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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., Comment: Accepted in the Proceedings of the IUTAM Symposium on Turbulent/Non-Turbulent Interface in Turbulent Shear Flows

Recently, Nagib et al (2024} utilized indicator functions of profiles of the streamwise normal stress to reveal the ranges of validity, in wall distance and Reynolds number, for each of two proposed models in DNS of channel and pipe flows. A method more suited to experimental data is proposed here, as establishing accurate indicator functions is a challenge. The new method is outlined and used with the two leading models that propose either a logarithmic or power trend, for the normal stresses of turbulence in a fitting region of wall-bounded flows. The method, which is simple and robust, is used to evaluate each model over a wide range of Reynolds numbers by applying it to two of the prominent experimental data sets in zero-pressure-gradient boundary layers (ZPG) and pipe flows. A somewhat larger exponent for the power trend equal to 0.28, instead of 0.25, is found to slightly extend its range of validity. Correcting for outer intermittency in ZPG data also extends validity of the power trend to around half the boundary layer thickness. Projecting near-wall peak normal-stress values using both models based on data from locations in the fitting region, yields nearly identical results up to the highest Re available from these two experiments.Neither model directly represents the inner peak values and differ by as much as 30%. However, the two relations provide projected values at Re=10M with a relative difference of approximately 6.2%; distinguishing between these values would require measurement accuracy well beyond our capabilities. Current results and those of Monkewitz and Nagib (2023) on the mean flow, suggest that considerations of nonlinear growth of eddies from the wall and accounting for some viscous effects, are important for modeling wall-bounded flows. Such effects also apply to potential refinements of the logarithmic trend along ideas advanced by Deshpande et al. (2021)., Comment: 15 pages, 15 figures

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 ($Re_{\tau}$) dependent large-scale motions and $Re_{\tau}$-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 $Re_{\tau}$). 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 $Re_{\tau}$. 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., Comment: Accepted in the Physical Review Fluids, with 13 pages and 3 figures

Analyzing human motion is an active research area, with various applications. In this work, we focus on human motion analysis in the context of physical rehabilitation using a robot coach system. Computer-aided assessment of physical rehabilitation entails evaluation of patient performance in completing prescribed rehabilitation exercises, based on processing movement data captured with a sensory system, such as RGB and RGB-D cameras. As 2D and 3D human pose estimation from RGB images had made impressive improvements, we aim to compare the assessment of physical rehabilitation exercises using movement data obtained from both RGB-D camera (Microsoft Kinect) and estimation from RGB videos (OpenPose and BlazePose algorithms). A Gaussian Mixture Model (GMM) is employed from position (and orientation) features, with performance metrics defined based on the log-likelihood values from GMM. The evaluation is performed on a medical database of clinical patients carrying out low back-pain rehabilitation exercises, previously coached by robot Poppy., Comment: European Conference on Mobile Robots (2023)

A phenomenological description is presented to explain the intermediate and low-frequency/large-scale contributions to the wall-shear-stress (${\tau}_w$) 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 ${\tau}_w$, ${p}_w$ and velocity fluctuation time series in the log region are considered, across friction-Reynolds-number ($Re_{\tau}$) range of $\mathcal{O}$($10^3$) $\lesssim$ $Re_{\tau}$ $\lesssim$ $\mathcal{O}$($10^6$). 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 ${\tau}_w$- 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 ${\tau}_w$-spectrum, which grow with $Re_{\tau}$ 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 $Re_{\tau}$ owing to the broadening and energization of the wall-scaled (attached) eddy hierarchy. This potentially explains the rapid $Re_{\tau}$-growth of the $p_w$-spectra relative to ${\tau}_w$, given the dependence of the latter only on the inactive contributions., Comment: Preprint submitted for review to the Journal of Fluid Mechanics, with 18 pages and 7 figures

Wall-scaled (attached) eddies play a significant role in the overall drag experienced in high-Reynolds-number turbulent boundary layers (TBLs). This study aims to delve into the underlying mechanisms driving this phenomenon by dissecting the active and inactive components of these attached eddies, as initially proposed by Townsend (1976). Employing a recently introduced energy-decomposition scheme, we analyze TBL datasets covering a wide range of Reynolds numbers ($Re_{\tau}$ $\sim$ $\mathcal{O}$(10$^{3}$)--$\mathcal{O}$(10$^{6}$)). This analysis provides empirical evidence of the distinct contributions of these components to drag generation, and reveals that while active motions are responsible solely for generating Reynolds shear stresses, inactive motions are crucial for transporting streamwise momentum from the logarithmic region to the wall, thus corroborating earlier hypotheses in the literature., Comment: Proceeding accepted for the Thirteenth International Symposium on Turbulence and Shear Flow Phenomena (TSFP13)

