arXiv subjects

Rahul Deshpande

Publications and source records attributed to Rahul Deshpande.

At least 19 recordsLinked to original sources

arXiv2026

Reynolds-number evolution of wall-pressure statistics beneath canonical turbulent boundary layers

This study investigates the Reynolds-number (Re_\tau) evolution of wall-pressure statistics beneath zero-pressure-gradient TBLs, and links their logarithmic variation to the increasingly energetic large-scale motions in the logarithmic region. The wall-pressure skewness is found to become more negative with increasing Re_\tau, owing to increasing contributions from large-scale wall-pressure fluctuations (that are negatively skewed) and their nonlinear interaction with the statistically invariant inner-scale fluctuations (that are positively skewed). The analysis draws on new, well-resolved simultaneous measurements of wall pressure and streamwise velocity spanning 5000 < Re_\tau < 11300 in the Melbourne tunnel, atmospheric surface-layer measurements at Re_\tau = O(10^6) and a published simulation dataset at Re_\tau = O(10^3). Particular attention is paid to the principal experimental limitations affecting wall-pressure statistics: spatial resolution, Helmholtz resonance, facility noise and statistical convergence. Helmholtz resonance is shown to contaminate inner-scale wall-pressure contributions even after conventional corrections, and reliable estimation of skewness is found to require acquisition durations of O(10^5) eddy-turnover times or longer. The inner-scaled wall pressure spectrum is Reynolds-number invariant over the small-scale regime, in contrast to turbulent channel and pipe flows, whereas at intermediate and large scales it grows substantially with Re_\tau, consistent with these internal flows. Linear and quadratic velocity--wall-pressure coherence link these intermediate- and large-scale contributions to two dynamically distinct coherent structures: the self-similar attached-eddy hierarchy and turbulent superstructures, respectively. These analyses establish the connection between the inertial region and the log variation of wall-pressure variance and skewness with Re_\tau

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arXiv2026

Measurements of non-linear energy transfer in canonical and drag-reduced turbulent boundary layers

Three-dimensional particle-tracking velocimetry (3D-PTV) measurements were used to compute the spectral transport of the Reynolds-stress tensor. The experimental framework is validated for a zero-pressure-gradient (ZPG) turbulent boundary layer (TBL) at a friction Reynolds number $Re_\tau = 1020$, demonstrating that the dominant non-linear energy transfer mechanisms are adequately resolved to draw flow physics-based conclusions. For the streamwise Reynolds stress in the ZPG TBL, a component-wise decomposition of the non-linear transport term is considered for the first time, which reveals distinct energy transfer mechanisms associated with the spanwise and wall-normal advection. The same experimental framework was applied to a drag-reduced ($\approx 38\%$) TBL flow, achieved by imposing a steady streamwise-alternating spanwise wall velocity. This wall forcing causes a strong attenuation of non-linear energy transfer and its shift away from the wall. The energy transfer mechanisms remain qualitatively similar to those of the canonical ZPG TBL, suggesting that the existing mechanisms simply readjust to their new low-turbulent-energy state.

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arXiv2026

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

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.

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arXiv2026

On the wake region of high-Reynolds-number turbulent boundary layers subject to adverse pressure gradients

The effect of a moderate adverse pressure gradient (APG) on the structure of a high-Reynolds-number turbulent boundary layer (TBL) was investigated experimentally using complementary multi-point measurements. Unlike many previous studies, the present work focuses on the wake region and aims to characterise the turbulent motions that are energised by local APG conditions. Simultaneous two-point hot-wire measurements of the streamwise velocity were used to estimate the linear coherence spectrum (LCS), quantifying the wall-normal coherence between a wake-region reference point and the rest of the TBL. LCS-based decomposition of the spectral energy and variance showed that motions coherent with the wake reference account for a significant part of the APG-induced increase at large time scales, but not all of the enhanced energy. The remaining increase is associated with relatively smaller-scale motions that are not correlated with the selected wake location. High-spatial-resolution snapshot PIV measurements were then used to examine this broader range of energetic motions, which are associated with spanwise vortices in the wake region. Spanwise vorticity statistics were evaluated over 0.2 < z/δ < 0.4, where the largest APG-induced change in spectral energy was observed. Under APG, both the mean and variance of spanwise vorticity increased significantly in this region, while swirling-strength distributions confirmed a relative increase in both the population and magnitude of spanwise vortices. Finally, dynamically significant clockwise rotating spanwise vortices were identified using different swirling-strength thresholds. Higher thresholds produced conditionally averaged velocity fields that best captured the key wake-region dynamics, motivating their use for vortex-based conditional averaging in future analyses.

