Large-scale Velocity Fluctuations Research Articles

Reynolds number dependence of turbulent flows over a highly permeable wall

Direct numerical simulations of turbulent flows over highly permeable porous walls were performed at various Reynolds numbers to examine the effects of the Reynolds number on permeable wall turbulence. The porous medium consisted of Kelvin cell arrays with porosity $0.95$ , and the permeability Reynolds number $Re_K$ ranged from approximately 7 to 50. Simulations with thin and thick porous walls were performed to investigate the effects of spanwise roller vortices associated with the Kelvin–Helmholtz instability. The results show that the effect of the Kelvin–Helmholtz instability becomes more significant with increasing the permeability Reynolds number, and spanwise rollers, for which length scale is an order of channel height, dominate turbulence when $Re_K \gtrsim 30$ . Spanwise rollers reinforce the negative correlation between the wall-normal and streamwise velocity fluctuations close to the porous/fluid interface, and intensify the turbulent velocity fluctuations away from the porous walls, leading to increased frictional resistance. An investigation of the Reynolds number dependence of the modified logarithmic law indicates that the zero-plane displacement and equivalent roughness height are proportional to the square root of permeability, whereas the von Kármán constant increases with the permeability Reynolds number because of the increased mixing length resulting from the relatively large-scale velocity fluctuations induced by spanwise rollers. We developed a model for the modified log law for permeable wall turbulence based on permeability, and confirmed that the skin friction coefficient obtained from the model reasonably predicts the skin friction coefficient for several types of high-porosity porous media. Hence, permeability is a key parameter that characterizes the logarithmic mean velocity profiles over a variety of porous media with high porosity.

The effect of turbulent motions on particle spatial distribution in high-Reynolds-number particle-laden flows

The spatial relationship between turbulent and particle concentration structures is investigated based on the turbulent velocity and particle concentration data obtained synchronously at the Qingtu Lake Observation Array site. In addition to the observation of particle concentration structures that contain not only large-scale coherence but also significant energy in the high-Reynolds-number atmospheric surface layer (ASL), the scale of turbulent motions that have the most significant coherence with particle concentration is found to follow a 1/2 power scaling law with the local height and ASL thickness. Moreover, large-scale turbulent velocity fluctuations have a significant amplitude modulation effect on particle concentration fluctuations, but the modulating influence is different for small dust particles and large saltating particles. Based on the interphase amplitude modulation, there exists a particle–turbulence structure phase difference that varies with height, which further makes the structure inclination angle of the particle concentration larger than that of the turbulence. In this scenario, a conceptual model reflecting the relationship between the two is proposed, and a quantitative formulation is further derived and found to be in good agreement with the experimental results. These findings and the proposed model contribute insights into particle–turbulence interactions, thereby providing theoretical support for a unified model of turbulence dynamics and particle kinematics.

Prediction of the phase difference between large-scale velocity and Reynolds stress fluctuations in wall turbulence

A resolvent-based model was used to predict the phase-difference profile between velocity and stress coherent motions measured in a high Reynolds number channel flow as a proxy for predicting large- and small-scale turbulent interactions. The resolvent model is based on the transfer-function approach for scale interactions in wall turbulence proposed in Jacobiet al.(J. Fluid Mech., vol 914, 2021, pp. 1–27), but incorporates a quasi-empirical weighting scheme to construct composite mode shapes that represent the realistic dispersion of convection velocities associated with the large scales of turbulence. The weighting scheme was derived from the observed similarity between the spectral region where the resolvent operator is low rank and the streamwise spectral energy density of wall-bounded turbulence, and was found to be superior to both single-convection velocity models and models based on linearly weighted modes, when compared with cross-spectral phase calculations from a channel flow computation. The ability to predict the phase relationship between large-scale coherent motions and their associated stress fluctuations allows for refining and extending resolvent-based models of turbulence to describe small-scale features of wall-bounded flows.

