Wall-scaled eddies and embedded shear layers in high-Reynolds-number moderate adverse-pressure-gradient boundary layers

Abstract: This study compares high-Reynolds-number turbulent boundary layers under zero and low-to-moderate adverse pressure gradients, showing similar scaling and energy contributions from the wall-scaled attached-eddy hierarchy and superstructures in both flows. The main differences occur in the outer/wake region, where APG-induced energisation is linked to an outer-scaled, embedded-shear-layer-type organisation that progressively penetrates the logarithmic region as the pressure-gradient strength increases. The analysis uses two complementary datasets for ZPG and APG boundary layers at matched friction Reynolds numbers of approximately 10,000, with minimal upstream pressure-gradient history: a new two-point hot-wire dataset and a previously published two-dimensional particle image velocimetry dataset. In the hot-wire experiment, one probe is fixed near the wall while the second traverses the full boundary layer, allowing estimation of the linear coherence spectrum. The results confirm the geometric self-similarity of the wall-scaled eddy hierarchy that remains coherent with the wall. Using the linear coherence spectrum as a spectral filter shows that most of the additional APG-induced energy is incoherent with the wall and is linearly superimposed on the wall-coherent component of the streamwise variance. This wall-incoherent contribution explains the departure of the variance profile from the inverse logarithmic law observed in canonical high-Reynolds-number boundary layers. Conditional averaging of the particle image velocimetry data identifies the structures responsible for this energy amplification. The results link enhanced outer-region Reynolds stresses, an outer inflection point in the mean velocity profile, and an ejection-sweep organisation of Reynolds shear stress, all characteristic of shear-layer dynamics.

• The paper demonstrates that moderate APG modulates turbulence by energizing wall-incoherent shear-layer-type motions while preserving the canonical wall-coherent eddy hierarchy.
• The study used two-point hot-wire measurements and high-resolution PIV to differentiate between wall-coherent and wall-incoherent contributions at a friction Reynolds number of around 10^4.
• Findings suggest improved turbulence models for non-canonical TBLs by capturing the impact of altered eddy aspect ratios and embedded shear layers.

Wall-Scaled Eddy Hierarchies and Embedded Shear Layer Dynamics in High-Reynolds-Number APG Boundary Layers

Overview

This paper presents an experimental study on the structure and scaling of turbulent boundary layers (TBLs) under moderate adverse pressure gradients (APGs) at high Reynolds numbers. A rigorous comparison is made to zero-pressure-gradient (ZPG) TBLs, utilizing matched friction Reynolds numbers ($Re_\tau \sim 10^4$) and minimizing upstream pressure gradient history effects. The investigation leverages two-point hot-wire measurements and high-resolution particle image velocimetry (PIV), enabling systematic exploration of wall-coherent and wall-incoherent turbulent motions, as well as the identification of embedded shear layers in the outer region.

Experimental Approach and Methodology

Experiments were conducted in a high-$Re_\tau$ boundary layer wind tunnel with adjustable APG conditions controlled via test-section ceiling air-bleed slots and outlet blockage. Both ZPG and various APG scenarios were generated and characterized by the Clauser pressure gradient parameter ($\beta$). Two complementary datasets underpin the analysis:

• Two-point Hot-wire Measurements: One probe fixed near the wall acts as a reference while another traverses the wall-normal direction, enabling spectral linear coherence spectrum (LCS) computation. LCS distinguishes wall-coherent from wall-incoherent contributions at each scale and position.
• Planar PIV: High-resolution velocity fields across the TBL allow conditional averaging based on vortex identification via the swirling strength criterion, focusing on the detection and characterization of spanwise-oriented vortical events consistent with shear layer dynamics.

The APG TBLs studied satisfy criteria for minimal upstream PG history and substantial scale separation between inner and outer regions, which is essential for accurate assessment of large-scale organizational effects.