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arXiv2026

Analog-Digital Quantum Computing with Quantum Annealing Processors

Quantum annealing processors typically control qubits in unison, attenuating quantum fluctuations uniformly until the applied system Hamiltonian is diagonal in the computational basis. This simplifies control requirements, allowing annealing QPUs to scale to much larger sizes than gate-based systems, but constraining the class of available operations. Here we expand the class by performing analog-digital quantum computing in a highly-multiplexed, superconducting quantum annealing processor. This involves evolution under a fixed many-body Hamiltonian that, in the weak-coupling regime, is well-described by an effective XY model, together with arbitrary-basis initialization and measurement via auxiliary qubits. Operationally, this is equivalent to implementing single-qubit gates at the beginning and end of an analog quantum evolution. We demonstrate this capability with several foundational applications: single-qubit and two-qubit coherent oscillations with varying initialization and measurement bases, a multi-qubit quantum walk with fermionic dispersion in line with theory, and Anderson localization in a disordered chain. These experiments open the door to a wide range of new possibilities in quantum computation and simulation, greatly expanding the applications of commercially available quantum annealing processors.

quant-ph↗

arXiv2026

High-Reynolds-number turbulent boundary layers under adverse pressure gradients. Part 1. Decoupling local and upstream pressure gradient effects

This study presents a controlled examination of the universality of the von Karman and additive coefficients in the logarithmic law of the mean streamwise velocity profile for high-Reynolds-number turbulent boundary layers under low-to-moderate adverse pressure gradients. The experiments use a method for prescribing pressure gradients along Melbourne's high-Reynolds-number boundary layer wind tunnel, combined with direct friction velocity measurements from oil-film interferometry. This allows systematic variation of upstream pressure-gradient history while maintaining locally matched Reynolds number and Clauser parameter at the measurement location. The configuration therefore separates the effects of Reynolds number, local adverse pressure gradient, and pressure-gradient history on turbulence statistics and energy spectra across the boundary layer. Owing to the high Reynolds number and moderate pressure-gradient conditions, the overlap region is sufficiently extended to assess the logarithmic law. The von Karman coefficient remains invariant within experimental uncertainty, whereas the additive coefficient varies systematically with both local pressure gradient and pressure-gradient history. Local adverse pressure gradients energize both large- and small-scale motions in the wake region around 0.4 delta, while pressure-gradient history also affects large-scale motions down to about 0.25 delta, just above the overlap region. In contrast to lower-Reynolds-number studies, neither effect extends into the inner region. These measurements provide a high-fidelity dataset for improving physical understanding and developing composite mean velocity profile formulations for adverse-pressure-gradient turbulent boundary layers.

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arXiv2026

High-Reynolds-number turbulent boundary layers under adverse pressure gradients. Part 2. A composite mean velocity profile

A robust composite mean velocity profile is developed for turbulent boundary layers (TBLs) subjected to adverse pressure gradients (APGs), extending the composite formulation for generic pressure-gradient TBLs proposed by \citeauthor{nickels} (\textit{J.\ Fluid Mech.}, vol.\ 521, 2004). Several modifications are introduced to capture key features of APG flows. A new parameter accounts for pressure-gradient history effects in the wake region, a velocity-overshoot function is incorporated in the inner region, and the wake function is reformulated using an independent, physically motivated definition of boundary-layer thickness. A compilation of APG TBL datasets from the literature, including the new dataset presented in Part~1, is used to assess and refine the formulation. The resulting composite profile contains three physically meaningful parameters that capture pressure-gradient effects on the mean velocity profile, determined through nonlinear curve fitting. These parameters provide a framework for identifying `well-behaved' APG TBLs and quantifying the strength of pressure-gradient history effects. The profile also enables reliable estimation of the friction velocity and boundary-layer thickness in well-behaved APG TBLs, providing a practical tool for scaling analyses when these quantities are not directly measurable. Its analytical form yields improved estimates of mean velocity gradients, facilitating evaluation of the indicator function and identification of inflection points. Finally, the formulation predicts both the coefficients and spatial extent of the logarithmic region of the mean streamwise velocity profile, enabling assessment of its universality in high-Reynolds-number APG TBLs. This shows that the von K'arm'an coefficient approaches an invariant value of $κ\approx 0.39$ at sufficiently high Reynolds numbers, independent of pressure-gradient effects.

