湍流壳模型的非线性动力学
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摘要
湍流的阵发性质是湍流运动的本质特征,是湍流理论研究的中心问题之一。上世纪七十年代以来,非线性动力学方法为考虑湍流问题提供了全新的视角。湍流和动力学系统的分岔、混沌、奇怪吸引子等概念有密切的关系。通过低维动力系统的研究,已经找到多种过渡到混沌吸引子的道路,比较著名的有Ruelle和Takens的Hopf分岔方式,Feigenbaum的倍周期分岔方式,以及Pomeanu和Menneville的阵发相变方式。现在人们普遍认为,湍流的发生与非线性系统中动力学的混沌吸引子的出现有密切联系。本论文研究Gledzer-Ohkitani-Yamada湍流壳模型的非线性动力学。
在22个壳GOY模型的数值模拟中,通过改变参数δ我们发现:惯性子区中各个壳阵发轨道的位相在其速度场相空间中的随机转动漂移、反转漂移和振荡漂移随壳数具有周期三的特点。与此对应,惯性子区中各个壳不稳定周期轨道部分的方位形成随机转动、反转和锁定的周期三级串。通过计算不稳定周期时间序列长度的统计平均值,拟合得到该平均值关于参数δ的临界标度律。我们计算了阵发湍流时间序列速度结构函数的相对标度指数,这个标度指数与kolmogorov线性标度律非常类似。此外针对阵发湍流中的不同时间序列,分别计算了不稳定周期时间序列部分和阵发时间序列部的结构函数的相对标度指数。结果表明阵发湍流中也具有充分发展湍流的速度结构函数标度律的非线性标度特征,它是由阵发时间序列中最强阵发和最弱阵发的部分贡献的。
通过改变外力参数f_0和耦合参数δ,研究GOY模型在f_0—δ参数空间中的动力学行为。我们通过临界标度指数,稳定和不稳定周期轨道对和Liyapunov指数等三个方面的计算确认了阵发混沌的相变类型为鞍结点分岔。并且给出了f_0—δ参数空间的相图,该相图将参数空间划分为周期、准周期和阵发混沌区域。通过改变泰勒微尺度雷诺数,我们计算得出速度结构函数相对标度指数随泰勒微尺度雷诺数的变化不大,并指出了速度结构函数相对标度指数的非线性标度律的参数范围。
The intermittent nature of turbulence is the essence character of turbulence and one of central problems in theoretical studies of turbulence.Since the seventies in last century,the nonlinear dynamical approaches have provided a new view on turbulence. Turbulence connects closely with the bifurcation,chaos and strange attractor in nonlinear dynamical systems.Through the studies on the low dimension of dynamical systems,many kinds of transition routes to the chaotic attractors have been found out. The three famous routes are Hopf bifurcation brought forward by Ruelle and Takens, doubling periodic bifurcation brought forward by Feigenbaum and intermittent transition found by Pomean and Menneville.Nowadays one has believed that turbulence occurrence has close relations with the dynamical chaotic attractor appearance in nonlinear systems.This thesis is about the studies on the nonlinear dynamics and scaling in shell model of turbulence,called Gledzer-Ohkitani-Yamada shell model.
In the numerical simulation for the 22 shells of GOY model,We find by varying parameterδ:For each shell in the inertial range,the phases of intermittent orbit parts in velocity phase space display clockwise rotation randomly,counter clockwise rotation randomly and oscillation randomly which have period three with shell number.The directions of unstable periodic orbit parts of each shell in inertial range form the rotational,reversal and locked cascade which also have period three with shell number.Through calculating statistical average of unstable periodic time series length,we obtain critical scaling of parameterδabout the average by fitting.We calculate relative scaling exponents of the whole time series of intermittent turbulence which is very similar to the kolmogorov linear scaling,and besides,we calculate the relative scaling exponents of structure function about unstable periodic time series parts and intermittent time series parts.The results show that intermittent turbulence also has the character of nonlinear scaling of velocity structure function of well-developed turbulence,which is contributed by the strongest and lowest intermittency parts of the intermittent time series.
