹格式与自由滑移的镜面反弹格式相结合的处理方法。利用上述的LBM模型对微尺度的Poiseuille流进行了数值模拟,得到了与解析解相吻合的模拟结果。然后,又从实际问题角度出发,考虑了带有粗糙元的弯曲通道的情况,利用LBM对复杂弯曲通道的流动情况进行了模拟。在该文中,对粗糙壁面进行了简化处理,采用三角形或矩形的粗糙元进行代替,得到了预期的模拟结果。
最后,为了使LBM能更有效地应用于实际工程问题中,本文对LBM模型应用于贴体坐标网格中进行了理论推导,介绍了将LBM应用于贴体网格中的一种常用方法——插值补充格子Boltzmann方法,然后运用这种方法模拟了贴体网格中常见的两个算例：斜方腔驱动问题和圆柱绕流问题,得到了与前人工作相吻合的模拟结果,证明了LBM用于贴体网格中的可行性。同时,应用插值格子Boltzmann方法对上述弯曲通道进行了模拟,并和直角坐标下的结果进行了对比分析。
Over the last few decades, a significant trend in science and engineering is towards miniaturization. The development of Micro Electro Mechanical Systems has indeed contributed to this trend. Under micro-scale condition, the inertial force becomes less important, while surface tension and viscous force has dominated. Therefore, there are many differences in transport phenomena between micro-scale and macro-scale. The research on theory and experiment has recently become a hotspot.
Lattice Boltzmann method (LBM) has become a competitive approach in Computational Fluid Dynamics (CFD) in the last decades. LBM, based on kinetic theory, has a distinct physical background. Moreover, LBM has advantages in parallelism, programming and dealing with the complex boundary. Besides having succeeded in simulating normal problems in fluid dynamics, LBM also achieved great success in simulations on porous media, multiphase flow, reactive-diffusive, interstitial flow, granular flow, and relative domain. Recently, some scholars have attempted to simulation fluid flow in micro-channel and gained great success.
In this paper, based on previous research, gas flow in slip regime is studied. Firstly, two key issues in simulating gas flow in micro-channel via LBM, the Knudsen number Kn and slip boundary condition, are discussed. For the former issue, Kn is introduced to LBM model by using kinetic theory and definitions in viscous coefficient. For the later one, a combined boundary scheme (CBC), which combines the no-slip bounce-back and the free-slip specular reflection schemes, is applied to boundary condition treatment. Then, to validate the model, poiseuille flow in micro-channel is carried out. The simulation results agree well with those of previous studies. Moreover, according to actual conditions, gas flow in a curved rough channel is studied. The rough wall of the channel is described by uniformly distributed rectangular or triangular rough elements.
Finally, to use LBM as a practical tool, an interpolation-supplemented lattice Boltzmann method is introduced on general body-fitted coordinate systems. The scheme is applied to a lid-driven skewed cavity flow and a steady flow around a circular cylinder. The numerical results are in good agreement with the results of previous studies. Also, the complex channel mentioned above is simulated with this scheme. The simulation results are compared with those obtained in cartesian coordinate system.
