微尺度矩形管道中气体滑移流的三维数值模拟
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摘要
随着微机电系统(MEMS)和生物工程技术的发展,微尺度流动和传热问题受到越来越多的关注。微尺度管道流动的研究对微机电系统(MEMS)、生物芯片等的应用有重要意义。
本文基于速度向前交错网格及带有压力修正格式的有限体积法,计算了滑移流动情况下,微尺度矩形管道中气体三维流动问题,讨论了管道中气体流动的速度、压力、质量流量、Darcy摩擦因子等动力学量,并与无滑移流动情况进行了对比。前人研究表明,当流动处于滑移区域( 10 ?3≤Kn≤0.1)时,N-S方程仍然适用,但需要考虑气体的稀薄效应,即采用速度滑移和温度跳跃的边界条件。本文以氮气作为研究对象。数值结果表明,当Kn = 0.055时,即处于滑移流动情况下,管道内气体的流动速度大于相同条件下无滑移流动的速度,且管道壁面上流动速度不再为零;管道内压力降较无滑移流动情况下的压力降平缓,这是由于压力降是用来克服壁面摩擦的,而滑移的边界条件导致较小的壁面摩擦;进一步研究发现,稀薄气体效应对微尺度矩形管道内气体流动的质量流量和Darcy摩擦因子会产生影响,随着稀薄效应的增强,质量流量增加而Darcy摩擦因子减小。
本文得到的三维微尺度矩形管道内气体流动的计算结果同二维计算结果以及实验结果保持了相当的一致,但同时也显示出了三维效应,为真实的微尺度管道中气体三维流动的研究提供了一些参考。
With the development of MEMS and biological engineering technology, more and more attention has been paid to the problems of micro-scale flow and thermal transmission. The study of flow in micro-scale tubes has great importance to the application of MEMS and biological chips.
To describe 3-D gaseous slip flow in rectangular microtubes, finite volume method with pressure correction scheme and velocity ahead uniform staggered grid is adopted. The dynamic parameters, including velocity、pressure、mass flow rate、Darcy friction factor, have been analyzed and compared with non-slip flow. Many results in the literatures have suggested that in the slip regime( 10 ?3≤Kn≤0.1), N-S equation with slip velocity and temperature jump boundary condition can be applied. Nitrogen flows have been studied in this paper. Numerical results indicate that, for the case of slip flow and Kn = 0.055,the velocity of slip flow in the microtubes is bigger than that of non-slip flow,and the velocity on the wall is no longer equal to zero; the pressure gradient of slip flow is smaller than that of non-slip flow,that is, the pressure gradient overcomes the friction on the wall and the slip boundary condition results in smaller friction. Moreover, the influence of rarefaction effect on mass flow rate and Darcy friction factor are also investigated. We find that rarefaction effect increases mass flow rate and decreases Darcy friction factor.
Our results of the 3-D gaseous slip flow in rectangular microtubes are consistent with 2-D numerical and experimental results. Besides,it also shows 3-D effect and gives some suggestion the study of 3-D gaseous flow in microtubes.
本文基于速度向前交错网格及带有压力修正格式的有限体积法,计算了滑移流动情况下,微尺度矩形管道中气体三维流动问题,讨论了管道中气体流动的速度、压力、质量流量、Darcy摩擦因子等动力学量,并与无滑移流动情况进行了对比。前人研究表明,当流动处于滑移区域( 10 ?3≤Kn≤0.1)时,N-S方程仍然适用,但需要考虑气体的稀薄效应,即采用速度滑移和温度跳跃的边界条件。本文以氮气作为研究对象。数值结果表明,当Kn = 0.055时,即处于滑移流动情况下,管道内气体的流动速度大于相同条件下无滑移流动的速度,且管道壁面上流动速度不再为零;管道内压力降较无滑移流动情况下的压力降平缓,这是由于压力降是用来克服壁面摩擦的,而滑移的边界条件导致较小的壁面摩擦;进一步研究发现,稀薄气体效应对微尺度矩形管道内气体流动的质量流量和Darcy摩擦因子会产生影响,随着稀薄效应的增强,质量流量增加而Darcy摩擦因子减小。
本文得到的三维微尺度矩形管道内气体流动的计算结果同二维计算结果以及实验结果保持了相当的一致,但同时也显示出了三维效应,为真实的微尺度管道中气体三维流动的研究提供了一些参考。
With the development of MEMS and biological engineering technology, more and more attention has been paid to the problems of micro-scale flow and thermal transmission. The study of flow in micro-scale tubes has great importance to the application of MEMS and biological chips.
