多孔介质孔隙尺度下流体流动与传热SPH模拟
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摘要
在自然界和很多的工业应用中,都存在流体在多孔介质内流动的问题。在石油开采领域,更是会经常遇到此类问题,对于我国的石油开采工业而言,迫切需要进一步提高石油的采收率(Enhanced Oil Recovery, EOR),这就需要详细研究多孔介质中流体流动的传质传热规律。在目前的研究中,主要涉及两种尺度,孔隙尺度(Pore Scale)和表征体元尺度(Representative Elementary Volume, REV),在实际工程中,常使用Darcy定律描述流体在多孔介质内的流动,渗透率k是其中的一个很重要的参数,然而在使用该定律时,流体和多孔介质复杂的交互作用只能体现在一个宏观的统计参数k之中,其中包含了流体和多孔介质的许多参数,而要确定这些参数,又必须建立在对孔隙尺度下流体流动的合理描述及流体性质的准确预测的基础之上。因而在孔隙尺度下详细研究流体在多孔介质内的流动与传热特性,建立物理背景清晰,数学基础牢靠的REV模型,无疑具有重大的工程意义。
因此,本文从孔隙尺度下流体在多孔介质内流动的研究出发,分析流体在多孔介质中的传质传热的规律,研究渗透率的微观机理,寻求提高采油效率的方法。基于孔隙尺度下的多孔介质模型,使用光滑粒子动力学(SPH)数值模拟方法研究多孔介质中不可压缩流体流动现象,分析流体在孔隙尺度下的流动特性。首先通过模拟经典的Poiseuille流和Couette流验证了方法的正确性。随后,针对流体在分别由圆柱和方柱为骨架构造规则的多孔介质内的流动情况,同时使用SPH和FEM方法进行模拟,得到的结果十分吻合。利用SPH方法模拟了流体在随机方法以同尺度的圆柱为基元构造的一种人工随机多孔介质模型和基于贝雷岩心获得的多孔介质模型,模拟结果表明平均流体速度和体积力具有良好的线性关系。然后将SPH方法拓展到了两相模拟的领域,验证了拉普拉斯定律。最后利用SPH方法研究了导热问题,从多个算例中验证了SPH方法的精确性,然后用该方法研究了多孔介质中的纯导热问题,得到了精度很高的结果。
In nature and many industrial applications, involve flow in porous media, such as oil exploration etc. For the industry of oil exploration in our country, Enhanced Oil Recovery (EOR) has great potency. In the present study about the fluid flows in porous media, two scales are mainly related, Pore Scale and Representative Elementary Volume Scale. In practice, modeling of fluid flow in porous media is generally conducted by using the Darcy's law. The Darcy's law is a macroscopic phenomenological model, which is largely based on experimental observations. In this law, the complex interactions between fluids and microscopic porous structures are all lumped in a macroscopic physical quantity—the permeability tensor, k, which contains many parameters of the porous medium and fluid. To determine these parameters, it is necessary to have a reasonable description and an accurate prediction of fluid flows in porous media at pore scale. Therefore, more research for the property of the fluid flow in porous media at pore scale, to establish a REV model with clear physical background and solid foundation of mathematics is undoubtedly has great significance.
This paper has studied the mechanism of the permeability, numerically simulated mass and heat transfer of the flows in porous media with the hope of helping to seek the most reasonable methods of oil recovery. In this paper, based on the porous media model, smoothed particle hydrodynamics (SPH) is employed to model the incompressible fluid flow in porous media and to study the flow characteristic at pore scale. First, the classic Poiseuille flow and Couette flow are investigated to validate of the SPH method. And then, the fluid flow in the regular porous media constructed by square and circle cylinders respectively, are simulated by both FEM and SPH. The simulation results yield to a good agreement. At last, the fluid flow in a artificial randomization porous media model constructed by the circular grains of same size, and in the Berea sandstone are simulated at the same time and a linear relationship between the average flow velocity and the body force is found.
And then SPH method is developed for two phase flow. The Laplace's law is verified. Finally, the SPH method is used to study the heat conduction problems. The accuracy of SPH method is proved by series examples. The thermal conductivity of porous media is studied, the results is good.
