具有固壁边界的RT不稳定性问题及其数值模拟
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摘要
Rayleigh-Taylor不稳定现象是ICF(InertialConfinementFusion)应用中一个很基础但很重要的问题。它在天体物理、海洋混合层、地质、爆炸等众多领域中都有着重要的作用和意义。近些年来,国内外的许多学者对Rayleigh-Taylor不稳定性现象做了一系列的研究和数值模拟,采用多种数值方法,并取得了许多重要的结果。
本文在一些近期文献的基础上对Rayleigh-Taylor不稳定现象做了进一步研究。文中利用一组控制方程组体系描述流场的运动状态,并且考虑固壁边界条件下整个流场形态的变化,发现在壁面附近的界面流体粘滞于初始位置,导致整个界面形状发生很大弯曲;同时,分别利用5rd-WENO,3rd-WENO,3rd-MCeno数值方法对Rayleigh-Taylor不稳定现象数值模拟,通过对数值结果的比较发现修正系数格式具有很大的计算优势。
第一章介绍计算流体力学的发展状况,对本文主要研究工作的背景及其发展进行简要的介绍。
第二章以描述高雷诺数流动为目的,将流场分为三个区域。通过对Navier-Stokes(NS)方程组的简化,得到一组描述流场的控制方程组体系。
第三章描述Rayleigh-Taylor不稳定现象的数值方法很多,本文对5rd-WENO,3rd-WENO,3rd-MCeno这三种数值格式进行介绍,最后通过数值例子来说明3rd-MCeno方法的有效性和优越性。
第四章在已有理论的基础上,首先利用高阶精度的5rd-WENO格式数值模拟滑移边界条件下的RT不稳定现象,让其与固壁边界条件下的数值结果比较,从而发现流场形态的巨大变化;其后通过利用3rd-MCeno和高阶数值格式5rd-WENO方法分别模拟此现象,通过比较图像结果发现3rd-MCeno格式的优势,其捕捉界面的能力接近于高阶精度格式,并且节约了相当多的CPU时间。
The problem of Rayleigh-Taylor instability plays a crucial role in ICF(InertialConfinementFusion). In many fields, such as astrophysics, ocean mixed layer,geology, explosion, this problem has important application and significance. Inrecent years, many scholars adopt various computational methods to simulatethis phenomenon, and receive series of numerical results.
In this paper, based on some recent methods, we present further research us-ing a system of control equations to describe the state of ?ow field; by consideringrigid boundary condition, we also find the interface nearby the well located to theinitial position, which leads to the whole interface shape changing largely. Onthe other hand, we compare three numerical methods―5rd-WENO, 3rd-WENO,3rd-MCeno, and obtain the advantageous of 3rd-MCeno, which has less runningtime than the other two.
In chapter one, we introduce the development of computational ?uid dynam-ics, and some background knowledge.
In chapter two, in order to describe high Reynolds number ?ows precisely,the ?ow field has been divided into three sections: inviscid region, viscid-inviscidregion, boundary region. By simplifying Naiver-Stokes equations, get a systemof equations.
In chapter three, by comparing with numerical methods 5rd-WENO,3rd-WENO, 3rd-MCenohas showed the validity and advantages with simulating theRT instability.
In chapter four, firstly, we compare the results of RT instability on thecondition of slip boundary condition with it on the condition of rigid boundarycondition, which show di?erent ?ow state. Then, numerical example shows thatthe 3rd-MCeno scheme has ability to capture interface and uses less runningtime.
