Probability distribution of power fluctuations in turbulence

We study local power fluctuations in numerical simulations of stationary, homogeneous, isotropic turbulence in two and three dimensions with Gaussian forcing. Due to the near-Gaussianity of the one-point velocity distribution, the probability distribution function (pdf) of the local power is well modeled by the pdf of the product of two joint normally distributed variables. In appropriate units, this distribution is parametrized only by the mean dissipation rate, $ϵ$. The large deviation function for this distribution is calculated exactly and shown to satisfy a fluctuation relation (FR) with a coefficient which depends on $ϵ$. This FR is entirely statistical in origin. The deviations from the model pdf are most pronounced for positive fluctuations of the power and can be traced to a slightly faster than Gaussian decay of the tails of the one-point velocity pdf. The resulting deviations from the FR are consistent with several recent experimental studies.

Figure 1

(Color online) The pdfs of the 2D power normalized by ${\sigma }_v{\sigma }_f$ for the inverse (○) and direct (∗) cascades in the Lagrangian frame and the inverse cascade in the Eulerian frame (◻). The solid line is Eq. (3). The insets show the corresponding pdfs of $\mathbf{f}$ (A) and $\mathbf{v}$ (B) normalized by their standard deviations.

Figure 2

(Color online) Comparison of the pdf of the ${\mathbf{v}}_1{\mathbf{f}}_1$ contribution to the 3D power with Eq. (3). Insets show the pdfs of ${\mathbf{f}}_1$ (A) and ${\mathbf{v}}_1$ (B) normalized by ${\sigma }_v$ and ${\sigma }_f$.

Figure 3

(Color online) Comparison of the pdf of the local 3D power with $e^{-I\left(x\right)}$. Inset shows the Kramer function, $I\left(x\right)$, given by Eq. (9), for $\rho =0.35$.

Figure 4

(Color online) Asymmetry of the pdfs of local power in 2D and 3D. Solid lines indicate the prediction of Eq. (12). Inset shows the decrease of the correlation coefficient for the 2D case, $\rho$, as the notional Reynolds number increases. We expect a similar trend in 3D.

Figure 5

(Color online) Fluctuation relation for the Lagrangian power averaged over time $\left(\tau \right)$ intervals in multiples of the correlation time $\left({\tau }_c\right)$ of the power signal for $\tau ∕{\tau }_c\equiv n=0$, 5, 10, 15, and 20. The solid line is the theoretical prediction from Eq. (12).