Through variation of parameter f_0 andδ,the dynamical behaviors on f_(0—δ) parameter surface is investigated for GOY model.We determine the type of intermittency transitions to chaotic attractor is saddle node bifurcation by calculating the scaling exponent,the pairs of stable and unstable periodic orbits and Liyapunov exponents.We plot phase diagram on f_(0—δ) parameter surface which is divided into periodic,quasi-periodic and intermittent chaos area.By means of varying Taylor-microscale Reynolds number,we calculate the relative scaling exponents of velocity structure function which change a little with Taylor-microscale Reynolds number,and we indicate f_(0—δ) the parameter range of nonlinear the relative scaling of velocity structure function.
在22个壳GOY模型的数值模拟中,通过改变参数δ我们发现:惯性子区中各个壳阵发轨道的位相在其速度场相空间中的随机转动漂移、反转漂移和振荡漂移随壳数具有周期三的特点。与此对应,惯性子区中各个壳不稳定周期轨道部分的方位形成随机转动、反转和锁定的周期三级串。通过计算不稳定周期时间序列长度的统计平均值,拟合得到该平均值关于参数δ的临界标度律。我们计算了阵发湍流时间序列速度结构函数的相对标度指数,这个标度指数与kolmogorov线性标度律非常类似。此外针对阵发湍流中的不同时间序列,分别计算了不稳定周期时间序列部分和阵发时间序列部的结构函数的相对标度指数。结果表明阵发湍流中也具有充分发展湍流的速度结构函数标度律的非线性标度特征,它是由阵发时间序列中最强阵发和最弱阵发的部分贡献的。
通过改变外力参数f_0和耦合参数δ,研究GOY模型在f_0—δ参数空间中的动力学行为。我们通过临界标度指数,稳定和不稳定周期轨道对和Liyapunov指数等三个方面的计算确认了阵发混沌的相变类型为鞍结点分岔。并且给出了f_0—δ参数空间的相图,该相图将参数空间划分为周期、准周期和阵发混沌区域。通过改变泰勒微尺度雷诺数,我们计算得出速度结构函数相对标度指数随泰勒微尺度雷诺数的变化不大,并指出了速度结构函数相对标度指数的非线性标度律的参数范围。
The intermittent nature of turbulence is the essence character of turbulence and one of central problems in theoretical studies of turbulence.Since the seventies in last century,the nonlinear dynamical approaches have provided a new view on turbulence. Turbulence connects closely with the bifurcation,chaos and strange attractor in nonlinear dynamical systems.Through the studies on the low dimension of dynamical systems,many kinds of transition routes to the chaotic attractors have been found out. The three famous routes are Hopf bifurcation brought forward by Ruelle and Takens, doubling periodic bifurcation brought forward by Feigenbaum and intermittent transition found by Pomean and Menneville.Nowadays one has believed that turbulence occurrence has close relations with the dynamical chaotic attractor appearance in nonlinear systems.This thesis is about the studies on the nonlinear dynamics and scaling in shell model of turbulence,called Gledzer-Ohkitani-Yamada shell model.
In the numerical simulation for the 22 shells of GOY model,We find by varying parameterδ:For each shell in the inertial range,the phases of intermittent orbit parts in velocity phase space display clockwise rotation randomly,counter clockwise rotation randomly and oscillation randomly which have period three with shell number.The directions of unstable periodic orbit parts of each shell in inertial range form the rotational,reversal and locked cascade which also have period three with shell number.Through calculating statistical average of unstable periodic time series length,we obtain critical scaling of parameterδabout the average by fitting.We calculate relative scaling exponents of the whole time series of intermittent turbulence which is very similar to the kolmogorov linear scaling,and besides,we calculate the relative scaling exponents of structure function about unstable periodic time series parts and intermittent time series parts.The results show that intermittent turbulence also has the character of nonlinear scaling of velocity structure function of well-developed turbulence,which is contributed by the strongest and lowest intermittency parts of the intermittent time series.
