有界速度梯度三维流动中标量扩散异常的约束条件
Constraints on scalar diffusion anomaly in three-dimensional flows having bounded velocity gradients
摘要：
本研究关注的是一个被动标量$θ$在三维流动有界速度梯度的衰减行为。给定一个初始光滑的标量分布，标量方差$<θ^2>$的衰减率$d<θ^2>/dt$被发现是有界的控制物理参数。此外，在零扩散率极限$κ\to0$中，如果存在与$κ$无关的$α_0\in(0,1]$，使得对于$α\leα_0$为$<|(-Δ)^{α/2}θ|^2><\infty$，则该速率作为$κ^{α_0}$消失。如果在极限$κ\to0$中，方差谱$Θ(k)$对于大的波数$k$保持比$k^{-1}$陡，则满足该条件。当不存在这样的正$α_0$时，标量场可以说变成虚拟奇异的。与Batchelor理论一致的一个合理的情况是，对于较小的$κ$，$Θ(k)$变得越来越浅，接近极限$κ\to0$中的Batchelor缩放$k^{-1}$。在这个经典的例子中，衰变率也会消失，尽管更慢--就像$(\ln P_r)^{-1}$，其中$P_r$是普朗特数或施密特数。因此，扩散异常被排除在一个广泛的标量分布，包括幂律谱不浅于$k^{-1}$。这意味着，为了有一个$κ$独立的和非零的衰变率，在小尺度的方差必须大于巴彻勒光谱所允许的。这些结果进行了讨论，在现有的文献中的渐近指数衰减$<θ^2>\sim e^{-γt}$，其中$γ>0$是独立的$κ$。
英文摘要：
This study is concerned with the decay behaviour of a passive scalar $θ$ in three-dimensional flows having bounded velocity gradients. Given an initially smooth scalar distribution, the decay rate $d<θ^2>/dt$ of the scalar variance $<θ^2>$ is found to be bounded in terms of controlled physical parameters. Furthermore, in the zero diffusivity limit, $κ\to0$, this rate vanishes as $κ^{α_0}$ if there exists an $α_0\in(0,1]$ independent of $κ$ such that $<|(-Δ)^{α/2}θ|^2><\infty$ for $α\leα_0$. This condition is satisfied if in the limit $κ\to0$, the variance spectrum $Θ(k)$ remains steeper than $k^{-1}$ for large wave numbers $k$. When no such positive $α_0$ exists, the scalar field may be said to become virtually singular. A plausible scenario consistent with Batchelor's theory is that $Θ(k)$ becomes increasingly shallower for smaller $κ$, approaching the Batchelor scaling $k^{-1}$ in the limit $κ\to0$. For this classical case, the decay rate also vanishes, albeit more slowly -- like $(\ln P_r)^{-1}$, where $P_r$ is the Prandtl or Schmidt number. Hence, diffusion anomaly is ruled out for a broad range of scalar distribution, including power-law spectra no shallower than $k^{-1}$. The implication is that in order to have a $κ$-independent and non-vanishing decay rate, the variance at small scales must necessarily be greater than that allowed by the Batchelor spectrum. These results are discussed in the light of existing literature on the asymptotic exponential decay $<θ^2>\sim e^{-γt}$, where $γ>0$ is independent of $κ$.
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