To describe 3-D gaseous slip flow in rectangular microtubes, finite volume method with pressure correction scheme and velocity ahead uniform staggered grid is adopted. The dynamic parameters, including velocity、pressure、mass flow rate、Darcy friction factor, have been analyzed and compared with non-slip flow. Many results in the literatures have suggested that in the slip regime( 10 ?3≤Kn≤0.1), N-S equation with slip velocity and temperature jump boundary condition can be applied. Nitrogen flows have been studied in this paper. Numerical results indicate that, for the case of slip flow and Kn = 0.055,the velocity of slip flow in the microtubes is bigger than that of non-slip flow,and the velocity on the wall is no longer equal to zero; the pressure gradient of slip flow is smaller than that of non-slip flow,that is, the pressure gradient overcomes the friction on the wall and the slip boundary condition results in smaller friction. Moreover, the influence of rarefaction effect on mass flow rate and Darcy friction factor are also investigated. We find that rarefaction effect increases mass flow rate and decreases Darcy friction factor.
Our results of the 3-D gaseous slip flow in rectangular microtubes are consistent with 2-D numerical and experimental results. Besides,it also shows 3-D effect and gives some suggestion the study of 3-D gaseous flow in microtubes.
引文
[1] Naomasa,Nakajima. Micromachines Open Up A New World. New World. Html-Microsoft Internet Explorer,1996,3(1)
[2] 刘静. 微米/纳米尺度传热学. 北京:科学出版社,2001
[3] Schaff S A,Chambre P L. Flow of Rarefied Gases. Princeton:Princeton University Press,1961
[4] Jang J,Wereley S T. Pressure Distributions of Gaseous Slip Flow in Straight and Uniform Rectangular Microchannels. Microfluidics Nanofluid,2004,1(1):41-51
[5] Ebert W A,Sparrow E M. Slip Flow in Rectangular and Annular Ducts. ASME J. Basic Eng.,1965,87(6):1018-1024
[6] Eckert E G R,Drake R M. Analysis of Heat and Mass Transter. New York:McGraw-Hill,1972. 467-486
[7] Arkilic E B,Breuer K S,Schmidt M A. Gaseous Flow in Microchannels. J. Microelectromech. Syst.,1997,6(2):167-178
[8] Liu J Q,Tai Y C,Ho C M. MEMS for Pressure Distribution Studies of Gaseous Flows in Microchannels. Micro Electro Mechanical Systems,1995. MEMS '95. Proc. IEEE,1995. 209-215
[9] Pfahler J N,Harley J C,Bau H H,et al. Gas and Liquid Flow in Small Channels. ASME DSC Symp. on Micromechanical Sensors,Actuators,and Systems. 1991. 32:49-60
[10] Choi S B,Barron R F,Warrington R O. Fluid Flow and Heat Transfer in Microtubes. ASME DSC Symp. on Micromechanical Sensors,Actuators,and Systems. 1991. 32:123-124
[11] Pong K C,Ho C M. Non-Linear Pressure Distribution in Microchannels. ASME FED Application of Microfabrication on Fluid Mechanics. 1994. 197:51-56
[12] Harley J C,Huang Y,Bau H H,et al. Gas Flow in Microchannels. J. Fluid Mech.,1995,284(2):257-274