The present study tests the efficacy of the well-known viscous drag reduction strategy of imposing spanwise wall oscillations to reduce pressure drag contributions in a transitional- and fully-rough turbulent wall flow. This is achieved by conducting a series of direct numerical simulations of a turbulent flow over two-dimensional (spanwise aligned) semi-cylindrical rods, placed periodically along the streamwise direction with varying streamwise spacing. Surface oscillations, imposed at fixed viscous-scaled actuation parameters optimum for smooth wall drag reduction, are found to yield substantial drag reduction (>25%) for all the rough wall cases, maintained at matched roughness Reynolds numbers. While the total drag reduction is due to a drop in both viscous and pressure drag in the case of transitionally-rough flow (i.e. with large inter-rod spacing), it is solely associated with pressure drag reduction for the fully-rough cases (i.e. with small inter-rod spacings), with the latter being reported for the first time. The study finds that pressure drag reduction in all cases is caused by the attenuation of the vortex shedding activity in the roughness wake, in response to wall-oscillation frequencies that are of the same order as the vortex shedding frequencies. Contrary to speculations in the literature, this study confirms that the mechanism behind pressure drag reduction, achieved via imposition of spanwise oscillations, is independent from the viscous drag reduction. This mechanism is responsible for weakening of the Reynolds stresses and increase in base pressure in the roughness wake, explaining the pressure drag reduction observed by past studies, across varying roughness heights and geometries., Comment: Manuscript accepted in the Journal of Fluid Mechanics comprising 20 pages, 7 figures

This study presents an experimental dataset documenting the evolution of a turbulent boundary layer downstream of a rough-to-smooth surface transition. To investigate the effect of upstream flow conditions, two groups of experiments are conducted. For the \emph{Group-Re} cases, a nominally constant viscous-scaled equivalent sand grain roughness $k_{s0}^+\approx160$ is maintained on the rough surface, while the friction Reynolds number $Re_{\tau 0}$ ranges from 7100 to 21000. For the \emph{Group-ks} cases, $Re_{\tau 0}\approx14000$ is maintained while $k_{s0}^+$ ranges from 111 to 228. The wall-shear stress on the downstream smooth surface is measured directly using oil-film interferometry to redress previously reported uncertainties in the skin-friction coefficient recovery trends. In the early development following the roughness transition, the flow in the internal layer is not in equilibrium with the wall-shear stress. This conflicts with the common practise of modelling the mean velocity profile as two log laws below and above the internal layer height, as first proposed by Elliott (\textit{Trans. Am. Geophys. Union}, vol. 39, 1958, pp 1048--1054). As a solution to this, the current data are used to model the recovering mean velocity semi-empirically by blending the corresponding rough-wall and smooth-wall profiles. The over-energised large-scale motions leave a strong footprint in the near-wall region of the energy spectrum, the frequency and magnitude of which exhibit dependence on $Re_{\tau 0}$ and $k_{s0}^+$ respectively. The energy distribution in near-wall small scales is mostly unaffected by the presence of the outer flow with rough-wall characteristics, which can be used as a surrogate measure to extract the local friction velocity.

This paper examines recovery of the wall-shear stress of a turbulent boundary layer that has undergone a sudden transition from a rough to a smooth surface. Early works of Antonia and Luxton questioned the reliability of standard smooth-wall methods to measure wall-shear stress in such conditions, and subsequent studies show significant disagreement depending on the approach used to determine the wall-shear stress downstream. Here we address this by utilising a collection of experimental databases at Re_\tau \approx 4100 that have access to both `direct' and `indirect' measures of the wall-shear stress to understand the recovery to equilibrium conditions to the new surface. Our results reveal that the viscous region (z^+\lesssim 4) recovers almost immediately to an equilibrium state with the new wall conditions, however, the buffer region and beyond takes several boundary layer thicknesses before recovering to equilibrium conditions, which is longer than previously thought. A unique direct numerical simulation database of a wall-bounded flow with a rough-to-smooth wall transition is employed to confirm these findings. In doing so, we present evidence that any estimate of the wall-shear stress from the mean velocity profile in the buffer region or further away from the wall tends to underestimate its magnitude in the near vicinity of the rough-to-smooth transition, and this is likely to be partly responsible for the large scatter of recovery lengths to equilibrium conditions reported in the literature. Our results also reveal that the smaller energetic scales in the near-wall region recover to an equilibrium state associated with the new wall conditions within one boundary layer thickness downstream of the transition, while the larger energetic scales exhibit an over-energised state for several boundary layer thicknesses downstream of the transition.