Through variation of parameter f_0 andδ,the dynamical behaviors on f_(0—δ) parameter surface is investigated for GOY model.We determine the type of intermittency transitions to chaotic attractor is saddle node bifurcation by calculating the scaling exponent,the pairs of stable and unstable periodic orbits and Liyapunov exponents.We plot phase diagram on f_(0—δ) parameter surface which is divided into periodic,quasi-periodic and intermittent chaos area.By means of varying Taylor-microscale Reynolds number,we calculate the relative scaling exponents of velocity structure function which change a little with Taylor-microscale Reynolds number,and we indicate f_(0—δ) the parameter range of nonlinear the relative scaling of velocity structure function.
引文
[1]G..Batchelor,The theory of homogeneous turbulence(Cambridge University Press,Cambridge 1953)
[2]Dyke,M.V An album of fluid motion(The Parabolic Press,Stanford,Califor-nia,1982)
[3]Richardson,L.F Weather Prediction by Numerical Process.Cambridge University Press,1922
[4]A.Kolmogorov,Dokl.Acad.Nauk SSSR 30,(1941)301(英文重印Proc.R.Soc.Lond.A,434,9(1991))
[5]A.Kolmogorov,Dokl.Acad.Nauk SSSR 31,538(1941)
[6]A.Kolmogorov,Dokl.Acad.Nauk SSSR 32,16(1941)(英文重印Proc.R.Soc.Lond.A,434,15(1991))
[7]Landau L.D and Lifshitz E.M Fluid Mechanics Pergmon Press London(1963)
[8]Kolmogorov A.N JFM 13:82-85(1962)
[9]Frish.U,Sulem P.L,Nelkin M.A,J.Fluid.Mech.87,719(1978)
[10]Anselmet,F.,Gagne,Y.,Hopfinger,.E.J and Antonia,R.A.J.Fluid Mech.140,63-89(1984).
[11]Parisi,G..and Frisch,.U.1985,in Turbulence and Predictability of Geophysical Fluid Dynamics,Proceed.Intern.School of Physics 'E.Fermi',1983,Varenna,Italy, 84-87,eds. M.Ghil, R.Benzi, and G.Parisi. North-Holland, Amsterdam
[12]Benzi,R.,Paladin,G.,Parisi.G,andVulpiani,AJ.Phys.A,17,3521(1984)
[13]She,Z.S, and Leveque Phys Rev.Lett,72,336-339(1994)
[14] She,Z.S, Waymire,E C. Phys Rev.Lett,74(2),262(1994)
[15] She,Z.S,Progress Theoretical Physics Supplement 130,87(1999)
[16] Desnyansky, V.N and Novikov, E.A., prinkl.Mat.Mekh.38, 507.(1974)
[17] Desnyansky, V.N and Novikov, E.A.,Atoms Oceanic Phys.10,127.(1974)
[18] Gledzer, E.B., sov.phys.Dokl, 18,216(1973)
[19] Yamada, M. and. Ohkitani, K., J. Phys. Soc. of. Japan56, 4210(1987)
[20]Frisch, U. Turbulence: Turbulence the legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 25(1995)
[21]Kadanoff, L. P., Lohse, D., Wang, J. and Benzi, R., phys. Fluids7, 617 (1995)
[22]D. Pisarenko, L. Biferale, D. Courvoisier and U. Frisch, M. Vergassoia, Phys. Fluids A5, 2533 (1993)
[23]Jensen, M.H, Paladin, D.,Vulpian, A., Phys.Rev.A.43 (2), 798(1991)
[24] Jensen, M.H, Paladin, D.,Vulpian, A., Phys.Rev.A.45 (10), 7214(1992)
[25]Ohkitani, K., Yamada, M. Progress of Theoretical Physics, Vol.8 (2), 329(1989)
[26]Biferale, L. , Kerr, R. Phys. Rev. E 52, 6113 (1995)
[27] R. Biferale, A. Lambert, R. Lima, and G. Paladin, Physica D 80, 105(1995)
[28] Sei. Kato and M. Yamada Phys. Rev. E 68. 025302, (2003)
[29]Mitra.D and Pandit.R, Physica A 318,179(2003)
[30] Mitra.D and Pandit.R, Phys.Rev.Lett93, 024501(2004)
[31]Benzi, R.,Biferale,L.,Kerr,R.M.and.Trovatore,E.,Phys.Rev.E53,3541(1996)
[32]Waleffe,F., Phys.Fluids A4,350(1992)
[33]L'Vov.V.S.,Podivilov,E.,Procaccia,I.,preprint in chao-dyn@xyz.lanl.gov#9706018
[34]Chkhetiani,O.,JEPT Lett.63,806(1996)
[35]Biferale,L.,Preiotti,D.,Toschi,F.Phys.Rev.E 57,2515(1998)
[36]L'vov,V.S,E.Podilov,A.Pomyalov,I.Procaceia and D.Vandembroucq.Phys.Rev.E58,1811-1822(1998).