[13] Barron R F,Wang X M,Warrington R O,et al. Evaluation of the Eigenvalues for the Graetz Problem in Slip-Flow. Int. Commun. in Heat and Mass Transfer,1996,23(4):563-574
[14] Beskok A,Karbuadajus G E. A Model for Flows in Channels、Pipes and Ducts at Micro and Nano Scales. Microscal Thermophysical Engineering,1999,3(1):43-77
[15] Guo Z Y,Wu X B. Further Study on Cmopressibility Effect on the Gas Flow and Heat Transfer in a Microtube. Microscale Thermophysical Engineering,1998,2(2):111-120
[16] Zhao Hanzhong. The Numerical Solution of Gaseous Slip Flows in Microtubes. Heat Mass Transfer,2001,28(4):585-594
[17] Aubert C,Colin S. High-Order Boundary Conditions for Gaseous Flows in Rectangular Microducts. Microscale Thermophysical Engineering,2001,5(1):41-54
[18] Graur I A,Meolans J G,Zeitoun D E. Gaseous Flows in Microchannels. AIP Conference Proceedings,2005. 762:755-760
[19] Barber,Robert W,Emerson,et al. Challenges in Modeling Gas-Phase Flow in Microchannels-From Slip to Transition. Heat Transfer Engineering,2006,27(4):3-12
[20] 郭照立,郑楚光,李青等. 流体动力学的格子Boltzmann方法. 湖北:湖北科学技术出版社,2002
[21] Loyalka S K,Hamoodi S A. Poiseuille Flow of a Rarefied Gas in a Cylindrical Tube-Solution of Linearized Boltzmann Equation. Phys. Fluids,1990,2(11):2061-2065
[22] Bird G A. Molecular Gas Dynamics. Oxford:Oxford University Press,1978
[23] Bird G A. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford:Clarendon Press,1994
[24] Chen C S,Lee S M,Sheu J D. Numerical Analysis of Gas Flow in Microchannels. Numerical Heat Transfer,1998,33(7):749-762
[25] Mohamed Gad-el-Hak. The Fluid Mechanics of Microdevices-The Freeman Scholar Lecture. Journal of Fluids Engineering,1999,121(5):5-33
[26] Morini G L,Spiga M. Slip Flow in Rectangular Microtubes. Microscale Thermophysical Engineering,1998,2(4):273-282
[27] Yu S P,Ameel T A. Slip-Flow Heat Transfer in Rectangular Microchannels. Heat and Mass Transfer,2001,44(22):4225-4234
[28] Dale A,Anderson,John C,et al. Computational Fluid Mechanics and Heat Transfer. Washington:McGraw-Hill,1984
[29] Piekos E S,Breuer K S. Numerical Modeling of Micromechanical Devices Using the Direct Simulation Monte Carlo Method. J. Fluids Engineering,1996,118(3):464-469
[30] Ho C M,Tai Y C. Review:MEMS and Its Applications for Flow Control. ASME J. Fluids Engineering,1996,118(3):437-447
[31] Flick M I,Choi B I,Goodson K E. Heat Transfer Regimes in Microstuctures. ASME J. Heat Tranfer,1992,114(4):666-674
[32] Barron R F,Wang X M,Ameel T A,et al. The Graetz Problem Extender to Slip-Flow. Int. J. Heat Mass Transfer,1997,40(6):1817-1823
[33] Marques W,Kremer G M,Sharipov F M. Couette Flow with Slip and Jump Boundary Conditions. Continuum Mech. Thermodyn.,2000,12(2):379-386
[34] 秦丰华,孙德军,尹协远. 长微管气体流动的近似解析解. 空气动力学学报,2003,20(1):48-55
[35] 秦丰华,孙德军,尹协远. 微管道气体流动的蒙特卡罗直接模拟. 计算物理,2001,18(6):507-510
