1. Introduction
Quantifying the interactions between turbulent structures of different size represents a long-standing problem in the turbulence research community. Both experiments (Kim & Adrian Reference Kim and Adrian1999; Hutchins & Marusic Reference Hutchins and Marusic2007) and numerical simulations (Lee & Moser Reference Lee and Moser2015, Reference Lee and Moser2019) have evidenced the presence of coherent velocity fluctuations (or eddies) populating the inertial layer of wall-bounded flows as the Reynolds number increases. Depending on their streamwise elongation, these coherent structures are classified as: wall-attached eddies, whose size grows linearly with their distance from the wall,
$z$
, (Townsend Reference Townsend1976; Perry et al. Reference Perry, Henbest and Chong1986); large-scale motions (LSMs), which are believed to arise as a concatenation of attached eddies and whose streamwise extent goes up to approximately 3 times the flow's outer length scale,
$\delta$
(Adrian Reference Adrian2007; Marusic et al. Reference Marusic, McKeon, Monkewitz, Nagib, Smits and Sreenivasan2010; Smits et al. Reference Smits, McKeon and Marusic2011; Jimenez Reference Jimenez2018); and very-large-scale motions (VLSMs), whose streamwise extent can reach 10–20 times the outer length scale (Kim & Adrian Reference Kim and Adrian1999; Guala et al. Reference Guala, Hommema and Adrian2006; Balakumar & Adrian Reference Balakumar and Adrian2007; Hutchins & Marusic Reference Hutchins and Marusic2007; Guala et al. Reference Guala, Metzger and McKeon2011). For laboratory flows,
$\delta$
is represented by the boundary layer height, channel half-height or pipe radius (Adrian Reference Adrian2007; Marusic et al. Reference Marusic, McKeon, Monkewitz, Nagib, Smits and Sreenivasan2010; Smits et al. Reference Smits, McKeon and Marusic2011; Jimenez Reference Jimenez2018), while for atmospheric flows, the outer scale of turbulence can be approximated by the height of the atmospheric surface layer (Hutchins et al. Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012; Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013; Puccioni et al. Reference Puccioni, Calaf, Pardyjak, Hoch, Morrison, Perelet and Iungo2023), which is roughly
$10\,\%$
of the atmospheric boundary layer height estimated through the geostrophic drag law (Stull Reference Stull1988). The ratio between the large scales dominated by inertia (proportional to
$\delta$
) and the small scales driven by viscosity (proportional to
$\nu /U_\tau$
, where
$\nu$
is the kinematic viscosity and
$U_\tau$
is the friction velocity) is the friction Reynolds number (
$\textit{Re}_\tau =U_\tau \delta /\nu$
).
The above-mentioned forward leaning nature of the structures, typically referred to as ‘inclination angle’, represents one of their most important features under investigation (Kovasznay et al. Reference Kovasznay, Kibens and Blackwelder1970; Hommema & Adrian Reference Hommema and Adrian2003; Marusic & Heuer Reference Marusic and Heuer2007). Arising from the balance between vertical turbulent diffusion and horizontal advection (Liu et al. Reference Liu, Bo and Liang2017), the inclination angle,
$\gamma$
, of turbulent structures gives rise to complex phase relationships between eddies of different sizes and, thus, is instrumental for understanding several flow features, such as Reynolds stress balance (Carper & Porté-Agel Reference Carper and Porté-Agel2004), amplitude modulation (Mathis et al. Reference Mathis, Hutchins and Marusic2009), large-to-small scale interactions (Chung & McKeon Reference Chung and McKeon2010) and aspect ratio of wall-coherent structures (Krug et al. Reference Krug, Baars, Hutchins and Marusic2019).
From the modelling standpoint, attempts have been made to numerically resolve the largest turbulent scales present in a wall-bounded flow and modelling their interactions with smaller scales (e.g. Speziale Reference Speziale1998; Liu & Shih Reference Liu and Shih2006) to reduce the computational cost. This strategy, however, is able to model turbulent scale interactions via the ensemble averaged approach, thus without resolving the effect of the single large-scale motion onto the smaller scales. Similarly, the inner-outer framework proposed by Mathis et al. (Reference Mathis, Hutchins and Marusic2011) to model the amplitude modulation of large structures is based on averaged values of VLSM inclination angles, thus not considering their variability within a flow and the subsequent effect on the spatial organisation of smaller scales. Finally, the velocity field induced by wall-attached eddies is typically represented by synthetic hairpin packets where each vortex filament is inclined by
$45^\circ$
in the downstream direction (Perry et al. Reference Perry, Henbest and Chong1986; Perry & Marusic Reference Perry and Marusic1995; Baidya et al. Reference Baidya, Philip, Monty, Hutchins and Marusic2014, Reference Baidya, Philip, Hutchins, Monty and Marusic2017; Chandran et al. Reference Chandran, Baidya, Monty and Marusic2017). Although this value is consistent with experimental evidence in an average sense, its variability within a wall-bounded flow has not been investigated so far.
Although strong consistency is found for the inclination angle of turbulent coherent structures across a wide range of Reynolds numbers (
$\textit{Re}_\tau =10^3$
to
$10^6$
) (Marusic & Heuer Reference Marusic and Heuer2007), any physical mechanism acting on vertical turbulent transport is expected to also affect the inclination angle. In this regard, large variability in the inclination angle of VLSMs is found for atmospheric surface layer (ASL) flows depending on thermal stratification (Hommema & Adrian Reference Hommema and Adrian2003; Jacob & Anderson Reference Jacob and Anderson2017; Liu et al. Reference Liu, Bo and Liang2017; Salesky & Anderson Reference Salesky and Anderson2018; Lotfy et al. Reference Lotfy, Abbas, Zaki and Harun2019; Li et al. Reference Li, Hutchins, Zheng, Marusic and Baars2022) (
$\gamma =12^\circ {-}15^\circ$
for neutral conditions,
$\gamma =20^\circ {-}60^\circ$
in convective conditions), the presence of sand particles (Wang et al. Reference Wang, Gu and Zheng2020) and distance from the wall (Liu et al. Reference Liu, Wang and Zheng2019). For wall-attached eddies, a value of
$\gamma =45^\circ$
for asymptotically high Reynolds numbers (
$\textit{Re}_\tau \gt 10^5$
) has been conjectured by several researchers (Deshpande et al. Reference Deshpande, Monty and Marusic2019, Reference Deshpande, de Silva and Marusic2023), but has never been experimentally verified. Finally, for small isotropic eddies, the inclination angle is estimated to be
$90^\circ$
(Cui & Jacobi Reference Cui and Jacobi2024) – a value consistent with isotropic turbulence. This variability in
$\gamma$
indicates that turbulent structures with different streamwise extensions have different values of inclination angle, whose precise quantification is still an open question.
In addition to the sources of variability mentioned previously, any event capable of modifying the instantaneous shear profile is expected to impact the inclination angle of coherent structures by modifying the streamwise advection. Hutchins et al. (Reference Hutchins, Monty, Ganapathisubramani, Ng and Marusic2011) showed that a major trigger event is the presence of VLSM velocity fluctuations that, by carrying a large portion of Reynolds stresses, induce high fluctuations of the wall shear stress and, in turn, increase the advection velocity. Thus, positive and negative VLSM momentum regions feature different inclination angles (Hutchins et al. Reference Hutchins, Monty, Ganapathisubramani, Ng and Marusic2011). This scenario has recently been corroborated by Cui & Jacobi (Reference Cui and Jacobi2024) for wall-bounded flows with
$\textit{Re}_\tau =530$
up to
$3070$
. The authors linked the observed variability of the VLSM and LSM inclination angle to different advection velocities of these structures, with larger advection velocity inducing higher shear that, in turn, reduces the inclination angle, and vice versa. The upper spectral limit of these shear-driven structures was found to be proportional to
$k_x\,z^{-1}$
(where
$k_x$
is the streamwise wavenumber), thus involving the attached-eddy spectral range.
The above-mentioned evidence of scale-dependent variability of the inclination angle call for further investigations for very-high-Reynolds-number flows. The thermally neutral ASL represents a compelling benchmark as it displays the largest turbulent scale separation currently achievable (
$\textit{Re}_\tau \approx \mathcal{O}(10^6)$
), yet providing strong analogies with classic laboratory flows (Hutchins & Marusic Reference Hutchins and Marusic2007; Metzger et al. Reference Metzger, Mckeon and Holmes2007; Guala et al. Reference Guala, Metzger and McKeon2011; Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013). However, the remarkable level of scale separation present in the ASL requires a careful design of the experimental strategy to resolve flow fluctuations that span several orders of magnitude in length (
$10^{-2}$
m to
$10^4$
m). Light detection and ranging (LiDAR) anemometry is a reliable instrument to resolve VLSMs and wall-attached eddies within the ASL (Calaf et al. Reference Calaf, Hultmark, Oldroyd, Simeonov and Parlange2013; Puccioni et al. Reference Puccioni, Calaf, Pardyjak, Hoch, Morrison, Perelet and Iungo2023) due to its spatio-temporal resolution (
${\sim} 10$
m,
${\sim} 2$
Hz) combined with the long measurement range (over
$3$
km). Furthermore, the recently developed super large particle image velocimetry (SLPIV), using snowfall as particle tracer in the atmosphere, has been used successfully to resolve small-scale fluctuations (between
$10^{-1}$
m and
$10$
m) in the ASL (Hong et al. Reference Hong, Toloui, Chamorro, Guala, Howard, Riley, Tucker and Sotiropoulos2014; Toloui et al. Reference Toloui, Riley, Hong, Howard, Chamorro, Guala and Tucker2014; Heisel et al. Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018; Iungo et al. Reference Iungo, Guala, Hong, Bristow, Puccioni, Hartford, Ehsani, Letizia, Li and Moss2024) and wind turbine wakes (Dasari et al. Reference Dasari, Wu, Liu and Hong2019; Abraham et al. Reference Abraham, Dasari and Hong2019; Abraham & Hong Reference Abraham and Hong2020) from close to the ground (i.e.
