[1]This work was supported by the Australian Research Council.

[orcid=0000-0003-2777-2919] [1]

[cor1]Corresponding author

# Data-driven enhancement of coherent structure-based models for predicting
instantaneous wall turbulence

Rahul Deshpande<EMAIL_ADDRESS>Department of Mechanical Engineering,
University of Melbourne, Parkville, Victoria 3010, Australia  School of
Mechanical and Manufacturing Engineering, University of New South Wales,
Sydney, NSW 2052, Australia  Combustion Research Facility, Sandia National
Laboratories, Livermore, California 94550, USA  Charitha M. de Silva
Myoungkyu Lee  Jason P. Monty  Ivan Marusic

###### Abstract

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

###### keywords:

boundary layer structure turbulence modelling wall-bounded turbulence

##  1 Introduction

Owing to its highly chaotic and random nature, an accurate prediction of the
instantaneous velocity in a turbulent flow remains the most challenging demand
from any turbulence model. Despite its inherent complexities, turbulence flow
modelling has remained an active area of research for over half a century
(Brunton et al., 2020), given the numerous incentives on offer. In case of
wall-bounded flows, for example, the ability to predict/replicate the
instantaneous flow phenomena can greatly facilitate the design of active flow
control techniques (Choi et al., 1994), and also aid in enhancing future high-
fidelity numerical simulations, by providing realistic inflow boundary
conditions and improving computational efficiency (Subbareddy et al., 2006;
Wu, 2017). One of the popular approaches of modelling wall-turbulence, amongst
many, has been by focusing on the recurring and statistically significant
‘coherent’ motions (Robinson, 1991; Choi et al., 1994) omnipresent in these
flows. These motions play a crucial role in both the kinematics as well as
dynamics of wall-bounded flows (Robinson, 1991; Jiménez, 2018; Lee and Moser,
2019) and have been directly associated with the behaviour of several flow
statistics – both averaged (Ganapathisubramani et al., 2005; Hutchins and
Marusic, 2007; Heisel et al., 2020) and instantaneous (Adrian et al., 2000; de
Silva et al., 2016a, 2018). Hence, several studies proposing coherent
structure-based models can be found in the literature (Theodorsen, 1952;
Grant, 1958; Willmarth and Tu, 1967; Townsend, 1976; Davidson and Krogstad,
2009), amongst which the attached eddy model (AEM; Perry and Chong (1982),
Marusic and Monty (2019)) based on Townsend’s attached eddy hypothesis
(Townsend, 1976), is one of the most cited. Here, the words ‘structures’,
‘motions’ and ‘eddies’ are used interchangeably and essentially conform to the
definition of a coherent motion given by Robinson (1991).

According to Townsend’s attached eddy hypothesis, an inviscid asymptotically
high friction Reynolds number ($Re_{\tau}$ $=$
$\frac{{U_{\tau}}{\delta}}{\nu}$) canonical wall-bounded flow can be modelled
via a hierarchy of geometrically self-similar attached eddies, having
population densities inversely proportional to their sizes, which vary over
${\mathcal{O}}$($z$) – ${\mathcal{O}}$($\delta$). The term ‘attached’ here
refers to any coherent motion whose geometric extent scales with its distance
from the wall, given by $z$, while the associated velocity fluctuations scale
with the friction velocity $U_{\tau}$, but doesn’t necessarily imply its
velocity signatures physically extending down to the wall (Marusic and Monty,
2019). Further, $\delta$ here corresponds to the outer scale of the respective
canonical flow geometry, which is the boundary layer thickness for a zero-
pressure gradient turbulent boundary layer (ZPG TBL), while it is the pipe
radius and channel half-height for internal flows. Increased access to high
$Re_{\tau}$ wall-turbulence datasets, over the past two decades, has confirmed
that the kinetic energy production and transfer within these flows is mostly
influenced by the inertial (inviscid) motions predominating in the inertial
region (Marusic et al., 2010; Lee and Moser, 2019; Hwang and Lee, 2020),
making AEM ideally suited for modelling these flows. These characteristics
also make the inertial motions a key target of active flow control schemes
(Abbassi et al., 2017). Studies in the past have already demonstrated that the
AEM is able to predict ensemble/time-averaged statistics, such as the Reynolds
stresses (Marusic and Perry, 1995; Baidya et al., 2014), higher order moments
(Woodcock and Marusic, 2015; de Silva et al., 2016b), two-point correlations
(Marusic, 2001) and the energy spectra (Baidya et al., 2017; Deshpande et al.,
2020a; Chandran et al., 2020) across the inertial layer (2.6$\sqrt{Re_{\tau}}$
$\lesssim$ $z^{+}$ $\lesssim$ 0.15$Re_{\tau}$; Klewicki et al. (2009)) in high
$Re_{\tau}$ canonical flows. While there have been a few studies focused on
predicting the instantaneous flow phenomena in both the inertial and the wake
region (de Silva et al., 2016a, 2018; Eich et al., 2020), success has mostly
been limited to the wake region, predominated solely by the largest hierarchy
of eddies ($\sim$ $\mathcal{O}$($\delta$)).

The primary challenge associated with accurately predicting the representative
instantaneous flow in the inertial region, is its high sensitivity to the
geometry of the coexisting motions, given that the region is populated by a
host of statistically significant eddies, with sizes varying from
${\mathcal{O}}$($z$) to ${\mathcal{O}}$($\delta$) (Lee and Moser, 2015;
Deshpande et al., 2020a; Yoon et al., 2020). These eddies are highly three-
dimensional (3-D) in geometry, and hence vary considerably in terms of their
spatial coherence, which also needs to be accounted for while attempting to
replicate the instantaneous flow. To this end, there are several studies in
recent literature (Yoon et al., 2020; de Silva et al., 2020; Chandran et al.,
2020) which have envisioned and subsequently characterized the inertial region
as a superposition of two types of inertial motions: (i) motions with their
velocity signatures (i.e. coherence) extending all the way down to the wall
(i.e. wall coherent or WC motions) and (ii) those with velocity signatures not
extending to the wall (wall incoherent; WI). Interestingly, both these eddy
types (WC and WI) have been shown by the same set of studies to depict
characteristics consistent with Townsend’s attached eddies, although with
certain caveats; for example, apart from the hierarchy of self-similar eddies,
the WC inertial motions also comprise $\delta$-scaled superstructures or very-
large-scale motions (VLSMs), which do not conform to Townsend’s attached
eddies (Deshpande et al., 2020a; Yoon et al., 2020). These results, thus,
showcase the prospect of predicting the instantaneous wall-bounded flow by
using the AEM framework to individually model the WC and WI motions in the
inertial region.

In the present study, we take the first step towards achieving this by
establishing an AEM-based methodology to model the inertial WC motions in all
three canonical wall-bounded flows. Specifically, to improve its predictive
capability, the geometry of the attached eddies is based on estimates
extracted directly from published experimental and numerical datasets. It is
worth noting here that although the inertial WC motions are only a subset of
the full flow, these motions are responsible for the $Re_{\tau}$-dependence of
the wall shear stress fluctuations (and hence, the skin-friction drag
characteristics) in any wall-bounded flow (Örlü and Schlatter, 2011; Deck et
al., 2014; de Giovanetti et al., 2016; Abbassi et al., 2017; Smits et al.,
2021). Further to that, the WC motions are also known to superimpose onto and
modulate the near-wall cycle (Hutchins and Marusic, 2007; Mathis et al.,
2009), which has a significant impact on the skin friction drag. This makes
the present modelling effort relevant for both fundamental as well as applied
research. We begin by first reviewing the current state of the AEM ($\S$1.1)
and its limitations ($\S$2), followed by proposal of the new methodology
($\S$3), which provides a data-driven basis to improve the AEM predictions.
Throughout this paper, we use the coordinate system $x$, $y$ and $z$ to refer
to the streamwise, spanwise and wall-normal directions respectively, with $u$,
$v$ and $w$ denoting the corresponding fluctuating velocity components.
${\langle}{\rangle}$ and capitalization indicates averaged quantities while
the superscript $+$ refers to normalization in viscous units (eg., $u^{+}$ $=$
$u$/$U_{\tau}$ and $z^{+}$ $=$ $z{U_{\tau}}/{\nu}$, where $\nu$ is the
kinematic viscosity).

###  1.1 Current state of the AEM

Figure 1: (a) Schematic depicting the representative $\Lambda$-eddy packet
used in the AEM (de Silva et al., 2016b; Eich et al., 2020), with the blue and
red iso-contours respectively denoting the negative and positive streamwise
velocity induced from the $\Lambda$-shaped vortex rods (indicated in green).
(b) Streamwise, (c) spanwise and (d) wall-normal velocity induced by the
$\Lambda$-eddy packet in the reference wall-parallel plane at $z$ =
0.01$z_{i}$ (highlighted in grey) shown in (a). Labels in (a) are used to
refer to the 3-D geometry of the packet, with ${\Delta}y_{i}$ representing the
spanwise half-width, owing to the symmetry of the $u$-distribution about the
$y$ = 0 plane. This figure has been adapted from de Silva et al. (2016b). Mean
flow direction is along $x$.

