Probing turbulence intermittency via autoregressive moving-average models

We suggest an approach to probing intermittency corrections to the Kolmogorov law in turbulent flows based on the autoregressive moving-average modeling of turbulent time series. We introduce an index ${\rm Y}$ that measures the distance from a Kolmogorov-Obukhov model in the autoregressive moving-average model space. Applying our analysis to particle image velocimetry and laser Doppler velocimetry measurements in a von Kármán swirling flow, we show that ${\rm Y}$ is proportional to traditional intermittency corrections computed from structure functions. Therefore, it provides the same information, using much shorter time series. We conclude that ${\rm Y}$ is a suitable index to reconstruct intermittency in experimental turbulent fields.

Figure 1

(a) Space structure function $F_p\left(\ell \right)$ of order 2 in black and of order 4 in blue (gray) at two PIV grid points of coordinates $R=-0.10$, $Z=-0.14$ (circles) and $R=0.98$, $Z=0.70$ (crosses). (b) Time structure function $G_p\left(\tau \right)$ of order 2 in black and of order 4 in blue (gray) at two LDV grid points of coordinates $R=0$, $Z=0.35$ (circles) and $R=0.52$, $Z=0.61$ (crosses). Lines represent the Kolmogorov predictions: solid is Eulerian with ${\ell }^{2/3}$ and ${\tau }^{2/3}$ in black and ${\ell }^{4/3}$ or ${\tau }^{4/3}$ in blue (gray) and dotted is Lagrangian with $\tau$ in black and ${\tau }^2$in blue (gray).

Figure 2

Plot of ${\zeta }_p^{*}$ computed for the LDV experiments. Different lines correspond to different measure points. Red (gray) spots mark the scaling exponents reported in [8].

Figure 3

(a) Index ${\rm Y}$ vs (b) the intermittency index $\mu ={\zeta }_2^{*}-\frac{2}{3}{\zeta }_3^{*}$. White crosses show measurement points. (c) Scatter plot of ${\rm Y}$ vs $\mu$. The red (gray) line shows a linear regression of the data.

Figure 4

(a) Index ${\rm Y}$ computed for a PIV experiment. The intermittency ${\rm Y}$ maxima are localized near the walls and trace the position of the time wandering shear layer at $R=0$. (b) Scatter plot of ${\rm Y}$ vs $\mu$, the former displaying a much greater sensitivity than the traditional intermittency index, despite the short length of the time series.