[37]L'vov,V.S,E.Podivilov,and I.Procaccia,Europhys.Let.,46,609-612(1999)
[1]Sei.Kato and M.Yamada Phys.Rev.E 68.025302,(2003)
[2]R.Biferale,A.Lambert,R.Lima,and G.Paladin,Physica D 80,105-119(1995)
[3]R.Benzi,L.Biferale,and G.Parisi,Physica D 65,163(1993)
[4]O.Gat,I.Procaccia,and R.zeitak,Phys.Rev.E 51,1148(1995)
[5]R.Benzi,S.Ciliberto,R.Tripiccione,C.Baudet,F.Massaioli & S.Succi,Phys.Rev.E,48(1):R29-R32,(1993)
[6]R.Benzi,G.R.S.Ciliberto,R.Tripiccione,Europhys.Lett,24,275(1993)
[7]M.Briscolini,P.Santangelo,S.Succi & R.Benzi,Phys.Rev.E,50,1745,(1994)
[8]Stolovitzky.G,Sreenivasan.K.R,Phys.Rev E48(1),33(1993)
[9]Benzi R,Ciliberto S,Baudet C,Chavarria G R.Physica D,80(4),385,(1994)
[10]Benzi,R.,Biferale,L,Ciliberto,S.,Struglia.M,Tripiccione,R.,Europhys Lett.32,709(1995)
[11]D.Pisarenko,L.Biferale,D.Courvoisier and U.Frisch,M.Vergassoia,Phys.Fluids A5,2533-2538(1993)
[t2]Kadanoff,L.P.,Lohse,D.,Wang,J.and Benzi,R.,phys.Fluids7,617(1995)
[13]L'vov,V.S,E.Podivilov,A.Pomyalov,I.Procaccia and D.Vandembroucq.Phys.Rev.E58,1811-1822(1998).
[14]Mitra.D and Pandit.R,Phys.Rev.Lett93,024501(2004)
[15]She,Z.S,and Leveque Phys Rev.Lett,72,336-339(1994)
[16]佘振苏 苏卫东 力学进展 vol.29,No.3,Aug.25.289-303(1999)
[1]R.Biferale,A.Lambert,R.Lima,and G.Paladin,Physica D 80,105(1995)
[2]ZHAO Ying-Kui,CHEN Shi-Gang and WANG Guang-Rui commun.Theor.Phys.(2007)
[3]Mitra.D,and Pandit.R,Phys.Rev.Lett.93,024501(2004)
[4]Frisch,U.Turbuience:Turbulence the legacy of A.N.Kolmogorov,Cambridge University Press,Cambridge,p61(1995)
[5]Y.Pomeau and P.Manneville Comm.Math.Phys.74.189(1980)
[6]R.Benzi,S.Ciliberto,R.Tripiccione,C.Baudet,F.Massaioli,and S.Succi,Phys.Rev.E.48(1993) 29
[7] R. Benzi, S. Ciliberto, G. Ruiz. Chavarria, and R. Tripiccione, Europhys.Lett. 24 (1993)275
[8] M. Briscolini, P. Santangelo, S. Succi, and R. Benzi, Phys.Rev.E.50 (1994) 1745
[9] R. Benzi, L. Biferale, S. Giliberto, M. V. Struglia, and R. Tripiccione, Physica. D. 96(1996)162
[10]She,Z.S, and Leveque Phys Rev.Lett,72,336 (1994)
[11] A.Arneodo, C.Baudet, F.Belin, R.Benzi, B.Castaing, B.Chabaud, R.Chavarria, S.Ciliberto, R.Camussi, F.Chilla, B.Dubrulle, YGagne, B.Hebral, J.Herweijer, M.Marchand, J.Maurer, J.F.Muzy, A.Naert, A.Noullez, J.Peinke, F.Roux,P, Tabeling, W.Vande Water, H.Willaime, Europhys.Lett.,34(6)(1996)411
[2]Dyke,M.V An album of fluid motion(The Parabolic Press,Stanford,Califor-nia,1982)
[3]Richardson,L.F Weather Prediction by Numerical Process.Cambridge University Press,1922