本文在一些近期文献的基础上对Rayleigh-Taylor不稳定现象做了进一步研究。文中利用一组控制方程组体系描述流场的运动状态,并且考虑固壁边界条件下整个流场形态的变化,发现在壁面附近的界面流体粘滞于初始位置,导致整个界面形状发生很大弯曲;同时,分别利用5rd-WENO,3rd-WENO,3rd-MCeno数值方法对Rayleigh-Taylor不稳定现象数值模拟,通过对数值结果的比较发现修正系数格式具有很大的计算优势。
第一章介绍计算流体力学的发展状况,对本文主要研究工作的背景及其发展进行简要的介绍。
第二章以描述高雷诺数流动为目的,将流场分为三个区域。通过对Navier-Stokes(NS)方程组的简化,得到一组描述流场的控制方程组体系。
第三章描述Rayleigh-Taylor不稳定现象的数值方法很多,本文对5rd-WENO,3rd-WENO,3rd-MCeno这三种数值格式进行介绍,最后通过数值例子来说明3rd-MCeno方法的有效性和优越性。
第四章在已有理论的基础上,首先利用高阶精度的5rd-WENO格式数值模拟滑移边界条件下的RT不稳定现象,让其与固壁边界条件下的数值结果比较,从而发现流场形态的巨大变化;其后通过利用3rd-MCeno和高阶数值格式5rd-WENO方法分别模拟此现象,通过比较图像结果发现3rd-MCeno格式的优势,其捕捉界面的能力接近于高阶精度格式,并且节约了相当多的CPU时间。
The problem of Rayleigh-Taylor instability plays a crucial role in ICF(InertialConfinementFusion). In many fields, such as astrophysics, ocean mixed layer,geology, explosion, this problem has important application and significance. Inrecent years, many scholars adopt various computational methods to simulatethis phenomenon, and receive series of numerical results.
In this paper, based on some recent methods, we present further research us-ing a system of control equations to describe the state of ?ow field; by consideringrigid boundary condition, we also find the interface nearby the well located to theinitial position, which leads to the whole interface shape changing largely. Onthe other hand, we compare three numerical methods―5rd-WENO, 3rd-WENO,3rd-MCeno, and obtain the advantageous of 3rd-MCeno, which has less runningtime than the other two.
In chapter one, we introduce the development of computational ?uid dynam-ics, and some background knowledge.
In chapter two, in order to describe high Reynolds number ?ows precisely,the ?ow field has been divided into three sections: inviscid region, viscid-inviscidregion, boundary region. By simplifying Naiver-Stokes equations, get a systemof equations.
In chapter three, by comparing with numerical methods 5rd-WENO,3rd-WENO, 3rd-MCenohas showed the validity and advantages with simulating theRT instability.
In chapter four, firstly, we compare the results of RT instability on thecondition of slip boundary condition with it on the condition of rigid boundarycondition, which show di?erent ?ow state. Then, numerical example shows thatthe 3rd-MCeno scheme has ability to capture interface and uses less runningtime.
引文
[1]高智.高雷诺数流动的控制方程体系和扩散抛物化Naiver-Stocks方程组的意义和用途[J].力学进展,2005,35(3):427-437.
[2]庄逢甘,张德良.扩散抛物化(DP)NS方程组的意义及其在计算流体力学中应用[J].空气动力学学报,2003,21(1):1-10.
[3] AIAA 2003-2005 Distance learning course. Advanced computational ?uid dynam-ics[J]. http://www.aiaa.org.
[4] Erikson L E. Numerical simulation of Compressible Flows[J]. VAC/CTH Themo:[PhD thesis]. USA,2001.
[5] Cebeci T,Cousteix J. Modeling and Computational of Boundary Layer Flows[J].2nd ed.New York:Springer,2001.
[6] Tannehill T C, Anderson D A,Pletcher R H. Computational Fluid Mechanics andHeat Transfer[J]. Second edition. New York:Taylor&Francis,1997.
[7] Ho?namm K A,Chiang S T.Computational Fluid Dynamics for EngineersVol(2)[J].A Publication of Engineering Education system.Wichita Kansas,1995.
[8] FPNS code Aerotechnologies, Inc.USA.http://www.aerotechnlilgied,com.
[9] VULCAN hypersonic airbreathing propulsion branch NASA. http://vulcan-cfd.larc.nasa.gov.
[10] VIPER software and engineering associates,Inc,USA.http://www/sealnc.com.
