# Unexpected Field Evaporation Sequence in $\gamma$-TiAl

Jiayuwen Qi  Fei Xue  Emmanuelle A. Marquis  Wolfgang Windl

###### Abstract

In atom probe tomography (APT), atoms from the surface of a needle shape
specimen are evaporated under a high electric field and analyzed via time of
flight mass spectrometry and position sensitive detection. 3D reconstruction
of the atom positions follows a simple projection law, which can sometimes
lead to artifacts due to deviation from an assumed ideal evaporation sequence.
Here, we revisit the evaporation behavior of [001]-oriented $\gamma$-TiAl
using a full-dynamics simulation approach empowered by molecular dynamics.
Without any knowledge of charge states or assumptions about evaporation
fields, we successfully reproduced the lack of distinct Al and Ti layers
observed in reconstructions of experimental data which is traditionally
attributed to the retention of Al on the evaporating surface. We further
showed that a step-wise bond breaking process of Ti in contrast to the
simultaneous bond breaking of Al explains the seemingly counterintuitive
preferential evaporation of the strongly bonded Ti atoms.

###### keywords:

atom probe tomography , APT , molecular dynamics , atomistic simulations ,
field evaporation , $\gamma$-TiAl

[inst1]organization=Dept. of Materials Science and Engineering, The Ohio State
University,city=Columbus, state=OH, country=USA

[inst2]organization=Dept. of Materials Science and Engineering, University of
Michigan,city=Ann Arbor, state=MI, country=USA

##  1 Background

Atom probe tomography (APT) is known as a very unique characterization
technique that possesses the ability to resolve both atomic structure and
chemical composition of materials in three dimensions. In APT, surface atoms
of a sharp needle-shape specimen are field-evaporated under a high electric
field, which is created by a high voltage applied to the specimen. A two-
dimensional detector records the sequence and positions of the arriving atoms.
Along with the chemical identity information recovered by mass spectrometry, a
3D distribution of atoms in the evaporated volume can be generated based on a
simple projection law [1].

The standard reconstruction protocol assumes that the evaporating specimen
always keeps a perfect hemispherical shape, which requires atoms to leave the
surface in an ideal sequence. However, the evaporation of materials with
layered structures often shows different rates of evaporation of the different
layers with one tending to stick to the surface while the other evaporates
readily [2]. This sort of behavior is typically explained by a hand-waving
argument: different atoms have different evaporation fields, which disturbs
the ideal evaporation sequence and results in positioning errors during the
reconstruction of certain atomic planes.

$\gamma$-TiAl is an excellent system for examining such errors in the observed
layer sequences in APT. Experimentally it is found that the subsequent,
distinct Al and Ti planes, as one would expect from the crystallographic
structure of TiAl (Fig. 1(a)) are merged by the reconstruction process leading
to half the number of planes, each containing Al and Ti (Fig. 1(c)).

Field ion microscopy observations [3, 4, 5] typically revealed that the Al
planes constitute the top layer, while the following Ti plane is only slightly
larger than the Al top layer but much smaller than the following Al layer as
shown in Fig. 2(a). These observations have largely been interpreted as the
result of Al being retained on the surface while Ti readily evaporates once
uncovered, due to the lower evaporation of Ti as compared to that of Al.
However, Lefebvre et al. came to the opposite conclusion proposing that Ti has
a higher evaporation field than Al based on based on APT reconstructed data
where Ti atoms were positioned slightly higher than Al atoms within each mixed
plane [6]. Subsequently, Boll et al. performed simulations in order to
reproduce their own experimental observation of Ti evaporating more easily
than Al [7]. They used Müller’s formula for the evaporation field, from which
they stated to have found a higher evaporation field for Al when assuming for
both Al and Ti a +2 charge state as observed experimentally. However, their
calculated evaporation fields are not reported in the paper, and it is not
possible to reproduce them since the bond energy $\Lambda$ of Ti-Al used to
calculate the sublimation energy is $\Lambda_{\rm Al-Ti}=0.70$ eV. Compared to
the energies of the homoelemental bonds, $\Lambda_{\rm Al-Al}=0.33$ eV and
$\Lambda_{\rm Ti-Ti}=1.07$ eV, the value of 0.70 eV results in an ordering
energy of zero, which suggests Ti and Al form an ideal solid solution rather
than an ordered intermetallic. Our own DFT-values for these bond energies,
which will be discussed later, are in stark contrast to either value, with
$\Lambda_{\rm Al-Ti}=0.86$ eV, $\Lambda_{\rm Al-Al}=0.58$ eV, and
$\Lambda_{\rm Ti-Ti}=0.91$ eV and an ordering energy of $-0.12$ eV in the
common range of intermetallics.

In order to sort through these contradictory findings, we revisit here the
field evaporation sequence of $[001]$-oriented TiAl by applying the full-
dynamics field evaporation simulation approach TAPSim-MD [8]. Our approach
evaporates atoms in the specimen as a result of competition between
interatomic forces and electrostatic forces, and gives an “ab-initio”
prediction of evaporation events as well as atom trajectories. Here for the
example of $[001]$-oriented TiAl, we show that without any knowledge of charge
states, TAPSim-MD successfully reproduces surface morphologies observed by FIM
and the mixed-layer reconstructed structure. Our results suggest that Al
indeed possesses a higher evaporation field than Ti, which is in agreement
with most previous conclusions from experiments. However, in contrast to
previous suggestions, we find from analyzing the atomic trajectories that Ti
is preferentially evaporated due to its two-step bond breaking process, which
is easier to happen in comparison to the evaporation process for Al, where all
bonds are simultaneously broken.

##  2 Methods

Simulations were conducted using the full-dyanmics APT simulation tool TAPSim-
MD [8], which combines the traditional finite element APT simulation package
TAPSim [9, 10] with the classical molecular dynamics approach as implemented
in LAMMPS [11, 12]. A [001] oriented virtual tip is built in cylinder shape
with 8 nm in radius and 24 nm in height. The virtual voltage in the simulation
is first increased by 10 V after a set of MD runs until 420 V, and then
increased by 2 V after a set of MD runs until an evaporation event is
triggered. The voltage never decreases to ensure a computationally efficient
data collection process. The employed interatomic potential is an embedded
atom model potential from Zope and Mishin [13].

For comparison, we also performed traditional TAPSim simulations for the same
tip structure. Since no MD is included in this method, an evaporation
criterion is used to determine the evaporation event which is based on the
magnitude of field-induced forces and empirical evaporation fields. Since the
hypothesis to test is if Al or Ti should have the lower evaporation field, we
use type 1 and type 2 to tag the atoms in the two layers. We set the
evaporation field of type 1 to 2.6 V/Å as the high field, and that of type 2
to 1.9 V/Å as the low field. Details about the TAPSim approach and
applications can be found in Refs. [9, 14].

Specimens for APT were prepared using a standard lift-out and sharpening
process performed with a Thermo Fisher Helios 650 Nanolab scanning electron
microscope/focused ion beam (SEM/FIB) from a Ti-52Al-2Nb (at.%) ingot in as-
cast condition. A $\langle 001\rangle$-oriented grain was identified by
electron backscatter diffraction (EBSD) using a Tescan RISE SEM. Data post-
processing used the Orientation Imaging Microscopy (OIM) software version 9 to
address the pseudosymmetry issue through spherical indexing and real-space
refinement. The orientation of the $\langle 001\rangle$ grain was further
verified by selective area diffraction (SAD) with a Thermo Fisher Talos F200X
G2 scanning/transmitting electron microscope (S/TEM) using the c/a ratio of
tetragonal $\gamma$-TiAl (L10) and the positions of the superlattice spots.
APT experiments were conducted using a Cameca LEAP 5000XR instrument. Data
collection was conducted with a base sample temperature of 50 K and a
detection rate of 1% in laser mode. A laser energy of 50 pJ and a pulse
frequency of 200 kHz were used. The APT data was reconstructed using the
Cameca AP Suite software 6.3. Values of the image compression factor were
estimated using the desorption map indexing by APS-integrated OIM and were
found to be 1.58-1.59. The field factor was optimized between 5.95 and 9.40 to
create a (001) plane spacing close to 0.4 nm comparable to that of
$\gamma$-TiAl. The measured compositions were Ti-(53.4-54.0 at.%)Al-1.7
at.%Nb, close to the nominal composition.

##  3 Results and Discussion

###  3.1 Reconstruction and Surface Morphology

In this section, we undertake a comparative analysis by juxtaposing various
simulation outcomes with experimental observations, focusing on the standard
reconstruction outcome and the surface morphology of the tip.

As mentioned before, the reconstructed structure of $[001]$ oriented TiAl has
mixed Ti/Al planes instead of alternate Ti and Al planes in the crystal as
shown in Fig. 1(a). Our simulated reconstruction result (Fig.1(b)) is
consistent with the experimental observation shown in Fig. 1(c). In both
cases, there is no discernible alternately layered structure.

Figure 1: (a) is the structure of $\gamma$-TiAl intermetallic compound. (b) is
the standard reconstruction result of simulation data from TAPSim-MD. (c) is
the experimental reconstruction result of TiAl.