Given a graph $X$ with a Hamilton cycle $C$, the {\em compression factor $\kappa(X,C)$ of $C$} is the order of the largest cyclic subgroup of $\operatorname{Aut}(C)\cap\operatorname{Aut}(X)$, and the {\em Hamilton compression $\kappa(X)$ of $X$ } is the maximum of $\kappa(X,C)$ where $C$ runs over all Hamilton cycles in $X$. Generalizing the well-known open problem regarding the existence of vertex-transitive graphs without Hamilton paths/cycles, it was asked by Gregor, Merino and M\"utze in [``The Hamilton compression of highly symmetric graphs'', {\em arXiv preprint} arXiv: 2205.08126v1 (2022)] whether for every positive integer $k$ there exists infinitely many vertex-transitive graphs (Cayley graphs) with Hamilton compression equal to $k$. Since an infinite family of Cayley graphs with Hamilton compression equal to $1$ was given there, the question is completely resolved in this paper in the case of Cayley graphs with a construction of Cayley graphs of semidirect products $\mathbb{Z}_p\rtimes\mathbb{Z}_k$ where $p$ is a prime and $k \geq 2$ a divisor of $p-1$. Further, infinite families of non-Cayley vertex-transitive graphs with Hamilton compression equal to $1$ are given. All of these graphs being metacirculants, some additional results on Hamilton compression of metacirculants of specific orders are also given., Comment: 11 pages

We investigate the underlying physics behind the change in amplitude modulation coefficient in non-canonical wall-bounded flows in the framework of the inner-outer interaction model (IOIM) (Baars et al., Phys. Rev. Fluids 1 (5), 054406). The IOIM captures the amplitude modulation effect, and here we focus on extending the model to non-canonical flows. An analytical relationship between the amplitude modulation coefficient and IOIM parameters is derived, which is shown to capture the increasing trend of the amplitude modulation coefficient with an increasing Reynolds number in a smooth-wall dataset. This relationship is then applied to classify and interpret the non-canonical turbulent boundary layer results reported in previous works. We further present the case study of a turbulent boundary layer after a rough-to-smooth change. Both single-probe and two-probe hotwire measurements are performed to acquire streamwise velocity time series in the recovering flow on the downstream smooth wall. An increased coherence between the large-scale motions and the small-scale envelope in the near-wall region is attributed to the stronger footprints of the over-energetic large-scale motions in the outer layer, whereas the near-wall cycle and its amplitude sensitivity to the superposed structures are similar to that of a canonical smooth-wall flow. These results indicate that the rough-wall structures above the internal layer interact with the near-wall cycle in a similar manner as the increasingly energetic structures in a high-Reynolds number smooth-wall boundary layer.

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 energy reduces with increasing friction Reynolds-number ($Re_{\tau}$). The trend is based on analyzing APG TBL data across 600 $\lesssim$ $Re_{\tau}$ $\lesssim$ 7000, and contrasts with the negligible variation in small scale energy noted for canonical wall flows. The datasets considered include those from a well-resolved numerical simulation (Pozuelo et al. 2022), which provides access to an APG TBL maintained at near-equilibrium conditions across 1000 $\lesssim$ $Re_{\tau}$ $\lesssim$ 2000, with a well-defined flow history, and a new high-$Re_{\tau}$ ($\sim$ 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 $Re_{\tau}$ is revealed via decomposing the streamwise normal stresses into small and large scale contributions, based on a sharp spectral cut-off. 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 $Re_{\tau}$ in the outer region. The conclusion is reaffirmed by investigating attenuation of streamwise normal stresses due to changing spatial-resolutions of the numerical grid/hotwire sensors, which reduces with increasing $Re_{\tau}$ and is found to be negligible at $Re_{\tau}$ $\sim$ 7000 in this study. The results emphasize that new scaling arguments and spatial-resolution correction schemes should be tested rigorously across a broad $Re_{\tau}$ range, particularly for the outer region of pressure gradient TBLs., Comment: Manuscript accepted in Physical Review Fluids, with 20 pages and 13 figures

As the impact of AI on various scientific fields is increasing, it is crucial to embrace interdisciplinary knowledge to understand the impact of technology on society. The goal is to foster a research environment beyond disciplines that values diversity and creates, critiques and develops new conceptual and theoretical frameworks. Even though research beyond disciplines is essential for understanding complex societal issues and creating positive impact it is notoriously difficult to evaluate and is often not recognized by current academic career progression. The motivation for this paper is to engage in broad discussion across disciplines and identify guiding principles fir AI research beyond disciplines in a structured and inclusive way, revealing new perspectives and contributing to societal and human wellbeing and sustainability.

