Lyapunov spectrum of forced homogeneous isotropic turbulent flows

In order to better understand deviations from equilibrium in turbulent flows, it is meaningful to characterize the dynamics rather than the statistics of turbulence. To this end, the Lyapunov theory provides a useful description of turbulence through the study of the perturbation dynamics. In this work, the Lyapunov spectrum of forced homogeneous isotropic turbulent flows is computed. Using the Lyapunov exponents of a flow at different Reynolds numbers, the scaling of the dimension of the chaotic attractor for a three-dimensional homogeneous isotropic flow is obtained through direct computation. The obtained Gram-Schmidt vectors (GSVs) are analyzed. For the range of conditions studied, it is found that the chaotic response of the flow coincides with regions of large velocity gradients at lower Reynolds numbers and enstrophy at higher Reynolds numbers, but does not coincide with regions of large kinetic energy. Further, the response of the flow to perturbations is more and more localized as the Reynolds number increases. Finally, the energy spectrum of the GSVs is computed and is shown to be almost insensitive to the Lyapunov index.

Figure 1

(a) Contour of the vorticity magnitude for the case solved with the spectral code using ${32}^3$ Fourier modes. (b) First 100 LEs obtained with the spectral code with ${32}^3$ Fourier modes and ${64}^3$ Fourier modes and the physical space code with ${32}^3$ grid points and ${64}^3$ grid points (, first 50 exponents only).

Figure 2

LEs rescaled by the first LE computed for case 1 ($⚊$), case 2 ($⚊$), case 3 ($⚊$), and case 4 ($⚊$). The shaded areas denote the statistical errors. The inset shows the magnified region near the first exponents for cases 3 and 4.

Figure 3

Kaplan-Yorke dimension obtained from cases 1–4 (variable ${\text{Re}}_{\lambda }$ and linear forcing scheme) plotted against the ratio $\frac{L}{\eta }$, along with a ${\left(\frac{L}{\eta }\right)}^{2.8}$ slope ($⁃⁃$). The error bars denote the uncertainty estimate for both the $x$ and $y$ axes due to statistical convergence and extrapolation uncertainty.

Figure 4

Contours of case 1 taken at the same instant. Shown on top is the instantaneous contour of turbulent kinetic energy $k$, enstrophy $\zeta$, strain rate $S$, and helicity density $H$ (from left to right). Shown on the bottom is the contour of $\delta {\mathit{\xi }}^2$ for the most chaotic first LV, the 27th LV corresponding to the near-zero LE, the 63rd LV, and the 100th LV (from left to right).

Figure 5

Physical space localization of the LV. (a) Time-averaged localization $C_{\theta }$ obtained from cases 1–4 with different thresholds $\theta =0.98$ ($▴$), $\theta =0.88$ ($\blacksquare$), $\theta =0.78$ ($▾$), and $\theta =0.68$ ($•$). The vertical error bars denote the rms fluctuations around the mean (as opposed to statistical uncertainty). The horizontal error bars denote statistical uncertainty in the average value of ${\text{Re}}_{\lambda }$. (b) Localization width plotted against the LE index obtained from cases 1 ($⚊$), 2 ($⁃⁃$), 3 ($⁃·⁃$), and 4 ().

Figure 6

(a) Conditional average of $\delta {\mathit{\xi }}^2$ conditioned on helicity density values $H$, for each LV. (b) Conditional rms of $\delta {\mathit{\xi }}^2$ conditioned on helicity density values $H$, for each LV, rescaled by the conditional average of $\delta {\mathit{\xi }}^2$.

Figure 7

(a) Conditional average of $\delta {\mathit{\xi }}^2$ conditioned on turbulent kinetic energy values $k$, for each LV. (b) Conditional average of $\delta {\mathit{\xi }}^2$ conditioned on enstrophy values $\zeta$, for each LV. (c) Conditional average of $\delta {\mathit{\xi }}^2$ conditioned on strain rate values $S$, for each LV. Results correspond to case 1.

Figure 8

(a) Conditional average of $\delta {\mathit{\xi }}^2$ conditioned on enstrophy values $\zeta$, for each LV. (b) Conditional average of $\delta {\mathit{\xi }}^2$ conditioned on strain rate values $S$, for each LV. Results correspond to case 3.

Figure 9

Conditional average of $\delta {\mathit{\xi }}^2$ at different rescaled enstrophy ${\zeta }_n$ for case 1 ($◯$), case 2 ($▵$), case 3 ($▿$), and case 4 ($+$). Error bars show the statistical uncertainty.

Figure 10

Field correlation of $\delta {\mathit{\xi }}^2$ and $\zeta$ ($⁃·⁃$), $\delta {\mathit{\xi }}^2$ and $H$ (), $\delta {\mathit{\xi }}^2$ and $k$ ($⚊$), and $\delta {\mathit{\xi }}^2$ and $S$ (). The curves are plotted alongside the index where the LS crosses zero ($⁃⁃$).

Figure 11

(a) Spatial correlation ${\rho }_{33}\left(r\right)$ of the LV (solid lines) and of the underlying flow field (dashed line). The lighter the solid line, the higher the Lyapunov index. (b) Spatial correlation ${\rho }_{33}\left(r\right)$ of the LV plotted against the LE index and the distance. Both plots are generated with the data of case 1. Here $L$ denotes the box length $2\pi$.

Figure 12

Time-averaged energy spectrum of the computed LV for case 1 (solid lines) and of the underlying flow field (dashed line). The lighter the solid line, the higher the Lyapunov index.

Figure 13

Field correlation of $\delta {\mathit{\xi }}^2$ and $\zeta$ ($⁃·⁃$) and $\delta {\mathit{\xi }}^2$ and $S$ () for (a) case 1, (b) case 2, (c) case 3, and (d) case 4. The curves are plotted alongside the index where the LS crosses zero ($⁃⁃$). The shaded areas indicate the statistical errors.

Figure 14

First 100 LEs for the linear forcing technique (case 1, □), the linear forcing applied to the large scales only (case 5, △), the stochastic forcing technique (case 6, +), and the linear forcing techniques with the same force across simulations (case 7, ▽).

Figure 15

Value of $\alpha$ obtained from the fit applied to the LE computed with cases 1–3.

Figure 16

Schematic of the uncertainty estimation method for the attractor dimension scaling. For each case, five points are considered using the uncertainty estimate of the dimension. The horizontal error bar represents the statistical error for the value of the average $\frac{L}{\eta }$; the vertical error bar contains the statistical uncertainty and the extrapolation uncertainty used to compute the dimension. The dashed lines represent examples of the fits considered.

Figure 17

Ratio $\frac{|E_{\text{corr},kl}\left(\mathit{\kappa }\right)|}{E_{\text{GSV}}\left(\mathit{\kappa }\right)}$ plotted for case 1 and the first 19 pairs of the LVs, plotted against the wave-number amplitude.

Figure 18

Estimated bounds for the energy spectrum of the second CLV.