基于改进LBM-BGK方程的高Knudsen数热流逸现象模拟研究

doi:10.11949/0438-1157.20250460

摘要/Abstract

摘要：

建立了适合模拟过渡流高Knudsen数（Kn）条件下的热流逸现象的LBM-BGK方程，并修正了松弛系数和速度滑移边界条件，将温度阶跃考虑到速度滑移边界条件中，采用D2Q9模型模拟了长直微通道内空气在过渡流态时的温度、压力、流线及速度分布情况。结果表明：随着Kn增大，微通道中心处空气的温度和压力分布将发生畸变，由线性分布转变为非线性分布，同时纵截面处流线由相互平行变为不断波动，即流场的随机性逐渐增强；Kn增大意味着空气处于更稀薄的流态，此时压力差也更大，即越稀薄的流态对于提升压力差越有利；Kn越大，微通道中心处空气的无量纲速度增长幅度就越显著地低于壁面处，速度滑移效应更明显；在高Kn区域，滑移速度无量纲斜率均值 $φ ¯$ 与Cercignani的结果更接近。模拟结果可为介尺度流动和稀薄气体动力学领域提供一定参考。

关键词: LBM-BGK方程, 过渡流区域, Knudsen数, 介尺度, 数值模拟

Abstract:

A lattice Boltzmann method with Bhatnagar-Gross-Krook (LBM-BGK) equation suitable for simulating thermal transpiration phenomena under high Knudsen number (Kn) conditions in transitional flow regime was established. The relaxation coefficient and velocity slip boundary conditions were modified as well. By incorporating temperature jumps into the velocity slip boundary conditions, the D2Q9 model was employed to simulate the temperature, pressure, streamline, and velocity distributions for air in a long straight microchannel under transitional flow regime conditions. The results show that: with increasing Kn, the temperature and pressure distributions at the center of the microchannel become distorted, transitioning from linear to nonlinear distributions. Simultaneously, the streamlines in the longitudinal cross-section deviate from parallel, exhibiting increasing randomness. A higher Kn signifies a more rarefied flow state corresponding to a larger pressure difference, which shows that a more rarefied flow state is more beneficial to raise pressure differences. With increasing Kn, the dimensionless velocity increase at the center of the microchannel is significantly lower than that near the wall, indicating a more pronounced velocity slip effect. In the high Kn regime, the average dimensionless slope of slip velocity is closer to Cercignani's results. The simulation findings provide valuable insights for the fields of mesoscale flow and rarefied gas dynamics.

Key words: LBM-BGK equation, transitional flow regime, Knudsen number, mesoscale, numerical simulation

中图分类号:

O 356

王博韬, 卢苇, 覃睿. 基于改进LBM-BGK方程的高Knudsen数热流逸现象模拟研究[J]. 化工学报, 2025, 76(10): 5057-5066.

Botao WANG, Wei LU, Rui QIN. A simulation investigation of thermal transpiration phenomenon at high Knudsen numbers by improved LBM-BGK equation[J]. CIESC Journal, 2025, 76(10): 5057-5066.

图/表 7

图1 长直微通道内不同Kn时分子碰撞与迁移过程示意图

Fig.1 Schematic diagramof molecular collision and streaming process in a long straight microchannel

图2 长直微通道物理模型示意图

Fig.2 Schematic diagram of physical model in a long straight microchannel

图3 计算流程图

Fig.3 Flowchart of calculation

图4 长直微通道内沿纵截面温度、压力分布云图

Fig.4 Temperature and pressure contours along longitudinal cross-section in a long straight microchannel

图6 长直微通道内沿纵截面无量纲速度分布

Fig.6 Dimensionless velocity profiles along longitudinal cross-section in a long straight microchannel

图7 长直微通道内无量纲滑移速度随Kn的变化

Fig.7 Variation of dimensionless slip velocity with Kn in a long straight microchannel

图8 基于不同模型计算的长直微通道内的滑移速度

Fig.8 Dimensionless slip velocity calculated according to different models in a long straight microchannel