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 for the first time 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 completely new 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.

We investigate the role of inter-scale interactions in the high-Reynolds number skin-friction drag reduction strategy reported by Marusic et al. (Nat. Commun., vol. 12, 2021). The strategy involves imposing relatively low-frequency streamwise travelling waves of spanwise velocity at the wall to actuate the drag generating outer-scales. This approach has proven to be more energy-efficient than the conventional method of directly targeting the drag producing inner-scales, which typically requires actuation at higher frequencies. Notably, it is observed that actuating the outer-scales at low frequencies leads to a substantial attenuation of the major drag producing inner-scales, suggesting that the actuations affect the non-linear inner-outer coupling inherently existing in wall-bounded flows. In the present study, we find that increased drag reduction, through imposition of spanwise wall oscillations, is always associated with an increased coupling between the inner and outer scales. This enhanced coupling emerges through manipulation of the phase relationships between these triadically linked scales, with the actuation forcing the entire range of energy-containing scales, from the inner (viscous) to the outer (inertial) scales, to be more in-phase. We also find that a similar enhancement of this non-linear coupling, via manipulation of the inter-scale phase relationships, occurs with increasing Reynolds number for canonical turbulent boundary layers. This indicates improved efficacy of the energy-efficient drag reduction strategy at very high Reynolds numbers, where the energised outer-scales are known to more strongly superimpose and modulate the inner-scales. Leveraging the inter-scale interactions, therefore, offers a plausible mechanism for achieving energy-efficient drag reduction at high Reynolds numbers., Comment: Manuscript accepted in the Journal of Fluid Mechanics comprising 16 pages, 5 figures

Here, we present measurements of turbulent drag reduction in boundary layers at high friction Reynolds numbers in the range of $4500 \le Re_\tau \le 15000$. The efficacy of the approach, using streamwise travelling waves of spanwise wall oscillations, is studied for two actuation regimes: (i) inner-scaled actuation (ISA), as investigated in Part 1 of this study, which targets the relatively high-frequency structures of the near-wall cycle, and (ii) outer-scaled actuation (OSA), which was recently presented by Marusic et al. (Nat. Commun., vol. 12, 2021) for high-$Re_\tau$ flows, targeting the lower-frequency, outer-scale motions. Multiple experimental techniques were used, including a floating-element balance to directly measure the skin-friction drag force, hot-wire anemometry to acquire long-time fluctuating velocity and wall-shear stress, and stereoscopic-PIV (particle image velocimetry) to measure the turbulence statistics of all three velocity components across the boundary layer. Under the ISA pathway, drag reduction of up to 25% was achieved, but mostly with net power saving losses due to the high-input power cost associated with the high-frequency actuation. The low-frequency OSA pathway, however, with its lower input power requirements, was found to consistently result in positive net power savings of 5 - 10%, for moderate drag reductions of 5 - 15%. The results suggest that OSA is an attractive pathway for energy-efficient drag reduction in high Reynolds number applications. Both ISA and OSA strategies are found to produce complex inter-scale interactions, leading to attenuation of the turbulent fluctuations across the boundary layer for a broad range of length and time scales.

Turbulent drag reduction through streamwise travelling waves of spanwise wall oscillation is investigated over a wide range of Reynolds numbers. Here, in Part 1, wall-resolved large-eddy simulations in a channel flow are conducted to examine how the frequency and wavenumber of the travelling wave influence the drag reduction at friction Reynolds numbers $Re_\tau = 951$ and $4000$. The actuation parameter space is restricted to the inner-scaled actuation (ISA) pathway, where drag reduction is achieved through direct attenuation of the near-wall scales. The level of turbulence attenuation, hence drag reduction, is found to change with the near-wall Stokes layer protrusion height $\ell_{0.01}$. A range of frequencies is identified where the Stokes layer attenuates turbulence, lifting up the cycle of turbulence generation and thickening the viscous sublayer; in this range, the drag reduction increases as $\ell_{0.01}$ increases up to $30$ viscous units. Outside this range, the strong Stokes shear strain enhances near-wall turbulence generation leading to a drop in drag reduction with increasing $\ell_{0.01}$. We further find that, within our parameter and Reynolds number space, the ISA pathway has a power cost that always exceeds any drag reduction savings. This motivates the study of the outer-scaled actuation (OSA) pathway in Part 2, where drag reduction is achieved through actuating the outer-scaled motions.

