Are extreme dissipation events predictable in turbulent fluid flows?

We derive precursors of extreme dissipation events in a turbulent channel flow. Using a recently developed method that combines dynamics and statistics for the underlying attractor, we extract a characteristic state that precedes laminarization events that subsequently lead to extreme dissipation episodes. Our approach utilizes coarse statistical information for the turbulent attractor, in the form of second-order statistics, to identify high-likelihood regions in the state space. We then search within this high-probability manifold for the state that leads to the most finite-time growth of the flow kinetic energy. This state has both high probability of occurrence and leads to extreme values of dissipation. We use the alignment between a given turbulent state and this critical state as a precursor for extreme events and demonstrate its favorable properties for prediction of extreme dissipation events. Finally, we analyze the physical relevance of the derived precursor and show its robust character for different Reynolds numbers. Overall, we find that our choice of precursor works well at the Reynolds number it is computed at and at higher Reynolds number flows with similar extreme events.

Figure 1

Time evolution of the kinetic energy (a) and the energy dissipation rate (b) for the near-wall minimal flow unit at $\text{Re}=2200.$

Figure 2

Time evolution of $E\left(t\right)$ and $Z\left(t\right)$ during the first laminarization event shown in Fig. 1. The vertical lines indicate the times that snapshots in Fig. 3 correspond to. The first and last snapshot correspond to the start and end of the time horizon shown above.

Figure 3

Snaphots of $Q$-criterion colored by axial velocity and wall shear stress. The axial velocity lies in the interval [0,0.3] and the wall shear stress lies in $[0,4\times {10}^{-4}]$.

Figure 4

A sketch of the proper orthogonal decomposition of the turbulent data. The attractor is approximated as an ellipsoid in the subspace spanned by the POD modes $\{{\mathit{v}}_1,\cdots ,{\mathit{v}}_n\}$. The origin is the mean flow $\overline{\mathit{u}}$.

Figure 5

The energy content of the POD modes. (a) Energy content of each POD mode. (b) Cumulative energy content of the POD modes. POD was computed with 1000 snapshots at $\text{Re}=2200$.

Figure 6

A sketch of the weight function $q_{\epsilon ,h}$ defined in Eq. (12).

Figure 7

Contours and isosurfaces of axial velocity $u$ for several POD modes. Isosurfaces are defined at $u=\pm 0.01$. Mode indicies are defined as in Fig. 5.

Figure 8

Convergence of the objective function for the same initial guess but two different choices of $\tau$. Solid dots represent instances when the optimizer was restarted.

Figure 9

POD mode weights for the initial guess and optimal solution.

Figure 10

Contours, isosurfaces, and spatially averaged profiles of axial velocity for the two most energetic POD modes in the optimal solution to Eq. (10). Isosurfaces are defined at $u\pm 0.01$.

Figure 11

Optimal solution ${\mathit{u}}_0^{*}$ to Eq. (10). (a) $Q$ criterion isosurface for $Q=0.0005$, colored by the streamwise velocity (b) Spatially averaged streamwise velocity profile with the mean axial velocity profile. (c) Spatially averaged streamwise velocity profile difference. Red-shaded boxes indicate regions from $y^{+}=20$ to $y^{+}=60$ units away from the wall.

Figure 12

Time evolution of the indicator $\lambda \left(t\right)$ with $E\left(t\right)$ and $Z\left(t\right)$ over the same time horizon as shown in Fig. 2. The horizontal lines indicate the mean $\left(\overline{X}\right)$ and the mean plus one standard deviation $\mathrm{[}\overline{X}+\sigma \left(X\right)]$ of $E\left(t\right)$ and $Z\left(t\right)$. The vertical lines indicate the times that snapshots in Fig. 3 correspond to. The first and last snapshot correspond to the start and end of the time horizon shown above.

Figure 13

Probability densities at $\text{Re}=2200$. (a) Joint PDF of the kinetic energy $E$, energy dissipation $Z$ and the indicator $\lambda$. (b) Conditional PDF $p_{E|\lambda }$. (c) Conditional PDF $p_{Z|\lambda }$.

Figure 14

Predictive conditional probability densities. (a) Top row: Conditional PDF $p_{\tilde{E}|\lambda }$ computed with $t_0=t_e$ and $\mathrm{\Delta }t=10t_e$ where $t_e$ denotes the eddy turnover time. The vertical dashed line marks the threshold $E_e$ of extreme events that is prescribed as the mean of $E$ plus one standard deviation. The horizontal dashed line marks the extreme event threshold ${\lambda }_e$ according to the indicator $\lambda$. Bottom row: The probability of upcoming extreme events $P_{ee}$. The horizontal dashed line marks $P_{ee}=0.5$ and the vertical dashed line marks the extreme event threshold ${\lambda }_e$ so that $P_{ee}\left({\lambda }_e\right)=0.5$. (b) Same as panel (a) but the figure correspond to the energy dissipation $Z$ versus the indicator $\lambda$.

Figure 15

Prediction quality for varying prediction times $t_0$. In all plots, the length of the future maximum time window in Eq. (14) is $\mathrm{\Delta }t=10$. First row shows the conditional PDF $p_{\tilde{Z}|\lambda }$ and the second row shows the corresponding probability of future extreme events defined in Eq. (17).

Figure 16

Probability densities at $\text{Re}=3000$. (a) Joint PDF of the kinetic energy $E$, energy dissipation $Z$ and the indicator $\lambda$. (b) Conditional PDF of $Y_t=\text{Indicator}$ and $X_t=\text{Energy}$. (c) Conditional PDF of $Y_t=\text{Indicator}$ and $X_t=\text{Energy}\text{Dissipation}$.