Causally coherent structures in turbulent dynamical systems

Abstract

The extraction of spatio-temporal coherence in high-dimensional, chaotic, non-linear dynamical systems, such as turbulent flows, remains a fundamental challenge in physics, mathematics and engineering. In this work, we employ Shannon transfer entropy (TE) to identify causally coherent motions in a zero-pressure-gradient turbulent boundary layer (TBL). This causality metric, rooted in information theory, enables the identification of sources and targets in dynamical systems using the corresponding time series. However, TE requires sophisticated tuning of various hyperparameters, such as the Markovian order of the source ($m$), which can spatially vary in wall-bounded turbulent flow. Here, we present an adaptive tuning and discuss the influence of $m$ across different TBLs. We introduce the concept of causally coherent structures (CCS), i.e. coherent structures interpreted as spatio-temporal patterns of causality. Moreover, the net transfer entropy flux is also utilised to identify boundary layer locations acting either as sources or targets. The standard viscous, logarithmic, and outer layers are characterised by information fluxes, highlighting, for example, dominant top-down interactions between the inner and outer layers, analogously to the classical energy cascade. This work extends techniques previously employed in the literature, such as correlation and spectral analysis, and presents an approach that is inherently general and applicable to a wide range of chaotic dynamical systems, with applications in cognitive sciences, systems biology and finance.

Figures (4)

• Figure 1: The Shannon transfer entropy (TE) from, or to, the wall-normal reference locations as a function of the maximum source time lag, $m$. The panel a) and b) refer to $Re_\theta = 2240$ and $Re_\theta = 8000$, respectively. Circle-solid lines indicate cases where the streamwise velocity at the reference wall-normal location $y_{REF}=y_A$ is considered as the source, with the target specified in the legend; the reverse configuration is shown by filled-dotted lines.
• Figure 2: Two-dimensional causally coherent structures of the streamwise velocity for a ZPGTBL at ($a$) $Re_\theta = 2240$ and ($b$) $Re_\theta = 8000$. From left to right, the wall-normal locations at $y^+ \approx 12$, $y^+ \approx 55$ and $y/\delta \approx 0.1$ are shown. In each panel, solid lines correspond to cases where the streamwise velocity at the reference point $y_A$ acts as the source, and dotted lines represent the reverse configuration. Three TE isolines are displayed; each enclosed region corresponds to TE values greater than or equal to the isoline value.
• Figure 3: The net Shannon transfer entropy flux (NTE) for a boundary layer at ($a$) $Re_\theta = 2240$ and ($b$) $Re_\theta = 8000$. From left to right, the wall-normal locations at $y^+ \approx 12$, $y^+ \approx 55$ and $y/\delta \approx 0.1$ are shown as reference wall-normal location, $y_A$.
• Figure 4: The Shannon transfer entropy (TE) from, or to, the viscous sublayer (the streamwise velocity at $y^+ \approx 1$ is considered) w.r.t. the wall-normal reference locations. TE is expressed as a function of the maximum source time lag for (top) $Re_\theta = 2240$ and (bottom) $Re_\theta = 8000$.