大密度比物质界面在激波作用下的R-M不稳定性研究
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摘要
当激波经过两种不同密度流体的界面时,界面获得一个有限速度,界面上的扰动随时间发展,最终导致两种流体强烈混合的现象称为Richtmyer-Meshkov(R-M)不稳定性。界面不稳定性的研究,因其应用背景和学术价值,在近二十多年受到了极大重视,国内外对此做了大量的研究实验。在界面不稳定性的数值研究中,核心是对运动的物质界面的精确描述。本文用MGFM方法定义虚拟流节点参数,采用二阶TVD-WAF格式结合HLLC求解器求解Euler方程得到流场参数分布,应用五阶WENO的空间离散以及三阶TVD Runge-Kutta的时间离散求解Level set方程捕捉物质界面。应用该方法对激波与物质界面相互作用系列问题进行了数值模拟,并给出了物质界面的演化过程。根据数值模拟结果,主要针对物质界面两边密度比以及物质界面形状对物质界面演化的影响进行比较分析,同时还对扰动增长率进行了具体分析。数值模拟结果表明,该方法在能精确捕捉到物质界面,并且高分辨率的捕获激波、物质界面和各类间断的相互作用。
A wide variety of fluid motions can be generated, following the interaction of a shock wave with an interface separating two fluids of different properties. Any perturbation initially present on the interface will, in most cases, be amplified following the refraction of the shock. This class of problems is generally referred to as the Richtmyer-Meshkov(R-M) instability. In the last two decades the R-M instability have attracted the attention of many researchers all over the world for its application background and academic value. The key point in the research of the instability is how to capture exactly the contact interface. In this paper, modified ghost fluid method(MGFM) was used to define node parameter of ghost fluid,and combined second-order TVD-WAF scheme with Riemann solver HLLC was applied to solve Euler equation for the distribution of flow field parameters. To capture the material interface, fifth-order WENO for spatial discretization and third-order TVD Runge-Kutta for time discretization were combine to solve the level set equation. The numerical simulation was accomplished with the methods mentioned above to study the interaction of shock wave with the material interface, the developing process of the material interface induced by shock wave was descrised in detail, for different density ratio and shapes of interface. The growth for amplification of perturbation was analysied with linear theord numerical method. The interaction between shock wave and the material interfac are accurately captured with high resolution.
A wide variety of fluid motions can be generated, following the interaction of a shock wave with an interface separating two fluids of different properties. Any perturbation initially present on the interface will, in most cases, be amplified following the refraction of the shock. This class of problems is generally referred to as the Richtmyer-Meshkov(R-M) instability. In the last two decades the R-M instability have attracted the attention of many researchers all over the world for its application background and academic value. The key point in the research of the instability is how to capture exactly the contact interface. In this paper, modified ghost fluid method(MGFM) was used to define node parameter of ghost fluid,and combined second-order TVD-WAF scheme with Riemann solver HLLC was applied to solve Euler equation for the distribution of flow field parameters. To capture the material interface, fifth-order WENO for spatial discretization and third-order TVD Runge-Kutta for time discretization were combine to solve the level set equation. The numerical simulation was accomplished with the methods mentioned above to study the interaction of shock wave with the material interface, the developing process of the material interface induced by shock wave was descrised in detail, for different density ratio and shapes of interface. The growth for amplification of perturbation was analysied with linear theord numerical method. The interaction between shock wave and the material interfac are accurately captured with high resolution.
引文
[1]李德元,徐国荣等.二维非定常流体力学数值方法[M].北京：科学出版社(第二版),1998.49-77.
[2]X.L.Li,Study of Three Dimensional Rayleigh-Taylor Instability in Compressible Fluids Through Level Set Method and Parallel Computation[J]. Phys. Fluids A,1993,Vol.5, Pp.1904-1913
[3]J. Grove, R. Holmes, D. H. Sharp, Y Yang and Q. Zhang, Quantitative Theory of Richtmyer-Meshkov Instability [J].Phys. Rev. Lett,1993, Vol.71No.21, Pp. 3473-3476.