[11] GASP virginia tech aerospace engineering USA.http://www.gasp.org.
[12] WIND the NPSRC alliance NASA glenn research center,clevelana,USAF amoldengineering development center, Tullagoma,Tennessee. http://www.npsrc.org.
[13] Miller J H,Tannehill J C,Lawrence S L. Parabolized Navier-Stokes Code for solvingsurpersonic ?ows[J]. AIAA J,2000,38(10):1837-1845.
[14]高智.摄动有限差分研究进展[J].力学进展,2000,30(2):200-215.
[15] R.T.Davis,S.G.Rubin. Computer&Fluid,1980,8(1):101-132.
[16] T.Cebeci,J.Cousteix. Modeling and computational of boundary layers[J].Springer,New York1999.
[17] Marinak M M,Remington B A,et al. Phys Rev Lett,1995,75-3677.
[18]刘儒勋,王志峰.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.
[19] X.D.Liu,S.Osher. Convex ENO high order multidimensional schemes without fieldby field decomposition or staggered grids[J]. J.Comput.Phys,1998,141:1-27.
[20]叶文华,张维岩,陈光南等.Rayleigh-Taylor和Richtmyer-Meshkov不稳定性的FCT方法数值模拟[J].计算物理,1998,15(3):277-282.
[21]唐维军,张景琳,李小林等.三维流体界面不稳定性的Ghost方法[J].计算物理,2001,18(2):163-169.
[22] Jing Shi,Yong-Tao Zhang,Chi-Wang Shu.Resolution of high order WENO schemesfor Computational Physics[J]. Journal of Computational Physics,2003,186:690-696.
[23] Chi-Wang Shu.ENO and WENO Schemes for Hyperbolic Conservation Laws[J].NASA/CR-97-20653 ICASE Report No.97-65.
[24]刘儒勋,舒其望.计算流体力学的若干新方法.北京:科学出版社, 2003.
[25] Yuyue Yang,Mingjun Li, Shi Shu,Aiguo Xiao. High order shemes based on up-wind schemes with modified coe?cients[J]. Journal Computational and AppliedMathematics.2006,195:242-251.
[26]王汝权,申义庆.扩散抛物化Naiver-Stocks方程数值解法评述[J].力学进展,2005,36(4):481-497.
[27]高智.流体力学基本方程组(BEFM)的层次结构理论和简化Naiver-Stocks方程组(SNSE)[J].力学学报,1988,20(2):107-116.
[28]高智,申义庆.粘性流动有限差分计算的新策略[J].中国科学(A辑),1999,29(5):433-443.
[29] Ming-jun Li,Shi Shu,Sheng-yuan Yang,Yu-Yue Yang. Modifying Coe?cient Es-sentially Non-oscillatory Schenes on Non-uniform Grid[J]. Advances in Rheologyand Its Applications,2005,191-194.
[30] Jianxian Qiu,Chi-Wang Shu. Hermite WENO schemes and their application aslimiters for Runge-Kutta discontinuous Galerkin method:one-dimensional case[J].Journal of Computational Physics,2003,193:115-135.
[31]彭国伦. Fortran 95程序设计.北京:中国电力出版社,2004.
[32]姜洋,赵宁,唐维军.界面不稳定性数值模拟中的虚拟流动方法[J].计算物理,2003,20(6):549-555.
[33]水鸿寿.一维流体动力差分方法.北京:国防工业出版社,1998.
[34] G.Jiang and C.W.Shu. E?cient implementation of weighted ENO schemes onTriangular Meshes[J]. J.Comp.Phy,1999,150:97-127.
[35]齐进,叶文华.三维激光烧烛瑞利-泰勒不稳定性并列计算[J].计算物理,2002,19(5):388-392.
[36] Xiaoyi He,Shiyi Chen,and Raoyang Zhang. A Lattice Boltzmann Scheme for In-compressible Multiphase Flow and Its Application in Simulation of Rayleigh-Taylor Instability[J]. J.Comp.Phy,1999,152:642-663.