As illustrated in the field-ion image in Fig.2(a), the size of the Ti layer
closely matches that of the Al layer situated above it, whereas it is
significantly smaller than the Al layer beneath. Our molecular dynamics (MD)
simulation, shown in Fig.2(b), indeed replicates this coupled plane
evaporation pattern. Remarkably, a traditional TAPSim simulation, without the
inclusion of MD, also yields the same pattern, as shown in Fig. 2(c), where
the low-field cyan plane has a similar size to the high-field red plane on top
of it.

Figure 2: (a) Field-ion image of the (001) pole of a Ti51Al46Cr3 sample,
reprinted from Ref. [3]. The brighter (dimmer) contrast is Al (Ti). (b) Top
view of a TiAl tip simulated by TAPSim-MD. Ti (Al) atoms are grey (violet).
(c) Top view of the tip from (b), simulated by traditional TAPSim. Atoms with
higher (lower) evaporation field are red (cyan).

The comparison between the simulation outcomes of TAPSim-MD and traditional
TAPSim reveals that Al atoms are likely to possess a higher evaporation field,
while Ti atoms are expected to exhibit a lower one. Consequently, Al atoms
tend to remain on the specimen, whereas Ti atoms have a greater propensity for
evaporation. This result is in agreement with most previous analyses in the
literature [15, 3, 4, 5].

###  3.2 Distinct Atom Trajectories and Bond-breaking Processes

As demonstrated previously, TAPSim-MD’s prediction of the bilayer surface
morphology along with the evaporation sequence for TiAl agree well with
experimental observations without any ad hoc assumptions about evaporation
fields or prior knowledge of charge states. The full dynamics of TAPSim-MD
allows us now to study the underlying processes of the evaporation sequence
from a dynamic perspective by analyzing forces and atom trajectories.

Figure 3: Field evaporation maps from the experiment and TAPSim-MD simulation
for Ti atoms ((a) and (c)) and Al atoms ((b) and (d)). The warmer the color,
the higher the intensity. Red circles in (c) and (d) depict a similar field of
view compared to the experimental results in (a) and (b). Further shown are
examples of (e) Ti and (f) Al atom trajectories during evaporation,
respectively. Ti (Al) atoms are shown in grey (violet). The equilibrium
positions of the evaporating atoms are indicated by smaller balls with the
same color scheme. The atom trajectories are described by a series of small
dots colored by the magnitude of net forces at different instances. The warmer
the color, the larger the net force.

Fig. 3(a)-(d) depict markedly different field evaporation maps for Ti and Al:
the central pole in the Ti map is more densely populated, while it is depleted
in the Al map, which can be observed in both experimental and simulated
results. Furthermore, in the simulated evaporation maps, the zone lines in the
Ti map (c) exhibit heightened intensity, whereas a slight depletion is
observed in the Al map (d). These pronounced dissimilarities in the
evaporation maps strongly suggest that Ti and Al follow distinctly different
launch trajectories. Indeed, our observed launch trajectories for Ti and Al,
as shown in Fig. 3(c) and (d), confirm this divergence: the evaporating Ti
atom initially ascends vertically along the $z$-direction, while the
evaporating Al atom departs at an angle of approximately 45∘ between the
$z$-axis and the $xy$-plane.

Considering that the electrostatic forces should not depend on the chemical
identities of evaporating atoms for equal charge states but on the geometry,
the distinct atom launch trajectories of Ti and Al indicate that the
desorption or bond breaking processes of the two species are distinctly
different. Based solely on bond strength, Al would be expected to have weaker
bonds than Ti. Nevertheless, it exhibits a higher evaporation field than Ti,
despite Ti possessing relatively strong bonds. As mentioned, we find the bond
strengths of $-0.58$ eV, $-0.86$ eV, and $-0.91$ eV for Al-Al, Ti-Al and Ti-Ti
bonds, respectively, from DFT calculations . While it is well-established that
a comprehensive analysis of real evaporation processes should consider
multiple factors, including total ionization energy and work function, our MD
simulation employs induced charges to calculate electrostatic forces and
otherwise solely depends on interatomic interactions. Consequently, the
preferential evaporation of strongly bonded Ti atoms can be attributed to the
manner in which these bonds are broken during the evaporation process.

Figure 4: (a) $xy$\- and z-components of interatomic forces (dashed lines) and
net forces (solid lines) vs. the magnitude of the electrostatic force during
field evaporation in TAPSim-MD simulation for an 8 nm-radius TiAl tip. Each
data point is an average over 3098 Ti atoms or 2871 Al atoms. The standard
deviations are indicated by shadows. The critical points are indicated by blue
or red arrows for Ti or Al. The schematic bonding condition and bond breaking
sequence of evaporating atoms are shown for a Ti atom (orange) in (b) and for
an Al atom (blue) in (c).

In order to understand the bond breaking process, we first examine the $xy$\-
and $z$-components of the interatomic force and net force as functions of the
electrostatic force induced by the applied voltage. As shown in fig. 4(a),
when the electrostatic force induced on an atom is rather small (approximately
less than 1 eV/Å), it is balanced by the interatomic force, resulting in an
almost zero net force in both the $xy$-plane and the $z$-direction, which
makes the atom vibrate around its equilibrium position. However, as the
electrostatic force increases, it begins to overcome the interatomic force and
consequently gives rise to a continuously growing instead of oscillating net
force, where the atom starts to evaporate. With regard to Al (blue lines), the
continuous growth of the net force (solid lines) in both $xy$-plane (upper
panel) and $z$-direction (lower panel) start when the electrostatic force is
about 2.2 eV/Å, as indicated by blue arrows. As for Ti atoms (red lines), the
continuous growth of the lateral net force starts around an electrostatic
force of 3.2 eV/Å, while the growth of the $z$-component starts around an
electrostatic force of 1.8 eV/Å, as indicated by red arrows. The simultaneous
growth of the lateral and $z$-components of the net force of Al suggests that
bond breaking in the $xy$-plane (2 Al-Al bonds) and $z$-direction (4 Ti-Al
bonds) should occur around the same time. In contrast, for Ti, the in-plane
and the out-of-plane bond breaking start at distinctly different times, first
in the $z$-direction (breaking 4 Ti-Al bonds at an electrostatic force of
$\sim$1.8 eV/Å), and second in the lateral direction (breaking 2 Ti-Ti bonds
at an electrostatic force of $\sim$3.2 eV/Å). Schematic plots of the bond
breaking processes for the two species along these findings are shown in Fig.
4(b)-(c).

Figure 5: (a) Magnitude of interatomic and net force vs magnitude of
electrostatic force. The thin black solid line indicates y=x. (b) Magnitude of
velocity vs magnitude of electrostatic force. Each data point is averaged over
3098 Ti atoms and 2871 Al atoms. The standard deviations are indicated by
shadows. The critical points are indicated by blue or red arrows for Ti or Al.

In order to show why the two-step bond-breaking process of Ti results in its
preferential evaporation, we examine the magnitude of both the total net force
and the resulting velocity as functions of the electrostatic force. Figure
5(a) clearly shows that the net force acting on the Ti atom (red line) doesn’t
exhibit continuous growth until the electrostatic force surpasses
approximately 1.8 eV/Å, indicated by a red arrow, while this point is reached
later for Al (blue line), at around 2.2 eV/Å, indicated by a blue arrow.
Consequently, when subjected to the same magnitude of electrostatic forces, it
becomes easier to initiate the evaporation of a Ti atom. Along with that, the
two-step bond-breaking process of Ti is also reflected on the growth of the
interatomic force (red dashed line), where two local maxima can be found at
around 1.9 eV/Å and 3.3 eV/Å electrostatic force, corresponding to the first
and second bond-breaking events of Ti respectively. In contrast, there is only
one interatomic force maximum of Al (blue dashed line) at around 2.3 eV/Å
electrostatic force, which is associated with the one-step bond-breaking
process of Al. As a side note, the electrostatic force and internal force can
not completely offset each other after the atom leaves its equilibrium
position as a result of the different anisotropic features of the interatomic
force and electrostatic force [16], which explains why the growth of the
interatomic force deviates from the line of $y=x$ at the initial stage of
evaporation.

The conclusion of preferential evaporation of Ti can also be drawn from
analyzing the velocities shown in Fig. 5(b). For both Al and Ti, the atoms
exhibit vibrational motion around their equilibrium positions with nearly zero
velocity before the electrostatic force starts to dominate over the
interatomic force. As the electrostatic force intensifies, a critical point is
reached at approximately 1.8 eV/Å for Ti and 2.2 eV/Å for Al, quantified by
the electrostatic force at the intersection of the linear-acceleration range
with the constant, low-velocity range. At this point, the interatomic force
cannot balance the electrostatic force anymore, and the velocity starts to
linearly grow with the increasing electrostatic force. Once again, when
considering the application of an equivalent magnitude of electrostatic
forces, we find that the evaporation of a Ti atom is easier than that of Al,
equivalent to a lower evaporation field of Ti compared to Al. Interestingly,
the velocity analysis and especially the determined intersection point
suggests that it is only the first bond-breaking event in Ti that seems to
correlate with the effective evaporation field rather than the second one.