[4]王淦昌,袁之尚.惯性约束核聚变[M].安徽教育出版社.102-130
[5]Osher S, Sethian J.A Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations [J].J.Comput. Phys,1988,79:12-49
[6]Sussman M, Smereka P and Osher S.A level set approach for computing solutions to incompressible two-phase flow[J].J, Comput, Phys., 1994,114,146-159.
[7]Hirt C W,Nichols B D.Volume of Fluid(VOF)method for the dynamics of free boundaries[J].J.Comput.Phys.,1981,39:201-225.
[8]Sethian J A.Theory,Algorithms and applications of level set methods for propagatinginterface[M].Acta Numerica,Cambridge UK:Cambridge Univ. Press,1995.
[9]Falcovitz J,Birman A.A singularities tracking conservation laws scheme for compressible duct flows[J].J.Comput.Phys.,1994,115:431-439.
[10]Hilditch J,Colella P.A front tracking method for compressible flames in one dimension[J]. SIAM J.Sci.Comput.,1995,16:755-772.
[11]Glimm J,Grove J,Li X L,et al.A Critical analysis of Rayleih-Taylor Growth Rates[J].J. Comput.Phys,2000,169:652-677.
[12]Glimm J,Graham M J,Grove J.Front tracking in two and three dimensions [J]J.Comput.Math.,1998,35(7):1-12.
[13]Mulder W,Osher S,Sethian J.Computing interface motion in compressible gas dynamics[J].J.Comput.Phys,1992,100:209-228.
[14]Adalsteinsson D,Sethian J A.A fast level set method for propagating interfaces[J].J.Comput.Phys.,1995,118:269-277.
[15]Sethian J A.A fast marching level set method for monotonically advancing fronts[J].Applied Mathematics,1996,93:1591-1595.
[16]Adalsteinsson D,Sethian J A.The fast construction of extension velocities in level set methods[J].J.Comput.Phys,1999,148:2-22.
[17]Fedkiw R,Aslam T,Merriman B,et al.A non-oscillatory Eulerian approach to interfaces inmultimaterial flows(the Ghost Fluid Method)[J].J.Comput.Phys. 1999,152:457-492.
[18]T G Liu,B C Khoo, and K S Yeo, Ghost fluid method for strong shock-impacting on material interface[J].J. Comput. Phys.190(2003), 651-681.
[19]Richtmyer R D. Taylor instability in shock acceleration of compressible fluids [J].Commun Pure Appl. Math.,1960,13:297-319.
[20]Meshkove E. Instability of the interface of two gases accelerated by a shock wave [J]. Fluid Dynamics,1969,4:101-104.
[21]Sadoto, Erezl, Alonu, etal. Study of Nonlinear Ev2olution of Single Model and Two2Bubble Interaction underRichtmyer2Meshkov Instability [J]. Physical review letters,1998,80(8):1654-1657.
[22]Brouillettem, Sturtevantb. Growth induced bymultiple shock waves normally incident on plangaseous interface[J]. Physica D,1989,37:248-263.
[23]Houasl,Chemounii.Experimental investigation ofRichtmyer2Meshkov instability in shock tube [J]. Physics of Fluids,1996, (8):614-627.
[24]Alonu,Hechtj,Oferd, etal. Power law and similarity of Rayleigh-Taylor and Richtmyer-Meshkov mixing fronts atall density ratios[J].Physical ReviewLetters,1995,74(4):534-537.
[25]王继海.二维非定常流和激波[M].北京：科学出版社,1994.348-470.
[26]Sturtevant B.Rayleigh-Taylor instability in compressible fluids[A].In:Proc 16th Intl Symp on ShockTubes and Shock Waves[C].Aachen W.Germany, 1988,26-32.
[27]Holmes R L, Grove J W,Sharp D H. Numerical investigation of Richtmyer-Meshkov instability using fronttracking [J].J Fluid Mech,1995, 301:51-64.
[28]叶文华,张维岩,陈光南,晋长秋,张钧.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.