格子Boltzmann方法(Lattice Boltzmann Method, LBM)是近几十年来发展起来的一种计算流体力学新方法。LBM基于分子动理论,具有清晰的物理背景。此外,格子Boltzmann方法还具有并行性,以及边界条件处理简单、程序易于实施等优点。目前除了在一般的流体力学问题中得到了成功的应用外,LBM在多孔介质、多相流、化学反应扩散、渗流、粒子悬浮流、磁流体力学等相关领域也得到了比较成功的应用。近几年,一些研究学者尝试将LBM用于模拟微通道内流体流动的模拟仿真,并且取得了一些卓有成效的成果。
本论文在前人的研究基础上,利用LBM对滑移区的微尺度流动进行了研究。首先,从LBM模拟微尺度流动的两个关键点——模型中Knudsen数(Kn)的引入和滑移边界条件的处理方法展开讨论。对于前者,本文依据分子运动论中的相关定义和结论,通过粘性系数的关系式,将Kn数引入到模型中；而对滑移边界的处理上,采用了无滑移的反弹格式与自由滑移的镜面反弹格式相结合的处理方法。利用上述的LBM模型对微尺度的Poiseuille流进行了数值模拟,得到了与解析解相吻合的模拟结果。然后,又从实际问题角度出发,考虑了带有粗糙元的弯曲通道的情况,利用LBM对复杂弯曲通道的流动情况进行了模拟。在该文中,对粗糙壁面进行了简化处理,采用三角形或矩形的粗糙元进行代替,得到了预期的模拟结果。
最后,为了使LBM能更有效地应用于实际工程问题中,本文对LBM模型应用于贴体坐标网格中进行了理论推导,介绍了将LBM应用于贴体网格中的一种常用方法——插值补充格子Boltzmann方法,然后运用这种方法模拟了贴体网格中常见的两个算例：斜方腔驱动问题和圆柱绕流问题,得到了与前人工作相吻合的模拟结果,证明了LBM用于贴体网格中的可行性。同时,应用插值格子Boltzmann方法对上述弯曲通道进行了模拟,并和直角坐标下的结果进行了对比分析。
Over the last few decades, a significant trend in science and engineering is towards miniaturization. The development of Micro Electro Mechanical Systems has indeed contributed to this trend. Under micro-scale condition, the inertial force becomes less important, while surface tension and viscous force has dominated. Therefore, there are many differences in transport phenomena between micro-scale and macro-scale. The research on theory and experiment has recently become a hotspot.
Lattice Boltzmann method (LBM) has become a competitive approach in Computational Fluid Dynamics (CFD) in the last decades. LBM, based on kinetic theory, has a distinct physical background. Moreover, LBM has advantages in parallelism, programming and dealing with the complex boundary. Besides having succeeded in simulating normal problems in fluid dynamics, LBM also achieved great success in simulations on porous media, multiphase flow, reactive-diffusive, interstitial flow, granular flow, and relative domain. Recently, some scholars have attempted to simulation fluid flow in micro-channel and gained great success.
In this paper, based on previous research, gas flow in slip regime is studied. Firstly, two key issues in simulating gas flow in micro-channel via LBM, the Knudsen number Kn and slip boundary condition, are discussed. For the former issue, Kn is introduced to LBM model by using kinetic theory and definitions in viscous coefficient. For the later one, a combined boundary scheme (CBC), which combines the no-slip bounce-back and the free-slip specular reflection schemes, is applied to boundary condition treatment. Then, to validate the model, poiseuille flow in micro-channel is carried out. The simulation results agree well with those of previous studies. Moreover, according to actual conditions, gas flow in a curved rough channel is studied. The rough wall of the channel is described by uniformly distributed rectangular or triangular rough elements.
Finally, to use LBM as a practical tool, an interpolation-supplemented lattice Boltzmann method is introduced on general body-fitted coordinate systems. The scheme is applied to a lid-driven skewed cavity flow and a steady flow around a circular cylinder. The numerical results are in good agreement with the results of previous studies. Also, the complex channel mentioned above is simulated with this scheme. The simulation results are compared with those obtained in cartesian coordinate system.
引文
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[38]Higuera F J, Succi S, and Benzi R. Lattice gas dynamics with enhanced collisions [J]. EPL (Europhysics Letters),1989,9:p.345.
[39]Chen S, Chen H, Martnez D, et al. Lattice Boltzmann model for simulation of magnetohydrodynamics [J]. Physical Review Letters,1991,67(27):p.3776-3779.
[40]Koelman J M V. A simple lattice Boltzmann scheme for Navier-Stokes fluid flow [J]. EPL (Europhysics Letters),1991,15:p.603-607.
[41]Qian Y H. Lattice BGK models for Navier-Stokes equation [J]. Europhys. Lett,1992, 17(6):p.479-484.
[42]郭照立,郑楚光.格子Boltzmann方法的原理及应用[M].2009：北京：科学出版社.
[43]何雅玲,王勇,李庆.格子Boltzmann方法的理论及应用[M].2009：北京：科学出版社.
[44]Chen S, Martinez D, and Mei R. On boundary conditions in lattice Boltzmann methods [J]. Physics of fluids,1996,8:p.2527.