##  4 Conclusion

In this paper, we have revisited the disrupted evaporation sequence in
[001]-oriented $\gamma$-TiAl, a phenomenon that often leads to erroneous
results in reconstruction due to the different evaporation fields exhibited by
different species. Employing our full-dynamics approach, we provide an ab
initio confirmation that Ti requires a lower electrostatic force along with a
lower evaporation field to evaporate, a finding that aligns well with the
majority of experimental observations documented in the literature. More
importantly, through a comprehensive examination of atom trajectories and
forces, we have unveiled a step-wise bond-breaking process for Ti, in contrast
to the simultaneous bond-breaking process observed for Al. This discovery
offers a compelling explanation for the seemingly counterintuitive
preferential evaporation of the strongly bonded Ti atoms.

##  5 Acknowledgement

This project has been sponsored by AFOSR (PM Dr. Ali Sayir) under Award No.
FA9550-14-1-0249 and FA9550-19-1-0378. Computing resources were supplied and
maintained by the Ohio Supercomputer Center [17] under grant number PAA0010.
Results of the simulations were visualized with the programs OVITO [18],
ParaView [19], and Python library Matplotlib [20].

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# Streamwise inclination angle of wall-attached eddies in turbulent channel
flows

Cheng Cheng1  Wei Shyy1  Lin Fu1,2,3<EMAIL_ADDRESS>1Department of Mechanical
and Aerospace Engineering, The Hong Kong University of Science and Technology,
Clear Water Bay, Kowloon, Hong Kong 2Department of Mathematics, The Hong Kong
University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong 3
Center for Ocean Research in Hong Kong and Macau (CORE), The Hong Kong
University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

###### Abstract

We develop a new methodology to assess the streamwise inclination angles
(SIAs) of the wall-attached eddies populating the logarithmic region with a
given wall-normal height. To remove the influences originating from other
scales on the SIA estimated via two-point correlation, the footprints of the
targeted eddies in the vicinity of the wall and the corresponding streamwise
velocity fluctuations carried by them are isolated simultaneously, by coupling
the spectral stochastic estimation with the attached-eddy hypothesis. Datasets
produced with direct numerical simulations spanning $Re_{\tau}\sim
O(10^{2})-O(10^{3})$ are dissected to study the Reynolds-number effect. The
present results show, for the first time, that the SIAs of attached eddies are
Reynolds-number dependent in low and medium Reynolds numbers and tend to
saturate at $45^{\circ}$ as the Reynolds number increases. The mean SIA
reported by vast previous experimental studies are demonstrated to be the
outcomes of the additive effect contributed by multi-scale attached eddies.
These findings clarify the long-term debate and perfect the picture of the
attached-eddy model.

###### keywords:

MSC Codes  (Optional) Please enter your MSC Codes here

##  1 Introduction

It is generally recognized that the high-Reynolds number wall-bounded
turbulence is filled with coherent motions of disparate scales, which are
responsible for the energy transfer and the fluctuation generation of
turbulence. Till now, the most elegant conceptual model describing these
energy-containing motions is the attached-eddy model (Townsend, 1976; Perry &
Chong, 1982). It hypothesizes that the logarithmic region is occupied by an
array of randomly-distributed and self-similar energy-containing motions (or
eddies) with their roots attached to the near-wall region (see Fig. 1). During
the recent decades, a growing body of evidence that supports the attached-eddy
hypothesis has emerged rapidly, e.g., Hwang (2015), Hwang & Sung (2018), Cheng
et al. (2020b), Hwang et al. (2020), to name a few. The reader is referred to
a recent review work by Marusic & Monty (2019) for more details. Throughout
the paper, the terms ‘eddy’ and ‘motion’ are exchangeable. It should be noted
that the terms of ‘wall-attached motions’ and ‘wall-attached eddies’ used in
the present study do not only refer to the self-similar eddies in the
logarithmic region, but also the very-large-scale motions (VLSMs) or
superstructures, as some recent studies have shown that VLSMs are also wall-
attached, despite that their physical characteristics do not match the
attached-eddy model (Hwang & Sung, 2018; Yoon et al., 2020).

Previous studies have established that the energy-containing eddies populating
the logarithmic and outer regions bear characteristic SIAs due to the mean
shear (see Fig. 1($a$)). As early as the 1970s, Kovasznay et al. (1970) found
that the large-scale structures in the outer intermittent region of a
turbulent boundary layer have a moderate tilt in the streamwise direction. On
the other hand, for the eddies in the logarithmic region of wall turbulence,
the wall-attached $\Lambda-$vortex was used by Perry & Chong (1982) to
illustrate them. According to Adrian et al. (2000), these $\Lambda-$vortexes
are apt to cluster along the flow direction and form an integral whole
(generally called as vortex packets). Further observations in channel flows
(Christensen & Adrian, 2001) demonstrated that the heads of $\Lambda-$vortexes
among the vortex packets tend to slope away from the wall in a statistical
sense, with SIAs between $12^{\circ}$ and $13^{\circ}$. Most additional
studies have shown a similar result, and it is widely accepted that the
approximate SIAs of eddies are in the range of $10^{\circ}$ to $16^{\circ}$
(Boppe et al., 1999; Christensen & Adrian, 2001; Carper & Porté-Agel, 2004;
Marusic & Heuer, 2007; Baars et al., 2016). Besides, the SIA is also found to
be Reynolds-number independent (Marusic & Heuer, 2007).

Figure 1: A schematic of the attached-eddy model: ($a$) $x$-$y$ plane and
($b$) $y$-$z$ plane view. Each parallelogram in ($a$) and circle in ($b$)
represents an individual attached eddy. Here, $x$, $y$ and $z$ denote the
streamwise, wall-normal, and spanwise directions, respectively. $y_{s}^{+}$
(100) and $y_{e}^{+}$ ($0.2h^{+}$) denote the lower and upper bound of the
logarithmic region, respectively (Jiménez, 2018; Baars & Marusic, 2020a; Wang
et al., 2021). $y_{o}^{+}$ is the outer reference height. $\Delta y^{+}$ is
the local grid spacing along the wall-normal direction. $\alpha_{m}$ and
$\alpha_{s}$ are the mean and individual SIA of attached eddies, respectively.
These two figures are merely conceptual sketches, and the eddy population
density is not in accordance with that of Perry & Chong (1982).

However, the SIA estimated by experimentalists using the traditional
statistical approach is the mean structure angle indeed (Marusic & Heuer,
2007; Deshpande et al., 2019). The common procedure to obtain the SIA is based
on the calculation of the cross correlation between the streamwise wall-shear
stress fluctuation ($\tau_{x}^{\prime}$) and the streamwise velocity
fluctuation ($u^{\prime}$) at a wall-normal position in the log region
($y_{o}$). The cross correlation can be expressed as

$R_{\tau_{x}^{\prime}u^{\prime}}(\Delta
x)=\frac{\langle\tau_{x}^{\prime}(x)u^{\prime}(x+\Delta
x,y_{o})\rangle}{\sqrt{\left\langle\tau_{x}^{\prime
2}\right\rangle\left\langle u^{{}^{\prime}2}\right\rangle}},$ (1)

where $<\cdot>$ represents the ensemble spatial average, and $\Delta x$ the
streamwise delay. The SIA can be estimated by

$\alpha_{m}=\arctan(\frac{y_{o}}{\Delta x_{p}}),$ (2)

where $\Delta x_{p}$ denotes the streamwise delay corresponding to the peak in
$R_{\tau_{x}^{\prime}u^{\prime}}$. Considering that an array of wall-attached
eddies with distinct wall-normal heights can simultaneously convect past the
reference position $y_{o}$, $\alpha_{m}$ in Eq. (2) should be regarded as the
mean angle of these eddies. Hence, the subscript ‘m’ in Eq. (2) refers to
‘mean’.

To estimate the SIAs of the largest wall-attached eddies, Deshpande et al.
(2019) introduced a spanwise offset between the near-wall and logarithmic
probes to isolate these wall-attached motions in the log region. They found
that their SIAs are approximately $45^{\circ}$. This observation is consistent
with several theoretical analyses. For example, Moin & Kim (1985) and Perry et
al. (1992) proposed that for the flows with two-dimensional mean flows, the
characteristic angles of the energy-containing eddies should follow the
direction of the principal rate of mean strain. More specifically, their SIAs
should be $45^{\circ}$ for a zero-pressure-gradient turbulent boundary layer
(Perry et al., 1992). Marusic (2001) found that the mean SIA of the induced
turbulence field by attached eddies is akin to the experimental measurements,
if the hierarchical attached eddies tilt away from the wall with individual
SIA being $45^{\circ}$ and organize like the vortex packets observed in
numerical and laboratory experiments.

