带有自由表面流动的N-S方程的数值解法
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摘要
自由表面流动问题普遍存在于化工、冶金、航空航天、材料科学等领域中,其过程中自由表面的位置随着液体的运动而变化,而且自由界面的运动对系统的行为具有重要的影响,其已成为流体动力学领域非常关注的问题。
本文研究的主要课题是研究自由运动界面的变化情况和与其相关的不可压缩粘性流动的Navier-Stokes方程组的解法,对自由界面的流动采用VOF (volume of fluid)来跟踪自由界面的运动变化情况,其基本思想是在整个流场中定义一个满足对流方程的流体体积函数,空单元时值为0；满单元时值为1；单元为界面单元时值介于0到1之间。讨论了自由界面重构的的四种主要的方法,H-N法、积分平均型TVD格式,FCT法和’Youngs方法,通过四种方法的比较,详细说明了四种方法在处理自由表面流动问题时在精度,效率方面的好坏。并且给出了四种方法在平移场和剪切场中的数值模拟结果。
通过理论分析,本文推导出了二维不可压缩粘性流动的Navier-Stokes方程组,介绍了四种主要的求解N-S方程组的数值解法,包括投影法、人工压缩性方法、压力泊松方程法和压力修正算法。介绍了交错网格在处理速度与压力耦合过程的具体应用,有效地解决了速度和压力存在于同一套网格中出现的棋盘式不合理压力场的问题。并且给出了在交错网格下N-S方程组的具体离散形式和相应的边界条件的处理方法,推导出了在交错网格下的压力修正方程的具体表达形式。在求解可压缩粘性流动的Navier-Stokes方程组时时间项采用向前差分,对流项-扩散项采用中心差分格式。为了加快速度和压力的收敛,在具体的求解过程中对相应的压力和速度的修正采取亚松弛技术。在最后,为了验证压力修正算法的有效性,给出了二维驱动方腔流动的数值模拟结果。
The free surface flow problems are commonly found out in the fields of chemical, metallurgy, aviation and aerospace and material science. In this process, the position of the free surface changes along with the movement of liquid, moreover, the movement of the free surface has a great influence on the system.it has become a great concern in the field of fluid dynamics.
The task of this paper is to study the movement of the free surface associated flow of an incompressible viscous Navier-Stokes equations solution. For the free interface flow volume,we use the VOF (volume of fluid) to track the free interface motion changes,the basic idea of which is that a volume fluid function which satisfies convection function, is defined in whole flow flied. When the cell is empty, the function equals 0; when the cell is full, the function equals 1; when the cell contains interface, the function between 0 and 1. We compare with four kinds of methods to solve VOF function:Hirt & Nichols method, Integral mean type TVD scheme, FCT method and Youngs method. By comparison of four methods, detailed description of the four methods in dealing with free surface flow problems in accuracy, efficiency, good or bad. And gives numerical simulation resultsOf four ways to cut farm field and translation field.
Through theoretical analysis,this paper deduceds two-dimensional incompressible vis-cous flow navier-stokes equations, and introduces four major numerical solution, including projection method, artificial compressibility method, pressure poisson equation method and pressure correction algorithm, solving the navier-stokes equations. Staggered grid in-troduced in the processing speed and pressure coupling process of the specific application, can effectively solve the speed and pressure on the same appear in a grid Chessboard prob-lem of unreasonable pressure field.Besides, we give the discrete form of the navier-stokes equations and the form of pressure correction equation in the staggered grid. Using time difference forwards and convection-dispersion central difference format. To speed up the convergence velocity and pressure, in particular the process of solving the corresponding amendments to the pressure and velocity relaxation techniques. In the last, we give an example of a driver of party cavity flow simulation results to verify the effectiveness of the algorithm.