In the original AEM conceptualized by Perry and Chong (1982), a single hairpin
or a simple arch-shaped ($\Lambda$) eddy, inclined forwards with respect to
the flow direction ($x$) at $\sim$45∘ (Head and Bandyopadhyay, 1981; Deshpande
et al., 2019), was considered as the representative coherent structure/eddy.
This shape was inspired from the seminal flow visualization studies of Head
and Bandyopadhyay (1981) in a low $Re_{\tau}$ TBL. The simplest version of
this eddy is essentially made up of two vortex rods, arranged in a
$\Lambda$-shape, with each rod containing a Gaussian distribution of vorticity
about its core. The corresponding velocity field for the eddy can be obtained
by performing the Biot-Savart calculations (schematically depicted in figures
2, 6 and 7 of Perry and Chong (1982)). Over time, the representative eddy
shape has evolved as more detailed quantitative measurements, revealing the
structure of wall-bounded flows were reported, such as the seminal PIV
measurements of Adrian et al. (2000). While this study highlighted the
presence of $\Lambda$-type eddies in the TBL, it was found that the eddies are
organized in the form of a packet in fully turbulent flows. The use of a
$\Lambda$-eddy packet (figure 1(a)), instead of a single eddy, as the
representative attached eddy was subsequently incorporated into the AEM by
Marusic (2001), and this change was shown to further improve statistical
predictions. Figures 1(b-d) depict the near-wall ($z$ = 0.01$z_{i}$) planar
flow field of the three velocity components associated with a single packet-
like eddy, which is simply a superposition of the velocity fields obtained
from multiple individual $\Lambda$-eddies. In case of the AEM, it should be
noted that $w$ = 0 $\neq$ $u$, $v$ at $z$ = 0 due to the impermeability
condition being the only condition imposed at the wall, which is achieved by
using packet structures with image packet pairs in the plane of the wall.

To model the inertially dominated (i.e. outer) region, the flow is simply
represented by the superposition of multiple such $\Lambda$-eddy packets
(referred as hierarchies), of varying sizes and population density, randomly
distributed in the flow domain. Following Townsend’s hypothesis, the geometry
of these hierarchies is considered to vary self-similarly, a claim which has
received both empirical (Baars et al., 2017; Deshpande et al., 2020a; Yoon et
al., 2020; Hwang et al., 2020) and theoretical (Perry et al., 1986; McKeon,
2019) support in the literature. The recent study by Eich et al. (2020) is the
latest published AEM flow-field configuration (henceforth referred simply as
AEM) used to replicate the instantaneous flow-field for a ZPG TBL at
$Re_{\tau}$ $\approx$ 3200\. In this study, the major axis of the
$\Lambda$-eddy packet was oriented at various angles (with respect to $x$)
along the $x$-$y$ plane, to incorporate the ‘meandering’ characteristics of
the eddies noted in experiments (Hutchins and Marusic, 2007; Kevin et al.,
2019). To summarize, while the meandering features and the eddy inclination
angles have been chosen based on the empirical estimates (Deshpande et al.,
2019; Kevin et al., 2019), the aspect ratios governing the representative eddy
geometry (marked in figure 1(a)) have never had an empirical basis to date,
which forms the motivation for the present study. The eddy geometry plays a
major role in the visual appearance of the instantaneous flow field generated
by the AEM, as will be highlighted in the next section ($\S$2).

The present study, hence, follows in line with previous studies (de Silva et
al., 2016b; Eich et al., 2020) with the aim of enhancing the AEM framework, in
order to improve spatial representation of a wall-bounded flow from the model.
We note that the typical shape of the representative eddy (of a $\Lambda$-eddy
packet) has been kept the same with the intention to preserve its low-order
complexity, and given its past success in replicating turbulent wall flow
statistics (Woodcock and Marusic, 2015; Baidya et al., 2014, 2017; Deshpande
et al., 2020a; Chandran et al., 2020). Further, the present shape is also well
suited to model the wall-coherent subset of the wall-bounded flows, given that
all three velocity components generated from the eddy extend down to the near-
wall region, (figures 1(b-d)).

Figure 2: Instantaneous (a,b) streamwise, (c,d) spanwise and (e,f) wall-normal
velocity fluctuations on a wall-parallel plane at the lower bound of the
inertial-region ($z^{+}$ $\approx$ 2.6$\sqrt{Re_{\tau}}$). Data in (a,c,e)
corresponds to the AEM of Eich et al. (2020) and has been plotted at $z^{+}$
$\approx$ 160, while that in (b,d,e) corresponds to the ZPG TBL DNS of Sillero
et al. (2014) comprising solely of wall-coherent motions (represented by
subscript wc). The DNS data has been plotted at $z^{+}$ $\approx$ 100\. Black
dashed lines are used to highlight the largest spatial features of low
momentum for a qualitative comparison. Note the difference in axis limits
between (a-d) and (e-f).

##  2 Motivation for a data-driven AEM

Here, we compare the instantaneous flow fields generated by the AEM (Eich et
al., 2020) with the corresponding flow fields from the direct numerical
simulation (DNS) of a ZPG TBL (Sillero et al., 2014) to motivate a data-driven
definition of the representative eddy geometry used in the former. Given that
the AEM simulates purely the WC portion of the TBL, a logical way to go about
this would be by comparing it with the subset of the full DNS fields
comprising solely the WC motions. Figure 2 presents this comparison between
the instantaneous velocity fluctuations for all three components, considered
along the wall-parallel plane at the lower bound of the inertial region. It
brings out significant differences in the geometry of the coherent motions,
with the velocity features in the DNS fields substantially longer than those
noted in the AEM fields. A plausible reason behind this difference may be the
geometry of a $\Lambda$-eddy packet not defined based on empirical estimates,
which we aim to facilitate in the present study. Here, we propose to obtain
the geometric estimates by reconstructing the 3-D statistical picture of the
wall-coherent turbulence from published datasets ($\S$3). Another noteworthy
limitation in the AEM is that all the coherent structures considered in the
current model correspond to the hierarchy of self-similar eddies, following
Townsend’s hypothesis. A true WC flow field, however, cannot be replicated
without also considering the $\delta$-scaled superstructures or VLSMs
(Deshpande et al., 2020a; Yoon et al., 2020), which we also intend to include
in the model. Interestingly, these motions can also be modelled via the same
representative $\Lambda$-eddy packet shown in figure 1(a), as demonstrated
recently by Chandran et al. (2020) for modelling the 2-D spectra.

Readers may note here that the DNS flow fields in figures 2(b,d,f), which
comprise solely the WC motions, have been computed using the publicly
available full (WC+WI) fields via a spectral linear stochastic estimation
(SLSE) based decomposition technique. Interested readers may refer to appendix
A for more details on how the WC fields were estimated, and also see figure
11, where the full flow field instants corresponding to the WC fields in
figure 2 have been plotted. Differences between the full (figure 11) and the
WC (figure 2) flow fields are substantial, especially for the lateral velocity
components, with large spatial coherence observed in case of the latter as
compared to the former. Further, the oblique features otherwise apparent in
the full spanwise velocity field in the log-region (figure 11(b); Sillero et
al. (2014),de Silva et al. (2018)), cannot be noted in its WC subset (figure
2(d)). These differences underscore the importance of comparing the AEM fields
with solely the WC motions, which the representative eddy models.

Table 1: Table summarizing the details of five previously published datasets comprising synchronized measurements of $u$-fluctuations by various near-wall fixed probes (placed at $x_{w}$ = 0,$y_{w}$,$z_{w}$) and a traversing probe ($x$ = 0,$y$,$z$), in turn separated by relative spanwise (${\Delta}y$ $=$ ${\mid{y}\;{-}\;{y_{w}}\mid}$) and wall-normal (${z}\;{-}\;{z_{w}}$ $\approx$ $z$) offsets. In case of dataset ${\mathcal{P}}_{1}$, the hot-film attached to the wall measures the instantaneous skin-friction velocity, ${\vec{U}}_{\tau}$ $=$ $\sqrt{{\nu}({{\partial}{\vec{U}}}/{{\partial}z})}$. The probe arrangements associated with each of these datasets have been schematically depicted in figures 3 and 4, with probes in blue and red respectively denoting the traversing and near-wall fixed probes. Dataset: |  |  |  |  |
---|---|---|---|---|---
$\bullet$ Label | ${\mathcal{T}}_{1}$ | ${\mathcal{C}}_{1}$ | ${\mathcal{T}}_{2}$ | ${\mathcal{T}}_{3}$ | ${\mathcal{P}}_{1}$
$\bullet$ Flow type | ZPG TBL | Channel | ZPG TBL | ZPG TBL | Pipe
$\bullet$ Facility | DNS (raw) | DNS (raw) | HRNBLWT | HRNBLWT | CICLoPE
$\bullet$ Reference | Sillero et al. (2014) | Lee and Moser (2015) | Baars et al. (2017) | Deshpande et al. (2020a) | Baidya et al. (2019)
$\bullet$ $Re_{\tau}$ ${\approx}$ | 2000 | 5200 | 14000 | 14000 | 40000
Near-wall probes: |  |  |  |  |
$\bullet$ Probe type | Grid point | Grid point | Hot-wire | Hot-wire | Hot-film
$\bullet$ Number | 38 | 40 | 1 | 1 | 20
$\bullet$ $z^{+}_{w}$ $\approx$ | 15 | 15 | 4.33 | 15 | Wall
$\bullet$ Spanwise/azimuthal | 0 $\leq$ ${y_{w}}/{\delta}$ $\leq$ 1 | 0 $\leq$ ${y_{w}}/{\delta}$ $\leq$ 1 |  $y_{w}$ $\approx$ 0 |  $y_{w}$ $\approx$ 0 | 0 $\leq$ ${y_{w}}/{\delta}$ $\leq$ 1
arrangement of probes | (log-spacing) | (log-spacing) |  |  | (log-spacing)
Traversing probe: |  |  |  |  |
$\bullet$ Probe type | Grid point | Grid point | Hot-wire | Hot-wire | Hot-wire
$\bullet$ Traversing direction | $z$ | $z$ | $z$ | $y$ | $z$
$\bullet$ Traversing range | 15 $<$ $z^{+}$ $\leq$ $Re_{\tau}$ | 15 $<$ $z^{+}$ $\leq$ $Re_{\tau}$ | 4.33 $<$ $z^{+}$ $\leq$ $Re_{\tau}$ | 0 $\leq$ $y/{\delta}$ $\leq$ $1$ | 0 $<$ $z^{+}$ $\leq$ $Re_{\tau}$
| (log-spacing) | (log-spacing) | (log-spacing) | (log-spacing) | (log-spacing)
$\bullet$ Position along non- |  |  |  |  |
traversing directions |  $y$ = 0 |  $y$ = 0 |  $y$ = 0 |  $z$ $\sim$ log-region |  $y$ = 0

