湍流壳模型中被动标量奇异性研究
摘要
在湍流研究中,对被动标量场的研究是当前湍流研究中的一个热点。最近十几年大量的实验测量、数值模拟和理论分析结果表明,被动标量场具有很多自身独特的性质,甚至有些性质并不依赖于速度场本身。被动标量所表现出的有别于传统认知的奇异性促使人们对经典理论进行重新认识、修正或者提出新的理论来取代。
我们取得了三方面的成果:
一、Kraichnan对被动标量标度指数的研究表明,即使选取没有任何阵发性的高斯型速度场,其输运的被动标量场的标度指数也表现奇异性。为了全面研究高斯型速度场下被动量的统计特性,我们取被动标量壳模型演化方程中的速度场为高斯型且时间上是δ相关的随机场。这时该方程为线性朗之万方程,由此可推得被动标量概率密度分布函数的福克-普朗克方程。再利用数值计算,得到已计入统计关联的被动标量的概率密度分布函数,它在壳数n较小的壳层上是近似的高斯分布,随着壳数越大,分布将有不同程度的翘尾现象。我们计算的高斯型速度场下与GOY壳模型速度场下的被动量的概率密度分布函数都有明显偏离高斯分布的特点,在小壳数层我们的计算结果比GOY计算结果奇异性较强,但随着壳层数n增大(n＞10),GOY计算结果反过来比我们的计算结果表现更强的奇异性。同时,由分布函数计算得到被动量的反常标度指数比kraichnan的结果更接近实验值。根据关联函数定义计算被动标量壳层间关联,结果表明,各壳层间最近邻壳的关联度最大,次近邻及再次近邻壳之间的关联度次之,表现明显的量级分布。
二、为了研究速度场的相空间特性对被动量奇异性的影响,我们首先计算了GOY壳模型下速度场的壳空间关联函数。然后选用具有空间关联的、满足多维高斯分布的速度场作为输运场,研究其下被动量的标度奇异性。这种速度场保证了壳模型计算中壳间关联的存在。与我们计算的没有任何关联的高斯随机速度场下被动量的标度指数相比,虽然两种不同速度场输运的被动量场的标度指数都存在奇异性,但多维高斯分布型速度场下的被动量标度指数具有较强的奇异性。同时还计算了多维高斯分布型速度场与独立高斯随机速度场下的被动标量场不同壳的概率密度分布函数,发现两者的分布函数都有“翘尾现象”,但随着壳数增加,前者的翘尾程度更高,说明速度场的壳空间关联对被动量标度指数奇异性的影响随着研究尺度的缩小而增大。
三、近年来,研究者对湍流速度场时间关联的研究,揭示了时间特性对整体流场统计性质的影响。为了研究速度场的时间与空间联合特性对被动量奇异性的影响,我们计算了GOY壳模型速度场的时空关联函数,发现其关联函数具有指数衰减的特性。因此采用时间上具有指数衰减特性,在时间和空间上都表现弱关联的色噪声型速度场作为输运场,通过壳模型计算,研究速度场时空关联对被动量标度奇异性的影响。与无任何关联的随机高斯速度场、SL模型非高斯分布速度场以及有空间关联无时间关联的多维高斯型速度场等相比,虽然它们的被动量扩展自相似标度指数都存在奇异性,但色噪声型速度场下的被动量扩展自相似标度指数具有较强的奇异性。非高斯修正后具有时空关联速度场的被动量扩展自相似标度指数呈现了更强的奇异性,但仍弱于GOY壳模型速度场下被动量的扩展自相似标度指数。另外我们还计算了有时空关联速度场的标度指数。
Since the 1980s,studies of passive scalar have been one of hotspots in turbulence research.Experiments and numerical simulations have provided lots of new phenomena and concepts,which put an impetus to reinterpret the classical turbulent theories and modify application models at present.