因此,本文从孔隙尺度下流体在多孔介质内流动的研究出发,分析流体在多孔介质中的传质传热的规律,研究渗透率的微观机理,寻求提高采油效率的方法。基于孔隙尺度下的多孔介质模型,使用光滑粒子动力学(SPH)数值模拟方法研究多孔介质中不可压缩流体流动现象,分析流体在孔隙尺度下的流动特性。首先通过模拟经典的Poiseuille流和Couette流验证了方法的正确性。随后,针对流体在分别由圆柱和方柱为骨架构造规则的多孔介质内的流动情况,同时使用SPH和FEM方法进行模拟,得到的结果十分吻合。利用SPH方法模拟了流体在随机方法以同尺度的圆柱为基元构造的一种人工随机多孔介质模型和基于贝雷岩心获得的多孔介质模型,模拟结果表明平均流体速度和体积力具有良好的线性关系。然后将SPH方法拓展到了两相模拟的领域,验证了拉普拉斯定律。最后利用SPH方法研究了导热问题,从多个算例中验证了SPH方法的精确性,然后用该方法研究了多孔介质中的纯导热问题,得到了精度很高的结果。
In nature and many industrial applications, involve flow in porous media, such as oil exploration etc. For the industry of oil exploration in our country, Enhanced Oil Recovery (EOR) has great potency. In the present study about the fluid flows in porous media, two scales are mainly related, Pore Scale and Representative Elementary Volume Scale. In practice, modeling of fluid flow in porous media is generally conducted by using the Darcy's law. The Darcy's law is a macroscopic phenomenological model, which is largely based on experimental observations. In this law, the complex interactions between fluids and microscopic porous structures are all lumped in a macroscopic physical quantity—the permeability tensor, k, which contains many parameters of the porous medium and fluid. To determine these parameters, it is necessary to have a reasonable description and an accurate prediction of fluid flows in porous media at pore scale. Therefore, more research for the property of the fluid flow in porous media at pore scale, to establish a REV model with clear physical background and solid foundation of mathematics is undoubtedly has great significance.
This paper has studied the mechanism of the permeability, numerically simulated mass and heat transfer of the flows in porous media with the hope of helping to seek the most reasonable methods of oil recovery. In this paper, based on the porous media model, smoothed particle hydrodynamics (SPH) is employed to model the incompressible fluid flow in porous media and to study the flow characteristic at pore scale. First, the classic Poiseuille flow and Couette flow are investigated to validate of the SPH method. And then, the fluid flow in the regular porous media constructed by square and circle cylinders respectively, are simulated by both FEM and SPH. The simulation results yield to a good agreement. At last, the fluid flow in a artificial randomization porous media model constructed by the circular grains of same size, and in the Berea sandstone are simulated at the same time and a linear relationship between the average flow velocity and the body force is found.
And then SPH method is developed for two phase flow. The Laplace's law is verified. Finally, the SPH method is used to study the heat conduction problems. The accuracy of SPH method is proved by series examples. The thermal conductivity of porous media is studied, the results is good.
引文
[1]Zhou G, Chen Z, Ge W, et al. SPH simulation of oil displacement in cavity-fracture structures [J]. Chemical Engineering Science,2010,65(11):3363-3371.
[2]董波.非混相驱替过程的格子Boltzmann模拟[D].大连：大连理工大学,2011.
[3]鲁建华.基于格子Boltzmann方法的多孔介质内流动与传热的微观模拟[D].武汉：华中科技大学2009.
[4]Jiang F, Oliveira M S A, Sousa A C M. Mesoscale SPH modeling of fluid flow in isotropic porous media [J]. Computer Physics Communications,2007,176(7):471-480.
[5]苏铭德,黄素逸.计算流体力学基础[M].北京：清华大学出版社,1997.
[6]刘儒勋.数值模拟方法和运动界面追踪[M].合肥：中国科学技术大学出版社,2001.
[7]何雅玲,王勇,李庆.格子Boltzmann方法的理论及应用[M].北京：科学出版社,2009.
[8]Sukop M C, Thorne D T. Lattice Boltzmann Modeling:An Introduction For Geoscientists And Engineers [M]. New York:Sringer, Heidelberg, Berlin,2006:172pp.
[9]郑兴.光滑质点流体动力学(SPH)算法研究[D].哈尔滨：哈尔滨工程大学,2005.
[10]王永鹏.光滑粒子流体动力学算法研究及其在流体力学中的数值模拟应用[D].天津：天津大学2009.
