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

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.

💡 Research Summary

This paper addresses the long‑standing challenge of uncovering the causal mechanisms behind turbulent skin‑friction drag (TSD) generation in wall‑bounded turbulent flows. While many previous investigations have relied on correlation analyses to link coherent structures—such as near‑wall streaks, quasi‑streamwise vortices, and large‑scale motions—to drag, correlation alone cannot establish directionality or asymmetry, and thus cannot unambiguously identify causation. To overcome this limitation, the authors introduce the Liang‑Kleeman Information Flow (LKIF) framework, an information‑theoretic causal inference method that is explicitly derived from the governing stochastic or deterministic dynamical equations of the system.

Methodology
The study uses a direct‑numerical‑simulation (DNS) database of a turbulent channel flow at friction Reynolds number Reτ≈183 (with additional results at Reτ≈548 presented in an appendix). The flow is solved with a pseudospectral code: Fourier discretization in the streamwise (x) and spanwise (z) directions, Chebyshev polynomials in the wall‑normal (y) direction, a third‑order Runge‑Kutta scheme for the nonlinear terms, and an implicit Euler scheme for viscous terms. Periodic boundary conditions are applied in the homogeneous directions, and no‑slip conditions at the walls. Time snapshots are stored every Δt⁺=1.002 viscous units, yielding a large ensemble of statistically stationary data.

Within the LKIF framework, the wall‑shear stress τw = μ∂u/∂y|{y=0} is taken as the “effect” time series, while the velocity fluctuations u′, v′, and w′ at various wall‑normal locations serve as “cause” series. The information flow from a given velocity component φ to τw, denoted T{φ→τw}(Δx,Δy,Δz,Δt), is computed by first estimating the sample covariance matrix of the four variables (φ, τw, and their time‑differences) and then applying the linear‑Gaussian approximation of LKIF (Equation 2.5). Statistical significance is assessed through a Wald test with a 99 % confidence level, ensuring that identified structures are not spurious.

Results – Positive and Negative Causal Structures
The authors find that the sign of the information flow carries a clear physical meaning: a positive T_{φ→τw} (denoted T⁺) increases the information entropy of τw, thereby promoting the occurrence of extreme drag events; a negative flow (T⁻) reduces entropy, suppressing such events. By scanning Δt, they locate distinct peaks: the positive peak T_p is associated with streamwise velocity fluctuations (u′) located just above the wall, while the negative peak T_n is linked to wall‑normal (v′) and spanwise (w′) fluctuations.

Spatial analysis reveals that these peaks are concentrated near Δx = 0, Δz = 0, i.e., directly above the wall point where τw is measured, and that their wall‑normal distribution mirrors the classic “streak‑roll” cycle. Specifically, when a low‑speed streak (characterized by a negative u′) approaches the wall, T⁺ reaches a maximum, indicating that the streak’s proximity makes τw more uncertain and thus more likely to produce a high‑drag event. Conversely, when quasi‑streamwise vortices (represented by v′ and w′) move away from the wall, T⁻ peaks, reflecting a stabilizing influence that reduces τw entropy.

Comparison with Correlation Analyses
When the same data are examined using conventional Pearson or two‑point correlations, the resulting maps are symmetric and cannot distinguish the direction of influence. LKIF, by contrast, yields asymmetric information‑flow fields (T_{φ→τw} ≠ T_{τw→φ}), directly exposing which variable drives the other. This asymmetry aligns with the physical intuition that streaks feed energy into the wall shear, while vortices can either amplify or attenuate that process depending on their relative phase and position.

Quadrant Decomposition
A further quadrant analysis partitions the (u′, τw) plane into four regions corresponding to the classic turbulent quadrant events (Q1–Q4). Positive information flow dominates the Q1 and Q3 quadrants (high‑speed streaks and low‑speed streaks, respectively), whereas negative flow concentrates in Q2 and Q4, where ejection and sweep events are prevalent. This decomposition reinforces the interpretation that the sign of the information flow is intimately tied to the underlying bursting dynamics.

Reynolds‑Number Robustness
The authors repeat the entire LKIF analysis for a higher‑Re case (Reτ≈548) and observe that the qualitative patterns of positive and negative causal structures persist, suggesting a degree of Reynolds‑number invariance. The magnitude of the flows scales modestly with Re, but the spatial locations of the peaks and their association with streak‑roll dynamics remain unchanged.

Conclusions and Implications
By applying the physics‑informed LKIF methodology, the study provides a clear, quantitative map of how specific velocity fluctuations causally affect turbulent skin‑friction drag. The identification of positive and negative causal structures offers a mechanistic explanation for the generation and suppression of extreme drag events, directly linked to the approach or recession of streaks and rolls. Compared with correlation‑based approaches, LKIF delivers directionality, asymmetry, and a direct connection to the governing equations, making it a powerful tool for turbulence research. The insights gained could inform targeted drag‑reduction strategies, such as active control of streaks or manipulation of vortex dynamics, by focusing on the variables that exert the strongest positive information flow toward τw.

Overall, the paper demonstrates that information‑flow based causal inference not only overcomes the limitations of correlation analysis but also yields physically interpretable, actionable knowledge about turbulent drag generation.