Spectral features of the wake and power fluctuations of model wind turbines under low-level jets

Laboratory experiments are conducted to inspect the modulation of model low-level-jet (LLJ) velocity profiles on the wake of a model wind turbine and power output fluctuations and a spectral structure of a simple two-turbine system in an aligned configuration. The scenario with a canonical incoming turbulent boundary layer profile is included for comparison. The results reveal a significant effect of the relative height of the LLJ peak velocity on the near and intermediate wake and power output fluctuations. Those effects are more notorious with the LLJ peak velocity coincident with the turbine top tip. The strong mean shear right above the wake promotes enhanced vertical transport and generation of energetic coherent motions. In particular, the near and intermediate wake velocity spectra showed a robust local signature with a streamwise length on the order of ∼4 times the rotor diameter. Although this shear layer did not significantly affect the spectrum of the power output fluctuations, the relatively large-scale velocity fluctuations may affect wind turbines' downwind in arrays.

Predictive models for near-wall velocity and temperature fluctuations in supersonic wall-bounded turbulence

Predictive models for near-wall velocity and temperature fluctuations in compressible wall-bounded turbulence are developed in the present study based on the model proposed by Marusic et al. (Science, vol. 329 (5988), 2010, pp. 193–196), which incorporates the superposition and amplitude modulation effects of the large-scale motions in the outer region on near-wall turbulence. The density variation is involved in the predictive model for velocity fluctuations to achieve Mach number independence. The predictive model for temperature fluctuations is derived to keep its consistency with the strong Reynolds analogy, in which the modulation effect is supposed to be cast as the quadratic function of the large-scale velocity fluctuations. An algebraic method is proposed to directly determine the modulation coefficients and extract the universal signals. A direct numerical simulation (DNS) of turbulent channel flow at the friction Reynolds number of $1170$ and bulk Mach number of $2.88$ is carried out for parameter calibration and validations. The variances and joint probability density functions of the predicted velocity and temperature fluctuations agree well with the DNS results.

Influence of aspect ratio on the wake dynamics of a pitching wing

The three-dimensional flow around a custom wing performing pitching oscillation is analyzed numerically using large-eddy simulation with the wall adapting local eddy-viscosity subgrid-scale model. The wake characteristics behind the pitching wing with three different aspect ratios, AR=2/3, 1, 4/3, is investigated in an upstream flow with Reynolds number, Re=2640. The Q−function is employed to identify the swirling structures in the flow field, and four primary vortices, namely, primary leading edge vortex (PLEV), secondary leading edge vortex, wing tip vortices, and trailing edge vortex, are identified. The evolution of vortices over the wing surface is presented in detail, and the arrangement of vortex rings in the wake is compared for three different aspect ratios. The PLEV develops into an arch vortex using the self-induction phenomenon, and the interlinking between the subsequent vortex rings depends on the arch vortex leaving the trailing edge. The force coefficient computed over the wing shows a positive thrust for AR=4/3. The mean velocity field contours indicate a spanwise loading on the wing, which is unaccountable in two-dimensional studies. The Reynolds stresses generated due to large-scale periodic velocity fluctuations in the wake are presented. The vorticity field in the midplane of the wing appears to be diffusive at higher aspect ratios. The comparison of vorticity from the three-dimensional and two-dimensional models suggests that a two-dimensional model will over predict the vorticity distribution and force coefficients in the wake.

Vertical and slanted sound propagation in the near-ground atmosphere: Coherence and distributions.

Atmospheric turbulence causes acoustic signals to fluctuate and diminishes their coherence. These phenomena are important in applications such as source localization and sonic boom propagation. This article provides formulations for the spatial, cross-frequency, and temporal coherences of narrowband acoustic signals propagating over vertical and slanted paths in the atmosphere. Formulations for single- and two-point distributions of acoustic signals are also overviewed. The theoretical formulations are compared with data from a comprehensive sound propagation experiment carried out in 2018 at the National Wind Technology Center (Boulder, CO). The theories for sound propagation in a turbulent atmosphere, when combined with turbulence models incorporating shear and buoyancy instabilities, correctly predict the measured spatial coherence, which is primarily affected by small-scale isotropic turbulence. For relatively small coherence times, this approach also correctly predicts the temporal coherence. However, the approach underpredicts the cross-frequency coherence and temporal coherence for relatively large coherence times, which are affected by large-scale anisotropic buoyancy-driven velocity fluctuations. For different regimes ranging from unsaturated to fully saturated scattering, the measured distributions agree well with the theoretical predictions.