We present experimental evidence that the superstructures in turbulent boundary layers comprise of smaller, geometrically self-similar coherent motions. The evidence comes from identifying and analyzing 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 analyzes the wall-normal velocity fluctuations within these very long motions, which reveals the constituent motions unambiguously. The conditional streamwise energy spectra of 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. Similar characteristics are also exhibited by the Reynolds shear stress co-spectra estimated for the superstructure region, suggesting 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 also lends empirical support to the concatenation hypothesis for the formation of the superstructures. Association between the superstructures and self-similar motions is reaffirmed on comparing the vertical correlations of the momentum carrying motions, 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 the prospect for modelling these dynamically significant motions via data-driven coherent structure-based models., Comment: Manuscript accepted for the Journal of Fluid Mechanics, with 25 pages, 15 figures

The predictions of resolvent analysis for turbulent channel flow are evaluated for a friction Reynolds number of Retau = 550. In addition to the standard resolvent operator with kinematic viscosity, a resolvent operator augmented with the Cess eddy viscosity is considered. Adding eddy viscosity significantly alters the low-rank behavior of the resolvent. Regardless of the wave speed selected, the eddy resolvent is low-rank for spanwise wavelengths of ly+ = 80 and ly/h = 3.5 in comparison to the standard resolvent whose low-rank behavior depends on the wave speed. The leading eddy modes have higher projections onto the leading mode from spectral proper orthogonal decomposition in comparison to standard resolvent modes. Neither analysis, however, reliably predicts the most energetic wave speed. The standard resolvent tends to overestimate it while the eddy resolvent underestimates it. When the most energetic wave speed is underestimated, the eddy modes are energetic too close to the wall. The eddy resolvent does, however, correctly identify the most energetic wave speed and mode shapes for structures associated with the near-wall cycle or that are most energetic at z/h = 0.5. These structures are likely to be correctly predicted for any friction Reynolds number due to the scaling of the Cess eddy viscosity profile. Finally, it is shown that the accuracy of eddy predictions relies on the right balance between positive and negative energy transfers. Even though eddy viscosity primarily adds dissipation, its wall-normal gradient injects energy in the near-wall region, resulting in mode shapes that are attached to the wall. For some scales the predicted positive energy transfer is too strong thus biasing structures towards the wall. The ability of the Cess eddy viscosity profile to model both positive and negative energy transfers suggests that it could be optimized for individual scales., Comment: 24 pages, 20 figures

For a permutation group $G$ acting on a set $V$, a subset $\mathcal{F}$ of $G$ is said to be an intersecting set if for every pair of elements $g,h\in \mathcal{F}$ there exists $v \in V$ such that $g(v) = h(v)$. The intersection density $\rho(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|\mathcal{F}|/|G_v|$ where $G_v$ is a stabilizer of a point $v\in V$ and $\mathcal{F}$ runs over all intersecting sets in $G$. If $G_v$ is a largest intersecting set in $G$ then $G$ is said to have the Erd\H{o}s-Ko-Rado (EKR)-property. This paper is devoted to the study of transitive permutation groups, with point stabilizers of prime order with a special emphasis given to orders 2 and 3, which do not have the EKR-property. Among other, constructions of infinite family of transitive permutation groups having point stabilizer of order $3$ with intersection density $4/3$ and of infinite families of transitive permutation groups having point stabilizer of order $3$ with arbitrarily large intersection density are given., Comment: 14 pages