##  3 3-D statistical picture of the wall-coherent turbulence

###  3.1 Datasets and methodology

The present study utilizes five previously published multi-point datasets,
spanning all three canonical flow geometries. Key parameters of the datasets
are summarised in table 1. Dataset ${\mathcal{T}_{1}}$ corresponds to DNS of
ZPG TBL, while ${\mathcal{T}_{2}}$ and ${\mathcal{T}_{3}}$ correspond to
higher $Re_{\tau}$ experimental data for the same flow geometry.
${\mathcal{C}_{1}}$ and ${\mathcal{P}_{1}}$ are respectively the DNS and
experimental datasets for a fully turbulent channel and pipe flow. The
selected datasets present a unique combination of synchronously acquired
$u$-fluctuations mapped across 3-D space in the wall-bounded shear flow (refer
figures 3(a,b) and 4(a-c)). ${\mathcal{T}_{1}}$, ${\mathcal{C}_{1}}$ and
${\mathcal{P}_{1}}$ each comprise $u$-fluctuations acquired using multiple
near-wall fixed probes (placed at $x_{w}$ $=$ 0,$y_{w}$,$z_{w}$), distributed
in log spacing along the spanwise/azimuthal directions, in conjunction with
those acquired by the probe traversing along the wall-normal direction ($z$).
On the other hand, ${\mathcal{T}_{2}}$ and ${\mathcal{T}_{3}}$ comprise
$u$-fluctuations from a single near-wall fixed probe (placed at $x_{w}$ $=$
0,$y_{w}$ $=$ 0,$z_{w}$), acquired synchronously with those measured farther
from the wall by a probe traversing along the $z$\- and $y$-directions,
respectively. Considering the fact that the canonical wall-bounded flows are
statistically homogeneous along the span, the cumulative 3-D space mapped by
combining the datasets ${\mathcal{T}_{2}}$ and ${\mathcal{T}_{3}}$ would be
equivalent to that mapped in each of the individual datasets,
${\mathcal{T}_{1}}$, ${\mathcal{C}_{1}}$ and ${\mathcal{P}_{1}}$. In case of
${\mathcal{P}}_{1}$, it should be noted that the near-wall probe is
essentially a hot-film sensor attached to the wall for measuring the
instantaneous skin-friction velocity ${\vec{U}}_{\tau}$ (= ${U}_{\tau}$ \+
${u_{\tau}}$), which is associated with the near-wall instantaneous streamwise
velocity (${\vec{U}}$) following ${\vec{U}}_{\tau}$ =
$\sqrt{{\nu}({{\partial}{\vec{U}}}/{{\partial}z})}$. Interested readers may
refer to the individual references for more specific details regarding the
measurements/simulations.

We now move on to proposing the methodology used to reconstruct the 3-D
statistical picture of the wall-coherent turbulence, from which the geometric
estimates for the representative eddy (of the AEM) would be extracted. There
have been several studies in the past (Ganapathisubramani et al., 2005;
Hutchins and Marusic, 2007; Lee and Sung, 2011) which have used multi-point
datasets to estimate the geometry of the coherent motions, by computing mostly
space-time cross-correlations. These cross-correlations, however, represent
cumulative contributions from eddies of various length scales at a specific
spatial offset (say streamwise offset, ${\Delta}x$). Here, since we are
interested in estimating the 3-D geometry of individual hierarchies/length
scales to be incorporated in the AEM, we compute the cross-correlations in the
spectral domain to get a scale-specific estimate of the coherent structure
geometry (i.e. as a function of the streamwise wavelength, ${\lambda}_{x}$).
Given our focus is on modelling the inertial wall-coherent motions coexisting
in the outer region, we consider the cross-correlations specifically between
$u$-signals in the outer region (at $z$) and those acquired close to the wall
(at $z^{+}_{w}$ $\lesssim$ 15), via the cross-correlation spectra ($\Gamma$)
defined as (Bailey et al., 2008; Deshpande et al., 2020b):

$\displaystyle{{\Gamma}}({z},{{\Delta}{y}},{\lambda_{x}})$
$\displaystyle=\frac{{\text{Re}}[\\{\widetilde{u}(z,y;\lambda_{x}){{\widetilde{u}}^{\ast}}(z_{w},{y_{w}};\lambda_{x})\\}]}{{\sqrt{\\{{\mid{\widetilde{u}(z,y;\lambda_{x})}\mid}^{2}\\}}}{\sqrt{\\{{\mid{\widetilde{u}({z_{w}},{y_{w}};\lambda_{x})}\mid}^{2}\\}}}}$
(1)
$\displaystyle=\frac{\text{Re}[{{{\phi}^{\prime}_{{u}{u_{w}}}}(z,z_{w},{\Delta}y;\lambda_{x})}]}{{\sqrt{\phi_{{u}{u}}({z},y;\lambda_{x})}}{\sqrt{\phi_{{u_{w}}{u_{w}}}({z_{w}},{y_{w}};\lambda_{x})}}},$

where ${\tilde{u}}$ = ${\mathcal{F}}$($u$) is the Fourier transform of $u$ in
either time or $x$ depending on the dataset with ${\lambda}_{x}$ $=$
2${\pi}/{k_{x}}$, where $k_{x}$ is the streamwise wavenumber. Further, the
curly brackets ($\\{\\}$), asterisk ($\ast$) and vertical bars ($\mid\mid$)
indicate the ensemble averaging, complex conjugate and modulus, respectively
while Re denotes the real component. ${\phi}^{\prime}_{{u}{u_{w}}}$ is, thus,
the complex-valued 1-D cross-spectrum between $u$-signals recorded at $z$ and
$z_{w}$, which quantifies the scale-specific cross-correlation, while
$\phi_{{u}{u}}$ and $\phi_{{u_{w}}{u_{w}}}$ respectively are the conventional
1-D energy spectra at ${z}$ and ${z_{w}}$ used for the scale-specific
normalization. Such a definition forces $\Gamma$ to vary between -1 to 1, with
the former and latter respectively indicating perfect anti-correlation and
correlation for each scale, ${\lambda}_{x}$. In case of dataset
${\mathcal{P}_{1}}$, notably, $\Gamma$ is computed by substituting the skin-
friction velocity fluctuations ($u_{\tau}$) measured by the wall-based hot-
films, in place of $u$($z_{w}$).

As can be noted from its definition in equation (1), $\Gamma$ is a function of
the spatial offsets in all three directions, which is facilitated by computing
the cross-correlations between $u$-signals recorded at various relative
spanwise offsets (${\Delta}y$ $=$ ${\mid}{y-{y_{w}}}{\mid}$) and wall-normal
offsets ($z$ $\approx$ ${z-{z_{w}}}$, given $z_{w}$ $\ll$ $z$). The offset
along $x$ is obtained in the form of a streamwise wavelength, which is
estimated by using the Taylor’s hypothesis in case of the experimental
datasets (where the local convection velocity, $U_{c}$ $=$ $U(z)$), or
obtained directly from the spatial domain for the DNS datasets.
$\Gamma$($z$,${\Delta}y$,${\lambda}_{x}$), thus, can be interpreted as a
coherence metric which associates each WC scale/eddy (${\lambda}_{x}$) with
its corresponding spanwise (${\Delta}y$) and wall-normal extent ($z$). This
information can be directly used to define the geometry of the representative
eddy, for the respective coherent structure-based model, corresponding to the
streamwise scale, ${\lambda}_{x}$. The benefit in case of models such as the
AEM, which incorporate vorticity-based structures, is that once the eddy
geometry of the $\Lambda$-eddy packet is defined, velocity field for all three
components can be simulated without needing to estimate $\Gamma$ for the other
(lateral) velocity components.

It is worth noting here that $\Gamma$ being used here is different from the
linear coherence spectrum (LCS; ${\gamma}^{2}_{L}$) computed previously by
Baars et al. (2017) and Baidya et al. (2019), since $\Gamma$ retains solely
the real part of the cross-spectrum while ${\gamma}^{2}_{L}$ uses the absolute
value of the cross-spectrum. Here, we prefer $\Gamma$ over ${\gamma}^{2}_{L}$,
to define the eddy geometry, given that the anti-correlated regions ($\Gamma$
$<$ 0) are indicative of the relative placement of low-momentum ($-u$) regions
with respect to the high-momentum ($+u$) regions, which otherwise can’t be
inferred from the LCS (0 $\leq$ ${\gamma}^{2}_{L}$ $\leq$ 1). For example, a
recent study (Deshpande et al., 2020b) focusing on the $\Gamma$-distribution
over very large spatial extents has confirmed the periodic distribution of
$\delta$-scaled $+u$ and $-u$ motions along the spanwise direction.
Investigating the $\Gamma$ distribution in 3-D space, thus, could provide
quantitative basis to the choice of the representative eddy for a coherent
structure-based model, as will be discussed ahead in this section.