[45]Ziegler D P. Boundary conditions for lattice Boltzmann simulations [J]. Journal of Statistical Physics,1993,71(5):p.1171-1177.
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[2]Gabriel K, Jarvis J, and Trimmer W. Small Machines, Large Opportunities:a report on the emerging field of microdynamics [M]. National Science Foundation, published by AT&T Bell Laboratories, Murray Hill, New Jersey,1988.
[3]Schaaf S A and Chambre P L. Flow of rarefied gases [M].1961:Princeton University Press Princeton, NJ, USA.
[4]Hohreiter V, Wereley S T, Olsen M G, et al. Cross-correlation analysis for temperature measurement [J]. Measurement science and technology,2002,13(7):p. 1072-1078.
[5]Meinhart C D, Wereley S T, and Santiago J G. PIV measurements of a microchannel flow [J]. Experiments in Fluids,1999,27(5):p.414-419.
[6]Ovryn B. Three-dimensional forward scattering particle image velocimetry applied to a microscopic field-of-view [J]. Experiments in Fluids,2000,29:p.175-184.
[7]Gad-el-Hak M. The fluid mechanics of microdevices-The Freeman scholar lecture [J]. TRANSACTIONS-AMERICAN SOCIETY OF MECHANICAL ENGINEERS JOURNAL OF FLUIDS ENGINEERING,1999,121:p.5-33.
[8]Kennard E H. Kinetic theory of gases [M].1938:McGraw-Hill New York.
[9]Smoluchowski M. Ueber w rmeleitung in verdunnten gasen [J]. Annalen der Physik und Chemie,1898,64:p.101-130.
[10]Tsien H S. Superaerodynamics, mechanics of rarefied gases [J]. J. Aero. Sci,1946, 13(12):p.653-664.
[11]Bird G A. Monte Carlo simulation of gas flows [J]. Annual Review of Fluid Mechanics, 1978,10(1):p.11-31.
[12]Oran E S, Oh C K, and Cybyk B Z. DIRECT SIMULATION MONTE CARLO:Recent Advances and Applications 1 [J]. Annual Review of Fluid Mechanics,1998,30(1):p.403-441.
[13]Fan J and Shen C. Statistical simulation of low-speed rarefied gas flows [J]. Journal of Computational Physics,2001,167(2):p.393-412.
[14]Lim C Y, Shu C, Niu X D, et al. Application of lattice Boltzmann method to simulate microchannel flows [J]. Physics of Fluids,2002,14:p.2299.
[15]Nie X, Doolen G D, and Chen S. Lattice-Boltzmann simulations of fluid flows in MEMS [J]. Journal of Statistical Physics,2002,107(1):p.279-289.
[16]Niu X D, Shu C, and Chew Y T. A lattice Boltzmann BGK model for simulation of micro flows [J]. EPL (Europhysics Letters),2004,67:p.600-606.
[17]Tang G H, Tao W Q, and He Y L. Lattice Boltzmann method for simulating gas flow in microchannels [J]. International Journal of Modern Physics C,2004,15:p. 335-347.
[18]Arkilic E B, Schmidt M A, and Breuer K S. Gaseous slip flow in long microchannels [J]. Journal of Microelectromechanical systems,1997,6(2):p.167-178.
[19]Niu X D, Shu C, and Chew Y T. Numerical Simulation of Isothermal Micro Flows by Lattice Boltzmann Method and theoretical analysis of the Diffuse scattering boundary condition [J]. International Journal of Modern Physics C,2005,16(12): p.1927-1942.
[20]Tang G H, He Y L, and Tao W Q. Comparison of gas slip models with solutions of linearized Boltzmann equation and direct simulation of Monte Carlo method [J]. International Journal of Modern Physics. C, Physics and Computers,2007,18(2).
[21]Tang G H, Tao W Q, and He Y L. Lattice Boltzmann method for gaseous microflows using kinetic theory boundary conditions [J]. Physics of Fluids,2005,17:p. 058101.