[37]申义庆,高智.双曲守恒型方程的二阶摄动有限差分格式[J].空气动力学学报,2003,21(3):342-350.
[38]陈材侃.计算流体力学.重庆:重庆出版社,1992.
[39]张梓雄,董曾南.粘性流体力学.北京:清华大学出版社,1999.
[40]孙祥海,周文伯,社振凡等.计算流体力学导论.上海:上海交通大学出版社,1987.
[41]刘导治.计算流体力学基础.北京:北京航空航天大学出版社,1989.
[42]吴望一.流体力学.北京:北京大学出版社,2004.
[43]李德元,徐国荣,水鸿寿等.二维非定常流体力学数值方法.北京:科学出版社,1998.
[44] X.D.Liu,S.Osher and T.Chan,Weighted essentially non-oscillatory schemes[J].J.Comput.Phys,1994,115:200-212.
[45] D.Levy,G.Puppo,G.Russo. Compact central WENO schemes for multidimensionalconservation laws[J]. SIAM J.Sci.Comput,2000,22,656.
[46] D.Levy,G.Puppo,G.Russo. A third order central WENO scheme for 2D conserva-tion laws[J]. Appl.Numer.Math,2000,33,415.
[47] G.S.Jiang and C.W.Shu. E?cient implementation of weighted ENO schemes[J].J.Comput.Phys,1996,126:202-228.
[48]高智.论简化Naiver-Stocks方程组[J].中国科学(A辑),1987,17(10):1058-1070.
[49]申义庆.广义扩散抛物化N-S方程组和大小尺度方程组的数值研究.博士论文,2001.
[50]高智.简化N-S方程的层次结构、它的力学内涵和应用[J].中国科学(A辑),1988,6:628-640.
[51] A.Sunahara,H.Takabe,K.Mima. 2D simulation of hydrodynamic instability in IFCstagnation phase[J]. Fusion Engineering Design,1999,44:163-169.
[52] J.Glimm,J.Grove,X.LI,W.Oh,Tan. The dynamics of bubble growth for Rayleigh-Taylor unstable interface[J]. physics of Fluids,1998,31:447-465.
[53] J.Glimm,J.W.Grove,X.LI,W.Oh,D.H.Sharpx. A Critical Analysis of Rayleigh-Taylor Growth Rates[J]. J.Comp.Phy,2001,169:652-677.
[54] M.M.Gupta. High accuracy solution of incompressible Navier-Stokes equations[J].J.Comput.Phys,1991,93:343-359.
[55] Taylor G I.The stability of liquid surface when accelerated in a direction perpen-dicular to their planes[J].Proc Roy Soc.London 1950,A201:192-196.
[56] He X,Zhang R,Chen S,Doolen C D. On the three-dimensional Rayleigh-Taylorinstability[J]. Phys Fluid,1999,11(5):1143-1152.
[57]张钧,赖来显,张维岩等.辐射烧蚀界面的单模Rayleigh-Taylor不稳定性分析[J].计算物理,1998,15(4):483-488.
[58]汤文辉,毛益明. Rayleigh-Taylor不稳定性的SPH模拟[J].国防科技大学学报,2004,26(1):21-73.
[59]张磊,刘儒勋. Rayleigh-Taylor不稳定性的MAC模拟[J].中国科学技术大学学报,2003,33(6):665-662.
[60]王丽丽,李家春,谢正桐.不可压缩流体三维Rayleigh-Taylor不稳定性的大涡模拟[J].中国科学(A辑).2002,32(2):158-169.
[2]庄逢甘,张德良.扩散抛物化(DP)NS方程组的意义及其在计算流体力学中应用[J].空气动力学学报,2003,21(1):1-10.
[3] AIAA 2003-2005 Distance learning course. Advanced computational ?uid dynam-ics[J]. http://www.aiaa.org.