Our studies on passive scalar compose of three parts:
1、The power law of the structure function is an important property of the turbulence.In this thesis,a shell-model version of passive scalar problem is introduced which is inspired by the model of K.Ohkitani and M.Yakhot.As in the original problem,the prescribed random velocity field is Gaussian andδcorrelated in time.Deterministic differential equations are regarded as nonlinear Langevin equation.Then,the Fokker-Planck equations of PDF for passive scalars are obtained and solved numerically.In energy input range(n＜5,n is shell number.),the probability distribution function(PDF) of passive scalars are near the Gaussian distribution.In inertial range(5≤n≤16) and dissipation range(n≥17),the probability distribution function(PDF) of passive scalars has obvious intermittence. And the scaling power of passive scalar is anomalous.The results of numerical simulations are compared with experimental measurements.
2、We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of passive scalars of turbulence.Different to the original problem, the distribution function of the prescribed random velocity field is the multi-dimensional normal and delta-correlated in time.Here,our random velocity field is spatially correlative.For comparison,we also given the result obtained by the Gaussian random velocity field without spatial correlation The anomalous scaling exponents H(p) of passive scalar advected by two kinds of random velocity above are determined for structure function up to p = 15 by numerical simulations of the random shell model with Runge-Kutta methods to solve the stochastic differential equations. We observed that the H(p)'s obtained by the multi-dimension normal distribution random velocity are more anomalous than those obtained by the independent Gaussian random velocity.
3、We consider temporal-spatial correlations of the turbulence's velocity here. We simply analyze the correlation of the velocity on the time-space serial.With numerical simulation on GOY shell model,we show the correlation function is exponential decay function.The advecting velocity field is regarded as colored noise field,which is spatially and temporally correlative.For comparison,we also given the scaling exponents of passive scalars obtained by the Gaussian random velocity field, the multi-dimensional normal velocity field and SL velocity field.We observed that anomalous RSH scaling exponents H(p)/H(2) of passive scalar obtained by colored noise field are more anomalous than those obtained by other three velocity fields.
我们取得了三方面的成果:
一、Kraichnan对被动标量标度指数的研究表明,即使选取没有任何阵发性的高斯型速度场,其输运的被动标量场的标度指数也表现奇异性。为了全面研究高斯型速度场下被动量的统计特性,我们取被动标量壳模型演化方程中的速度场为高斯型且时间上是δ相关的随机场。这时该方程为线性朗之万方程,由此可推得被动标量概率密度分布函数的福克-普朗克方程。再利用数值计算,得到已计入统计关联的被动标量的概率密度分布函数,它在壳数n较小的壳层上是近似的高斯分布,随着壳数越大,分布将有不同程度的翘尾现象。我们计算的高斯型速度场下与GOY壳模型速度场下的被动量的概率密度分布函数都有明显偏离高斯分布的特点,在小壳数层我们的计算结果比GOY计算结果奇异性较强,但随着壳层数n增大(n＞10),GOY计算结果反过来比我们的计算结果表现更强的奇异性。同时,由分布函数计算得到被动量的反常标度指数比kraichnan的结果更接近实验值。根据关联函数定义计算被动标量壳层间关联,结果表明,各壳层间最近邻壳的关联度最大,次近邻及再次近邻壳之间的关联度次之,表现明显的量级分布。
二、为了研究速度场的相空间特性对被动量奇异性的影响,我们首先计算了GOY壳模型下速度场的壳空间关联函数。然后选用具有空间关联的、满足多维高斯分布的速度场作为输运场,研究其下被动量的标度奇异性。这种速度场保证了壳模型计算中壳间关联的存在。与我们计算的没有任何关联的高斯随机速度场下被动量的标度指数相比,虽然两种不同速度场输运的被动量场的标度指数都存在奇异性,但多维高斯分布型速度场下的被动量标度指数具有较强的奇异性。同时还计算了多维高斯分布型速度场与独立高斯随机速度场下的被动标量场不同壳的概率密度分布函数,发现两者的分布函数都有“翘尾现象”,但随着壳数增加,前者的翘尾程度更高,说明速度场的壳空间关联对被动量标度指数奇异性的影响随着研究尺度的缩小而增大。
三、近年来,研究者对湍流速度场时间关联的研究,揭示了时间特性对整体流场统计性质的影响。为了研究速度场的时间与空间联合特性对被动量奇异性的影响,我们计算了GOY壳模型速度场的时空关联函数,发现其关联函数具有指数衰减的特性。因此采用时间上具有指数衰减特性,在时间和空间上都表现弱关联的色噪声型速度场作为输运场,通过壳模型计算,研究速度场时空关联对被动量标度奇异性的影响。与无任何关联的随机高斯速度场、SL模型非高斯分布速度场以及有空间关联无时间关联的多维高斯型速度场等相比,虽然它们的被动量扩展自相似标度指数都存在奇异性,但色噪声型速度场下的被动量扩展自相似标度指数具有较强的奇异性。非高斯修正后具有时空关联速度场的被动量扩展自相似标度指数呈现了更强的奇异性,但仍弱于GOY壳模型速度场下被动量的扩展自相似标度指数。另外我们还计算了有时空关联速度场的标度指数。
Since the 1980s,studies of passive scalar have been one of hotspots in turbulence research.Experiments and numerical simulations have provided lots of new phenomena and concepts,which put an impetus to reinterpret the classical turbulent theories and modify application models at present.