${\sim} 30$
m) to
$100$
m.
In this work, the results from a synergistic deployment of a scanning pulsed Doppler LiDAR and SLPIV are presented in the context of a neutral ASL flow developing over a relatively flat terrain. The goal of this analysis is twofold: first, to provide insights into the wall-attached eddy inclination angle at a very high Reynolds number; second, to investigate the variability of the wall-attached eddy inclination angle as a function of the VLSM-induced velocity fluctuations. The inclination angle, probed by the SLPIV close to the ground (
$z\leq 20$
m), is conditionally quantified based on the occurrence of positive or negative VLSM-induced velocity fluctuations simultaneously probed by the LiDAR. This approach allows us to assess whether the high- and low-shear occurrences induced by VLSMs influence the wall-attached eddy inclination angle for an ASL flow.
The remainder of this work is organised as follows. In § 2, the experimental dataset is presented, followed by statistical analysis of the turbulent flow in § 3. In § 4, methods for calculating the inclination angle and relative results are discussed. Conclusions are reported in § 5.
2. Experimental set-up
The data presented in this work were collected on 11 December 2021 from 02 : 00 to 02 : 22 UTC time (UTC: local time + 6 hr) at the EOLOS research site in Rosemount, MN (US) (Toloui et al. Reference Toloui, Riley, Hong, Howard, Chamorro, Guala and Tucker2014) (figure 1
a), during the Grand-Scale Atmospheric Imaging Apparatus (GAIA) experiment (Iungo et al. Reference Iungo, Guala, Hong, Bristow, Puccioni, Hartford, Ehsani, Letizia, Li and Moss2024). The experimental site (inset of figure 1
a) is surrounded by almost
$2$
km of flat terrain with obstacles below 5 m height (trees and bushes) creating conditions for the development of a homogeneous streamwise flow over a fully rough surface (Heisel et al. Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018).
In this work, the instantaneous velocity components
$\boldsymbol {u}=(u,\,v,\,w)$
are oriented along the mean wind (
$x$
), transverse (
$y$
) and vertical (
$z$
) directions, respectively. Uppercase letters denote time-averaged velocity components, while the prime quantities
$(u',\,v',\,w')$
refer to temporal zero-mean fluctuations, bold letters indicate vectors, overbar indicates time-averaged statistics of order higher than one and the ‘+’ superscript is used to indicate viscous-scaled quantities.
Overview of the GAIA field campaign: (a) aerial view of the EOLOS site (credits, Google Earth) and inset of the instruments’ location; (b) SLPIV apparatus; (c) streamline XR scanning Doppler LiDAR.
During the GAIA campaign, a Streamline XR scanning pulsed Doppler LiDAR (figure 1
c), manufactured by Halo Photonics and owned by the University of Texas at Dallas (UTD), was deployed at the EOLOS site (GPS:
$44^\circ 43^{\prime}33.17^{\prime\prime}$
N;
$93^\circ 2^{\prime}37.14^{\prime\prime}$
W) to perform measurements with constant azimuth (
$\varTheta$
) and elevation (
$\varPsi$
) angles (hereafter named fixed-point measurements). The azimuth angle is measured on the horizontal plane in the clockwise direction from North, while the elevation angle is measured on the vertical plane from the horizontal direction. Based on the Doppler shift between the emitted and backscatter frequencies, the LiDAR collects the along-beam (or radial) wind speed,
$v_r$
, at
$2$
Hz sampling rate in a quasi-simultaneous fashion at different radial distances, the latter separated by a constant spacing (or gate length,
$l=18$
m) along the line-of-sight with unity vector
$\boldsymbol {n}$
. For each probed location, the instantaneous radial wind speed represents the projection of the three-dimensional velocity vector,
$\boldsymbol {u}$
, along the LiDAR line-of-sight (Zhan et al. Reference Zhan, Letizia and Iungo2020):
\begin{equation} v_r(t)=\boldsymbol {u}(t)\boldsymbol{\cdot }\boldsymbol {n}=u(t)\cos (\varTheta -\varTheta _w)\cos \varPsi +v(t)\sin (\varTheta -\varTheta _w)\cos \varPsi +w(t)\sin \varPsi , \end{equation}
where
$\varTheta _w$
is the mean wind direction.
The azimuth angle is aligned along the mean wind direction with an elevation angle of
$\varPsi =6.8^\circ$
to record streamwise velocity time series. The elevation angle combined with a gate length of
$l=18$
m leads to a vertical resolution of
$l\sin \varPsi \approx 2.1$
m. Wind direction is monitored in situ using a co-located CSAT3–3D sonic anemometer (manufactured by Campbell Scientific and owned by The University of Texas at Dallas) facing north and mounted at
$z=2$
m probing the wind velocity vector and temperature with a
$20$
Hz sampling frequency. For the data presented in this paper, the mean wind direction measured by the sonic anemometer (
$26^\circ$
from North, reported in the inset of figure 1
a with black arrow) is in good agreement with the orientation of the LiDAR line-of-sight (red dotted line in figure 1
a), i.e.
$\varTheta =22.5^\circ$
from North.
After being recorded, the quality of the LiDAR time series is assessed to ensure the reliability of the subsequent statistics. First, based on (2.1), the instantaneous radial wind speed at each height,
$v_r(z,t)$
, is converted into streamwise velocity by neglecting the misalignment between the line-of-sight and the wind direction (
$\varTheta -\varTheta _w\approx 0$
) and correcting the error introduced by the LiDAR elevation angle through
$u(z,t)=v_r(z,t)/\cos \varPsi$
(Zhan et al. Reference Zhan, Letizia and Iungo2020; Puccioni & Iungo Reference Puccioni and Iungo2021; Puccioni et al. Reference Puccioni, Calaf, Pardyjak, Hoch, Morrison, Perelet and Iungo2023). Subsequently, each gate signal showing a time-averaged signal-to-noise ratio (SNR) outside of the interval
$[-15,\,10]$
dB is excluded. This limited the available wall-normal range between
$11.7$
m and
$37.3$
m above the ground. Due to the relatively short wall-normal range investigated in this study, the change of wind direction along the
$z$
-direction, i.e. wind veer, is deemed to be negligible.
For each remaining LiDAR gate, instantaneous velocity records outside of the interval
$U(z)\pm 3.5\sqrt {\overline {u'u'}}$
(where
$U$
is the mean velocity) are marked as outliers (i.e. non-physical records due to instrumental errors) and replaced by linear interpolation (Puccioni et al. Reference Puccioni, Calaf, Pardyjak, Hoch, Morrison, Perelet and Iungo2023). Lastly, the LiDAR signals are corrected in the Fourier domain based on the algorithm proposed by Puccioni & Iungo (Reference Puccioni and Iungo2021) to take into account the energy reduction associated with the LiDAR spatial averaging within each range gate.
An SLPIV apparatus (figure 1
b), owned by the University of Minnesota (UMN) (Toloui et al. Reference Toloui, Riley, Hong, Howard, Chamorro, Guala and Tucker2014; Hong et al. Reference Hong, Toloui, Chamorro, Guala, Howard, Riley, Tucker and Sotiropoulos2014), was deployed at a distance of
$248$
m upstream of the LiDAR (GPS:
$44^\circ 43^{\prime} 41.35^{\prime\prime}$
N,
$93^\circ 2^{\prime} 34.11^{\prime\prime}$
W) approximately along the mean wind direction. Natural snowfall is used as flow tracer in the SLPIV measurements. The tracer fidelity is evaluated for each event using a snow particle analyser, which provides snowflake density and aerodynamic properties. From these parameters, we estimate the particle response time,
$\tau _p$
, and compute the Stokes number
${St}=\tau _p/T$
for the turbulent time scales
$T$
relevant to this study. Restricting our analysis to scales with
${St}\lessapprox 0.1$
ensures that snowflakes behave as effective tracers for wall-attached eddies and VLSMs in the logarithmic region. The snow concentration is sufficiently dilute that the system is one-way coupled, so the snow does not measurably alter the atmospheric flow statistics. The SLPIV station is made up of a light emitter (placed roughly
$0.5$
m above the ground), a curved mirror reflecting the light over a vertical field of view (FOV) and a Nikon D610 camera placed
$87$
m from the FOV with positive tilt angle recording the in-plane streamwise and wall-normal velocity components with a sampling frequency of
$20$
Hz. The FOV is oriented at
$21^\circ$
from north, i.e. at only
$5^\circ$
difference from the mean wind direction recorded by the sonic anemometer, providing a good approximation of the streamwise and vertical velocity components over the
$x{-}z$
plane for the measured north-north-east wind.