[4]A.Kolmogorov,Dokl.Acad.Nauk SSSR 30,(1941)301(英文重印Proc.R.Soc.Lond.A,434,9(1991))
[5]A.Kolmogorov,Dokl.Acad.Nauk SSSR 31,538(1941)
[6]A.Kolmogorov,Dokl.Acad.Nauk SSSR 32,16(1941)(英文重印Proc.R.Soc.Lond.A,434,15(1991))
[7]Landau L.D and Lifshitz E.M Fluid Mechanics Pergmon Press London(1963)
[8]Kolmogorov A.N JFM 13:82-85(1962)
[9]Frish.U,Sulem P.L,Nelkin M.A,J.Fluid.Mech.87,719(1978)
[10]Anselmet,F.,Gagne,Y.,Hopfinger,.E.J and Antonia,R.A.J.Fluid Mech.140,63-89(1984).
[11]Parisi,G..and Frisch,.U.1985,in Turbulence and Predictability of Geophysical Fluid Dynamics,Proceed.Intern.School of Physics 'E.Fermi',1983,Varenna,Italy, 84-87,eds. M.Ghil, R.Benzi, and G.Parisi. North-Holland, Amsterdam
[12]Benzi,R.,Paladin,G.,Parisi.G,andVulpiani,AJ.Phys.A,17,3521(1984)
[13]She,Z.S, and Leveque Phys Rev.Lett,72,336-339(1994)
[14] She,Z.S, Waymire,E C. Phys Rev.Lett,74(2),262(1994)
[15] She,Z.S,Progress Theoretical Physics Supplement 130,87(1999)
[16] Desnyansky, V.N and Novikov, E.A., prinkl.Mat.Mekh.38, 507.(1974)
[17] Desnyansky, V.N and Novikov, E.A.,Atoms Oceanic Phys.10,127.(1974)
[18] Gledzer, E.B., sov.phys.Dokl, 18,216(1973)
[19] Yamada, M. and. Ohkitani, K., J. Phys. Soc. of. Japan56, 4210(1987)
[20]Frisch, U. Turbulence: Turbulence the legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 25(1995)
[21]Kadanoff, L. P., Lohse, D., Wang, J. and Benzi, R., phys. Fluids7, 617 (1995)
[22]D. Pisarenko, L. Biferale, D. Courvoisier and U. Frisch, M. Vergassoia, Phys. Fluids A5, 2533 (1993)
[23]Jensen, M.H, Paladin, D.,Vulpian, A., Phys.Rev.A.43 (2), 798(1991)
[24] Jensen, M.H, Paladin, D.,Vulpian, A., Phys.Rev.A.45 (10), 7214(1992)
[25]Ohkitani, K., Yamada, M. Progress of Theoretical Physics, Vol.8 (2), 329(1989)
[26]Biferale, L. , Kerr, R. Phys. Rev. E 52, 6113 (1995)
[27] R. Biferale, A. Lambert, R. Lima, and G. Paladin, Physica D 80, 105(1995)
[28] Sei. Kato and M. Yamada Phys. Rev. E 68. 025302, (2003)
[29]Mitra.D and Pandit.R, Physica A 318,179(2003)
[30] Mitra.D and Pandit.R, Phys.Rev.Lett93, 024501(2004)
[31]Benzi, R.,Biferale,L.,Kerr,R.M.and.Trovatore,E.,Phys.Rev.E53,3541(1996)
[32]Waleffe,F., Phys.Fluids A4,350(1992)
[33]L'Vov.V.S.,Podivilov,E.,Procaccia,I.,preprint in chao-dyn@xyz.lanl.gov#9706018
[34]Chkhetiani,O.,JEPT Lett.63,806(1996)
[35]Biferale,L.,Preiotti,D.,Toschi,F.Phys.Rev.E 57,2515(1998)
[36]L'vov,V.S,E.Podilov,A.Pomyalov,I.Procaceia and D.Vandembroucq.Phys.Rev.E58,1811-1822(1998).