Our studies on passive scalar compose of three parts:
1、The power law of the structure function is an important property of the turbulence.In this thesis,a shell-model version of passive scalar problem is introduced which is inspired by the model of K.Ohkitani and M.Yakhot.As in the original problem,the prescribed random velocity field is Gaussian andδcorrelated in time.Deterministic differential equations are regarded as nonlinear Langevin equation.Then,the Fokker-Planck equations of PDF for passive scalars are obtained and solved numerically.In energy input range(n＜5,n is shell number.),the probability distribution function(PDF) of passive scalars are near the Gaussian distribution.In inertial range(5≤n≤16) and dissipation range(n≥17),the probability distribution function(PDF) of passive scalars has obvious intermittence. And the scaling power of passive scalar is anomalous.The results of numerical simulations are compared with experimental measurements.
2、We propose a new approach to the old-standing problem of the anomaly of the scaling exponents of passive scalars of turbulence.Different to the original problem, the distribution function of the prescribed random velocity field is the multi-dimensional normal and delta-correlated in time.Here,our random velocity field is spatially correlative.For comparison,we also given the result obtained by the Gaussian random velocity field without spatial correlation The anomalous scaling exponents H(p) of passive scalar advected by two kinds of random velocity above are determined for structure function up to p = 15 by numerical simulations of the random shell model with Runge-Kutta methods to solve the stochastic differential equations. We observed that the H(p)'s obtained by the multi-dimension normal distribution random velocity are more anomalous than those obtained by the independent Gaussian random velocity.
3、We consider temporal-spatial correlations of the turbulence's velocity here. We simply analyze the correlation of the velocity on the time-space serial.With numerical simulation on GOY shell model,we show the correlation function is exponential decay function.The advecting velocity field is regarded as colored noise field,which is spatially and temporally correlative.For comparison,we also given the scaling exponents of passive scalars obtained by the Gaussian random velocity field, the multi-dimensional normal velocity field and SL velocity field.We observed that anomalous RSH scaling exponents H(p)/H(2) of passive scalar obtained by colored noise field are more anomalous than those obtained by other three velocity fields.
引文
[1] A. N. Kolmogorov, Dokl. Akad. Nauk SSSR, Vol. 30,301-305(1941)
[2] A. N. Kolmogorov, Dokl. Proc. R. Soc. Lond. A 434, 9 (1991)
[3] G. Falkovich, K. Gawedzki and M. Vergassola, Rev. Mod. Phys., Vol. 73, 913 (2001)
[4] L. D. Landau, and E. M. lifshiz, Fluid Mechancis, Addison-Wesly, (1959)
[5] A. N. Kolmogorov, J. Fluid. Mech., Vol.13, 82-85(1962).
[6] A. S. Monin, and A. M. Yaglom, Statistical fluid mechanics, The M. I. T. Press, Cambridge (1971).
[7] A. M. Obukhov, J. Fluid. Mech, Vol. 13,77-81 (1962).
[8] E. A. Novikov, Prikl. Mat. Mech, Vol.35, 266-277(1971).
