Intermittency and Non-Gaussian Fluctuations of the Global Energy Transfer in Fully Developed Turbulence

We address the experimentally observed non-Gaussian fluctuations for the energy injected into a closed turbulent flow at fixed Reynolds number. We propose that the power fluctuations mirror the internal kinetic energy fluctuations. Using a stochastic cascade model, we construct the excess kinetic energy as the sum over the energy transfers at different levels of the cascade. We find an asymmetric distribution that strongly resembles the experimental data. The asymmetry is an explicit consequence of intermittency and the global measure is dominated by small scale events correlated over the entire system. Our calculation is consistent with the statistical analogy recently made between a confined turbulent flow and a critical system of finite size.

Figure 1

PDF of global energy transfer for experimental data, taken from Pinton et al. [5] (symbols) and from cascade models of intermittency. The cascade models are the ${\chi }^2$ model, $\mathrm{R}\mathrm{e}={10}^5$ and $\lambda =2.0$ (solid line) and the log-normal model, $\mathrm{R}\mathrm{e}={10}^5$ and $\lambda =1.08$ (dashed line).

Figure 2

PDF of global transferred energy for Re varying between ${10}^4$ and ${10}^7$ (legend). The microscopic distributions are ${\chi }^2$ ones and $\lambda =2.0$.

Figure 3

Variation of the global PDF with the magnitude of the intermittency parameter $|\tau (2)|$ for $\mathrm{R}\mathrm{e}={10}^6$ and $\lambda =2.0$. The microscopic distributions are ${\chi }^2$ statistics.