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Intermittent turbulence means that the turbulence eddies do not fill the space completely, so the dimension of an intermittent turbulence takes the values between 2 and 3. Turbulence diffusion is a super-diffusion, and the probability of density function is fat-tailed. In this paper, the viscosity term in the Navier-Stokes equation will be denoted as a fractional derivative of Laplatian operator. Dimensionless analysis shows that the order of the fractional derivative α is closely related to the dimension of intermittent turbulence D. For the homogeneous isotropic Kolmogorov turbulence, the order of the fractional derivatives α=2, i.e. the turbulence can be modeled by the integer order of Navier-Stokes equation. However, the intermittent turbulence must be modeled by the fractional derivative of Navier-Stokes equation. For the Kolmogorov turbulence, diffusion displacement is proportional to t3, i.e. Richardson diffusion, but for the intermittent turbulence, diffusion displacement is stronger than Richardson diffusion.
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Keywords:
- intermittent turbulence /
- fractional derivatives /
- dimension /
- diffusion
[1] Frisch U 1995 Turbulence (Cambridge: Cambridge University Press)
[2] Liu S D, Liang F M, Liu S K, Xin G J 2008 Atmospheric Turbulence (Beijing: Peking University Press) (in Chinese).[刘式达, 梁福明, 刘式适, 辛国君2008 大气湍流(北京: 北京大学出版社)]
[3] Hu F 1995 Turbulence, Intermittency, Atmosphere Boundary Layer (Beijing: Science Press) (in Chinese)[胡非, 湍流, 间歇性1995 大气边界层(北京: 科学出版社)]
[4] Uchaikin V V 2013 Fractional derivatives for physicists and engineers (Beijing: Higher Education Press)
[5] Chen W, Sun H G, Li X C 2012 Fractional derivative modeling in mechanical and engineering problems (Beijing: Science Press) p11-54 (in Chinese) [陈文, 孙洪广,李西成2012 力学与工程问题的分数阶导数建模(北京: 科学出版社) 第11–54 页]
[6] Chen W, Holm S 2004 J. Acoust. Soc Am. 115 1424
[7] Landau L D, Lifshitz E M 2013 Fluid Mechanics (5th edition) (Beijing: Higher Education Press) (in Chinese)[朗道, 栗弗席兹2013 流体动力学(第5 版) (北京: 高等教育出版社)]
[8] Chen W 2006 Chaos 16 023126
[9] Dewar R L, Detering F (eds) 2010 Complex physical, Biophysical and Econophysical systems (Singapore: World Scientific Publishing)
[10] Liu S D, Liu S K 2013 Fractals in Physics (Beijing: Peking University Press) (in Chinese) [刘式达, 刘式适 2013 物理学中的分形(北京: 北京大学出版社)]
[11] Chen W 2005 Chin. Phys. Lett. 22 2601
[12] Wang D LYu Z G Anh V 2012 Chin. Phys. B 21 080504
[13] Shang P J, Zang B J 2007 Chin. Phys. B 16 565
[14] Kang Y M, Jiang Y L 2008 Chin. Phys. Lett. 25 3578
[15] Chen W, Sun H G, Zhang X D, Korosak D 2010 Computers Math. Appl. 59 1754
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[1] Frisch U 1995 Turbulence (Cambridge: Cambridge University Press)
[2] Liu S D, Liang F M, Liu S K, Xin G J 2008 Atmospheric Turbulence (Beijing: Peking University Press) (in Chinese).[刘式达, 梁福明, 刘式适, 辛国君2008 大气湍流(北京: 北京大学出版社)]
[3] Hu F 1995 Turbulence, Intermittency, Atmosphere Boundary Layer (Beijing: Science Press) (in Chinese)[胡非, 湍流, 间歇性1995 大气边界层(北京: 科学出版社)]
[4] Uchaikin V V 2013 Fractional derivatives for physicists and engineers (Beijing: Higher Education Press)
[5] Chen W, Sun H G, Li X C 2012 Fractional derivative modeling in mechanical and engineering problems (Beijing: Science Press) p11-54 (in Chinese) [陈文, 孙洪广,李西成2012 力学与工程问题的分数阶导数建模(北京: 科学出版社) 第11–54 页]
[6] Chen W, Holm S 2004 J. Acoust. Soc Am. 115 1424
[7] Landau L D, Lifshitz E M 2013 Fluid Mechanics (5th edition) (Beijing: Higher Education Press) (in Chinese)[朗道, 栗弗席兹2013 流体动力学(第5 版) (北京: 高等教育出版社)]
[8] Chen W 2006 Chaos 16 023126
[9] Dewar R L, Detering F (eds) 2010 Complex physical, Biophysical and Econophysical systems (Singapore: World Scientific Publishing)
[10] Liu S D, Liu S K 2013 Fractals in Physics (Beijing: Peking University Press) (in Chinese) [刘式达, 刘式适 2013 物理学中的分形(北京: 北京大学出版社)]
[11] Chen W 2005 Chin. Phys. Lett. 22 2601
[12] Wang D LYu Z G Anh V 2012 Chin. Phys. B 21 080504
[13] Shang P J, Zang B J 2007 Chin. Phys. B 16 565
[14] Kang Y M, Jiang Y L 2008 Chin. Phys. Lett. 25 3578
[15] Chen W, Sun H G, Zhang X D, Korosak D 2010 Computers Math. Appl. 59 1754
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