An investigation of particles effects on wall-normal velocity fluctuations in sand-laden atmospheric surface layer flows

Based on the high-quality observational data in the Qingtu Lake Observation Array (QLOA), the difference in the energy distribution, the scale of the coherent structures, and the amplitude modulation effect of the wall-normal velocity fluctuations between particle-free and particle-laden flow in the atmospheric surface layer are analyzed. The results show that the presence of particles enhanced the wall-normal turbulence intensity, especially the increase at the top of the logarithmic region is more significant though the particle mass loading decreases with the wall-normal distance. A further insight indicates that the increase in the length scale of the wall-normal fluctuating velocity coherent structure by particles is more significant further from the wall, which is supported by the premultiplied energy spectra and the two-point correlation. This leads to a drastic increase in kinetic energy of the large-scale coherent structures by the particle away from the wall and thus results in increased amplitude modulation effects of large-scale wall-normal velocity fluctuations onto small-scales.

Experimental investigations of the turbulent/non-turbulent interface over surface with spanwise heterogeneity

The sharp but irregular interface that separates the instantaneous turbulent and irrotational flows is termed as the turbulent/non-turbulent interface (TNTI). TNTI can be widely observed in various types of flow, such as turbulent boundary layers, jets and combustion flame fronts. Due to its importance on the intermittency and entrainment process, TNTI has been widely explored in its geometry and dynamic properties (da Silva et al., 2014). Most of the studies focus on the TNTIs in smooth plane boundary layer, while few investigate the effects of wall shapes. However, the wall conditions in many engineering applications are complex and heterogeneous, which will induce large-scale heterogeneity (Barros and Christensen, 2014) and require further investigations. To shed new light on the intermittency and entrainment above complex surfaces, the TNTI over spanwise heterogeneity are investigated here with time-resolved stereoscopic PIV (TR-SPIV). The model and TR-SPIV experimental set-up are shown in Fig. 1. The experiments are conducted in the low-speed water channel at Beijing University of Aeronautics and Astronautics. The spanwise distance S between two adjacent ridges is S/(δ) = 1.35, where (δ) is the spanwise-averaged boundary layer thickness. This spanwise distance is selected to induced strong secondary vortices (Vanderwel and Ganapathisubramani, 2015; Wangsawijaya et al., 2020). The Reynolds number based on the streamwise location x is Rex = 7.2×105. The field of view is around 2S×1.8S, and is captured by two CMOS cameras (2048×2048 pixel) with sampling rate as 500Hz. The averaged resolution is about 8 pixels per Kolmogorov scale (calculated at y/(δ) = 0.6), which is high enough for TNTI-related research (Borrell and Jimenez, 2016). The ´TNTI is detected by the magnitude of local enstrophy ω2/2, and the threshold is selected to be the value where changing the threshold has the smallest influence on the TNTI-mean-height (Watanabe et al., 2018). The time-mean velocity and TNTI location are present in Fig.2(a). A pair of counter-rotating largescale secondary vortices (SVs) are induced over the ridge-type roughness. At the position where SVs induce upwash flow, a low-momentum pathway (LMP) can be observed, while the time-mean height of TNTI (yI) is brought higher. As a contrast, where downwash flow induces high-momentum pathway (HMP), (yI) is lower. TNTI properties are further discussed from two aspect. The geometry properties are firstly investigated. The fractal dimension of the TNTI keeps as 2.3 along the spanwise direction. This value is consistent with the result over smooth plate (Borrell and Jimenez, 2016; Wu et al., 2020) and riblets plates(Cui et al., 2019),´ which indicates that the wall shapes do not influence the multiscale properties of the TNTI. The streamwise wavelength of the TNTI (λI) is further obtained by calculating the streamwise pre-multiplied spectrum of the TNTI. It is found that at each spanwise location, λI is identical to the wavelength of streamwise velocity fluctuation at the TNTI mean height. This shows that the large-scale fluctuation of TNTI is controlled by the large-scale streamwise velocity fluctuation structures. Secondly, the p.d.f. of TNTI instantaneous height is investigated, as shown in Fig. 2(b). It can be observed that the p.d.f. of TNTI height above LMP shows a negative skewness, while the p.d.f. above HMP skews positively. A closer look at instantaneous structures shows that the skewness is attributed to the different probability of Q2/Q4 events in LMP and HMP.