Streamwise velocity fluctuations resolved by (a) the LiDAR and (b) the SLPIV along the streamwise–vertical plane. For the LiDAR, the streamwise coordinate is reconstructed via Taylor (Reference Taylor1938) hypothesis of frozen turbulence. In panel (b), the vertical dashed line indicates the chosen vertical array of SLPIV measurements. Note that the two panels are reported with different spatial scales.
Each SLPIV image is dewarped to remove image distortions due to the camera's tilt, and de-noised. The velocity components
$u$
and
$w$
are estimated in space and time through a cross-correlation algorithm that tracks the motions of snowflakes carried by the wind (Hong et al. Reference Hong, Toloui, Chamorro, Guala, Howard, Riley, Tucker and Sotiropoulos2014; Toloui et al. Reference Toloui, Riley, Hong, Howard, Chamorro, Guala and Tucker2014; Nemes et al. Reference Nemes, Dasari, Hong, Guala and Coletti2017; Heisel et al. Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018; Iungo et al. Reference Iungo, Guala, Hong, Bristow, Puccioni, Hartford, Ehsani, Letizia, Li and Moss2024). As a result, the in-plane velocity is reconstructed over a
$x{-}z$
domain of
$15\,\mathrm{m}\times 19\,\mathrm{m}$
(starting from
$3.5$
m above the ground) with a spatial resolution of
$0.34$
m both in the
$x$
and in the
$z$
directions.
A sample of the streamwise velocity fields simultaneously resolved by the LiDAR and the SLPIV are reported in figures 2(a) and 2(b), respectively. High quantitative agreement is found between the velocity fluctuations resolved by the two instruments. From figure 2(a), an overlap (
$z = 12$
m to
$19$
m) is observed between the height probed by the LiDAR and the SLPIV. Within this interval, only SLPIV records are considered due to their higher temporal and vertical resolution with respect to the LiDAR measurements. In figure 2(b), the streamwise range probed by the SLPIV goes from
$x=-7$
m to
$x=6$
m, i.e.
${\sim} 0.14\delta$
assuming
$\delta =94$
m as ASL height (see § 3.1). This domain size is not sufficient to resolve neither wall-attached eddies nor VLSMs, as they both feature streamwise elongations greater than
$\delta$
. Thus, the SLPIV streamwise domain cannot be used to directly quantify wall-attached eddies or VLSMs, which would require a much larger FOV. In contrast, the time interval probed by the SLPIV (
$\Delta T=22$
minutes) is sufficient to probe the streamwise fluctuations induced by VLSMs and wall-attached eddies in space, as it corresponds to nearly
$79$
eddy turnover time (
$U{\Delta}T/\delta$
) based on the three-dimensional (3-D) sonic mean velocity. Hence, in the remainder of this work, the SLPIV streamwise velocity signal is extracted from the vertical array located at
$x=0$
within the FOV (vertical dashed line in figure 2
b), and the streamwise coordinate (when required) will be reconstructed leveraging Taylor's hypothesis of frozen turbulence (Taylor Reference Taylor1938; Zaman & Hussain Reference Zaman and Hussain1981; Del Alamo & Jiménez Reference Del Alamo and Jiménez2009; Moin Reference Moin2009):
$u(x,t)=u(x-\textit{Ut},0)$
, using the local mean velocity as advection velocity.
3. Analysis of the background ASL flow
3.1. Flow statistics
Before moving on with data analysis, it is necessary to assess whether the first- and second-order statistics of the measured ASL flow are consistent with those typically reproduced through numerical simulations or laboratory experiments of wall-bounded flows. First, thermal stability is quantified from the sonic anemometer data through the Obukhov length:
\begin{equation} L = -\frac {U_\tau ^3 T_v}{g\kappa \overline {w'\theta '}}, \end{equation}
where the friction velocity,
$U_\tau$
, is calculated through the eddy covariance method (Stull Reference Stull1988):
\begin{equation} U_\tau = \big(\overline {u'w'}^2+\overline {v'w'}^2 \big)^{0.25}, \end{equation}
and quantified as
$0.342$
m s-1 for the dataset under investigation. In addition,
$T_v=272$
K is the mean virtual potential temperature,
$g=9.81$
$\mathrm{m\,s}^{-2}$
is the acceleration of gravity,
$\kappa =0.41$
is the Von Kármán constant and
$\overline {w'\theta '}$
is the vertical heat flux (being
$\theta '$
the temperature fluctuation). The Obukhov length for the present dataset is
$L=851$
m, which leads to a Monin–Obukhov stability parameter (
$z/L$
) within the interval
$[2.3\times 10^{-3},\,4.5\times 10^{-2}]$
, which classifies the present flow as thermally neutral (Metzger et al. Reference Metzger, Mckeon and Holmes2007; Guala et al. Reference Guala, Metzger and McKeon2011; Liu et al. Reference Liu, Bo and Liang2017).
According to the Monin–Obukhov similarity theory, the mean velocity profile within a neutral ASL is modelled through the logarithmic law of the wall:
\begin{equation} U^+=\frac {1}{\kappa }\log \left (\frac {z}{z_0}\right )\!, \end{equation}
where
$U^+=U/U_\tau$
is the inner-scaled mean velocity and
$z_0$
is the aerodynamic roughness length introduced to quantify the momentum loss due to terrain roughness (Stull Reference Stull1988). The mean velocity profile obtained from the time average of the sonic anemometer, SLPIV and LiDAR signals is reported in figure 3(a) against the inner-scaled wall-normal height (
$z^+=zU_\tau /\nu$
), assuming
$\nu =4.4\times 10^{-5}\,\mathrm{m}^2\,\mathrm{s}^{-1}$
based on the mean temperature (Picard et al. Reference Picard, Davis, Gläser and Fujii2008). The previous mean velocity profiles obtained from various experimental sites are reported for reference (Kunkel & Marusic Reference Kunkel and Marusic2006; Hutchins et al. Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012; Wang & Zheng Reference Wang and Zheng2016; Heisel et al. Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018). The slope of the mean streamwise velocity in semi-logarithmic scale reported in figure 3(a) agrees well with previous studies, including three statistically converged datasets obtained by Heisel et al. (Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018) at the same experimental site.
Mean streamwise velocity and variance estimated from sonic anemometer, LiDAR and SLPIV data: (a) inner-scaled mean velocity plotted against
$z^+$
; (b) streamwise velocity variance along outer-scaled wall-normal coordinate; grey cross symbols refer to LiDAR measurements collected outside of the logarithmic region. Figure reproduced from Heisel et al. (Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018).
To quantify the effect of terrain roughness on the mean velocity, (3.3) is fitted to the experimental mean profile using
$U_\tau$
calculated from the sonic records obtaining
$z_0=1.8$
mm, which classifies the EOLOS site with the considered wind direction as fully rough, consistently with the results of Heisel et al. (Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018) obtained at the same site. The fitted logarithmic law, reported in figure 3(a) with a black line, agrees well both with SLPIV and LiDAR data (R-square of
$0.966$
). However, a significant shift is found between the current mean velocity profile and the one obtained by Heisel et al. (Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018), who quantified the aerodynamic roughness as
$z_0=40$
mm. In this regard, Heisel et al. (Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018) reported a partial snow coverage leaving several terrain obstacles uncovered that could have increased the soil roughness. In contrast, the present dataset was collected after several hours of snowstorm leading to uniform snow coverage of the area surrounding the EOLOS site, which is characterised by a lower aerodynamic roughness. Additionally, Heisel et al. (Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018) collected the data at a nearby location characterised by higher terrain slope and terrain irregularities than the present study. The latter, in contrast, is characterised by a flatter terrain with larger streamwise homogeneity. This difference further explains the lower aerodynamic roughness obtained for the present study.
The ASL height,
$\delta$
, which is used as the reference length scale in the rest of the analysis, is estimated based on the geostrophic drag law. In particular, we assume
$\delta \approx 0.1z_i$
(Stull Reference Stull1988; Wyngaard Reference Wyngaard1992; Garratt Reference Garratt1994), where
$z_i$
is the height of the atmospheric boundary layer. The latter is calculated based on the following relationship (Rossby & Montgomery Reference Rossby and Montgomery1935; Zilitinkevich Reference Zilitinkevich1972):
\begin{equation} z_i=C\frac {U_\tau }{f_C}, \end{equation}
where
$f_C=7.3\times 10^{-5}$
Hz is the Coriolis frequency and
$C$
is a constant related to thermal stability (
$C=0.1{-}0.5$
), assumed equal to
$0.2$
under neutral conditions (Peña et al. Reference Peña, Gryning and Hasager2010). Based on these assumptions, the ASL height is estimated as
$\delta =94$
m, which allows to quantify the friction Reynolds number for the present flow as
$\textit{Re}_\tau =U_\tau \,\delta /\nu =7.9\times 10^5$
.