[9] B. Mandelbrot, J. Elers, et al, Eds., No.12, Springer-Verlag, Berlin, 333-351(1972)
[10] S. A. Orszag, J. Fluid. Mech, Vol.41, 363-386 (1970).
[11] R. H. Kraichnan, J. Fluid. Mech, Vol.62, Part 2, 305-330 (1974).
[12] Nelkin, J. Statistical Phys, Vol.54, 1-15 (1989).
[13] E. A. Novikovand and R. W. Stewart, Izv. Akad. Nauk. SSSR, Ser. Geophys., Vol.3, 58-69 (1972)
[14] U. Frish, P. L. Sulem and M. Nelkin, J. Fluid. Mech, Vol.87, 719-736 (1978)
[15] H. A. Rose and P. L. sulem, J. Physique, 39, 441-484 (1978)
[16] G. Paladin and A. Vulpiani, Phys. Reports, Vol.156, 147-225 (1987)
[17] B. Mandelbrot, A. Dold and B. Eckmann, Eds., No. 565, Springer-Verlag, Berlin, 121-145(1976)
[18] R. Benzi, G. Paladin, G. Parisi and A. Vulpiani, J. Phys. A. : Math. Gen., Vol.17, 3521-3531(1984)
[19] D. Schertzer and S. Lovejoy, L. J. S. Bradbury et al, Eds., Springer, Berlin, 7-33 (1985)
[20] C. Meneveau and K. R. Sreenivasan, Phys. Rev. Lett. , Vol. 59, 1424-1427, (1987)
[21] She. Z. S, Phys. Rev. Lett. , Vol.66, 600-603 (1991)
[22] She. Z. S and E. Leveque, Phys. Rev. Lett. , Vol. 72, 336-339 (1994)
[23] She. Z. S and E. C. Waymire, Phys. Rev. Lett. , Vol. 74, 262-265 (1995)
[24] V. N. Desnyansky and E. A. Novikov, Atoms Oceanic Phys. 10, 127 (1974)
[25] V. N. Desnyansky and E. A. Novikov, prinkl. Mat. Mekh. 38, 507 (1974)
[26] K. Ohkitani, M. Yakhot, Phys. Rev. Lett. 60, 983 (1988)
[27] K. Ohkitani, M. Yakhot, Prog. Theor. Phys. 81, 329 (1988)
[28] T. bohr, etc. Dynamical Syatems Approch to Turbulence, Cambridge University Press, Cambridge (1998)
[29] A. M. Obukhov Izv. Akad. Nauk. SSSR, Vol.13, 58-69 (1949)
[30] S. Corrsin, J. Appl. Phys, Vol.23, 113-118 (1951)
[31] R.A.Antonia and A.J.Chambers, Boundary-Layer Meteorol., Vol.18, 399-410 (1980)
[32] U. Frish, J. Fluid. Mech, 87, 719 (1978)
[33] C. W, Van Atta, Phys. Fluids, 14, 1803 (1971)
[34] N. Z. Cao and S. Y. Chen, CD/9704011
[35] F. X. Witkowski, etc., Nature, 78, 392 (1998)
[36] M.Yamada and Koij. Ohkitani, J. Phys. Soc, 56, 4210 (1987)
[37] D.Pisarenko, Phys. Fluids A5(10), 2533 (1993)
[38] A. Brandenburg, Phys. Rev. Lett, 69, 605 (1992)
[39] R. A. Antonia and K. R. Sreenivasan, Phys. Fluids 20, 1800 (1977).