A numerical study on the interaction between two cross-flow turbines in tandem configuration to support a simplified turbine model approach

In the planning of hydrokinetic turbine array deployment and for the performance prediction of its constituent turbines, the use of simplified turbine models is essential to alleviate the computational costs. The Effective Performance Turbine Model (EPTM) introduced in 2018 is a promising tool for that purpose, allowing array analysis and optimization. Its performance predictions scale with a local flow velocity characterization, which ensures to take into account inherently blockage effects and mean, local flow conditions. However, the characteristics of the local flow within an array also include different types of perturbation such as shear, large-scale temporal fluctuations, and turbulence. To ensure that the model is still reliable in those conditions, this paper presents a validation of the EPTM through the analysis of a large-scale tandem turbine configuration. In fact, three unsteady-Reynolds-averaged-Navier–Stokes simulations of a cross-flow turbine tandem configuration with a longitudinal spacing of six diameters have been conducted in this study, in addition to a simulation of a single turbine with turbulent ambient conditions. This set of simulations allows us to study independently the different types of perturbations associated with array deployment. Whereas the slow-varying large-scale upstream velocity fluctuations do not seem to affect significantly the turbine operation, the upstream non-uniform velocity distribution affects appreciably the extracted power. We find also that the value of the effective power coefficient associated with the EPTM needs to be adapted to the array simulation. Compared to the case of a single turbine in a uniform flow with a low turbulence level, we show that a smaller effective power coefficient value must be used in array simulations. An important result is that the different types of perturbations are found to yield similar effective performance coefficients, which suggests that the same set of values can be used throughout the array. With the appropriate set of values, we show that the EPTM succeeds to predict accurate downstream turbine performance.

Simultaneous Stereo PIV and MPS3 Wall-Shear Stress Measurements in Turbulent Channel Flow

An extended experimental method is presented in which the micro-pillar shear-stress sensor (MPS 3 ) and high-speed stereo particle-image velocimetry measurements are simultaneously performed in turbulent channel flow to conduct concurrent time-resolved measurements of the two-dimensional wall-shear stress (WSS) distribution and the velocity field in the outer flow. The extended experimental setup, which involves a modified MPS 3 measurement setup and data evaluation compared to the standard method, is presented and used to investigate the footprint of the outer, large-scale motions (LSM) onto the near-wall small-scale motions. The measurements were performed in a fully developed, turbulent channel flow at a friction Reynolds number R e τ = 969 . A separation between large and small scales of the velocity fluctuations and the WSS fluctuations was performed by two-dimensional empirical mode decomposition. A subsequent cross-correlation analysis between the large-scale velocity fluctuations and the large-scale WSS fluctuations shows that the streamwise inclination angle between the LSM in the outer layer and the large-scale footprint imposed onto the near-wall dynamics has a mean value of Θ ¯ x = 16.53 ∘ , which is consistent with the literature relying on direct numerical simulations and hot-wire anemometry data. When also considering the spatial shift in the spanwise direction, the mean inclination angle reduces to Θ ¯ x z = 13.92 ∘ .

