Despite its great scientific and technological importance, wall-bounded turbulence is an unresolved problem in classical physics that requires new perspectives to be tackled. One of the key strategies has been to study interactions among the energy-containing coherent structures in the flow. Such interactions are explored in this study using an explainable deep-learning method. The instantaneous velocity field obtained from a turbulent channel flow simulation is used to predict the velocity field in time through a U-net architecture. Based on the predicted flow, we assess the importance of each structure for this prediction using the game-theoretic algorithm of SHapley Additive exPlanations (SHAP). This work provides results in agreement with previous observations in the literature and extends them by revealing that the most important structures in the flow are not necessarily the ones with the highest contribution to the Reynolds shear stress. We also apply the method to an experimental database, where we can identify structures based on their importance score. This framework has the potential to shed light on numerous fundamental phenomena of wall-bounded turbulence, including novel strategies for flow control.

Townsend (J. Fluid Mech., vol. 11, issue 1, 1961, pp. 97–120) introduced the concept of active and inactive motions for wall-bounded turbulent flows, where the active motions are solely responsible for producing the Reynolds shear stress, the key momentum transport term in these flows. While the wall-normal component of velocity is associated exclusively with the active motions, the wall-parallel components of velocity are associated with both active and inactive motions. In this paper, we propose a method to segregate the active and inactive components of the two-dimensional (2-D) energy spectrum of the streamwise velocity, thereby allowing us to test the self-similarity characteristics of the former which are central to theoretical models for wall turbulence. The approach is based on analysing datasets comprising two-point streamwise velocity signals coupled with a spectral linear stochastic estimation based procedure. The data considered span a friction Reynolds number range is the streamwise/spanwise wavenumber), lending empirical support to the attached eddy model of Perry & Chong (J. Fluid Mech., vol. 119, 1982, pp. 173–217).

The streamwise inclination angle of large wall-attached structures, in the log region of a canonical turbulent boundary layer, is estimated via spectral coherence analysis, and is found to be approximately $45^{\circ }$ . This is consistent with assumptions used in prior attached eddy model-based simulations. Given that the inclination angle obtained via standard two-point correlations is influenced by the range of scales in the turbulent flow (Marusic, Phys. Fluids, vol. 13 (3), 2001, pp. 735–743), the present result is obtained by isolating the large wall-attached structures from the rest of the turbulence. This is achieved by introducing a spanwise offset between two hot-wire probes, synchronously measuring the streamwise velocity at a near-wall and log-region reference location, to assess the wall coherence. The methodology is shown to be effective by applying it to data sets across Reynolds numbers, $Re_{\unicode[STIX]{x1D70F}}\sim O(10^{3})$ – $O(10^{6})$ .

We present experimental evidence that the superstructures in turbulent boundary layers comprise smaller, geometrically self-similar coherent motions. The evidence comes from identifying and analysing instantaneous superstructures from large-scale particle image velocimetry datasets acquired at high Reynolds numbers, capable of capturing streamwise elongated motions extending up to 12 times the boundary layer thickness. Given the challenge in identifying the constituent motions of the superstructures based on streamwise velocity signatures, a new approach is adopted that analyses the wall-normal velocity fluctuations within these very long motions, which reveals the constituent motions unambiguously. The conditional streamwise energy spectra of the Reynolds shear stress and the wall-normal fluctuations, corresponding exclusively to the superstructure region, are found to exhibit the well-known distance-from-the-wall scaling in the intermediate-scale range. It suggests that geometrically self-similar motions are the constituent motions of these very-large-scale structures. Investigation of the spatial organization of the wall-normal momentum-carrying eddies, within the superstructures, also lends empirical support to the concatenation hypothesis for the formation of these structures. The association between the superstructures and self-similar motions is reaffirmed on comparing the vertical coherence of the Reynolds-shear-stress-carrying motions, by computing conditionally averaged two-point correlations, which are found to match with the mean correlations. The mean vertical coherence of these motions, investigated for the log region across three decades of Reynolds numbers, exhibits a unique distance-from-the-wall scaling invariant with Reynolds number. The findings support modelling of these dynamically significant motions via data-driven coherent structure-based models.

We study the Reynolds-number effects on the outer region of moderate adverse-pressure-gradient (APG) turbulent boundary layers (TBLs) and find that their small-scale (viscous) energy reduces with increasing friction Reynolds number (${\mathrm{Re}}_{\tau }$). The trend is based on analyzing APG TBL data across $600\lesssim {\mathrm{Re}}_{\tau }\lesssim 7000$ and contrasts with the negligible variation in small viscous-scaled energy noted for canonical wall flows. The data sets considered include those from a well-resolved numerical simulation [Pozuelo et al., J. Fluid Mech. 939, A34 (2022)], which provides access to an APG TBL maintained at near-equilibrium conditions across $1000\lesssim {\mathrm{Re}}_{\tau }\lesssim$ 2000, with a well-defined flow history, and a new high-${\mathrm{Re}}_{\tau }$ ($\sim 7000$) experimental study from the large Melbourne wind tunnel, with its long test section modified to permit development of an APG TBL from a “canonical” upstream condition. The decrease in small-scale energy with ${\mathrm{Re}}_{\tau }$ is revealed via decomposing the streamwise normal stresses into small- and large-scale contributions, based on a sharp spectral cutoff. The origin for this trend is traced back to the production of turbulent kinetic energy in an APG TBL, the small-scale contribution to which is also found to decrease with ${\mathrm{Re}}_{\tau }$ in the outer region. The conclusion is reaffirmed by investigating attenuation of streamwise normal stresses due to changing spatial resolutions of the numerical grid or hotwire sensors, which reduces with increasing ${\mathrm{Re}}_{\tau }$ and is found to be negligible at ${\mathrm{Re}}_{\tau }\sim 7000$ in this study. The results emphasize that new scaling arguments and spatial-resolution corrections should be tested rigorously across a broad ${\mathrm{Re}}_{\tau }$ range, particularly for pressure gradient TBLs.

In this paper, we present the two-dimensional (2-D) energy cross-spectrum of the streamwise velocity (, where this self-similar trend is obscured by coexisting scales. The present analysis reaffirms that a self-similar structure, conforming to Townsend’s attached eddy hypothesis, is ingrained in the flow.