[22]Tang G H, Tao W Q, and He Y L. SIMULATING TWO-AND THREE-DIMENSIONAL MICROFLOWS BY THE LATTICE BOLTZMANN METHOD WITH KINETIC BOUNDARY CONDITIONS [J]. International Journal of Modern Physics. C, Physics and Computers,2007,18(5).
[23]Guo Z, Zhao T S, and Shi Y. Physical symmetry, spatial accuracy, and relaxation time of the lattice Boltzmann equation for microgas flows [J]. Journal of Applied Physics,2006,99:p.074903.
[24]He X and Doolen G. Lattice Boltzmann method on curvilinear coordinates system: flow around a circular cylinder [J]. Journal of Computational Physics,1997,134(2): p.306-315.
[25]Imamura T, Suzuki K, Nakamura T, et al. Acceleration of steady-state lattice Boltzmann simulations on non-uniform mesh using local time step method [J]. Journal of Computational Physics,2005,202(2):p.645-663.
[26]Chew Y T, Shu C, and Niu X D. Simulation of unsteady incompressible flows by using Taylor series expansion-and least square-based lattice Boltzmann method [J]. International Journal of Modern Physics C-Physics and Computer,2002,13(6):p. 719-738.
[27]Reider M B and Sterling J D. Accuracy of discrete-velocity BGK models for the simulation of the incompressible Navier-Stokes equations [J]. Arxiv preprint comp-gas/9307003,1993.
[28]Cao N, Chen S, Jin S, et al. Physical symmetry and lattice symmetry in the lattice Boltzmann method [J]. Physical Review E,1997,55(1):p.21-24.
[29]Mei R and Shyy W. On the finite difference-based lattice Boltzmann method in curvilinear coordinates [J]. Journal of Computational Physics,1998,143(2):p. 426-448.
[30]Wang Y, He Y L, Zhao T S, et al. Implicit-explicit finite-difference lattice boltzmann method for compressible flows [J]. International Journal of Modern Physics C,2007,18(12):p.1961-1984.
[31]Lee T and Lin C L. A characteristic Galerkin method for discrete Boltzmann equation [J]. Journal of Computational Physics,2001,171(1):p.336-356.
[32]Mavriplis D J. Multigrid solution of the steady-state lattice Boltzmann equation [J]. Computers & Fluids,2006,35(8-9):p.793-804.
[33]Peng G, Xi H, Duncan C, et al. Finite volume scheme for the lattice Boltzmann method on unstructured meshes [J]. Physical Review E,1999,59(4):p.4675-4682.
[34]Hardy J, Pomeau Y, and De Pazzis 0. Time evolution of a two-dimensional model system. I. Invariant states and time correlation functions [J]. Journal of Mathematical Physics,1973,14:p.1746.
[35]Frisch U, Hasslacher B, and Pomeau Y. Lattice-gas automata for the Navier-Stokes equation [J]. Physical Review Letters,1986,56(14):p.1505-1508.
[36]McNamara G R and Zanetti G. Use of the Boltzmann equation to simulate lattice-gas automata [J]. Physical Review Letters,1988,61(20):p.2332-2335.
[37]Higuera F J and Jimenez J. Boltzmann approach to lattice gas simulations [J]. EPL (Europhysics Letters),1989,9:p.663.
[38]Higuera F J, Succi S, and Benzi R. Lattice gas dynamics with enhanced collisions [J]. EPL (Europhysics Letters),1989,9:p.345.
[39]Chen S, Chen H, Martnez D, et al. Lattice Boltzmann model for simulation of magnetohydrodynamics [J]. Physical Review Letters,1991,67(27):p.3776-3779.
[40]Koelman J M V. A simple lattice Boltzmann scheme for Navier-Stokes fluid flow [J]. EPL (Europhysics Letters),1991,15:p.603-607.
[41]Qian Y H. Lattice BGK models for Navier-Stokes equation [J]. Europhys. Lett,1992, 17(6):p.479-484.
[42]郭照立,郑楚光.格子Boltzmann方法的原理及应用[M].2009：北京：科学出版社.
[43]何雅玲,王勇,李庆.格子Boltzmann方法的理论及应用[M].2009：北京：科学出版社.
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