Reviewing the work of predecessors, it can be found the SIAs of attached
eddies at a given length scale are ambiguous. Traditional measurements are
only applicable for the assessment of the mean SIA (Brown & Thomas, 1977;
Boppe et al., 1999; Marusic & Heuer, 2007). Moreover, the technique adopted by
Deshpande et al. (2019) can only isolate the largest wall-attached motions in
the logarithmic region. Considering the characteristic scale of an individual
attached eddy being its wall-normal height as per the attached-eddy model
(Townsend, 1976; Perry & Chong, 1982), it is self-evident that it is of great
importance to assess the SIAs of attached eddies with any heights in the
logarithmic region, not only for the completeness of attached-eddy hypothesis,
but also the accuracy of turbulence simulations (Marusic, 2001; Carper &
Porté-Agel, 2004). In the present study, we aim to achieve this goal by
leaning upon the modified spectral stochastic estimation (SSE) proposed by
Baars et al. (2016), and dissecting the direct numerical simulations (DNS)
database spanning broad-band Reynolds numbers. We will also discuss the
relationship between the mean SIA and the scale-based SIA.

##  2 DNS database and methodology to calculate the SIA

###  2.1 DNS database

The DNS database adopted in the present study has been extensively validated
by Jiménez and co-workers (Del Álamo & Jiménez, 2003; Del Álamo et al., 2004;
Hoyas & Jiménez, 2006; Lozano-Durán & Jiménez, 2014). Four cases at
$Re_{\tau}$=545, 934, 2003 and 4179 are used and named as Re550, Re950, Re2000
and Re4200, respectively ($Re_{\tau}=hu_{\tau}/\nu$, $h$ denotes the channel
half-height, $u_{\tau}$ the wall friction velocity and $\nu$ the kinematic
viscosity). All these data are provided by the Polytechnic University of
Madrid. Details of the parameter settings are listed in Table 1. Note that the
relatively smaller computational domain size of Re4200 may influence the
estimation of SIAs of the attached eddies populating the upper part of the
logarithmic region. This limitation will be discussed in section 3 and
Appendix A.

$Case$ | $Re_{\tau}$ | $L_{x}(h)$ | $L_{y}(h)$ | $L_{z}(h)$ | $\Delta x^{+}$ | $\Delta z^{+}$ | $\Delta y_{min}^{+}$ | $\Delta y_{max}^{+}$ | $N_{F}$ | $Tu_{\tau}/h$
---|---|---|---|---|---|---|---|---|---|---
Re550 | 547 | $8\pi$ | 2 | $4\pi$ | 13.4 | 6.8 | 0.04 | 6.7 | 142 | 22
Re950 | 934 | $8\pi$ | 2 | $3\pi$ | 11.5 | 5.7 | 0.03 | 7.6 | 73 | 12
Re2000 | 2003 | $8\pi$ | 2 | $3\pi$ | 12.3 | 6.2 | 0.32 | 8.9 | 48 | 11
Re4200 | 4179 | $2\pi$ | 2 | $\pi$ | 12.3 | 6.2 | 0.32 | 10.6 | 40 | 15
Table 1: Parameter settings of the DNS database. Here, $L_{x}$, $L_{y}$ and
$L_{z}$ are the sizes of the computational domain in the streamwise, wall-
normal and spanwise directions, respectively. $\Delta x^{+}$ and $\Delta
z^{+}$ denote the streamwise and spanwise grid resolutions in viscous units,
respectively. $\Delta y_{min}^{+}$ and $\Delta y_{max}^{+}$ denote the finest
and coarsest resolution in the wall-normal direction, respectively. $N_{F}$
and $Tu_{\tau}/h$ indicate the number of instantaneous flow fields and the
total eddy turnover time used to accumulate statistics, respectively.

###  2.2 Spectral stochastic estimation

According to the inner-outer interaction model (Marusic et al., 2010), the
large-scale motions would exert the footprints on the near-wall region, i.e.,
the superposition effects. Baars et al. (2016) demonstrated that this
component (denoted as $u_{L}^{{}^{\prime}+}(x^{+},y^{+},z^{+})$) can be
obtained by the spectral stochastic estimation of the streamwise velocity
fluctuation at the logarithmic region $y_{o}^{+}$, namely by

$u_{L}^{{}^{\prime}+}\left(x^{+},y^{+},z^{+}\right)=F_{x}^{-1}\left\\{H_{L}\left(\lambda_{x}^{+},y^{+}\right)F_{x}\left[u_{o}^{{}^{\prime}+}\left(x^{+},y_{o}^{+},z^{+}\right)\right]\right\\},$
(3)

where $u_{o}^{{}^{\prime}+}$ is the streamwise velocity fluctuation at
$y_{o}^{+}$ in the logarithmic region, and, $F_{x}$ and $F_{x}^{-1}$ denote
the FFT and the inverse FFT in the streamwise direction, respectively. $H_{L}$
is the transfer kernel, which evaluates the correlation between
$\hat{u^{\prime}}(y^{+})$ and $\hat{u_{o}^{\prime}}(y_{o}^{+})$ at a given
length scale $\lambda_{x}^{+}$, and can be calculated as

$H_{L}\left(\lambda_{x}^{+},y_{o}^{+}\right)=\frac{\left\langle\hat{u^{\prime}}\left(\lambda_{x}^{+},y^{+},z^{+}\right)\overline{\hat{u_{o}^{\prime}}}\left(\lambda_{x}^{+},y_{o}^{+},z^{+}\right)\right\rangle}{\left\langle\hat{u_{o}^{\prime}}\left(\lambda_{x}^{+},y_{o}^{+},z^{+}\right)\overline{\hat{u_{o}^{\prime}}}\left(\lambda_{x}^{+},y_{o}^{+},z^{+}\right)\right\rangle},$
(4)

where $\hat{u^{\prime}}$ is the Fourier coefficient of $u^{\prime}$, and
$\overline{\hat{u^{\prime}}}$ is the complex conjugate of $\hat{u^{\prime}}$.
$y^{+}$ is set as $y^{+}=0.3$, and the outer reference height $y_{o}^{+}$
varies from $100$ to the outer region $0.7h^{+}$ according to the wall-normal
grid distribution. Once $u_{L}^{{}^{\prime}+}$ is obtained, the superposition
component of $\tau_{x}^{{}^{\prime}+}$ can be calculated by definition (i.e.,
$\frac{\partial u_{L}^{{}^{\prime}+}}{\partial y^{+}}$ at the wall) and
denoted as $\tau_{x,L}^{{}^{\prime}+}(y_{o}^{+})$.

Analogously, to eliminate the effects from the wall-detached eddies with
random orientations, which contribute significantly to the streamwise velocity
fluctuations at $y_{o}^{+}$, we can also use the near-wall streamwise velocity
fluctuation in viscous layer $y^{+}$ to reconstruct the wall-coherent
streamwise velocity fluctuation in the logarithmic region $y_{o}^{+}$ by
spectral stochastic estimation (Adrian, 1979), i.e.,

$u_{W}^{{}^{\prime}+}\left(x^{+},y_{o}^{+},z^{+}\right)=F_{x}^{-1}\left\\{H_{W}\left(\lambda_{x}^{+},y^{+}\right)F_{x}\left[u^{{}^{\prime}+}\left(x^{+},y^{+},z^{+}\right)\right]\right\\},$
(5)

where $u_{W}^{{}^{\prime}+}$ is the wall-coherent component of
$u_{o}^{{}^{\prime}+}$. The wall-based transfer kernel $H_{W}$ can be
calculated as

$H_{W}\left(\lambda_{x}^{+},y_{o}^{+}\right)=\frac{\left\langle\hat{u_{o}^{\prime}}\left(\lambda_{x}^{+},y_{o}^{+},z^{+}\right)\overline{\hat{u^{\prime}}}\left(\lambda_{x}^{+},y^{+},z^{+}\right)\right\rangle}{\left\langle\hat{u^{\prime}}\left(\lambda_{x}^{+},y^{+},z^{+}\right)\overline{\hat{u^{\prime}}}\left(\lambda_{x}^{+},y^{+},z^{+}\right)\right\rangle}.$
(6)

Figure 2: ($a$) Variation of the statistic $\langle
u_{W}^{{}^{\prime}2+}\rangle$ as a function of $y_{o}/h$, and the full-channel
data $\langle u^{{}^{\prime}2+}\rangle$ are included for comparison; ($b$)
variations of the indicator functions $\Xi$ as functions of $y_{o}/h$. The red
line in ($a$) denotes the logarithmic decaying Eq. (7) with $C_{1}=0.54$.