A large-scale spanwise and wall-normal array of sonic anemometers in the atmospheric surface layer is used to acquire all three components of instantaneous fluctuating velocity as well as temperature in a range of stability conditions. These data permit investigation of the three-dimensional statistical structure of turbulence structures. The present work extends the view of a self-similar range of wall-attached turbulence structures to the atmospheric surface layer under unstable and near-neutral stability conditions, and includes the statistical structure in both the wall-normal and spanwise directions in relation to the streamwise wavelength. Results suggest that the self-similar wall-attached structures have similar aspect ratios between streamwise/wall-normal scales and streamwise/spanwise scales such that $\lambda_x/\Delta z : \lambda_x/\Delta y \approx 1$ for both near-neutral and unstable conditions. By analysing the phase shift between synchronized measurements, in the spectral domain, it is quantified how the structure inclination angle varies with stability. Under the most unstable conditions, coherent structures of $\lambda_x/\delta = 1$ are inclined at angles as high as $65^\circ$ relative to the solid boundary, while larger scales of $\lambda_x/\delta = 6$ exhibit inclination angles of approximately $35^\circ$. For near-neutral stability conditions, the angle tends towards $12^\circ$ for all scales. It is noted that in the near-neutral condition, the structure inclination angle and the aspect ratio -- and thus the statistical modeling of coherent structures in the ASL -- are highly sensitive to the value of the stability parameter.

For a permutation group $G$ acting on a set $V$, a subset $I$ of $G$ is said to be an intersecting set if for every pair of elements $g,h\in I$ there exists $v \in V$ such that $g(v) = h(v)$. The intersection density $\rho(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|I|/|G_v|$ where $G_v$ is a stabilizer of a point $v\in V$ and $I$ runs over all intersecting sets in $G$. If $G_v$ is the largest intersecting set in $G$ then $G$ is said to have the Erd\H{o}s-Ko-Rado (EKR)-property, and moreover, $G$ has the strict-EKR-property if every intersecting set of maximum size in $G$ is a coset of a point stabilizer. Intersecting sets in $G$ coincide with independent sets in the so-called derangement graph $\Gamma_G$, defined as the Cayley graph on $G$ with connection set consisting of all derangements, that is, fixed-point free elements of $G$. In this paper a conjecture regarding the existence of transitive permutation groups whose derangement graphs are complete multipartite graphs, posed by Meagher, Razafimahatratra and Spiga in [J.Combin. Theory Ser. A 180 (2021), 105390], is proved. The proof uses direct product of groups. Questions regarding maximum intersecting sets in direct and wreath products of groups and the (strict)-EKR-property of these group products are also investigated. In addition, some errors appearing in the literature on this topic are corrected., Comment: 13 pages

Two elements $g$ and $h$ of a permutation group $G$ acting on a set $V$ are said to be intersecting if $g(v) = h(v)$ for some $v \in V$. More generally, a subset ${\cal F}$ of $G$ is an intersecting set if every pair of elements of ${\cal F}$ is intersecting. The intersection density $\rho(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|{\cal F}|/|G_v|$ where $G_v$ is a stabilizer of $v\in V$ and ${\cal F}$ runs over all intersecting sets in $G$. Intersection densities of transitive groups of degree $pq$, where $p>q$ are odd primes, is considered. In particular, the conjecture that the intersection density of every such group is equal to $1$ (posed in [ J.~Combin. Theory, Ser. A 180 (2021), 105390]) is disproved by constructing a family of imprimitive permutation groups of degree $pq$ (with blocks of size $q$), where $p=(q^k-1)/(q-1)$, whose intersection density is equal to $q$. The construction depends heavily on certain equidistant cyclic codes $[p,k]_q$ over the field $\mathbb{F}_q$ whose codewords have Hamming weight strictly smaller than $p$., Comment: 8 pages

Predictions of the spatial representation of instantaneous wall-bounded flows, via coherent structure-based models, are highly sensitive to the geometry of the representative structures employed by them. In this study, we propose a methodology to extract the three-dimensional (3-D) geometry of the statistically significant eddies from multi-point wall-turbulence datasets, for direct implementation into these models to improve their predictions. The methodology is employed here for reconstructing a 3-D statistical picture of the inertial wall coherent turbulence for all canonical wall-bounded flows, across a decade of friction Reynolds number ($Re_{\tau}$). These structures are responsible for the $Re_{\tau}$-dependence of the skin-friction drag and also facilitate the inner-outer interactions, making them key targets of structure-based models. The empirical analysis brings out the geometric self-similarity of the large-scale wall-coherent motions and also suggests the hairpin packet as the representative flow structure for all wall-bounded flows, thereby aligning with the framework on which the attached eddy model (AEM) is based. The same framework is extended here to also model the very-large-scaled motions, with a consideration of their differences in internal versus external flows. Implementation of the empirically-obtained geometric scalings for these large structures into the AEM is shown to enhance the instantaneous flow predictions for all three velocity components. Finally, an active flow control system driven by the same geometric scalings is conceptualized, towards favourably altering the influence of the wall coherent motions on the skin-friction drag., Comment: 16 pages, 11 figures, accepted for publication in the International Journal of Heat and Fluid Flow