Wall-Coherent Eddy Hierarchy: Scaling and Robustness

LCS-based analyses confirm that the canonical wall-coherent eddy hierarchy—comprising viscous-scaled near-wall cycle, wall-scaled attached eddies (region II), and outer-scaled superstructures (region III)—exhibits geometric self-similarity and consistent energy distribution in both ZPG and moderate APG flows. In particular:

• Aspect Ratio and Inclination: The characteristic aspect ratio ($AR$) of wall-coherent eddies systematically decreases with increasing $\beta$, which is accompanied by a monotonic increase in mean inclination angle. This indicates more steeply inclined structures under APG conditions, corroborated by elevated mean wall-normal velocities.
• Spectral Invariance: The energy spectra of wall-coherent motions remain nearly unchanged with increasing APG strength, and their profiles continue to be well-modelled by the inverse logarithmic law ($u^2 \sim B_1 - A_1 \ln(z/\delta)$) with $A_1 \approx 0.98$.
• Self-Similarity Extent: While the hierarchy persists, the wall-normal range over which geometric self-similarity is observed reduces with increasing APG. Penetration of APG-energized wall-incoherent motion into the logarithmic region drives this contraction.

Wall-Incoherent Outer-Region Dynamics and Shear Layer Organization

The primary effect of moderate APG at high $Re_\tau$ is the energization of wall-incoherent, outer-scaled motions concentrated near $z/\delta \sim 0.3$ (region IV). The majority of additional turbulence kinetic energy (TKE) due to APG is incoherent with the wall and superimposes linearly onto the canonical wall-coherent hierarchy. Key findings include:

• Spectral Decomposition: LCS filtering demonstrates that APG-induced deviations in streamwise variance profiles are dominated by wall-incoherent energy, responsible for departures from the classical inverse logarithmic scaling in $u^2$ and a reduction in the extent of the logarithmic region in mean velocity profiles.
• Identification of Shear Layers: Conditional PIV analyses focused on intense spanwise-vortical events reveal spatial and dynamical characteristics consistent with embedded shear layers:
  • Q2-Q4 Reynolds Stress Organization: Enhanced ejection and sweep events (Q2/Q4) are observed, supporting the association of APG-amplified motions with shear-layer-type organization.
  • Inflectional Mean Profile: The outer energetic peak in TKE and Reynolds shear stress coincides with an inflection point in the mean velocity profile, a necessary condition for shear layer instability.
  • Strouhal Numbers: The representative Strouhal number for APG-energized vortices is estimated as $Re_\tau$0, matching typical values for shear layer flows.

Practical and Theoretical Implications

The results establish a dual-pathway framework for TBL dynamics at high $Re_\tau$1 under moderate APG: a wall-coherent pathway maintained by the attached eddy hierarchy and superstructures, and a wall-incoherent pathway dominated by shear-layer-type motions. APG increases the prevalence and energetic contribution of the latter, particularly in the outer/wake region.

• Modeling and Prediction: The explicit separation and quantification of wall-coherent and wall-incoherent contributions provide a foundation for improved predictive models of turbulence statistics in non-canonical TBLs. The invariance of the wall-coherent hierarchy under moderate APG supports extensions of data-driven modeling frameworks (e.g., Baars & Marusic 2020).
• Physical Interpretation of Velocity Profiles: The findings support interpretations of composite mean TBL profiles, where the wake region reflects the dynamical impact of embedded shear layers.
• Future Directions: The study raises questions regarding the universality of shear-layer-type motions across PG conditions (including FPGs and equilibrium flows), their precise relationship to energy transfer mechanisms, and the potential for three-dimensional characterization. There is also a strong motivation to systematically map the interplay between attached eddy hierarchies and wake structures across a continuum of PG strengths and Reynolds numbers.

Conclusion

This work demonstrates that moderate APG primarily modulates outer-region TBL dynamics through the energization and penetration of wall-incoherent, shear-layer-type motions, without fundamentally altering the scaling or organization of the canonical wall-coherent hierarchy at high $Re_\tau$2. The identification of embedded shear layers as the physical mechanism underpinning APG-induced deviations in turbulence statistics provides a clear direction for future modeling and experimental studies. The dual-pathway concept established here is critical for understanding complex boundary layer behavior in engineering and geophysical applications involving non-canonical pressure gradient conditions.