Fig. 2($a$) shows the variation of $\langle u_{W}^{{}^{\prime}2+}\rangle$ as a
function of $y_{o}/h$ in the case Re2000. The full-channel data are included
for comparison. It can be seen that $\langle u_{W}^{{}^{\prime}2+}\rangle$
roughly follows the logarithmic decay for $0.09\leq y_{o}/h\leq 0.2$, i.e.,
the logarithmic region. To quantify the logarithmic decay systematically, we
define the indicator function $\Xi=y(\partial\langle
u^{{}^{\prime}2+}\rangle/\partial y)$, and display their variations in Fig.
2($b$). Comparing with the full-channel data, a comparatively well-defined
plateau is observed for $\langle u_{W}^{{}^{\prime}2+}\rangle$. The
logarithmic variance of $\langle u_{W}^{{}^{\prime}2+}\rangle$ shown in Fig. 2
is the consequence of the additive attached eddies (Townsend, 1976), and can
be expressed as

$\langle u_{W}^{{}^{\prime}2+}\rangle=C_{2}-C_{1}\ln(y_{o}/h),$ (7)

where $C_{2}$ and $C_{1}$ are two constants, and $C_{1}$ is approximately
equal to 0.54. Actually, the magnitude of the slope of the logarithmic
decaying is affected by the Reynolds number, the configuration of the wall
turbulence, the methodology for isolating the signals carried by the attached
eddies, and the effects of the VLSMs. The indicator function $\Xi$ of the
fully-channel data shown in Fig. 2($b$) suggests that the logarithmic region
of case Re2000 is not full-developed, as the slope value of the logarithmic
decaying is smaller than the Townsend-Perry constant $1.26$ reported at high-
Reynolds number experiments (Marusic et al., 2013), and close to the magnitude
of $C_{1}$ observed here. Furthermore, Baars & Marusic (2020b) reported that
$C_{1}=0.98$ in turbulent boundary layers by analyzing the streamwise velocity
fluctuations carried by the attached eddies in the logarithmic region, while
Hu et al. (2020) and Hwang et al. (2020) showed that $C_{1}=0.8$ and 0.37 in
channel flows, respectively. Hu et al. (2020) adopted a scale-based filter to
extract the streamwise velocity fluctuations associated with the attached
eddies in the logarithmic region and did not take their imperfect coherence
with the near-wall flow at each scale into account. The wall-based transfer
kernel $H_{W}$ in Eq. (6) employed here can achieve this. Hwang et al. (2020)
utilized the three-dimensional clustering method to identify the wall-attached
structures in a channel flow. The differences among these decomposition
methodologies may be the reason why the magnitude of $C_{1}$ for turbulent
channel flows reported by Hu et al. (2020) and Hwang et al. (2020) is not
identical to that of the present study. Besides, it is noted that the effects
of VLSMs are also retained in $\langle u_{W}^{{}^{\prime}2+}\rangle$, and
their impacts on the logarithmic decaying are non-negligible. By the way, the
methodology introduced in section 2.3 to estimate the SIAs of attached eddies
at a single scale can effectively diminish the effects originating from the
VLSMs (see Fig. 4). In summary, these observations demonstrate that
$u_{W}^{{}^{\prime}}$ can be approximately considered as the streamwise
velocity fluctuations carried by the multi-scale wall-attached eddies. We will
focus on the statistics in the logarithmic region in the following sections.

###  2.3 Methodology to isolate targeted eddies

Apparently, the SIAs of attached eddies at a single scale ($\alpha_{s}$) can
not be pursued by Eq. (1)-(2). It is worth noting that in Eq. (1)-(2), the
input parameter and signals are $y_{o}$, $\tau_{x}^{\prime}$ and
$u^{\prime}(y_{o})$. Thus, to obtain an accurate $\alpha_{s}$, $y_{o}$ should
be set reasonably, and $\tau_{x}^{\prime}$ and $u^{\prime}(y_{o})$ should also
be properly processed, to characterize the properties of the attached eddies
at the targeted scale. Our new approach is based on this understanding.

According to the hierarchical distribution of the multi-scale attached eddies
in high-Reynolds number wall turbulence (see Fig. 1($b$), also Fig. 14 of
Perry & Chong (1982)), $\tau_{x,L}^{{}^{\prime}+}(y_{o}^{+})$ represents the
superposition contributed from the wall-attached motions with their height
larger than $y_{o}^{+}$. Thus, the difference value
$\Delta\tau_{x,L}^{{}^{\prime}+}(y_{o}^{+})=\tau_{x,L}^{{}^{\prime}+}(y_{o}^{+})-\tau_{x,L}^{{}^{\prime}+}(y_{o}^{+}+\Delta
y^{+})$ can be interpreted as the superposition contribution generated by the
wall-attached eddies with their wall-normal heights between $y_{o}^{+}$ and
$y_{o}^{+}+\Delta y^{+}$. Here, $y_{o}^{+}+\Delta y^{+}$ is the location of
the adjacent wall-normal grid cell of that at $y_{o}^{+}$, as $\Delta y^{+}$
is the local grid spacing along the wall-normal direction, in viscous units,
and determined by the simulation setups. The similar numerical framework has
been verified by our previous study (Cheng & Fu, 2022). Correspondingly, the
difference value $\Delta
u_{W}^{{}^{\prime}+}(y_{o}^{+})=u_{W}^{{}^{\prime}+}(y_{o}^{+})-u_{W}^{{}^{\prime}+}(y_{o}^{+}+\Delta
y^{+})$ is the streamwise velocity fluctuation carried by attached eddies
populating the region between $y_{o}^{+}$ and $y_{o}^{+}+\Delta y^{+}$. In
this way, the SIAs of these eddies can be assessed by

$\alpha_{s}(y_{m})=\arctan(\frac{y_{m}}{\Delta x_{p}}),$ (8)

where $y_{m}=\frac{y_{o}+(y_{o}+\Delta y)}{2}$, and $\Delta x_{p}$ is the
streamwise delay associated with the peak of the cross correlation

$R_{LW}(\Delta
x)=\frac{\langle\Delta\tau_{x,L}^{{}^{\prime}+}(x,y_{o}^{+})\Delta
u_{W}^{{}^{\prime}+}(x+\Delta
x,y_{o}^{+})\rangle}{\sqrt{\left\langle\Delta\tau_{x,L}^{{}^{\prime}+2}\right\rangle\left\langle\Delta
u_{W}^{{}^{\prime}+2}\right\rangle}}.$ (9)

Figure 3: Variations of $\Delta y^{+}$ as functions of $y_{o}^{+}$ in the
logarithmic region for all cases.

As the statistical characteristics of an individual attached eddy being self-
similar with its wall-normal height as per the attached-eddy hypothesis
(Townsend, 1976), $y_{m}$ is just the characteristic scale of the wall-
attached motions within $y_{o}^{+}$ and $y_{o}^{+}+\Delta y^{+}$. Fig. 3 shows
the variations of $\Delta y^{+}$ as functions of $y_{o}^{+}$ in the
logarithmic region for all cases. It can be seen that the maximum values of
$\Delta y^{+}$ are less than 7 in the case Re4200. In this regard, treating
$y_{m}$ as the mean height of the attached eddies populating the region
between $y_{o}$ and $y_{o}+\Delta y$ is reasonable, as the zone between
$y_{o}$ and $y_{o}+\Delta y$ is narrow compared to the spanning of the
logarithmic region. The new procedure isolates the attached eddies at a given
scale from the rest of the turbulence. The cross correlation, i.e., Eq. (9),
gets rid of the influences originated from other scales, and preserves the
phase information of the wall-attached motions with wall-normal height
$y_{m}$.

At last, the critical assumptions of the present approach and its realization
merit a discussion. Our methodology is based on the hierarchical distribution
of the attached eddies, and the hypothesis that the characteristic velocity
scales carried by the attached eddies with different wall-normal heights are
identical with their scale interactions omitted. That’s to say, the attached
eddies in each hierarchy contribute equally to the streamwise wall-shear
fluctuations on the wall surface and the streamwise turbulence intensity in
the lower bound of the logarithmic region. Only in this way, both
$\Delta\tau_{x,L}^{{}^{\prime}+}$ and $\Delta u_{W}^{{}^{\prime}+}$
approximately reflect the characteristics of the attached eddies at $y_{m}$.
In fact, these assumptions are also the key elements when developing the
attached-eddy model (Townsend, 1976; Perry & Chong, 1982; Woodcock & Marusic,
2015; Yang et al., 2016; Mouri, 2017; Yang & Lozano-Durán, 2017), and some of
them may be valid only in high-Reynolds number wall turbulence. For example,
the hierarchical distribution of the multi-scale attached eddies is prominent
at high-Reynolds number turbulence (De Silva et al., 2016; Marusic & Monty,
2019; Cheng et al., 2019). However, when the DNS data listed in Table 1 are
utilized to study the characteristics of the attached eddies, the finite
Reynolds-number effects and the intricate scale interactions would take
effects inevitably. Besides, the VLSMs, which can not be depicted by the
attached-eddy model, would also impose non-trivial impacts (Perry & Marusic,
1995; Hwang et al., 2020; Baars & Marusic, 2020a). Accordingly, the
subtraction between $u_{W}^{{}^{\prime}+}(y_{o}^{+})$ and
$u_{W}^{{}^{\prime}+}(y_{o}^{+}+\Delta y^{+})$ can not achieve a sharp cut-off
at the targeted scale in the spectral space, and hereby the spectrum of
$\Delta u_{W}^{{}^{\prime}+}(y_{o}^{+})$ would be comparatively small but not
negligible at the smaller and larger scales of the targeted one. The
finiteness of $\Delta y^{+}$ is another factor, which is worth attention in
some scenarios. Due to the limitations of numerical simulation, $\Delta y^{+}$
is a finitely small quantity. When assessing the SIA of the attached eddies at
a given wall-normal height, treating $y_{m}^{+}$ as their characteristic
scales (therefore, neglecting the effects of the narrowband between $y^{+}$
and $y^{+}+\Delta y^{+}$) is acceptable, because $\Delta y^{+}$ is rather
small compared to the spanning of the whole logarithmic region. The linear
growth of the typical length scales of $\Delta\tau_{x,L}^{{}^{\prime}+}$ and
$\Delta u_{W}^{{}^{\prime}+}$ shown in Fig. 6($b$) can verify this validity.
On the other hand, when the spectral characteristics of $\Delta
u_{W}^{{}^{\prime}+}$ are considered, $\Delta u_{W}^{{}^{\prime}+}$ should be
interpreted as the additive outcomes of the attached eddies with their wall-
normal heights within $y^{+}$ and $y^{+}+\Delta y^{+}$, strictly speaking.
Under this circumstance, the spectral energy distribution that corresponds to
the self-similar attached eddies within this range should be observed to peak
around the dominate wavelength and vary continuously and locally. The results
shown in Fig. 5 confirm our proposition. Details will be discussed in the
following section.