[4] Erikson L E. Numerical simulation of Compressible Flows[J]. VAC/CTH Themo:[PhD thesis]. USA,2001.
[5] Cebeci T,Cousteix J. Modeling and Computational of Boundary Layer Flows[J].2nd ed.New York:Springer,2001.
[6] Tannehill T C, Anderson D A,Pletcher R H. Computational Fluid Mechanics andHeat Transfer[J]. Second edition. New York:Taylor&Francis,1997.
[7] Ho?namm K A,Chiang S T.Computational Fluid Dynamics for EngineersVol(2)[J].A Publication of Engineering Education system.Wichita Kansas,1995.
[8] FPNS code Aerotechnologies, Inc.USA.http://www.aerotechnlilgied,com.
[9] VULCAN hypersonic airbreathing propulsion branch NASA. http://vulcan-cfd.larc.nasa.gov.
[10] VIPER software and engineering associates,Inc,USA.http://www/sealnc.com.
[11] GASP virginia tech aerospace engineering USA.http://www.gasp.org.
[12] WIND the NPSRC alliance NASA glenn research center,clevelana,USAF amoldengineering development center, Tullagoma,Tennessee. http://www.npsrc.org.
[13] Miller J H,Tannehill J C,Lawrence S L. Parabolized Navier-Stokes Code for solvingsurpersonic ?ows[J]. AIAA J,2000,38(10):1837-1845.
[14]高智.摄动有限差分研究进展[J].力学进展,2000,30(2):200-215.
[15] R.T.Davis,S.G.Rubin. Computer&Fluid,1980,8(1):101-132.
[16] T.Cebeci,J.Cousteix. Modeling and computational of boundary layers[J].Springer,New York1999.
[17] Marinak M M,Remington B A,et al. Phys Rev Lett,1995,75-3677.
[18]刘儒勋,王志峰.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.
[19] X.D.Liu,S.Osher. Convex ENO high order multidimensional schemes without fieldby field decomposition or staggered grids[J]. J.Comput.Phys,1998,141:1-27.
[20]叶文华,张维岩,陈光南等.Rayleigh-Taylor和Richtmyer-Meshkov不稳定性的FCT方法数值模拟[J].计算物理,1998,15(3):277-282.
[21]唐维军,张景琳,李小林等.三维流体界面不稳定性的Ghost方法[J].计算物理,2001,18(2):163-169.
[22] Jing Shi,Yong-Tao Zhang,Chi-Wang Shu.Resolution of high order WENO schemesfor Computational Physics[J]. Journal of Computational Physics,2003,186:690-696.
[23] Chi-Wang Shu.ENO and WENO Schemes for Hyperbolic Conservation Laws[J].NASA/CR-97-20653 ICASE Report No.97-65.
[24]刘儒勋,舒其望.计算流体力学的若干新方法.北京:科学出版社, 2003.
[25] Yuyue Yang,Mingjun Li, Shi Shu,Aiguo Xiao. High order shemes based on up-wind schemes with modified coe?cients[J]. Journal Computational and AppliedMathematics.2006,195:242-251.
[26]王汝权,申义庆.扩散抛物化Naiver-Stocks方程数值解法评述[J].力学进展,2005,36(4):481-497.
[27]高智.流体力学基本方程组(BEFM)的层次结构理论和简化Naiver-Stocks方程组(SNSE)[J].力学学报,1988,20(2):107-116.
[28]高智,申义庆.粘性流动有限差分计算的新策略[J].中国科学(A辑),1999,29(5):433-443.
[29] Ming-jun Li,Shi Shu,Sheng-yuan Yang,Yu-Yue Yang. Modifying Coe?cient Es-sentially Non-oscillatory Schenes on Non-uniform Grid[J]. Advances in Rheologyand Its Applications,2005,191-194.
[30] Jianxian Qiu,Chi-Wang Shu. Hermite WENO schemes and their application aslimiters for Runge-Kutta discontinuous Galerkin method:one-dimensional case[J].Journal of Computational Physics,2003,193:115-135.
