微气体表面粗糙效应的格子Boltzmann模拟
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摘要
随着微机电系统工业和微制造业的发展,对微流体的研究受到越来越多的关注。对于微通道中的液体流动的研究主要集中在表面张力、壁面可湿度等因素的影响,在理论或实验上的研究也取得了不少的进展。对于微通道中的气体流动的研究,情况又有些不同。因为在微通道的气体流动中,由于连续介质假设不成立导致基于Navier-Stokes方程(NSE)的解的可靠性受到了质疑。在微气体流动或稀薄气体流动的数值模拟中,我们需要求解一些基于分子(伪粒子)的模型,包括分子动力学(MD)模拟,Boltzmann方程(BE),直接Monte-Carlo模拟(DSMC)、信息保存(IP)方法等。
近年来,格子Boltzmann方法(LBM)提供了一个有效的研究微流体系统的新工具。关于格子Boltzmann在微流体或稀薄气体上的研究已经取得了一些很好的成果。与分子动力学、直接Monte-Carlo模拟或其它同类型的方法相比,格子Boltzmann方法在计算效率上,在处理复杂边界问题上具有相当的优势。然而,尽管在小尺度中粗糙效应重要性增加,但由于缺乏有效的模拟工具,到目前为止对高Knudsen数的微气体流动中的粗糙效应的研究成果仍相对较少而且不能统一。为了研究氮气和氦气在微通道中的流动,本文设计了一个微通道模型,其高度为H=1.2μm,长度为L=5H或L=10H,对应氮气和氦气的Knudsen数分别为Kn_N_2=0.055和Kn_(He)=0.16。微通道粗糙表面的几何形状被分别设计成方形、Sin函数形、三角形和分形形状。利用这个模型,我们做了下面主要工作:
1)在微流体格子Boltzmann模型的状态方程方面,我们从气体运动论出发,重新推导松弛时间因子以及它和Knudsen数的关系。通过设置具有物理意义的网格时间步长,使得Knudsen数的表达式与网格精度无关。同时,通过与实验比较我们验证了这一修正的松弛因子跟原模型相比可以消除高Knudsen数的微气体的人工粘性,得到和实验相比更吻合的而且具有明确物理意义的结果。
2)在微流体格子Boltzmann模型的边界处理方面,我们考虑在微通道中由于摩擦或者流体杂质沉淀而形成的表面粗糙具有分形性质。所以我们在微流体格子Boltzmann方法中引进分形粗糙面并使用具有物理意义的参数如分维来控制粗糙面,我们认为它比其他学者使用的规则型,如矩形或三角形粗糙面更接近真实情况。此外,我们还计算并讨论了漫反射边界条件中粗糙表面的切向动量容许系数的影响,得到其在格子Boltzmann模型中对应的取值。
3)利用修正的微流体格子Boltzmann模型,我们模拟了微通道中的氮气和氦气的流动,捕捉到了最小Knudsen数现象。我们计算并分析了固壁表面粗糙效应和粗糙面形状对流体速度、压力、质量流量以及摩阻系数的影响,讨论了微通道中粗糙效应与稀薄效应与可压缩效应的耦合作用,得到一些有意义的结论。对氮气和氦气的模拟表明表面粗糙不但影响微通道中的速度分布,而且影响其压力分布。随着相对粗糙度的增加质量流量快速下降,摩阻系数快速增加。微通道中稀薄性效应、可压缩性效应和表面粗糙效应的作用强烈耦合,综合影响宏观物理量的变化趋势。在微通道中氮气和氦气的稀薄性、可压缩性和表面粗糙效应强烈耦合。在模拟分形粗糙度时,我们也分析了分形层次和网格精度的影响。除了相对粗糙度之外,所使用的粗糙几何形状对结果的影响也很大。这些不同于宏观流动的特性对了解微流气体流动有着重要的意义。
Surface roughness is identified as an important factor in Knudsen flows of liquid or gas.As the development of Micro-Electro-Mechanical Systems(MEMS) and micro fabrication,micro flows have been receiving more and more attention.Liquid flows in microchannels are most concentrated on the effects of surface tension and wall wettability,and their experiments and theoretical analyses have made considerable successes.However,the way to study the roughness effect of micro gas flows(also rarefied gas flows) is quite different from that of micro liquid flows.Since the result based on the Navier-Stokes equation(NSE) is highly dubitable due to the invalidation of the continuum hypothesis in micro gas flows,the molecular-based models including molecular dynamics(MD) simulations,Boltzmann equation(BE) simulations,direct simulations Monte-Carlo(DSMC) and information preservation (IP) simulations have to be employed for numerical studies.