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arXiv2026

On the wall-normal velocity variance in canonical wall-bounded turbulence

The variance and spectra of wall-normal velocities are investigated for direct numerical simulations of turbulent flow in a channel, pipe, and zero-pressure-gradient boundary layer across a decade of friction Reynolds numbers. Spectra along the spanwise wavenumber have a pronounced peak well described by the turbulent dissipation rate and the local shear stress throughout the bottom half of the boundary layer. Deviations in the local stress from the surface shear velocity $U_τ$ account for almost all of the differences in wall-normal velocity variance observed across different canonical flows, including for plane Couette flow. The dependence on the local stress is attributed to the fact that wall-normal motions are predominately `active' per Townsend's attached eddy hypothesis and directly contribute to the local shear stress, noting this hypothesis assumes simplified ideal conditions with constant turbulent shear stress. A semi-empirical fit applied to the Reynolds number dependence of the variance matches the simulations across the lower half of the boundary layer and aligns with observed values in the literature. The fit extrapolates to a value between 1.45 and 1.65 times the local shear stress in the high-Reynolds-number limit, consistent with previous predictions relative to $U_τ$ including for the vertical velocity in the near-neutral atmospheric boundary layer. However, universality in the exact proportional constant is precluded by small discrepancies in the variances corresponding to dissimilarity in the low-wavenumber contributions across different flow configurations and wall-normal positions. We speculate the dissimilarity is due to relatively weak `inactive' wall-normal motions that are excluded from Townsend's original hypothesis.

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arXiv2025

Beyond-classical computation in quantum simulation

Quantum computers hold the promise of solving certain problems that lie beyond the reach of conventional computers. However, establishing this capability, especially for impactful and meaningful problems, remains a central challenge. Here, we show that superconducting quantum annealing processors can rapidly generate samples in close agreement with solutions of the Schrödinger equation. We demonstrate area-law scaling of entanglement in the model quench dynamics of two-, three-, and infinite-dimensional spin glasses, supporting the observed stretched-exponential scaling of effort for matrix-product-state approaches. We show that several leading approximate methods based on tensor networks and neural networks cannot achieve the same accuracy as the quantum annealer within a reasonable time frame. Thus, quantum annealers can answer questions of practical importance that may remain out of reach for classical computation.

quant-ph↗

arXiv2025

Defining the mean turbulent boundary layer thickness based on streamwise velocity skewness

A new statistical definition for the mean turbulent boundary layer thickness is introduced, based on identification of the point where the streamwise velocity skewness changes sign, from negative to positive, in the outermost region of the boundary layer. Importantly, this definition is independent of arbitrary thresholds, and broadly applicable, including to past single-point measurements. Further, this definition is motivated by the phenomenology of streamwise velocity fluctuations near the turbulent/non-turbulent interface, whose local characteristics are shown to be universal for turbulent boundary layers under low freestream turbulence conditions (i.e., with or without pressure gradients, surface roughness, etc.) through large-scale experiments, simulations and coherent structure-based modelling. The new approach yields a turbulent boundary layer thickness that is consistent with previous definitions, such as those based on Reynolds shear stress or `composite' mean velocity profiles, and which can be used practically e.g., to calculate integral thicknesses. Two methods are proposed for estimating the turbulent boundary layer thickness using this definition: one based on simple linear interpolation and the other on fitting a generalised Fourier model to the outer skewness profile. The robustness and limitations of these methods are demonstrated through analysis of several published experimental and numerical datasets, which cover a range of canonical and non-canonical turbulent boundary layers. These datasets also vary in key characteristics such as wall-normal resolution and measurement noise, particularly in the critical turbulent/non-turbulent interface region.