[31]彭国伦. Fortran 95程序设计.北京:中国电力出版社,2004.
[32]姜洋,赵宁,唐维军.界面不稳定性数值模拟中的虚拟流动方法[J].计算物理,2003,20(6):549-555.
[33]水鸿寿.一维流体动力差分方法.北京:国防工业出版社,1998.
[34] G.Jiang and C.W.Shu. E?cient implementation of weighted ENO schemes onTriangular Meshes[J]. J.Comp.Phy,1999,150:97-127.
[35]齐进,叶文华.三维激光烧烛瑞利-泰勒不稳定性并列计算[J].计算物理,2002,19(5):388-392.
[36] Xiaoyi He,Shiyi Chen,and Raoyang Zhang. A Lattice Boltzmann Scheme for In-compressible Multiphase Flow and Its Application in Simulation of Rayleigh-Taylor Instability[J]. J.Comp.Phy,1999,152:642-663.
[37]申义庆,高智.双曲守恒型方程的二阶摄动有限差分格式[J].空气动力学学报,2003,21(3):342-350.
[38]陈材侃.计算流体力学.重庆:重庆出版社,1992.
[39]张梓雄,董曾南.粘性流体力学.北京:清华大学出版社,1999.
[40]孙祥海,周文伯,社振凡等.计算流体力学导论.上海:上海交通大学出版社,1987.
[41]刘导治.计算流体力学基础.北京:北京航空航天大学出版社,1989.
[42]吴望一.流体力学.北京:北京大学出版社,2004.
[43]李德元,徐国荣,水鸿寿等.二维非定常流体力学数值方法.北京:科学出版社,1998.
[44] X.D.Liu,S.Osher and T.Chan,Weighted essentially non-oscillatory schemes[J].J.Comput.Phys,1994,115:200-212.
[45] D.Levy,G.Puppo,G.Russo. Compact central WENO schemes for multidimensionalconservation laws[J]. SIAM J.Sci.Comput,2000,22,656.
[46] D.Levy,G.Puppo,G.Russo. A third order central WENO scheme for 2D conserva-tion laws[J]. Appl.Numer.Math,2000,33,415.
[47] G.S.Jiang and C.W.Shu. E?cient implementation of weighted ENO schemes[J].J.Comput.Phys,1996,126:202-228.
[48]高智.论简化Naiver-Stocks方程组[J].中国科学(A辑),1987,17(10):1058-1070.
[49]申义庆.广义扩散抛物化N-S方程组和大小尺度方程组的数值研究.博士论文,2001.
[50]高智.简化N-S方程的层次结构、它的力学内涵和应用[J].中国科学(A辑),1988,6:628-640.
[51] A.Sunahara,H.Takabe,K.Mima. 2D simulation of hydrodynamic instability in IFCstagnation phase[J]. Fusion Engineering Design,1999,44:163-169.
[52] J.Glimm,J.Grove,X.LI,W.Oh,Tan. The dynamics of bubble growth for Rayleigh-Taylor unstable interface[J]. physics of Fluids,1998,31:447-465.
[53] J.Glimm,J.W.Grove,X.LI,W.Oh,D.H.Sharpx. A Critical Analysis of Rayleigh-Taylor Growth Rates[J]. J.Comp.Phy,2001,169:652-677.
[54] M.M.Gupta. High accuracy solution of incompressible Navier-Stokes equations[J].J.Comput.Phys,1991,93:343-359.
[55] Taylor G I.The stability of liquid surface when accelerated in a direction perpen-dicular to their planes[J].Proc Roy Soc.London 1950,A201:192-196.
[56] He X,Zhang R,Chen S,Doolen C D. On the three-dimensional Rayleigh-Taylorinstability[J]. Phys Fluid,1999,11(5):1143-1152.
[57]张钧,赖来显,张维岩等.辐射烧蚀界面的单模Rayleigh-Taylor不稳定性分析[J].计算物理,1998,15(4):483-488.
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