The velocity fluctuations are detrended via spectral filtering to remove non-turbulent fluctuations (Metzger et al. Reference Metzger, Mckeon and Holmes2007). The spectral gap, i.e. the frequency dividing turbulent and non-turbulent motions (Van der Hoven Reference Van der Hoven1957), is assumed as
$f_{\textit{gap}}=0.008$
Hz, close to the value typically used for neutral ASL measurements (
$0.006$
Hz) (Guala et al. Reference Guala, Metzger and McKeon2011; Metzger et al. Reference Metzger, Mckeon and Holmes2007; Puccioni et al. Reference Puccioni, Calaf, Pardyjak, Hoch, Morrison, Perelet and Iungo2023).
The subsequent inner-scaled streamwise variance profile (
$\overline {u'u'}^+=\overline {u'u'}/U_\tau ^2$
) calculated by sonic, SLPIV and LiDAR is reported in figure 3(b) against the outer-scaled wall-normal coordinate (
$z/\delta$
). The variance profile shows good consistency between the different instruments used during the GAIA campaign, as well as against previous datasets collected in the same outer-scaled wall normal range. However, a clear deviation from the wall-normal decay is found for LiDAR data as
$z\geq 0.3\delta$
(corresponding to
$27$
m). A similar discrepancy was found by Heisel et al. (Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018), who noticed a deviation from the mean velocity logarithmic profile starting from
$z=70$
m for a wind direction of
$70^\circ$
from North and thus assigned it as the lower limit of the boundary layer depth. In the absence of further information about the local wind direction (and the related effect on streamwise variance), the LiDAR gates above
$27$
m height will be excluded from further analysis.
3.2. Streamwise velocity decomposition based on the linear coherence spectrum
After removing the non-turbulent velocity fluctuations, the model proposed by Puccioni et al. (Reference Puccioni, Calaf, Pardyjak, Hoch, Morrison, Perelet and Iungo2023) is used to distinguish among the spectral signatures on the streamwise velocity due to wall-attached eddies, VLSMs and Kolmogorov small-scale eddies, which are referred to as type
$\mathcal{A}$
,
$\mathcal{B}$
and
$\mathcal{C}$
eddies based on the structure classification proposed by Perry & Marusic (Reference Perry and Marusic1995). This method is based on the calculation of the linear coherence spectrum (LCS) of the streamwise velocity using the lowest height probed by the SLPIV as reference (
$z_R=0.04\delta$
). The streamwise wavelength,
$\lambda _x$
, is estimated via Taylor's hypothesis of frozen turbulence (Taylor Reference Taylor1938):
$\lambda _x=U(z)/f$
, where
$f$
is the frequency vector obtained from the fast Fourier transform (FFT) of the velocity time series.
Given two streamwise velocity signals (
$u_1'(z,t),\,u_2'(z_R,t)$
) simultaneously measured at a reference and generic heights (
$z_R$
and
$z$
, respectively), the LCS is defined as
\begin{equation} \textit{LCS}(z,\,z_R;\,f) = \dfrac {\left |\phi '_{uu}(z,\,z_R;f) \right |^2}{\phi _{uu}(z;\,f)\,\phi _{uu}(z_R;\,f)}, \end{equation}
where
$\phi '$
is the cross-spectral density of the two streamwise velocity signals and
$\phi _{uu}$
is the auto-spectral density. In this work, we only consider probe locations,
$z$
, above the height of the reference velocity signal,
$z_R$
. Physically, the LCS represents the fraction of common energy shared as a function of frequency by two signals sampled simultaneously at two different wall-normal locations (Bendat & Piersol Reference Bendat and Piersol1986).
Previous studies argued that small eddies (typically associated with the
$k_x^{-5/3}$
inertial sub-layer of the streamwise velocity energy spectrum) give negligible contributions to the LCS as they are only locally coherent (Baars et al. Reference Baars, Hutchins and Marusic2017; Krug et al. Reference Krug, Baars, Hutchins and Marusic2019; Baars & Marusic Reference Baars and Marusic2020a
,
Reference Baars and Marusicb
; Li et al. Reference Li, Wang and Zheng2021; Puccioni et al. Reference Puccioni, Calaf, Pardyjak, Hoch, Morrison, Perelet and Iungo2023). In contrast, wall-attached eddies contribute to the LCS through the superposition of geometrically similar single-eddy contributions limited to the wavelength range
$\lambda _x\geqslant {AR}z$
, where AR is the (constant) aspect ratio of wall-attached structures, i.e. the characteristic streamwise-to-vertical ratio of the single attached eddy (Baars et al. Reference Baars, Hutchins and Marusic2017).
LCS spectrum from SLPIV with respect to the lowest height probed by the SLPIV reported against (a) outer-scaled wall-normal coordinate and (b) outer-scaled wavelength.
At a certain height, which is referred to as
$z_{{max}}$
, wall-attached eddies cease to grow; thus, the maximum streamwise wavelength achievable by wall-attached eddies is
$\lambda _{\textit{th}}= {AR}z_{{max}}$
. Therefore, structures characterised by
$\lambda _x\geqslant \lambda _{\textit{th}}$
correspond to concatenations of LSM packets that form VLSMs (Kim & Adrian Reference Kim and Adrian1999; Deshpande et al. Reference Deshpande, de Silva and Marusic2023).
In this scenario, the LCS for wall-bounded flows is modelled by a piecewise-defined function of
$\lambda _x$
and
$z$
having
${AR}, z_{{max}}$
and
$\lambda _{\textit{th}}$
as free parameters (Baars et al. Reference Baars, Hutchins and Marusic2017; Puccioni et al. Reference Puccioni, Calaf, Pardyjak, Hoch, Morrison, Perelet and Iungo2023):
\begin{equation} \textit{LCS}(z,\,z_R;\,f)=\left \{\begin{array}{@{}lc} C_1\log \left (\dfrac {\lambda _x}{\textit{AR}z}\right ) & \lambda _x\leqslant \lambda _{\textit{th}} ,\\[10pt] C_3-C_1\log \left (\dfrac {z}{\delta }\right ) & \lambda _x\gt \lambda _{\textit{th}},\\ \end{array} \right . \end{equation}
where
$C_1$
and
$C_3$
are constants. It should be noted that for
$\lambda _x\lt \lambda _{\textit{th}}$
, LCS is proportional to
$\log (\lambda _x)$
and
${-}\log (z)$
, while for
$\lambda _x\gt \lambda _{\textit{th}}$
, LCS is no longer a function of
$\lambda _x$
but only of
${-}\log (z)$
.
For the present experiment, the LCS calculated for the SLPIV signals with respect to the lowest height measured by the SLPIV (
$z_{R}/\delta \approx 0.04$
) is reported against
$z/\delta$
and
$\lambda _x/\delta$
in figures 4(a) and 4(b), respectively. The LCS is calculated using the Welch algorithm (Welch Reference Welch1967) over four signal subsets, each with a duration of
$307$
s with
$90\,\%$
overlap, followed by a moving average algorithm to smooth the LCS (Baars et al. Reference Baars, Hutchins and Marusic2017). The LCS profiles as a function of
$z$
(figure 4
a) indicate, as expected, a reducing coherence with increasing vertical distance, while this reduction has a larger slope with reducing
$\lambda _x$
. Furthermore, the LCS profiles as a function of
$\lambda _x$
(figure 4
b) indicate a small-scale region (i.e. up to
$(\lambda _x/\delta )\approx 0.1$
) where the LCS is very small (
${\sim} 0.2$
) and practically constant with variations of
$z$
and
$\lambda _x$
, indicating only a marginal local coherence in proximity of
$z_R$
due to small-scale eddies. With increasing
$\lambda _x$
, increasingly larger values of LCS are observed in figure 4(b), due to the increasing presence of larger structures contributing to the LCS. Finally, at larger
$\lambda _x$
, an asymptotic value of LCS is achieved at each
$z$
, which is ascribed to the wavelength range dominated by VLSMs.
The best fit of the calculated LCS with the model expressed through (3.6) returns the following fitting parameters:
$C_1=0.45,\,C_3=-0.03,\,\lambda _{\textit{th}}=5.8\delta$
(
$547$
m),
$z_{{max}}=0.95\delta$
(
$90$
m) and
${AR}=6.1$
, which is in line with the aspect ratio evaluated by Li et al. (Reference Li, Wang and Zheng2021) (
${AR}=7.8$
) for a neutral ASL flow over a vegetation canopy.