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arXiv2025

Convection velocities and velocity coupling of outer-scaled wall-pressure fluctuations in canonical turbulent boundary layers

This study shows that the turbulent velocities most strongly correlated with outer-scaled ($\delta$-scaled) wall-pressure fluctuations beneath a zero-pressure-gradient boundary layer reside within the logarithmic region. Even though contributions from the wake region are present, they are found to be statistically less dominant than those from the logarithmic region. The findings are based on bespoke measurements using an array of 63 microphones spanning 5$\delta$ in the streamwise direction (where $\delta$ is the boundary layer thickness), which synchronously captures space-time $p_w$ data alongside streamwise velocity fluctuations ($u$) from a single hotwire probe at the array's downstream end. The array is designed to spatially filter $p_w$ signals to uncover outer-scale contributions, by accurately resolving the large-scale portion of the frequency-wavenumber $p_w$ spectrum while avoiding aliasing of small-scale energy. This design, and its effectiveness in anti-aliasing, is validated against previously published low-Reynolds-number simulation datasets of turbulent boundary layer flow. Present experiments span a friction Reynolds number range of $1400 \lesssim Re_{\tau} \lesssim 5200$, over which the large-scale energy in the boundary layer grows significantly. This growth is reflected in both the frequency-wavenumber $p_w$ spectrum and the space-time $p_w$ correlations, both of which show scaling trends reflective of the large-scale pressure field convecting at an outer-scaled velocity of $0.75U_\infty$, where $U_\infty$ is the freestream velocity. The linear coherence between streamwise velocity and large-scale $p_w$ is directly quantified through space-time $p_w$--$u$ correlations, which show increasing magnitudes across the inner region with rising $Re_{\tau}$.

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arXiv2025

Upstream history quantification and scale-decomposed energy analysis for weak-to-strong adverse-pressure-gradient turbulent boundary layers

The present study delineates the effects of pressure gradient history and local disequilibration on the small and large-scale energy in turbulent boundary layers (TBLs) imposed with a broad range of adverse-pressure-gradients (APG). This is made possible by analyzing four published high-fidelity APG TBL databases, which span weak to strong APGs and cover dynamic conditions ranging from near-equilibrium to strong disequilibrium. The influence of PG history on TBL statistics is quantified by the accumulated PG parameter ($\overlineβ$), proposed previously by Vinuesa et al. (2017) to study integral quanitites, which is compared here between cases at matched local PG strength ($β$), Reynolds number ($Re$) and ${\rm d}β/{{\rm d}{Re}}$ at nominally similar orders of magnitude. While the effects of local disequilibration (${\rm d}β/{{\rm d}{Re}}$) are investigated by considering TBL cases at matched $β$, $Re$, and fairly matched $\overlineβ$. It is found that $\overlineβ$ cannot unambiguously capture history effects when ${\rm d}β/{{\rm d}{Re}}$ levels are significantly high, as it does not account for the delayed response of the mean flow and turbulence, nor the attenuation of the pressure gradient effect with distance. In two comparisons of APG TBLs under strong non-equilibrium, the values of $\overlineβ$ and ${\rm d}β/{\rm d}{Re}$ expressed using Zagarola-Smits scaling were found to be consistent with the trends in mean velocity defect and Reynolds stresses noted previously for weak APG TBLs. While an increase in $\overlineβ$ is associated with energisation of both the small and large scales in the outer regions of APG TBLs, it affects only the large scales in the near-wall region. This confirms the ability of near-wall small scales to rapidly adjust to changes in PG strength.

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arXiv2025

Response of a turbulent boundary layer to steady, square-wave-type transverse wall-forcing