Structural inclination angle of near-field scalar fluctuations in a turbulent boundary layer

Two-point concentration measurements are obtained in a meandering passive-scalar plume released at five different heights within the fully-turbulent region of a high-Reynolds number turbulent boundary layer (TBL). Mean statistics of two-point concentration measurements are found to agree very well with the single-point measurements previously reported in Talluru et al. (2017a). The two-point correlation results of concentration indicate strong coherence in the scalar field similar to the large-scale coherence observed in the streamwise velocity fluctuations in a TBL (Marusic and Heuer, 2007). Particularly, the isocontours in the two-dimensional correlation map of concentration fluctuations illustrate that the scalar structures are inclined at 30∘ to the direction of the flow; such a trend is consistently observed for all the elevated plumes below z/δ ≤ 0.33. This observation of steeper inclination angle of scalar structures relative to the inclination angle of large-scale velocity fluctuations in a TBL is explained using the physical model put forth by Talluru et al. (2018). Most importantly, these results provide insights on the differences in the structural organisation of a passive scalar plume in the near- and the far-field regions.

Electron Heating by Kinetic Alfvén Waves in Coronal Loop Turbulence

Abstract A test-particle model describing the energization of electrons in a turbulent plasma is presented. Parameters are chosen to represent turbulence in a magnetic structure of the solar corona. A fluctuating electric field component parallel to the background magnetic field, with properties similar to those of Kinetic Alfvén Waves, is assumed to be present at scales of the order of the proton Larmor radius. Electrons are stochastically accelerated by multiple interactions with such fluctuations, reaching energies of the order of 102 eV within tens to hundreds of seconds, depending on the turbulence amplitude. For values of the large-scale plasma velocity fluctuation of the order of tens of kilometers per second, the power absorbed by electrons per surface unit is of the order of that typically necessary to heat the corona. The power that electrons absorb from fluctuations is proportional to the third power of the large-scale velocity amplitude, and is comparable with the power associated with the turbulent cascade. Therefore, this mechanism can be considered as an equivalent kinetic dissipation for turbulence, and it can play a relevant role in the heating of electrons in the corona.

Экспериментальное исследование оптическим и статистическим методами крупномасштабных пульсаций потока при обтекании цилиндра

Using an optical method for measuring the velocity fields Particle Image Velocimetry (PIV) and a statistical method for analyzing coherent structures in turbulent flows Proper Orthogonal Decomposition (POD), an experimental study of the spatial structure of large-scale velocity fluctuations in the precavitational and cavitational flow past a circular cylinder with a Reynolds number of 280 000 was carried out.

Impact of Environmental Turbulence on the Performance and Loadings of a Tidal Stream Turbine

A large-eddy simulation (LES) of a laboratory-scale horizontal axis tidal stream turbine operating over an irregular bathymetry in the form of dunes is performed. The Reynolds number based on the approach velocity and the chord length of the turbine blades is approximately 60,000. The simulated turbine is a 1:30 scale model of a full-scale prototype and both turbines operate at very similar tip-speed ratio of λ ≈ 3. The simulations provide quantitative evidence of the effect of seabed-induced turbulence on the instantaneous performance and structural loadings of the turbine revealing how large-scale, energetic turbulence structures affect turbine performance and bending moments of the rotor blades. The data analysis shows that wake recovery is notably enhanced in comparison to the same turbine operating above a flat-bed and this is due to the higher turbulence levels generated by the dune. The results demonstrate the need for studying in detail the flow and turbulence characteristics at potential tidal turbine deployment sites and to incorporate observed large-scale velocity and pressure fluctuations into the structural design of the turbines.

Spatiotemporal Organization of Correlated Local Activity within Global Avalanches in Slowly Driven Interfaces

We study the jerky response of slowly driven fronts in disordered media, just above the depinning transition. We focus on how spatially disconnected clusters of internally correlated activity lead to large-scale velocity fluctuations in the form of global avalanches and identify three different ways in which local activity clusters may organize within a global avalanche, depending on the distance to criticality. Ouranalysis provides new scaling relations between the power-law exponents of the statistical distributions of sizes and durations of local bursts and global avalanches. Fluid fronts of imbibition in heterogeneous media are taken as a case study to validate these scaling relations.