Energy spectra of the streamwise velocity calculated from SLPIV and LiDAR data: (a) pre-multiplied energy spectra (red line refers to the coherence-based spectral decomposition between isotropic eddies and wall-attached eddies, while the red dashed line indicate the separation between wall-attached eddies and VLSMs); (b) attached eddy (green symbols) and VLSM (blue symbols) streamwise velocity variance resolved by the SLPIV.
The pre-multiplied energy spectra of the streamwise velocity (
$f^+\phi _{uu}^+=f\phi _{uu}/U_\tau ^2$
) are reported in figure 5(a) against the outer-scaled streamwise wavelength and wall-normal height, both reported on logarithmic scale. It is worth noticing that good agreement is found for the spectral energy in proximity of the overlap area between the LiDAR and SLPIV measurements domains (i.e.
$z/\delta \approx 0.2$
). The spectral boundary between the region associated with small Kolmogorov wall-detached eddies and wall-attached eddies is reported with a solid red line, which is identified by their aspect ratio
${AR}=6.1$
, while the spectral boundary between wall-attached eddies and VLSMs is indicated with a dashed red line, which is identified by
$\lambda _{\textit{th}}=5.8\delta$
. A distinct region of high spectral energy is identified within the range
$0.7\lt \lambda _x/\delta \lt 3$
for
$z/\delta \lesssim 0.1$
. This spectral energy can be associated with the spectral footprint of wall-attached eddies and LSMs (Kim & Adrian Reference Kim and Adrian1999; Hommema & Adrian Reference Hommema and Adrian2003; Guala et al. Reference Guala, Hommema and Adrian2006; Hutchins & Marusic Reference Hutchins and Marusic2007). Another high-energy region located at
$\lambda _x\approx 6\delta {-}10\delta$
is observed both from SLPIV and LiDAR spectra throughout the measured wall-normal range. This is deemed to be the spectral signature of the VLSMs, consistent with the spectral range indicated by Hutchins et al. (Reference Hutchins, Chauhan, Marusic, Monty and Klewicki2012) for neutrally stratified ASL flows (
$\lambda _x\approx 6\delta {-}20\delta$
).
The contributions associated in the streamwise velocity due to wall-attached eddies and VLSMs are extracted via bandpass filtering using as thresholds for the streamwise wavelength of
$6.1z\leqslant \lambda _x\leq 5.8\delta$
and
$\lambda \gt 5.8\delta$
, respectively. Subsequently, the filtered spectral energy is integrated across frequency to obtain the related wall-normal profiles of the energy of the streamwise velocity. A log-linear decrease with increasing
$z$
of the streamwise variance associated with wall-attached eddies (green symbols) is observed in figure 5(b), consistent with previous laboratory (Baars & Marusic Reference Baars and Marusic2020b
, for example) and ASL results (Marusic et al. Reference Marusic, Monty, Hultmark and Smits2013; Puccioni et al. Reference Puccioni, Calaf, Pardyjak, Hoch, Morrison, Perelet and Iungo2023). In contrast, the variance associated with VLSMs (blue symbols in figure 5
b) is nearly constant for the measured wall-normal interval. The logarithmic decay of the wall-attached-eddy variance of the streamwise velocity is validated against the classic logarithmic law proposed by Townsend (Reference Townsend1976):
\begin{equation} \overline {u'u'}^+=B_1-A_1\log \left (\frac {z}{\delta }\right )\!. \end{equation}
For a neutral ASL flow (encompassing all turbulent motions), Marusic et al. (Reference Marusic, Monty, Hultmark and Smits2013) estimated
$A_1=1.33$
, while for a pure attached eddy flow,
$A_1$
ranges from
$0.80$
(Hu et al. Reference Hu, Yang and Zheng2020) to
$0.98$
(Baars & Marusic Reference Baars and Marusic2020b
; Puccioni et al. Reference Puccioni, Calaf, Pardyjak, Hoch, Morrison, Perelet and Iungo2023). The best fit of (3.7) on the wall-attached-eddy variance profile returns
$A_1=1.36\pm 0.09$
for the present flow.
4. Methods to evaluate the inclination angle of the turbulent coherent structures
4.1. Bulk inclination angle
The inclination angle of coherent turbulent structures is estimated based on the two-point correlation,
$R_{uu}$
(Marusic & Heuer Reference Marusic and Heuer2007; Jacob & Anderson Reference Jacob and Anderson2017; Liu et al. Reference Liu, Bo and Liang2017; Heisel et al. Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018; Salesky & Anderson Reference Salesky and Anderson2018; Deshpande et al. Reference Deshpande, Monty and Marusic2019; Volino et al. Reference Volino, Schultz and Flack2007):
\begin{equation} R_{uu}(z_R,z;\tau )= \frac {\overline { {u}'(z_R,t)\,{u}'(z,t+\tau ) }}{\sqrt {\overline {{u'(z_R,t)^2}}\,\overline { {u}'(z,t)^2}}}, \end{equation}
where
$\tau$
is the temporal shift,
$z_R$
is the reference height and the overbar indicates ensemble average that, for the present work, corresponds to the time average. To obtain the inclination angle of turbulent structures with respect to
$z_R$
, the temporal lag,
$\tau ^*$
, corresponding to the maximum
$R_{uu}$
at
$z$
, is multiplied by the local mean velocity to obtain the spatial lag
$\Delta x^*=\tau ^*U(z)$
. The inclination angle is then estimated using the following relationship (Liu et al. Reference Liu, Bo and Liang2017; Marusic & Heuer Reference Marusic and Heuer2007):
\begin{equation} z-z_R = \tan \gamma \,\Delta x^*, \end{equation}
and performing a best fit among wall-normal heights spanning from
$z=0.04\delta$
to
$z=0.27\delta$
, where the correlation peak is larger or equal to
$0.6$
. Since the two-point correlation is averaged in time,
$\gamma$
should be interpreted as an overall value characterising structures within a certain spatio-temporal interval. In other words, information on the intra-flow variability of the inclination angle (both in space–time and across wavelengths) is lost when
$\gamma$
is quantified through (4.2). Thus, in the remainder of this work,
$\gamma$
defined by (4.2) will be referred to as the bulk inclination angle.
It is noticed that (4.1) can be used to quantify the bulk inclination angle of all turbulent fluctuations, or it can be used to estimate the inclination angle of a subset of velocity fluctuations by using a bandpass-filtered version of
$u'$
. In particular, the VLSM and wall-attached-eddy fluctuations are extracted from the flow and used as input to (4.1) to estimate the respective inclination angle.
For the data under investigation, the bi-dimensional correlation maps are reported in figures 6(a), 6(b) and 6(c) for detrended, VLSM and wall-attached-eddy filtered signals, respectively. In particular, the wall normal interval below
$0.2\delta$
is measured by the SLPIV, while the heights above are probed by the LiDAR; the separation between the two measurement domains is reported in figure 6 with a horizontal grey line. A discrepancy is noticed in the streamwise position of the cross-correlation peaks (red circles) associated with the LiDAR measurements with respect to those probed by the SLPIV. This discrepancy is introduced by multiplying the temporal lags for the local mean velocity (resolved either by the LiDAR or the SLPIV) to reconstruct the streamwise coordinate. For the detrended signals (figure 6
a), the bulk inclination angle is estimated as
$32.9^\circ$
, i.e. higher than the values reported in literature; for instance, for neutral ASL flows, Li et al. (Reference Li, Hutchins, Zheng, Marusic and Baars2022) reported
$\gamma =12^\circ$
, while Liu et al. (Reference Liu, Bo and Liang2017) reported
$\gamma =12.1^\circ {-}15^\circ$
, Carper & Porté-Agel (Reference Carper and Porté-Agel2004) estimated
$\gamma =16^\circ$
, and Heisel et al. (Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018) quantified
$\gamma =10^\circ$
.
For the present experimental site, Heisel et al. (Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018) used an alternative method originally proposed by Christensen & Wu (Reference Christensen and Wu2005). Considering a certain interval of correlation iso-contours (between
$0.6$
and
$0.9$
with
$0.05$
increase), for each iso-contour, the position of the farthest point from the reference is assumed as representative of the streamwise elongation of the coherent structures. When all the farthest points from the reference are evaluated for each iso-contour, the inclination angle is found as the slope of the fitting line across the farthest points. The inclination angle evaluated with this method for the present work is reported in figure 6 with a dashed line. The obtained numerical value of
$10.3^\circ$
is closer to the value obtained by Christensen & Wu (Reference Christensen and Wu2005) (
$11^\circ$
), as well as Volino et al. (Reference Volino, Schultz and Flack2007) (
$13.2^\circ {-}15.8^\circ$
) and Heisel et al. (Reference Heisel, Dasari, Liu, Hong, Coletti and Guala2018) (
$10^\circ$
), who used this method.
