Causal links between inner and outer turbulent motions in wall-bounded flows

Determine the causal links among energy-containing coherent turbulent motions in wall-bounded flows, specifically between inner motions (the universal near-wall cycle) and outer motions residing in the logarithmic layer that exhibit a near-wall footprint, and resolve whether their interactions are bottom-up, top-down, co-supporting, or independently self-sustained across scales.

Background

Wall-bounded turbulent flows contain coherent structures across scales, including near-wall streaks and vortices (inner motions) and larger structures in the logarithmic region (outer motions). How these structures causally interact has been debated, with proposed mechanisms including bottom-up, top-down, co-supporting, and independent self-sustaining processes at all scales.

The paper defines inner motions as the self-sustained near-wall cycle and outer motions as those in the logarithmic layer that leave a footprint in the near-wall region. To investigate causal relationships, the study applies transfer entropy and Liang–Kleeman information-flow methods to linear and nonlinear test systems, a low-dimensional near-wall turbulence model, and DNS of turbulent channel flow at Re_tau = 1000.

References

An open question is what are the causal links of these turbulent motions, especially in the context of inner and outer motions, either bottom-up \citep{Adrian2000}, top-down \citep{Hunt2000}, co-supporting \citep{Toh2005Interaction,Zhou2022Interaction} or independently self-sustained at all scales \citep{Cossu2017Selfsustaininga}.

— Causal analysis of inner and outer motions in near-wall turbulent flow  (2404.08907 - Zhang et al., 2024) in Section: INTRODUCTION

Quantitatively determining the effect of outer-layer energy on near-wall temporal scales requires further analysis and is left for future work.

— Coherent-structure dynamics in wall turbulence from state-space trajectories  (2609.04515 - Lenz et al., 3 Sep 2026) in Section 3.3, subsection “Temporal scales of dynamics”

Since trajectories are computed by projecting the flow field onto POD modes without subtracting or filtering large-scale outer structures, we do not distinguish the effects of footprinting and amplitude modulation, and determining the relative importance of these mechanisms is left for future work.

— Coherent-structure dynamics in wall turbulence from state-space trajectories  (2609.04515 - Lenz et al., 3 Sep 2026) in Section 4, Discussion