[40] S. Tavoularis and S. Corrsin, J.Fluid. Mech, 104, 311-347 (1981)
[41] B. Castaing, et al., J. Fluid. Mech, 204, 1-30(1989)
[42] Ya. G.Sinai, V. Yakhot, Phys. Rev. Lett, 63, 1962 (1989)
[43] B. Castaing and G. Gunaratne, etc. J. Phys Mech. 204, 1 (1989)
[44] S. B. Pope and E. M. Ching, Phy. Fluids A 5, 1529 (1993)
[45] A. Pumir, et al., Phys. Rev. Lett, 75, 3114-3117(1991)
[46] A. R.Kerstein, J. Fluid. Mech, 23, 1474-482(1991)
[47] A. R. Kerstein and P. McMurtry, Phys. Fluids, A 5, 2264-2277(1993)
[48] Gollub J. P, et al., Phys. Rev. Lett, 67, 3507-3510(1991)
[49] Jayesh and Z. Warhaft, Phys. Rev. Lett, 67, 3503-3506(1991)
[50] M. R. Overholt and S. B. Pope, Phys. Fluids., 8, 3128-3148(1996)
[51] Z. Warhaft, Annu. Rev. Fluid. Mech, 32, 203-240(2000)
[52] F.A. Jaberi, et al., J. Fluid. Mech, 313, 241-282(1996)
[53] E. S. C. Ching, CD/0111030
[54] Y. Cohen and T. Gilbert, Phys. Rev. E, 65, 026314 (2002)
[55] Tomas. Gilbert, Phys. Rev. E, 70, 015301 (2004)
[1]V.Belinicher and V.L'vov,Sov.Phys.JETP 66,303(1987).
[2]V.S.L,vov,I.Procaccia,and A.L.Fairhall,Phys.Rev.E,50,4684(1994)
[3]R.H.Kraichnan,V.Yakhot,and S.Chen,Phys.Rev.Lett.75,240(1995)
[4]M.Chertkov,etc.Phys.Rev.E,52,4924(1995)
[5]R.H.Kraichnan,Phys.Rev.Lett.72.1016(1994)
[6]E.S.C.Ching,CD/0111030
[7]A.Wirth,L.Biferale,Phys.Rev.E,54,4982(1996)
[8]C.Meneveau,etc.Phys.Rev.A,41,894(1990)
[9]M.H.Jensen,G.Paladin,and Vulpiani,Phys.Rev.A,43,798(1991)
[10]Michio Yamada,K.Ohkitani,J.Phys.Soc.Jpn.56,4210(1987)
[11]M.H.Jensen,G.Paladin,and Vulpiani,Phys.Rev.A,45,7214(1992)
[12]H.Riskem,The Fokker-Planck Equation,(Springer-Verlag Berlin Heidelberg 1984,1992)
[13]R.A.Antonia,etc.Phys.Rev.A,30,2704(1984)
[14]Zhao Ying-kui,Chen Shi-gang and Wang Guang-rui,Chin.Phys.2007(Accepted)
[15]现代应用数学手册,p653,清华大学出版社,北京(2005)
[1]R.H.Kraichnan,Phys.Rev.Lett.72,1016(1994).
[2]J.C.R.Hunt,O.M.Phillips,and D.Williams,Turbulence and Stochastic Processes,(Proc.R.Soc.London A 434,1-240 1991).
[3]U.Frisch,Turbulence,(Cambridge University Press 1995)
[4]T.Bohr,M.H Jensen,G.Paladin,and A.Vulpiani,Dynamical Systems Approach to Turbulence,(Cambridge University Press 1998).
[5]G.Falkovich,K.Gawedzki,and M.Vergassola,Rev.Mod.Phys.73,913(2001).
[6]K.Gawedzki and A.Kupiainen,Phys.Rev.Lett.75,3834(1995).
[7]M.Chertkov,G.Falkovich,I.Kolokolov,and V.Lebedev,Phys.Rev.E 52,4924(1995).
[8]M.H.Jensen,G.Paladin,and A.Vulpiani,Phys.Rev.A 45,7214(1992).
[9]Michio Wamada,K.Ohkitani,J.Phys.Soc.Jpn.56,4210(1987)
[10]Zhang Xiao-Qiang,Wang Guang-rui and Chen Shi-gang,Commun.Theor.Phys.2007(Accepted).
[11]Zhao Ying-kui,Chen Shi-gang and Wang Guang-rui,Chin.Phys.2007(Accepted).
[12]D.E.Knuth,Seminumerical Algorithms,3rd ed.,vol.2 of The Art of Computer Programming(Reading,MA:Addison-Wesley,1998).
