论文标题
强迫湍流中远程相关性和热光谱的出现
Emergence of long-range correlations and thermal spectra in forced turbulence
论文作者
论文摘要
最近的数值研究表明，强迫，统计上的各向同性湍流会发展出“热平衡”频谱，$ \ Mathcal {e}（k）（k）\ propto k^2 $在大尺度上。这种行为提出了一个难题，因为它似乎暗示了非零saffman积分的增长，这将需要纵向速度相关函数，$χ（r）$，以满足$χ（r \ to \ to \ infty）\ propto r^{-3} $。众所周知，Saffman积分是腐烂的湍流的不变性，这正是因为非本地相互作用（即通过压力波交换而相互作用）太弱而无法产生此类相关性。受强迫性质的某些限制，我们认为强迫湍流应该是正确的。我们表明，由于线性动量的湍流扩散，出现了远距离的相关性和$ k^2 $频谱，并且仅扩展到随着时间而缓慢增长的最大尺度。这张照片有许多有趣的后果。首先，如果强迫产生具有明显线性动量的涡流（如所谓的Saffman湍流），则未达到热光谱 - 相反，会形成较浅的光谱。其次，湍流的能量被迫持续一段时间，然后允许衰减服从Saffman的衰减法律，比强迫阶段的持续时间更长。
Recent numerical studies have shown that forced, statistically isotropic turbulence develops a `thermal equilibrium' spectrum, $\mathcal{E}(k) \propto k^2$, at large scales. This behaviour presents a puzzle, as it appears to imply the growth of a non-zero Saffman integral, which would require the longitudinal velocity correlation function, $χ(r)$, to satisfy $χ(r\to \infty)\propto r^{-3}$. As is well known, the Saffman integral is an invariant of decaying turbulence, precisely because non-local interactions (i.e., interactions via exchange of pressure waves) are too weak to generate such correlations. Subject to certain restrictions on the nature of the forcing, we argue that the same should be true for forced turbulence. We show that long-range correlations and a $k^2$ spectrum arise as a result of the turbulent diffusion of linear momentum, and extend only up to a maximum scale that grows slowly with time. This picture has a number of interesting consequences. First, if the forcing generates eddies with significant linear momentum (as in so-called Saffman turbulence), a thermal spectrum is not reached - instead, a shallower spectrum develops. Secondly, the energy of turbulence that is forced for a while and then allowed to decay obeys Saffman's decay laws for a period that is much longer than the duration of the forcing stage.