Recently,the lattice Boltzmann method(LBM) provides a new effective way to study the micro flow systems.The researches indicate that the results by lattice Boltzmann method are consistent with the theoretical analyses and experiments. Compared with molecular dynamics simulations,direct simulations Monte-Carlo simulations and other related methods,the lattice Boltzmann method gains the advantages of computational efficiency and flexibility for complex boundary geometries.However,few studies up to now present a deep understanding of the effects of surface roughness on the micro gas flow systems at high Knudsen numbers. To understand the effect of various roughnesses and different geometries on the micro gas flow systems,we here design a lattice Boltzmann model and a microchannel with various roughness boundaries,and then a series of numerical simulations are taken. The surface roughness of the microchannel is modeled to be square,sinusoidal, triangular and fractal,respectively.By this model,the roughness effect combining with the effects of rarefaction and compressibility in different boundaries on the micro gas flows of nitrogen and helium are studied and analyzed.The main contents of this paper are as follows:
1) The relaxation time and its relationship with the Knudsen number are re-deduced from gas dynamics.The relaxation time is determined from the physical properties of the investigated gas and cannot be freetly adjusted.The expression of the Knudsen number is independent of the grid size by setting a certain relaxation time. Compared with theoretical and experimental studies the effectiveness of the lattice Boltzmann model is validated.It is shown that the modified relaxation time is able to eliminate the unrealistically high value of the bulk viscosity and is in good agreement with experiments.
2) The surface roughness is modeled by a self-affine fractal which is a reasonable tool to produce suitable surface of controlled roughness to describe the wall boundary of the microchannel.The height of the rough microchannel is H=1.2μm and the length is L=5H or L=10H.Correspondly,the Knudsen numbers of nitrogen and helium are Kn_N_2=0.055 and Kn_(He)=0.16.Importantly,the thoee boundary conditions are discussed and the influence of the tangential momentum accommodation coefficient(TMAC) on the slip velocity near the rough boundary is measured.
3) The effects of relative roughness and Knudsen number in the micro flows of nitrogen and helium are studied and analyzed.Numerical simulations show that the surface roughness influences not only velocity distribution but also pressure distribution.In addition,the relative mass flow rate decreases greatly and the relative resistance coefficient rises rapidly as the relative roughness increases.The effects of rarefaction,compressibility and roughness are strongly coupled.To measure the effect of roughness geometry,the result of fractal roughness is also compared with that of square,sinusoidal,triangular roughness.The comparisons and analyses of velocity and resistance coefficient show that the roughness geometry is an important factor besides the relative roughness in studying the effects of surface roughness.These phenomena above different from macroscopic flows are of great importance in understanding the characteristics of micro gas flows.
近年来,格子Boltzmann方法(LBM)提供了一个有效的研究微流体系统的新工具。关于格子Boltzmann在微流体或稀薄气体上的研究已经取得了一些很好的成果。与分子动力学、直接Monte-Carlo模拟或其它同类型的方法相比,格子Boltzmann方法在计算效率上,在处理复杂边界问题上具有相当的优势。然而,尽管在小尺度中粗糙效应重要性增加,但由于缺乏有效的模拟工具,到目前为止对高Knudsen数的微气体流动中的粗糙效应的研究成果仍相对较少而且不能统一。为了研究氮气和氦气在微通道中的流动,本文设计了一个微通道模型,其高度为H=1.2μm,长度为L=5H或L=10H,对应氮气和氦气的Knudsen数分别为Kn_N_2=0.055和Kn_(He)=0.16。微通道粗糙表面的几何形状被分别设计成方形、Sin函数形、三角形和分形形状。利用这个模型,我们做了下面主要工作:
1)在微流体格子Boltzmann模型的状态方程方面,我们从气体运动论出发,重新推导松弛时间因子以及它和Knudsen数的关系。通过设置具有物理意义的网格时间步长,使得Knudsen数的表达式与网格精度无关。同时,通过与实验比较我们验证了这一修正的松弛因子跟原模型相比可以消除高Knudsen数的微气体的人工粘性,得到和实验相比更吻合的而且具有明确物理意义的结果。
2)在微流体格子Boltzmann模型的边界处理方面,我们考虑在微通道中由于摩擦或者流体杂质沉淀而形成的表面粗糙具有分形性质。所以我们在微流体格子Boltzmann方法中引进分形粗糙面并使用具有物理意义的参数如分维来控制粗糙面,我们认为它比其他学者使用的规则型,如矩形或三角形粗糙面更接近真实情况。此外,我们还计算并讨论了漫反射边界条件中粗糙表面的切向动量容许系数的影响,得到其在格子Boltzmann模型中对应的取值。
3)利用修正的微流体格子Boltzmann模型,我们模拟了微通道中的氮气和氦气的流动,捕捉到了最小Knudsen数现象。我们计算并分析了固壁表面粗糙效应和粗糙面形状对流体速度、压力、质量流量以及摩阻系数的影响,讨论了微通道中粗糙效应与稀薄效应与可压缩效应的耦合作用,得到一些有意义的结论。对氮气和氦气的模拟表明表面粗糙不但影响微通道中的速度分布,而且影响其压力分布。随着相对粗糙度的增加质量流量快速下降,摩阻系数快速增加。微通道中稀薄性效应、可压缩性效应和表面粗糙效应的作用强烈耦合,综合影响宏观物理量的变化趋势。在微通道中氮气和氦气的稀薄性、可压缩性和表面粗糙效应强烈耦合。在模拟分形粗糙度时,我们也分析了分形层次和网格精度的影响。除了相对粗糙度之外,所使用的粗糙几何形状对结果的影响也很大。这些不同于宏观流动的特性对了解微流气体流动有着重要的意义。