###  3.2 Application to high $Re_{\tau}$ ZPG TBL datasets

We begin the empirical analysis by first reconstructing
$\Gamma$($z,{{\Delta}y},{{\lambda}_{x}}$) for high $Re_{\tau}$ ZPG TBL using
the two experimental datasets, ${\mathcal{T}}_{3}$ and ${\mathcal{T}}_{2}$. In
case of ${\mathcal{T}}_{2}$, the traversing probe is constrained to move
vertically above the near-wall fixed probe (i.e. maintaining ${\Delta}y$ $=$
0), owing to which this dataset yields the statistical picture solely in the
streamwise wall-normal plane, given by $\Gamma$(${{\Delta}y}$ $=$
0$;z,{{\lambda}_{x}}$) in figure 3(d). This plot brings out the range of
energetic WC scales/motions coexisting in the outer region, which correspond
to the wavelength range, ${\lambda}_{x}$ $\gtrsim$ 0.1$\delta$. For a specific
energetic length scale, say ${\lambda}_{x}$ $\sim$ 4$\delta$, it indicates the
presence of a positively correlated region up to $z$ $\sim$ 0.3$\delta$,
followed by a negatively correlated region between 0.3$\delta$ $<$ $z$ $<$
0.6$\delta$. When interpreted physically in terms of the spatial extent of the
$u$-velocity distributions along the mid-span of a $\Lambda$-eddy packet of
length $\sim$ 4$\delta$ (figure 1(a)), the aforementioned $z$-ranges may be
respectively associated with the wall-normal extent of the $-u$ region
(represented by +$\Gamma$) and of the $+u$ region (represented by -$\Gamma$).
Consequently, the height of the $\Lambda$-eddy packet may be nominally defined
based on the region where $\Gamma$ $\approx$ 0, which is approximately $z$
$\sim$ 0.3$\delta$. If we consider this for the entire range ${\lambda}_{x}$
$\gtrsim$ 0.1$\delta$, figure 3(d) indicates that the height ($z$) of the
inertially dominated eddies varies self-similarly with respect to their
streamwise length scale (${\lambda}_{x}$), which is given by the linear
relationship ${\lambda}_{x}$ $=$ 14$z$ (indicated by a dashed-dotted green
line fitted to $\Gamma$ $\approx$ 0). Interestingly, the same linear
relationship between ${\lambda}_{x}$ and $z$ was noted by Baars et al. (2017)
and Baidya et al. (2019) for a ZPG TBL and a pipe respectively, although based
on ${\gamma}^{2}_{L}$ as the metric. The $\Gamma$-contours, however, can be
noted to be deviating from the linear relationship for $z$ $<$ 0.02$\delta$,
which corresponds to the nominal lower bound of the log-region ($z/{\delta}$
$\sim$ 2.6$(Re_{\tau})^{-0.5}$) as per Klewicki et al. (2009). This deviation
suggests that only ${{\lambda}_{x}}/{\delta}$ $>$ 0.3 ($\sim$
14$\times$2.6$(Re_{\tau})^{-0.5}$) conform to the truly self-similar hierarchy
at this flow $Re_{\tau}$.

Figure 3: (a,b) Experimental setups corresponding to datasets (a)
${\mathcal{T}}_{3}$ and (b) ${\mathcal{T}}_{2}$. Hotwire sensors marked in
blue correspond to traversing probes, which move along their respective
traversing directions also marked in blue, while sensors marked in red
correspond to the fixed near-wall probes. (c) Cross-correlation spectra
computed as a function of ${\Delta}y$ and ${\lambda}_{x}$ (i.e.
$\Gamma$(${\Delta}y$,${\lambda}_{x}$)) for $z$ $\approx$ 0.02$\delta$ from the
dataset ${\mathcal{T}}_{3}$. (d) $\Gamma$($z$,${\lambda}_{x}$) computed for
${\Delta}y$ $\approx$ 0 from the dataset ${\mathcal{T}}_{2}$. Black contour in
(c,d) corresponds to $\Gamma$ = 0.1. (e-g) Reconstruction of the 3-D cross-
correlation spectra, $\Gamma$($z$,${\Delta}y$,${\lambda}_{x}$) across the
inertial region (0.02$\delta$ $\lesssim$ $z$ $\lesssim$ 0.15$\delta$) by
fusing the empirical estimates from datasets ${\mathcal{T}}_{2}$ and
${\mathcal{T}}_{3}$. (e) is obtained by simply combining data in (c) and (d).
(f) and (g) are similarly obtained by fusing data in (d) and that estimated
from dataset ${\mathcal{T}}_{3}$ over a range of $z$: (f) 0.02$\delta$
$\lesssim$ $z$ $\lesssim$ 0.05$\delta$ and (g) 0.02$\delta$ $\lesssim$ $z$
$\lesssim$ 0.15$\delta$. Black iso-contour in (e-g) also corresponds to
$\Gamma$ = 0.1. Dash-dotted lines in green, orange, indigo and grey colours
represent the scalings: ${\lambda}_{x}$ = 14$z$, ${\lambda}_{x}$ =
20${\Delta}y$, ${\Delta}y$ = 0.17$\delta$ and ${\lambda}_{x}$ = 4$\delta$,
respectively.

The same analysis is next conducted along the wall-parallel plane, at $z$
corresponding to the inertial-region, by using the dataset
${\mathcal{T}}_{3}$. Figure 3(c) plots $\Gamma$($z$ $=$
0.02$\delta$;${\Delta}y$,${\lambda}_{x}$) as an example, which corresponds to
the 2-D statistical picture at the lower bound of the inertial region. This
picture, however, represents the spanwise coherence only along the
$+{\Delta}y$ axis and would be complete by considering a mirror image of the
same plot about ${\Delta}y$ = 0, owing to symmetry (Hutchins and Marusic,
2007). Given that the $\Gamma$-distribution in figure 3(c) is of the same
nature as in figure 3(d), the physical interpretation discussed for the latter
is now extended to the former. By fitting a dashed-dotted orange line along
$\Gamma$ $\approx$ 0, we can interpret from figure 3(c) that the spanwise
half-width (${\Delta}y$) of the $\Lambda$-eddy packet also varies self-
similarly with respect to ${\lambda}_{x}$, which is represented by the linear
relationship ${\lambda}_{x}$ $=$ 20${\Delta}y$. Consistent with what was noted
from figure 3(d), the self-similar trend is observed for $\lambda_{x}$ $>$
0.3$\delta$ but is only valid up to $\lambda_{x}$ $\sim$ 4$\delta$, as all the
larger length scales are found to have a constant half-width, ${\Delta}y$
$\sim$ 0.17$\delta$. These large eddies, hence, do not conform to Townsend’s
hierarchy of attached eddies but, in fact, correspond to the $\delta$-scaled
very-large-scales or superstructures noted previously in the ZPG TBL (Hutchins
and Marusic, 2007; Lee and Sung, 2011). The spanwise widths of these
structures, which is found from the present analysis to be 2${\Delta}y$ $\sim$
0.35$\delta$, is consistent with previous findings based on PIV experiments
(Dennis and Nickels, 2011; Gao et al., 2011).

Following the analysis on the individual 2-D planes, the plots in figures
3(c,d) can now be stitched together to reconstruct the 3-D statistical picture
of the WC motions in the ZPG TBL, as shown in figure 3(e). By estimating the
cross-correlation spectra at various $z$: 0.02$\delta$ $\lesssim$ $z$
$\lesssim$ 0.15$\delta$ from dataset ${\mathcal{T}}_{3}$,
$\Gamma$($z$,${\Delta}y$,${\lambda}_{x}$) can be reconstructed across the
inertial region, as depicted in figures 3(e-g). These figures, thus, describe
the scale-specific geometry of the range of energetic motions coexisting in
the inertial region: the flow at the beginning of the inertial region ($z$
$\sim$ 0.02$\delta$) comprises contributions from the widest hierarchy of
self-similar eddies, spanning 0.3$\delta$ $\lesssim$ ${\lambda}_{x}$
$\lesssim$ 4$\delta$, which reduces to 0.8$\delta$ $\lesssim$ ${\lambda}_{x}$
$\lesssim$ 4$\delta$ at $z$ $\approx$ 0.05$\delta$, and finally negligible
contributions from the self-similar hierarchy at the upper bound ($z$
$\approx$ 0.15$\delta$). The contributions from the $\delta$-scaled
superstructures (${\lambda}_{x}$ $\gtrsim$ 4$\delta$), on the other hand, is
consistently present throughout the inertial region. The corresponding
spanwise and wall-normal extents for each eddy (${\lambda}_{x}$) can also be
inferred based on the scalings shown in figure 3(g), and have been summarized
as a function of the flow $Re_{\tau}$ below:

${{\lambda}^{+}_{x}}{\gtrsim}4{Re_{\tau}}\;\begin{cases}{z^{+}}={{\lambda}^{+}_{x}}/14;{\Delta}y^{+}=0.17{Re_{\tau}}&\text{if
true}\\\
{z^{+}}={{\lambda}^{+}_{x}}/14;{\Delta}y^{+}={{\lambda}^{+}_{x}}/20&\text{otherwise}.\end{cases}$
(2)

This information can be directly used to facilitate definition of the
representative eddy geometries in any coherent structure-based model. It is
worth noting here though, $\Gamma$($z$,${\Delta}y$,${\lambda}_{x}$) in figure
3(g) looks consistent with the shape of a $\Lambda$-eddy packet when halved
along the span (refer to figure 1 or 5), thus providing quantitative support
to the choice of the representative eddy shape used in the AEM and making a
strong case in favour of implementing equation (2) into AEM.