##  3 Results

Before investigating the SIAs of attached eddies, it is important to study the
characteristic scales of $\Delta\tau_{x,L}^{{}^{\prime}}$ and $\Delta
u_{W}^{{}^{\prime}}$ first. Figs. 4($a$) and 4($b$) show their streamwise
premultiplied spectra at $y_{o}=0.1h$ and $y_{o}=0.2h$ for Re2000,
respectively. The spectra of $\tau_{x}^{{}^{\prime}}$ and $u^{\prime}$ of the
full-channel data are also included for comparison. Each spectrum is
normalized with its maximum value. It can be seen that the spectra of
$\Delta\tau_{x,L}^{{}^{\prime}}$ and $\Delta u_{W}^{{}^{\prime}}$ are roughly
coincident, and peak at $\lambda_{x}=2.1h$ for $y=0.1h$, and
$\lambda_{x}=4.2h$ for $y=0.2h$, respectively. By contrast, the spectra of
$\tau_{x}^{{}^{\prime}}$ and $u^{\prime}$ do not share similar spectral
characteristics. It is noted that $\Delta u_{W}^{{}^{\prime}2}$ and
$\Delta\tau_{x,L}^{{}^{\prime}2}$ only account for very little energy of the
full-channel signals at the same wall-normal positions. For example, $\Delta
u_{W}^{{}^{\prime}2}$ at $y_{o}=0.1h$ and $y_{o}=0.2h$ occupies 0.0034$\%$ and
0.002$\%$ of $u^{{}^{\prime}2}$ at the corresponding positions, respectively,
whereas $\Delta\tau_{x,L}^{{}^{\prime}2}$ for $y_{o}=0.1h$ and $y_{o}=0.2h$
occupies 0.012$\%$ and 0.0045$\%$ of $\tau_{x}^{{}^{\prime}2}$, respectively.
Moreover, comparing with the spectra of the full-channel data, the spectra of
$\Delta u_{W}^{{}^{\prime}}$ decay rapidly when $\lambda_{x}\geq 4h$ (see
Figs. 4), which indicates that the effects of VLSMs on $\Delta
u_{W}^{{}^{\prime}}$ are rather limited.

Fig. 5 shows the streamwise premultiplied spectra of $\Delta
u_{W}^{{}^{\prime}}$ around $y_{o}=0.05h$ and $y_{o}=0.1h$. Each spectrum is
normalized by the energy of $\Delta u_{W}^{{}^{\prime}}$ at a given $y_{m}$.
Clear plateau regions can be observed around the spectral peaks. For
$y_{o}=0.05h$, the region is between $18\leq\lambda_{x}/y_{m}\leq 30$, and for
$y_{o}=0.1h$, it is between $17\leq\lambda_{x}/y_{m}\leq 31$ , which
corresponds to the $k_{x}^{-1}$ region in the spectrum predicated by the
attached-eddy model, and can be considered as the spectral signatures of the
attached eddies (Perry & Chong, 1982; Perry et al., 1986; Hwang et al., 2020;
Deshpande et al., 2021). Besides, the spectra shown here resemble the spectrum
of the type A eddies hypothesized by Marusic & Perry (1995), i.e., the energy
fraction captured by the attached-eddy model. These observations support the
proposition that the $\Delta u_{W}^{{}^{\prime}}$ signals are the streamwise
velocity fluctuations carried by the self-similar attached eddies
predominantly. Moreover, they also indicate that the streamwise length scales
of the dominant eddies increase with $y_{o}$, as the self-similar range is not
altered significantly with increasing $y_{o}$.

Figure 4: Streamwise premultiplied spectra of $\Delta\tau_{x,L}^{{}^{\prime}}$
, $\Delta u_{W}^{{}^{\prime}}$, $\tau_{x}^{{}^{\prime}}$ and $u^{\prime}$ for
($a$) $y_{o}=0.1h$ and ($b$) $y_{o}=0.2h$ in the case Re2000. Each spectrum is
normalized with its maximum value. The vertical dashed lines in ($a$) and
($b$) are plotted to highlight the corresponding $\lambda_{x}/h$ of the
maximum values of the premultiplied spectra of
$\Delta\tau_{x,L}^{{}^{\prime}}$ and $\Delta u_{W}^{{}^{\prime}}$.

To further investigate the scale characteristics of
$\Delta\tau_{x,L}^{{}^{\prime}}$ and $\Delta u_{W}^{{}^{\prime}}$, the
autocorrelation function of $\Delta u_{W,p}^{{}^{\prime}+}$ (the signals that
are extracted from the spectral peaks shown in Fig. 5, i.e., filtered $\Delta
u_{W}^{{}^{\prime}}$ with wavelength larger than $17y_{m}$, but smaller than
$31y_{m}$) is considered, which takes the form of

$R_{\Delta u_{W,p}^{{}^{\prime}}\Delta u_{W,p}^{{}^{\prime}}}(\Delta
x,y_{o})=\frac{\langle\Delta u_{W,p}^{\prime}\left(x,y_{o},z\right)\Delta
u_{W,p}^{\prime}\left(x+\Delta x,y_{o},z\right)\rangle}{\langle\Delta
u_{W,p}^{\prime 2}\left(x,y_{o},z\right)\rangle},$ (10)

and the counterpart of $\Delta\tau_{x,L}^{{}^{\prime}+}$ can also be defined
similarly. Fig. 6($a$) shows the variations of $R_{\Delta
u_{W,p}^{{}^{\prime}}\Delta u_{W,p}^{{}^{\prime}}}$ as functions of $\Delta
x/h$ for two selected $y_{o}$. The larger $y_{o}$, the broader the $R_{\Delta
u_{W,p}^{{}^{\prime}}\Delta u_{W,p}^{{}^{\prime}}}$. As a measure of the
typical length scale, we employ $\Delta s/h$, which is the streamwise delay
corresponding to $R_{\Delta u_{W,p}^{{}^{\prime}}\Delta
u_{W,p}^{{}^{\prime}}}=0.05$ or
$R_{\Delta\tau_{x,L,p}^{{}^{\prime}}\Delta\tau_{x,L,p}^{{}^{\prime}}}=0.05$
(here, 0.05 is an empirical small positive threshold). Fig. 6($b$) shows the
variations of $2\Delta s/h$ as functions of $y_{m}/h$ for
$\Delta\tau_{x,L,p}^{{}^{\prime}+}$ and $\Delta u_{W,p}^{{}^{\prime}+}$. For
both $\Delta\tau_{x,L,p}^{{}^{\prime}+}$ and $\Delta u_{W,p}^{{}^{\prime}+}$,
$2\Delta s/h$ increases linearly with $y_{m}/h$ throughout most of the
logarithmic region. This observation is consistent with the attached-eddy
hypothesis, which states that the length scales of the attached eddies grow
linearly with their wall-normal heights (Hwang, 2015; Marusic & Monty, 2019).
Moreover, both the streamwise length scales of
$\Delta\tau_{x,L,p}^{{}^{\prime}+}$ and $\Delta u_{W,p}^{{}^{\prime}+}$ follow
$2\Delta s=10.8y_{m}$ (consider the symmetry of the autocorrelation function
with respect to $\Delta x=0$, $2\Delta s$ truly represents the streamwise
length scale of the signals). This scale characteristic agrees well with some
previous studies. For example, Baars et al. (2017) showed that the
streamwise/wall-normal aspect ratio of the wall-attached eddy structure is
$\lambda_{x}/y=14$ in turbulent boundary layers, which is close to the result
here. Hwang et al. (2020) reported that the spectra of the self-similar wall-
attached structures agree with the attached-eddy hypothesis at
$\lambda_{x}=12y$, which is consistent with the estimation of the present
study. All these observations indicate that $\Delta\tau_{x,L}^{{}^{\prime}+}$
and $\Delta u_{W}^{{}^{\prime}+}$ are representative of the attached eddies at
a certain wall-normal height, though the minor influences of VLSMs still
exist, and treating $y_{m}^{+}$ as their characteristic scales is reasonable.

Figure 5: Premultiplied one-dimensional streamwise spectra of $\Delta
u_{W}^{{}^{\prime}}$ around ($a$) $y_{o}=0.05h$; ($b$) $y_{o}=0.1h$ in Re2000.
The horizontal dashed lines represent the plateaus or peaks of the spectra.
The vertical lines are plotted to highlight the self-similar regions of each
spectrum. Figure 6: ($a$) Variations of $R_{\Delta u_{W,p}^{{}^{\prime}}\Delta
u_{W,p}^{{}^{\prime}}}$ as functions of $\Delta x/h$ for two selected $y_{o}$;
($b$) variations of $\Delta s/h$ as functions of $y_{m}^{+}$ for
$\Delta\tau_{x,L,p}^{{}^{\prime}+}$ and $\Delta u_{W,p}^{{}^{\prime}+}$. The
line in ($b$) denotes the linear variation $2\Delta s=10.8y_{m}$.

