Causal analysis of inner and outer motions in near-wall turbulence

TL;DR

This work investigates the causal interactions between near-wall inner motions and outer motions in wall-bounded turbulence using three non-intrusive methods: transfer entropy, information flow, and SURD. After validating these methods on simple linear, nonlinear, and low-order turbulence models, the authors apply them to a DNS-based inner–outer decomposition to reveal that inner and outer motions are self-sustained and largely independent, while outer motions exhibit bidirectional top-down and bottom-up influences on their footprints. Pressure emerges as an active mediator bridging inner and outer dynamics, with both local (one-point) and nonlocal (two-point) coupling observed. The study demonstrates the value of combining TE, IF, and SURD to obtain a detailed, mechanism-oriented causal picture of scale interactions in wall-bounded turbulence, with implications for flow control and drag reduction.

Abstract

In this work, we study the causality of near-wall inner and outer turbulent motions. The inner motions are defined as the self-sustained near-wall cycle, and the outer motions as those living in the logarithmic layer exhibiting footprints on the near-wall region. Causal inference with three typical methods is performed, i.e. transfer entropy, information flow, and SURD (synergistic--unique--redundant decomposition of causality). The causal inference methods are first applied to several canonical problems to illustrate their abilities and differences, including a linear problem, a non-linear problem, and a low-dimensional model of near-wall turbulence. It is demonstrated that all three methods can produce consistent causal findings. Furthermore, we study the causalities between the inner and outer turbulent motions in a channel flow using the three methods with an improved inner-outer decomposition method. It is revealed that both the inner and outer motions are self-sustained and independent of each other, supporting the self-sustaining mechanism of turbulent motions at all scales. We also find that there are top-down and bottom-up influences in the outer motions and their near-wall footprints, challenging the traditional sole top-down view. More interestingly, pressure is identified to play an active role in the inner-outer causalities and may act as a bridge in linking the inner and outer turbulent motions.

Figures (13)

• Figure 1: Causal maps of the linear problem. (a) Transfer entropy; (b) Information flow; (c) SURD. Red boxes: the ground-truth causal links in the system.
• Figure 2: Causal maps of the nonlinear problem. (a) Transfer entropy; (b) Information flow; (c) SURD. Red boxes: the ground-truth causal links in the system.
• Figure 3: Causal maps of the low-dimensional model of near-wall turbulence. (a) Transfer entropy; (b) Information flow; (c) SURD. Red boxes: the causal links associated with the self-sustaining cycle of streaks and vortex.
• Figure 4: Wall-normal profiles of the Reynolds shear stress and its decomposed components at $Re_\tau=1000,2000$, and $5200$. The line colours correspond to the cases listed in table \ref{['detailinfor']}.
• Figure 5: Velocity fluctuation intensities of inner motions at friction Reynolds numbers $Re_\tau=1000,2000$, and $5200$. (a) Streamwise velocity fluctuations, (b) wall-normal velocity fluctuations, (c) spanwise velocity fluctuations, (d) cross components of the decomposed Reynolds shear stress. The line colours correspond to the cases listed in table \ref{['detailinfor']}.