本文研究的主要课题是研究自由运动界面的变化情况和与其相关的不可压缩粘性流动的Navier-Stokes方程组的解法,对自由界面的流动采用VOF (volume of fluid)来跟踪自由界面的运动变化情况,其基本思想是在整个流场中定义一个满足对流方程的流体体积函数,空单元时值为0；满单元时值为1；单元为界面单元时值介于0到1之间。讨论了自由界面重构的的四种主要的方法,H-N法、积分平均型TVD格式,FCT法和’Youngs方法,通过四种方法的比较,详细说明了四种方法在处理自由表面流动问题时在精度,效率方面的好坏。并且给出了四种方法在平移场和剪切场中的数值模拟结果。
通过理论分析,本文推导出了二维不可压缩粘性流动的Navier-Stokes方程组,介绍了四种主要的求解N-S方程组的数值解法,包括投影法、人工压缩性方法、压力泊松方程法和压力修正算法。介绍了交错网格在处理速度与压力耦合过程的具体应用,有效地解决了速度和压力存在于同一套网格中出现的棋盘式不合理压力场的问题。并且给出了在交错网格下N-S方程组的具体离散形式和相应的边界条件的处理方法,推导出了在交错网格下的压力修正方程的具体表达形式。在求解可压缩粘性流动的Navier-Stokes方程组时时间项采用向前差分,对流项-扩散项采用中心差分格式。为了加快速度和压力的收敛,在具体的求解过程中对相应的压力和速度的修正采取亚松弛技术。在最后,为了验证压力修正算法的有效性,给出了二维驱动方腔流动的数值模拟结果。
The free surface flow problems are commonly found out in the fields of chemical, metallurgy, aviation and aerospace and material science. In this process, the position of the free surface changes along with the movement of liquid, moreover, the movement of the free surface has a great influence on the system.it has become a great concern in the field of fluid dynamics.
The task of this paper is to study the movement of the free surface associated flow of an incompressible viscous Navier-Stokes equations solution. For the free interface flow volume,we use the VOF (volume of fluid) to track the free interface motion changes,the basic idea of which is that a volume fluid function which satisfies convection function, is defined in whole flow flied. When the cell is empty, the function equals 0; when the cell is full, the function equals 1; when the cell contains interface, the function between 0 and 1. We compare with four kinds of methods to solve VOF function:Hirt & Nichols method, Integral mean type TVD scheme, FCT method and Youngs method. By comparison of four methods, detailed description of the four methods in dealing with free surface flow problems in accuracy, efficiency, good or bad. And gives numerical simulation resultsOf four ways to cut farm field and translation field.
Through theoretical analysis,this paper deduceds two-dimensional incompressible vis-cous flow navier-stokes equations, and introduces four major numerical solution, including projection method, artificial compressibility method, pressure poisson equation method and pressure correction algorithm, solving the navier-stokes equations. Staggered grid in-troduced in the processing speed and pressure coupling process of the specific application, can effectively solve the speed and pressure on the same appear in a grid Chessboard prob-lem of unreasonable pressure field.Besides, we give the discrete form of the navier-stokes equations and the form of pressure correction equation in the staggered grid. Using time difference forwards and convection-dispersion central difference format. To speed up the convergence velocity and pressure, in particular the process of solving the corresponding amendments to the pressure and velocity relaxation techniques. In the last, we give an example of a driver of party cavity flow simulation results to verify the effectiveness of the algorithm.
引文
[1]Ghia U.Ghia KN,Shin CT. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method[J], J Comput Phys,1982,48:387-411.
[2]Guj G,Stella F, Numerical solutions of high-Re recirculating flows in vorticity-velocity form.[J] Int J Numer Meth Fluids,1976,8:405-416.
[3]Patankar S V,Spalding D A. A calculation procedure for heat,mass and momentum tranfer in three-dimensional flows. Int J Heat at Mass Transfer Math.1972,15:1787-1806.
[4]Paolucci S,Chenweth DR., Transition to chaos in a differentially heated vertical cav-ity., J comput phys,1989,201:379-410.
[5]Abdullah S., Numerical solutions for the incompressible Navier-Stokes equations inprimitive variables using a non-staggered grid[J], J comput phys,1990,86:193-202.
[6]Schreiber R,Keller H B, Driven cavity Flows by Efficient Numerical Techniques J, comput phys,1983,49:310-333.
[7]Thompsoan M C,Ferziger J H, An adaptive Mutigrid Technique for the incompressible Navier-Stokes Equations[J], J comput phys,1989,82:94-121
[8]Gupta M M, High Accuracy Solution of Incompressible Navier-Stokes Equations, J comput phys,1991,93:343-359.
[9]Li M,Tang T,Fomberg B, A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids 1995,20:1137-1151.
[10]Sahin M,Owens G, A novel fully implicit finite volume method applied to the lid-driven cavity problem-part 1:High Renolds number flow calculations, Int,J Number,Meth,Fluids,2003:42:57-77.
