turbulence的多重分形模型与纳维-斯托克斯方程的对应
A correspondence between the multifractal model of turbulence and the Navier-Stokes equations
摘要：
通过比较多重分形湍流模型和$d$空间维纳维-斯托克斯方程的耗散长度尺度，研究了它们之间的对应关系.在Kolmogorov的1941年理论中，关键参数$h$是Navier-Stokes不变性标度中的指数，固定为$h=1/3$，但在多重分形理论中允许值的谱。考虑到所有的纳维-斯托克斯方程的导数，它被发现，为了保持多重分形谱$C(h)$必须从下面有界，使得$C(h) \geq 1-3h$，这是符合五分之四定律。此外，$h$还必须从下面限定，使得$h \geq (1-d)/3$。当$d=3$时，$h$的允许范围由$h \geq -2/3$给出，从而使$h$远离$h=-1$。这一影响进行了讨论。
英文摘要：
We study a correspondence between the multifractal model of turbulence and the Navier-Stokes equations in $d$ spatial dimensions by comparing their respective dissipation length scales. In Kolmogorov's 1941 theory the key parameter $h$, which is an exponent in the Navier-Stokes invariance scaling, is fixed at $h=1/3$ but is allowed a spectrum of values in multifractal theory. Taking into account all derivatives of the Navier-Stokes equations, it is found that for this correspondence to hold the multifractal spectrum $C(h)$ must be bounded from below such that $C(h) \geq 1-3h$, which is consistent with the four-fifths law. Moreover, $h$ must also be bounded from below such that $h \geq (1-d)/3$. When $d=3$ the allowed range of $h$ is given by $h \geq -2/3$ thereby bounding $h$ away from $h=-1$. The implications of this are discussed.
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