[13]A.M.Obukhov,Izv.Akad.Nauk.SSSR,Vol.13,58(1949)
[1] P. J. Diamessis and K. K. Nomura, Phys. Fluids, 12, 1166-1188 (2000)
[2] R. Fox and P. K. Yeung, Phys. Fluids, 15, 961-985 (2003)
[3] M, Gonzalez and P. Paranthoen, Phys. Fluids., 16, 219-221(2004)
[4] Zhang Xiao-Qiang, Wang Guang-rui and Chen Shi-gang, Commun. Theor. Phys. 2007(Accepted).
[5] Zhao Ying-kui, Chen Shi-gang and Wang Guang-rui, Chin. Phys. 2007(Accepted).
[6] Zhang Xiao-Qiang, Wang Guang-rui and Chen Shi-gang, Commun. Theor. Phys. 2008(Accepted).
[7] V. L'vov, E. Podivilov, and I. Procaccia, Phys. Rev. E 55, 7030 (1997).
[8] F. Hayot and C. Jayaprakash, Phys. Rev. E 57, R4867(1998).
[9] F. Hayot and C. Jayaprakash, Int. J. Mod. Phys. B 14, 1781(2000).
[10] D. Mitra and R. Pandit, Phys. Rev. Lett 93, 024501(2004).
[11] D. Mitra and R. Pandit, Phys. Rev. Lett. 95, 144501(2005).
[12] Z. S. She and E. Leveque, Phys. Rev. Lett. 72, 336(1994)
[13] A.N. Kolmogorov, Dokl. Akad. Nauk SSSR, Vol.30, 301(1941)
[14] A.M. Obukhov, Izv. Akad.Nauk. SSSR, Vol.13, 58(1949)
[2] A. N. Kolmogorov, Dokl. Proc. R. Soc. Lond. A 434, 9 (1991)
[3] G. Falkovich, K. Gawedzki and M. Vergassola, Rev. Mod. Phys., Vol. 73, 913 (2001)
[4] L. D. Landau, and E. M. lifshiz, Fluid Mechancis, Addison-Wesly, (1959)
[5] A. N. Kolmogorov, J. Fluid. Mech., Vol.13, 82-85(1962).
[6] A. S. Monin, and A. M. Yaglom, Statistical fluid mechanics, The M. I. T. Press, Cambridge (1971).
[7] A. M. Obukhov, J. Fluid. Mech, Vol. 13,77-81 (1962).
[8] E. A. Novikov, Prikl. Mat. Mech, Vol.35, 266-277(1971).
[9] B. Mandelbrot, J. Elers, et al, Eds., No.12, Springer-Verlag, Berlin, 333-351(1972)
[10] S. A. Orszag, J. Fluid. Mech, Vol.41, 363-386 (1970).
[11] R. H. Kraichnan, J. Fluid. Mech, Vol.62, Part 2, 305-330 (1974).
[12] Nelkin, J. Statistical Phys, Vol.54, 1-15 (1989).