Contribution of large-scale motions to the skin friction in a moderate adverse pressure gradient turbulent boundary layer

Direct numerical simulation of a turbulent boundary layer (TBL) subjected to a moderate adverse pressure gradient (APG,$\unicode[STIX]{x1D6FD}=1.45$) is performed to explore the contribution of large scales to the skin friction, where$\unicode[STIX]{x1D6FD}$is the Clauser pressure gradient parameter. The Reynolds number based on the momentum thickness develops from$Re_{\unicode[STIX]{x1D703}}\approx 110$to 6000 with an equilibrium region in$Re_{\unicode[STIX]{x1D703}}=4000$–5500. The spanwise wavelength ($\unicode[STIX]{x1D706}_{z}$) spectra of the streamwise and spanwise velocity fluctuations show that the large-scale energy is significantly enhanced throughout the boundary layer. We quantify the superposition and amplitude modulation effects of these enhanced large scales on the skin friction coefficient ($C_{f}$) by employing two approaches: (i) spanwise co-spectra of$\langle v\unicode[STIX]{x1D714}_{z}\rangle$and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$; (ii) conditionally averaged$\langle v\unicode[STIX]{x1D714}_{z}\rangle$and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$. The velocity–vorticity correlations$\langle v\unicode[STIX]{x1D714}_{z}\rangle$and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$are related to the advective transport and the vortex stretching, respectively. Although$\langle v\unicode[STIX]{x1D714}_{z}\rangle$negatively contributes to$C_{f}$, the positive contribution of the large scales ($\unicode[STIX]{x1D706}_{z}>0.5\unicode[STIX]{x1D6FF}$) is observed in the co-spectra of weighted$\langle v\unicode[STIX]{x1D714}_{z}\rangle$. For the co-spectra of weighted$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$, we observe an outer peak at$\unicode[STIX]{x1D706}_{z}\approx 0.75\unicode[STIX]{x1D6FF}$and the superposition of the large scales in the buffer region, leading to the enhancement of$C_{f}$. The magnitude of$\langle v\unicode[STIX]{x1D714}_{z}\rangle$and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$depends on the large-scale streamwise velocity fluctuations ($u_{L}$). In particular, the negative-$u_{L}$events amplify$\langle v\unicode[STIX]{x1D714}_{z}\rangle$in the outer region, and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$is enhanced by the positive-$u_{L}$events. As a result, the skin friction induced by$\langle v\unicode[STIX]{x1D714}_{z}\rangle$and$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$increases in the present APG TBL.

Influence of large-scale motions on the frictional drag in a turbulent boundary layer

Direct numerical simulation data of a turbulent boundary layer ($Re_{\unicode[STIX]{x1D70F}}=1000$) were used to investigate the large-scale influences on the vortical structures that contribute to the local skin friction. The amplitudes of the streamwise and wall-normal swirling strengths ($\unicode[STIX]{x1D706}_{x}$and$\unicode[STIX]{x1D706}_{y}$) were conditionally sampled by measuring the large-scale streamwise velocity fluctuations ($u_{l}$). In the near-wall region, the amplitudes of$\unicode[STIX]{x1D706}_{x}$and$\unicode[STIX]{x1D706}_{y}$decreased under negative$u_{l}$rather than under positive$u_{l}$. This behaviour arose from the spanwise motions within the footprints of the large-scale low-speed ($u_{l}<0$) and high-speed structures ($u_{l}>0$). The intense spanwise motions under the footprint of positive$u_{l}$noticeably strengthened the small-scale spanwise velocity fluctuations ($w_{s}$) below the centre of the near-wall vortical structures as compared to$w_{s}$within the footprint of negative$u_{l}$. The streamwise and wall-normal components were attenuated or amplified around the modulated vortical motions, which in turn led to the dependence of the swirling strength on the$u_{l}$event. We quantified the contribution of the modulated vortical motions$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$, which were related to a change-of-scale effect due to the vortex-stretching force, to the local skin friction. In the near-wall region, intense values of$\langle -w\unicode[STIX]{x1D714}_{y}\rangle$were observed for positive$u_{l}$. By contrast, these values were low for negative$u_{l}$, in connection with the amplification of$w_{s}$and$\unicode[STIX]{x1D706}_{y}$by the strong spanwise motions of the positive$u_{l}$. The resultant skin friction induced by the amplified vortical motions within$u_{l}^{+}>2$was responsible for 15 % of the total skin friction generated by the change-of-scale effect. Finally, we applied this analysis to a drag-reduced flow and found that the amplified vortical motions within the footprint of positive$u_{l}$were markedly diminished, which ultimately contributed to the total drag reduction.