[11]胡晓燕.计算流体力学中的SPH方法和MLSPH方法研究[D].北京：中国工程物理研究院,2006.
[12]Lucy L B. A numerical approach to the testing of the fission hypothesis [J]. The Astronomical Journal,1977,82:1013-1024.
[13]Monaghan J J. Smoothed particle hydrodynamics [J]. Reports on Progress in Physics, 2005,68(8):1703.
[14]Monaghan J J. An introduction to SPH [J]. Computer Physics Communications,1988, 48(1):89-96.
[15]Monaghan J J. Why Particle Methods Work [J]. SIAM Journal on Scientific and Statistical Computing,1982,3(4):422-433.
[16]Swegle J W, Attaway S W, Heinstein M W, et al. An analysis of smoothed particle hydrodynamics. PBD:Mar 1994,1994:142 p.
[17]Rasio F A, Lombardi Jr J C. Smoothed particle hydrodynamics calculations of stellar interactions [J]. Journal of Computational and Applied Mathematics,1999, 109(1-2):213-230.
[18]J. H, A. P. Hierarchical, Dissipative Formation of Elliptical Galaxies:Is Thermal Instability the Key Mechanism?Hydrodynamic Simulations Including Supernova Feedback Multi-Phase Gas and Metal Enrichment in cdm:structure and dynamics of elliptical galaxies [M]. Les Ulis, France:EDP Sciences,1999:769-798pp.
[19]Berczik P, Kolesnik I G. Gasodynamical model of the triaxial protogalaxy collapse [J]. Astronomical & Astrophysical Transactions,1998,16 (3):163-185.
[20]Berczik P. Modeling the Star Formation in Galaxies Using the Chemo-DynamicalSPH Code [J]. Astrophysics and Space Science,2000,271 (2):103-126.
[21]Monaghan J J. Modelling the universe [J]. Proceedings of the Astronomical Society of Australia,1990,8:233-237.
[22]Potapov A V, Hunt M L, Campbell C S. Liquid-solid flows using smoothed particle hydrodynamics and the discrete element method [J]. Powder Technology,2001, 116(2-3):204-213.
[23]Morris J P. Analysis of Smoothed Particle Hydrodynamics with Applications:{M}onash {U}niversity,1996.
[24]Monaghan J J, Kocharyan A. SPH simulation of multi-phase flow [J]. Computer Physics Communications,1995,87(1-2):225-235.
[25]Monaghan J J. Simulating Free Surface Flows with SPH [J]. Journal of Computational Physics,1994,110(2):399-406.
[26]Monaghan J J. Gravity currents and solitary waves [J]. Physica D:Nonlinear Phenomena, 1996,98(2-4):523-533.
[27]Zhu Y, Fox P J. Smoothed Particle Hydrodynamics Model for Diffusion through Porous Media [J]. Transport in Porous Media,2001,43(3):441-471.
[28]Shao S. Incompressible SPH flow model for wave interactions with porous media [J]. Coastal Engineering,2010,57 (3):304-316.
[29]Ovaysi S, Piri M. Direct pore-level modeling of incompressible fluid flow in porous media [J]. Journal of Computational Physics,2010,229(19):7456-7476.
[30]Chen J K, Beraun J E, Carney T C. A corrective smoothed particle method for boundary value problems in heat conduction [J]. International Journal for Numerical Methods in Engineering,1999,46 (2):231-252.
[31]Cleary P W. Modelling confined multi-material heat and mass flows using SPH [J]. Applied Mathematical Modelling,1998,22(12):981-993.
[32]Xu R, Stansby P, Laurence D. Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach [J]. Journal of Computational Physics,2009, 228(18):6703-6725.
[33]Monaghan J J, Gingold R A. Shock simulation by the particle method SPH [J]. Journal of Computational Physics,1983,52 (2):374-389.
[34]Swegle J W, Hicks D L, Attaway S W. Smoothed Particle Hydrodynamics Stability Analysis [J]. Journal of Computational Physics,1995,116(1):123-134.
[35]Monaghan J J. On the problem of penetration in particle methods [J]. Journal of Computational Physics,1989,82(1):1-15.
[36]Dyka C T, Ingel R P. Addressing Tension Instability in SPH Methods.1998.