Two-point correlation function from SLPIV and LiDAR data using as reference SLPIV data collected at
$z_R=0.04\delta$
(separation height between the SLPIV and LiDAR measurement area is indicated with a grey line): (a) detrended signals; (b) VLSM only; (c) wall-attached eddies only. Contours with continuous lines refer to positive levels (0–1 with 0.1 increment), while dashed lines refer to negative contours (−1 to 0 with 0.1 increment). Continuous and dashed black lines report the inclination angle calculated either by the linear fit of the two-point correlation peaks, or by fitting ellipses to the iso-contours, respectively. Red circles report the two-point correlation peaks at each height.
Focusing on the VLSM inclination angle (figure 6
b), a value of
$\gamma =7.4^\circ$
is estimated for
$z_R=0.04\delta$
by fitting the two-point correlation peaks. This value is comparable to that previously estimated by Liu et al. (Reference Liu, Bo and Liang2017) (
$13^\circ$
) at similar Reynolds number. Using the method of Christensen & Wu (Reference Christensen and Wu2005), a similar value is found (
$6.2^\circ$
). Finally, the wall-attached-eddy inclination angle is quantified as
$\gamma =41.1^\circ$
(figure 6
c). This value is close to the asymptotic limit of
$45^\circ$
for very-high-Reynolds-number flows argued by several studies (Perry & Marusic Reference Perry and Marusic1995; Baidya et al. Reference Baidya, Philip, Hutchins, Monty and Marusic2017; Chandran et al. Reference Chandran, Baidya, Monty and Marusic2017) and quantified by Deshpande et al. (Reference Deshpande, Monty and Marusic2019) for a large Reynolds number range (
$\textit{Re}_\tau =2\times 10^3{-}7.2\times 10^5$
). Considering the fitting uncertainty with
$95\,\%$
confidence level (
$2^\circ$
), we confirm that, for a neutral ASL flow, the wall-attached eddies inclination angle reaches the asymptotic value hypothesised in the literature. However,
$\gamma =15.1^\circ$
is found when the method of Christensen & Wu (Reference Christensen and Wu2005) is applied to the velocity signals filtered within the wall-attached-eddy spectral range. This value is close to the interval of LSM inclination angles found in the literature (
$12^\circ {-}15^\circ$
) (Salesky & Anderson Reference Salesky and Anderson2020), thus it reflects the packet organisation of the wall-attached eddies into LSM structures.
Based on the agreement between the literature and the present estimates of inclination angles via both methods, we conclude that the discrepancy in the inclination angles observed in figures 6(a) and 6(c) is primarily due to the method chosen to estimate this quantity. Since both methods are equally common in the literature to quantify the inclination angle of wall-bounded turbulent structures, it is beyond the goal of the present study to indicate which method yields to a more accurate estimate of the inclination angle for a given range of turbulent scales.
4.2. Scale-dependent conditional inclination angle of coherent turbulent structures
4.2.1. VLSM signal as conditional event
When the inclination angle is evaluated for high-Reynolds-number flows, different values are found across various turbulent scales, namely
$10^\circ {-}15^\circ$
for VLSMs (Liu et al. Reference Liu, Bo and Liang2017) and wall-attached-eddy packets,
$45^\circ$
for isolated attached eddies (Deshpande et al. Reference Deshpande, Monty and Marusic2019), and
$90^\circ$
for small isotropic wall-detached eddies (Cui & Jacobi Reference Cui and Jacobi2024). Although these values represent scale-specific estimates within a turbulent flow, they should be assumed as overall time-averaged values; therefore, not representative of the instantaneous inclination angle of individual structures. For a boundary layer flow, for instance, Hutchins et al. (Reference Hutchins, Monty, Ganapathisubramani, Ng and Marusic2011) obtained conditional VLSM inclination angles in the
$x{-}z$
plane evaluated based on the occurrence of either low- or high-shear stress at the wall. Eventually, they discovered that high wall stress is triggered by shallower VLSMs, and vice versa. Similarly, Cui & Jacobi (Reference Cui and Jacobi2024) reported a reduction in the inclination angle by
${\sim} 20^\circ$
for
$k_xz\leq 0.5$
caused by an increase in the advection velocity of large scales, and vice versa. The link between advection velocity and inclination angle was found in the increased shear induced by fast-travelling structures that reduce the inclination angle of smaller coherent structures.
Inspired by these approaches, the conditioning event chosen here is the presence of local extreme values in the VLSM streamwise velocity fluctuations measured by the LiDAR, with the goal of investigating their effects on the wall-attached-eddy inclination angle. The procedure for obtaining the conditioning event is illustrated in figure 7(a). The reference VLSM signal is obtained using the LiDAR measurements collected throughout the entire vertical range from
$18$
m to
$27$
m (grey lines) that are low-pass filtered using the wavelength range of
$\lambda _x\geq 5.8\delta$
, which has been previously identified through the LCS (cf. with § 3.2). All VLSM velocity signals are then averaged along the vertical direction, thus generating a single reference signal (black line in figure 7
a).
VLSM-filtered velocity signal measured by the LiDAR: (a) time series for each height between
$12$
m (
$0.13\delta$
) and
$27$
m (
$0.28\delta$
) (grey lines) and vertically averaged (black line); the red and blue lines report the intervals around each conditioning event utilized to calculate the SLSE; (b) probability density function of VLSM maximum (red triangles) and minimum (blue triangles) streamwise velocity fluctuations.
Over a duration of approximately 80 eddy turnovers, the mean VLSM record has
$13$
maxima (red triangles) and
$13$
minima (blue triangles) corresponding to VLSM-induced high- and low-momentum occurrences, respectively, around which the scale-dependent conditional inclination angle will be calculated. The statistical characterisation of these events is reported in figure 7(b) as the probability density function of positive and negative VLSM fluctuation peaks. Positive VLSM fluctuations are observed to have smaller absolute mean value and similar standard deviation with respect to the negative VLSM fluctuations.
Vertical profiles of the mean streamwise velocity measured with SLPIV and conditionally averaged with the VLSM high- and low-momentum fluctuations: (a) physical values; (b) mean velocity scaled by
$U_\tau$
.
To assess the relationship between VLSM maxima or minima and the shear variability for the ASL flow under investigation, the friction velocity is conditionally estimated in the presence of positive or negative VLSM-induced velocity fluctuations. In particular, the VLSM extrema shown in figure 7(a) are used to conditionally average the velocity profiles probed with the SLPIV. For each vertical velocity profile, the conditional friction velocity is estimated using the best fit with (3.3). In figure 8(a), the vertical profiles of the mean streamwise velocity are calculated only considering high speed (
$u'_{\textit{VLSM}}\gt 0$
) or only low speed (
$u'_{\textit{VLSM}}\lt 0$
) events, as defined in figure 7(a), then compared with the overall average velocity profile, i.e. the baseline case. It is evident, as expected, that the mean velocity profiles associated with high- (low-) speed events (i.e. maxima (minima) in figure 7
a) have larger (smaller) mean streamwise velocity at all heights.
When the vertical profiles of the conditioned mean streamwise velocity are corrected for shear (
$U^+=U/U_\tau$
), then all profiles collapse on the same curve, as shown in figure 8(b), thus indicating that the differences between the profiles in figure 8(a) are due to the different shear level induced by VLSM fluctuations. In particular, high-momentum VLSMs induce larger shear quantified by a higher friction velocity of
$U_\tau =0.356\,\rm {m}\,{s}^{-1}$
(obtained through the best fit of the mean velocity profile with 3.3) with respect to the baseline (
$0.342\rm \,{m}\,{s}^{-1}$
). In contrast, low-momentum VLSMs induce lower shear (
$U_\tau =0.327\rm \,{m}\,{s}^{-1}$
).
4.2.2. Scale-dependent inclination angle based on the spectral linear stochastic estimator
The concept of scale-dependent inclination angle has already been used in previous studies (Baars et al. Reference Baars, Hutchins and Marusic2016; Cheng et al. Reference Cheng, Shyy and Fu2022; Cui & Jacobi Reference Cui and Jacobi2024). To quantify the scale-dependent phase shift between two streamwise velocity signals, the spectral linear stochastic estimator (SLSE),
$H$
, is introduced to quantify the ratio between the cross-spectrum of the two signals divided by the spectrum of one of the input signals (Baars et al. Reference Baars, Hutchins and Marusic2016):
\begin{equation} H(z_R,z;f)=\dfrac {\langle \hat {u}(z_R;f)\,\hat {u}^*(z;f)\rangle _T}{\langle \hat {u}(z;f)\,\hat {u}^*(z;f) \rangle _T}, \end{equation}
where
$\hat {\boldsymbol{\cdot }}$
indicates the Fourier transform and
$*$
indicates the complex conjugated. The symbol
$\langle \boldsymbol{\cdot }\rangle _T$
indicates an ensemble average calculated over a certain number of temporal subsets
$T$
, each of which encompasses
$N_t$
samples. Baars et al. (Reference Baars, Hutchins and Marusic2016) used a uniform spacing of temporal subsets with
$50\,\%$
overlap to ensure statistical convergence of (4.3). In contrast, for the present work, the distribution and width of the subsets
$T$
are chosen based on the presence of high- and low-momentum events induced by VLSMs, which constitute the conditioning events (figure 7
a).