[36] 孙德军,秦丰华,尹协远. 微管道气体流动. 机械强度,2001,23(4):460-464
[37] 赵汉中. 计算流体力学微细圆管中气体流动的稀薄效应和可压缩效应. 华中科技大学学报,2001,29(10):96-98
[38] 柏巍,王秋旺,王娴等. 矩形微通道内滑移区气体流动换热的数值模拟. 上海理工大学学报,2003,25(2):139-142
[39] 王福军. 计算流体动力学分析-CFD软件原理与应用. 北京:清华大学出版社,2004
[40] 陶文铨. 数值传热学. 西安:西安交通大学出版社,1988
[41] 江小宁,周兆英,李勇等. 微流体运动的试验研究. 光学精密工程,1995,3(3):51-55
[42] 李万平. 计算流体力学. 武汉:华中科技大学出版社,2004
[43] 吴江航,韩庆书. 计算流体力学的理论、方法及应用. 北京:科学出版社,1988
[44] 马铁犹. 计算流体动力学. 北京:北京航空学院出版社,1986
[45] 顾尔柞. 流体力学中的有限差分法基础. 上海:上海交通大学出版社,1984
[46] 张涵信,沈孟育. 计算流体力学-差分方法的原理和应用. 北京:国防工业出版社,2003
[47] 中国人民解放军总装备部军事训练教材编辑工作委员会. 计算流体力学及应用. 北京:国防工业出版社,2003
[48] Jain V,Lin C X. Numerical Modeling of Three-Dimensional Compressible Gas Flow in Microchannels. Journal of Micromechanics and Microengineering,2006,16(2):292-302
[2] 刘静. 微米/纳米尺度传热学. 北京:科学出版社,2001
[3] Schaff S A,Chambre P L. Flow of Rarefied Gases. Princeton:Princeton University Press,1961
[4] Jang J,Wereley S T. Pressure Distributions of Gaseous Slip Flow in Straight and Uniform Rectangular Microchannels. Microfluidics Nanofluid,2004,1(1):41-51
[5] Ebert W A,Sparrow E M. Slip Flow in Rectangular and Annular Ducts. ASME J. Basic Eng.,1965,87(6):1018-1024
[6] Eckert E G R,Drake R M. Analysis of Heat and Mass Transter. New York:McGraw-Hill,1972. 467-486
[7] Arkilic E B,Breuer K S,Schmidt M A. Gaseous Flow in Microchannels. J. Microelectromech. Syst.,1997,6(2):167-178
[8] Liu J Q,Tai Y C,Ho C M. MEMS for Pressure Distribution Studies of Gaseous Flows in Microchannels. Micro Electro Mechanical Systems,1995. MEMS '95. Proc. IEEE,1995. 209-215
[9] Pfahler J N,Harley J C,Bau H H,et al. Gas and Liquid Flow in Small Channels. ASME DSC Symp. on Micromechanical Sensors,Actuators,and Systems. 1991. 32:49-60
[10] Choi S B,Barron R F,Warrington R O. Fluid Flow and Heat Transfer in Microtubes. ASME DSC Symp. on Micromechanical Sensors,Actuators,and Systems. 1991. 32:123-124
[11] Pong K C,Ho C M. Non-Linear Pressure Distribution in Microchannels. ASME FED Application of Microfabrication on Fluid Mechanics. 1994. 197:51-56
[12] Harley J C,Huang Y,Bau H H,et al. Gas Flow in Microchannels. J. Fluid Mech.,1995,284(2):257-274
[13] Barron R F,Wang X M,Warrington R O,et al. Evaluation of the Eigenvalues for the Graetz Problem in Slip-Flow. Int. Commun. in Heat and Mass Transfer,1996,23(4):563-574
[14] Beskok A,Karbuadajus G E. A Model for Flows in Channels、Pipes and Ducts at Micro and Nano Scales. Microscal Thermophysical Engineering,1999,3(1):43-77
[15] Guo Z Y,Wu X B. Further Study on Cmopressibility Effect on the Gas Flow and Heat Transfer in a Microtube. Microscale Thermophysical Engineering,1998,2(2):111-120
[16] Zhao Hanzhong. The Numerical Solution of Gaseous Slip Flows in Microtubes. Heat Mass Transfer,2001,28(4):585-594
[17] Aubert C,Colin S. High-Order Boundary Conditions for Gaseous Flows in Rectangular Microducts. Microscale Thermophysical Engineering,2001,5(1):41-54