###  3.3 Extension to various $Re_{\tau}$ and flow geometries

Having described the methodology to reconstruct
${\Gamma}$($z$,${\Delta}y$,${{\lambda}_{x}}$) and applied it to high
$Re_{\tau}$ ZPG TBL data, we now extend the same to datasets at different
$Re_{\tau}$ (${\mathcal{T}}_{1}$) and other flow geometries
(${\mathcal{C}}_{1}$ and ${\mathcal{P}}_{1}$) to check for the universality of
the scalings noted in figure 3. As discussed previously, each of these
datasets are self-sufficient to reconstruct the 3-D statistical picture across
the inertial region. Figure 4(d) plots the same for the low $Re_{\tau}$ ZPG
TBL data using similar plotting style as in figure 3(g). Interestingly, all
the scalings noted previously from the high $Re_{\tau}$ experimental data
(expressed in equation (2)) are also observed for the low $Re_{\tau}$ case,
confirming the $Re_{\tau}$-invariance of these empirical estimates and
consequently, their direct utilization in data-driven coherent structure-based
models for flows at any $Re_{\tau}$. Further, the spanwise half-width
(${\Delta}y$) is also found to tend towards a constant ($\sim$ 0.17$\delta$)
at ${\lambda}_{x}$ $\gtrsim$ 4$\delta$ for the DNS dataset, confirming that
the wavelength range estimated for the $\delta$-scaled superstructures is
independent of the Taylor’s hypothesis assumption. It is worth noting here
that the low $Re_{\tau}$ for the dataset ${\mathcal{T}}_{1}$ reduces the
thickness of the inertial region (0.05$\delta$ $\lesssim$ ${\lambda}_{x}$
$\lesssim$ 0.15$\delta$) and consequently the range of energetic scales
(${\lambda}_{x}$) corresponding to the self-similar hierarchy. This can be
confirmed from $\Gamma$($z$ $\approx$
0.05$\delta$;${\lambda}_{x}$,${\Delta}y$) plotted in figure 4(g), where the
contours can be seen to follow the linear relationship only for
${{\lambda}_{x}}/{\delta}$ $>$ 0.8, which is consistent with the
$Re_{\tau}$-dependance of the lower limit of the self-similar scaling, as
noted previously in the high $Re_{\tau}$ dataset in figure 3(c).

Figure 4: Reconstruction of the (d-f) 3-D cross-correlation spectra,
$\Gamma$($z$,${\Delta}y$,${\lambda}_{x}$) across the inertial region and (g-i)
$\Gamma$(${\Delta}y$,${\lambda}_{x}$) at the lower bound of the log-region
($z^{+}$ $\approx$ 2.6$\sqrt{Re_{\tau}}$) for datasets (d,g)
${\mathcal{T}}_{1}$, (e,h) ${\mathcal{C}}_{1}$ and (f,i) ${\mathcal{P}}_{1}$
by utilizing the $u$-fluctuations associated with the probe placements
depicted for the respective flow geometries in (a-c). Probes marked in blue
correspond to traversing probe which move along their respective traversing
directions also marked in blue, while those marked in red correspond to the
fixed near-wall probes. Here, only two near-wall probes are shown for
representative purposes, with the actual number given in table 1. Black iso-
contour in (d-f) corresponds to $\Gamma$ = 0.1. Dash-dotted lines in green,
orange, indigo and grey colours represent scalings indicated in each figure.

Figures 4(e,h) and (f,i) respectively depict
${\Gamma}$($z$,${\Delta}y$,${{\lambda}_{x}}$) reconstructed using the high
$Re_{\tau}$ channel (${\mathcal{C}}_{1}$) and pipe flow (${\mathcal{P}}_{1}$)
datasets. Remarkably, the same self-similar scalings for the WC motions, noted
previously for ZPG TBL, are also noted for the case of internal flows.
Similarly, it is found that the contribution from the self-similar hierarchies
becomes statistically insignificant beyond the upper bound of the inertial
region, in both the internal flows. Although not shown here,
${\Gamma}$($z$,${\Delta}y$,${{\lambda}_{x}}$) (similar to figure 4(f)) was
also reconstructed for the relatively low $Re_{\tau}$ data ($\approx$ 10000
and 20000) acquired at the same facility by Baidya et al. (2019), which
exhibited the same scalings/behaviour. This universality in the self-similar
scaling, across all three canonical flows, is consistent with the observation
of Hwang et al. (2020) based on low $Re_{\tau}$ ($\approx$ 1000) DNS datasets.
Given that the estimation of the exact linear scalings at low $Re_{\tau}$ may
be limited by the narrower range of the self-similar hierarchy, the present
study confirms the existence as well as the $Re_{\tau}$-invariance of these
self-similar scalings across all three canonical flows. Further, given the
channel and pipe datasets are at differing $Re_{\tau}$, the $\Gamma$ contours
(figure 4(h,i)) also exhibit the $Re_{\tau}$-dependance of the lower-limit of
the self-similar range. Consistent with the observations noted for the ZPG
TBL, we can see that an increase in $Re_{\tau}$ extends the linear scaling up
to much smaller scales, where the limit is governed by
${{\lambda}_{x}}/{\delta}$ $\sim$ $(Re_{\tau})^{-0.5}$.

In case of the internal flows, however, the ${\lambda}_{x}$ = 20${\Delta}y$
relationship is found to be valid up to a larger length scale ($\lambda_{x}$
$\approx$ 6$\delta$) than the ZPG TBL ($\lambda_{x}$ $\approx$ 4$\delta$),
beyond which the constant spanwise half-width gradually plateaus to a
constant, ${\Delta}y$ $\approx$ 0.3$\delta$. The scalings describing the 3-D
spatial extent of the inertial motions, in case of the internal flows, can
thus be expressed as a function of $Re_{\tau}$ following:

${{\lambda}^{+}_{x}}{\gtrsim}6{Re_{\tau}}\;\begin{cases}{z^{+}}={{\lambda}^{+}_{x}}/14;{\Delta}y^{+}=0.30{Re_{\tau}}&\text{if
true}\\\
{z^{+}}={{\lambda}^{+}_{x}}/14;{\Delta}y^{+}={{\lambda}^{+}_{x}}/20&\text{otherwise}.\end{cases}$
(3)

These trends suggest that the $\delta$-scaled VLSMs (${\lambda}_{x}$ $\gtrsim$
6$\delta$) conforming to the internal flow geometries are relatively wider
than the superstructures in ZPG TBL (Monty et al., 2009). These differences
between the internal and external flows have been previously noted by
Balakumar and Adrian (2007), Monty et al. (2007) and Lee and Sung (2013) and
were attributed to the ‘persistent growth’ of the energetic motions, beyond
the inertial region, in case of internal flows, which is otherwise inhibited
by the turbulent/non-turbulent interface (T/NTI) in the ZPG TBL (Monty et al.,
2007; Lee and Sung, 2013).