The phase of SLSE,
$\varPhi$
, is related to the scale-dependent temporal shift,
$\tau _{\textit{SD}}$
, induced by forward-leaning turbulent structures:
\begin{equation} \tau _{\textit{SD}}(z_R,z;f)=\dfrac {\varPhi (z_R,z;f)}{2\pi f}. \end{equation}
Hence, the scale-dependent inclination angle,
$\gamma _{\textit{SD}}$
, is calculated in physical space as
\begin{equation} \gamma _{\textit{SD}}(z_R,z;f)= \arctan \left [\dfrac {z - z_R}{U(z)\,\tau _{\textit{SD}}(z_R,z;f)}\right ]\!. \end{equation}
To obtain the conditional values of
$\gamma _{\textit{SD}}$
, for each time record, the temporal subsets
$T$
are centred around the conditional events (i.e. the occurrence of high- or low-momentum VLSMs in figure 7
a) and
$N_t$
samples are isolated around each of them to calculate the SLSE from (4.3). The ensemble average of the SLSE across these conditionally identified subsets then leads to the conditionally averaged value of
$\gamma _{\textit{SD}}$
. Within the wall-attached-eddy spectral region, the proper quantification of the scale-resolved inclination angles depends on the correct choice of the number of samples (
$N_t$
) within each subset
$T$
. In particular,
$N_t$
time samples lead to a spectrum made by
$N_t/2$
frequencies when the spectrum is evaluated via FFT. Thus, a large enough number of time samples should be entailed by each subset to provide a frequency bandwidth large enough to cover the wall-attached-eddy spectral region. However, the variability on the streamwise velocity due to the VLSMs should be filtered out, thus,
$N_t$
should not be too large. The best trade-off is reached assuming the time subset (
$T$
) equal to
$1.5$
times the vertical-averaged VLSM integral time scale. The latter is evaluated through the auto-correlation function of the reference VLSM signal and is equal to
$31$
s. This corresponds to length scales between
$3.2\delta$
and
$3.9\delta$
by leveraging the Taylor's hypothesis for frozen turbulence. The time scale of
$1.5\times 31$
s sampled by the SLPIV (with
$20$
Hz sampling rate) returns
$N_t=927$
points for each subset
$T$
and, finally, to
$462$
frequencies uniformly spaced between 0 and the Nyquist value. The minimum strictly positive frequency (
$0.043$
Hz) allows to resolve streamwise fluctuations up to
$1.6\delta {-}1.9\delta$
(via Taylor hypothesis), which is lower than the low-wavelength limit of the attached eddy spectral region (
$\lambda _x=6.1z$
) for the considered wall-normal interval (
$z\lt 0.3\delta$
). The averaging intervals around each conditioning event are reported in figure 7(a) with red and blue lines.
Estimate of
$\gamma _{\textit{SD}}$
from SLPIV data collected at
$z=0.06\delta$
with reference height
$z_R=0.04\delta$
through conditional sampling during high- and low-momentum VLSM (red and blue lines, respectively). (a) SLSE phase; the shaded intervals report the statistical uncertainty. (b) Inclination angle. The vertical dashed line marks the spectral boundary between small-scale isotropic wall-detached eddies and wall-attached eddies (
$\lambda _x=6.1z$
).
Following this criterion, in addition to the bulk inclination angle obtained from the entire velocity signals, two values of
$\gamma _{\textit{SD}}$
are calculated at each height using (4.5), which quantify the scale-dependent inclination angle induced by high- and low-momentum VLSMs, respectively. An example of phase shift,
$\varPhi$
, across wavelengths is reported in figure 9(a) relative to the velocity signal collected at
$z=0.06\delta$
. The shaded regions report the statistical uncertainty around
$\varPhi$
calculated assuming Gaussian variables. Across the scanned heights, the maximum statistical uncertainty around
$\gamma _{\textit{SD}}$
is below
$3^\circ$
. First, it can be observed that for small wall-detached eddies, which belong to the spectral region to the left of the vertical dashed line reported in figure 9(a), the variation of the SLSE phase is inconsistent, as already noted by Cui & Jacobi (Reference Cui and Jacobi2024), for both baseline and conditionally sampled velocity signals. The corresponding inclination angle is very close to 90
$^\circ$
, as shown in figure 9(b), which is typical for Kolmogorov isotropic eddies (Baars et al. Reference Baars, Hutchins and Marusic2016; Cheng et al. Reference Cheng, Shyy and Fu2022).
In contrast, for wavelengths approximately larger than
$0.2\delta$
(i.e. within the wall-attached eddy spectral region at the right of the dashed line reported in figure 9), the presence of VLSM maxima induces a larger phase shift (red continuous line) with respect to the baseline (black dotted line), whereas the VLSM minima induce smaller phase shifts (blue dashed line).
In physical space, the distribution of
$\gamma _{\textit{SD}}$
associated with the wall-attached eddy spectral region, that is, for
$\lambda _x\gt 6.1z$
, shows a clear distinction between the baseline
$\gamma _{\textit{SD}}$
and the conditional values. Within the attached-eddy spectral band, negative VLSM fluctuations steepen the streamwise inclination (blue dashed line) relative to the baseline (black dotted line), while positive VLSM fluctuations reduce it, consistently with shallower wall-coherent eddies. This trend, singled out through the ensemble averaging, represents the first evidence of a spatio-temporal variability of the wall-attached eddy inclination angle triggered by VLSMs in an ASL flow.
Scale-dependent inclination angle (
$\gamma _{\textit{SD}}$
) based on SLSE: (a) baseline (unconditional) scale-dependent inclination angle; (b)
$\gamma _{\textit{SD}}$
conditionally evaluated based on
$u'_{\textit{VLSM}}\gt 0$
; (c)
$\gamma _{\textit{SD}}$
conditionally averaged based on
$u'_{\textit{VLSM}}\lt 0$
. Panels (d) and (e) report the difference between conditional and baseline inclination angles (
$\Delta \gamma$
) for (d) positive and (e) negative VLSM-induced velocity fluctuations.
The above-mentioned procedure to calculate the scale-dependent inclination angle,
$\gamma _{\textit{SD}}$
, is now applied to all the detrended streamwise velocity records resolved by the SLPIV assuming the lowest measured height as reference (
$z_R=0.04\delta$
). The unconditional (baseline) scale-dependent inclination angle is depicted in figure 10(a), while figures 10(b) and 10(c) report
$\gamma _{\textit{SD}}$
evaluated based on the occurrence of VLSM-induced high- and low-momentum events, respectively. Furthermore, the difference between conditional and baseline values of
$\gamma _{\textit{SD}}$
is reported in figures 10(d) and 10(e) for
$u'_{\textit{VLSM}}\gt 0$
and
$u'_{\textit{VLSM}}\lt 0$
, respectively.
For all cases, the boundary between small scales and wall-attached eddy spectral regions (
$\lambda _x=6.1z$
) is reported with a black line. In general, this spectral limit shows that on the left-hand side of these maps, the inclination angle is generally larger than 70
$^\circ$
, which is a typical value for small isotropic eddies that belong to that spectral region. Furthermore, the iso-curves related to
$\gamma _{\textit{SD}}$
equal to 70
$^\circ$
or 80
$^\circ$
are roughly parallel to the spectral boundary
$\lambda _x=6.1z$
, especially for the VLSM-based conditional cases in figures 10(b) and 10(c). This feature corroborates the fidelity of the estimate of the spectral boundary between small wall-detached and wall-attached eddies obtained through the linear coherence spectrum.
For all three cases in figure 10, a decrease from
$\gamma _{\textit{SD}}\approx 90^\circ$
to lower values of
$\gamma _{\textit{SD}}$
(
$40^\circ \leqslant \gamma _{\textit{SD}}\leq 55^\circ$
) is observed with increasing streamwise wavelength, thus transitioning towards the spectral region dominated by wall-attached eddies. For the events with VLSM-induced high-momentum (
$u'_{\textit{VLSM}}\gt 0$
), the wall-attached-eddy inclination angle attains values between
$30^\circ$
and
$60^\circ$
(figure 10
b), thus lower than the baseline case. Additionally, with respect to the baseline case, the iso-lines in figure 10(b) are shifted towards larger wavelength throughout the measured
$z$
-interval. This is consistent with the assumption of higher shear induced by positive VLSM fluctuations, which reduces the heights attained by wall-attached eddies and, in turn, increases the spectral bandwidth of isotropic eddies. In contrast, for low-momentum events (
$u'_{\textit{VLSM}}\lt 0$
), conditioned values of
$\gamma _{\textit{SD}}$
are higher than the previous cases (
$50^\circ \leqslant \gamma _{\textit{SD}}\leq 80^\circ$
, figure 10
c).