[37]L'vov,V.S,E.Podivilov,and I.Procaccia,Europhys.Let.,46,609-612(1999)
[1]Sei.Kato and M.Yamada Phys.Rev.E 68.025302,(2003)
[2]R.Biferale,A.Lambert,R.Lima,and G.Paladin,Physica D 80,105-119(1995)
[3]R.Benzi,L.Biferale,and G.Parisi,Physica D 65,163(1993)
[4]O.Gat,I.Procaccia,and R.zeitak,Phys.Rev.E 51,1148(1995)
[5]R.Benzi,S.Ciliberto,R.Tripiccione,C.Baudet,F.Massaioli & S.Succi,Phys.Rev.E,48(1):R29-R32,(1993)
[6]R.Benzi,G.R.S.Ciliberto,R.Tripiccione,Europhys.Lett,24,275(1993)
[7]M.Briscolini,P.Santangelo,S.Succi & R.Benzi,Phys.Rev.E,50,1745,(1994)
[8]Stolovitzky.G,Sreenivasan.K.R,Phys.Rev E48(1),33(1993)
[9]Benzi R,Ciliberto S,Baudet C,Chavarria G R.Physica D,80(4),385,(1994)
[10]Benzi,R.,Biferale,L,Ciliberto,S.,Struglia.M,Tripiccione,R.,Europhys Lett.32,709(1995)
[11]D.Pisarenko,L.Biferale,D.Courvoisier and U.Frisch,M.Vergassoia,Phys.Fluids A5,2533-2538(1993)
[t2]Kadanoff,L.P.,Lohse,D.,Wang,J.and Benzi,R.,phys.Fluids7,617(1995)
[13]L'vov,V.S,E.Podivilov,A.Pomyalov,I.Procaccia and D.Vandembroucq.Phys.Rev.E58,1811-1822(1998).
[14]Mitra.D and Pandit.R,Phys.Rev.Lett93,024501(2004)
[15]She,Z.S,and Leveque Phys Rev.Lett,72,336-339(1994)
[16]佘振苏 苏卫东 力学进展 vol.29,No.3,Aug.25.289-303(1999)
[1]R.Biferale,A.Lambert,R.Lima,and G.Paladin,Physica D 80,105(1995)
[2]ZHAO Ying-Kui,CHEN Shi-Gang and WANG Guang-Rui commun.Theor.Phys.(2007)
[3]Mitra.D,and Pandit.R,Phys.Rev.Lett.93,024501(2004)
[4]Frisch,U.Turbuience:Turbulence the legacy of A.N.Kolmogorov,Cambridge University Press,Cambridge,p61(1995)
[5]Y.Pomeau and P.Manneville Comm.Math.Phys.74.189(1980)
[6]R.Benzi,S.Ciliberto,R.Tripiccione,C.Baudet,F.Massaioli,and S.Succi,Phys.Rev.E.48(1993) 29
[7] R. Benzi, S. Ciliberto, G. Ruiz. Chavarria, and R. Tripiccione, Europhys.Lett. 24 (1993)275
[8] M. Briscolini, P. Santangelo, S. Succi, and R. Benzi, Phys.Rev.E.50 (1994) 1745
[9] R. Benzi, L. Biferale, S. Giliberto, M. V. Struglia, and R. Tripiccione, Physica. D. 96(1996)162
[10]She,Z.S, and Leveque Phys Rev.Lett,72,336 (1994)
[11] A.Arneodo, C.Baudet, F.Belin, R.Benzi, B.Castaing, B.Chabaud, R.Chavarria, S.Ciliberto, R.Camussi, F.Chilla, B.Dubrulle, YGagne, B.Hebral, J.Herweijer, M.Marchand, J.Maurer, J.F.Muzy, A.Naert, A.Noullez, J.Peinke, F.Roux,P, Tabeling, W.Vande Water, H.Willaime, Europhys.Lett.,34(6)(1996)411