[29]王晨星,唐维军,程军波.应用多介质PPM方法计算斜激波与物质交界面的相互作用[J].计算物理,2007,21(6)：531-537
[30]欧阳良琛,马东军,孙德军,尹协远.单模大扰动的Richtmyer-Meshkov不稳定性[J].爆炸与冲击,2008,28(5)：407-414
[31]张镭,袁礼.用改进的耦合型Level Set方法计算一维双介质可压缩流动[J].计算物理,2001,18(6)：511-516.
[32]卓启威,施红辉.气／液界面上Richtmyer-Meshkov不稳定性的实验研究[J].实验流体力学.2007,21(1)：25-30
[33]马东军,何兴,孙德军,尹协远.柱状交界面Richtmyer-Meshkov不稳定性的数值研究[J].爆炸与冲击,2003,23(5)：398-404
[34]张瑗,李寿佛,刘玉珍,屈小妹,杨水平.高阶FD-WENO格式用于 Richtmyer-Meshkov不稳定性数值模拟[J].系统仿真学报,2007,19(18)：4122-4125
[35]张学莹,赵宁,朱君.多流体界面不稳定性的守恒和非守恒高精度数值模拟方法[J].爆炸与冲击,2006,26(1)：65-70.
[36]姜洋,赵宁,李晓林,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).551-555.
[37]唐维军,张景林,李晓林,赵宁.三维流体界面不稳定性的Ghost方法[J].计算物理,2001,18(2)：163-169.
[38]Birkhoff G. Taylor instability and laminar mixing[M]. LA1862,Los Alamos; Los Alamos Sciontific Lab.1954.76.
[39]Birkhoff G. Helmholiz and Taylor instability[J]. In:Birkhoff G. Bellman R. Lin C. C. ed. Hydrodynamic instability,Am.Math. Soe.,New York,1962, 55-76.
[40]蔚喜军,尤迎玖.流体界面不稳定性数值模拟中不同介质界面的处理方法[J].计算物理,2001,18(1)：23-26.
[41]唐维军,赵宁,李晓林,张景林,蔚喜军.随机扰动下三维流体界面不稳定性的并行计算[J].计算物理,2001,18(6)：539-543.
[42]严长林,孙德军,尹协远,童秉纲.激波加速的密度分层流体交界面的演化[J].空气动力学学报,2001,19(2)：228-233.
[43]Q. Zhang and S. Sohn.Nonlinear theory of unstable fluid mixing driven by shock waves[J].Phys.1997:1106-1124.
[44]Dr.Eleuterio F.toro. Riemann Solvers and Numerical Methods for Fluid Dynamics[J]. Department of Computing and Mathematics.1997.492-500.
[45]J Glimm,J Grove,X Li and N Zhao.Non-Oscillatory Front Tracking and Strong Shock-Contact Interaction in the Richtmyer-Meshkov Instability[J]. (preparing).
[46]E E Meshkov.Instability of two gases accelerated by a shock wave[J]. Fluid Dyn,1969,43(5):101-104.
[47]Hawley J F,Zabusky N J.Vortex Paradigm for Shock-Accelerated Density-Satisfied Interfaces [J]. Physical Review Letter,1989,63(12): 1241-1244.
[48]张斌.Level Set方法在多介质可压缩流的应用.哈尔滨工程大学硕士学位论文[D].2008.9-43.
[49]王晨星.利用多介质PPM方法数值模拟斜激波与物质界面的相互作用.中国工程物理研究院硕士学位论文[D].2004.33-45.
[50]姜洋.界面不稳定性数值模拟中的虚拟流动方法.南京航空航天大学硕士学位论文[D].2002.4-7；19-24.
[51]周袁媛.复杂界面的界面追踪法.南京航空航天大学硕士学位论文[D].2004.3-7.
[52]杨玉月.修正系数格式及其在Rayleigh-Taylor不稳定性数值模拟中的应用.湘潭大学硕士学位论文[D].2005.1-6.
[2]X.L.Li,Study of Three Dimensional Rayleigh-Taylor Instability in Compressible Fluids Through Level Set Method and Parallel Computation[J]. Phys. Fluids A,1993,Vol.5, Pp.1904-1913
[3]J. Grove, R. Holmes, D. H. Sharp, Y Yang and Q. Zhang, Quantitative Theory of Richtmyer-Meshkov Instability [J].Phys. Rev. Lett,1993, Vol.71No.21, Pp. 3473-3476.
