Reynolds number of transition and self-organized criticality of strong turbulence

A turbulent flow is characterized by velocity fluctuations excited in an extremely broad interval of wave numbers $k>{\Lambda }_f$, where ${\Lambda }_f$ is a relatively small set of the wave vectors where energy is pumped into fluid by external forces. Iterative averaging over small-scale velocity fluctuations from the interval ${\Lambda }_f<k\le {\Lambda }_0$, where $\eta =2\pi /{\Lambda }_0$ is the dissipation scale, leads to an infinite number of “relevant” scale-dependent coupling constants (Reynolds numbers) ${\text{Re}}_n\left(k\right)=O\left(1\right)$. It is shown that in the infrared limit $k\to {\Lambda }_f$, the Reynolds numbers $\text{Re}\left(k\right)\to {\text{Re}}_{\text{tr}}$, where ${\text{Re}}_{\text{tr}}$ is the recently numerically and experimentally discovered universal Reynolds number of “smooth” transition from Gaussian to anomalous statistics of spatial velocity derivatives. The calculated relation $\text{Re}\left({\Lambda }_f\right)={\text{Re}}_{\text{tr}}$ “selects” the lowest-order nonlinearity as the only relevant one. This means that in the infrared limit $k\to {\Lambda }_f$, all high-order nonlinearities generated by the scale elimination sum up to zero.

Figure 1

Normalized moments of the dissipation rate $\frac{\overline{{\mathcal{E}}^n}}{{\overline{\mathcal{E}}}^n}$ in homogeneous and isotropic turbulence. References [7] and [8]. DSY stands for Donzis, Sreenivasan, and Yeung.

Figure 2

Normalized moments of velocity derivatives in isotropic and homogeneous turbulence. Numerical simulations from Refs. [7] and [8]. Right: Schumacher et al. [ 7]: homogeneous turbulence driven by a force $\mathbf{F}(\mathbf{k},t)=\mathcal{P}\frac{\mathbf{u}(\mathbf{k},t)}{{\sum }^{\prime }{\left|\mathbf{u}(\mathbf{k},\mathbf{t})\right|}^{\mathbf{2}}}{\delta }_{\mathbf{k},{\mathbf{k}}^{\prime }}$ (for more details, see above). Left: results of Donzis, Yeung, and Sreenivasan [8]; HIT driven by a Gaussian large-scale noise. In the range $R_{\lambda }\le 9.0$, we see a clear Gaussian behavior of a few moments with $S_3=0$, $M_4\approx 3$, $M_6\approx 15$, $M_8\approx 105$, typical of Gaussian distribution. At $R_{\lambda }\ge 9-10$, all moments obey anomalous scaling of fully developed turbulence.

Figure 3

Length scales and turbulent kinetic energies. $\mathcal{K}$ and ${\mathcal{K}}_r$ denote energy contents in inertial and dissipation ranges, respectively. $L$ and $\mathcal{L}\approx \eta$ stand for integral scale, corresponding to the top of inertial range and the dissipation scale, respectively. Straight line: $E\left(k\right)\approx 1.6{\mathcal{E}}^{\frac{2}{3}}{k}^{-\frac{5}{3}}$.