Reynolds number trend of hierarchies and scale interactions in turbulent boundary layers.

Small-scale velocity fluctuations in turbulent boundary layers are often coupled with the larger-scale motions. Studying the nature and extent of this scale interaction allows for a statistically representative description of the small scales over a time scale of the larger, coherent scales. In this study, we consider temporal data from hot-wire anemometry at Reynolds numbers ranging from Reτ≈2800 to 22 800, in order to reveal how the scale interaction varies with Reynolds number. Large-scale conditional views of the representative amplitude and frequency of the small-scale turbulence, relative to the large-scale features, complement the existing consensus on large-scale modulation of the small-scale dynamics in the near-wall region. Modulation is a type of scale interaction, where the amplitude of the small-scale fluctuations is continuously proportional to the near-wall footprint of the large-scale velocity fluctuations. Aside from this amplitude modulation phenomenon, we reveal the influence of the large-scale motions on the characteristic frequency of the small scales, known as frequency modulation. From the wall-normal trends in the conditional averages of the small-scale properties, it is revealed how the near-wall modulation transitions to an intermittent-type scale arrangement in the log-region. On average, the amplitude of the small-scale velocity fluctuations only deviates from its mean value in a confined temporal domain, the duration of which is fixed in terms of the local Taylor time scale. These concentrated temporal regions are centred on the internal shear layers of the large-scale uniform momentum zones, which exhibit regions of positive and negative streamwise velocity fluctuations. With an increasing scale separation at high Reynolds numbers, this interaction pattern encompasses the features found in studies on internal shear layers and concentrated vorticity fluctuations in high-Reynolds-number wall turbulence.This article is part of the themed issue 'Toward the development of high-fidelity models of wall turbulence at large Reynolds number'.

Extended self-similarity in moment-generating-functions in wall-bounded turbulence at high Reynolds number

In wall-bounded turbulence, the moment generating functions (MGFs) of the streamwise velocity fluctuations $\left<\exp(qu_z^+)\right>$ develop power-law scaling as a function of the wall normal distance $z/\delta$. Here $u$ is the streamwise velocity fluctuation, $+$ indicates normalization in wall units (averaged friction velocity), $z$ is the distance from the wall, $q$ is an independent variable and $\delta$ is the boundary layer thickness. Previous work has shown that this power-law scaling exists in the log-region {\small $3Re_\tau^{0.5}\lesssim z^+$, $z\lesssim 0.15\delta$}, where $Re_\tau$ is the friction velocity-based Reynolds numbers. Here we present empirical evidence that this self-similar scaling can be extended, including bulk and viscosity-affected regions $30<z^+$, $z<\delta$, provided the data are interpreted with the Extended-Self-Similarity (ESS), i.e. self-scaling of the MGFs as a function of one reference value, $q_o$. ESS also improves the scaling properties, leading to more precise measurements of the scaling exponents. The analysis is based on hot-wire measurements from boundary layers at $Re_\tau$ ranging from $2700$ to $13000$ from the Melbourne High-Reynolds-Number-Turbulent-Boundary-Layer-Wind-Tunnel. Furthermore, we investigate the scalings of the filtered, large-scale velocity fluctuations $u^L_z$ and of the remaining small-scale component, $u^S_z=u_z-u^L_z$. The scaling of $u^L_z$ falls within the conventionally defined log region and depends on a scale that is proportional to {\small $l^+\sim Re_\tau^{1/2}$}; the scaling of $u^{S}_z$ extends over a much wider range from $z^+\approx 30$ to $z\approx 0.5\delta$. Last, we present a theoretical construction of two multiplicative processes for $u^L_z$ and $u^S_z$ that reproduce the empirical findings concerning the scalings properties as functions of $z^+$ and in the ESS sense.