This study investigates the spatial evolution of a zero pressure gradient turbulent boundary layer (TBL) imposed by a square-wave (SqW) of steady spanwise wall-forcing, which varies along the streamwise direction ($x$). The SqW wall-forcing is imposed experimentally via a series of streamwise periodic belts running in opposite spanwise directions, following the methodology of Knoop et al. (Exp. Fluids, vol 65, 2024), with the streamwise extent increased to beyond $\sim 11$ times the boundary layer thickness ($δ_o$) in the present study. This unique setup is leveraged to investigate the influence of viscous-scaled wavelength of SqW wall-forcing on the turbulent drag reduction (DR) efficacy for $λ^+_x = $ 471 (sub-optimal), 942 (near-optimal), and 1884 (post-optimal conditions), at fixed viscous-scaled wall-forcing amplitude, $A^+ = 12$, and friction Reynolds number, $Re_τ= 960$. The TBL's response to this wall-forcing is elucidated by drawing inspiration from established knowledge on traditionally studied sinusoidal forcing (SinW), based on analysis of the streamwise-phase variation of the Stokes strain rate (SSR). The analysis reveals the SqW forcing to be characterized by a combination of two markedly different SSR regimes whose influence on the overlying turbulence is found to depend on the forcing waveform: sub-phase-I of local and strong impulses of SSR downstream of the half- ($λ_x$/2) and full-phase ($λ_x$) locations, associated with a reversal in spanwise forcing directions, leading to significant turbulence attenuation, and sub-phase-II of near-zero SSR over the remainder of forcing phase that enables turbulence recovery (when wall-forcing magnitudes and direction remain constant).

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arXiv2024

Effects of Reynolds number and spatial resolution on the pressure source terms in turbulent boundary layers

The increase in wall-pressure fluctuations with increasing friction Reynolds number ($Re_τ$) of a turbulent boundary layer (TBL) is well known in the literature. However, very few studies have investigated the $Re_τ$-variation of the source terms of the pressure fluctuations, which are solely a function of the spatial velocity gradients within the TBL. This study quantifies the pressure source terms in a zero-pressure gradient TBL by utilizing a published direct numerical simulation (DNS; Sillero et al. 2013, Phys. Fluids) database across 1000 $\lesssim$ $Re_τ$ $\lesssim$ 2000. It is found that the magnitude of all source terms increases with $Re_τ$ across the entire TBL thickness, with the turbulence-turbulence (non-linear) interaction terms growing faster than the mean-shear (linear) source terms. Further, we use the simulation database to mimic the scenario of particle image velocimetry (PIV) experiments that are typically spatially under-resolved compared to DNS data. It is used to quantify the effect of spatial resolution on the accuracy of pressure source terms, which are estimated here for two common PIV scenarios: (i) planar PIV in the streamwise-wall-normal plane, and (ii) stereo-PIV in the spanwise-wall-normal plane of a ZPG TBL. This exercise reveals significant attenuation of all pressure source terms compared to those estimated from the original DNS, highlighting the challenges of accurately estimating these source terms in a high $Re_τ$ PIV experiment.

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arXiv2024

Turbulent/non-turbulent interface in high Reynolds number pressure gradient boundary layers

We report two-dimensional particle image velocimetry experiments in high Reynolds number turbulent boundary layers imposed with a moderately strong streamwise pressure gradient. The unique aspect of these data are the highly resolved measurements across the outer region of a physically thick boundary layer, enabling accurate detection of turbulent/non-turbulent interfaces (TNTI). The present dataset is used to detect the TNTI of an adverse pressure gradient turbulent boundary layer and compare its characteristics with that for a zero-pressure gradient boundary layer, at a nominally similar friction Reynolds number. It is found that the TNTI exists across a broader range of wall-normal distance in presence of an adverse pressure gradient, as compared to the zero-pressure gradient case. Implications on conditionally averaged turbulence statistics are discussed based on detection of the TNTI.

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arXiv2024

A note on the amplitude modulation phenomenon in non-canonical wall-bounded flows

The amplitude modulation phenomena, defined originally by Mathis et al. (J. Fluid Mech., 628, 311-337; 2009), corresponds to a unique non-linear interaction between Reynolds number ($Re_τ$) dependent large-scale motions and $Re_τ$-invariant inner-scale motions observed in canonical wall-bounded flows. While similar non-linear interactions have been quantified previously in non-canonical wall-bounded flows, linking them solely to amplitude modulation is questionable due to the fact that each non-canonical effect is associated with distinct variations in the energies of both the large and inner scaled motions. This study revisits analysis of non-linear triadic interactions, with consideration to various non-canonical effects, by analyzing published hot-wire datasets acquired in the large Melbourne wind tunnel. It is found that triadic interactions, across the entire turbulence scale hierarchy, may become statistically significant with increasing intensity of non-canonical effects such as wall roughness, pressure gradients, and spanwise or wall-normal forcing (when compared relative to their respective canonical baseline cases at matched $Re_τ$). This stands in contrast to previous observations made in canonical flows, where only the interaction between inner scales and inertia-dominated large scales was considered dynamically significant for increasing $Re_τ$. The implications of these findings are discussed for near-wall flow prediction models in non-canonical flows, which should take into account \emph{all} non-linear interactions coexisting in wall-bounded flows.