##  4 Data-driven AEM

###  4.1 Incorporating the empirically-obtained scaling laws into the AEM

Figure 5: (a) A discretized representation of $\Gamma$($z$,${\Delta}y$,${\lambda}_{x}$) for a ZPG TBL as the superposition of six discrete eddy hierarchies ($i$ = 1 to 6), following the attached eddy model of Perry and Chong (1982). Light-shaded boxes represent the self-similar hierarchy while the dark-shaded boxes represent the $\delta$–scaled superstructures. Dash-dotted lines in green, orange, indigo and grey colours represent the same scalings as described in figure 3. Blue and red colour contours plotted along the ${\Delta}y$ = 0 and $z$ $\approx$ 0.01$\delta$ planes are from figure 3(e) and are used simply for reference. (b) The spanwise half of the same $\Lambda$-eddy packet as in figure 1(a), representing the $i^{th}$ hierarchy with its geometry defined in accordance to the discretized plot of $\Gamma$($z$,${\Delta}y$,${\lambda}_{x}$) shown in (a). Table 2 discusses the geometry of each of the six eddy hierarchies interpreted from (a). Table 2: Comparing the geometry of the various eddy hierarchies in the AEM (Eich et al., 2020) and dd-AEM. Here, SS and NSS respectively denote self-similar and non-self-similar. Other terminologies have been defined in figure 5(b). $\Lambda$-eddy packets corresponding to the AEM and dd-AEM comprise of 7 and 30 $\Lambda$-eddies respectively, to account for the differing ${\lambda}_{x_{i}}$/$z_{i}$. |  | AEM (Eich et al., 2020) |  | dd-AEM (present study)
---|---|---|---|---
Hierarchy, $i$ | $z_{i}/{\delta}$ | ${\lambda}_{x_{i}}$ | ${\Delta}y_{i}$ | SS/NSS |  | ${\lambda}_{x_{i}}$ | ${\Delta}y_{i}$ | SS/NSS
1 |  $z_{1}$ = 2-5 |  ${\lambda}_{x_{1}}$ = 3$z_{1}$ |  ${\Delta}y_{1}$ = ${\lambda}_{x_{1}}$/12 | SS |  |  ${\lambda}_{x_{1}}$ = 14$z_{1}$ |  ${\Delta}y_{1}$ = ${\lambda}_{x_{1}}$/20 | SS
2 |  $z_{2}$ = 2-4 |  ${\lambda}_{x_{2}}$ = 3$z_{2}$ |  ${\Delta}y_{2}$ = ${\lambda}_{x_{2}}$/12 | SS |  |  ${\lambda}_{x_{2}}$ = 14$z_{2}$ |  ${\Delta}y_{2}$ = ${\lambda}_{x_{2}}$/20 | SS
3 |  $z_{3}$ = 2-3 |  ${\lambda}_{x_{3}}$ = 3$z_{3}$ |  ${\Delta}y_{3}$ = ${\lambda}_{x_{3}}$/12 | SS |  |  ${\lambda}_{x_{3}}$ = 14$z_{3}$ |  ${\Delta}y_{3}$ = ${\lambda}_{x_{3}}$/20 | SS
4 |  $z_{4}$ = 2-2 |  ${\lambda}_{x_{4}}$ = 3$z_{4}$ |  ${\Delta}y_{4}$ = ${\lambda}_{x_{4}}$/12 | SS |  |  ${\lambda}_{x_{4}}$ = 14$z_{4}$ |  ${\Delta}y_{4}$ = ${\lambda}_{x_{4}}$/20 $\approx$ 0.17$\delta$ | SS
5 |  $z_{5}$ = 2-1 |  ${\lambda}_{x_{5}}$ = 3$z_{5}$ |  ${\Delta}y_{5}$ = ${\lambda}_{x_{5}}$/12 | SS |  |  ${\lambda}_{x_{5}}$ = 14$z_{5}$ |  ${\Delta}y_{5}$ $\approx$ 0.17$\delta$ | NSS
6 |  $z_{6}$ = 20 |  ${\lambda}_{x_{6}}$ = 3$z_{6}$ |  ${\Delta}y_{6}$ = ${\lambda}_{x_{6}}$/12 | SS |  |  ${\lambda}_{x_{6}}$ = 14$z_{6}$ |  ${\Delta}y_{6}$ $\approx$ 0.17$\delta$ | NSS

With the 3-D statistical picture of the wall-coherent turbulence now
characterized for all three canonical flows, we move onto discussing how the
empirically obtained scalings can be used in the data-driven AEM (dd-AEM). In
the remainder of this manuscript, we concentrate our efforts on the attached
eddy modelling of solely the ZPG TBL flow. The methodology, however, can also
be applied to the internal flows by using the appropriate empirical estimates
(given in equation (3)). Although a real ZPG TBL flow comprises of a
continuous distribution of statistically energetic motions over a broadband
spectrum (figure 3(g)), for the case of the AEM, we consider a discretized
distribution of eddies across $n$ hierarchies for convenience in modelling.
Here, $n$ depends on the $Re_{\tau}$ of the flow being modelled, with higher
$Re_{\tau}$ requiring the inclusion of more hierarchies/scales. While the
wall-normal height of the largest hierarchy, say $z_{n}$ = $\delta$ by
definition, we follow the convention adopted by Perry and Chong (1982) of
defining the wall-normal extent of the smallest hierarchy (say $z_{1}$) equal
to the nominal lower bound of the outer region, i.e. $z^{+}_{1}$ $\sim$ 100\.
Perry and Chong (1982) further defined the heights of the subsequently larger
hierarchies to be double that of the relatively small hierarchy, which can be
expressed mathematically as $z^{+}_{i}$ $=$ ${{\delta}^{+}}{2^{i-n}}$, with
$i$ being the hierarchy number. For $z^{+}_{1}$ $=$ 100, ${\delta}^{+}$ $\sim$
$Re_{\tau}$ $=$ 100($2^{n-1}$), meaning the number of hierarchies $n$
essentially define the $Re_{\tau}$ of the flow.

Figure 6: Cross-correlation spectra computed along (a-c) XZ-plane at
${\Delta}y$ $\approx$ 0, $\Gamma$(${\Delta}y$ $\approx$
0;$z$,${\lambda}_{x}$), (d-f) XY-plane at the lower bound of the inertial
region, $\Gamma$($z^{+}$ $\approx$
2.6$\sqrt{Re_{\tau}}$;${\Delta}y$,${\lambda}_{x}$) and (g-i) XY-plane at the
upper bound of the inertial region, $\Gamma$($z$ $\approx$
0.15$\delta$;${\Delta}y$,${\lambda}_{x}$) from the (a,d,g) AEM of Eich et al.
(2020), (b,e,h) data driven-AEM and (c,f,i) datasets ${\mathcal{T}}_{2}$ and
${\mathcal{T}}_{3}$. Dash-dotted lines in green, orange, indigo and grey
colours represent the same scalings as described in figure 3. The plots in
(c,f) represent the same data as in figures 3(d,c) respectively, with the only
difference being the change in axis limits to match with the plots from the
AEM.

Here, we follow Eich et al. (2020) and use $n$ = 6 for the dd-AEM, yielding
$Re_{\tau}$ $\approx$ 3200 for the simulations which is also close to the DNS
at $Re_{\tau}$ $\approx$ 2000\. To extract the geometric estimates for the
eddy packet (figure 5(b)) corresponding to each of the six hierarchies, figure
5(a) shows the same 3-D statistical picture as in figure 3(g) discretized into
six individual blocks. Here, the $x$-axis location of the $i^{th}$ block
represents the streamwise length scale (${\lambda}_{x_{i}}$) of the $i^{th}$
hierarchy, while the corresponding extents along the $y-$ and $z-$axis
respectively indicate its spanwise half-width (${\Delta}y_{i}$) and the wall-
normal height ($z_{i}$). It is worth noting here that the eddy packet in
figure 5(b), with seven $\Lambda$-eddies, is simply shown for representative
purposes and the actual number of $\Lambda$-eddies in the packet may vary
depending on ${\lambda}_{x_{i}}$/$z_{i}$ (table 2), such that the inter-eddy
spacing is maintained constant for both AEM and dd-AEM.

Table 2 records the geometric estimates of all six hierarchies for the dd-AEM,
and compares them with those considered in the previous AEM by Eich et al.
(2020). The size of hierarchies 1 to 4 grows self-similarly in all three
directions, in accordance to the empirically obtained scalings, which
associates them with Townsend’s attached eddies. Hierarchies 5 and 6, on the
other hand, have a constant spanwise half-width (${\Delta}y$ $\sim$
0.17$\delta$). The two largest hierarchies, thus, represent the
$\delta$-scaled superstructures, which were found to be the sole energetic WC
motions coexisting beyond the upper bound of the inertial region (figure
3(g)). The presence of these $\delta$-scaled eddies, which do not conform to
the self-similar hierarchy, is another improvement of the dd-AEM over the AEM,
given that all six hierarchies in the latter conformed to the self-similar
hierarchy (table 2).

The best way to quantify the impact of the changed eddy aspect ratios (table
2) in the AEM would be to compute the cross-correlation spectra ($\Gamma$)
using both AEM and dd-AEM fields and comparing it with the empirical estimate.
This comparison is showcased in figure 6, where $\Gamma$ is considered along
the XZ plane (figures 6(a-c)), and that along two wall-parallel planes at the
lower (figures 6(d-f)) and upper bound of the inertial region (figures
6(g-i)). Here, for convenience in interpretation, $\Gamma$ computed from both
the AEMs is considered only in the ${\lambda}_{x}$-range encompassing all six
hierarchies (table 2). We can note a significant difference between $\Gamma$
estimated from the AEM and the dd-AEM fields, with the latter following the
scaling trends consistent with those obtained empirically. Based on the
comparison along both the wall-parallel planes (figures 6(d-i)), we can expect
the dd-AEM to perform much better than the AEM throughout the inertial region.

Figure 7: Instantaneous (a,b,c) streamwise, (d,e,f) spanwise and (g,h,i) wall-
normal velocity fluctuations on a wall-parallel plane at the lower bound of
the inertial-region ($z$ $\approx$ 0.05$\delta$ for the present $Re_{\tau}$).
Data in (a,d,g) corresponds to the AEM of Eich et al. (2020), that in (b,e,h)
corresponds to the data driven-AEM developed in the present study, while that
in (c,f,i) corresponds to the ZPG TBL DNS of Sillero et al. (2014) comprising
of solely the wall-coherent motions (represented by subscript wc). Black
dashed lines are used to highlight the largest spatial features of low
momentum for a qualitative comparison. Note the difference in axis limits
between (a-f) and (g-i). Figure 8: (a,b,c) Streamwise, (d,e,f) spanwise and
(g,h,i) wall-normal velocity fluctuations conditioned for ${u}^{+}$ $\lesssim$
-1 and ${w}^{+}$ $\gtrsim$ 1\. The conditional average is computed for the
flow on a wall-parallel plane at the lower bound of the inertial-region, which
is $z$ $\approx$ 0.05$\delta$ for the present $Re_{\tau}$. Data in (a,d,g)
corresponds to the AEM of Eich et al. (2020), (b,e,h) corresponds to the dd-
AEM developed in the present study, while that in (c,f,i) corresponds to the
ZPG TBL DNS of Sillero et al. (2014) comprising of solely the wall-coherent
motions (represented by subscript $wc$).