In summary, all the observations mentioned above indicate that
$\Delta\tau_{x,L}^{{}^{\prime}}$ and $\Delta u_{W}^{{}^{\prime}}$ are the
outcomes of the energy-containing motions with the wall-normal heights
approximately equal to $y_{m}$, and the cross correlation, i.e., Eq. (9),
truly reflects the phase difference between the streamwise velocity
fluctuations carried by these motions and their footprints in the near-wall
region. Other wall-normal positions and DNS cases yield similar results and
are not shown here for brevity.

Figure 7: ($a$) Variations of $R_{LW}$, i.e., the cross correlation between
$\Delta\tau_{x,L}^{{}^{\prime}+}(y_{o}^{+})$ and $\Delta
u_{W}^{{}^{\prime}+}(y_{o}^{+})$, as functions of $\Delta x$ for some selected
$y_{o}$ in the case Re2000; ($b$) variations of the normalized $R_{LW}$ as
functions of $\Delta x/y_{m}$ for some selected $y_{o}$ in the case Re2000.
The $R_{LW}$ profiles are normalized with their maximum values in ($b$). The
vertical dashed lines in ($a$) are plotted to highlight the maximum values of
$R_{LW}$ and their corresponding $\Delta_{x}/h$.

Fig. 7($a$) shows the variations of $R_{LW}$ as functions of the streamwise
delay for some selected wall-normal positions in the case Re2000. Since the
streamwise length scales of the energy-containing motions are increased with
their normal heights (see Fig. 4), $R_{LW}$ becomes wider about the peak with
increasing $y_{o}$. $\Delta x_{p}$ can be identified obviously from the cross-
correlation profiles, and the SIAs of the attached eddies at a given wall-
normal height can be calculated according to Eq. (8). Fig. 7($b$) plots the
variations of the normalized $R_{LW}$ as functions of $\Delta x/y_{m}$ for
some selected $y_{o}$ in the case Re2000. The $R_{LW}$ distributions are
normalized with their maximum values $R_{LW,max}$. It can be seen that the
profiles of $R_{LW}/R_{LW,max}$ for different wall-normal heights coincide
well with each other, which indicates the self-similar characteristics of the
energy-containing motions in the logarithmic region. We have checked that the
correlations calculated from the raw data, i.e.,
$R_{\tau_{x}^{\prime}u^{\prime}}$ in Eq. (1) can not coincide if normalized in
this manner. Again, it demonstrates that the new methodology is capable of
capturing the main properties of the attached eddies.

Fig. 8 plots the variations of $\alpha_{s}$ as functions of $y_{m}^{+}$ for
all cases. $\alpha_{s}$ increases approximately from $27^{\circ}$ for Re550,
to $40^{\circ}$ for Re4200. For a given case, $\alpha_{s}$ changes little
spanning the logarithmic region except for the upper part of logarithmic
region in Re4200. Deshpande et al. (2019) isolated the large wall-attached
structures in a DNS of turbulent boundary layer at $Re_{\tau}\approx 2000$,
and found the corresponding SIAs to be $32^{\circ}$ (see Fig. 4($a$) of their
paper). Their observation is consistent with the results of the present study.
However, Deshpande et al. (2019) only calculated the SIAs of the largest wall-
attached motions in the logarithmic region due to the limitation of the
methodology adopted in their study, whereas we make a thorough investigation
on the SIAs of attached eddies with any wall-normal heights in the logarithmic
region. Moreover, Deshpande et al. (2019) reported that the SIAs of the large
wall-attached motions identified in a wind-tunnel boundary layer with
$Re_{\tau}=14000$ are approximately $50^{\circ}$. They ascribed the result
difference between DNS and experiment to the limited streamwise scale range
owing to the DNS domain size selected for analysis. Our results reveal that
the Reynolds number effects play a non-negligible role in the formation of
SIAs of attached eddies. To the authors’ knowledge, this is the first time
that the Reynolds-number dependence of SIAs of the wall-attached motions at a
given length scale has been clearly shown. At last, it should be noted that
$\alpha_{s}$ of Re4200 decreases rapidly for $y_{m}^{+}>500$ (not shown here).
This diversity is due to the small computational domain size along the
streamwise direction in this database. Thus, in the discussion below, the
statistics of $\alpha_{s}$ in the range of $y_{m}^{+}>500$ in Re4200 will not
be taken into account. The sensitivity of the presented results to the number
of instantaneous flow fields employed for accumulating statistics is examined
in the Appendix A.

Figure 8: $\alpha_{s}$ as functions of $y_{m}^{+}$ for all cases, and the red
dashed lines denote the mean $\alpha_{s}$ across the logarithmic region of
each case. Figure 9: ($a$) Variations of the mean $\alpha_{s}$
($\alpha_{s,m}$) statistic in the range of logarithmic region as a function of
the friction Reynolds number, and the experimental results of turbulent
boundary layers (Deshpande et al., 2019) are also included for comparison;
($b$) $\alpha_{m}$ and $\alpha_{SSE,m}$ as a function of $y_{o}^{+}$ for
Re2000. The solid black line in ($a$) denotes the theoretical prediction angle
$45^{\circ}$, and the dashed lines in ($a$) indicates the asymptotic behavior
of $\alpha_{s,m}$.

Fig. 9($a$) shows the mean $\alpha_{s}$ ($\alpha_{s,m}$) distribution in the
range of logarithmic region as a function of the friction Reynolds number. It
can be seen that the SIA may reach the theoretical prediction angle
$45^{\circ}$ (Perry et al., 1992) when $Re_{\tau}\sim O(10^{4})$. The results
of DNS of a turbulent boundary layer and wind-tunnel experiment of Deshpande
et al. (2019) are roughly agreed with the tendency. The minor differences may
result from the distinct configurations of the wall-bounded turbulence.

##  4 Disscussion

###  4.1 Effects of near-wall and detached motions

To clarify the effects of near-wall and detached motions on the SIA
assessment, we calculate the mean SIA based on the predictive signals, i.e.,

$\alpha_{SSE,m}=\arctan(\frac{y_{o}}{\Delta x_{p}}),$ (11)

where $\Delta x_{p}$ is the streamwise delay associated with the peak of the
following cross correlation

$R_{\tau_{x,L}^{\prime}u_{W}^{\prime}}(\Delta
x)=\frac{\langle\tau_{x,L}^{{}^{\prime}}(x)u_{W}^{\prime}(x+\Delta
x,y_{o})\rangle}{\sqrt{\left\langle\tau_{x,L}^{{}^{\prime}2}\right\rangle\left\langle
u_{W}^{{}^{\prime}2}\right\rangle}}.$ (12)

Fig. 9($b$) shows the variations of $\alpha_{SSE,m}$ as a function of
$y_{o}^{+}$ for Re2000, and the statistics of $\alpha_{m}$ are also included
for comparison. We can see that $\alpha_{SSE,m}$ distribution is very closed
to that of $\alpha_{m}$. It highlights the fact that the phase information
embedded in the raw signals $u^{\prime}(y_{o}^{+})$ and $\tau_{x}^{\prime}$ is
preserved by SSE. It also suggests that the near-wall and wall-detached
motions, which can not be captured by SSE, have a negligible impact on the
magnitudes of SIA.

###  4.2 $\alpha_{s}$ versus $\alpha_{m}$

Reviewing the approach to obtain the $\alpha_{m}$ (i.e., Eq. (1)-(2)), the
proposition that $\alpha_{m}$ being the mean SIA of attached eddies manifests
in three aspects: ($1$) the generation of $\tau_{x}^{\prime}$ is not only the
outcome of the near-wall motions, but also the footprints of all the wall-
attached eddies (Cho et al., 2018; Cheng et al., 2020a); ($2$) $u^{\prime}$ in
logarithmic region results from a sum of random contributions from the wall-
attached eddies with distinct characteristic length scales (Yang et al.,
2016), and a portion of contributions from the wall-detached eddies (Baars &
Marusic, 2020b); ($3$) $y_{o}$ is a wall-normal position located in the
logarithmic region and chosen arbitrarily. As mentioned above, an array of
wall-attached eddies with distinct wall-normal heights can simultaneously
convect past this reference position.