[4]王淦昌,袁之尚.惯性约束核聚变[M].安徽教育出版社.102-130
[5]Osher S, Sethian J.A Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations [J].J.Comput. Phys,1988,79:12-49
[6]Sussman M, Smereka P and Osher S.A level set approach for computing solutions to incompressible two-phase flow[J].J, Comput, Phys., 1994,114,146-159.
[7]Hirt C W,Nichols B D.Volume of Fluid(VOF)method for the dynamics of free boundaries[J].J.Comput.Phys.,1981,39:201-225.
[8]Sethian J A.Theory,Algorithms and applications of level set methods for propagatinginterface[M].Acta Numerica,Cambridge UK:Cambridge Univ. Press,1995.
[9]Falcovitz J,Birman A.A singularities tracking conservation laws scheme for compressible duct flows[J].J.Comput.Phys.,1994,115:431-439.
[10]Hilditch J,Colella P.A front tracking method for compressible flames in one dimension[J]. SIAM J.Sci.Comput.,1995,16:755-772.
[11]Glimm J,Grove J,Li X L,et al.A Critical analysis of Rayleih-Taylor Growth Rates[J].J. Comput.Phys,2000,169:652-677.
[12]Glimm J,Graham M J,Grove J.Front tracking in two and three dimensions [J]J.Comput.Math.,1998,35(7):1-12.
[13]Mulder W,Osher S,Sethian J.Computing interface motion in compressible gas dynamics[J].J.Comput.Phys,1992,100:209-228.
[14]Adalsteinsson D,Sethian J A.A fast level set method for propagating interfaces[J].J.Comput.Phys.,1995,118:269-277.
[15]Sethian J A.A fast marching level set method for monotonically advancing fronts[J].Applied Mathematics,1996,93:1591-1595.
[16]Adalsteinsson D,Sethian J A.The fast construction of extension velocities in level set methods[J].J.Comput.Phys,1999,148:2-22.
[17]Fedkiw R,Aslam T,Merriman B,et al.A non-oscillatory Eulerian approach to interfaces inmultimaterial flows(the Ghost Fluid Method)[J].J.Comput.Phys. 1999,152:457-492.
[18]T G Liu,B C Khoo, and K S Yeo, Ghost fluid method for strong shock-impacting on material interface[J].J. Comput. Phys.190(2003), 651-681.
[19]Richtmyer R D. Taylor instability in shock acceleration of compressible fluids [J].Commun Pure Appl. Math.,1960,13:297-319.
[20]Meshkove E. Instability of the interface of two gases accelerated by a shock wave [J]. Fluid Dynamics,1969,4:101-104.
[21]Sadoto, Erezl, Alonu, etal. Study of Nonlinear Ev2olution of Single Model and Two2Bubble Interaction underRichtmyer2Meshkov Instability [J]. Physical review letters,1998,80(8):1654-1657.
[22]Brouillettem, Sturtevantb. Growth induced bymultiple shock waves normally incident on plangaseous interface[J]. Physica D,1989,37:248-263.
[23]Houasl,Chemounii.Experimental investigation ofRichtmyer2Meshkov instability in shock tube [J]. Physics of Fluids,1996, (8):614-627.
[24]Alonu,Hechtj,Oferd, etal. Power law and similarity of Rayleigh-Taylor and Richtmyer-Meshkov mixing fronts atall density ratios[J].Physical ReviewLetters,1995,74(4):534-537.
[25]王继海.二维非定常流和激波[M].北京：科学出版社,1994.348-470.
[26]Sturtevant B.Rayleigh-Taylor instability in compressible fluids[A].In:Proc 16th Intl Symp on ShockTubes and Shock Waves[C].Aachen W.Germany, 1988,26-32.
[27]Holmes R L, Grove J W,Sharp D H. Numerical investigation of Richtmyer-Meshkov instability using fronttracking [J].J Fluid Mech,1995, 301:51-64.