[11]葛永斌.田振夫.吴文权基于时间的驱动方腔流的高精度多重网格方法数值模拟.工程热物理学报.2003,24(2)：216-219.
[12]葛永斌.高精度紧致差分格式的多重网格算法和平面驱动方腔问题的数值模拟.[硕士论文].银川：宁夏大学,2000.
[13]Erturk E,Corke T C,Gokcol C, Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds Number, Int J Number Meth Fluids 2005,48:747-774.
[14]何安定,鹿院卫,李斌等.两相流动中自由界面的数值模拟,E油气储运,2000,19(10)：15-18.
[15]陈大宏,李炜等.自由表面流动数值模拟方法的探讨.水动力学研究与进展,2001,16(2)：216-224.
[16]陶文铨. 数值传热学.西安交通大学出版社,1988.
[17]帕坦卡S V,传热与流动的数值计算.张政译.北京：科学出版社.1988.149
[18]Patankar S V Numerical heat transfer and fluid flow ; New York:McGraw-Hill,1980
[19]陶文铨.计算传热学的近代进展,北京：科学出版社.2000.6
[20]Anderson D A,Tannehill J C,Pletcher R H. Computational fluid mechanics and heat tranfer. Washington:Hemisphere.2nd ed. 1997
[21]孔祥谦.有限元法在传热学中的应用.北京：科学出版社.1986
[22]Gottlieb D,Orszang S A., Numerical analysis of spectral methods: theory and appli-cations. Philadephia: society for Industrial and Applied Mathematics,1978
[23]Harlow F H, WelchJ E, Shannon J P,et al. The MAC method, a computing technique for solving viscous, incompressible, transient fluid problems involving free surface[R], Report LA-3425, Los Alamos Scientific Labratory,1965.
[24]Harlow F H, WelchJ E., Numerical calculation of time-dependent viscous incompress-ible flow of fluid with free surface[J]. Physics of Fluids,1965,8:2182-2189.
[25]Hirt C W and Nichols B D. Volume of fluid(VOF) method for the dynamics of free boundaries[J], J. Comp. Physics, 1981,39:201-225.
[26]Toorey M D and Hirt C W., NASA-VOF2D: a computer program for incompressible flows with free surface[R], LA-10621-MS,1985.
[27]Young D L. Time-dependent multimaterial flow with fluid distortion[J]. Numerical Methods for Fluid Dymamics, 1982,4(4):337-360.
[28]Rudman M. Volume-tracking methods for interfacing flow calculations[J], Int J Numer Meth Fluids, 1998,24:671-691
[29]Grilli S T. Free surface nonlinear flows[A]. In 1999 McGraw-Hill Yearbook of Sciences and Technology (a yearly supplement to the Encyclopedia of Sciences and Technol-ogy)[C]. Parker S P(ed), McGraw-Hill,New York,1997.134-136.
[30]Rochafellar, R.T. and Uryasev, S., Optimization of conditional value-at-risk, J. Risk 2 (2000), 21-41.
[31]Liao S-J. High-order stream-vorticity formulation of 2D steady state Navier-Stokes equations. Int J Numer Methods Fluids,1992.15:595-612.
[32]Harlow F H, WelchJ E. Numerical calculation of time-dependent viscous incompress-ible flow of fluid with free surface[J]. Physics of Fluids,1965,8:2182-2188.
[33]Patanka S V, Spalding D H. A Calculation Procedure for two-dimentional elliptic situation[J], Numer.Heat Transfer, 1981,4(4):409-425.
[34]Da-Ren Chen,David Y H Pui. Numerical and experimental studies of particle deposi-tion in a tube with a conical contraction-Laminar flow regime[J]. Journal of Aerosol Science, June 1995,Vol.26(4):563-574.
[35]王彤,谷传纲,杨波,等.非定常流动计算的PISO算法[J].Ser.A,2003,18(2)：233-239.
[36]刘儒勋,刘晓平,张磊,等.运动界面的追踪和重构方法[J].应用数学和力学,2004(3)：279-290.
[37]刘儒勋,王志峰.,数值模拟方法和运动界面追踪[M].安徽：中国科学技术大学出版社,2001.