[13] E. A. Novikovand and R. W. Stewart, Izv. Akad. Nauk. SSSR, Ser. Geophys., Vol.3, 58-69 (1972)
[14] U. Frish, P. L. Sulem and M. Nelkin, J. Fluid. Mech, Vol.87, 719-736 (1978)
[15] H. A. Rose and P. L. sulem, J. Physique, 39, 441-484 (1978)
[16] G. Paladin and A. Vulpiani, Phys. Reports, Vol.156, 147-225 (1987)
[17] B. Mandelbrot, A. Dold and B. Eckmann, Eds., No. 565, Springer-Verlag, Berlin, 121-145(1976)
[18] R. Benzi, G. Paladin, G. Parisi and A. Vulpiani, J. Phys. A. : Math. Gen., Vol.17, 3521-3531(1984)
[19] D. Schertzer and S. Lovejoy, L. J. S. Bradbury et al, Eds., Springer, Berlin, 7-33 (1985)
[20] C. Meneveau and K. R. Sreenivasan, Phys. Rev. Lett. , Vol. 59, 1424-1427, (1987)
[21] She. Z. S, Phys. Rev. Lett. , Vol.66, 600-603 (1991)
[22] She. Z. S and E. Leveque, Phys. Rev. Lett. , Vol. 72, 336-339 (1994)
[23] She. Z. S and E. C. Waymire, Phys. Rev. Lett. , Vol. 74, 262-265 (1995)
[24] V. N. Desnyansky and E. A. Novikov, Atoms Oceanic Phys. 10, 127 (1974)
[25] V. N. Desnyansky and E. A. Novikov, prinkl. Mat. Mekh. 38, 507 (1974)
[26] K. Ohkitani, M. Yakhot, Phys. Rev. Lett. 60, 983 (1988)
[27] K. Ohkitani, M. Yakhot, Prog. Theor. Phys. 81, 329 (1988)
[28] T. bohr, etc. Dynamical Syatems Approch to Turbulence, Cambridge University Press, Cambridge (1998)
[29] A. M. Obukhov Izv. Akad. Nauk. SSSR, Vol.13, 58-69 (1949)
[30] S. Corrsin, J. Appl. Phys, Vol.23, 113-118 (1951)
[31] R.A.Antonia and A.J.Chambers, Boundary-Layer Meteorol., Vol.18, 399-410 (1980)
[32] U. Frish, J. Fluid. Mech, 87, 719 (1978)
[33] C. W, Van Atta, Phys. Fluids, 14, 1803 (1971)
[34] N. Z. Cao and S. Y. Chen, CD/9704011
[35] F. X. Witkowski, etc., Nature, 78, 392 (1998)
[36] M.Yamada and Koij. Ohkitani, J. Phys. Soc, 56, 4210 (1987)
[37] D.Pisarenko, Phys. Fluids A5(10), 2533 (1993)
[38] A. Brandenburg, Phys. Rev. Lett, 69, 605 (1992)
[39] R. A. Antonia and K. R. Sreenivasan, Phys. Fluids 20, 1800 (1977).
[40] S. Tavoularis and S. Corrsin, J.Fluid. Mech, 104, 311-347 (1981)
[41] B. Castaing, et al., J. Fluid. Mech, 204, 1-30(1989)
[42] Ya. G.Sinai, V. Yakhot, Phys. Rev. Lett, 63, 1962 (1989)
[43] B. Castaing and G. Gunaratne, etc. J. Phys Mech. 204, 1 (1989)
[44] S. B. Pope and E. M. Ching, Phy. Fluids A 5, 1529 (1993)
[45] A. Pumir, et al., Phys. Rev. Lett, 75, 3114-3117(1991)
[46] A. R.Kerstein, J. Fluid. Mech, 23, 1474-482(1991)
[47] A. R. Kerstein and P. McMurtry, Phys. Fluids, A 5, 2264-2277(1993)
[48] Gollub J. P, et al., Phys. Rev. Lett, 67, 3507-3510(1991)
[49] Jayesh and Z. Warhaft, Phys. Rev. Lett, 67, 3503-3506(1991)
[50] M. R. Overholt and S. B. Pope, Phys. Fluids., 8, 3128-3148(1996)
[51] Z. Warhaft, Annu. Rev. Fluid. Mech, 32, 203-240(2000)
[52] F.A. Jaberi, et al., J. Fluid. Mech, 313, 241-282(1996)
[53] E. S. C. Ching, CD/0111030
[54] Y. Cohen and T. Gilbert, Phys. Rev. E, 65, 026314 (2002)
[55] Tomas. Gilbert, Phys. Rev. E, 70, 015301 (2004)
[1]V.Belinicher and V.L'vov,Sov.Phys.JETP 66,303(1987).