[37]Johnson G R, Beissel S R. Normalized smoothing functions for SPH impact computations [J]. International Journal for Numerical Methods in Engineering,1996, 39(16):2725-2741.
[38]Liu W K, Jun S, Zhang Y F. Reproducing kernel particle methods [J]. International Journal for Numerical Methods in Fluids,1995,20(8-9):1081-1106.
[39]Chen J K, Beraun J E, Jih C J. Completeness of corrective smoothed particle method for linear elastodynamics [J]. Computational Mechanics,1999,24(4):273-285.
[40]Chen J K, Beraun J E, Jih C J. An improvement for tensile instability in smoothed particle hydrodynamics [J]. Computational Mechanics,1999,23(4):279-287.
[41]Vignjevic R, Campbell J, Libersky L. A treatment of zero-energy modes in the smoothed particle hydrodynamics method [J]. Computer Methods in Applied Mechanics and Engineering,2000,184(1):67-85.
[42]Dilts G A. Moving-least-squares-particle hydrodynamics—Ⅰ. Consistency and stability [J]. International Journal for Numerical Methods in Engineering,1999, 44(8):1115-1155.
[43]Gomez-Gesteira M, Dalrymple R A. Using a Three-Dimensional Smoothed Particle Hydrodynamics Method for Wave Impact on a Tall Structure [J]. Journal of Waterway, Port, Coastal, and Ocean Engineering,2004,130(2):63-69.
[44]Randles P W, Libersky L D. Smoothed Particle Hydrodynamics:Some recent improvements and applications [J]. Computer Methods in Applied Mechanics and Engineering,1996, 139(1-4):375-408.
[45]Y. M. Lo E, Shao S. Simulation of near-shore solitary wave mechanics by an incompressible SPH method [J]. Applied Ocean Research,2002,24(5):275-286.
[46]陈海舟.不可压自由表面流的SPH法数值模拟研究[D].天津：天津大学,2009.
[47]Liu M B, Liu G R. Smoothed particle hydrodynamics (SPH):An overview and recent developments [J]. Archives of Computational Methods in Engineering,2010, 17 (Compendex):25-76.
[48]Liu G R, Liu M B. Smoothed particle hydrodynamics--a meshfree particle method [M]. Singapore:World Scientific 2004.
[49]Liu M B, Liu G R. Meshfree particle simulation of micro channel flows with surface tension [J]. Computational Mechanics,2005,35(5):332-341.
[50]Morris J P, Fox P J, Zhu Y. Modeling Low Reynolds Number Incompressible Flows Using SPH [J]. Journal of Computational Physics,1997,136(1):214-226.
[51]Zhu Y, Fox P J. Simulation of Pore-Scale Dispersion in Periodic Porous Media Using Smoothed Particle Hydrodynamics [J]. Journal of Computational Physics,2002, 182(2):622-645.
[52]龚凯.基于光滑质点水动力学(SPH)方法的自由表面流动数值模拟研究[D].上海：上海交通大学,2009.
[53]Monaghan J J. SPH without a Tensile Instability [J]. Journal of Computational Physics, 2000,159(2):290-311.
[54]Sigalotti L D G, Klapp J, Sira E, et al. SPH simulations of time-dependent Poiseuille flow at low Reynolds numbers [J]. Journal of Computational Physics,2003, 191(2):622-638.
[55]赵越超.多孔介质中CO_2与油(水)两相渗流的MRI研究[D].大连：大连理工大学,2011.
[56]张一夫.楔形流道内气泡动力学行为数值模拟及实验研究[D].大连：大连理工大学,2010.
[57]Brackbill J U, Kothe D B, Zemach C. A continuum method for modeling surface tension [J]. Journal of Computational Physics,1992,100:335-354.
[2]董波.非混相驱替过程的格子Boltzmann模拟[D].大连：大连理工大学,2011.
[3]鲁建华.基于格子Boltzmann方法的多孔介质内流动与传热的微观模拟[D].武汉：华中科技大学2009.
[4]Jiang F, Oliveira M S A, Sousa A C M. Mesoscale SPH modeling of fluid flow in isotropic porous media [J]. Computer Physics Communications,2007,176(7):471-480.
[5]苏铭德,黄素逸.计算流体力学基础[M].北京：清华大学出版社,1997.
[6]刘儒勋.数值模拟方法和运动界面追踪[M].合肥：中国科学技术大学出版社,2001.
