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中国物理学会期刊

间歇湍流的分数阶动力学

CSTR: 32037.14.aps.63.074701

Fractional derivative dynamics of intermittent turbulence

CSTR: 32037.14.aps.63.074701
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  • 间歇湍流意味着湍流涡旋并不充满空间,其维数介于2和3之间. 湍流扩散为超扩散,且概率密度分布具有长尾特征. 本文将流体力学的Navier-Stokes(NS)方程中的黏性项用分数阶的拉普拉斯算子表达. 分析表明,分数阶拉普拉斯的阶数α 和间歇湍流的维数D相联系. 对于均匀各向同性的Kolmogorov湍流α=2,即用整数阶NS方程描述. 而对于间歇性湍流,一定用分数阶的NS方程来描述. 对于Kolmogorov湍流,扩散方差正比于t3,即Richardson扩散. 而对于间歇性湍流,扩散方差要比Richardson扩散更强.

     

    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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