[38]Hitoshi Fujimoto,Yu Shiotani, Albert Y. Tong et al. Three-dimensional numerical analysis of the deformation behavior of droplets impinging onto a solid substrate, International Journal of Multiphase Flow,2007,33:317-332.
[39]George Strotos,Manolis Gavaises,Andreas Theodorakakos et al. Numerical investiga-tion of the cooling effectiveness of a droplet impinging on a heated surface. Interna-tional Journal of Heat and Mass Transfer.51(2008):4728-4742.
[40]Hitoshi Fujimoto,Yu Shiotani, Albert Y. Tong et al. Three-dimensional numerical analysis of the deformation behavior of droplets impinging onto a solid substrate. International Journal of Multiphase Flow,2007,33:317-332.
[41]N.Nikolopoulos,A.Theodorakakos,G.Bergeles.A numerical investigation of the evapo-ration process of a liquid droplet impinging onto a hot substrate.International Journal of Heat and Mass Transfer.50(2007):303-319.
[42]G.E.Cossali,M.Marengo,M.Santini. Thermally induced secondary drop atomisation by single drop impact onto heated surface. International Journal of Heat and Mass Transfer.29(2008):167-177.
[43]RUMDAN M. Volume-tracking Methods for Interfacial Flow Calcula-tions[J].International Journal for Numerical Methods in Fluids,1997,24:671-691.
[44]BORIS J P,BOOK D L.Flux-corrected transport I. SHASTA, A Fluid transport Al-gorithm That Works[J]. J Comput Phys,1973(11):38-57.
[2]Guj G,Stella F, Numerical solutions of high-Re recirculating flows in vorticity-velocity form.[J] Int J Numer Meth Fluids,1976,8:405-416.
[3]Patankar S V,Spalding D A. A calculation procedure for heat,mass and momentum tranfer in three-dimensional flows. Int J Heat at Mass Transfer Math.1972,15:1787-1806.
[4]Paolucci S,Chenweth DR., Transition to chaos in a differentially heated vertical cav-ity., J comput phys,1989,201:379-410.
[5]Abdullah S., Numerical solutions for the incompressible Navier-Stokes equations inprimitive variables using a non-staggered grid[J], J comput phys,1990,86:193-202.
[6]Schreiber R,Keller H B, Driven cavity Flows by Efficient Numerical Techniques J, comput phys,1983,49:310-333.
[7]Thompsoan M C,Ferziger J H, An adaptive Mutigrid Technique for the incompressible Navier-Stokes Equations[J], J comput phys,1989,82:94-121
[8]Gupta M M, High Accuracy Solution of Incompressible Navier-Stokes Equations, J comput phys,1991,93:343-359.
[9]Li M,Tang T,Fomberg B, A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations, International Journal for Numerical Methods in Fluids 1995,20:1137-1151.
[10]Sahin M,Owens G, A novel fully implicit finite volume method applied to the lid-driven cavity problem-part 1:High Renolds number flow calculations, Int,J Number,Meth,Fluids,2003:42:57-77.
[11]葛永斌.田振夫.吴文权基于时间的驱动方腔流的高精度多重网格方法数值模拟.工程热物理学报.2003,24(2)：216-219.
[12]葛永斌.高精度紧致差分格式的多重网格算法和平面驱动方腔问题的数值模拟.[硕士论文].银川：宁夏大学,2000.
[13]Erturk E,Corke T C,Gokcol C, Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds Number, Int J Number Meth Fluids 2005,48:747-774.
[14]何安定,鹿院卫,李斌等.两相流动中自由界面的数值模拟,E油气储运,2000,19(10)：15-18.
[15]陈大宏,李炜等.自由表面流动数值模拟方法的探讨.水动力学研究与进展,2001,16(2)：216-224.
[16]陶文铨. 数值传热学.西安交通大学出版社,1988.
[17]帕坦卡S V,传热与流动的数值计算.张政译.北京：科学出版社.1988.149
[18]Patankar S V Numerical heat transfer and fluid flow ; New York:McGraw-Hill,1980
[19]陶文铨.计算传热学的近代进展,北京：科学出版社.2000.6
[20]Anderson D A,Tannehill J C,Pletcher R H. Computational fluid mechanics and heat tranfer. Washington:Hemisphere.2nd ed. 1997
[21]孔祥谦.有限元法在传热学中的应用.北京：科学出版社.1986
[22]Gottlieb D,Orszang S A., Numerical analysis of spectral methods: theory and appli-cations. Philadephia: society for Industrial and Applied Mathematics,1978
[23]Harlow F H, WelchJ E, Shannon J P,et al. The MAC method, a computing technique for solving viscous, incompressible, transient fluid problems involving free surface[R], Report LA-3425, Los Alamos Scientific Labratory,1965.
