Fig. 3.

(A) The PDFs of $p/\epsilon$ at three different Reynolds numbers $R_{\lambda }=170$, 430, and 690 for 3D turbulence. For comparison, the PDFs of negative power $\left(p<0\right)$, shown by the dashed lines, are reflected around the vertical axis. The characteristic power is much larger than ε and increases with the Reynolds number. Moreover, large negative values of p are more frequent than large positive values, indicating that most of the violent energy exchange events that a fluid particle experiences are energy-loss events rather than energy gaining. Data at $R_{\lambda }=170$ and 430 are from DNSs and data at $R_{\lambda }=690$ are from experiments. (B) PDFs of $p/\epsilon$ from 2D turbulence simulations at $R_{\alpha }=26$, 51, and 102. Similar behavior as for 3D turbulence is observed. (C–F) Statistical properties of the instantaneous power p acting on fluid particles. (C) Variation of $-\langle p^3\rangle /{\epsilon }^3$ vs. $R_{\lambda }$ for 3D turbulence. Its increase is close to $R_{\lambda }^2$. (D) Variation of $-\langle p^3\rangle /{\epsilon }^3$ vs. $R_{\alpha }$ for 2D turbulence, which increases approximately as $R_{\alpha }^2$. (E and F) Variation of $\langle p^3\rangle /{\epsilon }^3$ for 3D and 2D turbulence, respectively. The variance increases rapidly with Reynolds numbers, close to $R_{\lambda }^{4/3}$ or $R_{\alpha }^{4/3}$ for 3D and 2D turbulence. This results in a skewness nearly independent of the Reynolds number: $\langle p^3\rangle /{\langle p^2\rangle }^{3/2}\approx -0.5$ in 3D and $\approx -0.20$ in 2D.