Here, an additive SIA is calculated to highlight the relationship between
$\alpha_{s}$ and $\alpha_{m}$, namely,

$\alpha_{add}=\arctan(\frac{y_{s}}{\Delta x_{p}}),$ (13)

where $y_{s}^{+}=100$ is the lower boundary of logarithmic region, and $\Delta
x_{p}$ is the streamwise delay associated with the peak of the following cross
correlation, i.e.,

$R_{add}(\Delta
x)=\frac{\langle(\tau_{x,L}^{{}^{\prime}+}(x,y_{s}^{+})-\tau_{x,L}^{{}^{\prime}+}(x,y_{o}^{+}))(u_{W}^{{}^{\prime}+}(x+\Delta
x,y_{s}^{+})-u_{W}^{{}^{\prime}+}(x+\Delta
x,y_{o}^{+}))\rangle}{\sqrt{\left\langle(\tau_{x,L}^{{}^{\prime}+}(x,y_{s}^{+})-\tau_{x,L}^{{}^{\prime}+}(x,y_{o}^{+}))^{2}\right\rangle\left\langle(u_{W}^{{}^{\prime}+}(x,y_{s}^{+})-u_{W}^{{}^{\prime}+}(x,y_{o}^{+}))^{2}\right\rangle}},$
(14)

where the reference position $y_{o}^{+}$ varies from $y_{s}^{+}+\Delta y^{+}$
(equals to 104) to $0.7h^{+}$. Fig. 10($a$) shows the variations of
$\alpha_{add}$ as a function of $y_{o}^{+}$ for Re2000. It can be seen that
$\alpha_{add}$ decreases from $37.8^{\circ}$ to $14^{\circ}$ as $y_{o}^{+}$
increases, which corresponds to $\alpha_{s}(y_{m}^{+}=102)$ and
$\alpha_{m}(y_{o}^{+}=100)$, respectively. In other words, $\alpha_{add}$
converges from the SIAs of attached eddies with wall-normal height
approximately $100$ in viscous units to the mean SIA at $y_{o}^{+}=100$. This
observation can be explained through the prism of the hierarchical attached
eddies in high-Reynolds number wall turbulence. The increase of $y_{o}^{+}$
indicates that
$\tau_{x,L}^{{}^{\prime}+}(y_{s}^{+})-\tau_{x,L}^{{}^{\prime}+}(y_{o}^{+})$
and $u_{W}^{{}^{\prime}+}(y_{s}^{+})-u_{W}^{{}^{\prime}+}(y_{o}^{+})$ are
contributed by more and more wall-attached eddies with their normal heights
larger than $y_{s}^{+}$, and gradually become equal to
$\tau_{x,L}^{{}^{\prime}+}(y_{s}^{+})$ and $u_{W}^{{}^{\prime}+}(y_{s}^{+})$,
respectively, when $y_{o}^{+}$ approaches $h^{+}$. Thus, $R_{add}$ would also
gradually converge to $R_{\tau_{x,L}^{\prime}u_{W}^{\prime}}$ in Eq. (12), and
$\alpha_{add}$ converges to $\alpha_{m}$ and $\alpha_{SSE,m}$ concurrently.

Figure 10: ($a$) Variations of the additive SIA $\alpha_{add}$ as a function
of $y_{o}^{+}$ for Re2000; ($b$) the mean SIAs $\alpha_{m}$ as functions of
$y_{o}^{+}$ for all cases.

Additionally, this study helps to understand the variation tendency of
$\alpha_{m}$. Fig. 10($b$) plots the variations of $\alpha_{m}$ for all cases.
It is clearly observed that $\alpha_{m}$ increases continuously with
$y_{o}^{+}$. Taking Re2000 as an example, $\alpha_{m}$ increases from
$14^{\circ}$ for $y_{s}^{+}$ to $15.3^{\circ}$ for $y_{e}^{+}$. Increasing
$y_{o}^{+}$ implies that fewer and fewer wall-attached eddies contribute to
$u^{\prime}$. In this way, $\alpha_{m}$ would converge to $\alpha_{s}$ as
$y_{o}^{+}$ increases, albeit more slowly.

###  4.3 Scale-dependent inclination angles of wall-attached eddies

An alternative approach for calculating the scale-dependent inclination angle
(SDIA) has been reported by Baars et al. (2016). The following are the primary
processes and outcomes. The scale-specific phase between $u^{\prime}$ at
$y^{+}$ and $y_{o}^{+}$ can be estimated as

$\Phi\left(\lambda_{x}\right)=\arctan\left\\{\frac{\operatorname{Im}\left[\phi_{u_{o}^{\prime}u^{\prime}}\left(\lambda_{x},y^{+},y_{o}^{+}\right)\right]}{\operatorname{Re}\left[\phi_{u_{o}^{\prime}u^{\prime}}\left(\lambda_{x},y^{+},y_{o}^{+}\right)\right]}\right\\},$
(15)

where Im($\cdot$) and Re($\cdot$) denote the imaginary and real parts of
$\phi_{u_{o}^{\prime}u^{\prime}}$, namely, the numerator of Eq. (4). The
scale-dependent streamwise shift can be calculated as

$l(\lambda_{x})=\frac{\Phi\left(\lambda_{x}\right)\lambda_{x}}{2\pi}.$ (16)

Accordingly, the SDIA can be estimated as

$\alpha_{sd}(\lambda_{x})=\arctan(\frac{y_{o}-y}{l(\lambda_{x})}).$ (17)

A positive $\alpha_{sd}$ value corresponds to a spatially forward-leaning
structure.

Figure 11: Variations of the scale-dependent inclination angles for three
selected wall-normal positions in the case Re2000. The vertical line denotes
$\lambda_{x}/y_{o}=14$.

Fig. 11 shows the SDIAs as functions of $\lambda_{x}/y_{o}$ for three selected
wall-normal positions in the case Re2000. For $\lambda_{x}/y_{o}>18$, the
SDIAs of the large-scale motions are shown to be approximately equal to
$14^{\circ}$ (in fact, this is not the real SIA of the large-scale wall-
attached structures, according to the study of Deshpande et al. (2019)).
However, for the smaller length scales, the SDIAs tend to be negative and vary
rapidly with $\lambda_{x}/y_{o}$. This is the range of self-similar structures
reported by previous studies, especially those with $\lambda_{x}/y_{o}=14$
(Baars et al., 2017; Baidya et al., 2019). Similar results have also been
reported by Baars et al. (2016) (see Fig. 5 of their paper). It indicates that
the phase spectrum shown in Fig. 11 cannot be interpreted with any physical
relevance at these scales, as the scale-specific phases of them are random
indeed. The contamination from the detached eddies with random orientations
could be the source of this problem. This is the main purpose of the present
study, i.e., to eliminate the corruption caused by the wall-detached motions
and appropriately measure the SIAs of the wall-attached eddies at a certain
wall-normal height.

##  5 Concluding remarks

In the present study, we develop a methodology to assess the streamwise
inclination angles of the wall-attached eddies at a given wall-normal height
in turbulent channel flows, by coupling the spectral stochastic estimation
with the attached-eddy hypothesis. Our results show, for the first time, that
the SIAs of the attached eddies are Reynolds-number dependent in low and
medium Reynolds numbers and tend to be consistent with the theoretical
prediction (i.e., $\alpha_{s}=45^{\circ}$) as Reynolds number increased. We
further reveal that the mean SIA reported by vast previous studies are the
outcomes of the additive effect contributed by multi-scale attached eddies.

The attached-eddy model has been the guidance for the reconstruction of the
velocity field in wall turbulence (Perry & Marusic, 1995; Chandran et al.,
2017; Baidya et al., 2017). Hierarchical vortex packets which consist of
$\Lambda-$vortexes with $\alpha_{s}=45^{\circ}$ are distributed on the wall
surface to mimic the attached eddies. The present results suggest that a lower
SIA of representative structures might be helpful for a more accurate
reconstruction when the Reynolds number is not high enough. Moreover, within
the state-of-the-art wall-modelled large-eddy simulation (WMLES) framework,
one may estimate the instantaneous $\tau_{x}$ based on the velocities carried
by the log-region eddies (Fu et al., 2021, 2022). The Reynolds-number
dependence of SIAs of these eddies should be accounted for by an advanced
model in this sense.

## Acknowledgments

L.F. acknowledges the fund from CORE as a joint research center for ocean
research between QNLM and HKUST, and the fund from Guangdong Basic and Applied
Basic Research Foundation (No. 2022A1515011779). We would like to thank
Professor Jiménez for making the DNS data available. We also express our
gratitude to the reviewers of this paper for their kind and constructive
comments.

## Declaration of interests

The authors report no conflict of interest.

## Appendix A. Statistic sensitivity to $N_{F}$

Figure 12: $\alpha_{s}$ as functions of $y_{m}^{+}$ for the cases Re2000 and
Re4200 with different $N_{F}$. The dashed lines denote the mean value of
$\alpha_{s}$ in the logarithmic region.

The influences of the number of instantaneous flow fields for accumulating
statistics are examined. Fig. 12 shows the effect of $N_{F}$ on the statistic
$\alpha_{s}$ for the cases Re2000 and Re4200. Alteration of the statistical
samples mainly affects the relative standard deviations (RSD) of the results.
To be specific, when $N_{F}$ increases from $48$ to $94$, RSD decreases from
$3.9\%$ to $3.3\%$ for Re2000; but for Re4200, RSD decreases from $6.5\%$ to
$3.7\%$ when $N_{F}$ increases from $20$ to $40$. Given the fact that the case
Re4200 has limited domain size, raising $N_{F}$ can effectively reduce the
wiggles in the outputs. Nevertheless, the mean value of $\alpha_{s}$ in the
logarithmic region seems to be insensitive to $N_{F}$.

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