Causal structures of turbulent skin-friction drag in wall-bounded turbulent flows

TL;DR

The study tackles the problem of distinguishing causation from correlation in turbulent skin-friction drag by applying the physics-informed Liang–Kleeman information flow ($T_{2\to1}$) to a turbulent channel flow. It constructs a velocity-to-wall-shear-stress causal map, showing that positive causal structures promote extreme $\tau_w$ events while negative structures suppress them, with the causal patterns closely tied to streamwise streaks and rolls. The method provides clearer causal interpretation than traditional correlation analyses and reveals consistent advection characteristics between causal and correlation descriptors. These findings open avenues for targeted drag-reduction strategies and can be extended to higher Reynolds numbers and compressible regimes.

Abstract

Understanding the mechanism of turbulent skin-friction drag (TSD) generation is of fundamental and practical importance for designing effective drag reduction strategies. However, many previous studies adopted correlation analysis to reveal the causal map between turbulent motions and TSD generation, an approach that is potentially risky as correlation does not necessarily imply causation. In this study, a novel causal inference method called Liang-Kleeman information flow (LKIF) is utilized for the first time to identify the velocity-induced causal structures related to TSD generation in a turbulent channel flow. The statistical properties of the causal structures are comprehensively investigated. The positive and negative causal structures, defined by their signs and respectively associated with an increase and decrease in TSD information entropy, promote and suppress the generation of extreme TSD. Particularly, we find that the underlying physics of causal structures is essentially associated with the processes of streamwise streaks and rolls approaching or receding from the extreme events. Results indicate that the physics-informed LKIF framework can reveal a more explicit and interpretable causal relationship than correlation analysis.

Figures (15)

• Figure 1: Schematic for computing LKIF ($T_{\phi\to\tau_\mathrm{w}}$) between the flow quantity $\phi$ and wall-shear stress $\tau_\mathrm{w}$.
• Figure 2: LKIF from $u'$ at five specific heights to the target $\tau_\mathrm{w}$. ($a$) Profiles of $T_{u'\to\tau_\mathrm{w}}$ versus $\Delta t^+$. ($b$) Profiles of the positive and negative peak values ($T_{\rm p}$ and $T_{\rm n}$) versus $y$, as well as their corresponding time lags.
• Figure 3: Spatial distribution of $T_{u'\to\tau_\mathrm{w}}$ at the time lag $\Delta t^+=-5$. ($a$) The three-dimensional view of $T_{u'\to\tau_\mathrm{w}}$. The positive and negative structures are displayed by the isosurfaces of $T_{u'\to\tau_\mathrm{w}}=\pm0.1$. Three slices show contours of $T_{u'\to\tau_\mathrm{w}}$ at $\Delta x=-0.8h$, $0h$, and $0.8h$, respectively. ($b$) Colored contour of $T_{u'\to\tau_\mathrm{w}}$ at the plane $\Delta z=0$. ($c$) Colored contour of $T_{u'\to\tau_\mathrm{w}}$ at the plane $y^+=20$.
• Figure 4: Spatial distribution of $T_{v'\to\tau_\mathrm{w}}$ at the time lag $\Delta t^+=-5$. ($a$) The three-dimensional view of $T_{v'\to\tau_\mathrm{w}}$. The positive and negative structures are displayed by the isosurfaces of $T_{v'\to\tau_\mathrm{w}}=\pm0.05$. Three slices show contours of $T_{v'\to\tau_\mathrm{w}}$ at $\Delta x=-0.8h$, $0h$, and $0.8h$, respectively. ($b$) Colored contour of $T_{v'\to\tau_\mathrm{w}}$ at the plane $\Delta z=0$. ($c$) Colored contour of $T_{v'\to\tau_\mathrm{w}}$ at the plane $y^+=20$.
• Figure 5: Spatial distribution of $T_{w'\to\tau_\mathrm{w}}$ at the time lag $\Delta t^+=-5$. ($a$) The three-dimensional view of $T_{w'\to\tau_\mathrm{w}}$. The positive and negative structures are displayed by the isosurfaces of $T_{w'\to\tau_\mathrm{w}}=\pm0.05$. Three slices show contours of $T_{w'\to\tau_\mathrm{w}}$ at $\Delta x=-0.8h$, $0h$, and $0.8h$, respectively. ($b$) Colored contour of $T_{w'\to\tau_\mathrm{w}}$ at the plane $\Delta z^+=18$. The horizontal dashed line denotes $y^+=8$. ($c$) Colored contour of $T_{w'\to\tau_\mathrm{w}}$ at the plane $y^+=8$. The upper horizontal dashed line denotes $\Delta z^+=18$.