Surface roughness is identified as an important factor in Knudsen flows of liquid or gas.As the development of Micro-Electro-Mechanical Systems(MEMS) and micro fabrication,micro flows have been receiving more and more attention.Liquid flows in microchannels are most concentrated on the effects of surface tension and wall wettability,and their experiments and theoretical analyses have made considerable successes.However,the way to study the roughness effect of micro gas flows(also rarefied gas flows) is quite different from that of micro liquid flows.Since the result based on the Navier-Stokes equation(NSE) is highly dubitable due to the invalidation of the continuum hypothesis in micro gas flows,the molecular-based models including molecular dynamics(MD) simulations,Boltzmann equation(BE) simulations,direct simulations Monte-Carlo(DSMC) and information preservation (IP) simulations have to be employed for numerical studies.
Recently,the lattice Boltzmann method(LBM) provides a new effective way to study the micro flow systems.The researches indicate that the results by lattice Boltzmann method are consistent with the theoretical analyses and experiments. Compared with molecular dynamics simulations,direct simulations Monte-Carlo simulations and other related methods,the lattice Boltzmann method gains the advantages of computational efficiency and flexibility for complex boundary geometries.However,few studies up to now present a deep understanding of the effects of surface roughness on the micro gas flow systems at high Knudsen numbers. To understand the effect of various roughnesses and different geometries on the micro gas flow systems,we here design a lattice Boltzmann model and a microchannel with various roughness boundaries,and then a series of numerical simulations are taken. The surface roughness of the microchannel is modeled to be square,sinusoidal, triangular and fractal,respectively.By this model,the roughness effect combining with the effects of rarefaction and compressibility in different boundaries on the micro gas flows of nitrogen and helium are studied and analyzed.The main contents of this paper are as follows:
1) The relaxation time and its relationship with the Knudsen number are re-deduced from gas dynamics.The relaxation time is determined from the physical properties of the investigated gas and cannot be freetly adjusted.The expression of the Knudsen number is independent of the grid size by setting a certain relaxation time. Compared with theoretical and experimental studies the effectiveness of the lattice Boltzmann model is validated.It is shown that the modified relaxation time is able to eliminate the unrealistically high value of the bulk viscosity and is in good agreement with experiments.
2) The surface roughness is modeled by a self-affine fractal which is a reasonable tool to produce suitable surface of controlled roughness to describe the wall boundary of the microchannel.The height of the rough microchannel is H=1.2μm and the length is L=5H or L=10H.Correspondly,the Knudsen numbers of nitrogen and helium are Kn_N_2=0.055 and Kn_(He)=0.16.Importantly,the thoee boundary conditions are discussed and the influence of the tangential momentum accommodation coefficient(TMAC) on the slip velocity near the rough boundary is measured.
3) The effects of relative roughness and Knudsen number in the micro flows of nitrogen and helium are studied and analyzed.Numerical simulations show that the surface roughness influences not only velocity distribution but also pressure distribution.In addition,the relative mass flow rate decreases greatly and the relative resistance coefficient rises rapidly as the relative roughness increases.The effects of rarefaction,compressibility and roughness are strongly coupled.To measure the effect of roughness geometry,the result of fractal roughness is also compared with that of square,sinusoidal,triangular roughness.The comparisons and analyses of velocity and resistance coefficient show that the roughness geometry is an important factor besides the relative roughness in studying the effects of surface roughness.These phenomena above different from macroscopic flows are of great importance in understanding the characteristics of micro gas flows.
引文
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