[2]V.S.L,vov,I.Procaccia,and A.L.Fairhall,Phys.Rev.E,50,4684(1994)
[3]R.H.Kraichnan,V.Yakhot,and S.Chen,Phys.Rev.Lett.75,240(1995)
[4]M.Chertkov,etc.Phys.Rev.E,52,4924(1995)
[5]R.H.Kraichnan,Phys.Rev.Lett.72.1016(1994)
[6]E.S.C.Ching,CD/0111030
[7]A.Wirth,L.Biferale,Phys.Rev.E,54,4982(1996)
[8]C.Meneveau,etc.Phys.Rev.A,41,894(1990)
[9]M.H.Jensen,G.Paladin,and Vulpiani,Phys.Rev.A,43,798(1991)
[10]Michio Yamada,K.Ohkitani,J.Phys.Soc.Jpn.56,4210(1987)
[11]M.H.Jensen,G.Paladin,and Vulpiani,Phys.Rev.A,45,7214(1992)
[12]H.Riskem,The Fokker-Planck Equation,(Springer-Verlag Berlin Heidelberg 1984,1992)
[13]R.A.Antonia,etc.Phys.Rev.A,30,2704(1984)
[14]Zhao Ying-kui,Chen Shi-gang and Wang Guang-rui,Chin.Phys.2007(Accepted)
[15]现代应用数学手册,p653,清华大学出版社,北京(2005)
[1]R.H.Kraichnan,Phys.Rev.Lett.72,1016(1994).
[2]J.C.R.Hunt,O.M.Phillips,and D.Williams,Turbulence and Stochastic Processes,(Proc.R.Soc.London A 434,1-240 1991).
[3]U.Frisch,Turbulence,(Cambridge University Press 1995)
[4]T.Bohr,M.H Jensen,G.Paladin,and A.Vulpiani,Dynamical Systems Approach to Turbulence,(Cambridge University Press 1998).
[5]G.Falkovich,K.Gawedzki,and M.Vergassola,Rev.Mod.Phys.73,913(2001).
[6]K.Gawedzki and A.Kupiainen,Phys.Rev.Lett.75,3834(1995).
[7]M.Chertkov,G.Falkovich,I.Kolokolov,and V.Lebedev,Phys.Rev.E 52,4924(1995).
[8]M.H.Jensen,G.Paladin,and A.Vulpiani,Phys.Rev.A 45,7214(1992).
[9]Michio Wamada,K.Ohkitani,J.Phys.Soc.Jpn.56,4210(1987)
[10]Zhang Xiao-Qiang,Wang Guang-rui and Chen Shi-gang,Commun.Theor.Phys.2007(Accepted).
[11]Zhao Ying-kui,Chen Shi-gang and Wang Guang-rui,Chin.Phys.2007(Accepted).
[12]D.E.Knuth,Seminumerical Algorithms,3rd ed.,vol.2 of The Art of Computer Programming(Reading,MA:Addison-Wesley,1998).
[13]A.M.Obukhov,Izv.Akad.Nauk.SSSR,Vol.13,58(1949)
[1] P. J. Diamessis and K. K. Nomura, Phys. Fluids, 12, 1166-1188 (2000)
[2] R. Fox and P. K. Yeung, Phys. Fluids, 15, 961-985 (2003)
[3] M, Gonzalez and P. Paranthoen, Phys. Fluids., 16, 219-221(2004)
[4] Zhang Xiao-Qiang, Wang Guang-rui and Chen Shi-gang, Commun. Theor. Phys. 2007(Accepted).
[5] Zhao Ying-kui, Chen Shi-gang and Wang Guang-rui, Chin. Phys. 2007(Accepted).
[6] Zhang Xiao-Qiang, Wang Guang-rui and Chen Shi-gang, Commun. Theor. Phys. 2008(Accepted).
[7] V. L'vov, E. Podivilov, and I. Procaccia, Phys. Rev. E 55, 7030 (1997).
[8] F. Hayot and C. Jayaprakash, Phys. Rev. E 57, R4867(1998).
[9] F. Hayot and C. Jayaprakash, Int. J. Mod. Phys. B 14, 1781(2000).
[10] D. Mitra and R. Pandit, Phys. Rev. Lett 93, 024501(2004).
[11] D. Mitra and R. Pandit, Phys. Rev. Lett. 95, 144501(2005).
[12] Z. S. She and E. Leveque, Phys. Rev. Lett. 72, 336(1994)
[13] A.N. Kolmogorov, Dokl. Akad. Nauk SSSR, Vol.30, 301(1941)
[14] A.M. Obukhov, Izv. Akad.Nauk. SSSR, Vol.13, 58(1949)