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arXiv2024

Active and inactive contributions to the wall pressure and wall-shear stress in turbulent boundary layers

A phenomenological description is presented to explain the intermediate and low-frequency/large-scale contributions to the wall-shear-stress ($τ_w$) and wall-pressure (${p}_w$) spectra of canonical turbulent boundary layers, which are well known to increase with Reynolds number. The explanation is based on the concept of active and inactive motions (Townsend, J. Fluid Mech., vol. 11, 1961) associated with the attached-eddy hypothesis. Unique data sets of simultaneously acquired $τ_w$, ${p}_w$ and velocity fluctuation time series in the log region are considered, across friction-Reynolds-number ($Re_τ$) range of $\mathcal{O}$($10^3$) $\lesssim$ $Re_τ$ $\lesssim$ $\mathcal{O}$($10^6$). A recently proposed energy-decomposition methodology (Deshpande et al., J. Fluid Mech., vol. 914, 2021) is implemented to reveal the active and inactive contributions to the $τ_w$- and $p_w$-spectra. Empirical evidence is provided in support of Bradshaw's (J. Fluid Mech., vol. 30, 1967) hypothesis that the inactive motions are responsible for the non-local wall-ward transport of the large-scale inertia-dominated energy, which is produced in the log region by active motions. This explains the large-scale signatures in the $τ_w$-spectrum, which grow with $Re_τ$ despite the statistically weak signature of large-scale turbulence production, in the near-wall region. For wall pressure, active and inactive motions respectively contribute to the intermediate and large scales of the $p_w$-spectrum. Both these contributions are found to increase with increasing $Re_τ$ owing to the broadening and energization of the wall-scaled (attached) eddy hierarchy. This potentially explains the rapid $Re_τ$-growth of the $p_w$-spectra relative to $τ_w$, given the dependence of the latter only on the inactive contributions.

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arXiv2024

Streamwise energy-transfer mechanisms in zero- and adverse-pressure-gradient turbulent boundary layers

The present study investigates streamwise ($\overline{u^2}$) energy-transfer mechanisms in the inner and outer regions of turbulent boundary layers (TBLs). Particular focus is placed on the $\overline{u^2}$-production, its inter-component and wall-normal transport as well as dissipation, all of which become statistically significant in the outer region with increasing friction Reynolds number ($Re_τ$). These properties are analyzed using published data sets of zero, weak and moderately strong adverse-pressure-gradient (APG) TBLs across a decade of $Re_τ$, revealing similarity in energy-transfer pathways for all these TBLs. It is found that both the inner and outer peaks of $\overline{u^2}$ are always associated with local maxima in the $\overline{u^2}$-production and its inter-component transport, and the regions below/above each of these peaks are always dominated by wall-ward/away-from-wall transport of $\overline{u^2}$, thereby classifying the $\overline{u^2}$-profiles into four distinct regimes. This classification reveals existence of phenomenologically similar energy-transfer mechanisms in the `inner' and `outer' regions of moderately strong APG TBLs, which meet at an intermediate location coinciding with the minimum in $\overline{u^2}$ profiles. Given that the wall-ward/away-from-wall transport of $\overline{u^2}$ is governed by the $\rm Q_4$(sweeps)/$\rm Q_2$(ejections) quadrants of the Reynolds shear stress, it is argued that the emergence of the $\overline{u^2}$ outer peak corresponds with the statistical dominance of $\rm Q_4$ events in the outer region. Besides unravelling the dynamical significance of $\rm Q_2$ and $\rm Q_4$ events in the outer region of turbulent boundary layers, the present analysis also proposes new phenomenological arguments for testing on canonical wall-turbulence data at very high $Re_τ$.

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