Two elements $g$ and $h$ of a permutation group $G$ acting on a set $V$ are said to be intersecting if $g(v) = h(v)$ for some $v \in V$. More generally, a subset ${\cal F}$ of $G$ is an intersecting set if every pair of elements of ${\cal F}$ is intersecting. The intersection density $\rho(G)$ of a transitive permutation group $G$ is the maximum value of the quotient $|{\cal F}|/|G_v|$ where ${\cal F}$ runs over all intersecting sets in $G$ and $G_v$ is a stabilizer of $v\in V$. In this paper the intersection density of transitive groups of degree twice a prime is determined, and proved to be either $1$ or $2$. In addition, it is proved that the intersection density of transitive groups of prime power degree is $1$., Comment: 10 pages

Townsend (1961) 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 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 analyzing datasets comprising two-point streamwise velocity signals coupled with a spectral linear stochastic estimation (SLSE) based procedure. The data considered span a friction Reynolds number range $Re_{\tau}$ $\sim$ ${\mathcal{O}}$($10^3$) -- ${\mathcal{O}}$($10^4$). The procedure linearly decomposes the full 2-D spectrum (${\Phi}$) into two components, ${\Phi}_{ia}$ and ${\Phi}_{a}$, comprising contributions predominantly from the inactive and active motions, respectively. This is confirmed by ${\Phi}_{a}$ exhibiting wall-scaling, for both streamwise and spanwise wavelengths, corresponding well with the Reynolds shear stress cospectra reported in the literature. Both ${\Phi}_{a}$ and ${\Phi}_{ia}$ are found to depict prominent self-similar characteristics in the inertially dominated region close to the wall, suggestive of contributions from Townsend's attached eddies. Inactive contributions from the attached eddies reveal pure $k^{-1}$-scaling for the associated 1-D spectra (where $k$ is the streamwise/spanwise wavenumber), lending empirical support to the attached eddy model of Perry & Chong (1982)., Comment: 22 pages, 8 figures, submitted to the Journal of Fluid Mechanics

Two-dimensional (2-D) spectra of the streamwise velocity component, measured at friction Reynolds numbers ranging from 2400 to 26000, are used to refine a model for the logarithmic region of turbulent boundary layers. Here, we focus on the attached eddy model (AEM). The conventional AEM assumes the boundary layer to be populated with hierarchies of self-similar wall-attached ($Type\,A$) eddies alone. While $Type\,A$ eddies represent the dominant energetic large-scale motions at high Reynolds numbers, the scales that are not represented by such eddies are observed to carry a significant proportion of the total kinetic energy. Therefore, in the present study, we propose an extended AEM that incorporates two additional representative eddies. These eddies, named $Type\,C_A$ and $Type\,SS$, represent the self-similar but wall-incoherent low-Reynolds number features, and the non-self-similar wall-coherent superstructures, respectively. The extended AEM is shown to better predict a greater range of energetic length scales and capture the low- and high-Reynolds number scaling trends in the 2-D spectra of all three velocity components. A discussion on spectral self-similarity and the associated $k^{-1}$ scaling law is also presented., Comment: 19 pages, 12 figures

Opposition-control of the energetic cycle of near wall streaks in wall-bounded turbulence, using numerical approaches, has shown promise for drag reduction. For practical implementation, opposition control is only realizable if there is a degree of coherence between the sensor--actuator pairs of the control system (these sensors and actuators should typically be wall-based to avoid parasitic drag). As such, we here inspect the feasibility of real-time control of the near-wall cycle, by considering the coherence between a measurable wall-quantity, being the wall-shear stress fluctuations, and the streamwise and wall-normal velocity fluctuations in a turbulent boundary layer. Synchronized spatial and temporal velocity data from numerical simulations at $Re_\tau \approx 590$ and $ 2000$ are employed. It is shown that the spectral energy of the streamwise velocity fluctuations that is stochastically incoherent with wall signals is independent of Reynolds number in the near wall region. Consequently, the streamwise energy-fraction that is stochastically wall-coherent grows with Reynolds number due to the increasing range of energetic large scales. This thus implies that a wall-based control system has the ability to manipulate a larger portion of the total turbulence energy at off-wall locations, at higher Reynolds numbers. Coherence values of 0.55 and 0.4, which are considerably lower than the maximum possible coherence 1, were found between the streamwise and wall-normal velocity fluctuations at the near wall peak in the energy spectrogram, respectively, and the streamwise fluctuating friction velocity. This suggests that a closed-loop drag reduction scheme targeting near wall cycle of streaks alone will be of limited success in practice as the Reynolds number grows.