[24]Harlow F H, WelchJ E., Numerical calculation of time-dependent viscous incompress-ible flow of fluid with free surface[J]. Physics of Fluids,1965,8:2182-2189.
[25]Hirt C W and Nichols B D. Volume of fluid(VOF) method for the dynamics of free boundaries[J], J. Comp. Physics, 1981,39:201-225.
[26]Toorey M D and Hirt C W., NASA-VOF2D: a computer program for incompressible flows with free surface[R], LA-10621-MS,1985.
[27]Young D L. Time-dependent multimaterial flow with fluid distortion[J]. Numerical Methods for Fluid Dymamics, 1982,4(4):337-360.
[28]Rudman M. Volume-tracking methods for interfacing flow calculations[J], Int J Numer Meth Fluids, 1998,24:671-691
[29]Grilli S T. Free surface nonlinear flows[A]. In 1999 McGraw-Hill Yearbook of Sciences and Technology (a yearly supplement to the Encyclopedia of Sciences and Technol-ogy)[C]. Parker S P(ed), McGraw-Hill,New York,1997.134-136.
[30]Rochafellar, R.T. and Uryasev, S., Optimization of conditional value-at-risk, J. Risk 2 (2000), 21-41.
[31]Liao S-J. High-order stream-vorticity formulation of 2D steady state Navier-Stokes equations. Int J Numer Methods Fluids,1992.15:595-612.
[32]Harlow F H, WelchJ E. Numerical calculation of time-dependent viscous incompress-ible flow of fluid with free surface[J]. Physics of Fluids,1965,8:2182-2188.
[33]Patanka S V, Spalding D H. A Calculation Procedure for two-dimentional elliptic situation[J], Numer.Heat Transfer, 1981,4(4):409-425.
[34]Da-Ren Chen,David Y H Pui. Numerical and experimental studies of particle deposi-tion in a tube with a conical contraction-Laminar flow regime[J]. Journal of Aerosol Science, June 1995,Vol.26(4):563-574.
[35]王彤,谷传纲,杨波,等.非定常流动计算的PISO算法[J].Ser.A,2003,18(2)：233-239.
[36]刘儒勋,刘晓平,张磊,等.运动界面的追踪和重构方法[J].应用数学和力学,2004(3)：279-290.
[37]刘儒勋,王志峰.,数值模拟方法和运动界面追踪[M].安徽：中国科学技术大学出版社,2001.
[38]Hitoshi Fujimoto,Yu Shiotani, Albert Y. Tong et al. Three-dimensional numerical analysis of the deformation behavior of droplets impinging onto a solid substrate, International Journal of Multiphase Flow,2007,33:317-332.
[39]George Strotos,Manolis Gavaises,Andreas Theodorakakos et al. Numerical investiga-tion of the cooling effectiveness of a droplet impinging on a heated surface. Interna-tional Journal of Heat and Mass Transfer.51(2008):4728-4742.
[40]Hitoshi Fujimoto,Yu Shiotani, Albert Y. Tong et al. Three-dimensional numerical analysis of the deformation behavior of droplets impinging onto a solid substrate. International Journal of Multiphase Flow,2007,33:317-332.
[41]N.Nikolopoulos,A.Theodorakakos,G.Bergeles.A numerical investigation of the evapo-ration process of a liquid droplet impinging onto a hot substrate.International Journal of Heat and Mass Transfer.50(2007):303-319.
[42]G.E.Cossali,M.Marengo,M.Santini. Thermally induced secondary drop atomisation by single drop impact onto heated surface. International Journal of Heat and Mass Transfer.29(2008):167-177.
[43]RUMDAN M. Volume-tracking Methods for Interfacial Flow Calcula-tions[J].International Journal for Numerical Methods in Fluids,1997,24:671-691.
[44]BORIS J P,BOOK D L.Flux-corrected transport I. SHASTA, A Fluid transport Al-gorithm That Works[J]. J Comput Phys,1973(11):38-57.
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