Finally, from the plot of the difference in inclination angle between conditional statistics and the baseline case in figures 10(d) and 10(e), it is first observed that the conditional statistics of the inclination angle of wall-detached eddies located to the left of the line
$\lambda _x=6.1z$
are practically not affected by the velocity fluctuations induced by VLSMs. In contrast, for the spectral region dominated by wall-attached eddies (i.e. at the right of the line
$\lambda _x=6.1z$
), conditioning the analysis to only high-momentum events leads to reductions in the inclination angle of wall-attached eddies in the range of
$\Delta \gamma =-25^\circ$
to
$-5^\circ$
(figure 10
d), while similarly low-momentum events lead to increase in the inclination angle in the range of
$\Delta \gamma =10^\circ$
to
$30^\circ$
(figure 10
e). Therefore, this study of a neutral ASL flow indicates that VLSM-induced velocity fluctuations modulate the wall-attached-eddy inclination angle: high-momentum VLSM events coincide with higher shear and smaller inclination angles relative to baseline; whereas low-momentum events coincide with lower shear and larger angles. However, no comparable effects are observed in the spectral range dominated by isotropic wall-detached eddies
$\lambda _x\leq 6.1z$
.
Vertically averaged VLSM signals for the streamwise (grey line) and vertical (black like) velocity components probed by LiDAR and SLPIV, respectively. Upward (downward) triangles report local maxima (minima). Empty symbols refer to maxima and minima of
$u^+_{\textit{VLSM}}$
, while filled symbols refer to maxima and minima of
$w^+_{\textit{VLSM}}$
.
$\rho$
stands for the correlation coefficient between
$u^+_{\textit{VLSM}}$
and
$w^+_{\textit{VLSM}}$
.
4.2.3. Role of vertical VLSM-induced velocity fluctuations
To gain further insight into the physical mechanisms that relate the VLSM-induced velocity fluctuations and the wall-attached eddy inclination angle, the vertical velocity component is bandpass filtered in the same spectral interval defined in § 3.2 for the horizontal VLSM-induced velocity (
$f\geq 8\times 10^{-3}$
Hz and
$\lambda _x\gt 5.8\delta$
), which is then averaged in the vertical interval probed by the SLPIV (
$z=3.5$
m to
$19$
m). The so-obtained reference signal of the VLSM-induced vertical velocity is reported in figure 11 overlapped in time with the reference VLSM-induced streamwise velocity signal (
$u^+_{\textit{VLSM}}$
, cf. with figure 7
a). The maxima and minima are identified for
$w^+_{\textit{VLSM}}$
and reported in figure 11 with filled symbols. The horizontal and vertical reference velocity records appear to be roughly anti-correlated, i.e. producing a correlation coefficient of
$\rho =-0.35$
.
Focusing on the VLSM maxima and minima, each maximum (minimum) in the streamwise VLSM signal roughly corresponds to a minimum (maximum) in the vertical velocity record, which indicates a downward (upward) VLSM motion associated with streamwise VLSM accelerations (decelerations). This leads to the hypothesis that, in addition to the increased shear induced by positive streamwise velocity fluctuations, the simultaneous vertical VLSM-induced motion could also contribute to the modulation in the inclination angle of wall-attached eddies. To assess this hypothesis, the conditional analysis detailed in § 4.2 is now repeated to obtain the scale-dependent inclination angle (
$\gamma _{\textit{SD}}$
) conditionally evaluated based on the vertical VLSM-induced velocity fluctuations. The results reported in figure 12 show that in the presence of positive vertical velocity fluctuations,
$\gamma _{\textit{SD}}$
increases from baseline values (
$40^\circ {-}60^\circ$
; figure 12
a) to
$50^\circ {-}85^\circ$
(figure 12
b) within the wall-attached eddy spectral range (
$\lambda _x\geq 6.1z$
, reported by a black line in figure 12), which corroborates the hypothesis that upward VLSM fluctuations increase the wall-attached eddy inclination angle in analogy with the negative streamwise VLSM-induced velocity fluctuations. Similarly, the occurrence of negative vertical velocity fluctuations reduces the inclination angle of wall-attached eddies from baseline values to
$\gamma _{\textit{SD}}=20^\circ {-}50^\circ$
(figure 12
c) in analogy with positive streamwise VLSM fluctuations.
Scale-dependent inclination angle (
$\gamma _{\textit{SD}}$
) based on SLSE: (a) baseline (unconditional) scale-dependent inclination angle; (b)
$\gamma _{\textit{SD}}$
conditionally evaluated based on
$w'_{\textit{VLSM}}\gt 0$
; (c)
$\gamma _{\textit{SD}}$
conditionally averaged based on
$w'_{\textit{VLSM}}\lt 0$
. Panels (d) and (e) report the difference between conditional and baseline inclination angles (
$\Delta \gamma$
) for (d) positive and (e) negative VLSM-induced velocity fluctuations. In panels (b)–(e) the black line reports the wall-attached eddy spectral boundary (
$\lambda _x=6.1z$
).
5. Conclusions
In this work, a neutral atmospheric surface layer (ASL) flow has been investigated by probing the streamwise velocity via the simultaneous deployment of a scanning Doppler LiDAR and a super large particle image velocimetry (SLPIV) apparatus. The goal of this study is twofold: first, to quantify the inclination angle of wall-attached eddies at a very high friction Reynolds number (
$\textit{Re}_\tau =7.9\times 10^5$
); second, to provide insights on the effects on the inclination angle of wall-attached eddies due to velocity fluctuations and shear variability induced by very-large-scale motions (VLSMs).
To address the first goal, the streamwise-velocity energy spectra are calculated to extract the wall-attached eddy contribution via bandpass filtering. The thresholds for the bandpass filtering of the streamwise velocity signals are selected using a procedure based on the linear coherence spectrum (Puccioni et al. Reference Puccioni, Calaf, Pardyjak, Hoch, Morrison, Perelet and Iungo2023). Subsequently, the two-point correlation between the resulting wall-attached-eddy fluctuating velocity fields is calculated to quantify the respective inclination angle, which results to be
$41.1^\circ \pm 3^\circ$
for a reference height of
$0.04\delta$
, thus close to the previously hypothesised value of
$45^\circ$
for an asymptotically high friction Reynolds number flow (Deshpande et al. Reference Deshpande, Monty and Marusic2019, Reference Deshpande, de Silva and Marusic2023).
The procedure introduced by Baars et al. (Reference Baars, Hutchins and Marusic2016) based on the spectral linear stochastic estimator (SLSE) between two streamwise velocity signals is used here to address the second goal of this analysis, i.e. quantifying the scale-dependent inclination angle (
$\gamma _{\textit{SD}}$
) conditioned on the occurrence of high- and low-momentum events induced by VLSMs. In particular, the phase of the SLSE, reflecting the shift between the two input signals in the Fourier space, is calculated in the presence of VLSM-induced high- or low-momentum fluctuations. The inclination angles of wall-attached eddies, which are estimated on the conditional analysis based on VLSM-induced velocity fluctuations, reveal that VLSM-induced high- (low-) fluctuations lead to lower (higher) inclination angles for wavelengths owing to wall-attached eddies, thus indicating a modulating effect of the VLSM onto the inclination angle of wall-attached eddies. The connection between these two phenomena is the increased (reduced) shear induced by VLSMs which, in turn, induces a smaller (larger) inclination angle of the wall-attached eddies.
In addition to the modulation induced by streamwise VLSMs, the wall-normal velocity component probed by the SLPIV is leveraged to calculate the vertical VLSM fluctuations and use the latter to re-evaluate the scale-dependent inclination angle based on the occurrence of maxima or minima in the vertical VLSM signal. In analogy with the streamwise VLSMs, the upward VLSMs are found to increase
$\gamma _{\textit{SD}}$
, while downward VLSMs reduce it. The quantitative differences between the conditioned and baseline values of
$\gamma _{\textit{SD}}$
are of the same order of the analogous quantity based on the streamwise VLSMs. This further corroborates the role played by VLSMs in modulating the streamwise inclination angle of wall-attached eddy structures.
Even though the effect of additional factors (e.g. variability in terrain elevation and roughness, changes in wind direction or thermal stratification, etc.) cannot be ruled out with certainty, the present results strongly support the conclusion that the variability of inclination angle within a wall-bounded flow is primarily determined by the occurrence of VLSM events. The current research emphasises the crucial role played by VLSMs in determining the structure of high-Reynolds-number turbulent flows across different scales. The scale-dependent variability of the inclination angle within a turbulent flow has been conjectured for a long time, but it has never been formalised so far, and moreover, it has never been linked to any phenomena occurring in an ASL flow. The present research emphasises an important interplay between VLSMs and attached eddies within an ASL flow, and could have major impacts on several nonlinear interactions between turbulent scales currently under study, including amplitude and frequency modulation, spanwise coherence, and momentum and energy transfer across different turbulent scales.
Funding
This research was primarily funded by the National Science Foundation, MRI program, Fluid Dynamics, Award 2018658. G.V.I. and C.M. are also funded by the National Science Foundation, Fluid Dynamics program, CAREER program, Award 2046160. A portion of M.P.’s work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 with release number LLNL-JRNL-2003042.
Declaration of interests
The authors report no conflict of interest.
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