###  4.2 Spatial representation in the inertial region

We now extend the comparison between the dd-AEM and the datasets to the
instantaneous velocity fluctuations in the inertial region. Figure 7 presents
this comparison between the instantaneous fields for all three velocity
fluctuations estimated from the dd-AEM, AEM and the ZPG TBL DNS dataset
(${\mathcal{T}}_{1}$) at the lower bound of the inertial region. Here, it
should be noted that the ‘meandering’ aspect of the large-scale motions
(Hutchins and Marusic, 2007) has been built into the dd-AEM framework in the
exact same way as in the AEM (Eich et al., 2020). Amongst these three
datasets, the plots associated with the AEM and ZPG TBL DNS are the same as
shown previously in figure 2, with significant differences noted between them
($\S$2). Now, with the geometry of the representative eddy defined based on
data in the dd-AEM, along with the inclusion of the $\delta$-scaled
superstructures, the corresponding instantaneous velocity fields (figures
7(b,e,h)) can be observed to closely match the DNS fields. In particular, the
extended coherence of the velocity features ($\gtrsim$ 2$\delta$) noted in all
three velocity components in the DNS have been well replicated by the dd-AEM,
highlighting the impact of basing the eddy geometry on the data. This
reaffirms the fact that all three components are essentially inter-dependent,
and can be modelled via a representative eddy comprising multiple vortex
structures (figure 1). The meandering very-large-scale motions or
superstructures which extend beyond 6$\delta$ in length (Hutchins and Marusic,
2007; Lee and Sung, 2011), as observed in the DNS (figure 7(c)), are also well
represented by the dd-AEM (figure 7(b)). The dd-AEM, however, appears to
overestimate the magnitude of the lateral velocity fluctuations (when compared
to the DNS), which is likely an artifact of choosing the simplest
shape/arrangement for the representative $\Lambda$-eddy packet in the present
study (its optimization is beyond the scope of the present study).

The qualitative agreement between the dd-AEM and the DNS, showcased in figure
7, can be confirmed quantitatively by generating averaged flow fields
conditioned on a statistically dominant feature in the flow. One such
statistical flow feature, which is predominant in the inertial region, is the
negative value of the instantaneous momentum flux ($uw$ $<$ 0; Wallace et al.
(1972), Deshpande and Marusic (2021)), which is carried by coherent motions
associated with ejection ($u$ < 0, $w$ > 0) and sweep ($u$ > 0, $w$ < 0)
events. Notably, the velocity field associated with the representative
$\Lambda$-eddy packet (figure 1) is also consistent with this notion and
models features associated with both ejection and sweep events. Hence, we
compute the conditionally averaged wall-parallel flow fields for the coherent
regions associated with strong ejections and sweeps by following the same
methodology as adopted previously by de Silva et al. (2018) and Eich et al.
(2020). This is performed by attaching a frame of reference to the centroid of
each flow feature representing strong $u$ and $w$ fluctuations, which are used
as the conditioning points. Here, the strong fluctuations correspond to
$u^{+}$ $\lesssim$ -1 and $w^{+}$ $\gtrsim$ 1 for ejections and vice versa for
the sweeps. The mean flow feature is obtained by averaging multiple such
strong features extracted from several flow fields.

Figure 8 presents the averaged velocity signatures conditioned for the strong
ejection features, at the lower bound of the log-region, from flow fields
obtained from the AEM, dd-AEM and the DNS. Firstly, it can be noted that the
conditionally averaged fields from the DNS (figures 8(c,f,i)) qualitatively
resemble the velocity fields around a single $\Lambda$-eddy packet (figure 1),
giving further support to the choice of the representative eddy. On comparing
the geometric extents of the velocity features generated from the two AEMs and
the DNS, a better match between the dd-AEM and the DNS can be noted,
confirming the enhanced performance of the dd-AEM. To list a few specific
improvements, the long streamwise coherence observed particularly in
${\langle}{w^{+}_{wc}}{\rangle}$ (figure 8(i)) is well replicated by
${\langle}w^{+}{\rangle}$ generated from the dd-AEM (figure 8(h)). Further,
the extent of spanwise coherence for all three velocity components, noted in
the dd-AEM, compares better with the DNS than that noted for the AEM. The
inclusion of relatively long hierarchies in the dd-AEM, which also considers
the meandering aspect of the large-scale motions, clearly brings out the
distinct X-pattern in the ${\langle}{u^{+}}{\rangle}$ similar to that seen for
${\langle}{u^{+}_{wc}}{\rangle}$ from the DNS, and noted previously in the
literature (Hutchins and Marusic, 2007; Elsinga et al., 2010). The streamwise
extent of the wall-parallel velocity features, however, are still falling
short of the estimates from the DNS. This may be an artefact of the assumption
of scale-independent yaw angles, imposed onto the $\Lambda$-eddy packets, to
replicate ‘meandering’ phenomena in the TBL (Eich et al., 2020). It seems
plausible that the scales with the longest streamwise extent (i.e.
${\lambda}_{x}$) ‘meander’ with lesser intensity than those with relatively
shorter extent, which may favourably alter the comparison being presented in
figure 8. This, however, remains a subject for future work.

It is also worth noting here that the close match of the conditioned fields
from the DNS (figures 8(c,f,i)), with the corresponding flow fields from the
$\Lambda$-eddy packet (figure 1), is owing to the consideration of solely the
WC motions. Interested readers may refer to figures 11(d,e,f) where full flow
fields (WC + WI) have been conditionally averaged based on the same criteria
as in figure 8. The significant mismatch between the velocity features from
the WC and full fields, for the DNS, suggests the present representative eddy
packet as a suitable choice to model solely the WC subset of the TBL (and not
the full flow as a whole).

##  5 Discussion

The present effort promises enhancement in the spatial representation of an
instantaneous wall flow by means of a data-driven coherent structure-based
model. One would hope that such models, which have the capability to predict
very high $Re_{\tau}$ flows in a computationally inexpensive way, will drive
future investigations into flow phenomena otherwise difficult to measure
experimentally, such as the instantaneous variation of the $w$-component over
the homogeneous wall-parallel plane (as in figure 7(h)) or that of
$v$-component over the wall-normal plane. While the data driven-AEM developed
here seems a promising prospect in this regard, more work needs to be done in
terms of also incorporating the WI motions to be able to predict the full
inertially-dominated turbulent flow field. These WI motions not only are as
statistically significant as the WC motions in the inertial region (Chandran
et al., 2020; de Silva et al., 2020), but are also the key driver of the flow
phenomena in the wake region. This is evident from the instantaneous
$u$-velocity fluctuations plotted in the wall-normal plane, which are compared
for the AEM, dd-AEM and the DNS in figure 9. As we would expect, the dd-AEM
predictions (figure 9(a)) compare well with the instantaneous flow field from
the DNS, comprising solely the WC motions (figure 9(c)). These motions,
however, lose their coherence in the wake region where the WI motions
predominate, which can be deciphered from the full DNS field in figure 9(d).
Another aspect of coherent structure-based modelling requiring further work is
the shape/form of the representative eddy. The present choice of the
$\Lambda$-eddy packet for the AEM, although consistent with the statistical
picture ($\Gamma$) obtained empirically, does not reflect the typical flow
structure observed in an instantaneous flow, leading to overestimation of the
instantaneous lateral velocity fluctuations noted in figure 7. Efficient
machine learning-based algorithms (Brunton et al., 2020), which have the
capability to analyze big datasets, may be better placed to propose a solution
for this.

Figure 9: Instantaneous streamwise velocity fluctuations on a wall-normal
plane. Data in (a) corresponds to the dd-AEM developed in the present study,
that in (b) is from the AEM of Eich et al. (2020), while that in (c,d)
corresponds to the same ZPG TBL DNS flow field Sillero et al. (2014), with
difference being (c) comprises of solely the wall-coherent motions
(represented by subscript wc) and (d) the full flow field (WC+WI).

Besides assisting with the modelling of the WC motions, the empirically
determined geometric scalings in (2,3) can also be used to enhance the active
flow control schemes operating based on real-time sensing. Figure 10 depicts a
conceptual sketch showcasing the coherent motions in a ZPG TBL, which can be
manipulated by cross-flow (wall-normal) jets controlled by a computer, based
on data from a spanwise array of skin-friction sensors placed at a
sufficiently upstream location (Abbassi et al., 2017). These skin-friction
sensors, which are used to detect the incoming high-momentum ($+u$) carrying
coherent motions, are also capable of detecting the streamwise extent
(${\lambda}_{x}$) of these motions. Utilizing the empirically estimated
scalings, the computer controlling the jet actuation can decide how much
momentum to inject through them, based on the estimated wall-normal extent
($z$) of the incoming motions (indicated as control system 1 in figure 10).
Alternatively, in scenarios where sufficiently long separations between the
sensors and the jets cannot be maintained, the spanwise sensor array can be
used to estimate the spanwise coherence (${\Delta}y$) of the incoming motions,
through which the wall-normal extent could be predicted (indicated as control
system 2 in figure 10). Such empirically-driven flow control systems, which
also consider the wall-normal extent of the incoming motions, could
potentially manipulate the turbulent inertial wall flow more efficiently, as
demonstrated previously by Yao et al. (2017). Given the significant
contribution of the attached eddy hierarchy to the mean skin friction at high
$Re_{\tau}$ (de Giovanetti et al., 2016), availability of their corresponding
3-D geometric scalings can also be used to decide the spanwise spacing between
the cross-flow jets targeting these eddies.