[18] Graur I A,Meolans J G,Zeitoun D E. Gaseous Flows in Microchannels. AIP Conference Proceedings,2005. 762:755-760
[19] Barber,Robert W,Emerson,et al. Challenges in Modeling Gas-Phase Flow in Microchannels-From Slip to Transition. Heat Transfer Engineering,2006,27(4):3-12
[20] 郭照立,郑楚光,李青等. 流体动力学的格子Boltzmann方法. 湖北:湖北科学技术出版社,2002
[21] Loyalka S K,Hamoodi S A. Poiseuille Flow of a Rarefied Gas in a Cylindrical Tube-Solution of Linearized Boltzmann Equation. Phys. Fluids,1990,2(11):2061-2065
[22] Bird G A. Molecular Gas Dynamics. Oxford:Oxford University Press,1978
[23] Bird G A. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford:Clarendon Press,1994
[24] Chen C S,Lee S M,Sheu J D. Numerical Analysis of Gas Flow in Microchannels. Numerical Heat Transfer,1998,33(7):749-762
[25] Mohamed Gad-el-Hak. The Fluid Mechanics of Microdevices-The Freeman Scholar Lecture. Journal of Fluids Engineering,1999,121(5):5-33
[26] Morini G L,Spiga M. Slip Flow in Rectangular Microtubes. Microscale Thermophysical Engineering,1998,2(4):273-282
[27] Yu S P,Ameel T A. Slip-Flow Heat Transfer in Rectangular Microchannels. Heat and Mass Transfer,2001,44(22):4225-4234
[28] Dale A,Anderson,John C,et al. Computational Fluid Mechanics and Heat Transfer. Washington:McGraw-Hill,1984
[29] Piekos E S,Breuer K S. Numerical Modeling of Micromechanical Devices Using the Direct Simulation Monte Carlo Method. J. Fluids Engineering,1996,118(3):464-469
[30] Ho C M,Tai Y C. Review:MEMS and Its Applications for Flow Control. ASME J. Fluids Engineering,1996,118(3):437-447
[31] Flick M I,Choi B I,Goodson K E. Heat Transfer Regimes in Microstuctures. ASME J. Heat Tranfer,1992,114(4):666-674
[32] Barron R F,Wang X M,Ameel T A,et al. The Graetz Problem Extender to Slip-Flow. Int. J. Heat Mass Transfer,1997,40(6):1817-1823
[33] Marques W,Kremer G M,Sharipov F M. Couette Flow with Slip and Jump Boundary Conditions. Continuum Mech. Thermodyn.,2000,12(2):379-386
[34] 秦丰华,孙德军,尹协远. 长微管气体流动的近似解析解. 空气动力学学报,2003,20(1):48-55
[35] 秦丰华,孙德军,尹协远. 微管道气体流动的蒙特卡罗直接模拟. 计算物理,2001,18(6):507-510
[36] 孙德军,秦丰华,尹协远. 微管道气体流动. 机械强度,2001,23(4):460-464
[37] 赵汉中. 计算流体力学微细圆管中气体流动的稀薄效应和可压缩效应. 华中科技大学学报,2001,29(10):96-98
[38] 柏巍,王秋旺,王娴等. 矩形微通道内滑移区气体流动换热的数值模拟. 上海理工大学学报,2003,25(2):139-142
[39] 王福军. 计算流体动力学分析-CFD软件原理与应用. 北京:清华大学出版社,2004
[40] 陶文铨. 数值传热学. 西安:西安交通大学出版社,1988
[41] 江小宁,周兆英,李勇等. 微流体运动的试验研究. 光学精密工程,1995,3(3):51-55
[42] 李万平. 计算流体力学. 武汉:华中科技大学出版社,2004
[43] 吴江航,韩庆书. 计算流体力学的理论、方法及应用. 北京:科学出版社,1988
[44] 马铁犹. 计算流体动力学. 北京:北京航空学院出版社,1986
[45] 顾尔柞. 流体力学中的有限差分法基础. 上海:上海交通大学出版社,1984
[46] 张涵信,沈孟育. 计算流体力学-差分方法的原理和应用. 北京:国防工业出版社,2003
[47] 中国人民解放军总装备部军事训练教材编辑工作委员会. 计算流体力学及应用. 北京:国防工业出版社,2003
[48] Jain V,Lin C X. Numerical Modeling of Three-Dimensional Compressible Gas Flow in Microchannels. Journal of Micromechanics and Microengineering,2006,16(2):292-302