参考文献 34

[1]	Reynolds O. On certain dimensional properties of matter in the gaseous state[J]. Philosophical Transactions of the Royal Society of London, 1879, 170: 727-845.
[2]	Muntz E P, Sone Y, Aoki K, et al. Performance analysis and optimization considerations for a Knudsen compressor in transitional flow[J]. Journal of Vacuum Science & Technology A: Vacuum, Surfaces, and Films, 2002, 20(1): 214-224.
[3]	Kugimoto K, Hirota Y, Yamauchi T, et al. A novel heat pump system using a multi-stage Knudsen compressor[J]. International Journal of Heat and Mass Transfer, 2018, 127: 84-91.
[4]	Vargo S E, Muntz E P. Initial results from the first MEMS fabricated thermal transpiration-driven vacuum pump[C]// AIP Conference Proceedings. New York: American Institute of Physics, 2001, 585(1): 502-509.
[5]	Zhang W J, Lu W, Wang B T. Performance analysis of a novel thermal transpiration vacuum cooling system[J]. International Journal of Green Energy, 2022, 19(2): 149-158.
[6]	An S, Gupta N K, Gianchandani Y B. A Si-micromachined 162-stage two-part Knudsen pump for on-chip vacuum[J]. Journal of Microelectromechanical Systems, 2014, 23(2): 406-416.
[7]	Lotfian A, Roohi E. Binary gas mixtures separation using microscale radiometric pumps[J]. International Communications in Heat and Mass Transfer, 2021, 121: 105061.
[8]	Baier T, Hardt S. Gas separation in a Knudsen pump inspired by a Crookes radiometer[J]. Microfluidics and Nanofluidics, 2020, 24(6): 41.
[9]	Nakaye S, Sugimoto H, Gupta N K, et al. Thermally enhanced membrane gas separation[J]. European Journal of Mechanics-B/Fluids, 2015, 49: 36-49.
[10]	Nakaye S, Sugimoto H. Demonstration of a gas separator composed of Knudsen pumps[J]. Vacuum, 2016, 125: 154-164.
[11]	Sugimoto H, Hibino M. Numerical analysis on gas separator with thermal transpiration in micro channels[C]// AIP Conference Proceedings. New York: American Institute of Physics, 2012, 1501(1): 794-801.
[12]	Klein T A. Energy conversion using thermal transpiration: optimization of a Knudsen compressor[D]. Cambridge: Massachusetts Institute of Technology, 2012.
[13]	王博韬. 热流逸效应的抽真空特性及其在真空制冷中的应用[D]. 南宁: 广西大学, 2018.
	Wang B T. Vacuum pumping characteristics and vacuum cooling application of thermal transpiration effect[D]. Nanning: Guangxi University, 2018.
[14]	柯杰坤. 热流逸效应下混氢天然气的传输特性及其分离研究[D]. 南宁: 广西大学, 2024.
	Ke J K. Study of transport characteristics and separation based on thermal transpiration effect in hydrogen blending natural gas[D]. Nanning: Guangxi University, 2018.
[15]	Zou Q S, He X Y. On pressure and velocity boundary conditions for the lattice Boltzmann BGK model[J]. Physics of Fluids, 1997, 9(6): 1591-1598.
[16]	D'Orazio A, Succi S. Boundary conditions for thermal lattice Boltzmann simulations[C]// Computational Science-ICCS 2003. Berlin, Heidelberg: Springer, 2003: 977-986.
[17]	Tian Z W, Zou C, Liu H J, et al. Lattice Boltzmann scheme for simulating thermal micro-flow[J]. Physica A: Statistical Mechanics and its Applications, 2007, 385(1): 59-68.
[18]	Tang G H, Zhang Y H, Gu X J, et al. Lattice Boltzmann model for thermal transpiration[J]. Physical Review E, 2009, 79(2): 027701.
[19]	Sheng Q, Tang G H, Gu X J, et al. Simulation of thermal transpiration flow using a high-order moment method[J]. International Journal of Modern Physics C, 2014, 25(11): 1450061.
[20]	Zhang Y H, Gu X J, Barber R W, et al. Modelling thermal flow in the transition regime using a lattice Boltzmann approach[J]. Europhysics Letters (EPL), 2007, 77(3): 30003.
[21]	Li L K, Mei R W, Klausner J F. Boundary conditions for thermal lattice Boltzmann equation method[J]. Journal of Computational Physics, 2013, 237: 366-395.
[22]	Sharipov F. Data on the velocity slip and temperature jump on a gas-solid interface[J]. Journal of Physical and Chemical Reference Data, 2011, 40(2): 023101.
[23]	Isfahani A H M, Soleimani A, Homayoon A. Simulation of high Knudsen number gas flows in nanochannels via the lattice Boltzmann method[J]. Advanced Materials Research, 2011, 403/404/405/406/407/408: 5318-5323.
[24]	Karimipour A. Provide a suitable range to include the thermal creeping effect on slip velocity and temperature jump of an air flow in a nanochannel by lattice Boltzmann method[J]. Physica E: Low-dimensional Systems and Nanostructures, 2017, 85: 143-151.
[25]	Shokouhmand H, Isfahani A H M. An improved thermal lattice Boltzmann model for rarefied gas flows in wide range of Knudsen number[J]. International Communications in Heat and Mass Transfer, 2011, 38(10): 1463-1469.
[26]	Karniadakis G, Beskok A, Aluru N. Microflows and Nanoflows Fundamentals and Simulation[M]. Berlin: Springer, 2005.
[27]	Beskok A, Karniadakis G E, Trimmer W. Rarefaction and compressibility effects in gas microflows[J]. Journal of Fluids Engineering, 1996, 118(3): 448-456.
[28]	陈浮, 宋彦萍, 陈焕龙, 等. 气体动力学基础[M]. 哈尔滨: 哈尔滨工业大学出版社, 2013.
	Chen F, Song Y P, Chen H L, et al. Fundamentals of Gas Dynamics[M]. Harbin: Harbin Institute of Technology Press, 2013.
[29]	Rahouadja Z, Madjid H. Microchannel fluid flow and heat transfer by lattice Boltzmann method[C]// 4th Micro and Nano Flows Conference. London: UCL Engineering, 2014: 1-8.
[30]	林建忠, 包福兵, 张凯, 等. 微纳流动理论及应用[M]. 北京: 科学出版社, 2010.
	Lin J Z, Bao F B, Zhang K, et al. Theory and Applications of Micro- and Nano-Scale Flow[M]. Beijing: China Science Publishing & Media Ltd, 2010.
[31]	胡立冰. 微尺度气体流动的格子Boltzmann模拟[D]. 沈阳: 东北大学, 2012.
	Hu L B. Lattice boltzmann simulation to micro-scale gas flows[D]. Shenyang: Northeastern University, 2012.
[32]	Cercignani C. Higher order slip according to the linearized Boltzmann equation: Institute of Engineering Research Report AS-64-19[R]. Berkeley: University of California, 1964.
[33]	Maxwell J C. On the dynamical theory of gases[J]. Philosophical Transactions of the Royal Society of London, 1867, 157: 49-88.
[34]	Toschi F, Succi S. Lattice Boltzmann method at finite Knudsen numbers[J]. Europhysics Letters (EPL), 2005, 69(4): 549-555.