Figure 10: Conceptualization of an active-flow control system aimed at
efficiently manipulating the wall-coherent inertial motions in a ZPG TBL by
utilizing the scalings determined from high $Re_{\tau}$ datasets. Concept of
the flow control system has been adapted from Abbassi et al. (2017).
$U_{\infty}$ denotes mean freestream speed.

##  6 Concluding remarks

The present study analyses a unique set of multi-point datasets to reconstruct
the 3-D statistical picture of the inertial wall-coherent (WC) turbulence in
all three canonical wall-bounded flows. Previous studies have found these
motions to be responsible for both, the $Re_{\tau}$-dependence of the skin-
friction drag (Örlü and Schlatter, 2011; Deck et al., 2014; de Giovanetti et
al., 2016; Smits et al., 2021) as well as the bulk production and the inter-
scale energy transfer in high $Re_{\tau}$ flows (Marusic et al., 2010; Lee and
Moser, 2019; Hwang and Lee, 2020). The aforementioned characteristics make
these motions a key target of coherent structure-based models, which the
present study attempts to enhance.

Here, the statistical picture is reconstructed by computing the cross-
correlation spectra ($\Gamma$) using the streamwise velocity fluctuations
mapped across the 3-D space in the wall-bounded shear flow. The intermediate-
and large-scaled inertial WC motions are found to exhibit geometric self-
similarity with respect to the distance from the wall $z$, expressible by
simple linear relationships, which are universal across all canonical flows
and independent of flow $Re_{\tau}$. The geometry of the very-large-scaled
motions, on the other hand, is found to exhibit $\delta$-scaling, associating
them with the superstructures and VLSMs for external and internal flows,
respectively. The present study also confirms the 3-D geometry of the VLSMs to
be much larger than the superstructures even in high $Re_{\tau}$ flows,
highlighting the role of the T/NTI in inhibiting the growth of these very-
large-scaled motions in external flows (Monty et al., 2007; Lee and Sung,
2013).

Alongside testing the universality of the 3-D statistical picture, the
geometric scalings brought out from the data (given by 2 and 3) are also
proposed to be used as a metric to estimate the spanwise and wall-normal
extent corresponding to each WC scale/eddy (${\lambda}_{x}$). Application of
these linear relationships to any coherent structure-based model can provide a
data-driven basis to the geometry of the representative structure, which is
demonstrated in this study using the attached eddy model (AEM; Perry and Chong
(1982)) for a ZPG TBL. Here, the choice of AEM (over other models) is also
driven by the empirical analysis, given the latter provides evidence of
geometric self-similarity of the inertial motions and also supports the
$\Lambda$-eddy packet as the representative eddy (both of which are built into
the AEM framework; Marusic and Monty (2019)). The present study further
extends the scope of the AEM by using its framework to also model the
$\delta$-scaled superstructures, which correspond to the very-large-scales
beyond the self-similar hierarchy. The data-driven AEM (or dd-AEM), which has
the representative eddies defined based on the data, is shown to improve upon
recent works on the AEM (Eich et al., 2020) in replicating instantaneous flow
phenomena associated with all three velocity components. This sets up the
platform for future work, which would include both the WC and WI (wall-
incoherent) motions modelled via the AEM framework, to replicate the full
turbulent wall-bounded flow that can provide realistic inflow conditions for
use in future numerical simulations (Subbareddy et al., 2006; Wu, 2017).

## Acknowledgement

The authors wish to acknowledge the Australian Research Council for financial
support. They are also grateful to the authors of Sillero et al. (2014), Baars
et al. (2017) and Baidya et al. (2019) for making their respective data
available. Sandia National Laboratories is a multimission laboratory managed
and operated by National Technology and Engineering Solutions of Sandia, LLC.,
a wholly owned subsidiary of Honeywell International, Inc., for the U.S.
Department of Energy’s National Nuclear Security Administration under contract
DE-NA0003525. This paper describes objective technical results and analysis.
Any subjective views or opinions that might be expressed in the paper do not
necessarily represent the views of the U.S. Department of Energy or the United
States Government.

##  Appendix A SLSE-based decomposition of the instantaneous flow fields

Figure 11: Instantaneous (a) streamwise, (b) spanwise and (c) wall-normal
velocity fluctuations on a wall-parallel plane at the lower bound of the log-
region ($z$ $\approx$ 0.05$\delta$ for the present $Re_{\tau}$) extracted from
ZPG TBL DNS dataset (${\mathcal{T}}_{1}$) of Sillero et al. (2014). Unlike
figures 2 or 7, these flow fields comprise contributions from both WC and WI
motions, i.e. they are the raw flow fields directly from the dataset. (d)
Streamwise, (e) spanwise and (f) wall-normal velocity fluctuations conditioned
for $u^{+}$ $\lesssim$ -1 and $w^{+}$ $\gtrsim$ 1 on the same wall-parallel
plane as in (a-c).

As discussed previously in $\S$1, the flow field in the inertial region
comprises contributions from both wall-coherent (WC) and wall-incoherent (WI)
eddies. Here, we demonstrate a methodology, based on the spectral linear
stochastic estimate (SLSE) approach, to estimate the WC subset of a velocity
fluctuation (say for $u$-component) at any wall-normal location $z$ in the
inertial region. This procedure has been utilized in previous studies (Baars
et al., 2016; Madhusudanan et al., 2019; Deshpande and Marusic, 2021) for
similar purposes (i.e. to obtain a subset of the full flow field comprising
selected coherent motions), which can be directly referred for an elaborate
introduction to this technique. According to SLSE, a scale-specific
unconditional input at any near-wall location, $z_{w}$ (with $z^{+}_{w}$
$\lesssim$ 15) can be used to obtain a scale-specific conditional input at $z$
following:

${\widetilde{u}}^{E}(z^{+};{\lambda}^{+}_{x},{\lambda}^{+}_{y})={{H^{u}_{L}}(z^{+},z^{+}_{w};{\lambda}^{+}_{x},{\lambda}^{+}_{y})}\widetilde{u}(z^{+}_{w};{\lambda}^{+}_{x},{\lambda}^{+}_{y}),$
(4)

where ${\tilde{u}}({z_{w}};{\lambda}_{x},{\lambda}_{y})$ is the 2-D Fourier
transform of the instantaneous wall-parallel flow field, $u$(${z_{w}};x,y$) in
space. Here, the superscript $E$ and $H^{u}_{L}$ respectively represent the
estimated quantity and the scale-specific linear transfer kernel (for
$u$-component). $u^{E}$ in equation (4), essentially, corresponds to the
energy distribution (at $z$) across wavelengths ${\lambda}_{x}$ and
${\lambda}_{y}$, which are coherent across $z$ and $z_{w}$. This is
facilitated by first computing $H^{u}_{L}$ from an ensemble of data as per:

${H^{u}_{L}}(z^{+},z^{+}_{w};{\lambda}^{+}_{x},{\lambda}^{+}_{y})=\frac{\\{\widetilde{u}(z^{+};{\lambda}^{+}_{x},{\lambda}^{+}_{y}){{\widetilde{u}}^{\ast}}(z^{+}_{w};{\lambda}^{+}_{x},{\lambda}^{+}_{y})\\}}{\\{{\widetilde{u}(z^{+}_{w};{\lambda}^{+}_{x},{\lambda}^{+}_{y})}{{\widetilde{u}}^{\ast}(z^{+}_{w};{\lambda}^{+}_{x},{\lambda}^{+}_{y})}\\}},$
(5)

where the curly brackets ($\\{\\}$) and asterisk ($\ast$) denote the ensemble
averaging and complex conjugate, respectively. Given that $z^{+}_{w}$
$\lesssim$ 15 and $z^{+}_{w}$ $\ll$ $z^{+}$,
${\widetilde{u}}^{E}(z^{+};{\lambda}^{+}_{x},{\lambda}^{+}_{y})$ thus
represents energy contributions at $z$ from solely WC motions, that are taller
than $z$ (Madhusudanan et al., 2019; Deshpande and Marusic, 2021), leading to:

${{{{\widetilde{u}}^{E}}(z^{+};{\lambda}^{+}_{x},{\lambda}^{+}_{y})}{{\big{|}}_{{z^{+}_{w}}{\lesssim}15}}}\;{\rightarrow}\;{{{\widetilde{u}}_{wc}}(z^{+};{\lambda}^{+}_{x},{\lambda}^{+}_{y})},$
(6)

where $u_{wc}$($z$) represents the subset of $u$($z$) comprising solely the WC
motions. On obtaining ${{\widetilde{u}}_{wc}}$, the corresponding flow field
in physical space can be obtained by:

${{{u}_{wc}}(z;x,y)}={\mathcal{F}^{-1}}({{{\widetilde{u}}_{wc}}(z;{\lambda}_{x},{\lambda}_{y})}),$
(7)

where ${\mathcal{F}^{-1}}$ represents the inverse Fourier transform. Here,
although the entire procedure has been demonstrated solely for
$u$-fluctuations, the same can be applied to other (lateral) velocity
fluctuations by estimating the respective linear transfer kernel ($H^{v}_{L}$
or $H^{w}_{L}$). To give an example, figures 11(a-c) plot the instantaneous
wall-parallel flow fields, comprising both WC + WI motions (i.e. the full flow
fields) at the lower bound of the inertial region. While, figures 2(b,d,f)
plot the subset of the corresponding full fields comprising solely the WC
motions, estimated using the aforementioned procedure.

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