[7]何雅玲,王勇,李庆.格子Boltzmann方法的理论及应用[M].北京：科学出版社,2009.
[8]Sukop M C, Thorne D T. Lattice Boltzmann Modeling:An Introduction For Geoscientists And Engineers [M]. New York:Sringer, Heidelberg, Berlin,2006:172pp.
[9]郑兴.光滑质点流体动力学(SPH)算法研究[D].哈尔滨：哈尔滨工程大学,2005.
[10]王永鹏.光滑粒子流体动力学算法研究及其在流体力学中的数值模拟应用[D].天津：天津大学2009.
[11]胡晓燕.计算流体力学中的SPH方法和MLSPH方法研究[D].北京：中国工程物理研究院,2006.
[12]Lucy L B. A numerical approach to the testing of the fission hypothesis [J]. The Astronomical Journal,1977,82:1013-1024.
[13]Monaghan J J. Smoothed particle hydrodynamics [J]. Reports on Progress in Physics, 2005,68(8):1703.
[14]Monaghan J J. An introduction to SPH [J]. Computer Physics Communications,1988, 48(1):89-96.
[15]Monaghan J J. Why Particle Methods Work [J]. SIAM Journal on Scientific and Statistical Computing,1982,3(4):422-433.
[16]Swegle J W, Attaway S W, Heinstein M W, et al. An analysis of smoothed particle hydrodynamics. PBD:Mar 1994,1994:142 p.
[17]Rasio F A, Lombardi Jr J C. Smoothed particle hydrodynamics calculations of stellar interactions [J]. Journal of Computational and Applied Mathematics,1999, 109(1-2):213-230.
[18]J. H, A. P. Hierarchical, Dissipative Formation of Elliptical Galaxies:Is Thermal Instability the Key Mechanism?Hydrodynamic Simulations Including Supernova Feedback Multi-Phase Gas and Metal Enrichment in cdm:structure and dynamics of elliptical galaxies [M]. Les Ulis, France:EDP Sciences,1999:769-798pp.
[19]Berczik P, Kolesnik I G. Gasodynamical model of the triaxial protogalaxy collapse [J]. Astronomical & Astrophysical Transactions,1998,16 (3):163-185.
[20]Berczik P. Modeling the Star Formation in Galaxies Using the Chemo-DynamicalSPH Code [J]. Astrophysics and Space Science,2000,271 (2):103-126.
[21]Monaghan J J. Modelling the universe [J]. Proceedings of the Astronomical Society of Australia,1990,8:233-237.
[22]Potapov A V, Hunt M L, Campbell C S. Liquid-solid flows using smoothed particle hydrodynamics and the discrete element method [J]. Powder Technology,2001, 116(2-3):204-213.
[23]Morris J P. Analysis of Smoothed Particle Hydrodynamics with Applications:{M}onash {U}niversity,1996.
[24]Monaghan J J, Kocharyan A. SPH simulation of multi-phase flow [J]. Computer Physics Communications,1995,87(1-2):225-235.
[25]Monaghan J J. Simulating Free Surface Flows with SPH [J]. Journal of Computational Physics,1994,110(2):399-406.
[26]Monaghan J J. Gravity currents and solitary waves [J]. Physica D:Nonlinear Phenomena, 1996,98(2-4):523-533.
[27]Zhu Y, Fox P J. Smoothed Particle Hydrodynamics Model for Diffusion through Porous Media [J]. Transport in Porous Media,2001,43(3):441-471.
[28]Shao S. Incompressible SPH flow model for wave interactions with porous media [J]. Coastal Engineering,2010,57 (3):304-316.
[29]Ovaysi S, Piri M. Direct pore-level modeling of incompressible fluid flow in porous media [J]. Journal of Computational Physics,2010,229(19):7456-7476.
[30]Chen J K, Beraun J E, Carney T C. A corrective smoothed particle method for boundary value problems in heat conduction [J]. International Journal for Numerical Methods in Engineering,1999,46 (2):231-252.
[31]Cleary P W. Modelling confined multi-material heat and mass flows using SPH [J]. Applied Mathematical Modelling,1998,22(12):981-993.
[32]Xu R, Stansby P, Laurence D. Accuracy and stability in incompressible SPH (ISPH) based on the projection method and a new approach [J]. Journal of Computational Physics,2009, 228(18):6703-6725.