[28]叶文华,张维岩,陈光南,晋长秋,张钧.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.
[29]王晨星,唐维军,程军波.应用多介质PPM方法计算斜激波与物质交界面的相互作用[J].计算物理,2007,21(6)：531-537
[30]欧阳良琛,马东军,孙德军,尹协远.单模大扰动的Richtmyer-Meshkov不稳定性[J].爆炸与冲击,2008,28(5)：407-414
[31]张镭,袁礼.用改进的耦合型Level Set方法计算一维双介质可压缩流动[J].计算物理,2001,18(6)：511-516.
[32]卓启威,施红辉.气／液界面上Richtmyer-Meshkov不稳定性的实验研究[J].实验流体力学.2007,21(1)：25-30
[33]马东军,何兴,孙德军,尹协远.柱状交界面Richtmyer-Meshkov不稳定性的数值研究[J].爆炸与冲击,2003,23(5)：398-404
[34]张瑗,李寿佛,刘玉珍,屈小妹,杨水平.高阶FD-WENO格式用于 Richtmyer-Meshkov不稳定性数值模拟[J].系统仿真学报,2007,19(18)：4122-4125
[35]张学莹,赵宁,朱君.多流体界面不稳定性的守恒和非守恒高精度数值模拟方法[J].爆炸与冲击,2006,26(1)：65-70.
[36]姜洋,赵宁,李晓林,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).551-555.
[37]唐维军,张景林,李晓林,赵宁.三维流体界面不稳定性的Ghost方法[J].计算物理,2001,18(2)：163-169.
[38]Birkhoff G. Taylor instability and laminar mixing[M]. LA1862,Los Alamos; Los Alamos Sciontific Lab.1954.76.
[39]Birkhoff G. Helmholiz and Taylor instability[J]. In:Birkhoff G. Bellman R. Lin C. C. ed. Hydrodynamic instability,Am.Math. Soe.,New York,1962, 55-76.
[40]蔚喜军,尤迎玖.流体界面不稳定性数值模拟中不同介质界面的处理方法[J].计算物理,2001,18(1)：23-26.
[41]唐维军,赵宁,李晓林,张景林,蔚喜军.随机扰动下三维流体界面不稳定性的并行计算[J].计算物理,2001,18(6)：539-543.
[42]严长林,孙德军,尹协远,童秉纲.激波加速的密度分层流体交界面的演化[J].空气动力学学报,2001,19(2)：228-233.
[43]Q. Zhang and S. Sohn.Nonlinear theory of unstable fluid mixing driven by shock waves[J].Phys.1997:1106-1124.
[44]Dr.Eleuterio F.toro. Riemann Solvers and Numerical Methods for Fluid Dynamics[J]. Department of Computing and Mathematics.1997.492-500.
[45]J Glimm,J Grove,X Li and N Zhao.Non-Oscillatory Front Tracking and Strong Shock-Contact Interaction in the Richtmyer-Meshkov Instability[J]. (preparing).
[46]E E Meshkov.Instability of two gases accelerated by a shock wave[J]. Fluid Dyn,1969,43(5):101-104.
[47]Hawley J F,Zabusky N J.Vortex Paradigm for Shock-Accelerated Density-Satisfied Interfaces [J]. Physical Review Letter,1989,63(12): 1241-1244.
[48]张斌.Level Set方法在多介质可压缩流的应用.哈尔滨工程大学硕士学位论文[D].2008.9-43.
[49]王晨星.利用多介质PPM方法数值模拟斜激波与物质界面的相互作用.中国工程物理研究院硕士学位论文[D].2004.33-45.
[50]姜洋.界面不稳定性数值模拟中的虚拟流动方法.南京航空航天大学硕士学位论文[D].2002.4-7；19-24.
[51]周袁媛.复杂界面的界面追踪法.南京航空航天大学硕士学位论文[D].2004.3-7.
[52]杨玉月.修正系数格式及其在Rayleigh-Taylor不稳定性数值模拟中的应用.湘潭大学硕士学位论文[D].2005.1-6.