We analyse the inter-scale transfer of energy for two types of plane Poiseuille flow: the P4U exact coherent state of Park and Graham (2015) and turbulent flow in a minimal channel. For both flows, the dominant energy-producing modes are streamwise-constant streaks with a spanwise spacing of approximately 100 wall units. Since the viscous dissipation for these scales is not sufficient to balance production, the nonlinear terms redistribute the excess energy to other scales. Spanwise-constant scales (that is, Tollmien-Schlichting-like modes with zero spanwise wavenumber), in particular, account for a significant amount of net energy gain from the nonlinear terms. We compare the energy balance to predictions from resolvent analysis and we show that it does not model energy transfer well. Nevertheless, we find that the energy transferred from the streamwise-constant streaks can be predicted reasonably well by a Cess eddy viscosity profile. As such, eddy viscosity is an effective model for the nonlinear terms in resolvent analysis and explains good predictions for the most energetic streamwise-constant streaks. It also improves resolvent modes as a basis for structures whose streamwise lengths are greater than their spanwise widths by counteracting non-normality of the resolvent operator. Eddy viscosity does not respect the conservative nature of the nonlinear energy transfer which must sum to zero over all scales. It is less effective, consequently, for scales which receive energy from the nonlinear terms., Comment: 23 pages, 14 figures

Since the introduction of the logarithmic law of the wall more than 80 years ago, the equation for the mean velocity profile in turbulent boundary layers has been widely applied to model near-surface processes and parameterise surface drag. Yet the hypothetical turbulent eddies proposed in the original logarithmic law derivation and mixing length theory of Prandtl have never been conclusively linked to physical features in the flow. Here, we present evidence that suggests these eddies correspond to regions of coherent streamwise momentum known as uniform momentum zones (UMZs). The arrangement of UMZs results in a step-like shape for the instantaneous velocity profile, and the smooth mean profile results from the average UMZ properties, which are shown to scale with the friction velocity and wall-normal distance in the logarithmic region. These findings are confirmed across a wide range of Reynolds number and surface roughness conditions from the laboratory scale to the atmospheric surface layer., Comment: Version 2 includes a new title and minor edits for clarification based on peer review. No changes were made to the primary results and conclusions. Version 3 corrected two important typos in the text related to the results

Machine learning algorithms are extensively used to make increasingly more consequential decisions about people, so achieving optimal predictive performance can no longer be the only focus. A particularly important consideration is fairness with respect to race, gender, or any other sensitive attribute. This paper studies intersectional fairness, where intersections of multiple sensitive attributes are considered. Prior research has mainly focused on fairness with respect to a single sensitive attribute, with intersectional fairness being comparatively less studied despite its critical importance for the safety of modern machine learning systems. We present a comprehensive framework for auditing and achieving intersectional fairness in classification problems: we define a suite of metrics to assess intersectional fairness in the data or model outputs by extending known single-attribute fairness metrics, and propose methods for robustly estimating them even when some intersectional subgroups are underrepresented. Furthermore, we develop post-processing techniques to mitigate any detected intersectional bias in a classification model. Our techniques do not rely on any assumptions regarding the underlying model and preserve predictive performance at a guaranteed level of fairness. Finally, we give guidance on a practical implementation, showing how the proposed methods perform on a real-world dataset.

This paper reports the first spin tune measurement at high energies (24 GeV and 255 GeV) with a driven coherent spin motion. To maintain polarization in a polarized proton collider, it is important to know the spin tune of the polarized proton beam, which is defined as the number of full spin precessions per revolution. A nine-magnet spin flipper has demonstrated high spin-flip efficiency in the presence of two Siberian snakes [1]. The spin flipper drives a spin resonance with a given frequency (or tune) and strength. When the drive tune is close to the spin tune, the proton spin direction is not vertical anymore, but precesses around the vertical direction. By measuring the precession frequency of the horizontal component the spin tune can be precisely measured. A driven coherent spin motion and fast turn-by-turn polarization measurement are keys to the measurement. The vertical spin direction is restored after turning the spin flipper off and the polarization value is not affected by the measurement. The fact that this manipulation preserves the polarization makes it possible to measure the spin tune during operation of a high energy accelerator.