[33]Monaghan J J, Gingold R A. Shock simulation by the particle method SPH [J]. Journal of Computational Physics,1983,52 (2):374-389.
[34]Swegle J W, Hicks D L, Attaway S W. Smoothed Particle Hydrodynamics Stability Analysis [J]. Journal of Computational Physics,1995,116(1):123-134.
[35]Monaghan J J. On the problem of penetration in particle methods [J]. Journal of Computational Physics,1989,82(1):1-15.
[36]Dyka C T, Ingel R P. Addressing Tension Instability in SPH Methods.1998.
[37]Johnson G R, Beissel S R. Normalized smoothing functions for SPH impact computations [J]. International Journal for Numerical Methods in Engineering,1996, 39(16):2725-2741.
[38]Liu W K, Jun S, Zhang Y F. Reproducing kernel particle methods [J]. International Journal for Numerical Methods in Fluids,1995,20(8-9):1081-1106.
[39]Chen J K, Beraun J E, Jih C J. Completeness of corrective smoothed particle method for linear elastodynamics [J]. Computational Mechanics,1999,24(4):273-285.
[40]Chen J K, Beraun J E, Jih C J. An improvement for tensile instability in smoothed particle hydrodynamics [J]. Computational Mechanics,1999,23(4):279-287.
[41]Vignjevic R, Campbell J, Libersky L. A treatment of zero-energy modes in the smoothed particle hydrodynamics method [J]. Computer Methods in Applied Mechanics and Engineering,2000,184(1):67-85.
[42]Dilts G A. Moving-least-squares-particle hydrodynamics—Ⅰ. Consistency and stability [J]. International Journal for Numerical Methods in Engineering,1999, 44(8):1115-1155.
[43]Gomez-Gesteira M, Dalrymple R A. Using a Three-Dimensional Smoothed Particle Hydrodynamics Method for Wave Impact on a Tall Structure [J]. Journal of Waterway, Port, Coastal, and Ocean Engineering,2004,130(2):63-69.
[44]Randles P W, Libersky L D. Smoothed Particle Hydrodynamics:Some recent improvements and applications [J]. Computer Methods in Applied Mechanics and Engineering,1996, 139(1-4):375-408.
[45]Y. M. Lo E, Shao S. Simulation of near-shore solitary wave mechanics by an incompressible SPH method [J]. Applied Ocean Research,2002,24(5):275-286.
[46]陈海舟.不可压自由表面流的SPH法数值模拟研究[D].天津：天津大学,2009.
[47]Liu M B, Liu G R. Smoothed particle hydrodynamics (SPH):An overview and recent developments [J]. Archives of Computational Methods in Engineering,2010, 17 (Compendex):25-76.
[48]Liu G R, Liu M B. Smoothed particle hydrodynamics--a meshfree particle method [M]. Singapore:World Scientific 2004.
[49]Liu M B, Liu G R. Meshfree particle simulation of micro channel flows with surface tension [J]. Computational Mechanics,2005,35(5):332-341.
[50]Morris J P, Fox P J, Zhu Y. Modeling Low Reynolds Number Incompressible Flows Using SPH [J]. Journal of Computational Physics,1997,136(1):214-226.
[51]Zhu Y, Fox P J. Simulation of Pore-Scale Dispersion in Periodic Porous Media Using Smoothed Particle Hydrodynamics [J]. Journal of Computational Physics,2002, 182(2):622-645.
[52]龚凯.基于光滑质点水动力学(SPH)方法的自由表面流动数值模拟研究[D].上海：上海交通大学,2009.
[53]Monaghan J J. SPH without a Tensile Instability [J]. Journal of Computational Physics, 2000,159(2):290-311.
[54]Sigalotti L D G, Klapp J, Sira E, et al. SPH simulations of time-dependent Poiseuille flow at low Reynolds numbers [J]. Journal of Computational Physics,2003, 191(2):622-638.
[55]赵越超.多孔介质中CO_2与油(水)两相渗流的MRI研究[D].大连：大连理工大学,2011.
[56]张一夫.楔形流道内气泡动力学行为数值模拟及实验研究[D].大连：大连理工大学,2010.
[57]Brackbill J U, Kothe D B, Zemach C. A continuum method for modeling surface tension [J]. Journal of Computational Physics,1992,100:335-354.